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2.5
Stabilization with Uncertainty
The fuB power of backstepping is exhibited in the presence of ullcel't,ail1 n011linearitics and unknown parameters, because for such applicat.ions no other systematic design pl'ocedul'o exist.Ii, Wo now begin the st.udy of sueh design problenls which are tho main subject of this book. The first of the design tools that will be llsed to COllnteract ullcertainty is nonlinear dam.ping.
2.5
73
STABILlZA'l'ION WITH UNCERTAINTY r
;r
.f
lL 'I'
"
..:
cp(. )
'C)j'
.6.(1) Figure 2.9: A system with "matched" uncertaintya(t.).
2.5.1
Nonlinear damping
'Vc int.roduce nonlinear damping for syst.t"ms ,,,ith "mat.ched'· ullcertainty, in which both the ullcertainty aud the control appear in the same equation. The simplest example is the scalar nonlinear system depict.ed in Figure 2.D: .1; =
11
+ !p(J:).6.(t)
(2.240)
I
where cp(x) is a known smooth nonlinearity, and .6.(/) is a bounded fuuC"t.iol1 of t. Let. us first examine the case when .6.(l) is an exponentially deeaying disturbance: .6.(t) = .6.(O)el.t. (2.2.n) Can such an innocentlooking unccrtainty eause harm? One might be tempteu to ignore it and usc the linear control·u = C.l:, which results in the closedloop system :[ = ex + cp(:r}.6.(O)e kl • (2.242)
While this design may be satisfactory when cp(.1.') is bounded by a constant or a lineal' function of :r it is inadequate if "'(:1') is allowed to be any smooth nonlinear function. For example, when cp(:1:) = :r2 we have J
,j;
= ex + .1·2 .6.(0)e kl
(2.243)
•
.As we saw in Chapter 1, equations (1.2D)(1.32), the solution :z:(1) of this system can he calculated explicitly using the change of variable lIJ = 1/:r: (2.244)
which yields
w(t) = [W(O)  .6.(O}.] eel c + h!
+ .6.(O} e kt • l'
+
(2.245)
fl'
The substitution w = 1/.1: gives
x(O)(c + k) :1'(1.) = [e + k  .6.(O).1.·(O)]ect + .6.(O).l:(O)e kl
.
(2.246)
74
DESION TOOLS FOR STABILIZATION
From (2.246) we see that the behavior of the closedloop system (2.243) depends criticAlly 011 t.lle initial conditions .!l(0), .x(0):
(i) If .!l{0).1:(0) < c + A", the solutions x(t) are bounded and converge asymptotically to zero. (ii) The situation cbanges dramatically when .!l(O)x(O) > c + k > O. The solutions x(t) which start from such initial conditions not only diverge to infinity, but do so in finite tim.e:
.1'(£)
+
1 { .!l(O)x(O) } 00 as t ..... t J = c + A: In .!l(O)a'(O) _ (c + k) .
(2.247)
Note f',hat this finite escape canllot be eliminated by maldng c larger: For any valnes of c and A: and for any nonzero value of .!l(0) there exist initial condiUons .1:(0) wbich satis(y the inequality .!l(O)x(O) > c + k. This example shows tbat in a nonlinear system, neglecting the effects of exponentially decaying disturbances or nonzero initial conditions can be catastrophic. To overcome this problem anel guarantee that x(t) Willl'eDltlin boullded for all bounded .!l{t) and for all :1'(0), we augment the control law lJ = ex with a nonlinear damping te1'1lJ. s(x).v:
u
=
ex  s(:r).1:.
'Ve design s(x) for the system (2.240), using the quadratic fUllction V'(.l·) whose derivative is
li = xu+.z;tp(x).!l(t) = c:r2  x!!s(.r) + x cp(:r).!l(t) .
= !.r!.'! 
(2.249)
The objective of guaranteeing global boundedness of solutions can be equivalently expressed as rendering ,7 negative outside a compact region. This is achieved with the choice (2.250) which yields the control (2.251) and tbe derivative
(2.252)
2.5
75
STABILIZATION WITH UNCERTAINTY
Comparing (2.249) with (2.252) we see that the nOll1inear damping term (2.250) is chosen to allow the complet.ion of squares in (2.252). In more complicat.ed situations we can use Young's ll1cq'llulitll, which, in ~l simplified form, st.ates that if the constant.s p > 1 and q > 1 are such that (]J  1)(1]  1) = 1. then for all c > 0 and all (.r,1/) E IR? we have 1
f:P
1:zoIP + qf:q lul q • p
.1'11 :$
Choosing P =
I]
= 2 and
f:2
(2.253)
= 2~, (2.253) becomes (2.254)
which is the incqualit:y lIsed ill (2.252):
(2.255)
Global boundedness and convergence. Returning to (2.252), we sec Umt li' is negative whenever Ix(t)1 ~ ::J:2:. Since d(t) is a bounded disturbance, we ('onclude that ,i' is negative outside the compact residual spt, (2.256) Recalling that. V'(x) = ~:l':'\ wc concludc that. 1:l'(t)1 decreases whellcver .1:(1) is outside the set 'R, and iwncc x(l.) is bounded:
IIxli oo
~
IId IlOJ } . ma.x { l:r(O)I, 2ViiC
(2.257)
~\'Ioreover, we can draw some conclusions about t.he asymptotic bchavior of :r(t). Let. us rewrite (2.252) as
~ (~.~2)
:$
c.t~ + d:~t)
.
(2.258)
To obt.aill c..xplicit bounds on .7:(1), we consider t.he signal J;(t)e d , Using (2.258) we get
(2.259)
76
DESIGN TOOLS FOR STABILIZATION
Integrating both sides over the interval [0, t] yields
.r"( f ) e"d
(2.260)
Mult.iplying bot:h sides with e:!rt ;:md using t,he fact that. Jlp· we obtain an explicit bound for .J:(f):
+ c!:! :s; Ibl + Icl.
,.1'(1) I (2.261) l::.
Sillce snp IA(T)I ~ sup IA(T)I = O:::;T~1
IIAlltXlt (2.261) leads 1:0
O$T
(2.262) which shows that :c{t) cOlwerges to t.he compact set 'R. defined ill (2.250);
lil11 dist. {.1:(t), 'R.}
100
= O.
(2.203)
\\Te l'eitera,1,e t.hat. these propert.ies of bonndedlless (d. (2.257)) and COIlvel'gence (cr. (2.263)) arc gna.rantced far any bounded disturbance AU) and for any smooth nonlinearity lP{a;}, including
=
2.5
77
STABILIZATION WI'l'H UNCERTAINTY
Finally, we should note that" if the disturbance ~(t) converges to zero in addition to being bounded, then t.he control (2.251) guarantees cOInergencc of .c(t) to zero in addition 1.0 global boulldedness. To show t.his, let. ~(t) he n continuous nonnegative m.onotonically dec1"(~a.si7).g function SUdl that 1~(1)1 ~ Ll(t) Md lilll , _ oo Li(l) = O. Then, st.arting wit.h the first inequality from (2.260) and using all argument. almost. identkal to thc proof of Lelllma 2.'24, we obtain I.l'(t) I ~ 1·1:(0}ler.L +
Since limtoo Li(t/2) =
~ (Li(O)e~1 + .6.(t/2)) .
2v"c
a, we se(' that. lim,_oo :r(t} =
(2.264)
O.
ISS interpretation.
For interpreting t.he effect of the nonliu(,al' damping term 1i..l·cp2(.r} in (2.251) from an inputoutput point, of vicw, it. is yel',\' COllvenient to use the eoncept of inputtnstate stability; (cf. Appendix C) This /i.term renders the closedloop system ISS with respect to the clist.urbauee input .6.(1.). To show that the ISS inequality (2.12) holds for our dmipdIoop system with v (T) replared by tIll" dist ur bance A ( T), we repeaL the tU'gll mell t that lerl from (2.259) to (2.261), t.his time integrat.ing over t.he interval llo, fl. The result is
I:r(t)l
~
l:c(to)lcC(/tu )
+ _1_
which is identical to (2.12) with .8(1', s) = s = t lo.
2ViiC 1'e
ca 1
[sup 1.6.(1")1], 'o~"'~1 1'(1') = 2~[''''
l'
(~.2G5)
= I:,.(lo)] a.nd
Operator gain interpretation, It. is also convenient t.o int.erpret. the ('.£fel't of nonlinear damping from an operator point. of view on t.he basis of (2.257) and Figure 2.10. For all initial conditions :1:(0) such that. 1:l'(O) I < ~~. we obtain
(2.266) which shows that the nonlinear operat.or 1\ maPl)ing the dist.urbau('e ~(l) to the output .7:(t) is bounded, and its .c~inrluced gain is
(2.267) The nonlinear damping term renders the operat.or 1\ bounded {or any positive values of n. and c. Note, however l that. (2.266) does not: provid(' a complet.e deseription of this opera,tor b('c'ause, unlike (2.257), it hides the effect of initial conditions, which can be quite dangerous for nonlinear systems. The following lemma recapitulates the properties achieved wit.h nonJinear damping as a design tool.
78
DESIGN TOOLS FOR STABlLlZA'rION
PLANT

A(t) u

CONTROLLER Figure 2.10: The bounded nonlinear Opel'll.l.ol· [(:
~{t.}
. xCi).
Lemma 2.26 (Nonlinear Damping) Let the sll.r;tcm (2.48) be pe1tm'bed as i11
(2.268) WhC7'C
cp(x) is a (p x 1) lJedoT of known smooth nonlinem' functiol1s, and t) is a (p x 1) 1Jecto7' of'll'llcc7tul1l. TAo71linea1'ities which a.J'C uniformly
~(.t', ll,
bounded f01' all 'lJal1J.e.r; of .1:,11, t. If Ass'lL'ITl,ptian 2.7 is satisfied with lY(a:) po.r;iti·lJc definite and 'I'U.diallll unbou.nded, then fhe control (2.269) when a.pplied to (2.268), 7"enders the closedloop system. ISS with respect to the distu,.bance inpu./. A(a', 'II, t) and hence !JlIurantees globaI1J.nifm"1n b01J.ndcil1l.ess of .r(t) lind COTlvergence to the residual set
(2.270)
where
')'11 "2, ')'3 (ITe
cla.s.IjIC oo functions such that 11
l' (:1:) $ 1'2 (1·1:1) 1'a(l:l:1) ::; lY(.1!) . 1', (I:r I)
$
(2.271) (2.272)
J I Since l'(x) lind H'(:l:) arc posit.ivc definite and radially ltllbouudcd ana l'(x) iii smoot.h, there exist classlC;x: fuuctious 71, ')'::1,,3 satisfying (2.271) and (2.272).
2.5
79
STABILIZATION WITH UNCERTAINTY
Proof. The derivative of V' (J') is
av all' T ax [J + gu] + a.~~ gep ll.
,1' =
01' [J + au] 7J x
by (2.269) = by (2.50)
by (2.254)
< 1 V{:t·) 
n
:::;
1F(.1')  n
:::;
IF(.L') +
It"
r·
('wF9
r J
(lW JJ D.I.
of 'r Ll Icp\ + {FOCP x
8lI
\cp12 + . ax 9l{JTll.
8.1' \et'I:! + IOV IIcpllIll.lloo (DI'r Dx 9
9
II~";, . Ii
(2.273)
From (2.273) it follows that ,1' is ncgativc whenever IV(.l:} t.his with (2.272) we cOllt'lnde that
J:r{l}l > 13"'
("~~!.)
=?
> II~~;;.. Combining
Ii < 0,
(2.274)
This llleans that if l.r{O)1 .$ ''iiI (jl~!;"'), then
V{x{t)) ::;
')':10,3"'
("~!!.)
,
(2.275)
which in t.urn implies tlu1t (2.2i6)
If, on the other band, 1.1:(0)1 implies
> 13]
(!I7.ll?s )
I
then 1/(:r(t» :::; \l{:r(O))! which

(" ....,,) (( Combining (2.270) and (2.277) leadR t,o the global uniform boU)ulcc1ness of
x{t);
11:"11", ::;
max {,,' 0 'Yo 0 1'3"
("~!;'
),
1'j"' 0 12 (I:r{O)I) } ,
(2.278)
while (2.274) and (2.271) prove the convergence of .r(t) to t.he residual set defined ill (2.270). Finally, the ISS property of the closedloop system with respect to the dis~urballce input ll.(J:, lJ, t) follows from Theorem C.2. 0
80
2.5.2
DEs[GN TOOLS FOR STABILIZATION
Backstepping with uncertainty
Lemma 2.26 deals with the case where the unrerblinty is in the span of the control '1/., i.e, t.he matching condition is satisfied. Combining Lemma 2.2B with Lemma 2.8 allows liS to go heyond t.he mat.rhing rase, as t.he following example ill us t, l"t1t es.
Example 2.27 Consider the system j'
~
= ~ + :l'~ arctan e6()(/.) = (1 + ~2)U + e:re~D(t) ,
(2.279a) (2.279b)
where 6.0(1.) is a bounded tim('~varyillg disturbance. Clearly, the Ullcert·ain terms in (2.279) arc not in the span of the control 'lI.. Therefore, we will design n static nonlinear controller in two steps, combining nonlinear damping and badcsteppjng.
Step 1. The staltillg point is equation (2.279a) and the choice of a virtual ('ont.rol vRriJ:luJc. Clearly, { is tlle only choice. The lact that E. is also present in the U1u'ert.aill term docs not. present a problem, since it enters that term through the bounded fundion aJ'ctan(·). In the notation of (2.268), we 11avc (2.280) The uncert.ain nonlinearity /);.1 (~,/.) is bounded: (2.281) Hence, Lemma. 2.26 can bo lIsed to design a stabilizing function for~. The unperturbed syst.em in this case would be the integrator ± = ~, for which 1:t elf is given by "(x) = !x2 and the conesponding control is o(x) CI'T.. From (2.269) we have (2.282)
=
which results in (2.283) with the error variable
z defined as in Lemma 2.8: (2.284)
The derivative of l,l(.:z') along (2.283) is
,i
=
by (2.280) by (2.254)
(2.285)
2.5
81
STABILIZA"rrON WITH UNCERTAIN'l.'Y
which confirms that if :; == 0, that is~ if ~ were the actual control. tben (2.282) would guarantee global uniform bOl1ndeduE'ss of x.
Step 2. Using the error variable
=from (2.284), the system (2,279) is l'cwritt.eu
as :i:
=

::::
(2,286a)
=
(2.286b)
where the partial ~ is comput,ed from (2.280) and (2.282): 80') 8:.1.'1
= c)  1i1..i!.(J:J:
[;l'tpi(:~')] ::::
Ct 
5Iil:c1.
(2.287)
If the ~o(t)terII1 "Were not present in (2.286b), then LeIllllltl 2.8 would dictate the Lyapunov [ullction
I"
112 ( x,) e = 2' .1;" +
1., '2:;
=
(2.288)
and the following choice of control:
1:
" = li("" {) =
£;2
[C2= + ';:' { :r] .
(2.289)
To compensate for the presence of the ~o(t)term in (2.28Gb), Lemma 2.26 is used again. From (2.269) we obtain (2.290)
which renders the derivative of 1l2(x, e) negative outside a compact set., thus gnaranteeing bounded ness of x(t) and ~(t):
V2 by (2.285)
by (2.286b) and (2.290)
=
Ii" + zz
::; "'x _ £.yr.2 ..,
=
Ct X2
..
+ 1I~11l~ + :.... 4lbl
+ II~I fI~ 4lil
+= { c!!z + [e.( 
h~2:: [ e:re 
;::£2
arctan
a.r x 2 arctan e]2
80'1
e] ~(t)}
82
DESIGN TOOLS FOR STABILIZATION
(2.291)
by (2.254)
o The combination of Lemmas 2.8 and 2.26, illustrated in t.he above example, is now formulated a..., a.nother design tool.
Lemma 2.28 (Boundedness via Backstepping) C07uJidcT Ihe system :i' = 1(.1:)
+ g(J.')u. + F(J;)~I (:1', Il, t) ,
(2.292)
where :,. E IRn. II E JR., F{:r) is an. (n x q) ma.tri:r: oj known smooth nonlinecu' Junctions, c1.1l·d ~1 (:1.', II, t) is a (q x 1) vectol' oj uncertain 1wnlineu7i.tie.i which 11.7'f. uniformly bounded J07' nil vallle.~ oj x, tt, t. Su.ppose lha.t i.here C:L'ists a. Jcedback contl'Dl 'lL = o{l') that 7enders x(/.) globall1J uniJ01'111.111 bounded, a.nd that this is est(J,blisheti 'uia positive definite and 'f'IJ.diallll U7l.bounded junclions ll(:l:),IF(x) and (]. constant h, such that all
D.'C (.1.') [f(:&)
+ g(.1·)(1(3:) + F(:r)d l (3:,11, t)]
~ ''I'(x)
+ b.
(2.293)
Now con.r;ider the aUflmentcci .'1y.r;tem .1: = E;.
=
j(.r} + g(x)~ + F(x).6 t (.1", u, i) 'U + cp(:z:, ~)1'~2(X, €, u, I.) ,
(2.294a) (2.294b)
when!
~2 (x, ~, 1l, t) is
u =
c (~  ll'(x)]
Bo·
+ a:c (.1') [J(.r) + g(.T)~J 
" [(  0:(x)] { I'P{ x. {W +
I::
(x )F(x)
guo.ronlees globlll 'lInijonn bOltTIde.dness of ,l;(t) and h'
> O.
~(t)
all'
n
a:r: (.r)g(J:)
(2.295)
'with any c > 0 and
2.5
83
STABILIZATlON WITH UNCERTAINTY
Proof. Using the error variable z =
~ a:(xL
(2.296)
the system (2.204) is l'ewritten as
;i'
=
., =
1(:1') + g(:l.') [a:(J') + =] + F(;v).6. (a',t" t) 1I + tp(.7·, e)T 6.:d:1:, t)
(2,297a)
e,u,
ao' (.1') [I(.1:) + g(:r){ + F{.r).6.{x, Il, t)] a x
.
(2.297b)
The derhrative of (2,208)
along Ula solutions of (2.297) wit,b the control (2.295) is
\1;
=
8"
ax (f + go' + F.6.) + all a.'£ g=
+= [u + 1"1·60.  :: (f + g~ + FAI)] 8\1 ] < aF a.}; (/+90 +F.1d+= [aa' 'u fJx(f+o{) + 8J,il
+= ['PTA, by (2.293)
:>
[v  :: (f + g~) + ~~g]
W{.t) + b + =
+= [opT A. by (2.295)
=
:~ FAr]
 II' (.T) +
~~FAI]
b c:'  ,,=' [1'P12 + I~: FrJ
+1=11'PIIIA21100 + 1=11:: FlllAtlioo by (2.25.1)
=
IV(x)  c:: 2 + b +
1I.611l~ + 116.2 11!: . 4h'
1 Ii
(2,290)
The radial ul1boundedness of li'(,}') combined with (2.2DD) implies lhat \~ is negative outside a compact set, which in turn implies that ,l:(t) and e(t) are globalJy uniformly bounded. 0
84
DESIGN TOOLS FOR Si'ABILJZATION
2.5.8
Robust strictfeedback systems
.Just as we generalized Lemma 2.8 to strictfeedback syst.ems in Section 2.3.1 and Lemma 2.25 to blockstrictfccdback systems in Sect.ion 2.3.3, we can generalize Lenmut 2.~8 to broader classes of Ullcertain nonlinear systems. We cOllsider syst.f'IDS ill the robu.st st1ictJeed1J(u:k J07"m:
= :r~ + cpT (xdd(~r:, lI, t) :1:2 :: x:i + cpI(,vl 1 x2)d(x, u., t) ";:1
(2.300)
xn + IPJl (.rl l •• • ,l'u_dA(.:r, 'lI, t) ;;:'1 :: {3(.I')'ll + IP;(:z:}d{x, 'u t) I
j'nl
::
1
where (J(:J') ::F 0, V:1: E .IR", IPi(Xh"" Xi) is a (p x 1) vector of known smooth l10nlinear functions, anrl d( x, u, I) is a (p x 1) vector of uncertain nonlinearitic5 wbich are lJ.71ijoT11J.ly bounded for tul values of X, u, t.
Corollary 2.29 (Robust StrictFeedback Systems) The .9tUte. :1'(1) oJthe 81Jstem (2.300) 1IJ'W be flloimJ,III1J.TJ.iforrnllJ bounded if the control is chosen as 1
u = {3{x) (J',,(x) , 'Ulhe7V~
i
the I.,m ctian an (.1:) i,&; defined by Ute fDllowin.g (yllhere 'we denote =0 == 0:0 == 0):
= 1, ... ,11
(2.301) 7"eC1tTlli1Je
C:L717Y!ssion.s JOT (2.302)
(2.303)
with C" Hi,
j ::
1. ... ,11 ]JOlJitive design CDnstants.
Proof. Using the definit.ions (2.302) and (2.303) and denoting Xo == 0'0 == ."l."n+l == {3(.J:)u., the derh'ative of the elTor variahle Zi, i:: 1, ... ,11, becomes
0,
2.5
85
STABILIZA'l'ION WITH UNCERTA1NTY
The choice of control (2.301) guanmtees that system can t.herefore be expressed 8S ":1
=
=n+l
== O.
The closedloop error
Cl=l + =:.! + cpT ~  h"l=II'P11 2
Z2 = C2=2 
=1 + =:" + ('P'J  aa~~l ~I) T~  h~2=2Icp'l  aa~J IPII~ .1'1
.ll
(2.305)
112
)=1
1
Zn
=
:CJ
T (
C,I=J1 
2
L 0'11')
lin_1 =nl 'PIIl 
•
00',,_2
n1
+ 'P,, L
ZnJ
0 0:1I _ 1
)
~
 f ) "
j=1
.,
:r J
fi.nZn
CPn 
III
L
j=1
80,,_1 1 8 ' 'P) "'J
Now we can use the quadratic Lyapllllov function \
1
n
_ ~_2 " 1.···' .._n ) ? L.J ':'i
' (_
.oJ
(2.306)
i=1
to prove global uniform boundedness, Indeed, the clerivat:ive of (2.306) along the solutions of (2.305) is
Vn
=
" L i=1 [
:::; L
') CjZj
,,[
i=L
<
'1
C:i=i
+ =i
(
+ I=il
iI OOil) L  "'P) )=1 8a.)
'P; 
L
iI a:tpj 8a:.;_1
j;1
T
~  li.i=;
II~IIIXI
iI 80';1
"
 tii=;
.1 J
t [c;..:; + /I~~I~l· ;=1
q]
;1 OO'i1 L  .. 'Pj )=1 OJ J
L
)=1
a:If')
2]
"'J
(2.307)
41l'i
The lnst inequality implies that .:(t) is globally uniformly bounded. But from (2.303) we see that, since t,he a/s are smooth functions, Xj can be expressed as a smooth funct:ioll of Zit ... ,Zj: (2.308)
Hence, .1:(1) is globally uniformly bounded and: furthermore, COllVC.lrges to the compact residual set: 'R =
{
:1: E JRII :
L Cj;;; :::; L / I n
j=J
;=1
11~112 } ~ I
(2.309)
4""
whose size is unknowll since the bound 1I~1I1X1 is unknown.
o
86
DESIGN TOOLS FOR STABILIZATION
Notes and References Integrator backstepping is an idea whose origins are difficult to trace, berause of its simultaneous appearance, often implicit, in UIe worles of Tsinias [193]. Koditschek [84], Byrnes and Isidori [12], and Sontag and Sllsslllann [175]. Kokotovic and Sussmann [85] viewed the stabilization t.hrough an integrator as a special case of stabilization through all SPR transfer funct.ion, as in the early adaptive designs by Parks [150], Landau [109], and Narendra. et aI. (142]. This passivity view was extended to nonlinear cascades by Ortega [145] and Byrnes, Isidori, and Willellls [1~1]. Integrator backstepping as a recursive design 1.001 [85, Corollary 3,2] was employed ill the cascade design of Saberi, Kokot.ovic, and Sussmann [163] and further dC\'eloped by Kn.nellakopoulos, Kolrotovic, and iVlorse [73]. The passivity aspect of this design was pointed out by Lozano, Bl'ogliato, and Landau [116]. A tutorial overview of backstepping was given in the HJ91 Bode lecture by Kokotovic [88), Among the current. applications of backstepping al'e electric machines in the monograph of Dawson, HUt and Burg [2GI and steering and braking control by Chen and Tomizuka [17]. An important. question, not addressed in the above references, is whether backstepping designs can incorporate optimization with respect to some meaningful cost funct.ionals. It is cleat' that minimizing a part;ial cost: functional at each step docs not imp}Yl and lllay in fact contradict, tbe overall optimality. A framework for a backsteppinglike recursive optimization was proposed in a Russian language paper by Kolesnil(Ov [89], which remained unnoticed in tlle English language literature. All the above references apply backstepping to systems without uncertainty. For s)'stems with uncertainty a robust nonlinear design was introduced for the makhed case in the works of Corless and Leitmann [20] and Barmish, Corless, and Leitmallll [G], and wId extended by variolls forms of "generalized matching condi1.ions·' [19, 20]. However, these extensions haven't led to a (.liscovery of hacksteppillg with uncertainty. After the development of adaptive backst.epping by Kanel1akopoulos, Kokotovic, and Nlorse {69, 87], backst.epping with uncert.ainty was pursued in the works of Qu [160], l\'lllJ'ino and Tomei [125], H'e~l11a.ll and Kolmtovic [37, 38]. and Slotille and Hedrick [168], Backst.epping designs for systems with unmeasured or unmodeled dynamics have been developed by Praly and Jiang [159], Jiang, Teel, and Praly (61], Khalil [82], WId Krstic, Sun, and Kokotovic [104]. Reeellt advances in backstepping, whidl are beyond the seope of this book, are due to Coron and Praly [21] and Praly, d'AndreaNovel, and Coron [158]. The restrict.ioll of backsteppillg to purefeedback systems has motivated alternative designs applicable to other classes of nonlinear syst.ellls, such as tbose by Teel [187], Teel and Pl'aly [192]. Qu [161], and Nlazenc and Praly [1261.
Chapter 3 Adaptive Backstepping Design The controllers designed in the prcceding chapter guarantee that in the presence of uncertain bounded nonlinearities the closedloop st.ate remains boundecl. In this chapter. and in the remainder of the book, the III1(:ertainties are more specific. They consist of unknown constant. parameters which appear linearly in the syst.em equat.ions. III t.he presen('e of sueh paramet.ric uncertainties we will be able to achieve both bounded ness of the closedloop states and convergence of the tracking errol' to zero. While all the controllers designed in Chapter 2 employ static fecdbaek, the controllers in this chapter will, in addition, employ a form of nonlinear integn\l feedback. The underlying idea in the design of this li1Jnamic part. of feedback is pa1nmeter e.r;tim.a.tion. The dynamic part of the controller is designed as a pammeter update law with which the static part is continuously mliJpteci to new parameter estimates, hence its name: AdaptitJf controllmn. In using this traditional terminology, however, we should keep in mind that so conceived adaptive controllers are but one type of nonlinear dynamic feE>dbaclc Adaptive controllers arc dynamic and therefore more romplcx than the static controllers designed in Chapt.er 2. \\That is achieved wit.h t.his additional complexity? As we will show in Section 3.1, an adaptive controller guarant.ees not only that the plant. state .1: rcmains bounded, but also that. it. t.ends t.o a desired constant value ("rcgulation") or asymptotically tracl\s a reference signal ("tracking"), The first results leading t.o a new syst.ematic design of adaptive cantrollers 81'e presented in Section 3,2, which introduces aclapti've backstcPIJin9. The recursive design procedure for pam.met7'ic strictfeedback .r;lJ,r;tems is then developed in Section 3.3. In its basic form, t.he adaptive backstepping design employs Ollerpammetdzutiol1, that is, more t.han one estimate per unknown parameter. This means that the dynamic part of the controller is not. of minimal order. In Chapter 4, a more int.ricate backstepping procedure is developedthe tuning functions methodwhich employs the minimal number of parameter
88
ADAPTIVE BACKSTEPPIN'G DESICN
estimates. The e~1:endedmatching design, presented in Sect.ion 3.4, is of int,erest. as a transition between overparametrizcd and minimalorder designs. It ah~o cont,nins the first. adaptive performance results.
3.1
Adaptation as Dynamic Feedback
The difference between a static and a dynamic (that, is, adaptive) design will .first be illust.ro,t.ed on t.he simplest nonlinear syst.em:
.7: = 11 + 8cp(:r) ,
(3.1)
This is the special case of the system (2.240), where the uncertaint.y 8(1) is the unknown (·onstant. parameter (). Even if we do lIot know a bound for 0, we can use Lemma 2.26 to design a statk nonlinear control1er which guarantees global boundedncss of ~r(t). The nonlinear damping design (2.251) applies also here. The corresponding static controller is (3.2) 11. c.r  /i,XCP"( .1: ) J
=
and the result.ing closedloop system is of first order:
x=
cx  1i.7:cp2(X) + Ocp(:c) .
According to (2.252), the derivative of V =
Y~
'1
c;r.
4:r
2
(3.3)
satisfies
()2
+
(3.4)
4n. '
which means t.hat 3:(t) converges to the interval
(3.5) This interval can be reduced by increasing the gains n. and c, but x(t) will not converge t.o zero if B is a nonzero constant. Excessive increase of these gains enlarges the syst.em bandwidth, which is undesirable. Our task is therefore to achieve lim/ooo x(t) 0 without increasing Ii, and c. In fact, we will first accomplish this t,ask with n. = 0, and then use Ii. > 0 for further improvement of transients. To achieve regulat.ion of x(t), we design a dynamic feedback controller, that is, we employ adaptation. If () were known, the control
=
'U
= B
Cl.1:, Cl
=
>0
(3.6)
=
2 would render t.he derivative of Vo{x) !x2 negative definite: Yo CIX • Of coursE' t.he cOlltrollaw (3.6) CRn not be implemented, since 0 is unknown.
3.1
8D
ADAPTATION AS DYNAMIC FEEDBACK
Instea.d, one Ctl.ll employ its ce1'i.aintIJeqltillalence form in which 8 is replaced by an est.imate 0: 11 = 8cp(.1:}  Cl:r. (3.7)
Sllbst,jtuting (3.7) illt:O (3.B),
W~
j:
where
obtain
=
+ iil;'(x) ,
CIX
(3.S)
ii is tbe pU'I'Umelel' e11"01'; (3.9)
The derhrative of ''0(:1'}
= ~.1:2 becomes li"o =
Cl x
2
+ O:rcp(:I:) ,
(3.10)
Since the second t.el'm is indefinite and contains the unknowll parameter ("rror ii, we can not conclude anything about. t;he st,ability of (3.6). 'Vc make the controller dynamic with A,n update law for To design t.his uprlate law, we augment Vo with a quadratic tP.l'fll in tbe parameter error 0,
e.
1 'I 1 " Vi(x. , 8) = _;rM + 02' 21" lvhere /' > 0 is the adc/.IJlation gain. .
V'l
Tlu~
(3.ll)
derivat.ive of tbis func'tioll is
1 :.
= .1::i + 88 1 =
'I 1 :. CIX + OXcp{.l') + 88
l'
=
ClX~
l)
+8 
[
"10
a'cp{;l'} + 1 :..]
(3.12)
e
The second term is sUll indefinii;e and conbuns as a fad.ol'. However, t,hc ~ituati(~)11 is much bet,tel' t.hall in (3.l0), because we now h~we t.11e dym:Ullics of
8 = 0 at
our disposal. With the appropriat,e choice of indefinite tcrm. Thus, we choose thc update law
jj = which yields
l~
0 =
I :rcp(:c) ,
= e)I?;!!::; O.
9 we can
callcel t.he
(3.13) (3.1<J)
The resulting adaptive system consists of (3.1) with the control (3.7) and the update law (3.13), and is shown in Figure 3.1. In Figure 3.2, t;hi~ system is redrawn in its closedloop form consisting of (3.8) and (3.13), nAmely
+ 8cp(x)
;1' =
C),T.
8
")' xcp(x).
=
(3.15)
90
ADAPTIVE BACI(STEPPING DESIGN
PLANT
I I I
I I
···: I
I
I I ~
•
___
_______ M ______
~
•• ____
~
~
__ •
_____ . _ . _ . ___ •
______ M _______________
~
________
~
__ • _________
_
cp(x) ADAPTIVE CONTROLLER
Figure 3.1: The closedloop adaptive system (3.15).
=
=
Because l~ ~ 0, the equilibrium :1; 0, 0 0 of (3.15) is global1y stable. In addition, the desired regulation property li1l1t_oo x{i) = 0 follows from the LaSalIeYoshizawa thC!orem (Theorem 2.1). The adaptive nonlinear controller which guarantees these properties is given by (3.8) and (3.13): 'U
=
8 =
C1X 
8cp(x) (3.16)
'Y xcp(x) .
One may think that the above adaptive design is so straightfonvard because (3.1) is a firstorder system. In fact, this is due to the matching condition: The terms containing unknown parameters in (3.1) are ill the span of the control, that is, they can be directly cancelled by u when f:} is known. To illustrate this point., let us consider the following secondorder system, where again the uncertain term is "matched" by the control u;
Xl :;:2
= =
+ 'Pl(X1) 8!.p2(X) + u. X2
If (J were known, we would be able to apply Lemma 2.8: First view the virt.ual control, design the stabilizing function
(3.17) :C2
as
(3.18)
3.1
91
ADAPTATION AS DVNAM1C FEEDBACK
(J
x
tp(x)
J
1+'
Figure 3.2: An equivalent representation of (3.15). and thon form tbe Lyapunov function
V~(x}= ~:l:i+4(X!! al(:t:d)2
(3.19)
t
whose derivative is rendered negative definite t~(x)
= clxi 
c!! (x!! 
Cll)2
(3.20)
by the control
Since 8 is unknown, we again replace it with its estimate iJ in {3.21} to obtaiu the adaptivo control law: 'il
== C2 (x:! 
0:1)  Xl
This results in the error system
80:1 + a (:V2 + CPI) 

f}CP2(X).
Xl
(Zl
=
Xl, Z!!
=
X2 
(3.22)
0'1):
(3.23)
Then we augment (3.20) with a quadratic term in the parameter error obtain the Lyapunov function: 
1 .,
1.,
1..,
1
~')
l'i(z,6) = If.: + 0 = ;;;;; + 2=2 + ;;0. 21

7
0 to
(3.24)
Its derivative is (3.25)
92
ADAPT[VE BACl(STEPPING DESIGN
/1+Figure 3.3: The closedloop adaptive sysLem (3.28).
The choice of update law (3.26)
plimint'l.tcs the jj~terll1 in {3.25} and reuders t.he derivativc of the Lya.punov fuuction (3.2"1) nonpositive: (3.27)
=
=
This implies tbo.t the z 0, jj 0 equilibriulll point of the closedloop adaptive systenl consisting of (3.23) and (3.26) (see block diagram in Figure 3.3)
:, [ ~~] = ij
=
[~i ~~] [~~ ]+ [ \02~X) ] fl 'Y [0 \02 J[ ~~ 1
is globally stable and, in addition, :c{t) ;. 0 as t
3.2 3.2.1
(3.28)
+ 00.
Adaptive Backstepping Adaptive integrator backstepping
The adapt:ive design iu the ltbove c..'\\:amples was simple because of the umtching: The pluametric nnccdainty wa.s ill the spall of the control. Vve now move to the more genera.l c.asc of extended matching, where the pnrametric uncertainty enters the system one integrator before the coutrol does:
.1:1 X2
= =
.1:2 'll.
+ 8tp{:I.'I}
(3.20a) (3.29b)
3.2
93
ADAPTIVE BACKSTEPPING
,\Ve use this e.xample to introduce ad{l.ptive 1m ckstepping. If (J were known, we would apply Lemma 2.8 t.o design the stabilizing function for 3:2 (3.30) 0'1 (.1;1,0) = elI)  8cp(J'd , wit.h the Lyapunov function {3.31}
whose derivative is rendered negative den.nit.(:'
(3.32) by the control II
=
C2 (X2 
(]'1(:J:l, 0))

D0
Xj
1 + ~(.1:2 + fJcp).
(3.33)
UX1
Since (J is unknown and appeam one C{luatioll before t.hl:' cont.rol docs, we can not apply Lemma 2.8 because the dl:'pendence of (tl (~r:I) = Ct.}:j  O'P(:CI) on the un}nlown parameter makes it impossible to contiuue t.he procedure. However, we can utilize the idea of int.egrator backstepping. Step 1. If .1.'2 were the control. an adaptive controlJer for (3.290.) would be given by (3.16): 0'1 (Xl!
t9d =
01
=
Cl=l
(3.34)
17 j 'P(J'd
(3.35)
1'=1CP(:Z:I)'
In the above equations we have replaced the parameter est.imatc
n
with the estimate th, which denotes the estimat.e generat.ed ill this design st.ep. As we will see, there will be another estimat.c generat.ed in the next. step. 'Vit.h (3.34) and t.he new error variable z:! = .l."::!  C\ I, t.he =Ieqnatioll becomes (3.36)
The derivative of the Lyapullov function
(3.37)
(3.38)
ADAPTIVE BACKSTEPPINC DESIGN
Step 2. The derivative of Z2 is
llOW
eJ..1l1·essed as
Substituting (3.29a) and the update law (3.35) results in =2
= =
oal
U'lL 
s; (x!! uXI
oat
+ Otp) 
at) 7tp Zl !
00'1
00']
DCXl
aXl
OVl
aJ~l
X:z  7tpZl  6tp.
(3.39)
At this point we need to select a Lyapunov fUllction and design II to render its derivative non positive. Onr first, attelnpt is the augmented Lyapullov function
It;(z!, =2t nd = Vi(z), '19 1 )
+ ~z~,
whose derivative, using (3.38) and (3.39), is
The control u should now he able to cancel the indefinite terms in l~. To deal with the terms containing the unknmvn parameter 0, we will try to employ t.he e..~sting estimate '19):
From the resulting derivative
.
'1"
)80'}
V:! = cl zi  c2 zi  (fJ  '191 8 'PI:;!!, ·'1:1
we see that we have no design freedom left to cancel the (8  iJ1)term. To overcome this difficulty, we replace () I in the expression for u with a ne'W estimate t?!!: (3.40) \Vith the choice (3.40), the =2e{luation becomes •
 = C'Jz.,  
':'J
%1 
(0 
EJo'!
'{)'J)Ill
 8xI .,...
(3.41)
3.2
95
ADAPTIVE BACI<STEPPING
The presence of the new paramet.er estimate 192 suggests the following augmentation of the Lyapunov function:
(3.42)
The deriv{1,tivc of
"2 is
(3.43)
Now tho (0  O!!)term can be eliminated with the update law .
02
an1
= l'a cpz'!. , Xl
(3.44)
which yields (3.45)
The equations (3.41) and (3.44) along wit.h (3.36) and (3.35) 1'01'111 I.ho error system representation of the result.ing closedloop adaptive syst:om:
+ Z2 + (fJ  D1)cp
=1
=
Cl=l
z~
=
C'l':'1  
ill = 172
=
=1 
(0  O.,)~cp  Dx,
(3.46)
"Y CP=l  "'I thncp8x, 2,
The matrix form of this system,
d dt
['17) ] {)2
=
"[
~ ~,,] [ ~~ ] ,
(3.47)
makes its properties more visible: • The constant system matrix has negative terms along it.s diagonal, while its offdiagonal terms arc skewsymmetric, and • the matrix that multiplieR the parameter errors in the iequat.ion is used in the update Jaws for the parameter estimates.
96
AOAPTIVE BACI(S'l'EPPING DESIGN
Tho stability properties of (3.47) follow from (3.42) and (3.45): The LaSalleYoshizawa theorem (Theorem 2,1) establishes tha.t =1! :2, {)I! 11'1 are bounded, and = io 0 as i  7 00. Sin<.'e':J :l'l, .1'1 is also hounded a.nd converges to zero. The boundedncs5 of J:2 then follows from the hounded ness of 0'1 (defined =2 + 0'1' Using (3.40) we conclude thai: the in (3.34)) and the fact. thai. .1'2 conf;rol 11 is also boullded. Finally, we note that the regula.tion of and :1:1 does nDt imply t.he regnlat.ion of :1'2: From =2 = X2  al and (3.34) we see that :1'2 + t?1'P(O) will converge t.o zero. Thus, .1:2 is not guaranteed to converge to zero unless cp(O) O. Howevcl', .}'2 will converge to a constant value:
=
=
=
=
lim = Hcp(O)
'~co
This can be seen .1:2
frolll
(3.29a): Since
Xl
b= :1'2 .
(3.48)
and j:1 (.'OllVerge to zero, so uoes
+ 8cp(0),
\Vith the above example we have illust.rated the idea of adapt.ive backstopping. To formulat.e it as it design tool analogous to Lemma 2.8, wo start wit.h the assumption thctt an adaptive l'ontroller is known for an init.ial system,
Assumption 3.1 COJ1,Sitlc1' the system .1: = f(x)
+ F(:r)O + g(J')u ,
(3A9)
whe1'f~
:,; E JR" is the ./itate., (J E HlP is a vector of unknown cOfl.st.a.nt p(l.7'Q.m.eter's, a.nd 11. E 1R is the r..ontml input, There exist (/.71. adaplil1e controller
= J = 'lJ.
0:(:,;,19)
T(x, tl),
(3.50)
with parnm.ete1· eBtima.te {) E IRq. and a 8mooth Junction V"(x, '19) : IR(r,+q) io R wMch is positit1e definite a.nd 1'adia1l11 'Unbounded in the va1'iables (:1', {)H) /:WI"/t that/oT all (.1:,,19) E lR.
81'
8:1: (:1.', l) [j'(:1:)
1IJhe1'f~
al'
+ F(:I.·)O + g(x)o{x, 19)] + atl (x, t9)T(:r;, tl) :5
H' : Dl,,+q
~
lR i.5 positivc semidefinite.
H'(:1:,O) :5 0, (3.51) 0
Under t.his assnmption, the control (3.50), applied to the system (3A9), guarantees global boundedness of .r(t), 'O(t) and, by the LaS al1 eYoshizRwa theorem (Theorem 2.1), regulation of IV (x(t) , iJ(I.)). Adaptive ba(~kst.cpping allows us to achieve the samp. properties for the augmented system.
Lemma 3.2 (Adaptive Backstepping) Let the systcm (3.49) be augm.ented b11 an integm.tor,
±
=
f(x)
~
=
u,
+ F(:v)(J + g(:c){
(3.52a)
(3.52b)
3.2
9i
ADAPTiVE BACl(STEPPINC
whe7'e
eE JR.
Consider for this .'f1Jste.m the dynamic feedback controlle7'
11. =
] + au a:t (:1', lJ) [/(3:) + F(;r)tJ+ g(:r:){
c(~  Q'(x, 17))
ao:
DV'
+ 8{)T(x, 19)  ax (;1:, t7}g(:l') , J
=
c> 0
(3.53)
T(x, iJ)
(3.54)
J = r [:: (x, 11)F(Xf (t;  a (.r., 11)),
(355)
where {j is a neul cstima.te oj fJ, r = r 1' > 0 is the adaptation gain maida:. Under Assumption 3.1, this adll.ptive cont1TJller g1l.c1.mntees global bo1t7l.dcdness oj x( t), ~(t), D(t), 0(1) a.nd 7"C!J'ull1.tion of fI'(J:( l), t?(t)) and ~(t) ll'(~l:(t), iJ( t»). These ProlJe7'tie..r; can be e.r;tablished 1JJith lile Lyapunov ju.nction 1 1 " 1 l~(:l:, ~t iJ, 17) = "(x, t~) + 2' [{  O'(:l~, ,a)] + 2(8  '0) r (8 19) . (3.56) I)
e
Proof. vVith t.lm error variable == 0:(;1:,19), (3.52) is rewritten as x = J(:I:) + F(x)() + g(x) [0:(.1;,11) + ~] (3.57a)
z
=
u  : : (x, iJ) [J(.'l:)
+ F(;r)O + g(.1:) (O'(x, ·O) + :;)]  :~'T{:r, .[)).
(3.57b)
Note that in (3.Sib) the derh'ative of {) was replaced by t.he update law (3.54). Introducing a new parameter estimate 11, we augment the Lyapullov function:
=
F(a:,·tJ) + _?1:;2 + !(8  ii)Tr 1 (8  11) . ... 2 Using (3.51), it is easy to Rbow that the deriva.tive of (3.58) satisfies l'o.(X'';1 0,19)
all
.
(3.58)
all'
Va = a3.~ (J + FIJ+ go + g:;) + {J.t) T +: [II 
=
81'" 8J.:
00' (J
8x
lY (x I 19)
+ cH) T
+ F8 + g(a: + z})  a~'T + av' 9] _,nTr 1 (fJ af) 8:r
+ z [u. 
 [:: F: + jT
:;T]  ilr'(9  ;i)
811
(f + FB + go')
+:: [u 5 
~: {f + F9 + 9(0' + =)) 
r']
i9)
oa (f + F';9 + g( a + .:») _ 80: T + at' gJ o.T at) ua' (9  0).
(3.59)
Tlle (8  79}term is now eliminated with the update law (d. (3.55))
~
=
r (ao~F)TZ' u:J.
(3.60)
98
ADAPTIVE BAC1{STEPPING DESIGN
and the control (3.53) is cbosen to make the bracketed term multiplying : in (3.59) equal to cz (ef. (2.54)): IJ
=
aa ( 
c=+ 8J.' f+FtI+g(a+=)
) + 80' al' 81?T a~.g·
(3.61)
This results in the desired l1on]>ositivity of l~:
\~ :::; rV(:L',1?)  c:2
:::;
o.
(3.62)
From (3.56) and (3.62) we conclude that "'(3:,0), i9 and z are bounded. By Assumption 3.t this means that. x(t) and l'(t) are bounded. Ilel1t'e,'; = z + a(.l:, ·0) and It are bounded. By Thorem 2.1, the boundedness of all the signals combined wit.h (3.62) proves the reglllat:ion of H'(x(t), t9(t)) and =(1).
o
3.2.2
Adaptive block backstepping
We now extend the Adaptive Baclc.stepping Lemma (Lemma 3.2) byaugmenting the initial system with a relativedegreeone nonlineaT system whose zero dynamics subsyst.em is ISS, justlike we did in Chapter 2, Lemmas 2.8 and 2.25. The adaptive counterpart of Assumpt.ion 2.7 was Assumption 3.1. "Ve now formulate the adaptive counterpart of Assumption 2,21, with analogolls changes in the properties of l/(x, t?) from Assumption 3.1,
Assumption 3.3 Suppose Assumpti.on 3.1 is 'Volid, b,J.t l'{x,19) is onl,} positi'Ve semidefinite. and the closedloop system {3.49} 'U1ith the atJ.apf.ilJC controller (3.50) h{t.s the lJropert1} that x{t) a.nd 11(t) are bounded if l!(x(t), tI(t)) is bounded. 0 Under this assumption, the control (3.50), applied to the system (3.49), guw'antees global bouudedness of .t·(t), t9(t) and, by Lemma A.6, regulation of
IV(x(t) , O{t)).
Lemma 3.4 (Adaptive Block Backstepping) Let the sJ}stem {3.49} be augmcnted by a 11.o71.linear system llJhich ill linear in the un/""1lo'Uln param.ete7' 'Vector 0, (3.63a) :1: = I(:,;) + F(x)8 + g(x)y (3.63h) € = 1n(:l:, {) + l\1(x, {)O + ,8(x, e)u, y = h(eL
e
where E :IRq, and suppose that {3.69b} has rclati11e degree 071·e 1.t.niforrnly in x and tllat its ze1TJ dynamics SUbsystem is ISS 1lJith. respect to y and x. Under' Assumption 3.3, the feedback C07l.trol tJ.
8h = [ a{ (';),8(:z:, e)
]1 {
ah C(l}  o'(x, 19))  8e (e) [m.(x, e)
+ l\l(x, {)i9]
] + aCt + 80: ax (x, 'D) [J(x) + F(x)tJ+ g(x)y af) T(x, ·0)  av ax (x, 11)g(x) } (3.64) I
3.3
99
RECURSIVE DESIGN PROCEDURES
with c
> 0 and 19 a ne'lIJ estimate of (J, along with I.he 1J.1Jda.te laws
iJ = T(:r,19) !..
f)
=
r
(3.65)
[8h a~ (~)}.J(X,~) 
with the adaptation go.in matrix r =
1 (1) 
80: u:r (.r, {})F(.r)
T
0:(:1:, 17)) ,
(3.G6)
r T > 0,
guam.ntees global bO'lJ.ndednes.fl oj .r(t), ~(t), 19(t), lj(t) and regulation oj l'F(:r(t), ·tI{t)) cwd ~(t)  a:(:r(t), {}(t)).
Proof. As in Lemma 2.25, we employ the change of coordinates (u, () = (h(~), 4>(:r:, ~)), with ~:/3 == 0, to transform (3.63b) into the normal form ;lj
= ah D~ U) [m.(:r:,~) + 1&1(:r:, ~)O + f3(x, ~)1I] a~
( = D:r (J',~) [1(.1') + F(x)O + g(:l:)'lJ) + l::,.
<po(x, y, ()
(3.67a)
a~ a~ (:t,~) [m(:r,~)
+ cJl(x, y, ()tJ .
Introducing a new parameter estimate " =
to rewrite (3.63a) and (3.67a) as x = f(x)
1j
(3.67b)
ii, we lise the feedback t.ransformat.jon
r {II  ~~
(~~p =
'U
[m +MD]}
(3.68)
+ F(.1·)9 + g(:I:)1/ fJh
+ AI(x, ~)B]
+ a~ (~)l'f(:r:, ~)(tJ 
(3.69a)

(3.69b)
17) .
We now apply Lemma 3.1 to (3.69). The only difference between (3.69) and (3.52) is the presence of the additional parameter error term ~~ 1&/(0  0) in (3.69b). This term can be eliminated in ,~'l by adding the term r(~~llJP'(y ex) to the update law (3.55). Combining this modifi('ation with (3.68), we see that the resulting adaptive controller is given by (3.64)(3.66). This guarantees the boulldedness of x, of), 19, z and the regulation of H'(:t,19) and z. Hence, 1J = Z + Q·(x, '19) is bounded. Then, from (3.67b) and the ISS property of the zero dynamics, ( is also bounded, and thus ~ and 1I are bounded. 0
3.3 3.3.1
Recursive Design Procedures Parametric strictfeedback systems
Through repeated application of Lemma 3.2, the bacl\stepping design procedure is now generalized to nonlinear systems which can be transformed! into IThe coordinatefree charncterizat.ion of I.bcse syslems in t.crms conditions is givcn in Appendix G, Corollary 0.15.
or differential geometric
100
ADAPTIVE BACKSTEPPINC DESIGN
Figure 3.4: Block diagram of it. thirdorder parametric strictfeedback systcm with
P(x) = 1. The nonlinearitics depend only on varia.bles which llre "fed back." the paramel1"i.c .drictfeedback j07m
:h =
3:2
.1:2 =
:t'3
+ V'I(xJ)O + I.{)I (x 1, x2)6 (3.70)
i:nl
=
:1:'1
=
Xu + tp~l (Xb" • ,xtldO ,8(x)u. + V'~'(X)O,
where j3 (x) :F 0 for all $ E 1Rn. The reason for the name "parametric strictfeedback" can be deducoo from tIle block diagram jn FigtlJ'e 3. t 1, whel'e, except for the integrators, there are only feedback paths. For systems in the fonn (3.70)1 the number of design steps required is equal to the degree 11 of the system. At each step, an error variable ::i: a stabilizing fUllction ll';, and a parameter estimate fJ, are generated. As a result, if a system contains p unknown parameters, the overparametrlzed adaptive cont:l'OIler may employ as many as p 11 parameter estimates. A schematic representation of this design procedure is given in Figure 3.5, and the resulting expressions axe summarized in the fonowing theorem:
Theorem 3.5 (Parametric StrictFeedback Systems) For the sy.'Jtem (3.70) with. {3(:c) :F 0 f07' all x E IRn , consider the adapU.1Je controller' (3.71)
i = 11 " when; iJ i E lRl' lIte mu.ltiple estimates
of () r
rT >
n1
'l'
(3.72)
0 is the adaptation gain m.at7ir., and the varia.bles ::; and the stabilizing fun.ctions Gi, i = 1, ... , 1l t 1
=
3.3
REcuRsIve DESIGN PROCEDURES
r =.
,
I
I
I I
I
101
I
_.J
0'1
' _____
~
___
Zq
I
~.J.
ll'::!
~a:'

11
U
Figure 3.5: Thc design procedure for overpa.rnmetl'i~ed schcmc.c:;. Each step generntes an error variable =i, a stabilizing fUllctioll Q;. nnd n. new cstimate 1?i of the unlmmvll pILT8metel' vedOl' 8.
defined by the following recu1'si'ue e:r.p'ressions (with C; > 0 being design constanl.s, and 4:0 :: ao == 0 '1I.S(1.(1 f01' notat.iDnal c01wenience):
aJe
This overparamct'li.=ed adaptive cO'1l,t1'(Jller ,Q1£lI1'llnlees global bO'Ulldetlness oj x(t), 'l?(t), ... ,'l?n(t), and regulation oj:t'.{t) ancl:r'i(1),l:Y, i = 2, ... ,n. where ':1'1 = OT tpil (0, J;~t·· . , a:l_I)' Proof. Using the definitions (3.73), (3.74) nnd denoting Xo == O:u :: 0, P(.l:)ll~ the derivative of the error variable =j, i = 1, , .. ,TI, becomes
:1'11+1
==
102
ADAPTIVE BACI{STEPPINO DrnSJGN
The choice of control (3.71) guarantees that system can therefore be expressed 88
':,11
=
Zn
=
fJ i
=
Zn+l
== O.
The closedloop error
or, equivalently, in the matrix form =1
Z2
d
==
dt
Cl
1
0
1
C2
1
.,.
0 1 Cnl 0 1
0 0 [ WI
+ ~
0
rlt
: tJ n
=n1
1
Zn
C n
T
 f)2 [ 8 9d,
W2
1
(3.77)
0
0
~ [:~ 1 = [1
:!:1 %2
0
Zn1 "'71
0
0
WI!
8  On 0
0
r 0
11
[I
0
1 [ , 1 ~ft :
W2
0
z..,
;,,'
3.3
103
RECURSIVE DESIGN PROCEDURES
where we have used the convenient notation il
WI
=
'PI ,
'Wi
= 'Pi 
L
00'i_l
a'Pj
I
i = 2, ... , n .
(3.78)
Xj
j;::;;;1
Tbis system has two important properties: (i) The zsystem matrix in (3.77) has negative diagonal and skewsymmetric offdiagonal t.erms, and (ii) t.he transpose of the matrix that mu1tiplies the parameter errors in the zequat.ioll appears ill the update law. This structure is a result of the design procedure, and it allows us to use the simple quadratic Lyapullov function 1 '1 lIn (.::1, ... ,Zn, Dh .. , ,t?,,) = ry + (8  OJ )'I'r] (8  iI;)] (3.79)
L [=;
... i==1
t.o prove stability and regulation. Its derivative along the solutions of (3.77) is
,i"n =
=T
z
n
~,t9lrI(8

t9 j )
1=1 n
= =
L
[CiZ~ + Zjw;(8  iJ j )
;=1
,.
L:CjZ;.

zi w ?,(8
19;)] (3.80)
i=I
The LaSalleYoshizawa theorem (Theorem 2.1) now guarantees t.he g10bal uniform boundedness of z(t), ill (t) •... , '!9 n (t), as well as the regulation of =(t). Since Zl = X], we see that :l'1 is also bounded and regulated. The hOl.lndedness of X2, •• • , Xn then follows from the boundedness of O'j (defined in (3.74)) and the fact that Xj = Zi + aiI Since x is bounded, {3(..:) is bounded away from zero. Combining this with (3.71) we conclude that the control 11 is a1so bounded. Finally, the regulation of .'Vi  xf is concluded as follows: Since Zj(t), i = 1, .. _,11 converge to zero, Jdt), i = 1, ... ,n also converge to zero and =i(t) is integrable over [0,00). Furthermore, the boundedness of all the signals and their derivatives guarantees the boundedness of =i(t) and hence the uniform continuity of Zj(t). From Lemma A.G, we conclude that lim/_ oo Zj(t) = 0, i = 1" .. , rI. Since x can be expressed as a smooth vector function of =1, • , • '=11 and '011' • , , {)n , we can express X as a linear comhination of Zj and 17i wit.h coefficients which are bounded because they are smooth vector functions of the bounded signals ZI," . ,Zn and 1)1,.' . I {In' Hence, the convergence of Z and tJ j to zero implies that :i: converges to zero. Combining this with (3.70) and the regulation of XI leads to the desired result. 0
3.3.2
Multiinput systems
The adaptive backstepping design procedure of Theorem 3.5 can be easily e}..'i;ended to nonlinear systems which have been transformed into the multiinput pam,metric strictfeedback 101m
104
ADAPTIVE BACKS1.'EPPING DESIGN
j.'I,1
';'1,2
= =
Xl,!!
+ cprl (3,'1,11'1"2,., ••• ·1"2,P2PI+h"
·1'1.3
+ cpI:ll.·1.1,J~I':!1.l~2.Jl'" •• , !·T.'rn.J,·"
J" l,p,]
., :L'm.lI ••• , ,l:m ,Pn,PJ+J)O
.1;2.p'.JPI+21
,X rn •p,,,Pl+2)9
= ·&.l.P1' ", + cp1'l.Pl1 (...."1.1,···, "l,Plh· :/. .... ,., 2.h··· "'2,P2h L
, •• I
·'I'm,], • , • 1 ,t m ,pml)6
m
i'l,111
I: f31J (.x) lJ.j + 'PT./)1 (:1:)0
=
j=l
(3.81) !i:i.)
=
Xi,j+]
+ cpl,j(xl,h' .. , 3.~l'P!P.+jl' •• ,3:;,1, ••. ,l'i,j, •.. , :t'lJI.l, ..• , X m ,p".PI+i}8
;i'm,l
,i'm,2
= =
+ It'!.,1 (.t'1.1,' .. ,.1'I,PlPrn+b .:r2,h ' , • X!!,p:tp",+b •.. .1'm.3 + fP;'.2(XI.lt., • ,3:1,PIP,"+21 ;]=2.1, , •• X2.P'.!Pm+2, .1: 111 ,2
• • • 1 :l'""l t
:i:m'PPI,J
=
Xm,Pm
:r:m,I)O
:l:m ,2)0
+ tp~'PfIl1 (.J;],., ...  •• I
1
1
1'l,PI]' X'J,l, ' •••1:2,{I2J 1
Xm.l1 ••• ,Xm,p,.._I)O
711
:i'm,p".
=
2: f3".J(:V)Ui + lP~'Pl (X)O
1
j=J
wberp 'Uh'"
, Um
are the inputs, and tbe input matrix is llollsingulru.' 'V:I:
e m,n
(ll=PJ+"'+Prn): clet B(l:) ::F 0, 'V ;1: E
m.n
(3.82)
I
The design procedure for this class of syst.ems cOllsists of applying the design proeedure of Theorem 3.5 to the fil'st Pi  1 equations of each of the m subs)'stems of (3.81), to obttUll the system CiJZi,j 
=iJl
+ ZiJ+J + tl1i ,)X, 191 " , , ,t),.l )({I llt) '('
;1
e= L:(Pk 1) + j ,I$. j S Pi  1, k=. = r Wi.j(X, 1111 , ,. , tlc)z;J' 1 $. e$. In 1]. 
1 S i $. m (3.83)
3.3
105
RECURSIVE DESJGN PROCEDURES
where t.ho functions Wi,j, rI> and HJ:II _ m +1 are defined appropria t.cly. Now let o"JII+I be a new estimate or 0 and define the cont:rol1J. as
lL
=
CI,PI ZI.Pl
B 1 (.1:)
{
[ Cm'PrrI7R,Pm

+ ':J ,PIl
_:
]
_
 ID(.1:, #1, ...• ,0 11  " , )
+ m,p... l
W~~m+l (x. 19" ...• U
n .n jU,,m+ 1 } •
(3.84)
and t.he updat.e law for 01lm+l as
(3.85)
The stability properties of the resulting closedloop system are analogous to those listed in Theorem 3.5, and can be similarly established Ilsing the Lyapunov fuuct.ion
{3.86}
3.3.3
Parametric blockstrictfeedback systems
Lemma 3.4 can be applied repeatedly to design adaptive controllers for nonlinear systems which can be transformed, after a change of coordinates, into the pa7·ameL1i.c blockstridfeedback form
Xl =
11 ttl) + FI (XI)6 + iit (Xl )Y2 11.1 (:\:1)
111
=
:\,'1
= h6:'1, :(2) + F':!(XIt X2)8 + 02 (X.I ,X:!)Ya
Y'l.
=
h 2 (X2)
X·
=
Yi
=
/;("Xl, ... ,Xi) + Fi(Xll ... ,Xi)6 + 9;(YI, ... ,Xi)Yi+l hi(Xi)
.
I
Xpl Ypl
:\,p YP
(3.87)
= j,ll (Xb ... ,Xpt} + Fp J (Xb ... , Xpl)9 + 9pl h'l' ... ,Xpl )yp = hp _ 1(Xp J) = jp(xJ + Fp(x,lO + 9p(X)U = hp(Xp) \
Ir06
ADAPTIVE BACKs'rEPPING DESIGN
lvhere each of the p subsystems with state Xi E m.nl , output IIi E lR, and input HI (for convenience we denote Yp+l == u) satisfies conditions (BSF1) and BSF2) (see Chapter 2, equation (2.198)), t.hat is, it has relative degree one niformly ill All' .. ,Xih and its zero dynamics subsystem is ISS with respect oX}'···, XiI! Yj·
Using the change of coordinates which transformed (2.198) into (2.201) in ection 2.3.3, we can now transform the system (3.87) into IiI
=
Ypl
=
11 (yt, (I) + CPI'1' (Yll (1)0 + Yl(Yh (1)Y2 Y2 = hhJl,(ldJ21 (2) + cpI (111 , (11 Y2,(2)(J + 92(1111 (ltlJ':!, (2)1/:1 /pl (y., (1, ... 1 IIpb (pI)
=
(p
=
I
1/pl. (p_I)8
+gp1(Yl, (h' .. , UpII (,,I )yP (3.BB) 1p(/l1, (1, ... , Yp, (,,) + qJ~ (Vll (I, ... , 1Jp, (p)O + 9p(YI , (1, ... 1 Up! (II)?/'
YP = (1
1' + 'Ppl (VI, (II ...
cI> 1,O(Yl, (1)
+i
l (Yl, (1)9
Then we employ anotber change of coordinates which replaces Vi by
t/JdUh (h'"
1
YiI! (ilt
=
YI ~ 1/'1 (Y1)
·'t·2
=
;;;(/1 (.lUI
:ti+l
=
+ 91112) =
II(y.,(I) + Ol(11I,(t}~J2
iI 8~J,
iI
=A '~'2(11I'(1'Y~)
tPi+ I (Yl t
(3.89)
81/Ji _
L ~(fj + gjYj+l) + L ar.~J WitO + 91 ... OiI!i + 91'" i=:1 ul/j
l:l
=
Hi), where
Xl
a~tl
Xi
9iYHI
j=1
'It ... ,
YEt (i I YiH)
i = 2, ... ,p  1 .
f
FinallYl we use the feedback transformation p 1
v =
L
j=J
8VJp
0. (h Y.1
pI
8t/Jp 
j=1
J
+ gjYi+l) + L
8(. tI>j,O + 91'" glll!p + gl'" 9p U. (3.90)
CondHion (BSFl) guarantees tbat UI,'" ,gp :F Q everywhere. Hence, the change or coordinates (3.89) relating [YI! ... dIp, (l t • • • , (JJT to [:CIt .. ,.1:/1< cT (J]T is a global diffeOJnorp11iSnl, and the feedback transformation (3.90) relat.ing II to u is llonsingular. It is now straighLfonvard La verify that (3.S9) and (3.90) transform (3.88) int:o a form l'Cmilliscent of tJle I ' •• t
3.3
107
RECURSIVE DESIGN PROCEDUR.ES
parametric strictfeedback form (3.70): !i'l
X2
=
+ 'PT(Xl1 (1)0 == Xa + \O~'(.Th X2t (it (2) X2
X,,l == XI' + cp~_] (:rl1 .•• , ·2:,,_11 (" ... t (Pl)O .i;p == v + 'P~'(XI ()6 (1 = Wl.0(l!I,(1)+ ID 1(Xlt(dB (" = y
=
tPp,O(a:h'"
(3.91)
,.l:p , (I," . ,(plt (,,) + tP p(:l!I •• •• toVPI (It ... ,(pll (,,)0
$1·
In (3.91) each (isubsystem is ISS with respect to 3'" ... , Xi, (1, ... , (;1 ~i,O, eDit i = 1, ... ,p are defined as
ru;
its
inputs, and IPi,
a'Pi L a.(YII (Jf .•. i
11,
j=l
I
YiI, (;1, lIi)
~J(Ylt'l" .. ,Yj,(j)
iIo,p.
+L
j=1
0(' bJh(j. .. ·,Yilt(ih1Ji) J
~AY1t (1, .... Hjt (j)
(3.D2) (3.93)
(3.94)
It is now clear t11at UlC class of parametric blockstric1.feedback nonlinear syst.ems strictly contains the dass of paramet:l'ic strictfeedback nonlinear systems, sill('e (3.70) can be obtained by setting ni 1, p 71, and v p(x)1J in (3.91). We now state and prove t.Ile generalization of Theorem 3.5 to blockstrictfeedback systems of the form (3.91).
=
=
=
Theorem 3.6 (Parametric Bloc1c.StrictFeedback Systems)
F07'
the
slJstem (3.91), consider U."e adaplive controller (3.95)
i
= 1"",p,
(3.06)
108
ADAPTIVE BACKSTEPPING DESIGN
whc7'e tJ j E 1RP a.re multiple e.r;tima,tes of (), r = r1' > 0 is t.he a,daptation .qain matri.:c, and the vmi.ables Zj anti fhe sta.bili:.iug jlJ.nctions O'j, i = 1, ... , P. are defined by the following 7'Ccursil1f C3.']J1'fJ8,1iions (with. Ci > 0 being design constants, and Zn :::;;: 0'0 == 0 1/sed fo'" 1J.ot(].tional convenience): (3.97)
D:j
=
Cj=j 
+
=il 
8a i  1 ..T), 8(, ':iJ,O
[
_
J
~ (DO;il T 80';1 )] 0; + ~ {80'iJ   ! . p j + Wj L J.'i+l 8x 8(j j=1 tJ.1:j
IfJi'1'  L
I
i=l
fJO'i1
8 11 • /'J
r
j
'1 ?' _ ~
[ 'PJ
L
k=l
(80;jl a
XI.'
]1'} (
(T») _,
f)a:jl f)r i:A' ',1.
J'
3.98
)
Thi.9 overparamct7'i::.ed adapli1.e controller gua,rnntces global boundedncss oj .,ch, ... , iJp a.nd reg'ulal.ion oJ:lJ = Xl'
.1:1,'" ,.J:p , (1,' " ,(pI
Proof. As one would expect from the similarities between the systems (3,91) and (3.70), t.he e.'\:pressions (3.95)(3.D8) are similar to (3.71)·(3.74). Using the same arguments as in the proof of Theorem 3.5, we write the derivative of the error variable Zj, i = I, ... , Pr as
3.3
109
RECURSIVE DESIGN PROCEDURES
The closedloop error system can thus be expressed in the matrix fonn ZI
d
dt
:2
=
Cl
1
0
1
c:!
1
0
':::1 .:~
0
0
(3.100)
with the not.ation iI "'"
[!'l
Uail
Wi = 'Pi  L." ..
+
(!'l
UO:i}
.(1).1
8(}
)or]
,
i = 2, ... , p. (3.101)
The struct.ural properties of this error system are once again apparent. The derivative of the nonnegative fUllction (3.102) along tlle solutions of (3.100) is p
l~
=
Lei=;'
(3.103)
i=l
The LaSaUeYosbizawa tllcorem {Theorem 2.1} now guarantees the global uniform boundedness of z(t), 0 1 (t), .. . ,vp(f), as well as the regulation of z{t). Since =1 = :ll, we sec that. y .t'} is also bounded and regulated. Tile boundedness of x:!, ... , x pt (It ... ,(p and of t.he trallsformed control variable v is then est.ablished vitt an induction argument for i = 2, ... I P + 1 (with Xp+1 == 1J): If x), ... , .1'il and (h .. " (12 are bounded, the ISS property of the (i_rsubsystD111 guarnntees t.hat (;1 is bounded. Hence, ail(Xl, .. " XIIt (1, .. " (iI, 0., •.. , Oi.) is bounded, which implies that :r.j = Zi + Q:il is also bounded, 0
=
I 10
~
ADAPTNE BACJ(S'I'EPPING DESIGN
\\Te should note that the adaptive controUer (3.95)(3.96), when apI)1ied o the system (3.87) using the expressions (3.89) and (3.90), guarantees the Iobal uuiform bouncledllcss of X and il, as weH as tbe l'egulutioll of y 11] (:\'1). This follows from the fact that ~gj :F 0, which guarantees tha.t tile transformation relating (Yl t • • • , Yn, (11 •.• , (,,) to (X I, ... I J'rl , (II ... 1(n) is a global diffeomorphislll, and the feedback transfo11natioll (3.00) from 11 to 11 :is nonsingular.
,3.4
=
Extended Matching Design
The increase ill the llumber of parameter est.jmat.es ealiRed by ovel'paramet.rization cau be all undesirable feiltnre, since jt. rapidly inCl'E"8SeS tlu~ dynamic order of t.he rcsulting adaptive controller. In Chapter ~l, the oYerparallletdzat.iol1 will be elillliImt;ecl by the tuning functions l11et])od. As prelim.inar.v to this development:, we now show how the ovcrpanunetrizatioll can be avoided in the case of ext.ended matrhing, that is, when the llucprtaill parnmetCl'f; are Duly one integrator away from t.he control.
I
3.4.1
Reducing the overparametrization
\Vc consider again the nonline~u: system (3.29),
.1: 1 =
:l!~
+ Btp(x)}
and modify it.s t.wostep design.
Step 1. Wit.h Z'1 = Xj and :t::! viewed as the virtual control hl the zlequation, we define the first stabilizing function 0') as ill (3.3..:1): (3.104)
Comparing (3.104) with (3.34), we see UUtt the parameter estimate 19 1 has been replaced by the paramcter esl:imate 0. The differellce in Jlotl:ltion iudict:ltcs thcl.t ill this design procedure ouly one estimate 0 of the unknown parameter will bc used. The first. Lyapullov fUnction is now chosen as (3.105)
where 8 = () Witb Z2 = .r2 
Bis 0'1,
the parameter error, and " the derivative of Vi is
>
0 is the adapt~tt:ioll gain.
(3.106)
3.4
111
EXTENDED MATCHING DESIGN
Vie postpone the choice of update law for () until the next st.ep. The first error su bsystt"m becomes
(3.107)
Step 2. The derivative of =2
= =
To design the ('ontro]
Z2
=
X2 
ao: I 'U 
l)XI (X2
0'1
is aO'l :. .
+ OIP)  00 0
ao
au) 1 80'1 DOl ':.. u :r..., 81.{)  (JI.{)   . B. ax! 0.1'1 OJ:. {)e 'U t
(3.108)
we consider t·he augmented Lyapullov function 1'11,)
:j
2
1...
+ .:; + (J . 2 
2"',
(3.109)
The only difference bet.ween (3.109) and (3A2) is the absel1ce of t,be new parameter error (8  O2 ) in (3.109). In view of (3.106) amI (3.108), the derivative
of l'2 is
(3.110) In the last equation, all the terms containing 0 ba.ve been grouped f·ogether. To eliminate them, the update law is chosen as
B= " (tpZl 
80:1
ox)
CP':2) .
(3.111)
Then. the last. bracketed term in (3.110) will be rendered equal 1.0 C2':~ wif:h tl1e control lL
=
(3.112)
where for 9 we usc the analyti('al expression of tbe update law (3.111). Substitutlng the expressions (3.111) and (3.112) illto (3.110) we obtain (3.113)
112
ADAP1.'IVE BACKSTEPPINO DESIGN
(_lu lP ]
'1'
U.r:1
J
1+'
Figure 3.6: The closedloop adaptive system (3.114).
and tile error system becomes (see block diagram in Figure 3.6)
~. [ ~ 1 = [ __~l _~] ~ [
iI =
1
]
+ ( _~~ ] 8
[~  ~~ 1[ ~~ ] .
(3.114)
Comparing (3.114) with (3.47), we see that the system matrbc in (3.114) has preserved the important structural properties it had in (3A7): Its diagonal terms arc negative and its offdiagonal terms are skewsymmetric. Furthermore, we see that, as in (3A7), the matrix that mUltiplies the parameter error 8 in the zequation is usecl (in its transposed form) in the update law for the parameter estimates. It is ruso instructive to compare tbe ex.pressions for the parametel' update laws ill (3.114) and (3.47): Even though the update law for iJ appears in the form of the sum of the update laws for t}. and il2 , the expressions (3.34) and (3.104) for 0'] depend on different parameter estimates (tJ 1 and 8, respectively), fUld thus Z2 and the partial derivative ~ will have different values in (3...:14) and (3.111). Due to the structure of the errol' system (3.114), its stability and convergence properties are derivE'd in a manner almost identical to those of (3.47) and are therefore omitted here. In the extended matching case we avoided the overparametrizati?D by postponing the choice of the update Jaw until the second step_ When 6 appeared in the second step, it was replaced by its known analytical ex.press~.on. Beyond the extended matching case we need more than two steps, so that iJ and bigher derivatives of 6 appear. Instead of the simple idea of postponing the choice
3.4
113
EXTENDED lVIATCHING DESIGN
of the update law, we need the more intricat.e tunin.q juncf;ions method, t.o be developed in Chapter 4.
3.4.2
Example: biochemical process
Extended matc:hing design is also applicable to purefeedback systemH, int.radured in Section 2.3.2, provided that the unknown parameters appenr linearly. 'iVhile the general case of these parmn.etric pu.refeedback s1}stc'rns is presented in Section 4.5.3, t.he extended matching design will be illustrat,ed on u simplified model of a biotechnological process which goes as far back as IvIonod [135]. In spite of its simplicity and somewhat unrealist,ic assumptions, this example is representative of several successful applications of adaptive nonlinear control to more complex processes described by Bast,in [7]. In a model of no fedbat.ch process. S is the concent.ration of the growt.h limitiug substrate, X is the concentration of the growing microbial population, ". is the yield const.ant, D is the dilutiollrate: and the control u is the substrate feed rate. In a batch process, that is, when both D = 0 and u = 0: the rate of microbial growth i; is modeled as i = p(5).¥, where p.(S) is t.he "spedfic growth rate." The nonlinear function p,(S) is usually poorly known. and for our illustrat.ive purpose we parametrize it using unknown parametel's: (3.115) Note that .X, 5, and J.L(5) are nonnegat.ive quantities. \Vith this panllnetrization the fedbatch process operating at constant temperature is modeled by the following two rnassbalance equa.tions:
..:t
=
S =
[IPo(S) + ()IIP1(S) + ()2IP:!(S)].Y  D ..Y k [IPo(S) + 81IP1(S) + 1J21P2(S)].\  DS + 'U.
(3.116a) (3.116b)
The control objective is regulation of X to Ute set point X r • To furt.her simplify the system (3.116), we use the change of coordinates Xl = In ...\'", X2 = S, which is welldefined and invertible since ~y > O. Then (3.116) becomes Xl = IPO(:C2) + ()lIPl(X2) + 82 IP2(X2)  D .1:2 = I;: [lPO(X2) + 814'1 (X2) + 82CP2(.l:2)] e:r'1

D.l:':]. + 'll.
(3.117a) (3.117b)
This system is clearly not in t.he paramet.ric strict.feedback form (3.70), since the nOl1linearities in (3.117a) depend on the second state variable :r:2. However, we can still apply the design procedure illustrated in Section 3.4.1 to this system. In Section 2.3 we saw that our recursive design procedures can be applied not only to strictfeedback systems (2.165), but also to purefeedback systems (2.180), whose nonlinearities are allowed to depend on one more state variable. The price to be paid is that the stability properties are no longer
114
ADAPTIVE BACI{S'l'EPPING DESIGN
Substrate feed (control u)
Heating/ cooling
Figure 3.7: Fedba.tch stirred tank rendar.
global, bu t regional: They are guaranteed only for a compad set of iuitial condit,ions. The same is possible for our adaptive designs. For t.he system (3.117), our design pl'Oceeds by choosing tpO(3:2) as tht:' virtual ('ontrol variable ill (3.117a) and dcsigning for it the stabilizing rnlletioll
(3.118) where Zl
=
=1
=
Xl
= IPO(X2) 
In X r • \'Vit.h Z2
C1Z1
at,
the error system becomes
+ =2 + (0  81)IPl + (fJ  92 )V'2
(3.119)
(aa~O + 0/a:,1 + 82 8aV(2) {  ~~ [V'o + OIIPl + lJ~tp2J eaJ
=2 =
X2
,1:2
X2
I

D:V2 + u}
+ct [Cl=1 + =2 + (0  6d'P1 + (0  82 )V'2] + B1V'] + 02V'!! .
(3.120)
Following the development. of Sectioll 3.4.1, we choose the update law
~
r [ : }{ =, + [c,
=
= [Ol 021.
wherc 01' u
!

+
(3.121)
Tbe corresponding control law
= (!!!!! 8 ~ I ~,~) O:r:l
 k ( : : + 8, :~; + 8, ::) ] =, } ,
I 8:1::1
{ 
C2=2  =1  Cl [C1Zl
+ z!!  D1V'l  92V'2]
 DJ:'J
[~ .. v>ol r [ ;. ] {., + [ct  k ( : : + 8, :~; + 8
2 ::.: )
+k [cpo + 81lP! + 62 V'2] e + DX2, Z
'
1o.}} (3.122)
3.4
115
EXTENDED MATCHING DESIGN
is feasibile only in the region in which ~ + 01 ~ choicps, the derivative of the Lyapunov function
+ O:!~ =/: O.
'Vith thcsp
(3.123) is nonposit.ivp: (3.12!)
" =
As we will see in Section 4.5.3, stability is guarant.eed for all initial conditiolls inside the largest level set of the Lyapunov function 3.123 (·ont.ained in the feasibility region.
3.4.3
Transient performance improvement
The llonlillem' damping with h:terms introduced in Section 2.5 can easily bE' incorporat.ed int.o the adaptive design procedures we have discnssed so far. The resulting adaptive controllers guarantee boundedness even when the adapt.ation is switched oIT, and their transient performance can be improved in a systemat:ic way t.hrough trajectory) initiuli:;ation and the choke of design paramelers. To illustrate the design with Ii.terms and t.1m process of trajedory init.ializatioll, we consider again the system {3.29} with the out.put /I = :1'1:
.1:1 = :1:2
:r2
=
+ 8cp( .1' J ) (3.125)
1I
Y =
.1'1'
The control objective is to asymptotically t.racl\: a reference out.put Yr(l) with the out.put JJ of the system (3.125). 'Ve assullle that not. ouly Yr. but also it.s first t.wo derivatives Yr, jjr are known and uniformly bounded. and, in addition, Yr is piecewise cont.inuous.
Step 1. The first errol' variable is now the t7'lJch71.g =1
= Y  Yr =
e7'1'07'
I1r ,
Xl 
(3.126)
whose derivative is ':1 = :1:2 + 0'1'91 (.t·l) 
Viewing
:1:2 ~lS
.llr·
(3.127)
t.he virtual control we define the st.abiliziug fUlldion 0'1
= c} =1

n.1 =1 cp2 
Ocp + Yr .
(3.128)
Comparing {3.128} with (3.104) we note two new t.erms in (3.128). The term which is intended to cancel t.he corresponding t.erm in (3.127), is due to the t.racking objective. The nonlinear damping tE'l'lll 11:1 =1 cp2 is motivated
Yn
116
ADAPTIVE BACI(STEPPJNG DESIGN
by Lemma 2.26. It contains the square of the term (
,,+ =~ + (hp. The derivative of the Lyapunov function Vi = 1=r + '3; 82 .
=1
=
•  2  \ r.1 _ ]
.   lil_] .  cP CII I ) "
I)
+ (J 
Step 2. As in (3.108), tho derivat.ive of =2 = =2
= =
U. 
(3.120)
CI=1  Iil=lcp·
(
CPl
X2 
bec!omes
,0 . 1 :..)
(3.130)
a:1 is
UOI aa:1 . aa:1 .. aa:l :.. (x:! +(JcP)  Yr   .Yr   ,(J
ao
aYr allr (JaO'l oerl • .. cP  cP  ~Yr  Yr aJ:l aXI aYr
[):rJ
[)a:l
u  X2 
(JA{JO']
{);,:]
!Jf;
where ill the last equalit.y we have lIsed the ident.ity and (3.131), the derivative of the Lyapunov fUllction
1 ')
1.,
2
2 
00'1 (}:.. ~
ao
(3.131 )
!
= 1. Using (3.130)
1.,
zi + :;:; + 0
(3.133)
2")'
is expressed as
Ii, =
=,::0  0,=;  Nl=i
=
...
{J01
aaal
80:.1
aXl
OXl
aX1
8a:l .
..
a::!  OcP  8cp  Yr  Yr .,.,
(~);:;i'  n.1=jCP·
+=2 [ =1 + u 
OYr
00'] + (J [ CP=l  =2cP 8
Xl
00'1
8 ;t':! :1:1
~{)O'l  I J  cP 
oa:.
00'1 :..]
an
A
(J
1 :..] l'
f)
aO'1 • a Yr Yr
•.
aO I
'!..]
Yr   . IJ. {J(}
(3.133)
As in (3.110), all the terms containing 8 have been grouped together. To eliminate them, the l1pdat,e law is chosen as
~ = y (
c2zi 
1i2=~ (~tp)!!,
instead of just equal to
£'2Z~
as in (3.112):
3.4
117
EXTENDED MATCHING DESIGN
f)
0
,.r. 'I'
0

[~~]
oJ
.f
1'1"\ ,,~
O:rL

[~~r "
=2
rr+
8.r1

[<""'~' 1
1
(',,/':'1 

(f!QJ. )~2 Or)
] f4
1
.

I
Figure 3.8: The closedloop lldaptive system {3.137}.
To implement this control law, we will replace iJ wit.h t.he analyt.ical expression of the update law (3.134). Su bst.ituting the expressiolls (3.134) and (3.135) int.o (3.133) we obt.ain (3.136)
while the romplete error syst.em becomes (sec block diagram ill Figurc 3.8)
,/ [ =, ] = [ c, =1"'1" c, _ n~ (~\O)' ] [ ~; ] + [ ;'1" ]H =2
dt
iJ
= .r[ ~ ~~ 1[ ~: ].
(3.137)
Comparing (3.137) with (3.114), we see that. the system matrLx ill (3.13i) is not constant: Its diagonal terms have been "fort.ified~' with additional nonlinear damping terms. These t,erms contain tho squares of the elcment.s of the vector that multiplies the parameter errol' O. Let us now study the properties of the error system (3.137):
Global stability and asymptotic tracking.
Using (3.132) and (3.136) we conclude that the (.:,8)syst.em has a globall), uniformly stable equilibrium at. the origin, amI lim .:::(t) = O. (3.138) too
In particular. t.his implies that Ule state of the syst.em (3.125) is globally uniformly bounded (since Ur, Yrl and fir are bounded) and t.hut. t.he tracking error Z1 Y  Yr converges to zero asympt.ot.ically.
=
118
ADAPTIVE BACKSTEPPING DESIGN
Boundedness without adaptation. It is also straightfonvard to see that. the designed controller guarantees global uniform boundedness eveu when the adaptation is turned off, that is, even with, = O. In that case, the closedloOI> system (3.137) becomes
A candidate Lyapullov Cundion for this system is given
_ ! (_2 + ""2.,.2) \/{).. _ !1_12 2'"  2 1
.
b)1
(3.140)
It.s derivative along the solutions of (3.139) satisfies
(3.142)
It is clear from (3.141) that., for allY positive values of Co and li:o, the state of the error system (and hence the state of the plant) is uniformly bounded, since lr < 0 whenever \zl:! > 1012/4noco, where 8 = 0  O{O) is constant. since adaptation is turned off.
Transient performance improvement with trajectory initialization. Let ns now investigate the transient performance of the adaptive dosedloop system (3.137). The derivative of the nonnegative fUllction \1(.::) defined in (3.140) along t.he solutions of (3.137) satisfies the same inequality as in (3.141):
(1, dt 2
2
d :; I")  ::; co Iz 1'1 + 0
4n:o
.
(3.143)
8 has already been established from (3.132) and (3.136), we can strengthen the inequality in (3.143) by replacing ij'J with its
Since the boundedlless of
3.4
119
EXTENDED lvIATCHING DESIGN
bound 11811~. This bound is est,imated from (3.132) using t.he fact that ~ is nonincreasing:
:5
1 ., 1  '1 2Iz(t)l + 21,8(t) = ''2(l)
:5 ~(O) = ~1=(O)12 + ')1 0(0)2. ...
(3.144)
....1'
This implies
IIOII~ $ 71=(0)1 2 + 0(0)2.
(3.145)
Combining (3.143) and (3,145) we obt.ain
d
at
(1;;12) :5  2col=l:! + _1 ["'11=(0)1 2 + 8(0)2] , 2nn
Multiplying both sides of (3.V!6) by results in
e'lcol
(3.l46)
and integrating over the int.erval [0, I]
(3.VJ7) The bound (3.l tl7) suggests that the transient beha.vior of the error system can be influenced through the choice of design constants Cn, n.o and 1', ~'hat is not clear, however, is that an increase of h~OCO alone may not reduce the ma..ximulll value of 1=(t)1 and \Vill certainly not reduce the comput.able .coobound of =. In fact, it may even increase this bound by increasing the initial value 1=(0)1. To clarify this point, let us recall the definitions of =. and =2: Zl
=
·rl  llr
Z2
=
X2 
0'1
=
;1:2
+ CI Zl + /i'IZ1cp2 + Otp 
lir·
Suppose now that ZI(O) is different than zero. In that case, all increase of and may increase the value of :~(O) and thus also the value of 1=(0)1. IvIOl'cover, this increase may more than offset the dccreasing efrect of the term 1/4n.ocn in (3.147), since Iz(O)1 2 will increase in proportion to ci and It would seem that the dependence of =(0) 011 the design constants Cl, C:Z, 11:1, n.:z eliminates any possibility of systematically improving the t.ransient performance of the error system through the choice of Co and 11:0' Fortunately, it is not so. The remedy for this problem is to use tmjector!J initialization to render =(0) = 0 independently of the choice of these design constants. The initialization procedure, present.ed for the general case in Section 4.3.2, is straightforward and is dictated by thc definitions of the =variables: Cl
li,
hi
• Stal'ting with
z},
set.
=1 (0) =
0 by choosing (3.148)
120
ADAPTIVE BACKSTEPPING DESIGN
• Since :'1(0) = 0, (3.128) shows that
where we use t.he notation !p(O) = CP(:t:I(O)). From (3.149) it is ("lear t.hat we can set. .::AO) = 0 with the c:hoice
(3.150) With the t.rajcct.ory initialization defined by (3.148) and (3.150), we have set. .:(0) = O. III t.he case of model reference control, this is achieved by adjusting the init.ial conditions of the reference model. If, on t.he other hand, the reference trajectory is given as a prccomputed function of t.ime, t.hen it can be iuit.ialized t.hrough the addition of exponentially decaying t.erms whicll define the refcl'Cnce l1Yl1t.rlients. Vve note t.hat. (3.148) and (3.150) are independent of tht" deRign ('onstnnts c., C:!. h'lt 1i2' This means t.hat. different choices of Co and KO will stilll'esult in =(0) = 0 with the same values of Yr(O) and 1jr(0). Ret.urning to (3.147), we substitute .:(0) = 0 to obtain 2 1=(t)1
::;
1

'l
8(0)~lli:oco
(3.151) j
which imp1ies
(3.152) Hence, the .coobound on the transient performance of the error system is direct.ly proportional t.o t.he initial parametric uncertainty and can be reduced arbitrarily by increasing the values of Co and 1\:0. In particular, this implies that the transients of the tracking error Zl = Y  Yr are directly influenced by the design const.ants Cj and Iii' This possibility of arbitrary reduction may seem peculiar, since it can be achieved for all initial conditions. vVe must. remember, however, that this elTor is defined with respect to the reference signaJs which have in tUrn been initialized to set =(0) = O. Hence, the effect of the plant iuitial conditions has been "absorbed" into the reference t.ransients. To provide some further insight into the process of trajectory initialization, let liS return to the Lyapunov function (3.132). When .:::(0) = 0, the initial value of this function is reduced to the initial value of the parametric uncertainty. 1£ we interpret the value of this function as a distance between the actual system trajectory and the reference trajectory, we see that trajectory initialization places I.he initial point of the reference trajeclory as close as possible to thc initial point of the system trajectory. If the parametric ullcertainty were zero, t.rajectory ini tiaJization would have placed the reference output and its derh'at.ives at the true values of the plant output and its derivatives. Tilis
NOTES AND REFERENCES
121
is easily seen if B{O) is replaced by () in (3.148) and (3.150):
Yr(O) = .1:1 (0) = !I(D) Yt(O) = l'2(0) + 81;'(0) = 1j(0) , However, since the paral11etpr () is unknown, trajectory initia.lizat.ion placed only the reference output at: the t.rut:' value of the plant output,. while ii,s derivo,th'es wore placed at the estimated values of the plant ont.put: derivat.iveF;. Through this process, the iniUI:11 value of the Lyapullov function is (3.153) \Vhich~
in the presence of parametric ullt"el't.ainty, is its smallest possihl(' valuo,
Notes and References Adaptive backstepping (Kallellakopoulos, Kokotovic\ and rvlorse [60]), first present.ed in a Grainger lecture [87], was a culminat;ion of an intellsivp eIrod of severol groups of ~1.uthors. The path to adaptive backst;cpping was not, as direct as it may appear from t.his chapter. It led ~,brough the makhed ease in Taylor, Kokotovic, rvIarino, a.nd f\:anel1akopoulos [186J, itnd then thp case of extellded mat,ching ill Kanellakopoulos, Kokot:ovic, and :Marino [65J, and Bastin and Campion [8]. Even t.hough nonadaptive backstcpping was ftvaHablc from Saberi, Kokotovic, and Sussmann [163}. the st.eps beyond the ext(:'uded matching case wore delayed lUld ruterlmtivo npproaehes were explored. The focus was on estimat:ionbased designs sllllullarizec1 in Pral)" Bastin, Pomet'l and .Jiang [157]" The rIass of paramet.ric strict:feedbal'k systems was characterizerl via coordinat.E"fl'ee geometric conditions by Kanellakopo ulm; ct 0.1. [63, 69), and the class of pru"ametl"ic }lurefeedback systelIlR by Akhrif tlud Blankenship The multiinput design £01' parametric pUl'L.1..feedback systems was prcs('l1t,ed
rlJ.
in [B7]. Teel {lBB] increased the feasibility region for parametric purefeedhack syst.ems by cast.ing the scheme [69J in an observerbaserl setting. Severa] extensions of [69] were proposed by Seto, AUllaswamy, and Brumettl [3, 167J.
Chapter 4 Tuning Functions Design
The adaptive baclcst.epping solution to the problem of nonlinea.r stabilization and tracking in the presence of unknown parametel's is a starting point for more elaborate adaptive designs which lead to new properties of the designed controller and the resulting feedback system. One of t.he improvcmcnts t.o be a.chieved with t.he tuning functions design in this chapter is the redudioll of t.he dynamic order of the adaptive controller to it.s minimum: The number of parameter estimates is equal to the number of unknowll pammet.ers. This minimumm'der design is advant,ageous not only for implement.ation, but also because it gunrantees the strongest achievable stability and convergence properties. In the tuning functions procedure the parameter updat.e law is designed recursively. At each consecut.ive step we design a tuning function as a potential update law. In contrast to adaptive ba("kstepping in Chapter 3, t.hcse intermediate update laws are not implemented. Instead, the controller uses them to compensate for the effect of parameter estimation t.ransients. Only the final tuning function is used as the parameter update law. We start this chapter with Section ~1.1, which introduces a g(meral framework for Lyapunovbnsed adaptive design via adaptive control L;l/ap U7/. 0 II functions (adf). VVe depart from the certainty equivalence principle and approach the problem of adaptive stabilization of the original nonlinear syst.em as a problem of nonadaptive stabilization of a modified system. In this setting, the tuning functions design is a method for recursively genera.ting aclf's. The design procedure is presented in Sections 4.2 and 4.3, which are independent from Section 4.1 and can be read first. In Sect.ion 4.4 we derive transient performance bounds on the error state of the adaptive system. An essential part. of t.he technique for improving the t.ransients is t.he trajectory initialization presented in Section 4.3.2. Several e..xtensions of the design are presented in Section 4.5. In Sect.ion 4.6 the tuning functions design is applied to suppress the wing rock jnst.ability in aircraft flying at high angleofattack
124
TUNING FUNC'rIONS DESIGN
4.1
Adaptive Control Lyapunov Functions
The basic: iclea of t:he Lyapunov approach to adapt.ive c:ollt.rol is t:o design a control law and a paramet.er update law to gnal'ant.ec that. t:lu' derivative of a suit.able Lyapunov function is nonposit.ive. \¥e are therefore sent. to search for a triple: Lyapunov function, ('ont.rol law. and update law. For a class of nonHnetu· systems caIled paramctric'st.ric~t.fceclback syst.ems we will be able to make til is search systel1la tic. To hegin with. let. liS investigate the possibility of adaptive design for the syst.("lll (4.1) i: = f(:l') + F(.l:)O + g(:t}u, :[' E JR.". 'U E 1R,
n
where E JR.]) is a vector or unknown constant. parameters, and ,((:1:), F(3;) and g(.1") are smooth. For simplicity let j(O) o. F(O) = 0, so that. x 0 is an equilibriulll of the uncontrolled plant..
4.1.1
=
=
Departure from certainty equivalence
Ivluch of the traditional adaptive cont.l'ol employs some form of "certa.inty equivalence" thinking. FoJlowiJlg t.his path one first performs a design fol' t.he C8.Cie when the exact. value of 0 is known. Suppose that. t.his noutrivial ta.."k is compJeted and that its result is a fecdb~lck contra} 'U = (JA~I:, 0) wliich stabilizes t.he equilibrium x 0 with respect to a known Lyapnnov function \~(x, 0). The suhscript 'r:' st.ands for "certaintyequiva]cllre.'· \Ve 1mow that V';,.(;v,lJ) is posit.ive definite and radially unbounded in .1' for all 0, and thnt there exists a funetion l.f!(x, 0), which is also positive definite in .1' fol' all 0, such that l
=
a,~
ax' [f(,T) + F(x)B + g(x)Q:r.(:c: 8)] ~ TV(:z:, H).
(4.2)
Hmv can we c>..1Jloit the knowledge of ll:c(:l', 8) and 1~(.1·1 0) for adaptive design when 0 is 110t. known? The certainty equivalence idea is to replace () by an estimate 9(1) obtained from a parameter Ilpual.e law (4.3)
r
where t.he adapt.ation gain matrb:: is posit.ive clefillit.e. \Ve want. to select 11and T 1.0 guarantee that the derivative of a Lyapullov fUllction is nonpositivc. For t.he syst.em (4.1), (4.3), a Lyapunov function candidate is
(4.4) 1 Throllghont. the chapter, we will drop t.he arguments in lJ\~.8) wId O\I/~~·/j) , nod write shortly ~~ and ~~. However, we will keep the flrgument.s in f(x), F(X)l g(x), and a(x. 0).
4.1
125
ADAPTIVE CONTROL LYAPUNOV FUNCTIONS
where the "certainty oquh7a]el1ce" form of \~ is augmented by a term quadratic ill the parameter estiIllation error
(4.5) Upon tbe substitution of F(:t)(J = F(x)9 along thE' solutions of ("1.1), (4.3) is
.
l'
+ F(.l')O,
the derh'at.ive of l'(.J:,8)
D1~ ( J(:I') + F(:z·)(J~ + g(x)u ) + ~'rT D1{, . (D'~. )T_ = 0' + Or ~F(.r)  liT T. ~ 00 ~
(...t.6)
To eliminate t;be indefinite c1epclldell(~e of ,:r on the unknowl1 parameter error 0, we select r to cancel the last ';wo terms ill (4.6):
)'1'
8V r(x,O) = ( a.l~ F(;.. )
(4.7)
With this choice of T, the expression (4.6) is reduced t.o
81~ (f{3.) . 1'F _ 8 X
•
...
. . al~ r (8\~ + F(.l)O + y(.t)lI) + . ~F(.l) )'T . 88 v.l'
(4.8)
=
Our nc..xt task is to select a control law u a'(x,8) to make 1> llonpositive. The "certaintyequiwtlenceH controll1. a·c (.t',8) fails to acllieve i.bis hecause then (4.2) and (4.8) yie1d
=
.
•
\f S 1'li(:c,6)
O\~ (D\~
+ D9 r
lJ.r F(:r;)
),r •
Clearly, l' is not nonposith'e because a. sigllillclelinit;e term is added to  (.t!(:r, 8). In search of a bettel' control law 0'(.1',8). we augment aA,'~, iJ) by 0.,. (x, 0), (4.10) The substitution of (·tl0) into (4.8) shows that the desired nonpositivit~t IV(x,8) will be achieved if aT CRn be found to saUsfy
al~
. + ~ aVe an r
~g(:t)o'l'(X' 0) vX
(81' c ) 0 F(:.:)
T
.1:
= O.
if s
(4.11)
This condition (or a'.,.. demonstrates tbe difficulty of adapt.ive designs for a general nonlinear system (4.1). It is casy to S(,E' that a r satisfying (4.11) is unlikely to c..xist: The scalar quantity ~g(:c) may be zel'O at: a set of points. StiU, the condition (4.11) is of iuterest because of an hUJJOl'f.ilUt. special caso, which will be the starting point of our recursive design. This special elISe is
126
TUNINC FUNC1'(ONS DESIGN
the "extended matching" studied in Section 3.4. In this case, a smooth vectorvalued fUllction cp : R n +1J + JRP is known such that ~ can be fadored as follows: a\~ aVe (}.'1: () 'P(.?", (j~)T . (~1.12)
ao = ax'
A
Then, irr~pective of the zeros of t~~.t}(X), an A
A
07"(:1',8) = 
T
ll:T
which sath;fies (lL11) js
T
a,~
r ( ax F(.];) )
A
T
A
= 
(4.13)
\Ve observe that, in addition to it.s "certainty equivalence" part G e , the adap:tive controllaw 0' contains a part aT which is proportional t.o i, that is, to iJ (see (4..3), (,1.10), and (~!.13)). In this way the adaptive control law t,akes int.o acconnt t.he parameter est,imation transients. \Vhcn t.he parameter estimate iFi constant, the control law reduces to the "certainty equivalence" cont,rol. Let us examine t\D example of a system for which (4.12) is satisfied. Example 4.1 Consider the problem of designing an adaptive controller for t.he system
=
:1: 2
(4.14)
It,
where () = [0 1 , 02J T is an unknown constant parameter vector, and tJle vectorvalued funct.ion cp(Xt) [cpt (.1:1), CP2(XI)F is known and smooth. VYe dealt with t.his system in Section 3.4.1. If the parameter f) were knowll, backstepping would result. in the 6dependent change of coordinates
=
Zl
'<::2
= =
Xt X2
+
(4.15)
and the (·ont.rol law
v with
= aAx,O) = =1 
el, C2
c,,,,  (~~~ + c,) (x, + 'I'(XI)TO) 8
(4.16)
> 0, which results in the closedloop system z=A=,
(4.17)
Due to the structure of A, an appropriate LyapllDov function is
t~(.l:, f))
=
~=(x, 8)T :;(a:, 8) ,
(4.18)
Observing from (4.1) and (4.14) that
j(x)
=[i
],
(4.19)
4.1
127
ADAPTIVE CONTROL LVAPUNOV FUNCTIONS
and evaluating
al~ _
;
.T [
..
fJ:J:
1
aT ~8+cJ
0 1
1,
(4.20)
with (4.19), (4.20), and (4.16), it is ensy to show that (4.21)
Let us now evaluate tIle partial derivatives ::tppearing in (4.11): al~
aD = 81{ g = ax
:;'re2lPT
;:;"e2
= =2tpT
= =2,
(4.22)
(£1.23)
where (~1.23) is immediate from (4.20) and (4.19). A comparison of (4.22) and (4.23) reveals that ~ = !!J:gcpTf so that 0'1' is given b~r ("1.13):
{4.24} Taking for simplicity U
r = I, the resulting adaptive control law is
=0:( •.,9) =
2, 
0,"2 
<,0"<,0
[I.
(01+ ~~~ II) (x,+ <,O(:r,)TO)
~~~ 11 + e1) : ,
(OS)
and the correspoDcUng parameter update law (4.7) is (~L26)
Note th::tt in (.£1.25) and (4.26) we llse =(x,9) instead of z(:v,O}. With the choice of a and T given by (4.25) and (4.26), the derivath'e li" of the Lyapunov fUDction 1'(:1:,8) 4z(x, O)T :(x, 9) + ~8Tjj is gllarallte~d to be llonpositive: V = Cl c2=i. This assures that both x and iJ are bounded. A standard argument llsing the LaSalleYoshizawa t,heorem {Theorem 2.1} proves that also x{i) + O. 0
=r 
=
In the above example the desired factorization (4.12) of ~ is 8. consequence of a. particular fea.ture of the system (4.14). The unknowll parameter R}'pears in the first, while the control appears only in the second equation. It is not
128
TUNING FUNCTIONS DESIGN
hard to sec tl1at. the same factorization (4.12) would be possible for ~t higherorder plant., provided that the unbwllm. par(l.mete7' i.s separated from the c071t1YJl inp1lt by a.t most one intcgroto7'. So {:he factorization (4,12) is not. a fmt.uitolls (went, hut a structural property. For systems wit.h this "extended matching" property, the above simple adaptive design is feasihle. However, most. systems fail to possess the "extended matching" property. A bellchmark example is the thirdorder syst.em
= = =
i' I j'2
i:a
:1'2
+ I;.7(Xl)'J'fj
.I,·j
(4.27)
u
which has t.he form of (4.14) augmented by an illt.egnttor. In {:his system, 0 and 1/ are sepantted by two integrators and we are unable to find 0 7 which satisfies (4.11). 'vVc will solve this problem with a recursive design which will drcnlllvent. the obst.acle posed by the rest.rict.ive condition (4.11).
4.1.2
Certainty equivalence for a modified system
Condition (4.11) was dictated by Olll' choice of t.he Lyapllnov function '~(.1~, 0) as I.he "certainty equivalence" form of '~(:r, O). The only good thing we know about. "~~(.r, 8} is that it \Vorles when the factorization (4.12) is possible. Othenvise we do not know how t.o remove the indelinite term preventing the n011positivit.y of Ii in (4.9). Having recognized t.hat a cause of our difficulties is '~(J:,8), we now embark on a search for Lyapunov functions mo~'c suitable COl' adaptive control. The key idea is to counteraet. the effect of 0 and thus prevent the parameter estimate transients from destroying the nonpositivity of t.he Lyapullov derivative.
We say that the syst.em
±=
J(x}
+ F(:r.)8 + g(.r}'U
(4.28)
is globally adaptively stabilizable if there exist a function 0:(.1.',8) smooth on (JR." \ {O}) x HlP wit,h 0:(0,0) == 0, a smoot.h function r(:I:, O}, and a positive definite symmetric p x 11 matrix r, such that t.he dynamic controller 'U
=
0'(:1:,8}
(4.29)
6
=
rr(.'l" O),
(4.30)
gua.rantees t.hat t.he solution (x(1), n(t)) is globally bounded, and .1:(t) + 0 as t + 00, for all () E IRP. Our approach is to replace the problem of adaptive st,abilization of t,he original system (4.28) by a problem of nonadaptive stabilization of a modified system.
4.1
129
ADAP'l'IVE CON'l'ROL LVAPUNOV FUNc'rrONs
Definition 4.2 A smooth Junction l~ : JRII x lR" + R+. positive definite and rudially unbounded in:1' for each 8, i,fj called an adaptive control Lyapunov function (aclf) for {4.28} if fJu~re exists a positi2Je definite s.lJrnmet1'ic mat,.i.:,: r E lRPxp such lIwt j07' each f} E lRP, '~(.l~, O) is a elf JOl' the modified S~Jlit.em ;i:
that is,
l/~
= /(.1:) + F(J:) (0 + r
(a; )T) +
g(:r)1I,
(4.31J
sati.#Jjies
We now show how to design aclf is known.
Theorem 4.3 The jollowing
all
t1t1O
adaptive ('ont.roller (4.29)(4.30) \Vhell an
stalemeni.s are equivalenl:
1. There exi.4Jts a t'li.plc (f1', \;;.., r) s'lJ.ch that a(J:, 8} globally (l.sym'1)loticaliy stabilizes {4.31} at :,; = 0 f07' ea.ch 0 E JRII 'with resIled to the Luap1J.nov junction ';;"(:1:, O). 2. The1"e e:J.i.9i.s an adf l~{J.', 6) jor (4.f!8),
}.{01"Cove1', if an aclf a.ble.
\~(.:l', O)
existS r then (4.. 28) is globtl.Uy adaptil.ely slabili:;
Proof. {1 => 2} Obvious because 1 implies that: t.herc exists a conthmous funct:ioll lV : lRlI x JR/I + 1R.+, positive definite in :1' for each 8, such that
8\/ [ 8.1~
f(x}
+ F(x}
((81/ )T) + +r 6
8~1
]
g(x}o(x, 6) :5 H'(J', 0) ,
(4.33)
Thus l~(x, 8) is a df for (4.31) for each 8 E lRP , and therefore it is an aclf for
(4.28). (2 => I) The proof of this part is based 011 Sontng's cOllstructive proof [171J of Artstein's theorem [4J. "Ve assume tbat 1~1 is an adf for (4.28), tha.t is, a elf for (4.31). Sontag's formula (2.19) appJied t.o (4.31) gives a rOllt.rollaw smooth all (JR" \ {O}} X lRP :
.I
a:(x,6}
+ =  !!Y.a.! 8:1'
(4.34)
o
!
Qiag(x Ox • t 6)
= 0I
130
TUNING FUNCTIONS DESIGN
where
(4.35) 'Vitb the c110ice (4,34), inequality (4.33) is satisfied with t:he ('Old.inuous function
IF(:Z',8) =
_ )2 + (tn~ ).a ax 1(3:,8) Dxg(J.·,O) , ( al~
(4.36)
which is positive definite ill x for each 0, because (4.32) implies that ~ j(:r, 8) < 0 whenever IlJ:g(~v, 8) = 0 alld 3' i= O. We note that tbe COlltrollaw o'(:L',8) will be continuous at .1' = 0 if and only if the aeIf \fa satisfies the following property, called the small controi,}roPC7"ty [171]; For each () E B" and for any E: > 0 there is a 6 > 0 such that, if x =1= 0 satisfies 13:1 S 0, then there is some 'u, witb luI ~ E: such that
all axu [J(:r:) + F(:r.} ((all 0 + r ao )T) + g(~l:)ll ] < o.
(4.37)
Assuming the existence of an aclf we now show that (4.28) is globally adaptively stabilizable. Since (2 => 1), there exists a t~rjple (a, Va, f) and a function ll' such that (4.33) is satisfied, that is,
~~ (f(x) + F(·l:l8+ g(xla{x, 8)1 +
:"r (~
F{''I:'f
~ W{x, II) .
r 1(0 
8) .
(4.38)
Consider the Lyapullov function candidate 1
~
4T
lI{.T.,O) = l~(:r:, 0) + 2(6  0) A
~
(4.39)
'Vitb the help of (4.38), the derivative of V along tbe solutions of (4.28), (4.29), (4.30), is .
V
= 81~ ax =
l)8'~ x
[
cJl~
]
;;J'
,.
f + FO + .qa:(x, 8) + 80 rr{x,8)  0 r(x, 8)
[J + PO + ga(x, 8)] + 81~ rT(X, 0) + ~~ F8 88
eTr(x, 8)
uX
,. al~ :::; H'(;r,O)  ,. f
80
"
+0m' (al~) ax F 
(aVa )T Dl~ ~ ;;F +  . fT(X,O) 80
uX
° T{:C,O). nT

(4.40)
Choosing
r(x, 0) =
al~ (
T
ax (x,6)F{x) ) A
,
(4.41)
4.1
131
ADAPTIVE CON'rnOL LVAPUNOV FUNCTIONS
we get
11 ::; 1"'(.1',8) ,
VB E ffiP.
(4.42)
Thus, the equilibrium x = 0,0 = 0 of (4.28), (4.20), (4.30) is globally stable, and by the LaSalleYoshizawa theorem (Theorem 2.1), :c(t) ~ 0, that is, (4.28) is globally adaptively st.abilizahle. 0 The adaptive controller const;l'ueted ill the proof of Theorem .4.3 consists
=
of a control law lJ. = a(:r,8) given by (4.3 l1), and an update law iJ fr(:1.',8) with (4.,U). It. is of int.crest to inl;erpret this controllcr a.1Oj a certaint.y equivalence COIltroller. The control law 0:(:1:,8) given by (4.34) is stabilizing for the modified syst.em (4.31) but may not; stabilizing for the original system (4.28). However, as t.he proof of Theorem 4.3 shows, its cel'tainty equivalencc form 0:(:1:,8) is an adaptive globally stabilizing control law for the original system (4.28). Hence, if ~t certainty equivalence approach is to be applied t,o a nonlinear system, the system is to be modified to require a cont.rollaw whi{'h anticipates the parameter estimation transients. In the proof o[ Theorcm 4.3, this is achieved by incorporating thc t'uning JUTlction T in t.he control law 0', Indeed, the formula (4.34) for a: depends on r via
ue
av;. . _ aVa .. T a,T. f(:,;, 0)  D."C J + r(.l, 8)
avo (0 + f ( 80 )T)
'
which is obtained by combining (4.35) and (4.41). Using (4A1) 1,0 rewrite the inequality (4.38) as a~
ax
[j(x)
+ F(x)8 + g(x)o'(x, 0)] + 8~ ao fr(x. 8) ~ H'(,l.', 8)
I
it is not. difficult. to see that the control law (4.34) containing (4.43) prevent.s
r fl'om destroying the nonpositivity of the Lyapunov derivativE'. Remark 4.4 A relevant question remains unanswel'ed: If there exists an aclf for (4.28), is this system globally asymptotically st,abilizable for each f) (and vice vel'sa)? In other wOl'ds, does the existence of a pair (a, l/~) satisfying (~J.33) for some r > 0 imply the existence of it pair (0:°, l~O) satisfying (4.33) [or r 0 (and vice versa)? Adaptive Lyapunov designs available in the literature [59, 65, 69, 94, 156, 157, 186] are all for systems whirh are not. only globally adapt.ively stabilizable, but. also globally asymptot.ically stabiliznble for each O. 0
=
As is always the case in adaptive control, in the proof of Theorem 4.3 we used a Lyapunov function li(.1\ 8) given by (4.39). which is quadratic in the parameter errol' (J  O. The quadratic form is suggested by the linear
132
TUNING FUNCTIONS DESIGN
dependence of (4.28) on 8, and the fact that (J cannot be used for feedback. VVe will now show that the quadratic form of (4.39) is both necessary and sufficient for the existence of an adf. Vve say that systcm (4.28) is globally adaptively quadratically stabilizable if it is globally adaptivellJ .litabili::able and, in addition, there exist a smooth function 1'1I(.1·,9) positive definit.e and radially unbounded in x for each 9~ and It continuous function lV(x,O) posit.ive definite in .T for each 0, such t.hat fat' all (x(O),8(0)) E lR"+P tl.lld all () E JRP, the derivat.ive of (4.39) along the solutions of (4.28), (4.29), (.t.30) is given by (4.42).
Corollary 4.5 The 51JS tern (4.28) is gio bally lI.dap ti1Jei1J ljualiratically stabilizCLble if and only if there exists an adf Va (x, 8). Proof. The 'if' part is contained in t.he proof of Theol'em 4.3 where the Lyapullov function F(x,O) is in the form (4.39). To prove the 'only if part, we start. by assuming global adaptive quadratic sta.bilizability of (4.28), and first show that 1"(:1:,0) must be given by (4.41). The derivatjve of l' along the solutions of (4.28), {4.29}, (4.30)1 given by (4.40), is rewrit.t.en ru;
This expression has to be nonpositive to sat.isfy (4042). Since it is affine in 0, it can be nonpositive for all (x,6) E lRn +p and ~Il (J E lll.P ouly if Ule last term is zero, that. is, only if T is defined as in (4.41). Then l it is straightforward to verify that
a;;
[J(Xl + F(x)
(0 + r (a;) T) + Y(X}l>{X,Ol]
=V+(6T~ir) (T(~:Fr) ~ lV(x,8) for all (~'I 0) E IR"+p. By (1
(4.46)
=> 2) in Theorem 4.3, Vo(:r, 0) is an aclf for (4.28). o
The above analysis applies also to t.he ('ase where t.he unknown parameters entcr the control vector field:
.i· = f(x)
+ F(x)8 + [g(x) + G(:r)O]u..
(4.47)
4.1
133
ADAPTIVE CON1'ROL LVAPUNOV FUNCTIONS
In this case the existence of an adf Va is equivalent to the existel1cc of a elf for the syst.cm
±=
/(x)
+ F(,,')
(OH (lJ~y) +
[9(X)
+ G(x)
(6+ r (~ f)]
II.
(4.48) Tl1e ext.ension 1:0 t.he multiinput case is also st.raightforward.
It. is of intercst. to examine the itl}llltoutput Pfopclties of I.he system resulting fro111 tbe application of the adaptive control Jaw Q·(.r.9) to the plant (4.1):
± = I(x) + F(x)9 + O(J:)O'(:I', 0) + F{a:)6.
(4.49)
In early Ly~tpUllDV designs for linear systems of relative dcgree DIU', an important pl"opert:y was the strict positive realness of t.he transfer I'ullctioll between the parameter error and tbe out;put error [142J. For an analogous passivity pl'operty of the nonlinear system (4.49), let us consider that it.R input is 0.
Corollary 4.6 (Passivity) Suppo.'Je a j'll1lcLion l~,(.r: 0) is hU:J1Im. to be an ad! 1lritll. (1.11. associated control la'lJJ o·{ X t 8). Then the By/stern
'llnth
+ F(:J:)O +g(:t)o'{J:, 0) + F(.")ii
j:
=
f(:L')
T
=
81'0 ~ )T ( 03: (:1:,O)F(:d
0 lIS the input and T
as the D1Jlp'Ut is
(4.50)
l;#'riCtly
passizlc.
Proof. Along the solutions of (4A9) we have .
1~1
= ~
01~
8
x
(
~
o\~
.
a\'~I
f + FO + go·(:r, 8) + . rT(:!:, 6) + 8 FO A
,
]
W
 T
ll"(:r:,O)+T(X,8) 0,
7
(4.51)
wbich, upon integration, yields
10' TT(J=(S), O(s)}6(s)ds ~ V:1(.t'{f), 6(t))  \~I(X{O), B(O)} + 10' l·{l(.r(s) , O(.~»d8. (4.52) Using V;,(.l~, 0) as a storage functioll and TV(x, 8) as fL dissipation rate, and r noting that. l~ and al'e positive definite in 3: [or each B~ tlle inequality (4.52) estnblishes strict passivity by Definition D.2. 0
n
Hence, our closedloop ada,pt.ivc system represents a llega.1.ivc feedback connection of the strictly passive system (4.50) and tbe int.egl'ator
 T, r
 9=
S
(4.53)
134
TUNING FUNC'fIONS DeSIGN
which is passive (positive real) because
!!.. (!orr 1o) = dt 2
_8'r T
(4.5~1)
implies that (4.55)
For snch a feedback connection, Theorem DA establishes that the equilibrium x = 0,9 = 0 is globally stable, and x{t) + 0 as t I> o. Thus, t,he problem of adapt.ive stabilization can be approached as the problem of finding an output T with respect to which it is possible to achieve strict passivity from 8 as the input.
4.1.3
Adaptive backstepping via aclf
\Vith Theorem 4.3, the problem of adaptive stabilization is reduced to the probJcm of findhlg an adf. We }lOW address the problem of systematic COllstruction of an adf. Our aim is a recursive approach because we already know how to find ac1f's for systems with the extended matching property, and expect to recursively enlarge this initial class of syst.ems with repeated use of backstepping. So, we assume that an aclf is known for an initial system, and construct a new aclf for the initial system augmented by all integrator.
Lemma 4.7 If the system
± = f(x)
+ F(l')6 + g(x)u,
(4.56)
is g{oball1J adaptivcl1J q1l.adratica1l1J stabilizable with a: E Cl, then the augmented system :i; = /(:1') + F(x)6 + g(x)~
{ =
(4.57)
'I/.,
is also globally a.daptively quadra.tically stabilizable. Proof. Since system (4.56) is globally adaptively stabilizable, then by Corolhuy 4.5 there exists an aclf {~(x,O), and by Theorem 4.3 it satisfies (4.33) with a control law 'U = 0:(.'&,6). Vve will now show that
l'i (:I:,~, 8) = V'a(.'&, 6) + ! (~  a:(x, 8))2
(4.58)
2
is an ac1f for the augmented system (4.57) by showing that it satisfies aVI
8(x,~)
[f + (8 + r (~)T) + F
0:)
(.1:, ~ t 6)
lJf. ] $ _ HI _ ({ _ a
f
(4.59)
4.1
135
ADAPTIVE CONTROL LVAPUNOV FUNCTIONS
with the contra] law
=
aa (/ + Fe + g~) ax
oVa   0  (~ 0') + 
ax
+ 80'r (8Vi F)T + av;,r (BIl'F)T olJ
OJ'
08
o:r
Let us start by introducing for brevity:; = t 0:(3:,8). With (4.58) we compute
alii [ J + FO + gf. ]
o{x, f.)
01 (x,
{, 8}
alii
=
81't 8x (/ + FIJ + g{)
=
(av0;  =ax ao:) (J+ FO+gf,)
=
8\'0 8d~ (1 + FO + go:)
+
o{
0:1 {x,
{, iJ}
+ZO:l
81~
Oet
+ ax gz  ;; aa: (J + FO + .Qf,) + =°1
80: ) = 01'a ax (/+F8+go:)+z ( 0]+ tn! 8:9 ox (J+F8+gf,)
. (4.61)
On tbe other hand, ill view of (4.58), we have
8111 O(x, {)

[Fr (!:!!i)T ] = Fr alii (alii) 80 8e l' 0 ox
(en;;. _:; au) Fr (8\!a _; an) T
=
ax
ax
ae
89
a)T
8V'", Fr ( 8V
=
ax
z
ao
(oar (8Viox. F)T + ollar (OO·F)"). 8S
80
83.:
(4.62)
Adding (4.61) and (4.62), with (4.33) and (4.60) we get
alii
[f+
F(0 + r (~ )T) + DC; ] 0'1 (x,
8(x f.) t
=
{,O)
aVa (/ + Fe + go:) + 81'", Ff
ox
a.r:
+Z
(0:
1
(aVn)T {)f}
+ al'a g _ 80' (/ + FO + gf.) ox 8x
_ Bar ( 81"i 00 ax ::; _HT(X,O)
F) T_tH~, r (00' F) T)
_.::;2.
01J
ax
(4.63)
136
TUNING FUNC'l'IONS DESIGN
This pI'oves by Theorem 4.3 that \'1 (x. {, 0) is an aclf for syst.em (4.57), and by Corollary 4.5 this system is globally adaptivcly qnadrat:ically stabilizablc.
o The new tuning function for system (4.57) is determincd by the new aclf
Vi and given by T] (:1', ~~O) =
[F ])T = (~Vj F)'f = [(aVa _(e _ a,}aa:) F]T ( B(~ .1.:, 0 o:r B.T. a3: ~)
(4.64) \Xle note that the new tuning function
I)
is obt.ained by augmenting the initial T
tuning IUllct:ion T with the term  (~~F) (~ 0') which accounts for the faet t hat the ac1f 1~, is augmented by ! (~  a(.1: I 8))2. The form of the controllJ:l.w Cl:l~(3:1 {, £J) in (4.60) is of part.icu1ar interest. It consist.s of two parts~ 0:1 = O'ltC + O'I.T' TIll" first part., ltl,c(Xt(,IJ)
a\~ = a' 9:r;
(~ 0')
ao
+~ cr + FIJ + g~) v.1.
,
{£l.GS}
would becomc the "certaintyequivalencc" controlla,w for the augmcnted system (4.57) if we were to set r = 0. 2 The set'olld part consists of t;wo terms,
aJ
,,(X,e,8)=:r(:Ff +~;r(~;Fr
Their role is to produce ~Fr (~)
'f
(4.66)
in t.he aclf inequality (4.59). Observe
that the first. term ill (4.66) incorporates TI = (~F) T. The controlln.w 0'1 (x, €, 0) in (4.60) is only one out of many possible control given by (4.58) is an ac1f for (4.57), we can laws. Once we have shown that uset for example, the CO control law 0'1 given by Sontag's forllluia (4.34) witl1 ...illi.. g   "'lld O(.:E'.~I 1  '" u
1'.
~
I
I
It can he shown that the following function, used as a. elf ill [1581, is a 1110re general aclf than (4.58): {tj (x,~, 6) :!Note, however, I.hat are also fUllctions of
r.
al.e
(a(.:E'IO)
= "11('1', 6} + .L n
is not. obtained by setting r
l1{s)ds,
(4.68)
= 0 in 01 since o'(x, O} and Va(xlO)
I
}
4.1
137
ADAPTIVE CONTROL LYAPUNOV FUNCTIONS
where TJ is a CO function sneh that 811(8) 7J f/. £1 « 00,0]) U £1 ([0, +(0)).
> 0 whenever s
The following example illustrat.es t.he use of Lemma
=f:. 0, 7]/(0)
> 0,
and
~L7.
Example 4.8 Let us consider the system Xl X2 ·'Va
=
x:!
=
X:i
=
+ cp(:r:} )T (1 (4.G9)
'/I.
"Ve will treat the state 3'a os an integrator added to the (:1:1, .r~)subsystem from Example 4.1. In that example, we have already designed an adaptive control law for the syst.em
Xl ·1:2
= =
.l'2
+ cp( X1 ) 'I'fJ
(4.70)
Xa,
considering X:i as a control input. With (4.18), (4.10), (4.20), (4.22), it ean bc shown t.hat
=
which means that l'a{xt, :1:2, 0) = '''=(XI, 3:2, 0) ~(zr + =i) is an aclf for the system (4.70) considering X:i as a control input. Therefore, Lcmma 4.7 is directly applicable. \Ve define z = X3  a(x, 8). By Lemma 4.7, the function
Vi (:r; fJ ) = '12 ( =j + =2 + '::3 'I
'J
.')
I
(4.72)
is an aclf for the system (4.69). With (~L60) and (4.64) we obtain 0'1
(.1',0)
= (4.i3) (4.74)
'\Tith the following adaptive cOlltrollaw and the paramet.er upclat.e law:
(x, 0)
(4.75)
B = T}(x,9),
(4.76)
·u. =
0'1
138
TUNING FUNCTIONS DESIGN
it is straightforward to verify that the closedloop adaptive system is
where ;;1, =2,'::3 are used with (j as all argument. The global stability of this system is established using the Lyapunov fuudion l'{x,O) = lIt(r,O) + !(JTO. ~
0
While in Lelllma "1.7 the initial system is augmented only by an integrator, a minot' modification is sufficient t.o obt.ain an ~Ulalogous result for the more general system
x =
f(x) + F(x)8 + g(.T,)c;
t =
u + F1 (x,c;)8.
(4.79)
Corollary 4.9 The Junction Vi (x, ,;,8) defined in {4.58} is an acl! Jor the system. (4.79) wit}, the control law and the tuning function gi'uen as (4.80)
(4.81) A repeated applica.tion of Corollary 4.9 will furtber extend t.he class of nonlinear systems for this type of adaptive design. With the knowledge of Va, i, alld 0: for the system (4.79), it is llot bard to see that by applying and 0:2 for the system Corollary 4.9 twice we can find 1'2,
'2.
.1:
{l {!!
= = =
f(:v) ~2 'U
+ F(x}8 + g(X)(l
+ Fl (3:, €l)8 + F2(X, elt ~!!)8.
{4.82}
4.2
139
SETPOINT REGULATION
In fact, it is clear that an nfold appJication of Corollary 4.9 ",ill provide us with \;;' , Tn, and Q' n for the system :i;
=
1(:1.') + F{.1:)6
~i
=
~2
+ g{J:)~l
+ FI (J:,~,)6 (4.83)
~r'l
=
~11
\Ve
WillllOW
4.2
€" + F,,l {x: {b' .. , {1Il)6 1/. + Fn(~t:,~1., ... '~II)8.
develop a detailed design procedure for such systems.
SetPoint Regulation
\Vith repeated use of Corollary 4.9, we can design an adaptive cOllj:!'ollel' to globally stnbilize a desired equilibrium .,;C of the paramel1i.c sf.1'iclJeedback system (3.70): :i:) = ·1:2 + !PJ (xd T 8
x!! =
X3
+ Y'2(X1, l':SJ'e
;r:n
+
(4.84) :i:,,_] = xn =
B(x)u + 'Pn(J·)T(J
where 0 E RP is a vector of unknown constant pru'ameters, {3 and
(·1.85) are smoot.h nonlinear (unct.ions taking arguments in [til, a.nd ,8(:1:)
m.n,
"f 0, 'rI:I.: E
In this section we develop a procedure I'or adaptive regulation o[ thc ontput Y = Xl to a given setpoint. Ys. \Vi1.h a constant control u~ the first. n 1 equilibrium equations of j.e = 0 in (4.&1.) can be successively solved for x; ,. , . ,X~I as fUllct.ions of xi and 8: t
x~ = XC:J =

x~
tp7l1(.1:j, . , . t x~,J )T(J
=
= =
',.e fl'0
t.p:.! (:1'(! • '1" 2
(4.86)
.i·~, 0 yields n rclat.ionship between .i:j I u'\ and O. \Vhen (J is known, then :~ 0 can be 50hrpd for U C needed to keep :l'j at a desired setpoint :z:Y = 1Is' The corresponding values 3:21' . , ,.T.~ will be dictated by (4.86). Therefore, for each value of 0 and a prescribed lis, the equilibrium :l'c
Then tlle 71th equation
140
TUNING FUNCTJONS DESIGN
and tIle corresponding control value lI. C are uniqucJy defined. In the special case whel'c tpl (0) = ... = 1,0,.1 (O) = 0, t.he choice Us 0 resu1ts in the equilibrium being .te = 0 for all values of B. Our problem now is to globally si'alJilize this equilibrium when Dis unlmowll and also to achicvf" setpoint regulation: J·(t) + :t.e as t .. 00. Comparing the syst.ems (4.84) and (4.79), w(' observe t.hat if Xa WCl'e the control variable, then Corol1lU}" 4.0 wouJd provide tlle desired adaptive control for the subsystcm made of the first t.wo cquations of (4.84). Thereforc, we can init,iate our recursive design procedure by augmenting this subsystem by the third equal.ion, as in (4.82). For convenience, we wi11 do l:llis in a selfcoutajuf>d fa.shion, independent, of Sed.ion 4.1. An 8.dditional feattu'e of the procedure in this scrt.ion is a set. of error coordinates in which the stability properties of the resulting closedloop ada.ptive system are c1en.rly displayed without an explicit, lise of the aclf concept,
=
4.2.1
Design procedure
Vve will start by adaptively stabilizing t.he first equation of (4.84) considering to he its (·ontrol. At c.:'lch subsequent step we will augment the designed subsyst.em by one equation. At the ith stepl rul ithorder subsystem is stabilh:ed with respect to a Lyapunov function V; by the design of l:\ stabil7:zing junction D:; and a tuning ju.nction. ri. The uprlate law fol' the pal'ameter estimate O(t) and the adaptive feedba.ck cont.rol '/l are designed at the final step. .1'2
The third step is crucial for understanding thp general design procedure. Step 1. Introducing the first two error variables (4.87)
=
4.2
we rewrit.e :;:1
.1'2 
0'1 ,
(4.88)
= :1:2 + 'PI {.1:!lTO, the first eqnation or (4.84), as :.
";'1

 + o· I + 'W 1·1 {a' )'rfJ ,
...2
(4.89)
whol'e, for uniformity with subse(Juent steps, we have defined the first regressor vector as (4.90) Our task in this step is to stabilize {4.89} with resped to the Lyapunov function
(4.91) whose derh'ative along tIle solutions of (4.89) js (4.92)
4.2
141
SETPOINT REGULATION
efrom l~ wi til the update law (j ==
We can pliminate
rTtl
where (4.93)
If X2 were our actual control, we would let. ma,)re l~ = CI we lVould choosc3
=2 ;:
=r,
0, that is, :r!! ==
0'1.
Theil, t.o
(4.94) Since J.'!,! is not our control: we h.c1.ve ':;21= 0, and \ve do nol USE' fJ = rn as an update law. Insf,cad, we retain 1') as our first. tun.in.g fU71ction and tolerate tbe presence of 6 in 1:"1:
(4.95) The second term =1=2 ill 1~ will be cAllcelled at the next step. \\lith a:l(Xl.6) as ill (4.94), the =rS)1St.em becomes (4.96) Step 2. Vile now consider that: :ra is the control variable in the second equation of (4.84). Introducing =:1
we rewrite j:~
=
X3 
(4.07)
0'2 ,
= a';.:j + 1f'2(:J:t,X2)TB a..c;; =2
=
Z3
00'1
+ ct2 
DXt x:!
+ tlJ2(Xl1 X:.!
~ T
1
9)
80'L :.
(J 
DO
(J:
(4.98)
where the second regl'Cssor vector w:! is defined as (4.99) Our task in this step is respect to
tlO
stabilize the
ti... = VJ 
(Zh
=2)S)'st.em (4.9G), (4.98) with
+ !..:;; 2 .'
(4.100)
whose derivative along the solutions of (4.96), (cl98) is •
1'2
=r + =2 I)
=:
Cl
+8T
(Tl
[
=1
+ Za + 0'2 
+ 'W2=:J 
rJo) .
80'. l ax} X2
T•
80: J ~J
+ W2 H ali ()
(4.101)
=
3The cbaructcr of the con(.I'Ol Jaw OJ is ilccrtnintyeqnivalence" because the ndf .\.:i l(J!l  YII)2, with respect to whid. t.he design is I)erformed, is illdependent. of 0. so it is also ;;, dC.
142
TUNING FUNCTIONS DESIGN
We can eliminate jj from
\i2 with the update law " = fT21 where {4.102}
were ou)' actual control and, hel1('e, =:. == 0, we would achieve li:! = c2=i by designing 0'2 t.o make t·he bl'ac1\eted term multiplying =2 in (4.101) equal to C2=2, namely4
If
X3
Cl='t 
Q'2(J:1, a:2,
We ret.ain
T:!
, 0) = :1 
{'2=2
+
80 1 {}J:I J:2 
T' w:! (J
+
an
lJal
rT2.
(4.103)
a.c; our second t.uning ft1l!ction in the term rr2 which repJaces
in (4.103), However, we do not use resulting \~ is
8 ;;: rr2
iJ
as all update law, so that the
Tile first. two terms in \~ are negative definite, the t.hird ~erm will be cancelled at t.he next step, while ill€' discrepanc,V bf'twE'(,]) 1"72 and iJ in t.he last two terms TQnUlins. By substituting (4.103) into (4.98), t.hc (=1, =2)subsystcm becomes
Step 3. Proceeding to the t.hird equation in (L1.84) we int.roduce (4.106)
(~1.107)
where tho third regressor vect.or IVa is
d~fincd
as
(4.108) "'\Vhile 0 I WRS .l IIcerletiut.y ~qui\'alcncc" ('ontrol, I he ('oot.rolla,w 02 is not. or the "L"eI'I.aint)' eqllh'fllence" type because it is designed wit.h respect. t.o the Odepcndcnt. nclf k:i+~':::;5' The
term ~rl'.! in Q!! corre.'iponds to the first lcnn ill th
III
i.:r
4.2
143
SETPOINl' REGULA"rION
Our task is to stabilize the (z\, Z2t Z3)system with respect to
1 'l V:... = If.. + :: 2 ....... ' wbose derivative along (4.105)
V:i. =
Rlld
').,
c'l:j  c2 zi +'::.1
(4.109)
(4..107) is
lJa:t (
:.)
8iJ rT,28 BCt'!
8Q''1
Va .. :.)
+=:i [ .":2 + .:., + a':~  ax,,1'2  aX2:l'!i + wJ8  ;;:.(} ao +OT (1;) + W:~Z:i  r1o) . Vva can ciimi.nate
efrom
~a with the update hnv
A
B = rT:",
\Vbel"e
(4.110) T3
is our
tuning fund.ion
If x:\ were OUf actual contl'ol, we would have ':., == 0 and achieve l~ = Cl =r C3Z~ hy designing 0:3 to make the hra.elected term multiplying =3 equal to ClZ3, Dft.nlcly
C:!=5 
(4.112) where "'J is a eorrectioll term yet to be chosen. 5 Substitutiug (4.112) into (4.110). and noting t.hat
jj 
1'T2
=
0
rT3
+ rT3 
==
e
rT3
+ rUJa=a,
rT;z
(!.113)
(4.110) is rewritten as
Ii;,
= <,=;  ""~ + =. (,';]  ~'r'»3'" ) +=3=.. + (="1 8C:1 + =3 aO:2) (rT3  08
DB
0) + 0"(T3  r10),
(4.114)
ftJt. is of iuterest to compare tMs cOlltl'ollaw with I.he cOllLrollaw (4.60) in LeU11Ua 4.7: The hlst two torms ill {J.1l2} aro analogous to the last two terms in (4,60).
TUNING FUNC'l'IONS DESIGN
and the (Zl, z:,h z3)subsystem becomes
=
0, and wi(;h the up
If ;r.1 were our control, we would have Z,I
.
.
We again postpone tIle decision about [) a.nd do nOE use 0 = law. The resulting is
Va
rT:. as an updatE'
(4.117) and the (=h Z21=3)subsystem becomes
(4.118) The 'system matrix' in (4.118) has a significant property: the sl,e\v symmetry of the nonlinear term ~ fWa ac.hieved by the choice of Va in (4.116). This term is analogous to the second term in (4.66) and tlle skew symmetry is crucial for stabilization.
Step i. Introducing • Xi • we rewr.1te •
Zj
=
Zi+l
= XHl + 'Pi (XJ r ••• , .ri)PJ'(J as + (Xi 
i 1 8 """ O'il L.; llX'k+l k=l UXk
+ UJt ( ~1'"
. , xh
o")T(J

aa:il (J'."
,,
86
,
(4.120)
4.2
145
SSTPOINT REGULAT(ON
whet'e the ith regressor vector is defined as
(.:1.121) Our objective is to stabilize the
,Zi)systelD with respect to
(ZI,'"
1 .,
V1 = V,  1 +Z=" 2! ,
(4.122)
\vhose derivative is
+.t'i [ Zil
+11""J' We can eliminate
+ ::i+l + 0i 
'T iI a0:11 a0lJ 'J L ~a. Xk+l + Wi 0  A(J 3'1' De A
A
k::::1
( T;_1
+ tlJI=i

r I:") (J
(4.123)
•
lii with the update law {j = rTiJ whel'e
8 from
= [w" •.. , IUil
=1 ] [
~
(4.124)
tii
Then, in tbe absence of Zi+l, we would achieve =  L~=l CI..=l, by designing a; to make the bracketed term mUltiplying =i equal to CjZit llRme]y
(4.125)
wbere
Vi
is a correction tenn yet to be chosen. Noting that
6
fTil
=
+ rTj  rTil == 0 Pii + rfJJjZi fJ 
fTj
(4.126)
I
we rewrite
\1 as iI 2 L Ckzk +
[ Zj
k=l
a0:;1 (rr;  8). ] + Vi  AA
Zi+l
~ ZI':+l" oak) (r1il + ( Lk=l
80
:.
89
ilT
0) + (J
(Ti 
r
J :.
fJ)
146
TUNING FUNCTIONS DESIGN
(4.127)
0"2.;1
o o
1 + 0"12.11
o
Cil
1
o
U:]
1
0"23
=
o o
1 
0'2,il
o
o +
WI] : 0+
[ Wi'J'
O'i2.il
o
1
~1 ~1
+[
+[
_ HI
O'jl,i=j
'Ii
(fT; 
9),
(4.128)
8n._t
no
where A
O"jI..(x: 0) = 
80';_1
88 r·w,..
(L1.129)
Now the correction term is chosen as ~ II; (Xl , ... , Xi,
i2
6) =
L
lJll"
ao
k=J
Because we do not use iJ = •
Vi = 
~2
i """
~ Ck":'k
rTj
iI
Zk+I;:PWi = 
L O'k.iZk.
as an updat:e law: tlle resulting li, is
(iJ 8)'8) + 6
_ _ + "'ii+l +
""" _
O'k
L ""l"+IA
k=l
(4.130)
1.:==2
k=l
80
(rTi 
•
ilT
(Ti 
.
.. r 1 OL
{4.131}
and the ('::1, ... , Zi)Sllbsystem becomes
U:]
=
CI
1
1
C2
o o
1 
o
0 1 + 0'23
0'23
1 O";I,i
{4.132}
4.2
147
SETPOINT REGULATION
Step n. At the final step, we introduce =
;;n
and rewrite the last. equat.ion :?n = /311
711
L 1..=1
+ cp~'(J 
:j"1
(4..133)
0'n1
:L'n 
+ IPn{.r)TB as
= (3{."C)u
ao'
a~1 (.1'k+1 ,'lk
+ cfJIo) 
ao' . ao
~o,
(4.131)
where the last regressor vector is defined as ~
III
aO',,_1
L D" 'P"" 1..=1 .[1.
'Wn(x,B) = 'P" 
(4.135)
In this eqnation, the act.ual control input. is at: our disposal. 'Ve are finally in the position to design our actual update law 0 = 1"TII and feedback cont:rol 1.1 to st.abilize the full =systelll with respect. t.o 1~,
=
, 1111
]
_2
+ 2""71
= !=T =+ !oTr1o, 2 2
(4,136)
Our goal is to make l~ nonpositive:
l~r
= 
L
III
'1
CI.:;;j;
+
(JI2 L
1.'=1
,.
ao'' _) (frll_1  tJ)'.
=k+l
A 
ae
k=l
+=n
[ =,,1
+8T
(
T,,1
+ f311. 
+ W n _n
aOnl 1 + 'Ill"T'(I  ~O a 'J L a.:rI.·+ .7:1._ ao Q'nl ~
nI 1.:=1 
r 1 B') .
(4.137)
:::
To clhninate
efrom l~, we choose the updat.e law o= =
rTn{:,O)
= rr"_1 + rw".="
rlV(.:, O)z,
(4.138)
where the regressor matrix H' is composed of t.he regressor vectors WI, •. , , W,,: (4.139)
'Vc choose the control u to mal{f t.he bracketed term multiplying .;;" equal t.o
u
= (31 ( =nl 
C'.=1l
f){Xn_1 +~ L ~.1.'I.·+1
~'=1
k

T /IJ"':,,
aO'IIl I'" ) + Tn + l.1n
ae
1
(4.140)
148
TUNING FUNCTION'S DESIGN
whcre
1)'1
is a rOn'€ction tel'm yct to be chosen. With (4.140), '{I becomes (4.141)
TllCll, noting that. (4.142) we rewrite l~1 as
l~1
=
111
L

~I
Now the correction term
(n2 L IJn 
~)
EJ Zk+l
G.J.· fwn
)
(4.143)
.
W
is chosen as
I)"
_ 1.111 (.1',0)
Cl.'=~ + =11
112
EJo'I.'
III
1.'=1
88
I..=:.!
= ~ ="'+1.. rWn =  L
(4.144)
O.,.·.f.Zk·
\Vc have thus reached our goal: 71
'7.1  L" Ck1...· I
_
_
_
(4.145)
k=l
The overall closedloop system is .:.
=
o=
A:(=, 8)= + H'(=, iJ)'I'o
(4.146)
fH'(=,8):,
(4.147)
where
A:(:::,O) =
Cl
1
1
C2
0 0
1
o
0 1 +0"2:)
.
0"2:i
1  0',,1,"
U2n
(4.148)
1 + unI,u en
The syst.em (4.14G) will be referred to as the enYJT system. It is important to note that a major portion of the design effort was invested into achieving
A.(z,li) +A,(=,ti)T = 2
[Cl ".
C
n
l'
'0'(=,8) E JR."'"
(4.149)
which yields (4.145) with the simple quadratic Lyapunov function (,1.136). 'Ve observe t.hat, as desired, the system (4.146)(4.147) has an equilibrium at (=,0) = (0,0). The stability properties of this equilibrium will be established in Section 4.2.2.
4.2
149
SETPoIN'r REGULA'nON
Remark 4.10 Slrew symmetry is not the only tool we can llse for achieving nonpositivity of l~. For instance, instead of (4.130) we can use (4.150)
which results in
A:(.:, iJ) =
Cl
1
1
C2
0
1
o
0 1 + 0"2~t C;i 
o
112
2
~0"2:l
o
It can be shown that with (4.150) we obtain
V(z,B) E lR"+'~
which yields
. ~ l~r
""
1 "
(4.152)
(4.153)
? L CkZ'k. ')
 k=l
We can carry this idea a step further. By replacing the st;abilizing function (4.125) by
(1.154) we arrive at the error system (4.146) with Cl
1
1
=
1 1 en
[
q~.l.
+ [
O"n,l
0'2,n
o
..•
1
q'
n.1I
o +
. (4.155)
150
It
CRn
TUNING FUNCTIONS DESIGN
be shown t.hat A: satisfies
'9'(=,6) E rn,n+p (4.156)
which yields (4..153). Instead of cancelling the tuning funct.ion Ti, the st.abilizing function OJ in (4.154) dominates it, \Ve ('onsider (4.130) preferab1e over (4.150) and (4.154) because the latt.er two lead to a much faster growth of nOlllincaritics in the ront.rol Jaw. 0
Exalnple 4.11 In applications of the t.uning functions procedure wc do not need to repeat t.he Lyapunov argument. All we need [01' a specific design al'e the nnal analyt.ical expressions provided by thc procedure. Let liS now iIlust.rate t.his by designing an adapt.ive cont.roller for the benchmark system from Example 4.8: Xl = .1:'2 +
(4..157)
·T.'3
11.
The design objective is the regulation of t.he output y =
.1:'1
to the setpoint
:'19' The first. threc exprcssions provided by the procedure are the definitions
(4.87), (4.88), and (4..97) of the error variables
where
0'1
and
02
=1
=
.1'1 
1Js
Z:!
=
:1:2 
at (:1:1,0)
Z:i
=
:Z:3 
02(Xl,
(4.158)
x~, 8) ,
are the stabilizing functions given by (4.94) and (4.103): TA
0'1
=
CI=t 
'P 8
0'2
=
C2=2 
=1
80'1
+ 8:1:1 (X2 +
T
A
8)
aa1
+ 80
T!,!.
(4.159)
The tuning funct.iol1s, determined from (4..03), (4.102), and (4.111), are T1 T2
= =
Ta =
=I
2 
(4.160)
 80 .. cp
"'31kt' .
vVit.h t.he above e.."\."Pressions and the choice r = I, the parameter update la.w and t.he feedbacl\: control are obtained from (4..138) and (4.112), respective1y. They are
4.2
151
SETPOINT REGUT..A'fION
This completes the design of the adaptive controller for (4.157). In t.he coordinates the designed system is
.. = 6
=
[io
1
~,
+ ~lrpP~
1
(=Ji)
~II"I' ] .: + [ k ]rpTO (4.163)
(~3
~
lP[l, ~, ~~~].::.
(4.16~1)
It is of interest to relate the stabilizing functions 0'1 and 0'2 and the ("ontrol law u to the material from Sect:iOIl 4.1. The stabilizing funct:ion o( has a "certainty equivalence" [orm. The sta bilizing function a::! has the t,erm ~ T2 which accounts for parameter estimation transiellts, while the rest of it is in the "certainty equivalence" [orm. The control law U departs from th~ '~certaillty equivalence" form ill its ]a.",t two terms whose role is the same as that of (4.66). The last term in u is particularly import~U1t. SinL'f:' ~ = _t.pT, this term contributes with +~lcpI2 in the 'Fiystem mat,rix' in (. t163) and achieves the skew symmetry, which is crucial for stability. 0
4.2.2
Stability and convergence
To investigate stability properties of the closedloop acb'pt.ive system (.1.146)(4.147), we express !.pi, ai, Tit and 'Wi in the zc0t?,rdiuat.es. Then, by Theorem A.5, the global stability of the equilibrium (=,0) = 0 follows from t.he fact; that the derivative Vn of along the solutions of (.1.146) (4.147) is given by (4.1(5). Fl'om LaSalle's Invariance Theorem (Theorem ~.2). it Curt.her follows that the ((n + 1J)dimensional) state (.:(I),8{t» C
v,.
Z == O. Setting
== 0, .: = 0 in (4.146) we obtain 0 = 0 and JV(z, O)T{O  8) = 0,
V(=, 0) E 111.
(.tHi5)
From (4.121) and (4.139) it is easily scell that
l'V(=, (})'l'
=
1
0
_!!U
1
8.1:1
0
F(.1:)T 0
_80.,._1 OZI
_80 11 _1 lb:,. .... 1
l!. T = N(z,O)F(:r) . ~
(4.166)
1
Since lV(=,6) is obviously nonsingular for all (=,8) E lll, then (4.165) and (4.166) imply (4.167)
152
TUNING FUNCTJONS DESIGN
Now we sl1mv that x
= x e on 1\1. Since Zl = Xl 
Ys then :':1 == Us  x~ on
.1\1.
In view of (4..167), wo get (4.168) Recall from (4.94) that
01
= C1Zl 
{)Tcpl'
Therefore, on ili we have 0:1 = Z:z = 0 = X2  0'] and (4.86),
6Ttpl(Xn = 8Tcpl(xi). Combining this with we get X2 = :r.~ on lIf. Using (.1.167), we obtain
(9  6)T cp2(X~ t a:g)  0 on lH .
fashioll, \Ve ])rove that i 1, ... , n. Thus, the largest invariant set AI j)) E is
Continuing
O)'l'IPi(x1, ...
ill
the
SaDIe
,xn  0 on AI,
AI = {(z,O)
=
=
fl
I
E lR +1) z= 0, n
{(x, 0) E lR
+1) ,
(4.169) Xi
:c1
=
and (6 
~re =o}
x = xe, FeTiJ == p;:1'e}
I
(4.170)
Fe = F(x C ). The two equivalent expressions [or ill and the convergence of (=(1), 8(t» to Al prove that x(t) I> xe as t + 00. where
An important property of 111 is its dimension, I}  rank{Fe}. When ranlL{Fo } P, then dim)ll = 0, that is, Jll becomes the equilibrium point x = 3:1.1, iJ = 8. This means that the parameter estimates converge to their true values, so that the equilibrium x xc, 9 = IJ is globally asymptotically stable. The ~1.bove facts prove the follO\ving result:
=
=
Theorem 4.12 The dosedloop adapti1Je .';ystem. consisting oj the plant (4.8,4), the controller' (4.140), and the 'update lUtlJ (4.138) has a globally stable equilibrirtm. (x, e) = (XC, 0). F'i~rthermoret its state (x(t), O(t)) confJerges to Ute (p  ff:1.llk{Fc })dimensianal equilibrium mCl.nijold AI given by (4,170), which. mea'lU1, in pa1'ticular, that lim xU) = XC • ' .... 00
If Ys = 0 and F(O) = 0, f.h.c1Ilim,_oo x(t} = O. The equilibrium x = x e I 0 = (} is glolJally asymptotically stable if and only iftallk{Fe} = p. As the dilllension of :AI reduces, the stability properties of the adaptive system improve. The most desitable case is when Ai is an equilibrium point, hl which case this equilibrium is globally asymptotically stable, and the par l'ameter estimates converge to the actual parameter values. Globa) asymptotic st,ability can be achieved with as mal1Y as p n unknown parameters. This is among the main advantages of eliminating overp'o.rametrizatioll. We now discuss the basic stability properties established in Theorem 4.12 on a simple example.
=
4.2
153
SETPOINT REGULATION
Example 4.13 'Ve consider the second order system \vith 8n unitnmyn parameter vector 8 E JRP:
Xl == X2 + IPI {XI )T8 i:2 == 11 + IP2(x)TIJ. The control ob.iective is to regulate .l~ to
ZE'ro (;l:~
(4.172)
== 0). 'Ve define the error
variables =1 =2
= =
Xl !t':.! 
0'1 (XI'
The controller is designed applying (4.94)
al1d
9) .
(4.173)
(4.103) as
while the parameter update law is
:. = r ['PI,
(J
DOl]
CPr.!  n:ct'l :;. v.lt
(4.175)
The resultiug error system is .:. _
... 
[ Cl
1
1] =+ [
C2
T
ifJ2 
'PI~¥'I an T ] B.
(4.176)
Now we illustrate and discuss the stabilif.y )ll'operties established hy Theorem 4.12. From (4.172) we see that :I:~ == 0, J;; ~J(OrrO. By Theorem 4.12,
=
the point
(4.177) is a globally st.a.hle equillbrilllTI, and the sl;atc of the dosedloop system converges to the equilibrium lllallifold
11'[ =
{(x, 0) E IffHp I [
~~ ] = [ _
A basic question that aile would ask is: Wlmt type of fignre ill m?+p is AI? Our further discussion will, without loss of generality, be limited t:o ]J S 2. In the simplest case where dim 0 = p = 1, the following two possibilities exist:
154
TUNING FUNCTIONS DESIGN
• If both J.:
1(il (0)
= 0 in ffi,:1,
= 0 and !p:!(0, 0) = 0, then the manifold J11 is the subspace that; is, 1H is the Oaxis.
• If either 9] (0) i 0 or Y2(01 !P1 (0)0) =I 0, then the manifold 1\1' is the single point :Z:I = 0, :1.:2 = Y1(0)fJ, 8 = fJ. This point is an equilibrilllll which is not only globally stable, but; also globally aBymptoUcally stable. we analyze the case p
~;g:: ~~:]
• Suppose [
= 2. =
[C~:~~l e~I].
Since
[~:i~i~] = [~ ~ 1
has full rank, the manifolcllH is the point :1;1 0, ;};2 = 821 O2 = 021 which is a globally asymptotically stable equilibrium,
81 =
OIl
• SIIp])ose [
[ 92(,1.'])
~1 ~],
T 'I'
1
[eos :);] .
e:1:1
0
sm :r1
the manifold AI is the linear
iJ'!.  @l = o',!  01 • Neither of the converge to the actual parameter to the line O2 = e] + 82  Ol in the
1 .
'""ll'lOT",,,
Xl
= 0,
:r2
=
(h 
0'21
estima.tes is guaranteed t.o but they are jointly cOllverging 0, :L~2 = 01  O2 
• Suppose
[~ ~ l,
the manifold M is the platle (lincar variety)
.~
=
o.
This is
the ease of the weakest convergence properties because one cannot guarantee that the parameter estimates converge to any submanifold in the plane 1'1. 0
4.2.3
Passivity
The dosedloop adaptive (4.146)( 4.147) has all important passivity property_ Rewriting this system in the following (z,O)form
0);:
[j
+ Hl(z, O)1'(}
fl'V(z, 0).::: 1
(4.179) (4.180)
we see that due t.o the structure of ;1::(z, 0) in (4.148), along the solutions of (4.179) we have n
L
Ci Z ;
+ z1'lV1'(}
i=l
(4.181 )
4.2
SETPorNT REGULATION
155
r Figure 4.1: Negative feedback connection of the strictly passive .:::system and the passive system ~.
By integrating (4.181) over [0, I), we get
By Definition D.2, (4.182) implies that the syste1l1
AAz, (})z n"(.:, 0)::
+ 11'(.:, ())T()
js strictly passive with fj as its input, T" as its output, V(.:) = k.: T z as the storage function, and 'IN.:) = L~~l CjZ; as the dissip(1tion rate. On the other hand, the integrator system

 () =
r
Til S
(4.184)
is passive from Tn to 0. Hence, the closedloop adaptive system represents a negative feedback connection of the strictl:\" passive system (4.183) with the passive By"stem (4.184), as shown in Figure 4.1. By Theorem D.4, the equilibrium (.:: (5) = (0: 0) of the feedback system in Figure 4.1 is globally stable: and z(t)  t 0 as I. i O. An interpretation of the tuning functions design procedure is t.hat at each st.ep i, a tuning function Ti is chosen as an output of tIlE.' (':1: ... , Zj )sllbs:vstem, and a control law Ctj is designed to make this subsystem strictly passive. A schematic representation of the recursive procedure is given in Fignre ~1.2. The design is completed at the nth step, where the last: tuning [unction Tn i~ used to close the adapt.ive feedback loop via the passive panuneter update law (J = fTu' As an illustration of the use of passivi ty at. each step of the design procedure Figure :1.3 shows the adaptive feedback connection at st.ep i = 2. The property that the output matrix is the transpose of the input matrix, a distinctive feature of passive systems, is clearly displayed for the upper syst.em.
156
TUNING FUNCTIONS DESIGN
.
ITI
I
J+
IT!)
I
I I
t;.
=1 r.J
0
Tn
1
:;:1) I
0'1
t~,~  =.J .
"
n
'a',. = lJ, r
r

I,
Figure 4.2: A schema.tic l'epl'csentation of the design procedure.
4.3
Tracking
The setpoint regulation design is readily extended to the task of tracking. The control objective is to force the output :11 = a:l of the system
:;;1 =
l!2
:i:!:,!
X3
=
+ 1,01 (x.)TfJ + 'P'2(X11 .l'2)TO (4.185)
+ !PnI (xt, . . , ,X !_I)TfJ = xn = fj{x)u + 'Pn(:t),rO
.:l:n_l
X'I
1
t.o asymptot.icaUy track the reference output Yr(t) whose first n derivatives are assumed to be known, bounded, and piecewise ('olltillUOU6. All alternative control objective \v01.l1d be to asymptotically track the output of a knowll asymptotically stablp linear reference morlel (4.186)
where SR + 1n,l_l='11 + ... + mo is Hurwitz, knz > 0, and 1·(t) is bounded and piecewise continuous, A realization of (4.1Si) which is of particular interest is
(4.187)
because, in this case, t.he derivatives of Yr arc available as the states of the reference model: y~i) = .l:1II 1i+l, i = 0, ... , n  1.
4.3
157
TRACJ{ING
1
c) [ 1
C2
1
r s Figure 4.3: The feedback connec.'tion of the strictly pnsRive passive update law.
4.3.1
(=., =2)system with a
Design procedure
The design for tracking is only tl minor modification of t.he setpoint design procedure. As before, the first =variable is the tracking error, =1 = :rl Yr' However, because the reference signal Yr(t) is not constant., its derivative y~iI)(t) appears in the definition of the it.ll error state :i, i = 1, ... ,71. The only change this creates in the design is the addition of the sum L~:A u(::~)y~~.) OYr in the definition of ai. As we showed in Section 3.'1.3, nonlinear dampirlll can be used to guarantee global bouncledness in the absence of adaptat.ion, as well as to enhall('e performance. Therefore, the general design also in('orporat.es t·he nonlinear damping terms (4.188)
in the definition of a/so It is sufficient that we now only give a complet.e set. of recursive expressions for the stabilizing functions ll:i and the tuning fllnctio,ns Tj leading to the final adaptive control Jaw 11 and the final npdate law for O. These expressions, organized in Table 4.1. give a succinct summary of the t.uning funct.ions design.6 It can be checked that the resulting errOl" system has the following form similar t.o t.he setpoint regulation case: : E UFor notational convenience we define =0 ~ O. ao ~ o. TIl ~
o.
lR"
(4.195)
158
TUNING FUNCTIONS DEStaN
Table 4.1: Tuning Functions Design for Tracking (4.189)
(4.190) (4.191) (,1.192)
i = 1, ... ,n ,(i) _
.Ilr 
Un Ilr, ... , Yr(i»
(.'
Adaptive coutrol1aw: 1L
= {3(x) _1_ [a
(x
JI"
Parameter update
0"'/r 'ii(nl» + y(I.)] r
(4.193)
law~
(4.U)4)
rWk,
the matri'l: ;1:; (z, iJ, t) has the form where, with the definition O'iI.. =  8';;01 of (4.V!8) with the addition of t.he nonlinear damping terms liiI1ll;j2Zj: Cl  Ii.llwl1 2 1
A:;
=
0
o
C2 
1 0 1i21w212 1 + 0'2:"
1 
0 0'21.
0'2:"
1 + UnJ.n 0'2n
1 
O',,l,n
C"  Ii ll lwn
l2
(4.196) alld 11/(.:,8, t) has the same form as ill (4.139). Although the functions O'iI•• and 'Wi lllay appear t.o be the same as ill the setpoillt regulation case, tbiR is not so, because uow they include y~i)(t) through the partial derivatives of aiJ, wbich is reflected in the dependence of A:;(.:, 8, t) and H'(.:, 6, t) on t.
4.3
159
TRACKING
The change of coordinates (4.189)(4.192), which we compact1l' write as
(4.19i)
is smooth in 3: and jj and bounded ill I. Note also that the invprse transrormation
x is smooth ill
= 4>(=,6,1)
(J.108)
=and jj and bounded in f.
Tbeorem 4.14 The closedloop (1.dClpti'l./c systcm. c01)"r;iBting oj the 1,lant {4.185}, I.lIe controller (4..193), and th.c 'U.pdatc law (..I. 194}· h.as a glol,ally ufl,ijo;'nly stable equ.ililnium at (:,0) = 0, and lim :(1) = O~ which meanB, in 100 particular, t/f.{I,t global asymptotic trrJ.cking is achic'ucd: (4.199)
lim fy(t)  Yr(1)] = O.
100
lIf()re.o1Jcr~ if lim y~i) (t)
'00
Proof. Denote
Co
Ol i = 0, ... ,11 1, and P(O) = 0, the1' ,lim x{t) = O.
00
= mi1l1SiSn Ci. 1l'n
The uerivative of tbe Lyapunov function
1 T = 2"= =+ 'lT 2 8 r1o
(4.200)
along t.he solut;ions of (4.105) aud (4.104) is (4.201)
=
which proves that the equilibrium (.:,6) 0 is globally unifOl'mly stable. Fl'om the LaSalleYoshizawa theorem (Theorem 2.1), it further follows that, as t ~ 00, all the solutions converge to the manifold = = O. From the definitions in (4.189)(4.192) we conclude that, if lim y~t)(1) = 0, i = 0, ... ,71  1, and
F(O) = 0, then a:(I)
~
0 as f
~ 00.
'00
0
The proof of Theorem .l1.V! reveals the st.abilization mechanism employed in the tuning functions desigIl. The update law is chosen so as to mal\:e t.he delivative of the Lyttpu110V fUllctioll llol1positive. The update law is fast because it does not use any fOl'm of normalization common ill t.raditional certaillty equivalence adi1.ptive C:Olltrol. The speed of adaptation is dictat.ed by the speed of the nonlinear behavior capt.ured by t.he Lynpunov fUllction. The tuning functions controller incorporates the knowledge of t.he update law and eliminates the disturbing effect of the parameter estimation transients on the error system. The controller and the update law designs are interlaced. The nai. theorem shows t,hat in the absence of adaptation the nonlinear damping terms guarantee boundedness. In addit.ion, global asymptotic stability is achieved for sufficiently small parml1eter error ii.
160
TUNING FUNCTTONS DESIGN
Theorem 4.15 (Boundedness and Stability Without Adaptation) Consider the closedloop adaptitw system con.sisting oj the plant (4.185), the contmlle1' (4.193), and the ·upda.te la'llJ (,1.19.0, with r = 0 and Iii > 0, i = 1, ... ,71. All the solu.tions are globalll1 lJ,'niforrnly bounded. Fu,'the.7inoTe, if F(O) = 0 and Yr(t) == 0, then there exist Rn > 0 sytch that /01' each constant iJ E ]R!), 10  01 :5 Ro, the equilib7'ittm:c = 0 is globl/·1l11 asymptotically stable.
(Ei~J
t)l.
r
= 0, the paramet.er est.imate 0 is consttUlt. Denote For the nonadaptive system (4.195) we have
Proof. Since
From Lemma C.5(i), taking
t1
IiO
=
= ;:2 and p = Fn 101, it follmvs that (4.203)
This proves that the solution .:(t) is globally uniformly bounded and, sin('e 3.' = q, ( :; I ij It) is smooth in :; and fJ and bounded ill t, this also proves that x( t) is globally uniformly bounded.
Figure 4.4: The solutioll enters the larger ball (solid) in finite time, Then, the equilibrium z 0, wbich is exponentially stable for 5ufficiently small 9, tal(es over and attracts the solution.
=
4.3
161
TRACKING
=
Now, suppose that F(O) = 0 and 1Jr(t) O. Since x = Dwhenever == 0, in view of (4.166), we have TV(D, 0, t} == O. Thus = 0 is nn equilibrium of (4.195). To prove asymptotic stability of == 0, we only need to consider t,ile case 0 "# O. The situation is depicted in Figure 4.4. From (4.203) we know
=
that =(t) converges exponentially t;o the ball of radius
4ink around == o.
This means that =(t) enters a larger ball, say the ball B of radius finite time t :::; T = rna..x {O l enters B it decays
~ In :!~I:(O)I}.
A, in
Note that before the solution
e~ponentially:
I $. T.
(4.204)
VVe now examine tho trajectories inside 8. Because H'(:;, 8, I.) is locally Lipsrhitz and vanishes at 0, there exists a finit;e positive number L sllch that. fol' 0.11 : E B we have jlV{:::, 0, t.}j :::; Lj=j, and therefore
==
!!.(! jzj:!) dt 2
:::;
:::;  (co 
LIB!) 1=12,
t?. T.
=
The equilibrium z 0 is (locally) m.ponelltially . st.able provided j81 z(O) E B we have T = 0 and
I=(t)j :::; 1=(0)je(co1J1fil)t . 'Vhen z(O)
rt B then T > 0 and, for t ;?: T 1=(1.)1 <
=
For
(4.206)
we have
~e(cllLIOI)(/7') JColio 8j 1
rnLli'l :l.li'iiToI={o)l n 161 e(C41LIOj)/,
  e  "oJCon.o _
I'IILI81
= J!L (2~1=(0)1) "1) e(col~I;;l>t . JCOI~O
Since z(a)
< f£.
161
rt B, it follows that 2~1::(O)1 > 1, which because of 0 <
(4.207) co:!JIOI
<1
yields
Iz(t)1
~
181 2Jco"·olz(O)j _ e (coLIO!)t Jcon.o 101
=
21=(O)le(COL1ol)t.
(4.208)
Combining this with (4.204), we get
1=(t)1
< {2IZ(0)le cnf
2jz(O)le(cuL loj)t
t ~T t ?:. T
'tit ;::: 0,
(4.209)
162
TUNING FUNCTIONS DESIGN
This proves that for each {) E R" such that ~
Co A
16  0\ < L = Ro ,
(4.210)
the equilibrium z = 0 is global1y exponentially stable, so the equilibrium x is globally as~rmptotically stab1e.
4.3.2
== 0 0
Trajectory initialization
One of the goals of adapttttioll is to reduce uncertainty, tbat is, to make B(t) smaller. Pi'om the proof of Tbeorem 4.14 we know that 8(t)Tr 10(t) $ 2Yn(1} ~ 21~(0)j that is, (4.211 )
This bound shows that a possibility for reducing O(t) lies in .:::(0). '¥e will nmv explain how =(0) can be set to zpro by an app1'oprhlote initialization of the reference trajectory. As we shall see in the next sec(:ion, all even 1110re important benefit of se(:t.ing =(0) = 0 is in reducing a bound 011 .:(£), that is, in improving the adaptive system's transient performance. Let us consider the change of coordinat.es (4.189) which defines:: in t.erms of Yr and its derivatives: Zl
=
Xl 
.32
=
X2 
Yr fir 
0'1 (3"1 ,
Z:i
=
:r3 
Vr 
a·']. (~'lt 3.'2,
Clearly, in order to set :.:;(0) the initial conditiollS
Yr(O) = Xl (0) lir(O) = 2:2(0) fir(O) = X:i(O) y!"1)(0) = ,'V,,(O) 
iJ, Yr)
e, Yr,Jjr)
= 0, tIt£.' l'eference trajectory 1Ir(t)
(4.212)
has to satisfy
0:1 (.1:} (0), 8(0), Yr(O))
0:2(X1 (0), J:2(O),
0',11
0(0), Yr(O), Yr(O»)
(Xl (0), ... "I: n l (0),6(0), Yr(O), ... ,y~n'.!)(o») .
( 4.213) It is true that Ur(O), Yr(O), ... ,y~'II)(O) are prescribed by Yr(i) and appea.r not to be free for us to chose. However, it is also true that the tracking objective y(t) = X1(t) = Yr(f) ca.nnot be achieved instantaneously, but only asymptotica.lly as t  t 00. We are therefore allowed to modify a prescribed
4.3
163
TRACI{tNG
refcrence signal YrO(t) by an additional signal 6r (t) which vanishes as 1 ~ So our system will be designed to track
00.
(4.214)
and the asyrnpt.otic tracking of YrOU) will still be achieved because t5r (l) ~ O. By selecting c\(O), 6r (0), ... t5~nl)(o), we arc free to choose Yr(OL 1}r(0), ... , ll~'II)(O) as desired, 1
Dr(O) !ir(O)
= =
Ur(O)  l/rO(O) fir(O)  YrlJ(O)
o~,,I)(O)
=
y~'Il}(O)  .IJ~l )(0) .
(4.215)
The signal ~At) can, for example, be generated as the first state 6r = 6) of tho exponentially stable lincar system in phasecanoniC' form,
(~L21G)
with init.ial conditions Oi(O) = ,WI)(O). Alteruatively, if the cont.rol object.ive is to track the output of a reference mode1 as in (4.187). t.hen the trajectory initialization amounts to setting the reference model initial conditions at X""I (0)
= :rl (0) xm,:z(O) = X2(0) :1:771 ,3(0) = :!':i(O) 
(:rl(O), 8(0), X""l(O» O:2(Xl (0), X2{O)~ 8(0): ,1: m .l (0), xm,:z(O»
3.'m.n (0) =
ltll_1 (Xl(O):
0;)
... , Xnl (0).0(0), x m ,ICO), ... , X""nI(O» . (4.217) Let us gain more insight into the initializat.ion algorithm (4.213). Since y .T.l, from the first equation ill (4.213), we see that the in.itial value of tbe reference output is placed at the initial value of the system output, .1: n(0) 
=
Yr(O) = yeO) .
(4.218)
Next., we evaluate the derivatives of y from the p1ant. modcl (4.185):
Y = y(i)
l::..
Xl = hO(.'l:l)
8h i _ 1(Xi, f) ) ( Xk+L + CPk (_ )T ) = ~ L ~ :l.·k 0
l::.. = hi (_ .:z.'i+I,8 ) ,
k=l U·'tk
i = 1, ... ,11. 1.
( ~1.219)
164
For 'i
TUNING FUNCTIONS DESIGN
= 1, we ha.ve
iJ = X2 + lPI(.l:tlTO = h t (X2,O).
(4.220)
On the other ba.nd, fmm (4.213), in view of (4.190), we have
= = = =
lir(O)
.1:2(0}  (}:1 (XI (0), 0(0), lIr(O» X2(0) + CIZ1(0) + lPI (.Ll(O»TO(O) :1:2(0) + 'PI (Xl(O})TO(O) hI (.1:2(0),0(0» ,
(4.221)
which means that Yr(O) is placed at an estimated value oC ~j(O). For i = 2, we have {)h 1
ii =
~(X2 VXI
= nv.Lt
8t.{Jl (
=
T w
X2
T )
']'
+ 'PI 0 + 3:3 + 'P2 fJ (4.222)
h 2 (.fa,O).
From (4.213), in view of (4.190) for i
Yr(O) = !ta(O) 
=
tJh
T
1 + lPt 0) + D (:t3 + cp., 8) X2
= 2~ we obtain
Yr{O) , ,ir(O» Bert Bal. %3(0) + =1(0) + C2 z2 (0)  ~X2(O)  ~Yr(O) Q·:)(.1:1 (0), x2(0),8{O),
VXl
uYr'
+~~(X2(0))T8(O)  ~al /PI (Xl (O))TO(O) u'''C1
Bal
 B8 • 0 Smce ~
= Cl 
D'I'TiJ
r1"l (Xl (0), 1/r(O))
Do.
O;J ' ~
(.:1.223)
.
= Cl, and 71 (Xl «0), Xl 0)) = 0, we ba.ve . ']'
jir(O)
=
xa(O) + 'P2(X!!(O)?O{O) +
=
h2 (x;i(D),9(O» ,
BX;lO (X2(0) + tpl(a:l(0}yr 9(0») (4..224)
which means that fir(D) is placed at all estimated value of yeO). By continuing ill the same fasbioll, we reach the following conclusion.
Lemma 4.16 For the coordinate change (4.189)(4.192)1 the following equivalence holds: (4.225) Remark 4.17 The import.ance of t.his conclusion is twofold. First., the trajectory initialization can be performed using IJ$i)(O) hi(Xi+l(O), 8(0» instead oC (4.213). Second, and much more important., the trajectory initial conditions do not depend on the design pa.rameters Ci, K;;, ttlld r but only on the initial state x(O) and the initial parameter estimate 6(0). <>
=
4.4
165
TRANS lENT PERFORMANCE
4.4
Transient Performance
III this section we will derive £2 and £00 transient performance bounds for the error state =, which incluue a bound for the tracking error =1 == 11  Yr' We will make use of tile constauts For simplicity, we let = 11.
r
4.4.1
Co
=
l11illl~i~f1 Ci anulio =
(E7:1
t) I,
£2 performance
We first derive a bOllud
all
the £" norm of the state of the error system.
Theorem 4.18 In the atlapti:ve Bt/stem. {4.185}. (4. 194}, (4.199), t./le /ollowi'flY inequality holds
11=112 ~
18(0)1
1
J?('01' + v:.!co m::I={O)I·
( 4.226)
Proof. From (4.201). we have
\~I $
co,:," ::; o.
(4.227)
Since lin = ~'=I:! + f16'j2 is nonincreasing a.nd hounded from below by zero, it 
118..'1
a. limit as t
Jo
i 00,
so
II=II~
(4.228)
o
which implies (4.226).
It may appear that tlle bound (4.226) can be reduced by increasing Co alone. This is not so. Althougb the ternl ~}O)I is reduced by illcreasing Co, , the initial condition z(O) may, in fact, iucrease by increasing Co (lzi(O)llllay ~CII"
grow as C~l). Fortunately, we can circumvent this difficulty using t:rajecf.ory initialization. By applying the initialization algorithm (4.213), wo set =(0) = 0
anel obtain (4..229)
At t.he same time, as we noted in Remark 4..17. the tn\jectory init.ial conditions are independent of Ci Therefore, the £2 transient performance of t·he =systCl11 can be systelnatically improved by increasing CD
166
TUNING FUNCTIONS DESIGN
Another way of improving the £2 t.ransient performance suggested by the bound (4..229) is by increasing the adaptation gain "y. The tuning functions scheme is unique in this respect because it directly compensates for the timevarying effect of parameter estimation. As we shalJ sec, the bounds for estimat.ionbased schemes are increasing functions of"( for large ,ralues of;.
4.4.2
£00 performance
Now we turll our attention to £.00 performance. For the adaptive system (4.195) with (4.196), the following bound is obtained from (4.227) and (4.200):
Iz(t)1
:s
1 ~IO(O)l
+ 1=(0)1.
(4.230)
Although this bound suggests that performance improvement. is possible through increasing l' and init.ializing =(0) = 0, it docs not capture the effect of the design coefficients Cj on the system transients. In the general tuning functions procedure summarized in Table 4.1 we introduced the nonlinear damping terms K:;lWi 12 =j for achielring boundedncss. With these terms, we are also able to derive the following L./Xj bound. 7
Theorem 4.19 In the adapti'l1e. s1}stem (4.185), (4.199), (4.19.0. with x.o > 0, the. following inequality holds (4.231) Proof. For the adaptive system (4.195) inequality (4.202) gives
d (~lzI2) d.t _
::; colzf + ~1912. 4#\'0
(4.232)
From Lemma C.5(7.), taking v = z2 and p = Fo191, it follows that (4.233) A bound on IIOII/Xj is obtained from (4.211) as 11911~ ::; 19(0)1 2 + 'Ylz(O)f, By substituting the latt.er e.xpressioll into (4.233), we complete the proof. 0
Remark 4.20 All elegant way of removing the term 'Ylz(O) 12 from t.he bound (4.231) wit.hout I;ntjectory initialization, is using t.he "observer"
z(O)
= z(O).
(4.234)
T As we shall see later, a similnr bonnd can be derived for linear system.! even without these t.erms.
4.4
167
TRANSIENT PERFORMANCE
VVe introduce the observer error

~ ==ZZ',
(4.235)
and instead of (4.194) we employ the updat.e law driven by the observer error: (4.236) Combining (4.234) and (4.235) with (4.105) we obtaiu the observer error system ~ A:(.:,O, t)z + HT(=, 6, t)TO.
=
Now, along the solutiollS of {4.236)(4.237} we have (~1.238)
which implies that (4.239) because =(0)
= =(O) 
=(O) = O. By substituting (4.239) into (4.233) we get (4.240)
instead of (4.231). Using this method, the £2 bound (4.226) and the £00 bound 0 (4.230) remain unchonged.
With our stal1dard initializat,ion ::;{O) = 0, combining t,hc bound (L1.231) with the bound (4.230), we get 1 1 } Iz{t)\ ~ min { ?~'" r::; \8(0)1· 
COliC
v'Y
(4.241)
It follows t,bat the £00 bound for the tuning functions scheme can be systematically reduced by increasing anyone of the design parametel's: Co, lio, Dr 1'. The tuning fUllctions scheme is uniqne ill its capabiHty to use the adaptatiol1 gain "y for improvement of transient performance. In Theorem 4.15 we proved that: nonlinear damping guarantees boundedness without adapta,tion for any (unknown) amount of parametric ul1(·ertaint.y. For the nonadaptive system we also estabUshed an Lee performance bound (4.203), which with =(0) 0 becomes
=
(4.242)
168
TUNING FUNC'l'IONS DESIGN
where 8 is a constant parameter error. Let us suppose that. tIle paramet.er estimate initial condition 8(0) in the adaptive design is the same as the const,ant parameter est.imate iJ in the nonadaptive design. (This a..'isumption is reasonable because the a priori lmowledge is the same.) Comparing the adaptive bound (4.241) wUh t.he nonadaptive bound (4.242) we come to an em;y but. fundamental conclusion: ''YIth adaptation gain "'Y >
4.5
Extensions
For t.he sake of clarity, the adaptive design in this chapter \Vus presentcd for the class of panullct;ric strictfeedbad: systems. We now givc three extcnsions of the tuning functions design. Vve first consider strict~feedbRck systems with unlmown virtual cont.rol coefficients. Second, we explain how the design is modified for parametric blockstrictfccdhack systems, Third, we extcnd thc design to parametric pllre~fcedback systems.
4.5.1
Unknown virtual control coefficients
\Ve consider systems of the form .1;j
i' n
== bi:Z: i + 1 + <,0;(;,;], • •. ,Xi)T(),
=
bn f3(.l·)U.
i
= 1, ... ,11 
+ CPn (:1:11 .•• "'CJl)T() ,
1 (4.243)
whcre, in addition to the unknown vector (), the constant coefficients bi are also unknown. "Ve refer to the cocfficients bi as the 'virtual control coefficients'. The occurrence of the unknown bicoefficients is frequent in applications ranging from electric motors and robotic manipulators to flight dynamics. Assumption 4.22
Tl/.(~
signs oj bi , i
= 1, ... , n,
are knoum.
\Ve consider two special cases of (4.243). The eAtension to the general case is straightforward but tedious.
4.5
169
EXTENS10NS
The first special case is when the ouly unknown virtual control coefficient is the 'highfrequency gain' (),,:
For this case the modification of the tUl1ing functiolls desigll is simple. In Table 4.1 we only need to chauge the cOlltrollmv (4.193):
e (n1» + l1r(JI))
fl [ (
A
u = /3(a;) an x, 'Yr where
,
(.:1.245)
f! is the estimate of e = 1/bn computed as
e= 1' sgn(b
n)
(a'" + !/~n») .:"
t
(4.246)
\Ve arc ouly using the knowledge of the sign of the unknown parameter b". In this simple case it. is not necessary to estimate bn itself. It call be checked that the result.ing error system has the form (4.195) with an tl.dditional term due to U= e  g:
Consider the Lyapunov function
(4.248)
Its derivatjve along the solutions of (4.2.:17) and (4.246) is
1; ~
coJ=f! ,
(4.249)
wbich leads us to the same conclusion as in Theorem 4.14: All the states are bounded and asymptotic tracking is achieved.
Now we move on to a more difficult case:
+ 'Pi(''VI!'" ,Xi)TO, i = 1, ... ,m. 1,711 + 1, ... ,711 xm = bmXm+l + 'Pm(X1 ,xm)TO, Xi =
Xi+l
1 '"
3:n
=
.8(x)'l/. + 'Pn (.'tlt ••• I Xn )TO t (4.250)
where bmt m < 7l t is the only unlcnown coefficient. From st.ep m· on, the design procedure for this case differs considerably from the procedure in Table 4.1.
170
TUNING FUNCTIONS DESIGN
We now need bm and il, the estimates of bm and fl = 1/bm • The estimat.e g is introduced to avoid the division by brll(t} which call occasionally truce value zero. The complete design procedure is given by the following expressions (with Zo 0, 0:0 = 0, To = 0):
=
Coordinate. tmnsJO'lTnation:
=,' =
x'  'I,(il) lIr
0",  1
• I
= x'J  .... Ol,UI) /r
=J'
i= l, ... ,m
,
(4.251)
j = m + 1, ... ,11
0:' J 1 ,
(4.252)
Regre,fUlor: i = 1, ... tn
Thning functions for
(4.253)
0: i = 1, ... ,11
(4.254)
Thni1t!J jtmctiofl.s f01" 'brn : ?Tnt
=
=m+l=m
.". = .... I'j IIjl

(4.255)
ao: . . . m+l .... am,.._
j,
Xm
j = m. + 1: ... , 11
(4.256)
= 1•... ,111 
(4.257)
Stabilizing functions:
((iI}) :ri, iJ ,llr = ii·It u(ml) A) a". (X m , iJ ,1Ir , fl = Bfi"1 fJ .(jI)· OJ, (. XJ' ,l1r ,1)m, flA) = a·
i
OJ
(4.258)
j = m + 1, ... ,n
J
iij
=
_
iI ""
T 
';;';1  CiZj  Wi
0+L
k=I
acriI
(ao: aiI
X &+l
iI OCikI
=

k=2
(4.259)
aa:) Y iI
+
(k)
(kI) r
aYr
XI.'
+.rTi + L .r·W;Zk'
ae
1
i = 1, ... ,m
ao
(4.260)
b m=m  Cm+lZm+l  'WT+ 0m I ml
a
a.
nl
!l
""" O·nI. bOm " " uOm (k) + k=L L a. Xk+l + ma' Xrn+I + L v;::r)Yr Xk ·"l:m k=10Yr
oexrnrTrn+1 + ((m) 80m ) ~ aO'kl +.Yr + a Q + L ~rWm+l=k '!.
8()
{}
k=2
ao
{4.261}
4.5
171
EXTENSIONS T~
:;;1 
a°jl aO"jl ;1 a b· "" O'j1 (k) Xk+ J + ma Xm+! + L u;=rrYr aXI.: XIII 1.=1 aYr
jI
""
+
"WJ ()
CjZj 
L
k=1
ki<: m
. + (, (jl) + an'iI) :. . + 80';1 ... r TJ .. [7r'J lIr [) .. fl + 8n'j_I aD ob B
\=! 80
m
k 1
+ L _rlJJjZk 80
k=2
~ OOkl Oo.'j_l L ."'(0:1:711+1=1.8b711 Xm j = m +2: .. ,,11
(4.2(2)
[
(4.263)
A'=rn+!
Adaptive control law: _
1
f3(x)
lJ. 
_
0"
(n)]
+ DUr
Pammetc1' update l(LW,If:
=
rTn
bm
=
,1r = ,
g
=
fJ
= rH';;:;
1l
(4.264)
[zm+1ZlII 
t
;=,"+1
1'sgn(bm ) (U~m)
°Oa:~l .l'm+l :;j] .1'JII
(4.265)
+ am) =m
(4.266)
Lengthy hut st:raightfol"ward calculations show that the design procedure (4.251)(4.266) results in the closedloop system
=i =
iI
L (J'J.'iZk 
CiZi 
"
L
+ =i+1 +
=il
1.:=2
O'ik=/..
+ w;O,
/'::;:;+1
i = 1, ... :1711
(4.267)
ml m
=
Cm=m 
L
n
O'km=l.· 
=1111
+ bm zm + 1 +
"'=2
+w!8' 
L
bm (y~m)
+ dm) U+ =m+tbm
(4.268)
m
=
Cm+l=m+l 
L
II
O'I.·,m+l=I.' 
bmzm
+ =111+2 +
80'm

+Wm +1 6  a X",+] .Lm
b
=
CjZ,; 
L
O'm+l,k=l.·
(4.269)
m
)1
Zj
L
k=m+2
1.:=2
'J'
UmkZI.'
k=m+I
II
O'kjZk 
=jl
U.:I m
L
(J'jl.,Zk
k=;+1
k=2 TOO'jJ +1Oj B  ~xm+lbm
+ :;j+l +
1
i=
171
+ 2, ... , 71 ,
(4.270)
172
TUNING FUNCTIONS DESIGN
where
Uil.
is defined for ~.. = i
+ 1, ... ,71 as i=1
0,
i = 2, ... ,m
Uil.·=
8a i_1 r.
~
f)O
Wk
+
UO'k1 • ,'"Ya.1 m+l, 8bm ."tJII OO:il
i = m
+1
+ 2, ... ,11 
(4.271)
1
A Lyapunov function for this system is (4.272) Its derivat,ive along t.he solutions of (4.2G4)(4.266) and (4.267)(4.270), (4.273) lcads us to t,he samc conclusion as in Tl)corem ~1.14: All the st.ates are bounded and as.ymptotic t.racking is achieved. For clarity, we lutve not included in the stabilizing flllldious the nonlinear damping t.erms which guarantee hOllndedncss in the abseuee of adaptation. To achieve this property, we augment the functions (4.260), (4.261), and (4.2G2), l'espp.ctively, by t.he nonlinear damping terms
and h"m
we
augment
sgn(bm )
the
stabilizing
function
(4.258)
by
the
term
(y~m) + a'm)2 =7111 (4.274)
It can be verified t.hat the resulting closedloop systcm is iI
=j
=
E
CjZi 
n
CII.·iZk 
=iI
+ Zi+] + L
k=!! 'J
n.dllJil~Zi
+ WiT8,
Uik=k
k=i+1
i
= 1, ... , m 
1
(4.275)
4.5
173
EXTENSIONS
=
Zm
ml
n
L
CmZm 
O'knr Z'" 
Zml
+ bmZnr+l +
"'=2
L
O"mkZI.'
h: 111+ 1
(1J~'") + amY'! ='111 +W!O  bm (y~nl) + am) g nml wml 2zm  Ii·mlbml m
ZI'II+l
:::
L
Ctn+lZm+l 
bmzm +
UI.· t7ll +1=1.· 
(4.276) "
'::",+2
+ L
k=2
I
I
t)
fi:m+l Ulm+l  Z",+l 
=
h:,n+1
(80'711)2 =m  8' ·'1: m +l ':m+l :I'm
+w~+16 + ( =.. Zj
::: Xm+l) bm
(<1.277)
jl
E UI.'jZk 
Cj'::j 
n
':jl
L
+ Zj+l +
t)
I I
T
Ujk=k
k::j+l
k=2
lij Wj :;j 
U m+l,k=l.·
N=m+2
n.j
(EJa:;_t 8x
3;1n+1
)
2
Zj
m
Oaj_l

+wjfJ.  xm+lbJrl1
';=m+2, ... ,n.
aXm
(4.278)
and then show that the following inequality is satisfied by the solutions of (4.275}{4.278):
d
dt
(11'2 z\"I) ::; co I.:\"I + 411'01 (1'1 " + fb 10  + b~l
,_I))
m IF
,
(4.279)
which proves that. a11 the signals are bouuded even when the adaptation is turned oft'.
4.5,,2
Blockstrictfeedback systems
'Ve consider systems of the form
+ !.p;(x!, ... ,J~i, (h ... , (i)'rO, = f3(:r, ()u + CP,J(x, ()TO
:h = ;i:p
Xi+l
i == 1, ... ,p1 (4.280)
i = 1, ... ,p wit.h the following notation: .fi
= [.LI, ... ,.l:JJ', (i = [(1, ... ,(l]T, X = .l:p, and
(=(P'
Assumption 4.23 Each (jsubsystem 0/ (4.280) Itas a bou1~dedi'nptl.t boun.dedstate (BIBS) p'roperty 'with 7'C.9pect to (Xi, 01) us the input.
174
TUNING FUNCTIONS DESIGN
A nonlinear system has a BIBS property if for any initial condition and fol' any uniformly bounded continuous input the solutioll exists and is uniformly bounded. s For this class of systems it: is quite simple to modify the procedure in Table 4.1. Because of the dependonce of !.pi on (j, thp stabilizing function (4..190) is augmented by the term E~:::\ 8~,;1 (Pk.lh namely,
1_"

llJ·(X· lJ t'( .. ) ! 'lsI 1".) = I • It '.'Ir A
_
til· T'
IJ
aQ';1
iI
"""'
{I'", T /t'
"'"" L..t
L..t
k=l
a.Xr.
k=l
(lJ
Q'iJ
«D.I [II".".
)T •
(4.282)
Note that the change in Wi in (4.192) also has an impact 011 the tuuing function (4.191). The control law and tbe paramet:er update law are fLCI in (4.193) alld ( 4.194), respectively. It can be checked that the resulting error system has the same form (4.195)t (4.196) as the basic tuning functions design. Since the parameter update law (4.194) is A)SO the same, the Lyapullov funct.ion
1 T IT I"2"" ... + 2(J r (J
T/" 
'p
(4.283)
has the silme derivative: (4.284)
whieh proves that (=,8) = 0 is globally uniformly stable. The boundedness of ZI implies that. Xl is bounded. Then, by Assnmption 4.23, {. is bounded. This along with the boundedness of 4:2 establishes the bOlludedness of :C2. Then, by Assumption 4.23, (2 is bounded. Continuing in t.he same fashion) ,ve arrive ai, the conclusion that a~~ (, and {) are all bounded. As a cont:it1ll0US function of its arguments, the control 11. is also bounded. The asymptotic tracking property is established as in I:he proof of Tbeorem 4.14.
8Illpntt.()state slability implies BIBS, but the ('ODverse is not true. This can be seell. x =  (1 sin!! 1I).:t' for t.be particnlar input u = H/2. see (1l4, Remark
(rom t.be syst.em 3.1.5).
4.5
175
EXTENSIONS
4.5.3
Purefeedback systems
Strictfeedback systems have a 'triangular' form. \Ve now consider a broader class of para,m,et1ic ]Ju.refeedback systems which include some upper t.riangular entriesU i = 1, ... , '11
1

(4.285)
where
= ... =
11'1(0)
(4,286)
tp'I(O) = 0,
and lPo, tpI, • , , , 'Pn, as well HoS Po, p, are smooth functions in B:J!' a neighborhood of the origin ~. = o. In this sectioll we consider ouly the case of regulation. In genera.l, stability is achieved only ill a welldefined 7"fgion around the origin. The f,uning functions design for parrunetric purefeedback systems (4.285) is summarized in Table 4,2. The differences from the design in Table 4.1 are: • ai, Ti, Wi depend 011 :CHI rat.her than ,7:;. This is because 'Pi depends all xi+h so the sums E 8~;:I.1.·k+] in (4.290) and  L U~~~I!Pk ill (4.292) run
to i rather than to iI. • The update law (4.29.:J) contains the term (1 t.o UlC term pTf) multiplying u ill (4.285),
• Afaetor (1 
lJU
n

.r"
1
OB,,I)·Pa=n which is due :I'll
appeal's in the denominat.or of t.he control Jaw in
)
(4.293). In additioll, instead of tJo + pTO, the dellominat,or of the control Jaw has a factor
Po + pT ( 8  Lk=2 r (O';;~_I) T Zk )
'
One can verify t.hat, with these modificat.ions, the closPrlloop system is = = A:z
+ [TV + 13e~
(1 _ Da:,nl) 0:] TO,
(4.287)
.1·'1
where
A:(z,O) =
CI
1
1
C2
0
1
o
0
1+
0"23
U:.m
1 + UII_I,II 0
1
U2n
!J A coordinal.efl'ee chnrn(:t.crizalioll or rem G.g.
t,llCSC
UIII,n
systems is given
en ill
AppE'ndix G, Theo
176
TUNING FUNCTIONS DESIGN
Table 4.2: Tuning Functions Design for PureFeedback Systems
Zi
adx i+lI 0)
=
Xi 
=
Z'il 
Qil
80'i]
+ftrTj
f)()
Ti(Xi+b O) 1LJdxi+lt 8}
Tft Wi ()
CiZi 
'i_I +WjZi
=
!Pi 
i
t
8a il
~:Z;.t'+L
k=1
XI..0
iI 80'kl
+L
(4.290)
ArWj=k
k=2
=
+
(4.289)
f)(}
(4.291)
f)ai_1
L 'Pk k=l 8Xk
(4.292)
i = I, ... ,n (0 _ (
Yr 
.'
(a»
Yrl Yr, ... , Yr
Adaptive control law: u· = a(x,8) =
(4.293)
Parameter update law: ()~ =
r
[' II + (I  ;,:aanI) {3az ] = r [t'V + {je.T ( n 1n
80"11)] x:a: z (4.294)
and
O'ik
is defined for k
= i + 1, ... ,n as i = 1
0, {)O:jl
APWk, 80
80:80
i 1  r [W A'II
i = 2, . , , I 'fJ
80.,11) + (1   {3a:] X'll
'

1,
k:f:. 11
k=n. (4.295)
4.5
177
EXTENSIONS
In deriving (4.287), (4.288), (4.295) it is belpful to rewrite t.ilC control law (4.293) as
(4.296)
Equation (4.296), which is implicit ill U = o(x, e)t is more cOllvenient for analysis, whereas (4.293) is used for implementat.ion. The design procedure from Table 4.2 is JcasilJlc only if the denomilUltor in the control law (4.293) never becomes zero, namely, only if
aonII > 0 Po + p'r (0  t 1
1
(t1.297)
Xn
k=2
r
(OD:k::l)T =k) 88
Let us firsf'. analyze condition (4.297). From (4.290)
WE.'
> O. compute
(4.2!J8)
178
TUNINO FUNCTIONS DESIGN
Thus, condition (4.297) becomes (4.300) By repeating the computations as in (4,209), condition (~1.297) is brought to the f01'm EJ'PiT ( 0'" )PI' )] '"'" r (0. £l:kl L...., IT 1 +a:r:i+l 1.'=2 ao .'k, 'j
nl [
i
;=1
>0
.
(4.301)
vVith (4.301) and (4.298), we have the following proposit:ion. Proposition 4.24 Let B z em." and Bo C lR" be open sets such that (.1:,8) = (0.0) E B.r x Bo, and f01' all (.1:,6) E B.r. X Bo,
)'f) >, L r (J:l ... )T) >
~ ( (J~ _ L...., . 8xi+ 1 '=2
1 + lJ'PiT
f30 + f3
or

(
0
Then :F = Bz Table
X
r ( ao:/._~ 1 lJ(J
U
UO:k_)
k=2
()f)
0
..
k
Zk
O.
i=1, .. &,111
(4.303)
Bs is a subset of the sel in which the design. p1vcedu1'e oj
4.2 is feasible.
Let us now examine the clulllge of coordinates in Table 4,2. For all. i
=
1t •• , ,n we have
(4.304) 'Ve showed above that (4.305)
4.5
179
EXTENSIONS
By subst.ituting (4.305) into (4.304), and inductively applying the implicit function theorem, we cOIu'lude that the change of coordinates
(.1·,0). (:,0)
(4.306)
is onetoone, onto, smooth, and has a smoot.h inverse on:F. !Vroreover, since IPo(O) 0, CPl (0) = .,. = 'P,. (0) = 0, by examining (,1.304), one can show
=
inductively that. x=O
<==>
(4.307)
==0.
Because of (4.307) it is not hard to show that. a feasibility condit.ion, equivalent. 1.0 Proposition 4.24, is that there exists a set :F' = B~ x B~ cont.aining (;1:,8) = (O,B) such thaI. for all (:c,O) E :F', 1 + a"'i(x)T 01 O:Ci+l
> 0,
; = 1", . ,n  1
1
IJ3o(x)
+ ,8(.1.),1'01 > O.
(4.308) (4.309)
It is then possible to find a subset. of :F' which is anot.her est.imate of the feasibility region. In general, the feasibility region is not global. However, t.his is not due to the adaptive scheme. Even when t.he paranl('t.cl's 0 al'e known, the feedbac1\: linearizat.ion of the system (~1.285) ran only be guarmlteed for lJ E Bo clAP, an open set such that for all (.1;, B) E Br x Bo, 1+
1
a;i(~·)T lJl > 0,
l,8o(:r)
i
:rj+l
= 1, ... , n 
+ ,8(x) TO I > O.
1
(L1.310)
(4.311)
Let us now return to the closedloop adaptive system (4.29..1), (,L287),
(4.312)
The derivative of t.he Lyapunov function
1
" = :; 2
'1'
IT =+ lJ r1o2
along the solutions of (lL312) and (4.288) is •
JI
"I
V ~c· ,...;. ~
i==1
(4.313)
180
TUNING FUNCTIONS DESIGN
=
This proves that the equilibrium z 0,0= 0 is stable. Since we showed above that the coordinate change (4.306) is a diffeomorphism which preserves the origin, we condude t.hat the equilibrium :£ = 0, {j = 0 is also stabJe. ""Ve HOW give an est.imate c :F of the region of attract.ion of t.his equilibrium. Let. n(c) be the inval'iant set defined by l"(.1:,6) < c, and Jet. c· be the largest constant c such t.hat O(c) c:F. Then! an ostimate 0 of t.he rogion of at.t.nu.·tion is
n
c* = arg sup {c}.
( 4.315)
O(c)CF
Finally, hy applying [81, Theorem 4.8] (a local version of the LaSalleVos hizawa t.heorem (T heorem A. 8») to (4.314.), it foHows that z (I.) ~ 0 as t  00. In view of (4.307), this means that x(t)  o.
Theol"mn 4.25 Suppose Ihat sllstem, (4.285) sati.sfies Propo.r;itioTi. 4.24. Then the closedloop adn~)tive sysfem CD7}"f~i8tirr.lJ oj the plant (4,285), the controllu'w (4.293), (J:ltd the u.pdatc ia.1IJ (4.294) has a stablc cquiiib7'ill.m.1: = 0,6 = 0, and its 1'Cgion of attra.ction in.cludes the sct fA defined in (4.315). F1J.7·the'lmo7'e, for all (:£(0),0(0)) E 0, we hn.IJC lim :l'(t) = O.
100
4.6
(4.316)
Example: Aircraft Wing Rocl<
'Ving rock is a limit cycling oscillation in the roll angle cjJ and t.he roll rate ri> which can occur in highperfofmance aircraft. with slender forebodies when Hying in high angleoratt.ack, Couvent.ional methods of eliminating wing rock include a redesign of the airframe configuration and limiting of the angleoCattack. These methods may reduce maneuverability of the aircraft., An eR'octive method for suppressing wing rock without. degrading nUl.l1cuverability is using feedback control. Several onedegreeoffreedoIll models of wing fock have been proposed in Nguyen, \Vhipple, and Brandon [143], HSll and Lan [47], and Elzebda, Na~feh, and 1vlook [32]. They are all nonlinear and contain parameters OJ which depend on th~ angleofattack, dynamic pressure, wing reference area, wing span, roll moment of inertia, and flight. velocity, VIe now present an ada.ptive controller of IVlonaheIlli, Barlow, and Krst.ic [133, 134] which allows these parameters to be unknown and eliminates tho wing rock phenomenon by achieving global stabilization. The model we consider here, (4.317) is hased all wind tunnel test.s [143] at NASA Langley Research Center. In these tests, physical scaled wings were mounted on an apparatus wlIich allows free
4.6
181
EXAMPLE: AIRCRAFT WING ROCK
rotation about t:he roll a:\:is. These lllodel wings are aerodymnnicaUy similar f.o the wings of an F18 HARV aircraft. There are no control surfa(:es in the model (4.317). Wit.h allerons modeled as firstordel' actna1:or dYllamics, the statespace form of the wingrock model is
=
c/J
p
p =
81 + 83¢ + 9:Ip + 9.d4Jlp + 95 1plp + bOA fJ A + 11 ,
T6A =
(4.318)
where fJ A is the aileron deHection angle, tI is t.he ('ont;rol input, , is the ailPI'Oll time constant, and b is an unknown COl1stant parameter. DE'llOt:iug tp(rp, p) = [1, cp, P, i¢\p, Iplp]T and f} = [8 1 , 92 , 9~h 0,(. Os]'l'. we l'f'write (4.318) as
(p =
1J jJ = bfJA + tp(rb,p)'I'O • 1 J DA = ;11.  ;.fJA •
(4.31!))
This model is ill the parametric strictfeedback form with unknown virtual control coefficient b, so we apply the design fl'OIU Section 4.5.1. Our control object:ive is to asymptot.icnlly t.rack a given referellce tPr(t) with the roll angle ljJ. Vvc use the error variables Zl =2 ::3
= t/J  rPr = P  ~r  a'l (4): lPr) = fJA 1J~r  tl'2(t/>, p, 1>,. tPr, 0, g) ,
and derive the Fitabilizing functions 0:1
=
G2
= Uii2'
Cl=l
(4.321)
The design procedure from Section 4.5.11'esults in an adapt.ive controller consisting of the control Jaw 11
=
T
1 ~ [ :rOA  b:;'J aa") .
acr:!,
C3=:i
aa2 (.
+ at/> P + ap (3)
80:") ..
oalf'/ :.
bOA
(..
+ tp
T )
(J
)
'J
+ a¢~ tPr + a;P~ cPr + BtPr + aiJ~ 8 + tPr + ii2 il
(4.322)
and the update laws
( an.) b = "( (=2  on.) "A =3
0
=
rtp
8p =3
=2 
£)p
1J
=
"(sgn(b)
(~r + 0:2) =2'
(4.323) (4.324)
(4.325)
182
TUNINC FUNCTIONS DESICN
3
2
o 1
2 3
:~.8
0.6
0.4
0.2
o
0.2
0.4
0.6
O.B
r/J Figure 4.5: (a) Uncont.rolled wing rock (). (b) Suppression of \ving rock by adaptive llonlinear control (am).
For the resulting error system
u
the equilibrium z = 0,8 = 0, b = 0, = 0 is globally stable and, moreover, ~ O. This means that the wing rock phenomenon is eliminated alld the tracking objective q,(t)  tPr(1) . 0 is acbieved. This is illustl'o.t:ed in Figure 4.5. \Vithout feedback control the response to an initial condition cjJ(O) = O.4,p(O) = OA(O) = 0 Is the trajectory (a) which represents limit. cycling oscillations typical for wing rock. They are obtained for t.he model (4.317) with wind tUllllel data provided in [143J at angleofattack a: = 30°: 61 = 0, 8'1. = 26.67. 83 = 0.76485, 8., = 2.9225, Os = O. The amplitude of the willg rock oscillations is about 35° and the frequency is about 1Hz. The traject:ory (b) in Figure 4.5 sho\vs the wing rock suppressing effect of the adaptive controller (4.322){4.325) acting through the aileron with b = 1.5
.;(t)
NOTES AND REFERENCES
183
and r = 1/15. Parameter estimates are initialized as 8(0) = 1.350, b(O) = = l/b(O). The controller coefficients are Cl = C2 = Ca = 5 and the adaptation gains are r = 0.021, "I = 0.02. For softer regulation t.o the origin, we have employed an exponentially decaying reference trajectory tPr(t) governed by the equation (8 + 10)(.'12 + 45 + 24.25)tPr(s) = o. 1.35b, and ,g(0)
Notes and References The adaptive bRckstepping design of Kanellakopolllos, Kokotovic, and lvlorse [69] required mUltiple estimates of the salliE:' paramet.er. This ovcrparametrization is impractical for highordcr 1l111Itiparamet.er systems. Jiang and Praly [59] were quick to not.ice t.hat. the extended matching idea can be employed to reduce the number of estimates by half. Still a significa.nt overparamet,rizut.ion remained uutil I(rst.ic, Kanellakopoulos, and Kokotovic [D~l] introduced the tuning functions Inethod which complet,ely eliminated overparametrization. The removal of overparametrization strengthened thc stability and convergence properties of the l'esulting adaptive syst.em. By studying global invariant manifolds it was further shown in Krst.ic [93] that, except for a measure zero set. of initial conditions iu the state space of the complet.e adaptive system, t.he parameter estimates converge t.o values with which a nonadal.Jtive controller would guarantee (at least local) asymptot:ic stability. Trajectory initialization was employed in Krst.ic, KanelJakopolllos, and Kokotovic (95] and in Kanel1akopoulos, Krstic, and Kokotovic [7D]. Krstic and Kokotovic [100J developed the adaptive control Lyapunov function framework which is behind the tuning functions method. A Lyapul1ovbased design by Praly [156] offered a solution alternative to [94] for st.rict feedback systems with nonlinearities of polynomial growth. Jiang and Po met [58] extended the LUlling functions design to a class of nonholollomic systems, while Jain and Khorrami [55] employed it in a decentralized design. Polycarpou and Ioannou [151) and Yao and Tomiz1l1m [199] presented robust eA'tensions of the tuning funct.ions design. Freeman and Kokotovic [40) used the tuning fUllctions technique in a design of partialstate feedback controllers for nonlinear systems linear in the unmeasured stat.es. The dcsigns in this chapter involve expressions with partial derivatives which are difficult to derive for systems of high order. Ghanadan [42J developed symbolic software for mechanizing the derivations of tuning functions contl'ollers and updat.e laws.
Chapter 5 Modular Design with Passive Identifiers The tuning fnnct.ions approach developed in the preceding chnpt,er has two dist.inguishing features: First, a single Lynpullov function encompasses the complet,e sl:a.tc (=,6) of t.he dosedloop syst:em. and second, the dynamic order of the l'el:!ult.ing cont.roller is as low as t.he number of IInknowu parameters. However, the tuning f~nctions controller is sOlllewhat complicated becnuse it cancels t,lle effects of () in the =systetn. Another drawback of this approach is that. the choice of n p~U'ameter updal.e law is limited to a Lyapullovl;ype algorit,hm. In the traditional adaptive linear control t.he rcstrietion t,o the Lyapunov update law is removed by estimationbased designs which achieve a m.orl1J.la1·ity of the controlleridentifier pair: Any st.abilizing controller can be combined with any identifier. The controller module is rapable of stabilizing the plant when all the paral11et,ers are known. This is its ce7·taint1J equivalence property. The identifier module, in turn, guaraut,ecs rertain boundedness properties independently of the controller module. The modularity of the est.imationbased designs mali:es them much more versatile than the Lyapunovbased designs, In this chapter we begin the development. of a modular approach to adaptive nOlliinecu control. \Ve first show t.hat a major obst.acle to earlier att.empts to apply estimationbased designs to nonlinear systems was the weakness of their certainty equivalence ('ant rollers. Such controllers cannot achieve any controlleridentifier sep8.l·ation wit.hont severely L'Cstricting the system nonlinearities. To overcome the weakness of certainty equivalence contro11ers, we develop a new controller with st.rong parametric robust,ness properties. In the presence of unknown parameters this nonlinear C"ontroller achieves boundedness without. adaptat.ion. It guarantees boulldedness not only in the presence of constant parameter errors, but also in the presence of timevarying parameter estimates. The strong controller is suitable for modular designs of adapt.ive
186
MODULAR DESIGN WITH PASSIVE IDENTIFIERS
control schemes for nonlinear systems. It can be combined with any stand81'd identifier. In this chapter we develop two passive identi/ier,r;. They employ 'observers' wit.h passive error systems and unnol"malized gradienttype update laws. The first identifier is based on the openloop plant model, while t,he second is based on the closedloop model. Vve derive bounds for transient performance of the new adaptive schemes and compare them with the tuning functions design. In addition to strong controllers, we also design controllers with a small gain property, analogous to those in adaptive ]inear control. These weaker controllers reduce some of the nonlinear complexity of the strong controllers, but their performance is not as high. This chapter also serves as an introduction to the next chapter where we develop a fully modulru' design with standard identifiers, including those with leastsqu81'es update laws. Section 5.2 introduces a general framework for modular adaptive design. The problem of adapt.ive stabilization is approached as a problem of inputtostate stabilization (ISS) with respect to the parameter estimation error considered as a disturbance input. This section introduces ISScontml Lyapunov functions (ISSelf) which are the main tool in constructing the control law in the modular design.
5.1
Weal{ness of Certainty Equivalence
A major obstacle to the certainty equivalence design for nonlinear systems is a fundamental difference between the inst.ability phenomena in linear and nonlinear systems. The states of an unstable linear system remain bounded over any finite time interval, so that there is enough time for the ident.ifier to "catch up" and generate stabilizing values of parameter estimates. The situation is dramatically different in systems with nonlinearities whose growth is faster tImn linear (x'.J., Xl X:!, e:Z:, etc.). Even it small parameter estimation error may drive the state to infinity in finite time. For tins reason, only a few nonlinear estimationbased results go beyond the linear growth constraints [152, 153, 154, 157]. III [157], nonlinear certainty equivalence designs are characterized by relationships between nonlinear growt.h constraints and controller stabilizing properties. In the absence of matching conditions, all these designs involve growth restrictions. Let us explain why the cert.ainty equivalence approach fails to achieve global stability for systems whose nonlinearities are not linearly bounded. Consider the scalar system (5.1)
5.1
187
WEAKNESS OF CERTAINTY EQUIVALENCE
where cp( x) is a vectOl' of smooth nonlinear fUllctions and () is a vector of unknown parameters. For this system, an obvious certainty equivalence controller is (5.2) wllere jj is the estimate of O. With this controller, the result.ing certainty equivalence feedback system is
(5.3) In Section 3.1 we have shown that the update law iJ =
x = x + tp(x),
e= iJ
~  tp(x)T 0,
X
X(O) = a {CO) = :r(O)
(5.4) (5.5)
= 1+'xT x (x_XTO~). A
(5.6)
(This identifier was used in the early adaptive designs for nonlinear systems [141, 166J.) To understand .t.his identifier, observe that; if t.he estimate 0 were correct, f) = 8, then x :::: X'rf) +~. This can be verified by direct substitution. When 8 ::/: 6, then x XTe ~ represents the state estimation error which is used to drive the parameter update law (5.6). The update is ill t.he direction of the regressor vector X, but its rate is 'normalized' by the factor (1 + XTX)J. This identifier is convenient for our iUustration, because it.s stability properties XT 8 so that are well known [51, 165]. It is easy to check that x  X'l'O  ~ (5.6) is rewritten as
=
'v,,/f .\'A.
_ (J
l+XT X . Using the inequality d
(5.7)
9T XXT O:5 OTX?x.O, we see that
(lT)0 f)
dt 2
(5.8) which implies that
(5.D)
188
:MODULAR DESIGN WITH PASSIVE IDENTIFIERS
Thus the convergence of this common identifier can be at best exponential. Let. us now connect this ident.ifier with the certaint.y equivalence cont.roller (5.2). To be specific, we let 0 and cp be scrum's:
cp(:r) =
2 .1. ,
O(t) = O(O)e' .
(5.10)
Then the certaint.y equivalence feedbaek syst.em (5.3) is described by j;
=
:1.'
+ .z::.!8(0)e t •
(5.11)
The analytical solution to this eqnat.ion was obtained in (2.243)", (2.2~16):
x(t) = :r(O)
C)
_ _ . .1·(0)8(0)e t + (2  :c(0)8(0»)e t
(5.12)
It is not 1..ul.l'd to see that whenever
.1'(0)6(0) > 2,
(5.13)
the solution (5.12) escapes t.o infinity in finite time, that. is 1
:I:(t)
j.
00
as
:c(O)O(O)
(5.14)
t + 2111X(o....:....)O=""(O)"_2
This cat.astrophic instability is due to cp(x) = ~.:2. If, instead. we had i:t nonliuef! rit.y 'P(.1:) with a linearly bounded growth, Icp(:I:) I $ 1..:lxl, the above instability would not have Dccuned. A farreaching conclusion to be drawn from the above example is that a modular design, with a certainty equivalence controller and a standard identifier as it.s modules, is not applicable 1;0 systenls with nonllnearities whose growth is higher than linear. The mechanism of instability is clear: The ident.ifier, whose speed of convergence is at. best. exponent.ial, is not rast enough to cope with potentially e..xplosive instabilities of nonlinear syst.ems. This simple example shows that; t.o achieve stability we need either a much fast.er identifier, as in our tuning functions design, or a stronger controller with a biggf'r stability margin for dist.urbances such os 0 in (5.3). \Ve consider the parameter estimation enol' and its derivative as two independent disturbance inputs and design stronger cOllt.rol1ers t.hat guarantee boulldednesA of all the states of the closedloop syst.em whenever 1. t:he parameter estimation error 2. its derivative
8 = 0 is
8 = 0  0 is bounded,
and
either bounded or squareintegrable.
These new controllers create a possibility for a complete identifier·cont.rol1er modularity. Such (.'omplete cont.rolleridentiliel' separation has been an unachieved goal of adnptive control, even for linear systems.
5.2
5.2
189
ISSCONTROL LYAPUNOV FUNCTIONS
ISSControl Lyapunov Functions
In Section 4.1 we introduced a framc\Vork for Lyapunov design of adaptive stabilizers where thc central role was played by adaptive ("ont.rol L)'apunov functions (aclf's). In this section we develop a framework for modular design of adaptive l1011lillC:'al' controllers. Let us consider t.be system
:i: = /(:I:)
+ F(:r}(J + g(:I~)11
(5.15)
where /(a:), F(3.·), and gel:} are smooth, and 1(0) = 0, F(O) = O. \Ve say that. syRtelll (5.15) is globally adaptively stabilizable l if tlu1J'e exist a function a(:l:, 8) cont.jnuons on (Hl" \ {o}) x IV with Cl (0,0) == 0, continuous functions T(a~, 0, 71} amI fl(:r:, 0, ',), and a positive dpfinitc symmetrie p x p matrbc r, such that the dynamic controller 'lL
=
6 =
O:(:l~, 0)
(5.16)
rT(X, 8, 17) fl(x. 8, 1])
(5.17)
= " guarantees tbat the solution (:c(i), OCt}, .,,(t» is global1y boundcd, and .t(t}
(5.18)
as t ;. 00, for all (J E W. We I'cler to (5.17)(5.18) as an identifier, wUh (5.17) being it.s update "Vc st. art by rewriting (5.15) with (S.lG) as .1; = j(.r)
+ F(:r)O + g(.1:)a:(X, {}) + F(;,:)8.
+
0
IRW.
(5.19)
Suppose we know how to find Ii ('ontl'olla\V a:(x, 8) UUIt fitabilizes this syst.em with 0 = O. As we saw in Section 5.1, ill tbe presence of t.)le dist'.tubance inpnt 0, this cOlltrollaw, in gencral, does not preserve st.ability evcn if 8 is bounded and exponentially decaying. To preserve stabilit.y, we need a strollgel' conj~rol1er. Sincc the standard parameter estimators guarantee thn t. jj is bou llded, \VC are int.erest.ed in designing controllers which call guarantee inputtostRt:C stabiHt:y of (5.19) with respect to 0 as input, in the sense of Dpfinition C.l. Howe\'er, ~he timevarying character of the paramet.er estimate OCt) forces us t.o consider
8(t) as another disturbance input, even though it. is not explicitl)' present. ill (5.19). Our goal is to find a controlla\v 0'(x,8) continuous on (R" \ {O}) x mp with 0:(0,8) == 0, such that the following inp1tlrlost.nte stabilitIJ (ISS) propert.y is satisfied:
Ix(t)\ ::; P(\J'(O)I, t) + "I (sup
O::;T::;I
[~(T) J) . O(r)
(5.20)
J'fhjs definition is more general than the one from Section 4.1.2 to suit the rnodulnr approach.
190
MODULAR. DESIGN WITH PASSIVE IDENTIFIERS
where {3 is a class JC£ fUllction and 7 is a class JC function (see Appendix C for detaHs). Lin, Sontag, and Wang [115, Theorem 3] recently proved that a necessary and sufficient condition for (5.20) is the exist.ence of an ISSLlIapunotJ function. (The proof of sufficiency is given in Theorem C.2.) 'Ve say that a smooth funct.ion l' : lRJ1 X lR.I) + 114, positive definite and radially unbounded in :c for each () is an ISSLyapullov function for (5.15) if there exist.s a class /Coo ~unction p such t.hat the following implication holds for all :z: =F 0 and all
0,8, ii E lRP :
(5.21)
JJ.
aV' [
8 f(:c) x
avo
 av' 8;. < O.
+ F(:c)8 + g(a:){l'(x, 8) + a F(x)O + A
A
]
x
W
A
If there exists such a triple (0, V',p) we say that system (5.15) is inputtostate stabilizable with respect t.o
(0, 0) .
\Ve have t.lms chosen 1:0 st.udy the problem of modular adaptive stabilization as a problem of input.tostate sta.bilization. Similar to Section_ 4.1 where we cast Lyapunov adaptive stabilization in the framework of control Lyapunov functions, in this section we develop a clf framework for modular adapt.ive stabilization.
Definition 5.1 A smooth function v· : [ttl x JRP ~ 1R.+. positive definite and radially 'imbounrled in x jor each 0, is called an ISScontrol Lyapunov function (ISSelf) jor (5.15) if there exists a class /Coo .function p such tlla t the following im.plication holds for all x # 0 and all 0,6, fJ E nlP :
1·1: 1
~p( [ ~] ) ./J
inf
uEIR
(5.22)
all' [j(x) + F(x)8] + 8V' av' ~} < o. + g(x)'U 8 F(x)8 + .8 { a. ~ x ao
We now show that. the existence of an ISSc1f is a necessary and sufficient condition [or inputtost.ate st.abilizahilit.y. The proof of sufficiency is construct.ivewe design a control law starting from a given ISSelf.
Lemma 5.2 (InputtoState Stabilization) System, (5.15) is inputtostate stabilizable with 7'Cspect to if and only if there e.vists an ISSelf.
(8,9)
5.2
191
ISSCONTROL LYAPUNOV FUNCTIONS
Proof. The 'only if' part is obvious because (5.21) implies that there exists a particular cOlltrollaw 'It = a'(x,9) which satisfies (5.22). \Vc \yillllOW proye the 'if' part by showing that the following control law achieves input.tostate stabilization:
~~O(x,8) # 0 l:J\'
(5.23)
•
o;:O(:r:,8) = 0 t where
.
w(a:,8) =
elf [f(x) + F(3:)8]. + [81' " 8\']1'_1 a 0 F,  . p (lxl) . .1' an
(5.24)
:1'
\¥e first sbow that (5.23) if; continuous on (lRfl \ {O}) x 1Rl'. In [1(1), Sontag provC'd that the fll11ct~ion (5.23) is slllooth provided that. its argUlllents wand ~~ 9 are such thitt
all"
g=O => w
(5.25)
8~c
\Ve show that l' being an ISSelf implies tlult (5.25) is saf.isfied. By Definition 5.1, if x # 0 and ~~g = 0, then
I·TI ~ p ([ :
]I) (5.26)
.ij.
aTi [
0 f(x}
x
av F(a:)9 + . av I):. < O. + F(~r)9~] + 8 x 00
Let us consider the particular input
[ilY F ~ 'I' [ IJ:. = I[m:. F: ill;.] 'l'1 B:r ' 8 ]
1):1:
t
DO
J
(5.27)
P (Ixl).
fJ(J
This input satisfies the upper part; of implication (5.26):
(5.28) Therefore, substit.uting (5.27) into the lower part of (5.26)! we cOllclude t.hat, if x;': 0 and ~~g 0, then
=
[81' 8""] ')' 08
ali . 8x [I(x) + F(x)8] + ax F,
1
p (Ix!)
< 0,
(5.20)
192
IvIoDULAR DESJGN WITH PASSIVE IOEN'l'IFTERS
that is, (5.25) is satisfied for.1" =f:. O. Therefore, (5.23) is a smooth function of w and ~~ 9 \V hencvcr .1: "# O. Si nce w(x, 6) is continuous and ~~ g( 3.', 0) is smoot, h. the control law (5.23) is continuous for x "# O. \Ve note that the control law a(J,,8) given by (5.23) is also continuous at oX = 0 if and only if the ISSelf \/ satisfies the following .r;rn,a,ll contml prope1'ly [171]: For each iJ E lR,P and for any
E
> 0 there is a 6 > 0 s\lch that, if ;1: " 0 satisfies p
then there is SOlUe
avo [
11
with
0 f(x) .1'
lu.I
~ E:
(I [: ] ) : ; 1.'1 ::;
6,
such that
. ] DF F(x)O + ~ at' :. + F(.J·)8 + g(x)1J. + 8 0 < O. :1' ao
(5.30)
Now we show that. t.he controlla,w (5,23) achieves inputtcrstat.e st.~tbiliza t.ion, Along t.he solutions of (5.19) and (5.23), the derivative of l' is
fJlI]T all F, ~ [a :z: aB
1.
p (Ixi)
av F(.t)B .  + av:./J + a ~. ao a
<
(~~ [.r(.• )+ F(x)O] + [~~>, ~~r P'(I.vl)r + (~~'g)' (5.31) In view of (5.29) this proves that 1i' < 0, 'rI.v " 0, whenever
1·r.1 ?: P ( [
~ ] I) ,
that is, V' is an ISSLyapunov func:tioD, which by Theol'em C.2 establishes that (5.15) is input.tostate stable with respect to
(0,0).
0
This lemma shows that, if an ISSelf is available, it is easy to design ~ cont.rol law which guarantees bOllndedness of the state x whenever 0 and ~ M'C bounded. Therefore, we neEd identifiers which can guarantee that, 0 nnd iJ are bounded. As.we shall see in Cha,pter 6, the swapping identifiers guarantee t.hat both 8 and iJ are boundcd. The passive identifiers that we present in this chapter can guarantee only.the bounded ness of O. Fortunate1y, even though they cannot guarantee that 0 is bounded, they can guarantee that it i~ squareint.egrable. The boundedness of 6 and the squareintegrability of jj will be enough to establish the boundedness of x because we will design controlJers
5.2
193
ISSCONTROL LVAPUNOV FUNCTIONS
which guarantee tbe following linearlike relationship;
i\ > O. Wit1~ inequality (5.32) we will show t.hat
.1:
(5.32)
is bounded whenever ij is bounded
and {} is squareintegrable. In contrast to the Lya.punov design, where it is not dear if i.he global asymptotic stabilizabilit.y for each (J is a necessary condition for tlu:> existence of an aclf (sce Remark 4.4), t.he global asymptotic stabilizabiJity for eat'll (; is a necessary condit.ion for the existence.of an ISSdf. This becomes ohvious by set.ting B(t) Ne~i;
== 0,
whi(~h implies 0(1) = B(t}
== 0, into
(5.20).
we give sufficient conditions under which a:(t)
+
0 as t +
00:
Duf"
to the ISS pI'operty (5.20), one sufficicnt condition is thnt both 6 and {} tend to zero. However, ill general, identifiers cannot guarantee that 8 goes to zero, so the lle>...i; lelIlma. gives a less demanding condit.jon. Both the passive (see Remark 5.15) al1d the s,vappil1g (see RClIuuk 6.8) identifiers will be able to guarantee that these conditions arc satisfied.
Lemma 5.3 (Regulation) Suppose f.he contro/lull1 U = Q·(.v,6) gUll1fl.nlees that system (5.1,5) is ISS with 1'eslJcct to (OtO). If a:(t) is bounded, and F(x(t»O(t) and 6(t) converge Lo zef"O a.s I ~
00,
then lim,_oo x(t} = O.
Proof. Since the system :i: = J(x} + F(x)O + g(x)a:(x, 0) is ISS with respect to
(5.33)
(0,9), i;hen the same system with 6 = 0= 0, name]y
the system
.1! = I(x) + F(a')O + g(:t:)a:{x, 8)
(5.34)
is globally asymptotically stable. Therefore by [174, Theorem 2] there exist f3 E K.£, "y E IC, al1d a continuous fUllction 0' : R+ + R+, O'(s) > 0 for" > 0 such that for each continuous and bounded input wet)
6. [ =
F(:r:(i))8(t) ~
OCt)
1, for
each x(to} E JR." t and for aU 1 ~ to 2: 0, t.he following implication holds;
(5.35)
lL
Ix{t}1
:s; ,B{lx(lo)l, t  to) + 'Y (suP~J$"':91'W(T)1)
.
194
l\1:0DULAR DESIGN WITH PASSIVE IOEN'fIFIERS
=
Let itl be sucb that /.l:(t)/ :5 1"[ for all t ~ O. Let e mil1{ u(r) I", :5 ill} > 0, and let. T 2: 0 be such that Iw{t)f ::; e for aU t ~ T. Then from (5.35) we obtain
Ix(t)l ::; ,8(I:r{to)l, t  to) + '1 ( sup
t~I$'T9
IW(T)I)
(5.36)
for all 1 2: to ~ T. TllUs, from time T onward, the system satisfies the ISS inequality (5.36). To complete the proof, we have to show that. tbe ISS property implies thl:1.t :ret) + 0 as t + 00. Our computations folloW' those in the proof of [173, Proposition 7.2]. First, we 11ot:C that there e.'1i.:ists 8. lllonoLonicaHy decreasing to zero I'unction "1 continuous all [T t (0) such thil.t
Iw(t)1 :5 T/ {t  t·o} ,
'tit ~ to
~
T.
(5.37)
Then we have
Ix{t}1
~ P (Ix C~ T)I, t ~ T) +y c;~s, IW{T}I)
~ P (p (Ix (T)I, t ~ T) + y (TS~~~ IIII{T)I) , t ~ T) +y
(¥~:9 '1 (T  T))
.
Not.iug that for any class 1C function 0, 6(a + b) nonnegative a ll.nd b, we llroceed from (5.38) with
which converges to zero as t
+ 00.
{5.38}
:5 b(2a) + 6(2b)
for any
0
As we shall see later in this chapter, as well as in Chapter 6, ~otb the passive and the swapping identifiers guarantee that F(x(t))ii{t) and O(t) converge to zero whenever x(t) and u(t) are bounded.
5.2
195
ISSCONTROL LVAPUNOV FUNCTIONS
The two lemmas in this section outline a framework for modular adaptive design. Lemma 5.2 shows how to design a control law once an ISSelf is known. While Lemma 5.2 gives sufficient conditions that the identifier has to saHsfy to gum'untee boundedness of the plant state x, Lemma 5.3 gives sufficient. identifier conditions for regulation of .1'. Identifiers satisfying t:hese conditions will be designed in this chapter and Chapt.er G. Therefore the main task is t.o find an ISSelf for a given system. For t.he simple scalar system
:?
i'
±=
I(x)
+ F(a;)B + g(.l:)l1 ,
where g(x) ¥= 0, a valid ISSelf is Vex) control law .
U
1
= x'J..
= 0(3:,0) = () ( J(x) g :c
(5AO)
.1: E 1R I
This is easy to see be('ause the
(5,41)
F(:I:) :r IF(x)I.1;)
yields
all'" [J(.1:) + F(.r.)O + g(x)a(.'l!lB). ] + F(a:)8 av  + Otl A ·0 a.... 0:1: [)fJ = _2x2  2IF(x)lx 2 + 2:I:F(:v)O $ _2:1.,2  2IxF(x)I(lxl 191) < 2:c 2 whenever IJ:I ~ 16L which implies that system (5.40)(5.41) is ISS with respect to
(5.42)
(0 0). 1
(To make the control law 0;(.1', ti) smoot.h, we replace IF(x)1 in (5A1) by JF(x)F(x)T + 1.) Since we know how t.o design ISSclf's for scalar systems, our approach is recursive: We a.':isume that. an ISSelf is knowll for an initial system, and construct a new ISSelf for the initial system augmented by an integrator using backstepping.
Lemma 5.4 (ISSBackstepping) If the system
.r. = J{:I:) + F(:c)8 + g(x)u is inputtostate stabilizable with respect to b01mded, then the augmenteci system .i'
=
I(x)
~
=
u
(0,0)
(5.43)
'using
+ F(.1:)O + g(.1:)~
is also inpu.ttastate stabilizable with 7'fSpect to ( 6,
B) .
o' E
C l J and
9 is
(5.44)
196
l\JI00ULAR DESIGN WITH PASSIVE IDENTIFIERS
Proof. Since (5.43) is in pli t t.ostate stabilizable with respect to
(0, 8), there
exist·s a. triple (0', V', p) and a class K. function 1" stich that
Ixl
~ p ([
r)
(5,45)
.lJ. •]
avo [ • aV'  aV' :. a f(:r) + F('l:)8 + g(.r)a.(x, 0) + a F(.l·)O + . (J :5 p(l:r/) . x x 00 In fact, without loss of generality we assume that 11· is class /Coo' It. was shown ill [173] t.hat if Jl is only in class /C, t.he given Lyapunov function V' can be modified so t.hat the new I' be in class /Coc . For p. E /Coo, it wa..1:j shown in [177] t.hat (5.45) is equivalent to the following 'dissipa.tion' t.ype of characterization:
aV' [ /(x) + F(:r.)O + g(:I:)a(.T 0) a x A
]
t
of  aF:.() ::; jl·(I:rl}+7f +'7lF(x)O+.
ao
uX
(I [0 ] ) :.
8
.
(5.4G) where 7r is a class IC Cunct.ion. Since the proof of the affine case considered here is simple, we give it for eompleteness. It is clear that (5.46) implies (5.45). To see the converse, one only needs to cOllsider the case
1."1 ::; p
(I [~ ] ).
Since
ois bounded, with Young's inequality one obtains a1' [ .  ] DV'  aV' :. a /(:r) + F(.l·)O + g(:t)a(x, 0) + a F(x)O + . (J + JL(lxl) x x 00
av [
::::;; a.1.: f(x)
+ F(x)1J. + g(x)a'(x,lJ). ] + 11·(/.1..'1)
+~ [~~ F(.r), ~~r' + [: ::; Mlxl) + [~ll' fi
:;
0
r
P ([:: )
+ [:
~ rr ([ : ] ) ,
r (5.47)
where p. is 8 dass /Coo fUllction. This completes the proof of (5,46). We will now use (5.46) to show t.lmt
.
Vi (x, ~, 0)
• = l/(x, 0)
1
+ 2(~ 
 '1 a(x, 0))
(5.48)
5.2
197
ISSCONTROL LVAPUNOV FUNCTIONS
is an ISSclf for (5.·U). \¥e do tbis by showing that, the controllA\\, 11
= a:1 (:r, e, 0)
=
811
  g  (~ 0')
a.l'

Do: ( +a.t f + F8 + .O~ 4
oa]T 2(e 
80: .. [F(;}:), a'I: 88
)
(SAg)
0:)
achieves inputtostat,e stnhilizut.ion of (5.44). Towards this cnd, consider
l~
=
~V [/(:1.') + F(:l,)jj + g(:r)o·(x. 8)] + av F(:r)9 + D\~ 0+ a\r !}(e 
ox
11.1;
+(€ 
~
a) (u  :: (f + FH g€) 
 JI(lxil + w
of}
a:l'
Q.)
~~8)
(I [~ ]I)
+«(0) (u+ ~:g :: (J+FIi+g~))
ao: 
00' ~
(e  0') a.~ Fe  (e  a) ao B
~
p·(I,eil + 7r (
[ : ]
I) ((  aV
_ [aa: F, D~]T 2 (e _ O,}2 _ (~_ Q.) faa: P, a~]']' [ ~ ] a.J;
~
a.l'
Df)
JL(lxi) + 1r
(I [: ]I) (~ 
of +
~
ao
[:]
l
0
•
=
(5.50)
Denoting 'iil{") 11'(1') + ~1·2 and picking a class /Coo function /II{l') min{JI.{1·), r!.!}, be(~ause of t.he boundedllcss of 0, we gct:
:5
where Ji'1 is a cla..IOiS }Coo fUllction. Thus, \Ii is an ISSLyapullov function. Bl' applying Theorem C.2 to (5.51) witll P = pi] 0 211"11 we prove that s)lstem
(5A4) with control law (5049) is ISS with respect, to
e,
(8,0).
0
The control law al(.r. 0) in (5.49) is ouly one out of many possible control laws. Once we have shown that Vi given by (5.48) is an ISSelf for (5.44) (with
198
lI!fODULAR DESIGN WITH PASSIVE IDENTIFIERS
=
P J.i2'J 0 27rd, we can use, (or example, the CO control lawa'l given by the fonnula (5.23). While in Lemmo 5.4 the initial system is augmented only by an int.egrat.or, minor modificat.ioll is sufiicient to obtain an analogous result for the more general system .f = fr:c) + F(x)f) + g(.t)( (5.52) ~ = 11 + FI(x, ~)(J.
1:1
Corollary 5.5 The Junction Vi (x,~, 6) defined i1l, {5.48} 7.s a.n ISS·clj Jor S1)Stern (5.52) 1l1i.lh I.he control law
QI (x, ~, 6)
= 0:1 (X,~, 6)(5.49) 
Fl (X, {}O IF1(:l", ~)12(~  a(x, 6)) .
(5.53)
An nfold application of Corollary 4. 9 will provide us with \~ and ar! for the system :& f(x} + F(x)fJ + D(X)~l
=
{I
=
{2 + PI (:1:, ~l)B
(5.54) ~nl ~n
= =
€n + F,,l(X, ~lt··· ,~,,d8 U
+ Fn(x, ~lt ••• '~fI)6.
We will now develop a detailed design procedure for such systems.
5.3
ISSController Design
Our goal is to develop a modular adaptive design for nonlinear systems in the pal'amet7;'c strictfeedback form
XI
=
X2
+ CPI (xd T 8
2;2
=
.'1:3
+ CP2(Xl! X2}T() (5.55)
:&111
Xu
:Z:1l
=
+ ¥'n1(Xl1"
• 1.7:n _I)T()
j3(:c)'U. +
where (J E lR.P is a vector of unknown constant parameters, p and F = [IPI," ·,tpn] are smooth nOJ1Jinear functions in lRn t and j3(x) i 0, \:Ix ERn. The coutrol objective is to £orce the output 11 Xl o[ the system (5.55) to asymptotically track the reference output Yr whose first n derivatives arc known, bounded wld piecewise continuolls. As we aUllounced in Section 5.2, our modular adaptive design for (5.55) will place the burden of achieving boundedness on the controller module. We
=
5.3
199
ISSCONTROLLER DESIGN
require that the control1er guru'antees that the s~at.e x .be bounded whenever the parameter error 6 8  fJ and its derivative 8 == 8 are bounded. The next lemma., wllich builds upon Lemma 2.26, is the main t,ool for the controller module design.
=
Lemma 5.6 (Nonlinear Damping) Assume that f07' the system :1:
(5.5a)
= 0'(x, t) gUB7'Untees
a feedback control u
av [lex, t) + 0(:1:, t)tJ:(x, t)J + 8 of ~ U(x, t),
~
t
~
,
e mn, u E 1R
'Ix e JR", 'V1
~ 0
(5.57)
where the junctions 1/, U are 110sitive defin.ite and l'adially unbounded in:r an.d V is decresccnt, I, 0, p, a: all'~ contin1l.o'luJly d·i.Jjerentiable in x and piecewise continuous fmd bounded in t, and cI i.9 piece'UJise continllo'WJ. Then the feedback control 'liJV {S.58} '\>0 U = o(x, t}  '\Ip(l:, t}l ax o(x, t) ,
guarantees that: 1. // eithe'"
of the
(a) dE £00
condition.,
01'
(b) dE £'}, and U(x, 1.)
~
cl/(x, t), c> 0
is satisfied, then x E £00' ~. If d E ~
n £00
and U{x, t) ~
clxr!,
then limt_oc x(t)
= O.
Proof. 1. Due to (5.57), the derivative of" along (5.56), (5.58) is
if =
~~ [I + ga: + 9 (,\p'fp:~g + pTrJ)] + :
i 0;':
< U _ A p of 9 
~
U + 2:..ldl 2 • ..:1,\
2
d1 + 2.l d l2 2. 2'\ 4,\ (5.59)
(a.) Since U is positive definite and radially unbounded, there exists a class A.oo function l' such that U(x, t) ~ 'Y(lxl), and tberefore {5.60}
200
MODULAR DESIGN WITH PASSIVE IDENTIFIERS
It follows that. if I:vl ~ II C!~\ IdI2 ), thon V ::; ~/(lxl). By Theorem C.2, system (5.56) with ('antral (5.58) is ISS with respect. to d. Hence, if d E £0::;, t.hen .1: E £0::;' (b) In the case when U(l.', 1) ~ cV(.t', t), from (5.59) we get .
1
~
V" ::; (:1' + 4. .\ Idl .
(5.61)
\Vit.h d E £'!.t by Lemma B.5, .1" E Coo. 2. Integrating (5.59) over la, t], we obtain
c 10' I·r(r)fdr < ::;
\ (' Id(r)1 2 dr + 11(0) Jot U(.t'(r), r)dr ~ 4A.k 4~\ IIdll~ + v"(0) ,
V"(t) (5.62)
which implies that. .J' E £2' By part 1 of this lClluna, x E £00' and therefore 'U E £rxJ' Hence:i: E L. oo • By Barbalat's lennml, (Corollary A.7), l;(t) ~ 0 as t
~ 00.
0
Our primary inl,erest. is in prut 1 of Lemma 5.6, which states that. :1' j~ bounded if d is either bounded or squareintegrable. In our design, 8 and ij play t,he role a! d. The ident.iIiers that we shall use will gnarantee that 8 is bounded, and [) is eit.her bounded or squareintegrable. Part 2 of Lemma 5.6 is also useful. l~. will help us to achieve tracking because our identifiers will guarantee t,hat. iJ is squareintegrable. Before we develop a general design of a cont.rollaw for the parametric strictfeedback systems (5.55) we illustrate the use of Lemma 5.6 on a secolldordpr example. Exrunple 5.7 Let us consider the system ·1:1 =
·1:2
x:! =
IJ.
+ 'P(.1:d'l'8
(5.63)
Viewing .1:2 as n cont.ro] input, we first, design a cont.ro] law at (Xl, 0) t.o guarantee that the state :l'1 in .i:, = .1:2 + 1p(:rd T Ois bounded whenever 0 is bounded. Following Lemma 5.6, we design (5.64)
Then we define the elTor vcuiable =2 = =1 = Xl. The first equation is now
J.·20'1(Xl,
8). and for uniformity denote (5.65)
If Z2 were zero, the Lyapunov fUllction l'I = ~zi would have the derivative
(5.66)
5.3
201
ISSCONTROLLER DESIGN
which would mean that ZI is bounded whenever this is no longer clear because then
8 is bounded.
\Vith
::2
:F
0
(5.67) The second equation in (5.63) yields •
•• 0:1
=2 = X2 
80') (
=
l/. 
lJx]
The derivative of the Lyapullov function
Tn)
+ cp

t12 = Vi +
~=i
!l':!
ae n:' .
80'1
(5.(8)
= ~I=I~ is
(5.69) Qur design is now led by Lemma 5.6. To make the bracketed term equal to ')
C2Z2 
U
=
wbere
#b2
'~CP\"" Z2
'1
g21~ T," =21 we design t.he control law

ea.}
4;1 
C.,Z", 
....
> O.
C2, 1i.'J"g'l •
V'2:S

'1
 1ax}"" 1 Z"  g.)
Ii...,
(I')
TI'1 Z., 
1DOl"
.
80

ao']
(X'1 8.'1.:1 
+ .,..
T·
,I')
n) ,
(5.70)
Thus, using completion of the squares as ill (5.59), we get ')
(1 ~'1 1 ~'1 + +  1 ) 181+ 191, 4Ii.1 41i2 492 'J
CtZi  C!!z:;
(5.71)
which means that the state of the error system
(5.72)
:I 'R
l
is hounded whenever the disturbance inputs
8 Rnd iJ are bounded.
<>
'Ve now consider the parametric strict.feedback systems (5.55). Tbe recursive design procedure is given in Table 5.1. 2 :!For Dotational convenience we define :0 ~ 0,
00
~ o.
202
l\'10DULAR DESIGN WITH PASSIVE IDENTIFIERS
Table 5.1: ISSController
(5.75 )
(5.76) i = 1: ... : n 'y' .vi
('1'"1,···
,(i) _ ( .
Yr
'j' ) , l'i

,
(i»)
?/r,!ln···)!lr
Adaptive controllaw: ,II _ 
1
(.)
/3
,r/
[. ( . B~ (111») 011 ,1., ']}r
+ ]}r(11)]
(5.77)
By l'Olnparing the expression for the stabilizing function (5.74) in the modular design with the expression (4.190) for the tuning functions design we see that the difference is in the second lines. \Vhile the stabilization in the tuning functions design is achieved using the terms iJ~Jit [Ti + 2:t:~ a(~;;l fWiZk: the stabilization in the modular design is achieved with the nonlinear clamping tenn Si:':j.
Clahn. The dosedloop system obtained by.' applying the design procedure (5.73)(5.77) to system (5.55) is (5.78) where }l:, Hl, and Q are matrixvalued functions of z,
A.:(z, 0,1)
C1 .51
1
0
1
c:!  82
1
0
1
0,
and t:
0
0
1 0
0
1
ell .5 11
5.3
203
ISSCONTROLLER DESIGN
o _ili:u. Of}
_
E Jff!>;p. (5.7D)
{h~'I_l i:J(j
Proof. For i = 1 we have
':2+0:1 + (('I + sd.:) +':2 + For i 2, ... , n  1, since (. '"» fJ, !iI" ...• yr' 1 wc have ~
O'i"l
is a function
wTo.
(5.80)
or only the variables :1'11 . . . :
( .1: /,._!_}
.rj_]
1
+ i.p,.T 0)
(5.81)
For i
n ,,,e have 11
(5.82)
in vedaI' form yields (5.
o
System 78) will be referred to as the el'ro1' Notc that the first. component of its 2:1 = :CJ  Yr = Y  ~Jr, represents tlH:~ tracking errOl', The ebange of coordinates (5.73)(5,75), which we compactly write as (5.83)
204
MODULAR DESIGN WrTH PASSIVE IDENTIFIERS
is smooth in x and iJ and is hounded in f. N ot.e also that the inverse transrormation x=iI>(z,6,t) (5.84) is smooth ill z and iJ and is houndp.d in t. Except for the term Q( z, iJ, t)T 0, tbe error system (5.78) is similar to the error syst.em (4.195) in which the t.erm
Q,(=, 8, trrO was accounted COl' by using
tuninlJ fund·ions. Here we let both ii and iJ appear as disturb!U1ce inputs. In the modu1al' design their bound~dl1ess will be gum'auteed by parameter identifiers. We now establish the basic inpllttoshtte properties of the error system (5.76), (5.78), (5.79), lllalullg use of the following cOllstil.nts:
Co
= min Cit
"·0
1:$i:$11
=
1) I L : ;=1 Ii; rI
(
t
90 =
(rlL1)1 ;=1
9r
(5.85)
Lemma 5.8 In the error system (5.78), (5.7!J), (5.76), the following inputtostate pmpc1'ties hold:
1
,=(t>l ~ 2 ~ (~IIOII~ + ~1I611~) 2 + Iz(O)lcCUL • veo
no
Yo
(ii) If 0 E £00[0, tf ) and 0 E £2[0, if), then
Iz{f.)'
~
(5.86)
=, x E £00[0, tf)' and I
(4 1_1I01/~ + !IIOII~)'!i + Iz(O)le~t. Colio ....go
(5.87)
Proof. Diffel'Cntiating ~1:::12 along the so1utions of (5.78) we compute
and arrive at
1(1,," 1:.
dtcI (I?) '2:::1 ~ Co I::: 1'1 + 4 Ko (J .. + 90 181'1) . *
(5.89)
5.3
205
ISSCONTROLLER DESIGN _
(i) From Lemma C.S(i), hiking 'IJ
~
1/2
= =2 and p = (:',1912 + ;;1(12) , it follows
that (5.90) which proves.: E £00[0, i/) and (5.86), and
(ii) By Lemma C.5, taking PI = 0 and P2 =
br (5.84), :t~ E £00[0, tJ). ii,
from (5.89) we get
1=(1)1 2 ~ 1.:(0)12e2Cl)t + 4c~no IIOII~ + 2~o 11911i, which proves
=E .coo [0, tl) and (5.87), and by {5.8'1}, 3; E .c00[0, 'I}'
(5.91)
o
As we can see fro111 Lemma 5.8, with 110nlinear damping we achieve not only inputtostate stability (5.86) in the sense of ~efinjtion C.1. but also the inputtostate propcrt:y (5.87). With respect to 0, this property ean be understood. as ".c2illPllt + .coooutput" stability, but it. can also be seen as ISS with
1I{jlb
considered as input.. \Vhile this property is not important in schemes, it is crucial in passive schemes whel'e boundedlless
sw~ppingbased
of ii C~tll be independcntly guaranteed by the identifier only in the .c2 sense. The quadratic form of the llo11lille~U' damping functions is only one out of many possible forms. Any power greater than one would yield an ISS property. but the proof with quadratic nonlillem' damping is by far thc simpJest. A consequence of Lemma. 5.8 is that, even when the ad!lptatioll is switched
off, tha.t is, when the parameter estiuul.te iJ is constant (8 = O) tlnd the only disturbance inpnt is 0, the state =of tbe elTor system (5.78), (5.79). (5.76) remains bounded and converges exponentially to a posit.ively invariant co Ill
TI2 =i are not needed when i1. = 0.) I Moreovel" when the adaptation is switched off, this boundedness result holds pact set. (Note that the terms OJ fJ~i'
even when the unknown parameter is timevarying.
Corollary 5.9 (Boundedness Without Adaptation) If 0 ; lR+ + R,f is piecewi,se continuo'us and bounded and iJ i.9 C01U;tn.ut, then .:, x E .ccr.t and 1 1=(t}1 ~ ?JCOiiO supIO(r} ...
co~o
T'~O
81 + 1=(O)leent .
Proof. Since B(i} == 0 1 (5.89) balds with 8(t) = 9(t}
8.
(5.92)
o
Thus, the controller module alone guarantees bounrledlless, and the task of adaptation is to achieve tracking. In fact, a stronger result given by Theorem 4.15 for the tuning functions scheme also holds: if tldaptatiol1 is discollnected and the parameter error is sufficiently small, not only will the global boundedness be achieved, but also the global asymptotic stability.
206
A10DULAR DESIGN WITH PASS1VE IDENTIFIERS
5.4
Observers for Strict Passivity
.
.
Having designed a controller module which achieves inputtostate stability with respect t.o
8 and iJ,
we turn our attention to ~he design of an ident.ifier
module which guarantees t.he boul1dedness of 6 and 8. This chapt.er deals on]:y with those modular schemes which use passive identifiers. Swapping schemes will be the subject of the next cbapter. In order to design identifiers which guarant.ee boundedness of 8, let us consider the negative feedback connection in Figure 5.1. It consists of a transfer
mat.l'ix
!:., 8 r = r"
> 0, which is passive, and ll.llOlllinear dynamical system E _
whose input is the parameter error lJ. If we can design the system E and select. ~).n output T so that E is strictly passive from 8 to T, then by Theorem D.4 the equilibrium at i;he origin of t.he interconnected system ill Figure 5.1 is globally uniformly stable (and, ill addition, the state of system E cOllverges to zero). Thus, ill order to guat'allte~ the ~oundedncss of 8, it suffices to desigul:t strictly
=
= rT.
pasRive system E and let 6 8 Now we present. the design of observers whose errol' systems ~tre st.rictly passive from ii as the input, to a judiciously selected output T. These error systems will play the role of E ill OUI' identifier design.
The parmnetric .:model Let
llS
first discuss t.he parametric model (5.93)
If the term Q{=, 8, t)Tij were not prcseut, we would have strict. passivity from t.he input 0 to the outpnt H/(Z, 6, t):. To see this, let, us consider the system
T
r s Figure 5.1: Negative feedback interconnection of the possive system rIB, r = We have to design tile system E nnd select an output T suell that 'E is strictly passive with input O.
rT > 0, lvith 8. dynamic nonlillear system E.
5.4
207
OBSERVEBS FOR STRICT PASSIVITY
which is a copy of (5.93) without Q(z, iJ, t)T8. In view of (5.79), system (5.94) satisfies d pr' ~ T(5.95) dt 21zl~ :5 cl=l~ +.: Hi(=, fJ, t) 8.
(1
'1)
.,
Integrating over [0\ t] we obtain
 ~1=(0)12 + C 10It 1=(T)j2clr, 10f' (HT=)TOdT ~ ~1=(t)12 _ _
(5.96)
which by Definition D.2 proves that (5.94) possesses a strict passivity property from 1:he input jj to the out:put HT(Z,8 f i)=, or ill ot.her words the nonlinear operatol' (5.97)
is strictly passive. To eliminate the term Q(=, 0, t)TiJ from (5.93), we jnt.roduce the observer ~ = A:(=, 6,/)= + Q(=, 8, t}TO (5.98) and define the observer error ==z':.
(5.99)
It is readily verified tbat. .: is gm'erned by (5.94). Heuce t with the addit.iou of an observer, we lutve generat:cd a strictly passive error system wboFie sj:ate is available.
The parametric xmodel Our goal is to design a parameter identifier for nonlinear systems in the parametric strictfeedback form (5.55): .i:i =
x"
=
+ IPi(Xlt ... , Xi)T6, 1:5 i :5 TI f3(x)u + IPn(:v)TIJ, Xi+l
1
(5.100)
wbere IJ E RP is the vector of unknown constant parameters, and the complete state x is assumed to be available for measurement. System (5.100) is a special case of the general affine parametric model: ;1:;
where vector
f
E JR.",
(5.l0l)
and tile "regressor" matrix F are defined by 3
X"
J(x,·u) =
:
[
1 ,
(5.102)
.7,1
{30(.1:)U though F ill (5.102) does not. depend on II, we llllow t.bis dependence in (5.101) because our identifier design will be applicable to general linearly parametrized nonlillenr syst.cms.
208
rVIODULAR DESIGN WITH PASSIVE IDENTIFIERS
It was easy t.o achieve strict passivity with the parametric zmodel because the undl'iven system was eA'"Ponentially stable. How can we bring t.he parametric xmodel (5.101) into a form similar to (5.94)1 First, we need the presence of 9 instead of 8; second, we need an e}..llonentially stable homogeneous part.; and, third, we must remove f(:r;, ·u.). Namely, we would prefer to have the model
.t =
A{.~, t):1: + F(x, u.)T6.
(5.103)
whose homogeneous part is exponentially stable:
PA(.r, t) + A(x, t) P:5 I, ')'
P= pT > o.
(5.104)
To obtain (5.103), we introduce the observer
i.: =
A(x, t)(x  :z:)
+ /(:1:, u) + F(x, ·u)T{).
(5.105)
It.s error state (5.106) is governed by the error system (5.103). This error syst.em satisfies (5.107) which upon int.egration turns into (5.108)
By Definition D.2, this establishes the strict passivity from the input 0 to the output F{x, u)Px, that is, the strict passivity of the nonlinear operator
Ex : 9 H F(:I:, u)Px.
(5.109)
It is important to note that ,wit.h passivity we can only claim the boundedness of O. The bounded ness of 0 is yet to be dealt ~ith. It turns out that with passive identifiers we can achieve boundedness of {) in the £2 sense but not in the Ceo sense, which means that we will depend on part (ii) of Lemma 5.B, We present two passive schemes: the zpassive scheme and the xpassive scheme. The zpassive identifier is based on the parametric zmodel, whereas the xpassive identifier is based on the paramet.ric xmodel.
5.5
209
=PASSIVB SCHEME
Figure 5.2: The =passive identifier.
5.5
zPassive Scheme
'Ve consider the parametric .:model
(5.110) and the observer
(5.111) which is a copy of the system (5.110) with the term I'F'(.:, 0, t)TO omitted. The observer error (5.112) is governed by an equation driven by . E
8:
= A::(z, 8,... t)€ + 111(z, 8,... t) T8.
(5.113)
As we have explained in Section 5.4, the observer error system possesses a strict passivity property from the input 8 to the output l'V(z, 9, t}f, that is, ~be operator E: defined in (5.97) is strictly passive. Therefore, we choose 6 = rE::{8}, that is,
iJ = rHf(z, 0, t)£,
r = rT > o.
(5.114)
The basic properties of the zpassive identifier (Figure 5.2) are as follows.
Lemma 5.10 Let the ma:r.imum inte'l'1Ja.l of existence of solutions of (5.110), (5.113) and (5.114) be [0, If). Then the following identifie'" properties hoM: (i) (ii) (ii.;)
8 E £00[0, tf} €
E
£2[0, tf ) n £00[0, tf}
9 E £2[0, tf) .
(5.115) (5.116) (5.117)
210
A10DULAR DESIGN WITH PASSIVE IDENTIFIERS
Proof. Let us introduce a Lyapullovlike function 1 ..,
1'" = IOIF' 2
1
')
+ ;;Ifl. ...
(5.118)
Its derivative a.1ong the solutiolls of {5.113)(5.114} is
,i"
=
oTr
=
cleF! 
6+ ~fT (A:: + A~) f + f'J'HiTO
1
t (liilWil2 +.Qi lI:Ja';:1 T12) f; + 8 r (rUiF.  9) T
,=1
1
f)8
/I
< clff~ 
L n:ilwd2f~ .
(5.119)
.=1
The nonposi t.ivity of Ii" proves that; 0 o,nd E nrc in £00 [0, t I ). Intflgrating (5.11 g)
\Vege!: c
,Ifl!!dr ~  £1.Vd, ~ V(O)  l"{t) ~ V{O) <
lao
00
.0
which proveR that. e: E £2[0, t J). To prove (iii), let
1812 = =
fT
H,Tr2 y,Vf
~
118
(5.120)
consider
X(r)2 f T H/T H' F
X(f)2It WiEil2 .=1 II
< X(rr!n L IWiI2E~ i=l
X(r):!n ~ _I
2 :s;   L.J/l,j Wi 12fi' lim
where
lim
(5.121)
i=l
~ nlJ,nlii' Substituting (5.121) into (5.119) we get lSf$lI
(5.122) which upon integration yields (5.123)
and hence
o
Ii E C:![O, t I)'
.c
The most important fact in this lemma is the 2property of i) achieved with the nonlinear damping terms n.llllJlI2, ... , 1i"lwu I2 •
5.5
211
=PASSrVE SCHEME
The properties established for t.he identifier hold only as long as t.lle solution to t:he plrult differential equat.ion exists. Because the right.hand side of this e(luatioll is locally Lipschitz in the state variables and piecewise cont.inuous ill till1e~ the existence of solutions over an open interval [0, t I) is asslll'ec.l (see, e.g.• [81, Theorem 2.2]). Lemma 5.10 establishes t.hat even if Ute plant state eseapes to infinity as t  t 1'1, the solutions of the identifier equatiou is unifonnlll bounded by a COllstant independent of t I' The same boundedness property of all the signals on (0, tl) can then be deduced wUh Lemma 5.8. The independence of the bouIlds of t I impli(3s that t I = 00. All OUl' proofs of boundedness for modular scbemes will follow this pattern because of UIC lack of Lyapullov functions encompassing both the states of the identifier and those of the plant. The Duly exceptiou is the .::passive scheme where we can COl1strud 11. single Lyapullov fUllction. For this scheme we establish a stronger st;abilH.y property than for other modular schemes: In addition to global uuiform bouucledness and asymptotic traclciug, we prove glohal uniform st;a hilUy of the referem'e trajcc(:ory.
Theorem 5.11 (zPassive) The closedloop aclapti'ue s1lsi.em con.sisting of the plll.ni. (5.55), controller (5.77), olJllerlJer {5.111}. and update latD (5.114) has a globa.lly tt1},i/o'lTll.111 stable equilibrittm at the origin:: = Ot f = 0, jj = O. an.d lim .::;(1.) = lim €(f) = O. This means, in pa'l1.icltlu'l·, t/uJ.t global IUlJJmpiotic 100 'CXl tracking is achie'vcd: (5.124) lim [yet) lJr(t») = O. 100
Moreover, if lim y~i)(t) = 0, i = 0, ... , n.l, lmd F{O) = 0, then. lim .r(t) = O.
'.CXl
''00
Proof. \Ve make use of a const.ant. J1 > 0 to be chosen later. Along the solutions of (5.111), (5.113), (5.1l.J)\ we have
212
l\10DULAR DESIGN WITH PASSIVE IDENTIFIERS
Substituting (5.121), we get
(5.126) • 4gonm Choosmg JJ < 11 X(rp we get
dld ('1IAI"} 2 =. + 2111'e ) + 12 1f)1r1 '1
)
~
J1co 1'" Z  ('0 '1 f '1 ,
(5.l27)
which proves that: the equilibriulll (£, f, 0) = 0 is globally uniformly stahle. From the LaSalleYoshizawa theorem (Theorem 2.1), it further follows t hat; all the so1ut:iollS converge to the manifold £ = f = 0 as t ~ 00. Since z = .:  E, the last two conclusions imply that the equilibrium (=, E, 8) = 0 is globally uniformly stable, and all solutions converge to the manifold z = E = 0 as t~oo.
From the definitiolls in (5.73){5.75) we conclude tbat, jf lim !J~i){t) = 0, i = 100  I, and F{O) = 0, t:hen x{t)  t 0 as t + 00. 0
0, ... 111
Note from (5.125) and (5.l26) that the nonlinear damping terms n.;IWil!'.! in the matrix A:; of the observer err?r equation (5.113) are crucial for counteracting the de.qtabiliziug effeets of {j.
5.6
xPassive Scheme
'Ve consider the parametric ."Cmodel (5.l28) which encompasses the class of parametricstrict feedback plants (5.l00). Following the passivity motivation from Section 5.4, we illlplenlcnt the observer
.i: = A{x, t)(x  :r) + /(x, u) + F(X,1/)TO, where A(x, t.) is exponentially stable (uniformly in A{x, t)T P ~ 1, P = p'J' > O. The observer error
.1:
and t):
(5.120) P A(x, t)
+
E=XX
(5.l30)
f = ;1(x, t)e + F(x, U)TO,
(5.131)
is governed by
5.6
213
xPASSIVE SCHEME
whirh is a system shown t.o be strictly passive from the input 0 to the output F(x, U}Pf in Section 5.4. Therefore, recalling the definition of the stri~t.ly passive nonlinear operator E;r from (5.109), we choose the update law as ii =  fE;r. {O}, that is, (5.132) When A(x, t} is a constant Hurwitz matrix, (5.129), (5.132) is a passive identifier wit.h standard properties: 8, e are bounded and e is squareintegrable. Neve.rt,heless, these propertics arc not enough for our purpose because we nped also 6 t.o l~c either bounded or squareintegrable. It is not known if the boundedness of iJ = r F(:r, u}Pe can be guaranteed irrespectively of the houndeducss of F(:J:, u}. However, ~ven when F(x, u} is growing unbounded, we can guarantee that the signal 8 = r F(x, U)PE is squareintegrable if wc choose
..4(:1.', t)
= AD 
..\F(~·, u,)'I'F(.l:, 1l.}P,
(5.133)
where A > 0 and Ao is an arbitrary constant rnat,rix such that PAa
+ Ari p
= 1,
P
= pT > O.
(5.134)
Thus, our strengthened observer error system becomes i:. = (AD  ..\F(x,u)'f F(:,., ll)p) e + F(.1:, 'lI)TO,
(5.135)
while the updatc law is
iJ = r F(J:, 11 )PE.
(5.136)
This sets the stage for the following lemma.
Lemma 5.12 SUPIJoSe the sol'utions of (5.128). (5.135)~ and (5.136) a.Te defined on [0, f./}. Then the jollo·wing identifie7' p1vper'tics hold: (i)
(1i)
(iii) Proof. Let
lIS
8 E £00[0, t/} f
E £!![O,t/)
n.coo[O,t/}
8 E £2[0, t J} .
(5.137) (5.138) (5.139)
introduc!e a Lyapunovlike function (5.140)
whose derivative along the solutions of (5.135), (5.136) is li =
28TrlO+fT(PAo+AJP)f.2 ..\eTPFTFPf+2eTPFTO
= 1e1 2 
2,,\IFPeI 2 ,
(5.141)
214
l\4:0DULAR DESIGN WITH PASSIVE IOEN1'lFJERS
.1'
x=f+Jfi8
~
J, = (.04 0  ApTFP) (Ii; 
iJ J
jj
X)
+f
+FTfJ
~
rFP
Figure 5.S: The xpassive identifier.
The nonpositivity of V proves that 0 and f are in £00 [OJ if). Integrating (5.141) we get
Jot Ifl2clr ~  10t·V'dr:5 YeO)  V'(t) :5 \/(0) < 00. which proves that
f
(5.142)
E £2[0, tf). As for (iii), noting that.
18\:! = fT P F T r2 F Pe :5 X(r)2IFPfl 2 and substituting into (5.141), we get •
'l
V :5 Ifl~
A:. ')
 A(r)2 IO I.
(5.144)
Upon integration W£l arrivc at
.fa' 1~12dT :S X(~'2 V(Ol < 00.
(5.145)
o Vve reiterate that all the £00 and £2 bounds (;hat we have established all
[0, tf) are il1clep£lndcnt of t J. Theorem 5.13 (~Passive) All the signal.9 in the closedloop adapti1Jf. ,'ystern consisting o/the plant (5.55), controller (5.77), obscrocr (5.12fJ), and the 1Lpdn.te law (5.136) (1.1"e globally uni/07m1ll bottnded, and lim =(t) = lim €(f) = too '00 O. This mea.ns, in particular, that .qlobal aS1Jmptotic tra(:.li1lg is achie'ucd: lim [y(1.) '00
Yr(t)]
= o.
(5.146)
Itlareover, if lim lI~i)(t) = 0, i == 0:, .. , nl, and F(O) = 0, then lim x(t) = O. toc
100
5.6
215
xPASSIVE SCHEME
Proof. Due to the piecewise continuit~, of Yr(f), . .. , y~n)(t) Rnd the smootbness of the non1inearities in (5.55)4 the solution of the closedloop adaptivc system exists alld is unique. Let its ma.ximuID jnt.erval of 7xistence be [Ot t /). From Lemma 5.12 we have lJ E £00[0,1/) alld 8 E £2[O,t/), wbich ill yie\\' of Lemma 5.8 implies that z E £oo[O,t/) and J; E £00[0,1/). Because by Lemma 5.12, f E £00[0, til, t,hen by (5.130), x E £oo[O,t/). "Ve have thus shown that aU of tlte signals of the closedloop adaptivc system are bounded 011 [0, tJ) by constants depending on the illitial conditions, design coefficients, and the extcrllal signals Yr(t), ... 1 y!n)(t), but not. on t J. The independence of the boul1d of tJ proves that t / = 00 (if t / were finite, then the solution would escape any compact set as t to. t /, which would contradict the existence of 0. bound independent of t /). Hence, all signals are globally uniformly bounded on lOt (0) . . To prove convergence of z to zero, we recall first from Lemma 5.12 that f,8 E £'2' Let us factor the regrcssor matrix H't l1siug (5.79) and (5.75), as
i1
o
1
1
F'r(",)
~
N(:.Ii./)FT(Z), (5.1,,17)
Considering now the state l:J.
(=.:::Nf,
(5.148)
consisting of the state of the error system z and the observer error are both driven by 8, we obtain . ,
=
[N + A::(.z,9, 1.)N 
A z (.=, 8, t)( +
t:,
N (Ao  ..\F(x)T F(:,;)P)]
• 'r +Q(z,8, t) 8. A
which
f
(5.149)
'What we have arriv~ at is a system which is not: driven by jj but., instead, by the £2 signals f and 8. 'Ve now compute 2
co 1(1 +(T
:5 
~
L.,9j i=1
18ai1 8a: A TI:!,...2 + ~ L .
[N +A::;N 
88 N
i  l (j'...""
~;
i=l
\i
80
(Ao  ,,\F'I'FP)] f
c01(f + ~IOI2 2 4go
+1IN + AzJV ~~
N
(Ao  "\F'rFP)I~ 1e1 2 •
(5.150)
Examining (5.74), we see that. the terms ~ appearing ill N in (5.147) are IJ%J continuous functions of z and iJ and bounded functions of t. Hence lV is
216
MODULAR DESIGN WITH PASSIVE IDENTIFIERS
bounded. Likewise, we can show. that ~,
~i are bounded. Since, in
z and 8 are bounded,
view of (5.78) and (5.136),
iT _ H
is also bounded. Thus
W'

aN ~ aNe:' aN a"'+ z ao~ + f) t
1" +A:N  N (Ao 
(5.151)
AFT F p) is b~unded. By applying
Lemma B.5 t.o (5.150) with v = I
Remark 5.14 \¥riting the .:system (5.78) in the form
and comparing it with (5.128), we see that the parametric zmodel could be considered to be a special case of the parametric .l:model. One can also see that the zpassive identifier is a special case of the xpassive identifier. However, this special case is also the most important one. "Ve have already seen this by going through the proofs of Theorems 5.11 and 5.13: The proof of convergence for the zscheme is much simpler than the proof of convergence for the .rscheme. In the sequel l we will encounter more instances where the pru'ametric zmodel will be more convenient than the parametric .1:model. 0
Remark 5.15 An alternative t.o the proof that .:(t) as follows. For convenience, we first copy (5.135): f =
~
0 in Theorem 5.12 is
(Ao  AF(x)T F(x)P) E + F(:v)T6.
(5.153)
Because of the boundedness of all the signals, i is bounded. Since f E £2, by Barbalat's lemma (Corollary A.7), f(t) + O. By virtue of the smoothness of F (which implies that its partial derivatives are bounded for bounded values of their arguments) and bOUlldedness of all the states, f is also bounded, and therefore i. is uniformly continuous. Since e(t) + 0, t.hen lim
Ieo
1'·i.(r)dr 0
= lim e(t)  E(O) = E(O) 100
< 00.
(5.154)
Then by Barbalat's lemma (Lemma A.6), i(t) + O. From (5.153) we conclude f(x(t))TO(t) + 0, and therefore, by (5.147), H'(=(t), iJ(t), t)T6(t) ~ O. Since
8 E £2,
using a Barbalat's lemma argument, we also have 8{t)
+
0, so
5.6
217
xPASSIVE SCHEME
Q(=(t)!B(t), t)T8(t) ~ Thus the input
O. (Note that while QT/J H!Tn + Q'rO to the error system
is in
£21
H!T8 may not
he.)
(5.155)
converges to zero. In view of
! (~Izr~)
: :;
col=f+='I'(H!T8+QTfj)
:::;;
_ c0
2
by a.pplying Lemma B.8 wit.h p
2 O + QTal::! 1=1 + ....!:....lnll' 2cn
= 0,
/31
,
= 0, /3'1 = ~ IlVTO + QT Ol2
(5.156)
(or [28,
Theorem N.1.9.(e)]), we arrive to the conclusion that =(t) ~ o. At. this point we remind the readel: of Lemma 5.3 whcre we proved that if F(.l:(t))O(l.) (as we just showed) and /J(t) cOllvcrge to zero as t ~
00,
then Iimt_oo :l.'(t) = O. 0
Remark 5.16 Even though Theorem 5.12 states only global boundedness and tracking, and its proof used an inputtostaf;e rather t.han a Lyapunov argument. we can prove that the equilibrium z = 0, € = 0, 8 ~ 0 is global1y uniformly stable, as we did in Theorem 5.11 for the =passivc scheme and Theorem 4.14 for the tuning fuuctions scheme. It is enough t.o lIol.c from (5.149), (5.151), (5.78), and (5.136) that
( =
A:(=, 0, i)(
8, t) + [ aN(=, 8=
(
_  _ ,. _ ~ '1' _. T ) A:(_, f}, t) .. + n (.. , (), 1) () + Q( .. , fJ, 1) rF{X)PE
8, t) rF( .1:')P €+ aN(=, + 81V(Z, at 8, t) aD A
+A:(=, 0, t)lV(=, 8, t)  N(z, b, t.) (Ao  >..F(:,;)T F(:l')P) +Q(=, 8,~ t) T r F(x)P ]
(5.157)
f,
which, in view of (5.148), can be expressed as .
( =
Ad"
€,

0, t)(
T + H',(, €, 0, t) f,
(5.158)
where lYe is smooth in (, € and 9, and bounded in t, and the homogeneous part of the system (5.158) is e.xponentially stable. Starting from (5.158), one CRn prove that the equilibrium ( = 0, f = 0, B= 0 is globally uniformly stable by Definition A.4, which along with (5.148) implies that the equilibrium z = 0, f = 0, 9 = a is globally uniformly stable. The difference from Theorems 5.11 and 4.1~J is that the stabiHty proof is not based on a quadratic Lyapunov function encompassing the complete state of the closedloop system. 0
218
MODULAR DESIGN WITH PASSIVE IDENTIFIERS
Remark 5.17 At the first glance, one may thinl,: that a passive identifier can be designed using only one of the state equations in (5.55), for example, (5.159)
(or a combination of several state equations). The observer (5.160)
would yield the observer error system (5.161)
where e =
Xl 
;i':l.
The update law would be (5.162)
A simple a.nalysis with a Lyapunov funct.ion V = e 2 + 161!:! would show that 8 is bounded, iJ is squareintegrable, and e is both bounded and square.integrablp. Therefore all the states are bounded. Unfortunately, we cannot prove that :;: converges to zero. If we apply the sa.me approach as in the proof of Theorem 5.13, namely, if we define ( = =1  e and subtract (5.161) from t;he ZlCquation (5.163)
then we get (5.164)
Since e can easily be shown to converge to zero, in Ol'der to show that =1 does so too, it, suffices to show that ( converges to zero. However, z!:! in (5.164), which we only know is bounded, keeps us from concluding that ( goes to zero.
o
5.7
Transient Performance
Transient performance of our adaptive system will be estimated by £.2 and £00 bounds for the error state z. Since ZI = Y  Yrl these bounds also bound the tracking error Y  Yr' Vle analyze first the zpassive scheme and then the J:passive scheme. To simplify the derivations, without loss of generality, we assume that £(0) = z(O) (in the zpassive scheme) and .i:(0) = x(O) (in the xpassive scheme), which implies f(O) = O. Such observer initializatiolls should be performed in practice to eliminate the disturbing effect of the initial observer error. For simplicity, we also let r = "(I.
5.7
219
TRANSIENT PERFORMANCE
Theorem 5.18 (zPassive) In the adaptive system (5.55), (5.77), (5.111), (5.114), the following inequalities h.old:
(i) (ii)
(1 +  (+ 10(0)1
11=112 :::; 19(0)1
wy:.! ) 1/2
JC?i
Iz(t)1
~
+ 11::{O)l ,
2goli,m
1
2v'
con.o
2nr~)I~ 90 lim
Fa
+ 1=(O)lecnl .
(5.165) (5.166)
Proof. (i) Since £(0) = 0, (5.119) implies that (5.167) Substituted into (5.122), E(O)
= 0 gives 1

IIElb:5 . ~18(0)1·
(5.168)
V"Co"Y
Now, for po <
4!~m
I
by integrating (5.127) over (0, co], we get (5.169)
and, sincc 2(0)
= z(O), then (5.170)
Letting 2g0N.m
(5.171)
11.=,.,, wy
and adding {5.16B} and (5.170) in
(5.172) lve arrive at (5.165). (ii) In a fasMon similar to (5.126), wc compute
'I
fI
i=l
i=1
+1' 'L" 2i W iT(J + L~ £iWjT0 _ ~ n
;=1
I
_ OO:i_1
LO.
A
B9
TI2 2 + ~ n f)niI n • __ L .. '7
... ;
i=1
of)
[7 ... ,
220
IvloDULAR DESIGN WJTH PASSIVE IDENTIFIERS
:s; co(/LI=1 2 + 1~12) 
11.
;1 L IWiI2~~ 11
... i=1
+2.:...101 + _1 1012 + L161~ 2
.:11\'0
4go
211.0
~ co(J1I=1 2 + If12) + LIOj2 + _1 lof 411"0 2h~O
 (n.:; _JJ nl'~) t IwJ!f.7 , 490
(5.173)
;=1
where for the last incquality we have used (5.121). Setting (5.174) in (5.173) and applying Lemma B.5 part (ii), we gct
which, by taking the supremum or 6(r), implies 1
.,) J/'!
(
I=(t)l ~ ?.jCOiiO 1 + .:. ... Calia /L
1101100 + 1=(O)feCoL •
Substituting (5.167) and (5.174) into (5.176) results in (5.166).4
(5.176)
o
\Vben we perform the trajectOl'Y initialization introduced ill Section 4.3.2, which sets =(0) = 0, the bounds (5.165) and (5.166) become
11=112
~
1=(t}1 <
1 ( 1 + .,nl'" )'" 10(0)1, vern _co')' ....gOn.m
( J'"
1 2nl'2JCOKO 1 + g Ii,
a a
.0 m
18(0)1·
(5.177) (5.178)
These bouuds depend only on the design parameters Ci, n.i, 9i, and "'I and the initial pal"clmeter error 0(0). It should be clear that, sil1ce the plant initial conditions and the reference model initial conditions are the same, the tracking error transient is caused only by the initial parametric uncertainty 0(0). While by increasing Do and /l'rn we can reduce t.he bounds somewhat, by increasing Co we can malre them arbitraril~' small. By increasing Ito we can arbitrarily reduce the £00 bound but: not t.he £2 bound. This general propert:y of the h~termR, "'Inequality (5.176) esf.lthHsbes nn ISS property from ii t.o =. Note that in cont.rn.C!t to (5.8o) Ilnd (5.87), 9 is absent from (5.176). It is also ensy to see thnt le(t)1 ~ :!ArnIl(]lIoc describes an ISS property from (] to
E.
5.7
221
TRANSIEN'r PERFORMANCE
of reducing the transient peaks while at the same time making the transient. tails longer by slowing down the adaptation, will be iUust,rated by shnulat·ions, The dependence of the bounds 011 the adaptation gain " is Jess clenr. The £00 bound is an increasing function of '1, and the £'}. bound is decreasing for small values of 'Y and increasing for large values of 'i'. \Vith regard to ,,(, the £00 bound does not seem to be t,ight because, as we shall see, t.he simulations do not corroborate its incn!fL..qing dependenc'e on "(. Let us now compare the bounds for the .:passive scheme wU,h the bounds for the tuning functions scheme. Examining the expressions (5.177) and (4.229), we see that the £2 bound for the .:passive scheme has the term 2;':"1 which is not present in the C2 bound for the tunillg functions scheme. Also, examining the expressions (5.178) and (.,;1.241), we see tbat the Coo bound for the zpossive scheme bas tbe term 91) 2n..,.2, which is not presellt in the £t::JO bound for li m tIle tuning functiOl~S scheme. These "(depeudenl: terms occur because ill tbe modull:1X approach () is llot eliminated from tbe zsysl;em, but: ouly "damped." In other words, by ignoring the timevarying character of the pArameter estimates, the modular approach results in performance inferior to that of the tuning functions design. Now \Ve give an Ceo bound for the xpassive scheme, for a specia.l choice of design parameters. 1
Theorem 5.19 (zPassive) In the '1.dapli'lJe system (5.55)~ (5.77), (5.129), (5.136) wilh. tILe choictf' Au = co, P = 1, . \. = no, I.lLe Jollowi11'!1 i'neq'ulIlitll holds: 
,.,
(1 + 9oh"o 21''') 2 VColio
1/'2
Iz(t)l::; 16(0)1
+ Iz(0)leCU1 •
(5.179)
Proof. With this choice of design parameters, we have
e
=
COE  li.opT Fe
{) =
+ pT&
,Ff.
(5.180) (5.181)
It follmvs that d (1IeI2) cit "'2
CO!EJ2 I
n.olFel 2 + 9T Fe
::; co IeI..?  n.o IFE: ,'..1 +  1 'I" IJ .. 2 21\:0 h"n:' '1 1  '1 = colel'J  191+ 191. 2')'2 2n.o
(5.182)
5Note tbllt this choice of AI) nnd P gives PAII+/1JP = 2coJ, which differs from (5.134). The proof i.alces t,his int,o ncCOUQt.
222
:MODULAR DESIGN WITH PASSIVE IOEN1.'IFIERS
Combining this with (5.89), we get
(5.183)
Setting p.
9011.0 = ,,
"r
(5.184)
in (5.183) and applying Lemma 13.5 part (ii), we get inequality (5.175) from the proof of Theorem 5.18. By proceeding in the same fashion as there, we arrive at (5.10). 0 Noting that n.m ::::; nKo, the £00 bound (5.179) for the .rpassive scheme becomes ident,ical to t,he £00 bound (5.166) for the =passive scheme. 'Ve can thus conclude that both schemes are equally capable of reducing the transient peaks. However, an £2 bound similar t.o (5.165) is not available for the xpassive scheme. At this point we recall that the proof of convergence of :z to zero was much more involved for the :t'passive scheme than for the zpassive scheme. Thus, the paramet.ric zmodcl is once agrun more convenient t·hnn the parametric xmodel.
5.8
SGScheme (Weak Modularity)
So far in this chapter we have been using the ISScontroller from Table 5.1. This controller achieves a complete controlleridentifier separation thanks to the strength of its nonlinear damping terms. Now we examine other possibilities for achieving bounded ness and tracking. In traditional adapt.ive linear control, the strong ISS controlleridentifier separation does not e..xist. Instead, boundedness and tracking are achieved employil1~ a smallgain (SG) stabilization mechanism. Ivlostly thanks to the fact that 8 is £2boullded, the loop gain in adaptive linear systems is asymptotically small, namely, a stabilizing smallgain propert.y is achieved as I ~ 00. It is this feedbaclc phenomenon that allows the nse of the certainty equivalence principle and makes adaptive lineal' control possible without nonlinear damping. When the parameter estimates of certainty equivalence adaptive controllers are frozen, these cont.rollers become lineal'. TillS is not t.he case with our ISScontrollers. Even whcn the plant is linear, the underlying nonadaptive controller is nonlinear because of the nonlinear damping tel'ms. There are advantages and disadvantages of nonlinear damping. It improves t.he tracking performance but, since it acts as high gain for large signals, it may result
5.8
223
SGSCHEME (WEAl( IvloDULARITY)
in large control effort (and could potentially llave an adverse effect 011 robustness). This prompts us to try to reduce the growth of 1l0111inearities ill our ("ont.rollers. The weaker ('ontroUers, which become linear wben applied to linear syst.ems, are also of interest because they will enable us to make a comparison between adapt.ive controllers designed using the linear certainty equivalence principle, and those designed using ISS.
5.8.1
Controller design
The only clifferen{'e between the stronger ISScontl'ollers and the weaker SGcont.rollers is that the SGc.'ont,rollers employ weaker nonlinear damping fUll(,,tions Sj. For example, for an ullcertain term IP(XI)() in 1:he first. equat.ion of the plant (5.55), tbe ISS and SG nonlinear damping functions are, respedively,  "1/' S 1ISS(". •• 1) " 1"('"1 )2
SG( Xl ) _ [tp(J!l)  CP(O)]:!
and
';1
~ W (,,)2 .11 •
(51St::) • il
Xl
s?G
In this way t.he growth of is rednced by a factor of a;j. In the process of backstepping, this reduction becomes more pronounced. However, the controller can no longer guarantee the ISS property wi t h respect to 0 and jj. Instead t it: relics on a small gain property achieved by ndaptal:ion. To derive t.he nonlinear damping expressions for the SGcontl'oller we rewd te the regressor vectors Wi as follows
sq
(5.186)
where Wi : lRi x m," x I~ .......,. JRixp is a matrixvalued functioll smooth in the mst t:wo arguments and continuous and bounded in t.he third argument., and z; g [=1",,: ':i]T, ('Vith a sligl1t abuse of notation relative t,o (5,75) we now e!\."'}Jress 'Wi as a function of =i, fJ, t.) Thus we have (5.187)
To reduce future notation, we dellote H'o from (5.79): A
Q(:,8.t)
g IV(O, 0, t).
TAT
Likewise, we rewrit.e Q
~o 1
=1
= Q(O,8,t) + .T: [
2
(5.188)
""1_1 6'1
where Oi : x ]RP X ~ + m.(il)Xp are matrixvalued fUllctions smooth ill the first two arguments and continuous and bounded in the thh'd argument. ll. To reduce notation, we denote Qo = Q(O, 6, t). lRi  1
A
IVloDULAR DESIGN WITH PASSIVE IOEN1'IFIERS
The SGcontroller has the same form (5.77) as the ISScontroller, but its nonlinear damping functions are different:
= h:ilwil} + 9doil}.
S;
(5.189)
The form of the en'or system (5.78)(5.79) is unchanged:
:: = A:(z, 8, t)z + H'(z, 6, t)Tjj + Q(=, 8. t)T8
(5.190)
where
1
CJ  81
1
c:! 
o
0
0 82
1
0
1
(5.191)
1
o
0
1
CIIB tl
Underlying nonadaptive controller Before we combine the SGcontroller with an identifier, we explain the uuderlying nonadap1;ive contro]]er. Suhstituting (5.187) and (5.188) into (5.1DO), we obtain
(5.192)
Then, along the solutions of (5.192) we compute
col=1 2 + $
t,=J
Zi
n
L (n'; IWi l} + gi!od}) z; i=]
(=T
Wi
8+ :;~1(ji6) + ZT (H/J'8 + Q~B)
_~1=12+ C~IOI2+ 4~1812) Izj2+ ~ IWJ·O+Q~8r· (5.193)
Now let 8(t) be COllstant and bounded. Then (5.193) becomes
") s;  (Co 1  ") Iz\" + 1 IH'oT8,2 . dtd (11=12  1612 4h~O 2co
(5.194)
Since H'o is a continuous runction of iJ bounded in t, (5.19~J) implies that z will be bounded provided that (5.195)
5.8
225
SGSCHEME (WEAK lVI0DULARJTY)
In other words, bounded ness of .: : can be assured in the absence of adapt.ation by using high gain. It. is crucial that, in order to satisfy (5.195), the designer
must knO\v a hound on the parameter error O. This is a weaker result than the one given in Corollary 5.9, where no such a priori knowledge is required. We will see ne~1; that with adapt.ation we circumvent both the requirement of a priori knowledge and the use of high gain.
5.8.2
Scheme with strengthened identifier
We now present an adapt.ive scheme which combines the SGcontroller with a strengthened passive identifier designed from the parametric =modcl: To achieve some sort of smallgain property, we need to have iJ E C2 • However, if we employ the observer (5.196) and the update law
fJ
= rH'(=, 0, t.)e ,
r = rT > 0
(5.197)
where thE' observer error is given by
€==.:,
(5.198)
we can prove t.hat 8 is .bounded and e is bounded and squareintegrable, but we cannot prove that 8 E £2. The reason for this obstacle is that, hecause of the weaker nonlinear damping functions (5.189) in A=, the stability of the observer error system (5.199)
is not as strong as before. We strengthen the observer (5.196) by augment.ing it with t.he term H,Trl'Ve:
.. = =
A:(z, 8, t)z + Hi(=, 8, ·t)'l'rHI'(z, 0, t)€ + Q(z, 8, t)TO
A:(=, 6, t)z + (HI'(z, 8, t.)
+ Q(z, 8, t)) TO,
(5.200)
and get the obsen'er error system (5.201)
We will show that this st.rengthening yields 6 E £2. As we mentioned above, the bounded ness of iJ and e is guaranteed by the identifier (5.200), (5.197), so what remains to be proven is the boundedness of either z or z. It turns out
226
l\10DULAR DESIGN WITH PASSIVE IDENTIFIERS
.:
Figure 5.4: The strengthened zpassive identifier.
that the boundedncs~ of =has to be proven first. It should be not.f'd tha.t the term lV'rrTYf = t.J'TO, which enhances the stability of the ~systell1, acts as a perturbation in the ':syst.em. "vVe deal with this in the proof of the following t.heorem.
Theorem 5.20 (SGScheme) All the signals in the closedloo]) a.daplive s1Jstem. consisting of lhe plant (5.55), tJu~ SGcon17nller (5.77) with {5.189}, the ObSeT1Jer {5.200}, and the upda.te law (5.197) are globa.lly uniformly bO'unded, and lim =(t) = lim f(t) = O. Thi.s means, in pa1·ticula.r, that !Jlobal. asymptotic 100 too tm.cking is achieved: (5.202) lim [y(t)  Yr(t)1 = o.
'00
A/oTCove1', if lim y~i)(I) = 0, 100
i = 0, ... ,n1, and F(O) = 0, thc.,! lim :r(t) = 100
o.
Proof. The solut.ion of t.he closedloop adaptive syst.em e.'\1.sts and is unique on its ma.x:imUlll interval of existence [0, t f). We first study the observer enol' system (5.201) and t:he update law (5.197). From
(5.203) we conclude that
8, € E £00[0, if). By integrating (5.203)
we get. (5.204) (5.205)
5.8
22i
SGSCHEME (WEAK MODULARITY)
wbich implies that. f,B E L2[O,tj). III order to prove that E: E Lco[O,t,), we first. rewlite (5.200) as
(5.20a)
and tben compute
~ colzl 2 
n
L: (#£ilwil} + 9iIDil}) ::; ;=1
{5.207}
Since (5.208)
t~'
from (5.207) we obtain
(1: + 1)I():"1')\z~I'l+ [(~ + !..) IIfll~ + !..UH/o + QolI~] 181 _I'" + co I= lio
go
90
Co
2
,
(5.209)
f is bounded, and note t11&t \I H'D + Qo 1\00 is finite because i:1.l1d Q(O, 0, t) are continuous fUllctions of iJ and bounded. functions
where we recall that
1"'(0,0, t)
no
228
l\10DULAR DESIGN Wl'l'H PASSIVE IDENTIFIERS
=
oft. Now we apply Lemma B.6 with v lil 2 to conclude that =E .coo[O,t/) n .c2 [0.t/). Hence Z E £oo[O,tl) n £2[0,t,). By the same argument as in tbe proof of Theorem 5.13, " = 00. The proof of convergence uses Barhalat'Fi lemma and the fact that ::, i E .coo. 0
sa
Remark 5.21 In contrast to the schemes with the ISScontroller, for the scheme we cannot derive transient performa.nce bounds similar to those from Section 5.7. The,.coo and .c'J, bounds tbat we get by applying Lemma B.6 to (5.209) involve 116n21 IIEI/oo and IIH'o + Qolloo· While for the first two qua.ntities we have bounds which depend ouly 011 f{O), 9(0) .and r, the third quantity, 111'1'0+ Qo II 001 depends on t.he reference signals and 8 in a complicated 1l0nlillCltr ~hloo.
0
The ~thove SGscheme employs a passive identifier designed from a parametric :model. It does llOt seem possible to obtain all anulogous result with the parametric xmodel. This adds to the list of advantages of the parametric zmodel. Tbe main advantage of the Sacontroller over the ISScontroller is that it has slowel' growth of Ilonlineat'ities in the control law. The main contl'ibutors to the growth of nonlillearit:ie; are the nonlinear damping functiolls Si. The strengthening nonlinear term HiTrH' f added in the observer (5.200) does not appear ill the contI"oller. The following e.'tRlllple illustrates the differences in the growth of nonlinearities between the ISS and tbe SO controllers.
Example 5.22 Consider the relativedegree two plant Xl =
X2
X2 =
tL,
+ 6cp{Xl),
cp(O)
=0
(5.210)
\Ve define the elTor variables ':1
=
''Vl
2
=
X2 
al(xt,D).
(5.211)
The twostep ISScontroller design is
H,]tp(xd
2
•
crl
=
C1Zl 
~
=
Cl 
=
=1  C2Z:!  fL2 (~)2 cp(.'Vt)2Z2 
11
1.':1 CP(·'l·1
f
':1
lhp(Xl)
2H.t
CP{Xl )CP'(Xl)X,  Dcpl(Xl) 02'P(x,)2 z!! + ~
(~'!! + Otp(."l:l») . (5.212)
The t'vostep Sacontroller design is 0:)
t11
=
(!p~~I)r':l  Dcp(x,) iii (!p~.})2  2Kl cp~~d (
Ct Z,  n.l
= Cl = ':1 
(5.213)
5.9
229
UNI
Let us now make a comparison of the nonlinear growths when the plant nonlinearit.y is (5.214) First, W~ compare the stabilizing functions 0'1' \Vhile the highest power in the ISS design is 3, in the SG design it is 2. The reduction of the nonlinearity growth with the SGcontroller is more dramatic whcn we compare t.he final control laws u. 'Vhile t.he highest power in thc ISScontrol Jaw is 17, in the SGcontrol law it is only 7! This differCll('e is even more pronollnced for systems of higher relativc degree where t.he l'ccursive cont.roller design ('onsists of more steps. 0 This disadvantage of the ISScontroller is more than compensated by a superior transient performance. 'Vc shall illustrate this in the next chapter by making a simulation comparison of two swappingbased schemes: one using the ISScontroller, and t.he other using thc SGeont:roller.
5.9
Unknown Virtual Control Coefficients
As in Section 4.5.1, we consider systems of the form i = 1, ... , m  1, Tn
+ 1, ... , 11 
1
(5.215) where, in addition to the unknown vector 8, the constant coefficients bm , m < TI, referred to as the "virtual control coefficient," is also unknown. The extension of t.he result of this section to more general systems of the form (4.243) is straightforward but tedious.
Assumption 5.23 In addition to sgn bm , a positive constant C;711 is known such that Ibml ~
'In'
5.9.1
Controller design
From step m. on, the design procedure considerably differs from the procedure in Table 5.1. It is also quit.e different from the tuning functions design of Section 4.5.1. Instead of introducing an additional parameter B to avoid a division by brll , in the modular design we do allow division by brnl because we employ parameter projection which keeps bm(t) fro111 becoming zel'O. The complete controller design is given by the following expressions (with Zo
6 IJ. IJ. = 0, 0'0 = 0, TO = 0):
230
MODULAR DESIGN WI'l'H PASSIVE IDENTIFIERS
Comtlinatc transformation: =i
= ........, 
=j
=
J.'j 
y,r(iJ) 
1 ("
~y/bm
0:'1 1
1)

,
O:j1 ,
i= 1, ... ,111.
j = m
(5.216)
+ 1, ... ITl
(5.217)
Regressor: ~ 80.;_1 'Pi  L.... (). 'Pk , k=l
i = 1, .. . ,n
(5.218)
Xk
Sta.bili:ing functions: Q;i(Xi, 0, jj~il»
=
(5.2IO) 
O'm ( X m ,
8A (ml)
'Yr
bJrt A
,
(5.220)
)
(5.221) 
aj ( Xj)
0 (;1) b ) ,1Jr I JII . 1
 8aj_l ~ +bma·Vm +1 + L....Xm
fJUj_l
(k)
a (kJ) Yr k=l Yr
j=m+2, ... ,n
(5.222)
Nonlinear damping Junctions: 8;
=
i = 1, ... ,mI
(5.223)
(5.224)
5.9
231
UNl\:NOWN VIRTUAL CONTROL COEFFICIENTS
(5.225)
j =
711
+ 2, ... , 11
(5.226)
A daptivc control law:
U
_1_ f3() X
[0'
It
+ ..!..ytr n )]
(5.227)
btu A
Lengthy calculations show tbat the design pl'ocedul'e (5.216)(5.227) results in the closedloop system
(5.228)
(5.229)
(5.230)
Lemma 5.24 In the error system (5.228)(5.231), {5.223}(5.226}, input· tostate properties hold as in Lemm.a 5.8, yDith
input.s.
~h~ same
(9 b,u) and (8, bm) as t
232
MODULAR DESIGN WITH PASSIVE IDENTIFIERS
Proof. Noting that Assumption 5.23 implies that
_Ib
rr
./
$ 1, b.y completing
squares as ill the proof of Lcnllna 5.8, we can prove that (5.232) and the rest of the proof is identical as for Lemma 5.8.
5.9 . 2
o
Passive adaptive scheme
Now we design an idcntifier which, ill addition to the standard properties of passi ve identifiers, guarantees tbat Ibrn (t) I 2: ~m' Let us start by Wl'itillg system (5.215) in the following compact; form:
i: = f{:v, 1/.) + F(:,:, U)Tf} ,
(5.233)
'Ve use our standard observer
:t = (}lo 
AP(X, t/)T F(z, u)p) (.i;  x) + f(X,ll}
+ F(X, u),r·O!
(5.234)
where A > 0 and An is an arbitrary constant matrh: such that P Ao + A;r P = / I P = p'l' > O. Tl1e estimation elTor
E=XX
(5.235)
is uscd to drive the update law for the estimate
iJ = Proj {rF(X,tl)PE}
,
bn•
.n of iJ:
h (0 )sgn bm > C;m , rl •
(5.23G)
where the projection operator is employed to guarantee that Ibm(t)1 ?: C;m > 0, 'Vi 2: O. For a detailed tl'catment of parameter pl'oject.ion the rearIer is referred to Appendix E.
Lemma 5.25 Suppo.fle th.e solutions of (5.283), (5.284) and (5.236) are defined on [0, tj). Then the following identifie1' properUe8 hold:
(0) (i) (ii) (iii)
Ib (t,)/2: C;III > 0, .0 E £oo[O,t,) nl
f
'Vt E [O,tj)
E £2[0, tj} n £00[0\ tj)
iJ E £2[0, t j) .
(5.237)
(5.238) (5.239) (5.240)
NOTES AND REFERENCES
233
Proof. The proof of this lemma is only slightly different from the proof of Lemma 5.12. The differences are due to projection whose propert,ies are summarized in Lemma E.l. First, from Lemma E.1 we conclude that Ibm(t)1 ~
Lemma E.1 implies that I'OF! :::; ifp,lrFPel:! ~ ~(?; \FPEI!:!. This fact is 110W ~sed as (5.143) in the proof of Lemma 5.12 to establish sqmU"eintegrability of
d.
0
By ('ombining Lemmas 5.24 and 5.25, we obtain the following result. Us proof is the same as the proof of Theorem 5.13.
Theorem 5.26 All the boundcdne,9s a.nd convergence pr01)criics from Theorem 5.13 hold also for the closedloop adaptive system c07l.sisti11..q of the plant (5.215). control law (5.227). obsel'Vc'I' (5.294). and thc 'Update ll1.w (5.286). Remark 5.27 The proof of Theorem 5.13 involves a discussion of the existence of solutions on [0, t/), whirh is a condition in Lemma 5.25. The projection operat.or we introduced in Appendix E is locally Lipschitz, as st.ated in Lemma E.l. Therefore, the conclusions about the existence and unklueness of solutions in the proof of Theol'em 5.13 remain unchanged. 0
Notes and References The foundations for estimationbased certaint,y equivalence adaptive control of linear systems were given by Egardt [31J and, in a more recent unified framework, by Goodwin and Mayne [43]. The estimationbased certainty equivalence design for nonlinear systems was summarized by Praly, Ba...,tin, Pomet, and Jiang [157]. All of the results in [157] involved some form of either matching or growth conditions. The modular adaptive nonlinear design which removed these conditions was introduced by Krstic and Kokotovic [98], while Borne of the pl:lSsive designs in this chapt.er were present.ed in Krstic and I(okatovic [97]. The usefulness of nonlinear damping for achieving bounded ness without adaptation was stressed by Kallellakopoulos (64]. Passive identifiers (also known as observerbased identifiers and equationerror filtering identifiers) have earlier been used by Campion and Bastin [IS], Teel, Kadiyala, Kokotovic, and Sastry [190], and Praly, Bast.in, POlllet, and Jiang [157]. Various restrictions were imposed in these early results: in [15] t.he system structure was restricted by extended matching conditions. in [190] the nonlinearities had to be linearly bounded, and in l157] the nonlinearities and the Lyapunov fUllction had to satisfy nonlinear growth conditions. Ghanadan
234
MODULAR DESIGN WITH PASSIVE IDENTIF!ERS
and Blankenslnp [41] obtained local results for 1tpproxi111ately feedback linearizable systems. The ISSelf formalism proposed in !(rstic and Kokotovic 1101] is suit~l.ble for modular adaptive design because it takes advantage of the fact that the system is aHine both in the control input and in the disturbance input. An ISSelf is a particular form of a robust conlrol Lya.punov function (rel!) introduced by Freeman and Kokotovic [39].
Chapter 6 Modular Design with Swapping Identifiers In the preceding chapt:er we initiated a modular approach in which the coutroller and t.he identifier are designed separately. The strong ISScontroller a.Jlowed the usc of any ident.ifier which can independently guarantee that the parameter error and its derivative are bounded. However, we IU:lVe not. y«:t employed the standard gradient and leastsquares updat.e laws. The leastsquares is of particular interest because it is considered to have better convcrgenee properties and is the only estimation algorithm that, guarant.ees t:hat the parameter estimat.es always converge 1.0 const.ant values. Instead, in Chapt.er 5 we retained simple passive identifiers similar to t,hose in Chapter 4, In this chapter we fully cxploit, the modular approach and develop adaptive schemes with standard gradient and leastsquares update laws. \Ve accomplish this by extending the wellknown swapping technique to nonlinear systems. V,Tith this technique we convert dynamic parametric models into static ones t.o which the standard update laws are applicable. For the swapping lllouular schemes we derive transient performance bounds and compare them with t,he bounds obt,ailled fOl" the tuning functions and passive designs. III addition to the strong ISScontrollers, we employ the SGcont.roller ti'OI11 Section 5.8,1 and design an adaptive scheme which achieves stabilization via a small gain property. \¥e further enrich the choice of modular controllers by a weaker ISScont.roller which expends less control effort for large signals.
6.1
ISSController
As in Chapters 4 and 5, our main focus is met7;c strictfeedba,ck form Xil
:e '
,...
ll
011
+ '('iI (XI • ••• ,:1:;1 )TO , + CPn(.'z;)'I'O,
=
Xi
=
f3(.l:)U
nonlinear systems in t,he i
= 1, ... ,11 
1
pa7YL
(6.1)
236
MODULAR DESIGN WITH SWAPPING IDENTIl"IERS
Table 6.1: ISSController
=1' O:j
=
(6.2)
,x.I _t,{il)  0" , I ,/r
 O· ,Yr (iI») = (Xi,
T ~
_
,,;1  Cj=i  IU'i (J
iI """'
+ L,
k=l
(a.......... j  I .
a..l"+l + Xk
a.......) .....il (Ii) (kl) Yr
BYr
(6.3) (6.4) (6.S) .j (i) Yr
= 1, ... ,11
= (11n.Yr, ••. ,1Jr(i»
Adaptive control low: lL
= _1_ [0; (x 0 y(fIl» + 1.'rJ(1I)] p{:,;) n'" r
(6.6)
where 6 E lRP is a vector of unknown constant panulletel'S, {3 and F :::: [
where A:a HE, and Q are matrixvalued functions of z, Cl 
1
A:(z, 0, t)
=
0
81
1 C2 
0 82
8 and t: 0
1
1
0
1
0
0
1 en 
Sn
6.2
SWAPPING AND STATIC PARAMETRIC
IvIoDELS
237
_Qru. iJfJ
o.
1.
(6.8)
11£:11_1 uo
Inputtostate properties achieved by this cont.roller were established in Lemma 5.8. As we shall see, t~lC swapping ident.ifiers will be able to independently guarantee that both /J and 8 are bounded. For this reasoH, in the swapping design only the property (5.86) is of intercst: (6.0)
where
6.2
Co
= m.in Ci, no = 1~I:S:n
1)
(
n 'E =i=l Iii
1 I
90 =
(,.
1)1
Li=l 9i
Swapping and Static Parametric Models
In Section 5.1 we considered the scalar s.ysLem (6.10)
and used the identifier \:(0)
=0
~(O) = jj
=
:t'{O)
(6.11) (6.12)
(6.13)
to illustrat;e the weakness of the cert.ainty equivalence approach. This ident.ifier was based on the swapping technique which is standard in adaptive linear control. In the swapping technique, the goal is to convert. a dynamic parametric model into a static form, so that standard parameter estimation algorithms can be used. To clarify this, let us consider the dynamic parametric model (6.10). Even though this model is linear in the parameter error 0, standard paramet.er est.imation algorithms are not applicable because .1: is not available for measurement. The swapping technique circumvents this obst.acle by employing the filters (6.11) and (6.12) which, as is easily verified, convert (6.10) into the stat.ic parametric model (6.14)
238
IHoDULAR DESIGN WITH SWAPPING IDENTIFIERS
The term 'swapping' is descriptive of the fact that we e..xchange the order of the transfer function in (6.10) and the timevarying parameter error O(l)! that is, we replace .1; from
s!1
.r. = _1_
8+1
[cp(x)TO]
(6.15)
by (6.16) 'Vit.h the help of t.he filt.er X, the signal f is available from the static e.'\.llressiol1 (6.14). The effect of adding the filters to compensate for the timev~u'yillg
1/' ~ xTiJ  ~, which
nat.ure of 0(1) is visible from
l/J
1 ['1':.] =X (J 8+1
is governed by
•
(G.17)
Details of the swapping technique and a nonlinear generalization of the swapping lemma [138] are given in AppendLx F. The swapping t.ec·hnique can also be interpreted by treating f = XT Bas t.he "predict:ion error. " Consider the signal Y ~ x  ~ governed by )J =
8!
1 [¥,(:z;)TO] = X"fJ.
(6.18)
Y
The sigmu ~ \.TiJ is I'efel'l'f.'d to as lprecliction' because t;he Unl\llOWI1 parameter (} is replaced by its estimate 0. Thus we obtain a st.atic expression (6.19) which is t.he srune as (6.14). The linear pa.ramet.ric: model (G.14) is suitable for any st.andard parameter update law including t.he normalized gradient (6.13) or t.he least squares:
iJ
=
t
=
( :l.':\:'I''' r 'X 8(. ) 1+XTr X .
r
X\," r. 1 + XTrX
(6.20) (6.21 )
In t.he next t.wo sections we present two swapping schE"mes! one using a =swappillg identHier derivC"d from the parametric =model (6.22) and t.he other using an .'l:swapping identifier derived from the parametric .1.'model j. = f(:r:, u) + F(.1:, 1I)'r () . (6.23) For each of the t.wo we can employ any standard update law: gradient. or leastsquares, normalized or un normalized.
6.3
239
zSWAPPING SCHEME
6.3
zSwapping Schelne
Let us consider the pal'Rmetric .:model (6.24) This model is ill the form (F.I) of the Nonlinear Swapping Lemma (Lemma F.I) wit.h g(;:, t) = 1(.:, I) = Ot 11(.:. t) = I", A(.:, t) = A,:(=, 0. t), and e(t) =:;; O. By directly I:tpplying Lemma F.I, with the filters
n'f
=
..4:;(=, 8, t)OT + IV(.:, 0, t)T ;
(6.25)
~
=
..4:(;;, il, t)1/J  OT9  Q(z, 8, 1.)T8,
{6.26}
we get the
~tlttic
lineal' pal'ametril' model (6.27)
where .:,.l/J, and goverllcd by
n are
Rl'ailable I:mel E ifi an exponentially decaying signal
l E ]R".
(6.28)
On the lefthand side of (6.27) we observe that the statc :: of the closedloop error system has becll augmented by ~J t·o get a static error ccluatiol1 linear in the parameter ~rror. To bring the augmented erl'01' f ~ .: + 1/J into tbe fOlm where the swapping terms explicitly appear, we introduce ~
•
~
TA
A
T~
no = A.:(Z, 0, t}no + H'(.:, OJ t) 0  Q(=,8, t) 8, and replace Tj; by
no  nTo, that is, we write the augmcnted error as f = Z + no  n'' o = nT 8 + E.
(6.29)
(6.30)
It is sometimes simpler to view the swapping idelltifier design as a method to generate a "pl'ediction error." Therefore, we give a prediction error interpretation of (G.30). First we note that J} ~ .: + no satisfies (6.31) which implies that (6.32) Hence~
Since error"
we ha,'e a st:atic 1110 del linear in 0, plus all e..'\:ponent.inIly decaying term.
n is available, we define t.he "prediction Y = nT 0 and the "prediction n
f
= Y  Y = n'rO + E.
(6.33)
240
MOOULAR DESIGN WITH SWAPPING IOENTIFIERS
We shall not dwell on the difference between the augmented error and the prediction error views of the swapping technique. We will adopt one or the other view depending all what is more appropriate for a particular paramet.ric model. Instead of either augmented or prediction errOl, f will be called
estimation.
en'OY',
Before we proceed to the selection of update laws, let us summarize the implemented filters and the estimation error:
o.T =
no
E
A,1
T
~
A:{ =,8, t)!l + H'(z, 8, t) A
::::
=
A
'J'~
(6.34) A
T:'
A.:(.:,8,t)rlo +H'(z,8,t) 8Q(=,8,t) 8 l'
~
.:+00 0 8.
(0.35) (6.36)
The choice of updatp laws is intimately connected with the fact that the estimation error (6.37) is linear in 6. This al10ws us to use either the gradient or the leastsquares up~ date laws. On the other hand, the enbanced exponential stability of A=(=, 8, t) allows the update laws to be either normalized or unnornuilized. We unify the notation by donating by r both the constant a.claptatioll gain matrix in f;he gradient algorithm, and the timevaryillg covariance ma.trix in the leastsquares algorithm, The update law for fJ is either the gradient: (6.38)
or the least squares:
iJ
= r
r
=
!le
1 + vtr{ flTfrl}
r
noT
1 + l.ltr{ flTrn}
r
(6.39)
,
where by allowing v = 0, we encompass unnormalized update laws. The complete zswapping identifier is shown in Figure 6.l. '\lith the regressor n being a matrix, our use of the Frobenius norm 10lF in the gradient update law and the "rweighted Frobenius norm" tr{!lTr!l} in the least.squares update law avoids unnecessary algebraic complications in the stability arguments that arise from applying the normalized gradient update law in tIle form = rn (In + v!lTn)l f or the normalized leastsquares with
6
r
the covariance matrix computed from = rOlIn +vnTrn)l nTr. It is even more important that this eliminates the need ior online matrix inversion.
6.3
241
z~SWAPPINQ SCHEMm
z =A:;=+ H/TO+QTD
no = A;:no + IVTe nT = ."l::nT +
Q'I'O

no
+ +
f.

~ ~
H:'I'
() L...

~
f
rn 1 + ,"In!}
f+
'
Figure 6.1: The :swapping identifier. Tbe followiug lemma establishes propert.ies of the ideutifier module which consist,s of tllf~ filt:ers (G.34}(6.35) and eit·her t1le gradient (6.38) or the:' leustSC111ares (G,39) update law. The propcrt,ies bold for botll llnllot'uulJized ('/ = 0) and normalized (11 > O) updnte htWfi. In contrast. t:o standard identificl's with u. n.normali~e,d npdate laws, which arp. able only to gllal'antee that jj iF; hounded ~nd e is sqnare integnlble. this ident.ifier also guarant,ees that cis boundeu and
ois bounded and sqmue integrable.
Lemma 6.1 Let the ma..7:imal 'interlml of exilltence 0/ sol'lJ,J.ioTl.S of (6.24). (6,34)(6.35) with either (6.38) or (6.99) be [0, t f following identifie1* p1"Operties hold:
(1) (ii) (iii)
}.
0 E £00(0. t,)
Then for
'I
~ O.
the
E £:2[0.1 f) n Loo(O, tf)
(GAO) (6...11 )
8 E L!![O, tf) n £00(0, t f} .
(6.42)
f
Proof. First we prove that the filter state n is uniformly bouuded on [0, If}, in'espectivcly of the bounderlness of u's input Hi'. Then we prove f.be stated properties separately for the gradient (6.38) and for the leastsquares (6.39) update 1aws. Along t.he solutions of (6.3£1) we 11ave
242
MODULAR DEsrCN WITH SWAPPlNG IDEN'flFlERS
and arrive at
d
rlt
(11l1F ,") :$cOOF+' I I') 1 2 41\0
(6.45)
n
This proves that is uuifol'mly bounded on [0, f J) even if TV is only in that is, even if lV is escaping to infinity as t + t J. In view of (6.28) and (6.8), which give
.cOOl! [0, tJ}'
(6A6)
it is clear that ~ E £2[0, tl) n .coo [0, tf). Gmdient updat,e law {6.38}. Let us consider the positive definite function 1 
li = 2 18 121'1
1
+ ?....co
,_,I) E •
(6.47)
Along (6.38) and (G.28) the derivative of l' is
\r
~ :=
< =
(6.48)
and hence (6,49)
6.3
243
ZSWAPPlNG SCHEME
The nonposit.ivity of ,; proves that. 8 E £00[0, t 1)' Recalling t.hat (. = nT6+ E. from the houndedncss oC n it. follows tha.t E e £00[0,1,/). This in turn shows
that
e= r 1 + ~I!ll} E £.0[0. t/).
Let
liS IlOW
prove the S
statements. Integrating (GA9) we get. 3 I I 12 E 4. Jo 1 + tI
,0," rlr :::; 
 f
j:
I
f VrlT:::; 11(0) .10
where we have used the fact that V(I) is
V(1):::; V(O} <
00,
(G.50)
nOllne~'1.tive. Thus
(6.51)
This, along with the boulldedness of 0, yields
10' IEI dT ::; 111 + vlnl}IL", 1.' 1+1~~n I} dT < 2
00 ,
( 6.52)
so E E £2[0 tJ). Finally 1
f' ~.,
rt
c7nT r!!ne
.., II
Jo IOIdT:S 10 (1 + l.l1111})2dT :s A(r)
,nl} II
1 + 1.lIOI}
ft
00
'E12
10 1 + l.1l n l} dT < 00, (6.53)
and we get
6 E £2[0, tf).
Leastsquares 'updlLte law (6. 39}. First we est,ablislt properties of tl1e coret). Since t :::s 0, then f(t) :::s reO), which implies t:lUlt r is
variance mntrL'\':
d . 1 n~ bounded. From dt = r= 1 + I.Itr{f)'rrn} ;;::: 0 ~1.11d the fad that r I (0) > 0, ,ve see that; r(t)l is positive defil1ite (or each t. Now we introd uce tbe fUl1ction (6.54)
1rr
(rl)
which is positive definit.c. It.s derivative along the solutions of (6.39) and (6.28) is 1;:S; _nTrlo+iir~ Iel~. (6.55)
(r1o)
Upon the
e.~aminatioll
of
!!.. (r1o) dt
=
= =
r1tr1o  r1(j nn"jj
Of
1 + vtr{nTrn}
1 + l.ltr{OTrn}
Oe
1 + IItr{n'l'rn}
(6.56)
244
MODULAR DESIGN WITH SWAPPING IDENTIFIERS
and its subst.itution in (6.55) we obtain
,:r
<
= <
f'l'nTo
eTnTB
1 + vt.r{rrrrO} 
1 + vtr{ fflTn}
Ifl2
1 + lJt.r{OTrO}
+
lil 2
1 + vt.r{nTrn}
Ifl2 1 + l.ltr{O'ITO} .
"
1(1_?
1f'1(6.57)
In view of the posit.ive dcfinitencss of r(t)I this proves that jj E .c1Xl [O,1 f ). It also proves that E E £,[0, t I). Now using the botmdedness of 1 + I.ltr{ rtf ro} rand 0, £ollo\ViI~g I:he same line of flrgulTIcnl. a.s for the gJ.'ndient update law, we prove that E, iJ E £2[0, ',) n £00[0 , 1f). 0
.j
The proofs for other swapping identifiers t.hroughout this chapt.er have similar steps and arguments and will therefore b(' given in much less detail. The fact tha.t all th~ £00 and £2 bounds that we have established on [0,1f} are independent of If will he used for e.xtending tf to infinit.y ill t.he stabilit.y proofs for the complete closedloop adaptive systems. As explained in [43}, varions modifications of the leastsquares algorithlllcovariance resf'tting, f'xponent.ial dat,a weighting, etc.do not. affect the propClties est.ablished by Lemma 6.1. A priori knowledge of parameter bounds can also be induded in t.he form of projection (see Appendix E). It is possible to est.ablish addit.ional properties of the leab"tsq~tares algorithm not stated ill Lemma 6.1: 0(1) converges to a constant., and {) E £1'
Remark 6.2 It is possible to note a very int.eresting connect.ioll between t.he .:passive identifier and t.he zswapping identifier. Denoting': = l/J, (6.2G) becomes
.::. = A:(.=, 0, 1)z + Q(z, B, t) T:'0 + n'1,:'B,
(6.58)
so that the est.ima.tion elTor is (a.59) Let us eompare (6.58) with t.he observer from the zpassive ident.ifier (5.111),
(6.(30)
nTo
The 'observer' ill t.he zswapping ident.ifier is augmented by the t.erm to achieve the static parametrization (6.59). It is important t.o underst.and that
6.3
245
zSWAPPING SCHEME
this para rnetrizatioll also leads to a strict: passhtit.y property t.hat is diffcrent from the strict passivit.y property for the operat.or (5.97) in thc zpassivE' scheme. Let llS C'onsirler t.hc operator ~::
with
f
defincd in (6.59) and.
fa'(fle)TOdr
=
1.'
OH Of,
n defined ill (6.25).
fl'flTOdr
= fa' ,T (f 
E)Jr
(a.Gl) In view of (6.59) we have
(I'  ~.r
= 1.'
~Ill')
~ ~ 10' Ii?J!!rlT. au t.he other hand. integrating (6A6)
dT
(6.G2) we get (6,63)
Substituting (B.G3) ill (6.62) we obtain
t (nf),"jjelT ~ ~lf(t)l~ .!CO
1D
_?1 li(0}1 2 + ~1 {' Ill:!dT _co ~ .10
I
(6.04)
whicb by DefinH:ion D.:! proves t.lutt thc operator (fUn) is stricl.Iy passivE'. This explains via. passivity why (6.38) and {6.39} are valid updat:c hlWH. The
{lTO
additional term ill ((j.58) is tIle price paid in the Rwapping apPI'oach to obtain not only passivity but. also ~t static paramctrization (6.59). 0 Lemma 6.1 sct:s the stage for the fonowing tl1eorem that charact:cl'izC5 st.ability properties of the closedloop adaptive syst.cm.
Theorem 6.3 (zSwapping ScheJne) All the signn.ls 1:11. the c!osetl.loop Q.(laptive sll.stem. consisting of the pl(lnt (6.1), cont7'Olle'l' (6.6). flUe7','; (6.34), (6.35), anti eithe7' '.he gradient (6.38) 01' the. lea8tsq'Uan~s (a.39) ·1I.pclate law are globally 1J.ni/07'1n./lI b01m.ded. and lim z(t) = lim c{t) = O. This mcan,t;, in 100 '00 particulm', tlw.t !llobczl as1Jm.ptotir. tmcking is achieved: (6.65)
lim [11(1)  1Jr(l)1 = O.
100
.Aforeover, if lim :V!i)(t) = 0, i = 0, ... ,111, and F(O} too
:=
0, then lim ;J~(I} = O. '~oc
Proof. Duo to the piece\Vise continuity of 11,(1), . .. ,y~n)(t} and the smoothness of the Ilonlinearities in (6.1), the solution of t.he closedloop adaptive system exists a.nd is unique. Let its nUl.."ICimum interval of existence be [0,1/). For b~t;h llormalized and unflonualized update laws, from Lemma 6.1 we obtain jjJ),e E £oo[O,iJ}. Therefore, by Lemma 5.8,
=,.1:
E
£oo[O,l,). In
246
MODULAR DESIGN WITH SWAPPING IDENTJJ"IERS
Lemma 6.1 we also proved that 0 E £00[0, if).
£ooro, 1,/).
Finally, by (6.36), 0 0 E
\Ve have thus shown that aU of the signals of t.he dosedloop adaptive system are bounded on [0, t,) by const.ants depending only on the initial COHditions, design gains l and the e:..t.ernru signrus IJr(t), ... , y~,,){t), but. not on t J. The independence of the bound of t f proves that t J = 00 (if t J were finite, then the solut.ion would escape any compact set as t ~ i" which would cont.radict. t.he exist.ence of a bound independent of t J). Hence, all signals are globally uniformly hounded on (0,00). Now we sct out. to pl'ove that. .:: E £2 and that =(t) ;. 0 as 1 00. Fa)' .bot.h 1'
normalized and unnol'lllalized update laws, from Lemma 6.1 we obtain
Ii, f
E
£'1.' Consequ«:'ntly OTjj E £2 because e E £2. ~'ith V' = ~ 1(1 2 , all the conditions of Lemmas P.] and FA are satisfied. Thus, by LClUu;a FA, .:  0'1'8 E £2' Hence Z E £2. To prove the cOllvergence of =t.o zero, we note tha.t. (6.7) implies that = E £00' Therefore, by I3arbalat's lemma (Corollary A.7) z(t) + 0 as I + 00. Fl'om the definitions in (6.2)(6.4) we conclude t.hat if lim lI~i)(t) = 0, i =
0, .. ,.71

1, and F(O)
= 0, then x(t) ~ 0 as t ~ 00.
100
0
The standard parameter estima/:ors cannot guarantee that iJ is bounded or squareintegrable unless they use normalized update laws. For this reaSOH, normalization is common in traditional adaptive linear contro1. l It is, thel'~ fore, significant t.hat Tllcol'em 6.3 holds also for unnormalized update laws. The proof of Lemma 6.1 reveals that normalization is not necessary because the nonlinear damping 1 which is built. into t.he errol' fiyst.em (6.7L guarantees that the filt.er st:ate n i~ bounded even if it.s input., I"he regressor l'Y, is growil1g unbounded. Thprefol'e fJ is boundf"d, which means that nonline81' damping :,.Is as some form of nOl'malizat:loll. Both the updat.e law normalization and r.I,.nonlinear damping act to slow down the adaptation. The slow ~n~pt.ation is a basic ingredient in the modular approach to adap~ive control. Thf' controller module tolerates the presence of the disturbance but: requires t.hat this disturbanee be bounded.
Controller with OJ
8 in
t.he error syst.em (a.7),
=0
In the swapping a.pproach we can eliminate the nonlinear dampillg t.erms ('OUI1teracting 0, tha.t is, we can set. 9i = 0 in (6.5) without dest.roying the stabilit.y result of ~heorell1 6.3. To achi.eve t.his, we need an ident.iIier which guar811tees not. only {} E £00' but also QT{) E £00' .
1 An
exception is the unnormwizcd lenstsqunres updaLc law which can gunrant.ee that
iJ E L.I (157. Lemma (264)).
6.3
247
zSWAPP1NO SCHEME
Let us first discuss the construction of such all identifier. It uses tbe sallie filters (6.34) and (6.35) but with some modifications in the npdate laws. The gnuiient update law (S.3B) is modified itS
:. fJ
r
!1E
,/ >0
= 1 + ''''\QIF 1 + '''Inl}:
(6.66)
so tbat the following ineqnaJity is readily established
(6.67) This illequality shows tbat 0 E £00' Since t.lw boundec1ness of n t'stablished by (6.45) is unaffected, then by (6.37), c E £oc. Now Ule boundedllcss of n and c, along Wit.Jl (G.6S) proves that QTiJ E £00' As ing (6.G7) we get
.j
rOf
e
the squareintegrability properties, by integratE £2. Because of the boundeclness
(1 + ''''/QI.'F)(l + Illnl})
of n, this implies that
J' E £.". FiIDIlly, 1 + 1J'lQIF
161 < rn  J{l + 1,"IQIF){1 + '''Inl}) wher~
f
Jl + vlQI.'F '
(6.69)
the first factor is bounded and the second is squareintegrable, proves
that iJ E £'2' Now, let us consider a modified leastsquares updtl.te law: A
O
r Of 1 + v'IQI.'F 1 + lItr{flTrfl} ,
=
t =
1
1 + ''''IQIF
r
nrzT
,
'" > 0
r
(6.70)
1 + vtr{fl'rrn} .
Like in the case of the original leastsquares update law (6.39), we readily derive
~ (1IiIfe'l' + 2~ 'f12) $
(1+
v'IQI.1")(~·~ /ltr{OTrn}) .
{6.71}
Follmving similar lines of argument as for the modIfied gradient update law, we 
::..
establish that 8, E, QTfJ are bounded and
vi
t:"
1 + ''''IQIJ''
,6 are square~integrable.
248
MODULAR DESIGN Wl'rH SWAPPING IDENTIFIERS
To summarize, for both the modified gradient (G.G6) and t.he modified least:squares (6.70) update laws. fol' y ~ 0 and y' > Ot t.he following properties hold:
Ii E LCCI[O,tI)
U) (ii)
f
E LCCI[O, if), T'~
(iii)
Q IJ
E
(B.7!:!)
J1 + v'IQI.:F E L2[O, if ) f
Loe[O, tIL
:.
(1
E '£:2(0, t/)'
(6.73)
(6. 7~1)
It is importl:tlll. t.o note that. t.he bounclerlness of Q'I'8 ean be achieved only witll swapping idclltHiers. Passhre identifiers do 110t: seem to be abl!? t,o guarantee this property independently oC the hounded ness of Q. Since we have S!?t gi = Ot Lemma 5.8 is no longer applicahle. However, ill a fashion analogous to the proof of Lemma 5.8, we arrive ttf. (6.75) which, in view of tho hOulldculless properties (6.72) and (6.71), means that The boundedness of ot.her signals follows rea.dily. and the convergeucc of :(t) to zero is esta.blished with the sqmucintegrability properties (6. 73) and (6.74). \Ve have thus proven the following result.
=and .c are bounded.
Theoreln 6.4 All the si.qnals in tile. closedloOl) a.tluptive sllslem consisting of the plant (6.1), c(Jntroller (6.a) with 9i = 0, i = 1, ... ,11, filters (6.34). (6.35), and either the modified gradient (6.66') 01' the modified lea,t;tsquo.7'p.8 (6.70) u.pdate la1l1, are !Ilobally lI.71ifoT7n.ly bounded, and IIE~ =(1) = 0.
6.4 Let
xSwapping Schelne
llS
consider i.he parametric :l'rnode]
.r =
1(.1:, 1/.)
+ F(x, U)T() ,
(6.76)
which encompasses tlle class of parametricstrict feedba.ck plants (6.1) with
J.'"
/(x,1J.) =
: [
1
(6.77)
'
:In
.ao(x)u
Vic introduce two filters
nu orr
= A(x, t)(no  11:)  .f(:1:, u) = A (;'C, t)nT + F('L; u)T 1
t
(6.78) (6.79)
6.4
240
xSWAPPING SCHEME
where A(x, t) is exponentia11y stable for each :1: continuous in t. Combining (6.76) and (6.78), we define Y = x + no so that.
Y=
A(:r, t)Y + F(x, u)T().
(G.SO)
Since (J is constant, it follows that
Y where l ~ x
= n'J'() + f,
+ flo  nTo is exponentially decaying because it l
(6.81)
is govcrned by
e E IR".
= A(x, t)l,
(6.82)
Now we introduce the "prediction" of Y as (6.83)
The "prediction enoe'
f
~ Y  jJ =
.1:
+ nn  nTo can be written in the form
Like the zswap~ing identifier~ the .1:swapping !dentifier should guarantee boundedness of 8, l, 0, and squareintegrability of E, {} witIl both normalized and un normalized update laws. However, if A(:l:, t) is constant and Hurwitz, the above properties would be achieved only by normalized update laws, whereas the unnorrnalized update laws would just guarantee that 6 is bounded and € is squareintegrable [165]. To enforcc the rest of the properties with unnol"malized update laws, we need to guarantee the bounrledness of fl even when F(x, v) is growing unbounded. "Ve do so by strengthening the stability of A(x, t), namely by choosing A(x, t) = Ao  )'F(x, u)T F(x,u)P
(6.85)
where). > 0 and Ao is an arbitrary constant matrix such that 'r PAo+AoP= I,
p=pT>o.
(6.86)
With tllis choice the filt.er and estimation error equations are
nT
=
no = f
=
(An  >..F(:c\ 'U.)T F(x, u)p) nT + F(x, u)T
(6.87)
(Ao  >..F(:L·, u)T F(x, u.)p) (flo  :z:)  f(:L"~ u)
(6.88)
X
+ no 
n1'{J .
(6.89)
The update law for iJ is either the gradient:
(6.90)
250
l\10DULAR DESIGN WITH SWAPPING IDENTIFIERS
x = J+F're
X
no = (Ao  AFTF p) (no  3:) + f
no
+ +
f

~
n= (An  AFT F p) n + F
~
r
iJ .f rL 1 +rnviOl} 
f4

Figure 6.2: Tbe xswapping identifier.
or the least squares:
jj =
r
Of
1 + vtr{nTrn}
(G.9l)
noT
t = r 1 + vtr{n'l'rn} r
t
reO) = r(o)T > 0 t
11
~
o.
Again, by aUowing II = 0, we encompass unuormalized gradient: and lenstsquares. The complete a:swapping identifier is shown in Figure 6.2. Lemma 6.5 Let the maxim.al intenlal of c:Lislence oj solutions of (6.76)! {6.87)(6.88} with cithc7' (6.90) OT (6.91) be (0, tJ). Then f01* lJ ~ D. the following identifie7' pmpe'rlies hold: (i) (ii)
Uri)
6 e £00[0, tJ)
(6.92)
E £2[0, LJ) n .coo [0, t,}
(6.93)
iJ e £1[0, tJ} n £00[0, tj) .
(6.94)
f
Proof (Outline). Along the solutions of (6.81) we have
c1t
(npn")
=
n(pAo+A~P)OT 2AOPpTFPOT +nPF1t + FPOT
=
noT  2A ( F PO T 
1 II' )T ( F pnT 2.,\

1 IS) ) 2,,\
1 II) • + 2,\
{6.95}
6.4
251
XSWAPPING SCHEME
Using the Frobenius norm we obtain
(6.96) In view of the fact. that d(P)IO,} $ tr {flPO'l'}, (6.96) proves t·hat 0 E £00[0, t J). From (6.82) and (G.85) it. follows that
~ < dt (,,2) f p _ _112 E,
(6.97)
which implies f E £2[0, tf) n t:.~[O, if). Gmclienl, update la:w (6.90). ''''e consider the posit.ive definitc fune'Hon \f =
" + '_f1'1P "21,,' {} r
(6.98)
I
whose derivative is readily shoWIl to satisfy
3 IEI2 41 + 1)101} .
(6.99)
The nonposith'ity of li proves that 9 E £oc[O, tf). Due to f = nTo + land the boundedness or n it fo]]ows that. f E £~ [0, f f) I which, in tnrn proves that :. Of f B= r 1 Inl:! E £00[0, t f)· Integrating (6.99) wc get E £2[0, t'I)'
+V
H
J1 + I)IOI}.
:F
Since n is hounded, then
f
E £2[0,1 f). The boundedness of n and the squa.re
integrability of f prove that Leastsl]1J.U7"eS
8= r 1+11fllfI!'j: E c,,[0, t f).
update law (6.91). \Vc consider the function (6.100)
which is posit.ivc definit.e becanse f(t)J is positive definite for cal'h I. After routine calculations we get. (6.101)
1 + vtr{OTrO} , which, due to the positive definiteness of r(1)I, proves that Integration of ineqnality (6.101) yields

J1 + vtr{OTrn} f
E
8E
C~(O, f f).
.c, [0, tf).
Now
using t.he boundedness of rand 0, followipg the same line of argument as for the gntdient update law, we prove that
f,O E £2(0, tf) n £00[0, tf).
0
252
l\!fODULAR DESIGN WITH SWAPPING IDENTIFIERS
Our xswapping identifier, unlike the standard lill~fl.r parameter estimators, guar811j,ees boundedness and squareintegrability of iJ even with ul1Ilormalized update laws. This is achieved by including nonlinear damping into filters (6.87)(6.88) to slow down t,he adapt.ation. Remark 6.6 As we noted in Remark 6.2 for the =swapping scheme, there is R. passivity interpretation of the xswapping idelltifier. The signal x = + OT8 is driven by the 'observer'
no
j. =
(A.n 
;\F{3:, 'lt)T F(x, u)p) (x
~ :1') +1(.1:, '11) + F(x, 'u)T8 + 0'1'0
l
(6.102)
which differs fr<;)fll t:he observer (5.129) in the xpassive identifier in the additional term o.TiJ. The filter employed ill t.his term is the cost of maldng t.he operator .ET : 81+ nf stl'ict.ly pflSsive aud achievil1g tho stat:ic parametrization (6.84). 0 Lemma 6.5 Rets the stage for the ronowing result.
Theorem 6.7 (:vSwapping Scheme) All the signal.q in th.e closedlDop adal1ti'lJe .'lystem, consi.sting of tlte plant (6.1), controller (6.6), }iltef's (6.87), (6.88), and eith.er the gm.dient (6.90) or the leastsquares (6.91) update la11J are globa1l11 unifm1nly bOftnded, Q.ntl lim =(f) = lint c(t) = O. This means, in too too pmticula.7·, that !Jlobal asl1mptotic trackin!l is achieved: lim [y(t)  Yr(t)] =
too
A1oreove,.,
if 100 lim y~j)(t)
o.
(6.103)
= 0, i = 0, ... , nl, and F{O) = 0, then. too lim x(i) = O.
Proof. As ill t.he proof of Theorem 6.3, we show that Of Of =, x E £00[0, tf) and hence U E £00[0, I J). From (6.87) and (6.88) it follows that nOt n, and therefore ( are in £00 [0, t J ), By the same argument as in the proof of Theorem 6.3 we conclude that. l f == 00. Now we set out to prove t,hat z E £'J,. From Lemma 6.5, we have {}, f E £2. COllS8
=
X'I'e E
£21
where
(6.104) In order to show tbat = E £2! we need to prove that o.To E £'1 implies X?n E L')" or, in the notation of Lemma F.7, that T(.tlo,\FTFP)[J"T]6 E £'1 implies 7:.t=fHi'f]O E £2' To apply this lemma to our adaptive system we note from
6.4
253
xSWAPPING SCHEME
(6.B) and (6.4) tbat
H,T(=, 6, t) =
1
0
_l!ru.
1
DZI
IJn,,_1 
0 'I'
~
6.
T
F (:r;) = lV (.::, (}, 1) F (.r). _00,,_1
CJ:l't
8.1'nl
0 1
(6.105) Since F(.J:(t» is continuolls and bounded, and N(.::(t), 8(t), /.) is hounded and hl:l8 a bounded derivative (due to t.he smoothness of the components of F(.!'», they satisfy the conditions of Lemma F. 7. Then x? ii E £'}, and hence .: E £.J,. The rest of the proof is the stune as for Theorem 6.3. D ~ 0 which avoids Lemma F.7 is as follows. Comhiuing (6.87), (6.88), and (6.89), we get
Remark 6.8 All alternat:ive proof of =(1)
(G.I06)
Because of the boundedness of all the signals, E is bounded. Since f E £2. by Barbalat's lemma (Corollary A.7), f(t) ~ O. By virtue of the smoothness of F (which implies t,hat its partial derivatives arc bounded for bounded values of their arguments) and boundcdness of all the states, f is also bounded, and therefore if. is uniformly ('ontinuol1s. Since f(t) ~ 0, then
lim
100
10t
e(T)dT =
liIH
100
f(t)  f{O)
= e(O) < 00.
Then by Barbalat's lemma (Lemma A.6), f(t) ~ O. Since
(6.107)
8 E £00 n £2
and
8E £00' it follows that 8(t) ~ O. Fl:om (6.106) we conclude f(.l:(t»,I'O(l) ~ 0, and therefore, ~y (6.105), It'{.::(t), OCt), t)T8(t) :+ o. Since OCt) t 0, we have Q(=(t), 0(1), t)TO(t) ~ o. (Note that while Q'l'iJ is in £2 H,'1'(j may not be.) Thus t.he input. H,T 6+ QT iJ to the error system (6.108)
converges to zero. By the same argument as in (5.156)_ fol1owcd b)' tlle application of Lemma B.B, we arrive ~tt the conclusion that z(t) ~ O. 0
Remark 6.9 As in the =swappillg identifier, we can modi(y tbe gradient,
:. r nf (} = 1 + 11'IQI.1' 1 + I/lnl} ,
,./ > 0
(6.100)
254
MODULAR DESIGN WITH SWAPPING IDENTIFIERS
and t.he leastsquares
{} =
r Of 1 + 71'IQIF 1 + vt.r{OTrn} , 1
t =
r
noT
1 + 7.1IQI.:F 1 + llt.r{OTro} update laws and obtain t.he following properties
v' > 0
r
(6.1]0)
(i)
(6.111)
(ii)
(6.112)
(iii)
(6.113)
This allows us to set gi = 0 in the controller ill Table 6.1 as we did in Theorem 6.4 with the :;swapping identifier. 0
6.5
Transient Performance
In this section we will derive transient performance bounds for the error state z. Since =1 = Y lIn these bounds also bound the tracking error .11  Yr. ''''Ie analyze first the zswapping scheme and then the :l:swapping scheme. To simplify the derivations without loss of generality, we assume that 0(0) = 0, os well as 0 0 (0) = =(0) in the zpassive scheme, and 0 0 (0) = J:(O) in the :rpassive scheme. Tllese filt.er initializations guarantee tlutt l(O) = 0 and should always be performed t.o eliminat.e the disturbing effect of the initial estimat.ion error. Vve will derive performullce bounds only for scbemes using gradient update law. For simplicity we let r = '"/1. \Ve will then briefly explain how the bounds are modified for the least.squares update law. 'Ve first give a lemma which establishes performance properties of the zswapping ident.ifier with the gradient update law.
Lemma 6.10 (zSwapping Identifier with Gradient Update) The identifier (6.34), (6.35), (6.38) guarantees
(1) (ii) (iii)
110110:1 = 16(0))
1II0IFiloo ~
':.
(6.1Vl)
1
{II} IlJ(O)/ 'l min {!, 1}16(O)/ 2 v 4Coli.o
IIlJlI lXI ~ ,'min , 4IJ
(iv)
(6.11S)
?..jCOiiO COIi(J
...
110112
~
lIelb ~ ~2 (1 +  II. 4) v &.1
(0.116)
Co"~o
Colio
1/2
18(0)1.
(6.117) (6.118)
6.5
TRANSIENT PERFORMANCE
255
= 0, we have £(t) == 0, so (6.37) becomes c = nTo.
Proof. Since £(0)
(6.119)
Tbus t (6.47) and (6.49) imply (6.114). From (GAS), by Lemma C.5 we conclude (6.115). '¥ith (6.38) we writ.e
~
'I
'l
f"nTOf
If/2Inl}
?
191 = '1 (1+11 lOr''j: r ~ 1' ( 1+1/{Or7F r <
'12 II
Iff
1 +'1lor' 'j:
(6.120)
By llsing (6.119) we get
:. ".p 18\2 < 
II
oTno:ro 1 + vlnl}
<

"(2 101210/2:F II
1 + '1In\}
(1)2IBf
< 
 v
'
(6.121)
which, in view of (6.114). proves that (6.122) By noting that 191 = ; Il+~nl}1 ~ 710fl, witb (6.119) we get 161 $ which in view of (6.115) and (6.114) yields
iIO\}IBI, (6.123)
Combining (6.122) and {6.123}, we ru. rive at (6.116). Since leO) = 0, the first row of (6. iB) gives L
.
Jff
V$
1 +vlnl}'
(6.124)
Integrating (G.124) over [0, (0) we obtain
I
J
•
1+
,
vlfllJ:"
2
~ JV(O) = v!:c18(O)!. 2,
(6.125)
Tlle integratioll of {G. 120) over [0,00) and substitution of (6.125) yields (6.126)
In view of (6.38) we write
(6.127)
256
l'\'lODULAR. DESIGN WITH S\VAPPING IDENTIFIERS
Illtegr
rr
~ _ 10(0)1· V2. <J. . JCOh.O
(6.128)
Combining (6.126) and (6.128) we get (0.117). A bound on the L'2 norm of f is obtained llsing
2
')
(1 I I/IIIOIFII~J
<
J1 I viOl):" f
(6.129)
'l
'2
o
By substituting (0.125) and (6.115) into (6.129) we prove (6.118).
Lemma 6.10 is now used to characterize t.he transient performance of the closedloop adaptive systom.
Theorelll 6.11 (zSwappillg, Gradient Update) In the (.ulapti've system (6.1). (6'.6), (6. ,eLf). (6.35). (6.38), the following 'inequah/:ies hold:
(i) (ii)
10 (0) I ( 1 1~(l)1 ::;   
2JCO
Ilzll')
<
10(0)1
  v'2Co I
hO
. {I + A/! n11n 0' 90
1
}) 1/2 'l
v ( 4COhO)
[_1)_ + ~I' (2. + _1_) min {~ 2/{·o7 go 2c5h~o
10(0)1 _1 1(0)1 h I r;;:. v~/
y Co
1/'
.
I 1::(O)le COI (6.130)
_1_}]1/2 4coli::o
(6.131)
Proof. By direct substitution of (6.114) and (6.116) into (G.9) we obtain (6.130). \Ve nmv set out to derive (6.131). \Ve will calculate the £.2 norIll bound for z as IIzl12 ::; IIfl12 I II'ljJlb where (6.132)
A bound on Ilflb is given by (6.118). To obtain a bound on 1101b, we examine (6.133)
By using (6.8) and repeating the completion of squares with gterrns as in (5.88), we derive d
(1 'l) I'~'I
dt 2
<
(6.134 )
6.5
257
TRANSIENT PERFOR!>.lANCE
which wit;h (6.115) gives ( 6.135)
By applying Lenuna B.5.(ii} to (G.135). we arrive at (6.136) Since we lIse initial conditiolIs 0(0) inequali ty (6.136) bccomcs
II'~'II~
1[
=
0, 0 0 (0)
=
.:(0), which irnply 1/'(0)
( 1 1)
:.. j ::; Fo 1.:(0)1 + ,', + , ",', I/~ IltJll~ 2.rJ() ~lc(iftf)
= 0:
(6.1:37)
By sllhstii".nting (6.117) into (6.131) we gd,
11'!'112::;
~ [1':(0)1 + (~, + .~,) 1/'2 ~1. mill {~. ]}lfJ(O)lj. 2go :lcOh~()
V Co
II
c!cOIl O
(6.1;38) Combining this Ivith (6.118): and rearranging the terms. Ive obtain (6.131.). 0 \\Fl1en we perform the standard initialization .:(0) = 0 int,rociu('ed in Section 4.3.2, the hounds (6.130) and (6.131) become
1 (1 + t)
I.: (t) I < .lJco/1~o 11:lh ::;
{
1Gqjl{,ogo
1/'2
18 (0)1
(6.139)
1)1 +fi
1 [/) "'/ (1 .2JC()h~(),' ++",2co go 2cij/{o
I/,'
1
}
IO(O}I·
(6.140)
Thesc bounds depcnd only on the design parameters el, /'i, gil ~J', and /), and the initial parameter error 0(0). By increasing Co and h~O we can make the L= bOllnd (6.139) arbitrarily small. Thesf' gains alone are not enough to arhit.rarily reduce the £.1 bound because of the last. term in (6.140). However, this bound can be s.vstematicall~y reduced, for example, by increasing "y simultaneously with Cn or hO(). As we shall see, the increasing dependence of the bounds on the adaptation gain "')' is not corroborated by simulations, which indicaj,es tital, the dependence 011 " may not be very t.ight. Let us now compare the bounds for the .:swapping scheme with the hounds for t.he tuning functions scheme and t.he :passive schelIle. Examining the Crx; bounds (';1.2.:11), (5.178) and (6.139), we note that, the bounds for the modular schemes (.:passive and zswapping) arc higher than the bounds for the tuning functions schemc. The additional ~r'dependent t.erms
258
A100ULAR DESIGN WITH SWAPPING IOENTIFIERS
in the modular schemes occur due to the fact that Bis not eliminated from the .:syst£lm, but only "damped." Because of these terms~ the r~sult from Corollary 4.21 does not hold for the modular schemes. Thus, we arrive at an important conclusion: The capability to outperform the nonadaptive design is a unique [eatUl'e of the tuning functions adttptive design. We will illust.rate tbis point in Example 6.15. Examining the £2 bounds (4.229), (5.177), and (6.140), we again see that, the bound for the t.uning functions approach is lower. In Examples 6.14 and 6.15 we illustl'ate the performance bounds with simulatiol1s.
Remark 6.12 For v = 0 the bounds from Lemma 6.10 t as well as those from Theorem 6.11, are readily modified to hold for the leastsquares update law (6.39). While the bOUllds (6.114), (6,115), and (6.116) remain the same, the bounds (6.117) and (6.118) increase by a factor of '\12. <> Finally we give all £G:) bound for the xswapping scheme, for a special choice of design parameters.
Theorem 6.13 (mSwapping, Gradient Update) In the adaptive syste111, (6.1), {6.6}, (6.87), (6.88), (6.90) with the choice.' AD the following inequality holds: Iz(t)l
1 '). {II}) r
( ~ ?\8(0)1 r;;:  + !.. min "'v Co
liD
go
;t (4
v
Co/;;o 
1/2
= co,
P
= I, A = pn.o,
+ Iz(O) Iecol .
(6.141)
Proof. With this choice of design parameters, we have
n=
(6.142)
8
(6.143)
coO  pK.OFT FO, + FT Of = 1'1 + "Inl} .
From (6.99), we obtain the same bound on 11"lIcc as in (6.114) for the zswapping scheme, In view of (6.96), we obtain the same bound on n10lFil00 as in (6.115) for tIle zswapping scheme. By repea,ting the third part of the proof of Lemma 6.10, we get the same bound 011 IIOIIG:) as ill (6.116). It then follows that the resulting bound (6.141) is the same as (6.130). 0 Since the bounds (6.141) and (6.130) are the same, we conclude that the zswapping Rnd the xswapping schemes are equally capable of reducing transient peaks. However, an £'}, bound similar to (6.131) is not avai lable for the xswapping scheme. 2Notc l,baL this choice oC Ao and P gives PAn+A1 P = 2coI, which differs from (6.86). The proof Lakes this int.o account.
6.5
TRANSIENT PERFORMANCE
6.S.1
259
Simulation examples
Vve now present simulation results for the swapping and the tuning functions schemes. They will illustrate the derived performance properties alld serve for a qualitative comparison between the tuning functions and the modular approach. The first example illustrat.es the performance bounds established in Theorem G.11.
Example 6.14 (Performance of the Swapping Design) VYe rOIlsider t,he secondorder plant
Xl =
X2
=
u.
:1;2
+ rp(Xl }8,
(6.144)
The ISScontroller for t,his system is
It results ill the closedloop system
(6.ViG)
The zswapping identifier uses the filters
The gradient update law is (6.149)
260
l\iIoDULAR DI]SION WITH SWAPPINO IDENTIFIERS
t?
ir
O
o
I
i
i
2
3 110
o
o
0.05
o
3
2
0.1
adaptat.ion
0
0.15
a) l' = 10
b) 1
1; Co = 1
1: 2: 3:
2: co=5 3: Co = 15 Figure 6.3: Dependellce of the transients expanded time scale for control tI..)
011 (."(}
where t.he estimation error is implemented
0.1
0.05
0.15
0 Co Co Co
=1 =5
= 15
\vitb nil = 00 = 1. (Note an
a.c;
TA
E==+{10!l O.
(6.150)
Simulations were carried out wit.h Ilomimu vtllues Cl = C2 = Co = I\.J = 1\0 = !J2 = 90 1, l' = 10, v = 1, (} = 5, and 9(0) = 0, which were judged (:0 give representat,ive reAponses. An simulations Rl'e with following initial conditiol1s: x(O) 0 0 (0) = [0, lOrI', ncO) = 0 (to set e(O) = 0). Figure 6.3n illustrates Theorem 6.11. The design parameter Co can be used for systematically improving the transient performance. Up to a certain point the error (,l'allsiellts llnd the control effort in Fig. 6.3a arc simultaneously decreasing as Cn increases. Beyond that point. the control effort starts increasing. The ('ont,rol u is given in an expanded time scale ill order to clearly display the main qualitAtive differences among the three cases. Figure G.3b illustrat:es Corollary 5.9. When adaptation is switched off: the st.ates arc uniformly bounded and converge to (or remain inside) a compact residual set. Corollary 5.9 does not describe the behavior inside t.be residual set, which may contain mult.iple equilibria, limit cycles, and so on. For this example (but not in general), there is an asymptotically stable equilibrium at
"2 =
=
=
6.5
261
TRANSIENT PERFORMANCE
o
2
3
0
2
3
1 : no=·2 2 : no = 1
3:
h~O
=5
Figure 6.4: Dependence of the transients on
RO
with
Co
= 00 = 1• ., = 10.
the origin for any vaJue of t.he parameter errol'. The origin is a."iymptotk·a.lly stable because CP(XI) = .1:i is quadrat.ic (around the ol'igiu)t hut it is not globally stable for nonzero values of the parameter errOl". For h:trgP valncs of the parameter en'or, this eqnilibrium has a region of attraction whit'll is strictly inside the residual set, as shown in Figure 63b, curve 1. For Sllutll Wtllles of the parameter error, according to Theorem 4.15, global asympt.oUc st~thility is achieved, Figure GA shows the influence of lio 011 transieIlts. A["["ording to Theorom 6.11, the peale values Cl:l11 he dccreased by increasing h'0 1 w]lich is ('011firmed by the plot. An important observation is that the h'terms slow down the adaptation and make the transients longer. The effect: of the gterms was shown to be significant only for very small Co and IiO or for very largt:" "(_ Figure 6.5 demonstrates the inBuence of the adaptation gain '"'I 011 transient.s. Due to the slow initial adaptation, which should be attribut.ed not only to the normalized gradient updat:e law but also to the fnet that the regressor is filtered, there is I:t clear separation of action bet.ween the nonadaptive controller, which at the begiuuing brings the stat.e :; quickly to t.he residual set, and the adaptive controller, which takes over to drive the stat.e to the origin. The property that the L.c:o bounds are increasing functions of 1', to be expected from Theorem 6.11, W8...'i e..xhibited in simulations only witb ext.remely l1igh Vttilles of "/. This indicat.es that some of the bounds derivpd are not very tight over the entire range of design parameter "alues. In our simulat,ions we used the plant initial condition :1'(0) = [0, 10]T and hence =(0) = [0, lOP', whi{'h is independent of the design gains Co, lio, go ThiA is why the peal\. of decreases monotonically as any of these gains increases.
=,
262
:MODULAR DESIGN WITH SWAPPING IDENTIFIERS
o
2
3
o
2
3
1: ')'= 0 2: 1'=1 3: l' =3 4: l' = 10 Figure 6.5: Dependence of the transients on 'Y with
Co
= !JO =
11.0
= 1
If, instead, we used .1:1(0) =f. 0, then, according to Section 4.3.2, we would have added an appropriately initialized reference traject.ory to set· z(O) = O. In this way, bad t.ransients would be eliminated by following a less aggressive path to the origin. In the plant (6.144) the unknown parameter appears only in the first equation. This opens a possihility for a reduction in the dynamic order of the identifier. Instead of t.he zswappillg identifier we can use an :rs'llJapping. identifier
n no
= (c + I\:tp2)O + l{J, = (c + Il:tp:!)(Oo  .l:1) + X2, Of
0
=
')' 1 + vO:!
f
=
''&1 
00

ne1R.
(6.151)
no Ern.
(6.152) (6.153)
00,
(6.154)
which is uncertaintyspecific and avoids unnecessary filtering. In simulations, the only difference between the zswapping and t.he .1:swapping approach was in the value of')' needed to achieve the same speed of adaptationhigher value was needed in the zswapping case. Since the responses were similar we show them only for the zswapping scheme. Simulations were also carried out for the zpassive scheme and were qualitatively the same as for the zswapping scheme and are therefore omitted.
o
6.5 =1
t~
2
t:s o
3
2
0
n
263
TRANSIENT PERFORMANCE
~~
1
2
3
2
3
~rtt
?
~
T
a
3
2
0
11} Tuning FUllctions
a) Swapping
1: 2: 3: 4:
,=0
1: 1'= 0 2: '1 =.1 3: 1'=.3 4: l' = 1
1'= 1
l' = 3 1'= 10
Figure 6.6: Comparison between the swapping and the tuning fl.lnctions designs Co = Yo = 11:0 = 1
with
The next example gives a sill1ubttioll comparison bet;ween the =swapping scheme and the tuning functions scheme.
Example 6.15 (Swapping VB. Tuning Functions) The tuning functions controller designed for system (6.144) is a:]
u
=
CJ=l  Klep'J=t 
= .:;1  c.,z" M
W
/il) 
8ep
2 (!!e.L)2 Bx! .,... ...,.l + ~8 + !l!u.(X., + 8/1'1) In
{}(J
DZI

.,...
(6.155) I
and the parameter estimator is
(6.15G) The resulting error system is
z=
[
Cl =:'l~
OJ _
"21(~)\". ] z+ [ f.cp ]Ii.
(6.157)
Simulations were carried mit for the tuning fUllctions scheme using the same desigl1 parametel's and initial conditions as for the zswapping scheme in Example 6.14. A comparison is given in Figure 6.6. The initial adaptation with the tuning functions scheme is ra.pid because it has no filters or normalization
264
MODULAR DESIGN WITH SWAPPING IDENTIFIERS
o
0.5
o
O.S
o
1.5
o
O.S
1.5
lit:!,: 0.1
1.5
0.3
1: co=l 2: co=2
o
0.5
1.5
O.S
1: Co =3 2: Co =5
Figure 6.7: Transients with tuning functions approach. Dcp(mdence of the transients on C1 = C2 = Co for "y = 1. (Note fI. different time scale for control u.)
to slow it. down, as the swapping ident.ifier does. Figure 6.6 corroborates the difference between the £00 bounds (4.241) and (6.139). The increasing of the adaptation gain 'Y reduces t.he £'01:) bound in the the tuning functions design but; not in the swapping design. This confirms our earlicr conclusion that the tuning functions design is unique in its capability to improve the performance of the underlying nonadapt.ive design. The importance of fast adaptation in the tuning functions design is even more pronounced when wc set the nonlinear damping terms to zero, /i.o = O. ShlluJation result,s that illustrate this point are shown in Fig. 6.7. The rapid changes of Bwhen the values of CI = C2 = Co are small reflect the fact that a fast identifier is indispensable for catching up with fast nonlinear inst.ability. The ident.ifier is as fast as necessary to lwcp the derivative of the complete Lyapunov function 112 = !(zf + =~ + (J2 l'"Y) Ilonpositive. It is important to understand that the respOlises in Figure 6.7 are neither the bost nor very typica1. They only show that when aU other means of achieving stability and performance are absent (high gain, for inst.ance), the fast adaptation, an inherent prot of the tuning functions design, comes to rescue the syst.em from instability. 'Vhile nonlinear damping is not necessary for stabilization in the tuning fUllctions design, it is helpful for transient peri'orm811ce improvement. However, it is important to keep in mind that the drawback of nonlinear damping is the rapid growth of the controller nonlinearity. For e.xample, while the tun
6.6
265
SGSCHEME
ing functions controllcr with no ::; 0 has the the higbest power of a nonlinear term equid to 7, with no > 0 the highest power is 17. Therefore, U1e tUlling funct.iolls scheme without llonlinear damping has a big aclvant.age in this desigll aspect over the modular schemes which require nonlinear damping. \Vhile very small values of the noulinear damping coefficients 1\., and gi will result in poor performance, very large values will result in excessive contra] effort (alld could potentially have an adverse effect. on robustness). Fortunately, there will always be a range of values of ni ~tnd 9, that will result in good 0 performance without large cont.rol effort.
6.6
SGSclleme
The SGcontroller introduced in Sect.ion 5.8.1 achieves a weak form of modularity using wea]\er nonlinear dl:tmping terms. This C'onl;l'OlIer was combined in Section 5.8.2 ,vith a .::passh'e identifier t.o design an SG adctpthre schenle which guarant.ees stability due to a smallgain property. In this section we follow the same idea and design an SGscheme with a zswapping ident.ifier. This scheme is of interest because in the case of linear plants it is a certainty ecluivalence schcme and sel'yeS for a comparison of the ISS modular design with traditioual certainty equivalence designs. The main benefit of this scheme is a reduction of tlle growth of nonlillearH:ies in the control law relative to the ISScolltrollf'l'. We first briefly review the controller design from Section 5.8.1. The only modification of the ISScontroller in Table 6.1 is in t.he nonlinear damping terms (6.5). Following the notation of Sectioll 5.8.1, we first re\vrite the matrices 1·{I and Q from (6.7) as
(6.158)
(6.159)
~.
snd, to reduce notation, denote H'O = IV(O, 0, t) and Qo modified nonlinear damping functiolls are defi11ed as S;
= l\.ilw;l} + oiJ6d}.
6.. = Q(O, 0, t).
The
(G.160)
AltlloUgb the nonlinear damping functions (6.160) are different from (6.5), the form of the errOl' syst.em (6.7}(6.8) is unchanged: (6.161)
266
MODULAR DESIGN WITH SWAPPING IDENTIFIERS
where Cl  81
1
0
1
C2  82
1
0
1
A:(=,9, t) =
0
0
(6.162)
1
0
0
1
c,,sn
As is usually the case in linear estimationbased (certainty equivalence) design, we usc an identifier with a normalized update law. \\fe employ the zswapping identifier consisting of the filters
rzT
= A::(z, 9, f)OT + f:V(=, 8, t)'1'
go =
(6.163)
A:;{z,O, t)no + H/(:, 8, t)TO  Q{z, 8, t)T8,
the estimation error
e=z+OoO",..8,
(6.164) (G.1(5)
and either the normalized gnu:lient,
8=r
at:
1
r=rT>O,IJ>O
+ JJlnl} ,
(6. luG)
or the normalized leastsquares update law
o=
r
t
r nf2 r 1 + J/If2I} ,
Of
1 + viOl}
(6.167)
T
=
r(o)=r(o)T>o,
1,1>0.
This identifier is different from the original zswapping identifier (6.34), (6.35), (B.38), (6.39), because of 1:he difference in the nonlinear damping fUllct:ions (G.1BO). The new nonlinear damping functiolls Si appearing in the matrix (6.162) are no longer strong enough to guarantee the boundedncss of n (which
implied parts (ii) tuld (iii) in Lemma 6.1). Therefore we cannot usc Lemma 6.1. Fortunately, if we use the normalization, v > 0, the identifier can gmU'o.ntee certain bounded ness properties sufficient for global stability. These properties are summarized in the following standard leml11a.
Lemma 6.16 Let the mazimal interval of existence of salution.'.i 0/ (6.161), (6.168)(6.164) with eitlw7' (6.166) or (6.167) be [0, tf)' Then the followin.g identifie7' properties hold: (i)
(ii)
(iii)
8 E £00[0, tf)
J1 +IJlnlj:. , E £2[O,tf)nL:
(6.16B)
f
8 E £2[0 tf) n £00[0, tf). 1
00
[0,tf)
(6.169) (6.170)
6,6
267
SGSCIIEME
Proof (Outline). The proof or boundedness of 0 and squareintegrability J l 11 ., is the same as in Lemma 6.1. To establish the bounded ness of
of
1+"1 1;.
V1+vfil;F' (; I :J' we first. recall that (6.171) where € is a bounded, cxponentially decaying signal. Therefore,
= $.
InTH + iJ < Inblol + lei <
Inl.:F
V1+ viOl}  Jl + 1.llnl}  JI + '''Inl} ~IOI + lei < 00.
IHI +
IfI
JI + IJ\OI} (6.172)
v'"
Then the boundcdness of 0 follows rcadily:
(6.173)
From (6.173) it also foHows that
Ii is square integrable.
o
The next theorem establishes the stability properties of the SGscheme,
Theorem 6.17 (SGScheme) All the signals in the closedloop af/aptine system consisting of the plant (6.1), the SGcontroller (6.6) with (6.160), the filters (6.163), (6,164), and either I.he n07'malized g'l'adient (6.166) or th.e n07'malized leastsqua1't~s (6.167) update law are globallu uni/o7mly bounded, and lim z(t} = lim e(t) = O. This mean.9, in pa1ticular, that globa.l asllmptotic 100 100 trackin.q is achieved: {6.174} lim [y(t)  Yr(t.)] = 0,
'00
MoreoVf1', if lim y~i)(t) too
= 0, i = 0, ... ,111,
and F(O)
= 0, then 100 lim x(t) = O.
Proof. Using (6.158) we write (6.163) as
(6.175)
268
:MODULAR DESION WITH SWAPPINO IDENTIFIBRS
In a fashion similar to (6.44) we compute d
dt
(12/ nl:1,,) (6.176)
On the other band, using (6.159) we write (6.133) as
(6.177)
and compute
The system (6.176), (6.178) is summarized as d,t d
(In/})
$
Co/n/}+ //.:/2 + 2:. llVo l} ..... lio Co
(6.179)
!!:.. (, .'I'J'I'.1) ::5 coI'l/J ,"1 + 2 I0:..,", nI":1 + ?1 I(J~I'll , Z ") + 2, Qo ''''F,:.,., 0. dl
Co
...go
Co
(6.180)
From Lemma 0.16 we have iJ E £00[0, i/)' Because the functions BTo and Qo are smooth in 8 and bounded in t, then IH/o'} < k and IQol} < k, where l.~ denotes a generic positive finite constal1t. From Lemma 6.16 we also have
6, J
f "E £,[0, tJ) n .c.., [0, t J). Let us denote by 11 a generic function 1 + viOl.1=' in .etto, t I) n .eo:: [0, t I)' (Note that this notation allows us to say k:lt ::5 11 for any finite k) Heuee
(6.181) Since
z=e'l/',
(6.182)
6.6
269
SGSCHEME
we have
Izl' ~ 2 1 +Ifl~(W (1+ vlnl}) + 21"'1' ~ 2t.(1+ vlnl}) + 211/11' . I) j:
(6.183)
Thus with (6.182) and (6.183). inequalities (6.179)(6.180) become
:t (Inl})
~ (co ldlnl} + I~{~ + k
(6.184)
~ (lif!I!!) d.t
< {co /dl1/J12 + 'llnl} +~'.
(G.185)
:0
These two differential inequalities are interconnected and form a loop with small gain because Inl} appears multiplied by 1t in (6.185). To finish the prool we define the "superstate" (6.1Ba)
differentiate it, and substitute (6.184)(6.185):
.¥
:$
coIOI}  "0 21l/Jf + I} (Inl} + ~I"'I:! + ~Inl}) + 2.11/f +k hOCO h'oCo liO
~
coIOI} 
~1¢12 + ll_\ + k j no
(a.1S7)
thus we get (6.188)
By applying Lemma B.6 we conclude that ..'\. is bounded on [0, t f), so nand 'l/J are also bounded. In view or (6.171), f is bounded, which, along with t.he bounded ness of l/J, proves that =is bounded. Hence all Ule signals are bounded, and thus tf = 00. The rest. of the proof is the same as for Theorem 6.3 and 0 again uses Lemma FA for proving convergence. Performance bounds comparable to those in Section 6.5 for the ISS design 1l0t available for the SG design. By examining the design procedure ill Table 6.1 wit.h the Illodifi('at.ion (6.160), one can see t.hat if t.he plaut. (6.1) is linear, then the SGcolltrollel' is nonlinear in 0 bllt linear in x. {To arrive at t.his ('onclusion, by carefully examining t.be nonlinear damping t.erms one should observe that they arc independent of x.) Because of the linear dependence of t.he cont.rollaw on J', the SG design for lineal' systems should have qualitat.ively the same behavior as the traditional celt.aint:y equhraloncp adaptive linear controllers. Therefore, the SG design will serve for comparison of the strong ISS design with t.radit.ional adaptive cont.rol. In t,lle next exmnple we compare t.he SG and ISS designs for a lincar plant. are
270
'A1'ODULAR DESIGN WITH SWAPPING IDENTIFIERS
Example 6.18 (ISS vs. SG) Let us consider system (6.144) witb
=
For this linear system we make a comparison between the ISS and SG '1 (') designs. The only difference is that the terms iiI t.p2, "'2 (~ )t.p2, 92 ~)  in Xl
the ISS design (6.145){6.1~!8) are, respectively, replaced by Ii',J, n.'l (~)2, 92 in the SG design. Tile same design coefficients and initial conditions are used as in Example 6.14, except for (J = 3. The adaptation gains, '1 = 5 for the ISS design, and 1 = 1.5 for the SG design. are chosen so tbat the rate of parameter convergence is the same for both designs. The control law of the SG design is linear in x and nonlinear in iJ.
ISG o
=1
I
I I
6
I
I
I
I
4
I I
I
0 0
9
2
~ 0
II
1
1
2
~::Esfo: '0
0.1
0.2
3
i
3
i
0.3
Figure 6.8: ISS design VB. SO design. The dashed lines sbow =1 (t) when "y = O. (Note the expanded time scale for the control u.)
6.7
SCHEMES WITH WEAK ISSCONTROLLER
271
Figure 6.8 shows t.he difference in performance between the two designs. The ISS design achieves a better attenuation of the =ltransicnt but. with ll. larger control effort. Hence, the ISS and sa designs offer a clear tradeof[ between performance improvement and control effort. Let us now focus attention the dashed responses in Figure 6.8, marked ISSo and SGo. They illustrate the underlying nonadaptive behavior h' = O} of t.he two controllers. According to Corollary 5.9, as shown hy curve ISSo in Figure 6.8, the ISS design results in global boulldedness (although the origin is unstable). In contrast, the SG design results in instability, as shown by curve SGo in Figure 6.8. The instability occurs beca.use the constant parameter estimate has a destabilizing value. The closedloop system with the SG controller is linear, and the instability is exponential. The tradeoffs between the ISS and SG designs revealed by this example also apply qualitatively if the SG design is replaced by any of the traditional 0 certainty equivalence adaptive linear designs.
6.7
Schelnes with Weak ISSController
The SGcontroller reduces the growth of nonlinearities in the control law. The weal\: ISScontroller designed in this section achieves a more sjgnificant red uction of the nonlil1earit~r growth. However, in contrast to the SGcontroIler, the weak ISScontroller is nonlinear in the pJant state even when the plant is linear. 'Ve start with a motivating example.
Example 6.19 Let us consider the scalar system (6.189)
The (strong) ISScontroller for this system is U
=
:z:  x·liJ _
(.1:4) 2 .T
=
:z:  x·IO 
x!l ,
(6.190)
We now show how the desired ISS propert.y is achieved with the following wealcer controller: u
= xx·IOI:z.··1Ix = .1:  :,;40  .z:5 .
(6.191)
The closedloop system with this cont.roller is (6.192)
I ... 2_1)
MODULAR DESIGN WITH SWAPPING IDENTJFJERS
Along t.lle solut.ions of (6.192) we have
(6.193) By Theorem C.2, it follows t.hat inputt,ostate st.nbility is achieved with resped t.o jj as input,: (6.194) l·l'(t)1 ~ la:(O)le 1 + 1101100 .
Other ISS bounds are also possible. 3 The ISS bound (6.194) is somewhat higher than t.he one that: would follow froll) the (st.l'ong) ISS contl'ollaw (6.190) substit.uted in (6.189): (6.196) The ISS property (6,194) allows us to combine the weak ISScontl'OlJer with any ident.ifier that independently guarantees boundedness of 0, The weak ISS controller (6.191) whose highestorder term is .'l'5 is expected to lise less control effort than the ISSeontrollcr (6.190) wit.h .1..tI , It is also important to note that the weak ISScontroller is 'less llonline8.1,1 than the SGcontroller where the highest.order t.erm is J: i : 1I
=
.J: 
:,,"0 _ (x:i) 2 .J;
=
:1.' 
.1:,18  :1.7 ,
(6.197)
This reduction wit.h the weak ISScontroller is even more pronounced when the plant nonlinearity is of higher order, say :en • While the highestorder terms in the strong ISScontroller and the SGcontroller axe :1..2ra +1 and X 2n  L, respectively, in the weak ISScont,roller it is l,1I+1. 0 Example 6.19 suggests that the nonlinear damping functions (6.5) should be modified to contain the norms, rather than the squarcs of the norms, of the columns of Hi and Q, namely, (6.108)
lxl 5 (1:r.1IOI). When Ixl ~ 101. this expression is nonpositive. ~Then Ixl : : ; 101 we have _/.T.1 5 (IxlIOI) : : ; Ixl6191 : : ; jjO, which, substituted into (6.193). gives 1, (~) :5 _x'J + Oa. The ISS bound hns 8 cubic gain with respect to 8: 3Considcrl Cor example, t.he expression
1.r(t)1 :5 Ix(O)le ' + 1I01l~ which is tighter than (6.194) fol' small
O.
I
(6.195)
6.7
273
SCHEMES WITH WEAI<: ISSCON'l'ROLLER
Table 6.2: Weal, ISSController (6.200)
(6.202)
(
(i 'l)
Xi, (J t Iir A
8j
..
=
'I \'1 + 1 + Oi \
_ Iii V Wi 
oail ']'12
(i)
Or
an
1
+1
(6.303)
i = 1, ... ,n
= (Yr,.Yrf .•.
(i)
I
Yr
Adaptive control law:
u = _1_ p( X)
[an'" (x fJ y{rIll) + "/('1)] r
(6.204)
,~r
A sHght problem with this form of nonlinear damping is that the function I . I is not differentiable, which mettllS that it would pose a problem ill t.he process of backstepping. Consequently, wo replace (6.198) with s·f =
oafl TI2 H,VlwI + 1 + g a+ 1' 80 "I
I
I
(6.199)
I
1
wbich is a smooth function that overbounds (6.198). To simplify the analysis we make an additional modification to the stabilizing functions (6.3)  we eliminate the term :.';1' The weak ISScontroller is summa.rized in Table 6.2. The closedJoop error system obtained by applying the design procedure (6.200)(6.204) to system (6.1) is
(6.205) where H' and Q are as in (6.8), and A: is without the 1'5 below the diagonal:
A:(z, 6, t)
=
[Cl  81
1
1
1 CJ1 
Sn
(6.206)
274
rvIODULAR DESIGN WITH SWAPPING IDENTIFIERS
Before we proceed to the identifier design, we establish the [SS property of the error system (6.205).
Lemma 6.20 The error .r;y.9tem (6.205) is ISS with respect to Proof. Along the solutions of (6.205), for i = 1, ... , '11

(0,0).
1, we have
80;il TI2 .+ 1 ~._2 80
I
I
Thus we have the following implication:
:::}
d ( =;") ::5 dt
Cj
z;.., .
(6.208)
By Theorem C.2, for all 0 ::5 s ::5 t., (6.209)
that is, each ziequatioll is ISS with respect to however, has only
(0,0) as input.
(Zi+l, 0, 0).
The znequation,
When the same arguments as (6.207)
(6.209) arc applied to the =nequatioll, we get (6.210)
Inequalities (6.209) and (6.210) define a set of cascaded ISS inequalities. By repeatedly applying Lemma C.4 we show that there exist positive real numbers f31, PI, and u independent of initial conditions such that (6.211)
which complet,es't.he proof.
o
Lemma 6.20 ~mplies that we can use any identifier that guarantees bOUlldedness of 0 and O. This means, in particular! that the result on boundedness
6.7
275
SCHEMES WITH WBAI{ ISSCONTROLLER
without adaptation given in Corollary 5.9 for the ISScontroller also holds for tbe weak ISScontroller. Vie now look for identifiers which guarantee the boundedness of 9 and O. We already know from Lemma 6.5 that these properties are guarant.ecd by the xswapping ide.nti.fie1· {6.87)(6.91} (see Section 6.4 for details). The =s·wa.ppinf} identiJier, as presented in Section 6.3, is lIot directly applicable because the matrix A: in (6.206), obtained with the wmtli: ISScontroller, is different from the matrix ((i.B) resuHing from the strong ISScontroller. The zswappillg identifier tbat we use here bas the filters and the update laws of the same form as in Section 6.3 t but with matrbc A: defined in (6.20G):
07 =
no
e
A:(z, 9, t}nT
+ H/(=, 9, l)" T = A:{=, 0, t)no + H/( =,0,1) () TA = z+non 6, A
A
A
(6.212) 'I':"
A
Q(:;, 8, t.) B
(6.213) (G.214)
with either the gradient.
r = rT > 0, 01'
II
~0
(6.215)
the least squares
iJ
=
t =
nf
r 1 + vh{O'l'rO}
r
{6.216}
noT r 1 + vtr{nTrn} ,
r(o) = r{o)'1' > 0,
v 2:: o.
The differences between this zswappil1g identifier and the one from Section 6.3 are emphasized in the proof of the following lemma.
Lemma 6.21 Let the maximal inlenJ(Ll of existen.ce oj solutions oj (6.205), (6.206), (6.212)(6.218) lIIUIL eithe1' (6.215) 01' (6.216) be [0, if}. Then. J01" v ~ 0, I.he Jollo'wing identifie1' properties hold: (i)
(ii) (iii)
Be£oo[O,tf) e E £2[0, tf) n £00[0, tf)
(6.21i) (6.218)
8 E £2[0, tf) n £00[0, t[} .
(6.219)
Proof (Outline). The main difference from the proof of Lemma 6.1 is in establishing the bOlllldedness of n. III view of (6.212) with (6.206), we haVe?
n i
=
On
=
+ Si)f2 i + 0;+1 + Wi, (c" + Sn)nn + 1Drr •
(Ci
i
= 1, ...
t
111
(6.220) (6.221)
276
I\10DULAR DESIGN WITH SWAPPINC IDENTIFIERS
'V'ith (6.203) we get
:t (111;12)
::;  ~ Il1 il'  ~ In, I (1141  ;;'ll1i+d) liilWjllflil (Iflil :;) ,
i =
~ cn;I') ::; c"ll1nl' I;:II1Onlll1.1 (1 I1nl
1~ ... , n 
:J .
1
(6.222)
(6.223)
From (6.223) it follows t:hat nn is bounded, which, by (6.222), implies that S"lnl is boullded. Cont.hming in the same fashion, (6.222) proves that n is bounded. Let us now define the positive definite diagonal mat.rix
P =
dia.g{PJI'" ,Pr.},
PIl = 1,
It can be shown that A: defined in (6.206)
8.lld
P defined in (6.224) satisfy (6.225)
whicb implies
g
mol.
(6.226)
For syst.em
i = A:(.:, Ot t)i
(6.227)
by viI1;ue of (6.226) it. foUows tbat (6.228)
The proof of the lemma is completed wit.h the same arguments as the proof 2 function 1" = ~liit2r1 + .,L of Lemma 6.1, with the Lyapullov • _ .rno lil for the gradient update law and the Lyapullov fUllction V" = 18If(l)] + 2r!lD Ill2 for the leastsquares update law. 0 Having established the properties of the :rswRpping and the =swapping identifiers, we arc ready t.o conclude t.he stability properties of the closedloop adaptive system.
6,8
277
UNKNOWN VIRTUAL CONTROL COEFFICIENTS
Theorem 6.22 (Schemes with Weak ISSController) Alilhe signals in the dDsedloop a.daptive system (~Onsi8ting DJ the plant (6.1)1 the weak ISScontroller (6.204), and either the .1:swapping 01' the :;swcLpping identiJic7' are globally uniJo'l'mly bounded and tex) lim z(t.) = !ex) lim f(t) = O. This means, in pari.icula.r, that ,qlobal a.sllm.ptotic tmckin!1 is achieved: l
II~~ [y(t)  llr(t)] = O.
(6.229)
}'loreover, if lim y~i)(/) = 0, i = 0, .. " n 1, and F(O} = 0, then lim 3:(/} = O. ico
Iex)
Proof. With Lemma 6.20 and either Lemma 6.5 or Lemma 6.21, the bOUlldedness of all signals is established as in Theorems 6.3 and 6.7. The convergence properties are established as in Theorem 6.3 (for the schemc with the zswappil1g identifier) and either Theorem 6.7 or Remark 6.8 (for the scheme 0 with the xswapping identifier). Remark 6.23 As in Theorem 6.4 and Remad: 6.9, 'ye can normalize the update law with 1 + IlIQi.?:·, which guarantees that QTiJ E L.ex) and al10ws us to set gj = O.
6.8
0
Unknown Virtual Control Coefficients
As in Sections 4.5.1 and 6.8, we ('onsider systems of the form
i
= 1, ... , m I,m. + 1, ... ,11 1 {6.230}
where b", is unknown, but, as statcd in Assumption 5.23, in addition to sgn bm : a positive constant (m is known such that Ibm I ~ c;",. We present swapping adaptive schemes with the controllel' (5.216)(5.227) from Sectio115.9.1, which, by Lemma 5.24 1 g~arantees input.tostate propert.ies with respect to the inputs (B,b rn ) and (O,b m ). The swapping identifiers usc parameter projection to keep bm (t) ti'om becoming zero.
The xswapping scheme Let us start by writing system (6.230) in the following compact form: (6.231)
278 '~Te
:MODULAR DESIGN WITH SWAPPING IDENTIFIERS
use our standard filters
nT no f
=
(Ao  AF(:c, u)T F(:r, u.)p) OT + F(x, (Ao  .\F(....·, ll)T F(x,u)P) (00  x) 
=
x+Oo  OTJ
=
1.I)T
J(X,ll)
(6.232) (6.233) (6.234)
and either t.he gradient.,
bin (0) sgn bm > C;m
r=rT>o,
v~O
(0.235)
or the least squares update law
(6.236)
where t,he project.ion operator is employed to guarant.ro that Ibm(t)1 ~ (m > 0, \:It ~ O. For a detailed treatment of parameter projection the reader is referred to Appendix E. Lemma 6.24 Let the maximal inte7'Vul of e.'l;istencc of .r;olu.tio11.5 of (6.231), {a.232}{6.234} wHh either' {6.235} or (6.236) be [0, tf). Then for I) ~ 0, the foll011ling identifier pmpcrtics hold:
(0)
Ibm(t)1 ~ 'm > a,
(i)
8 E £o.:l[O,tf)
(ii) (iii)
f
E £2[0, tf)
'Vt E [0. tf)
(6.237) (6.238)
n £o.:l[O! tf)
Ii E L2[O,tf)n£o.:l[O,if)'
(6.239) (6.240)
Proof. The proof of this lemma is only slightly different from the proof of Lemma 6.5. The differences are due to projection. First, from Lemma E.1 we conclude that Ibm (1)1 ~ (m > 0, \:It E [a, tf). To estahlish (6.9~) in preseu('e of projection, we use the fact from Lemma E.l that 1 fit } < :ii'1'r"lr Hlllfll:}' fit F'ma11y, we note  {JTr1iJ:"   {)Tr p rOJ. {r l+Jllfll} _ "U that Lemma E.1 implies that I';; I::; ~ r I+~~t} This fact is now used as in ~he proof of Lemma 6.5 to est.ablish boundedness and squareintegrabililY oft?' 0
I
I,
By combining Lemmas 5.24 and 6.24, we obtain the following result.. It.s proof is the same as the proof of Theorem G.7.
6.8
279
UNI(NO\VN VIRTUAL CONTROL COEFFICIENTS
Theorem 6.25 All the bountledness and convergence properties from Theorem O. 7 hold also Jor the closedloop 1I,daptifJe system C011..5isting of the plant (6.230), control law (5.227), filtC7'S (0.232)1 (6.233), and eillwr the gmc/icnt (6.235) 07' the leastsquares (6.296) update la1l1. The =swapping scheme Consider the closedloop system (5.228){5.231). Our =swapping identifier is
not direct.ly applic~l.ble to this parametric model bectl.use the torlns ~(crn + ~'" 8 ,11 );:;m and bm=m+l ill (5.229): as well as the term ll"'=rn in (5.230), depend on the unknown pal'ametel' bm • HowevE'r, writing bm in (5.229) and (5.230) as bm + blU , in view of (5.220) we obtain
Zm
= (6.242)
(6.2..13)
Let us rewrite the system (6.241)(6.244) in the following compact form
{) = [b;
],
(6.245)
where 1
. (6.246)
1
280
MODULAR DESIGN WITH SWAPPING IDENTIFIERS
Since we will use projection to guarantee that Ibm(t)1 ~ <;"m and sgn b",(t) = sgn bm, which implies bm sgn bm ~ (m, the matrix A: will sat.isfy (6.2~G)
With the parametric =model (6.245), the form of the filters is the same as in Section 6.3:
o.T
nO f
=
A:(=, 0, t)OT + HI{)(z,.o, t)T
=
A:(=: {), t)Oo + H/I'J(z, fJ, t) 8  Qo(z, {), t) {) •
=
_
A
(G.2£18) 'r:'
~
TA
'1'.
_+00 0 iJ,
(6.249)
(6.250)
with either the gradient
:.
{) = P.roj b...
{nf} r + viOl:! , 1
bm(O}sgnbrn > ~"" v>O
:F
r
=
rT > 0 {6.251}
or the least squares update law
~ =
t =
P[.~.i { r 1 +~InI:} } , onT
r 1 + vlnl} r ,
bm {O} sgn bm > (11& (6.252)
r{O}
= r(O)T > 0,
II>
0,
where the projection operator is employed to guarantee that Ibm(t}1 ~ (711 > 0, "It ~ o. The reader should note in (6.251) and (6.252) that v> 0, which means that we allow only normalized update laws. This is in contrast to the zswapping identifier from Section 6.3, where the normalization is not necessary because the nonlinear damping substitutes the normalization in the task of slowing down the adaptation. From the proof of Lemma 6.1 we recall that the nonlinear damping slows down the adaptation by guaranteeing the boundedness of 11 for any input Hr. It is important to understand why we have to use normalization here. To this end, let us note from (6.241)(6.244) that the regressor HI'17 is given by i
= 1, ... ,ml (6.253)
8a'j_l
Xm+1 8x m
,
wJT
j
= m+2, ... ,TI
6.8
281
UNKNOWN VIRTUAL CONTROL COEFFICIENTS
We recall from (5.223}(5.226) that the relevant terms in the nonlinear dnmp~ ing functions are 81
8m
Sm+l
Sj
= Kil w il2 + ... =
J
r b+am ((y<m
li':m
m
= I<m+l
((
(
= I<j (
r ")
+ l1lJm l + ...
8X
m Xm+l)
(6.255)
+ Iwm+d" ) + ... + IUI;! + .... ")
8o'm
Zm  83'm Xm+l)
aO'il
(6.254)


')
(6.256)
(6.257)
By comparing the nonlinear damping terms (6.255) and (6.256) with the mth and the (m + l}st rmvs of (6.253), we see that the the nonlinear damping
terms cannot be expected to guarantee the boundedlless of n when HId grows unbounded because they no longer correspond to the columns of the regl'essor H't). Therefore, we have t.o use normalization. This is the price paid for replacing the original nonimplemelltable parametric model (5.228)(5.231) by the parametric model (6.241)(6.244).
Lemma 6.26 Let the maximal inle7'7Jal of existe1lce of solutions of (6.245), {6.248}(6.249} with either (6.251) or (6.252) be [0, tf). Then Ute follo'wing identifier properties hold:
(0) (i)
(ii) (iii)
Ibm(t)1 ;::: (m > 0, UE £00[0, t I)
Vt E [0, t/)
(6.258) (6.259)
J1 + vlfll} E £:1[0, t/) n £00[0, tl)
(6.260)
J E £2[0, t l ) n £00[0, tf).
(6.261)
f
Proof. Because of the update la\v normalization, the proof of this 1emma is similar to the proof of Lemma 6.16, which, in turn, connects t,o the proof of Lemma 6.1. We emphasize only the differences which are due to projection. From Lemma E.1 we conc1ude that Ibm(t)1 ;::: (m > 0, Vt E [0, tl)' Therefore, inequality (6.247) holds, and so does (6,46). To establish (6.49) in presence of projection, we modify (6.48) with the following fact implied by 1p  _.iTrDe } < nTr1r 1+1I101} De . Lemma E.1.·1_.oTrdi 1 II V rOJ. {r l+,"IOI} _ y . TlllS
establishes the boundednes. of
a. The proof of J1 + 1)lnl} E C:![O, tf) i. as in f
282
MODULAR DESIGN WITH SWAPPING IDENTIFIERS
Lemma 6.1, while the proof of
V· E £",,[0, til is as in Lemma 6.16. 1+ Illnl}
With Lemma E.1 we have
wbich proves that ~ E £,,[01/'j) n £oo[Ot tf)'
o
Theorem 6.27 All the boundr..dness (Ind convergence pmpe'l'ties }rom, Theorem 6,3 hold. also /0'1' the dosedloop adapti've system c(Jnsisting oj the plant (6.230), control law (5.227), filtenl (6.248), (6.249), awl eithe7' the gradient (6.251) or the leastsqu(l7'es (6.252) update law. Proof. As we comment.ed in Remark 5.27, because of the local Lipschitzness of the projection operat.ol', the argument ctbout the exist;ence of solutions l'emains as in tbe proof of Them'PIll 6.3. The cHfferellce between this proof and tlutt of Theorem 6.3 is that the identifier here does llot indepelldently gmu'Rntee the boundedness of n. By combining Lemmas 5.2"1 and 6.26 t we conclude that .: is bounded, This implies the boulldedness of H'., and QUt so n and no are also bounded, To prove cOllvergence. we point out that Lemma 6.26 establishes that. ~ E £2, so, due to the boulldedness of 0, we have f E £2. The l+. O:;r proof of the convergence properties is finished as in Theorem 6.3, 0 J
Notes and References IV.hmy early approaches to adaptive control of nonlinear systems were estimationbased and employed swapping identifiers: Sastry and Isidori [166], Nam and Arapostathis [141], IvHddletoll and Goodwin [130], Bastin and Campion [8], Teel, Kadiyala, Kokotovic, and Sastry [190J, Pomet and Praly [152, 153, 154J, Kanelhikopoulos, Kolcotovic. and Middleton [67, 68], and Prruy, Bastin, Pomet, and Jiang [157], Various restrictions were imposed in these results: [8, 130] contained structural conditions, [141, 166, 190) imposed linear growth conditions, [67, 68] had both structural and growth conditiolls, Ouly in [152, 153, 154, 157] did the growth conditions have a less demanding nonlinear form. Both the structural and the growth conditions were removed in •. he modular design with swapping iden1.ifiers developed by Krstic and I(okotovic [98].
Part II Output Feedback
·Chapter 7 OutputFeedback Design Tools All the results in Part I have bccn obtained under the aSBumption that. the full state of the system is measured. 'Ve now remove this assumption and consider more realist.ic problems where only a patt of the state or just. t.he scalar plant output is available for mel:lSurement. In the case of linear systems, the separation principle allows outputfeedback problems to be solved hy combining statefeedback controllers with stat.e observers. However, the separa.tion principle does not hold for nonlinear systems. In this chapter we introduce backstepping procedures for outputfeedback and partialstatcfeedback designs for nonlinear systems, whose nonJinearities depend only on the measured signals. For these systems, we buiJd exponentially convergent nonlinear observers and replace the unmeasured states by their estimates. We show that global resnIt.s can be obtained jf nonlinear damping is used to counteract t.he destabilizing effect of the observer errors. As we did in Chapters 2 and 3, we start with systems whose parameters are known, and then present adapt.ive schemes fol' systems with unknown const.ant parameters. Such adaptive schemes are further developed by the t.uning functions design in Chapter 8 and the modular design in Chapter 9.
7.1 7.1.1
Observer Backstepping Unmeasured states
To illustrate the difficulties that arise when some of the states of the system nre not measured, as well as the means to overcome these difficulties, we use two simple examples.
State feedback revisited.
First, let us consider the system
:i'
=
i.
=
:r + x·1 + :C2~
ke +"U,
(i.la) (7.1b)
286
OUTPUTFEEDBACI{ DESIGN TOOLS
_!I__~H__~
J ~__~e__~ ·· ·· ···
· :
( . )4
I I I
! ... __________ ~~
M •• _ • • _. _______________ •• __________
!
Figure 7.1: The block diagram of system (7.1).
with k > O. Us block diagram is given iu Figure 7.1. \\Then both stat.es x and are measured, this system is st:abilizable via hacl(stepping. Usil1g as the virtual control in (7.1a), an obvious choice of stabilizing function is 0:1 (3:) = _:,;2. Indeed, if _:z:2, then (i.la) is reduced to j; ::::: .c. Then, with the error variable:z 0'1 (x) e+.z:2 , the system (7.1) becomes
e
e
e: : :
= e
x
=
.: :::::
=
+ .r,2:; ke + 'U. + 2a: (J: +.:z.4 + x:'\~)
(7.2a)
;1:
=
k~ + u + 2:& (x + x
2
:;).
(7.2b)
The derivative of F(x, e) = ~(.r:! + z!,!) along the solutions of (7.2) is
t:r
=
_x2 +;; [.v: ke + 'U + 2.1: (3: + :r;2z)] . i
(7.3)

Hence, the choice or control U :::::
c:;x:i+ke2x(:z:+a;2z)t
with c > 0 a design constant, yields \! ::::: _x2 the GAS equilibrium of the closedloop system:
= :: =
Ai:
Cz,2
(7A)
and renders x ::::: 0,
x + 3;!!:! cz  x3 •
e= 0 (7.50.) (7.5b)
State estimates as virtual controls. Suppose now that ill (7.1) the state ~ is not measured. This implies that == ~ +.1:2 is not available for feedback, and hence the control (7.4) can not be implemented. \Va see that the unmeasured state ~ can not be chosen as the virtual control, and has to be replaced by a measured variable. Fortunately, (7.1b) suggests that the state C~tn be estimat.ed by ~! where
e
(7.6)
7.1
287
OBSERVER BACI{STEPPING
Indeed, subtracting (7.6) from (7.lb) shows that t.he stat.e estimation error = €  ~ converges exponentially to zero:
€
(7.7) To utilize this estimate, we replace ~ by
t + ~ in (7.la): (7.8)
In this equation, the only possible choice of virtual control is ~, which means that in the backstepping procedure we must replace (7.1b) with (7.6). In essence, we start with the syst.em composed of (7.1) and t.he observer (7.6)1 shown in Figure 7.2, and manipulate it il1t;o the form shown in Figure 7.3:
+ .1.'.1 + x'2t + x'J~
€
= =
k~ + 11
(7.Db)
~
=
kf,.
(7.0c)
:i;
x
(7.ga)
The ne..,'t step is to design a control law for (7.9). Since ( is exponentially convergent, one might be tempted to ignore its effect on the rest of the system, as is often done in linear control desigll. This leads to the error variable (7.10)
and the control (704), which now yields the c1osedIoop system .j;
=
:1' + x:; + .z;e
~
IJ
i

=
c=  x:
(
=
k~.
(7.1 la) (7.l1b)
+ 2x € 3
(7.11c)
The reader familiar with Section 2.5 of Cha.pter 2 will recognize the flaw of this "certaintyequivalence" design, namely the presence of t.he t.erms .1:2~ in (7.1Ia) and 2x 3 in (7.llb). As we saw in Section 2.5, even an exponentially decaying disturbance like can destabilize a nonlinear system and lead to finite escape time from certain initial conditions. In (2.242)(2.246) we c..xamined the system
t
t
which is identical to (7.IIa) when
=== O.
There we showed t.hat this syst;em
has the solution x(t) =
;r(O)(1 + k) (1 + k  ~(O);z:(O)lel + t(O)J'(O)e kl
which escapes to infinity in finite time
'
288
OUTPUTFEEDBACI< DESIGN TOOLS
x
I;
·:
i..,.IlIoo •
·...• . ... ..
..~.
..... ....
Figure 7.2: Adding the observer (7.6) to tbe system (7.1).
t/ =
1 I { t(O)x(O) } 1 + A: n {(O)x(O) _ (1 + k)
from all initial conditions for which t(O)x(O) > 1 + k. This implies that irrespective of bow fast ~(t) converges to 7..ero or how small its initia.l condition ~(O) is, there always exist initia.l conditions x(O) from which the system escapes to infinity in finite time.
u
Figure 7.3: Replacing
eby eas tile virtual control in (7.9).
7.1
289
OBSERVER BACI{STEPPING
Nonlinear damping. The remedy for this problem is, again, nonlinear damping. Starting with (7.9a) and using i, as the virtual control, we modify the stabilizing function O'I(X) by adding to it a nonlinear damping term s(x).r.: (7.12)
Follmving the developll1cnt ill Section 2.5, equations (2.248)(2.252), we design sex) using the function F(:r) = ~x::!, whose derivative is
,i"
=
_3.;2
+ 3:3 .:; 
x 2 s(x)
+ :1:(x!!)t.
(7.13)
The choice of nonlinear damping
s(.t:) = el j (x 2 ):!, d1 > 0
(7.Vl)
yields the stabilizing fUllction a I (J:) = _:r 2
d 1.1: 5 ,

(7.15)
the error variable (7.16)
and the closedloop mqnession foJ' (7.9a)
x  d1x
:i: =
5
+ ;r= +x~. '1
")
(7.17)
The derivative of \i becomes
V = =
_x2

.) .V
+ x 3 :;; 
:!
:5
:r;
d}x6
+ X:i:; + x 3e dl
("
.1:" 
1 ) '). 1'1 2d ~ + 4d ';t
.3_
+.1 ... +
1
12 /old ~ .
(7.18)
l
The last inequality illlplies that if z == 0, .1: will remain bounded if ~ is bounded. Here, bowever, wc can achieve more than that by exploiting tile fact that €(t) is the error of an exponentiaJly converging observer. To this end, we augment tIle function ll"(x) with a quadl'atic term ill
e:
1") 1 '"2 _ 1.2 1 Z2 1 (x +?d k';  ?,.:I +?d I .. ~ • 
1
(7.19)
. . . . . . . Ill.
Using (7.18), we see that the derivative of 'Ii satisfies .
Vi
=
. 1... Vdt
<
.2 3,
e3_
1
2
~1l: 1
3_. x + x ;:  ~2 . 4d 1 '1
1
+:r. "" +  l ~  d ~2 3
1
(7.20)
290
OUTPUTFEEDBACI{ DESJGN TOOLS
Hence, jf === 0 the control (7.15) would render (0,0) the GAS equilibrium of the (:1:, t) syst.em. The derivative of z is now expressed as
., = =
k~ a
+ 1J. 

k~ + 'u 
'1) + ..,;~ ( x + x 4 + ..,;~'1 A) 
anI ax (:1:) ( .'t + X I aal
ax (x)
'1 ax (o't)x~.
EJCXl
(7.21)
In this equation, the state estimation erl'or appears again, so its effect will have to be compensated by another nonlinear damping term. This is reflected in the corresponding Lyapunov function, which is augmented not only by a :2term, but also by an additional ~2term:
Its derivative is •
23
ll2 :5 x
+ J;
.:; 
3., 4d ~
1., d ~

2
1
+z [/of. + u  ~:I (x + .". + x'€)  ~:I x't] =
') 3:>J :r;  4d ~
11)

1
dll~
+z [x"  k€ + u ~:I (x + ",. + x'€) ] z~:I .,,'t.
(7.23)
The choice of control
_ _(80ax
1 ( ) 2) 2
u. = c...  d2 .:.
X.7:

X
3
+ k~ + Bat ax (x) ( x + x,I + x 2~)
(7.24)
yields
(7.25)
Hence, the origin is the GAS equilibrium of the resulting closedloop system:
.i·
=
5
z =
cz 
t
k:~.
=
.,
x  d1x +XZ
.\
J.o' 
d2z
.,+x~
(a"l ax:r:
r '1
2
(7.26a)
a"l. 
 EJx x~
(7.26b) (7.26c)
7.1
291
OBSERVER BACJ(STEPPINC
7.1.2
Outputfeedback systems
The above example illustrat.ed t.he design t.ool of observer backstepping: First a nonlinear observer is designed which provides e.xponentially convergent estimates of the unmeasured states. Then, backstepping is applied to a new system, in which the equations of the unmea..c;ured states have been replaced by the corresponding equat.ions of their estimates from the observer. At. each step of the procedure, observation errors are treated as disturbances and accounted for using nonlinear damping. Observer backstepping can be used to constrnct systematic design procedures applicable to nonlinear systems for which e.xponential observers are available. One such class consists of outputfeedback system,s, whose output. y is the only measUl'ed signal. These systems can be transformed into the outputfeedback form., in which nonlineoarities depend only on the output. y: 1 :i: 1 X2
1·:'1
.'tpl :i; p
Xnl
X" y
+ 'PI (Y) X3 + 'P2(Y)
= =
.'l;~
= =
.Tp + 'PpI (]))
=
= =
.1: p+l
+ !pp(Y) + bm /3(Y)1J.
(7.27)
XJI + 'Pnl (y) + bIP(y)u CPn(Y) + bo/3(Y)u
Xl·
We assume that (7.27) is minimum phase, t.hat is, b",snl + ... + bIB + bo is a Hurwitz polynomial, and f3(y) '# 0 Vy E R. The syst.em (7.27) has relative degree p = 11  171, and its mdimensional zero dynamics are linear: (7.28)
The eigenvalues of the m. x m mtttrix Azd are the roots of the Hurwitz polynomial bms nl + ... + bls + bo, and the elements of 1l'(y) are linear combinat.ions 'PI til), ... ,11'71(11). \Ve first derive an observer for the system (7.27), rewritten as
:i: = Ax + 11'(1)) + bf3(y)u 1/ =
(7.29)
cT."C
IThe coordinntefree characterizntion of thcsc systems in t.crms of differential geometric conditions is given in Appendix G, Corollaries G.6 and C.7.
292
OUTPUTFEEDBACI( DESIGN TOOLS
o (7.30)
110 An exponential observer for (7.29) is
.~ =
A:i~ + ~'C1J
1i
cTi;,
=

+ 'P{Y) + b /3{y)lt
y)
(7.31)
lvhere /(0 is chosen so that Ao = A  kcT is Hurwitz. Subtracting (7.31) from (7.29) shows tlUl.t the observation error i: = 3;  j: exponentially decays: a:
= A0:1:,
(7.32)
Using the observe I' (7.31) and Lemmas 2,8 and 2.26, we now design a feedback controller to forco the output 7J of (7.27) to track a reference signal Yr{t),
Theorem 7.1 (OutputFeedback Systems) For the non.linear ,f11Jstem (7.B7), (1.8S'UmC that bms m + ... + bls+ bo is a lIu11nit:; pol1lnomial, and that Yrt Yn , .. ,y~fI) a1'e It:nown and bounded on [0,00) and y~p){t) is piece111ise conti'1l'U01J.S. Then, I.hc1Y~ e:,;ists a feedback control'which guarantee.9 glol)al boYJ.ndcd1,CJj!:i of .1~(t) and ;i'(l) and 7'egulatiDn Df the tmc/""in,q erl'Dr: lim [yet}  Yr(t)] = 0,
(7,33)
too
One choice Jor thi.r; control is ~()
7IIJJ
with. Zit O'h i = 1, . , . ,p defined di > 0, i = 1, ... t p): :;1 Zi 0:1 O:i
=
= = =
[0: _ x·p+l 
1
U, 
p
Y
•
y{fI)] r
(7.34)
,
uy the. following 7'ec'lJ.rsiue expression.'.i (Ci > 0,
Y  Yr Xi  0';1 (y, .f}, ... I Xil,llr, . .. ,y!i:l})  y~il}, i CIZI 
= 2, ... , P
d1=1  "'1{Y)
(7.35) (7.36) (7.37)
"l
CjZi 
Zil 
80'iI
+~ [X2 UY
A
d.; (
iI
j=1
OYr

J
I~i
(
Y
8ail _

Xl
)

'Pi ( Y)
+ 'PI (Y)] + L 8.... [Xj+l + It'j{1I 
OO'i_l. (j+l) L." u) Yr +~ j=l
T
OO:il) Zi
1
• 1
?
XJ
= , ... , p •
..
Xl) + 'Pj(1/)] (7.38)
7.1
293
OBSERVER BACI<STEPPING
Proof. From (7.27), (7.31), and the defillitions (7.34)(7.38), we can eAllress the derivatives of the error variables J • • • J ::p as follows:
=.
Zl
.
Zi
= iJ  fir = X2 + 'PI (y) tir = :1;2 + :1:2 + tpl (Y)  Jir = =2 + at + c,ol(Y) + x!! = CI Z I + =2  d1 z 1 + X2 =
•
xi+1
~ ) + c,oi () + k4i (Y  3:1 Y 
~ 8a:il [ftXj+l + J..111 ( 3
oail [..r2 + !PI () 11
• ) Xl
 { Oi. 3=1
au
zp
CiZi 
= Xp+l
Zil
+ Zi+l 
+ bmP(y)u. + ~~p(y 
E°0a:~~1 j=l
[.Tj+l
:1:J
80;_.) lJ]J
Zj 
J=l
A

Yr
oO:;_t  . { . a y X 2 ' 'I = 2, ... ,p  1, u:lO}
:i"l) + !Pphl)  8a'pI [&2 + !PI (Y) 8 11
+ ~~(y 
5: 1)
+ tpj(U)] 
Eaa~~)ty~j+l) j=l
= bm f3() Y U + Xp+l =
di (
] + :r::!
~ Oa:il (J'+t) 1J (i) + tpj ()] Y  ~ lJ U) Yr r 'I
=
(7.30)
O:p  Cp"'P 

"'pl 
dp
Ollr
+ .V2] 
y~p)
.,
(oo'p_t)" OOp_1 :. ay :'p  ay3.'2  Ilr(p)
CpZ,  =,_1  d 8i~1 fzp  8';;;1 ft.. p (
(7.41)
The resulting error system is
(7A2)
Due to the piecewise continuity of y~p}(t) and the smootlmess of the nonlinearities, the solution of the closed1oop system (7...:12) exists. Let its ma."{imulll illtel"V1u of existence be [0, tf). On this interval, the nOllnegative function V'p defined by V;J(=,X) =
t [~zJ +
j=l 
dl.XTPOX] ,
(7.43)
J
wbere Po is the positive definite symmetric solution of the Lyapunov equation PoAo + A6'Po 1, is llonincreasing, since its derivative along the solutions
=
294
OUTPUTFEEDBACK DESIGN TOOLS
of (7.42) satisfies
.
Ifp
<
EP [c·:;=') + 1]13 '1] 4d.
j=l
'J J
J
< O.
(7.44)
Thus, ZI, ••• I zp are bounded on [0, tr) by some constants depending only on the initial conditions of (7.27) and {7.31}. The boundedness of all other signals on [0, tr) is established as follows. Since =1 and Yr are bounded, y is bounded. The boundedncss of x and .tl = Y  Xl imply that .1:1 is bounded, Since z:! is bounded, X2 is bounded. In the same manner, it. can be shown t.hut :l:),. , . ~ xp are bounded. Hence, :r 11 ••• ,:r p are bounded. To provo t.he bounded ness of x p+ l! ... , XII' we note that the boundedness of y implies t;he boundeclness of ( from (7.28). Since [XII'" ,:l'p, (r]T = Tx, wbere T is a nOllsingulal' matrix [164, Theorem 2.1], we conclude that. x is hounded. Finally, the feedback control 'It defined in (7.3,,1) is bounded because bm {3(y) is bounded away from zero. We have thus ShOWll that the state of the closedloop system is bounded on its ma.ximal iuterval of existence [0, tr). Hence, tr = 00. The convergence of the trucking orror to zero now follows Crom (7,43) and (7A"1) and the LttSalleYoshizawa theorem (Theorem 2.1). 0
7.2
MIMO Design: Induction Motor
Heretofore we have not dealt with multiinput systems in much detail, because the baclcstepping procedure is essentially the same, provided that the system is in an appropriate form. 'Vhen it comes to baclcstepping with two or more controls, there are additional degrees of fl'cedom which designers can skillfully employ. This additional flexibility is particularly welconle when some states are not measured but instead estimat,ed through an observer, Rather than develop n general theory, we will illustrat.e such a !vIlMO design on an induct.ion motor control problem. The induction motor is indispensable because of its ruggedness and low cost, but, unt.il recently, its control applications have been restricted by the difficulties in controlling its torque and speed. Advances in power electronics and the field orientation design methodology promise that induction motors will replace less reliable dc motors in industrial drives and servo systems. With fullstate feedback, field orientation can emulate the dc machine dynamics. Since fullstate feedback requires expensive and unreliable sellsors for rotol' filL',:, flux observers arc used in place of sensors. In this design we combine nonlinear damping and observer backstepping with a simple flux observer. Under the assumption that tbe motor parameters are known, our design achieves stability with a guaranteed region of attraction. Simulation results indicate that this observerbased design recovers the performance of the fullstate design.
7.2
295
MIMO DESIGN: INDUCTION MOTOR
The wellknown twophase equivalent model of the induction motor is ~ 't1 p Al (I
dw
dt = 2 J Lr
. _
'Pra 7'sb
J
•
~ rb 1siI
) _
TL
J .
dl ga .11JRr 11 JJ AI 1 = Y:;'l/Jra + LWl/Jrh 1'1 83 + USii elt u ~ U r a 1lp lal Af Rr . 1 disb y;wt/Jra + y:VJrb  ')'1sb + 1/ s b rlt = U r a r U Rr Rr . d"l/Jra =  Lr 1/Jr;a.  l1)Jw1/Jrb + Lr .1\11sl1 dt dl/Jrb
=
dt.
(7.45)
Rr • Lr l/Jrb + Lr Af 1'g b ,
Rr 71 pW lPra. 
where w, i, .,p, Us are the angular speed, current, flux linlcage and st.at.or voltage input to the machine; R, L, ilf, J, T L , and np are the resistance, induct.ance, mutual inductance, rotor inertia, load torque, and number of pole pairs; the subscripts sand r stand for stator and rotor; (a, b) denote the components of a J L ; R~ . vector wi th respect to a fixed reference frame; and (f = Ls  lIL/:J , l' = lIf Rri'J cr r "Ve assume t.hat the torque load is a known function of the rotor speed, Tdw) = k n + k,w + k2W 2 , whieh is realistic for loads slIch as pumps and compressors. Denoting I
Xl
=w
X2 =
iSR
.'T.a = isb ·"t4 = :1:5 =
all,.M
1/1
= lIso.
OI=2"JLr
11.2
='Usb
a2 !!D.  J
b= !IT
',pra ,prb
0'3 =
.h J
o4  h:2. J a  MR. '5 
lTL~
o (I 
U7 =
11,,111 tTL.
'Y
a·8 .&  l. r
(7.46)
09 = np
o 10 
RrM Lr
we rewrite (7.45) as :i: 1
=
al (.1'3:1:4  x!!.'J:S)  02 
aa:Vl 
+ 06XIXS  07X2 + bUI 3:3 = 06.TIX4 + 0.5.7.'5  ajXa + b1l2 ,i'4 = o.SX4  0'!J X ).'J:5 + aID X 2 3;5 = 00XIX.I  a.S:r,5 + alO·'l:'a •
.i=2
=
a..I.Ti
0S.'V4
(7047)
The control objective for Yt = Xl is to asymptotically track a given speed reference wrer(t) while keeping all the signals in the closedloop system bounded. It is also required that Y2 = x~ + .'J:~ = ',p~L + ',p;b track a reference .,prer(t) to prevent excessive flux magnitudes or demagnetization. The first two derivatives of wrcr(t) and of 1/Jrcr(t) are assumed to be known and bounded.
296
OUTPUTFEEDBACK DESIGN TOOLS
A simple flux observer is constructed as a copy of the last twost.ate equations:
(7A8) The resulting error system is (:C.I =
:l:4 
i'4, :rs = :Z:s  1:s)


a8: r 'l  aH:r 1:r5 a8:i:[i
+ (]!j:rl:C,1
.
(7.49)
Since {]8 > 0, this secondorder system is globally exponentia.lly stable. This is est.ablished via the LYC1punov function . ( _) \lobs:C
_ ="21 (.];~ + ;7: 5 'J
_ ') )
(7.50)
J
whose derivative along the solutions of (7.49) is
l~Jbs
u'I/i:~ (1.H(:r~
a!J.1:1: r S:r 4 
CJ8:r;
+ CJ!J:1:d:.j:r5
+ :c~)
 20 8 \/~)bs
::;
0.
(7.51 )
\\lith an exponentially convergent observer ava.ilable, we proceed to our backstepping design. Step 1. Let LI, C'2, C;i, L.I, ell, d2 , and d. 1 be positive design constants. \Vc first consider the speed tracking objective. Vle define ZI = :Z:I  Wref and write i l (7.52) To initiate backstepping, we need to choose our first virtual control. The form of (7.52) suggests that instead of a single state variable \ve choose 0,1 (xa:cI :7:2:7:5). For an electric machine expert, this choice of virtual control is natural, beca.use it represents t.he torque produced by the rotating magnetic Jield. If .7;':! and ,T.1 were measured, the first stabilizing fUIlction would be (7.53) Since .7:4 and :r5 are not measured but only estimated [rom (7.48), we apply observer backstepping: \Ve rewrite (7.52) as (7.54) and choose the est:inwtcd torque 0.1 (.7:3:i;.1.7~,:!:1~5) as our virtual control. Viewing :l:4 and i:5 in (7.54) as unknown disturbances, we apply the Nonlinear Damping Lemma (Lemma 2.26) to design the stabilizing function (7.55 )
7.2
I"HMO
297
DESIGN: INDUCTION IvIoTOR
Step 2. Defi.ning the error variable (7.56) we uugrnent (7.54) with the =2equation Z2
(0.1;1;.1 + 2d,ai:c:i Z j)( aGJ~).1~·1 + 05;Z;5 +( lll:i:5 + 2d J )(0.5.7:.1 + a6;Vj:rfj
Q:i 
0.2 
a.a:Vi
+ 1m2) + bnd
(J.!):r,X5 I 0.10·7:2)  aj:r~(a!):rl
+alJ:3( oH:i:. j +[Cl 
(/i:r~1
 (/7.7:2
20.. IJ:1
+ :1;~)H(J.l
+ rl,
(J..t:ri  Wl'd]  Wrcf  (aa
 o.8:i: 5
4  .1:2:1: 5 )
+ alOXa)
+ (/.j (;ra:l:'1

+ 2(LI:r! )Wrof·
(7.57)
The resulting system is then rewritten as '::1
22
_
_
)
') ( ' )
'1 )
+ al :z:;;:r:,j :1.>2:1:5  d 1rrj' :r 2 + :1:3 ':1 + ':2 (1l,:l:4 + 2d 1o}l:3 Zd hv'2 + ((JI + 2d1oT'?:2ZdbUj +(al:l;,j + 2dlo.~;r:3::;d( aoxd:.1 + (15:t:5  (/7.1:3) Cl':l
(
(7.58) +(UI:r5 + 2r1 j o.i:1;2 Z))(a.,'j:h + a(j:rd:5 a7:!:::!) +al:r;;(Us:l:"  0n:r l:i '5 + (1.10:1:2)  at 1:2 (O!).Tl:J;.1  ag:i: s I (1,1O:r :.I) +[Cl (13  20 4:rl + d1(1T(:r~ + :1:5)][C13!  d1o.~(.T~ + :rj)Zl +  ((J,:i
+ 2a.,:1; 1)Wrcf
+{( 01.1':5 +
2d j aI:r2 Z 1)05
(ad:'l + 2d\oi:c:i Zd o (jJ;1
+[CI  0,3 2a4:CI + dlaT(·1;~ + :r~)la':Z::i};1:.! + {(aI'?:'1 + 2rl 1ai:r:lzd a s +( + 2d 1ai:L2::da(J:1:1  [CI  lla 20_I :rl + dlni(:r~ + ;I:~)]a.I:D2}.i5
Lemma 2.28 results in the ehoice of feedback (7.59)
Of course, (7.59) is only one equation to be satisfied by the two controls lIt and 'H2' This leaves us with an additional degree of freedom which we will use for flux tracking in the following steps. Step 3. 'Ve now turn our attention to the flux tracking objective. The flux error is .7:~ + :c~  4J ref' However, since :1:,1 and .1:5 are not measured, we replace them by their estimates and define the error variable ':a = + i;~ 7!'l'er. This error variable represents the e8iimaied jl'll:1: errol', and its derivative contains no unknown terms such as :r., or i5 :
2( o.8X~  aO.1:I:v5:r,. + 2a8(x~
+
a.lOX::!X.j
+ (J.!).1;1.1':.!.f: 5
+ 2alO(:r25;,1 + :[;3.1':5)

')
08:1:5
+ (J1O:r3:t:5) 
.
1!J re f
(7.60)
298
OUTPUTFEEDBACI< DESIGN TOOLS
Once again, we have to choose a virtual control. Since differentiation of X2 and Xa produces the actual cont.rol variables 11) a.nd ll:h we choose the term 2aJO(x:d:I + XaX5) as the virtual control, and we design for it the stabilizing funct.ion (7.61 )
Step 4. Applying Lemma 2.8, we define the error variable (7.62) and augment (7.60) with ':;.1 =
20. 10 3;.I(aS.1:.1 + 0a J:I X 5  0.7.1:2 + bud +2a lO x5( a6 X l X " + 05 X 5  arXa + bU2) +2(UlO:1:2  208:1:.1)( osi'"  agxlXij + aJOx2) +2(aIO X3  2o.a:i':S)(au XI.i .1  08.1:5 + al0~':i) +c3[208(.i:~ + X~) + 20 10(3:2:1:.1 + Xa X5)  ¢rer] ~TCr.
(7.63)
The resulting system is then rewritten as =a = =4
=
+ .:;. . 2a.lD:l:4 bu ] + 20lD. j;sbu~ + 2alOX.l(a5X,1 + Ua X I X5  a7~'2) +2a·1O.1:S( 06:CI.i., + o,SXS  a7.1:3) +2(0.10:1:2  Oa:1:.,)( OSX.l  a9XIXS + aIOX2) +2(0'IOX3  Ua.1:S)(a9:t'lx"  os.i·5 + aloXa) + ca( ca=a + +201Q(aS.T..l  Oa3:IXs)X" + 2alO(aaXIX4 + as:cs)i's C:iZa
(7.6 Ll)
z",)  ;Prer
From Lemmas 2.8 and 2.26, the choice of feedback is (7.65)
This is the second equation to be satisfied by the two controls U,l and U2. Combining (7.59) and (7.65) we obtain t,hc following expression for UI and 1L:!:
This concludes our design procedure, whose schematic representation is given in Figure 7.4.
7.2
299
:MIMO DESIGN: INDUCTION MOTOR
Flux Observer
O'~Ubll'l +0.22b"2
o',u/JUI +0'.12bu2
...Jl .......................Jl .., I
I
!~ [11 1] ='b1 tt2
IL
I
[0'21
0'22]1 [
0:11 a.12
*] ~~
*
_______________________________ ~, I
Figure 7.4: MIMO backstepping design for all induction motor with a flux observer.
Feasibility and Stability.
From (7.66) we see that our control law is well
defined only in the region where the matri.."'\: D
= [a0:412L
0'22] a.12
is invertible.
The determinant of this matrix is given by
det(D) = =
= =
0'21a.12  0'220',n
(at.i 5 + 2d1 o·I:l:2 Z1)2o lO:i:5  (OlX,1 + 2dlai.:z:3Z1)2alOX4 2alalO(X~ + :t~) + 4dl oio'lO=1 (X2Xs  :Vax.)) 2al0'10Za  2alalOIPreC(t) + 4d1oiaIOZl(X2:l:5  xax.• ). (7.67)
Hence, D is not globally invertible, and (7.66) does not achieve tbe speed and flux tracking objectives globally. Nevertheless, under the natural assumption that the fiLL"'\: reference does not dictate demagnetizatioll, l/Jrer(t) ;::: {) > 0 for aU t 2:: 0, we can guarantee that (7.66) achieves the control objectivcs from a set of initial conditions that can be determincd a priori. To see this, we first combine (7.,,19), (7.58), (7.64), and (7.66) to compose
300
OUTPUTFEEDBACK DESIGN TOOLS
the system il Z2
+ at (X:4 X.1  X2 XS)  d1ai(x; + X5)=1 + =2 = C2=2 + W:!I.T..1 + W2S XS  d2 (wi.. + W~5)~  Zl =
2:3 =
=. . =
Cl=l
CaZa + z.. C4=<:I
(7.68)
+ W.14:EI + W45 X5
X.I
== UijX.,  2:1:]xs
X5
=

d4(W~..
+ W~5)Z4 
Z3
aaX5 + 2.1:lX4 •
= Z2
This system bas an equilibrium at =1 Furthermore, the derivative of the function
=
=3
= =.. = X4
= Xs
'} 1 (1 1 1 ) (_'J 2) " = ?1 (') Zi + Zi + =; + zJ. +? d + d + d X:j + Xs _08 ·1 'l
.')
!!
= o. (7.69)
<:I
along the solutions of (7.68) is llonpositivc: if =
clzl c2zi  c:i=i  c4zl ~ (d 4 '1 121 1
d1 (2d
:i)  d
.1:.1  Zl a I X J
1
cJ" (2~2 5:< 
Z,W24 )' 
1 d... ( 2d4. X.I 
z"W",..
)~

1
l
+ d + d ) (x~ + x~) 2
'.1
2
(2d Xs  :'la].'t2) 1
cJ" (2~2 5:5 """"')' ( 1 d4. 2d" i5 
Z.lW45
)2
::; c,zr  c2 z i  Ca Z;  C4Z~  ~ (:'1 + :.:! + :4) (X~ + x~).
(7.70)
Now if r/Jror(t) ~ fJ > 0, we see from (7.67) that dateD) ::; 2a·tatofJ < 0 Z3 = O. Fl'om this fact, the definltions of z], :'2, Z3, Z", and whenever =1 the Implicit FUllction Theorem (see, e.g., [B1}) , we conclude that there e..x:ist an open set :Fo containing the equilibrium 21 = =2 = Z3 = Z" oX" = .is = o and a function c,h(Zh =21 Za, .:", X", 3;5, WreC(/.),Wrof(t),f.{1reC{t), ~}ror(t», which is continuous 011 :Fn, such that on :Fa we have
=
=
det(D) ::; 2alaUI=3  2alOalO'l/Jrer(t) + 4d]aialOzliJJ. Then, ,ve define
(7.71)
.r as the largest subset of .1"0 on which
sup {20.1£l.lOZ3  2alolo'l/Jrcr(t) + 4d1aialOZtiJJ} < 0,
(7.72)
t~O
where in (7.72) we treat =h =21 Za, z."x.. , and X5 as independent variables and not as functions of time. Since wror(t}, 1/JreC(t) and their first and second derivatives are bounded, the supremum ill (7.72) exists. From (7.69), (7.70), alld the definition of:F, we see tbat any set of the form
U(c) = {(Zl' Z2, =3, :'4, x", Xs) I" < c} , where c is such that U(c) c :F, has tile following properties:
(7.73)
7.3
301
ADAPTIVE OBSERVER BACKSTEPPING
(i) U(c) is an inV"c1.riant, set of the closedloop syst.em, and (ii) det(D) < 0 011 U(c).
From these properties, the definitions of =11 =!:!, ;:;::11 =.j, Ill, and U2, and tbe bounded ness of wrer(t). 'l/Jrcf(t) and their firsl', and second derivatives, we conclude that for any initial conditions snch that (=J, =2, =3, :::,,11,1'4, Xs) E U(c'*), c· = sup {c}, all of the signals in the closedloop syst'~111 are bounded (note U{r.)cF
that the boundedncss of :i;~ + xg = ':3 + 'l/Jref implies the boulldedness of bot.h X,. and xs). Furthermore, from (7.69)(7.70) tlnd Theorem 2.1 we see that Zh::::hZ':·hZ'.J,l'.J,XS + 0 as t to 00. In pttrticular, this implies that =1 = W  Wtei + O. ::::1 = i:~ + x~  ""rof + 0 as t + 00, Since X.I = .1:,1  :r4 + 0, X4 = I5  5:s  0 as t + 00, Rlld X.s, :C4,.rS, fU'e bounded, we conclude that x~ + J~~ = 'l/J;a + 'ljJ;b  "'fref + 0 as t + 00. Hence, the speed and flux tracking objectives are achieved for auy initial conditions such t,hat. (Zit z!!, =;~, Z .. , .c." X5) E U( c·).
Simulation Results. We simulated the cont.roller designed in this section with the flux observer and compared its performance with the performance obtained when flux measurements are used. Itl silllulatiolls tile fllL,\: reference 0, the speed reference steps from sj;andsti1l to steps to its rated vaJue at: l the rated spoed at, t = 0.5 se(~, Rlld fiually the load torque steps from no load to 40 Nm (roughly 60% of the rt'Ltfld value) at t = 1.0 sec. Figure 7.5 shows the speed tracking for tile controller ,vitll flux measurement (a) and with the fllL,\: observer (ll). It is clear tht1.t the observerbased controller recovers the performance of the fullstatefeedback controller. Tbe beneficial effect of nonlinear dampillg is clear from Figure 7.5(c), which shows the performance of the controller without nonlinear damping, that is, with dt = d'1. = dol = O. The system now shows high sensitivity to flux error, which results in det.erioration of transient performance.
7.3 7.3.1
Adaptive Observer Backstepping An introductory example
Our udaptive design ror outputfeedback systems will be first illustrated plant with one unknown parameter:
Xl = 3;2 =
xa
=
'y =
+ 6IP} (y) ,1:3 + 6!.p';}. (y) + 1..1
011
a
;z.'2
tL
Xl·
(i.7.!1)
302
OUTPUTFEEDBACK DESIGN TOOLS
Speed 200
100
O~~~~~
a
L5
0.5
2
(a)
Speed 200
100
O~==~~~~~
a
0.5
1.5
2
(b)
Speed
200
100
O~===+
a
0.5
__~__ 1
1.5
2
(e) Figure 7.5: Speed tracking with (a) flux measurement, (b) flux observer and linear damping, (c) flux observer without nonlinenr damping.
nOD
7.3
303
ADAPTIVE OBSERVER BACKSTEPPING
'¥e assume that only the output y = .'1:1 is measured, that is, the states X2 and X3 are not available for feedback. We want to design an adaptive nonlinear outputfeedback controller that guarantees asymptotic tracking of the reference signal tlr(t} by the output y while keeping all the states of the closedloop system bounded. \Ve will attempt to reconst.nlct the full state oC the system through the use of two filters, one for the part. of the plant that does not contain 0 and one for its unknowll part. Denoting the states of the two :6.lters by €o and el, our virtual estimate is ~o + (}€l, which depends on the unknown parameter 8. The state is reconstructed as (7.75)
This expression is good enough for our backstepping design, provided t.hnt the errol' g tends to zero asymptotically. To ensure this, we construct t.he two filters as follows:
€01 eO!!
eos
= =
+ ~o2 + ~03 + /l. ~01) + 1J,
~!l(Y  ~01) 1'!2(Y  ~Ol)
= ka(y 
~11 ~1!!
= =
eUI
=
kl~ll + ~12 ~:2ell + ~la k3~ll'
+ tpl (y) + 'P2(Y)
(7.76)
The gain vector k = (kJ, k2' ~:3]T is chosen so t.hat the matrix 1
°° is Hurwitz. We then rewrite the plant and the filters x
= AD.'}; + ~~y + blL + O",(y),
The filters
b=
0] [~
1
(7.77)
~o
tp(y} =
and
~I as
[tpl(Y)] tp2~Y}
follows: (7.78)
.
€o and ~1 are rewritten in the more transparent form: ~o = Ao~o + ~:y + bu, ~1
= Ao~l + cp(y).
(7.79)
Combining these equations, we derive the equation for the error g =
e= =
=
=
~o  8~1 Ao.'}; + 1~11 + bu + (}cp(y) Ao(x  ~o  O~tl Aoe.
x~O(}€I:
:i; 
Ao~o
 ky  b1J. 
Ao8~J
 Otp(y} (7.80)
Thus g converges to zero exponentially since AD is Hurwitz. We are now ready to design our adaptive outputfeedback controller.
304
OUTPUTFEBDBACI< DESIGN TOOLS
Step 1. The control objective is to track the reference signal Yr(t) with the output y, so the first error variable is the tracking errol': (7.81) The derivative of =1 is (7.82) Since ,1'2 is not. measured, it cannot be our virtual control. We replace it with the sum of its ''vil'tual estimate" and the corresponding error: (7.83) Substituting into (7.82) we obtain
Zl =
e02
+ 9 ['PI (1/) + ~J2] Yr + e2· "
4J
(7.84)
~
W
Now we must choose one of the known variables appearing in the above equation as the virtual control. Looking at the filter equations, we see that, only t,lle equation for ~02 contains the control 11., so it is the only candidate. VVe introduce the second error v81'iable as (7.85) and substitute it iuto (7.84):
ZI =
Z2
+ 0'1 + Ow + C2.
(7.86)
Denoting the first estimate of 9 as '17 1, the first stabilizing function O't is chosen to be (7.87) 0'1 = CIZI  d,=t  t9 l w. \Vith t.his choice, the ztequation becomes (7.88) Our first Lyapunov fUllction is (7.89) where l' > 0 is the adaptation gain, and Po = Pl > 0 satisfies PoAo + A~ Po =
7.3
305
ADAPTIVE OBSERVER BACl{STEPPtNG
 I. The derivative of Vi is
,~
=
.!.(O  t9 1 )J 1 + !... ~
ZlZl 
d l dt
'Y
(eT Poe)
= "I ..  clzr  d,=~ + (8  171) (ZIW  ~.?I) + ZI<2  :1 eTe =
d1 ::;;
clzi + (0  Dd '1
ZlZ2 
(1:Y'"1.) =lW 
1]2 + e;:; 1 1  e e
[
2d1
T
'J
e.,
ZI 
4d 1
~
dl

CI"; + (8 171) (z,w  ~.dl)  4!/Te.
ZIZ!! 
(7.90)
The (8  t?.)term is now eliminated from (7.90) by the choice of update law (7.91)
Step 2. The derivative of =2 is expressed as Z2
= eo:!  al  iir = 'u + "~2(Y  e01) + (03  ~aliJ UY oat   (k:zen
ae],l
oO:t •
OJ
L
00'1'
+ e13 + Y'2(Y)) Yr  ilt OYr ofJ 1
•
tt2
= 'U + k:dY 
e01) + e08 
00:1
oy
••
Yr
~e02 + ~W + e2), fJ
Oa1 ( ,_   lV.!ell Oel!!
= u + k2(y 
(» + e13 + '1'2 Y
(0).
00'1
 Yr  OYr afJ1
001
eOl) + e03  ay e02 
.,
'Y WZ I Yr "'v"
61
oar
eel2 (k:zeu + el3 + 'P2{Y))
aat . aal oal oa1 .. Yr  'YWZI  Ow  e2  Yr'
of) J
OYr
ay
oy
(7.92)
Since the unknown parameter 8 appears in (7,92), we employ a new estimate t'J'J,. To counteract the disturbance e2, which is now multiplied by the nonlinear term ~t we employ nonlinear damping. The control is thus chosen as U
=
""%:I oa) ( + V~12 !3& 
ZI 
k
1:
d" (:: ) 2z.  k,(y 
12~1l
~Ol) ~oa + ~:I (eo. + 172w) 
1: aa:J. au} .. + ",13 + '1'2 (» Y + 8 Yr + of) 'YWZI + Yr, Yr 1
(793)
.
306
OUTPUTFEEDBAC({ DESIGN TOOLS
,vhich results in (7.94)
URing (7.90), (7.91) and (7.94), the derivative of the Lyapunov function
1., 1( ., = lli. + ":':; +  8 17.,)+ d'J,IT E PoE 2 2, 
tJ;,
(7.95)
is computed as
The (fJ  d 2 )term is then eliminated with the update law
. Ocrl 19., = "Vwz"

which yields
.
tJ;, < 
(7.97)
'oy'
.,  C'l=:;'l3(1    + I),. e
C] zl
~

4. dl
£.
~
(7.98)
'Vith (7.95) and (7.98), the LaSalleYoshizawa theorem (Theorem 2.1) guarantees the boundedness of .';;1, =2, 19 1, iJ 21 £ and the convergence of zl, =2,£ to zero. In particular, this implies that the tracking errOl' =1 11  y, converges to zero ~'Lnd that Y XI = %1 + Yr is bounded. From the definition of ~t ill (7.76) and the boundedlless of y we conclude that ~l is ruso bounded. To prove the boundedlless of ~o, Xa and X:J I we fust recall the definition e = x  ~D  (J{l, which implies that e01 = Xl  £1  fJ6 = Y  Cl  (J~1 is bounded. Then we note that w = CPl(Y) +';12 is bounded. Recalling (7.87), we see that 0:1 is also bounded. Combining (7.85) with the boundedness of Z2, D:l t ii, shows that ~D:l and ,7.2 = £2 + eO!! + ()~12 are bounded. To show the boundedness of X3, we define the variable, = X3  .1:2 + Y as suggested by (7.28). Its derivative is
=
=
( = = =
X3  :~2 +.1:1 .1:3  Ocp!!(y)  'U + :1:2 + (JCPt (V) ( + y + 8[IPI(1I)  '1'2(1/)] • 1I 
(7.99)
7.3
ADAPTIVE OBSERVER BACKSTEPPINO
307
Since y is bounded, , is also bouuded. Hence, ·'t3 = (+ X2  1J and e03 = are also bounded. Finally, from (7.93) we see that the control 11. is bounded. The effect of the nonlinear damping terms is illustrated by the mat.rix form of tbe error system:
3:3  C3 1J€13
(7.100)
We see tbat the llonlineax damping terms strengthen the negativity of the diagonal entries by including the squares of the factors multiplying the estimation error C2
7.3.2
Parametric outputfeedback systems
The above procedure is now generalized to the class of pa.ramet77c outpu./.feedback 811stems
x] =
p
X2
+ /PO,l(Y) + L IJj/Pj,l (J}) ;=1 1)
X2 =
X3
+ /PO.2(U) + L IJj /Pj,2(Y) j=1
p
:i:pl
=
:c,J + lPO,pl(Y}
=
Xp+1
+ L IJj«Pj,pl(Y)
(7.101)
j=l 1)
.Tp
xn = Y =
+ CPo,p(Y) + L Bjcpj,p(Y) + b"a/3(U)1I j=1
p
CPO,n(Y)
+ L IJj/Pi,n(Y) + bo/3(Y)'U j=1
Xl,
where 1J1t ••• IlJp and ho, ... ,bill are unknown constant parameters.!:! \¥e make the fallowing assumptiol1s about the syst,em (7.101): 2CoordinatcCrcc characterizatious of outputreedbuck syst.ems are given ill A))PClldix G, Corollaries G.6 and G.7.
308
OUTPUTFEEDBACI{ DESIGN TOOLS
Assumption 7.2 The sign of bna is
~".1I.o'Um.
ASBumption 7.3 The polynomial B(s) = b1l1sm + ... + bis + bo is Hurwitz. Assumption 7.4 f3(y)
# 0 'r/y E JR.
Assumption 7.5 The reference si.qnal Yr(t) and its fi"st p derivatives are known and bounded, (Ind, in addition, y~p)(t) is piecewise continuous. Assuming that oni,} the output Y is measured, the cont.rol objective is to trade the given reference signal Yr(t) with the output y of t.be system (7.101), while keeping all of the signals in the closedloop system globally bounded. The first step ill our design procedure is the choice of filters which will provide Uvirtual estimates" of the Ullmeasured sttl.te variables x!;!! ••• , X". As in the preceding suhsection, we rewrite (7.101) in the form p
x = Y
A=[~
j=l
=
(7.102)
cTx
 [
[(oI).(nl) ] ,
...
L BjcpAy) + bf3{y)tJ.
Ax + CPo(y) +
b
O(Pb~" 1, .
0
c= [
O(n~I)'l 1
(7.103)
110
IPj(Y) = ('Pitl (y) I . . . ! CPj,1I (IJ)) T,
(7.104) O'5.i'5.p· T VYe then choo.qe a gain vector k such that. Ao = A  kc is Hurwitz, and define the filtel's
';0 =
~j
=
'Uj
=
Ao{o + ~:y + 'Po (y) AO{j + CPj(Y) , 1'5.j~p Aovj + f,l_j f3(71)'U, 0~i:51n)
where ei is the 7th coordinate vector in can be obtained from the single filt.er
.\ =
nR.
(7.105)
Note that the signals Vo, ..• , t'm
Ao" + e,.,B(y)u.
(7.106)
through the algebraic expressions Vj
= (Ao)j...\, i = 0, ... 1m..
(7.107)
From (7.102) and (7.105) it £ol1ows that (7.108)
7.3
309
ADAPTIVE OBSERVER BACl(STEPPINO
which implies that c converges exponentiolly to zero. Therefore, the virtual estimate of x is + E~=l ()jei + Ej=o bj llj. In particular I the derivative of JJ call be e.."<pressed as
eo
m
P
iJ = ~O,2 +
(7.109)
The design procedure for the system (7.101) starts witb this equation and backstops all tile variables of tile filter '11m defined in (7.105).
Theorem 7.6 (Parametric OutputFeedback Systems) Cortside?' the nonlineaY' system (7.101) with Assumptions 7.27.5 and the filley's (7.105). The adaptive cont7'Olier 'U
fJ 1 iJ2 il"
°1
= = =
Q2
=
=1
Zi.
= /3ty} [a: p =
+ t?l,lY~P)]
(7.110)
sgn(bm)r [Wl(YI~(2)"D(2),y~J)}1jrel]=1
= r =
vm.p+l
r
(7.111)
[W2(Ylet2),i;(2),19(2),y~1)} + ZICp+m+l] %2 W'('ll c(i) v(i) .o(il) I • .H ~"
yOl)} . , i r ,
'J , ••• ...
(7.112) I
P
(7.113)
310
OUTPUTFEEDBACK DESIGN TOOLS
wi
[CJ Zl
+ d1z 1 + eo,::! + 'P0.11 4'1,1 + 6.2, ...
+ et),'::., VO,2,' .. ) V m l,2]
'Pl),l
1
(7.119)
w!
DQ'i1 [ a'PI,1 + 6,2, ... ,!Pp,1 + Y
~(i)
c [",0,1,'
'u(i)
[t10,1, ... 1 '00,11' .. , PmI.l, ... , 11111 1,1, 1.1 111 ,1, ...
1
i = 2, .. . ,p ..
ct.] 1 '::.0,'" .. 1 <"'1',1,· •• ,1.,,1',1 C
.
i = 1, ... , p
,
['I) f, ... 19T] ,
.fjU) r
[Url1irl' .. 1 y~i)] ,
 1
(7.121)
/,,1/11,i_l] ,
= 1, ... ,p
i. ,15(i)
1
(7.120)
(7.122) (7.12;3)
1, ... ,p
1
(7.124)
1, ... ,p ~o(l)
guarantees global bOll'Iulcdness of regulation of the tra.cking en'or:
... f.,Jt) , and vo(t), ... ,'lJm(t) and 1
[yU)  Yr(t)] = o. Proof. Design :::J
=
~lT'f'f'""d'Jj1r''''
!J  lln we write
=1
the definition (7.114) of the output error
as p
:'::J
:7:'2
+ lpO,l(Y) + "L))jipj,1(Y)
(7
 lir·
j=l
Since
.1:2
::1 =
is not measured, we use (7.109) to rc\vrite (7.126) as
~O,2
P
+ 'PO,1 (Y) + L
III
OJ
['Pj,l
(Y)
j=1
+ (j,2] + L
bj'Uj,2 
IiI' + "2'
(7
J::::O
The choice of virtual control in (7.127) is 11 111 ,2, because (7.105) reveals that the can trol '/l. appears in the p t.h deri vat.ive of ''Ill ,2, sooner than for any of the other variables in (7.137). If v m ,:! were the control and the parameters OI, ... 10,1, lJ m , .. 1 bo were knmvll then our choice of control law would be j
I)
1 =1
+ ell ': l + ~f),~ + ipO,l

lir] 
() ,
I: t
j=l
Tn
/1]1
[ip,;,l
+
b. b
"\' .Lv· ') . L.;
j=(J
J.
Ttl
(7.128) To deal with the presence of the unknown coefficient bm in front of v m ,:.!, \ve add and subtract t.he righthand side of (7.128) to the righthand side of (7.127),
(7.129)
7.3
311
ADAPTIVE OBSERVER BACKSTEPPlNG
where we have used the definitions (7.119)(7.124) and
06' = [~, ~ ..... /)111
bIT!'
Op 'b m
l
bo .... , bm '
bm
I
(7.130)
].
DClloting the estimate of 8u hy 19 11 we obtain the stabilizing function (7.116): . z::! = 'U m ,::!  crl VI,!,!)r II • " • (  ] It::) . ((.... '1')9) as = "f) lWl. USlllg as III {., u , we rewnl:e
0:,
Exploiting the fact thaI; I;he sign of /}m is known, we use Uw function
I  "21_::! + 2Ibm I ({f0 
V_I
(7,132)
.46
where P o..4 o + Po = I. The law (7.111) is then chosen to eliminate the (tio  '0] )term from the derivative of l): (7.133)
'Um,:l 
k2 v m ,1
Octl 
aU
oal
 aE,o [AoE,o + ky + 'Po(U)]
(7.134 ) where for the la13t equality we used the definition (7.120) for
W)2
and (7.135)
Using the estimat,e 172 of the definition 23 = V m,:!
ti,
the stabilizing function 0'2 defined in (7.117), and 191.1ijr from (7.115), we rewrite (7.13~1) as
0'2
(7.136) The derivative of the nonnegative funetion
l /1'
1 2
+ 1 + (fJ :2
(7.137)
312
Ou'rpuTFEEDBACl( DESIGN TOOLS
is computed using (7.133), (7.136), and the update law (7.112): (7.138) From here on the design proceeds along the lines of the example of Section 7.3.1. At the final step, tlle derivative of tbe nonnegative function VP1
(7.139) is rendered nonpositive: (7.140) Stability and convcrgence. Due to the piecewise continuity of y~p) (t) and the smoothness of the nonlinearities in (7.101), the solution of the closedloop adaptive system exists. Let its maximum interval of existence be [0, tr). On this interval, the npllnegl:l.~i"e function is nonincreasing because of (7.140). Thus, Zl,"" zP' 19 11 "" t9 p , and hence "'1, ••• iJ p , are bounded on rO, tr) by constants depending only 011 the initial conditions of the adaptive system. In addition, (7.108) shows that e is bounded. The boundedness of all other signals on [0, tr) is established as follows. Since Zl and Yr are bounded, it follows that y is bounded. This implies that f3(y) is bounded away from zero and, from (7.105), that eo? .. are bounded. From (7.105) we also see that
v;,
,e"
Vnt;.i
= [e; (sI  Ao)l e,,+;] f3(y)U,
0 S; j ~ m,
(7.141)
where ei is tIle ith coordinate vector in Rn. Then, \Ve express (7.101) in the differential equation Corm (D = dJdt) (7.142) Since y is bounded and, by Assumption 7.3, the polynomial B(s) = bms nl + .. ·+b1s+bo is Hurwitz, we conclude from (7.142) that Hp(s) [f3(y)u] is bounded, where Hi(s) denotes any e>"'Ponentially stable transfer function of relative degree greater than or equal to i. By (7.141), this in turn implies that the vectors v~~}), 0 ~ j ~ 111, are bounded, where _(i)
Vj
=
[ Vi,l,'"
]
I
V;,i •
(7.143)
7.3
313
AOAPTIVE OBSERVER BACl{STEPPING
In particular, by (7.122), this implies that vC!) is bounded. Combining (7.119) and (7.116) we conclude that WI and 0:1 are bounded, which ill turn implies that 'v m.:! = ;;2 + a1 + iJ1.1iJr is bounded. Hence, by (7.141), H p l (s)[,8h/)U], and thus 'jj~::;}, a ::; j :5 m, are bounded. This in turn implies that W!!, a'!! and 'Umt3 are bounded. Continuing in the same fashion, we use (7.115), and (7.J.cll) to show that Hi(s)[P(Y)u], p  2 ?:: i ~ 1, are bounded, wbich impJies that t' is bounded. Since {3(y) is hounded away from zero, we conclude from (7.110) that u is bonnded. Furthermore, from (7.108) we see that x is bounded. "ve have thus shown that all of the signals in the closedloop adaptive system are bounded on [0, tr) by constants depending only on initial conditions. Hence, tr = 00. The convergence of the tracking error to zero can now be deduced from the LaSalleYoshizawa theorem (Theorem 2.1), since =h"" =P and e converge to zero as t  t 00. 0
7.3 . 3
Example: singlelink flexible robot
As all example, we consider a singlelink robotic manipulator coupled to a de motor with a nOll rigid joint. When tile joint is modeJed as a linear torsional spring, the dynamic equations of the system are J1iil
+ F1cil + I\,(ql  ~) + mgdcosq1
=
a
q2)
=
/\Li
J" F!' /(( 2q2 + 2q2  N 91  N
(7.144)
LDi + Ri + /\bq:! = 'u, where ql and q2 are the angular positions of the link and the motor shaft, i is the armature current, and 'U is the armature voltage. The inertias .It, J!!, the viscons friction constants FIt F2t tile spring constant 1(, the t.orque constant J(Lt the backemf constant J(b, the armature resistance R and inductance L, the link mass wI, the position of the lin]<;'s center of gravity d, the gear ratio N and the acceleration of gravity 9 can all be unknowll. 'Ve assume that only the link position q} is lneasured so that n = p = 5 (m = 0). Let us examine whether our design procedure is applicable to this system. We first try the natural choice of state variables (1 = qlt (2 = f.b, (3 = Q2, (.. = lb, (5 = i. The dynamic equations (7.144) become
(2
= =
(3
= ( ..
(1
'.1
(2 mgd FI  I( ( (1  (3 ) cosy  (!! J1 J1 J1 N
(7.145)
Po .. + (5 1(, =  1\ ( ('I  (3 )  =(' J 21V
N
J2
J2
314
OUTPUTFEEDBACK DESIGN TOOLS
. (5 =
R
[(b
 L (5  T(4
1
+I U
Y = (I. Clearly, (7.145) is not in the outputfeedback form (7.101). However, there exists it different choice of coordinates which brings (7.Vl~..I:) into t:hat form. To sbow this, we derive an inputoutput description of (7.145). Differentiating y twice, we obtain (2 = Dy (D = is the difrereutitttion operator) and
1t
(3)
'1 Tngd FI [( ( Dy=cosyDy Y  J1 J1 J1 lV
t
which implies that (7.14i) (7.148) Differentiating (7.148) and substituting (3 and (.1 from (7.147) and (7.148), we obtain
(7.149)
Using (7.150), it is tedious but straightfofWll.rd to lind a. choice of state variables
7.4
315
EXTENSIONS
which bring (7.144) into the form (7.101), Xl
3:2 3.'3
X4 2:5
= = = = =
Y =
X2
+ fhy
+ (JJ.Y + 83 cos 11 :CI + 9.l y + 85 cos y Xs + 8ay + 07 cosy 68 cos Y + bOll X3
(7.151)
Xl,
where the unlalo,vn parameters 8ft , •• , Os, bo are defined as
81
=
82
=
(J3
=
6.,
=
85
=
86
=
87
=
6s
=
bo
:::::
7.4 7.4.1
Extensions Interlaced controllerobserver design
In the observerbacksteppillg procedures presented ill this chapter the obsprver design is independent of the controller. 'Ve first design a st.ate observer (or filters ,vhen ullcertaint.ies are present) and then use nonlinear damping in the
316
OUTPUTFEEDBACt<: DESIGN TOOLS
cOlltrol1er design to counteract the effect of observation errors (.r or e). In particular, the quadratic nonlincar damping terms dominate the crossterms containing observation errors in the derivative of the Lyapunov function, thereby rendering it nonposit.ive. III this subsection we present an alternative approach, which introduces c(Jl1"ection te.7'1r1.S ill the observer in order to cancel (instead of dominating) t.he crossterms in the Lyapunov derivative. As a result, the design of the observer is intc7'laced with that. of the controller. Inspired by the tuning functions of Chapter 4, this approach introduces inte7'lacing junctions at each illtermediat,e step of the design procedure. These functions are analogs of the tuning funct.ions for state estinul.tioll. At the final step, the last interlacing fUllction becomes t.he correct.ion term for the observer. As an example, consider the syst.em .i:) = .T!! =
x~
+ (1 + Xi),l:2
.1:2
+U
(7.153)
I
where Xl is mea..lOjured but :l'2 is not. The observer for term X yet to be designed:
.1:'2
includes the correction
(7.154)
If we were to choose X == 0, we would need nonlinear damping terms to guarantee stability, Here we avoid those terms by designing :\ to cancel all indefinite terms in t.he derivative of the Lya.pllnov fmIction. The equat.ion for t.he observcr error is (7.155) Instead of applying backstepping t.o the system (7.153), we apply it to the system
x)
= .1.'1 + {1
.1:2 =
,i~2
+ :r.i).T2 + (1 + Xi)X2
+ 11 + A '
(7.156a) (7.156b)
This form suggests how to employ the tuning functions t.echnique. While the state estimation error '~2' which appears linearly in (7.156a), plays t.he same role 3..<; the pw'ameter estimation error 8 in the tuning rUllc~ions ~esign, t.he observer correction term X = .i2  X2 should be viewed as iJ = 8 = rTp. Using z = X2 + Xl and (7,155), we compute the derivative of Vi = ~:z:i + )
_'1
2do ,1:: 2:

(7.157)
7.4
317
EXTENSIONS
If this were the 1ast step of the design procedure, we would eliminate .1':2 from (7.157) with the choice X = dn/.I! where (7.I5S)
Since we have one more design step to go, we do not use \' = dnl., as our correction term. Instead, we retain ',1 as our first inte·,.[acing Junction and 1'ewrite (7.157) as •
Vi
=
'l
:L:i
+ :1:1 1 + xi)= (
The derivative of :
'J
1
_'l
do·l·~
_
+ J.·2
[
II 
1]
dnY .
(7.159)
= :1:2 + .1.'1 is (7.lBO)
Since t.he actual control u. nppears ill (7.1BO), we deRign 'It and the actual cm'reeting ternl. X to st.abilizc the {Xli =, .r2)syst.em wit.h respect t.o the augmented Lyapunov fUllct.ion V2 = Vi + ~=:!. Using (7.159) and (7.1BO), the derivative of 112 is computed as 
xi  :0.r5+ ={Xl (1 + .1·i) 
112 =
+X2
X2
+ 11 + h' + [.1::t + (1 + .Tn.i::!]}
[(1 + =(1 + xi)  ell n] .
(7.161)
To eliminate J:2 from (7.161), the correction teI'Ill X is chosen as
(7.162) Then, the choice "II
=
=  .1:1{1 + .1.'i) + :i':! 
(2  [3'~ + (1 + .1:i)X:!]
(7.163)
yields l~ = :ci  ;/.  Jx~, which implies that the (3:h =..1:2)system (or! equivalently, the (3:1, .T.2, x2~systelll) has the odgin as a globally exponent.ially st.able C'quilibriulll. This interlaced design can be generalized to outputfeedback nonlinear systems of t.he form (7.27) as follows:
Corollary 7.7 (OutputFeedback Systems) For the n071linea.r system (7.27), assu.me that IJ m ,s7n + ... + b1s + bo is a. Hu.nlJit= polllTlornia/, and that Yr, Yr, ... , y~p) a1'e J.:no'llJn and bounded on [0,00) and !J~p){t) is piece1lJise continuou.s. Then, the contml t1
=
(7.164)
318
OUTPUTFEEDBACK DESIGN TOOLS
and tile observer
with
+ ~~(y 
x =
Ai~
iJ =
cTj;,
li) + lP{y)
+ bP(y)u + DI,p
(7.1G5)
ai, I.i, i = 1, ... t p defined by the following recursive expressions (Po = 21, dfJ > 0, Ci > a, i = 1, ... , p)
Zi)
pl > 0, PoAo + A~ Po = =
=1
Y  Yr (7.1GG) Xi  O:i_I(Y,Xt, •.. , XiI, Yrt ... , y~i2})  y~il), i = 2, .. , 1 P (7.1G7)
. =
...., 0:1
=
ai
=
CIZI 
c,,,, iI
'PJ{,/)
(7.1GB)
erdo', + 8:~' [X2 + 'PI(Y)]
k,(y  XI)  'Pi(Y) 
ZiI 
lJ "
i::! {) '.
+ E {)O~~l [Xj+l + kAy  Xl) + epAy) + eJdol'i] + L: a~;)l Vp+l) x,
j=l
+dOPD 1C2
j=l
E(e&.  E{)aa:~~l
k=2
=
1,1
ia =
j=I
OYr
i = 2, ... , P
Cj) T 80/:J'il =kt
(7.169)
Y
''C J
(7.170)
PO Ie2=1
8a'il Ra 1e2 ay=i'
"il 
.?
I = .... , ...
(7.171)
,p
guarantees global boundedness of x(t), x{t), and regu.lat.ion of the tracking er1'01':
Jim [yet)  1Jr(t)] = O.
(7.172)
Ico
Proof. From (7.27)1 (7.165), and the definitions (7.164)(7.171), we call express the derivatives of the error variables ZI
%1, ••• , =p
as follows:
= iJ  ilr = X2 + epl{Y)  ilr = X2 + X2 + lPl{Y)  ilr =
Zi• =
Z2
+ itl + 'Pl ClJ) +:1:2 =
A
il
L
+ Z2 + X2
(7.173)
T
A) + eTdotp + t.fJ( () y  aa:iJ [A3:2 i
r ( + h:j ;rJ 
Xi+I
CIZI
Xl
] + ~l () Y + 3:2
a
~'~~1 [Xj+l + kj(y  Xl) + qJj{Y) + eJdoLp]
i=1
uX J
i2 a
U+1)
' " UO:il
 .t u>11r j=I
=
81ft
cz.  z· •
1
1
1
(i)
 Yr
(e. _~ EJO:8 . . c.)
+ ""+1  80:;1 aY XII + do •
iI (
d R l ' " +OOe2L."
k==2
1
L."
j=l
i 1 _
XJ
J
T
"1 n ) T 8 O'il _ '"" UOkl ' _ ? CkL.,,••.a Cj {)';'k,1., ... j=I''CJ Y
(t p _ t·) I
,p
(7.174)
7.4
319
EXTENSIONS
The resulting error system is
(7.1i6)
For t.his error system, consider the Lyapunov function
~ 2 V( z,fi ) = ?1 ~Zj + ?d1 a:T Pox . ... j=l

0
(7.177)
320
OUTPUTFEEDBACK DESIGN TOOLS
Its derivative along the solutions of (7.176) is llonpositive:
(7.178) Thus, Zt, ... I zP' i are bounded and converge to zero. The remainder of the 0 proof is similar to that of Theorem 7.1.
7.4.2
Design with partialstate feedback
All t.he nonadaptive and adaptive schemes we have presented so far as well as those to be presented in the remainder of the book, assume either that the full state is measured, or that only one scalar output is available for feedback. In this section we present a natural generalization of these results which bl'idges the g~tP between fullstate and singleoutput feedback. Combining the design tools we have developed for these two cas~., we call easily incorporate information available from additional sensors and accommodate systems whose nonlinearities depend on measured state variables only. Such designs are applicahle to systems in partia.lstatefeedback form which have k groups of measured state variables, denoted by XlI'" I XmJ and Xn .+1, . .. ,xmi+I' i = 1, ... ,Ii:  I, and k groups of unmeasured variables, denoted by xTlli+lt .•• ,xn • , i = 1, ... k, where 11 k = nand mIt ~ p. For notational convenience, we adopt the following definitions of the vectors of measured variables for 1 ~ i ~ k : I
As we can see, the nonlinearities in (7.180) are strictfeedback nOlllinearities, since they depend only on measured state variables which are fed back: In the
7.4
321
EXTENSIONS
xiequation, the nonlineal'ities depend only on measured state variables up to Xi An example of such a system is3 P
Xl =
X2
+ 'Po•.(xa) + l:lJj'Pj,l(:t'I) j:::::1
:C2
=
,1:3 + 'Po.:;! (Xl I :1"2) +
1l
L 8j rpj,2(J;t, :J::!)
j=l
,1
Xml
=
Xm1+l
=
:1"n 1 +1
=
,1:'n,+2
+ 'PO,ml (Xfllt) + L
OJ'Pj,mt (x'" I )
j=1
p
Xnt
+ lPO,n
(xm I )
I
+L
0jl.pi,n I (:e" I )
i==l p
X,. 1 +1
+ <;'0,111+1 (;Z;nl(, XJl1+l) + .L: IJj
1Z 1+1
(Xliii, J~JlI+I)
j=1
J)
Xm :!
=
J:nI:I+1
=
.1: n2 +1
=
3:nl1.+l
+ 'PO,m:! (x m.,) + L Bj!.pj,m2 (X IJI") 
(7.180)
j=l 1)
XJI :!
+ 'PO,"2(X
rll
:l) + L 8j l.pj.lI:1 (3;m:!) j=l
p
Xmk
+ lPO.mA:(Xm) + LBjlPj.n&,,(:z:m) j=1 p
Xp 1 =
Xp + 'PO,pl (X nl )
+L
(Jj'{JJ,f1 1(Xm)
j=1 l'
Xp
=
Xp+l
+ lPo,p(X + L ()j'Pj,,.(x + bn_pfJ(xln)'lJ, fll
Ul
)
)
j=l
l'
Xn = 1,00,7.(3:111 )
+ L (}j'Pj".(X + boP(x Ol
)
lll
)1t
j=1
y =
Xl 
3DHfercnlial geometric charact.crizlltions DC partialstat.eCeedhack systems are given in Appendix G t Theorems G.2 and G .5.
322
OU1.'PUTFEEDBACI( DESIGN TOOLS
In (7.180), x E nn is the state, U E lR is the input, y E lR is the output, 0 ::5 j ~ p, 1 ::5 i ~ ll, and {3 are smootl1 nonlinear functions, and fJ = [OJ, ... ,fJp]T E lRP and b = [bn _ p,"" bolT E nnp+l are vectors of unknown constant parameters. 'Pj,i,
UndE"r Assumptions 7.2, 7.3, and 7.4 (with y E lR replaced by k
fn.
= ml + L)m.i 
a;m
E
m.n"
nil », we can design adaptive controllers which guarantee
1=2
global stability and tracking of reference signals satisfying Assumption 7.S. The design procedure starts with the choice of filters which will provide "virtual est.imates" of the unmeasured state variables. In (7.180) there are 1.:: groups of such variables: Xm.+l, •. " XII" 1 ~ i :5 k, with mk :5 p and nk n. Following the development of Section 7.3, for each of the first k  1 such groups we consider the subsystem
=
p
:i: nt•
=
:l:'m,+l
+ 'Po,lJI.(xmi ) + L
m (Jj'Pi,m, (x ,)
j=l
(7.181) p
=
:i:nl
Xni+l
+ IPO,ni(.1;7Iti) + 'L,8j CPj,Ri(Xrni ) j=l
where x na, and in the form
Xni+l
:i:ni
=
are measured. The subsystem (7.181) cnn be rewritten p
AiXR,
+ cp~i(xmj) + L {}j cPj (x I
nli
)
+ bi.T.n.+J
j=l
(7.182)
where, denoting ki
= ni 
mi,
we have for 1 :5 i :5 k (7.183)
Ai
=
[°
: I!.:jXk,],
c'.  [ 
0 ... 0
c,oji(xmi)
= (CPj,mi(Xmi ), •.. , rpj,Ri(Xmi)T,
o~ j
1
Oki X l
1
~ p.
(7.184)
(7.185)
For each of the k  1 subsystems of the form (7.182), we choose a gain vector I(i such that A~ Ai  J(iCiT is a Hurwitz matrix and we define the filters
=
e~ ~;
= =
A~~~ + I(i xmi + 'Po'(.'l:'mi) + bix"i+l A~~; + !pj' (.1: m .), 1 5: j ~ p.
(7.186)
7.4
323
EXTENSIONS
It is then straightforward to verify that Ej
~ X""  (e~ + tOleJ) .
(7.187)
]=1
Ab
Since is Hunvitz, (7.187) implies that the signals Ei converge exponentially to zero. Furthermore, we note that the derivative of .'t'm can be expressed as l
!inti = e~,2 + CPO.nli(X"'I)
P
+L
OJ [~j,mi(xml)
+ EJ,2] + €'it~!'
1 '5:i '51,:1.
j=l
(7.188) For the last group oC unnleosured variables we consider the subsystem (recall tllat ?nlt :5 p, 'nk = n, xml: = xm) 7J
.1:111 1: =
X mll +!
+ lPO,mk(Xm ) + L iJjCPj,Ulk(X m ) i=l
P
X,Jl =
Xp
+ CPO,p_l{X m ) + I: Bj CPj,pl (.r: m ) j=1 p
Xp =
Xp+J
+ 'PO,p{x ,n ) + 2: OjIPj,p(•.,;m) + b
tl _
p{3{x Jn )U
(7.189)
j=l
Xn = CPO,n(Xm) +
p
L Bj'Pj.n{xru ) + boB(xm)u, j=1
which can be rewritten in the form p
j;n,.
=
AkXnk
+ CPOI:(Xm ) + 2: 9j 'Pj'" (xm) + bk B(xm)u ;=1
(7.190)
where
(i.191)
Again, we choose a gain vector and we define tl1e filters
= ij = ~~
](k
such that A~ = Ak _](krJt.T is Hurwitz,
A~e~ + ](kxmk + 'Pol: (xm) A~eg + cp.?(xm), 1 '5: j ~ p
~ = A~A + ~{3(xm)u,
(7.192)
c,· = [0,. .. 10, IJT .
'Ve also define the signals Vj
== (A~)j..\, j = 0, ... ,n  p.
(7.193)
OUTPUTFEEDBACl< DESIGN TOOLS
It is then stl'aightfonvard to verify that these signals satisfy the differential equat.ions
. = 'OJ
AkO"j + e,,mj.,+lj f3( :rRl) 'u,
05i=:;71p,
(7.194)
where fi is the ith coordinate vector in .Dl"Pl~.+l. From (7.190), (7.192) ~Lnd (i.194) it follows that (7.195)
which implies that eA convprges exponentially to zero. 'Ve also note that, the derivative of :J:"'k call be expressed as :i:",J. = {~.2 + IPO,fII1.(:t·ln )
,I
+ L 8j
up
[lPj,1IIkC.1:
j=1
Ul )
+ e;;2] + L: bj'l'j,2 + Ek.2·
(7.196)
j=O
Once tihese filters have been designed, the design proceeds as follows: Starting with = x.  Yr, any of the fullstaiefeedbt:tck design procedures can be applied to the first group of measured st.ates .1:1, ••• , ;1: 1111 1 For etl.ch of the subsequent measured subsystenls of (7.180), addit.ional nonlinear damping terms are needed, since unmeasured states J:tppear in the corresponding =iequat;io115 and have to be replaced by their "virtual estimat.es." AB for the unmeasured subsystems of (7.180), folImvillg Section 7.3.2, their equations are replaced by the {'orresponding equations from the filters (7.186), amI then allY of the design procedures presented in this or the rollm~,..ing chapters call be R))}lliecl.
=.
Notes and References EVE'n before the development of adaptive backsteppiug with fnllstate feeclback, the more challenging out.put.feedback problem was addressed by Kanellakopoulos, Kokotovic t and Middleton under restrictive st.ructural and grQwth conditions on the llonHllcm'ities via an extension of adaptive linear techniques [67, 68]. Subsequently, the growth rest.rictiolls were removed by Kanellalmpoulos, Kokotovic, mId lvlorse [70 t 71], applying all early adaptive scheme for linear systems by Fene.r and Morse [35]. Still, the output nOlllinearities wel'E' not allowed to precede the control input. Adaptive backstepping [69] creat.ed the possibilit:y for removal of this structural restriction. Combining adaptive backst.epping with their adaptive observers [121] aud outputfeedback linearization with filtered tl'ansfol'matiol1s [120], lVlarino and Tomei [122, 123] developed a new adaptive scbeme for systems in the output feedback form. This is still the largest class of nonlincar systems for which global boulldedness and tracking of arbitrary trajectories can be achieved llsing
NOTES AND REFERENCES
325
output feedback. A more general class of syst.ems was considered in [124], but only for the setpoint regulation problem. The design summarized in Theorem 7.6 is an alternat.ivc approach for t.he same class of systems due to Kanellakopoulos, Kokot.ovic and iVlorse [72]. Tbis design employs the filt.ers (7.105) patterned after Kreissehneier [01). Bot.h of these outputfeedback schemes, (122, 123] and [72], inherited ovcrparametrization from [69]. Partialst.atefeedback designs were first developed by Kallellalmpoulos, Kokot.ovic, Ivlarino, and Tomei [66] using adapt.ive backst.eppillg and filtered transformations. The design we present.cu in Sec·tion 7A.2 was developed hy Kanellakopoulos, Kokotovic, and IvIorse (74]. The idea of interlacing 1:11c design of t.he observer with that of the ('011tl'oller was developed by Kancllakopoulos. Krstic, and Kokotovi{' (78] and was subsequently used for the design of passive arJapl:ive systems by Kanellalmpoulos [64]: as well as for induction motor control by Kanellakopolllos, Krein: and Disilv(lo:;tro (77, 75]. An out.putfeedback result fo1' systems with UllcE'rtain parameters (of known bound) ent.ering nonlinearly was developed by Ivlarino and Tomei [12'J] for t.he case of set. point regulation. Praly and .liang [150] Rolved the stabilization problem for a class of systems broader than the out.putfeedback form: where the nonlinearities also depend on the ullmea.surecist.at:c of nonlinear ISS inverse dynamics. Teel and Praly [191] e.xtended the !'esuJt. of [159] for the ('ase of uncertain nonlinearities. Induction motors served as testing ground for mu]t.ivariable nonlinear designs. The physically motivated fieldorientation design by Blaschl\:e [9] was followed by feedback linearization designs of Krzeminski [106] and Nlarino, Peresada, and VaIigi [119]. \Vhile these schemes assume flux measurement, most implemented controllers use fllLx observers such as thoRe proposed by Verghese and Sanders [194]. The scheme present.ed in Sect.ion 7.2 was developed by Kanellakopoulos, Krein, and Disilvestro [76) and subsequently improved by Bu, Dawson, and Qian [46], who obtained global st.ability and tracking results through a modification of t:he flux reference signal. A different, physically motivated approach, was recently developed by EspinoRa and Ortega. [33].
Chapter 8 Tuning Functions Designs In Chapter 7 we presented an outputfeedback design for systems in the outputfeedback form, and started the development of adaptive schemes by designing output;feedback adapt.ive backstepping controllers with overparametrizat.ion. In this and the l1e~t chapter we develop outputfeedback adaptive nonlinear controllers without overp81'ametrization. This chapter extends the tuning functions design of Chapter 4 to the out.putfeedback case. The next chapter e:3l1;ends the modular designs of Chapters 5 and 6 t·o the outputfeedback case. We present designs that use different HJter stnlct.ures and identifiers. This chapter can be read immediately after Chapter 4. As in Chapter 7 we consider systems in the outputfeedback form.: q
Xl = :1:2 + 'Po, I (y)
+ L Djtpj.1 (y) j=l
q
:i:p _ 1 = X(I + 'PO,pl(Y)
+ L OJ'Pj,pl(Y) j=1 q
,1:p
=
Xp+J
+ 'PO,(I(Y) + L Dj!t'i,p(Y) + brrau(y)·u
(8.1)
j=l
q
. Xnl
=
Xn
+ IPO,nl(Y) + L OJ<'oj,II1(Y) + b1u(y)u j=1 q
xn
=
'PO,n(Y)
+ L aj!{Jj,n(Y) + bou(y)'U j=1
Y =
Xl,
where :r E mn is the state, u E m. is the input, ?J E lR is the output, 'Pj,i, 0:::; j ~ q, 1 S i :::; n, and u are smooth nonlinear fUllctions, and a
= [a'1I'"
,a.q]T E m,q, b = [b m , ••• ,bolT E lRnI + 1
(8,2)
32B
TUNINC FUNCTIONS DESIGNS
are vect.ors of unlmown constant parameters. Only t.he output y is available for measurement. We rewrite (B.1) as
:i: =
Ax + q,(y} + if>(y}a. + [
Y 
err". 1"°1
~ 1a(y}u.,
(8.3)
where
~ 1"_1] [ o .. ·0
A
rp(y) =
[
(8.'1)
'aOl,l(Y) ••• 'aOqt~~(IJ)] •
'PO,l{Y) ] : ,
tll(y)
=: [
'PO,n{Y)
_
tp) ,11 (y)
(8.5)
'aOq,n (y)
The control objective is t;o track a given reference signal Yr{t) with the out.put y, while keeping all the signals ill the closedloop system globally bounded. \Ve mal{e the following assumptions about the system (8.1):
Assumption 8.1 The sign of bm is known. Assumption 8.2 The polynomial B(s) = bmsm be Hu·rwitz. Assumption 8.3 a(y)
+ ... + bis + bo is 1.....,l,own to
# 0 Vy E JR.
The class of systems (8.1) is restrictive because the llonlinearities depend only all tbe output. However, this restriction, is not imposed because of adapM tatiOll. It is needed evell when the parameters are known. It was showll in l127] tbl1.t the system
Xl X2 y
= =
=
X2
:r;~
+u
(8.6)
Xl
cannot be globally stahilized by dynamic output feedback for n > 2. In other words, the clclSS of systems which are globally stabilizable by output feedback is not much broader than the class of output feedback systems (8.1). In the absence of full state measurement we need filters, both in the design of an outputfeedback control law and in the design of an identifier. We present designs which employ two different sets of filters:
• I(filters, originally proposed for adaptive observer design by Kreissebneiel' [91] . •
ltIT~filters,
developed by :Marino and Tomei [121, 123].
8.1
329
DESIGN WITH KF1LTERS
the designs with K~:fi.lters, and briefly present Ule designs with for completeness, There are considerable differences between tIle two families of designs, and we will comment on t:heil' relative merits. In this chapter we develop output feedback designs using the tUlling functions approach.
'Ve focus
all
A1T~filters
8.1 8.1.1
Design with KFilters Filters and observer
We strut by re\vriting {8.a} as
where the p = q + m
x =
A:t: + r/J(y} + F(y, alTO
y =
e'}'x,
(8.7)
+ Idimensional parameter VectOl' 8 is defined by
8=[!], a.nd F(y,u.?
= [[ O(p;~;~+I) ]O'(y)u,
(8.8)
iJ)(y)] .
(8.9)
If 8 were know II, we would design an observer
.f =
Aoi; + ky + ¢(y) + F(y, u.)'f () ,
(8.10)
with the vector h": = [k1 , ••. , kn]T chosen so tbat. the matrix
Ao
=A 
keI
(8.11)
is Hurwitz, that is, PAo+A~P=l,
Theu, the observer error stable system
x=
a= 
p=pT>O.
(8.12)
x would be governed by the ex.ponentially ~;:
= Aox.
(8.13)
Since () is not known, the observer (8.10) is not implementlthle but it provides motivation for the subsequent development. We define the state estimate (8.14)
which employs the filters ~
n
T
= Aoe; + b':y + q,(y) = AOOT+F(y,u)T.
{B.15} (8.16)
330
TUNING FUNCTIONS DESlONS
The state estimation error e=X!i;
(8.17)
i=Aoe:.
(8.18)
is readily shown to satisfy The llomninimal observer (8.14), (8.15), and (8.16) is still nonimplementable because it depends oulJ. However, it has a key property not present in (8.10): The state :c satisfies a static relationship with OJ that is, (8.19) This is easily verified by substituting (8.14) into (8.17). Remark 8.4 The certainty equivalence counterpart of the est,imate (8.1"1) is (8,20)
It can alternatively be generated via (8.21) This obs~rver is 110t a certainty equivalence version of (8.10) because of the term 0
nTo.
To reduce the dynamic order of the n:6.1ter (8.16), we e..xploit the structure of F(y, u). Denote the first 11l.+ 1 columl1s of T by Vm , ••• , VI, vo. Due to the special dependence of F(y, u) 011 a(y)u, the vectors 't'm, ••• ,?lIt vo satisfy the equations (8.22) j=O, ... ,m. i 1j AO'lJj + Cnj a{y)1J.,
n
=
It easy to show that j = 0, . , . 1 n.  1 .
(8.23)
Therefore, the vectors Vj are generated by only one input filter ..\ = AoA + enuu ,
{8.24}
with the algebraic e:\."P1'essions 1tj
= A~'\,
j=O, ... ,m.
(8.25)
While we always implement the filter (8.24), fm' analysis we use the equations (8.22). No\V, in vie\V of (8.9), (8.16), 8lld (8.22), is obtained as
n
(8.26)
8.1
331
DESIGN WITH KFILTERS
Table 8.1: KFilters Kfilters:
~
~\
= = =
Aoe + kg + c1J(y) Ao::: + cD(y) AOA + fnU{Y)U
Vj
=
A~A,
.i =
nT
=
[Vm1
'1'1, 'VO t
••• ,
(8.28) (8.29) (8.30)
(8.31)
0, ... , m
(8.32)
where tIle matri."'( ::: is generated by
3 = A D3 + «!l{y),
(8.27)
The implemented filters are summarized in Table 8.1. The tot.al dynamic order of the Kfilters is n{q + 2). As explained in [63], a further reduction is possible by using the reducedorder observer technique, so that the total filter dynamic order becomes (n  1)( q + 2). To prepare for the hackstepping pro("ed urc ill the next su bsectioll, we consider the equation for the output y = Xl rewritten from (8.1): (8.33) We need to replace the unavailable state (8.19) we have X2
X2
by available filter signals. From
=
€:!
+ n~)8 + C:!
=
€2
+ [Vmt2t vm l.2,· .. ,")0.2,
=(2)]0 + E."2
= bm V m ,2 + €2 + [0, 'Vm 1,2,""
VO,2,
3(2)]0 + C2.
(8.34) (8.35)
Substituting first {8.34} and then (8.35) into (8.33), we obtain the following two important e.xpressions for y:
iJ
=
Wo + w T 8 +
(8.36)
C2
= bm V m .2 +wo+WTO+c::h
(8.37)
where the 'regressor' wand the 'truncated regressor' ware defined as W
=
[v m .,2, Vm l,2, ... ,Vo,:!,
(j)
=
[0, 'VrrI l,2, ••• ,'VO,2,
q,(l)
«;P(l)
+ =(2)J T
+ 3(2)]T ,
(8.38) (8.39)
332
TUNING FUNCTIONS DESIGNS
and Wo
8.1.2
=
+ ~2 •
(8.40)
Adaptive controller design
The tuning functions design with Kfilters has many similarities with the st.atefeedback design presented in detail in Section ...1..2.1. It also uses the same technique for dealing with unknown high frequency gain brll as in Section 4.5.1. vVe present the design for the case p > 1. The first obstacle to backstepping with ontput feedback is t.hat the st.ate .1:2 is not measured. For this reason, (8.37) is written in a form which suggests that the filter signa] 'lIm ,2 be used for bacl\:st.epping. Indeed. by comparing (8.1) with UrI. = Aovrn + cpu(Y)'/J. (S,41) (the latter is obtained from (8.22) for j = 1n), we see that, both J:2 and V rn .:! a.re separated from t.he control u by p 1 integrators. A closer examination of tLte .filters in Table 8,1 reveals that more int.egrators stand in t.he way of 8lly other variable in (8.37). Therefore, the system to which we will apply backstepping is
if = bna'v 11m ,i = 'um,p =
+ Wo + (jJT8 + ~2
rtI ,:!
Vrl.,i+l 
i = 2, . , . , p  1
kjvm.1 ,
U(lI)U. + l'm,p+l
(8.42)

(8.43)
(8.44)
kpVrrJ,l •
Our analysis will show that once we stabilize the system (8.42)(8,44), alIt,he closedloop signals remain bounded. The second outputfeedback difficulty is the presence of the disturbance ~:! in (8,42). In the process of backstepping, £2 will be multiplied by nonlinear terms, which can destabilize the system even though ~2 is bounded and exponentially decaying. To prevent this, we will include nonlinear damping terms in our stabilizing functions. To avoid repetition, the backstepping de.~igll will not be presented in a stepbystep fashion as in Section 4.2.1. We present only the first step in detail and then proceed to the complete error system, for which we choose the st.abilizing functions and the tuning functions to achieve a desired slcewsymmetric form. For system (8.42)(8.44) we define a change of coordinates 2:1
=
, "". where
(8.45)
Y  Yr
v111,1. 
ny(iI) ~. r
Bis au estimate of {! = l/bn
0:' 1 1
.
,
7.
?
= ... , ... ,p,
(8,4G)
l'
Step 1. The equation for t.he tracking error =J obtained from (8.45) and (8.42) is
(8.'17)
8.1
333
DESIGN WITH KFILTERS
By substituting (8.46) functioll 0:1 as
fOf
= 2 into
i
(8.47), and scaling tIle first stabilizing (8.48)
we rewrite (8.47) in t.llC form =1
= al + WI) + w'rO + c::! 
bill
Wr + ill) B+ bm =,!.
(8.49)
Then Ute choice of the stabi1izing fund.ion (8.50) results in
(Two posif:ive constants Cl and d1 arc introduced for uniformity with subsequent steps ill which d j is a. coefficient of a nonlinear damping term counteracting c:!.) \Vith (8.46), (8.48), and (8.38), we have 1'W f)
+ bm ':2
=
Tn+bW C1 m=2
=
iiJTB + (vm ,:!  Ulir  nd eiO + bill'::! T
+ I"}m=:! T
W 0  fJ (Yr + iiI) eI 0 + b",':2 = (W  (Yr + d:d edT 8 + 6",':2 •
=
4
e
(8.52)
Substituting (8.52) into (8.51) we get
Zl =
C1Z1
d1=1 +E2+{W 
g(Yr + ih} Ct}'r 8 iJm(Yr + d'l) B+bm~'
(8.53)
As in Section 4.2.1, the cboice of the update law for {) is postponed unt.il the last step a.nd we only select the first tuning function (8.54) Howevef 1 the updat;e lnw for
g is designed
at, the first step as
" > O.
(8.55)
\Ve pause to clarify tlle ru.'gllment.s of the fUllction 0'1 By examining (8.50) along with (8.39) and (8.40), we see that 0'1 is a fUllction of 7J,~,S,6,e,.Vo.2, ... ,'LIm).2 and Yr' For brevit.y, we denotr ,x = (y,~,::.,8,B). From (8.31) one can show that l'iJ can be expmssed as
(8.56)
334
TUNING FUNCTIONS DESIGNS
where Ak ~ 0 for k > n, and * denotes entries that can have any values. 'Vith (8.56) we conclude tbat III is a function of y,~, ~t 0, il, Ah"" Am+l' Yr Deuoting Xi = ("'\1,' •• ,Ai), we write O:I("~t Xm+b Yr). In the backstepping procedure, ai is a fUllction of Xm+il y~il»). To make the expositions shorter, this fact is postulated here, and then verified once the a:i'S have been selected.
(X,
Step i =2, .•• , p. Differentiating (8.46) for i = 2, ... t P 1, with the help of (8.43) we obtain
~~1~
 L
~(kj'\I+Aj+l)~1 a~
Noting from (8.46) that
Zi =
Zi+l
'Um,i+l  By~i) = Zj+! aOil ( ay \WO
+ O:i  k"i1Jl'lr,l 
_ 8ail (A
a~t
C
o~
+ k·Y + .J..) _ rp
_mfl a;~~1 j=l
l+
(kj A
~
~
•
'i::ljj_~B. 00 ~
(8.57)
+ O:i, we get
+ W T (J + e2 )
8O:i  1 (A = 6.) _ ~ 80'il (i) 8= 0 .... + '.I:' ~ Ul)Yr ......
Aj+l) _
j=18Yr
(y~il) + a~i:l) B. (8.58)
80;;:10 _ a(J
J
(l
From (8.44) it follows that the final step i = P can be encompassed in these calculations if we define O:p = u(y)u + 'urn,pH  ey~p) and ZP+l = O. To make the choice of a stabilizing function easier, we add and subtract _D':;;1WT9 lJ~i;l rTj ill (8.58) and get
Zi
=
OJ 
~":iVm,l
_ 80:;1 (A
at ~

rn+il

t'
o~
OCtil ( Wo + ay
+
k
.J..) _ y + If'
80:i  1
'~ " OA'  ( kAl J
J=l
80:il
J
8a:il
aOil
a=.....
=
(A
0
+
tP) _ ~
.LJ
j=l
aO: i  1 + AJ'+1)  fT.' ali
aye ayw 2 
'r ..)
W (J
Tf}
I
11
al1:il
80
(0':.

r) 1i
(i
aO: i  1 (j) (jI) 'Yr
8 1
Yr
aai 1 ):"(! an
Y(r  } +  t:
_
+ ""i+l •
(a.59) ,
The stabilizing fUllction 0:; will be chosen to cancel all the terms except those e2 in the last line of (8.59). The potentially destabilizing disturbance 
87;1
8.1
335
DESIGN WITH KFILTERS
(8';;;1) 2Zi
is counteracted by includillg the nonlinear dampillg term d j in G:j. Dealing with the remaining three terms in the last line of (8.59) is no different from the way we dealt with similar terms in Section 4.2.1:
• The term _O';IIlwT8 determines the ith tuning function ? 1' ~, •••
• The term 
0';;01
(8.(0)
,po
(6  rTi) represents the mismatch between the actual
update law and the tUlling fUllction, and it appears because at step i only Ti can be cancelled in (8.59). As u~ual in the tuning functions design, our choice of the update law will be we ha.ve •
8
0=
rTp.
p
rTi
= L r j=i+l
Therefore, in view of (8.60),
a a:jJWZjf ay
(8.61)
which yie1ds Oa:il
ao

(0:' 
r)
=
1'.' I
(8.62) The stabilizing function Q:i will include the term  L~:~ O'jiZj to achieve skewsymmetry in the error system. • The 'abovedia.gonal' term diagonal' 2il
is compensated by adding the 'below
Zi+l
To summarize, our choice of the stabilizing fUllctions for i
., G:i
=
Zil  Ci 2 i 
di
ay
aO'i  1 ) 
(
a
Zi
= 3, ...
I
P is
( T A) + kj'Una,l + Oa:il By Wo + W (}
a
iI
a
a:il (A '=' 4» '" Qil (j) + lti_1 ac"e (A Dc"t' + k11 + If'.I.) + a= 0 ..... + + j=l L.J (iI) Va..... BYr
ao
k \ \ ) +iI r + L OA'iI  ( "'1\]+1\'+1 . 'J :J 00' T,'+ J=1 3 m+il
!:la:
U
I
a ac:til.,.. _ ' " a:jI r L;=2 ao 8"'" y
(
(i I)
Y r
a....)
"'~il ~ +an (} r::
iJ
A
(8.63)
336
TUNING FUNCTIONS DESIGNS
For i = 2 the stabilizing fUllction differs because of the term compensate for brn32 in (8.53):
bm 3]
needed to
(8,G4)
At the end of t.he recursive procedure, the last stabiJizing function used in the actual control law:
O:p
is
(8.65)
and the last tuuing function
Tp
is used in the updittc law:
By sUbstituting (8.63) and (8.64) along with (8.62) int.o (8.59)1 the error system becolnes d1 z1 + bm z2 + e2 + {w  U(Iir + 0'1) et )"
Cl =1 
bm (lir + 0:) ) 0 Z2
=
d'},
C2Z2 
(~~'l) '}, ':2  btnZ1+ Za + vY
aCXl_
t
0 (8.67)
O';}..jZj
j=3
T ii
aCX1
(8.68)
e"lW u
ay 
Xi
=
C;=i 
ay
di
(aao'~l) '},
By
zp+J
M
By
E
lTjiZj 
Z;_I
+ =i+l +
j=2
OO'il,,
OCXil
C'lW
where
Zi 
1J
8,
t
lTijZj
j=.+1
i =3, ... ,p,
= O. This system is compactly written as
(8.69)
8.1
337
DESICN WITH KFILTERS
\vhere the system matri.'1t A;(z, t) is given by
CJ
bm
d]
brn
o
1
o
0
d.,. ( ~ )!.! 1+0'23
C2 
0'24
0'23
1 + Upl,P
o
1Up _I,p _ Cp dP
U'l,P
(OaiJJJe )2 1
(8.71) and H'E"(.z, t) and H'o{z, t) are defined as
(8.72)
To prepare for the sta.bility analysis in the lla.t subsection, along with the elTor system (8.70) we consider the erl'or equations for parameter estimators (8.66) and (8.55), as well as the state estimation error (8.18):
8
=
rTJVo(z, t)z
(8.74) (8.75) (8.76)
g = 1'sgn(bm) (ilr + 0:1) eI ;;
e
=
Aoe.
The candidate Lyapullov function for system (8.70), (8.7.!!), {8.75}, (8.76) is 1 tr 2
1~ 2
Ihml_'J
~ 1
1" = z z + _9' r 8 +  r [ + L..J  e Pe. 1
2')'
i=l
T
(8.77)
4di
Recalling {8.12}, the derivative of 1" is
l:r
=
zT
(A; + A;) ;; + .;:TvVeE:2 + z'l'Hilb 
8T H'oz + Bbm (ilr + {l.t> el'1 z z T ( A:::
.f. 4d.1 E:" E I
~ ZiE2 aO:i  1  ~ 1 I 1"1 + A:T) z  L..J ' . e  , , i=l
ay
+ 0:1) elB
L..J i=1
=
Z Tbm(Yr
i=l
4d
(8.78)
338
TUNING FUNCTIONS DES10NS
where for notational convenience we have introduced ~ ~ 1. The skewsymmetry in (8.71) gives us
C.., + d.,.. (!!!ll)2 Uu
(8.79)
CP+ dP(80.,'By_1)2 which substituted in (8.78) yields
" = p
:s; 
L CjZ;.
(8.80)
i=l
From this inequality we can conclude that z, 8, g, and e are bounded. In the next subsection we use tllis along with the minimum phase Assumption 8.2 to establish the boundedness of all signals and asymptotic tracking. To guarantee bounded ness witbout adaptation we add (strong) nonlinear damping terms which counteract the parameter estimation error. To motivate the choice of these terms, we first rewrite the error system (8.70). After replacing (B.67) by (8.51) and adding ±bmz1 in (8.68), we arrive at
(8.81) where the only difference between A:: and A; is that (2,1) is replaced by bmt
bm at positions (It 2) and
A;(z, t) =
o
bm 0 cld1 cl)d~(~ )~ bm 1 + 0'23 8u 
0
M
10'23 U24
O'p'J.,p 1+O'pl,p
0
(J2,p
O'p'J,p
1Up _l,p
Cp dp
(B8"yl)2
8.1
339
DESIGN WITH K·FYLTERS
and TV; is given by
(8.83) _lJo.,,_1 WT
Ou
While the system (8.70) is more ade
KJ!wI 2 z1 Ba:)
(8.84)
I 811 It, la~, Wl'=i' Ii';!,
=Iel
W 
12
(8.85)
=2
;=
(8.86)
3, ... ,p
and added to (8.50), (8.64), (8.63)1 respect.ively. In addition, to counteract (8.48) is replaced by
Ut
(8.87)
With these terms, the error system becomes
z} =
C1ZI 
d 1 =1

1i 1\W\:!ZI 
Ii'l \brn \ (fir
+ Q'd:! ZI + lJ",Z2
+c2 + w'rO  brn (Yr + Q.]) ~ Z2
=
C2Z2 
_
Z, =
d2
(Bact l ) 2 Z2 Y
n: 2 1
~a:1 w  =1 ell::! =:1  b =1 + Z3 + m
Y
~~Jl E,  (~:1 w _ z,e, ) T8
c,z,  d, (a:~l )'
1<,
(J2j Zj
j~
(8.89)
Ia;;;, wI' =i P
iI
L
Z; 
t
(8.88)
UjiZj 
Zil
+ Zi+1 + L
j=2
80'il 8a:il TO aye 2  ayw,
Uij=j
j=i+1
.=
1
3t
••• ,
p.
(8.90)
The system (8.88)(8.90) satisfies tho Lyapunov inequality (8.80) beca.use the nonlinear damping t.erms enter it as l1onpositive. Ivloreover, for t.his syst.em we
340
TUNING FUNCTIONS DESIGNS
can also prove that
(8.91) where
Co
= 19:5:p min c"
PI) 1
do = (L~ i=l d,
= i=l L Ii,~. p
IiO
(
)1 .
(8.9~)
Inequality (8.91) shows that, thanks to the nonlinear damping terms, z remains bounded even when adaptation is turned off, in which case 8and p are constant, and E2 is bounded and exponentially dec~tying. In t.he next subsection we prove tlmt the boundedlless of =implies t.he boulldec1ness of all other signals in the adaptive system. The complete tuning functions design with Kfilters is summarized in Table 8.2. It employs the filters given in Table 8.1.
8.1.3
Stability
For the adaptive scheme developed in the previolls subsection, we establish the following result..
Theorem 8.5 (Tuning Functions with KFilters) The closedloop adap1.ive system consisting of the plant {B.3}, the control and update laws in Table B.2, and the filters in Table B.l has tILe following p1'Operties: 1. If r, "( > 0 and ni ~ 0, i = 1, ... , p, t.hen all the signnl.f1 are globally uniformly bounded, and Mymp1.otic tracking i.5 achieued:
lim [y(t.)  Yr(t)]
toc
2. 11 r,'Y = 0 a.nd Iii > 0, i 'U.niformlll bounded.
=
= O.
(8.105)
1, ... ,PI then all the si.CJnals are .qlobally
Proof. 1. Due to the pieccwise continuity of Yr(t), . .. , y~'I)(t) and the smoothness of the nOlllinearities in (8.3), the solution of the closedloop adaptive system exists and is uniquc. Let its ma.ximum interval of existence be [0, t /). Along the solutions of (8.70), (8.74), (8.75), (8.76), we established in (8.80) that (8.106)
=,
By (8.77), this implies that 8,~, E arc bounded on (0, tf). Since ZI and Yr are boundcd, y is also bounded. Then, from (8.28) and (8.29) we conclude that ~ and oS are bounded. We have )'et. to prove tl1e bounded ness of ..\ and x. Our
8.1
341
DESIGN WITH KFILTERS
Table 8.2: Tuning Functions Design witll KFilters
=1
=r'
=
11  11r
=
1tmil,  ~ riy(il} r
(8.93) 0:'1 I
i =2, ... ,p
,
(8.94)
O'J
=
(8.95)
ih
=
(8.06)
0:2
=
Tl
=
U(Yr + (1) el)':l
(W 
Oail
811
Til 
W=i,
(8.100)
,; =2, ... ,p
(8.101)
Adaptive cOlltrollaw: 1J.
1 (
= 0'(11)
O'p  'Um,p+1
_ (p»)
+ eYr
(8.102)
Parameter update laws:
o= U
=
rTp
(8.103)
,sgn(bm) (fir + 0:1) =1
(8.104)
342
TUNING FUNCTIONS DESIGNS
main concern is Abecause the boundedness of x follows from the boundedness of E,~, =: and A. The input filter (8.30) gives '._ SiI+~~ISi2+ .. ·+ki_] [ ( )] /((8) u y u
"', 
where ]«(8) = sri + kls,,l one can show that
dny
"dn 
n 
d,t
E d'1. i= 1
i
ni
+ ... + ko.
= 1, ... ,n ,
i
,
(8.107)
On thc ot.her hand, for the plant (8.1)
[CPO,i(Y) + ~(i)(y)a]
di
m
= l:bi l i [u(y)u]
.
t
l
i=O
(8.108)
Not,iug that t.he last sum is B(s) [a(Y)'u,1 and subst,ituting (8.108) into (8.107)
we get,
Ai
=
Sil
+ ~~lsi2 + ... + l.:i  1 J{(s)B(s)
{dnJJ
df" 
I.
dn 
i
~ dtni i
[
'PoAy)
+ cI>(i)(Y)o.
= 1, ... ,11 •
]}
,
(8.109)
The boundedncss of y, t:he smoothness of r/J(y) and ~(y), Assumption 8.2, and (B.109) imply that "\1, .. , ''''\m+1 are bounded. We now return to the coordinate change (8.94), which gives vtII, ;
_
_

':'j
• (iI)  (JA + (}Y + ltil (& y, ~ ,=, r
I
~ '\ O::l») tl, "'m+il! 'Yr ,
.
')
l .  ... , ...
,p.
(8.110) Let i = 2. The boulldedlless of Xm+1! along with the boundedness of Z2 and y, {,.:=, 0, B, ;IJr' Yn proves t.hat Vm,::l is bounded. Then from (8.56) it fol1ows that Am+2 is bounded. Continuing in the same fashion, (B. 110) and (8.56) recursively establish that. ,\ is bounded. Finally, in view of (8.19), (8.31), (8.32), and the bOllndedness of ~,=:,,,,\, and E, we conclude t,hat l' is bounded. Since u(y) is bounded away from zel'O, 'u. is bounded. VVe have thus shown that an of the signals of the closedloop adaptive system are bounded on [0, t f) by const,ants depending only on the initial conditions, design gains, and the e)..i;crnal signals Yr(t), ... , y~n)(t.), but not on tf. This proves t.hat t I = 00. Hence, all signals al'e globally uniformly bounded for aU t ~ o. By applying the LaSalleYoshizawa theorem (Theorem 2.1) to (8.106), it further follows t,bat =(t) ~ 0 as t  t 00, which impJies that limtoc [yet)  Yr(t)] = o. 2. The boundedness of all the signals without adaptation follows from (8.91), by repeating the argument from point 1. 0 TlleOl'C1ll B.5 established global uniform boundedness of all signals but not g10bal uniform st.ability of individual solutions. To refer sucb a stability property to the origin, we now determine an error system such that all of its states
8.1
343
DESIGN WITH KFILTERS
except the parameter error cOllverge to zero. 'Ve start with the subsystem (z, E, 8, 0) whose 2n + q + 2 states are encompassed by the Lyapunov fUIlction (8.77). Then we derive additional equations to complete the error system. For filt.er states we introduce the reference signals f/ and ;::r defined by
~r
sr
= =
Aoer + k.lJr + q,(Yr} Ao:::' + lP(IJr) •
This allows us to define the error states { = t;. by
~
=
Aol + li'::1
(8.111) (8.112)
for and 3; = :::i  =f governcd
+ ¢(.:::ltYr)ZI
(8.113)
..... = AoE + (;(ZI, 1Jr)=J ,
(8.IV1)
wbere ~ and i are SIJlooth functions defined by the meanvalue theorem. The system (::, E,~, 2,8,0) hus (q + 3)11 + q + 2 states~ while the original (.7:, f., S, A, 6, 0) system has (q + 3)n + q +Tn + 2 states. Vve recover the missing m erl'or states ill the inverse dynamics of (B.3). Let. us consider the similarity transformation
°PX1h ] .".
T
(B.115)
''',
where
T = [Atel," . ,Abel, In.] Ab =
[
bnatlbm
bo/hm
(8.116) lml
0
(8.117)
o] .
Tbe following two identities are readily verified:
(8.118) With these identities the inverse dynamics of (8.3) arc expressed as
(8.119) Introducing the reference signal (r as
"
= A.C' + T (AP [
ny, +
¢(Yr) +
,
(8.120)
344
TUNING FUNCTIONS DESIGNS
we see that the error state, = ,
 (I' is governed by
<: = Ai + T ( AP [ ~ ] %1 + r,6(z" Y')=I + 41{zJ, Y,)Zla) 8
A b( + cp,,(Zlt Yr)=l .
(8.121)
We have thus constructed the error system
= e = I; =
A:(=, t)z + H7f:(::, t)E2
{
z
bm (fir + (1) elg
AOE
(8.122) (8.123)
Ab( + CPb(Zh Yr)=1
(8.124)
=
Aoe + kZl + ~(Zl\ Yr)Zl
(8.125) (8.126)
=
Ao2 + i(Z1, Yr)Zl rl,{'o(z, t)Z 'Ysgn(bm ) (ill' + atl ei z
0 == 0
+ H',(z, t)'rO 
(8.127) (8.128)
whose stability and regulation properties are established in the following corollary.
Corollary 8.6 The error system (8. 122}(8.128} has a glo/)ally unifonnly stable equilib,"i:u.m at the origin. Its (q+3)n+q+m+2dimensional state converges to the q + 'In + 2dimensional manifold
(8.129)
Proof. From (8.106) and (8.77) it follows that there exists a positive number v such that 'Vt
~
to
~
O.
(8.130) Let us now consider (8.124). Due to the boundedness of Yr and the smoothness of cpr" in view of (8.130), there exists a class x:.oo function 'Yr. such that
Icp,,(Z)(t),y.. (t)Zl(t)/ ~", (/(z(to),E(to),6(to),O(to))1) TheIl, since Ab is Hurwitz, there exist positive numbers
I
Vl
'Vt ~ to ~ O. (8.131) and 1I~ such 'Vt
~
to
~
o.
(8.132) By the same reasoning, we establish bounds for ~ and ::: analogous to (8.132). Hence, for tbe complete error state E = (z, e, (, {, 2, 6, 0) we have proven that
IE(t)/ ~ 'Y (IE(ta)!) ,
(8.133)
8.1
345
DESIGN WITH KFILTERS
where'Y is a class IC oo function. By Definition AA, the equilibrium E = 0 is globnlly uniformly stable. To establish convergence to the manifold Ill, we recall that in Theorem B.5 we showed that z converges to zero, and so does E because of (B.1B). Since the vector cp"(z,, Yr)Zl in (8.124) converges to zero, then by [28, Theorem IV.1.9] ( converges to zero. The same reasoning proves tbat §: converge to zero. 0
€,
Corollary B.6 establishes global uniform st.ability, a property stronger than global uIliform bounded ness. Moreover, it proves the regulation of all the error states e.~cept possibly the parametel' estimation error. However, COI'ollary B.6 does not estabHsh a correspondence between the two systemsthe errOl' system (=, E, (, E, 6, jj) and the original system (x,'\, €, S, 8, B). Vve need to show that the coordillate change
e,
(8.134) is a global Coodiffeomorphism for each t 2::: 0, whenever B(.s) and 1«8) m'e coprime. Although the coefficients of B(s) are unknown, this condition is satisfied with probability one. Vve only indicate the main idea of the proof that (8.134) is smoot.hly invertible for each t 2::: O. Since = ~  er(t), S = S  :::f(t), 8 = 8  8, and B = {}  g, it is clear t.hat we only need to consider t.he portions (x,'\) and (z, E, () in (8.13~1). In view of (8.19), (8.31), and (8.32), we have
e
x
= =
E E
+ e+ Sa + B(Ao)'\ + ~ + Ba + er + Sa'o. + B( Ao)'\ ,
(B.135)
where B(Ao) = Ei::o biA~. By multiplying (8.135) with T, recalling (8.115), we get (8.136) TB(Ao).\ = (+(" + €+ Ba+~r + Sfa).
T(e
It is not hard to prove that (8.137)
where Tm E lRmxm. Then, since T is full rank, Tm is nonsingular if and only if B(Ao) is nonsillgular. By the fact tbat the eigenvalues of B(Ao) satisfy Aj{B{Ao)} = B('\i{Ao}), i = 1, ... tn, Tm is nonsingular iff B(s) and ]«(8) are coprime. Consequently, under this condition, Am is a smooth fUllction of (, e, E, for each t. Multiplying the identity (B.135) by eJ from the left and noting that e"f B(Ao) = [*, ... 1*,1,0, ... 10], where 1 is the (711 + l)st entry, for each t. The rest one can see that Ant+1 is a smooth function of ZI, (, E, of the proof exploits (8.110) along with (8.56) to establisb that ..\ is a smooth function of z,(,c,t,B, for eacb t. Thanlts to (8.135), so is x. Tbus (8.134) is onetoone, onto, smooth, and has a smooth inverse for each t, iff B(s) and 1( (8) are coprime.
€,
e,
346
TUNING FUNCTIONS DESIGNS
Note that the singularity in tbe coordinate transformation (8.134), which occurs when B{s) and ](.'1) are coprime, is not in contradiction with the boundedness result of Theorem 8.5 where a diffel'eut arguIIlent was used (cf.
(8.107)(8.109».
8.1.4
Transient performance
\Vhile the stability and convergence results obtained thus Car prove that the designed adaptive system has desired asymptotic pelformance, we now analyze its transient performance by deriving L'l Rnd Loo transient performance bounds for the error state z. For simplicity, we let r = 71.
Theorem 8.7 (Tuning Functions with KFilters) In the adaptive system (8.3), (8. 28}, (8. 29}, (8. 3D}, (8.102), (8.103), (8.104), tile /ol101lJing in.equalities hold;
Proof. The bound (8.138) follows immediately by integrating (8.106) over [0,00), IIzlI~
=
oo
L 1=(r)ldr:5   looo . o
<J
1
COO
1 V"(r)dT:5 V(O) , Co
(8.140)
and by recalling that the Lyapunov fUllction II is given by (8..77). The derivation afthe bound (8.139) starts with (8.91). From Lemma C.5(i), it follows that (8.141) From (B.18) and (8.12)
we have ftlEI~ :s; Iel'l , which gives
"e:211~ :s; !(~) 1e:(0)1~ , au the other band,
(8.142)
(8.106) and (8.77) yield
(8. VIa)
8.1
347
DESIGN WITH KFILTERS
By substituting (8.1::12) and (8.143) into (8.141) and expressing 1!(0) from (8.77), we get (8.139).
0
=
0, that 1s t without, the While the £2 bound (8.138) holds even for Il.o nonlinear damping terlllS (8.84)(8.87), the £00 bOllud (8.139) is valid only if 1\:0 > O. It is of interest to compare these bounds with the bounds (4.226) and (4.231) for the state feedback case. As expected, the bounds obtained with output feedback have an additional term due to the initial state cstimat,ion error c(O). The initial condition =(0) in the above bounds is, in general, dependent 011 t.he design parameters Co, dOl 1\:0, , . However, as explained in Section 4.3.2 for state feedbaclc, with trttjectory initialization we can set =(0) = O. Following (B.93) and (8.04), z(O) is set to zero by selecting Yr(O)
=
yeO)
y~i)(O)
=
utO) [lJ",.i+l(O)O'i (y(0),{{O),=(0),8(0)'l?(O),Xm+i(0),il~il)(0»)],
(8.144)
i = 1, ... ,p  1.
(8.145)
Since bm f:. Ot it is reasonable to cboose bm(O) :f:. O. Then the choice ()(O) = 1/bm (0) makes (8.145) welldefined. After =(O) is set to zero, t:be bounds (8.138) and (8.139) become
11=112
(8.146)
Iz{t)1
(8.147) Both of these bounds can be systematically reduced by increasing Co. Ot:her options for reducing the values of the bounds are as follows. The £.3 bound (8.146) can be reduced by simultaneously increasing f and do. The £00 bound (8.147) can be reduced by simultaneously increasing 1\.0 ancl do. In fact, the £00 bound (8.147) can also be systemaUcally reduced by simultaneously increasing 'Y and do. This is easy to sec by noting that (8.106) guarantees that !=(I)f"! $ 21'(0), which implies
(8.148) where z(O) can be set to zero with trajectory initialization (8.1441)(8.145).
348
8.2 8.2.1
TUNINO FUNC'l'IONS DESIGNS
Design with MTFilters Filtered transformations and observer
The design with IvITfi]ters is motivated by the idea of llsing an adaptive observer fOl" outputfeedback control. The nuun tool in Chapter 5 was the passivity property of the observer error system. This property was easily achieved there because the full state of the plant was measured. \,yhen only the output is measured, this propert.y is difficult to achieve. This difficulty is overcome by filtered t:ransforma.t.ions introduced itl [123], which bring the system (8.7) int;o R. special form. We follow this idea, but we introduce significant modificat.ions which make the filtered transformations easier to uuderstand and implempnt. The system (8.7) is rewritten here for convenience:
A:l' + t/J(y) + F(y,'lJ)T8
.1: =
1I
=
(8.1~19)
:1'1 ,
where
[~ ]
0 =
F(11, u)T =
(8.150)
[[ O(I'I)x(m+1) ]
a(y) .. , t)(y)].
Im+l
(8.151)
In this design \ve employ the filters
e = A,e + B,cP(Y), n = A,nT + B1F(y, 'U)T T
eE R
1
n

I
n E Rpx(711) ,
(8.152) (8.153)
where A, and B, are given by
AI
=
[I
/R.~2 0
(8.154)
] ,
and tIle vectors Tand l are defined via the coefficients of the (n 1 )dimensional Hur\Vitz polynomial L(8) = 8"1 + 1. 18,t2 + ... + Inl:
(8.155)
The filtered transformation, defined as
X=
:V 
[
~ +Of/TO ]
,
(8.156)
8.2
349
DES[GN WITH 11TFILTERS
is readily shown to bring the system (8.149) to the a.daptive obsenJer farm
i = Ax+1 (WI) +wTO) y
=
Xl,
Wo
=
(8.157)
where
W
+ ~l = F(u + 0(1). CPO.l
(8.158) (8.159)
The system (8.157) is minimum phase and, considering Wo+w"8 as its input, its relative degree is Ol1e. This "reduction" ill relative degree is a property which makes it possible to employ passivity in the design of an adaptive observer. AB before, all important reduction ill the dynamic order of 1:he Ofilter is possible by exploiting the stI'ucture of F(y, 1I) in (8.151). Denote the first; m+l columns of OT by V m , ••• , L'h va. Due to the special uependence of F{lI. u) on O'(y )'u, the vectors Um , •.• , t'1 , 1'0 arc goverllPd by
j =0, ... ,711.
(8.160)
It easy to show tbat j
= 0, ... ,11. 1.
(8.161)
Therefore, the vectors V, can be obt.ained from only one implemented filter
(8.162) through the algebraic eApreSSiolls
j = 0, ... , 71]••
{8.163}
While we always implement the filter (8.162), for analysis we lise the equations (8.160). Now! ill view of (8.151) and (B.153), we have
(8.164) where (8.165) The implemented filters are summarized in Table 8.3. Vvc recall that the dynamic order of the Kfiltel'S is n(q + 2) and can be reduced to (n 1}(q + 2) using the reducedorder observer techuique. TIle total dynamic order of the MTNfilters is (n l)(q + 2). However, the design with l\IITfilters also employs an observer of X, which makes the dynamic order of the complete adaptive scheme wit.h NITfilters higher than with Kfilters.
350
TUNING FUNCTIONS DESIGNS
Table 8.3: MTFilters IvlTfilters:
e=
+ B,tP(y) .... = A,=: + Blil?(y) j = AlA + en_ld(;,,)'U
Vj
[IT
Observer.
= =
A,~
AjA,
(8.166) (8.167) (8.168)
j = Ot .•• ,m
(8.169)
=J
(8.170)
[Vn." ••• , 'VI, VO,
The simplest observer for (8.157) is (8.171)
where (8.172)
It is casy to show tha.t, with this choice, Ao (8 + ca)L(s), namely,
= A R'oeT satisfies det( s1 
Ao) = (8.173)
The observer error
e=xx
(8.174)
is governed by (8.175)
Since
el
= _1_ rwTij] 8+ Co I:
is a strictly passive tl'ansfcr fUllction, the update
law (8.176)
guarantees tile boundedness of nand e, as we]] as the squareintegrability of e. This update law,vilI be tile starting point in building tile tuning functions update law in the next subsection. To prepare for the backsteppillg design in the next subsection, we rewrite the output y == Xl from (8.157) as
iJ =
X2
+Wo +w1'O.
(8.177)
8.2
351
DESIGN WITH MTFILTERS
Noting frOln (8.151) that
= ~,«J>(l}]T, eApressions (8.159) and
F(l)
m+l
(8.170) yield w=
_
[vm.It Vml,lt ... ,'lIO.I, q,(l)
+ '::'(J}]
'1' •
(8.178)
Combining (8.177) and (8.178), we obtain the following two important e."{pressions for y: :~ =
:\:2 +wo +wTe + €::!
(8.179)
= b"I'Vm,l + X'l + Wo + w' r() + C2,
(8.180)
where (8.181)
8.2.2
Adaptive controller design
Equation (8.180) is written in a form to suggest that the filter signal 'l'm.! will play the role of a virtual control. TIle system to which we apply bacl(steppiug is
iJ = Vm,i = V m ,pl
=
bmVnI,l
+ X2 + Wo + wT () + C2
Vm.i.... l 1~i'um.1 ,
a(y)'lJ. + 'Vm,p

j
= 1, ... , p  1
kp'l'rrl.! •
(8.182) (8.183) (8.184)
The similarity between the systems (8.182)(8.184) and (8.42)(8,44) indicates that the backstepping procedure with IvIT filters will be similar to the backstepping procedure with Kfilters. The most important difference is in the the way the state estimation influences the design of the update law. While with the Kfilters the system (8.18) is autonomous and exponentially stable, with the IvlTfilters the observer errol' system (8.175) is driven by the parameter error. The parameter update law has to take this into account. The strict pa.l:isivity of E'} = _1_ .~+C(l
[WT~
dictates the choice of the tuning function
TO
=
I.lWE'l ,
1.1
> 0,
(8.185)
which is the starting point in the design of the update law. The coefficient 1.1 is a weight assigned to WE'l in the final update law. Following the idea in Section 4.5.1, for system (8.182)(8.184) we define = Y  Yr  _ v lII,il .:.i
(8.186)
=1
ny(il)  "', t!' r L.l.il ,
where Uis an estimate of l! = l/b m •
i = 2, ... ,p,
(8.187)
352
TUNING FUNCTIONS DmSIGNS
Step 1. The equation for the tracking error (8.182) is
ZJ
obtained from (8.186) and (8.188)
By substituting (8.187) for i = 2 into (8.188) and scaling the first stabilizing fUllction 0'1 as (8.189) (8.188) becomes (8.190) Then the choice (8.191) results in (8.192) Substituting (8.52) into (8.192) we get
ZI =
CIZldJ z1
+E:2+(W  b{Yr + a1) el)T 6bm (Yr + al) e+bmZ"2. (8.193)
To select the first tuning function, we recall that the parameter error appearing in the observer errOl' system (8.175) already determined the tuning function 'To in (8.185), which we now augment with a term corresponding to (8.193): (8.194) Wllile the choice of the update law for 0 is postponed until the last step, the update law for iJ is selected at this step: (8.195) We pause to clarify the arguments of 01' By examining (8.191) along witb (8.158) and (B.181), we see tllat 0:1 is a function of Y, 0, D, t'o,!" .. t'Vm l,1 and Ur' For brevity, we denote X: (Yt X, ~, E t 8, From (8.169) one can show that ViJ can be expressed as
=
e).
x, e,:,
(8.196) where .Ak ~ 0 for k > n  1. With (8.196) we conclude that 0'1 (X, Xm , Yr). Now we present a baci(Stepping procedure in which O:j (X t Xm +i  1t y~il»).
B.2
353
DESIGN WITH MTFILTERS
Step i =2, •.• ,p. We now proceed with the design by differentiating (8.187) for i = 2, ... ,p  1 with the help of {8.183}: Zi
=
Vm,il 
=
Vrn,; 
:. 01) gU r
•
A
,
[A·,\ +
 UO'il aX
1.' (
J\o
•
0';1
:. (&1)
 (i) {}Yr 
h'j'Um.l 

gllr
80;1
8=
'l.
Noting from (8.187) that •
Zi
=
+ Qi 
=;+1

28 ,"+i"" O'il ( k \
L."   j=l a"\j
I
h:jVm ,1 
 8a:il ax. [A"x + _ 8O:i _ 1 (A e {)~
l's
~
r.. (
1\0
+
ft
\"
8y
.,.:. +
+
Bifi) ~ OO'i1 (j) l'~  L." u=f)lIr j=l aUr
D0';1 .'" uQ'i1 n :. 0  Ag· 81J ag
 . ..
(8.197)
+ ail we get
80';1 (Tn) BY A2 + Wo + w + C2 .. ) Y  Xl
+.l (Wo + w'I'(})]
B "") _ 8a:il (A :: B w) _ ~ 1Of' 8= 1 + I L." 
j::d
_rnf2 8~1 (k Al + "'\j+1) _ j
~1
tJ'(}) e~
+ w'I'IJ")]
(:4 
Bu~i) = =1+1
12 + Wo + W
8Qi1 ("

\ ) + Aj+1 
i""'l
Urn.; 
", e ...... 8 ~ (i2)) II. X, 'l.t'::', ,Q, Am+ilt lIr
") ,y :\'I + I (Wa
8 CX i1 (4 e B)  Be "'~ + IcP 

(
J
80:i ::
00
80';_1 (j) UI} Ilr
aUr
18 _ (y~il) + a~i::l) g. (8.198) Q
From (8.184) it (o]1ows that the final stop i = P call be encompassed in tbesc u(y}v + v""p  ey~p) and =11+1 O. To prepare calculations if we define O:p w'l'iJ  I)~;;t rTj for choosing a stabilizing function we add and subtract. in (8.198) and get
=
Z,'
=
0t 
rn,
J,,:"U I
1 
{)CXiJ 8y
= 0"0;1
(Av ... +~o +wTO) ,\w
 a~~1 [Ai + [(o(Y  XI) + 1(wo +wTb)] _
8 Qil (A e
ae ~
l'l.
+ Bl'I'A.) _
aO'il (A ':' 8=
....
, .... +
B rp) _ I
iI
a
~ ~ (j)
~ 8 {iJ}Yr
~l
~
(8.199)
354
TUNING FUNCTrONS DESIONS
The term _8~~IWT8 determines the itb tuning fUllctioll ') z. =_,.",p.
Since the final update law will be aail
 
aiJ
(O~. 
(8.200)
8 = rTp , in view of (B.20Q) we have
r) r,.
=
I
~
aa:il r&O:j1 WZj
L.,   . j=i+1 &0
8y
p
L:
(8.201)
UijZj.
j=i+l
Then ouI' choice of the stabilizing fUllction for i = 3, ... , p is O'j
=
.,.. 1  C·.,.'  d·r (fJCfil) ~ ..... + k·vI m.1 + aail 11[ fJy ay
(,,~ + Wo + wTO) ,\.~
+ 8~~1 [AX + ](o(Y :ttl + I (We> +w'l'iJ)] aail (A OO'il +a& ,{ + B'/") I'll + a= I."
(A
.....
'='
l .....
+ B ,oj}.T,) + ~ L...,
fJO'i1 Ul (iI) Yr j=1 aYr
OO:il ) 8CXil (i]) + a=ao:i  1 ) :fl. + 7n~2 ~ 7)'f':""( kj ..\] + Aj+l + .rTi + l1r
3=1
fJfJ
J
_L: aO'j;:1 r 8 O:fl =j • j=2 80 ay
fl
il
(B.202)
Only for i = 2 the stabilizing function differs from (8.202) because its first term is 1,m z 1 rather than ZI. The last stabilizing fUl1ction O:p is used in the actual control law: u= and the last tuning function :.
8 = rTp =
r
I W
uty) (Q Tp
p 
vJ1I ,p + BY~P»)
.
is used in the update law:
aQpl [1.  aa:l ay •••.• ""8iI]
[ ~l Z2
;p
••
(8.203)
I
 g (1Jr + iii) e,zl + !lWEI
.
(8.204)
By substituting (8.202) along with (8.201) into (8.199), the error system becomes (8.205) with A:(z, t), 14"1'£(z, t) and I¥,(z, t) defined as in (8.71), (8.72), and (8.73), respectively.
8.2
355
DESIGN WITH MTFILTERS
Example 8.8 COllsider the relativedegreetwo nonlinear plant:
= = =
,1: 1 X2 Y
x!!
+ Oy3 (8.206)
'U
Xl,
where only y is measured and 0 is unknown but constant. In this case only the scalar filters (9.218) and (9.217) are needed:
v
it! + 'u 1'£  IIi' ,
=
..... =
(8.207) (8.208)
and the regressor (9.224) is scalar: w=y3+B.
(8.209)
The control objective is the regulation of y to zero, namely, YI·(t) == O. The error coordinates (9.238) and (9.239) are Zl
=
11
=2
=
V 
(8.210)
0:1 •
The observer (8.171) i.s given by
:. = [Oil 0 0 X+
X
A
[
CoCol +l
1 + [ 11 1(v + wOA) ,
(8.211)
E}
where E] = Y  Xl. The control law is generated via (ll
=
(c + d)=l 
u = 
X2 
Bal)2] [c + d ( BU
we
(8.212)
+ OO'} 011 (x!! + V + wO) A
Z2 
;:1

lCoE) 
~
wO, (8.213)
and the parameter update law is (8.214) Figure 8.1 sho\vs the system response with 0 = 2\ C = Co = 1, d = 0.5, 1 = 1, "y = 0.4, v = 1, and all initial conditions zero except for .h1(O) = X1(0) = 0.8. The main feature of the responses is that they are fast. The adaptive transient settles in about 0.5 sec, when the parameter error 0 becomes very small. After that, the system is practically feedback linearized, and responses look "linear." The convergence of 6 to zero is fast because the initial swing of the error states and the regressor accelerate the parameter convergence. This is pronollllced because of the cubic nonlinearity in the plant which is responsible
356
TUNING FUNCTIONS DESIGNS
y
0 11
2
If: 0
0
U 0
i
s====:
3
4
S
i
i
,
,
2
3
4
S
I
i
i
2
3
4
S
Figure 8.1: Response of tlle tuning functioIlS scheme \vith MTflltcrs.
for the sharp peak ill y. (Ill fact, without adaptation, this nonlinearity would cause an "explosive" escape to infinity; see the dashed curve in Figure 8.1.) \Vil:h a considerably lower adaptation gain, the parameter estimate still COllverges fast because the initial swings become larger and further accelerate the cOllvergence. vVith higb values of c, Co, and/or ci, which !Day increase the COlltrol effort, the regulation of 11 to zero is achieved ,,·itbout 0 converging to zero, that is, witbout feedback linearization. . 'Ve revisit this example to illustrate EL modular design in Example 9.23. 0 To guarantee boulldedness without adaptation we add nonlinear damping terms (8.84)(8.87) to tbe stabilizing functions, as we did in the design with Kfilters. The stabilizing functions are now stabilized in Tal)Je S.4. The resulting error system has the same form (8.88)(8.90) as in the design with Kfilters. COllsequently, it.s stat~ satisfies t.he inequality (8.215) with Co, do, and lio defined in (8.92). When adaptation is turned off, 8 wld B are constant and bounded. HO\vever, unlike in the design with Kfilters t
B.2
357
DESIGN WITH MTFILTERS
C2 is not guaranteed to be bounded because the input wTO to t.he observer error system (8.175) cannot be guarallt(~ed to be hounded before est.ablishing the houndedness of y. For this reason, we use nonlinear damping to also strengthen the system (8.175). The observer (8.171) is strengthened by adding the nonlinear damping term h~olwl!:!l(y  :\'d:
(8.21G)
Thus the observer error system becomes (8.217)
The inclusion of the term ~nlwfl(y  id in (8.216) forces us to include this term also in the stabilizing function Q'j in (8.202). The cornplete tuning functions design wit.II lVITfilters is summarized in Tables 8.4 and B.5. It employs the filters given ill Table 8.3.
8.2.3
Stability
The stability properties of the adaptive scheme with MT£ilters are as follows.
Theorem B.9 (Tuning Functions with MTFilters) The clolleel/oop adaptive system consisting of the plant (8.8), the controllml1, the update /(I.UJs. the observer in Tablcs 8.4 and 8.5, a'1'ld the ftlte7's in Tablc 8.3, has the Jol101lfing properties: 1. If r, 1 > 0,
#i o , "~i ~
O. i
= 1, ... ,P,
and (8.218)
then all the ,r;ignals are globa1l11 uni/o7'1n.l.lJ bountied, and a.sym.ptotic l'racking is achieved: (8.219) lim (y(t)  Yr(l)] = O. ,~oo
2. 1/ r, I
= 0 and ~o, "~i > 0, i uniformly bo·unded.
= 1, ...• p.
'.hen all the signa.l.r; are globally
Proof. 1. As argued in the proof of Theorem 8.5, tlu~ solut.ion of the closedloop adaptive system exists and is unique on iLs m8..ximulll interval of c..\:istence
{O, tf). Let us introdure the similarity transformatiou
[
~ ] ~ [ ;~ ] = [ ~ ]
E
(8.220)
358
TUNING FUNCTIONS DESIGNS
Table 8.4: Tuning Functions Design with MT~Fi1ters (cont'd in Table 8a5)
=
~ ':'j
al
't ? ... , ...
ny(il) ,." tf r .....1 1 ,

=
(C1
+ d. + 1\:] Iw12) Zl
A
[
brnZl 
+
a; =
C2
(aa

+ d2 8yl )
(y, + aa~l) h
X2 :!
wTfJ
Wo 
(8.224)
80
+ 1\:2 1 ay'1 W 
Zlel
12] ':2
a~lrT" + P2
(8.225)
[c; + (Ii. + "llwl') (~~I)·] "I + (y!I_I) + a:~l )~
:1_1 
~lD
D
alJ
r.I
(8.222)
(8.223)
uO'il r "" Uajl r altil r.I +.T;  £  .    Z j + pi,
pi
,p
glil  Ii)sgn(bm ) (fir + lh)2 z)
=
ih = 0:2
(8.221)
= Y lIr 'um,l. 1
':j
j=2
8fJ
ay
= Bail (A:2+ W O+WTIJ)
8y
i =3, . .. ,p
(8.226)
80i  1 (A  + B.;r,) +T ,{ + B1'1'A.) + Bail 8S (A ,=.. l'j}
+ ~I [A~ + J(.(y  XI) + H.• lwl"l(y ;ill + I (Wo +wTIi)] iI 80 i  1
+L
.
u=:r>y~) + kj'Um ,1 +
j=18Yr
L
j=1
= =
'IWet
Tl Ti
=
Til aWZi,
TO
where T
1II+i280'i_l
E R(nl)xn
(w 
~(kj)..l U~\J
+ )..j+.)
(8.228) (8.229)
e(Yr + lit> el) Zl 8ai1
y
(8.227)
i
= 2, ...
t
P
(8.230)
is defined as (8.231)
The followillg t\VO identities are straigbtforward to verify: Tl=O,
(8.232)
B.2
359
DESIGN WITH MTFrLTERS
Table 8.5: Tuning Functions Design wI MTFilters (cont'd from Tab. 8.4) Adaptive control law: (8.233) Parameter update laws:
iJ =
rTp
(8.234)
~ =
')'sgn(bm)(Yr + Q'd Z1
(8.235)
Observer: (8.236)
(the latter is immediat.e froln (8.172)). Using (8.231)(8.232), we compute
=
[Aie},
AI] = A, [A,e},
InI]
= A,T.
(8.237)
Combining (8.217) and (8.220) and using (8.237) we have
Ti = T Aoc  ~olwl2TleT e + Tlw'l'8 = TAoE' = A,Tc I
i] =
(8.238)
which because of (8.220) gives (8.239) This system represents the e,'1lponelltially stable inverse dynamics of (8.217). Since they are not controllable by Cl, they are also the zero dynam.ics of (8.217). Now we examine the clequation: (8.240) The second identity in (8.232) gives eTI\Q = EI =  (co + li:olwl2) el
Co
+ ll'
Therefore
+wTO + £2 [lel.
(8.241)
Subst.ituting (8.231) into (8.220) we see that (8.242)
360
TUNING FUNCTIONS DESIGNS
and obtain (8.243) Now we al'e ready for a Lyapullov stability analysis for the closedIoo}> system consisting of the error systiem (8.205)\ the error equations for parameter estimators (8.20Ll) and (8.195), and the observer error system (8.243), (8.239)~ :;
=
A=(z, t)= + l'l'e;(=, t)C2
+ H'o(.:, t)T8 
bin (Yr
+ lh) elU
r(B'o(z, t)z + '.lWEI) g = "( sgn( bm) (Yr + iiI) eI.:
8 =
E"t
iJ
(8.244) (8.245) (8.246)
=  (Co + n.olw\9) el + WT(J + 711
(8.247)
= Am·
(8.248)
(The system matrix A;: in (8.2M) is oftbe C01'Dl (8.71), but it also incorporates the lIonlinear damping terms (S.84)(8.87).) A candidate Lyapunov function for t.his system is
V'
1 T 1T 1 ~ Ibml lJ = ;; Z + f) r IJ + U + £i + 1117 2 2 2"( 2 t)
t)
"
where J1 is a positive constant to be chosen later, and solution to the Lynpunov equation
P'17
/1
'
= Pl
/1Ar + All1 = I.
(8.249)
> 0 is the (8.250)
vVith calculations similar to (8.78)(8.80), we a.rrive at (8.25] )
By applying Young's inequality to the term . li :5
co/zl.., + 4(11E2 IJ
0
Noting that: (8.242) implies E~
V. :5 co IZ 12

(vC
o
2

o
Ile .., Ei 
2
IJEttl17
(
we obtain
po  
1.1) 117I" •
2eo
(8.252)
:5 21i£I + 21Jr, we finally get 
Ii) El 
2do
IJ
(
p.   IJ   1 ) 1111'1 • 2co 2do
(8.253)
By choosing v as in (8.218), and f.L = 2:.,. + ~fI' we prove that " :5 O. This implies that z, 8, U, E1 t T/ are hounded. We now use this to establish the boundedllesS of all other signals ill the adaptive system. In view of the similarity transformation (8.220), the bounded ness oC El and 11 establish that £ is bounded. The boundedness of Yr and ':::1 implies that y is
8.2
361
DESIGN WITH MTFILTERS
bounded. Therefore ~ and first re\Vrite (8.236) as
:e: are bounded.
To prove bounded ness of :\:, let us
(8.254) and note that the boulldedlless of Cl and y implies that Xl is bounded. To prove that. the remaining componcnts of Xare bounded, we employ the similarity transformation
[ Xl ',p
1~ [ Tx. X: 1= [ eTT ] ,.
(8.255)
.\.1
and, from the observer equation (8.254), obtain the system (8.256) which shows that tP is independent of the input Ii'D Iwl:! (1/  XJ) + Wo + w T fJ. To arrive at the last equation, we have used the identities (8.232) and (8.237). Because of the boundedness of y and the Hurwitzness of AI, (8.256) proves that 'l/J is bounded. By (8.255), the boundcdness of Xl and l/J establisbes that X is bounded. We have yet to prove the bounded ness of A and ;t'. Our main concern is ,,\ because the boundedness of .r will follow from the boundedlless of E I X, ~,3, and A. The proof of boundedness of ,,\ is similar to the corresponding part of the proof of Theorem 8.5. From (8.168) it follows th~tt ,A.
= .'Jil + It,si2 + ... + lit [ L(5}
I
J
(, )
a II
U
i
,
= 1, .. 'In 1.
(8.257)
Substituting (8.108) into (8.257) we get \
I
,\,: =
+ llsi 2 + ... + 'il
Sil
L(.'1) B( s)
{d
RlI
dt,n 
d t; dt"; r'Po.iCl1) + n
n i
}
i = 1, ... ,n  1.
(8.258)
~.
The boundedlless of y, the smoothness of ,p(y) and cIi(y), Assumption 8.2, and (8.258) imply that At" .. , Am are bounded. We now return to the coordinate change (8.222) which gives Vm,i
= ZHl + IlYr(i) + D:i (y, X, '!oj =, fJ ,Il,.. xm+il, Y(il») r A
A
C 
,
i
= 1, ... , p 
1.
(8.259) Let i = 1. The boundedness of :\rn+l, along wit.h the boundedness of Z2 and y, x,~, 2, fJ, il, Yr, Yrl proves that urn,l is bounded. Then from (8.196) it follows that A",+1 is bounded. Continuing in the same fashion, (8.259) and (8.196) recursive]y establish that .,\ is bounded. Since e and X are hounded, from (8.174) it follows that X is bounded. Finally, in view of (8.156), (8.169),
362
TUNING FUNCTIONS DESIGNS
(8.170), and the boundedlless of €,:, A and X, we conclude that x is bounded. Since 0'(11) is bounded away from zero, 'll is bounded. By the same argument as in tIle proof of Theorem 8.5 we conclude that t I = 00. By applying the LaSalleYoshizawa t.heorem (Theorem 2.1) to (8.253), it. further follows that :z(t) ~ 0 as t + 00, which implies that lilll t _ oo [y(t)  Yr{l)] = O. 2. Not.ing that (8.243) yields d (1 ')) Co ') 1 II') 1 ... d ~£i < ~£j + 4 0  + 2 (i, ~
t

~
'h~o
Co
(8.260)
we conclude that el is bounded when t.he adaptat.ion is switclled off. In view of t.he similarity transformation (8.220), the boundedness of £1 and '1 establish that £ is bounded. The boundedness without adaptation now follows from (8.215), by repeating the above argument which deduces the boundedness of all the signa1s fl'om the boundedness of z, 0, g, £It TJ. 0 The requirement l.Jcodo > Ii given by (8.218), not. found in the design with Kfilters, is needed in the design with IvIT:6lters because the parameter error 8 appears not only ill the zsystem (8.205) but also ill the observer error system {8.217}. The need for each of the factors to be large CRn be explained as follows: should be large because the term weI needs to be given a sufficient weight relative to the other terms in the update law (8.204). In other 'Words, t.he adaptation with respect to the esystem needs to be sufficiently fast. because £2 appears as a disturbance in the .:systcm.
• 1'
• do should be large to prevent the destabilization of the :;system by £2 if the adaptation with respect to the esystem is slow.
• Co should be large to maIm the part of the £system controllable by w T6 fast enough if the adaptation with respect to the £system is slow. Hence, if codo is small, l.I should be large enough t.o satisfy condition (8.218). However, v must not be too large because, as our performance analysis in Section 8.2.4 shows, an increase in 1' may cause a deterioration of pel'fol'mance of z.
1
As in Corollary 8.6 for t.he design with K:61ters, we can show that the error system ."
= A.:(z, t)z + IVE(z, t)e2 + H'o(z, t)"l'6  bm (fir
E =
;p
(Ao  olwl leT) £ + lw 8 2
fL

2
= A,'l/J + A,elzl
T
+ (j'I) fIB
(8.261)
(8.262) (8.263)
.~
~ ~
8.2
363
DBSIGN WITH MTFILTERS
,
= Ab( +
(8.264) (8,265) (8.266) (8.267) (8.2GB)
has a globally uniformly stable equilibrium at the origin, and its {q +4)n +m.dimensional state converges t.o the q + m + 2dimensional manifold 111
= {z = 0, E = 0,
.1ft = 0,
(= 0, l = 0, 2 = o} .
(8.269)
The systems (8.263), (8.265), (8.266), arc defined from (8.256), (8.16B), (8.167), in analogy with Corollary 8.6. One can also show that (8.270)
is a global Coodiffeomorphism for each t 2: 0, whenever B(s} and L(s} are coprime.
8.2.4
Transient performance
We derive £2 and £00 transient perrOl'mance bounds for the error state z. Theorem 8.10 (Tuning Functions with MTFilters) In the ada.ptivc system (8.3), (8.166). (8.167), (8.168), (8.233), (8.284), (8.235), (8.236), the
following inequalities hold:
Proof. We first derive the bound (8.271). For v satisfying condition (8.218) 1 and J.t = _c., "J~ + 'Jd  D , inequality (8.253) becomes
(8.273)
364
TUNING FUNC'fIONS DESIGNS
By integrating (8.273) over [O,OO)t
IIzlI~ =
rco 1=(T)f!dT :5 .!.Co ioroo V{r)dr :::; .!.v'(O) , Co
./0
(8.274)
and recalling from (8.249) that the Lyapunov fUllction V' is given by
i
V = [1=12 + ~ (i,jI2 + Ibmlf) + vei + (: + ~) 1111~] , we arrive at (8.271). The derivation of the bound (8.272) starts with (8.215). Lemma C.50), it follows that
1=(t)12::; Iz(0)1 2e2COI + 4~0 [:0 lIe211~ + :'0 111812+ Ibmlo211col .
(8.275)
From
(8.276)
From (8.275) and (8.273) we get (8.2;7)
21rer + 2TJf, we have (8.278) IIe211!o :::; 2liIJclII!o + 211TJIII!o·
On the other hand, noting that (8.242) implies £~
::;
From (8.275) ~l,nd (8.273) we get
"£lll~
CJ
:::; ::v v(O) ,
and from (8.239) and (8.250) we have f,ITJI~
(8.279)
::; ITJI2, which gives
lI'7dl~ ::; ~(~) II1(O)I~ .
(8.280)
Substituting (8.279) and (8.280) into (8.278), and then along with (B.277) into (8.276) we arrive at the bound (8.272). 0 While the £2 bound (8.271) bolds even for KO = 0, the Coo bound (8.272) is valid only if lio > O. The initial condition z(O) in the bounds in Theorem 8.10 is, in general, dependent on the design parameters Co, do! "'0, co; 1, v. However, with trajectory initialization we call set =(0) = O. Following (8.221) and (8.222), z(O) is set to zero by choosillg
= =
yeO) 1 b(O) [Vm,i(O) (ri
(8.281)
(y(O), ;((0), {(D), E{O), 8(0), b{O), Xnl +i  1 (0), jj~il)(O))] 'i = 1, ... ,p  1.
(8.282)
8.3
365
EXAMPLE: LEVITATED BALL
After z(O) is set t·o zero, the bounds (8.271) and (8.272) become
Iz(t)1 (8.284)
Both of these bounds cau be systematically reduced by increasing Co. Other possibilities for reducing t.he values of the bounds are easy t.o fiud by examining the dependence of the bounds on t.heir arguments. In fact., alt.ernative £00 bounds can be dorived which emphasize the role of 1, do, Co, no. The dependence of tho bounds on II is such that. they may increase with 1). On the other hand, II should be sufficiently large to sat.isfy t.he stability condition (8.218). A way to guarautee stability \Vith small I) is 1.0 satisfy the condition (8.218) by increasing do. Duo to the vdependent terms, the bounds with NITfiUet's are strictly larger than the bounds with K:6.1ters. The performance with IvITfjlters is not as good as with Kfilters because the design wit.h IVITfilters results in the observer error syst.cm (8.217) dist.urbed by the parametcr errol', rather than in the autonomous observer errol' system (8.18). Although this disturbance is easily compensated for in the parameter update law, it. causes the performance bounds for the design with :NITfilters to be higher.
8.3
Example: Levitated Ball
In this sect.ion wo consider the magnetic levitation syst.em in Figure 8.2 and design a controller which keeps the iron ball a.t a. desired posit.ion Yr' The position of the ball 11 is sensed by the photoelect.ric sensor S and fed back t.o the controller C. The controller is to adjust the electromagnet current J t.o attract the ball with the force F and thus counteract the gra.vitation force 111g. Many versions of this system e.."
111 jj = ill9  F
(8.285)
I
and the key modeling tasl( is to describe the force F as a function of the position y and the current. I. \Ve assume that F is proportional to J2 and that it is a known deca.ying st.rictly positive function of 11, that is, ,\(y)
> 0, A'(JJ) < 0 'V1J
~
o.
(8.286)
366
TUNING FUNCTIONS DESIGNS
I
c o x
s Figure 8.2: Levitated boll system.
The model is meaningful only for y > 0, and therefore the design will not yield a global result. To complete the model, its electric part should describe how the electromagnet current 1 is produced by the control voltage u. Here we will Msnllla tl. simple RL relationship
Li= Rl+Gu.
(8.287)
Tbus, our complete thirdorder model is given by the state equations
= = xa = Xl
X2
X2
9  6J A(xJlxj 62X3 + 6au .
(8.288)
with state variables Xl = V, X2 = il, and Xa = I and unknown parameters 1/AI, 62 = R/ L, and 63 = G / L. The unknown parameters are all 61 positive. The system (8.288) is not in any of the required forms, but, as we shall see, this does not preclude the backstepping design. While the states .1:1 and X3 are measured, the velocity X2 iJ is not mearsured. Hence, our design will use partial state feedback. Led by the reducedorder observer idea, we employ the filters
=
=
~ = , =
+9
(8.289)
k(  A(xdxi ,
(8.290)
kf.  k2xl
where k > 0, and define the estimate (8.291)
so that the estimatioll elTor e
= :":2 
.i2 sat.isfies the equation (8.292)
8.3
367
EXAMPLE: LEVITATED BALL
Thus, we apply backstepping to the system
xJ = , = xa =
81 (
+ e+ ~:Xl + £
k(  A(xdx~ 82:C3 + 83 u.
(8.293)
The design objective is to regulate Xl to a given set point Yr III the first equation, , is considered as vidual control. III tbe secolld equation, rather than Xa alone, the force A(Xtlx~ is the virtual control. Thus, our goal is to stabilize system (8.293) ill the error coordinates Zl
=
Xl 
=2
=
,  0'1
Za
=
Yr (8.294)
A(Xl}Xj  Q'2.
The parameters (h and 83 act as unknown virtual control coefficients. Applying the backstepping design with tuning functions, we obtain stabilizing functions al
=
gl (CIZI 
0'2
=
BIZ)
d1 z1 
kXl 
e) ~
glal
(8.295)
A~(  8aXl1 (e + 8  ~~' (k~  k'x, + g)  ::: b, + C2Z2 + d2 (aaa 1 ) 2 Z!! 
1(
0:
Xl
+ kXl) (8.296) (8.297)
and the control law U
=
i?a 2A(}
XI3:a
[ 4:2 
ds
DdZa 
(80: a  A 2
Xl
I
(Xl)Xa
)
2
Z:i
+ 28• 2 A(Xl)X..,3
+ (::  '\'("'1)"'3) (( + Ii,( + k",,) + ~~2 (( + 0,( + ka:,)
+ 00:2 a(
(
2)
k'  .f\(Xl}Xa
80:2:"] + G0:2 a·01 {h:. + aO.81 l
ea _
6
= 'J~( } 03· (8.298) Xl xa w.f
The singularity of this control law at X3 = I = 0 is not an impediment, because the set point value of the current is well above zero. A calculation shows that with the control (8.298) the error system is
368
TUNING FUNCTIONS DESIGNS
o
o
] [ ~a 01
o
B~
2"\(Xl)X5
1. (8.299)
The parameter update laws obtained using the tuning functions technique nrc Ql = Q3 =
,),01=3
61
')'
=
(8.300) (8.301)
,),0'1=1
82 =
(
ZIZ!! 
Oat all. +..\ a:' ='1  a:(=:i x I
·'Vl
I(
•
J:I}:r3C=a
)
(8.302) (8.303)
",(2A(:J:l )xi=a .
The closedloop adaptive system has an e(luilibrium at. the point Xl
·'l.'3
=
Yr
9 ;:kYr
f.
=
(
=:'JL ~~(Jl
(8.304)
9
=
Before we discuss the stability properties, let us introduce the sigl1al ~ + kYn which is governed by
e
t
=
(8.305) The derivative of the Lyapunov function l' =
1 (.,
? ...
'J)
.,
1.,
Cl;::Z
1
(_'l
_., 'J 'l)
zi + zi +::j + 4"h·(J.O .1 E + ')k { +?: 8 1Di + 8aga+ 8j + 82' _ 3 ••/)
(8.306)
along the solut.ions of (8.299), (8.292), (8.305), and (8.300)(8.303) can be easily shown to satisfy .
Ii < 
Cl ..,
ZI 
2
.,
c.,=:; 
 
., 1" CI.,  _EM  _ t : :.. ., a 4d{J 2k2 \ t
C.• Z 
(8.307)
which means that the equilibrium (8.304) is stable. Tbis property is not global because of the physical constraint confining the solutions to the feasibilit.y set :F defined by Xl > 0 and j:3 > O. The Lyapul10v function (8.306) WillllOW he used to estimate the region of attraction within:F. Let O( c) be the invariant set defined by l'{.'J:l,X2,X3,~,(,Bl,{?at61,62) < c. Theil an estimate n of the stability region is
n = {. Y.. = (xt, X2, ,1:3, e", ~1t Ba, OJ, 62 ) Il'(...Y) < arg ll(c)C.r sup {c}}.
(B.30B)
36D
NOTES AND REFERENCES
From (8.307) we also conclude that all the states except possibly the parameter estimates converge to values given by t.he equilibrium (8.304). Using LaSalle's invariance theorem one can prove from (8.299) t,hat Q1 and 81 convel'ge to zero, which means t.hat t.he estimates of ill and 1/1'1 converge t,o their true values. This is due to the gravity 9 which causes t.he force of the elect.romagnet to be nonzero even when Yr = 0, thus introducing in the regressor tenns which do not vanish at the equilibrium. AB we explained in Example 4.13, nOllvanishing elements in the regressor contribute to convergence of parameter estimates. From LaSalle's invariance theorem it also follows that. a linear combination of U3 and 82 converges to zero. Since V' converges to a constant, we conclude that and O2 con.verge t.o constant values.
ea
Notes and References Both the outputfeedback scheme of Marino and Tomei [122, 123) and the modi.fied scheme of Kanellakopoulos, Kokotovic, and Morse [72} inherited overparametrization from the original adaptive backstepping design 169]. The designs presented in this chapter avoid Dverparametl'ization using the tuning functions technique which WiiS first employed for outputfeedback adaptive designs in Krstic, Kanellalmpoulos, and Kokotovic (95] and in Krstic and Kokotovic [99]. \Vhile the I(filters employed in this chaptel' arc the same as those in [72] patt.erned after Kreisselmeier [91], the !vrTfilters in this chapter are simpler than those proposed for liltered transformations in [122, 123]. The use of nonlinear damping as a tool to improve performance and guarantee boundedness without adaptation was suggested in Kanellakopoulos [64] and in K811ellakopoulos, Krsti6, and Kokotovic [80]. Teel [189J proposed an outputfeedback design where high gain is employed to allow the regressor to depend only on the reference signals, which means that the adaptation is active only in the case of tracking. Khalil [83] and Jankovic [57J developed semiglobal adaptive designs for a class of nonlinear systems which includes some systems not transformable into t.he output feedback form. Their designs employ highgain observers and control saturation. While Khalil's identifier is Lyapunovtype, Jankovic employs a passive identifier. Tao and Kokotovic developed adaptive designs for systems with unknowll backlash [184] and deadzone [185].
Chapter 9 Modular Designs In this chapter we ext.end the modular approach of Chapters 5 and 6 to the case of output feedback. The outputfeedback modular approach results in separation of three design modules: the control law, the identifier, and the state estimator. With the Kfilters we design outputfeedback control laws which guarantee inputtostate stability with respect to the parameter error, its detivative, and the state estimation enor as the inputs. At the end of t.his chapter we briefly present some schemes with NITfilt.ers. Following the ideas introduced in Chapters 5 and 6, we develop outputfeedback forms of passive and swapping identifiers. The schemes in this chapte.r are simpler than the tuning functions schemes because they do not eliminate iJ from the error system, but inst.ead use stronger ISScontroUers. As in Chapter 8 we consider the outputfeedback systems :i;
=
Y =
A,,' + ",(y) +
[ ~ ]
(9.1)
eIx.
\Ve retain Assumptions 8.2 and 8.3 and modify Assumption D.l as follows:
Assumption 9.1 In addition to sgn(b711 ), that Ibr,.1 ;: : ~m.
{I,
positive constantC;m is known stLch
Assumpt.ion 9.1, which is st.ronger than Assumption S.l, is st.andard in 'indirect' adaptive control. It allows the control law to contain a division by the estimate bnl! which is kept away from zero by parameter projection. In the tuning functions designs it was possible to avoid this at. the e.xpeuse of an additional estimate of u = 1/bm • The Kfilters introduced in Section S.1.l are repeated for convenience in Table 9.1. \Vith these filters the state x can be represented as (9.2)
372
lVloDULAR DESfONS
Table 9.1: KFilters
Kmtel's:
+ ky + (p(y) Ao + (y) Ao;\ + c a(lJ )1£
Ao~
(9.5)
=
(9.6) (D.7)
7l
A~;\,
[V m ) •••
.i 1
(9.8)
0, ... , 'In
(9.9)
1'11 /)0,
where (9.3) \Vc recall that the 'vector It: A  kef is Hurwitz, that is,
PAo
[k 11 ••• ,k,Jr is chosen so that the mat.rix .,4 0
+ ,A~rp =
p
= pT > O.
(9.4)
'\Ie also repeat the two crucial expressions for li: Wo
+ wT () + C2
bm Um ,2
(£L 10)
+ Wo + wTf) + C2
(9.11)
!
where the 'regressor' wand the 'truncated regressor' ware defined as W
W
= ["(, m,:.h 'L' ml,2 , ···, V 0,2, [0,
Vm l,21 ... ,VO,2,
(T) ± (1)
(P(l)
+ '= ]T ~(2)
+ =(::!)p'
1
(9.12) (9.13)
and Wo
9.1
= IPO,l + 6 .
(9.14)
ISSController Desigll
The output feedback ISScontroller design has many similarities with the statefeedback design in Section 5.3. Also, the technique for dealing with unknown
9.1
373
ISSCONTROLLER DESICN
high frequency gain is the same as in Section 5.9.1. As in Section 8.1.2, our task is j'O develop a backstepping procedure for the system
+ Wo + W'1'0 + ':2
bm 1)111,2
!1
'Um.i+1 
a{y)'ll
i
k(Vm,l,
+ Vm ,p+l
(9.15)
=2
"
" ,p1
kpVm.l.
To achieve modularit.y, we seek control laws which
gl1aranl',e~
(9.16) (9.17) inputtostate
stability with respect to the error 8, it,s derivativc 0, and the state estimation error.:. Our main tool is nonlinear damping of Lemma 5.6. Our presentation assumes familiarity with the backstepping procedure o/" Chapter 8 1 and the ISScontroller of Section 5.3. Following Section 5.9.1, for the system we define (9.18)
Y  IJr V
'
m,!
1 (' 1) '1/ 1 
b' r

0.' i1,
2, ... ,fl,
(9.19)
In
wherc the division by the estimate bill poses no problem because bm will be kept away from zero by employing the parameter projection in the update law. Step 1. The equation for the tracking error':l obtained from (9.18) and (9.15) is (9.20) ':1 = bm 'U m ,2 + Wo + wT(} +':2 :Ijr' Substituting (9.19) for i
= 2 into (9.20), we
bm;:'!.
T
I
+ bmO.'I + Wo + W 0 +':2 + b::J/I'l>m .
(9.21)
111
In selecting the Hrst stabHizing function 0'1 we can disregard the term ::'2, but we have to deal with the difficulty created by the unknmvn bm multiplying 0'1. \Vriting 0.'1 in the fornl
(9.22) where 81 is the first nonlinear damping function, yet to be chosell, from (9.21) we get
Even though
the term
Wo
+
appears together with the parameter eITor bm , it could cancel if () were known. This is a convenience provided by
The choice
(9.24 )
374
MODULAR DESIGNS
yields (9.25) or, noting that
bm
is the first component. of 0, (9.26)
In (9.26) we see one 1l1Ore benefit of using sgn(hr,,) rather than ~ in (9.22): the appeal'tlllce of 81 in the bracketed term ill (9.26) is avoided. Now, to achieve inputtostate stability with respect 1;0 e2 and 8, the nonlinear clamping SI is chosen as ""
", = d, + ", I'" + b~ bi. + 0;,)
As
lI",
c,r .
(9.27)
Section 8.1.2, with (8.56) we conclude tbat 0:1 is a function of 11, (,,3,0, Xm +11 1J", and postulate that ai (111 {, 2, 0, Xm+il1i~il»). ill
Step 2. Differentiating (9.19), with the help of (9.16), we get
Noting from (9.19) that vm 3 i!!u. wT iJ wo got J
Oy
fUr =
':3
+ 0:21 WId by adding and subtracting
ftl
1
(fJ.29) The stabilizing fUIlction
0:2
is chosen to cancel all the terms except those in
9.1
375
ISSCONTROLLER DESIGN
the last line of (9.29):
where the term b",Zl is added to compensate for bm =2 ill (9.26). Substituting (9.30) into (9.29) we get
(0.31)
The uncancelled disturbance inputs ill the second line of (9.31) are counteracted, t:erlll by term, with the nonlinear damping function
(9.32)
Step i =3, •.. ,p. \'lve now proceed with the design by differentiating (9.19) = 3, ... , p 1 witb the help or (9.16):
for i
.
1
(i)
Vrrl,i  rUr
. (~  ii \' (i2») + ~1 b:'mYr(iI) ai1 Y,.",,::',U,,/\m+il'Yr
na
=
Vm,i+l 
In
kivm.l 
b1 Yr(i) m
80"_1
   (Aoe
8e
+ ky + q,) 
8a:i1 ( l' T Wo + w 0 +
C2
8o
iI
_
_
)
8 a il
.
 i1 (An + tIl)  L __ yU) 83 i=l ay~jl) r (0.33)
Noting from (9.19) that t'm.i+l  ty~i)
= Zi+l + ai, we get
376
MODULAR DESIGNS
=
From (9.17) it follows that the final step i p can be encompassed in these calculations if we define up = U(1J)U + v m ,p+1  t,;y~p) and Zp+l O. To prepare
=
for choosing a stabilizing function we add and subtract 8';j;IWT6 in (9.34) and get
(9.35) The stabilizing function the last line of (9.35):
a:j
is chosen to cancel all the terms except those in
(9.36) Substituting (9.36) into (9.35) we get
(9.37) The uncancelled disturbance inputs in the second line of (9.37) are counteracted, term by term, with the nonlinear damping function
8I
8a:i_1
8a:il
I
lJO 
i 1 T = d·'Ch  ) 2 + fL'I 18,  w12 + g'I A( 8f)
Y
Y
1
ry( L'l
Uj'n
r
il
lei 12
(9.38)
The complete design of the control law is summarized in Table 9.2. It employs the filters given in Table 9.1. The error system (9.26), (9.31), (9.37) is now compactly written as
z = A;(z, t)z + liJlll::, t)E2 + t¥;(z, t)TO + Q(z, t)"'iJ,
z E IV,
(9.39)
9.1
377
ISSCONTROLLER DESICN
Table 9.2: ISSController Design with
Zl
=
Y  Yr
~I'
=
V

. 
JII,!
I{~Filters
(9.40)
~y(il) b'r 
I"IJ.
i =2, ... ,p
1
..... , ,
(9.41)
m
0:1
=
al
=
02
=
OJ
f3i
=
_.
Adaptive control law: u = _1_
lI{Y)
(0: p
V
m.p
+1
+ J:y(p» b ' m
(9.50)
r
where
f A;(=, t) =
ll!m.l(Cl + 5]} lim
bm
0
bm
(C2 + S2)
1
0
1
0
a 1
0
0
1 (Cp
+ Sp)
(9.51)
378
A10DULAR DESIGNS
=
[
~ By
(9.52)
_ 0.:;'1
(9.53) _BOp1WT I:JU
(9.54)
Vie next establish the most important inputtostate properties of the error system (9.39), (9.51}(9.54), making use of the following constants: Co
= 11l~1l Cit IStSp
do
=
(t i=l
1 d .) I t l
n.o
=
(t :.)
1,
i=1 1\..
Do
= (t~) 1 • i=2 g,
(9.55)
Lemma 9.2 In the error system (9.39), (9.51)(9.54) with (9.47)(9.1,9), the lalla111ing inputtastate properties hold:
(i)
116,8 E .coo[O,tJ)'
th.en z,X,€,:::,A E £OO[Oltj), and
Proof. Differentiating ~lzl2 along the solutions of (9.39), (9.51)(9.51), using the definitions of the nonlinear damping functions (9.47)(9.49), and noting that .l!!!d > 1I we compute 1;'111 
9.1
379
ISSCONTROLLER. DESIGN
where for notational convenience we have defined ~ ~ 1. Now, with (9.52)(9.54) we get
(9.59)
By completing the squares in (9.59), we obtain 1 oail 1 P P P 1 dtd (Izl) ::;  L Cjz;2  E di (  Z j + ;;E.'l )" + L: £2 2 i=l i=l fJy ... ~ i=l 4d. I)
I)
380
MODULAR DESIGNS
which with (9.55) becomes dt' d
(211=12)
$;
col~f +!4 (2c~ + ~1812 + .qo ~IOI2) do Ii.Q
(0.61)
By applying Lemma C.5 to (9.61) we est.ablish the following two inequalities:
1=(t)1 2 Iz(t)12 whic11, in view of the boundedl1ess of c, prove that z E £:00[0, tl) and (9.56)(9.57). It. remains to prove that the boundedlless of = and 8 implies that x,~, and A are also bounded. A proof of this implication was already given for the tuning functions design in Theorem 8.5 (cf. (8.107)(8.110». The same argnment is applicable here with (8.110) replaced by
=,
.VnI,i = ..._j + ~Yr 1 (iI) +O:il bnl
The bounds on
(&  0" '\ (i2)) y,~,.=., '''m+il, Yr
,
=, x,~, =:, and A are independent of tl'
i = 2, ... , p. (9.64)
o
A consequence of Lemma 9.2 is that even in the absence of adaptation 011 the closedloop signals remain bounded, as we stated in Corollary 5.9 for the state feedback case. "Vith Lemma 9.2 at, haud, our next task i~ to design t.he identifier module which guarantees t.hat 0 is bounded, and iJ is either bounded or squareintegrable.
9.2
yPassive Scheme
'''ith the Kfilters in Table 9.1, we have obtained the parametric ymodel (9.10): (9.65)
9.2
381
VPASSIVE SCHEME
where Wo and ware measured and defined in (9.14) and (9.12), respect.ively, Except for the state estimation error ;2, the parametrk ymodel displays little difference from the parametric :t'model (5.128) we used in the st.ateff'edback design. What malres (9,65) desirable is the relativedegreeonc pl'operty between (J as the input and y as the output. \Ve introduce t.be setuar observer
ii =  (co + n.olwI2) (;Q where
Co
and
lin
are as
ill
y) +wo +wTiJ.
(9.66)
(9.55). The observer error
{9.G7}
f=yiJ is governed by
e= 
(co + IiO\W\2) f + wTO + e2' (9.68) It call be shown tlmt. this system and t11e system g = AoE form a syst.em wit.h a strict passivH.y property from the input 8 to the output WE:. This determines our choice of the paramet.cr update law: fJ = Proj
{rWf} ,
(9.69)
h". where the projection operator is employed to guarant.ee that fbm(t)1 ~ <;m > 0, \:It 2: 0. (For a detailed tl'eatment of parameter projection, see Appendi.'\: E.)
Lemma 9.3 Let the ma:r:imal interval of existence oj solutions oj (9.65), (9.66) and {9.69} be (0, t J). Then the following identifier prope7'ties hoM: (o) (i)
Ihm(t)\ ~ <;'m > 0, \:It. E [0, t J}
(9.70)
8 E Loo[O, tJ)
(9.71)
(ii)
EEL2IO,tI)n£oo[O,tJ)
(9.72)
{) E L2[0, t J) •
(9.73)
(iii)
Proof. First, rr~lll Lemma E.1 we conclude that the parameter projecl:ioll guarantees t.ha.t Ibm(t)l ~ <;nI > 0, 'Vt E [O,t/)' Let liS illtroduc~ a. Lyapul1ovlike fUllctioll \f
= ~ (18\~_, + IEI2 + ..!:.Iel~) , Co
(9.74)
whose derivative along the solutions of (9.69), (9.68) and (9.3) is
\i
=
~
_OJ r
1~
Co ::; 2"C "I
=
_ Co e2 _
:s
~ £'1. 
2 
1., (}  Cof~  nolwlC + fWT() + ee2  Iel2co lJ
\"1 2
Knl w f
., ..
+ BT ( Wf  r 1 6':..)
Co
 2 f + £G2 "I

1
_Co2 (f _~)e:!)2 oIWI2€2 + 1fT' (we  r 8) .
n.o\wI2e2 + jjT (we _r1o) li
1
\'1
2co le2 
~
(9.75)
382
MODULAR DESICNS
Using Lemma E.!, we have
 err 1o= 8'1'r 1 Proj {rWE} ::; tP'rlrWE = UfWE
1
(9.76)
so (9.75) becomes • Co '1 1 12 f~. 'l 1'::;2fKOW
8, f
The non positivity of li proves that we have
1 1 (J::'12 t 10:"1?~ ::; ~(rl) r
<

(
0.77 )
E £00[0, tl)' Now due to Lemma E.1
1 Ir 12 _ 1 I 12 .d.(r1) WE rI  ~(rl) we r
X(r) I 12 _ ( 21 12 ::; .d.(r1) WE. ..\ r) WE ,
(9.78)
,vhich, substituted into (9.77), yields (9.79) Upon integratioll, we get Co
r' .
ft., "".0 f':'.., E(T)dr + X(r)2 10 IfJ(T)IdT ::;  10 Vdr::; V'(O)  Vet)
'2 10
::; V(O) < 00 which proves that
f,O E 4[O,t/)'
1
(9.80)
o
This proof reveals that the nonlinear damping term
nolwl 2e .included ill
the observer (9.66) is crucial for achieving squareintegrability of O. It is important to remember that the bounds in this lemma are all independent of tl, which will allow us to extend the maJcimal interval of existence of solutions t I to 00 in the proof of the next theorem.
Theorem 9.4 (yPassive) All the signals in the closedloop adaptive system consisting of the plant {9.1}, f},e control law in Table 9.2, the filters in Table 9.1, the obsc7'1Je7' {9.66}, and the update law {9.69}, a·re globally unifo7mly bounded and global aS1Jmptotic tra.cking is achieved: tll~ [yet)
 Yr(t)]
= O.
(9.81)
Proof. The projection operator in Appendix E is locally Lipschitz, as stated in Lemma E.1. Therefore, as argued ill the proof of Theorem 8.5, the solution of the closedloop adaptive system exists and is unique 011 its ma:l\:imUDl interval of existence [0, t/)'
9.2
383
V·PASSIVE SCHEME
From Lemma 9.3 we have 6 E £oo[O,tJ} and 8 E £2[0,t / ), which in view of Lemma 9.2(ii} implies that z,x,~,:=:,A E £00[0, tJ). Furthermore, Lemma 9.3 guarantees the bounded ness of f, which in view of the boundedness of y establishes the boundedness of y. Since bm(t) is bounded away from zero r the control 'U is also bounded. By the same argument as in the proof of Theorem 8.5 we conclude that tJ = 00. Now we set out to prove the tracking. In view of (9.41) and (9.42) we have
Ibml
 
(CI
t;m
_
1
.
T
_

+sdzJ +bm =2+ ( W+. (Yr+al}el ) 0 b'll
=
=
=
=
(9.82)
On tbe other hand,
aY W + alll

brnZl 
T(J
T
Zt el

8 = bmz 1 
'raW Y 8C'tl
(9.83)
(J.
With (9.82) and (9.83), the errOl" system (9.39), (9.51)(9.54) is rewritten as
Z = A=(z~ t)z + H'e:(z, t)
(wTD + E!!) + Q(z, tyro,
(9.84)
where A= (\vithout *) is given by
"n sgn(lJm)(c1 + atl
b,n
0
bna
(C2 + 82)
1
0
1
1;1'11
A:=
0
0
(9.85)
0
0
1 1 (cp +
8p )
z
and H'e: and Q are as in (9.52) and (9.54). We ]cnow that z and are bounded, so to prove that z(t) ; 0, we need to shmv that Z E £'].. Since we also know that f E £", let us consider (9.86)
384
:MODULAR DESIGNS
which, in view of (9.84), (9.68), satisfies
(; = A:(z, t)( +
[((CO + lLolwl2)I + A=(z, t)) H't:(z, t)  ~t'e(Z, t)] f + Q(z, t))8
(9.87) where the bracketed expression and Q are bounded due to the bounded ness of all the signals and the smoothness of the plant nonlinearitics. Noting that Lemma 9.3 guarantees that b".sgn(b ... ) ~ 1, with the help of (9.85), it is now 'i,n straightforward to derive
d (.') 1 + A: ) d 1(1 :5 co I(1'1 + 1 1( (co + lio IW 1')1 ~
t
HIt: 
. 12 If 1 + 161· 1:. H't: 'J
'J
~
(9.88) Since f,O E £2, it. follows by Lemma B.5(H) that ( E £2. Tlms, by (9.86), z E £2' By Barbalat's lemma (Corollary A.7), z(1,) ~ 0 as t + 00. Since Zl = Y  Yu this proves the tracking. 0 The lJpassive scheme consisting of the identifier (9.66)(9.69) and the ISScontroller in Table 9.2 is simpler than the tuning funct.ions scheme in Table 8.2 because the ISScontroller does not incorporate tuning functions t.erms. The cost of this simplification is very low: only one extra integrator for the observer (9.66).
9.3
ySwapping Scheme
Since the ypassive scheme is limited to gradienttype update laws we develop a swapping scheme which also allows the leastsquares algoritlun. \Ve again start with the parametric ymodel (9.10) (9.89) and introduce the swapping filters
mo = rl:1 =
(co + liolwl2) (roo  y)  Wo,
 (co +
11:01"'12) + w, r;:J
Em.
(9.90)
tv E IRP
(9.91)
'iVo
and the estimation error f
= Y + tvo 
fA] T
6.
(9.92)
Substituting (9.89), (9.90), and (9.91) into (9.92), we get (9.93) where € is governed by (9.94)
9.3
385
7JSWAPPING SCHEME
The update law for
8 is either the gradient:
'm
{at:} ():. = Proj r 1+1I'W" I I" , b
bm(O) sgn bIll > r=rT>o, ,,;:::0
m
(9.95)
or tIle leastsquares:
iJ = Proj &,,.
{r wi r}' 1
+IJwY.
wtv"
t = r1 + lItv I rr, f
(9.96)
r(Q) = r(O)T > O!
II
2:: 0
'III
where the projection operator is employed to guarantee that Ibm(t)1 ;::: > 0, Vf ;::: 0, and by allowing II = 0 we encompass unnormalized gradient and leastsquares. (For details of parameter projection, sec Appendi.x E.) Lemma 9.5 Let the maximal interva.l 0/ existence of ,90l7J.ti0l1S of (9. 89}, (9. 90}{9. 91 ) with either' {9.95} or {9.96} be [0, i/)' Then f07" IJ ;::: 0, the following identifie1' properties h.old:
IbmO.)1 ;::: ~nl > 0, Vi 6 E £oc[O,t/)
(0) (i) (if)
(9.98)
E £2[0, t / ) n £00[0, tJ)
(9.99)
{} E £!![O, tl) n £00[0, tl) .
(9.100)
f
(iii)
~
(9.97)
E [0, tl)
Proof. First, from LemmaE.1 we conc1ude that Now we consider
=
co Iw 1'1  lio
Ibm(t)l2;: C;m > 0, Vt
(T
E [O,t/)'
1)2 +  1 2n:o 4n~0
tV W 

~ colml!! + .!:.... ,
(9.101)
41\.0
~.
which proves that
tr1
E £00[0, t I)' \Vith (9.4), along the solutions of (9.94) and
(9.3) we have
I'))
d (., 11 1E+ep
2dt
Co
= ~
.J} COE
   1 IeI"lio IW I"""" f + ee:! 2co
Co _') Co f
2
2
(_
1
fe2
Co
)
:3
1 1 I I') +e2te'l
2cQ
2co
(9.102)
386
MODULAR DESIGNS
which shows that f E £2[0, t,) n £00[0, tf). Gmdient update law (9.95). We consider the positive definite function 1 (2 = 21 1(),'1r1 + 2co E + 1 I€ 12) P • Co
V
(9.103)
Using (9.102), the derivative of V is
,i' $ _il'rJ~  ~t! .
(9.104)
By Lemma E.l t \Ve have
_ iJI'rle = iJI'rJ Pro' {r WE 2 } < rrr 1 + IJIWl
J

(9.105)
mf
1 + vlml:!
1
so (9.104) becomes i' \
(}nT
:s; 
wf
1 + vlm/:!
1_,)  2c .
(9.106)
In view of (9.93), we have
,i' ::; _OT m(m
~ =
T
9+i) _ !f2 = (wT9)2 _ eTrol _ 1 + IJlml:! 2 1 + vlml 2 1 + vlml:! 2 1 (W'l'iJ)2 1 (mTO) 8Twi _ ~l2 21 + IJlwl!! 2 (1 + vlwl2)2 1 + v\wPz 2 1
!l2 2
(wT O)2 _! ( mTe +f)2
21+vlm1 2
(9.107)
1+v\mI2
2
and alTive at (9.108) which implies
ii E £..[0, tf)
J1 +evlwl:! E £2[0, tf)·
and
Using this and the
boundedness of wand e, we readily show that f,B E ~[Ottf) n C,o::r[Ottf). Lea8tBqua1Y~B update law (9.96). We consider the function 2
V = 19tr(t)_1
1_,) 1/') + e" + ;; (Ii>, I
Co
CO
(9.109)
wllOse derivative is readily shown to be
v< _ 
which proves that
ii E £ ..[0, tf)
and
2 f
(9.110)
1 + vlmlr. t
J
e
• E L2[O, tf ). Using this and
1 + vltrJlf
the boundedness of m and f t we show that
f,
BE £2 [0, tJ) n Coo [0, tJ ).
0
9.3
387
ySWAPPING SCHEME
Theorem 9.6 (ySwapping) All the signals in the closedloop adaptive system consisting of the plant (9.1), the control law in Table 9.2, the filters in Table 9.1, and the filters (9.90)(9.91), 1lJiUI, either lhe gradient (9.95) 07' the leastsquares upda1.e law (9.96), are globally uniform.ly bounded, and global asymptotic tracl.:i.ng is achieved: lim [y(t)  Yr(t)]
100
= O.
(9.111)
Proof. The projection operator in Appendix E is locally Lipsrhitz, as stated in Lemma E.1. Therefore, as argued in the proof of Theorem 8.5, the solution of the closedloop adaptive system exists and is uniqne on its ma.'ti.mum interval of existence [0, t f). From Lemma 9.5 we have 0, iJ E £00[0, tl), which in view of Lemma 9.2(i) implies that z, :c, {, 3,'\ E £00[0, t f). Equations (9.90) and (9.91) imply that to'o and 1AJ are in L.oa [0, t f). Since bm(t) is bounded away fronl zero, the control 11. is also bounded. By the same argument as in the proof of Theorem 8.5 we conclude that tf = 00. To prove the tracking, let us consider the error system (9.84),
(0.112) along with the error equation governing
i =  (CD + Kolwl'.!)
£:
+wTO + c2  mTO.
f
(9.113)
We could finish the proof of convergence of z to zero by an argument similar to (9.86)(9.88) in the proof of Theorem 9.4. Instead, we present an alternative proof based on the idea in Remark.6.B. As in Remarl~ 6.8, we show that i(t)
:+ o.
it follows that B(t)
+
Since,
O.
by
therefore, H'E(=(t), t)T (w(t)1'6(t)
Q(z(t), t)T8(t)
Lemma 9.5, {) E £DCJ
o.
+ C2(t))
Thus the input H'E (9.112) converges to zero. In view of +
:t (~lzI2)
n
£2 and
From (9.113) we conclude w(t)T8(t) +
O.
Since O(t)
+
",
£001 and
°
0, we have
(wTO + C2) + QT8to the error system
:::; colzl 2 + ZT (I'T'e (wTO + C2) + QTO) :::;  ~Izl' + ~ IIV, (wT8+£2) + QT01'
m
8E +
by applying Lemma B.B, we arrive at the conclusion that z(t) = Y  Yn this proves the asymptotic tracking.
Zl
(9.114) +
O. Since 0
Remark 9.7 In Theorem 6.4 we showed that the state feedback ISScontroller can be simplified by setting 9i 0, i 1, ... ,n, provided that a particular
=
=
3BB
MODULAR DESIGNS
form of normalizat.ion is introduced in the parameter update law. 'The output feedback case is no different. It is possible to show that the modified gradient
o= ProJ' {:...'_r__
wE
1 + v'IQI.F 1 + IJIWl 2
iJ",
}
1/
>0
(9.115)
'
and the modified leastsqua,res {)
=
Proj bm
t
=
{r + I)'IQIF 1
1
1 + J)'IQI.F
r
me} , 1 + vlwlr.
(9.116)
l'
ww r 1 + vlmlr~
update Jaws, which guarantee the properties (0)
(i) (ii)
(iii)
Ibm(t)1 ~ ~m > 0, 'Vt E [0, if) 8 E £00[0, tl) f
E .coo[D,tl), '1':'
E
/(1 + zlIQIF) :..
(9.117)
(9.11B) E
Q 0 E .coo [0, tf)' 0 E £2[0, t/),
.c2 [0,t/)
(9.119) (9.120)
allow one to set gj = 0, i = 1, ... ,71 in the ISScontroller in Table 9.2, while 0 retaining the result. of Theorem 9.6. 'Vhile the 1}passive identifier uses an e~tra integrator to generate fj, the yswapping identifier employs p + 1 integrat.ors for wand woo Therefore, the price paid fol' having flexibility in the selection of the update law is the increase of the dynamic order by p.
9.4
xSwapping Scheme
In this section we design a swapping identifier which uses the Kfilters alrea.dy employed for state estimation. Instead of the parametric ymodel we consider the parametric J'model (9.2): (9.121)
This parametric model is already in the static form due to the use of the Kfiltet's from Table 9.1. However, this parametric model may seem unimplementable because .7: is not measured. Only ~ n and the first component of x (the output y = XI) are available. Fortunately, it suffices to consider only the first row of (9.121) where all the signals are measured except for Ct: I
(9.122)
9.4
389
xSWAPPING SCHEME
It is crucial that el is bounded and exponentially converging to zero. We introduce the "prediction" of y as 'r~
A
Y=
so that, the "prediction enor"
f
f.l
+ 0 1 (J ,
(9,123)
~ Y  iJ is implemented as (9.124)
and satisfies the following eqnation lincar ill the parameter error: f
'1'= !l18 + Cl'
(9.]25)
fJ is either the gradie11t:
The update law for
bill (0) sgn bm > 'III r= r T > 0, v> 0
(9,126)
or the leastsquares:
iJ
=
t
= f
P.roj 11",
{r 1
~;r fl
+1.1
I
0
Ot~I
1
} , (9.127)
f~
1 + vf11foO I
r(o) = f(O)T > 0,
II>
0
lvltere the projection operator is employed to guarantee t.hat Ibm{t)1 ~ \m > 0, Vt 2:: 0, MatrL'\: ro in the leastsquares update la\v only indicates t:hat we = + 1'0/' 1'0 > 0, 1.0 lwep the either use covariance resetting for or let normalization positive definite, The above update laws are normali~ed. '~Titb ullnonnalized update laws we are not able to guarantee bonndedlless of n governed by (8.16),
r
ro r
o'T = AoO"I' + F(y,u) T
!
(9.128)
independently of the boundedness of F{y, 1I).
Lemma 9.8 Let the maximal inten,"l oj existence of solutions of (9.1), (9.5)(9.7) with either (9.126) 07' (9.127) be [0, tf). Then the following ide71.tifie7· propertie/i h.old:
(0) (i) (ii)
(iii)
Ibm {t)l2::
'm > 0
1
(9.129) (9.130)
Vt E [O,tf)
9 E £o;,[O,tf) ..;
E
. ' ..;
1 + "10 1 1
f
1 + I.lOIfoO l
8 E £2[0, tf} n .coo [0, tf)·
E
4[O,tjl n£~[O,t,l
(9.131) (9.132)
390
MODULAR DESIGNS
Proof. First t from Lemma E.1 we conclude that Ibm(t)1 ~ C;m > 0, 'It E [0, tJ). Gradient update lauJ {9.1B6}. We consider the positive definite function
V
= 2"111" () r 1 + 2"11 e 12p.
By virtue of Lemma E.l we have ffI'r 19~ show that
.
V ~
1 21
(9.133)
_{jT 1+e1hll::l' which enables us to
13.2
+ IJln11!l .
(9.134)
The nOllpositivity of li proves that jj E £00[0, tJ). Integrating (9.134) we get f
r==== E £2[Ottj). From
JI + vln l l
2
(9.135)
in view of the bOllndedness of jj and
g,
we establish..; E E ,c",,[0, tJ). 1 + vlnll!!
With Lemma E.l we have
iJ E £2[0, tJ) n £00[0, tj). Least·squa1es update law {9.127}. We consider the function
which proves that
(9.137) " = l{jl~(t)I + lel~ which is positive definite because r 1 (t) is positive definite for each t. Using Lemma E.l and the fact that 1, (r1o) = 1+11~1f~onl' it is straigbtfonvard t
to arrive at
(9.138) In view of the positive definiteness of rl(t) this proves that jj E Coo[Ot tj). It also proves that
J1 +vnlron .. f
E C2 [0, tj). From 1
(9.139)
9.4
391
xSWAPPING SCHEME
in view of the boundedness of
esta.blish
8 a.nd
E
and positive definiteness of
J1 + IIlnllrn E .c",,[O,tl)· With Lemma E.1 we ha.ve f
r 0,
we
•
o Combining tbe xswapping identifier with the ISScontroller with I(filters, we obtain the following result.
Theorem 9.9 (:vSwapping) All the signals in the closedloop adaptive system consisting of the plant (9.1), the control law in Table 9.2, and the filters in Table 9.1, with either the gradient (9.126) or the leastsquares u.pdate law (9.127), arc. globall]l uniformly bounded, and global asymptotic t"acking is achie'lled: (9.141) lim [yet)  Yr(t)] = 0. ico Proof. Tile projection operator is locally Lipschitz, so the solution of the closedloop adaptive system exists and is uniq}le all its ma.~imum interval of existence [0, 1,f). From Lemma 9.8 we bave 8, 8 e £00 [0, t f), whicll in view of Lemma 9.2(i) implies that z,X,e,S,A e £oo[O,tf)' Hence, tf = 00. To prove tbe tracking, let us consider the ell'or equation (9.125): f
=
"
(0.142)
+ O2 + FJ(y,u).
(9.143)
{llO+EI&
First, from (9.128) we note that
fh
= kin!
=
Recalling from (9.9) that f22 [Um .2, Vm l,2, .•• , Vo,:!, =(2)]'1', and from (8.9) that F1(y,u) = [0, ... ,0, l)(l)]T, we conclude that O2 + F1{y,u) = w, so (9.144) With (9.142), (9.144) and (9.3), we now get (9.145) The rest of the proof follows the lines of (9.112}{9.114) in tile proof of Theorem 9.6. The centraJ part. of the argument is to show that (9.145) implies that w T 8(t) . 0. 0
392
l\10DtlLAR DESIGNS
Remark 9.10 As in Remark 9.7 for tile yswapping identifier, we can modify t.he update laws (9.1.26) and (9.127) by a normalization with 1 + v'/QIF, which guarantees that QTfJ E L oo , and allows us to set gi = 0 in the ISScontroller in Table 9.2. 0 The xswapping scheme is the only modular scheme whose dynamic order is as low as that of the tuning fUllctions scheme. The xswapping scheme is simpler than the tuning functions scheme, but, as we shall soo, its performance propert;ies are a little less strong.
9.5
Schemes with Parametric zModel
The lllod1l181' schemes we designed in the last three chapters were based on the parametric IJll1odel (9.10) and tIle parametric xmodel (9.122) rather tban 011 the parametric zmodel
(9.V16) with A:, l·F" and Q defined in (9.B5), (9.52), and (9.54), respectively, and defined in (9.47}(9.49). (Note that (9.39) cannot be a parametric model because 11m in A~ in (9.39) is not known.) Even though the parametric :;model was central in the statefeedback design, our attention was devoted to other parametric models because of the lower dynamic order of the resulting adaptive schemes. For example, while the vpassive scheme employs a scalar observer that generates ii, the :;passive Bchenle would use an observer of order p for :. The differt:llce is more drastic between the xswapping scheme which does not use any extra fiIt.el'S, while the zswapping scbeme uses additional filters of total dimension P(1l + 1). For the sa,lre of completeness and continuity with our state foedback designs, we briefly llre.C3ent t'VQ schemes based on the parametric zmodel: tIle =passive scheme and the .;;swapllil1g scheme. We omit tbe stability proofs. Sj
zPassive scheme In analogy with the statefeedback ::;passive design in Section 5.5, starting from the paramet.ric model (9.146), we consider the identifier
z = A:(z,t)z + Q(=,'l)T8 f
iJ
= =
(9.147)
zz
~roj {rwH'7 E} ,
(9.148)
bm(O) sgn bm > ;m 1
r=rT>o.
(9.149)
brA
If bm were known, in which case we would not have the estimate bm (and would not use projection), then one could prove the same result as in Theorem 9.4
9.5
393
SCHEMES WITH PARAMETRlC zIvloDEL
(cr. [103]). With unknown bm , the identifier (9.147}(9.149) is not directly applicable. To see this, recaU that t~e crucial property for establishing global boundedness in passive schemes is 0 E £2. This property was in the statefeedback zpassive design guaranteed automat.ically by t.he nonlinear damping terms built into A=. When bm is unknowll, the nonlinear damping terms (9.47)(9.49) are capable of guaranteeing inputtostat.e stability with respect. to 8 in the system (9.39), but in the observer error system f
= A=(z, t)f + H/£(z, t) (w T 6 + C2)
(0.150)
with the paral~eter update (9.149), they canllot guarantee the squareintegrability of {J as they would if bm were known (cf. Lemma 5.10). To remove this difficulty we strengthen t.he observer error system (9.150) by including the additional nonlinear damping term diag { li:llwj:!, "~21~wI2 10, ...
,o} (£  z) in the observer (9.147):
t,= A:(z, t)zdiag {"dWI" "'" 1<;;;>"
,0, ...
,o}
(z.o)+Q(z,t)TO. (9.151)
It is possible to show that the adaptive scheme consisting of the control law in Table 9.2, the filters in Table 9.1, the observer (9.151) and t.he updatE' law (9.149) achieves global uniform boundedness, as well as asymptol:ic tracking.
zSwapping scheme The difficulty with inadequate nonlinear damping terms due to unknown bm in the zpassive scheme is much easier to deal with in the zswapping sche!ne. A key property of swapping identifiers is that they guarantee that iJ is bounded. VVe often achieve this with nonlinear damping terms which guarantee that the filtered regressor is bounded. However, when bm is not knowll the nonHnear damping terms built into A= are not capable of guaranteeing the boulldedness of the filtered regressor (cf. Section 6.8). Fortunately, in the swappin~ design we can use normalizat.ion which can guarantee the bounded ness of 9 even when the filtered regressor is growing unbounded (cf. Lemma 6.26). With this observation one can show that the following zswapping identifier guarantees the global ulliform boundedness and asYlllptotic tracking:
UT = A=(=, t)U T + H'E(Z, t)wT , UO = A:;(z, t)Uo + 1,Vf:(z, i)w'r(j  Q(z, t)TiJ, f = z+UoU 8,
lJ E IRP>:P (9.152)
U E HlP
T~
(9.153) (9.154)
with either the normalized gradient:
{Uf}
~ = Pl'oj r lUI'" b l+v 7F
(J
m
bm (0) sgn bill > C;m
r=rT>o,
1.1>0
(9.155)
394
MODULAR DESIGNS
or the normalized leastsquares update law:
0
r 9.6
=
=
p[~j {r 1 +~~UI}} , f
b,.,. (0) sgn bm > C;m (9.156)
T
tRJ r 1 + llIUI}
f(O) = r(o)T > 0 I
II> O.
I
Transient Performance
In this section we derive £2 and £00 transient performance bounds for the error state z, which include bounds for the tracking error Zl = Y  Yr' We first consider the passive schemes (f}passive and zpassive), and then the swapping schemes (yswappillg, .~swapping, and zswapping). For simplicity, we Jet r=;I.
9.6.1
Passive schemes
First we derive an £00 performance bound for the 1}passive scheme. To eliminate the effect of the initial condition of the estimation error f = Y  fJ, we initialize the observer with y(O) = y(O), which sets c(O) = o.
Theorem 9.11 (yPassive Scheme) In th.e adaptive system {9.1}, (9.5), (9.6), (9.7), {9.50}1 {9.66}, and {9.69}, the following inequalit1J holds:
Iz(t)1 ~ . ~ (AlI8(0)12 + Nle(O)I~) 1/'!. + Iz(0)leCOI / 2 I v Co
(9.157)
111here
l t l =1 (l +"(2 ) 2lio li.oDo
N
=
(9.158)
"(3 ) +1 (1 +"(2) "( (1+ , 2coli.o
li.ogo
2.a( P)
do
Conago
(9.159)
Proof. To obtain an £fXl bound on z, it would s~em that inequality (9.56) could be used along with £00 ,bounds on C2, 8 and 0. However, it is not clear how to obtain a bound on 1191100 depending only on design p81'ameters and initial cOllditi~ns. Therefore, we apply a different approach which eliminates the need for 1181)00' First, we note from (9.68) that =
coe:!  li.o/wI 2 e2 + f
~
fa< 
Co 2
I)
(w T 8+ e2 )
lio 12 + c2 1 2 + 1 IWE 2
2co
2li.o
II" f) ,
(9,160)
9.6
395
TRANSIBNT PERFORMANCE
From Lemma E.1(H) and (9.69) with substituted into (9.160) yields
r
= ,It ,ve have
161 ::; l'lwel,
which
(9.161) In this inequality 161 2 appears with an opposite sign of that ill (9.61). Theref~re! by adding these t,vo inequalities with an appropriate scaling t \ve eliminate
1812:
By applying Lemma C.5, we get
Iz(t}I' + 2~~:(t}2
~ 2~ [Uo + Co~:Yo) 11£.11;' + :0 (1+ ":110) 11811!'] + (lz(0)1 2 + i
2~o!1o
e(O)2) ecot .
(9.163)
Since f(O) = 0, we have 2 ) 2 ) Iz{t)1 :s .~ [ ( d1 + _7_ lIe211~ + 2:.. ( 1 +.:::L 11611~] 1/2+lz(O)le Col/ 2 •
v 2co
0
COliD9o
From (9.3) and (9.4) we have 1ilel~
lio
Ko90
::; lef2, which gives
(9.164)
(9.165)
396
MODULAR DESIGNS
To obtain a bound on 116!1~ we recall (9.77) and (9.74), which give
11611~ :5 10(0)12 + 1.Ie(0)1~.
(9.166)
Co
Substituting (9.165) and (9.166) int.o (9.164), we obtain (9.157) with (D.158)(9.159). 0 The initial condition z(O) in the bound (9.157) is, in general, dependent ou the design parameters CO, dO,lio, go. However, as explained in Section 4.3.2 for state feedback, with trajectory initialization we can set z(O) = O. Following (9.40) and (9.41), :(0) is set to zero by selecting Yr(O) =
y~i)(O) =
y(O)
(9.167)
bna (0) [Vrn'i+l (0) 
O'i
(y(O), e(O), 3(0),0(0), Xna+i(O), ii~iI)(O))] , i
= 1, ... ,p 
1.
(9.168)
Upon setting z(O) to zero, the bound (D.157) can be systematically reduced by increasing Co. By examining (9.158){9.159) one can see t,hat the bound (9.157) can also be systematically reduced by simultaneously increasing liD and
do. A careful comparison of (9.157) with (8.139) reveals that the bound for the tuning functions schcmc is lower. Also, an advantage of the tuning functions scheme is t.hat an L.2 bound like (8.138) is not available for the yobserver scheme. We now give performance bounds for the zpassive scheme. The observer initial condition is set to z(O) = z(O). For comparison with the ypassive 6. scheme, we select ~1 = ... = n.p = ~o.
Theorem 9.12 {zPassive Scheme} In the adaptive system (9.1), (9.5), (9.6), (9.7), (9.50), (9.1,/9), and (9.151), the following inequality holds:
Iz(t)1
~ ~ (M",,19(0)1 2 + NooIE(0)I~f/2 + Iz(0)1.",'/2,
(9.169)
where
M"" = N"" =
2~o (1+ ~~~) 8~u [(1+ ,,~~) (:. + ~(~)) + ~(~)l·
(9.170) (9.171)
AforeDver, if bm is kno'Wn, then the zpassive design with the observer (9.147) results in the L.2 bound 1 ( 1\1 10(" IIzlb:5 . r;:: 0)1 + N'J. IE(O)I:p., ) 1/2 + 2
vCo
1 ~lz(O)I, v.:.co
(9.172)
9.6
397
TRANSIENT PERFORMANCE
M2
=
H1+ 4::Uo)
(9.173)
N2
=
~ (1+ 4~Uo)·
(9.174)
While the Coo boullds (9.157) and (9.169) are similar, the £2 bound (9.172) is available only for the zpnsshre scheme.
9.6.2
Swapping schemes
First we derive an £00 performance bound for tbe vswapping scheme. For simplicity, we consider only the gradient. update law. To eliminate the effect of the initial condition of the estimat.ion enol' f = y + ron  ro TO, we initialize it with ro(O) = 0, wo(O) = y(O), w1Iich sets l(O) = O.
Theorem 9.13 (ySwapping Scheme) In the auapti1Je slJstem (f).l), (9.5), (9.6), (9.7), (9.50), (9.90), (9.91), and (9.95), the Jollo'wing ineqlJ.fJ.lily holds 1=(t)I =s;
~ (1\118(0)12 + lVle(O)I~) J/'2 + 1=(O)lecllt •
(9.1 (5)
yeO
whe1"e
ill
= 1
N =
(1 +
.))
'1
411:0
Bc5noDo

1+
l[2 (
4 cfi~o
"'r'1
8CB~oDo
(9.176)
1]
+2.. ) +_...,4{Jo
doLl( P) .
(9.177)
Proof. 'Ve derive an £00 bou!ld all :; using (9.56). It rcmltillS '.0 det.ermine bounds on
lIe:zlloo, nOnce itnd lI'ince'
First, from (9.3) and (9.4) we have (9.178)
In view of (9.108) and (9.103), using e(O) = 0, we get (9.179) 'Vith the help of (9.95) we write
e2 "11 II" f2 '111 III) II 11'1 1B::"1" =s; '1"IW\2 (1 + vlwl 2 )2 ~,. to' ;;.;, (1 + vlrol 2)2 ~ ')' ro ;;.;, e ;;.;,
(9.180)
398
MODULAR DESIGNS
and by substituting (9.93) we obtain (9.181) From (9.101), using m(O) = 0, it follows that (9.182)
To obtain a bound on d dt
11;IICXl, along the solutions of (9.94)
(111')  f  + 1 Ie I')) p 2
4eo
~ CaE2
+ f~2 

1
~1ca
and (9.3) we have
IE 12 ~ 0
(9.183)
which yields
IIfll~ ~ 2 Ic(O)I~· Co 1
(9.184)
By substituting (9.179), (9.182), and (9.184) into (9.181), and then, along with (9.178) and (9.179), into (9.56), we arrive at (9.175) with (9.176) and (9.177).
o The initial condition z(O) ill the bound (9.175) can be set to zero by the trajectory initialization procedure (9.167)(9.168). Upon setting z(O) to zero, the bound (9.175) Crul be systematically reduced by increasing Ca. By examining (9.176)( 9.177) one can see that the bound (9.175) can also be systematically reduced by simultaneously increasing Ka and dll . The £00 bound (9.175) for the yswappillg scheme is lower that the bound (9.157) for the Vpassive scheme but higher than the bound (8.139) for the tuning functions scheme. Nmv we derive an £00 performance bound for the xswapping scheme.
Theorem 9.14 (xSwapping Scheme) In the adaptive 81/stem (9.1), (9.5), (9.6), (9.7), (9.50), (9.126), the Jollo'wing inequality Il.olds: (9.185)
where (9.186) (9.187)
9.6
399
TRANSIENT PERFORMANCE
Proof. We derive an £a;J bound on z using (9.56). The b01:1nd on in (9.178). It remains to determine bounds on and (9.133) we get
1\81100 and 110\100'
1Ie211co is as
From (9.134) (9.188)
By sUbstituting (9.135) into (9.136) we get
18\2
~ 21'2 (.!.IBI2 + led2) . 1.1 1/
(9.189)
Since (9.190)
then (9.189) yields (9.191)
By substituting (9.188), (9.178), and (9.191) into (9.56), we arrive at (9.185) with (9.186) and (9.187). 0 The initial condition z(O) in the bound (9.185) call be set to zero by the trajectory initialization procedure (9.167)(9.168). Upon setting =(0) to zero, the bound (9.185) can be systematically reduced by increasing Co. By examining (9.186)(9.187) olle can see that the bound (9.185) can also be systematicaHy reduced by simultaneously increasing "0, go, and do. We now give performance bounds for the zswapping scheme. The filter initial conditions are selected as U(O) = 0 and Uo{O) = zeOl, to set leO) z(O) + Uo{O)  UT(O)8(0) o.
=
=
Theorem 9.15 (zSwapping Scheme) In U"e adapti1Je system (9.1). (9.5), (9.6), (9.7), (9.50), (f).152), (9.159), (9.155), the jollo'llJing inequality holds Iz(t)1
~ ~ (J1/0018(0)12 + NoaIE{O)I~) 1/2 + Iz{O)\eCOi , veo
(9.192)
where {9.193} (9.194.)
Moreover, if bm is I.."nown, then the zswapping scheme results in the £'}, bound
a
1 ( 1112 19(O)I ") + N2 1e(0)lp 2 ) 1/2 1 IIzlb::; t;:: + n;:lz(O)1 + 3 16(0)1, v~
v~
7
{9.195}
400
MODULAR DESIGNS
where
AI, N,
= =
H2;~n +;(~ +2~Ko)] 3~, [1+ ~~ + ~V (~ + 2~Ko)]'
(9.196) (9.197)
'Vhile the £00 bound for the xswapping is similar to that for the :;swapping scheme, the £2 bound (9.195) is available only for the =swapping scheme.
9.7
Swapping Schemes with Weak ISSController
In Chapters 5 and 6 we discussed the possible undesirable effects of the strong nonlinear damping terms. In Sect.iDll 6.7 we introduced a weal<:er ISScontrDlJer with a much slower growth of nonlinearit.ies. In this section we develop a weak version of the outputfccdbacl\: lSScontroller in Table 9.2. Table 9.3 summarizes the design of the weak ISScontroller. There are two differences between the ISScontroller in Table 9.2 and the weak ISScontroller in Table 9.3: • Instead Df the quadratic nonlinear damping functions in (9.47)(9.49), the nonlineru' damping functions in (9.207)(9.209) are 'linearlike.' • The t.erms bmz1 and Zih present in (9.44) and (9.45), respectively, are eliminat.ed from Table 9.3 to make the analysis simpler. The result.ing closedloop system has a form similar t.o (9.39): (9.198) where H'el H'D', and Q are as in (9.52), (9.53), and (9.54), rcspcct;ively, and A~ is without bm , 1, ... ,1 below the diagonal:
A~(z, t)
=
(0.199)
Before we select an identifier, we establish an ISS propcdy of the errol' system (9.198).
9.7
401
SWAPPING SCHEMES WITH WEAI( ISSCONTROLLER
Table 9.3: Weak ISSControl1er with KFilters (9.200)
Y  Yr
=
ZI
i = 2, ... ,p 0'1
=
0:]
= =
0'2
(9.201)
= Pi =
OJ
, (9,207)
(9.208)
I
~
UO:il T
1
(iI)
+gH A  ;::Yr '88 b~
el
12
+ 1,
i
= 3, .... p (9.209)
Adaptive control law: u = _1_
u(y)
(0: P
11
mtp
+1
+ Jy(P») b r m
(9.210)
402
MODULAR DESIGNS
Lemma 9.16 The error system (9.198) is ISS with respect to Proof. For the states
Zi,
'i
= 3, ... , P 
(e::!1810).
I, of the system (9.198), one readily
shows that
d (z7) 2

dt
.!...
< 
~zr  ¥IZil (IZil ~IZi+ll)
_d+a:11 (1z,1 ~;le2!)
,,+ a:~lwl gi
8a'i1 I ( 86 Zi
.
T
(lz;I ~Iiil) 
1 ("1) Cl ) ::Y/ b~
I(IZil 161 1 ~) . 9j
(9.211)
Thus we have the following implication:
2
1
1 
IZil ~ ( ~Izi+d + di le21 + fi)OI By Theorem C.2, for all 0 :5 s
IZi(t)1
~
1:')
+ 9i 161
t,
1 ~ IZi(s)lecif12 + S:5T$f sup {~IZi+l(7")1 + d 1£2(7")1 + ~ 19(7")/ + .!.16(T)I} i 9i Cj
I
~;
(9.213)
that is, each zjequation is ISS with respect to (Zi+ h e2, 0,
6) . Illequality
(9.213) can also be established for i = 1,2, and P, with Zp+1 ~ O. Inequalities (9.213) define a set of cascaded ISS inequalities. By repeatedly applying Lemma C.4 we show that there exist positive real numbers {3lt Pit u independent of initial conditions such that
(9.214)
o
which completes the proof.
Since e2 is b,:mnded, we now look for identifiers which guarantee the boundedness of 0 and {). Two such identifiers are the vswapping identifier introduced in Section 9.3 and the xswapping identifier introduced. in Section 904.
Theorem 9.17 (Schemes with Wealc: ISS~Controller) All the signals in the closedIDop adaptive system consisting of the plant (9.1), the weak 18Scontroller in Table 9.3, the filters in Table 9.1, and either the vswapping or the xswapping identifier are globally uniJonnly bounded, and global asymptotic tracl..ing is achieved: lim [y(t)  Yr(t)] = o. (9.215) too
9.8
SCHEMES WITH
403
lVITFILTERS
Table 9.4: MTFilters lVITIilters: ~
.... ......
=
= l =
Vj T
A,~ + B,r/J(y)
(9.216)
A,S + B,iP(y) A,A + en _10'(Y)U
(9.217)
AlA,
=
n =
(9.218)
j = 0, ... ,111
[Urn"'" 'Vl,'I'O,
3]
(9.219) (9.220)
Proof. \¥itb Lemma 9.16 and either Lemma 9.5 or Lemma 9.B, the proof is as ill Theorems 9.6 and 9.9. 0 Remark 9.18 For a further reduction of the growth of nOlllinearities we can eliminate the OJterms from the nonlinear damping fUllctions (9.207)(9.209), provided that lve normalize the update laws with l+lllQIF, as ill Remarks 9.7 and 9.10. 0
9.8
Schemes with MTFilters
We briefly present some schemes with MTfilters. lVlost of the t.echnical details are left out. The MTfilters introduced in Section 8.2.1 are repeated for convenience in Table 9.4. The matrices .fll and B, are defined in (8.154). With the filters in Ta.ble 9.4, the filtered transformation X= x  [ {
+~T8 J
(9.221)
brings the system (9.1) to the form
X = AX+l(wo+wTtJ) Y = Xl,
(9.222)
where l is defined in (8.155), and
+ ~l
Wo
=
'PDt I
W
=
[Vrn, 1 , Vnrl.l, .•. , VO.1 t q,(l)
(9.223)
+ 3(1)f .
(9.224)
404
9.8.1
MODULAR DESIGNS
ISSobserver
In the modular design we design controllers which guarantee inputtostate stability with respect to the parameter error, its derivative, and the state estimation error C2. We also need identifiers which independently guarantee the boundedness of the parameter error and its derivative, as well as observers which guarantee the bounded ness of the state estimation error independently of the choice of the identifier. The observer (8.171), which we used in the tuning functions design, is not strong enough to guarantee the bounded ness of the state estimation error. We strengthen it by adding the nonlinear term li'·olwI 2 /(y  :\\): (9.225)
In Table 8.5 we have suggested that this strengthened observer may also be used in t,he tuning functions design. However, while nonlinear damping was not necessary for stabilization in the tuning functions design, and was only used for improving t.ransient performance and guaranteeing boundedness without adaptation, in the modular design the nonlinear damping is indispensable. In the next lemma we show that the observer error system (9.226)
is inputtostate stable with respect to the input
O.
Lemma 9.19 Let the ma.:cimal interval oj existence of solution.Ii oj (9.226) ',e [0, tf). IJ 0 E .coo [0, tf), then e E £00[0, tf). Proof. Since ej(.91  Au)II = _1_ is strictly positive real, then by the 8+ Co
PopovKalmanYakubovich lemma (Lemma D.6) there exist Po qo > 0 such that
AJPo + PoAo Pol
$;
qoI
=
el'
= PaT> 0 and (9.227)
Therefore, along the solutions of (9.226) we have
(9.228)
o
9.8
405
SCHEMES WITH MTFILTERS
Before we go on to the design of the controller, we remjnd the reader of the two crucial expressions for iJ derived in Section 8.2.1:
iJ
= X2 + Wo + wT (J + S:2 =
bmVnl,l
(9.229)
+ X2 + Wo + iiJT(J + S:2,
(9.230)
where (9.231)
9.8.2
ISScontroller
Applying the backsteppiug procedure to the system
1j Vm,i VrJLtpl
=
bmvrnJI +X2+ W O+ W'1'f)+e!!
=
Vm,i+l  kj'IJ71t ,J ,
=
a(y)u. + l'm,p
 kp'Vmtl
i
(9.232)
= I, ... , p 
1
(9.233) (9.234)
I
with the IvITfilters given in Table 9.4, we end up with the design summarized in Table 9.5. Oue can now show that this results ill the same error system (9.39), (9.51)(9.54) as tbe ISScontroller \vith Kfiltcrs: (9.235) Therefore the error system (9.235) possesses Ule same inputtostate properties in Lemma 9.2.
85
Lemma 9.20 The e'lror system {9.1135} satisfies the inp'uttostate p7'Operties stated in Lemma 9.2. (i)
Ij6,6 E .coo[D,t!),
then
z,x,x,€,.:,;\ E £oo[O,t!), and
1 1 Iz(t)\ $ [4 (d 1IS:211!, + ~1I811~) + JIIOII~]:i + 1=(O)leCo 0 lio ...go 1
co/
•
(9.237)
The bounds (9.236) and (9.237) are derived ill the proof of Lemma 9.2. The rest of the proof uses Lemma 9.19 and t1le bOllndeduesB argument from the proof of Tl1eorem 8.9. With Lemmas 9.20 and 9.19 at .haud, it remains to design identifiers which guarantee that 6 is bounded, and iJ is either bounded or squareintegrable.
406
MODULAR DESIGNS
Table 9.5: ISSController Design with MTFilters
Zl
=
(9.238)
Y  Yr . 1  .2.. y (it) 77'.'b r
.:.• =
1)
, UI
=
sgll(bJ,.)
(
 ....;;;......;....;.. Cl
i = 2, ... ,p
11 ,
0;'
m
(9.239)
1 _ + 8 1) =1 + :::0'1
(9.240)
bm
<;"m
(9.241) = X2  Wo  (jjT{) a2 = bm zl  (C2 + 82) Z:! + 112 (9.242) ai = Zil  (e; + Si) Zi + Pi, i = 3, ... P (9.243) a cril (A t: B A.) 8cril (A B ",,,) l1i = aail T (X2+Wo+WTO) + T lr.,. + ,.~ + as I::' + 1':1'
al
t
+ 8:;1 [AX + [(.(11 XI) + Kolwl'/(Y  x!l + / ("'0 + wTti) 1 i1 a..., m+;:! an. " uil (j) k " ....il ( k \ \ ) (9.244) + "V=OY r + iVm.1 + " 7:J>::"  4:1 Al + Aj+l j=18yr
;=1
J
~ (Yr + 0:1) el12 b,n 12 + 92 lao' ay1)2 + 18aJ ay a6l (80'
81
=
d 1 + n.1 1w +
S:,!
=
d2
8,'
8o'il T aai_l) 2 18 Cf.il 12 1 (iI) Cl 12 = d'( w +g' 1ao .  ft ay +Ii:' ay b2 Yr
K2
I
W 
I
Zl el
(9.245) T

12
1. b~ lIre} (9.246)
.1
i = 3, ... , p
TTl
(9.247)
Adaptive cOlltrol law: u
= _1_ (0:p  11m,p + ~y(P)) u(y) b m
r
(9.248)
Observer:
(9.249)
9.8
407
SCHEMES WITH IvITFILTERS
9.8.3
XPassive scheme
Let us consider the adaptive observer form
X
=
Y =
AX + I
(Wo + WTO)
(9.250)
Xl'
This system is a useful parametric model because of the minimum phase and relativedegreeone properties. The observer (9.249) results in the error system (9.226) which possesses a strict passivity property fro.In th~ input 8 to the output WCl' Because of this we choose the update law iJ = 8 in the form
o=
Proj {rwcd ,
(9.251)
£".
where the projectiou operator is employed to guarantee t.hat 0, Vt ~ O.
Ibm(t)1
~
C;m
>
Lemma 9.21 Let the maximal interval oj existence of ,r;olutions of (9.226) and (9.251) be [0, t j
).
Then the following identifie1' properties hold:
(0) (i) (ii) (iii)
Ibm(t)1 ~ C;m > 0, Vt E [O:lj)
8 E Loc[O, tj) c E L2(O, t J)
n Loe[O, t j)
iJ E £2[0, t j) .
(9.252) (9.253) (9.254) (9.255)
One can use the Lyapunov function " =
181il + Icl~Q
(9.256)
along with (9.227) and Lemma E.1 to show that (9.257) ~nd
draw the conclusions stated in Lemma 9.21. This lemma guarantees that iJ is squareintegrable, which by Lemma 9.20, is su~cient to establish the boundedness of all signals. The squareintegrability of iJ is due to the nonlinear damping term ~olwI2leI in (9.226). Hence, this is the second role of nonlinear damping in the observer. Its first role was to guarantee inputtostate stability with respect to 8 in Lemma 9.19.
Theorem 9.22 (xPassive) All the signals in the closedloop adaptive
8118
tem consisting of the plant (9.1), the control law in Table 9.5, the filters in Table 9.4., the obseruer (9.249), and the u.pdate law (9.251) are globally uniform.ly bounded, and global asymptotic tracl.:in.g i.s a.chie'lIed:
lim [y(t)  Yr(t)} = O.
toe
(9.258)
408
MODULAR DESIGNS
The boundedness is obtained by combining Lemma 9.21 with Lemma 9.20. One way to establish the convergence of z(t) to zero is to follow the idea in the proof of Theorem 9.4. Instead of (9.86), one should consider
iT
b.
(= z  We(z, t) Ill2e
(9.259)
and combine the systems (9.84) and (9.226). The X·passive scheme is the simplest among the schemes with MTfiltcrs. Moreover, its dynamic order is as low as the dynamic order of the tuning functions scheme with MTfilters because the observer (9.249), already used for state estimation, is employed to design an identifier. This is a property the Xpassive scheme has in common with the xswapping scheme (with Kfilters), which employs in the identifier only the filters from Table 9.1, already used for state estimation.
Example 9.23 (Example 8.8, cont'd) For the system (8.206) we now design a Xpassive scheme. The strengthened observer (9.249) is given by
:. [OIl· + I 1el + 0 0 X + [ col
X=
Co
li.oW 2 [
1l
1el + [ 1l 1(v + w0·) ,
(9.260)
and the stabilizing function (9.240) and the control law (9.248) are al
U
= (c + d + ~2)Zl  X2  wB = 
(80.1)2 'l] 8y + ay w + gw[ + d (8al)2 C
i(co + l\: ow2 )e]
2
Ii.
+ 2ln.w2 y.
(9.261) 80.1
Z:z 
ZI
+ 8y
_
(;\:2
•
+ V + wO) (9.262)
The update law (9.251) is
8 = "(weI'
(9.263)
Responses with the Xpassive scheme are shown in Figures 9.1 and 9.2. The values of 0, c, Co, d, I and all the initial conditions are as for the tuning functions scheme, whereas Ii. = ~o = 9 = 1. Without adaptation and with 8 constant and bounded, Lemma 9.20 establishes that all the states of the closedloop system remain bounded. Hmvever, regulation of y to zero is not achieved. Instead, without adaptation the closedloop system has an asymptotically stable periodic orbit with ymagnitudes as large as ±1.2. The projection of the periodic orbit to the (w, y) plane is shown in Figure 9.1n. With slow adaptation, "( = 0.01, the response is a s]owly "shrinking" family of periodic orbits parametrized by 8; see Figure 9.1b. When 8 reduces enough, the trajectory is attracted by an asymptotically stable equilibrium at the origin. An increase of the adaptation gain to a moderate value of "( = 0.1 results in the response in
9.8
409
SCHEMES W1TH l\4:TFILTERS
y
y
y
I
1
o
I
o
w
I
(a) ')' = 0 no adaptation
(b) l' = 0.01 slow adaptation
w
l
o
w
(c) " = 0.1 moderate adaptation
Figure 9.1: Tile xpassive scheme. (a) Without adaptation, the closedloop system has an asymptotically stable periodic ol'bit. (b) With slow adaptation, the trajectory passes through n "sltrinking" family of periodic orbits parametrized by O. (c) With moderate adaptation, the trajectory starts by approaching t.he periodic orbit, but then the adaptation takes over.
Figure 9.1c, which initially approaches the periodic orbit, but when the adaptation takes over, it rapidly converges to the origin. For the st1.me adaptation gain, "( = 0.1, the time responses of the output y, control u" and parameter error iJ are given in Figure 9.2. They show t1ut.t the regulation of y is achieved without iJ converging to zero, that is, without feedback linearization. Comparing Figure 9.2 with Figure 8.1, we see that the transient in the Xpassive scheme takes considerably longer to settle than in the tuning functions scheme, but, thalllcs to nonlinear damping, witb much less cont.rol effort. 0
I
eSwapping scheme
"r,'
9.8.4
,"
Let us consider the observer error system (9.226) as a parametric model. The :filters
U E lRPxn Uo E mn
(9.264) (9.265)
410
~IODULAR DESIGNS
y:
o
10
20
o
10
20
o
10
20
Figure 9.2: Responses with the xpassive scheme
and the estimation error vector (9.266) result in the static parametric model (9.267) where f ~
g
+ Uo 
UT (} is governed by (9.268)
Since only E'] = 11  :b is measured, only the first component of implemented as the estimation error:
f
would be
(9.269) Filters (9.264) and (9.265) have dynamic order n{p + 1). Fortunately, the filter dynamic order can be reduced to p + 1. The reduction is based on recognizing a decomposition of the observer error system (9.226) used in the proof of Theorem 8.9. We remind the reader that the similarity transformation El [ 1]
1=~ [Tee) 1= [ l,eTI 1e n 1
(9.270)
'9.. 8
411
SCHEMES WI'l'H MTFILTERS
couverts the system (9,226) into
= 
il
(co + Kolwl2) El + wTj + 711
(9.271) (.9.272)
'1 = Am·
The scalar system (9.271) is a candidat.e for a paranletric model because all the quwltities in this system ure available, except for the parameter error 8 and the msturbance 1'/1 which is el..ponentially decaying. We intr.oduce the filters
 (Co + ll:o lwf2) m + w ,
tb =
.
WO
mElR"
(Co + n.o Iw I")  'Wo + w'1'. 8,
= 
i:i:70
E R
(9.273) (9.274)
a.nd the estimation errol' f
= e1 + wo 
T m {).
(9.275)
Substituting (9.271), (9.273), (9.274) iuto (9.275), we get
f=WTO+f,
(9.276)
where f is governed by (9.277) The update Jaw for fJ is either the gradiellt:
:.
(J
=
PL
Oj
{'We} 2 r 1 + 1I/tt71
bm(O)sgnbru > ~m
r=rT>oJ
1
1)2:0
(9.278)
or the least quares:
iJ = Proj {r 1 b,n
t
=
r 1
r}
'Wi
+ 'W r
1I1111T
I
I)
12 r
+1I'W'r
f
bm(O) sgn bm > r;m
(9.279)
r(O)
t
= r(O)T > 0,
II ;:::
0,
where the projection operator is employed to guarantee that Ib7ll (t)1 ;;::: ~m > 0, 'Vt 2: 0, and by allowing II = 0 we encompass unnormalized gradient and leastsquares.
Lemlna 9.24 Let the maximal intenm.l of existence of solutions oj (9.271), (9.273)(9.214) with eithc7' (9.278) 07' (9.279) be [0, tJ). Then Jor" ;::: 0 the following identifier properties hold:
(0)
Ibm(t)1 ~ <;7tr > 0,
(i) (ii)
8 E £oo[O,t.})
(9.281)
c E £2[0, tJ) n £00[0, i})
(9.282)
(iii)
oE £2[0,tJ) n£co[O,tf)'
(9.283)
\:It E
lOt tJ)
(9.280)
412
MODULAR DESIGNS
Theorem 9.25 (eSwapping) All the signals in the closedloop adapt'i1}e system consisting of the plant (9.1), the controlla'UJ and the obseroer in Table 9.5, the filters in Table 9.4. the filters (9. 273){9. 274). and eilher the gm.dient (9.278) or the leastsquares update lau} (9.279) are .qlolJally uniformly bounded, a.nd global asymptotic tro.cking is achieved: lim [y(t)  Yr{t)J = O.
100
(9.284)
The boundedness is obtained by combining Lemma 9.2,1 with Lemma 9.20. The convergence of =(t) to zero is proven as in Theorem 9.6, by replacing the estimation error equation (9,113) by
(9.285) As mentioned in Remark 9.7 for the yswapping scheme, we can eliminat.e the g;t.erms from the nonlinear damping functions (9.245)(9.247) in Table D.5 by normalizing the update laws with 1 + '/IQI.F' The gl'owth of nonlinearities in the controller design in Table 9.5 can be reduced by following the weak ISS approach employed in Section 9.7.
Transient performance
9.8.5
In this section we derive £00 t.ransient performance bounds for the error state =, which include bounds for the tracking error =1 = Y  Yr' £'], performance hounds Bl'e not available. V·le first consider the xpassive scheme, and then the cswappillg scheme. To simplify the analysis, we let r = ,,/1. For the same rea.~on, we implement Xl (0) = y(O) to get el (0) = O. '1t.Te first ('onsider the Xpassive sehenle.
Theorem 9.26 (XPassive) In the adaptive system (9.1). (9.216), (9.217), (9.218), (9.248), (9.249), and (9.251), the following inequality holds (9.286)
wh.ere
C
= min{ Co, co} and (9.287)
(9.28B)
Proof. To obt.ain an £00 bonnd 011 =, it would see.ll1 that inequality (9.236) could be used along with £00 bounds 011 E'2, 8, and 8. However, it is not clear
9.8
l\1:T~FILTERS
SCI:lEMES WITH
413
110W to obtain a bound 011 1101\00 depending only on design pnrameters and initial condit.i~ns. Therefore, we apply a different apprmtCb which eliminates the need for 1181100' Although in Section 9.8.3 we did not usc the deC'omposition (9.270) of (9.226) into (9.271)(9.272), wc herc exploit it in order t.o make c = min{ Co, co} appear explicitly in the bounds. First, we note from (9.2;1) that
d (1?Ej,,) ~ 2 Cu " I;;() I /".., Ej  ;;W Ej dt. .... Combining (9.Gl) and (9.289) we compute
(I"
,2,,)
1 d 1=  + Ej' 2 al
2"'090
~
1
v 2c
0
(9.289 )
'I
o
:'1")
1 col=l + 1 (1 E2 + 1 11" 0  + I(J 'I
?
~1
do
and since by Lemma E.1(H) and (9.251), obtain
Iz(t)1 :5 . ~ [ d
1? 111 + ;;:1 1'" + _co (J  . .... li
]
Jle21\!, + (  + lio
+
,2 )
floYo
go
liD
181
~
,lwEI,
with Lemma C.S wc
IfOIl~ + _"Y1It711l!' :1
JJ/2 +1=(O)lcct/2,
Coh.oOO
(9.291)
where we have uscd E1(0) = O. Now we determine bounds and I/TlllIlXl' Fh'st, fTOll1 (9.2;2) we have d d .t
whidl implies that
(I 711p,I)) = 1 111
all
lIe2111Xl,
1181\00
(9.292)
IJ
t
IIndl!' :'5 ~(~) 111(0)1~ .
(9.293)
Along the solutions of (9.271)(9.272) and (9.251) we have
1 I \2 1 1 12 ) < Co::! 21 dtd (2 e1 + Co 'T} Pt + :y 0   2 e 1 
Ii 181:1 ,2 '
(9.294)
wbich yields
11811~
(9.205)
I\cllI~
(9.296)
414
MODULAR DESIGNS
To obtain It bound on lIe2l1oa, from (9.270) we recall that e2 = i ll:: 1 + 7]11 which by virtue of (9.293) and (9.296), shows that
2) 17J(0)1~ .
2 2ir I{ )1 2 (2iI II e2 11 oa :::; :y 8 0 + ;;; + ~(11)
2
(9.297)
Substituting (9.293), (9.295), and (9.297) into (9.291), we arrive at (9.286) with (9.287)(9.288). 0 The initial condition =(0) ill the bound (9.286) is, ill general, dependent on the design paramet.ers Ca, do, liD, and go. However, as explained in Section 4.3.2 for state feedback t with trajectory initialization we can set =(0) = O. Following (9.238) and (9.239), z(O) is set to zero by selecting
Ur(O) = yeO) 1/~i)(0) = ;}"r(O) [Vmti(O) 0';
(9.298)
(y(O), X(O), ';(0), :S(O), 8(0). ~\'n+il (0), jj~iJ)(O»)] , i = 1, ... , p  1 . (9.299)
Upon settiug z(O) to zero, the bound (9.286) can be systematically reduced by increasing c. By examining (9.287}(9.288) one can see that the bound (9.286) can also be systematically reduced by simultalleously illcreasing n.o, liot and do. A compa.rison of the bound (9.286) with the bound (9.157) for the ypassive scheme (with Kfilters) re,reals that the latter bound is lower. This is because the design with lvITfilters results ill all observer el'l'or system disturbed by the parameter error, unlike the design with Kfilters. Now we consider the o£swapping scheme with a gradient upda.te law. It is straightforward to e.xtend this result to a leastsquares update law. We initialize wo(O) 0£1 (0) = 0 and w(O) = 0, which set ;(0) to zero.
=
Theorem 9.27 (eSwapping) In the. udapti11e system (9.1), (9.216), (9.217), (9.218), (9.248), (9.249), {9.273}, (9.274), and (9.278), the following i1l.eq11.ality holds
(9.300) where (9.301)
(9.302)
9.8
415
SCHEMES WITH MTFILTERS
Proof. We derive an £00 bound on z using (9.236) rewritten as
11811001 aud 11811to. First, along the solutions of (9.278), (9.277), and (9.272), ,vith (8.250) we readily arrive at
It remains to determine bounds on 11e211oo1
d
(111" 1 f2 + ;; 1 I11 I?)  f) rI + P, Co Co
dt2
(9.30~1)
~
Then, using f(O) = 0, (9.304) yields
11811~ ~ 18(0)12 + ~ 11I(O}I~ .
(9.305)
IJ
Noting that (9.271) gives
'1) ~ Ei +  1 1'8 '1~+ 1]j, 1..,
d (1Ei dt
2
CD?
2
4ltD
2co
(9.306)
we obtain (9.307) Recalling that
E2
= lIE)
+ '1]11 using (9.293) and (9.307) we get
2 2 II 1111 Ii) 2 12 ( II e2 11 to ~ Coli. 8 00 + 1 + ~ £i(f1) 17](0) Pr • o
(9.308)
Fi'om (9.278) we write 2 Iwl f2 I8  ~ 1 (1 + vlml 2 )2
~I?
2
=::; 1
2
lit) f2 "II? 2 11m ; .;, (1 + vltoI 2 ):! ~ 1· mll~lIflloo'
(9.309)
and by substituting (9.276) we obtain (9.310) For the filtered regressor (9.273) we readily show (9.311) wbich, using w(O) = 0, implies
. II ml1 002 < _1_ 4colio
(9.312)
416
MODULAR DESIGNS
To obtain a bound on d
I!llllXl, along (9.277) and (9.272), we consider
(111"  f!  +  1 1 T1 I") P,
dt 2
4co
I) ~ CDt:
+ f111 


1
4co
I1]I" :5 0
(9.313)
which yields (9.314) By substituting (9.305) into (9.308), and also (9.305), (9.312), (9.314.) into (9.310), and then the two results, along with (9.305), into (9.303), we arrive at (9.300) with (9.301){9.302). 0 The initial condition .:(O) in the bound (9.300) can be set to zero by the trajectory initialization procedure (9.298)(9.299). UpOll setting =(0) to zero, the bound (9.300) can be systematically reduced by incre~lSing Cn. By examining (9.301){9.302) one can see that the bound (9.300) can also be systematically reduced by simultaneously increasing 1£0, N.o, and do. A comparison of the bound (9.300) with the bound (9.] 75) for the yswapping scheme (with Kfilters) reveals that the latter bound is lower.
Notes and References The early results on estimationbased outputfeedback adaptive control by Kanellakopoulos, I(oli:otovic, and Middleton [67, 68] imposed both structural and growth restrictions all system nOlllinearities. Except for the material in Section 9.8 which appeared in Krstic and Kokotovic [99], as well as t.he ;;passive scheme presented in Krstic, Kokotovic, and Kanellakopoulos [103], the results in this ch~tpter have not been previously published.
r. Chapter 10 Linear SysteIns For more than two decades, a.daptive control research has dealt exclush'ely with linear systems, and has developed controllers which guarant,ee global boundedness and tracking [5, 44, 51, 142, 165]. How do these traditional adaptive contl'ollers compare wit,h the now controllers developed in this hook? How do the controllers developed in Chapters 8 and 9 behave when applied to linear systems? Do they become variants of t:raditiollal adaptive l'ontrollers or do t.lw resulting feedback systems possess some llew properties? From the preceding two chapters we already kllow that olle of the Ilew controllers' properties is t.heir capa.bility to systematically improve t.he transient performance. Other llew properties include tbe guaranteed stability without adaptation a.nd passivity of the adaptation loop irrespective of the relative degree. When the controUers designed in this book are applied to linear systems, these new properties come at no increase of the dynamic order of t.he adaptive schemes. The task of this chapter is to introduce the new adaptive controllers for linear systems in a selfcontained manner I so that a comparison with t.raditional adaptive control can be made independent. of the rest, of the book. After introducing the filters used for state estimation in Section 10.1, in Section 10.2 we develop the design procedure, estctblish the resulting stability and passivi t.y properties, and provide a design example. III Section 10.3 we reveal the structure of the underlying nonada,ptive controller and determine conditions for stability without adaptation. In Section 10.4 we derive transient performance bounds, and in Section 10.5 we give a simulation comparison with a traditional linear scheme. Then in Section 10.6 we preseut modular schemes which are closer to certainty eCluivalence adaptive linear designs. Finally, in Section 10.7 we summarize the main properties of the new ada.pthre designs for linear systems. We consider linear singleinput singleoutput syste1lls
yes)
B(s)
= A(s) u(s)
=
(10.1)
418
LINEAR SYSTE1IS
where the coefficients a/s 'and bi lS are consta11t but unknown. The control objective 'is to asymptotically track a reference signal 'Yr(t) with the output y. Our assunlpmons about the plant and the reference signal are the same as in traditional model referel10e adaptive control: ASSUlllption 10.1 The/plant 'is m:inimumphasc, 'i.e., the polynomial B(s) = t'bms m + '... + bl S + bo is il1'lll'witz. Assumption 10.2 Vhe sign oj the .h:(qhtfTequcncy gain (sgn(bm )) is known. Assum;ption 10.3 TThc relative d:Og'ree {vp = n  177,) and an upper bound for :'the ,pima order ('/1,) ,m'e known. iASSUlnption 10.4 lThe rcjcTcncc .'signal Yr(t) and its jiTSt p derivatives are .'kn.own, 'em,a b01/,nddl,' and, 'in add:i't.ion, u}p) (t) 'is p'iecewise cont;i,77.'u.ou,s. In pa'/',ltimila.l'" Yi(/;) may 1be the outp'll,1, IQf.a :rdJeTence model oj Telah:ve degree Pr 2:: (J wlth' a, piecewise 'c071.iin'U.o'l1.s inp'llt i7:~t;).
:illO.l
)~Stcite
Estin:lfatien Filters
\\Ve statt: byl r~presenting the il>lant (10.1) in the I{!)'bserver Icanonical form
Xp":"J
J: p 

;r:p =
(10.2)
am+l'Y
xp+l  amY
+ b,n:u
xnatylblll.
J;,,1
,ao!)
X II
Y
+ bou,
Xl
or, :more compactly, as
( 10.3)
where m
111 
1 !
o
]
1 ] a = all :. ,
[
ao
b=
[
b: •
bo
]
(lOA)
10.1
419
STATE ESTIMATION FILTERS
Although the linear system (10.3) is a special case of the nonlinear system (8.3) with ¢(y) 0, (:D(y) = ly, and a(y) = 1, and the designs from Chapters 8 and 9 are applicable, we proceed with an independent derivation. 'Ve first re'\vrite (10.3) as .i' = A:r + F(y, 11 (10.5 )
VB
where
F(y, '11)'1' = [[ and the]J = n
O(pl)x(mIl) 1111 + 1
1
'11,
IlIn] ,

(10.6)
+ In + 1Mdimensional parameter vector () is defined by
O=[b]. a
(10.7)
For state estimation we employ the filters .Ao~
t; ~yr
where the vector k = [kl' ...
=
+ k~lj
.AOOT
(l0.8)
+ F(Ylll)T ,
,h:nP' is chosen so that; Ao
= A'
l· II;
(10.9) tflC
e'IC1
mat.rix (In.IO)
is Hurwitz, and hence P exists such that
P Ao + A5 P = I,
pT > 0'.
P
'''ith sodesigned filters 0UT state estimate is i = ~ + f2 T O
(lo.n)
I
and it is easy to show thail tile state estimation error (lDH3)
vanishes exponentially because it satisfies
i: =
(lO,lY}
ADE.
'!\That has been achieved tlms far is a static relationship between the statbx ancl~ Hie unknown parameter 8:
x
+o7o+c:.
A fmther practical step is to ,lower!' the' d)rnnmic order of the Ofilter·by
exploi1iing the stHlcture of F(y, 'l/.) iil '('10)6): \Ve dEmote the first 171 + 1 colu:mns of Om by vm1 .. , '"'VI, 'Vo and the remaining n,eolmnns by 3, nT ol[;
['v m " , , ! ,'U II"/"·0, =]
~,j
(10.16)
420
LINEAR SVSTE.MS
and show that due to the special dependence of F(y, '1/.) for the first. n + 1 columns of nT arc governed by
j=o, ... ,m.
011 'U,
the equations (10.17)
This means that thanks to the special strueture of AD,
j = the vectors
l'j
o, ... ,n 1,
(10.18)
can be obttuned from only Due input Ii.lt.cr
~ = Ao'" + enlt,
(l0.19)
through the algebraic expressions 'lIj
= A~"\,
j = 0, ... , 111 •
(lO.20)
Similarly, S, governed by
E: =
B E IRnxn ,
Ao:=:  Ill,
(10.21)
call be obtained from only OIle output filter
r, =
AUTI + CnY ,
(10.22)
through the algebraic expression
B= 
[A8 17],
Finally, with the identity
A8e n
••• ,
Aol1, 17] .
(10.23)
= k,
(10.24) the vector e in (IO.S) can be obtained from the filter (10.22) through tile algebraic e.xpression (10.25) The implemented input and out,put. filters are summarized ill Table 10.1. For comparison with tradit.ional adaptive controllers, we note that the tott1.1 dynamic order of these Kfilt.ers is 211 and can further be reduced t.o 2(11  1) by exploiting the fact that'll and yare available.
Remark 10.5 From (10.15) and the expressions in Table 10.1 all equivalent expression for the virtual estimate .t is nl
X =
i=O
=
m
Ao·q Ea;A~71+ LbiA~'" ;=0
B(Ao)'"  A(Ao)17 ,
(10.32)
\vhere A{·) and B(·) are matrixvalued polynomial functions. The generation of the virtual estimate i is described pictorially in Figure 10.1. With (10.32) we get all explicit relationship among ,\, '1']\ and e and .7:: :t =
B(Ao)'"  A{Ao)rJ + e.
(10.33)
o
10.1
421
STATE ESTIMATION FILTERS
Table 10.1: KFilters Kfilters:
i] = AOl1 + en 1} ,\ = Ao . \. + en'll.
(10.26) (10.27)
r
*'I
..j
..... =
 [AOITJ, ... ,A0 11.
e=
Ao11
Vj = n'r =
y,'
~
Ab'\,
17]
(10.28) (10.20)
j=O, ... ,m
'l'o, ::::]
[117711 ••• , 'VI,
(l0.30) (10.31)
The bacltstepping design for the plant; (10.1) sf,art.s witlt it.s ont.put. IJ. which will be the only plant state allowed to appear in t.he control bnv. For this reason, (10.2) is rewritten as:
t. "
(10.34) From the algebraic e."CprcssioIl (10.15) we have %2
=
'r
e2+!l(2)8+c2
= {2 + [Vm .2, 11rn 1.21"·' VO.2, =(20)]0 + €2 = bmvm,'J + e2 + [0, 'Um l.2,···, "'0,2. :::(2)]0 + e2'
(10.35) (10.36)
Suhst:ituting both (10.35) and (10.36) into (10.3.:J). we obtain the following two important expressions for y: (10.3?) (10.38)
where the 'regressor' wand the 'truncat.ed regressor'iiI arc defined as
I
l'ml.::!~ ..•
W
=
['Urn,!!!
iii
=
[0, t'mI,:!! ••. , "0,2, ::::(2)  lJeJ]T .
I
t'O,21 =(:!)
Uef]T
I' I
We bave thus prepared the ground for a backstepping design.
(10.39)
(lOAO)
422
LINEAR SYSTEMS
y
B(s} A(s)'
•
I~SJ 
Ah)'IIen iI
I I '17i
i
Figure 10.1: Virtual estimate x genera1Jed iwith·input filter· "'\and output
filte~71~
Functions·~~~sig!l.'
10.2
Tuning
10·..2.1
Design procedure'
'\Te now present the tuning fUllctions desigll'lpJ'oeedure fopr the case p" > L Tbe design for the case p = 1 can be easilY.dednned from',th~ first. step of the' recursive procedure. Thanks to the minimum pbase Assumptirinl'l0~1, the design is· restricted to the first p equations in (10.2):
Xl Xpl
Xp
=
= =
X2 
anlY!
xp 
am+I'Y."
(10.41) xp+1  amY + brn u •
We will return to the behavior of the last m equa.tions in the.stability proof. In the backstepping approach we view the. state variable Xi+l as a control· input to tbe subsystem consisting of the states Xl,'" ,Xi, and we design a stabilizing function 0:, which would achieve the control objective·if Xi+l were available as a control input. The control law for the actual control input 'U is obtained at the pth step of the recursivedesign. Becauseonly tbe system output y = .'1:1 is measured, we replace (10.41) with a new system·whose states are available. We start with (10.38), which is just an alternative form of tbe first equation ill (10.41). Equation (10.38) suggests that 'Vm ,2 is chosen instead of the unmeltSUl'ed X'2 to be tlte 'virtual control' in pu t for backstepping. The reason fOl" this choice is that both X2 and V m ,2 are separated by only p  1
10.2 ''DnNI:Nc IFiuNCmICdNS 1l!)~SIGN
423
in~gv81hors !ham lUbe .aatmill contrulJlu, which
is clear {from' (10.17) for j
= m: (10.42)
A (dl(!)sEJr!exaniina1iiollICl{lthe i fi.ltersiain Table 10.1n'e,re8:ls that more integra.tors stand iin ithe .way ·of any· other\va.r.ia:ble. Therefore,! bhe design system chosen t'fl Tftplace{(10.f:.I!lJ)Js ,iJ = :lb,rivm,2 + ~2 +:(iil)6 + E:2
"'Um ';2 :::
'vm ,3 
ki11rn,1
(10.43)
k j V na ,1 ··,vm ,p+1 1t:'"Vm.li+ u.
'iJrn.p":"l ::: '":Vm,p 
'il""p .:::
~ll'
of its states are, avit1l8lble
for feedback. ' Our design tasle is to forooe the
ontpmt/lll' to 8&1'lllPttlltto8ll1y. track :the 'reference outlmt Yr while keeping all nhe
closedL.}(!lCllp signals, .bounHdd. . As in' the tuning:fu.nctions desigll'in Chapter 4, we employ the change of !
coorainates .E:l
:= :,1/ Yr
.,., := ",,,.. ~,l
ny(il) 
r:: r
a·. 1 t
i
,(10J14) ,(10,t15)
= 2, ... "p,
\vhere '0 is an estimaltc\of (! = lIbRa' Our goal is to regulate:; = [=h .... , zp]T to.zero because by regitlating z to:lzC!ro we will achieve asymptotic tl'a
%1
obtaillel:l from
(10.44) and (10.43'):
= bnt'Um .2 +';2 + iiJT(J + E2  Yr.
(10.46)
= 21irnto (10.46), we get = bmz2 + bmlll1 + bmOYr + e2 + iiJT8 + E2  Yr = bmll:1 + e2 +'iiJT 8 + E2  bmOYr + bmz2 •
(10.47)
Zl
By subst.ituting (10.45) Zl
for i
Scaling the first stabilizing function
III
as (10.48)
we obtain
':1 = 0:1 + ~2 + iiJT0 + Then the choice}
E2 
bm (ilr
+ oJ) 0 + bmZ2 •
(10.49) (10.50)
IThe reason for using two positive constants Cl and d 1 is to have uniformity with subsequent steps of the backstcpping procedure where ~ is a coefficient of a nonlinear damping term counteracting E2.
424
LINEAR SYSTEMS
results in the system
=1 =
Cl=l 
d1z1 + E~ + {i)TO  bm (ilr + a'l) 0 + bmz~.
(10.51)
\Ve stress that (10.51) along with (10.14) would be globally asymptotically st.able if 0, ~, and Z2 were zero. With (10.45), (10.48), and (10.39), we lutve =
 "l'() + l'1711 =2+ b'"mZ~ w
=
(jj (J TW (J 
=
=
'r
+ (V
71l .2

T
".
BYr  al) el (J + bmz2
e(fir + 0:
')'
1)
e1 (J
'I' 
+ bm =2 A
(w  b(Yr + Q'd el) 0 + bm=2' A
(10.52)
Substituting (10.52) into (10.51) we get =1 = Cl=l d1 z1 +E2+(W  iJ (ilr
+ 0'1) el)T Obm (ilr + 0:1) O+bmz'J.
{10.53}
Tbis systcm along with (10.V!) is to be stabilized by selecting update laws for the paramet.er estimatcs 0 and fl. These update laws will be chosen to achieve stability with respect. t.o the Lynpunov fUllction
V:I
1 1 "J' 1 mI " l I T = =+ 0 r 0+ Ib2y { !   +  E Pe 2 1 2 4d • '1
(10.54)
l
We examine the derivative of Vi:
,~
=
=2 [CtZt  d1=1
ii1'rl~f1 
(I
=
+ £2 + (w 
lbml_:.
(!{! 
y
clzr + b.m =t=2
iJ (Ur + 0:1) edT 0  bm {lir + al}
e+b z
m 2]
I_T
e £ 4d1
Ibmlo..!:. bsgn{bm ) (Yr + al)'<:1  u] 'Y
+8Tr [r {w  g(lir + 0:1) ed =1  0]  d1zr + .lIE:!  4~1 ETe. (10.55) 1
To eliminate the unknown indefinite
6, gterms in (10.55) we choose y>0
and
(10.56)
8 = rrl, where (10.57)
If z!! where zero, these choices would yield the following e:\."Pl"essiol1 for the derivative of Vi:
l~
=
CIZ;  dl (=.  ~E2)'l ~ (Ei +ei + ... +E!) 2d 4d 1
1
(10.58)
10.2
425
TUNING FUNCTIONS DESIGN
Since Z2 '# 0, we do not use {j = rTI as the updat.e law for fJ! because fJ will reappear hl subsequent steps. However, [! will not reappeal', so we do use (10.56) as the actual update law for~. We retain (10.57) as our first, tuning junction for O. Substituting (10.56) and (10.57) into (10.55), we obtain (10,59)
\Ve pause to determine the arguments of the function al. By examining (10.50) along wit.h (lOAD), we see that 0'1 is a function of y, '11, 8, g, 'Uo,::!,' •• , UrnI,:!, and :tJr' In view of (10.30), ViJ can be expressed as (10.60) 
T
~
~
where Ak = ['\1, ... , A".j and Ak = 0 for k > n. With (10.60) we conclude that 0:1 is a function of y, 7J, 8, 'xrn+l, and Yr' As we shall see in t.hc subsequent . a fuIIe t'Ion 0 f Y,7], oA ,[!, A Am+i' '\ (iI) . 5 t eps, O:i IS an d Yr
e,
Step 2. Differcntiat.ing (10.45) for i in (10.43) we obtain Z2
=
1'm,2 
OYr 
=
Vm ,3 
h:2 V m,1 
c:h (y, '11, 0, U, Xrn+lt 1Jr)
bYr 
,
= 2, with the help of the second equation
• ..
80'1 fJ
80:1:"
~
where
f32
V
m,
3 
ay
a8alYr Yr. 
8' (Ao1] + eJ1y) 
 8iJ
8a.t (~
:. .
gYr  OUr 
~2
+ W TO + C2 )
m+l 8at
L
8A' (kp\1
j=1
+ Aj+d
J
80:1:'
aU
(J 
/3., 
g
gAy" 
r
B0 1

ay
(
W
0 + C'J)
T 
80'1 :..
(10.61)
 . (J

80'
is a function of available signals:
(10.62)
Noting from (10.45) that
.:" = ..
0:., 
eiir = .33 + 0:2,
V m ,3 
t:I" _
JJ~
80:1
8y
we get
1 (wT8 + c,,)  8C: 8+ .3., . 80
Since our system is augmented by the new state function (10.54) as
v.,.. = Vi
(10.63)
oJ
Z2,
we augment the Lyapullov
1" 1 T + z:; + d e Pe, 2 4 ,2
(10.64)
426
LINEA'R ~S1t.Sr.t!EMS
:\vhere another E..Ite]j01\\Vas.in~lu!ded to account for the presence,df c2iin!(o10.63). ;In view of (llOi"B9) I {lm63),..81ud' (10.14), tbe derivative of ~~ iis
, (10.65) The ~limination ofth~ unknown indefiuite Oterm from (10.55) can be acbieved . witiI the update law iJ = rT.!h where (10.66) ,Thell, if
2:3
were zero, the stabilizing function
(10.67)
would yield (10.68)
=
; However, since Z3 :j:: 0, we do not use 9 rT2 as an update law. Instead, .\Ve retain T2 as Qur second tuning function and 0:2 as our second stabilizing {ullction. Upon ,the substitution into (10.65), we obtain
li:J:5 C1Zr 
Co.; +
The mismatch term
Z,Za
+ or (T.  r 1 + z, 8;'0'
Z2~ (rT2 
9)
(rT2  0).
(IO.69)
0) will be dealt with in subsequent steps.
10.2
427
TUNING FUNCTIONS DESIGN.
Step 3. This step is crucial for understal1diil'g, tIre tuning functions technique. By differentiating (10.45) for i = a:;·.\vitli. tlie help E)£ the third cquat,ion in '10..43'):, we have
(10.70)
Noling'.from (10J45) that 1Jm,4

Uy~3) =
Z4 +0'3,
we get (10.72)
Since'
OUf'
system is 8lBgmented by the new state
Z3,' we
also ,augment tbe
LyapuDov, f6n~tion!{10.64): IT
1"3
=
tJ"
V2
1
'1
1
Tb:..:
+ 2Zi + 4r13 E
rI:!.
(10.73)
Itl'view of '(la.60), (10112), and (10.V1), tbe derivative of Y3 is
l~:, ~
(10.74)
As, in the previous steps, for the elimination c:f the unknown indefinite 8term" from (10. 74)~ we can· choose the tlpdate la.w (j = r1"a, where (10.75)
Noting that (10.76)
J
428
LINEAR SYSTEMS
(10.74) becomes
(10.77)
Then, if
=. were zero, and if 0 ,vere ellosen to be rTa, the stabilizing function (10.78)
would yield (10.79)
=
However, '::4 :/: 0, and we do not usc fJ f'T3 as an update law. Instead, we retain T3 as our third tuning function and 0:3 as our third st.abilizing functioll. Upon the substitution iuto (10.77), \ve obtain
As we pointed out, this step is crucial for ullderstanding the general design procedure. Unlike the first two stabilizing functions, the third stabilizitlg function 0:3 in (10.78) contains the term Z2~r~W. The role of this term is to cancel the indefinite term Z2~r(T2  T3) in the Lyapunov function derivative V:1. Tllis is achieved by recognizing in (10.16) that T2T3 has Z3 as a factor. We pl"Ovide additional interpretat.ioll after equa.tion (10.118). For further insight, the reader is referred to Sections .l1.1 and 4.2.
Step i =4, ••. ,p. We now proceed fast in presenting the remaining steps of the design procedure. By differentiating (10.45) for i = 4, ... t P 1, \vith the
10.2
429
TUNING FUNCTIONS DESIGN
belp of (10.43) we obtain Zj
=
bm.i  by~i)  Dy~iI) 
=
1J
p.I 
"+1 
111,'
(Utll, 8, g, X1)t+ih ii~i2»)
10 ay (w'J'O + eil) _ aO:DI)'
i 1 aO' 
jjy(i) r

Q'iI
i
(10.81)
:
where [jj =
oa·
~a"
~Jo~
U II ( T ) II ( ) '" + De2 + w () +  a A01] + en 1J +.t
~~illrll.l
Y
11
j=1
a0i] . (i fI ) ltiI ) (h!'''\1 + A'+l) + y( ) +   n
m+il
8;\ .
J
;
.
=
OJ 
80';1 (T()ay + )
B
• i 
aD'" .
r
J
Noting from (10.J5) t.hat U",.i+1  ~y!i) = =i+l Zi
DJ/r
~
+ !~ ;=1
C2
W
(10.82)
..
+ lXi' we get aO'i16~
afJ

','11 (j) (ii) Yr
+ =i+} •
(10.B3)
From (10.43) it follows that the .final step i == P call be encompassed ill these calculations if we define O' {J
= II + 'tlm.p+1 
I1~Y(P) r'
(10.84)
Now consider t.he Lyapullov function
1
'lIT
111 1 + 2;:;; + 4d E Pe
\~ =
= ~
t
2 1t=1
j
(=~ + ~cT Pc) + !nTrlo + Ibml e2 , 2
2d,.
2,
(10.85)
where, ill analogy with (10.80), ViJ satisfies
With (10.86). (10.83), and (10.14), the derivative of l~ is iI
_ \i:I <
2:
Ck Z:
+ =;=i+1
k=1
T 1
+()
r
+Zi [0°;
( Til

Bro. ...·11
aWZi 
1/
+ =il  Pi 
(;1
~ o'0kl ) (fTil  6') r 1 6~ + L Z,,A
Oa.i_1 :..] A8 8(J
')
«=2
80
1:2  z;aC2  4d eO') • aail
y
'j
(10.87)
430
LINEAR SYSTEMS
We can n~w eliminate the indefinite 8term from (10.87) by choosing the update law
iJ =
rTi, where (10.88)
Noting that
rT;_1  d= rTil 
fTi
+ fT.  d= fa:;;l w::; + (fT' 
0) ,
(10.89)
(10.87) becomes _ ti:I <
(10.90) Then, if Zi+l were zero and if 9 were chosen as ai
= CjZj 
di
aail
(
8 ) y
" Zi  =i1
80: 
rTj,
+ Pi +  . i r1T i 80
the stabilizing function
aO: I: k=2 80
1 1 ( ' I Z k k..  )
80: i  1 raw Y (10.91)
would yield (10.92)
=
However, Zi+1 ~ 0, and we do not use 9 rTj as an update law. Instead, we retain Tj in (10.88) as our itb tuning function and llj in (10.91) as our itb stabilizing function. Upon the substitution into (10.90), we obtain
10.2
431
TUNING FUNCTIONS DESIGN
Comparing (10.93) with (10.86), we see that the design at step i has resulted in the Lyapunov derivative of the same form as in step iI. At the end of the design procedure, that is, when i = P, we se1ect the actual update law as
n=
(10.94)
rTp.
This choice mal\:es li'p negative semidefinite: p
l~
L C&.z~ ,
:=; 
(10.95)
k=1
because from (10.84) we have Zp+l = O. The control law (10.84) which has helped us to achieve (10.95) is our actual control law: 'U 
vm,p+l
", ""'p
+ ny(P) r •
(10.96)
t!'
The complete tuning functions design employing the filters in Table 10.1 is summarized in Tab1e 10.2. The resulting error system is Z)
=
d1 z1 + bm z2 + £2 + (w 
CIZ1 
e(Yr + ii:d e])T ii
bm (Yr + (1) B
'" =
(10.109)
C2 Z ' 
d.
(~:l) \,  bmz
_ aal £.,
_
aal wTO
1
t z, 
~~l (0  rT,) (10.110)
ay  ay
Zj
=
di (
CiZj 
8a: i 
ay 1 ) 2
E ao
aaj,:l r 8a:i 
ay
j=2
aail
aO:i 
ay 
ay
Zp+l
= O.
all i:::1
1 Zj _
8(J 1
1'0
g,,W
where
=il + Zi+l
Zj 
(0  rTi) i = 3, ... ,P,
,
(10.111)
In view of (10.105) we have ~
0 rTi
p
= E
8a:.j_l
faw=j.
(10.112)
Y
j=i+l
Defining l::t.
aO'i.l f811j1 w
an
(J"IJ 
ay
(10.113)
1
(10.112) yields
_ aO:i  1
80
(0  fT.') I
=
p
L j=i+l
UijZj.
(10.114)
432
LINEAR SYSTEMS
Table 10.2: Tuning Functions Design for Linear Systems
0'1
0) 02
Cl:j
=.
=
Y  Yr
.:.i

V
fII,i 
(10,97) o~y(i))  a: iI, .. r
1,• 
I)
_, •••
= = =
,p
(10.98) (10.99) (10.100)
(10.101)
=
TI
=
(W  fl (Yr
T;
=
lil 
+ 0'1) el).2:1
(jo;il
ay
W=i,
(10.104) 1,'  ') _, •••
,p
(10.105)
Adapt.ive contl'ol law:
(10,106)
Parameter update laws:
o= g
Pip
= "'( sgn(bm) (Yr + al) .2:1
(10.107) (10.108)
10.2
433
TUNING FUNC'fIONS DESIGN
By substituting (10.114), we bring the error system (10.109)(lO.111) into the cOlnpact form (10.lI5) where the system matrL~ A:::(=, t) is given by
cld j
;'711
1.,711
C2d2 (~)2
o
10"23
o
0 1+0'23
02,p
Up'2,p
a
l(J'pl,p
l+O'p_],,, cpdp ( D~;1)2
(l0.116)
H'!!'(z, t)
=
[ ~ 1 u
87
11
H'o(z, t)T
=
E lR.P
,8
(10.117)
1
H'~(z, t)wT 
il (ilr + 0:1) eleT E RP)(P.
(lO.lIS)
The structure of the matrb: A:(z, t) is very important. The terms (J'ij above the diagonal are due to the terms  0';;;1 r( Tj  Tjl) in the ziequation of the error system. Since i < j, these terms cannot be eliminated from the error system. However, by introducing their negative images uij below the diagonal in the matrix A:(z, t), we achieve ake1J18ymmetTJj off its negative diagonal. This Inakes the homogeneous part of the error syst.em (10.115) exponentially stable and yields nonpositivity of the Lyapunov derivative (10.95). The closedloop system which satisfies (10.95) consists of (10.115) along with
e=
rIVo(z, t)=
e=
!,sgn(bm) (Jjr + 0:1)
g
AOE,
=
eT z
summarized here from (10.107), (10.108), and (10.14), respectively.
(10.119) (10.120) (10.121)
434
LINEAR SYSTEMS
10.2.2
Stability analysis
For the adaptive scheme developed in the previous subsection, we establish the following result.
Theorem 10.6 (Tuning Functions) All the signals in the closedloop adaptive system consisting of the plant (lO.t), the control and update law8 in Table 10.2, and the filters in Table t 0.1 are globally uniformly bounded, and asymptotic tracking is achieved:
lim (y(t)  lJr(t)] = O.
(10.122)
Ioa
Proof. Due to the piecewise continuity of lJr(t) , •.. ,y~n)(t) and the smoothness of the control law, the update laws, and the filtel'S, tile solutioll of the closedloop adaptive systenl exists and is unique. Let its ma.'timum interval of
existence be [0, t J}. Let us consider the Lyapunov function
v;, =
1 .•'
:;.1 Z
2
1 T I1Ibml 2 + 1 ~ L..J  e Pe + 0T r 8 + B . 2 k=l 4dr:;
2
2"'(
(10.123)
In (lO.95) we established tbat V'p(t) is nonincreasing. Hellce t z, 9,0, and e are bounded on lO, t / ). Since';:1 and Yr are bounded, y is also bounded. Then, from (10.26) we conclude that 11 is bounded. We have yet to prove the bounded ness of A and x. OUf main concern is A because the boundedness of x will be immediate {rom the boundedlless of Ell1, and A. From (10.27) it follows that
i where [((.s)
::=
sn
= 1, ... ,n,
(10.124)
+ k1sn  1 + ... + ko. By substituting (10.1) we get
\.( ) = (Sil + kt S i  2 + ... + A~il) A(s) "'. S
J{(s)B(s)
( )
i = 1\ ... ,1)..
Ys )
(10.125)
In view of the boulldedness of y and Assumption 10.1, the last expression proves that AI, ... ,Am+l are bounded. We now return to the coordinate cbange (10.98) which gives 't'm.i
"\ (i21) = "'i• + IlY~ r(iI) + Q'il (Y,1], 0" ,0, "'m+ih lIr
,
i
== 2,,,. ,p.
(10.126)
Let i == 2. The boundedness of Xm+1, along with the boundedness of Z2 and y, TJ, 0, Yr, and Yn pl'oves that vml~ is bounded. Then, from (10.60) it fallows that Am+2 is bounded. Continuing in the same fashion, (10.126) and (10.60)
e,
10.2
435
TUNING FUNCTIONS DESIGN
recursively establish that ,,\ is bounded. FinaJly, in view of (10.33) and t.he boundedness of '1,", and E, we conclude that J.' is bounded. We have thus shown that all of the signa1s of the closedloop adaptive system are bounded on [0, t f} by COllst.ants depending only on the initial conditions, design gains, and the external signals Yr(l), ... ,!J~")(t), but not on tf. The indepBndence of the bound of t f proves tbat t f = 00. Hence, all signals are globally uniformly bounded all [0, (0). By applying the LaSaUeYoshizawa theorem (Theorem 2.1) t.o (lO.95), it further follows that z(t) + 0 as t + 00, which implies that lim,_cc [yet)  Yr(t») = o. 0 Theorem 10.6 establishes global uniform bouudedlless of all signals but not global uniform st.ability of individual trajectories. We now determille all error system which translates the investigated system to the origiu. Then we prove tllat the equilibrium at the origin is globally ulliformly stable, and alJ the error states except the parameter error are regulated to zero. We start with the subsystem (z, E, 8, u) wl10se 311+2 states are encompassed by the Lyapunov function (10.123), and const.ruct. additional equations t.o form a complete error system. \Ve first introduce the equation for the reference signal rt
(10.127) so that tbe error state 1'j = 11  fJr is goverlled by
~ = Ao7J + e,lzI .
(10.128)
The system (z,e,ij,9,u) has 4n + 2 states, \Vlli1e the original (X,'11,AJ/ t U) system has 411 + m + 2 states. We find the remaining m error sttttes in the inverse dynamics of (10.3). Let us consider the similarity transformation
[~1 1 Xp
=
(
[~l 1 [Ip xp

°PXJII
T
1x"
(10.129)
Tx
wbere (10.130) (10.131)
With the belp
01 the readily verifiable identities
T[
n
= 0,
TA
= A!,T + TAP [ ~ ] eL
(10.132)
436
LINEAR SYSTEMS
we represent the inverse dynamics of (10.3) as
( = l:1
Ab(
+ T ( AP [ ~ ]
A b(
+ bbY •
 a ) 1/ (10.133)
The equation (or the corresponding reference signal (r is
I;r so thttt the error state (
=
Ab(f + b"Yr ,
(10.134)
= ,  (r is governed by (10.135)
\\Te have now characterized the error system = A::(z, t)z + H'e(s, t)E:2 + H'n(z, I.)T8  bm (1ir + ad elg e = Aoc .;;;
,
Ah{ + bh=1
= ij =
8
e
Aolj
(10.136) (10.137) (10.13S) (10.l39)
+ en=t
= rTT'o{z, t)= = l' sgn{bm ) (1ir + a'l) ei:;
(10.140) (10.141)
wbich possesses the desired stability and regulation properties.
Corollary 10.7 The errol'sllstem (10.137)(10.1./.1) has a globall'1 uni/o1'7n.l'1 stable equilibl'ium at the origin. lvI01'eo11e'l', it.'l 471. + m + 2dimensional state converges to the n + m + 2dimensio11.al manifold (10.142)
Proof. "Ve augmented the Lyapunov function and {: V
V;,
by the terms quadratic ill ij
1'1' 1 , = V;, + ;:ij pjl + k ( 1 Ph( h"."
=
1
T
z 2
,
~ 1 'J' Ibm I ' l l T 1  1 T 1 'r .. =+ j=l £c PE:+ g + 6 r 6+ ij Pfj+  ( Ph(, i:td 2')' 2 ~':q k( j
(10.143)
where Pb satisfies
P,. = Pi! > 0,
(10.144)
10.2
437
TUNING FUNCTIONS DESICN
and k'l1 k( are positive constants to be chosen. In view of (10.95), (10.138), (10.139), (10.11), and (10.144), the derivative of l' is given by
(10.145)
Thus, if we choose k'l and k, as k
> 41Pc n l2
" 
Cl
(10.146)
'
the derivative of" will be nonpositive,
1i":5
~lijf2  ~1'f2, 2A.'l 2k,
(10.147)
which proves the global uniform stability. By applying the LaSalleYoshizR\VEL theorem (Theorem 2.1) to (10.147), it further follows that; il(l},CU} p 0 as t p 00. The convergence of z(t) t.o zero follows from (10.95), and the convergence 0 of c follows from the exponential st.ability of the system (10.137). Corollary 10.7 establishes stability properties which are stronger than the properties guaranteed by traditional certainty equivalence adaptive controllers [5, 44, 51, V12, 165]. In addition to the usual global uniform boundedness and asymptotic tracking, the etTOr system possesses a stronger property of global uniform stability. Moreover, all the states converge to zero, except possibly the parameter estimation errors ~ and O. As uSlUtl in adaptive control, in order to also have the C01lvergence of 0 and U to zero, one needs to assume some form of persistency of excitation (PE). Thus! the achieved stability and convergence properties are as strong as possible without PE. Corollary 10.7 bas not dealt with a correspondence between Ule original system (x, A, 77, 0, e) and the error system (z, E, (, ii, 6, g), which can be done by analyzing the coordinate change
(x, A, 7], 0, b)
H
(=, e, C. ij, 6, u) .
(10.148)
It can be shown that whenever B( s) and 1( s) are coprime, this coordinate change is a global COOdiffeomorphism for each t ~ O. The proof of a similar
438
LINEAR SYSTEMS
claim is given in Section 8.1.3. Although.t'he ,co,pnimeness condition cannot be guaranteed by design because the coefficients of B(s) are unknown, it is satisfied wit.h probabilit.y one. We stress that the singularif,y of the coordinat.e transformation (10.148) is not in contradict.ion ~v.it.h the lboundedness result of Theorem 10.6 (cf. (10.124)(10.125)) whose promlliemains valid when B(s) and /\'(s) are coprime.
10.2.3
Passivity
An additional novelty of the tuning functions design is a passivit.:r property for any relat.ive degree of t.he plant transfer function. At each step of the backstepping procedure, the corresponding error system is strictly passive from 9 as the input t.o the tuning function as the output.. We will show this ttBsumillS, for simplicity, that the highfreq1lency gain brrl is known (in which case b",(t) = 1/ ~(t) == bTll ). \Ve consider the error system at step i, obtained by setting Zi+l
= 0 and {J = [
~]
[A~tt) w:(~t)er][;] + [Wl(Zot)wT]1i
=
Ti
rTj:
[11';(=0 t)wT
=
r[;],
(10.149)
where Cl 
c') d'J
bin
A~ =
bm
d1
0
1 
0
(~t By
0
o
1 + 0"23
0"')"
,'
0"23
U2,1
1 
O"il.i
~ 1 IV! = [
_to
1 + O"il,i co _ d ,
I
(Ba._ )2 8y 1
(10.150)
iJy
To show that this system is strictly passive from the input 6 to the output Til by Definition D.2, we need to find a nonnegative storage funct.ion Ui(Zj, e) and a positive definite dissipation rate 1h(Zi, e) such that
10t

T;rOdu ~ U.{t)  Ui(O)
+ 10f'" 'l/Jdu)du.
(10.151)
1(j)~.2'
439
TUNING FUNCTIONS DESIGN
We employ the positive definite andlraclib.llY· unbOlmded functions
(10.152) (li(}.153) Along the solutions of the systODl: (~l!(ih.1i49), tIle derivative of U, is
(10.154) By integrating over [o~ t], we arrive at the inequality (10.151), which proves the strict passivity between 9 and 'i' The strict passivi'ty. results from a special choice of the output (tuning function) ii and the stabilizing function ai At the end of the recursive procedure, for i = p, we obtain the error system
[~ J Tp
= =
[Att)
W~(zot)ei] [ : ] + [W'("ot)W
T
[W. (%0 t)w
T
r[:].
]
ii (10.155)
Because of the strict passivity from jj to TPI by Theorem DA, the feedback system will be stable if the feedback path from ' i to 0 is passive. The simplest implementable choice is an integrator:
 r'po
8=
(10.156)
s
The resulting feedback system, shown ill Figure 10.2, is
[;] =
8
[A=X:t)
r ][: 1+ [H'.(:ot)W
T
W'(rit)e T
=  [ W'(:o t)w
r[:].
]
8 (10.157)
440
LINEAR SYSTEMS
,....
oL
[WOWT]
to+ 
e
J
[A=An W.0er ]
[w:T
Tp
f4
r

.S
Figure 10.2: The 'feedback connection of the strictly passive (Z, e)system with possive update law.
8.
By Theorem D.4, the system (10.157) has a globally uniformly stable equilibrium z = 0, E: = 0, and = 0, and, in addition, z(t), e(t) ~ 0 as 1. + D. The bounded ness of aU the signals is argued as in the proof of Theorem 10.6. The fact that the passivity property of the error system is achieved for plants of higher relative degree does not contradict the relative degree restriction: From 8 to Tp the relative degree is still one.
e
10.2.4
Design eXaInple
We illustrate the new design procedure on an unstab1e relativedegreethree plant
y(s)
1
= s'1(s  a) u(s),
(10.158)
where a = 3 is considered to be unknown. Simulations of this example are presented in Sections 10.4.2, 10.5, and 10.6.4. The relativedegreethree design contains the main features of the general design procedure. The control objective is to asymptotically track the output of the reference model
Yr(s)
=
1 (8 + 1)3 "(8).
(10.159)
To derive the adaptive controller resulting from our nonlinear design, the
10.2
TUNING FUNCTIONS DESIGN
plant (10.158) is first rewritten in the statespace fOfm (10.2):
= = xa = Y = :1:.
X:J
X2
Xs
+ aXI (10.160)
'U
Xl •
The filters from Table 10.1 are implemented as ::: =  [A~71' Aor/t ,,] !
iJ = Ao17+ eaY! ~\
=
An
=
AoA + e:itL, [ k] 1
1.'
= A,
(10.161) (10.162)
!].
k:2 0 ka 0
3
~ = Ao1J
(10.163)
The signals Yr, ilr, Yn and 11$3) are implemented from the reference model (10.159) as
= = = =
Yr fir
iir
(8) r
Y
T1
T2
(10.164)
Ta 3T3 
31':! 
+ 7' ,
"'1
by writing the reference model (10.159) as follows:
1') =
=
72
1'3 =
T2
(10.165)
Ta
3"'3  31'2 
T)
+T •
Since ill this example the bighfrequency gain is known, in the first step we can directly treat V2 as a virtual control and do not need the additional parameter estimate b. The virtual estimate (10.12) is x =  (Ag  aAfi) T/+ A, and by defining W
=
....
'::'2,1 
TA'l
Y = e2 '0' 1  11 ,
(10.166)
the results of the three steps of our design procedure are:
Step 1. Z)
= Y  Yr
(10.16i)
Tl
=
(10.168)
a1
WZI
= eel + d1)ZI ~2 
wo..
(10.169)
442
LINEAR SYSTEMS
Step 2. Z2
=
1'2
=
0:2
=
Step 3. (10.173)
(10.174)
(10.175)
The adaptive control law (10.106) and the parameter update law (10.107) are
=
'U
a. = Tile matrix form of the
+[
a=
"II
r
(Z11 Z2,
0'3
+ y~3}
17'3.
Z3, a)system is
~] (iiw+C2)
~ B1J
[1
,
Q,gJ, lJ,I
1
(10.176) (10.177)
_!2:.]wz Bu
(10.178)
I
(10.179)
where 0' = 7~ B:" w. Note again the skewsymmetry oCthe offdiagonal entries and the stabi1izin~ role of the diagonal entries. The block diagram ill Figure 10.3 shows tha.t the overall structure or the new adaptive system has the familiar form of the input and output filters
10.3
443
PROPERTIES OF THE NONADAPTIVE SYSTEM
Plant 1
11
1
8 2 (8
JJ
 a)
1 7J~filter
A~filter
~1
+/"i'\ ,~

Ref. model Yr l' 1 "'""~ (8 + 1)3

•
••
(3)
Ur, Jlr, Un Yr
Nonlinear controller
.:
.  _ _ .. _ _ _ _ _ _ _ •
Step 1
.
_ _ '1
:
;~~~[I~;~; :
Step 2
r
:
~ ~ ~~~ ~ ~ ~[~~ ~ ~:
::
Step 3
iIIo _ _ _ ...
___ •
Parameter update I' 8
:
___ ...
Figure 10.3: The distinguishing feature of the tuning functions design is the "Non· linear controller" block. In contrnst to certaintyequivalence designs which result in linear·like control laws, the tUlling functions design produces a control law in which both par&meter estimates and filter signals enter nonlinearly.
feeding into an estimator/controller block. The fund81nental difference, hO\v· ever~ is that this b10ck is now a nonlinear controller. \Vhereas in traditional schemes this block would be a "certaillty~equivalence" linear controller, the t.uning functions design produces the control law (10.176) in which both parameter estimates and filt.er signals enter nonlinearly.
10.3
Properties of the Nonadaptive System
10.3.1
Underlying linear controller
The underlying nonadaptive 'detuned' controller is obtained by setting the a.daptation gain r to zero, that is, by freezing the parameter estimates at. some 'detuned' values. In traditional certainty equivalence adaptive control, there is a clear sep· aration between the controller and the identifier. The identifier 'tunes' the
444
LINEAR SYSTEMS
pnramet:ers of the linca'T' controller. Although the control law is a nonlinear funct.ion of t.he paramci.er estimates, it is linear in the output y and the filter states. In the tWlillg functions design the control law is a nonlinear function orboth the parameter estimates, and the output and filter states. The nonlinearity in tIlc output and filter states comes from illcluding the parameter updat:c law in the control Jaw. The resulting enor system is also nonlinear. One would expect. that the controller and the error system obtained after set.t.ing = 0 are a1so nonlinear. 'Ve show that this is not so: The ulldcl'lying nonadaptive eontroller llsed in the tuIling functions design is lincU1'. To mal{e the analysis simpler, \Ve aSSUOle that the highfl'equency gain bm is kllown,2 and without loss of generality, let bm = 1. With blll knowll, the first
r
components of tlle Ul1knOWll paramct.er vector () and tIle filt.el·ed regressor w m'e dropped from (10.7) and (10.39), respectively:
o = [bm  1•.• · lbo, a n l, ••• ,00] '1' w = [r!m]t~" .. ,'OO,!h 3(2)  UeT]T
(10.180) (10.181)
Thus, tbe closedloop syst.em (10.115)(10.121) becomes
.., = A;:(.:,t)z+Hl"e(=,t)(wT 8+E2)
=
5
jj =
(10.182)
AOE
(10.183)
rwl'FeT (:, t)z,
(10.184)
wlwra t:bo system matrix .4;(.:, t) is given by
1 0 cld1 1 C!!d2(~)2 1 +028 0
1 
0 0'2.}
(1''2,p
0'23
"24
0
O'p2,fI
"'.!,p
""p2"
lO'p_l,p
1 + O'pl.p cP dP (80,.1)2 iJg (10.185)
and
TFs(':; It) and
(Jij
are defined as
l"e(=, i) =
[
~ t'll
i' I'
(10.186)
_Do,._1 8u
2Wlum bm is unknown,
It
slight modificaf.ion in the design leads to t.he same conclusions.
~ . .
I I
.L:~
.
,
.
10.3
PROPERTIES OF THB NONADAPTIVE SVS'rEI\·,
..
_
G"J
r
00"_1
ao

rCJc~'JJ
all
(10.]87)
W.
Now we set out to det.ermine the det.uned nonadaptive cont.roller. \Vith = 0, the zsystem (10.182) becomes (10.188)
1
cld1 1
A:
=
(''J _


Uy
[
1
1
0
0
0
0
b:; =
0
0
d" (~)2
(10.189)
1 c _ d (Uo.U,r _ 1 )2 1 P P
_!!!ll
1811
(10.190)
_D~II_t l:Jy
The nonlinear terms G'ij have disappeared from the matrLx .4: because t.hey have as a factor (cf. (10.187)). Examining the expressiolls [01' 0·], .... Q'pI in Table 10.2, it call be established t.hat when r = 0, all the derivat.ives ~ are ull
r
known constants depending on Ci, di , and {) only. Hence. t.he matrix A.:: and the vector b: a.re constant. Since A.:: is Hm'witz (as a snm of a s]mwsymmetrir matrix and a negat.ive diagonal matri."'() , the transfel' function from B'J'w to =" (10.191) is stable, its relaf,ive degree is one, and hath 13:(8) and Q;:(8) are monic polynomials. Disregarding the initial conditions and the exponentially decaying input £2, the lineal' en'or syst;em (10.188) is rewritten as
=1
=Y 
Yr
,8:;(8) 1'
= ( )0 0'.: 8
/.IJ,
(10.192)
which is one part of the detuned feedback system. However, this system alolle does not reveal the underlying linear controller because 11. does not appeal' in (10.192). In order to have 11 appe~lI, let us examine 8'rw along with (10.180),
446
LINEAR SYSTEMS
(10.181), and the filter expressions from Table 10.1:
(10.193)
=
where A(s) = A(s).A.(s), A(s) sll+tin _ 1S n  1 +.. '+ao, B(s) and H(s) = 8 711 + bm _ 1s m  1 + ... + boo Combining (10.192) and (10.193) we obtain the system
y Yr
=
iP'w
=
P=(s) jjl'  w O',:{s) s + kl ( [(.5)
= B(s)B(s),
(10.194)
) A{s)y + B(s)u .
(10.195)
Substituting (10.195) into (10.194) we get
Y  Yr
+ J.~1) ( ) = p:;(s) ,()8 (8.r() A(s)y + B(s)u = \ S G:.=
P::{S) (8 + kd ( ~ A) ,() .N'() A(s)y 8(8)U 0;:: 8 \ S
I
(10.196)
whicb shows that the underlying linear controller is defined by
(8 + ~~l)!3::(s)B(s)'lL = (s + ~~d/3::(s)A{s)  O!:(s)/(s») Y + Q::(s)l(s)Yr. (10.197) In the block diagram of the underlying linear system in Figure lOA the
transrer function (iI+kl)t=SK is strictly proper because /3::(s), /(s), and B(s) are monic. To prove that the transfer fUllction o=I((~11.:I)fI.:A is proper, we examine the control law 'u (Y,'fJ, A, jj~p), 0). Since 'U is linear in V, 11, A, jj~p), from the definitions of the filter states 'fJ and A ,ve conclude that (10.198)
10.3
447
PROPERTIES OF THE NONADAPTIVE SYSTEM
r
Bm
+
Am
u
~ t

y
B
A
..... (s + k J ){3;:B  I( ~ A
a=!(  (s + l:l){3:A I(
Figure 10.4: The underlying detuned linear controller.
where degqn_l(S) :::;; n I, degpnl(S) that (10.197) has the following form:
~ n
1, and deg rp(s)
~ p.
[«s)  Pnl(S) _. ( ) qn(s) I( (s) U  1 P S Yr + !( (s ) Y t
This shows
(10.199)
where
q,.(s) = Qnl(S) + ijI«s) = ll::(S)/((S) + (s + kl){3:(s)A(s),
(10.200)
and degqn(s) = deg [qnl(S) + q/(s)] = n. Hence, from (10.200) we conclude that deg [a::(s)]«s)  (8 + k 1 ){3=(s)A(s)] = 71. Thus t the transfer function
O::K(8/~kl).B::A is proper and the system in Figure lOA is implementable. From the block diagram in Figure 10.4 we obtain the transfer function of the detuned system from the reference output Yr to the output y: (10.201)
The matching condition for yes) = Ur(S) is
B(s) B(s) li(s) = A(s) ,
(10.202)
which becomes
..4(8) = A(s) ,
B(s) = B(s)
(10.203)
when A(s) and B(s) are copriIne. When the matching is achieved, the characteristic polynomial of (10.201) is D::(s)]«s)B(s), so that the closedloop poles consist of:
448
LINEAR SYSTEMS
1. the roots of o::(s}, i.e., the eigenvalues of the error system matrix A:, 2. the roots of J((.c;}, i.e., the eigenvalues of the filter matrix Ao, 3. the roots of B(s), i.e., the zeros of the plant.
It is of interest to compare t.hese closedloop poles with those of the traditional .M.RAe whose characteristic polynomial is Am (s)J«(s)B(.r;). The design here replaces the reference model denominator polynomial Am (s) with the error system denominator Jlolynomial 0'.::: (s). In this way, our controller allows the closedloop poles to be placed independently of the reference model poles.
10.3.2
Parametric robustness and nonadaptive performance
'Ve now show thaI; the clesigll panlmeters call be chosen to make the detuued closedloop system stable, that is, tbe underlying nonadaptive controller has a parametric robustness property. In addition, the design parameters can be used for systematic performance improvement. Traditional certaiuty equivalellce adaptive linear controllers do not possess these properties. 'Ve express the regl'essor w as the response to the output y:
w
=
. . .  yeTJT [V l.2, ••. , VO,2, ':'(2)
=
(S+k1)[m_l 1 S+kl[n_l , ... ,s,ly , ... ,s,ltt, ]((s) s [ ]({s) S
=
S+kl
ll.
l
m
l]T
[(s)
[r
Hw(S)11.
8
1111
, ...
,s,
11 A (s) [tIl l]]T y B(s)' s , ... ,.~, (10.204)
The (71.+m) X 1 tr811sfer matrix H,As) is proper and stable, and its coefficients depend only on the p]ant parameters 8 and the filter coefficients k1 , ••• , kn • The expressions (10.192) and (10.204), rewritten here as (10.205) (10.206) define the feedback systems shown in Figure 10.5. The closedMloop transfer function of the detuned system in Figure 10.5 is (10.207)
10.3
449
PROPERTIES OF THE NONADAPTIVE SYSTEM
Yr
y
+
OTW
6TR~(s)
+
13:.(8) Q:;(S}
=1
Figure 10.5: The detuned linear feedback system. The system is tuned when the parameter error is zero! iJ = 0, in which case the transfer fUllction is YrV(8)B = 1. We will establish tbe parametric l'obustness property by showing that the controller parameters call be selected to guarantee st.a.bility when 0 =I O. Both bloclts ill Figure 10.5 are stable, and the transfer function i3::({S)) iJT Hw(s) is Ct. s strictly proper. Thus, the feedback system is welldefined. OUI= proof of parametric robustness will proceed using the small gain theorem. In addit·ioll to the panul1etric robustness property, we will est.A.blish transient performance bounds for the state of the closedloop system expresspd in t be enor coordinates: z
=
A:;..;,_ + b:JJ~tW
t
=0
( 10.208)
ij(O) = 0 (0) = O.
(10.209) (10.210)
:;
( =
AbC + bb=1 , ij = Ao71 + Cn =l ,
(0 )
e = AoS' is not included because e:(0) = O. The init.ial conditions ,}(O) ~.l1d (0) can always be set to zero because the initial cOllditiol1f; of t.he
The system
conceptual filters (10.127) and (10.134) are free for us t.o chose. From (10.209) and (10.210) we define transfel' fUllctions
HFij(5) A (s1  Ao)le71 H/({S)
l::.
(10.211)
,
(51  Ab)lbb,
(10.212)
and denote the respective impulse responses by UJrj(t) , 111,(1). In the sequel, we will malce use of the following e()uivalent gains: Co
= l,$;iSp min Ci,
I'
do
1)1
= ( ~: 1=1 d,
Theorem 10.8 (Stability and loo Performance) Thc
(10.213) nona.nlJ.pti'lJe .'iystem
(10.208)(10.210) is 081lmptoticalll1 stable /07'
J
2 codil
> 18111hwlh .
(10.214)
450
LINEAR. SYSTEMS
Tile C. OO nonns of the :;tate., of this system
a7Y~
bounded by (10.215) (10.216) (10.217)
Proof. Differentiating ~lzl2 along the solutions of (10.208), (10.189), and (10.190), lve get ~
(10.218)
By applying Lemma C.S(i), we obtain (10.219) On the other ha.nd, since Hw(s), 8B defined ill (10.204), is a stable proper tl'ansfer matrix, then by Theorem B.2{i} we have
IIwllog
S;
IIh",/llllyl/oa
=s; =s;
IIhwlhllzllloo + I/hwlhl/urlloa Ilhwlltl/z/loo + I/hwlhl/urllc.:l·
(10.220)
To apply the small gain theorem (Theorem B.10) to (10.219){10.220), we note that 2:};a; in (10.219) can be ma.de arbitrarily small by a choice of Co and do. Since I/hwl/l is finite and indepelldent of Co a.nd dOl the loop gaiu 2~181Ilhw'h can be lnnde less than one. Thus, by Theorem B.lO, the £0;:.sta'tifity of the feedback system in Figure 10.5 is guaranteed. Ne~tt we show that the Loastability also guarantees the internal asymptotic stability of this system. Consider the closedloop system (10.201):
Y= D::;I\:B + (s
+ k1 )/3:; ("BA  AB)Yr.
(10.221)
A
Since 0::(8), ICes), and B(s) a.re all HUl'witz, if there are cancellations in the transfer function ill (10.221), they are all in the open left halfplane, so that
10.3
451
PROPERTIES OF THE NONADAPTIVE SYSTEM
the denominator in (10.221) is also Hurwitz. We have thus shown tbat for sufficiently large cOdD the linear system (10.221) is asymptotically stable. The bound (10.215) follows by substituting (10.220) into (10.219):
11.11", ~ 18~ 11011"" + 19~ IIUrll. 2coo 2coo
(1O.222)
Finally, from (10.209)(10.210) and (10.211){10.212), by Theorem B.2(i} we llave
lliilloo
"'"00
~
lI'Wiillt U=dlo:> ~ IIw,lh 11=1110:> ,
(10.223)
(lO.224)
o
which, in view of (10.215), proves (10.216) and (10.217).
Theorem 10.8 is a highgain result: The coefficients Ci and d i should be high not only to improve performance, but also to satis(y the stability condition (10.214). However, tbis result is sigllificant beC'ause it shows that the cantroller enlpioyed in tIle tuning functions design can be used both as an adaptive controller and as a parametrically robust highgain controller. Tradit.ional adaptive linear controllers do not possess tIns property. They can be used only as adaptive, and parameter adaptation is the only tool they can usc to guarantee stability ill the presence of unknown parameters. This nonadaptive controller does not, in general, achieve asymptotic tracking, so we cannot talk about its £2 performance. However, it is possible to prove tllat meansquare performance can be made arbitrarily good. In addition t.o tllis, the following theorem also provides an alternative stability condition. Theorem 10.9 (Stability and MeanSquare Performance) The nonadaptitle system (10.208)(10.210) is asymptotically stable f07' (10.225)
The meansquare 'Values of z, fi,' are bounded by
Proof. By applying Lemma B.5 to (10.218), we get Iz{t)1 2 5
2~ lot e~co(IT) (iFW(T}) dT 2
1
(10.229)
452
LINEAR SYSTEMS
which, upon integration over [0,
/'I=(r)rldr
./0
~
tl becomes
.Jf' [IT C2~(T8) (OTW(S») cls] (lr. . . do 10 .10 2
(10.230)
Changing the order of integration, (10.230) becomes
.J t
(It
f '\z(r)1 2dr S e2ClJB (eTw(s»)2 e 2COT dr) ds Joo .... d o .10 .S S 1
'2do
1t
e'Jcos
(T 1 .., (I WeB) )2 c·C()l'd"
0
2co
(10.231)
becmlSC
r' e
.Iii
2CIlT
dr =
2.... (e2co
2cn ... 
e2CU')
S .Je 2t:o!l • _co
(10.232)
Now, the cC:111cellatioll e2~8c2r~ls = 1 in (10.231) yields (10.233)
all the other hand, since Hw(.9) is stable and proper, then by Tbeorem B.2(ii) the t:nmcated £'}. norms of wand ;; tl.l'e relat.ed. as (10.234) From (10.233) and {10.'2~1), by the small gain theorem (Theorem B.lO), £2 stability is guru'ant:eed ror 2Jcndo > /81l1Hwlloc:u and a.':iYlllptotic stability follows as in the proof of Theorem 10.8. Substituting (10.23,1) iuto (10.233) and I
solving for
IIzlb.1 = (J~ 1=(r)12dr)!i, we get
)1 2d ) ~ < 18l11Hwiloo (I II ( io('1( ... r T  2..Jcodo 19111 Hwlloo 1Jr 2,1'
(10.235)
JJ
and (10.226) follows because IIYrll~,1 = IlJr(r)1 2 dT 5 IIIJrll~t. From (10.209)(10.210) and (10.211)(10.21'2), by Theorem B.2(ii) we have
l\ i111b :5
1/(,112
s
IttVij/lool/(Zl)tl/2 I/H',lIooII(=d,lb,
which 1 in view of (10.226), proves inequalities (10.227) aud (10.228).
(10.'236) (10.237)
o
Theorems 10.8 and 10.9 provide two different stability conditions, (10.188)
a.nd (10.225), of which (10.225) is directly computable [16] and less conservative because IIHwl/oo S IIhwllt (see Theorem B.2(iii)). Another way of expressing the performance properties is by comparing the detllned closedloop transfer function (10.207) ,vit.b the desired transfer function YrII(CRR») 1.
=
10.4 TRANSIENT
453
PERFORMANCE WITH TUNING FUNCTIONS
Theorem 10.10 (Frequency Domain Performance) In the nonadaptive .'lysf.em (10.207), the design IJa1'O:mete7's Ci and di, 1::; i :s; p, can be chosen t.D satisj1J the following tracking pe7jonnance .'fpccification far any Dc > 0: Vw E lR. Proof. By setting t =
00
in (10.233) we sec that the induced £'.}, norm of ::~:~
is 110 grpatcr tluul ~. This, ill tllrl1~ means that ~""O(IO implies
B:(jW)
IlX::(jW) From (10.20i) we
G .
I c(Jw)
llOW
_ 1 _
1
(10.238)
1<
1  2vcodo '
II &11 Lt:
CCI
~
Vw E IR.
whirh
(10.230)
ha.ve {J=(iw)OTH (. ) o.:(jw) w JW
1
li=(~W)jjT H w (jw) Q=(jw)
I<
I liilll!1.W II co :hrcocln
 1
lIOIIIH II 2VCUdO W
(10.240) 00
which is less than any given Dc provided that corlo is sufficiently large:
(10.241}
o As e......:pect.cd, the tracking rondition (10.241) is more stringent tJum t:he corresponding stability condition (10.225). The required value of 2vcodo is increased by the factor 1 + and tends to infini ty as Dc I> O. In t:his sense the underlying lincar controller is a "highgain·' controller which achieves R. good tracking pel'fOrnlRnce at the expense of all increase of the bandwidth.
t
10.4
Transient Performance with Tuning FunctiollS
III the abscnce of disturbances and umnodelcd dynamit·s, the traclciug error of most adaptive control schemes converges to zero, that is, they moet the stat:ed asymptotic performance objecth'e. In applicat:ions, however t the syst.em's transient performance is often more important. Analytical quantification and improvement of transient performance have been longstanding open problems in adaptive control. For tnldit.ional adaptive linear contI'oUers there are virtually 110 results which allow the designer to a priori compute bounds on the transient behaviOl', let alone to meet a given transient performance specification. 'll·ansient. responses of traditional adaptive schemes suffer from lnrge initial swings [202] because the certaillty equivalence controller does not take into account t.he parameter estimation transients. III additioIl I the ident:ifier which j
454
LINEAR SYSTEMS
is designed separately from the controller, is driven by an 'estimation error' signal that is unrelated to the control objective. We will shmv that the transient performance can be improved by letting the controller and the identifier exchange information during the operation of t.he adaptive system. The tuning functions controUer takes into account the effect of the. parameter estimation transients by incorporating the parameter update law fJ and it.s fast unnormalized update law is driven by the tracking error.
10.4.1
= rTp1
Transient performance of the adaptive system
\Ve now derive computable bounds on both £2 and £IXJ norms of the states z, ij, and, of the adaptive system, and we show how they can be made arbitrarily small by tl. choice of t.he design parameters Cj, di , and r.
Theorem 10.11 (£2 performance) The £2 norms of I.he states Z, 77, ( of the adaptive system (J0.182)(JO.184), (10.209), (10.210), a7"e bou.nded by
where
Ilzll:! ::; Jco';v.(O) Co
(10.242)
IlTilb ::; Jco';V.(O)IIWjjll~
(10.243)
';V;'(O)/lW{lIoo , "(112 < Jco Co
(10.244)
IItv'Filioo an.d 111'1',1100 are independent of co, do, and r.
Proof. As shown in (10.95), the derivative of t'p along the solutions of (10.182)(10.184) is Since lip is nonincreasing, we have (10.246) which implies (10.242). From (10.209) and (10.211), by Theorem B.2(ii), we get (10.247) and, from (10.210) and (lO.212) we get (10.248)
o
10.4
455
TRANSIEN'l' PERFORMANCE WITH TUNING FUNCTIONS
The in.itial value of the Lyapunov function is
V,,(O)
1 = ~lz(0)12 + 4d le{O)I~ + ~IO(O)If,J . .... 0 
(10.249)
From (10.242) and (10.249) it may appear that by increasing Co we reduce the bound on Jlzlb. This would be so only if e:(0), 8(0), and z(O) were iudepcndent of Co. Whi1e e:(0) , 8(0), and 21 (0) = y(O)Yr{O) are clearly indepcndent of Cj, d jJ and r, the initial values =2(0), ... , =p(O) depend on Ci, eii, and r. Fortunately, we can set z(O) to zero by appropriately initializin.g the reference tmjectonJ. Following (10.97) and (10.98), z(O) is set to zero by selecting Yr(O)
=
1/(0)
y~i)(O}
=
u~) [I.'nI.i+l (0) 
(10.250)
a:j (y(O), nCO), 8(O}t £j{O), );m+i(O), y~il)(O»)] i = 1, ... ,p 1.
(10.251)
Sincc bm ;!:. 0, it is reasol1able to choose bm(O) 't= O. Then the cbaice li{O) = l/bm(O) makes (10.251) welldefined. A detailed pl'esentation of the initialization procedure, given in Section 4.3.2 for the Cl1se of st.ate feedback, explains how to modIfy a prespecified reference trajectory, so that the implemented trajectory is properly initialized. Thus, by setting z(O) = 0, we make
V'p(O)
= 4.~ Ig(O)\~ + ~le(O)I~_l and r independent of Co
(10.252)
t This means that t.he a decreasing fuuct.ion of do bounds resulting from (10.242)(10.244) and (10.252) far r yl,
IIzlb ~
Ilijlb :::; 11(112 :::;
1 (1  + 11s:(O)Jp (1 _ ., + J2Co 1 (1 8 + 1 'J) ~
v'2cO 1
l)
::y16(0)/M
1,8(0)1'l
;1 (0)1
<») 4d o 1 ., ) 4d le(O)\'P o 4d le(O)\p o
=
1/2
1/2
1/2
(10.253)
II H'ii 1100
(10.254)
II H', II
(10.255)
OJ ,
can be systematically reduced either by increasing Co or by simultaneously increasing do and 'Y. The possibility to impt'ove performance with the adaptation gain y is particularly clear in the case e(O) = 0, when the £'), bounds of Theorem 10.11 become
11=112
~
~IO(O)I Co1'
\1';112
~
",;
11(112
~
co'Y
J:
Ijj{O)IIIH'iill~
H 6 _Co.,. 1 (O)III ',1I00'
(10.256)
(10.257) (10.258)
456
LINEAR SYSTEMS
These p...xpressiollB provide insight, although, of course, c(O) canllot be set to zero by design because it depends 011 the initial value of the unmeasured state
.1:(0). Another advantage of the derived bounds is that they are computable. The bound for 11=112 is explicit, while the bound for Iliilb involves IltVijlloo which is known from (10.211). Only the factor IIHi,Jloo in the bound for the zero dynamics 11(112 depends on the unknown parameters ho, ... ,bm  1 • \iVhen these parameters belong t.o l,nown intervals, !lH',lloo can be computed using lIG]. For a further characterization of the achieved performance, we proceed to derive £00 norm bounds for the states of the adaptive system (10.182)(10.184), (10.209), (10.210). These bounds are also useful for a comparison with nonadaptive systems. \Ve first give simple bounds 011 Ilzlloo and 1181100:
11=1100 ~
1101100 :5
V2Vp(0) VX(r)V2Vp(0).
(10.25D) (10.260)
Since l~ $ 0, t.he bound (10.259) follows immediately from (10.261) and the bound (10.2GO) is obtained by noting that (10.262)
For
r = ,I, it further follows from (10.260)(10.261) that (10.263)
In this way, eters.
1101100
is e.xpliciUy related to initial conditions and design param
Theorem 10.12 (.coo Performance) The states z, f/, and ( of the adaptive sysf.em (10.182)(JO.184), {1 O. 209), (10.210) arc bounded by 1
_
1:(t)1 <   A i + Iz(O)le co" Jcodo
liHt)1
$
I((t) 1 <
(10.264)
(v'~ M + IZ(O)I) IIwiilh
(10.265)
(v'~do M + IZ(O)I) Ilw, I!. ,
(10.266)
IDA
457
TRANSIENT PERFORMANCE WI'l'H TUNING FUNCTIONS
whe7Y~
M~ ~ {..jX(r)..j21'p(0) [111>.. 111 (..j2v.(0) + IIY,II",,) + K..] + ..j:CP) le(O)lp}. (10.267)
IIw;;lb, 1I'W,IIt, IIhwl/ b and Kw an.! independent of co, do, and r. Proof. Differentiating !1=I2 along the solutions of (10.182) and (10.185), and
w£'
get
=  tCkZ~ tdkZ~ (a~:l)' f>/~';1 (ii"w+o.)

~
k~
Y

~l
U
~ eol=r~ + 4~o (bTw + E2)2 .
(10.268)
By applying Lemma C.50), we obtain
1=(tW~ ~
2
Iz(0)1 e
From (10.VI) and (10.11) we ha.ve
2cot + 4~dn 1I0T w + E211~ .
(10.269)
ftle:l1.. ~ \c:f\ which gives
IIE211!' ~ d(~} Ic(O)\~ .
(10.270)
With (10.269) and (10.270) we obtain
\.:::(1)1 ~
1 ? r:Td v~uo
(\\Ollocllwlloc 1 ) + I\7"D\je(O)\p
VA(P)
+ 1=(O)lecnt .
(10.271)
It was shown in (10.204) that
w = Hw(S)lJ + wo(l) ,
(10.272)
where Iwo(t)1 ~ liweat is the response due to the initial conditions of 11(0) and ;\(0), and n,w and a depend only on the plant and filter parameters and not 011 co, do, and r. Now, using y = Z] + Yr and (10.259), we get
IIwll oo ~ II hwlh (llzllioo + IIYrlloo) + li:wettt ~ IIhw\h ( J211j;(0) + IIYrlloo) + liw • (10.273) Substituting (10.273) into (10.271) and using (10.260) we obtain
1=(t)1 :::;
v'~ +
H..jX(r)..j2v,,(0) [1Ih..lh ( ..j2v.(0) + 1Iy,.1I",,) + "'"J
~IE(O)lp} + /.:::(O)leQ)t
Vt\(P)
=
_l_}.{
Jeodo
+ {z(O)leQ)t
(10.274)
458
LINEAR SYSTEMS
From (10.209) and (10.211), by Theorem B.2(i), we get
lii(tll ~ IIw.lhl\=dl... ~
(v'~do 111 + I:(OJI) IIw;;lh,
(10.275)
and from (10.210) and (10.212) we get (10.276)
o \lVitb the iuitialization =(0) = 0, the expression (10.252) for "p(O) and (10.267) SI10W that lIf is a decreasing function of elo independent of co. With the bounds 011 =,11, and , tbat can be reduced by design parameters, and fixed bounds on £ and 8, the entire state of the adaptive system is guaranteed to have a good £00 performance. Since III in (10.267) depends on IIhw lh, the bounds (10.264)(10.266) require computation of IIhwl/J, I!lVijlh, and /lW,I/I' AIUlOUgh IIhwl/. and /I'w,11I depend all Ullcertain parameters, we can employ the procedure of [16} to compute their 1f.o:J 1101'ms a.nd tben apply the wellknown inequality Ilg/h S (2n + l)IIGllocl1 wbere G(.q) is a stable transfer function, n is its McMillan degree, aud get) is its impulse response (see Theorem B.2(iii».
A special form of the above 1:.00 bounds is more revealing. Corollary 10.13 In the case z(O) = 0, e:(0) tile £00 bounds of Theorem 10.12 become
(lIy.ll
= 7](0) = A(O) = 0, and r = 'YI,
II=/Io:J <
18(0)11Ih...lh
/171/100 S;
18~~~~1I1 (11"."00 + ;'18(0)1) IIwqlh
(10.278)
18~OJ"~1I1 (11".11... + ;'J8(OIi) IIw,lh.  CD 0 'Y
(10.279)
1/'1/00
~
2 Jcodo
oo
+ _1 18(011) .J1
=
=
(10.277)
Tbe assumption =(0) = 0, e:(O) = 7](0) A(O) 0 is satisfied ill the particular case where x(O) = 71{0) = "\(0) 0 and the trajectory initialization is performed. In this case the system is driven only by the reference trajectory. The form of bounds in Corollary 10.13 clarifies tbe dependence of the £00 performance 011 the parameter uncertainty 10(0)1 and the design par81netel's COt do, and 'Y. Any increase in those parameters results in an improvement of the £00 performance. It is of interest to observe that dot present in the £00 bounds (10.277)(10.279), is absent from the £2 bounds (10.256)(10.258).
=
IDA
TRANSIENT PERFOR.MANCE WITH TUNINC FUNCTIONS
459
Remark 10.14 The bound (10.260) can be useful for ensuring stability o[ the system in case of an accidental intelTuption o[ adaptation. By substituting (10.252) into (10.260) we get 11811CXl
~
( 18(0)12 + 2~o Ic(O)I~)
1/'2
(10.280)
Suppose that the design paramet.ers Co and do are chosen so that.
2vcodo> 18(0)IIIHwIl00,
(10.281)
which, by Theorem 10.9, means that without adaptation the result.ing closedloop linear system would be stable. Let. us now suppose that this system is running 'with adaptation until time T, when the adaptation is disconnected. The bound (10.280) indicates that the parameter error 18(T) 1 may be larger thau the initial value 18(0)1. Therefol'e, 18(T)lmay violate the stability condition (10.281), and the resulting lineru' system may be unstable. It may therefore appear that the parametric robustness results of Theorems 10.8 and 10.9 hold only if adaptation is disconnected at T = O. Fortunately, the bound (10.280) shows that adaptation can be disconnect.ed at any time T ~ 0 without destroying the system stability provided t.hat. Co and do arc chosen so t.hat
~ 2V codo >
(
'l
l'
'))1/2
(10.282) 18(0)1 + 2d Ic(O)lp II.FI",IICXl' o Suppose, for simplicity, that t.he filter initial conditions are ''7(0) = A(O) = 0, which, in view of (10.33)1 means that c(O) = x(o). It is reasonable to assume that, even t.1lOugb x is not measured, we know a bound on its initial condition x(O). This bound, along with a bound on the initial parameter error, can be used in (10.282) to select Co and dfl which guarantee stability. In ot.her words, with sufficiently high Co and do, each 'frozen controHer' is stabilizing. 0
10.4.2
Performance improvement due to adaptation
In the literature, robust and adaptive designs coexist as two separate approaches with little quantitative evidence for performance comparison. One would expect adaptive controllers to perform better because they are using additional knowledge about the uncertaint.y acquired online, while the robust controller design is based only on a priori knowledge. However, for t.raditional adaptive controllers this is true only asymptotically because they go through big initial transient swings. With the performance bounds derived in Sections 10.3.2 and 10A.1, we have assembled a data base for a quantitative performance comparison of the nonlinear adaptive system and its linear nonadaptive counterpart.. Before we present a comparison of transient performance, we review the basic differences in boundedness properties and asymptotic performance between the adaptive and the nonadaptive controllers:
460
LINEAR SVSfEMS
• Boundedness of the adaptive system is guaranteed to be global for any No a priori inpositive values of the design parameters Co, do, and formation is required about the parameter uncertainty. In contl'ast 1 the linear controller guarantees boulldedness only if a bound on the panuneter uncertainty is known so that the value of codo can be set large enough to satisfy the stability condition (10.225) .
r.
• AS1Jmptotic tmcking is achieved by the adaptive controller for allY partl.meter unC'..crtainty and any positive Co, do, and r. The tracking elTor of the lineal' system can be reduced but, ill general, does not converge to zero. To make the tracking error small, t.he value of coclo is required to be high. It can be shown that the increase of codo increases tbe bandwidth, which nuty be undesirable.
\X/e show now that. the transient performance. of the adaptive sysf:em can be improved over that of the nonadaptive system witbout an increase of codo. We usc the superscript.s A and N to denote the quantities in the adaptive and in the nonadaptive system, respect.ively. For convenience, we repeat the nonadaptive bound (10.215):
II=NI'~ ~
/jjNIIIJz~lh
2Jc~~ IONHlhwlh
I/Yrl/oc ~ BN.
Under the conditions of Corollary 10.13, because type of bound holds for the adaptive system:
IIzAlioo ~
IjjA{O)I!llzwlll
I/0Alloo
2Jc~~ 19 (O)/lIhwlh
~
/8 A (O)I,
/lUrl/oc.
A
(10.283)
the
SaDlE:'
(10.284)
In additioll, for the adaptive system we have the bound (10.259),
IIzAUoo ~ ~IBA(O)I,
(10.285)
VI'
and the llo11nd (10.277),
:"1 (1Ivr"oo + ..tY IBA(O)I).
I/=Alloo ::; /OA{O)IIl h 2
Jet do
_1
(10.28G)
Thus, the tightest adaptive bound we have is the smallest of the above three bounds, (10.284), (10.285), and (10.286):
IIzA/loo ::;
{'9 (0)lIl h:/ll (Ihlr/loo + ~19A(O)j) , ~\6A(0)/, 2 Jt$do ...;:y J1 A
mill
} t::t. B /9A (0)llIhw // 1 2Jc&d~ IBA(O)IUhwllt IJYrl/ca = A·
(10.287)
IDA:
461
TRANSIENT PERFORMANCE WITH TUNING FUNCTIONS
A good mea.15ure of the performance improvement due to adaptation is the performance rat.io (10.288) between the £00 bounds (10.287) and (10.283). The improvement is a('hieved ir the performance ratio is small: RJ.:'lO :5' Rc,oa < 1. For the sa.me parameter
ullcertainty in the adaptive and the nonadaptive cases, following corollary is es1.ablished by direct calculation. Corollary 10.15 Let the i'flitial conditions of
fiN
=, c, '1, A be
= nA(O) ~ 0, :;C1YJ.
the
Then with
adaptation gain
(10.289)
~ and IOI\\hwllt + 'ly.CiCiA Co do Rc,oc. > 2V Co db > IOlllhwlh, the perfo11Twnce ratio Rc.:.:. is no greate7' than B.C,,,,, < 1. Fl'om this corollary we ean deduce two furt.her advantages of the adaptive controller. • First, the adaptation gain l' provides an addit.ional degree of frel"dom wit.h lvhich the performance can be improved when thc adapt.ivE'" and the nonadaptive gains arc the same, c~d{} := c~d~ ~ codol and sat.is(y the stabilit.y condition 2Jeodo IBII!hwlli > O. In this case the adaptive bound is lower than t.he nonadaptive bound provided t,hat. (10.290) and the bounds are the same when l' ::; tive plot of tbe quantity
BN
,*.
Figure 10J:i shows a qualita1
Qh') =In = I n BA R.c. x
(10.291)
obtained using the bounds (10.283) and (10.287). ''''hile Corollary 10.15 demonst1'8,tes a performance improvement due to adaptat.jon ouly fol' 'Y > ,'''' t.he simulations, some of which are shown in Example 10.16, exhibit a performance improvement for all 'Y > o. • Second, and more important, performance improvement can be achie,red even with c8c1~ sma.ller than c~d~A. In the presence of a, large parameter
462
LINEAR SVS'l'EMS
Q(y) = In ~: = performance improvement
,,"
," ," ,,"
,,'"
__ 
..... ...........
, ,'"" simulations,' , "
,,"
, ,,' "
,/
,,
Corollary 10.15
"
Figure 10.6: Performance improvement due to adaptation.
uncert.ainty 0, the nonadaptive controller must use c~d~A sufficiently large to satisfy 2Jc~d~ 18'llIhw ll l > 0, thus increasing the bandwidth. From Corollary 10.15 it is clear that with the adaptive controller such an undesirable bandwidth increase can be avoided, because when both 8' and c~d~ are large, the condition 2Jc~d~ Rt.r:c > 2J~d~ IOlllhw!1t can be satisfied with c~d~ much smaller than c~d~. This analytically confirms that adaptation is an efficient tool for reducing the effects of large parametric ullcertainty without unacceptable widening of system bandwidth. For small parametric uncertainty, the linear controller is effective.
Example 10.16 The improvement of performance due to adaptation is now briefly illustrated with the example introduced in Section 10.2.4. We consider the unstable relativedegreethree plant
y(s) =?( s 8
1 
a
) u(s) I
a
> 0 unknown.
(10.292)
The control objective is to asymptotically track the output of the reference model 1 (10.293) Yr(s) = (8 + 1)3 1'(8) . The tuning functions design for this problem was showll in detail in Section 10.5. To illustrate the parametric robustness (Theorems 10.8 and 10.9), we switch off the adaptation ('Y = 0) at a constant estimate O. = 1, when
10.4
463
TRANSIENT PERFORMANOE WITH TUNING FUNC'l.'IONS
Tracking error y  Yr 0.1
0.1
o
10
20
30
o
40
10
20
30
40
I.rr_
0
10
20
30
40
Control u 2
2
2iirr'T'_ o 10 20 30 40
2
2ii__r........ . . ...... o )0 20 30 40
211...r, o 10 20 30 40
Parameter estimate a 4 2
o
a fixed 10
20
30
40
o
10
20 'Y
30
40
o
10
20
30
40
= 0.3
Figure 10.7: Adaptation improves the tracking error transients without an increase in control effort. The plant is driven by r{t) = sin 1'1 and the plnnt parameter is a = 3.
the parameter error ii = 2 is significant, With Cl = C2 = Ca = 3 and d1 = d2 = da = 0.1, the resulting detuned linear system is unstable. \Vith an increase to Cl = C:l = Ca = 5, the system is stabilized. However, without adaptation, the tracking error, shown ill Figure 10.7, is about 12% of the reference input, \vhich is not acceptable in most applications. The adaptive controller is sbnulated with tbe same coefficients Cl = C!! = Ca = 5 and d1 = d2 = d3 = 0.1. The effectiveness of the adaptive scheme is demonstrated by the fact that even with slow adaptation h' = 0.3), tbe tracldng error is reduced to zero after a few periods of the reference input, as shown in Figure 10.7. It is remarl{able that even dill'iug the adaptation transients, the tracldng errol' is smaller than in the nonadaptive system, while the control effort is about tbe same. When the adaptation gain is increased to 'Y = 1, the tracking performance is further improved with about the same control effort,
464
LINEAR SYSTEMS
vVhile Coronary 10.15 shows the performance improvemellt only beyond a certain 1', the simulations indicate that the performance improvement is 0 present for any "y ~ o. As a conclusion to tllis section, we point out that the improvement of performance due to adaptation is the first such result in the literature. TheI'e al'e two reasons for this. First, the traditional certainty equivalenc!e adaptive controllel's do not possess the para.metric l'obustness property, so they do not have nonadaptive counterparts which can achieve stability, let alone a given level of performance. Second, even if they stabilize the plaut with some C011stant estimate, the adaptation is likely to make the performance worse during the transient because it is not based on tbe control objective (the identifier is not driven by the tracking error) I and the controller docs not account for the parameter estimation transients,
10.5
Comparison with a Traditional Sclleme
The tuning functions scheme is now compared using simulations with a standard certaintyequivalence scheme on the basis of tral1sient performance and control effort, The comparison is made for the relativedegreethrec unstable system from Section 10,2.4.
10.5.1
Choice of a traditional scheme
The comparison with a direct MRAC scheme is llOt pursued because such a. scheme updates at least tll,ree parameters. This is clear frOll1 its control la,v
(10.294) where .s2 + 'm18 + n12 is a Hurwitz polynomial. A calculation using the Bezollt identity gives 8
5
+ s4[ml 
83

a] + s3[m:!  8.1 
a(ml  83 )]  s2[Oo + (n1.2  8.. )a] = (8 + 1)3(82 + mls + m'2) + IJJs + 62 I (10.295)
which shows that 60 ,O;:s, and 84 have to be updated, while 81 and 82 can be 711.)  3m'!h 82 = m2' Simulations showed that the upd~tte of three parameters results in transient performance infel'ior to indirect linear schemes which update only one parameter estimate. Therefore t we compare our new controller to a standard indirect scheme [43 t 129}, ill which the plant equation 8 2 (8  a)y(s) = 1/.(s) is filtered by a Hurwit.z
fixed at 61
=
10.5
465
COMPARISON WITH A TRADITIONAL SCHEME
observer polynomial s!a
+ kl s2 + k',!s + ka to obtain the estimation equation
4J
=
1jJa.
tb
=
1/J
=
+ k 1 s 2 + k'ls + ka y{.s) .., s:::y( s) S!i + /0:18 2 + k:2 8 + ~~:i '
1
S3
S:i
S3
+ kJs2 + k2 s + ka u(,s)
(10.296)
and the parameter update law is a normalized gradient:!: •
1/Je
il = 'Y :Jil, 1 + .jJM
e
= l/J  1/Jo, •
(10.297)
The control law (10.294) is implemented by replacing a with a. in (10.295) and then solving it for the controller parameters: 83 = (3+&), (J.J = (3+3'111.] + a(ml 8a )], 00 = [1+ 3m l +3m2+(m.204)aJ, 01 = ml3m'2, ()2 = 7J}·2. The indirect. adaptive linear scheme and the tuning functions scheme were applied to the plant (10.158) with the trne parameter a = 3. In aU tests the initial parameter estimate was a(O) = 0, so that, with the adaptation switched oH, both closedloop systems were unstable. The reference input was r(t) = sin t. For a fair comparison, our first taslc was to adjust the design parameters of the indirect scheme to achieve the best transient performance wit.h a prescribed control effort. This was done in detail in [95, Section VII]. The tradeoff between transient performance and control effort was examined for various initial conditions. To reduce the transients due to the mismatch of initial conditions, the initial condition of the reference model output was set in all tests to be equal to the initial value of the plant output. The available design constants were the adaptation gain 'Y and the coefficients of the observer polynomial ,r;3 + k1s 2 + l:28 + k3 and of the controller polynomial s2+mJs+m'2' All the roots of the observer polynomial were placed at s = 2 with kl = 6, k2 = 12, k3 = 8, while the roots of the controller polynomial were placed in a Butterworth configuration of radius 3 with ml = 4.2426, m2 = 9. These were judged to yield the best tradeoff between transient performance and control effort for different initial conditions. On the basis of the simulation results shown in [95, Figures 2,,:1] the best compromise between transient performance and control effort was judged to be for the value of the adaptation gain '1 = 1000.
10.5.2
Comparison of the schemes
For a comparison of transient. performaJlce, the tuning functions scheme was adjusted to employ about the same control effort as the indirect linear scheme. 3Thc simulation results \Vith a leastMsquares update law were virtually identical and are therefore omitted.
466
LINEAR SYSTEMS
Tracking error y  Yr
o
4
2
10
B
6
Parameter estimate
o
a.
2
4
6
Control u 20
o
~Ir~~~~
2
4
Indirect linear
6
o
2
..
6
Tuning functions
Figure 10.8: Comparison for 11(0) = o. The tuning functions controller improves performance with about the some control effort by incorporating the update law
a.
10.5
467
COMPARISON WITH A TRADITIONAL SCHEME
Tracking error y  Yr
ludirect linear 10rr.rr,
o
6
4
10
8
P81'amcter estimate 0
2
s 3
4
0
Control
2.
3
4
'U
200
lao
100
200 0
Indirect linear
200 2.
3
4
0
2
3
I
Tuning fUllctions
Figure 10.9: Comparison for lI{O) = 1. The parameter estimate ill the tuning functions scheme lS smootber because its update law is driven by the state of the error system z(t).
468
LINEAR SYSTEMS
Tracking error 11  Yr I.S
1.5
1.5
0.5
0.5
0.5
0.5 ;r""..,..., o 2 3 4
0
2
3
i).5 ;rr~.,
i)j +......,.
0 1 2 4 Parameter estimate ci
4
0
2
Control
1000,
4
3
0
2
3
4
0
2
3
4
0
2
3
4
1J.
0 1
1000· 0
d 1 = d2
2
3
4
0
= da = 0.001
2
3
4
d1 = d2 = d:t = 0.2
d. = d"].
= da = o.a
Figure 10.10: Nonlinear damping fol' 1J{O) = 1. The effect of state estimation error is attenuated.
=
=
=
Tbis was achieved with kl 6, "'2 = 12, h:3 = 8, Cl C2 = Ca 1, d. = d2 = da = 0.1, and the adaptation gain "'( = 0.5. The plots ill Figures 10.8 and 10.9 show that. the transient performance of t.he tuning functions scheme was far superior for both sets of illH:ial conditions. .Measured by any norm, the tracking error with the tuning fUllctions scheme is only a fraction of the indirect linear scheme error. The simu}i;ttiollS presented here confirm the strong transient performance properties of the tuning fUllctions design derived ill Section 10.4.2. As we explained, the distinctive feature of the tuning functions design is tbat the COlltroller incorporates the parameter update law = 'YT:'1t with which it accounts for parameter estimation transients. The effect of this additional information about ii. is tbat the settling time of the tracldng error is much shorter for the tuning functions scheme. Figures 10.8 and 10.9 show that the settling time of the tracking error is closely coupled to that of the parameter error. III contrast, tIle tracldllg elTor of t.he indirect linC!ar scheme continues to grow even aft.er the parameter estimate has converged to its true value.
a
10.5
469
COMPARlSON WITH A TRADITIONAL SCHEME
Tracking errol' 11  Yr 1
1
1
I
0
1
:~ soo[ 0
1
2
3 4 0 Paramet.er estimate iJ.
2
3
1
0
3
4
2
3
4
2
3
4
Control u.
1500 0
4
2
2
3
n011initialized
.It:
4
0
initialized
Figure 10.11: Reference model initialization fol' 1/(0) = 1. The large initial value of control is eliminnted, and the parameter transient is made wrnosL monotonic.
Two other most important factors which contributed to the superior performance of t,he tuning functio11s scheme are nonlinel:lI damping and referelu'e model initialization.
):!
Nonlinear damping. The nonlinear damping terms di (ad;1 Zj contributed to a significant reduction of the effect. of initial conditions on the new adaptive system. Its attenuating effect is displayed ill Figure 10.10. If the damping is increased over an optimum rate, the tracking elTOI' continues to decrease, but the control effort increases. Re.ference model initialization. In contrast to the indirect scheme, the new tuning functions scheme provides clear guidelines for reference model initialization, which follow from the design objective of driving the zval'iables to zero. According to (10.167), (10.170), and (10.173), the initial values of :;variables are set to zero by choosing 1'1 (0) = y(O), "2(0) = 112(0)  (tl (0), and T3(0) = va(O)  0;2(0). In general, it is always possible to set z(O) = 0 by e."\:pressions (10.250)(10.251). In all tests, the reference model initialization was
470
LINEAR SYSTEMS
found to significantly improve both the transient performance and the control effort. A typical e.."Cample is Figure 10.11.
10.6
Modular Designs
The tuning functions controller and update law are interlaced in an intricate fashion, which makes t11e design complex. In Chapter 9 we introduced modular outputIeedbaclL schemes with independently designed controllers and identifiers. Vle pursue here the same idea and design modular backstepping schemes for linear systems. However, for lineal' systems we do not use the strong ISScontrollers because their underlying nonadaptive controllers are nonlinear even for linear systems. Instead, we employ an SGcontrollc7' which is a certainty equivaJence version of tIle llOllo.ch1.ptive linear controller in Section 10.3.1. This controller is different from traditional certainty equivalence controllers because its backstcpping st,ructure endows it with design coefficient.s which are useful in shaping the transient behavior. Because of their certainty equivalence nature, the modular schemes will serve nicely for a qualita.tive comp81ison between the traditional certainty equivalence sehemt's and the tuning functions sehelDt'. In addition to Assumptions 10.110..4, we make the following assumption about a lower bound on the highIff'C!llency gain, standard in 'indirect' adaptive control.
Asswnption 10.17 In addition to sgn{bmL a positive conodant fmch that Ibm I ~ C;m'
c;'m
is known
Assumptioll 10.17 stl'engthens Assumption 10.2 ill the tuning functions design. It allows the control law to contain a division by the estimate bill! wbich is kept away from zero by paraJneter projection. In the tuning functions designs it was possible to avoid this assumpt.ion by intl'oducing an additional estimate of !l == 1/b,rt. This section is organized as follows. In Section 10.6.1 we present the SGcontroller design using the knmvledge of the tuning funct.ions controller from Section 10.2.1. Then in Sections 10.6.2 and 10.6.3 we present two identifiers and stability analyses for corresponding closedloop adaptive systems. Finally ill Section 10.6.4 we compare the modular designs wit.h the tuning functions design.
10.6.1
SGcontroller
The SGcontroller, given in Table 10.3, is a modification of the tuning functions controller ill Table 10.2. 'Ve briefly discuss the modifications leading to the controller in Table 10.3.
10.6
471
MODULAR DESIGNS
Table 10.3: BGController for Lloear Systems (10.298)
i == 2, ... ,p
= a] = =
0:1
a'2
a, = rll fJ.
(10.299)
1
(lO.aOO)
:0:1
b",
 (Cl
+ d1) Zl 
" b,n=l Zil 
_TA
[ (n c2
(10.301)
{2  W (J
+ d2
aat 811
[ + d; (8a;1 a:yr] Ci
Z2
Zj
+ /32
(10.302)
+ Pi ,
i = 3, ' .. P (10,303) I
8aj_l ( TO) + a (A0'11 + cnY) + ~ ~ 8ail (j) = ae2 + W r=T)Yr Y 1] ;=18yrJ [JOil
(10.304)
Adaptive control law: (10.305)
• The adaptation gains r and 'Y are set to ~eJ:ob ,~bjch ~leans that we eliminate ~r'T2 from (lO.IOl), ~il Ej;1 '6;' r~;' Zj Il'om (10.102),
rT. 
aud (y~il)
+ 8';:i 1 )
~ from (10.I03) .
• With Assumption 10.17, 0 is replaced by crossing zero by parameter project.ion.
l/bm, and bm{t)
is l(ept from
Straigbtforward but lengtllY calculations show that the resulting error system is
(10.306)
472
LINEAR SYSTEMS
where
bm
cld1
A;
0
bm
c.., d.. (808u )?
0
1
=
!!!ll 
~
0
1 0
(10.307)
1
0
0
HIe
=
[
~ 1
1
Cpdp(88"u1 )2
E 1R.P
_1:_.
(10.308)
l!1I
t
+ (Yr + itt) e'f _~lwT +ZteT
(ijT
IV,...T 0
_fJs!:.1wT
=
E RPxp
8U
110.,._1
(10.309)
T
lfiIW
Q'l' =
~
[
+ ""21 iJr eT1
8IJ"1II
_~ + ~'fPl)eT 0
80
b~·r
1
RPxp
(10.310)
E
1
By examining the expressions for ~ is a function of only 0,
Cit
ll:i in Table 10.3, it is not hard to see that and di, which means that A; and loVe depend all
0, but not on other states of the closedloop system.
Thus, in contrast to the tuning functions design and all the other outputfeedback designs so far (cr. Chapters 8 and 9), the matrix A~ and the vector I,VE are functions of fJ only. This is due not only to the SGcontroller design, but also to the linearity of the plant. In view of (10.98) and (10.99) we have
(10.311)
10.6
473
MODULAR DESIGNS
On t.he other haud,  bmz 1 
Bal a Y
TW (J
+ zleTt /J =

bmzl

Bat Ta W (}.
(10.312)
Y
With (10.311) and (10.312), the errol' syst.em (10.306)(10.310) is rewl'iUcn as (10.313)
where Cl 
bm AA8) =
0
d1 c,:! 
bm
0
d 2 (~)2 lly
1
0
1
0 1
o
0
1 c  d P
P
(OQP_I)2 Oy
(10.314) The error system (10.313) is similar in form to the error syst.em (9.84) in the modular nonlinear design. However, because of the absence of the Iii and ,qi nonlin~ar damping terms, (10.313) is not. inpnt.tostate stable with respect to
8 and 8. .
For a brief comment on the underlying nonadaptive SGcontroller, we let fJ == 0, so that the error system (10.313) becomes
iJ = ('onst.
(10.315)
Let, for simplicit.y, bm = bm = 1. The detuned error system (10.188) in the tuning functions design and the detuned error system (10.315) in the modular design are identical. Therefore, the parametric robustness and nonadaptive performance properties established in Theorems 10.810.10 for the tuning functions design hold also for the modular designs. These properties distinguish our modular designs from the traditional estimationbased certainty equivalence designs. The backstepping approach results in a nonadaptive controller which has parametric robustness and performance properties, and it can be made adaptive in two different waysusing either a Lyapunov methodology (tuning functions) or a modular inputoutput methodology with identifiers designed separately from the SGcontroller.
10.6.2
yPassive scheme
Consider the parametric ymodel (10.37)
iJ =
€2
+ wT /J + c::! ,
(10.316)
where {2 and ware available and defined in (10.29) and (10.39), respectively. \Ve introduce the simple scalar observer (10.317) where error
KO
is a positive constant, and
as
defined in (10.213). The observer
f=y:Q
(10.318)
Co
is
is governed by the system
.=
f

(Co + li'llIWI'1) f + W T8 + e2 .
(10.319)
The parameter update law is chosen to be
iJ = Proj {fWf} , bm
bm(O) sgnbm >
r=rT>o
'III
(10.320)
where the projection operator is employed to guarantee that Ibm(t)1 ~ <m > 0, \It 2: 0, (see Appendix E).
Lemma 10.18 Let the m.aximal inie71JaI of existence oj sol'utions oj (10.316), (10.317), and (10.320) be [0, tf). Then the following identifier properties hold:
(0)
Ibm (t)l2: (;m > 0, \It E [0, If)
(i) (ii)
9 E Loc[O, tf) f E '£:2[0, tf) n .coc[o, tf)
(10.321) (10.322) (10.323)
(iii)
wf,8 E £'2[0, tf) .
(10.324)
TIle above identifier is the same as t.he identifier used in Section 9.2. Therefore, the proof of Lemma 10.18 is identical to that of Lemma 9.3. Hmvever, the stability proof for the resulting error system is different from that of Theorem 9.4 because the SGcontroller makes the proof of the following theorem considerably more involved.
Theorem 10.19 (yPassive) All the signals in the closedloop adaptive .'lystern consisting of the plant (10.1), the control law in Table 10.3, the filters in Table 10.1, the observer (10.317), and the update law (10.320) are globally uniformly bounded, and asymptotic tmcking is achieved:
t1!n! [lI(t) 
t/r(t)] = O.
(10.325)
Proof. The projection operator in Appendix E is locally Lipschitz, as stated in Lemma E.!. Therefore, as argued in the proof of Tlleorem 10.6, the solution of
10.6
475
MODULAR DESIGNS
the dosedloop adaptive system exists and is unique on its ma.."\imum interval
of existence [0, t, ). Let us consider the systems (10.313) and (10.319), (10.326) (10.327) and define the signal
.
~
( = :;  IV£(8)f.
(10.328)
Equations (10.326), (10.327), and (10.328) yield the system
(. =
[. ] • 81l' (9) :. A::(8)( + A::(O) + col H'e;(O)e 8'0 Oe
+Q(z, t)'rO + 1l0 HT£(O)WT Wf • From Lemma 10.18,
9 E .coo [0, t,).
(10.329)
Since ~ are smooth functions of iJ when
ever bm =F 0, and Lemma 10,18 guarantees t.I~at brn(t) 1= 0, then H'~, A: and
flJ/t
are bounded. Since, by Lelnma 10.18, we conclude that
f,O
E £2[0,t,) alld e E .coo[O,tJ),
• ] 8HT£(ti):. ~ 80 Oe = Ll E 4(0, t,). [A:(8) + col H'£(O)e 
(10.330)
Tlms, system (10.329) bccomes
( = A:(O)( + Ll + Q(z, t)T ~ +li.oHl~(O)WT ~ . E£2
f
(10.331)
E£:',!
Since A:(8) is e~llollential1y stable unifomlly in 0, we view (10.331) as a perturbed lineax timevarying system which cannot be destabilized by ~he squareintegrable disturbance L 1 • So we focus our attention to Q(:;, t)'J'iJ + IiO HIE (O)w T we.
Claim 10.20 There e:r.istfYJ,nctionsp~ E £2[0,tl) andqt E L2[0,tf}, i,j = 1, ... , p, k = 0, ... , '11. + m, such that Q(z, t)T8 + li oHTE (8)w'l'we = B(t, s)[z]
+ G(t, s) [U!Pl)] ,
(10.332)
where the linea,' operators Hand G are n+nt
B(t, s) = {hij(t, s)} pxp ,
hij(t, s) =
{Oij(t,S)}pxp'
gij(t, s)
=
SN
(10.333)
r.; q~(t) 1\(S~B(8)
(10.334)
n+m
G(t, s) =
..
~ pf(t) ]\'(s)B(s) k
476
LINEAR SYSTEMS
The proof of t.hiR claim is leugt.hy but straightforward. It relies 011 tbe fact that: the O'i '5 are linear ill'll, 71, ,,\ and ti~pl) and nonliuear only in 8. SubsLitut.ing (10.328) into (10.332) fmd then into (10.331), one obtains
<= A;: (ti)( + LI + B(t, 8)[(1 + JI(t, s) [tVe (6)f] + G(t, s) [y~Pl)].
(10.335)
SincE" HrE(iJ)f and y~pl) ar~ bounded and tlle coeffidents in fJ(1, 8) and G(t, s) are squareintegrable! then
(10.336) Heu<.'~,
am' problem is reduced to t.hE" study of the syst.em
(10.337)
Let. HE; denote by 11 a generic fUlU.'tion iu £1 [OJJ ). (Note that t:his notation allows us to say kil ::; 11 for any fiujt.p k) III view of (IO.333), we have ~
V,I
alii, + I(r~ (10.338)
fol' some fJ (10.31(1) is
> O. The derivative of K/2/2 along the solutiolls of (10.337) with
; (~I(I~) ~ col('!! + (TH(i, 8)[(1 + (T L2 ~
_ CO/(/2 2
+ ~/}I(l,8)[(]12 + .!..IL2 12 • Co
Co
(10.339)
Combining (10.330) with (10.338), we get
:, (1(12) ~
(Co
/1)I<:f + 11 l'lt + /.1
1~1 ~ 6Vh + '(I:! .
(10.340) (10.341)
Tbese two differential inequalities are illtercoDnccted and form a loop "ith small gl:Ull because lli, appears IIlultilJlied by 11 ill (IO.340). To finish the ploof we define t.he "superstfLte" v ~ ~." 
differentiate it
SlId
\7.I,
?
 1'12 , + ., Co
(10.342)
substitute (10.340)·(10.341),
(10.343)
10.6
477
MODULAR DESIGNS
so we get
By applying Lemma B.B we conclude that ),; is bounded and integrRble 011 [0, t J }. The houndcdness of J: implies the boundedlless of (, which, ttl turll, implies that z is bounded. It remains to prove that the boundf>dncss of:, ,j and y~p) implies that ~', 1} and . \. are also bonndf>d. A proof of tllis implication \Va!; already given for the tuning functions design in Theorem 10.6 (cf. (10.124)(10.126)). The same argument is applieabJo ho1'O wit'.h (10.126) I"oplnccd by
Vm,i
=
_ 1 (iI) (  .
brn
,
i
= 2, .... p.
(10.345)
All t.he bounds are independent of t f. By the same argument as in the proof of Theorem 10.6 we coneillde that tf 00. To prove the tra.cking. we first note t.hat the integrability of X implies that. , is squareintegrable. Thus z E £2. The bOlludedness of all the signals and (10.313) prove that. ::. is also bounded. By Barbalat's lel1lma (Corollary A.7) .:::(t) + 0 as l + 00, aud ther~fore asymptotic tracldng is achieved. 0
=
The simple updnt.e law (10.320) is b~L'icd on pnssivify of the (system (10.319). The term tiolwl2 in (10.317) serv,cs as II f01'111 of :normalization I which slows down the a.daptat.ion and achieves iJ E c.'),. The proof or Theorem 10.ID reveals t.hat tIlls property is crHcial for stability. It is easy to sce t.hat TheOl·cml0.19 holds even when d] = ... = rip = O. We have chosen to propose the design with cia > 0 because it can also guarantee sta.bility withont. adaptation.
10.6.3
xSwapping scheme
Of thc two swapping ident.ifiers presented in Chapter 9, we choosc the :1'swapping ident.ifiel' beea.use it. uses only t.he KfilteJ.·s wir.h no additional swappillg Iill:ors. COllsider the parametriC' .1:model obt.ained by substituting (10.12) into (HI.13): (10.3~lG)
In
HlP
first. row of (10.346),
(10.347)
all the signals are measul'ed c.xcept. for the bounded, c."{pollPutiaily deCDying e 1. The
(~predict.ioll
error" is hn plcment.ed as
(10.318)
478
LINEAR SYSTEMS
and its relationship with the parameter error is linear: (10.349)
The update law for
8 is either the gradient: bm (0) sgn bm > <;m r=rT>o, 11>0
(10.350)
or the leastsquares:
(10.351)
f(O) = r(O)T > 0,
LJ
> 0,
where the projection operator is employed to guarantee that Ibm(t)1 ;::: C;m
>
0, Vt ~ O.
The proof of t.he following lemma is t,he same as that of Lemma 9.8.
Lemma 10.21 Let the ma:cima.l interval oj existence of solutions of (10.2) and (10.26)(10.27) with either (10.350) OT (10.351) be [O,if)' Then the Jollowing identifier properties hold:
'm
(D) (i)
Ibm(t)1 ~ > 0, jj E £00[0, if)
('ii)
J1 + 1110 1., E£2[O,tf)n£00[O,tf)
(10.354)
oE £2[0, if) n £00[0, tf).
(10,355)
vt E [O,tf)
f
(10.352) (10.353)
1
(iii)
In contrast t,o the ypassive identifier, the, normalization is employed here ill the update law. The normalization malees 8 not only squareintegrable, but also bounded. It slows down the adaptation sufficiently so that the following stability result holds.
Theorem 10.22 (xSwapping) All the signals in the dosedloop adapti?le system consisting of the plant (10.1), the cont7'Dlla1u in Table 10.3, and the filters in Table 10.1, with either tile gradient (10.350) Dr the leastsq1J.o,res 1Jpdate law (10.(51), are globally u1J.ifonnly bounded, and asymptotic lra,eking is achieved:
lim [y(t)  Yr(f)] = O.
Loo
(10.356)
10.6
479
MODULAR DES1GNS
Proof. The projection operator in Appendix E is locally Lipschitz, so the solution of the closedloop adaptive system exists and is unique 011 its maximum interval of existence [0, t I)' In the proof of Theorem 9.9, we showed that
o} == /;;}fh +W
(10.357)
I
which, along with (10.349) and (10.14), means that T
i. = kJf: + W
Let us
1l0W
(J
+ e2 I"lel 
T:'
(10.358)
OJ ().
define the signal
, ~ z  HleUi)f: .
(10.359)
Combining (10.359) with (10.313) and (10.358), we get
, =
A:;(8)( + ~ + [Q(z,t)T + HlE
(9)nr].l. E~
A =LIE'c:I
+ {[A:(8} + col] Ttll£(D)  8T1'".(8) •
EJ(J
e£CIC
o}.;·+ "10 ~
1
L
E£.:a
2 11
';1+ IIl nd' ,
J
... (10.360) where the £2 and £00 signal properties are established using Lenuna 10.2l.
Claim 10.23 There exist junctions p~ E £2[0, t/) and q~ E £2[0, tJ), 'i, j :: 1, ... , p, k 0, ... , n + m, 81tch that
=
(10.361) Inhere the linear operators Hand G are n+m
H(t,s)
{hij(t, 8)} pxp
,
G(t, s) = {gij(t, s)}pxp
,
::
"
sk
= k=O L p~(t) }.( )B( ) \8 S n+m .. it, 9i;(t, s) = E q~(t) J( )B( ) . k==O ('s S hij(t,s)
(10.362) (10.363)
The proof of this claim is omitted. With (10.361) and (10.359), syst.em (10.360) becomes ,
=
A=(8)( + Ll + B(t, s)[(J + R(t, s) [H/£(B)f]
+L2V1 + v\n1\2.
+ G(t, 8) [y~pl)] (10.364)
480
LINEAR SYSTEMS
Since ii~pl) is bounded and the coefficients in G(t: s} are squareintegrable, theu Ll
( =
+ G(I,s) [y~Pl)] ~ L:t E £2[0,I'f)'
Thus (10.364) becomes
A~(O)(+L:~+H(t,s)[(l+H(l,s) [We(O).j1 +•vlfl •.jl + V1fl IP] 1\
+L'l.Jl + v/!1d 2 • ~
\Ve not.e that lVe(8}
{l
(10.365) E
E
.coo (0 tf), which implies that, there exists 1
1 + v(01/2 a function BI E .coo(O,t,) and functions r~ E 0, ... ,71
+ m, such that
H(tt s) [W.(Ii)
.c [0,tJ), 2
i,j
= 1, ... ,p, ~: =
J1 +
l
/2
2
(lO.36G)
where the linear operator F is
F(t,s) = {li(t,sH pXl
n+m
,
1;(1, s)
.
s"
= k=O E 7'k(t) F(\ 8 )B(8 ) .
(10.367)
vVith (10.366), we write (10.365) in tbe form
t = A~(O}( + La + R(t, 8)[(} + F(t, s) [EI VI + 1)/f2 1 2] . 1
(10.368)
By combining (10.357) with (10.204), we get 01
= ~k H,As)[y] 8+ "J 1
+ lId
=
 k  H",(8)[Z)
=
1 ,H,As)[() 
s+
"1
8+":1
E
+ Yr]
= ~H",(s) [CI .j1 +':;.jl + vlfll12 + y,], s + ,,:] vl!1 I
(10.369)
I
where
have
J l+EIVn ll2
is squareintegrable. Since
.1kI H",(8) is stable and proper, we
...
10.6
481
MODULAR DESIGNS
where 6n and kn are positive constants. Let k denot.e a gen(lric posit.ive constant, and 12t denote a generic function in £.1 [0, tJ). Then (10.370) is rewritten 8S
'1
;/~ (10 1 12) ~ 
(On I.) 10 1 \2
+ ko l(12 + ~'.
,1" <
+ I~f
(10.371)
In view of (10.362), we have 611,
11/(t, .5)[(11 2 :5 11 '11
(10.372)
+ '11(1 2 ,
and 111 view of (10.3G7), we have
li'f $; ol'i
IF(t, s)
[BI J1 + 1.1101 r:!J 12
+ Br (1 + vln l 12) (10.373)
:5 ' 1 FJ + ltB; (1 + 1}IOd!!)
.
The derivative of 1(12/2 along the solutions of (10.368) and (10.3I~1) is
! (~I
:5
col
S;

~K12 + ~ IH(I,s)[(W + ~ IF(I.,s) [BI V1+ v1od'll'
1 I". +IL3
(10.374)
Co
Thus (l0.37I)(10,374) define a system of different.ial inequalities
:~ (1(12)
<
 (r; _II) I(F~ + ' 10 r! +
! (lnI12)
$
 (60  ' 1 ) lOll:!
1
1
'I \11
+ ~"n\(12 + k
'/"
:5 6'~, + 1(1
"f
:5 6\1"J + kln 1 +~'.
+ 111'J + II (10.375)
2
r!
These four ineqnalities are intel'conne
we get. (10.377)
482
LINEAR SYSTEMS
where
6X = mill {
~, 6;, ~~, 6} > 0.
(10.378)
By applying the Lemma B.6 we conclude that .X. is bounded on [0, t J). The bonndedness of X implies t.he boundedness of ( and 0 1, which, in turn, implies that. f and z are bounded. The boundedness of x, TJ and .,\ is argued as ill Theorem 10.19. All the bounds are independent of tJ, so tJ = 00. To prove the tracking, we first not.e from (10.360) that, due to the boundedness of all the signals, all the inputs to the (system are squareintegrable. Then, with Lemma B.5 we show that ( itself is squareintegrable. Due to t.he boundedness of Oil f E L.'.).! which establishes that z E £2' The bounded ness of all the signals and (10.313) prove t.hat ~ is also bounded, so by Barbalat's lemma (Corollary A.7) z(t) + 0 as t ~ 00. Hence, asymptot.ic tracking is 0 achieved. The st.ability proof for the xswapping scheme is more involved than for the ypassive scheme. However, the xswapping scheme is of lower dynamic order because it uses only the Kfilters in the identifier, as opposed to an additional observer in the yobserver scheme. The dynamic order of the xswapping scheme is the same as in the tuning functions scheme. Like Theorem 10.19, Theorem 10.22 also holds even when d] = ... = dp = 0, but we have presented the design with do > 0 because it can also guarantee stability without adaptation. Our presentation in this subsection was with normalized update laws. It is also possible, however, to prove the result of Theorem 10.22 with the unnormalized leastsqua7"es update law. T!Je proof exploits the properties of the unnormalized leastsquares algorithm: iJ E £1, f E £2 (see, e.g., [157]).
10.6.4
Comparison: modular vs. tuning functions design
Even though the tuning functions design and the modular design both nse the backstepping approach with the same underlying nonadaptive controller, they result in fundamentally different adaptive schemes. The tuning functions scheme is designed using a single Lyapunov function, and results in a simple and direct stability analysis. The stability analysis of modular schemes is far more involved. Even though one can derive transient performance bounds for the modular schemes, they are neither as simple nor as insightful as the tuning functions performance bounds. 'Ve now illustrate the difference in performance behavior between the tuning functions and modular schemes on the example introduced in Section 10.2.4. Of the two modular schemes, we present simulations only for the xswapping scheme. The responses with the yp8SBive scheme are qualitatively
10.6
=1
483
MODULAR DESIGNS
O'sD:
ZIO.s~
O.S
0
'lJ,
~J=
2
3
4
5
i
I
i
i
2
3
4
5
t::: 1
0
2
,
i
i
4
S
tuning functions (,.
5
3
4
i
i
3
4
S
~jt i
0
ii
3
2
0
u
0
a
0.5
2
b 0
= 0.5)
1
2
3
xswapping (g
I
i
4
5
= 500)
Figure 10.12: Comparison of responses with the tuning functions and the xswa.pping schemes. While the tuning functions update la\v (driven by the tracking error Zt) monotonically reduces the parameter error at the xswapping update law generates an 'overshoot' in iit whicb results in a. considerably higher peak in the response of the control u. similar. The SOcontroller used in the modular schemes is obtained by' setting "'( = 0 in the tuning functions controller (10.167)(10.176). The xswapping scheme employs tbe update law !
'::::1 11 f
a = gI+===2 , .... ].1
£
= y
~1
+ a':: 1•1 
'Vt •
(10.379)
For both schemes we use the design parameter values Cl = C2 = ('3 = 1, d 1 = d2 = da = 1, h:1 = 6, k'J. = 12t and k3 = 8. A comparison of responses with the tuning functions and the :z:swapping schemes is given in Figure 10.12. While the tuning functions update la\v (driven by the tracking error %1) reduces tbe parameter error a almost monotonically, the xswappillg update law generates an 'overshoot I in iit whicb results in a considerably higher peak in the control u. It is important to stress that the responses given in Figure 10.12 are not the best possible with eitber of the schemes. They are only chosen to illustrate a fundamental difference in transient properties between the tuning functions and modular scbemes. Even though the modular schemes may not have performance properties as strong as the tuning functions scheme, tbey are simpler to design and offer flexibility ill the selection of an update law.
LINEAR
SYSTElvlS
\Vhile tilE' rnodular schemes presentecl in this cbapter have a lot in common with the certaiut;y equivalence adaptive linear schelnes, the.y renlOve soniE' of tile shortcOlnings of the cf'rtainty equivalence sciIelnes. As shown in Section 10.3.21 t.heir underlying nonadapt.ive controller can achieve s1:(lbilir,y even without adaptation, and t.heir dcsign paramci.ers can bc uscd for systematic improvelllcnt. of llonadapt:ive performance.
10.7
S UITll1.1.ary
\Ve have presented two classes of adaptive designs for lincar systellls: t.uning functions and modular. These clesigns have the same underlying nonadaptive linear controller based on backstepping. This nonadaptivc con1Toller ca.n guarantee st.abilit.y without adaptation when a b01lnd on the pararnetric uncert.ainty is known 1 and in aclcli hon l achieve a prescribed level of tracking pcrfonllililce. This is an improvement over tradit.ionnl adaptive designs which cannol: gllarant.ee stability w}1<:'n the adapt.ation is disconnected. The tuning fundions design removes several other obstacles from adaptive liucar conlTol. Since the design is based 011 a single Lyapunov function inc01'poratillg both the state of the crror system anel the update law, tllc proof of global uniform stability is direct and silllple. 1'\"loreover, all t.he elTor st.at:es except; for the paranwtcr error converge to zero. This is the strongest convergence property without persistency of excit:ation. The main advantage of t.llC tuning funci.ions design over t1'adit.ional cc1'taint.y equivalence adaptive designs is in transient performance. The nonlinear control law which incorporates the paramcter update law keeps the pararncter est.imation transient.s from callsing bad tracking transienl:s. The performallce bouuds obt.ained for the tuning fUllctions scheme arc computable ancl call b<:, used for systematic iInprovcmclll, of trallsicnt performance. The modular schemes arc simpler to design than the I"'tlning ftlllCl",ions scheme, but do not. possess as strong performance properties (altbough their desigl1 paramet.ers Ci and di Ciln be llser! to influence the performance). Their underlying nonadaptive controller offers an advantage over traditional certainty cquivalence schemes in guaranteeing st,ability without, adaptation. The dynamic order of the adaptive schemes in this chapter ean be as low as 3n + 171 1 which is no higher (and in most eases is significantl:')' lower) than with any other scheme tilat, achieves tracking.
Notes and References Aclaptive control of linear systems is thoroughly covered in the books of Goodwin and Sin [4.:J], Astrom and "YVittenrnark 1"5], l\arendra and Annilswamy [142], Sastry and Boelson [lG5L and Ioannoll and SUll [51] and in the research 111ono
N 01'£8
AND REFERENCES
485
graphs by Egal'clt [31L Landau [109], Ioannou and Kokotovi(~ [50], and Anderson ct al. [2]. Transient performance of adaptive s~Jstems has recently reccinx] a great deal of attention in the works of IOt:lI1110U and Tao [52L Zang and Dii:lUl'ad [202]. Krause, Khargonckar, and St.ein [90J. Ydstic [200L l'\aik, Kt.ll11
stic [VWL Ortega
traditional 8daptive schemes. Inil'.ial progress on developing a robust adaptive design with tuning functions has been reported by Li, \Von, and Soh [l11. 11 2]. \Ven [196] solved the decentralized adapt.ive regulation problem b~r emplo~'ing the tuning functions design. Sections 10.110.5 are based on Krstic, KanellakojJ01Ilos\ aud 1\01\:0tovic (05, 96] and Krstic, Kokotovic, and I\:al1011akopoulos [102]. The results of Section 10.6 have not been previously published.
Appendix A Lyapunov Stability and Convergence In addition to the main Lyapunov stability notions and theorem reviewed in Section 2.1, we noW' briefly review the main invariance theorems. For CODlpleteness ~tlld CODvollienee, we l'Cpeat some of t.ho definitions and theorems already presented in Section 2.1. COllsider the llonautonomons system
x = f(x, t) where
J : 1R" x 1R+
~
m." is locally Lipschitz in x
(A.I) l:U1d
piecewise continuous
in t.
Definition A.l The o1igin x = 0 is the. c.qu.ilib,;'um point fo'1' (A.1) if f(O, t)
= 0,
Vt~O.
(A.2)
Scalar compmiso'JI. Junctions are iUlporto.nt. stability tools rrequellt.ly used in this book.
Definition A.2 A continuot/.s jflnction i' : [0, a) l> R.r is said to belong to class 1(, if it is strictly increasing and ,(O) = O. It is said to belong t.o cla.ss IGDQ il a. = 00 and 'Y(r)  t 00 as r  t 00. Definition A.S A conti.nuous function (3 : [0, a.) x 1R+ ...... 1Rr is said to belo1£g to class IC£.. if/0'1' each fi:ced s the mappiny (j( T, s) belongs to class K 1vitIL respect to r I and for each fia;ed.," the mappi.ng (1( r, s) is decreasing "mtb. f'ClllJect to 8 and {3(7·, s) ...... 0 as "'i ...... 00. It is said to belong to class JC£(XJ if, in. addition, for each fixed s the. mapping P('\ s) belongs to class /Ct:XJ witll. respect to 7. Using comparison functions we can restat.e the stability definitions from Sectio1l 2.1.1 in a more practical form:
490
ApPENDIX A
Definition A.4 The equilibrium. point x = 0 of (A.1) is • uniformly stable, if there exists a class IC function l' ( .) and a positi'lJC constant c, independent of to, such that
I:z;(t)I :5 'Y(lx(to)f),
'Vt ~ to ~ 0, 'Vx(to) Ilx(to)1 < Cj
(A.3)
• uniformly asymptotically stable, if there exists a class IC£. /unciion{3(',·) and a. positive constant c, indeperulent of ta, such that
Ix(t)1 :5 {3(I:r,(to)I, t  to),
'Vt ~ to ~ 0, 'Vx(to) Ilx(to)1 < c;
(AA)
• exponentially stable, if (A.4) is satisfied with {3(1', s) = k1'e os , k a, 0: > 0;
>
• globally uniformly stable, if (A.:1) 1.8 satisfied with 'Y E Koo fOT any initial
state .'I:(to); • globally uniformly asympt.otically st.able, if {A ...I} is so.tisfted with j3 E K£oo for an1J initia.l state :c(l.o); and • globally exponentially stable, if (A ...1) is satisfied for any initi.al state .1:(to) and wilh {3(1", s) = k,'e OB , k > 0, a: > O. The main Lyapul10v stability theorem is then formulated as follows (see, for example, [81, Theorem 4.1, Coronaries 4.1 and 4.2]):
Theorem A.5 (Uniform Stability) Let x = 0 be an equilibrium point of {A.1} and D = {x E lRn Ilxl < ·r. Let V : D X lRD + ~ be a continuously differentiable junction such that 'Vt ~ 0, 'Vx E D, (A.5) (A.6)
Then the equilibrium x = 0 is • uniformly stable, if 'Y] and 'Y2 a·re class IC functions on [0, r) and 'Y3 (,) ~ 0
on [0, r); • uniformly asymptotically stable, if 1'11 'Y2 and 'Y3 are class JC ju,nctions
on [O,T); • e.'\."Ponentially stable, if 'Yi(P) = kiPQ on [0,7"), ki
> 0,
ll'
> 0, i = 1,2,3;
• globally uniformly stable, if D = IRn, 'Yl and 'Y2 are class /Coo functions,
o.nd 'Ya(') ~
a on IR+;
491
LVAPUNOV STABILITY AND CONVERGENCE
• globally uniformly asymptotically stable, if D = n" t "il and 12 are class A.oo functions, and 1'3 is a etas,., JC function on 1R;..; and
= n"
• globally e>"''']lonentially stable, if D kj > 0, o· > 0, i 1,2,3.
=
and 7i(P)
= kiPQ
on lR+1
In general, 01.11' goal is to achieve convergence to a set. For timeinvariant systems, the main convergence tool is LaSalle's Invariance Theorem (Theorem 2.2). For timevarying systems, a more refined tool was developed by LaSalle [110j and Yoshizawa [2011. For pedagogical reasons, we will introduce it via fl technical lemma due to Barbalat [155]. These key results and their proofs are of utmost importa.nce ill guaranteeing that all adaptive system wi1l fulfill its tracking task.
Lemma A.6 (Barbalat) Consi(ler the function tP : R+ i' lR. If cP is 1/.nifO'lm11J continuous and lim,_oo f;o q,(T)dT exists and is finite, then lim q,(t) too
= O.
(A.7)
Proof. Suppose that (A.7) does not hold; that is, either the limit does not e..ust or it is not equal to zero. Then there exists e > 0 such that for every T > 0 one can find tt ~ T with 14t(tdl > e. Since q, is ulliformly continuous, there is a positive constant r5(e) such that Iq,(t)  ,p(t})1 < £/2 for a11 tl 2: 0 and all t such that It  tIl ~ b(e), Hence, for all t E (tll tl + 6(e)], we have 1q,(t)1 =
Iq,(t)  ¢(tl) + ,p(tl)1
2: f4>(tdl\q,(t)  4>{l1)\ > which ilDplies that
/1
'1+6'(£)
e
£
(A.B)
£2=2'
I
q,(T)dT =
1"1+6(£)
'1
il
eo(c) 1q,(T)/dT>, 2
(A.9)
where the first equality balds since 4>(t) does not change sign all [tlJ tl + bee)). Noting that fJl+ 8(S) q,(T)dT = f~l q,(T)dT + .r,'11+6(E) q,(T)dT, we conclude that t/J(T)dr cannot converge to a finite limit as t ~ 00, which contradicts the assumption of the lemma. Thus, HmtIX! ,pet) = o. 0
fJ
Corollary A.7 Conside7' the function q, E L.p for some. p E [1, (0), then
4> : lR.t
lim q,(t) =
too
o.
I
JR. If,pl ~ E £00' and (A.I0)
492
ApPENDIX A
Theorem A.S (LaSalleYoshizawa) Let x = 0 be an equilibrium. point of (A.i) antll:11lppOSe J is locally Lipschitz in :r. 'lLnij'ormI1J in t. Let V : 1Rn x R+ + lR+ be a continuO!tsly diffe7Y:!nliable j·u11.ction such ULat 11 (Ixl) ~ V(:r, t) ~ "2(lxl) «
11' =
(A.II)
av 81' t) < H'(1') < 0 at + j(x ax «, "
(A.I2)
~ 0, T/:l: E JR.'I, 11Jherc 11 an.d 1'2 are class /Coo junction/; and 11' is a COfl.I.inuous Junction. Th,en, all solll.tions oj (A.l) a1'e globally unijo1.,nly bounded and sa.ti,r;jlJ lim Tl'(:r(t)) = 0 . (A.I3)
Vt
too
In addition, ij Tl'(x) is posili1Je definile, then the equilibt'iu.m. J; = 0 is globull1J 1tnijormly asymptotically stahle, Proof. Since l:r ~ 0, V' is Ilollincreasing. Thus. ill view of the first inequality in (A,ll), we conclude that :r: is globally ulliformly bounded, that is, 1:1:(l) 1 ~ B, Vt ~ O. Since 11'(x(t,), t.) is nonincrellsillg and bounded from below by zero, we conclude t.lu·..t. it has n limit VIX) as t + 00. Integra.ting (A.12), WE' have
lim
1,
too In
It'(x(r»dr
l' ll'(:r:(r}, .
~
 lim
=
L·10
=
V(:r.(io}, to)  VIXl ,
100 to
r)dr
lim {'i(:z:(tO)' to)  V(x(t),
tn (A.I4)
which means that ftC: H'(x(r»dr e.'Cists and is finite. Now we snow that ll'(x(t)} is also unifOl'lllly continuous. Since Ix(t)1 ~ Band j is locally Lipschitz in x uniformly in i, we sec that for allY t ~ to ;::: 0,
Ix{l)  x(to)1 =
11~ j(:r(r), r)drl ~ L
l:
Ix(r)ldr
~
LBlt  tol , where L is the Lipschitz constant of fall {Ix 1 ~ B}. Choosil1g tS(e)
(A,I5)
= tB' we
have
la:(t)  x(to)1 <
E,
V It  tol ~ t5(e),
(A.16)
which means that x(t) is uniformly continuous. Since HI is continuollS 1 it is uniformly cOl1tinuous 011 the compact set {Ix I ~ B}. From the uniform COlltinuity of Hl(X) and x(t), we conclude that Hl(x{t)) is uniformly continuous. Hence, it satisfies the conditions of Lemma A.6, which then guarant,ees that H'(x(t)) r 0 as t + 00. If, in addition, H'(x) is positive definite, there exists a class K: function "3('«') such tbat Hl(X) ~ 13(lxl). Using Theorem A.S, we conclude that x = 0 is globally uniformly asymptotically stable, 0 In applications, we usually have Hl(X) = xTQ:c , where Q is a symmetric positive semidefinite matrL'C. For this case, the proof of Theorem A.S simplifies using Corollary A.7 with p = 1.
1 j
Appendix B InputOutput Stability In addition to a review of basic inputoutput stctbilit.y results, \ve give several technical lelll mas llsed in the book. For a fUllction .J: : 114 + lR" we define t:he c. p norm, p E [1,00], as
I\:vll p =
( I ( I
rOO Ix(t)IPdt)lllJ p E [1,00)
10
(B.1)
sup I:v(t)!
p = 00
I~(}
and the
£IJ,e
norm (truncated £p norm) as
1I~l:tllp =
r /X(T) IPdT)
l/p
10
sup Ix{r)1
}J
E [1,00) (B.2)
p=
00.
Te[O,/)
'X.'e consider an LTI causal system df\scribed by the convolut.ion 1J(t)
= h * It = ./0r h(t 
T)u(T)dr!
(l?3)
: lI4 + m. is the jll])ut, y : 1R+ + lR is the output., and h : ill + lR is the system's impulse rcspOllse t which is defined t·o he zero for negative values of its argument. Let H(s) denote the Laplace t.ransform of h(t).
where 1.1.
Tbeorem B.t [28] A stn:ctly proper' 1Yl.liona.ll.ran...'Jfer jmwtio71 If(s} is o·null/ti.c in. ~e{ s} ~ 0 if and only if h E £1The quantity
\lH\loo :.:
sup IH(jw)1 we1R is referred to as the 1f.00 norm of the transfer fUllction H (s). Tbe follO\villg theorem is known from [10, 23, 28 t 30].
(BA)
494
ApPENDIX
B
Theorem B.2 For system {B.3}, if It E £1 and H E 1i0tJ1 then
(1) (ii)
(iii)
IlhlllllUtllco 1I1JLlb ::; IIH1I0tJ1lu.t112 IIHIIOtJ ~ IIhlh ~ 271.IIHIl00,
(B.5) (B.6)
111111100 ::;
(B.7)
where n is the McMillan degree of H(s). Lemma B.3 (Holder's Inequality) If p, q E [1,00] and ~
II(jg),lh
~
IIftl/pllg,lIqJ
+ ! = 1,
'It ~ O.
then (B.B)
The following theorem is referred to as Young's cOllvolution t.heorem [197].
Theorem B.4 If he
.c1,e, then PE
Proof. Let y
= h * 'U.
Iv(t}1 < =
[1,00].
(B.9)
Then, for p E [1,00), we have
fot Ih(t ft
Jo
T)II·u( T)ldT
IIJ(t  T)I
l?:! p
1
Ih(t  r)lii l'U.(T)ldr J!=.!.
< =
(Ia! Ih(t  T)ldT) (lo' Ih(t  r)1 11J.(T)I dT) IIh'lI~ (I.' Ih(t  r)llu(r)IPdr/ ' JI
P
! p
(8.10)
where the second inequality is obtained by applying Holder's inequality. Raising (B.lO) to power p and integrating from 0 to t, we get
!I Yt I!;
10' IIh,/lf (loT 111(T  s)IIu.(s)IPds) dT t = IIh,II~l 10' (I Ih(r  s)llu(s)IPdT) (/s 1
$
la' (lot Ih(1"  s)1 lu(s) IPdT) ds IIh,lIf lu' lu(s)IP (lot Ih(r  s)ldT) ds
= IIh,lIr1
=
1
IIh,IIr 1 lot lu(s)IP (la' Ih(T)ldr) ds ::; IIh,lIf1Ilhll}l!udlr $
~
IIh,lIfllu,lI~ I
(B.II)
where the second line is obt.ained by changing the sequence of integration, and the third line by using the causality of h. The proof for the case p = 00 is immediate by taking a supremum of hover [0, t] in the convolution. 0
495
INPUTOUTPUT STABILITY
Lemma B.5 Let v and p be realvalued functions defined on 14, and let b and c be positive constants. If the1/ satisnl the differential inequality iJ
5 cv + bp(t)2,
(B.12)
(i) then the following integ1ul inequality h.olds:
(ii) If, in addition, p E £2, then v E £.1 and
\lvlll :$ ~ (11(0) + bllplI~) .
(B.14)
Proof. (i) Upon mUltiplication of (B.12) by eel, it becomes (B.15) Integrating (B.15) over [0, t], we arrive at (B.13). (ii) By integrating (B.13) over (0, t], lve get
lui 11(r)dr
:$
lot v(O)eCTdr + b lot [foT eC(T.!I)p(s)2ds] dr
::; ;v(O) + b
10' [foT eC(TS)p(s)2ds] dr.
Noting that the second term is bll(h*p2)tlh where h(t) Theorem BA. Since IIhlb = ~, we obtain (B.14).
(B.16)
= e ct , I, ~ 0, we apply 0
Lemma B.6 Let v t 11, a:",d 12 1)e realvalued functions defined on 114. and let c be a positive constant. If 11 and '2 are nonnegative a.nd in £1 and satis/11 ilLe differential inequality iJ
S cv + 11 (t)t, + 12(t) t
v(O) ~ 0
(B.17)
then v E £00 n £1 and v(t) :$ (v(O)e d I\v/It
+ II 12 I!. ) ellllih
(B.IS)
~ .!: (v(O) + IIl2lh)elllllh .
(B.19)
c
=
Proof. Using the fact tbat fact that w(t) .::; vet), w cw + 11(t)W + 12(t) , w(O) = v(O) (the comparison principle; see, for exa.mple, [108, 132]), and
496
ApPENDIX
B
applying the variation of constants formula, the differential inequality (B .17) is rewritten as
v(t) ::; 'lI(O)e.r:ic+ll(ll))ds oo
$ lI(O)e ct eIn
::;
['lI(o}e CI
h(s)dll
+
la' eI:Ir.+lt(s»)tlSI'!(T)dr
+ 10' e ·(tr)l2( T)dre.r~""C It{s)ds t
+ fa' e (tr)12(r}dr] eilldh. C
(B.20)
By t,akiug a supremum of eC{IT} over [0, 00]' we obtain (B.18). Integrat:ing
(B.20) over [0,00], we get
.Io
t
tt(r)dr
~ (~u(O) + la' [foT eC{TS)lAs)ds] dr) e ll1dh . (B.~I)
Applying Theorem B.4 to the double integral, we a.rrive at (B.19).
0
Remark B.7 An alternative proof that v E Coo n C 1 ill Lcmma B.G is using tlIe Gl'onwalllelllma (Lemma B.l1). Howevpr. with the Gronwall lemma, the est.imat.es of thc bounds (B.18) and (B.19) are more conservative:
·v(t) :$ (u(O)e ci
+ 1112111) (1 + IIhIbelll' III )
IlvllI ~ ~ (u(O) + 1112 111) (1 + Ilhlllelllllh),
(B.22)
(B.23)
+ .1;~), 'Vx > O. Note that. the ratio between the bounds (B.22) and (B.18), and the ratio between the bounds (B.23) and (B.19), are of order 11/1111 when IIl111t ~ 00. 0 because rr < (1
For cases where II and l2 are functions of time j:ho.t converge to zero but are not in Cp for any }J E [I, 00) we have the following lemma.
Lemnla B.B Con.sider the differentia" ineqllality v{O)
= 1'0 ~ 0,
(B.24)
c> 0 and TO ~ 0 a1"f constants, and /31 anll f3'J. are class IC£ functions. Then there e:r.ist.i a cla,ss ICC function f3v and a class IC Junction "I,. such that
whe7'i~
1J(t) ~ f3v(vo
+ "a, t) + l'v(P),
'tit ~ O.
ltforeo Vf7', if f3i(7', t) = 0'; (l')eO',1 , i = 1,2, where O'j E K. and Uj the,.,~ exisls lXv E IC and U v > 0 such that f3v{1', t) = O:v(1")eD'II/.
(B.25)
> 0, then
497
INPUTOU'l'PUT S'rABILI·.ry
Proof. Vve start by introducing 'U = v 
~ and rewriting
(B.24) as
e
~ ::; [c  PI {l'O, t»)ii + p1 (1'0, t) + fi:!{1'O, t} . C
(B.26)
It then follows that
'l'{t,) ::; '110tJ;rPJ(rOJ8)clds +
t [!!./31 (1'0, 'T) + /32 (J'u, 'T}] c.f;18J(l'o,JI)c],lSdT + !!.. . c r.
10
(B.27)
"Ve l'ecall the standard result (see, for example, [81, Lemma 4.6J):
(B.28) where ~: is a positive, continuous, increasing function. To get an est.imate of the overshoot coefficient k(ro), we provide a proof of (B.28). For each c there c."'{ists a class JC function Tc : R+ + lI4 such that
(D.29) Therefore, for 0 ::; 'T $ Tc(1'0) $ i, we have
so the ovm'shoot coefficient in (B.28) is given by
k(ro) ~ eTc(ro).t:JI(ru.O} •
(B.3l)
For the other two cases, t $ Tc(ro) and Tc(1'O) $ r, getting (B.28) with k(ro) as in (B.31) is immediate. Nmv substituting (B.28) into (B.27), we get
v{t) ::; Vok(1'o)e;1
+ ~~(ro) lot [~/31 (1'0, T) + /32 (ro, 'T)] eI(t'T}dr + ~.
(B.32)
To complete the proof, we show that a class JC£ function /3 ('onvo]ved with an exponentially decaying kernel is bounded by another class IC£ fUllction:
498
ApPENDIX
B
Yl
Figure B.l: Feedback connection.
Thus, (B.32) becomes
vet)
~
k(1"o) {[vo + ~~,81 (7'010) + ~P2(1'O' 0)]
+ 2~ PI (ro C'"
l
t/2) +
e~L
~8.!(1·OI t/2)} + e . C C
(B.34)
By applying Young's inequalit.y to the terms k(7'o)~,Bl(ro, O)e 1' and ":(7'a)~PI (rol t/2), we obtain (B.25) with
=
k(r)
{[7' + k(:) pt(r, 0)2 + ~,82(7'I 0)] eit c c~
k(r) ,81(1', t/2}2 + :'(j,.z(r, ? +.) t/2) } c,'" c
{B.35} (B.36)
The last statement of the lemma is immediate by substitution into (B.35). 0 All the above results describe inputoutput properties of individual syst.ems. To prepare for a basic result on feedback connections of systems, we first give a definition of L.p stability.
Definition B.9 A mapping H : £p,e exist finite positive numbe7's l' and u E £p,e and all t E [0, 00 ).
f3
~ L.p,e is said to be L.p stable if the7'e such that U(HU}Lll p :5 'YIiULli p + P Jor all
The following, theorem known as the small gain theorem [28] I gives sufficient conditions for L.p stability of the feedback connection in Figure B.l.
Theorem B .10 Consider the s;IJstem in FigU7'e B.1. Let H 11 H2 : L.p,c ~ C.P,£I P E (1,001, be two L.p,c slable operat07'S wit), finite gains 1'1, 'Y!! and asSDciated constants f31,{32 • Let the operator HIH2 be stlictly ca1Jsal. IJ (B.37)
499
INPUTOUTPUT STABILITY
lI'uull" :::;
1
lIu:ullp :::;
1
1 ')')')'2
1
 '11'2
(livull" + 121\ v2fllp + 1'2 + '/2{jl)
(B.3S)
(lI l12Jllp+"YJII Vull,.+,BJ
(B.go)
+1'1P2)
for all t E {O,oo). II, in addition, VhV2 E £p, th.en (B.38)(B ..99) /told with all subscripl.s t dropped, which ifllplie.r; that lJ.1r Ilz! Yh and 1/2 have bounded L..
"
norms.
Now we give a version of Gronwall's lemma. For the proof, the readpl' is referred to lSI]. Similar lemmns can be found, among other references, ill [28, 81, 165].
Lemma B.II (Gronwall) Consider the contirlU01J.S functions A: lR+ + JR., . lI4, and II : 1R+ . ~, where J.l and II are also nonnegat.ivf. If a co'n.tinflOUB function y : 1R+ + If{ sa.tis./ies the inequality
p. : 114
y(t) ::; A(t) + IJ{t)
t
ltD
(BAO)
11(.9)y(s)ds,
then 'TIt In particular,
if A(t) == A is
a constant and jJ.{t)
\.
J:
1J (t) • < _ ",e
0
V(r)dT
•
== 0,
~
to
~
O.
(BAl)
then
'tit ?::. to ?::. o.
(B.42)
Appendix C InputtoState Stability Inputtostate stability introduced by Sontag [173] plays a cruciaJ role in our modular adaptive nonlinear d~signs. We e~..t.end Sontag's definition to timevarying syst.em.c;:
Definition C.l The sy.IIlem .1: ::::: f(t, :J:, u)
(C.l)
I
1.IJhere f LIJ piecewise conlin1J.ou.'l in t and IDcally Lipschitz in .1: and 1I, i.s said to be inputtostate stabJe (ISS) if there e.a:ist a class IC£ f1J,Tlction t3 and (l clas,fj K:, function y, such that, ID1' any :c(O) a.nd for any input u(.) cDl1tinnow; anti bounded on [0,00) I,he solu.lion exists fo'l' aU t ~ 0 a.nd sal,isji.es
l:v(t)1 ::; ,B(lx(ta)/, t  to) + I'
(sup "U(T)I)
(C.2)
fo5;T:st
for all to a.nd t such t.hnl. 0 ::; to ::; t. The following theorem estabHshes t.he connection betwf"ell the existence of a Ly8~)UnD\'like function and the inputt.ostate stability.
Theorem C.2 [173, Claim on p. 441] Suppose that f07' flle slpdem (C.l) there ea:ist8 a C 1 Junction \1' : ~ x lR,R 0 114 such I.hat jor all .1' E lRn and tl
E
m.m,
where 'j'l, i'!!, and p a71~ cla,,,,, K.o:J Junctions anrl1'3 is a cla,c;,'l1C function. Then the system (C. I) is ISS with l' = I'll 0 "(2. 0 p.
502
ApPENDIX C
Proof (Outline). If :r(to) is in the set (C.5)
then x(t) remains within the set
St" =
{x E JR" 1.,,1 :'0 1'1' 01',0 P (~~r.IIl(T)I) }
(C.6)
1
for an f. ~ to. Define B = [to! T) as the timc interval before :z:(t) enters Rto for the first t.ime. In vicw of the definition of Rio! we have
'Vt E B.
(C.7)
Thcll, by [173, Lemma G.1], there exists a class K£ function /3" such that
'Vt E B,
(C.B)
which implies
'tit E B.
(C.g)
On the ot.her hand, by (C.G), we conclude
Then, by (C.g) and (C.IO),
1·1:(t) I :5 /3(I."v(to)I, t.  to) + "( (sup 1u.(T)I)
'tit
I
1"~to
'2:. to '2:. O.
(C.II)
By causality, it follows that
Ix(t)1 $ P(I·l:(to)l, t  to)
+ I' (sup
'o$r9
IU(T)I)
I
'Vt ? to
~
o.
(C.12)
o A function l' satisfYing conditions of Theorem C.2 is called an ISSLyapunov function, Sontag, Wang, and Lin recently proved that the inverse of Theorem C.2 is also true. They also introduced an equivalent dissipativitytype characterization of ISS.
503
INPUTTOSTATE STABILITY
Theorem C.3 [115, 177] For the system
x = f(x,u) , the Jollo'wing properties are equi'tJa.lent: 1. the system is ISS, 2. there exist,rJ a sm.ooth ISSL,/apunov function, 9, tl&e1e exists a smooth posiUve definite 'radially unbounded function 1; and class ICoo functions Pl and P'l such that the following dissipativity inequalit7/ is satisfied:
The following lemma establishes a useful property tbat a cascade of two
ISS systems is itself ISS.
Lemma C.4 Suppose that in the system Xl = !I(t,xl,X2,U)
X2
=
h(t, X2, 'u)
the Xl subsystem is ISS with respect to with respect to lJ" that is,
3:2
(C.l3) (C.14)
and 11., and the X2subsystC'ITI, is ISS
(~~~l {lX,(T) I + IU(TlI})
1",(t)1
$
.8,(lXI(8)1. t  s) + '11
IX2(t)1
~
.82(1x2(8)\, t  s} + 1'2 (sup IU(T)I) , Ii~T!;t
(C.15) (C.16)
where /31 and fJ2 are class ICC. junctions and 1'1 and 12 arc cla,9s JC fUTtdions. Then the complete x = (Xl,X2)system is ISS with
Ix(t)\ :$ ,8(lx(s}I, t  s) + 1
(sup IU(T)I) ,
(C.17)
S'5T~t
where (j(r, t) =
"(7')
=
.8. (2,81 (r, t/2) + 271 (2{j2(r, +1'1 (2{j2(r, t/2)) + f3J(r, t)
a», t/2} 1
(C,18)
tJl(2"1(2')'2(7') + 21'), 0) + ')'1(2'Y2(~") + 21") + '"Y2(r), (C.19)
504
ApPENDIX
Proof. With (8, t)
= (l/2, t), (C.I5) is rewritten as
~ 131 (/.1!I(tj2)f, t/2) + "'It ( tJ'J$T$,t sup {lx!!(T)1 + /1/(T)/}) •
IXt(t)l
C
(C.20)
From (C.lo) we have
,ft.~~SI 1·:'(r) I
<
'/~~~SI {13.( 1'''2(0) I, r) + 'l':! (o~~~, lu(0') I) }
~ ~(I:r2(O)/, t/2) +;2 (sup /U(T)/) , O$T'$I
{C.21}
and from (C.15) ,va obtain IXt(t/2)1 5: ,Bl(l x l{O)I, t/2) + 1'1 ( sup
O$r$tI2
~
Pl(lxl(O)(, t/2) +1'1 (
:$
{IX2(T)1 + 1'll(T)!})
sup
D$1"$.1 I'
{,82(1'~2(O)I' 'T) + '12 (sup \'lJ(a)l) + IU(TlI}) 0$.17:$1'
I't(lXl (0)1, t/2) +1'J (f12(lx:!(O}I, 0) + sup {1'2 (IU(T)I) O$.T$.1/2
~
+ IU(T)I})
Pl{l;"1 (o)l, t/2) + 71 (2,62(l x 2(O)/, O}) +11
(2 O:~/2 h
(Iu( r) I) + lu(rll} ) ,
(0.22)
where in the last inequality we have used the fact that 6(a+b) ~ 6(2a) +5(2b), for a.ny class lC function 0 and any nonnegative a and b. Then, substituting (C.21) and (0.22) into (C.20) we get
IXI (t)l
~
/31 (.81 (lXl (O)I, tj2) + 1'1 (2.8.l{lX2{O)I,0)) +1'1
+1'1 ~
(2
sup (1'2 (ltt(r)/) D:$T'5 i /2
+ /U(T)f}))
(P2{/X2(O)/, t/2) + 1'2 (sup Itt(T)I) + sup {ltt(i)l}) O$.l"~;t
'/2$1":$.1
PI (2f31 (Ix) (0)\, £/2) + 2')'1 (2132(13:2(0)1,0», t/2) +71 (2132{\x2(O)" t/2))
+P, (2,),1 +1'1
(2 o~~, h
(2 ~~~t
(lu( rJl) + lu(r)l} )
{'l':! (la(rl/) + lu(r)!})
.
,0) (C.23)
505
INPUTTOSTATE STABILITY
Combining (C.23) and (C.16) we arrive at (C.17) witb (C.18)(C.19).
0
The proof of Lemma C.4 foUows the Jines of the proof of [173, Proposition 7.21. An alternative proof has been givell ill [176J by using part 3 of Theorem C.3. Since (C.13) and (C.14) are ISS, then there e.xist ISSLyapullov fUllctions \'1 and 11'2 and dass ~ functions G:}, p},1l2r and P'J. such t.hat. (C.24) (C.25)
It is sIlown in [176) that Fi, V!!, 01 t Pl, Q:.h fJ2
call
be found such tha.t (C.26)
Then the ISSLyapunov £uuctiOll for tbe complete system (C.13)(C.14) can be defined a..s; (C.27) and its derivative (C.28) establishes the ISS property of (C.13)(C.Vl) by part 3 of Theorem 0.3.
In many of our applications of inputtostate stahility, we usc the following ]emma. which is much simpler than Theorem C.2. 7"eo.ivalfled junctions defined on m+. and let b and c be positive constants. If they salisfy the differential inequality
Lemma C.5 Let v and p be
(C.29) then iJ&e following holds:
(i) 1/ p E £OCH then 'v E £00 and v{t)
~ v(O)ed + ~lIplI!'. c
(C.30)
:s u(O)ect + b!lpUi .
(C.3l)
(ii) If p E £2, then v E Coo and v(t)
506
ApPENDIX
Proof.
0) From Lemma B.5
1
C
we have
vet) ::; 'v(O)e cl + b r' ec{/'J'} p(T)2dT
./0
::; tl(O)e ct + b sup {p(T)2} ::;
r
ec (t'J'}d7
In d 'v(O)c + bllpll~ ~ (1  ed) TErO.i!
::; u(O)e ct +
~"plI~ . c
(C.32)
(ii) From (B.13) we have vet) ::; 'u(O)e ct + b sup {ec{/T)} rerO,t)
r t
p(T?d7
./0
= 'l,(O)cct + hI/pili.
(C.33)
o Remark C.6 From Lemma. C.5, it fallows that if
v(O) !:: 0
(C.34)
and PI E Loo and P2 E £2, thell v E £00 and (C.3S)
This, in particular, implies the inputtostate stability with respect to two inputs: PI and IIP2lb. <>
Appendix D Passivity Now we briefly review some basic passivity results. The concept of passivitYt which was first used in network synthesis, became a. fundamental feedback control concept ill the seminal work of Popov [155]. Its applications to adaptive control were pioneered by Pad{s [150] and Landau [109], and to ot.her areas of systems and cont.rol tbeory by \ViUems [198J and Hill and Moylan [45]. \Ve use the passivity definitions of Byrnes, Isidori, and \Villems [14] extended to timevarying nonlinear systems. Consider systems of tbe form
x
=
y =
j(x, t)
+ g(x, t)u
(D.1)
h(3;,t) ,
with x E m. n, y E JRm, U E lR,'\ and I, g, h cOlltinuous ill t and smootb ill :1:. Suppose 1(0: t) = D and h(D, t} = D for all t ;::: O.
Definition 0.1 The .'iystem. (D.1) is said to be passive if there exists a continuous nonnegative ('Istorage") function V' : lRn x R+ + lli+, 'Iuhich satisfies V(O, t) = a, 'Vt ;::: 0, such that far all·u E Co, x(O) E m.n ,t;::: to ;::: 0
l'
yT(u)u(u)da ;::: V(x(t), t)  V'(x(t.o) , to).
(D.2)
In
Definition D.2 The system (D.l) is said to be strictly passive if there exisl a continuous nonnegative (storage) fttnction V : m.n x R+ + m+ 1 1lJhich satisfies V(O, t) = 0, 'Vt ;::: a, a.nd a positive definite function (dissipation rate) V) : mn + lR+, such tJl,at for all'il E Co, x(o) E m.n , t ;::: to ;::: 0 [' yT(u)lI.(u)du ;::: "(x(t), t)  V(x(t o), to)
k
+ f' ¢(x(u»da.
k
(0.3)
Passivity and LyapuDov stability are cJosely related concepts.
Lemma 0.3 Suppose the system (D.l) is (stli.ctI1J) passive. If V i,t; positive definite, radially unbounded, and decTescent, IJtat is, if there exist class /Coo
50S
ApPENDIX
D
jUlIcUonsl'] and"l2 such that "(](I:rl) ::; 1/(:r,t)::; "fAI:rl)) V(:r:n E IHH x 1li+ ~ then.. JOT '/I ::::: 0, lhe equilihri /1.117. :z: = oJ (D.1) ,is glo bally u'Iujorm,lll (asyrnptolically) stable.
°
Proof. \\Then
II. :::::
0, in the case of strict passivit.y) differen!:iating (D.:3), we
have
(DA) Tlms, the equilihrium .1: = 0 is globally uniformly asymptotically stable. The case of passivily is annJogolls. 0 l\'iany problems in paramet,er identification and adaptive control can be studied as feedback int:erconnections of passive systems (see Figure D.l):
B] ~2
COll11cct.ed by
:1.'1
.rd:D) t)
YI
h] (:1:) t,)
.1'2(;7:, t) + g2(.T) t)'/I,'2 h'2(:r, t)
.r2
/1'2
1'1
(D.5) (D.G)
n1C relat.ions l/'2 +vl
(D.7)
.1/1 )
(D.S)
'//]
where
I g1 (.1:, t)111
is a new input Lo the syst,em.
TheorelTl D.4 Suppose the 8U8te.'f71, ~1 'is (8tridly) ])(],8sivc with storage junction V'1 (a.nd dissipation T(].lc 1/']) independent of :1.72 _ Likewise, 8'11ppOSe I,he sl}sie'm. B2 is (st,'ricl.ly) passive willi. slorage funct.ion \/~ (and (i'iss'ipa.t;i 0 'II. ral.e 'l,b'2) 'independent O}':Ll. Then t.he 'in,t.erconneded syst,em, (D.5)(D.8) willi. input, VI lind output 'lJl is 1. strictI:v passive:' if both ~1 and:B 2 aTC Htrictly ]Hlssivc,
2. passive if al. Ica..'){ one oj {he .systems
~]
and:B 2 i.s pa.",8ivc hut not st.'rictly
]}(J,SS'l/JC.
AforcolJer. when 0\ ::::: 0, 'Jj ~l is s/'riclly pas.'31:vc a.nd 1::2 'is pa.8SiVC, then the equililJ'riu'JII,.r = 0 is globally uniformly stahle and limt"x:.1."l(t) = O.
~l
;t}2
E2
;III
'U';2
Figure D.l: Feedback interconnection of two passive systems.
509
PASSIVITY
Let us first assume t.hat ~t and L:2 arc both strictiy passive. ]'hC11 , in view of (D.7)(D.8) we have
Proof.
lJ2]dCJ
>
(D.9)
(VrlJlr/a
>
(D.10)
.hl/' Unv, Jl o
Adding inequalities (D.9) and (D.10), we obtain (D.ll) where the storage fUllctioll F and the dissipatiou rate :rs~rstem are defined as
F(t, .1')
I/{J:)
'I' for
the C'LHnplE'te
+ V2(:C2:t) 1/ 1(;l'd + 1!':.!(·1:2).
(D.12)
Fi(:rj,t)
(D.l:3)
1
Since V is positive definite, radially unbounded and decreseent, and I/; is positive definite, this proves the strict passivity. If at. least; one of the systems EI and is passive but not. strictly passive, thell its dissipation rate 1/'1 is at best: positive selnidefinite but. lIot positive definite, and the ovt'rall s~rsl('nl is onl,v passive. Finally, when 'L't 0, if::S l is strictly passive and 2:::2 is passive, then 'I/!':!, is positivE' scmjrlefinitc, and b~T differentiating (D.ll) we get
(D.14) Thus, by Theorem 2.1, :r = () is globa.lly uniformly stable al1d lim,_c:c :/'j (t)
o.
o Now we turn our attention to linear timeinvariant passive sysi:ellls,
Definition D.S A Hlt'iO'll.nl t"'(l1I.~rCl' IuncUon 0(.5) 'is sai.d to be positive real if 0(8) is Teal for all Tcal8, (/TIlPRe{G(.9)} 2: a for all!}ce{s} 2: o. fl, in addition., G(8  f) is positive. real fO'l' some [ > 0 then G(8) is 8ahl to be sLridly positive 1
real. For eompldeness, we quote the celebrated PopovKalmanYakubovich lelllma. A recent \'('rsion of its proof can be found ill Tao and Ioannoll (183].
Lenllna D.6 LeJ the strictly posit,iuc Tea.l iraTl.sjc'!' jllnci.io7/. G( s) have the slalespace l'epreseTdation (A, h, c, d), d 2: O. Then . fOT {InlJ given L L'1' > 0, til C1'C e:t:ists (J, scalar 1) > 0, a Vf;cf,l)'l' q, an.d {J, P = pT > 0 81/.ch til at ATp+PA = Pb
ql/I'IJL
= qV2d.
(D,15) (D.16)
510
ApPENDIX
D
Witb this lemma a. Lyupunov function l' = x T Px can be constructed such that P satisfies 110t only the Lyapullov equation (0.15) but also the inputoutput condition (D.IS) from which the restriction to rela.tive degree zero (d > 0) or one (d = 0, cb > 0) is apparent. The maiu utility of tIns special Lyapunov function for adaptive and c8Bcade designs is that the indefinite term in its derivative l' depends on the output y and not 011 the whole state :1:.
Appendix E Parameter Projection The modular adaptive controllers have a point. of singularity bill = 0, where bm is the estimate of the highfrequcncy gain (virtual control coefficient) bnl" In order to prevcnt bm from taking the value zero, we use the parameter projection in our identifiers. For this, we need to know the sign of the actual highfrequency gain bm • We first give a treatment of projection for a general ('onvex parameter set and then specialize to the case where only the highfrequency gain is constrained. Let us define the following convex set
n=
I
{{) E RP P(8)
:5
o} ,
(E.1)
where by assuming that the convex function P : IRP + rn. is smooth, we assure o that the boundary an of n is smooth. Let. us denote the interior of n by n and observe that VoP repl'esents an outward normal vector at 8 E an. The standard projection operator is
I
T,
ProHT} =
(
VfJ PT ) 1 r \/.P1T'V _p
(E.2)
ViJP o
T,
0
where r belongs to the set g of all positive definite symmetric p x p matrices. Although Proj is a function of three arguments, T, {) and r, for compactness of notation we write only Proj {T}. The meaning of (E.2) is that, when {) js in the int.erior of n or at the boundary with T pointing inward, then Proj{T} = T. When {) is at the boundary with T pointing outward, then Proj projects T on the hyperplane tangent to 811 at 6. In general, the mapping (E.2) is discontinuous. This is undesirable for two reasons. First, the discontinuity repl'esents a difficulty for implementation in continuolls time. Second, since the Lipschitz continuity is violated, we cannot
512
ApPENDIX
E
usc standard theorems for e.\':istellce of solutions. Therefore, we need to smooth the projection operator. Let us consider the following COIlVe.\,: set
rr~ = {(j E
lR,P
I1'(B) ::;
(E.3)
E} ,
which is a union of the set. TI and an Q(e)boundary layer around it. We now modify (E.2) to achieve continuity of the t.ransition from the vector field T on V·'p 't'pT) r V;1'rr~Ii'P r
the boundary of IT to the vector field ( 1 
on t.he boundary of
II!":
ProHT}
=
[(T'
_
VPVP'I") J  c(O)f V :p'I'r~ 07'
T,
c(O) = min { 1, P~iJ)} _
(E.5)
It is helpful to not.e that. c(8TI) = 0 and c(aTI E ) = 1. In the proofs of stability of identifiers we need the following technical properties of the projection operator (EA).
Lemma E.l (Projection Operator) The following are the p1"Ope1·ties oj the pl'Ojcction ol}eralo'f' (E.4): (i) The mapping Proj : lRP x lIe: x Q + JRP is locally Lipschitz in its a7'guments r, 0, r.
VO E TIE'
(ii) ProHr}'1'r 1 proj{r}:5 rTr1r,
(iii) Let r(t), ret) be continuously differentiable and
8=
Proj{r},
Then, on its domain oj definition, the solution (jet) remains in TI~.
(iv) B'rr I Proj{ r} ::; o'rrIr,
VB E n~,f~ E TI.
Proof. (i) The proof o[ this point is lengthy but straightforward. The reader is referred to [157, Lemma (103)]. (ii) For {} or VfJp'r r ::; 0, we have Proj{.,.} = rand (ii) trivially holds
En
with equality, Othenvise, a direct computation gives
~ (v apTT)
Proj{r}Tr I Pro.i{r} = rTrIr _ 2c(O)
0
vfJPTrviJp
IVPvpTr I
2
2 A
+ C(O)2
0 0 r (Vo pTrVOp)2 2
=
r
T
r
1
(VOpTr) r  c(O) 2  c(O) VopTrvoP •
(
.)
(E.6)
513
PARAMETER PROJECTION •
~
a
where the last inequality (ollows by notillg that c(6) E [0,1] for 6 E TIE:\ n. (iii) Using the defillitioll of t,he Proj operator, we get
iJ
V,/pTT, VopTproj{r}
=
{
err
or V;/PTT:5 0
(E.7)
c(iJ») Vjj1'
(1 
T
iJ
T,
E
IIE\ IT and ViJ1"J'T > 0,
which, in view of the fact that c(O} E [0, 1) for
ii, implies that 8 E an£ ,
iJ E TIe: \
V o1'T Proj{ T} :5 0 whenever
(E.B)
that. is, the vector Proj{ T} either points inside lls or is tangential to the hyperplane of ane: at Since O(O} E lli:' it follows that O(t) E TIt: as long as the solut.ion exists. 0 0 (ivJ For (J En, (iv) trivially holds with equality. For 8 E lli! \ sincE' 8 E n and P is a convex [unctiol1, we have
e.
A
n.
A
(8  8)TVoP :5 0 whenever
0 E II!\ Ii .
(E.g)
\¥ith (E.g) we now calculate
 8Tr 1 Pl'oj{r} =
oTrIT
! 0,
+
Q
4
c(8} (Ol'vIP),eVi/pT.,.)
vo'pJ rv,,"
(J E
,
TIE \ II
V;/pTT >
~tnd
a
(E,lO)
o
which completes the proof.
Since we intend to use the projection operator only to keep the estimate of the highfrequency gain bm from becomiIlg zero, we now specialize the projeetioll operat:or for t.his l'1:l.fie. We assume that Ibm l ; :;: C;m > 0, where sgn b", and C;m ate known. Recalling that bm is the first element of the parameter veetor 8, i.e.~ 8 [bRl I O!;!, ••• ) 8,l]T, we define P(6) C;1I1  bm sgnb m and 11ol.e that VoP = sgnbmei'. Let us denote the nominal vector field fOl' the parameter update la~v by T = [Tl' T:h ••• ,1j,]T and choose E E (0, C;m). The updatp law of
brn
=
tbe form
8=
=
Proj{'T} using the projection operat.or (EA)(E.5) is given hy
! ! 1,
bm
=
Tl
a,X rnao
{a,
m }. Ec;m+b,:SgUb .. .
A
bm sgn bnI
_
(E.Il)
an d TI sgn I)m
bm sgn b", > C;m
Ti,
8i = . Ti 
< ~I!I
{l
. 7"1 ill r] I ll11n,
C;,JIb",SgUb ... } E
b b < ' I l l sgll m _ 4
~m
or
Tt
<0
sgn bill ;;:;: 0
(E.12) d b an Tl sgn m <
0
514
ApPENDIX
E
where i = 2 " ,p, and ru is the (I, i) element of the positive definite symmetric matrix " This update law achieves bm (t) sgn bm ~ t;m  e, 'Vt ~ 0 \V henever b",(O) sgn bm ~ C;m  e. Fl'om expression (E.12) one ~hould observe that when the update law is
r.
gradient it call l)e simplified. to iJi = Ti by the choice of the adaptation gain matrix: r == diag{"Yl, r2}, where 1'1 is a positive scalar and r 2 is a symmetric pos.itive defiuite (p  1) x (p  1) matrix. Thus, projection is applied only to bill' This simplification is not possible with the leastsquares update law because r( t) is not a constant matrbc, so it does not maintain a blockdiagonal structure even if r(O) is blockdiagonal. By setting e = 0 in (E.II) and (E.12), we obtain update laws with the stand81'd discontinuous projection, which in the case or gradient estimation simplifies to
(E.13)
For the most; part, tbe properties of the projection operator given in this appendix are recapitulated from [157] (see also [43, 51, 165]).
Appendix F Nonlinear Swapping The wellknown Swapping Lemma [138) is ubiquitous in adaptive linear control. Here we provide its nonlinear counterpart.
Lemma F .1 (Nonlinear Swapping Lemma) Conside7' the nonlinear time1Ja1"Ying system Z
:::
Yl
=
A(=, t)z + g(z, t)H'(z, t)T8  Q(z, l)T8 + E(z, t)e h(z, 1,)z + l(=, t)lV(z, t)'r&
(F.l)
where 8 lR+ ~ lllP i.CJ differentiable, e : m.+ + lR is continuous and limt_oc e(t) = 0, the matrixvalued junctions A : R" x 1R+  t m.ltxn. 9 :' 1R7& x IR+  j . m,nxm, H' : .IRR x 1Rt. + ffi.pxm , Q : m," X lR+ ~ lRPxn , E : n x 1I4 + nn, and I : lR,R X IR+ + lRrxm are locally Lipschitz in z and continuou.s and bounded in t, and h : nn x ll4 + lRrxn i.II IWl1nded in z an.d t. Along with (F. 1) consider the linear timeva7ying system.II
m.
E2
:
1::3:
n,T = Y2
=
,p = 11&
=
A(=, t)OT + g(:;, t)H'(=, t)T h(z, t)O" + l(z, t)H'(=, t)T A(z, t)1/J + OT8 + Q(z, t)T8 h(=, t),,p.
(F.2)
(F.3)
Assume that =(t) is contin1J.ouB on lO,oo) and there exists a c07ltinuously differentiable junction V : m.n X IE4 + 1I4 such that (FA) and for each
Z
E Co,
(F.5)
516
ApPENDIX
F
Vt ~ 0 I 'r/( E JRfI, a}, 0'2, 0'3, 0:" > o. Then for Vz(O) , '~b(O) E JR.", 'Vn(O) E R,)XrI, Vt 2: 0 the outputs of S11stems (F.l)(F.8) n.re related by Yl
= Y2 0+ .lJa + y,
10he7'e 11£ is 'lJ.ni/ol'7nly bounded and Yt.(t)
+
(F.B)
0 as t +
00.
Proof. Due t.o the t'OllLinuity of :;(t), the matIi..'\:valued functions g(=(t)~ f). H'(=(/,),1), Q(=(I), t), and E(=(t), t) are continuous ill I. Situ f1 the input 910FT o
to the liuea!' l.imevatying system E!! is continuolls on [0,00), then
n E L.
OCJc '
0 E L.f)O(! and systenl E;i is linear timevarying, then 'I/J E L.OCJ£' Hencc t.he solution of tbe nonlinear sys1:cm (F.I}(F.3) is defined on [0,00).
Sincc also
Differentiating f
=
=+'1/1  nTe, we compute : + rb  nT 0  n1'o
=
A.: + gU,TO  QTfl + Ee + A'I/J +
(=
AnTe  .qH:TiJ = A(z +
nTD + QTB
0.'1'0
't"  o.TO) + Ee
(F.7)
and obtain
~ = A(z, 1)e + E(=, t)e.
(F.B)
III view of (F.5) we havf1 =
<
av n
uE
_
[A(=, t)e + E(::, l)e] + _ '1
a'3Iel  a',J
av at
(av· )~ all' Be. E(=, t) + De. E{z, t)e 'J
1.1) = 0:3 If.~'"  0:/1 (av =E (z,t )   1)  + e~ DE 20:.1 40,1 I.) , < 0:;1 Ie_I'.." + e
(F.9)
40',1
which, ac('orciing 1:0 (FA), yields •
v'
~
0:3 1.., V + e.
o::!
4(t4
(F,lO)
°
By Lemma B.B we conclude that F is uniformly bounded and V'(t) ~ as t *' 00. III view of (FA) we have \£(t)1 :::; vV{l)/al' which implies that e. is uniformly bounded and converges to zero. Noting that
lJr(t)
=
YI 113  Y2 8
11:; + ht/J + IH,T8  11070 lH,1'O + h#, = h(: + '1/.'  nTe) == 11.(:, t)e, =
since h(z, t) is bounded, y, is bounded and converges to zero.
(F.II)
o
517
NONLINEAR SWAPPING
Remark F.2 If instead of [0, (0), the ma..~imal interval of existellce of =(t) is [0, t J ), then the lemma holds all this interval, st:ating that lit is unjformh' bounded on [OJj). 0 Remark F.3 'Vhen Q(:::, t) == 0 and E(=, t) == O. the result. of Lplllma F.1 is remiuiscent of Ivlorse~R linear Swapping Lemma [138]. To see this, we rewrite (F.6) as (F.12) In this notatioll T:; : 11"1'0 H Yl is the nonlinear operat.or defined by (F.l) with D(z, t) == 0, wItile t.he system
e= y =
A(=(f), t)~ + g(=(t), I)u
fl(=(l), t)~ + l(=(t). Ou
(F.13)
is used to define the linear timevarying operators: T: 11 H 11, Ttl: 1I 110 .lJ for h = 1 and 1 = 0, T,.: U H 11 for 9 I and I = O. vVhen A, ll, " and I are constant, then the operator T:(s) = T(.5) :::; h(sl  A)lg + I is fl proper stab1e rational transfer function, Tg(s) = (81  A)lg, Th(s) = h(a1  A)I, and Lemllla F.l reduces to Lemma 3.6.5 from [105]. 0
=
In som~ texts on adaptive linear control, an e.."'{teuded result which gUa1'antees that 0 E £2 =} T::[n>T6]  T[H1T]B E £2 is a.lso referred t.o as Swapping Lemma. Our next lemma is a nonlinear timevarying genel'~uiz(Ltioll of this result.
Lemma F.4 Consider syste'ms (F.l}(F.9) willi. the sam.e set of (lSs1l1!,ptions as in Lem.ma F.l. Jilmthe''71W1'e, assume that:; E £00 and e E £2 11 jj E £2t then 111  Y2 0 E £'! .
(F.14)
If 0 E £2 n £001 then
(F.15)
Proof. Since z E £001 then gH!T E £00 and Q E £00' Due 1:0 the eXpOllf.'lltial stability of ;.1{=,1), it fo]]ows t.hat E £00' \¥e need t.o prove that Ya E £2 and 11£ E £2. The solution of (F.3) is
n
(F.16)
518
ApPENDIX F
where (FA)(F.5) guarantee that the state transition matrix «}): : :IJ.=4 x R+ t is such that IIP:(t, r)/2 :S keo(fT), k, a: > O. Since nand Q are bounded, then
m.nxn
I'!/J(t) I <
t
~:eDtl1jJ(O)l + kiln + Qllee 10
eo (tT)18{r)ldr
)1(1'
.
:5 ~'cQ'I,p(O)1 + ~:lIn + Qlloo (10It ea(l')dr  10 e D {lT)18(r)1 2 dr :5 ~~eatl1jJ(O)/ + kiln + Qllee
Ja (L e (tr)/9(T)/2dr ) t
O
)i
I
(F.17)
lj ,
where the second inequality is obtained using the Schwartz inequality. By squaring (F.17) and integrating over [0, t] we obtain
Changing the sequence of integration, (F.lS) becomes
la'I1/1(r}fdr :::; ::IV}(O)1 2 + ~ lin + QII~ lot e 16(s)12 (1' eOIl
2
:5 :2/1/1(0)/2 + k lin + QII!, ~a:
a
because J$rt eordr = 1a (e OB 1 in (F.19) yields k

OT
r' eaB/8(s)12.!.eOSds a:
k
dr) ds (F. 19)
as . Now the cancellation eOllcas = eot) < lea I
k
!.
\11/1112 :::; J2(X11J1(0)1 + ;;11 0 + QIla.lIlBlb < 00,
(F.20)
which proves 1/1 E £2' Due to the uniform boundedness of It, it follows that Ya E £2' As for Yc, by applying Lemma B.S to (F.10), we arrive at (F.21) Because h E £eet then ill view of (FA) we have
and hence UFo E £2' This completes the proof of (F.14). When 0 E £'}. n £a.l' then 'I/J E £ee n £2 and .,p E £00' Thus t by Barbalat's lemma, "p(t) 10 0 as t t 00. Therefore Ya(t) I> 0 as t 10 00. Tbis proves (F.15) because Yf.(t) t 0
as t
t 00.
0
519
NONLINEAR SWAPPINC
Remark F.5 When D(z, t)
== 0, we rewrite (F.14)
as
T:;[H/T 01 T(lVT]8 E £2
(F.23)
and (F.15) as
(F.24) with T:; and T as in Remark 4.2. For constant. A, 9, h, and I, t.he operat.or T:; = T is a proper stable rat.ional transfer fUllction, and Lemma FA reduces
to LemmJ:t 2.11 from [142].
0
Comparing the zparametric model (6.24) to (F.I), we see that Lemma F.l is directly applicable as a design tool. By .filtering, we C8Jl transform the dynamic parametric model (F.l) into the static parametric model (F.G), where y£(t) converges to zero and Ylt Y2 and 113 are available. Hence, with the NonHneur Swapping Lemma at hand, the design of swappingbased identifiers for the zparametric models is easy. A natural qnestion arises~ How to use this lemma, for example, fo1' the :epal'ametric model .1: = J(x, u) + F(:V,1.l.)T(J, (F.25) where the complete state x is available? This system obviously does not match the fOfm of the system (F.l) in Lemma F.1. What is needed to bring the xmodel into the form (F. I}? First, we need a presence of jj instead of 0, we need an exponential1y stable homogeneous part, and we must remove J(x, 11) because only "disturbances" converging to ~ero (represented bye) are allowed in (F.l). Namely, we would prefer to have the model .
T
X = A(x, t)x + F(x, u) 0 I
(F.26)
where A(x, t,) is exponentially stable for each x continuous in t. To ohtain this, we define .1: = x  X and introduce
1: = A(x, t)(x 
x) + f(x\ u)
+ F(x, u)TiJ.
(F.27)
Now with system (F.2G) in the form (F.1) we can perform OUI' ident.ifier design according to the filters and the static parametric model in Ule Nonlinear Swapping Lemma.
Remark F.6 Lemma F.l is given in the form convenient for analysis. For design, instead of the ""filter we use
no
=
+ 0(=, t)Hr(z\ t)TiJ Y4 = h(z, t)Oo + l(z, t)H'(Z, t)"1'O, A(z, t)Oo
(F.28)
and instead of (F.6) we use
Yl
+ Y4
 Y'l9
= Y2 0+ y" •
(F.29)
o
520
ApPENDIX
F
Ifhc following lemma is essential in proofs of convergence in swappingbased schemes where the identifier is not designed directly from the error system but from the plant model.
Lemlna F.7 Let T; : u ~ (il i = 1,2 be linea7' timevarying operator,li defined by (F.30) where Ai : lR+ + m.nxn a1Y~ conlinu,mul, bOllnded and exponentially stable. SUP1)oSe 8 : lE4  rnp is diffe,,.entiable, qJ : m+ + lRp xm is piecewise continu.ou.s and bO'lmded,. a.nil Al : rn.+ ~ ffi.nxn is bounded and has a bounded de11.vati1le on R+. If 9 E £2, then
(F.3l) If, m07'Covc1', 1'1(1.) is non.r;ingular "Vt and 111 1 is bounded and has a bounded dC7'ivative on JR+ then. (F.31) holds in bot.h directions. 1
Proof. Suppose that Tl[tjJT]O E £2' By Lemma FA, TdljJTO]  TdrbT]O E £2 b. b. and therefore (I = TdrpTO] E £2. \Ve will show first that (2 = T2[1I1ljJT9] E £'1' By substitut.ing t/J1'O = (1  Al (t)(1 into the variat.ion of constants formula and applying partial integration, we calculate
(2(t) =
= =
cD 2(t,0)(2(0)
+ .10f'
+ 10' f!l2(t,r)llJ(r) [(I(r)  A1 (r)(I(r)] dT rI>:!(t, 0)(2(0) + 1I1(t) (I (t)  ID2 (t, 0)1I1(0)(J (0) 1 + 10
(F.32)
where c!'2(t, 7) is the stat.e transitiolllllatrix of A2(t) that satisfies IcD 2(t, T)b :::; ~:en(t7), ~:J 0° > O. It is clear that
110' iJ!2(t, r) [il(r) + A,(r)M"(r) $;
IlkI + A2~{ 
M(r)A,(r)] (I(r)drl'
l'J A111~k~2
10' e 2o (tT) 1(1 (r)1 2dr,
(F.33)
then simiJarly to (F.18)(F.20) from the proof of Lemma FA, we can show that the expression (F.33) is in £2. Thus (2 = T2[1I1¢T8] E C2 • By Lemma FA, T2 [1I1ljJT8]  T2 (1I1t/>TJO E £2 and therefore T2 [1I1c/i1'8] E £2. The proof of the other direction of (F.3l) is ident.ical (whel1l1f(t) is nonsingular 'fit and 1\11 is bounded and has a bounded derivative on lE4). 0
Appendix G Differential Geometric Conditions
For completeness, we now derive necessary and sufficient conditions under which a nonlinear system can be transformed into one of the canonical forms considered iu this book. These conditions represent coordinatefree charaderizations of classes of nonlinear systems suitable for our bacl{stepping designs. The differential geometry baclcgl'OUlld required for a full understandiug of this appendix is contained in Isidori [53] and Nijmeijer and van del' Schaft [144].
G.l
PartialStateFeedback Forlns
\Ve first give geometric: conditions which are necessary and sufficient COl' singleinput singleoutput nonlinear systems of the form
( =
Io(()
y =
h(C)
+ f.Ojfi(() + ~«() + t.0jDJ«()]
II
(G.I)
to be transformable via a parameterindependent diffeomorphism into tbe canonical forms of Chapters 3, 4 and 7. The first form considered is obtained
522
ApPENDIX
by setting k = 1 in (7.180) (so that Xl
a:2
= [a:1,."
I
x mt ]):
l'
=
X2
=
Xg
:i: mt =
.'em
G
+ /;'0 1 (Xl) + EOj/;'J,l(Xl) 1
j=1 P
+ !PO,2(Xll.1:2) + L: ()j/;'j,2(Xb 3:2) j=1
X m1 +l
+ 'PO,flI, (Xl t· •.
P
t
X ml )
+ L ()j'Pi,rrIl (Xl \ ...
1
X mt )
J=1
p
XliII +1
:=:
X ml +2
+ /;'O,ntl+l(X"') + E8j !pj,ml+1(Xm )
(G.2)
j=1
p
Xp l =
'''C(I
+ /;'O,pl (Xm) + L Bj'PJ,pl (Xm) i=l II
.1::p
=
.'ep+l
+ tpO.p(Xm) + E ()jtpj'P(XIll) + bm/3{xm)U j=l
p
.i~n =
tpo,n(Xm)
+ L OJ/;'i. (X
fII
I1
)
+ boP(xm)'U.
j=l
We start with a result for the case when 0 = 0 a.nd u
= O.
Proposition G.1 The system
( =
fo(() y = h«)
(G.3)
can be transformed via a global diffeomorphism ."C feedback farm
Xl = :i: mt  1 = xn1, =
3;11
=
11 =
if and only
if the fallowing
= ¢( () into the partialstate
X2
X ml '''C m1 +l
+ 'PO,mJ(Xl
/PO.n (.x 11 ••• ,
1 '"
,xm1 )
x m1 )
Xl
conditions
B1'e
globally satisfied:
(G.4)
523
DIFFERENTIAL GEOMETRIC CONDITIONS
(i) rank {dh, d{L/oh), ... , d (Lj;lh)} = n
(ii) [ad}nT\ ad}!l,.] "') tlte vect· ( nz 07
t
0 ~ i ~ n  m'l  1, a.nd
= Ot
fi eIds 7,• adJor"
2l
•. t a d/n
ma
T,
n
a.d10
ml
+ 1r, , .. ,adn/0 11" are com
plete, 10he,'e .,' and fa are the vector fields define.d by
L Li h _ { 0, i = r
/0

1, i
o... ,'n 
=n 
2
10 = fa  (Lj./I.) 1',
1,
(G.5)
Proof. Sufficiency. Condition (i) implies that the change of coordinat.es Xi = L};l h() I 1 ~ i ::; 71., is a local diffeomorphism transforming the system (G.S) into the system
:b
x:!
=
(G.G)
\:01 = Xn .:\'n = p(X) 1/ =
Xl,
where p(x) is the function LjolJ e)..llressed in the coordinates of (G.G). _The definitions (G,5) imply that in these coordinates the vector fields r ~LDd fo are e...xpressed as
a
1"=, aXn

a
a
aX.1
0Xnl
(G.7)
/O=X2+"'+Xo .
Hence, the vector fields ad~o7" 0 ~ i ~ 'n  1, become
. , 0 adjf' = (1)1tl aX'Oi
I
0~
i::; nl.
(G.B)
vVe now show by induction that condition (ii) implies that X,"I+1I"·' Art can be replaced by new coordinates x mt +1t ••. I X'n such that ill the xcoordinates (with Xl = :Xt,. .. I X ml = Xml) tbe system (G.6) takes on the form (G.4) and, moreover, the vector fields ad}o 1" 0 ~ i ~ n  m It become
(G.9)
• First induction step (i = n): In the coordinates of (G.B), the vector field ad/oT is expressed tlS ad/oT
'11
8
a
= [ j=l L:Xi+la" +t.t(X)a, ' AJ Xn
a1
8 = I
X'O
a
a :\:'01 I

ap
a
An
An
a, hJa , . (G.lD)
524
ApPENDIX
= 0 implies that
Then, the condition [r, adIn '1']
Hellce~
G
the fUIlction #(X) can be e.'Cpresscd as (G.12)
Let us tben define the new coordinate
Xn
as (G.13)
In the coordinates {:\h ... , X,,!, Xu} the system (0.3) becomes
Xj =
Xi+l,
l::;j::;n2
= Xli + P(Xh"" Xn]) :\,. = #1(X1, •. " \n.)
Xn1
1J
and the vector fields
=
(0.14)
XI,
Min"" become
7',
a aXn ,
7'=
a
adlo 1' =    . DXnl
(G.15)
• Induction hypothesi.s (i = k + 1, ml + 1 $ k ::; 71.  1): Assume that we have repla.ced \k+l ... Xn by new coordinates ';k+l,' .. ,.;" such that ill the (XII"" X,·,';"·+II·" ,c;fI)coorclinates the systenl (G.3) is c.'Cpressed as I
Xi = Xk = ~k+l =
en = Y = and, moreover, the vector fields
Xi+1 ,
l::;j::;kl
+ ",,{\:I, ... ,Xk) ~1.·+2 + lIk+lC\),"" xd ~k+l
",. (Xl, ... , Xk) Xl,
a.d}oT, 0::; j
::; n  ~~, are expressed. as
(G.16)
525
DIFFERENTIAL GEOMETRIC CONDITIONS
=
• Induction proof (i I.'~ 1111 + 1 ~ k ~ the vector field adioI.·+l'1' is expressed as
11
1): In the coordinat:es of (G.16),
..J = [f.JO, ad,lk 10'
rl  k +1 l' ad10
ri: U=I
=
+
Xl+1 111
2:
aa. + ({k+l + Vk) aa AJ
:\k
(';j+1
+ l l j )  + '''n, (_l}lI"'_ lJ~i 8f." 8,,,.
j:: 1.'+ 1
(_l}rI"'+1
=
8
(_a_ + {)'\A'J
8
81)k
~+
8;(1,.8x)..
{) ]
t
8Vj~).
(G.18)
i=k+l Otk 8f,j
Then, the condition [arljc~"'1" adj;"'+J 7] = 0 implies that {)'!.,,,.
;f = lJX"k
0
=> '''it\}, ... , \.k) =
Ili,l (\11"
 ''''1) 1
+Vi,2(:\.1," ., .\'1.'1 h}. k ~ j ~
Let us then define {k
(.j
= =
\k 
';j 
r 10
U10 110W
Xk  1
(G.19)
coordinates {I.', ... I en as 6
_
'''k,::{n, .. · ,:\"2, s)ds = :\1.'  V"'1 t\:l! ... 1 'I.·d
(XI:t
Jo
'J!.
s)cls,
lIj.2(Xh ••• 1 \1.'2,
k + 1 ::; j ~
(G.20)
fl .
In tbe coordinates (XII"" Xj~h(J.·, ... ,en) the system (G.3) becomes ~'j
=
:h'1 =
111 oreover ,
,.li ...
lur/oI
{It
t
1~j ~ k 2
+ OkI (Xl t·· ., Xkl)
(/.: =
ek+l
€I) =
Vn(Xh ... , XItI)
11
and,
:\:j+l
=
+ iil.·{Xl,""
:\1,
the vector fieJds adi) 1', 0 ::; j ~ n.  k
= (l)il!8t: . \.nJ
t
(G.21)
XItI)
+ I, become
O n l+1,. = (_l}Uk+l_ 0< J' < n'"I t ' lad ??) l) 8 ' _ , (G ._:\:1'1
Thus, we have shown that. ('ouditions (i) and (ii) guarantee the local existence of tl diffeomorphism :l' = r/J(() transforming (G.a) into (GA). Furtbermore, (G.B) and (G.22) imply that in the :vcoordinates we have
(_1)i_8_ , 8.r n i = (_1)i_8_, =
OXni
(G.23) on 
1111
+ 1 :5 i ~ 11 
1.
526
ApPENDIX
G
Then, from condition (iii) and [162, Corollary 2.4] we conclude that this diffeomorphism is global. Necessity. If there e.'tists a diffeomorphism x = t/J«) that tra.nsforms (G.3) into (GA), Olle can directly verify that the coordinatefree conditions (i)(iii) arc satisfied for the system (GA), and hence for the system (G.3). 0
Theorem G.2 The system (G.l) can be tmns/armed via a global parometerindepefJ.dent dilJe01n01phism x = ¢( () into the partia1.9tatefeedback fonn (G.B) if and on111 if, in addition lo the conditions a/ Proposition G.l, the following conditions
(iv)
[h
1
an~
globallll satisfied:
ad}ol'] = 0, 0:5 i :511 ml I, 1:5 j:5 p,
(v) d (LJJL}oh) E spall {dh, ... , d (L}oh)} , O:5·j:5
m'l 
2, 1:5 j :5 p,
('IIi) [gj,ad}or] = 0, 0:5 i:5 n.  ml 1, 0:5 j :5 p, and 11
m
j=l
i=O
(vii) 90 + I:8j gj = fj(.) L bi (I)iad}o", 10here (3(.) is a lJrn,ootil nonlinear function and 7' is the 'lJector field defined by
(G.5).
Proof. Sufficiency. In the xcoordinates of Proposition G.l, condition (iv) becomes (cr. (G.23)):
8
_] =0, 0:5i:5nmt1, l:5i:5p. [h,(_I)i_ 8X n i Hence, the vector fields
!; are e.\.pressed in the xcoordinates as 8
n
!;
(G.24)
= L'Pj,i(XIt,,,,xm1)a.' i=l
1:5; :5p.
(G.25)
Xl
Furthermore, since in the xcoordinates we have .T.i+l 'Pj,i+l (XI, ... ,.'e rnl )
= L}oh,
0:5 i ::; ml

= L/JL}oh, 0::; i :5 7nl
1 
1, 1:5 j :5 p,
(G.26)
condition (v) becomes
d'Pj,i E span {d.1!h ••• , dxd, 1:5';:5 771.1 Combining (G.25) with (G.27), we obtain

1, 1:5 j ::; p.
(G.27)
527
DIFFERENTIAL GEOMETRIC CONDITIONS
Similarly, conditions (vi) and (vii) imply that in the xcoordinates we have (G.29) From (GA), (G.28) and (G.29) we conclude tbat in tbe xcoordinates, which are globally defined, the system (G.I) is el:pressed as (G.2) with 'PO,i == 0, 1 :5 i ::; m'l  1Necessity. Again, it is straightforward to directly verify that condit.ions 0 (iv)(vii) are satisfied for the system (G.2). We now proceed to another partialstatefeedback form, ill which the measured variables are XIII = [Xh ... , x ml , xr]: P
Xl =
X2
+ 'PO,l(XJ) + I.: 8j 'Pj,1(xd j=l p
X2 =
Xa +
+ L 8j 'Pi.2(:Z:1, :r2} j=1
Xql
=
P
Xq
+ 'PO,qJ (Xit.· . ,Xq1) + L: 8j 'Pj,91 {Xb'"
t
.1: q_l}
j=l p
.fq =
Xq+l
+ 'Po,q(Xt, •••
t
r X q , :t }
+L
8j
j=l
p
.i:m1 =
Xml+I
+ !.pO,J1II (Xl1 ••• , X mI , ."tt) + L
(Jj'Pj,nll (XII' •• ,J.· ml , Xr)
j=l p
XmJ+l
=
X ml +2
+
(G.3D)
i=I p
Xpl
=
Xp
+
m (Jj<'oj,Jl_l(X )
j=l p
Xp =
IPO.p(x
m} + E (JjIPj,p(x m) + bmP(xm)u j=1 p
•r
X
= 4iO(Xh'"
I
X q , Xr)
+ 2: O/Pj(X}, .. _,Xq , Xr} j=l
Y =
Xl
Comparing (G.30) with (G.2), we note several differences: In (G.30) the zero dynamics subsystem (xr) is nonlinear, its states are measured. and they
528
ApPENDIX G
enter t.he nonlineaIities of the :i:q, . •• ,.i;p equations. In contrast, the states XP+ll .•• ,.T.n of the linear zero dynamics subsystem of (G.2) are Hot measured and do not enter the .7.11' , . I xp equations,
Proposition G.3 The sll.r;tcm =
fo(()
Y =
h(()
(
+ 9o(()U.
(G.31)
can be transf07'111·ed via a difTeomorphi.flm a; = t/>() , 'which is satisfied in a neighb01'lwod U of a point (01 into the partialstatefeedback fO'l1n :i: 1 =
:i.'mll
=
xrnl = :Cpl
.i:p l·r .,
y
where q ~ Ut ~U:
111.1 J
= = = =
X:z
x ml .1: ml +l
+ IPO,nJl (·1:11' , ,
IPO,pl (Xl,
cJl O( XI, '
'''[77111 :r:r)
(G.32)
... ,a:"'I' .1;r)
'PO,p(Xlt •.•
,1:1
!
+ f3(X1t • , •• X ml
,X77I11 xr)
t
.l:r)u
•• , X ql xr)
I
if and only if the following conditions are va.lid in a. neighborhood
(ii) the distri.bu.tion QPq = span {gO! ad/n90' ... , ad~;qgo} is iTwolulive and of cons/.eml rank p  q + 1, (iii) [ad~ojjo, ad}!llio] (iv) [go,ad~ugo]
= O.
=0
t
0~i~p
0 ~ i ~ p
Inl
7nl 
1, and
I,
where the vector field Yo is defined by
go
I = L LP1/ 90 . IJO
/0
(G.33)
1
Pl'oof. Sufficiency. It was proved in [118, Proposition 10] and in [13] that conditions (i) and (ii) guarantee the existence of a local diffeomorphism ~ =
529
DrPFERENTIAL GEOMETRIC CONDITIONS
~«(), with Xi = L}~lh«}, 1 ~ 'i. ::; p, which transforms the system (G.31) into the system
i, = ~'pl
= Xp = :\:r = 11 =
X'l
,
M
(G.34)
Xp
Il(X} + .BhJ1l O(:\:b ••. , \:ql \:T) XI,
where Xr = [X,,+h ... ,x,llT and Il(X), .B{X) are the functions Lin" and Lg(lL~~l h expressed in the coordinates of (G.34). The definit.ion (G.33) implies that in these coordinates the vector fields 90 and Yo are e.'\.'"J>ressed as 90
_
a
= {3{X}7j Xp
90=
I
a
8 Xp '
(G.35)
We now show by induction that conditions (iii) and (iv) imply that XmI+11 ••• , Xn can be replaced by Ilew coordinates x m1 +11 ... I.'1'n such that in the xcoordinates (with 3:1 = XIt ... , X'nlI = Xm I ' xr = Xr) t.he system (G.34) takes 011 the form {G.32} and, moreover, the vector fields ad~)go, 0 ~ i =::; p  mIl become
.
adinOo =
, a
(_1)1~ UXp_i
1
0~i
~
p  ntl.
(G.36)
• First induction step (i = p): In the coordinates of (G.34) the vector field adinDo is expressed as
(G.3T)
Then, the conditions [Yo, adlilDo]
= 0 and [go,90] =
0 imply that
(G.38)
(G.39) Hence, {1(.} is independent of Xp and the function 1lC\:) can be expressed as (GAO)
530
ApPENDIX
Let us then define tbe new coordinate Xp
= 'Xp )0r
Xp  1
G
X'p as
Jl.:1(Xl,"" Xp21 s, ;{)ds
A == X"  P(XI, .. ' Xpl, X'). (G.41) t
In the coordinates (Xh« .. , XPl: XPt ){)~ the system (G.31) becomes
:\:i
=
Xi+1,
XpI
=
Xp + j2(Xl,« .. , Xp_I,X r)
:\p
=
.£1,1 (Xlt ... ,Xplt Xr)
it
=
tI>O(:Xll ' • Aq, x'}
Y
=
Xl,
1 $. i $. p  2
+!3(Xll'
« •
,'Xpl, xt)u
(GA2)
a
(G,43)
« ,
Do, ad/Duo become
and tIle vector fields
 = _a axp
go
ad10Yo =    . aXpl
I
:s
• Induction hypothesis ('i = k + 1, m 1 + 1 k ~ p  1): Assume that we bave replaced Xk+I"" Xp by new coordinates ele+1,«'" p such that in the (Xli' .. ,Xk,';"+b'" '';Pl xr}caardinates tbe system (G.31) is expressed as
=
Xj XL:
ek+l =
e
1$;j~k1
Xi+ 1 ,
e"'+l + VIt(;\:l~ , •. ,XJ.:, X') el.:+2
+ Vk+l (Xl, ...
I
Akl
X') (G.44)
fop =
l/p(X) , ' .. , Xk, A.r)
Xr
=
(lO(XI, ' .. I Xql X')
Y
=
Xl,
+ P('tJ, ... ,Xk, Xf)U
and, moreover, the vector fields ad}oUo, 0 $. j ~ p  k, are expressed as
fJ adi 9a == (_l)i_ _ 10
,
a~p_j
05 j
~ p
ad P
k 1,
k
/0
7'
= (l)Pk~ 8Xk"
(G 45)
• Induction p'l'OoJ (i = k, ffll + 1 ::; k ::; p  1): In the coordinates of (G.44), tbe vector field adi;'=+llio is expl'essed as
adj;k+llio
= [/0
~
i
ad'};kgo J
a
l
E Xj+l 8 AJ,. + (~k+l + lit) '=1
+11 
vAk
pI
+
E
{)
({j+l
j=J.'+l
+ Vi)
£\t; .
V~J
a + L ()O', a (_l}Pk_ a] i=p+l J ax.j aXk
pa~p
=
a A.,
(_l)pl+l
n
(_8_ + 0"X.kl
Bilk
~+
OAk 8Xk
t
j=k+l
8Vj~). 0XI: fJ{j
(G.46)
531
DIFFBRBNTIAL GEOMETRIC CONDITIONS
Then l the conditions [adj;kjjo 1 ad~;k+lgo] = 0 and rOOt that a'll)'
~ 8 XI;
= 0 :::} IIj(Xh .. ·
1
adj;kgoJ =
0 imply
Xk, XT) = lIi,J{:\:h ..• I AJ.·_l,Xr ) +lIj,2(Xl," . ~ XI:l, Xr)Xk, k'5j '5P (G.47)
tJ{3 8 (:\1, .. , I XI.·, :{) = XI.
o.
(GAB)
Let us then define the new coordinates (1:1"" {p as /XkI
(k
:=
~
:=
Xk10
~kI
eJ.10r
IIk,2(;\:1,. Vit 2(:\'I,
Of
.'Ic'l, s, xT)ds
,) •
... 'Ak!!I s, x')ds,
6 = XI: 
iikl(Xh ... , XlIt X')
(GA9)
I~ + 1 '5 j '5 n .
III the coordillates (XJ,' .. ,XkJ I (kt' .. I {PI X'), the syst.em (G.31) is expressed as
Xi
=
:\:;+1,
1'5j'5~~2
T :h'l = {It + VkJ (\:1, ..  , Xkll X ) (k = {1'+1 + Vk(XI, ... Xkl, XT) I
(G.50)
ep = X'
vpCn, ... ,Xklt X + ,8h~ll ... f
)
XlI, Xf)71
= tPO(Xl \ •.. 1 XIJ1 X'}
Y =
Xl,
and, moreover. the vector fields ad}ooot 0 adi10 90
I
= (lV!} a!~ . ~PJ
I
0
'5 j '5 P  A: + I, become
< J. < __IPk+J =  P  k , tLU 10 go 
8_ (_1)Pk+l_
8 '
:\:1.'1
. (051) .
o The necessity is again straightforward. The COllditions of Proposition G.3 are necessary and sufficien1: on1), fot" the local existence of a. diffeomorphism tral1sforming (G.31) into (G.32). At this time there are 110 necessary and sufficient conditions [or the global existence of such a diffeomorphism. Of course, the global validity of conditions (i)(iv) of Proposition G.3 is necessal)" as are the completeness of the \'ector £elds adJp 90, 0 :::; i :5 p  7111  1, and tile connectedness of the manifold ill = {( E lR": h«() =L/oh(C.) = ... = L,/;lh(() = O}, as proved in [13J. Therefore, to formulate the cOlluterpart of Theorem G.2, we make tbe following assumption: Assumption G.4 Tile system. (G.31) can be transformed via a global diffeomorphism x = 4>«() into (G.9S).
532
ApPENDIX G
Theorem G.5 Under Assumption G.4: the s1/.dem (G.l) can be f.ransJonned via a global pmnmetc1''illdependenl diffeomorphism .1~ cjJ( () into the partialstatcfeerlbll.ck Jorm (G.30) if and oni!/ iJ the foliD'll/ing conditi()n.r; arc globallll satisfied:
=
(i)
[Ij ad}olio] 1
= 0, 0 $ i ;;;; p 
m,1 
1, 1 $ j $ p,
(ii) t1 (LfJL}uh) E span {dll , .«. ,n (L~oh)}, 0 $ i $" q  2, 1 $ j $ p, (iii)
[li, ad~.go] E (l = span {YOl adfnyo,., . 1$ j
~
f
acl}oyo}, 0 $ i $ p  q  1.
p,
(itl) [9j' a dj.. 90] = 0, 0 $ i $" p 
1, 0 $ j ~ p, and
7111 
1)
(v) 'L,6j Uj = (b m  1)yo, i=l 1JJhe7'C
Yo is the vector field
defincd in (G. 93}.
Proof. Sufficiency. In the xcoordinates, which are defincd glohally by APrsumption GA t we have
ad}ooo
0 ~ i $ p  ml
(G.52)
f)
(G.53)
= (1)i_ a , aTpi
f) g'. = spall { 8
I • ,
'l'p
«,
}
!:}
,
vXpi
0 '5:. i '5:. p  q.
Hcnce, condition (i) becomes (G.54) which implies that the ,'cctor fields
Ii are e..'\."}ll'cssed in the xcoordimttes as (G.55)
Since in the xcoordinates we have a;H.1
'Pi,i+l(.t'j" •• , X m1 ,
xr)
=
L}oh,
0 $; i '5:.
= LfJL}lIh,
0
1111 
Si 5
1
711) 
1, 1 $ j ~ p,
(G.56)
condition (ii) becomes
d'Pj,; E spall {dx., .... dXi} , 1 '5:. i
5
q  1, 1 5 j ~ p,
(G.57)
533
DIFFERENTIAL GEOMETRIC CONDITIONS
or, equivalently, 8CPj,i _ 0 8:Vk 
i t
+ 1 $ ~. $
11,
1 $ i $ q  I, 1
~j
$
]J.
(0.58)
From (G.53) ~lncl (G.55) we see that condition (iii) is equivalently expressed in the J'coordinates as
[8 a.' IJl E span {a~.c '''C _ 1 P
1""
p
a~ .}, 3.· P_ 1
0 $ i $ p  q, 1::; j $ p! (G.59)
which implies that
Dcpj.i
=
D.J:I.' 8tpj,; 8:r:/r
=
0, i + 1 $ II! $ p, q $ i ::; p  1, 1::; j ::; P
(G.GO) 0, q + 1 ::; ~: $ p, p + 1 ::; i $
11,
1
~
.i $
p.
Combining (G.55), (G.58), and (G.GO), we sec that t.he vettor fields fj, 1 $ j ::; p, are expressed in tile .1:coordinates as
Ij
=
Similarly, conditions (iv) and (v) imply that in the .1!roordinatcs we have
(G.G2) From Assumption G.4, (G.Gl), and (G.62), we conclude t.hat: ill the .Tcoordinates t.he systcm (G.1) is expressed as (G.30) with CPo,; == 0, 1 $ i $ m'l  1. The proof of necessity is straightforward. 0
G.2
OutputFeedback Forms
Setting 1111 = 1 in (G.2), in Proposition G.1, and in Theorem G.2, we obtain thc following corollary, which was proved in [122, 121]:
Corollary G.6 The system (G.3) can be tronsj01'7ncd via. (/ global param,ete1'independent diffeom.orphism x = !/J(() into the paramet1i.c outp'utjeedbackjo11n (7.101) if and only if the following conditions hold globally:
(i) rank {dh, d (LJoh) I ' " ,d (Lj;lh)} = n,
534
ApPENDIX G
(ii) [ad}or,
ad}!1,,] = a, a ~ i ~ n 
2,
0,
0 ~ i:5 n 
2,
(iv) [gi' ad}o"] = 0,
0:5 i ~ n 
2, 1 ~ j :5 p,
(iii)
[fj,ad}o1']
=
p
1:5 j:5 p,
m
(1Ji) 90 + L,(Jjgj = f3( ,)~bi( l)iad}o", and j=l
(vii) the 1Jeclo7' fields
i=O 7',
ad/07', ... ,adji~Jl' are com.plete,
whe1'C f3 is a smooth nonlinear junction anti" is the 1Jector field defined by L Li h = { 0, ~ = 0, ... ,n  2 r 10 1, 1 = 11  1 .
(G.G3)
Corollary G.G gives necessary wId sufficient conditions for (G.3) to be globally transformable into the parametric outputfeedback canonical form (7.101) via a pU7umeterindependent diffeomorphism. However, this would unnecessarily exclude numy systems such as the robotic example of Section 7.3.3, for which a ]Jaramete1'dependent diffeomorphism is needed to go from the physical coordinates into the outputfeedback form. In the fullstate feedback case, we need parameterindependent diffeomorphisms, because we want to be able to calculate the new state variables from the measurements of the original ones. \\Then only the output is measured, the dependence of the diffeomorphism 011 the unknown parameters is acceptable because t.he states do not appeal' ill the control law . Therefore, we now give necessary and sufficient conditions for the system ( = f((;0)+g((;8)11 (G.64) y = h((dJ) , where 8 is a vector of unlmowll parameters, to be globally transformable into (7.101) via diffeomorphism which is allowed to depend on the unknown parameters. The following result was first given in [72J:
Corollary G.7 The system (G.64) can be transformed via a global diffeomorphism x = ljJ((; Ii) into the outputfeedback canonical jorm. (7.101) ij a.nd only if the following conditions a7'(~ satisfied f07' all ( E JR." and jor the true 1}o.luc of the pam.meter 'llcctD1' ij:
535
DIFFERENTIAL GEOMETRIC CONDITIONS
(iii) adjr
= nl t; [ tp~.nj(y) + ~P 6j fPj,ni(Y)]
(I)ni adjr,
'whe7'e 'Pj,ni(Y) == iou fPj,n_i(s)ds, 0 $; i $; n  1, 0 $; .i ::; p,
(iv)
[u, adir] = 0,
O:S; i ~ n  2,
fit
(11) 9 = {3(.) 'l)i(I)iad}r, and i=O
(vi) the 'llector fields 'r, 2:tdfl', . . , ,adjI r an~ com.plete, where {3 is a smooth nonlinear junction and r is the 1Ject07' fielll defin.ed by i I _ { 0, i == 0 , ... , n  2 L r L f)'. 1, I=TJ1.
G.3
(G.65)
FullStateFeedback Forms
III this section, we consider the fullstate feedback case and, hence, we require the diffeomorphisms to be parameterindependent. The result.s in this sectiol1 were first obtained in [69], except for Theorem G.9, which was first given in [1]. Setting k = I, ml = pin (G.30)(G.32), in Proposition G.3, and in Theorem G.5, we obtain the following corollary: Corollary G.S There exists a parameterindependent diffeomorphism x = r/J( (), satisfied in a neighbol'hood U of a point (0, which tmnsfonns {G. 1) into the form
.1:1 = 2:2 = Xql Xq
X2
+ IPf(Xl)B
X:J
+ 'P2(JT (Xl, X2)(J
+ tp~_l(Xll'" = Xq+l + 'P:(Xl1'" =
Xq
,Xql)6
(G.66)
, Xql ;Jl)9
+ t.p~1 (:'Cl' ••• Xpl , xr)9 .ip = 'Po,p(X) + t.p;(x)6 + f3(x)u P if = q,o(Xlt ... ,xq , x') + L 8j lbj (x., ... ,3:1'/1,1:')
Xpl
=
Xp
I
j=1
Y =
Xl,
if and only if the following
conditions are valid in a 11 eighb01'hood Ut ;2 U :
536
(i)
ApPENDIX
LgnL}oh == 0
I
0~i~
G
2, L!/n L'j;lh # 0 I
P
(ii) the distribution gPq = span {gO, ad/ago, ... ,adj;qgo} i8 in'IJol·uf.ille and oj con.r;iant rank p  q + 1 , (iii) d (L/JL}/l) E span {dh,
(iv)
[h, ad)n90]
E
... ,d (L}ah)},
0
~ i:S;
q  2, 1:S; j $ p,
gi = span {go! ad/uUol"" adJoYo}, O:S; i :s; p  q  1,
1 $j:5 p,
(v) 9j
:E
0, 1 '5: j S; p.
For the diffeomorpbism of Corollary G.B to be globally valid, it is ll·ecessa1lJ that the above ('onditions (i)(v) be globally valid and that the manifold A{ = {( E 1Rn : h«) = L/uh«() = ... = L1;1 h«} = o} be connected. As ca.n be shown nsing tbe results of [13], these conditions, together with the completeness of the vector fields 90, ad/uUo! ... , adj;l YOt where go is defined in (G.33) and = In  L 10 90, are sufficient for x = ,pee} to be a global diffeomorphism. In the case q = 1, these conditions are actually necesso,nJ and sufficient [13, Corollal'Y 5.7]. However, for q > 1, the completeness of ad/ugo, ... , adj;lyO is not necessaly. For example, consider the system
10
Xl
=
:1:2
3:2 =
(G.67)
'U
3 Xa 
X:i = y =
')
a
X1X
Xl'
This system is already in the form (G.66), but the vector field
(x~ + x~xi) aBXa a a = +X38Xl 8xa
adiolio =
[X2 J::i8 UXt

1
!ol0
UX2
1
2
(G.68)
is not complete, since the solutions of the system .t! = 1 0 X2 ..,
=
.T3
=
(G.69)
.T5
starting from any point with X3(O) > 0 escape to infinity in finite time. "Ve now turn our attention to nonlinear systems of t.he form (G.70)
537
DIFFERENTIAL GEOMETRIC CONDITIONS
where h, gj1 0 :5. j ~ p, are smootll vect.or fields in a neighborhood of the origin ( 0 with h(O) = 0, 0 ::s; j ::; ]), 90(0) =F 0, and give necessary and sufficient conditions for (G.70) to be locally transformable via ~l. parameterindependent diffeomorphism x =