Perspectives in Robust Control (Lecture Notes in Control and Information Sciences)

OO

Ilf(~+Jw)ll2d~

< cc

} .

Thus, for any f E T/~ and g E 7-/2, (f, g) = 0. We caution that for each of these normed spaces, we use the same notation I1" 112 to denote the corresponding norm; however, use of each of these norms will be clear from the context. Finally, we denote by I~?Yoo the class of all stable, proper rational transfer function matrices. 4.2

Problem

Formulation

We shall consider the finite dimensional linear time-invariant system depicted in Figure 1, which represents the standard unity feedback, one-parameter

44

J. Chen and S. Hara

control scheme; the more general t w o - p a r a m e t e r control structure will be t r e a t e d elsewhere. In this tracking scheme, P represents the plant model a n d K the compensator. We shall denote by P(s) and K ( s ) their transfer function matrices; more generally, from this point onward we shall use the same symbol to denote a system and its transfer function, and whenever convenient, to omit the dependence upon the frequency variable s. T h e signals r, u, and y are,

K

I

U

~,

p

Fig. 4.1. The unity feedback system respectively, the reference input, the plant input, and the system o u t p u t . For any given reference r, a c o m p e n s a t o r K is to be designed so t h a t the o u t p u t y tracks r, while preventing the energy of u from being excessive. We a d o p t an integral square criterion to measure b o t h the tracking error and the plant input energy. This leads to the performance index J := (1 - e)

/0

Ily(t) - r(t)ll2dt + e

/0

Jlu(t)ll2dt.

Here e, 0 < e < 1, is a p a r a m e t e r to be determined a priori at one's choice, and it m a y be used to weigh the relative i m p o r t a n c e of tracking objective versus t h a t of regulating the input energy. Suppose, t h r o u g h o u t this paper, t h a t the system is initially at rest. Then, it follows from the well-known Parseval identity t h a t J = (1 - e ) l l ~ - ~'ll2 + e]l~]]2. Furthermore, let the system sensitivity function be defined by

S(s) := (I + P(s)K(s)) -1. It is immediate to find t h a t g = (1 - ~)llSr~l~ + ellgSr~l ~.

(4.1)

For the rational transfer function m a t r i x P , let its right and left coprime factorizations be given by

p = N M - 1 = ]~/-1/~r,

(4.2)

4

Tracking Performance

45

where N, M, N, M E 11~7-/or and satisfy the double Bezout identity [ )(-N~]

[MY]=/'

(4.3)

for some X, Y, )~, 17" E I~7-/~. It is well-known that to stabilize P every compensator K is characterized by the Youla parameterization [7] /C:= {K:

K=-(Y-

M Q ) ( X - NQ) -1

= _(f( _Qf~)-l(fz -Q2fl), Q ~ ]l~-l~}. For any given r, we want to determine the optimal performance achievable by all stabilizing compensators from the set/C, defined by J* := inf J. KEtC

Hence, for a nonzero e, the optimal compensator attempts to minimize jointly the tracking error and the plant input energy. In the limiting case, when e =- 0, J* defines the minimal tracking error with no regard to input energy [6], and the minimization problem coincides with one of cheap control [13,18]. For e = 1, on the other hand, it reduces to an optimal energy regulation problem [17]. We shall assume throughout that P is right-invertible, by which we mean that P(s) has a right inverse for some s. For a right-invertible P, it is wellknown (see, e.g., [1,2]) that each of its nonminimum phase zeros is also one for N(s). In other words, a point z E I!~+ is a nonminimum phase zero of P if and only if rlHN(z) ----0 for some unitary vector ~, where y is the (output) direction vector associated with the zero z, and the subspace spanned by 71 is termed the (output) direction of z. Let z~ C ~+, i = 1, -.. , Nz, be the nonminimum phase zeros of P. It is then possible [2] to factorize N(s) as

N(s) = L(s)Nm(s),

(4.4)

where Nm (s) represents the minimum phase part of N(s), and factor. A useful allpass factor is given by

L(s)

::

IIn{(s), i=l

Li(s) = [~?i Ui]

[

z~ ~+s ~_~=A 0

[~

L(s)

an allpass

(4.5)

We refer the details of this factorization to [2]; for the present purpose, it suffices to point out that the unitary vector y~ can be sequentially determined from the zero direction vectors of P , and U/forms together with r/i a unitary matrix. It is useful to note that Arm E ~ admits a right inverse analytic in r and hence is an outer factor of N.

46

J. Chen and S. Hara

4.3

Main Results

T h r o u g h o u t this p a p e r we consider the step reference i n p u t

r(t) =

v

0

t>0

t<0

where the c o n s t a n t vector v specifies the m a g n i t u d e a n d the direction of the input. W i t h o u t loss of generality, we take v as a u n i t a r y vector, I]v[] = 1, a n d call the subspace s p a n n e d by v the input direction. T h e Laplace t r a n s f o r m of r(t) is ~'(s) = v/s. In view of (1), it is clear t h a t in order for J to be finite, the sensitivity function S(s) m u s t have a zero at s ---- 0; t h a t is, either t h e plant or the c o m p e n s a t o r must c o n t a i n an integrator, a n d thus to p r e v e n t hidden instability, none m a y have a zero at the origin. This necessitates t h e following assumption. Assumption

1 P ( 0 ) has no zero at s ----0.

O n the other hand, to m a i n t a i n a finite e n e r g y cost, also clear f r o m (1), precludes the possibility t h a t K m a y have an integrator; instead, the required integrator must be in P . Thus, it is also necessary to p o s t u l a t e Assumption

2

P(s)

has a pole at s = 0.

We note t h a t A s s u m p t i o n 1 is s t a n d a r d in step reference t r a c k i n g problems, while A s s u m p t i o n 2 can be met in m a n y cases of interest, especially for, e.g., mechanical systems which typically exhibit r e s o n n a t e m o d e s [9,11]. We will now a t t e m p t to derive an analytical expression for the o p t i m a l p e r f o r m a n c e J*. To proceed, we first rewrite J as

J = (1 - e)

(X - NQ) MVs 22+ ~ (Y - M Q ) - ~ -

],

or equivalently,

J

=

[vq-4X-NQ)]M ] x/~(Y - MQ)

- -s

.

(4.6)

These follow from s t a n d a r d algebraic m a n i p u l a t i o n . F u r t h e r m o r e , we p e r f o r m an inner-outer factorization [7] such t h a t x/qM

] = O~Oo,

(4.7)

where Oi C ]~T/o~ is an inner m a t r i x function, a n d Oo c II~7-/o~ is outer. O u r m a i n result concerns plants which have no pole in C+.

4 Tracking Performance

47

T h e o r e m 1 Suppose that P(s)-

Po(s)

(4.8)

8n

for some integer n >>_1 and Po E R~-t~, such that Po(s) has no zero at s = O. Let z~ E ~+, i = 1, . . . , Nz, be the zeros of P ( s ) with output direction vectors w~. Define f ( s ) := (1 - e ) V T N m ( S ) O o l ( S ) O o T ( o ) N T ( O ) v

(4.9)

and factorize f ( s ) as ~=1 s i ( S i ~ )

fro(s),

(4.10)

where si E I1~+ are the nonminimum phase zeros of f ( s ) and fro(s) is minimum phase. Then, f ( s ) , fm(s) E I I ~ , and f(O) = f~(O) = 1. Under Assumption 1, N~

J* = 2(1 - e) Z

Ns

i cos2 Z(r/i, v) + 4(1 - e)Tr E

i = 1 Zi

1

(4.11)

-si- + Y o

i=1

where Jo := - 2 ( 1 - r

~ l~

(4.12)

0) 2

Furthermore,

Jo _> ( 1 - ~) ~

( 1 ~log

(4.13)

1 + ( 1 - ) , ~ ,)p[ ~) _,3e ,, ) d~.

Theorem 1 demonstrates that unlike in the standard tracking problem, the optimal performance herein generally depends on, in addition to the nonminimum phase zeros, also the minimum phase part of the plant. The effect from the latter is captured by the second and the third terms in (11), and is made especially transparent by the lower bound (13). Since Jo is nonnegative, this effect will in general lead to an increase in the minimal achievable cost. Indeed, while f ( s ) may or may not have zeros in C+, which is partly due to the input direction and hence the corresponding effect may disappear, the lower bound (13) shows that Jo will not vanish unless e = 0, that is, when the tracking error is taken as the sole performance objective. Note

also that when

We number

e -~ I, J* -* O.

now proceed to prove Theorem I. For this purpose, we shall need of preliminary results. Consider the class of functions in

IF :=

If : f ( s ) analytic in I1~+, lira R~

max

oeI-~/2,~/21

I'~ R

-- 0

1

a

48

J. Chen and S. Hara

The following fact is well-known, and can be found in, e.g., [19] (pp. 40). Lemma

1 Let f ( s ) 9 E and denote f ( j w ) = h(w) + jg(w). Then for any

oJo E (o,

~),

g(wo) _ 1 [ ~ h(w) -_h(___wO)dw. ~o

~j_~

(4.14)

~2_~o 2

L e m m a 2 Suppose that f ( s ) is analytic and has no zeros in ~+, and that log f ( s ) c IF. Then for any z E r

f'(z) f(z)

1/_ ~ l~ ~ ( j w _ z)2

(4.15)

-

Furthermore, if f ( s ) is also conjugate symmetric, i.e., f ( s ) = f(~), then

f'(O) _ f(O)

1 f~

log[f(jw)[ dw

27~ J _ ~

~-2

(4.16)

"

Proof. The formula in (15) follows from a direct application of Cauchy's theorem to the first order derivative of log f ( s ) , with an integral contour consisting of a semicircle of a sufficiently large radius in C + , and the i m a g i n a r y axis with appropriate indentions where f ( s ) may have zeros or poles. The second equality is obtained by taking the limit of f f ( z ) / f ( z ) with z -* 0, and by noting that the integrand in (15) converges uniformly to that in (16). 9 Proof of Theorem 1. Using the allpass factorization (4), we first obtain [ lVff-L~-c(L-IX - N m Q ) ] IVIv z

f =

L

vq(Y - MQ)

J

8

2"

It was shown in [6] that there exists some R C II(?-/~ such that

L-1X]~I = L -1 + R. Hence,

[ 14r:-~-~(L-1+ R - gmQi)] v : J =

L

v ~ ( y - MQ)IVI

J

Here Q c ]~T/~ is to be selected so that

[

v~-

~(I + n - NmQM)

v ~ ( r - MQ)2f4

1

J

V

-

8

E~2.

4

Tracking Performance

49

For such a Q, it thus follows that j=

+ [vff-e(I+R-NmQM)] x/~(Y- MQ)M 2

-

v 2 J s 2"

With L given by (5), it is known from [6] that Nz

(i-1 -- I) V-- i = 2 E ECOS2z(/]i, V). i = 1 Zi

8

Consequently, we have Nz

j. = 2E

1 cos2 Z(~i, v ) + J%,

(4.17)

i = 1 zi

where ~.:=

inf

QcRU~

07,

[x/1-e(I+R-NmQf4)] vf~(Y - MQ)M

~:=

v :

"

I

We now evaluate J. Notice first that from the Bezout identity (3), it is straightforward to show that Xl~

= I + PYI~.

This gives

R = L-I(X_IVI - I) = L-1PYI~ = NmM-1y]~, and in turn

j=

{[ X / 1 - e ( I + N m M - 1 Y M )

Define

F

E(s) := 1I I

E'(j~,)E(j~)

Then

j=

v / l - eNm]

oT(--S) - o~(s)oT(-~)J

v u

1

= I. As a result,

E{[x/1-e(I+N.~M-1YM)]

v~Y ]~/

[x/1-eNm]~

J-[

v 2

~/~M j Q]~/j s 2"

Denote

WI : = oH [ x/I - e(I + NmM-1YJ~/I) ] v~YM W2 : = ( I - O i O H) [ lx/T-~7--e(I+ gmM-1Yf4) 1 L x/~Y~l J"

50

J. Chen and S. Hara

We t h e n o b t a i n -

V

2

A direct calculation shows t h a t

W1 = Oo"

((1 - e ) N H + (1 - e ) N H N m M - 1 Y I ~

+ eMHyIFI)

e)OoUN H + Oo H ((1 -- e)NHNm + cMHM) M - 1 Y M = (1 - e)OoHN H + OoM-1Yl~, W2 = [V/I-c(I + NmM-1yI~) ] = (1 --

~yM

- e~Wl

Ix/1-e(I+NmjVI-1YI~)v~YM] - [j V,/-cNm] ~M 1 OolW1 [vfl-- s =

(1-ff)Nrn~)olOoHNg)]

_v~(1 -

e)MOolOoUNHm

[ i ~ - ~ ( x + ~pmP~-) -1 ] z

Since

1--e

H

1--e

H

--1

L-7-P~ (x + -~P,,,P~) J

OoM-1yI~I E ]~TI~, it J*

=

(l-e)

9

is clear t h a t

(0 o- H N~H - OoH(O)NH(o)) SV 22 +

2

W2v

2"

2

It then follows from a direct calculation t h a t J* = (1 - e) ~-/~oo 2 - 2(1 - c)Re = -2(1

- e)

(vTNm(j03)Ool(jw)ooH(o)NH(O)v)032 dw

--f/ Re{f(j03)} - ld~ ' oo

032

where f(s) is defined by (9). It is trivial to verify t h a t f ( 0 ) = 1. Hence in light of L e m m a 1, we have J* = - 2 ( 1

- e)Tr lim

Im{f(jw)}

w----~0

Since

f(s)

f(s) c IF,

and that

= - 2 ( 1 - e)TrIm{jf'(0)}.

03

is conjugate symmetric, this gives rise to

J* = - 2 ( 1 -

e)Tvf'(O).

(4.18)

It is easy to check t h a t

f'(O)- dlogf(s)

~s

I~=o= - 2 ~

Ns 1 + f m ( 0 ) -

i=1

s~

(4.19)

4 Tracking Performance

51

According to Lemma 2, however,

fm(O ) = ~ 1 / / ~

log[fm(jov)ldw 2

~1//~

log If(jov)l &v~_2

.

This expression together with (17-19) gives (11). To establish (13), it suffices to observe that

]f(Jw)l <_ (1 - e)y (Nm (jW)Ool(jw)Oo H (O)NH (0)) <_ ~

~ (Nm(jW)Ool(jw))

(I

j i)

~1/2 (Ym (jo2)Oo 1(jw)N H (jW)Oo H (jw))

= ~

= X 1/2

I +~

(Pm(jw)pH(jw))--1

1 1 + 1-e ~(Pm(jcv)) This completes the proof.

9

The above derivation yields also a byproduct useful for computing the optimal performance. The following expression for J* becomes clear instantly from (19). C o r o l l a r y 1 For f(s) defined by (9), let its zeros and poles be ai, i = 1, ... , Na, and bi, i = 1, ... , Nb, respectively. Then, J* = 2(1 - e) E 1 cos2/(r/i, v) + 2(1 - e)Tr i=1 Zi

1

i=l

aU -

1

i=1

~

. (4.20)

Thus, Theorem 1, with the aid of Corollary 1, serves to provide also a rather computable expression for the optimal performance, hence solving analytically an important case of the general "two-block" 7-12 optimal control problems. Additionally, for single-input single-output systems, Theorem 1 can be further strengthened, as shown in the following corollary. C o r o l l a r y 2 Let P be a scalar transfer function. Under the assumptions in

Theorem 1, J* -- 2(1 - e) N~2~__1 zi + ( l - e ) i=l

log

1+

(1- e)lP(j )l 2)

(4.21)

52

J. Chen and S. Hara

Proof. For a SISO system, f(s) = 1VT-ZT-eNm(s)Ool(s), which is minimum phase. Furthermore, If(jw)[ 2 = ( l - e )

1 ~ ip(jw)]2

~N(Jw) 2 = ( 1 _ { )

1 + ~i~lP(jw)l 2" The result then follows from Theorem 1. The implication of Theorem 1, and in particular that of Corollary 2, are now worth noting. First, while it has been long well-known that nonminimum phase zeros impose fundamental performance limits, these results show that certain zeros in the left half plane are also likely to play a significant role, when the performance objective takes into account of the input energy constraint. Indeed, with the objective under consideration herein, the achievable performance can be seriously constrained as well by those minimum phase zeros close to the imaginary axis. This is seen by noting that in the vicinity of such zeros [P(jw)t may be rather small, thus rendering the integral in (21) large. Due to the weighting factor 1/0; 2, the zeros close to the origin have more negative an effect. For the same reason, while in general the performance is affected by

-5(P(jw)) at all frequencies, the effect is particularly dominating in the low frequency range. It is clear from both (13) and (21) that a low plant gain at low frequencies will lead to a large value of J*. Consider the frequency band [0, wo], where s

(1 - {)-~2(p(jw)) <- 1, Assume also that for some constant

w e [0, w0].

a > 0

~(Po(jW)) < a,

w C [0, w0].

Then using the inequality log(1 + x) > x/2, valid for /0~

0 < x < 1,

we have

1 log ( 1 + (1 - e)-~2(p(jw)) e ) dw >_ /0~~ ~-2 1 2(1 - e)~2(p(jw))dw.

This leads to an estimate for J*, and in turn for J*: Nz 1 J* > 2(1 -

-

ew2n- 1 cos

+ 2(2n

1)a "

(4.22)

i~l zi

Thus, a low plant gain will duely limit J*. Intuitively, it then appears that the plant bandwidth would also limit the performance, since high loop gain and bandwidth are both required for tracking. This intuition is confirmed by the following result. C o r o l l a r y 3 Let wc be the crossover frequency for -~(P(jw)), i.e.,

-~(P(jw)) < 1,

w E [wc, c~).

4 Tracking Performance

53

Then~

J* > 2 ( 1 - e) ~N* zU1cos2/(~i, v) -- (1 -- e)log(1 -- e) 1.we

(4.23)

i=1

Furthermore, if for some k > 0 k

then 1 CO82 J* _> 2 ( 1 - e ) ~Nz z-~

L(~]i,

V) "~- 2 k e ~ .

(4.24)

Proof. To show (23), we weaken Jo to Jo >_ ( l - e )

log c

--

(

1 + (1-e)-52(p(jw))

-(1 - e)log(1 - e) --/~ __ld c 032

)

dw

"

Similarly, to establish (24), we note that

o

log

The desired lower bound may then be obtained by performing integrationby-part on the integral. 9 From Corollary 3, we conclude that the plant bandwidth necessarily imposes a constraint on the best achievable performance, which will only become nonexistent when either e --* 0 or e --~ 1. Hence, the result brings to light another class of fundamental design constraints, one that cannot be found in usual "one-block", "single-objective" optimal control problems, some of which constitute the extreme cases of the present formulation. Yet these constraints do result when more realistic, multiple design goals are taken into consideration. Under latter circumstances, the lower bound (23) indicates that the performance is in general inversely proportional to the plant bandwidth, while the bound in (24) shows that it is proportional to the high frequency rolloff rate of the plant; the faster does the plant gain decrease, the more difficult is to achieve a good performance. 4.4

Conclusion

In this paper we have studied an 7-/2 type optimal control problem which attempts to minimize jointly the tracking error in response to the step reference, and the energy to the plant input. Our work is partly motivated

54

J. Chen and S. Hara

by practical applications found in the control of mechanical systems, and is partly driven by our goal in search of fundamental design limitations beyond those known to be imposed by nonminimum phase zeros, unstable poles, and time delays. For this purpose, we have derived an analytical solution to the problem, which describes explicitly how the best achievable performance may depend on the plant nonminimum phase zeros, and more importantly, on its frequency response as a whole. The result enables us to conclude that in addition to the nonminimum phase zeros, the performance may also be significantly constrained by the bandwidth as well as the gain of the plant, and that certain minimum phase zeros are also likely to have an undesirable effect. These constraints will vanish when energy consideration is discarded from performance goal, and thus find no place in the standard tracking problems. We have focused solely on stable plants and one-parameter tracking scheme. It is nevertheless possible to extend this work to unstable plants and twoparameter control structures. Past work [6] has shown that two-parameter compensators improve tracking performance, especially in countering the negative effects by plant unstable poles. While this remains true with the performance objective being considered herein, it can be shown, however, that they cannot lessen the constraints resulting from plant gain and bandwidth. Thus, the constraints in this category are intrinsic of the conflict between output tracking and input regulation, but independent of control structure. Moreover, it is also possible to extend the present work to disturbance rejection problems. In particular, in use of a two-parameter compensator, it can be shown that a similar performance objective is constrained by unstable poles of the plant, and additionally, by its stable dynamics; the former is known from the standard energy regulation problem [20,17], while the latter can be characterized by a similar integral term. It can be found that plant bandwidth remains to be a negating factor, and that stable, lightly damped poles of the plant also have a serious effect. Finally, other extensions may be pursued for time-delay systems (see, e.g., [6]), and for tracking strategies with the so-called preview control [5]. These extensions will be reported elsewhere.

References 1. S. Boyd and C.A. Desoer, "Subharmonic functions and performance bounds in linear time-invariant feedback systems," I M A J. Math. Contr. and Info., vol. 2, pp. 153-170, 1985. 2. J. Chen, "Sensitivity integral relations and design tradeoffs in linear multivariable feedback systems," I E E E Trans. Auto. Contr., vol. AC-40, no. 10, pp. 1700-1716, Oct. 1995. 3. J. Chen, "Multivariable gain-phase and sensitivity integral relations and design tradeoffs," I E E E Trans. Auto. Contr., vol. 43, no. 3, March 1998, pp. 373-385.

4

Tracking Performance

55

4. J. Chen, "Logarithmic integrals, interpolation bounds, and performance limitations in MIMO systems," IEEE Trans. on Automatic Control, vol. 45, no. 6, June 2000, pp. 1098-1115. 5. J. Chen, Z. Ren, S. Hara, and L. Qiu, "Optimal tracking performance: preview control and exponential signals," Proc. 39th IEEE Conf. Decision Contr., Sydney, Australia, Dec. 2000. 6. J. Chen, O. Toker, and L. Qiu, "Limitations on maximal tracking accuracy," IEEE Trans. on Automatic Control, vol. 45, no. 2, pp. 326-331, Feb. 2000. 7. B.A. Francis, A Course in Ho~ Control Theory, Lecture Notes in Control and Information Sciences, Berlin: Springer-Verlag, 1987. 8. J.S. Freudenberg and D.P. Looze, "Right half plane zeros and poles and design tradeoffs in feedback systems," IEEE Trans. Auto. Contr., vol. AC-30, no. 6, June 1985, pp. 555-565, 9. S. Hara and N. Naito, "Control performance limitation for electro-magnetically levitated mechanical systems," Proc. 3rd MOVIC, Zurich, 1998, pp. 147-150. 10. S. Hara and H.K. Sung, "Constraints on sensitivity characteristics in linear multivariable discrete-time control systems," Linear Algebra and its Applications, vol. 122/123/124, pp. 889-919, 1989. 11. T. Iwasaki, S. Hara, and Y. Yamauchi, "Structure/control design integration with finite frequency positive real property," Proc. 2000 Amer. Contr. Conf., Chicago, IL, June 2000, pp. 549-553. 12. P.P. Khargonekar and A. Tannenbaum, "Non-Euclidean metrics and the robust stabilization of systems with p a r a m e t e r uncertainty," IEEE Trans. Auto. Contr., vol. AC-30, no. 10, pp. 1005-1013, Oct. 1985. 13. H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, New York, NY: Wiley-Interscience, 1972. 14. N. Levinson and R.M. Redheffer, Complex Variables, Baltimore: Holden-Day, 1970. 15. R.H. Middleton, "Trade-offs in linear control system design," Automatica, vol. 27, no. 2, pp. 281-292, Feb. 1991. 16. M. Morari and E. Zafiriou, Robust Process Control, Englewood Cliffs, N J: Prentice Hall, 1989. 17. L. Qiu and J. Chen, "Time domain performance limitations of feedback control", in Mathematical Theory of Networks and Systems, eds. A. Beghi, L. Finesso, and G. Picci, I1 Poligrafo, 1998, pp. 369-372. 18. L. Qiu and E.J. Davison, "Performance limitations of non-minimum phase systems in the servomechanism problem," Automatica, vol. 29, no. 2, pp. 337-349, Feb. 1993. 19. M.M. Seron, J.H. Braslavsky, and G.C. Goodwin, Fundamental Limitations in Filtering and Control, London: Springer-Verlag, 1997. 20. M.M. Seron, J.H. Braslavsky, P.V. Kokotovic, and D.Q. Mayne, "Feedback limitations in nonlinear systems: from Bode integrals to cheap control," IEEE Trans. Auto. Contr., vot. 44, no. 4, April 1999, pp. 829-833. 21. G. Zames and B.A. Francis, "Feedback, minimax sensitivity, and optimal robustness," IEEE Trans. Auto. Contr., vol. AC-28, no. 5, pp. 585-600, May 1985.

5 Linear Quadratic Control with Input Saturation Minyue Department of Electrical and Computer Engineering, The University of Newcastle, N.S.W. 2308 Australia A b s t r a c t . This paper studies a new approach to linear quadratic control for linear systems with input saturation. Our work presents an optimal sector b o u n d to model the mismatch between the unsaturated controller and saturated one and an optimised control design associated with this sector bound. This leads to a new characterisation of invariant sets and new switching controllers. The main outcome of this paper is that better performance can be guaranteed for the same region of attraction, or equivalently, a larger region of attraction is given for the same level of guaranteed performance.

5.1

Introduction

In this paper, we consider the problem of linear quadratic control for linear systems with input saturation. This problem has been widely studied and many design methods are available; see, e.g., [1-4]. When an optimal control input exceeds a given level of saturation, it is wellknown that optimal performance can not be achieved by simply saturating the control input, unless the level of over-saturation is sufficiently small [4]. To socalled anti-windup technique is commonly used to overcome the saturation. The key to most anti-windup methods is to "de-tune" the optimal controller in some way. That is, a lower control gain is used when the state is large and the control gain is gradually increased when the state becomes small. Many ad-hoc methods were used in early days, but with little theoretical guarantee on stability. However, many rigorous design methods are available now to provide some guaranteed properties on stability [2-4]. To assure stability, most recent anti-windup design methods use the idea of nested ellipsoidal invariant sets. More precisely, a sequence of ellipsoids are given in the state space along with a sequence of controllers. The design is done such that each ellipsoid is an invariant set, the corresponding controller is asymptotically stabilising, and the ellipsoids are nested. The overall control law is of a switching type, i.e., the selected controller corresponds to the smallest ellipsoid in which the state resides. A common approach used to compute these ellipsoids and controllers is to "de-tune" the optimal controller. That is, the ellipsoids and the controllers are constructed by adding cost penalty on the control. The larger the cost penalty, the lower the control gain and the larger the ellipsoid. This idea is supported by the observation that low-gain controllers tend to improve the stability at the cost of performance. Although this idea is intuitive and practical, it is not clear in general how to design these ellipsoids (and the controllers) to give the best performance bound.

58

Minyue

Despite t h e differences in various a n t i - w i n d u p design m e t h o d s , m o s t of t h e m , if not all, use a sector b o u n d on the m i s m a t c h b e t w e e n an u n s a t u r a t e d controller and a s a t u r a t e d one. Different design m e t h o d s use different sector b o u n d s a n d use t h e m in different ways. No rigorous s t u d y has been done on how to o p t i m a l l y choose a sector b o u n d and how to o p t i m a l l y use a given sector bound. In this research, we consider t h e p r o b l e m of designing a linear controller to optimise a given q u a d r a t i c cost function. We also use a sector b o u n d to m o d e l t h e m i s m a t c h between t h e u n s a t u r a t e d controller and t h e s a t u r a t e d one. However, we aim to use a least conservative sector b o u n d and use it in a least c o n s e r v a t i v e way. T h e main o u t c o m e of this research is t h a t b e t t e r p e r f o r m a n c e can be g u a r a n t e e d for the same region of attraction, or equivalently, a larger region of a t t r a c t i o n is given for the same level of g u a r a n t e e d performance. This p a p e r is organised as follows: Section 5.2 deals w i t h t h e p r o b l e m of designing a linear time-invariant state feedback controller to give t h e best p e r f o r m a n c e bound. Section 5.3 studies the key p r o p e r t i e s of this o p t i m i s e d linear t i m e - i n v a r i a n t controller. Section 5.4 deals with t h e p r o b l e m of designing switching controller for the purpose of i m p r o v i n g t h e performance. Section 5.5 gives a simple illustrative example. T h e conclusions are given in Section 5.6.

5.2

Linear

Time-invariant

Control

T h e system we consider in this p a p e r is given by

= A x +b(r(u),

x(0) = x0

(5.1)

where x E R n is the state, u E R is the input, A E R '~x'~ and b E R "~ are c o n s t a n t , and a(.) is a s a t u r a t i o n function with s a t u r a t i o n level equal to 1. We assume t h a t (A, b) is a controllable pair. Given a control input u, the level of over-saturation is defined to be

d(u) -- max{O, luI

-

1}

(5.2)

Suppose t h e control law is such t h a t t h e level of o v e r - s a t u r a t i o n is b o u n d e d by p _> O. Our first problem is to d e t e r m i n e how to b o u n d the nonlinearity caused by the saturation by a sector. More precisely, we rewrite a(u) as

~(u) = p ~ + 5(~)

(5.3)

where

~(,~) ~(~) =

-

(5.4)

p~,

We seek t h e optima] value for Pl so t h a t 6(u) has t h e smallest sector bound, i.e., p2 below is minimised:

16(u)l _< p2[u[, Vlu [ _< 1 + p Lemma

(5.5)

1. The optzmal value [or pl and the coT~esponding m i n i m u m pu are given

below:

2+ p

pi -- 2(1 + p ) '

p

p~ -- 2(1 + p)

(5.6)

5

Linear Quadratic Control

59

Proof. This is verified straightforwardly. Next, we consider the following quadratic cost function

J(xo, u) =

(xT Qx + ra(u)2)dt

(5.7)

for some Q -- QT > 0 and r > 0, and linear control input

u ---- kTx

(5.8)

for some k C R n. Ideally, we would like to provide an optimal control law, i.e., an optimal k, for each given initial state x0 such that the cost function J ( x o , u ) is minimised. However, the optimal k is generally dependent on x0, and the solution is difficult to give. To relax the problem, we aim to characterise an ellipsoid Xp={x:xTppx~#p2},

p p _ _ p T > 0 , #p > 0

(5.9)

and an associated suboptimal linear control gain k with the following properties for any parameter p > 0: 9 The level of over-saturation d(u) < p; 9 The set Xp is an invariant set, i.e., x(t) E Xp for all t _> 0 if x0 E Xp; It is well-known that if the control is not saturated, the optimal solution is given

by k = -r-lpob

(5.10)

where P0 solves the following Ricatti equation:

AT po + PoA + Q - r - l PobbT po ----0

(5.11)

Moreover, the optimal cost is given by xTpoxo. We can rewrite (5.1) as follows:

5~ = A x + b(plu + 5(u))

(5.12)

In view of (5.5), we relax the optimal control problem to designing a control gain k to minimize the worst-case cost for all 5(.) satisfying the sector b o u n d (5.5). Now we give some analysis on the relaxed optimal control problem. Denote

T(

f0

J(x0, u, T) ----

x T Q x + r(T(u)2)dt

(5.13)

and consider the Lyapunov function candidate

V ( x ) -~ xT ppx

(5.14)

Also, define

F2p = AT pp + PpA + Q - r - l PpbbT pp

(5.15)

60

M i n y u e Fu

and

u* = --r-lbT ppx

(5.16)

Given any initial state x0 a n d a n y 5(.) satisfying (5.5), it is easy to check t h a t

x(t) is finite for a n y t > 0 (i.e., there is no finite escape) a n d J(x0, u, T) ----V(xo) - V ( x ( T ) ) +

(V(x)

+ x T Q x + ra(u)2)dt

f T T <_ V(xo) + .], (x Y2pX + r(plu + 5(u) - u*)2)dt = V(xo) +

/:

f ( x , u, 5(u))dt

where

f(x, u, 6(u)) = xT,f-2pX + r(plu + 5(u) -- u*) 2

(5.17)

It is clear t h a t if f ( x , u , 5(u)) < 0 for all x E R '~ a n d (f(.) satisfying (5.5), t h e n

J(xo,u) < V(xo)

(5.18)

From the analysis above, we formulate the following relaxed o p t i m a l control problem: P 1 . Design Pp a n d u to minimise Vo(xo) subject to f ( x , u, 5(u)) <_ 0 for all x E R n a n d 5(-) satisfying (5.5). Further, d e t e r m i n e the (largest) invariant set Xp for which the cost J(xo, u) is b o u n d e d by Vo(xo). 1. Consider the system in (5.1) and the cost function in (5.7). For any level of over-saturation p >_ O, the optimal Pp and u for Problem P1 are given by

Theorem

u = k T x = p~lu* = - - p ~ l r - l b T p p x AT pp + PpA + Q - (1 - p~)r-lPpbbT pp = O, Pp = p T > 0 and the invariant set by Xp =

{ x : x T p p x <_

r2

(1 -- p ~ b r p p b

}

(5.19)

(5.20)

where p2 _

po - pl

p

2+p

(5.21)

Proof. Using the well-known S-procedure [5], t h e above holds iff there exists ~- > 0 such t h a t ] ( x , u , 6) = f ( x , u , 5) W T(p~u 2 - 5 2 ) <_0, V x E R n , 6 E R For the above to hold, it is necessary t h a t v > r. A s s u m i n g this a n d m a x i m i s i n g f ( x , u, 5) with respect to 5 yields 5 =

r

T--r

(plu-u*)

5

Linear Q u a d r a t i c Control

61

which in turn yields

f ( x , u, 5) = xT ~ , x + rp]u 2 +

rr T--r

(plu - u*) 2

Then, minimising f ( x , u, 5) with respect to u results in U -~

rpl

u*

and / ( x , u, 5) = x r ~ , x +

Trp 2

O

- ~)p~ + ,'p~ (~*)~

Note t h a t pi > P:. Finally, m i n i m i s i n g / i x, z, 6) with respect to ~- yields ~- ----r,

u

=

p~iu*

and --

2

P2

i

*\2

Pl

where

~2p ----AT pp T PpA -t- Q - (1 - p2)r-ippbbT pp with po given by (5.21). To assure ] ( x , u, 5) > 0, it is necessary t h a t ~p < 0. It is easily verified t h a t Pp is a monotonically decreasing function of ~p. Therefore, the optimal ~p = 0. Consequently, we obtain the optimal solution in (5.19). Next, we try to characterise an invariant set Xp as in (5.9) for which the control law above applies. All we need to do is to find the largest #p in (5.9) such t h a t m a x 16(u)l = p21ul

xEXp

Equivalently,

max p ~ l r - l b T p p x

= 1+ p

xEXp

The solution is given by x~

# bTv/~-~pb b

and r(2 + p)

"-

2 b~V'~-E-~b -

r

(1- p0) b~JF-P-~.b

That is, Xp is given by (5.20). Since f ( x , u, 6(u)) <_ 0 for all x E Xp,

dv(x)

_< ~ ~ x T Q x + r(~(u)2),

Vx E XR

Hence, we know that Xp is an invariant set.

(5.22)

62

MinyueFu

Remark 1. In Theorem 1, we assume that the sector bound characterised in Lemma 1 is used. This assumption can be relaxed to any sector bound satisfying the following condition: Parameters pi and p2 are such that pi > p2 _> 0 and ]5(u)l ~ p21ul for all u with the level of over-saturation bounded by p, where 5(u) is defined in (5.4). It can be shown (with somewhat more effort) that the optimal solution to Problem P1 is still given by (5.19) as in Theorem 1, with p0 ---- p2/pi. It can be further shown that the parameters pl and p2 that minimise V(xo) and maximise Xp are those given in Lemma 1.

5.3

Properties

of the

Proposed

Controller

In this section, we study two key properties of the proposed controller in Theorem 1. The first property shows the improvement of the saturation control compared with an unsaturated controller. The second property is to do with nesting of invariant sets and monotonicity of Lyapunov matrices. Returning to (5.19), we see that the Ricatti equation for Pp corresponds to the solution to an optimal control where the weight (or penalty) for the control in the cost function is changed to ( 1 - p~)-lr. To achieve the cost J(x0, u, T) = J* (xo, T), u must be such that (r(u) = --(1 -- p2)r-lbTppx

(5.23)

To make the above feasible (i.e., to avoid saturation), the invariant set must be

{

r2

f(p ---- x : xT ppx ---- (1 -- p~)2bTppb

}

(5.24)

Compared with (5.20), we have

xp = (i +

po)2p

(5.25)

This illustrates that the control law in (5.19) gives a substantially larger invariant set for the same cost, compared with an unsaturated control law. On the other hand, we may consider choosing the control law such that 2

a(u) = -- 1 -- POr_lbT ppx 2

(5.26)

and choosing an invariant set such that the above control law is feasible (i.e., no saturation). Note that this control law is stabilising, due to the well-known gain margin of an optimal linear quadratic control (also seen directly from the Ricatti equation in (5.19)). In this case, the invariant set is given by 4r 2

x" =

~: ~ P ' ~

= (1 - po:):b~P~b

}

(5.27)

and we have

Xp - 1 + Po f(p

(5.28)

5

Linear Quadratic Control

63

Although Xp < X'p, no performance grantee can be delivered by the controller in (5.26). To make the comparison fair, we take p --~ cx) and note that Xor = 2 ~

(5.29)

This gives a somewhat surprising result: C o r o l l a r y 1. The largest invariant set given by the controller in (5.19) is the same as the largest invariant set given by an unsaturated controller (5.26). One implication of the results above is that the saturated controller can bring a good benefit when p is not close to 0 and not too large. Next, we study the nesting property of Xp and monotonicity of Pp. To this end, define Sp = (1 - Po)Pp

(5.30)

We then rewrite Ricatti equation in (5.19) as

AT Sp + SpA + (1 - po)Q - (1 + p o ) r - l SpbbT Sp = 0

(5.31)

and the invariant set Xp as

Xp =

x : xT Spx <

(5.32)

L e m m a 2. The solution Sp to (5.31) is monotonically decreasing, i.e., Sp+~ < Sp if 0 ~ p < p T e. Consequently, Xp are nested in the following sense:

Xp c Xp+~,

V0
(5.33)

Further, the solution Pp to the Ricatti equation in (5.19) is monotonically increasing, i.e., Pp+~ > Pp if O <_ p < p + c. Proof. The monotonicity of Sp is a basic property of the Ricatti equation (5.31). We only need to show this for sufficiently small 9 > 0. Denote E = Sp - Sp+~ and ~2p to be the left hand side of (5.31). Also define 9 o --

P+( -

-

2+p+ 9

P 2+p

>

0

Then, 0 = ~p - ~p+~

= EA T + AE + 9

- (1 + po)SpbbTSp

+(1 + Po + eo)(Sp - E)bbT(Sp - E)

= E ( A - (1 + Po + eo)SpbbW) T + (A - (1 + po + eo)SpbbT)E +9

+ (1 + po + 9

E + (oSpbbT Sp

LFrom (5.31), we know that A - (1 + po)bbTSp is Hurwitz. Therefore A - (1 + p0 + 9o)bbTSp is also Hurwitz when 9 (or equivalently, e) is sufficiently small. Hence, the equation above implies that E > 0. Therefore, the inonotonicity of Sp is established. The nesting property of Xp then follows naturally from (5.32). The monotonicity of Pp is proved similarly.

64

Minyue Yh

R e m a r k 1. If p ---- 0, the control in Theorem 1 recovers the optimal control without saturation. In this case, the invariant set is given by

Xo -~

x : xT pox ~_

R e m a r k 2. The "largest" invariant set, called region of attraction, is given by taking p --~ cx~ (or equivalently, p0 --* 1) and solving for Pp in (5.19). T h a t is, the region of attraction is given by

X~ =

x : x T p o x < (1 -- po)2bTpob'

p0 --+ 1

(5.34)

Note that the solvability of Pp for any p > 0 is guaranteed by the controllability of (A, b) and positive definiteness of Q. R e m a r k 3. Suppose A is either Hurwitz or marginally unstable (i.e., the only unstable eigenvalues are the ones with a zero real part). Then, the solution to the Ricatti equation in (5.19) is such that the directions of Pp approach to either a constant (corresponding to stable eigenvalues of A) or o ( ~ / r - L - ~ ) (corresponding to marginal eigenvalues of A). In either case, the limiting invariant set is the whole space, i.e., X1 = R n.

5.4

Switching

Control

A common control strategy for combating the control saturation is to start with a small gain when the state is "large" (to avoid or reduce saturation) and then gradually increase the gain when the state is "small" (to improve the performance). This strategy can be easily applied to the controller in the previous section due to the nesting properties of Xp and monotonicity of Pp. More precisely, a switching control strategy is simply formed by choosing a sequence of saturation indices 0 ---- p(0) ~ p(1) ~ ... ~ p(N) and solving for the corresponding Lyapunov matrices Pi, invariant sets Xi and control gains ki. The control law simply selects the control gain ki when x E Xi and x ~ Xi-1 (unless

i=

0).

T h e o r e m 2. Suppose xo E X N and we apply the switching control law above by

starting with kN (or pN equivalently). Denote the switching control law by u8 and the switching time from p(i) to p(i-1) by Ti. Then the cost of this switching control is bounded by J(xo,u~) ~ x T p N x o -- EN=I xT(T~)(P~ - P~-I)X(T~) < xTpNxo,

(5.35) VXo E X N , XO # 0

Proof. Follows directly from Theorem 1 and the monotonicity of Pp. The advantage of the switching law is clearly seen from the theorem above where the negative terms in (5.35) are a result of the switching. However, it is somewhat difficult to express the cost explicitly in terms of x0 and p(i). More work needs to be done to study this issue and also on how to choose the sequence {q(i)} to optimise

J(xo, u).

5

5.5

Linear Quadratic Control

65

Illustrative E x a m p l e

,

i -0.2 -0.4 -0.6 -0"I xl

F i g . 5.1. Nested Ellipsoids and State Trajectories To illustrate the design approach presented in this paper, we consider the following simple system: ic -=

[01] [:] x(t) +

a(u(t) )

(5.36)

-1.25 1 The cost function has r = 1 and

0=[::]

Choose p = 0, 2, 5, 10, 20, 40, 70, 100. The corresponding invariant sets X o are shown in Figure 5.1. We take xo = [0.87 0] I. Figure 5.1 also shows the two state trajectories corresponding the switching controller and nomswitching controller. Their performances and control inputs are given in Figures 5.2-5.3, respectively. It is seen clearly that the switching controller significantly outperforms the non-switching controller.

5.6

Conclusion

In this paper, we have presented a new approach to designing linear quadratic controllers for systems with input saturation. The key contribution of the p a p e r

66

Minyue Yh x 10' i

J

J

i

J

i

i --

Switching Non-switchin

9

'

0.5

,

,

~

1

15

2

, 2.5 Time

,

,

,

,

3

35

4

45

Fig. 5.2. Performance Costs

y

1

0.8

i

i

I

--

i

Switching Non-switching

0.6

0.4

0.2

-0.2

-0.4

i

0

o15

1

i

i

115

4 Time

Fig. 5.3. Control Inputs

415

5

Linear Quadratic Control

67

is of two-fold: 1) We optimise the sector bound which models the mismatch between the unsaturated controller and the s a t u r a t e d one; and 2) We determine the largest invariant set for the given sector bound above and the associated optimal controller. The invariant sets and the corresponding Lyapunov matrices have the nice properties of nesting and monotonicity, respectively. These properties allow a switching controller to be designed easily to yield substantially lower quadratic cost (in comparison to non-switching controllers) while guaranteeing stability. The sector bound used for control design can be generalised to include integral quadratic constraints. This allows a dynamic relationship between the linear control input and the saturated control input. It is expected t h a t this approach can yield some improvement in the performance at the expense of somewhat more complicated control design. More specifically, the state of the system needs to include the dynamics of the integral quadratic constraints, which implies t h a t the control gain will be dynamic. More work on this topic will be carried out by the author. Finally, it should be noted t h a t the design approach given in this p a p e r can be easily generalised to discrete-time systems. Acknowledgement The author appreciates the constructive communications with Professor G . . C . Goodwin on this paper. The author also appreciates discussions with Professor K. Hollot on the optimal sector bound that have resulted in Remark 1.

References 1. D.S. Bernstein and A. N. Michel (1995). "A chronological bibliography on saturating actuators, Int. J. Robust and Nonlinear Contr., vol. 5, pp. 375-380. 2. P. O. M. Scokaert and J. B. Rawling (1998). "Constrained linear quadratic regulation," IEEE Trans. Auto. Contr., vol. 43, no. 8, pp. 1163-1169. 3. A. Saberi, Z. Lin and A. R. Teel (1999). "Control of Linear Systems with Saturating Actuators," IEEE Trans. Auto. Contr., vol. 41, no. 3, pp. 368-378. 4. J. A. De Dona (2000). Input Constrained Linear Control, Ph.D. thesis, University of Newcastle. 5. V. A. Yakubovich (1971). "S-procedure in nonlinear control theory," Vestnik Leninggradskogo Universiteta, Ser. Matematika, pp. 62-77.

6 R o b u s t n e s s Issues A s s o c i a t e d w i t h t h e P r o v i s i o n of Integral A c t i o n in N o n l i n e a r Systems Graham C. Goodwin and Osvaldo J. Rojas The University of Newcastle, Callaghan NSW 2308, Australia

A b s t r a c t . A key motivation for feedback control is that of disturbance compensation. In the case of linear systems, this is a very well understood problem. It is known, for example, that the straight forward inclusion of integral action gives compensation for constant disturbances and off-sets. Moreover, in the linear case, no interactions occur between disturbances and the underlying dynamics since the principle of superposition holds. However, in nonlinear systems, superposition does not hold, and this implies, inter-alia, that non trivial interactions can arise between disturbances and plant dynamics. As a result, a disturbance that is incorrectly interpreted can destabilise an otherwise stable system. Thus, disturbance compensation and, in particular, the provision of integral action, presents non-trivial challenges for nonlinear systems. The aim of this paper is to raise awareness to these issues and suggest possible strategies for attacking the problem.

6.1

Introduction

Since feedback mechanisms were first designed, disturbance compensation has been a key consideration [1]. For example, most feedback mechanisms incorporate some form of integral action to give steady state compensation for constant disturbances. Also, integral action is a core ingredient in the celebrated PID controller [2], which is used in 95% of all real world control systems. This paper will review aspects of disturbance compensation for both linear and nonlinear systems. For simplicity, we treat only the single input, single o u t p u t case. It will be seen that, in the case of linear systems, all methods essentially lead to the same end result. Moreover, no significant stability or robustness issues arise in the linear case irrespective of the true nature of the disturbance. In the nonlinear case, even in the absence of disturbances, there exist non-trivial stability issues - see for example [3]. In the presence of disturbances, non-trivial interactions can occur between disturbances and plant dynamics as well. Thus, an incorrectly modelled disturbance, will mean that the plant dynamics are incorrectly interpreted, leading to robustness issues. This, in turn, can lead to serious stability problems. If we specialise to the class of constant disturbances, then we see that for linear systems, all methods reduce essentially to the one underlying principle of including an integrator in the controller. However, for nonlinear systems, no single answer exists to the question of integral action. Indeed, multiple solutions are needed which specifically take account of the precise disturbance injection point. Several different strategies will be discussed below.

70

Goodwin and Rojas

6.2

Brief

review

of the linear

case

In the linear case, there are several ways of thinking about the design of a disturbance compensating controller. However, it turns out that, for this case, the end result is often identical, irrespective of the design route used. As an illustration, let us assume that disturbance compensation is achieved by using an observer to estimate a constant input disturbance and the estimated disturbance is then cancelled from the input. To briefly expand on this idea, let the system be modelled as

xp(t) -- A p x p ( t ) + Bp (u(t) + d(t) )

(6.1)

where u(t) and d(t) are the manipulated plant input and disturbance respectively. Also, let the disturbance be modelled by the following homogeneous equation:

Jzd(t) = Ad Xd(t) d(t) = Cd x d ( t )

(6.2) (6.3)

In this way, the composite model takes the form:

~(t) = A x(t) + B u(t)

(6.4)

y(t) = c x(t)

(6.5)

where,

C -- [Cp 0]

(6.6)

An appropriate linear observer for this composite system takes the form:

x(t) = A ~(t) + B u ( t ) + J (y(t) - C &(t) )

(6.7)

where J = [JTvJ[ ] T, and the corresponding control law is given by:

u(t) = -Kp ~p(t) - Ca ~ ( t )

(6.8)

where the second term on the right hand side uses the estimate to cancel the disturbance from the input applied to the system. It is readily seen that the controller transfer function is

C(s) = [Kp Cd]

[

sI-A

v+BpKp+JvCp JdC v

0 s I - Ad

Jp Jd

(6.9)

It follows, immediately that d e t ( s I - Aa) appears in the controller denominator. Thus, we have simply described a special form of the Internal Model Principle [4]. Note that, in the linear case, there is no interaction between the form of the actual disturbance and the issues of closed loop stability or robustness. This is because, in a linear system, disturbances are simply external signals and the principle of superposition implies that their response can be separately evaluated. In particular, it does not matter, in terms of stability or robustness, whether the disturbance actually appears at the plant input or output. Of course, the closed loop responses

6

Robustness Issues of Integral Action in Nonlinear Systems

71

to input and output disturbances can differ because different transfer functions are involved [5]. In particular, if S(s) denotes the closed loop o u t p u t disturbance sensitivity function, then the input disturbance sensitivity, Si(s), is given by:

Si(s) -- S(s) G(s)

(6.10)

where G(s) is the open loop plant transfer function. One simple implication of this relationship is that, whilst output constant disturbances are fully compensated in steady state if the plant has an integrator, this is insufficient for input disturbances. A related point is that the open loop dynamics of the plant (including slow poles) will always be part of the input disturbance response, unless they are included as zeros of the sensitivity function S(s). Thus, in general, one should avoid cancelling slow plant poles (even if stable) in the controller transfer function .

6.3

Input s a t u r a t i o n

As a first step towards extending the above ideas to the nonlinear case, consider a linear system subject to input saturation constraints. In this case, the equivalence between having an explicit integrator in the controller or using an implicit formulation such as the one achieved through equations (6.1) to (6.9), no longer holds. In particular, if one has an explicit integrator and the controller output saturates, then the state of the explicit integrator will continue to grow, leading to the well known problem of integrator wind up [6]. On the other hand, if one implements the integrator implicitly by an input disturbance observer, as in equations (6.1) to (6.9), then the observer will continue to operate correctly, irrespective of input saturation. Hence, the latter formulation has clear advantages. Indeed, this is the basis of many of the existing strategies for achieving antiwindup protection [7]. The key point here is that, whilst in the linear case, superposition allows one to rearrange the blocks that form the controller in order to make each strategy equivalent, this is clearly not possible in the case of even a simple nonlinearity such as a saturation function. Moreover, the issue of stability of antiwindup strategies is highly non trivial, especially in the presence of disturbances, precisely because of the nonlinear behaviour of the system taken as a whole [8,9]. Going beyond the simple case of saturation, as discussed above, the problem becomes even more interesting. Specifically, when dealing with nonlinear systems, we suggest that the disturbances should be explicitly modelled. Moreover, their injection point into the control loop becomes an important issue. We will illustrate these points for some special cases below.

6.4

Special c a s e : s t a b l e and stably invertible nonlinear systems

If the nonlinear system is stable and has a stable inverse, then it is possible to extend the concepts associated with the Internal Model Control strategy for linear systems to nonlinear systems. Since, the principle of superposition does not apply for nonlinear systems it is necessary to treat the problem of having input and output disturbances in a separate way.

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6.4.1

Output

disturbance

design

Consider a nonlinear system which has o u t p u t disturbances and is b o t h stable and stably invertible. Then it is possible to compensate for o u t p u t disturbances following the strategy illustrated in Fig. 6.1. Here G~ (o) represents the system nonlinear dynamics. In Fig. 6.1, HaCo) is, ideally, an inverse for Ga(o). For most nonlinear systems, an exact inverse does not exist, and therefore some approximation technique is necessary. One way of building an approximate inverse for stably invertible systems is via feedback linearization. In order to briefly show how this can be done, let Ga Co) be described in state space form by:

~(t) = / ( x ) + g(x)u(t) y(t) = h(x)

(6.11)

We introduce a stable differential operator,

p(p) = prp ~ + p r _ l p ~-1 + . . . + 1

(6.12)

where r is the relative degree of the nonlinear system. It is easily seen t h a t p(p), applied to the system output y(t) can be written as:

p(p)y(t) = b(x) + a(x)u(t).

(6.13)

When a(x) ~ 0, the application of the following input signal

u(t) - y*(t) - b(x)

a(x)

(6.14)

leads to the result

p(p)y(t) --~ y* (t)

(6.15)

where y*(t) represents any external signal. Specifically, y*(t) can be a reference signal. Thus, HaCo) can be implemented using equation C6.14) together with an appropriate embedded nonlinear model for the real plant Ga Co). If we choose y* (t) to be the signal e(t) in Fig. 6.1, then u(t) in C6.14) will be the plant input necessary to achieve y(t) = ~p(p)" We can see t h a t the control strategy for o u t p u t disturbances presented in Fig. 6.1, basically mirrors the well known Internal Model Control strategy for the linear stable case [5]. The assumption of stably invertibility of the nonlinear plant Ga Co) is needed to ensure that Ha Co), designed via feedback linearization, is stable.

Remark 1. Note that it is possible to show that, provided the loop DC gain from u(t) back to u(t) via ym(t) is unity, then the output y(t) will equal a constant reference r(t) in steady state, irrespective of the nature of the plant or the disturbance injection point. Thus the circuit gives a form of integral action. However, a key issue is that the loop dynamics (and hence stability) is determined by b o t h the n a t u r e of the disturbance and its injection point. Specifically, if there is an unmeasured input disturbance, we are not able to construct the parallel model Ga(o) since its dynamics will depend on the precise nature of the input signal.

6

Robustness Issues of Integral Action in Nonlinear Systems

~

~(t)

+

.~~

I

73

do(t) y(t)

~(t) &(t)

Fig. 6.1. Control strategy for output disturbances: stable and stably invertible case

I di(t)

:t) ~ + +

,~(t)

Fig. 6.2. Control strategy for input disturbances: stable and stably invertible case

6.4.2

Input

disturbance

design

Motivated by the comments made at the end of the last sub-section, we can conceive of an alternative scheme for input disturbances (see Fig. 6.2). H~(o) is again an approximation to G~-1(o), achieved, for example, via a feedback linearization design. The filter F in the scheme, has been included to avoid an algebraic loop. However, the design of F actually contains some subtle issues which mirror points made in the linear case. Specifically, let us assume that Ha(o) is a very good approximation to G~ 1(o), so that d~(t) ~ di(t). Then the output response to an input disturbance only, is y = Ga((1 - F ) d ~ > . Thus, as in the linear case, the open loop dynamics will appear in the disturbance response, unless (1 - F ) is appropriately designed. This aspect raises some interesting design issues in the nonlinear case.

Remark 2. When comparing the two schemes of Fig. 6.1 and Fig. 6.2, it is clear that, in the linear case, they are roughly equivalent. This can be easily checked if we move the Ha block in the feed forward path of Fig. 6.2 along the loop, based on superposition. However, in the nonlinear case, each scheme is different from the other, because Q(a + b) ~ Q(a) + Q(b) for nonlinear operators.

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Goodwin and Rojas

A simulation example: pH neutralisation

As an illustrative example of the application of the above ideas, we will analyse the pH neutralisation problem. This is a very common control problem in many industrial processes with many, well known, difficulties, e.g. the large dynamic range needed in the controller [5]. However, for illustration purposes, we will ignore these practical issues and focus on the nonlinear features, as raised in section 6.4. In order to control the pH variations in a liquid flow, this flow is usually introduced into a tank, where it is mixed with a certain amount of concentrated reagent with a different pH. A common assumption used to obtain a model for the pH neutralisation dynamics is that the tank is well-stirred, generating a uniform pH throughout the tank. Under these conditions, an appropriate state space model of the system, where a strong acid - strong base type of pH neutralisation has been considered, is the following:

do(t) -- (u(t) +vd~(t)) (c~, - co(t)) + ~(c~ - co(t))

(6.16)

po(t) ---- -log (X/0.25 co(t) 2 + 10 -14 + 0.5 co(t)) + do(t)

(6.17)

where the following notation has been used:

ci, Co, c~ : excess of hydrogen in inlet stream, effluent and reagent stream, respectively (~--).m~

Po : effluent pH. V : tank volume (t). q flow rate of inlet stream (~). u flow rate of reagent (~). di disturbance in the reagent stream (input disturbance),(~). do disturbance in the true pH of the effluent (output disturbance). For illustration purposes we use the following values for the parameters listed above (they are not representative of a real application): V : 80 (l). q: 1(~). c i : - 2 . 8 4 6 . 1 0 -~' ( - ~ ) . 005

We have chosen ci in order to avoid the initial transient of the system response to the set-point r ----7.5. Since the process is clearly stable and stably invertible, we are able to use the schemes of Fig. 6.1 and Fig. 6.2 in order to reject output disturbances and input disturbances, respectively. The approximate inverse H~(o) will be designed based upon the feedback linearization ideas presented in section 6.4. First of all, we notice that the nonlinear system has relative degree r = 1, hence we can use the following first order differential operator:

p(p) = f~p + 1

(6.18)

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Robustness Issues of Integral Action in Nonlinear Systems

75

8

7.5 7

~6.s 6 __

with filter F 1 ] with filter F 2

5.5 5

0

50

1O0

150

200

250

300

350

400

450

time [s]

F i g . 6.3. Input disturbance response for the scheme of Fig. 6.2, with different design of the filter F with a suitable choice of ~. We have that an approximate inverse is obtained if: u ( t ) = V l n ( l O ) ~ / ~ o ( t ) 2 + 4 . 1 0 -14 c~ - & ( t ) ( & ( t ) - c~) ( u ( t ) - 1~o(t)) + ~o(t) - c~ " q

(6.19)

where u(t) is the driving input to the inverse. ~o(t) and g o ( t ) are obtained from a model of the system running embedded in H~(o) and having u ( t ) in (6.19) as the driving signal. It is worth noting that it is useful to choose different values of ~ for each of the approximate inverses H~(o} of the scheme in Fig. 6.2. For example, we want the H a ( o ) block at the output of the plant to be a fast approximation of the plant inverse, since it is desirable that di(t) be a fast approximation of the input disturbance as well. On the other hand, H~(o) in the feed forward path of the scheme in Fig. 6.2, takes account of the achieved closed loop bandwidth for reference tracking. Thus, if limitations in the control energy exist, it will be desirable to choose comparable to the open loop dominant time constant of the nonlinear system. In our case, the dominant time constant of the nonlinear system is approximately V / q = 80[s], thus we select ~1 = 10 [s] for the first block H~(o}, and ~2 ----0.5 [s] for the second approximate inverse in Fig. 6.2. We will first illustrate the effect of the filter F in the design shown in Fig. 6.2 for input disturbances. If the only intention is to avoid an algebraic loop, then the following selection for F will suffice: FI(S) -

1 - ~-s+l

(6.20)

However, we suggest to use an approximate linear design for an alternative F in an effort to improve the system response to input disturbances. As was explained in section 6.4, the idea is to design F such that (1 - F ) approximately cancels the plant dynamics. This leads to the following alternative filter: Y2(s) -

flS+l

(T S ~_-I-~

(6.21)

where f l must be selected appropriately. Note that, in order to preserve the integral action feature of the input disturbance rejection scheme of Fig. 6.2, the DC gain

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Goodwin and Rojas

of both Fl(S) and F2(s) must be unity. To achieve the approximate cancellation of the open loop plant dynamics, we choose:

T2q fl

=

2T-

(6.22)

--V--

for T ---- 2 [s]. Fig. 6.3 compares the responses to an input disturbance, using Fz (s) and F2 (s) in the control scheme. An input step disturbance of magnitude 5.10 -3 [I/s] is applied at t --- 50 Is]. Inspection of the transient responses clearly shows the advantages of using the second alternative design for F . We see, for example, that when using F1 (s), the input disturbance response is dominated by the open loop dynamics of the system, whereas for F2 (s), the input disturbance transient response has been modified. In the sequel we will use the filter F2(s). Next, we compare the response of the strategy in Fig. 6.1 when the disturbance is actually at the output or at the input. Also, we compare the response of the strategy in Fig. 6.2, when the disturbance takes these two forms. The results are shown in Fig. 6.4 and Fig. 6.5, respectively. Input disturbance response

7,5

-

Output disturbance response

7.5J h

-

7

71

:Z6,5

~8.st

6

el

5.5

5.sl

s~

0

so

1~0

150

time Is]

2~0

2~0

30o

0

~

1~0

is0

time [s]

2~

2~0

30O

Fig. 6.4. Design for input disturbance compensation

We see from Fig. 6.4 and Fig. 6.5 that, irrespective of the disturbance point, zero steady state error is achieved. However, the simulations show precise nature of the transient response does depend on the disturbance point. In particular, best results are obtained when the true disturbance point corresponds to the one assumed for the purpose of design.

6.6

injection that the injection injection

General nonlinear systems

Finally, we consider general nonlinear systems which are not necessarily stable nor stably invertible. In this case, we need to depart from the simple feedback linearization scheme used above. So as to retain the same general methodology we will use a related scheme which we have called Generalized Feedback Linearization (GFL) [10]. We will begin by describing the key features of the GFL strategy and then, we will show how this control strategy can be used, in combination with a suitable state and disturbance observer, to achieve integral action.

6

Robustness Issues of Integral Action in Nonlinear Systems Input disturbance response

77

Output disturbance response

8

7.5 - -

J

f 7

=:6,

16.5 o.

o.

6

5.5

50

1O0

150

time [s]

200

250

300

50

1O0

150

time Is]

200

250

300

Fig. 6.5. Design for output disturbance compensation

6.6.1

The Generalized

Feedback

linearization

Strategy

As explained in section 6.4, the usual feedback linearization technique can be useful in obtaining an approximate inverse of a nonlinear system. However, one important drawback of the scheme is that it cancels the zero dynamics of the nonlinear system, hence stably invertibility is required. Here we will show how the scheme can be generalized to cover classes of non stable invertible systems. Recall that the basic feedback linearization scheme achieves

p(p)y(t) = y* (t)

(6.23)

where p(p) is a differential operator. To prevent the control signal becoming unbounded in the case of non stable invertible systems, it seems desirable to match (6.23) with a similar requirement on the control signal. Thus, we might ask that the input satisfies a linear dynamic model of the form:

l(p)u(t) = u* (t)

(6.24)

where l(p) is a suitable differential operator of degree rz, such that l(0) = 1 and u* (t) is the input needed to achieve y*(t) in steady state. It is clear that, in general, conditions (6.23) and (6.24) are not simultaneously compatible. This suggests that we could determine the control signal u(t) as the value that satisfies a linear combination of (6.23) and (6.24) of the form: (1 - A) (p(p)y(t) - y*) + A (l(p)u(t) - u*) = 0

(6.25)

where 0 < A < 1. Equation (6.25) constitutes the Generalized Feedback Linearization (GFL) control law. It is clear that this strategy will handle all stably invertible systems (take A = 0) and M1 stable systems whether or not they are stably invertible (take A = 1). By continuity, various combinations of stable and stably invertible dynamics will also be able to be stabilised by this class of control law. To develop the control law implicitly defined in (6.25), we introduce a d u m m y variable ~(t), as follows

l(p)u(t) = ~(t)

(6.26)

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G o o d w i n a n d Rojas

It is now clear that, if the original n o n l i n e a r system has relative degree r, t h e n t h e nonlinear system between ~(t) a n d y(t) has relative degree r + r~. Hence, if we use a r + rl degree operator p(p), p(p)y(t) will d e p e n d explicitly on ~(t). Following t h e development t h a t led to (6.14), we can write:

p(p)y(t) = b(x') + a(x')ft(t).

(6.27)

where x' is now an extended state vector which includes the n o n l i n e a r system states a n d the states introduced by the l(p) polynomial. S u b s t i t u t i n g (6.27) into (6.25), gives the following nonlinear control law: ~(t) = (1 - A)(y*(t) - b(x')) + Au* (1 - A)a(x') + A

(6.28)

The signal u(t) is t h e n o b t a i n e d via (6.26). T h e success of the preceding control law d e p e n d s on being able to make a judicious choice of p(p) a n d I(p) in order to achieve closed loop stability. To show how the G F L strategy works, we present below, a n illustrative example with an u n s t a b l e a n d n o n m i n i m u m phase n o n l i n e a r system. As a first approach to the problem we will assume complete knowledge of the system states. Later we will examine the o u t p u t feedback case. Consider the following u n s t a b l e SISO n o n l i n e a r system: xl = 10xl - 10x2

(6.29)

52 = 9.9Xl - 10x2 + 0.1xz3 + 0.1u

(6.30)

y = x2

(6.31)

We note t h a t the system relative degree is r = 1 a n d t h a t the system zero d y n a m i c s are unstable. We choose the following differential operators p(p) a n d / ( p ) :

p(p) = O.lp 2 + 0.4p + 1 l(p) = - 4 . 1 p + 1

(6.32)

In this way, p(p)y(t) can be expressed as in e q u a t i o n (6.27), in terms of the e x t e n d e d state vector x' = [Xl x2 xl] T a n d the d u m m y variable fi(t):

p(p)y(t) = b(x') + a(x')ft(t) = = p2 _(9"951 - 1052 + 0.3x~ x2 - 0"lxz'~ll ]+ Pl (9.9Xl -- 10X2 + 0.1X 3 + 0.1X/) + X2 + 0.1 P2-~-aU

(6.33)

where p~, pl a n d 11 are the coefficients of the p(p) a n d l(p) operators in (6.32), respectively. Furthermore, xl(t) = u(t) is the state i n t r o d u c e d by the first order l(p) polynomial:

u(t) ----/~p) u(t)

(6.34)

If we consider, for the m o m e n t , t h a t we have complete knowledge of the system states, t h e n using the G F L control law as in (6.28), (6.26), it is possible to stabilise

6

Robustness Issues of Integral Action in Nonlinear Systems

79

1.5

~" 0.5

0 I

I

i

I

I

0.5

1

15 time [s]

2

2.5

Fig. 6.6. GFL strategy assuming complete state knowledge. Response to a step reference of 1.5 the closed loop for certain values of )~. Since the plant is open loop unstable, we can not choose )~ ----1, for this leads to an open loop strategy. Instead, the value of ~ must be reduced until the closed loop becomes stable. Via simulations we have observed that this happens when ~ becomes smaller t h a n 7 9 10 -3. Eventually, if we keep reducing the value of A, the closed loop becomes unstable again. This is because the system is also non stably invertible. Fig. 6.6 shows the system closed loop response to a step reference of 1.5 with )~ ----3 . 4 . 1 0 -3. Note that both undershoot and overshoot occurs in the closed loop step response. This is a consequence of the fact that the system is both open loop unstable and is not stably invertible [5].

Remark 3. Note that the GFL control strategy presented in equation (6.28) only achieves perfect tracking for constant reference signals and with no disturbances or modelling errors. In the presence of input or output constant disturbances the strategy fails to ensure perfect tracking, the reason being that we have not explicitly considered the disturbances when designing the control strategy. We will see in the next section that using an appropriate observer for the disturbance, we will be able to incorporate the available information about the nature of the disturbance and its injection point into the GFL controller, achieving the desired integral action feature. 6.6.2

System

states and input disturbance

estimation

In many practical applications, the system states will not be directly measurable. Hence, a key point in the practical implementation of any nonlinear control law (in particular the GFL strategy described in the previous sub-section) is the issue of state estimation for nonlinear systems. This is necessary in order to approximate a(x') and b(x') in (6.28). There are still many open problems related to state estimation for nonlinear systems [11]. To illustrate the application of the GFL strategy in the control of nonlinear systems, particularly with regard to input and o u t p u t disturbance rejection, we will use a nonlinear observer for the system states and for the disturbance, based on a linearized design (a simple form of the E K F method). We will first analyse the case in which an input disturbance d~(t) is assumed. Hence, mirroring the linear case presented in section 6.2, we have that the nonlinear

80

Goodwin and Rojas

system can be modelled as:

&(t) = f ( x ) + g ( x ) ( u ( t ) + di(t))

(6.35)

y(t) = h(x)

(6.36)

where the input disturbance di (t) is given by the homogeneous differential equation: di(t) = 0

(6.37)

A suitable (EKF like) state and disturbance observer is then given by:

&(t) = f ( ~ ) -t- g(&)[u(t) +

d~(t)]

+ Jl[y(t) - h(&)]

(6.38)

A

d~(t) ---- J2[y(t) - h(&)]

(6.39)

where J1 and J2 are designed in order to ensure the stability of the nonlinear observer dynamics. One possibility is to design the observer gains J1 and J2 based on a linearized model of the system around the current operating point. Hence, consider the linearized model :

C

[ Oh 0]

(6.40)

we can design g -- [JTJ2T] T such that A - J C is stable, around the given operating point. Having an appropriate estimation of the system states, we are able to implement the GFL strategy in an output feedback setup. Moreover, to compensate for the unmeasured input disturbance di(t), we can use the estimated value di(t) and subtract it from the control signal applied to the plant. The resulting control scheme is depicted in Fig. 6.7. It is worth noting that the inclusion of the observer for an input constant disturbance, is the key point which ensures that the control strategy of Fig. 6.7 achieves integral action. This can be seen as follows: if the system settles to a steady state, then equation (6.39) implies that y(t) must equal h(&), irrespective of the presence of disturbances or model errors. Hence, provided that the control law ensures that the model output h(~) is taken to y *(t ) , then the plant output, y(t), will also be taken to y* (t). Thus, integral action has been achieved via inclusion of the input disturbance observer. Next we consider the case of output disturbance.

6.6.3

System states and output

disturbance

estimation

If we know beforehand that an output disturbance do(t) is likely to occur, we can design an appropriate observer to estimate its value and make the controller react accordingly, based on that information. In this case the nonlinear system can be modelled as:

&(t) = f ( x ) + g(x)u(t)

(6.41)

y(t) = h(x) + do(t)

(6.42)

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Robustness Issues of Integral Action in Nonlinear Systems

81

u(t)

D

Nonlinear Observer

d~(t) r

xl(t) I

I

~t :

(1--)~)(u--b)+)~us

(1-- ;9a+ ), Nonlinear feedback control law

F i g . 6.7. G F L strategy with states and input disturbance estimation

and, again, if we assume a constant disturbance do(t), its model is given by:

do(t) = 0

(6.43)

The appropriate observer for both the plant states and the disturbance is then:

~:(t) = f ( ~ ) + g(Jz)u(t) + Jl ( y(t) - h(~) - do )

(6.44)

.

(6.45)

do(t) = J2(y(t) - h(2) - do )

We can now determine the observer gain vector J = [ j T j T ] T based, again, on a linearized model of the system around the current operating point: A=

[ ~176176176 ~] '

C=[

~

1]

(6.46)

In order to compensate for the output disturbance do, we basically proceed by modifying the reference signal y* (t), used in the G F L strategy, to the value y'* (t) = y* (t) - do, which now becomes the new reference signal for the system. In this way, the plant output y(t) will be taken to the desired value y*(t), despite the presence of the output disturbance do(t). This is, in turn, a consequence of the integral action property achieved using the disturbance observer in (6.45). The latter can be explained in the same fashion as we did for the input disturbance case: provided a steady state is reached, then equation (6.45) implies t h a t the plant o u t p u t y(t) must equal h(~) + do. Hence, since the control law takes the model o u t p u t h(~?) to the "compensated" reference y'*(t) = y*(t) - d o , then it is clear that y(t) will reach the desired reference y* (t).

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Goodwin and Rojas

6.7

Comparison between input disturbance design and output disturbance design, using the GFL strategy

We are now able to compare the performance of the control strategies proposed in the previous section, under both input and output disturbance conditions. We simulate the unstable and non m i n i m u m phase nonlinear system given by equations (6.29) to (6.31), with the controller defined by (6.32). Fixed observer gains J are considered, designed with the corresponding linearised models (6.40) and (6.46), around the operating point xl = x2 = 1.5. A linear optimal filter design is adopted, choosing appropriate values for the the assumed spectral densities Q and R of the state and measurement noises, respectively. For both input and output disturbance designs we use R = 10, whereas Q = diag[0.2, 0.2, 100] is assumed for the input disturbance case, and Q = diag[O.1, 10, 100] for the output disturbance case. For both designs, the observer quickly converges to the real plant states and the real disturbance being applied to the system. In this way, the response of the system to a step reference, for example, stays very similar to the response presented in Fig. 6.6, where complete state knowledge is assumed. In Fig. 6.8 we show the response of both strategies to a step input disturbance of magnitude 1 applied at time t = l[s]. Similarly, the responses to a step output disturbance of-0.5 at time t = l[s], are presented in Fig. 6.9.

Input disturbance response

Input disturbance response

1.6

1,6

1.5

1.5

o 1.4

~t.4

c=

1.3

1.3

1.2

1.2

1.1

2

4

6 time [s]

8

10

1.1 {

2

4

6

8

10

time [s]

Fig. 6.6. I n p u t disturbance response: GFL design for input disturbance compensation (left) and output disturbance compensation (right)

Comparing the results shown in Fig. 6.8 and 6.9, we can see that having modelled the disturbance injection point correctly, we achieve a better transient response than in the case in which incorrect modelling occurs. This is particularly clear in the case of an output disturbance. Moreover, the responses in Fig. 6.8 and 6.9 clearly show the nonlinear nature of the system: the oscillating response to the o u t p u t disturbance, for example, reveals the interaction occurring between the disturbance and the system dynamics. It is also worth noting that, for this nonlinear system, an output disturbance observer design becomes especially troublesome, since the linearised system becomes unobservable for x2 ~ 0.58, whereas no such difficulty

6

Robustness Issues of Integral Action in Nonlinear Systems Output disturbance response

83

Output disturbance response

2 1.8

1.8

1.6

1.6

~

1.4

o c

--~1.2

1

1

0.8

0.8

0.6

2

4

6

time [s]

8

10

0.6

2

4

6

8

10

time [s]

Fig. 6.9. Output disturbance response: GFL design for input disturbance compensation (left) and output disturbance compensation (right)

arises in the case of the input disturbance model. This is an extra consideration in the robustness of these various strategies. The results presented here, confirm that we are able to include a certain form of integral action when dealing with nonlinear systems, which ensures, inter alia, zero steady state error, even when the disturbance injection point has been incorrectly modelled. However, it is, in general, difficult to predict the precise nature+of the system transient response, as a result of the complex behaviour of nonlinear systems. It is also interesting to notice that the particular implementation of the control strategy, using state observers, allows a rather straightforward inclusion of a form of anti-windup protection in the presence of constrained input. This mirrors the linear case discussed in section 6.3. In fact, the key point of any anti-windup strategy, is to drive the controller states (in this case, the observer states) by the actual plant input [5]. This can be easily achieved by passing the plant input signal through an appropriate limiting circuit, before feeding the observer. The results are presented in Fig. 6.10 where we have employed the output disturbance design of section 6.6.3, and where an input saturation of 7 has been applied. The control strategy having anti-windup protection performs remarkably better than the one without any antiwindup protection.

6.8

Conclusions

This paper has shown that provision of integral action for nonlinear systems appears to require multiple strategies depending on the injection point of the disturbance. In particular, an incorrectly modelled disturbance can lead to poor transient response or even instability. Many open problems remain in this area. The purpose of this paper has been aimed at introducing the difficulties so as to stimulate further work in this important aspect in nonlinear feedback control.

84

Goodwin and Rojas 2.5

2 .~1.5 o

0.5 0

~

0

v

I

I

I

i

I

1

2

3

4

5

time [s]

Fig. 6.10. Output response of GFL control law to a unity reference step: (1) nominal response with no constraints, (2) input saturation and no anti-windup protection, (3) input saturation with anti-windup protection

References 1. Mayr, O. (1970) The origins of feedback control. MIT Press, Cambridge, Mass. 2. Minorsky (1922) Directional stability of automatically steered bodies. J. Am. Soc. Naval Eng. 34, 284 3. Mazenc, F., Praly L. (1996) Adding integrations, saturated controls and stabilization for feedforward systems. IEEE Trans. Aut. Control 41, 1559-1578 4. Francis, B., Wonham, W. (1976) The internal model principle of control theory. Automatica. 12, 457-465 5. Goodwin, G., Graebe, S. and Salgado, M. (2000) Control System Design, Prentice-Hall 6. Teel, A.R. (1998) A nonlinear control viewpoint on anti-windup and related problems. In Proceedings of the Nonlinear Control Symposium. Enschede, The Netherlands. 7. Kothare, M.V., Campo, P.J., Morari, M. and Nett, C.N. (1994) A unified framework for the study of anti-windup designs. Automatica. 30(12), 1869-1883 8. Teel, A.R. (1995) Semi-global stabilization of linear controllable systems with input nonlinearities. IEEE Trans. Aut. Control 40, 96-100 9. De Dona', J.A., Goodwin, G.C. and Seron, M.M. (1999) Anti-windup and model predictive control: reflections and connections. European Journal of Control (accepted for publication) 10. Goodwin, G., Rojas, O. and Takata, H. (2000) Nonlinear control via Generalized Feedback Linearization using Neural Networks. Asian Journal of Control. (submitted for publication) 11. AllgSwer, F., Zheng, A. (editors)(2000) Nonlinear Model Predictive Control. Birkh/iuser.

7 Robust and Adaptive or a Free Relationship?

Control

--

Fidelity

Per-Olof G u t m a n Faculty of Agricultural Engineering Technion - - Israel Institute of Technology Haifa 32000, Israel peo~tx, technion, ac. il

A b s t r a c t . Robust and adaptive control are essentially meant to solve the same control problem: Given an uncertain LTI model set with the assumption that the controlled plant slowly drifts or occasionally jumps in the allowed model set, find a controller that satisfies the given servo and disturbance rejection specifications. Specifications on the transient response to a sudden plant change or "plant jump" are easily incorporated into the robust control problem, and if a solution is found, the robust control system does indeed exhibit satisfactory transients to plant jumps. The reason to use adaptive control is its ability, when the plant does not jump, to maintain the given specifications with a lower-gain control action (or to achieve tighter specifications), and also to solve the control problem for a larger uncertainty set than a robust controller. Certainly Equivalence based adaptive controllers, however, often exhibit insufficient robustness and unsatisfactory transients to plant jumps. It is therefore suggested in this paper that adaptive control always be built on top of a robust controller in order to marry the advantages of robust and adaptive control. The concept is called Adaptive Robust Control. It may be compared with Gain Scheduling, Two-Time Scale Adaptive Control, I n t e r m i t t e n t Adaptive Control, Repeated Auto-Tuning, or Switched Adaptive Control, with the important difference that the control is switched between robust controllers that are based on plant uncertainty sets that take into account not only the currently estimated plant model set but also the possible jumps and drifts that may occur until the earliest next time the controller can be updated.

7.1

Introduction

Plant uncertainty was always at the heart of feedback control theory. Various robust control methods were developed to handle extended plant uncertainty. Based on classical control, Quantitative Feedback Theory (QFT) was invented, see [10,11]. Based on modern control theory, Hor /-/2, and the #-methods were developed, see [14]. Robust pole placement is described in [1]. All robust methods are also strongly supported by and dependent on design software, e.g. [8]. The robust control problem is in general NP-hard. Still, the available computational tools have proved to be very useful. If successful, every robust design results in an LTI controller that controls any "frozen" plant in the plant uncertainty set,

86

Per-Olof G u t m a n

d~

Fig. 7.1. The closed loop control system for robust control.

satisfying the given specifications. It is not possible to tell, a p r i o r i , if a given robust control problem has a solution without performing (a part of) the design. Adaptive control emerged as an alternative to handle uncertain plants. The idea is to combine an on-line identification algorithm with a control design method to yield a controller that follows the changing plant, see [3]. In spite of their obvious conceptual appeal, and an impressive development effort, adaptive controllers, in particular those based on the certainty equivalence principle, have not become as ubiquitous in industry as expected. The reason for this seems to be that closed loop stability cannot be guaranteed on the same level of confidence as with linear controllers, that adaptive controllers often seem to have unsatisfactory transient behaviour during adaptation to a plant change (e.g. during a start-up when the adaptive controller is initially wrongly tuned), and that they demand highly skilled and educated personnel for tuning and maintenance. This chapter is not meant to be a review of or a comparison between various robust and adaptive control design methods. Instead we try to view adaptive control from a robust point of view, and suggest a remedy for some of their respective shortcomings by marrying them to each other in a suitable way. Because of its graphical nature, Q F T is used as a tool to illustrate the concepts. The chapter is organized as follows: In section 2 the control problem is defined. Sections 3 and 4 contain very brief descriptions of robust control and adaptive control, respectively. In section 5 an illustrating example is given of the transient

7

Robust and Adaptive Control

87

behaviours after a sudden plant change of a robust and adaptive control system, respectively, with a summary of their advantages and disadvantages. The argument is made clear in section 6 where adaptive control is seen from a robust perspective. The rSle of adaptation is discussed in section 7, leading up to the suggested paradigm of Adaptive Robust Control in section 8. A short conclusion is found in section 9, followed by an Acknowledgement and a bibliography.

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7.2

Problem

definition

For simplicity, we consider the SISO case only. Given an uncertain, strictly proper LTI plant

n(s,p) -pqs,t P(s) 9 {Pi(s)} -- d(-7,~,p)e LI + M(s))

(7.1)

with the uncertain parameter vector p 9 H 9 R q, and the multiplicative unstructured uncertainty satisfying ]M(jw)l < re(w). M(s) is assumed to be stable and proper, and the high frequency gain sign of (7.1) is known. The index i in (7.1)

88

Per-Olof G u t m a n

only denotes membership in the set and not enumeration. Input, state, and o u t p u t disturbances, D~ (s), are assumed to act on the plant. The closed loop specifications are given as a servo specification,

a(jw) ~ IY(jw)/R(jw)l ~ b(jw)

(7.2)

where Y(jw) and R(jw) are the Laplace transforms of the controlled output and reference, respectively, or sensitivity specification,

IS(jw)l ~ x(w)

(7.3)

or any other disturbance rejection specification. It is assumed that the plant P(s) "slowly" drifts among {Pi(s)}. Such a drift may be caused by wear, change of operating point, or a change in the outside environment. Moreover, it is assumed that P(s) "occasionally" jumps within {P~(s)}, i.e. suddenly changes from one LTI plant instance to another. Such a j u m p may be caused by a change of plant equipment, or a partial failure, or unknown loading (e.g. when a robot arm picks up an unknown load), or when switching on the control system without knowing which member of the set (7.1) describes the plant at that moment. It is assumed that the frequency of the j u m p s is considerably lower than the bandwidth of the closed loop system and that the speed of the drift is considerably lower than e.g. the speed of the step response transient of the closed loop system, see e.g. [3]. One could say that the "product of plant change and time is small" [5]. The control design problem is then to find a controller that makes the closed loop system satisfy the specifications.

7.3

Robust Control

The robust control problem is to find a feedback controller G(s) and a prefilter F ( s ) , see Figure 7.1, such that the specifications (7.2), (7.3), etc, are satisfied for each member in the plant set {Pi(s)}. Note that the actual controller implementation may differ from the canonical form shown in Figure 7.1. I f a solution is found, then the specifications are also satisfied for slow plant drift, due to the quasi-LTI assumption in section 7.2. If, in addition, the specifications include the rejection of disturbances equivalent to the envisaged "occasional jumps", and a solution is found, then the specifications are also satisfied during plant jumps. Some robust control methods were mentioned in the introduction. Here a few details about Q F T will be given, since Q F T will be used for illustration. For the reader familiar with the H ~ - m e t h o d , we would like to point out that the SISO robust sensitivity problem for a plant with unstructured multiplicative uncertainty only, is identical for H ~ and for Q F T (cf. Figure 2.17 in [14]). In QFT, the specifications and the plant uncertainty set {Pi (s)} give rise, for each frequency w, to a complex valued set Bc(jw) such that G(jw) E Bc(jw) r the specifications are satisfied for each member in {Pi(s)}. Then, BL(joa) : BG(jw)Pnom(jw), where Pnom(8) E {Pi(s)} is an arbitrary nominal plant, such that Lnom(jw) E BL (jw) r the specifications are satisfied for each member in {Pi(s)}, where Lnom(S) = Pnom(s)G(s) is the nominal compensated open loop. In general, the Horowitz bound OBL(jw), defined as the border of BL(jw), is displayed in a Nichols chart. With Horowitz bounds for several frequencies displayed, L . . . . (jw) is manually loopshaped in order to satisfy the specifications. See Figure 7.2.

7

Robust and A d a p t i v e Control

89

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7.4

Adaptive

Control

The ideal adaptive control would be dual control [3], in which the control signal is optimal for both plant estimation and control. Unfortunately dual control is computationally prohibitive. LTI based adaptive control should have the following desired capabilities: estimate the current plant model, redesign the controller, and decide when to estimate/redesign [5]. Many types of adaptive controllers do not have all these features; instead, a practical definition of adaptive control could be: "an adaptive controller is a controller with adjustable parameters, and a mechanism for adjustment" [3]. See Figure 7.3. The parameters are adjusted such t h a t after convergence, the specifications (7.2), (7.3), etc, are satisfied. The Certainty Equivalence (CE) adaptive controller combines p a r a m e t e r estimation (RLS, LMS, etc) with some control design method (pole placement, MRAS, etc). The controller parameter are computed as if the current p a r a m e t e r estimate is true. Under ideal conditions (no noise, no disturbances, no undermodelling, minimum phase plant) MRAS gives boundedness and convergence [3]. To handle some of the non-ideal conditions, detuning from CE, such as dead-zone, a-modification, back-stepping, etc, have been suggested [5]. A functioning CE a d a p t i v e controller handles slow plant drift very well. The a d a p t a t i o n transient after the occasional plant j u m p is however often very unsatisfactory, as illustrated in the next section.

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.

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F i g . 7.4. Simulation of the robust control system. The upper and lower graphs show step responses and the control signals, respectively. On the left, the plant gain k changes from 1 to 4 at time t = 15 s while T = I . On the right, the time constant T changes from 1 to 0.5 at time t ---- 15 s while k ---- 1.

An auto-tuner [3] is an adaptive controller where the control law is automatically updated only after the convergence of the parameter estimate, and only on human operator demand and under operator supervision. Thus the disadvantages of the CE adaptive controller are avoided, but an auto-tuner is not able to handle neither plant drift nor plant jumps without human intervention. Gain scheduling [3] is a scheme where you apply different pre-computed controllers for each operating condition. The operating condition is given in a separate identification loop based on measured external signals or process variables. Often soft transfer between the different controllers is implemented. Since the system is almost LTI at all times, there are no stability problems. Cain scheduling handles plant drift well, if for each operating condition the controller is robust. Plant j u m p s within an operating condition are also handled if the local controller is designed appropriately. The difficulties with CE adaptive control seem to be due to the interference between the identification and control loops [5]. Therefore a new type of adaptive controllers have been suggested and researched lately, under names such as Two time scale adaptive control, Intermittent adaptive control, or Switched adaptive control, attempting to combine the advantages of Auto-tuning and Gain Scheduling.

7

Robust and Adaptive Control

91

The identification and control loops are separated with the control parameters being updated only when necessary and when the identification has converged. Hence the closed loop is LTI at (almost) all times and there is no local stability problem. It is however not clear how these controllers behave during slow plant drift and occasional plant jumps. This is a main issue of this paper.

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7.5

R o b u s t vs. A d a p t i v e

This example is found in [2]. Consider the uncertain plant

P(s)-

k (l+Ts) 2

with

k 9 [1,41 and T 9 [0.5, 21.

(7.4)

92

Per-Olof G u t m a n

A Q F T design was performed with a servo specification having a bandwidth of 2 rad/s. Simulation results are shown in Figure 7.4. The resulting controller was

G(s) = 4 . 1 0 7 .

(s+0.25) (s+1.5) 1 s " (s + 30) (s 2 + 5008 + 250000)'

and

1 F ( s ) = 2.89. s2 + 1.87s + 2.89"

(7.5)

As a comparison, an explicit second order adaptive pole placement controller, with RLS as the parameter estimator was implemented with a sampling interval = 0.3 seconds. The required pole location had a natural frequency = 1.5 rad/s and the relative damping = 0.707. Thus the specifications of the two designs were similar. Simulations of the adaptive control system are found in Figure 7.5. It is clear from the simulations that the adaptive controller exhibits unsatisfactory transients during adaptation, while the robust controller works fine. However, after convergence, the adaptive controller has the same reference step response for different plant cases. The same conclusion is drawn from Example 10.1 in [3]. Robust control is stable and satisfactorily controls the plant during occasional plant jumps and slow plant drift. A C E adaptive controller is stable only under restrictive assumptions, exhibits ugly transients during plant jumps, but controls the plant more uniformly during slow or no drift. In fact, an adaptive control solution may be found (with the exception of plant jumps) for a larger uncertainty set or for tighter specifications, when no robust solution exists. The challenge is how to marry robust and adaptive control to get the best of both.

7.6

A d a p t i v e control from a robust p e r s p e c t i v e

Figure 7.6 shows the trade-off in all feedback design: If the plant uncertainty increases, more feedback gain is needed to maintain the same specifications. If the specifications get tighter more feedback gain is needed if the plant uncertainty remains unchanged. The trade-off will be illustrated with a scalar example. Referring to Figure 7.1, let P(s) = k C [kmin,kmax] be a scalar gain plant, and G(s) = g > 0 a scalar compensator. Let the plant uncertainty be defined by kmax/kmin. Referring to (7.2), let the specification for the closed loop uncertainty A be given by Z~ ~-- maxk I~'l _ kmax mink ISI - kmin

1 ~-kming < b 1 § kmaxg -- a

(7.6)

where b > a are given, and S = P G / ( 1 + P G ) is the complementary sensitivity function. A plot of A as a function of g is shown in Figure 7.7 for two different values of kmax/kmi~. Clearly the trade-off mentioned above holds. With QFT, the tradeoff is apparent at each frequency. This is illustrated with the following example. Consider the uncertain plant

sTa P ( s ) = k - l + 2r

+s2/w 2e-~''

k E [2,5], a C [1,3], r e [0.3, 0.6], T E [0, 0.05].

(7.7)

The uncertainty at each frequency, w, is defined as the value set [14], or template [9], {P~(jw)}. In Figure 7.8, ( P ( 2 j ) } is illustrated. Notice that the maximum gain of

7

Robust and A d a p t i v e Control

93

Closedloop specs Feedbackgain

Trade-off Plantuncertainty F i g . 7.6. The trade-off in feedback design.

{P(2j)} is 27 dB and the minimum gain is 13 dB, i.e. the plant gain uncertainty at 2 r a d / s is 14 dB. Referring to (7.2), assume that the original servo specification in Figure 7.9 has to be satisfied. From the figure it transpires, t h a t the remaining closed loop gain uncertainty at 2 r a d / s must satisfy Zl(2) -- max~

]S(2j)]

< 3.73 dB

(7.8)

mini [S(2j)[ where S(jw) is the complementary sensitivity function and i is taken as the plant cases. The resulting Horo@itz bound, OBL(2j), is shown in Figure 7.10. Notice that the gain of the bound depends on the phase. Assume now t h a t a tighter specification is desired for which A(2) < 2.13 dB. Then it follows t h a t the new Horo@itz bound for 2 r a d / s is about 5 dB higher than before, see Figure 7.10, implying t h a t the feedback controller gain must increase by about 5 dB. If, however, at the current operating condition e.g. the plant zero is less uncertain, such t h a t a E [2.5, 3], and this can be detected by an on-line parameter estimation algorithm, then the templates and Horowitz bounds can be recomputed. In Figure 7.8 the reduced uncertainty template for 2 r a d / s is displayed. We notice t h a t the template is considerably thinner and shorter than the original fat and tall one. Therefore, the ensuing Horowitz bound for the tighter specification will have lower gain (Figure 7.10) and almost become equal to the original Horowitz bound valid for the original plant with the original uncertainty and the original specification. Hence

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the trade-off in Figure 7.6 is apparent: A tighter specification can be off-set by increased plant knowledge. But why not increase the feedback gain when larger plant uncertainty or tighter specifications require it? This can of course be done to a certain extent, and that is the basis of robust control. There are however fundamental limitations in feedback control [15]. Any real-life plant includes delay or is non minimum-phase, or has large phase uncertainty at high frequencies. Then the phase margin requirement together with Bodes gain-phase relationship imposes a bandwidth limitation, and hence a limit on the allowed feedback gain. Moreover, the sensor noise is amplified at the plant input by - G / ( 1 + PG), see Figure 7.1. High feedback compensator gain is not wanted at the sensor noise frequencies, since it may cause actuator saturation and wear, or require the use of a more expensive, low noise sensor. By decreasing plant uncertainty, adaptation fights the fundamental feedback gain limitation, and shifts the trade-off in favor of tighter closed loops specifications. See Figure 7.11. The landmark paper [16] presents an algorithm how to adapt the parameters of a robust controller when more plant knowledge becomes available, and demonstrates the benefits. Gain adaptation of a robust controller is described in [6,17].

7

Robust and Adaptive Control

95

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7.7

T h e r61e of a d a p t a t i o n

The r61e of adaptation is to identify the plant templates {P~(jw)} at those frequencies that constrain the design, i.e. in Q F T parlance, the frequencies at which Lnom(jw) 9 OBL(jW) or, if the design failed, Lnom(jw) ~ BL(jw). The identification should be helpful for the design purpose to satisfy the specifications with least possible feedback gain, and could therefore e.g. be selective with respect to what plant parameters to estimate. In Figure 7.8 the identification of one parameter only was sufficient to decrease the template size so that the feedback gain requirement decreased by 5 dB. Figure 7.7 illustrates the case when there is a sensitivity specification, S < 6 dB that is satisfied tightly for some frequency. The identification of a smaller template (shaded area) is helpful only if it increases the distance from the template to the 6 dB sensitivity locus. Moreover, the adaptive controller should be able to redesign or retune the robust controller on which it is based, and switch the controller from one robust controller to another, in order to keep the closed loop system robust and quasi-LTI.

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7.8

A d a p t i v e robust control

It is however not sufficient to base the redesign or re-tuning of the robust controller on only the helpfully identified templates. Certainty equivalence is not suitable. The currently applied robust controller must be based on templates that incorporate plant cases to which the plant may drift or j u m p during the period to the next controller update. We call this paradigm Adaptive Robust Control. A block diagram is found in Figure 7.7. Composite templates can e.g. be constructed as follows, see Figure 7.7. Before taking into account template extension due to possible plant drift and plant jumps, the identified template may be based on probing at selected frequencies [4] or may even be a frequency function point estimate, e.g. from some common recursive estimation algorithm [3]. A probabilistic template, e.g. the point estimate and its 1 a ellipse (in the Nyquist diagram) could constitute the template on which a high performance design is based, e.g. satisfying a servo specification (7.2). A worst case template, e.g. identified with set membership methods [7,13] could serve as the template for stability design only. Finally, the trade-off between specifications, design template size, speed of plant drift and size of plant jumps, and the speed of on-line identification and controller u p d a t i n g can be illustrated as in Figure 7.7.

7

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7.9

Conclusions

A lot of research remains to be done before Adaptive Robust Control is in place whereby Adaptive Control solidly lies on Robust Control. Current on-line identification routines have to be developed further to identify templates in the required, selective, way. One useful idea may be found in [4]. Few algorithms for switched adaptation of robust controllers have been published, [16] being a landmark exception. Nevertheless one should mention [12] as a very successful application of the ideas presented in this paper. So, what about fidelity or a free relationship in the marriage between Robust and Adaptive Control? Obviously, Adaptive Control should stick to fidelity. Robust Control may however fiddle around, but not with too fat templates.

Acknowledgement I warmly thank my friends and colleagues Bo Egardt, Arie Feuer, Izchak Lewkowicz, Leonid Mirkin, and H~ctor Rotstein for helpful discussions during the preparation of this paper.

98

Per-Olof Gutman

Fundamental Limitations

Adaptation

J Plant uncertainty Feedbackgain

~ Trade-off Closed loop specs Fig. 7.11. The trade-off in feedback design with adapatation.

dB

isLl d

]

~ not helpful

L

dei

he ,ful

Fig. 7.12. Helpful and unhelpful identification of a smaller template.

7

i--~

Template

I~

I

I ldenti~catiOnI~

I

I Design

I'

Robust and Adaptive Control

99

Externalmeasurement

P

~7o' I

adiustment

I

):

Fig. 7.13. The block diagram for Adaptive Robust Control.

dB

O

f Fig. '/.14. The identified template (solid), the probabilistic template for performance design (dashed), and the worst case template for stability design only (shaded), before template extension due to possible plant drift and plant jumps.

100

Per-Olof Gutman

SDeedofDiantdrift ~ { Speedofiden~ ~ ofcontrollerupd~ Fig. 7.15. The Trade-Off in Adaptive Robust Control. A long identification time may give accurately identified templates, but will still require large design templates due to possible plant drift and jumps.

References 1. Ackermann J. (1993) Robust Control: Systems with Uncertain Physical Parameters. Springer, New York. 2. _~strSm K. J., Neumann L., Gutman P.-O. (1986) A comparison between robust and adaptive control of uncertain systems. Proc. 2nd IFAC Workshop on Adapative Systems in Control and Signal Processing, 37-42, Lund, Sweden, July 1-3. 3. /~strSm K. J., Wittenmark B. (1995) Adaptive Control - - Second Edition. Addison-Wesley, Reading, Ma. 4. Galperin N., Gutman P.-O., Rotstein H. (1996) Value set identification using Lissajou figure sets. Proc. 13th World Congress of IFAC, vol I, San Fransisco, California, USA, 30 June - 5 July. 5. Goodwin G., Feuer A., Mayne D. (2000) Adaptive Control: Where to Now?. Preprint. 6. Gutman P.-O., Levin H., Neumann L., Sprecher T., Venezia E. (1988) Robust and adaptive control of a beam deflector. IEEE Trans. Aut. Contr., 33 (7), 610-619. 7. Gutman P.-O. (1994) On-line parameter interval estimation using Recursive Least Squares. Int. J. Adaptive Control &=Signal Processing, 8, 61-72. 8. Gutman P.-O. (1996) Qsyn - - the Toolbox for Robust Control Systems Design for use with Matlab, Users Guide. E1-Op Electro-Optics Industries Ltd, Rehovot, Israel. 9. Horowitz I. M. (1963) Synthesis of Feedback Systems. Academic Press, New York. 10. Horowitz I. M., Sidi M. (1972) Synthesis of feedback systems with large plant ignorance for prescribed time domain tolerances. Int. J. Control, 16, 2,287-309.

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101

11. Horowitz I. M. (1993) Quantitative Feedback Theory (QFT) vol. 1. QFT Publications, Boulder, Co. 12. Nordin M. (2000) Nonlinear Backlash Compensation for Speed Controlled Elastic Systems. Doctoral Thesis, Division of Optimization and Systems Theory, Royal Institute of Technology, Stockholm, Sweden. 13. Rotstein H., Galperin N., Gutman P.-O. (1998) Set membership approach for reducing value sets in the frequency domain. IEEE Trans. Aut. Contr., 43, no 9, 1346-1350. 14. Sdnchez-Pefia R. S., Sznaier M. (1998) Robust Systems Theory and Applications. Wiley Interscience, New York. 15. Seron M. M., Braslavsky J. H., Goodwin G. C. (1997) ~ n d a m e n t a l Limitations in Filtering and Control. Springer Verlag, London. 16. Yaniv O., Gutman P.-O., Neumann L. (1990) An Algorithm for the Adaptation of a Robust Controller to Reduced Plant Uncertainty. Automatica, 26, 4, 709720. 17. Zhou T., Kimura H. (1994) Robust-control of the Sydney Benchmark Problem with intermittent adaptation. Automatica, 30, 4, 629-632.

8 E x p e r i m e n t s in S p a t i a l Piezoelectric Laminate Beam

Control of a *

D u n a n t Halim and S.O. Reza Moheimani Department of Electrical and Computer Engineering, University of Newcastle, Callaghan, NSW 2308, Australia

A b s t r a c t . This paper is aimed to develop a feedback controller that suppresses vibration of flexible structures. The controller is designed to minimize the spatial 7-/~o norm of the closed-loop system, which guarantees average reduction of vibration throughout the entire structure. The spatial 7-/o0 control problem can be solved by finding an equivalent system representation that allows the solution to a standard T/~ control problem to be used. Feedthrough terms are incorporated into the flexible-structure model to correct the locations of the in-bandwidth zeros. The controller is applied to a simply-supported piezoelectric laminate beam and is validated experimentally to show the effectiveness of the proposed controller in suppressing structural vibration. It is shown that the spatial 7-/~ control has an advantage over the pointwise ~ control in minimizing the vibration of the entire structure. This spatial T/~ control methodology can also be applied to more general structural vibration suppression problems.

8.1

Introduction

Vibration is a natural phenomena that may occur in all dynamic systems, such as flexible structures. Vibrations in flexible structures can be detrimental to structural performance and stability. Thus, it is important to find a means of suppressing structural vibrations. In this paper, we describe a controller design framework for suppressing the unwanted structural vibrations in flexible structures. Flexible structures are distributed parameter systems. Therefore, vibration of each point is dynamically related to the vibrations of every other point over the structure. If a controller is designed with a view to minimizing structural vibrations at a limited number of points, it could have negative effects on vibration profile of the rest of the structure. The concept of spatial 7-/~ control was first introduced by the second author in [1] for the purpose of suppressing structural vibration over the entire structure. This paper presents experimental implementation of this concept on a piezoelectric laminate beam for the first time. Based on this concept, the controller is designed such that the spatial 7-/~ norm of the closed loop system is minimized. Minimizing the spatial T/~ norm of the system will guarantee vibration suppression over the entire structure in an average sense. This spatial T/~ control problem can be solved by finding an equivalent * This research was supported by the Centre for Integrated Dynamics and Control and the Australian Research Council. Work of the first author was supported via a University of Newcastle RMC scholarship.

104

Dunant Halim, S.O. Reza Moheimani

system representation that allows a standard 7-Ur control optimization problem to be solved instead. The spatial 7-/~ control produces a controller with similar dimensions as t h a t of the plant. If the modal analysis technique is used to develop a model, a direct truncation can be used to obtain a finite dimensional model of the system. However, it is known that the locations of the in-bandwidth zeros are not accurate because of truncation [2-5]. This inaccuracy can have negative effect on the closed loop stability. To fix this problem, we add a feedthrough term to the model to correct the locations of the zeros as discussed in [2-5]. This technique is known in the aeroelasticity literature as the mode acceleration m e t h o d [6]. To demonstrate our proposed controller, a SISO (Single Input, Single O u t p u t ) spatial 7"/oo controller is designed for a piezoelectric laminate beam to suppress the vibration of the first six bending modes of the structure. Th e controller is applied to a real structure, a simply-supported beam with a collocated piezoelectric actuator-sensor pair. Piezoelectric devices have shown promising applications in active vibration control of flexible structures [7-12]. The ability of piezoelectric materials to convert mechanical strain into electrical voltage and vice versa allows them to be used as actuators and sensors when placed on flexible structures. This paper is organized as follows. Section 2 describes the dynamics of flexible structures such as those with collocated piezoelectric actuator-sensor pairs. Section 3 briefly describes the notion of spatial norms that are used as performance measures for flexible structures. Section 4 deals with the model correction to reduce the error in the location of zeros of the model. Section 5 describes the concept of spatial T / ~ control for flexible structures. Section 6 discusses the design of a feedback SISO controller for suppressing the vibration of the first six modes of a simplysupported piezoelectric laminate beam. Section 7 presents experimental validations on the application of the developed controller to a beam structure. The last section concludes the overall of the paper.

8.2

Models of flexible structures

SeNSOrS

IZ

actuators

Fig. 8.1. A simply-supported beam with a number of collocated piezoelectric patches.

In this section, we briefly explain how a model of a beam with a number of collocated P Z T actuator-sensor pairs can be obtained using a modal analysis technique. Interested readers can refer to [9,10] for more detailed derivations.

8

Experiments in Spatial Hr

Control

105

Consider a homogeneous Euler-Bernoulli beam with length L, width W and thickness h as shown in Figure 8.1. The piezoelectric actuators and sensors have length Lpx, width Lpy and thickness hp. In this paper, we assume that hp << h, which is true for the patches that are used in our experiments. Thus, the assumption of uniform beam properties can be justified. However, we can also use approximation methods such as the finite element method to deal with more general non-uniform structures. Suppose there are M collocated actuator-sensor pairs distributed along the structure. Piezoelectric patches on one side of the beam are used as sensors, while patches on the other side are used as actuators. Voltages that are applied to actuating patches are represented by Va (t) = [Val ( t ) , . . . , YaM (t)] T. We assume that a model of the structure is obtained via the modal analysis procedure. This procedure requires finding a solution to the partial differential equation (PDE) which describes the dynamics of the composite system. The governing P D E of a flexible beam is as follows [8,13],

EI ~04z(x't)Ox

+ pAb 02z(x't)-~ -- 02Mp*(x't)Ox z

(8.1)

where the beam transverse deflection at point x and at time t is denoted by z(x, t). Also, p and Ab represent the density and the cross-sectional area of the beam, while E and I are the Young's Modulus and the moment of inertia about the neutral axis of the beam respectively. The right-hand-side term represents the forcing function produced by the piezoelectric actuator. In this case, Mpx is the forcing moment acting on the beam. The PDE can be solved independently for each mode by using the orthogonality properties of its eigenfunctions, Wk, which are as follows for the case of a uniform beam,

fo ~ Wk(x) w~(x) dx =

~k~

fo L E1 d4Wk pAb dx 4 Wp dx = w~ 5kv

(8.2) (8.3)

where wk describes the natural frequency of the beam at mode k. Here, 5kp is Kronecker's delta function, that is, 5kp = 0 for all k # p, and equals one if k = p. The MIIO (Multiple-Input, Infinite-Output) transfer function from the applied actuator-voltages, Va(s), to the transverse structural deflection z(s, x) at location x is, G(s, x) = P

s 2 + 2r ~k s + ~

(8.4)

k=l

where ~k = [k~kl,. 9 , ~kM] T and the mode n u m b e r is denoted by k. k~ki is a function of the location of the i th piezoelectric actuator-sensor pair and the eigenfunction Wk(x) (see [12,14,15]). The damping ratio is denoted by ~k respectively and P is a constant that is dependent on the properties of the structure and the piezoceramic patches. Such models have the interesting property that they describe spatial and spectral properties of the system. The spatial information of these models can then be

106

D u n a n t Halim, S.O. Reza Moheimani

used to design controllers, which guarantee a certain level of damping for the entire structure. Furthermore, the MIMO (Multiple-Input, Multiple-Output) transfer function of the flexible structure with piezoelectric actuator-sensor (collocated) pairs can be determined in a similar manner. The transfer function from the applied actuatorvoltages V~(s) to the induced voltages at the sensor Vs(s) ---- [Vsl (8),... , VsM(S)] T is,

Gw(s) = Pw

s2 +2r

(8.5)

k=l

where Pvs = T P > 0 is a constant based on the properties of the structure and the piezoceramic patches.

8.3

Spatial

norms

This section presents an overview of the spatial norms of signals and systems. For a more detailed review of this concept, the reader is referred to [5]. Consider the transfer function G(s,x), where x E X , which maps an input signal w(t) E R m to the output signal z ( x , t ) C R • X (see Figure 8.2). Hence, z ( x , t ) is spatially distributed over the set X.

w(t)

-1 G(s,x)

z(x,t)

Fig. 8.2. System G(s, x)

The spatial 7/2 norm of the signal z(x, t) is defined as,

<( z >>2 -=

Z(X, t)Tz(x, t)dxdt.

(8.6)

The spatial 7/2 norm of the signal z can be interpreted as the total energy of the signal z(x, t). Now, let G be the linear operator which maps the inputs of G ( s , x ) to its outputs. The spatial induced norm of G is defined as (see [5]), << G >>2 =

sup

<< z >>22

0~w~2t0,o~)

Ilwll~

(8.7)

where IIw1122= f o w(t)Tw(t) dt" Moreover, following [5], the spatial 7/0~ norm of G(s, x) is defined as, <>2=

~eRsup)~maX(/xG(jw'x)*G(jw'x)dx)

w h e r e ~max (F) represents the maximum eigenvalue of the matrix F.

(8.8)

8

Experiments in Spatial H~o Control

107

Theorem (3.1) of [5] gives a representation of the spatial 7-/~ norm of G(s, x) in terms of total energy of z(x, t) and energy of the input signal w(t). The theorem states that, <<:G>>-- < < G > > ~ .

(8.9)

It is possible to add spatial weights to all of the above definitions, to emphasize the regions that are of more importance. This issue will be further explained in Section 5.

8.4

Model Correction

In practice, dynamical models of a flexible structure as described in (8.4) and (8.5) have to be truncated to represent the system by a finite-dimensional model. The model can be truncated so to include only the modes within the frequency bandwidth of interest. However, the truncation of the model produces additional error in the locations of the in-bandwidth zeros. This is due to the fact that the contribution of the out-of-bandwidth modes is generally ignored in the truncation. One way to improve the truncated model dynamics is to include a feedthrough term to correct the locations of the in-bandwidth zeros. This technique is known in the aeroelasticity literature as the mode acceleration method [6] and has been re-visited in [2-5]. Thus, the infinite-dimensional model of the collocated system in (8.5) can be approximated as GNu(s), GvN~(s)

2 + K,~

(8.10)

k~l

where N is the number of modes included in the model, and Kv8 is a M • M matrix added to correct the location of in-bandwidth zeros. For a general multivariable system, Kv8 can be found using the method proposed in [4]. For a SISO system, Kv~ will be a scalar. Similarly, we describe the approximate spatial transfer function of G(s, x) in (8.4) by,

Gg(s'x) --~ P ~

k=l

s 2 -~2-~k-~;s-+ 2 + K(x) 02k

(8.11)

where K(x) is a 1 x M vector. K(x) is a function of the spatial variable, x. It has to be estimated from the modal model of the system. One technique that can be used is to find the feedthrough term that minimizes the weighted spatial ~r162norm of the error between the infinite-dimensional and truncated models is presented in [5]. The term K(x) is determined such that the following cost function is minimized,

J = << Wc(s,x) (G(s,x) - GN(s,x)) >>~.

(8.12)

Here, We(s, x) is an ideal low-pass weighting function distributed spatially over X with its cut-off frequency wc chosen to lie within the interval wc C (WN, WN+I).

108

D u n a n t Halim, S.O. Reza Moheimani The cost function (8.12) is minimized by setting [5],

K(x) ---- ~

K kovtWk(x)

(8.13)

k=N+l

where, k

1(1

_-- 2

~

09 k

+

02 2

p~T.

(8.14)

Note that in practice we can only include a finite number of modes to calculate the feedthrough term, K(x).

8.5

Spatial 7-/oo c o n t r o l of a p i e z o e l e c t r i c l a m i n a t e beam

w+~+ Controller

1

-x Z

F i g . 8.3. Spatial ~

control of a flexible beam

This section is concerned with the problem of spatial T/~ control for flexible structures. Consider a typical disturbance rejection problem for a flexible structure such as the one shown in Figure 8.3. The system consists of only one piezoelectric actuator-sensor pair for the sake of clarity. Here, the purpose of the controller is to reduce the effect of disturbance, w(t), on the entire structure, using piezoelectric actuators and sensors. The concept of spatial 7-/oo control was introduced in [1] to address problems of this nature. A spatially-distributed linear time-invariant dynamical system such as the beam in Figure 8.3 can be defined in its state-space form as,

~(t) = A2(t) + Blw(t) + B2u(t) z(x, t) = C1 (x)~2(t) + Dll (x)w(t) + D12 (x)u(t) V~(t) = C22(t) + D21w(t) + D22u(t)

(8.15)

where 9 E R ~ is the state, w E R is the disturbance input, u E R is the control input, z is the performance output, V~ E R is the measured output. For a flexible structure, z(x, t) represents the spatial displacement at time t, where x E X.

8

Experiments in Spatial H ~ Control

109

The system matrices in (8.15) can be obtained from transfer functions (8.10) and (8.11). Note t h a t for the system shown in Figure 8.3, D22 ----D21 in (8.15) is the feedthrough term gv~ described in (8.10), while D n (x) = O12 (x) is K(x) in (8.13). Moreover, B1 --- B2 since disturbance is assumed to enter the system through the actuator. The spatial 7-/oo control problem is to design a controller,

5ck(t) = Akxk(t) + Bk V~(t) u(t) = Ckxk(t) + DkV~(t)

(8.16)

such that the closed-loop system satisfies, inf

sup

KEU wEs

J ~ < 72

(8.17)

where U is the set of all stabilizing controllers and,

j ~ = f o f x z(x, t)TQ(x)z(x, t)dxdt f o w(t)Tw(t) dt

(8.18)

Here, Q(x) is a spatial weighting function. The purpose of Q(x) is to emphasize the region where the vibration is to be d a m p e d more heavily. The numerator in (8.18) is the weighted spatial 7-/2 norm of z(x, t) [7,16]. Therefore, J ~ can be interpreted as the ratio of the spatial energy of the o u t p u t of the system to the energy of its input. The control problem is depicted in Figure 8.4. w(t)

z(x,t)

G(s,x)

u(t)

Vs(t)

-I K(s) F i g . 8.4. Spatial 7-/o~ control problem It can be shown by the method in [1] t h a t the above problem is equivalent to a standard 7-/oo control problem for the following system,

~(t) ----A2(t) + Blw(t) + B2u(t) 2(t) = H 2(t) + Ow(t) + Ou(t) Vs(t) = C22(t) + D21w(t) + D2eu(t) where D21 = D22 and [H

"Cl(x)r]

I'TF-~ f ax

D11(x) T

.D12(x) T

O

(8.19)

@] = F . Here, F is any matrix t h a t satisfies,

Q(x) [Cl(X)

Dl~(x)

D~2(x)] dx,

(8.20)

110

Dunant Halim, S.O. Reza Moheimani

and D l l (x) ----D12 (x). Hence, the system in (8.19) can be solved using a standard T/~r control technique [17,18]. The spatial ~ controller can be regarded as a controller that reduces structural vibration in an average sense. T h e resonant peaks will be particularly targeted by this controller, which is desirable for our purpose of minimizing structural vibration. It can be observed that the T / ~ control problem associated with the system described in (8.19) is non-singular. This is due to the existence of feedthrough terms from the disturbance to the measured output and from the control signal to the performance output. Had we not corrected the location of in-bandwidth zeros, the resulting T/ ~ control problem would have been singular. Designing a 7-/~ controller for the system (8.19) may result in a very high gain controller. This could be attributed to the fact that the term O in (8.19) does not represent a physical weight on the control signal. Rather, it represents the effect of truncated modes on the in-bandwidth dynamics of the system. This problem can be fixed by introducing a weight on the control signal. This can be achieved by re-writing (8.19) as,

~(t) = A2(t) + Blw(t) + B2u(t)

Vs(t) = C22(t) + D21w(t) + D~2u(t)

(8.21)

where R is a weighting matrix with compatible dimensions. What makes this system different from (8.19) is the existence of matrix R in the error output, ~. The matrix R serves as a weighting matrix to balance the controller effort with respect to the degree of vibration reduction that can be achieved. This can be shown to be equivalent to adding a term, f o u(t)TRTRu(t) dt' to the numerator of the cost function, J ~ in (8.18). Setting R with smaller elements might lead to higher vibration reduction but at the expense of a higher controller gain. In practice, one has to make a compromise between the level of vibration reduction and controller gain by choosing a suitable R.

8.6

Controller design

In this section, effectiveness of the spatial ~ control method will be demonstrated on a laboratory scale apparatus. A simply-supported flexible beam - such as the one shown in Figure 8.1 - with a collocated piezoelectric actuator-sensor pair attached to it is used in the experiments. The apparatus is shown in Figure 8.5. Th e structure consists of a 60 cm long uniform aluminum beam of a rectangular cross section (50 m m • 3 ram). The beam is pinned at both ends. A pair of piezoelectric ceramic elements is attached symmetrically to either side of the beam, 50 m m away from one end of the beam. The piezoceramic elements used in our experiment are PIC151 patches. These patches are 25 m m wide, 70 m m long and 0.25 m m thick. Th e physical parameters of PIC151 are given in Table 8.1. A model of the composite structure is obtained via modal analysis. We use the equivalent standard 7-/~ control problem described in (8.21) for our spatial 7-/~ controller. Here, V~ is the output

8

Experiments in Spatial H~o Control

111

F i g . 8.5. A piezoelectric laminate beam T a b l e 8.1. Properties of PIC151 piezoceramics Piezoceramic Young's Modulus, Ep

6.70 x 101~ N / m 2

Charge constant, daa

- 2 . 1 0 x 10 - l ~ m / V

Voltage constant, g31

- 1 . 1 5 x 10 .2 V m / N

Capacitance, C

1.05 x 10 . 7 F

Electromechanical coupling factor, k31

0.34

voltage from the piezoelectric sensor, while u is the control input voltage from the controller. Here we wish to control only the first six vibration modes of the beam via a SISO controller. Hence, the model is truncated to include only the first six modes. The 7-/oo control design procedure will then produce a 12 th order controller. This means that the controller complexity can be reduced effectively. The effect of outof-bandwidth modes has to be taken into consideration to correct the locations of the in-bandwidth zeros of the truncated model as discussed in the Section 8.4. Based on the experimental frequency-response data from actuator voltage to sensor voltage, the feedthrough term in (8.10), D21 = D22 = K v ~ , is found to be 0.033 if the first six modes are considered in the model (see also [19]). Since the disturbance is assumed to enter the system through the same channel as the controller, the SISO transfer functions from w and u to the transverse deflection of the beam, z ( x , t ) , are the same, i.e. G N ( s , x ) . Incorporating (8.13) and (8.14) in (8.11), we have, N

Wk (x) r

Nrr~ax9

=

K k

Wk(x)

(8.22)

k=N+l

k=l

Notice that N ~-- 6 since we wish to find a controller of minimal order to control the first six modes of the structure. The feedthrough term is calculated by considering modes N + I to N , ~ , = 200 to obtain a reasonable spatial approximation to the feedthrough term. Similarly, the SISO transfer functions from w and u to the collocated sensor voltage, V~, are denoted by GNv~(S) (8.10),

N k=l

~

+ KV,

(8.23)

112

D u n a n t Halim, S.O. Reza Moheimani

The state-space model of the spatial 7-/~ control problem can be defined as in (8.15), with:

IN•

[0N•

A = L A21

A22 J

where, A21 = -diag(w~,

A22 =

...

- 2diag( ~lwl,

,w~v) . . . , ~NWN )

and, B1 = B2 = P [0 C1 (x) :

[Wl i x)

... -..

C2 = T [q'11

""

0

~11

WN ( x ) ~N1

"'"

0

0

~]gl] T

...

.--

0]

0]

Nmax I~k Wk(x)

=

D21 = D22

k:Nq-1 ---- Kvs.

(8.24)

The spatial weighting function Q ( x ) is set equal to one, which means that all points along the beam are weighted equally. Based on (8.20), we can obtain the error output in (8.21), 2, using the orthogonality property in (8.2), w i t h / / and (9 as follows,

[INxN ON• I~ ~ [ON• ON• I [.01•

O~•

-02Nxl 0

=

('~-,N .....

] ( K O P t ~ 2 - ~ 89

.

(8.25)

\z-~k=N+l~ k J J The scalar weighting factor, R, can then be determined to find a controller with sufficient damping properties and robustness. Matlab #-Analysis and Synthesis Toolbox was used to calculate our spatial 7-/oo controller based on the system in (8.21) via a state-space approach.

8.7

Experimental validations

The experiment was set in the Laboratory for Dynamics and Control of Smart Structures at the University of Newcastle, Australia. The experimental set-up is depicted in Figure 8.6. The controller was implemented using a dSpace DSl103 rapid prototyping Controller Board together with the Matlab and Simulink software. The sampling frequency was set at 20 KHz. The cut-off frequencies of the two low-pass filters were set at 3 KHz. A high voltage amplifier, capable of driving highly capacitive loads, was used to supply necessary voltage for the actuating piezoelectric patch. An HP89410A Dynamic Signal AnMyzer was used to obtain frequency responses of the piezoelectric laminate beam. A Polytec PSV-300 Laser

8

Experiments in Spatial H ~ Control Amplifier

LP Filter

113

LP Filter

r

dSPACE

Signal Analyzer F i g . 8.6. Experimental set-up

Doppler Scanning Vibrometer was also used to obtain the frequency response of the beam's vibration. This laser vibrometer allows accurate vibration measurement at any point on the beam by measuring the Doppler frequency shift of the laser beam that is reflected back from the vibrating surface. Important parameters of the beam, such as resonant frequencies and damping ratios, were obtained from the experiment and were used to correct our model. Our simulation and experimental results are presented in the following. The frequency response of the controller is shown in Figure 8.7. It can be observed that the controller has a resonant nature. This is expected and can be attributed to the highly resonant nature of the beam. That is, the controller tries to apply a high gain at each resonant frequency. Figure 8.8 compares frequency responses of the open-loop and closed-loop systems (actuator voltage to sensor voltage) for both simulation and experimental results. It can be observed that the performance of the controller applied to the real system is as expected. The resonant responses of modes 1 - 6 of the system have been reduced considerably once the controller was introduced. A comparison of the loop gain up to 1.6 KHz from simulation and experiment is shown in Figure 8.9. Our simulation gives a gain margin of 11.3 dB at 1.55 KHz and a phase margin of 89.0 ~ at 79.3 Hz. T h e experiment gives a gain margin of 10.7 dB at 1.55 KHz, and a phase margin of 87.1 ~ at 79.6 Hz. Some reduction of the stability margin in the real system is expected because of the phase delay associated with the digital controller and filters used in the experiment as seen in Figure 8.9. Moreover, there may be a slight difference between our model and the real plant, i.e. modal damping ratios and resonant frequencies. This can contribute to the loss of robustness. Figures 8.10 and 8.11 show the simulated spatial frequency responses of the uncontrolled and controlled beam respectively. Here, x is measured from one end of the beam, which is closer to the patches, while the frequency response is in terms of the beam's transverse displacement (displacement in Z-axis). It is clear that vibration of the entire beam due to the first six bending modes has been reduced by the action of the controller.

114

Dunant Halim, S.O. Reza Moheimani

60

50

4O

~' -o

30

"5 20

10,

100

200

300

400

500 600 Frequency [Hz]

700

000

900

1000

700

800

900

1000

(a) magnitude

-80

-100

f

-120

l

-140

- - -160

"o - 1 8 0

-200

-220

/

-240

-260

-280

i

100

200

300

400

500 600 Frequency [Hz]

(b) phase Fig. 8.7. Frequency response of the controller (input voltage to output voltage

[v/v])

8

Experiments in Spatial H ~ Control

115

o +s

I'

II

,I

I

Jf

-is I

-25

401 451

-SSl

1oo Frequeo~y [HZ]

(a) magnitude - simulation

~o

3oo Fn~W

4c~ EHzI

5r

eoo

700

(b) m a g n i t u d e - experiment

i+o

-ior

~

,

+,++

i

:!

i:

2~

(c) phase - simulation

(d) p h a s e - experiment

F i g . 8.8. Simulation and experimental frequency responses (actuator voltage to sensor voltage IV/V])

Next, a Polytec PSV-300 Laser Scanning Vibrometer was used to obtain the frequency response of the beam's vibration at a number of points on the surface. The results allow us to plot the spatial frequency responses of the uncontrolled and controlled beam using the experimental results as shown in Figures 8.12 and 8.13. It can be observed that the resonant responses of modes 1 - 6 have been reduced over the entire beam due to the controller action, which is as expected from the simulation (compare with Figures 8.10 and 8.11). The resonant responses of modes 1 - 6 have been reduced by approximately 27, 30, 19.5, 19.5, 15.5 and 8 dB respectively over the entire beam. Thus, our spatial 7-/~ controller minimizes resonant responses of selected vibration modes over the entire structure, which is desirable for vibration suppression purposes. To demonstrate the controller's effect on the spatial 7-/~ norm of the system, we have plotted the pointwise 7-/~ norm of the controlled and uncontrolled beam as a function of x in Figure 8.14. The figures show that the experimental results

116

Dunant Halim, S.O. Reza Moheimani

~F Fr~cy

I~z]

F,,,q.,,,,c~ IHzJ

(a) magnitude - simulation

(b) magnitude - experiment

ii

-eov

(c) phase - simulation

(d) phase - experiment

F i g . 8.9. Loop gain IV/V]: simulation and experiment

are very similar to the simulations. Furthermore, they clearly show the effect of our spatial T/~ controller in reducing the vibration of the beam. It is obvious that the 7-/~ norm of the entire beam has been reduced by the action of the controller in a uniform manner. The highest T/~ norm of the unc.ontrolled beam has been reduced by approximately 97%, from 3.6 • 10 -5 to 1.1 • 10 -6. The effectiveness of the controller in minimizing beam's vibration in time domain can be seen in Figure 8.15. A step disturbance signal was applied through the piezoelectric actuator. The velocity response of the beam, at a point 80 m m away from one end of the beam, was observed using the PSV Laser Vibrometer. The velocity response was filtered by a bandpass filter from 10 Hz to 750 Hz. The settling time of the velocity response has been reduced considerably. To show the advantage of the spatial 7-/~ control over the pointwise 7-{o0 control, we performed the following experiment. A pointwise ?-/~ controller was designed to minimize the deflection at the middle of the beam, i.e. x = 0.3 m. The controller had a gain margin of 14.3 dB and a phase margin of 77.9 ~ It was implemented on

8

Experiments in Spatial H ~ Control

117

-o_

r:

;E-

0 x-location [m]

100

ZUU

~vv Frequency

-- [Hz]

Fig. 8.10. Simulation spatial frequency response: actuator voltage - beam deflection (open loop) [m/V]

-o_

)

)0 x-location [m]

100 Frequency

[Hz]

Fig. 8.11. Simulation spatial frequency response: actuator voltage - beam deflection (closed loop) [m/V]

118

Dunant Halim, S.O. Reza Moheimani -90 -100 -110

-120 -130 -140 ~ -150,

~ -160. -170, -180. -190, -200, 0.6 0.5 O. U.~o 2 ~.

-

0.10 x-location [m]

0

100

200

300

400

500

800

700

Frequency [Hz]

Fig. 8.12. Experimental spatial frequency response: actuator voltage - beam deflection (open loop) [m/V]

~0 x-location Ira]

0

1O0

;~UU

our

-- -

Frequency [Hz]

Fig. 8.13. Experimental spatial frequency response: actuator voltage - beam deflection (closed loop) [m/V]

8

E x p e r i m e n t s in Spatial Hor C o n t r o l

~ 0 -i

119

XI~ 6

(a) simulation

(b) e x p e r i m e n t

F i g . 8 . 1 4 . Simulation and e x p e r i m e n t a l 7-/o0 n o r m plot - spatial control

15

-1So

i

2

3

4

5 Tlml (lj

e

(a) open loop

7

8

~

o

i

2

3

4

5 T I ~ [I]

8

7

(b) closed loop

F i g . 8 . 1 5 . V i b r a t i o n at a point 80 m m away from one end of t h e b e a m

the b e a m using the set-up in F i g u r e 8.6. In Figure 8.16, we have p l o t t e d 7-/~ n o r m of the controlled and uncontrolled b e a m as a function of x. Figure 8.16 shows t h e effectiveness of the pointwise control in local r e d u c t i o n of the 7-/o~ n o r m at and around x = 0.3 m. T h i s is not surprising as t h e only p u r p o s e of the controller is to m i n i m i z e v i b r a t i o n at x = 0.3 m. In fact, t h e pointwise controller only suppresses the odd n u m b e r e d m o d e s since x = 0.3 m is a n o d e for even n u m b e r e d modes. C o m p a r i n g Figures 8.14 and 8.16, it can be c o n c l u d e d t h a t the spatial ~oo controller has an a d v a n t a g e over t h e pointwise 7-/~ controller as it minimizes t h e v i b r a t i o n t h r o u g h o u t t h e entire structure.

120

4

D u n a n t Halim, S.O. Reza Moheimani

10 ~

X 10"

3S

3

Z ' 1 Si

,I ~

01

02

03 x-~ocal~n [mj

04

05

08

(a) simulation Fig. 8.16. Simulation and experimental ~

8.8

0}

02

03 X-~VaV~ [ml

04

05

00

(b) experiment norm plot - pointwise control

Conclusions

A spatial 7-/~ controller was designed and implemented on a piezoelectric laminate beam. A feedthrough term was added to correct the locations of in-bandwidth zeros of the system. It was observed that such a controller resulted in suppression of the transverse deflection of the entire structure by minimizing the spatial 7-/~ norm of the closed-loop system. The controller was obtained by solving a standard 7-/~ control problem for a finite-dimensional system. A number of experiments were performed, which demonstrated the effectiveness of the developed controller in reducing the structural vibrations on a piezoelectric laminate beam. It was shown that the spatial 7t~o controller had an advantage over the pointwise %goo control in minimizing structural vibration of the entire structure. The application of this spatial 7-/oo control is not confined to a piezoelectric laminate beam. It may be applied to more general vibration suppression problems.

References 1. S.O.R. Moheimani, H.R. Pota, and I.R. Petersen. Broadband disturbance attenuation over an entire beam. In Proceedings of the European Control Conference, Brussels, Belgium, July 1997. 2. R.L. Clark. Accounting for out-of-bandwidth modes in the assumed modes approach: Implications on colocated output feedback control. Transactions of the ASME, 119:390-395, September 1997. 3. S.O.R. Moheimani. Minimizing the effect of out-of-bandwidth dynamics in the models of reverberant systems that arise in modal analysis: Implications on spatial 7-/~ control. Automatica, 36:1023-1031, 2000. 4. S.O.R. Moheimani. Minimizing the effect of out of bandwidth modes in truncated structure models. ASME Journal of Dynamic Systems, Measurement, and Control, 122:237-239, March 2000.

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121

5. S.O.R. Moheimani and W.P. Heath. Model correction for a class of spatiotemporal systems. In Proceedings of the American Control Conference, pages 3768-3772, Chicago, Illinois, USA, June 2000. 6. R.L. Bisplinghoff and H. Ashley. Principles of Aeroelasticity. Dover, New York, 1975. 7. S.O.R. Moheimani and T. Ryall. Considerations in placement of piezoceramic actuators that are used in structural vibration control. In Proceedings of the 38th IEEE Conference on Decision 84 Control, pages 1118-1123, Phoenix, Arizona, USA, December 1999. 8. R.L Clark, W . R Saunders, and G.P. Gibbs. Adaptive Structures: Dynamics and Control. Wiley, Canada, 1998. 9. E.K. Dimitriadis, C.R. Fuller, and C.A Rogers. Piezoelectric actuators for distributed vibration excitation of thin plates. ASME Journal of Vibration and Acoustics, 113:100-107, January 1991. 10. C.R. Fuller, S.J. Elliot, and P.A. Nelson. Active Control of Vibration. Academic Press, London, 1996. 11. H.T. Banks, R.C. Smith, and Y. Wang. Smart Material Structures: Modeling, Estimation and Control. Wiley - Masson, C h i c h e s t e r - Paris, 1996. 12. H.R. P o t a and T.E. Alberts. Multivariable transfer functions for a slewing piezoelectric laminate beam. ASME Journal of Dynamic Systems, Measurement, and Control, 117:352-359, September 1995. 13. L. Meirovitch. Elements of Vibration Analysis. McGraw Hill, New York, 1975. 14. T.E. Alberts and J.A. Colvin. Observations on the nature of transfer functions for control of piezoelectric laminates. Journal of Intelligent Material Systems and Structures, 8(5):605-611, 1991. 15. T.E. Alberts, T.V. DuBois, and H.R. Pota. Experimental verification of transfer functions for a slewing piezoelectric laminate beam. Control Engineering Practice, 3(2):163-170, 1995. 16. S.O.R. Moheimani and M. Fu. Spatial 7-/2 norm of flexible structures and its application in model order selection. In Proceedings of the 37th IEEE Conference on Decision 8~ Control, pages 3623-3624, Tampa, Florida, USA, December 1998. 17. K. Zhou, J.C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, Upper Saddle River, N.J., 1996. 18. I.R. Petersen, B.D.O. Anderson, and E.A. Jonckheere. A first principle solution to the non-singular 7-/~ control problem. International journal of robust and nonlinear control, 1(3):171-185, 1991. 19. S.O.R Moheimani. Experimental verification of the corrected transfer function of a piezoelectric laminate beam. IEEE Transactions on Control Systems Technology, 8(4):660-666, July 2000.

9 On Establishing Classic P e r f o r m a n c e Measures for R e s e t Control Systems* C.V. Hollot, Orhan Beker, Yossi Chait, and Qian Chen College of Engineering, University of Massachusetts, Amherst MA 01002, USA

A b s t r a c t . Reset controllers are linear controllers that reset some of their states to zero when their inputs reach a threshold. We are interested in their feedback connection with linear plants, and in this context, the objective of this paper is twofold. First, to motivate the use of reset control through theory, simulations and experiments, and secondly, to summarize some of our recent results which establish classic performance properties ranging from quadratic and BIBO stability to steady-state and transient performance.

9.1

Introduction

It is well-appreciated that Bode's gain-phase relationship [1] places a hard limitation on performance tradeoffs in linear, time-invariant (LTI) feedback control systems. Specifically, the need to minimize the open-loop high-frequency gain often competes with required high levels of low-frequency loop gains and phase margin bounds. Our focus on reset control systems is motivated by its potential to improve this situation as demonstrated theoretically in [2] and by simulations and experiments [3]-[5]. The basic concept in reset control is to reset the state of a linear controller to zero whenever its input meets a threshold. Typical reset controllers include the socalled Clegg integrator [6] and first-order reset e l e m e n t (FORE) [3]. The former is a linear integrator whose output resets to zero when its input crosses zero. The latter generalizes the Clegg concept to a first-order lag filter. In [6], the Clegg integrator was shown to have a describing function similar to the frequency response of a linear integrator but with only 38.1 ~ phase lag. Reset control action resembles a number of popular nonlinear control strategies including relay control [7], sliding mode control [8] and switching control [9]. A common feature to these is the use of a switching surface to trigger change in control signal. Distinctively, reset control employs the same (linear) control law on both sides of the switching surface. Resetting occurs when the system trajectory impacts this surface. This reset action can be alternatively viewed as the injection of judiciously-timed, state-dependent impulses into an otherwise LTI feedback system. This analogy is evident in the paper where we use impulsive differential equations; e.g., see [10] and [11], to model dynamics. Despite this relationship, we found existing theory on impulse differential equations to be either too general or broad to be of immediate and direct use. This connection to impulsive control helps to draw * This material is based upon work supported by the National Science Foundation under Grant No.CMS-9800612.

124

C.V. Hollot et. al.

comparison to a body of control work [12] where impulses were introduced in an open-loop fashion to quash oscillations in vibratory systems. Finally, we would like to point other recent research and applications of reset control found in [13]-[15]. The objective of this paper is twofold. First, to motivate reset control through theory, simulations and experiments, and secondly, to summarize some of our recent results ([2], [16]-[20]) which establish properties ranging from quadratic and BIBO stability to steady-state and transient performance. The paper is organized as follows. The next section provides three examples to demonstrate the advantage in using reset control. After that, Section 3 writes out the dynamical equations of our reset control systems and in Section 4 we present one of our main results giving a necessary and sufficient conditions for quadratic stability. In Section 5 we give internal model and superposition principles. Specializing to first-order reset elements, we then go on in Section 6 to establish results concerning BIBO stability. We then restrict attention to a class of reset control systems whose linear dynamics are second-order dominant. For this classic situation, we will show that the associated reset control system is always stable, enjoys steady-state performance akin to its linear counterpart and can be designed for improved overshoot in its step response.

9.2

Motivation

In this section we give three examples comparing reset to linear feedback control. The first gives an example of control specifications not achievable by any linear feedback control, but achievable using reset. The second example shows how the simple introduction of reset in a control loop reduces step-response overshoot without sacrificing rise-time. Lastly, we describe an experimental setup of reset where we again demonstrate reset-control's potential.

9.2.1

O v e r c o m i n g l i m i t a t i o n s o f linear c o n t r o l

Consider the standard linear feedback control system in Figure 1 where the plant P(s) contains an integrator. Assume that C(s) stabilizes. In [21] it was shown

r ~ _ ~

C(s)

*

P(s)

F i g . 9.1. Linear feedback control system.

that the tracking error e due to a unit-step input satisfies

f0

1

e ( t ) d t = K----~

where the velocity constant K~ is defined by K~ ~=lims--,osP(s)C(s).Alone, this constraint does not imply overshoot in the step response y; i.e., y(t) _> 1 for some

9

Reset C o n t r o l S y s t e m s

125

t > 0. However, introduction of an additional, sufficiently s t r i n g e n t t i m e - d o m a i n b a n d w i d t h constraint will. To see this, consider t h e notion of rise t i m e t~ i n t r o d u c e d in [21]:

tr=sup{T:y(t)<

_ ~t , t E [ O , T ] } .

T h e following result (see [2]) is quite immediate. 2 . F a c t : If t~ > -R--j, i.e., the rise time is sufficiently slow, then the unit-step

response y(t) overshoots. To illustrate this result consider t h e plant P(s) in F i g u r e I as a simple integrator. In a d d i t i o n to closed-loop stability suppose the design objectives are t h e following: 9 S t e a d y - s t a t e error no greater t h a n 1 w h e n t r a c k i n g a u n i t - r a m p input. ,, Rise t i m e greater t h a n 2 seconds w h e n t r a c k i n g a unit-step. 9 No overshoot in the step response. To meet the error specification on t h e r a m p response, this linear feedback s y s t e m must have velocity error c o n s t a n t K~ > 1. Since tr > 2 > __2 t h e Fact indicates t h a t no stabilizing C(s) exists to m e e t all t h e above objectives. However, these specifications can be met using reset control w i t h a first-order reset e l e m e n t ( F O R E ) described by

itr(t) ---- -bu~(t) + e(t); u r ( t +) = O;

e(t) ~ 0 e(t) = 0

where b, t h e F O R E ' s pole, is chosen as b = 1. Indeed, in Section 6.2 of this p a p e r we will show t h a t this reset system is a s y m p t o t i c a l l y stable, has b o u n d e d response y

t F i g . 9.2. Reset control of an i n t e g r a t o r using a first-order reset element.

to b o u n d e d input r and zero s t e a d y - s t a t e tracking error e to c o n s t a n t r. T h i s reset control s y s t e m is given in F i g u r e 9.2. F i g u r e 9.3 shows a simulation of this control s y s t e m ' s t r a c k i n g error e to a u n i t - r a m p input. T h e s t e a d y - s t a t e error is one. In Figure 9.4 we show its response y to a u n i t - s t e p i n p u t and see t h a t its rise t i m e t~ is greater t h a n 2 seconds and has no overshoot 1. Thus, this reset control s y s t e m meets the previously stated design objectives t h a t were not a t t a i n a b l e using linear feedback control. 1 T h e step response in Figure 4 is d e a d b e a t . T h i s occurs since (u, y) = (0, r) is an equilibrium point.

126

C.V. Hollot et. al.

~o

0.4

5

10

F i g . 9.3. Tracking-error e to a unit-ramp input for the reset control system.

y(t) = o.5 t

e

os

1

1s

2

,

25

F i g . 9.4. Output response y to a unit-step input for the reset control system.

9.2.2

Reducing

overshoot

Another motivation to use reset control is that it provides a simple means to reduce overshoot in a step response. For example, consider the feedback system in Figure 5 where the loop transfer function is: 1 L ( s ) -- s ( s + 0.2) and where the FORE's pole is set to b = 1. Without reset, the linear closed-loop system has standard second-order transfer function

Y(s)

1

R(s)

s~ +2(0.1)s+ 1

The damping ratio is r = 0.1 and the step response exhibits the expected 70% overshoot as shown in Figure 6. The step response of the reset control system is also shown in this figure and it has only 40% overshoot while retaining the rise time of the linear design. Moreover, as in the previous example, this reset control system

9 r

Ur

Reset Control Systems _[

s+l

127

Y

"[s(s + 0.2) Fig. 9.5. Resetting can reduce overshoot in response to step reference inputs.

can be shown to be asymptotically and BIBO stable, and to asymptotically track step inputs r; see Section 6.2. Also, the level of overshoot can be quantitatively linked to the FORE's pole b. This will also be discussed in Section 6.2. Thus, the performance of a classical second-order dominant feedback control system can be significantly improved through the simple introduction of reset control. 1.a

1.e

~!

12

~

::

0.e

o6

:'

~.

'

:~

O4

O.2

0

o

~o

20

t (~1

4~

5o

8o

Fig. 9.6. Comparison of step responses between reset (solid) and the linear control system (dotted).

9.2.3

Demonstrating

p e r f o r m a n c e in t h e lab

The benefits of reset control have also been realized in experimental settings. Here we describe a laboratory setup in which we applied both linear and reset control to the speed control of the rotational flexible mechanical system shown in Figure 9.7. This system consists of three inertias connected via flexible shafts. A servo motor drives inertia J3 and the speed of inertia J1 is measured via a tachometer. The controller was implemented using dSPACE tools [22]. A more complete description of this experiment can be found in [19]. T r a d e o f f s i n l i n e a r f e e d b a c k c o n t r o l A block diagram of a linear feedback control system is shown in Figure 9.8 where the plant P(s) was identified from frequency-response data of the flexible mechanical system as: 46083950

P(s) -= (s + 1.524)(s 2 + 3.1s + 2820)(s 2 + 3.628 + 9846)"

128

C.V. Hollot et. al.

J2

J3

J1

I Servo

motor

flexibleshaft

Fig. 9.7. Schematic of the rotational flexible mechanical system. We posed the following specifications to illustrate the limitations and tradeoffs in LTI design and their subsequent relief using reset control:

1. Bandwidth constraint: The unity-gain cross-over frequency we, defined by [PC(jwc)l = : must satisfy wc > 37r. 2. Disturbance rejection: Low-frequency disturbances are to be rejected; specifically, y(jw)

<0.2,

when w < T r ;

3. Sensor-noise suppression: High-frequency sensor noise is to be suppressed; i.e., y(jw) n(jw)

< 0.3, when w > 10~r; -

4. Asymptotic performance: Zero steady-state tracking error to constant reference r and disturbance d signals. 5. Overshoot: Overshoot in output y to a constant reference r should be less t h a n

20%.

P(s)

F i g . 9.8. The linear feedback control system.

In terms of Bode specifications, the first two constraints translate into minimumgain requirements on the open-loop gain IPC(jw)I at low frequencies while the

9

Reset Control Systems

129

third specification places an upper b o u n d on this gain at high frequencies. The fourth specification requires C(s) to contain an integrator and the fifth specification requires a phase margin of approximately 45~ assuming second-order dominance. Using classical loop-shaping techniques we were unable to meet all of the above specifications. To illustrate the tradeoffs, consider two candidate, stabilizing LTI controllers: 1281489(s + 4.483)(s 2 + 3.735s + 2851)(s ~ + 5.158s + 10060)

Cl(s) = s(s2 + 295.1s + 22330)(s 2 + 126.2s + 8889)(s 2 + 239s + 27560) and

C~(s)=

1075460(s + 7)(s 2 + 3.662s + 2798)(s 2 + 5.419s + 9876)

s(s + 209.6)(s + 35.8)(s 2 + 132.8s + 1 2 0 5 0 ) ( s 2 + 3 7 5 . 9 s + 6 6 9 3 0 ) "

Figure 9.9 compares the Bode plots of the corresponding loops L1 (jw) = P(jw)C1 (jw) and L2(jw) = P(j~v)C2(jw). Loop L1 fails to satisfy the sensor-noise suppression

:L l I fill}If l i........,~ l ] ~ --'I----~TH-&Lii[I i [ ~-:'~71] ,I I I III ~ IZL[] .L__ I L L Z L L [ ~ Ill ~

| ;I

{,,,,I

fill II

t-t}ft- .............f-t--t--i-trill

I

I-"t'~i

I

I I t l 1711

ill *I

~lll ~

Illi l

i

i~llll

I

I I I IF~,,,I i

I

Pl!imll~ll

Fig. 9.9. Bode plots of L1 and L2.

specification at w ---- 107r. This specification can be met by reducing the gain of Ll(jov) as done with L2(jw). This is verified by the time response y to 5 Hz sinusoidal noise n in Figure 9.10. Since both designs stabilize and since both lowfrequency gains are constrained by the first two specifications, Bode's gain-phase relationship [1] dictates that L2(jw) must have correspondingly larger phase lag as verified in the phase plot of Figure 9.9. The reduced gain in L~(jw) comes at the expense of a smaller phase margin and hence larger overshoot as shown in the step responses in Figure 9.11. Extensive t u n i n g of these controllers failed to yield a design meeting all specifications. R e s e t c o n t r o l d e s i g n Now we t u r n to reset control design where we exploit its potential to satisfy the above specifications. The design procedure consists of two steps as developed in [3]-[5]. First, we design a linear controller to meet all the specifications - except for the overshoot constraint; C2(s) is a suitable choice. The second step is to select the FORE's pole b to meet the overshoot specification. In

130

C.V. Hollot et. al.

u

.,

^ P.A

A AAI

~ tl

.. l| gll fit t~1 Lil fll i~ .11 I~ fl fl il I'1 ~1

~A , ' f h

9

U

It

H

ll

I

U

Il

Id

~i

lumumm

Fig. 9.10. Comparison between LTI designs Lz and L2 of output y to n ( t ) sin(10zrt).

-

-,!

I

"I

ILl

40

,.d,e,,

m

U

m

P

1.1~

~

U

Fig. 9.11. Comparison between LTI designs L1 and L2 of output y to r ( t ) -- 1.

this respect, [Figure 5, 3] provides a guideline for this choice. Using this tool, we select b ---- 14. The resulting reset control system is shown in Figure 9.12. Later in this paper we show that this reset control system is quadratically and BIBO stable and asymptotically tracks constant reference inputs r.

d

"1~l (s+ 14) C2(s) n Fig. 9.12. Reset control system for the flexible mechanism.

9

Reset Control Systems

131

Finally, we compare the performance of the LTI (using L1) and reset control systems. Figures 9.13 and 9.14 show that the reset control system has better sensornoise suppression to a 5 Hz sinusoid and to white-noise.

UL,~.

d~

E~.lr,

Fig. 9.13. Comparison of steady-state response y to r(t) - 1 and n ( t ) = sin(101rt).

i -.u

Fig. 9.14. Comparison of output y power spectra when n is white sensor noise.

However, unlike the LTI tradeoff experienced by controller C2(s), the reset control system has comparable transient response as shown in Figure 9.152 .

2 The steady-state noise in Figure 9.15 is due to ripple in the the tach-geuerator.

132

C.V. Hollot et. al. 0.7

reset controldesign] ............

LTI design

1

O.e 05

~

o.4

o,3

o.2

o.1

i

i

os

t t

i

i

1 .s

2

25

(seconds)

F i g . 9.15. Comparison between reset and LTI control (using L1) of o u t p u t y to r(t) -- 1.

9.3

T h e D y n a m i c s of R e s e t C o n t r o l S y s t e m s

The reset control system considered in this p a p e r is shown in Figure 9.16 where the reset controller R is described by the impulsive differential equation (IDE) (see [10]) ~ ( t ) = A~x~(t) + Bre(t); x r ( t +) = An~x~(t);

e(t) ~ 0 e(t) = 0 (9.1)

u~(t) = c ~ ( t )

where xr(t) E R '~ is the reset controller state and ur(t) E R is its output. The matrix A n t E R n~ xn~ identifies that subset of states xr t h a t are reset. For example, in this paper we will assume that the last nr~ states x ~ are reset and use the P

"1

0 / " Illustrations of ( 9 . 1 ) i n c l u d e the Clegg integrator structure An~ = ] I'~-'~r~ 0 L

J

described by A~=0;

B~=I;

C~=I;

An~=O

C~=I;

Ann~-0.

and the F O R E having A~=b;

B~=I;

(9.2)

The linear controller C(s) and plant P ( s ) have, respectively, state-space realizations: ~c(t) = Acxc(t) + B~u~(t) ~(t) = cc~(t)

and 2 , ( t ) = Apxp(t) + Bpuc(t) y(t) =

cp~p(t)

9

Reset Control Systems

133

where xc(t) E R TM, xp(t) C R ~p and y(t) E R. The closed-loop system can then be described by the IDE ic(t) = A~ex(t); x ( t +) = A R x ( t ) ;

x ( t ) ~t M4;

x(O) = xo

x(t) E A4

y(t) = C~ex(t)

(9.3)

where x=

xc

; Ace~

xr

[I~,0 0] An~

L-BrCp

[ ~ 1~r An~O ;

Ac

BcC.

0

Ar

;

Ccez~[CpO0]

and where the reset surface .A/[ is the set of states for which e = O. More precisely,

_•

I Ur

J

"J C(s) II ur "l'l P(s) ~-l---"y "1

F i g . 9.16. Block diagram of a reset control system.

M 2 {~: Cce~ = 0; (I - A~)~ #

0}.

As a consequence of this definition, x(t) E A~

~

x ( t +) ~ 2~4.

The times t ----ti at which the system trajectory x intersects the reset surface A/~ are referred to as reset times. These instants depend on initial-conditions and are collected in the ordered set: T ( x o ) z~ {ti : ti < t i + l ; x ( t i ) C • , i

-----1 , 2 , . . . ,c~}.

The solution to (9.3) is piecewise left-continuous on the intervals (ti, ti+l]. We define the reset intervals 7"~ by A

71 = t l ; A

Ti+l = ti+l -- ti,

i E N.

We make the following assumption on the set of reset times: R e s e t t i n g A s s u m p t i o n : Given initial condition xo E R n, the set of reset times T ( x o ) is an unbounded, discrete subset of R+.

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Unboundedness of the set of reset times implies continual resetting. If this condition is not satisfied, then, after the last reset instance, the reset control system behaves as its base-linear system. We avoid these trivial cases. Discreteness of q-(x0), together with this unboundedness, guarantees the existence and continuation of solutions to (9.3). Finally, in absence of resetting; i.e., when AR = I , the resulting linear system is called the base-linear system. We denote the loop, sensitivity and complementary sensitivity transfer functions of the base-linear system by: L ( s ) = P(s)C(s)R(s),

S(s)-

1 l + L(s)'

T(s)--

L(s) l + L(s)

where R(s) is the transfer of (9.1) when AR = I.

9.4

Quadratic S t a b i l i t y

In this section we give a necessary and sufficient condition for (9.3) to possess a quadratic Lyapunov function. First, we state some general Lyapunov-like stability conditions for our reset control systems which are similar to the analysis in [10] and [23]. Their proofs are relegated to the Appendix. As usual, V is the time-derivative of a Lyapunov candidate V(x) along solutions, while / i V z~ V ( x ) - V ( A R x ) , is the j u m p in V(x) when the trajectory strikes A/t. P r o p o s i t i o n 1: (Local Stability) Let ~2 be an open neighborhood of the equilibrium point x = 0 of (9.3) and let V ( x ) : Y2 --* ]~ be a continuously-differentiable, positive-definite function such that I/(x) < 0; A V ( x ) < 0;

x C ~2/AA

(9.4)

x C f2A.h4.

(9.5)

Then, under the Resetting Assumption, x = 0 is locally stable. Moreover, if either

V(x)<0;

xE~/{0}

(9.6)

or A V ( x ) < 0;

x E t2 N J~4,

(9.7)

then x -= 0 is asymptotically stable.

P r o p o s i t i o n 2: (Global Stability) Let V ( x ) : R n ---+ R be a continuouslydifferentiable, positive-definite, radially-unbounded function such that

?(x)<0;

x~tM

A V ( x ) <_ O; x E M . Then, under the Resetting Assumption, x ----0 is stable. Moreover, if either

y(x) < 0; x c Rn/{0), or z~V(x)<0;

xEM,

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then x = 0 is globally asymptotically stable. We now specialize to quadratic Lyapunov functions. D e f i n i t i o n : The reset control system (9.3) is said to be quadratically stable if there exists a positive-definite symmetric matrix P such t h a t V ( x ) ----x t P x satisfies the asymptotic stability conditions of Proposition 2. T h e o r e m 1: Under the Resetting Assumption, the reset control system described in (9.3) is quadratically stable if and only if there exists a 13 E R n~ such that

Hz(s) ~ [13Cp 0 I , ~ ] (sI - Ac~) -1

I , 0~

]

(9.8)

is strictly positive real ( SPR) 3. P r o o f " (Sufficiency) We first define V(x) = x ' P x . By Proposition 2 the reset control system described in (9.3) is quadratically stable if there exists a positivedefinite symmetric matrix P such t h a t

x' (A'c~P + PAc~) x < 0;

x E R~/{0}

(9.9)

and

x' ( A ~ P A R - P) x <_ O; x E A 4 .

(9.10)

Let O be a matrix whose such t h a t its columns span the nullspace of Cc~. Using this, we can express A4={O~:(I-An)

O4#0;

~#0}

and define JQ as

Therefore,

O' (A~PAR - P) 0 < 0

(9.11)

implies that (9.10) holds. (9.11) holds for some positive-definite P if there exists a 13 E R n ~ such that [0 I,~]P----[13Cp 0 In~].

(9.12)

Thus, the proof reduces to finding a positive-definite symmetric m a t r i x P such t h a t (9.9) and (9.12) hold. From the Kalman-Yakubovich-Meyer ( g Y M ) lemma; e.g., see [24], such P exists if there exists a 13 E R n'r such t h a t H z ( s ) in (9.8) is s P a . 3 A transfer function X ( s ) is said to be strictly positive real if X ( s ) is asymptotically stable, and Re[X(jw)] > O, Vw >_ O.

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(Necessity) Suppose (9.3) is quadratically stable. Then, from Proposition 2 there exists a positive-definite symmetric matrix P such that

x'(A'ceP + P A ~ ) x < 0;

x ~ 0

(9.13)

x ' ( A ' R P A n - P ) x ~ 0;

x CAd.

(9.14)

The continuity of AV, together with (9.14), implies that

x ' ( A ' n P A n - P ) x < 0;

x 9

which in t u r n implies that (9.12) holds for some /3 9 R ' ~ . The strict positive realness of H z in (9.8) then follows from (9.12), (9.13) and the KYM lemma. This concludes the proof. [] Theorem 1 gives an easily-testable condition for the quadratic stability of the reset control systems described by (9.3). This condition is also key in showing that reset control systems enjoy other properties. Before we present these results, we formally introduce first-order reset elements.

9.5

Steady-state performance

In this section we study the steady-state performance of the reset control system in (9.3) and show that it enjoys an internal model principle and steady-state superposition property.

9.5.1

An internal model principle

We introduce an internal model principle for the reset control systems by considering a model of the reference signal r inside the loop as part of P ( s ) C ( s ) . We can state the following theorem. T h e o r e m 2: Under the Resetting Assumption, if P ( s ) C ( s ) contains an internal model of r, and if there exists a 13 E R "~r~ such that H~(s) in (9.8) is SPR, then the reset control system described in (9.3) achieves asymptotic tracking of the reference input r. P r o o f : We first adopt the realization {Ape, Bpc, Cpc} for P ( s ) C ( s ) , which contains an internal model of the reference input. Since r is realized within P ( s ) C ( s ) , there exists states z(t) and Czr E R '~p+nc, such that

k(t) = Apcz(t);

z(O) = r(O),

r(t) = C r u z ( t ) . Then using the state transformation &pc(t) ~ xpc(t) - z ( t ) we obtain the following:

x(t) ---- Ace~(t);

2(t) • All;

~c(t+) = An~c(t); ~(t) e M , y(t) = c c ~ ( t ) + r(t),

5c(0) = xo - r(O),

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where 5: = [Xrr . Hence, the asymptotic tracking problem is now expressed as the asymptotic stability problem for the unforced system. By Theorem 1, asymptotic stability of the unforced system is guaranteed if there exists a ~ E R n~r such that H~(s) is SPR. This concludes the proof. []

9.5.2

A s u p e r p o s i t i o n principle

We now introduce an additional input to the control system as shown in the Figure 9.17. This system can be described by

5c(t) = Acex(t) + Bcerl(t) + Bcer2(t); x(t +) = ARx(t);

x(t) r M ;

x(o) = xo

x(t) ~ M , (9.15)

y(t) = c o a x ( t ) where .h4 ~ {~ : r l ( t ) + r2(t) -- Cce~ = 0; ( I - AR) ~ 7~ 0}.

The next result gives a steady-state superposition result. If the loop contains an internal model of one of the inputs signals, say r2, then this result claims that the steady-state response to rl -}- r2 is simply the steady-state response to rl. C o r o l l a r y 1: Consider the reset control system with two inputs rl and r2 described in (9.15). Suppose the Resetting Assumption is in force, P(s)C(s) contains an internal model of r2 and there exists a /~ C R n~ such that H~(s) is strictly positive real (SPR). Then, the steady-state error, l i m t ~ e(t) is independent of r2. r2

r

l

-

-

~

Y

F i g . 9.17. Block diagram of a reset control system with two inputs.

9.6

Specialization to First-Order R e s e t E l e m e n t s

As illustrated in Section 2.3 the design of reset control systems, as developed in [3] and [5], involves the synthesis of both linear compensator C(s) and reset controller R in Figure 9.16. Typically, C(s) is designed to stabilize the base-linear system and shape the loop L(s) = P(s)C(s)R(s) to satisfy classical Bode specifications at high and low frequencies. The reset controller is then designed to meet overshoot specifications. In this subsection we focus on F O R E described by (9.1) and (9.2) where b > 0 is the F O R E ' s pole.

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9.6.1

BIBO

stability

Consider the reset control system with a reference input as shown in Figure 9.18 and described by the following IDE

5:(t) = A~ex(t) + B~er(t); x(t +) = ARx(t);

x(t) ~ M ; x(O) = O, x(t) E .A4,

y(t) = Ccex(t)

(9.16)

where

and M~{~:r(t)-C~t~=0;

(I--AR)~-~0}.

In this section we analyze the BIBO stability of (9.16) which requires every bounded

Ur

.I C ( S )

1

I uc

I

.I P ( s )

1

'1

F i g . 9.18. Block diagram of a reset control system with reference input r.

input 4 r to produce a bounded output y. To begin this analysis we let xe be the state of the base-linear system; that is:

~e(t) = Acexe(t) + Bc~r(t);

x(0) = 0

A

and take z = x - xe. We partition X

~

Xr

-

'

Z ~

Zr

9

~

X

~

[ Xer J

.

Applying the transformations: z~(t) = x~(t) - x~(t); z~(t) =

x~(t) - x~(t)

to (9.16) and expressing the reset rule with the set of reset times 7-(0) we obtain: ~L(t) = AzL(t) q- Bz~(t) ~(t) = z~(t$)

-CzL(t)

= -xe~(ti);

-

bz~(t);

t ~t 7-(0) t C 7-(0).

(9.17)

4 A signal z is said to bounded if there exists a constant M such that Iz(t)l < M for all t.

9

Reset C o n t r o l S y s t e m s

As an i n t e r m e d i a t e step, we show t h a t b o u n d e d n e s s of

ZL

139

implies t h a t y is b o u n d e d .

L e m m a 1: Assume Ace is asymptotically stable and r is bounded. I f z i is bounded, then output y is bounded. P r o o f : Suppose z i is b o u n d e d . We have

ly(t)l = ICxL(t)l

<_ ICz~(t)l + I C ~ ( t ) l . Since Act is stable and r is b o u n d e d , t h e n XtL is b o u n d e d . O u t p u t y is t h u s b o u n d e d . [] Before establishing B I B O stability, we need t h e following lemma. L e m m a 2: I f Ac~ is asymptotically stable and r is bounded, there exist constants M1 and M2 such that Izr(t+)] < M1 and ICzi(t~)l < M2 f o r i = 1 , 2 , . . . ,oo. P r o o f : Because Ac~ is a s y m p t o t i c a l l y stable and r, t h e n xt~ and XgL are bounded. F r o m (9.17), z~(t +) ---- -xt~(t~). Therefore, there exists an M1 such t h a t Iz~(t+)[ < M1 for i --- 1, 2 , . . . , ~ . By definition, CzL(ti) ----r(t~) -- C x t i ( t i ) . Since r and XeL are b o u n d e d , t h e n t h e r e exists an M2 such t h a t [CzL(ti)l < M2 for i ---- 1 , 2 , . . . ,co. [] We can now state our B I B O stability result. T h e o r e m 3: Under the Resetting Assumption the reset control system (9.16) is B I B O stable if it is quadratically stable; i.e., there exists a/3 E R such that H~(s) in (9.8) is SPR. P r o o f : Since H~(s) in (9.8) is strictly positive-real, then, from t h e K Y M l e m m a , there exists a positive-definite m a t r i x P , a vector q and a positive c o n s t a n t ~ such that

P A d + A~lP = -q~q -- r P[0 ..- 0 1]' = [ t i c

1]'.

(9.18)

Hence, P can be written as

where P1 ~ N nxn is positive-definite. A l o n g t h e piecewise left-continuous solutions of (9.17) we define

v(t) =

[z~(t), '

z~(t)]P[zL(t),zr(t)]'

= Z~L(t)PlzL(t) + 2ZCzL(t)z~(t) + z2(t) over t E (t~,t~+l]. At t h e reset instants t ----ti we t h e n have

V ( t +) = ZL(~t)PzlL(~t)'

+ 2~CzL(ti)z~(t +) + z~(ti2 +)

= V(ti) + 213CzL(t~)z~(t +) + z~(t~ 2 + ) -- 2j3CzL(ti)z~(ti) --z~(2 ti).

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C.V. Hollot et. al.

Since --2~CzL (ti)zr(t~) -- z~(t~) <_ (~CzL (t~)) 2, 2 + V ( t +) <_ V(ti) + 2~CzL(t~)z~(t +) + z~(t~ ) + (BCzL(ti)) 2

= V(ti) + [z~(t +) + ~CzL(t~)] 2.

(9.19)

Because r is bounded, it follows from L e m m a 2 t h a t there exists a c o n s t a n t M > 0 such t h a t [z~(t +) + ~CzL(ti)] 2 <_ M for i ---- 1, 2 , . . . ,oc. Thus, from (9.19):

Y ( t +) < Y(t~) + M,

i -- 1, 2 . . . . . c~.

(9.20)

Differentiating Y(t) along solutions to (9.17), we use (9.18) to o b t a i n

[ZL y ( t ) = [zL(t),z~(t)](PAc~ ' + AclP) ' ' (t), z~ (t)] ' i l I i = [zL(t), z~(t)](--q q -- eP)[ZL(t), z~(t)] < - ~ [ z ' ~ (t), z~ (t)]P[z L ' (t), Zr it)] '

= -~u(t) for all t 6 (t~, t~+~]. The non-negativity of V(t) implies

v(t) <_ e-~(~-~)v(t+, )

(9.21)

whenever t 6 (t~,ti+l]. Since ti+l - ti > a,

u(t,+~) < ~-~(~'+~-~')v(t, +) <_

~-~~

<_ e - ~ [ V ( t , ) + M]. Combining (9.20) with (9.21) and r e p e a t e d l y a p p l y i n g the above gives

V(t) <_ e - ~ ( t - t ~ ) [ e - ~ V ( O ) + M + e - ~ M + . . . + e-ei~ M] for all t 6 (t~,t~+l]. Since V(0) = 0, V(t) <_ M / ( 1 - e - e ~ ) . Therefore, V is b o u n d e d . Because P is positive-definite, it follows t h a t ZL is bounded. Finally, from L e m m a 1, y is b o u n d e d . This completes the proof. [3

9.6.2

When

the

base-linear

system

has classical second-order

form In this section we focus on a class of first-order reset control systems shown in Figure 17 where (s + b)w~

(9.22)

P ( s ) C ( s ) -- s(s + 2~w,~)

As a result, the associated base-linear system has classical second-order (complem e n t a r y sensitivity) transfer function

T(s)- v ( s ) _ R(s)

2

~n s2 + 2~ns + ~ "

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This setup allows us to compare the reset control system's performance against a linear control system with dominant pole-pair. We will show that this class of reset control systems is always quadratically stable and, by virtue of Theorem 3, BIBO stable. We will characterize the step response of the reset control system in terms of standard measures such as rise-time, overshoot and settling time, thus allowing a direct comparison to its base-linear system. First, we establish that the key SPR condition in (9.8) is always satisfied. Hf~ is a l w a y s p o s i t i v e - r e a l We begin with a lemma that removes the standing Resetting Assumption. L e m m a 3: For the reset control system in (9.16) utilizing F O R E and L ( s ) given in (9.22), the set of reset times T ( x o ) is unbounded and discrete f o r all positive b, ~r w,~ and ~ E (0, 1). Moreover, the reset action is periodic with period T~ -- ~1V/~_~2 . P r o o f : To prove the theorem it suffices to show that the reset time interval is A

~r constant; i.e., ~'i -= T -- ,~nlX/~_r 2 for all integer i > 1. Without loss of generality we

start with an initial condition xo E A/I; i.e., [w~ 0 0] Xo = O. Again, without loss of generality we only consider xo such that [IXpcolI -- 1, where Xpco E R 2 denotes the initial state of P ( s ) C ( s ) . Since Xo E A,I we have Xpco = [0 1]' and therefore Xo = [0 1 X~o]', where X~o E R is the initial state of the FORE. For Ti to qualify as a reset time interval, the following condition must be satisfied:

where Ace = [ - C p c

and AR =

. This gives Ti ~ ~nX/l_r

The next

step in the proof is to show that T is a valid reset time interval. To do this we need to show that X~(T) ~ 0; i.e.,

which is equivalent to saying

b2 - 2~wnb + w~

This is true for all positive b, wn and ~ E (0, 1) and hence concludes the proof,

t~

Our next theorem shows that this class of reset control system is quadratically and BIBO stable and enjoys the previously described internal-model and superposition properties for all b, r and w,~. T h e o r e m 4: For the reset control system described in (9.16) utilizing F O R E with P ( s ) C ( s ) given in (9.22), there exists a ~ E R such that H ~ ( s ) in (9.8) is S P R

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C.V. Hollot et. al.

for all positive b, ~ and wn. Consequently, such reset control systems are always quadratically stable, BIBO stable and enjoy the internal model and superposition properties of Theorem 2 and Corollary 1. P r o o f : First we note t h a t if ~ > 1, the reset control system becomes equivalent to its base-linear system, which is stable. Therefore we only need to consider the case when the system is u n d e r d a m p e d ; i.e., ~ < 1. By L e m m a 3, the R e s e t t i n g A s s u m p t i o n is satisfied for reset control systems with P ( s ) C ( s ) given in (9.22) for all positive values of b, w~ a n d ~ E (0, 1). To show a s y m p t o t i c stability we use T h e o r e m 1 and show there exists a t3 > 0 such t h a t s 2 + (2~w~ + ~ ) s + b~ Hz(s) = (s + b) (s 2 + 2~wns + w2~) is SPR. T h e remainder of the proof deals with finding a 13 C ]R such t h a t Ha(s ) is S P R for all positive b, ~ a n d wn. We consider three cases: C a s e 1: (b > 2~wn) First, we form the p a r t i a l fraction expansion 1

Ha(s) = w~ - 2@wn + b2 b(b-2~wn)

/,

I

s Y- b

w~s+(w~-2@wn+b2)/~+w2(2~w,~-b) +

s2 + 2~ns

1

+ w~

1

= w~ - 2r

+ b2 [hH(s) + h~2(s)].

Next, since b > 2~wn, t h e n w 2 - 2@wn + b2 > 0. Hence, it suffices to show t h a t b o t h h11(s) a n d h12(s) are SPR; e.g., see [24]. Since b ( b - 2 ( w n ) > 0, t h e n hll(s) is SPR. Finally, since s 2 + 2(wns + wn2 is stable a n d the zero of h12(s) can be arbitrarily placed via j3, there exists a ~ r e n d e r i n g h12 (s) SPR. C a s e 2: (b = 2~a;n) In this case it is clear t h a t s+3

H~(s) = s2 A- 2~WnS q- ~2n is S P R for sufficiently small a n d positive t3. C a s e 3: (b < 2~wn) In this s i t u a t i o n we write H~(s)

1 s+(b-a)

= i "4- sA_(b_t3 ) /',

s2..k(2~Wnq_t3)sq_bl 3

h21 (8)

1 + h21(s)h22(s)" Now it suffices to show t h a t b o t h h21 (s) a n d h22 (s) are SPR; again, see [24]. Clearly, h21 (s) is S P R for a n y / 3 < b a n d a straightforward calculation shows t h a t h22 (s) is S P R for sufficiently small positive/3. Hence, there exists a j3 C (0, b) such t h a t H~(s) is SPR. This proves Case 3 a n d the theorem. []

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H ~ ( s ) is p o s i t i v e - r e a l f o r a l l t h e e x a m p l e s i n S e c t i o n 2 In Section 2 we gave three examples motivating the use of reset control systems. We claimed that each was quadratically and BIBO stable and asymptotically tracked constant reference signals. This is certainly the case for the first two of these situations since Theorem 4 applies. To establish these properties for the third case we explicitly check the satisfaction of (9.8). Since C2(s) stabilizes, then H z ( s ) in (9.8) is asymptotically stable for all /3. A simple search and computation shows t h a t Re[H~(jw)] > 0 for all w _> 0 when 13 ----0.008.

O v e r s h o o t , r i s e t i m e a n d s e t t l i n g t i m e In this section we analyze the reset control system (9.16) when r(t) =_ ro and prove t h a t the step-response m a x i m u m occurs during the time interval (tl, tl + ~-o). The proof of the following can be found in [18]. Theorem

5: Consider the reset control system described in (9.16) utilizing

F O R E with L ( s ) given in (9.22) and r(t) -- ro. Let M r = ~ supt>0 [y(t) - ro[ denote the step-response m a x i m u m . Then, Mr =

max [y(t) te[tl,tl+rO]

- ro[.

From Theorem 5 the step response maximum M~ is equal to the peak response in the first reset interval [ti, tl + TO). In [3], this overshoot value has been explicitly computed in terms of b, (, and w,~ as repeated below: Mr = e

- Al

where

I R[4M242e-4P--2(~M(1--442M)e-V'/~M] l_4(~2M+442M2 ; /~ : R[M2e_~ --M(1--2r l_24M+M2

- q

R = e V ~1 ~

cos- i r

;

,

wc

M=y;

~-

~ ~ 0.5 ~ ~ 0.5

71" - - C O S - 1

~/1-r

and where wc is the unity-gain crossover frequency of P ( j w ) C ( j w ) . Since the reset control system (9.16) behaves as a linear system before its first reset, then its rise time is t h a t of its base-linear system ( ~ 1.s). The 2% settling time ts can be computed using [18] adjacent intervals of y are shown to be scaled copies of each other. Indeed, using this, the settling time is computed as ts

k7v

where k is the smallest integer satisfying ]pn(To)l k M~ < 0.02.

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C.V. Hollot et. al.

9.7

Conclusion

This p a p e r has given a s u m m a r y overview of reset control. It p r o v i d e d a n u m ber of m o t i v a t i n g examples, b o t h t h e o r e t i c a l and e x p e r i m e n t a l , and a f r a m e w o r k for establishing basic feedback loop properties such as stability, s t e a d y - s t a t e and transient performance. One area of present s t u d y is the response of reset control systems to high-frequency sensor noise. Such results could help give a c o m p l e t e a description of classical properties of reset control systems.

A

P r o o f of P r o p o s i t i o n 1

We will first i n t r o d u c e some notation. G i v e n an r E R, we define

and given a / 3 E R and Br as above, we define

~ g {~ E ~ : V(~) _ 0, choose r E (0,r and let a - minll~ll=rV(x ). Since V is positive-definite we have a > 0. We t a k e / 3 E (0, ct). Inequalities (9.4) and (9.5) imply t h a t V(x(t)) <_ V(xo) <_ /3 for all t, and hence any t r a j e c t o r y s t a r t i n g in f2Z will remain in ~2~ for all t. Since V(x) is continuous and V(0) = 0, t h e r e exists a 6 > 0 such t h a t V ( x ) 0, there is a T > 0 such t h a t Iix(t)l] < a for all t > T. It suffices to show t h a t V(x(t)) --* 0 as t --* cx). By (9.4) and (9.5), V(x(t)) is non-increasing. We now assume t h a t it is decreasing when x E ~ / { 0 } by (9.6), or w h e n x E Y2 M A/[ by (9.7). Since V ( x ) is b o u n d e d from below by zero, t h e n there exists a c _> 0 such t h a t V ( x ( t ) ) ~ c _> O;

t --~ oo.

(9.23)

To show t h a t c ----0, we proceed by c o n t r a d i c t i o n and suppose c > 0. By c o n t i n u i t y of V(x), there exists a d > 0 such t h a t 13d C f2c. T h e inequality (9.23) implies t h a t the t r a j e c t o r y x(t) lies outside the ball /3d for all t. We proceed w i t h the p r o o f in two cases.

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Reset Control Systems

145

C a s e 1: ((9.6) holds) Let - 7 -- maxd_ 0 by (9.6) and because V ( x ) is continuously differentiable . It follows t h a t for ti+l _> t :> ti,

V(x(t)) = V(xo) + ~

V ( x ( T ) ) d r + AV(x(t,~))

dT+ 7 "~=1

= V(xo) - V =

V(xo)

--1

f

+

(Z(X(T))d~-

d~"

t~ - t,~-l) + t - ti

-

Since the right-hand side will eventually become negative, the inequality (9.23) contradicts the assumption that c > 0. C a s e 2: ((9.7) holds) Now let -~/---- maxd<_llxll<~,xe~.~AV(x). Then, ~, > 0 by (9.7) and because V ( x ) is continuously differentiable. It follows t h a t for ti+l :> t > ti, ~

tn

V ( x ( t ) ) -= V(xo) + n=l i

<

V(Xo)

t

(/(X(T))dT + A V ( x ( t ~ ) )

+

V(xO-))dT

--1

-

= V(xo) - i% Since the right-hand side will eventually become negative, the inequality (9.23) contradicts the assumption t h a t c > 0. Therefore, V ( x ( t ) ) ---* 0 as t ~ c~, and proof is completed. []

B

P r o o f of P r o p o s i t i o n 2:

Given any point q C R n, define fl = V(q) > 0. Since V ( x ) is radially unbounded, given any ~3 > 0, there exists an r > 0 such that V ( x ) > /~ for all ]lxll > r. Given r, we d e f i n e / ~ ~ {~ E tg: I1~11-< r} and given/3 and B~ as above we define ~2~ =~ {~ E B~ : V(~) ~ ~}. The rest of the proof is similar to t h a t of Proposition 1. []

References 1. Horowitz I.M., Synthesis of Feedback Systems, Academic Press, New York, 1963. 2. Beker O., Hollot C.V. and Chait Y., "Plant with Integrator: An Example of Reset Control Overcoming Limitations of Linear Feedback," ECE D e p a r t m e n t Technical Note #ECE07.13.2000, University of Massachusetts Amherst, also submitted to I E E E Transactions on Automatic Control, 2000.

146

C.V. Hollot et. al.

3. Horowitz I.M. and Rosenbaum P., "Nonlinear Design for Cost of Feedback Reduction in Systems with Large Parameter Uncertainty," International Journal of Control, Vol. 24, No. 6, pp. 977-1001, 1975. 4. Zheng Y., Theory and Practical Considerations in Reset Control Design, Ph.D. Dissertation, University of Massachusetts, Amherst, 1998. 5. Zheng Y., Chait Y., Hollot C.V., Steinbuch M. and Norg M., "Experimental Demonstration of Reset Control Design," IFAC Journal of Control Engineering Practice, Vol. 8, No. 2, pp. 113-120, 2000. 6. Clegg J.C., "A Nonlinear Integrator for Servomechanism," AIEE Transactions Part II, Application and Industry, Vol. 77, pp. 41-42, 1958. 7. Tsypkin Y.Z., Relay Control Systems, Cambridge University Press, Cambridge, UK, 1984. 8. Decarlo R.A., "Variable Structure Control of Nonlinear Multivariable Systems: A Tutorial," IEEE Proceedings, Vol. 76, No. 3, pp. 212-232, 1988. 9. Branicky M.S., "Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems," IEEE Transactions on Automatic Control, Vol. 43, pp. 475-482, 1998. 10. Bainov D.D. and Simeonov P.S., Systems with Impulse Effect: Stability, Theory and Application, Halsted Press, New York, 1989. 11. Haddad W.M., Chellaboina V. and Kablar N.A., "Nonlinear Impulsive Dynamical Systems Part I: Stability and Dissipativity," Proceedings of Conference on Decision and Control, pp. 4404-4422, Phoenix, AZ, 1999. 12. Singer N.C. and Seering W.P., "Preshaping Command Inputs to Reduce System Vibration," Transactions of the ASME, Vol. 76, No. 3, pp. 76-82, 1990. 13. Bobrow J.E., Jabbari F. and Thai K., "An Active Truss Element and Control Law for Vibration Suppression," Smart Materials and Structures, Vol. 4., pp. 264-269, 1995. 14. Bupp R.T., Bernstein D.S., Chellaboina V. and Haddad W.M., "Resetting Virtual Absorbers for Vibration Control," Proceedings of the American Control Conference, pp. 2647-2651, Albuquerque, NM, 1997. 15. Haddad W.M., Chellaboina V. and Kablar N.A., "Active Control of Combustion Instabilities via Hybrid Resetting Controllers," Proceedings of the American Control Conference, pp. 2378 2382, Chicago, IL, 2000. 16. Hu H., Zheng Y., Chait Y. and Hollot C.V., "On the Zero-Input Stability of Control Systems Having Clegg Integrators," Proceedings of the American Control Conference, pp. 408-410, Albuquerque, NM, 1997. 17. Beker O., Hollot C.V., Chen Q. and Chait Y., "Stability of A Reset Control System Under Constant Inputs," Proceedings of the American Control Conference, pp. 3044-3045, San Diego, CA, 1999. 18. Chen Q., Hollot C.V., C h a r Y. and Beker O., "On Reset Control Systems with Second-Order Plants," Proceedings of the American Control Conference, pp. 205-209, Chicago, IL, 2000. 19. Chen Q., Reset Control Systems: Stability, Performance and Application, Ph.D. Dissertation, University of Massachusetts, Amherst, 2000. 20. Chen Q., Hollot C.V. and Chait Y., "BIBO Stability of a Class of Reset Control Systems," Proceedings of the 2000 Conference on Information Sciences and Systems, p. TP8-39, Princeton, N J, 2000. 21. Middleton R.H., "Trade-offs in Linear Control System Design," Automatica, Vol. 27, No. 2, pp. 281-292, 1991.

9

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147

22. dSPACE Experiment Guide, dSPACE Inc., Paderborn, Germany, 1999. 23. Ye H., Michel A.N. and Hou L., "Stability Analysis of Systems with Impulse Effects," I E E E Transactions on Automatic Control, Vol. 43, No. 12, pp. 17191723, 1998. 24. Slotine J-J.E. and Li W., Applied Nonlinear Control, Prentice-Hall, New Jersey, 1991.

10 Generalized Quadratic Lyapunov Functions for Nonlinear/Uncertain Systems Analysis Tetsuya Iwasaki University of Virginia, Charlottesville, VA 22904, USA

A b s t r a c t . We consider the class of discrete-time n o n l i n e a r / u n c e r t a i n systems described by the feedback connection of a linear time-invariant system and a "troublesome component," i.e. either a static nonlinearity or a time-varying parametric uncertainty. We propose a generalized quadratic Lyapunov function for stability analysis of such systems. In particular, the Lyapunov function is given by a quadratic form of a vector that depends on the state in a specific nonlinear manner. Introducing a quadratic-form model of the troublesome component in the spirit of integral quadratic constraints, we obtain sufficient conditions for the existence of such Lyapunov functions that proves global/regional stability. The conditions are given in terms of linear matrix inequalities that can be numerically verified in polynomial time.

10.1

Introduction

~

While the linear control system theory has reached a mature state, there is still much work to be done for analysis and synthesis of nonlinear a n d / o r uncertain control systems. Recent progresses in nonlinear control theories (see e.g. [16,21]) are nontrivial, but it appears difficult to have a universal design procedure that is guaranteed to "work" for a general class of nonlinear systems. A seemingly more tractable, yet practically important problem is to consider a special class of nonlinear systems that can be described by the feedback connection of a nonlinear component r and a linear time-invariant (LTI) system G (see Fig. 10.1). This restricted class of nonlinear systems have been extensively studied in the literature [24,25,35]. Moreover, such feedback systems have constituted a paradigm for linear robust control theory [7,17,27] where r acts as an uncertainty in the system rather than as a nonlinearity. The representation of n o n l i n e a r / u n c e r t a i n systems as in Fig. 10.1 has lead to a powerful tool for stability and performance analysis - input-output multipliers. Another powerful tool for analysis and synthesis of nonlinear a n d / o r uncertain systems is the Lyapunov function. The problem of quadratic stabilization 0 We use the following notation. The sets of n x m real matrices and n-dimensional vectors are denoted by R nxm and R n, respectively. For a matrix (or a vector) A, A t means the transpose. For vectors vi (i = 1 , . . . , k), we denote the composed vector v : = [ vl --- v~ ]~ by v = [ v l ; . - - ;vk ]. For matrices A and B of the same dimensions, A 9B denotes the entry-wise multiplication. The set of nonnegative integers is denoted by Z.

150

Tetsuya Iwasaki

U

F i g . 10.1. Nonlinear/uncertain system

was extensively studied in the 1980's (e.g. [22,1]), where a single quadratic Lyapunov function is used to prove stability of a family of systems. It has been shown [2,8,11,32] t h a t the notion of quadratic stability leads to robust control design techniques t h a t are conservative but numerically efficient, with the aid of interior point methods [26] for solving linear matrix inequalities (LMIs) [3,12]. More recently, attention has been paid to the so-called parameter-dependent Lyapunov functions [6,9,10,13,20,33] t h a t give less conservative stability conditions than the quadratic stability approach. Some of the Lyapunov approaches can be interpreted within the framework of input-output multipliers, and vice versa. For example, satisfaction of the (multiloop) circle criterion [31] implies the quadratic stability, while the Popov criterion for systems with sector-bounded nonlinearity is related to the existence of an affine parameter-dependent Lyapunov function for the corresponding uncertain systems [14,3]. The Lyapunov interpretations of the frequency domain conditions given in terms of input-output multipliers have been found useful for developing regional (rather than global) stability conditions for nonlinear systems of the form in Fig. 10.1; For instance, the references [15,28] consider saturating control systems and exploit the circle and the Popov criteria to give an estimate of the region of attraction, that is, the set of initial state vectors t h a t generate trajectories converging to the origin. In this paper, we propose a new class of Lyapunov functions for discrete-time systems described by the feedback connection in Fig. 10.1 where r is either a static nonlinearity or an uncertainty. In particular, we generalize the class of parameterdependent Lyapunov functions in [20] in order to allow for the analysis of both nonlinear and uncertain systems with reduced conservatism. The main idea can be explained as follows. In [20], uncertain systems of the form in Fig. 10.1 is considered where G is a stable LTI system with a state space realization C ( s I - A ) - I B + D and r is the function defined by u ---- Aly with A being a matrix consisting of uncertain timevarying parameters. For this class of systems, the following parameter-dependent Lyapunov function is shown to provide a reasonable trade-off between the degree of conservatism and the computational burden to check robust stability: I

t

Note that this Lyapunov function can also be given by

[:]',[:]

I

10

Generalized Quadratic Lyapunov Functions

151

where u is uniquely determined from x as the solution 1 to u = A ( C x + D u ) . Note that the Lyapunov function in (10.1) does not depend explicitly on the uncertain parameter Al and thus can also be defined for the system in Fig. 10.1 with n o n l i n e a r component r Here in this paper, we consider discrete-time feedback systems and propose the class of Lyapunov functions specified by

Y(xk)

Xk := [Xk ; Uk 9, "'"

= x k' P x k ,

;

Uk+q--1 ]

(10.2)

where Xk is the state at time k, uk is the signal in Fig. 10.1 evaluated at time k, and q is a given positive integer. We call V in (10.2) a generalized quadratic L y a p u n o v f u n c t i o n since V is quadratic in the auxiliary vector x but not in the original state x. Clearly, V is dependent upon either the uncertain parameters or the nonlinearity. Based on the generalized quadratic Lyapunov function, we shall give sufficient conditions for global stability and for regional stability of the feedback system in Fig. 10.1. The quadratic-form model (QFM) of the "troublesome component" r will play a crucial role in our analysis. The QFM of r is the set of weighting matrices whose quadratic form with the vector vk is nonnegative for all time k where vk consists of the i n p u t - o u t p u t graphs of r at time instants k through k + q. Two specific QFMs are developed; One is for the class of nonlinear functions r with a given first-order information2. The other is for the time-varying parametric uncertainty for which bounds on the magnitude and the rate of variation are known. We show that QFMs can be used as "multipliers" in our Lyapunov analysis, leading to stability conditions given in terms of LMIs.

10.2 10.2.1

Feedback

system

Problem

formulation

analysis

Consider the discrete-time system in Fig. 10.1 described by xk+l = A x k + B u k yk = Cxk + Duk Uk

= r

(10.3)

yk)

where xk E R n is the state vector, uk E R p and yk C R m are the signals in the feedback loop (see Fig. 10.1), and r : Z x R m --* R p is a nonlinear a n d / o r uncertain function. The following assumption is enforced throughout the paper. Assumption 1 (a) T h e f u n c t i o n r is c o n t i n u o u s and satisfies r 0) ----0 f o r all k C Z. (b) F o r each k E Z a n d x E R n, there exists a u n i q u e u C R p such that u ---r Cx + Du).

With this assumption, the existence and uniqueness of the solution to system equation (10.3) is guaranteed, and the origin is an equilibrium state. The condition (b) in Assumption 1 is often referred to as the well-posedness of the feedback connection. To state our objective, we need the following: 1 The existence and the uniqueness of such u is guaranteed by the well-posedness d e t ( I - AD) ~ 0 of the feedback system in Fig. 10.1. 2 The slope-restricted nonlinearity is an example of such r

152

Tetsuya Iwasaki

D e f i n i t i o n 1. Consider the system (10.3) and let A be a subset of l:t'~ containing the origin. The system is said to be globally stable if the state trajectory xk (k E Z) converges to the origin for any initial state x0 E R '~. We say that the system is regionally stable in A if the state trajectory xk (k E Z) converges to the origin whenever xo C A. Such A is called a region of attraction. Given matrices A, B, C, D and certain information on the function r to be specified later, our objective is to develop numerically tractable methods for analyzing global/regional stability of the system (10.3). In particular, we would like to give a (sufficient) condition for global stability of the system, and a characterization of the region of attraction. The notion of global stability is sometimes too strong to ask for nonlinear systems. On the other hand, the notion of local stability is too weak to provide an engineer with a confidence since the set of initial states converging to the origin, the region of attraction, can be arbitrarily small. Hence, it is often the case in practice that the notion of regional stability, which lies in between the notions of global stability and local stability, is the right property to be questioned. Before proceeding, let us introduce some notation. In view of Assumption 1, the vector u satisfying the relation u ----r C x + Du) is uniquely determined from k and x. We denote this mapping by u ----~(k, x). Clearly, ~(k, x) ----r Cx) when D = 0. Further introducing the nonlinear function

f ( k , x) :--- A x + B ~ ( k , x), the system (10.3) can be described by Xk+ 1 =

f ( k , xk),

uk = 7~(k, xk).

For integers k0 and kl such that 0 ~ ko _< kl, we see that the state at time kl is uniquely determined by the state at time ko. Hence xkl can be expressed as xkl ---- F(ko,kl,xko), where the function F : Z • Z x R ~ --* R n is the "state transition mapping" and is defined by recursive applications of the function f. For instance, F(4, 7, x) -- f(6, f(5, f(4, x))). Similarly, Ukl is also a function of k0, kl and xk0, and can be given by ukl : ~(kl, F(ko, kl, Xko)) --~: G(ko, kl, xko). 10.2.2

Generalized

quadratic

Lyapunov

function

We use the Lyapunov function candidate described in Section 10.1 for our stability analysis. A precise definition of the class of such Lyapunov functions is as follows:

V ( k , x ) :-~ x'Px,

x := Tk(x)

(10.4)

where P is a fixed symmetric matrix and Tk (x) is a function defined by T k ( x ) : : [ x ; u0 ; . . -

; uq_l],

ui :----G(k,k + i,x),

(i = 0 , . . . , q - 1)

for a given integer q _> 1. The function V is well defined if the system (10.3) is well posed. Clearly, V(-, 0) = 0 holds due to the assumption r 0) ----0. Below, we shall develop a related auxiliary system, that has • := Tk(xk) as its state.

10

Generalized Q u a d r a t i c L y a p u n o v F u n c t i o n s

153

Consider t h e s y s t e m (10.3) and let xk, uk and yk be any signals t h a t satisfy t h e system equations. T h e n , it can be shown t h a t t h e signals xk E R nq , wk E R p, and vk 9 R ~q, where nq :~- n + qp, Eq :---- ( m + p)(q + 1), defined by

E

h]

:-----

Wk

Vk

,

:~

,

Uk

Uk

~

;uk+q],

y~:=[yk;

;y~§

satisfy xk+l = .AXk + Bwk vk = Cxk + 7)wk

(10.5)

where

A:=

0 0

, B:=

C:= [CD0],

c

i01

, C:=

77:=

/7,

/)y : =

[~

CA

C~ :=

,

C~

,

(10.8)

[0]

10.0,

T h e above e q u a t i o n s are o b t a i n e d by recursive applications of t h e first two e q u a t i o n s in (10.3) and n o t i n g t h e following identities: C,AkB = 0

(k = 0 . . . . . q - 2),

C f i [ q - l B = D.

T h e third e q u a t i o n uk -- r Yk) in (10.3) imposes a static constraint on t h e signal vk as follows. Let ~ : Z x R re(q+1) --. R p(q+l) be t h e function consisting of r on the diagonal; more precisely, u ----(~(k, y) m e a n s + i, yi)

ui ----r

[~o ;

.

.

.

(i = 0 , . . . ,q),

; ~q] :=-,

[yo ; . .

(10.10)

; y~] :=y.

T h e n it can be verified t h a t t h e signal vk is constrained to be a g r a p h of t h e f u n c t i o n at t i m e k; vk 9

Gk(r

(10.11)

154

Tetsuya Iwasaki

where the set Gk(r Ok(C)

:----

is defined by

(~b(k,y)

E R lq : y 9

By construction, any solutions xk, uk and yk of dynamical equations (10.3) satisfy equations (10.5) and constraint (10.11) for appropriately defined signals Xk, Wk and v~. The converse statement also holds true as shown below. Consider the auxiliary system described by (10.5) and (10.11). The system has an algebraic constraint (10.11) and hence there may be no solution for some initial state xo. From the construction above, we see that the initial state is constrained to satisfy x0 E T0(r where Wk(r

:----{ Tk(z) C R nq : x E R n }

if equations (10.5) and (10.11) are to generate a trajectory of the original system (10.3). The following lemma provides existence and uniqueness of the trajectory of the auxiliary system starting from xo E T0(r L e m m a 1. Consider the set Gk(r in (10.12) and the augmented matrices in (10.6)-(10.9). Suppose that Assumption 1 holds. Then the following statements hold true: (a) Given x E R "~q and k E Z, there exists a unique vector w E R p such that Cx + / ~ w E ok(C) /f and o n l y / f x E Wk(r holds. (b) Givenx E R '~q, w E R p a n d k E Z, supposeCx+T)w E Gk(r Then.Ax+Bw E Wk+l(r Proof. See Appendix A. It follows from this lemma that, if x0 E To, then the resulting trajectory is unique and satisfies xk E Ta(r for all k E Z. Therefore, for each xk, there corresponds a vector xk such that xk ----Tk(xk). It can readily be verified that this xk will be a solution to the original system (10.3). Note from Lemma 1 that there exists a unique vector w such that Cx 4- T)w E Gk(r for each k E Z and x E Tk(r Let us denote such w by ~(k,x). T h a t is,

w=~(k,x)

~

C x + / ) w E Gk(r

Then the auxiliary system can be identified as a nonlinear time-varying system

xk+1 = f(k, xk),

f(k,x) := Ylx+ B(p(k,x)

(10.13)

with algebraic constraints xk E Tk(r for all k E Z. As noted above, if the initial state satisfies the algebraic constraint, i.e. x0 E T0(r then so does the state xk for all k E Z. In summary, the auxiliary system described by (10.5) and (10.11) is equivalent to the original system (10.3) in the sense that one generates a trajectory of the other. Thus the global/regional stability of the original system can be examined by studying the stability property of the auxiliary system. Moreover, the Lyapunov function in (10.4) is quadratic in the state xk of the auxiliary system, although it is not quadratic in terms of the original state xk. This fact enables us to consider the non-quadratic Lyapunov function using tools from the quadratic Lyapunov function analysis.

10 10.2.3

Generalized Quadratic Lyapunov Functions

155

Global stability analysis

In this section, we shall give a sufficient condition for the existence of a Lyapunov function of the form (10.4) t h a t proves global stability of the feedback system (10.3). As noted in the previous section, stability of the original system (10.3) is equivalent to stability of the auxiliary system described by (10.5) and (10.11), and a quadratic Lyapunov function for the auxiliary system will act as a (non-quadratic) Lyapunov function for the original system proving its stability. To this end, let us define the following sets: T(r

U

:=

Tk(r

G(r

keZ

0(r := { o = o' e R ~q•

U

:---Gk(r keZ

v'Ov > o / v e a ( ~ ) } .

(10.14)

Notice that, given any O E O(r and vk in (10.11), the quadratic form vkOvk is nonnegative for all k E Z. Thus the set O(r may be considered as a "model" of the nonlinear/uncertain function r in the spirit of IQC analysis [25]. We shall call the set O(r a quadratic-form model (QFM) of r T h e o r e m 1. Let an integer q >_ 1 be given. Consider the system (10.3) and define the set O(r as in (10.14) and augmented system matrices as in (10.6)-(10.9). Suppose that Assumption 1 holds and there exist symmetric matrices P and k~, 4~ E O(r such that (10.15)

< o

(lO.16)

Then the system is globally stable. Moreover, V ( k, x) defined by (10.4) is a Lyapunov function that proves stability. Proof. We show that V(x) :-- x'Px is a Lyapunov function that proves stability of the auxiliary system defined by (10.5) and (10.11). The result then follows from the equivalence of the original system (10.3) and the auxiliary system as discussed in the previous section. Recall that every trajectory xk of the auxiliary system is algebraically constrained to be xk E Tk(r for all k E Z. Thus V is positive definite along the trajectory if V(x) > 0,

v nonzero x E T ( r

From statement (a) of Lemma 1, we see t h a t this condition is equivalent to x'Px>0,

Vxr

such t h a t

Cx+:DwEG(r

for s o m e w .

156

Tetsuya Iwasaki

Now, applying the S-procedure (Lemma 8 in Appendix B) with

we obtain (10.15) as a sufficient condition. We can show that (10.16) is a sufficient condition for the function V(x) to be monotonically decreasing along the trajectory of the auxiliary system as follows. Recall from Lemma 1 that there exists a unique vector w such that CxTT)w E Gk(r for each k E Z and x 9 Tk(r and such w is denoted by ~(k, x). We seek a sufficient condition for the following: V(.Ax + Bcp(k,x)) < V(x),

v nonzero x 9 Tk(r

k 9 Z.

It can be verified that this condition is equivalent to V(.AxTBw)
Vnonzero [ : ]

s u c h t h a t C x + Dw 9 G(r

Then using the S-procedure again, we obtain (10.16). 10.2.4

Regional stability analysis

In this section, we give a characterization of the region of attraction for the system (10.3). We consider for simplicity the case where the function r in (10.3) is time-invariant. In order to perform regional stability analysis, we shall restrict our attention to the class of Lyapunov functions3 V ( x ) given in (10.4). As in the previous section, we consider the auxiliary system described by (10.5) and (10.11). Then the analysis problem reduces to the search for a quadratic Lyapunov function V(x) ----x'Px that is positive definite and monotonically decreasing along each state trajectory xk E Tk(r of the auxiliary system in some region A of the state space. The following lemma provides a basis for our regional stability analysis. The essence of this approach has already appeared for example in [30,15,28]. L e m m a 2. C o n s i d e r a n o n l i n e a r s y s t e m xk+l = f(xk) where f : R" ~ R" is a c o n t i n u o u s f u n c t i o n satisfying f(0) = 0. L e t X be a subset o f the state space X C_ R". I f there exists a f u n c t i o n V : R " ---* R such that V(x) > 0,

v nonzero x C X ,

V(f(x)) -- V(x) ( 0,

and

v nonzero x C X

A C_ X

V(0) = 0

(10.17) (10.18) (10.19)

where

A:--{xER":

V(x)_
(10.20)

t h e n A is a region o f attraction, i.e., xk approaches the origin as k ~ co w h e n e v e r

x0 E A . a With a slight abuse of notation, we omit the first argument k in V since V is not an explicit function of the time k due to the time-invariance of r We use similar notation for r Tk(r etc.

10

Generalized Quadratic Lyapunov Functions

157

Proof.

Let xk be the state of the system at time k in response to the initial condition x0 9 A. First we show that xk 9 A for all k >_ 0. If xkl ~ A for some kl > 0, then there exists k0 < kl such that

V(xko) _< 1,

V(xko+l) > 1,

hold. This is a contradiction because Xko # 0 due to V(xko+t) > 1 and Xko C A _C X, and hence we must have

V(xko+t) = V(f(Xko)) < V(Xko) _< 1. Thus we see that A is an invariant set, that is, xk does not leave A for all k _> 0. In this case, the function V(xk) is monotonically decreasing, justifying that V(x) is a Lyapunov function to prove the regional stability. The first step for the analysis is to choose the set X. A possible criterion for the choice is such that X allows for regional modeling of the nonlinearity ~b. Suppose that the function r (or ~b) can be modeled, in a certain region Y C R re(q+1), as follows:

f

o r ( C ) := / e =

O' e R~qXeq : Vt~V :> 0,Vv 9 C V ( r

.

(10.21)

If we choose the set X as

X : - - - - { x e R ~q : 3 w e R

ps.t.Cx+:DwCGv(r

},

then conditions (10.17) and (10.18) can be characterized as in Theorem 1 by replacing O with O v. In general, the set Ou162 is larger than 8(r in the previous section due to the restriction of the "operating region" of the function r Hence, if we seek ~p,q5 E 8u162 in (10.15) and (10.16), then the conditions become less restrictive. Below, we consider the following set for Y a m o n g others: Y : = { y c R '~(q+`) : v : = [ y ; ,/,(y) ],

Li

[1]

~0,

Vi=l,...,r

}

where L~ E R (Q+l)x(eq+O are given symmetric matrices. Partition Li as

L~=[U~viJ V~ Wi

where wi is a scalar. Considering the additional condition (10.19), we have the following regional stability result.

158

Tetsuya Iwasaki

T h e o r e m 2. Let an integer q > 1 be given. Consider the system (10.3) and define the set 8Y(r as in (10.21) and augmented system matrices as in (10.6)-(10.9). Suppose that Assumption 1 holds and there exist positive scalars Ti, cri symmetric matrices kV,r C 8v(r ~2i E 8(r and P satisfying (10.15), (10.16), and

[C/0D]'[ZiO] [C/0D] > 0

Ti

(10.22)

(7i

for all i = 1 , . . . , r. Then the system is regionally stable in A where

A:={=eR~:

V(=) < 1}

with V(x) given by (10.4). Proof. If conditions (10.17)-(10.19) with

h:={xeRnq:

xCW(r

are satisfied, then the auxiliary system (10.13) is regionally stable in A, which in turn implies that the original system (10.3) is regionally stable in h. As noted above, conditions (10.17) and (10.18) can be verified as in the proof of Theorem 1. Condition (10.19) holds if (and only if) x 9 T(r

x'Px _< 1

:=~ Cx + Dw 9 GY(r for some w.

Using the definition of G v (r and Lemma 1, we see that this condition is equivalent to

Cx+/)w9162

x'Px_< 1

=~

C~x+D~w9

Furthermore, noting the definition of Y, another equivalent condition is given by: For each i = 1,... ,r, [ Cx ~ T)w 1 'Li [ Cx +1T)w] _ > 0 ,

v x Cand x + :wD such w 9 1 6that 2

x ' P x < 1.

Applying the S-procedure, this condition holds if there exist Ti > 0 and ~2i 9 8(r such that 01]C' 0 T) Ti [D0 ' 0 Li IC 0 01] > [C~] ' ~i [C :D O] + [-00P0!] 0 . 0 It is now straightforward to show, using the Schur complement and introducing slack variables aij, that this condition is equivalent to (10.22) and (10.23).

10

Generalized Q u a d r a t i c Lyapunov ~ n c t i o n s

159

Theorem 2 characterizes a region of a t t r a c t i o n for system (10.3) in terms of LMIs. An estimate of the maximal region of a t t r a c t i o n may be found by maximizing the "volume" of the set A subject to the conditions in Theorem 2. However, the set A is not an ellipsoid in R n and it is difficult to measure its volume in a c o m p u t a t i o n a l l y tractable manner. One thing we could do is to maximize the volume of an ellipsoid that is contained in A. Let us explain this point in more detail. Let a symmetric positive definite m a t r i x Q -- Q' 9 R n be given and consider the ellipsoid defined by x'Qx _< 1. This ellipsoid is contained in A if and only if x'P•

1,

x'Qx< 1,

V•

or equivalently,

x'Px <_l,

Vx, w s . t . C x + i D w 9 1 6 2

x'J'QJx < 1

where J := [ In 0 ] 9 R '~xnq. Then, using the S-procedure, it can be shown t h a t a sufficient condition for this is given by the existence of A 9 8 ( r such t h a t

[Ci lff ] ' [A p _ 0,j oj ] ICIly]

<

O.

(10.24)

Thus, approximating the volume of the ellipsoid x'Qx < 1 by t r ( Q ) , an estimate of the maximal region of attraction can be found by minimizing t r ( Q ) subject to (10.15), (10.16), (10.24), and the conditions in Theorem 2 over the variables T~, ai, ~P,~ 9 OY(r ~2~,A 9 O(r P , and Q > 0. This is an eigenvalue problem [3] and can be solved efficiently via interior point methods [26]. 10.2.5

Connection

to IQCs

In this section, we briefly discuss relationship between our Lyapunov-based stability condition in Theorem 1 and the IQC-based stability condition in [25]. For brevity, we consider the case where the function r in (10.3) is time-invariant. From the discrete-time K Y P lemma [29], there exists a symmetric m a t r i x P satisfying (10.16) if and only if the following frequency domain inequality holds:

~(e'~)*~G(e j~) < 0,

v 9 [0, 2~]

where

6(z) := C(zI - A)-113 + D, provided .A has no eigenvalues on the unit circle. Noting the identity:

C(zI-.A)-IB-F T) = F(z) [ G~z) ] where, with ~ :--- m or p,

[F~(z) 0] F ( z ) := 0 Fp(z) '

I z-qIa F,~(z) := I z-lI'~ , L I,~

160

Tetsuya Iwasaki

G(z) := C(zI - A)-I B + D, the above inequality can be rewritten as

G(~)

u(e~) a ( ~ )

<0,

v 9

(10.25)

where

H(z) := F(1)'q~F(z). Thus condition (10.16) is equivalent to the frequency domain inequality (10.25). Now, condition 45 9 8 ( r basically means t h a t

for any integer k and any signal y 9 ~2, where lf~ is the set of square summable (vector-valued) sequences, vk is the value of the signal v evaluated at time k, and F and r denote, with a slight abuse of notation, the operators on ~2 represented by the transfer function F(z) and the static function r respectively. Using the Parseval's equality: k=-o~

V k ~ k - - - - 1 ~02~r ~ v(e3~)*r

we see t h a t the above inequality implies the following IQC: LCy(e3~) j

LCy(e3~)

y 9 g2

(10.26)

where ^ denotes the discrete-time Fourier transform. Thus, the conditions (10.16) and q5 9 8 ( r implies the existence of a finite impulse response (FIR) multiplier H(z) satisfying the IQC stability condition (10.25) and (10.26). This IQC condition is slightly different from the discrete-time counterpart of the s t a n d a r d IQC condition [25] as follows. In the s t a n d a r d case, not only r but also z r are required to satisfy the IQC of the form (10.26) for all T 9 [0, 1]. This property is then exploited to prove stability via a homotopy t y p e argument. On the other hand, we require the IQC (10.26) for r only. The price we paid to avoid the homotopy type argument is the additional condition (10.15) in Theorem 1, which guarantees t h a t the Lyapunov function V(k, x) in (10.4) is indeed positive definite and thus allows us to conclude stability.

10.3

Specific quadratic-form m o d e l s

In this section, we consider some specific functions r and provide Q F M s for them. It is difficult to have an exact characterization of the set 8 ( r in a numerically tractable manner. Here, we t r y to develop, at the expense of conservatism, tractable QFMs that are suited for numerical computations involving LMIs. In particular, we show some inner approximations ~ of the exact set ~9(r Replacing 8 ( r by 9 in the stability result (Theorem 1), we will have a computationally tractable sufficient condition for global stability. A similar comment applies to the regional stability result of Theorem 2.

10 10.3.1

QFM

Generalized Q u a d r a t i c L y a p u n o v F u n c t i o n s

of diagonally

composed

161

functions

Consider t h e function r : Z • R m ---, R p consisting of several functions r 1 , . . . ,e) as follows: r = diag (r where r

(i ----

,r

: Z • R mi ---, R pl and

t

t

Z

mi = m,

~

i=l

p~ = p.

i=1

Define t h e function ~b as in t h e previous section, i.e. u = ~b(k, y) m e a n s (10.10). D e c o m p o s e the vector u and form a new vector fi as

U----

,

Uj

----

,

LI :----

'

,

/i i :----

Lu j and similarly for y, where u~ E R p~ and y} E R ml for i = 1 , . . . ,~ and for j = 0 , . . . , q. Let F~ and F~ be the orthogonal matrices such that 6 ---- F~u and ~ = Fyy, and define F := d i a g (F~, F,,). Suppose, for each i E { 1 , . . . , ~ }, the function r is modeled by ~ C O ( r T h a t is, for any ~)~ e R mi(q+l) and Oi 9 ~i,

holds where q~i : Z x R m~(q+l) --~ N p~(q+l) is defined similarly to 4), i.e. a i =

4}i(k, f/~) means

^' r "j=

3, Yj) (j=O,...,q).

T h e n a Q F M for r can be given 4 by t h e following ~5:

4~:=

{

o |diag(O~) diag(O22)| [diag(Oh) drag( 12)]

F' /i=l:t L i=l:t

i

i=1:~ i=l:t

i

/ F:

[Oh 012]

[O~ O~2J 9

(i=l,.

"'

f)

)

J

T h u s t h e function r can be m o d e l e d by using Q F M s for r m o d e l i n g of a single function r (i.e. g = 1) in t h e sequel.

Hence, we consider t h e

4 For matrices Ai 9 R k~xt~ i = 1 , . . . , m , d i a g ( A i ) is t h e k • g block diagonal i=l:m m

m

m a t r i x w i t h A~ on t h e i t h diagonal, where k = '~-~k~ and ~ = ~--~f~. /=1

i=1

162

Tetsuya Iwasaki

10.3.2

Repeated

static nonlinearity

Consider the class of nonlinear, time-invariant functions r : R --~ R satisfying

X12 X22

r Lr162

y21 y22 /

[1

yll y~l zll z~ i r . y ~ y ~ zl~ z ~ j Lr162

> 0

(10.27)

-

for all ~,r E R where xij, y~j and z~j are given scalars. This class of functions includes several important classes of nonlinearities that have been studied in the literature. For instance, the sector-bounded nonlinearity can be captured by the constraint of the form (10.27). In particular, the sector-bound is given by the following condition: For any ~ E R, (r

- a~)(r

- j3~) ~ 0

(10.28)

holds where a and /3 are given real scalars. It is easy to see that this function satisfies (10.27) with -2a~

0

o

-2

o

0

-2

/~+/~

a+~

a +/3

as the weighting matrix of the quadratic form. Recall that the gain-bounded nonlinearity r satisfies Ir

-< ~l~[

for all ~ C R where "y is a given scalar. This is a special case of the sector bounded nonlinearity with a = -/3 = -y, and hence can be captured by (10.27) as well. Another example is the class of nonlinear functions r satisfying the following slope restriction; For any scalars ~, ~ E R such that ~ ~ ~, a -< r

~ _ -r r162 -< p,

(10.29)

holds where a, p C R are given scalars. It can readily be verified that the above inequality is equivalent to r162162

0 <#, -

~.-- p- ~ 2 '

0.-- p + ~ 2

Squaring the both sides, we have (10.27) with

_#2+0 2 0 -0

#2_0 2 -0 -0 -1 0 1-

1

as the weighting matrix of the quadratic form. The Lipschitz nonlinearity r satisfies

Ir

- r

-< ~1~ - r

10

Generalized Q u a d r a t i c L y a p u n o v F u n c t i o n s

163

for all scalars ~, ( E R , where # E R is a given scalar 9 T h i s is a special case of t h e slope-restricted nonlinearity w i t h p ---- - a ----#. T h e s e e x a m p l e s are s u m m a r i z e d in the following table. Nonlinearity S e c t o r - b o u n d e d (r Gain-bounded Slope-restricted Lipschitz

Condition - a~)(r - / 3 ~ ) _< 0 [r _~ ~[~[ ~ -< r162162 ~-r -< p [r - r ~ #[~ - r

In t h e sequel, we consider the class of nonlinear t i m e - i n v a r i a n t f u n c t i o n s r satisfying (10.27). It is also assumed t h a t r crosses t h e origin, i.e. r = 0, and is possibly an odd function, i.e. r = -r for all ~? E R . O u r o b j e c t i v e here is to find a Q F M for r where m is a given positive integer. Below, we consider the special case where 1 1 1 ~ 122, Yll : y22, and z u ---- z22 for brevity. T h e case w i t h o u t this a s s u m p t i o n can also be t r e a t e d in a similar m a n n e r . It t u r n s out t h a t applications of S - p r o c e d u r e s lead to a Q F M of r characterized by t h e following set of diagonally d o m i n a n t matrices:

sii>E[sij[,

S:--{SeRr•

s~j=sji,

(i,j=X,...,r)}.

(10.30)

This class of matrices has been used [4,5,23] to define a class of multipliers for t h e analysis of systems with r e p e a t e d nonlinearities. In this regard, our result can be considered as a generalization of these previous results. To s t a t e t h e result, let us define

[ r

::

[y21

Y11

/ yll y12 ... y121! [y 1

,

(10.31)

y21 y u . I

and X and Z similarly. 3. Consider the static time-invariant nonlinearity r that satisfies (10.27) with x u -~ x22, y u = y22, and z n = z22, and the set O ( r given by (10.1~). Define Y by (10.31) and X and Z similarly. Let S be the set of diagonally dominant matrices given by (10.30) with r := m(q + 1). If r crosses the origin, then

Theorem

~:={

S-Y S.Y'S.Z

: SESAP}

is such that q5 C_ O ( r where P is the set of symmetric matrices with positive entries. If in addition r is odd, then ~o:={

S.Y S.Y'S.Z

is such that q5o C_ O ( r Proof. See Section 10.4.1.

: SES}

164

Tetsuya Iwasaki

10.3.3

Time-varying

parametric

uncertainty

Consider a static, linear, time-varying, uncertain function r : R -~ R given by r ~) -- 5k~ for all k E Z and ~ E R where 5k E R is an uncertain time-varying parameter satisfying

I,~kl _< %

I,~,,;+,-

- ~kl ~ P

(10.32)

for all k E Z. In this section, we develop a Q F M for r We first consider the simple case where the parameter q in Section 10.2.2 is equal to one, and then generalize the result to the case q > 1. T h e o r e m 4. Consider the time-varying parametric uncertainty r ~) -- 5k~ that satisfies (10.32) for all k C Z. Define the set O ( r by (10.14) for q --- 1. Let

~:=

[

C'

-D

: C+C'=O,

R12 - - 0 , S12 ----0, Q : - - [ a l 2 D22

where D i j , G i j , R ~ j , S i j E R m• (i,j -- 1,2) are partitioned block matrices of D, G, R and S, respectively. Then ~ C_ O ( r Proof. See Section 10.4.2. The above Q F M reduces to the standard D-G scaling when the rate of parameter variation p is either zero or infinity. If p = 0, then S -- 0 and R = # I with sufficiently large # > 0 can be chosen without loss of generality, and all the constraints that define ~ reduce to D > 0 and G + G ~ ---- 0. Thus we have the standard D-G scaling for a single uncertain parameter. On the other hand, when p approaches infinity, R must approach zero and we have D > S > Q in the limit, implying that G12 -- 0 and D12 ----0. Thus we have the standard D-G scaling for two independent uncertain parameters. We now consider the case where q > 1. To state the result, we need to define, for each i = 1 , . . . ,q, a mapping E~ : a 2mx2m "-'-* 1~m(q-bl)• aS follows: For a matrix U, Ei(U) is a matrix given by Ei(U) ----d i a g (0, U, 0) where 0 on the left of U is a square matrix of dimension (i - 1)m while 0 on the right is a square matrix of dimension (q - i)m. Using the mapping E~, we have the following result. Th e proof is straightforward and hence omitted. T h e o r e m 5. Consider the time-varying parametric uncertainty r satisfies (10.32) for all k E Z. Define the set O(r by (10.14). Let

~=1 [E,(V~)' E~(W,)J : LV~ W~

----5k~ that

e ~

where ~ is defined in Theorem 4. Then ~P C_ O(r This Q F M has been obtained by "superposing" the previous result. For example, if q -- 3, using

U~:=

Sf

'

(,=123)

10

Generalized Quadratic Lyapunov Functions

165

we have

R1

S1

I[o '

3

S~ Q1 +

~=1

10.4 10.4.1

0

R2

]

$2

~ Q2+R3S~J

0

S~

Q3

Proofs Proof of Theorem

3

B a s i c Q F M ( m ---- q -~ 1) First we consider the case where m -- q ---- 1 and develop a QFM q5o C O(r If r crosses the origin, i.e., r = 0, then 00

r

1 0 Zll

Lr162

r

Lo 0 0

~ 0,

Lr162

Lr162

by letting r = 0 or ~ ----0. If r is odd, i.e., r ------ r

]~(~)] Lr162

|-x12 |

r

x22-y21

yll-y~1

L-Y~2

r

u2~ 0 z ~

Lr

___~0

then

y22]

>0

Z~l-Z~:]

y22-z~2

0 0 0

z22j

~(~)

-

LC(r

by replacing ~ with - r Hence, if r is odd, crosses the origin, and satisfies (10.27), then

I bx12 cx22 by21 cy22 [

>0

[r [ ayll by2~ az~l bz~2 [ r Lr162 J L b~,~ cy~ bz~:cz2~ j Lr162 J

-

holds for all q l , q 2 , q 3 , q 4 >_ 0 and ~,~ E R where a :--- ql + q3 -t- q4,

b : ~ q3 -- q4,

C : ~ q2 q- q3 + qa.

This class of multipliers is given by

where 9 denotes the entry-wise multiplication and LX12 x22

'

qa--qe

1

Ly21 y 2 2 j '

q2+qa+q4

LZ12 z 2 2 j '

: ql,q2,qa,q4>O

.

(10.33)

It turns out that the set Q is exactly the class of static multipliers (of dimension 2) proposed in [4,5,23] for the analysis of systems with repeated nonlinearities.

166

T e t s u y a Iwasaki

3. The set Q in (10.33) can be characterized as the set of diagonally dominant matrices of dimension 2, i.e., Q = S where

Lemma

=

: 80

sl

82

>

Is01,

s2 >

--

Isol

-

--

Proof. F i x Q E Q a n d let qi (i = 1 , . . . ,4) b e t h e scalars t h a t g e n e r a t e t h i s Q. T h e n , for so:--q3-qa,

sl :=ql+q3+qa,

s2 : - - q 2 + q 3 + q a ,

we see t h a t si - [so[ = qi + q3 + qa - [q3 - q41 _> qi > 0 for i = 1, 2. H e n c e Q 6 S. Conversely, fix S 6 S a n d let s~ (i =- 0, 1, 2) b e t h e s c a l a r s t h a t g e n e r a t e t h i s S. T h e n c h o o s i n g

{

ql =sl - so q2 =s2-so

{ ql =8171-80 (when so >0),

q3 =So

--

q4 = 0

q2 = s 2 + s o q3 = 0 qa = --So

( w h e n so < 0),

we see t h a t S 6 Q. I n s u m m a r y , we h a v e t h e following Q F M for r w h e n q = 1. 4. Consider the static time-invariant nonlinearity r that satisfies (10.27) and the set 8 ( r defined by (10.14) with q = 1. Suppose that r is odd and crosses the origin. Let

Lemma

S-2 S.]:~]

~o:={

s.?'s.2

Then ~o C O ( r

c__o(r ~:={

for

: s~}.

Moreover, i f r is not necessarily an odd function, then we have

s.? [s.~ s.?'s.2 1 : s ~ n P }

where P is the set of 2 • 2 symmetric matrices with positive entries. Proof. T h e r e s u l t for t h e case w h e r e r is o d d follows f r o m t h e a b o v e discussion. W h e n r is n o t necessarily o d d , a n a r g u m e n t s i m i l a r t o a b o v e c a n also b e u s e d t o show t h e result; I n p a r t i c u l a r , t h e o n l y difference is t h a t t h e p a r a m e t e r q4 in (10.33) is n o w c o n s t r a i n e d t o b e zero.

10

Generalized Q u a d r a t i c L y a p u n o v F u n c t i o n s

167

General QFM We now develop a Q F M for r e p e a t e d scalar n o n l i n e a r i t y r with q ~ 1 using the result of the previous section. T h r o u g h o u t this section, we consider the special case where x l l = x22, y l l -- y22, and z n -- z22 for brevity. The case without this a s s u m p t i o n can also be t r e a t e d in a similar manner. We shall generalize the basic Q F M in L e m m a 4 to a Q F M for the case m _> 1 and q _> 1, through the two lemmas given below. The following is our first p r e l i m i n a r y lemma. To s t a t e the result, let us define the following matrices and the sets: 5

[

y l l y l : . . . y12]

r

x ~ x12 --. x~2 1 X12 X l l

X:----

,

Y:----

i'-.

X12 /

" ' 9

I,:={(i,j) 9

•

,

Z12 ' " " Z12 1

2:12 Z l l

Z:=

9

Y21 y l l a

i=l,...,r-1;

:={S=S' 9

~!

Ly 1

[-X12 ' " " W12 X l l J

Zll

' "9

LZl2

Z12 | / Z12 Z l l J

j=i+l,...,r}.

shh=

s 2i h,

s~ ~ + '

j=hq-i

sis = So ,

(h = 1 . . . . , r)

i = I

~r

80

S

C S }.

(10.34)

L e m m a 5. Consider a function r : R --~ R that is odd, crosses the origin, and satisfies (10.27). Then, the set ~o :=

S.Y S.Y' S.Z

is such that ~o C O ( r

: S9 where r := m ( q + 1).

Proof. Fix ~] 9 l:t ~ and S 9 S arbitrarily. Let s~ J be the m a t r i x in (10.34) t h a t generates S. We see from L e m m a 4 t h a t , for each (i,j) 9 It,

,7~

r

[ s~J . 2 s " . ? ]

LS~J? ' s , . 2 j

,TJ

r

Lr

> o

-

Lr

holds 9 Noting t h a t ~

e~ 0

.

r

Lr

~)

= i]I1o e~

r

~/ '

0 e;

5 For example, if r ----4, 27r ---- { (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4) }.

168

T e t s u y a Iwasaki

where ei and e s are t h e i th and the jth s t a n d a r d basis vectors in R ~, we can c o m b i n e these inequalities for different choices of t h e pair (i, j ) E 2"~ to get Z

~

r

[EO (S's')f)E~s

q -

'

[ Eq(S ,S .Y- , )Eq, E~s(S~3.Z)E~s ]

r

> - 0

where E q : = [ ei e s ]. It is straightforward to verify t h a t ~ ! EiS (S is .Y)Eq

=

S.Y

(i,s)ez~ where

(i,j)e:Zr and similarly for t h e X and the 2 terms. Moreover, it can be shown t h a t S = S. Hence

holds and we conclude t h e result. T h e second p r e l i m i n a r y result is t h a t t h e set S coincides w i t h t h e following set of diagonally d o m i n a n t matrices:

S:={Sear•

s._>)--~ls,jl,

s,~=sS,,

( i , j = l . . . . . r)}.

S#i

Lemma6. S=S.

Proof.

Suppose 5: C S. Let S is E S be t h e m a t r i x t h a t g e n e r a t e S as in t h e definition (10.34) where S is is p a r t i t i o n e d as in (10.35). T h e n , for each h = 1 , . . . , r, h--1

shh =

+ j=h+l

sP i=1 h--1

_>

IsohSl+ ~ IsPI S=h+l

i=l h--1

IshSl+ ~ Isihl

=

j=h+ l

= ~

i=1

Ishsl

j#h

holds and hence S E S. Conversely, suppose S E S. For ( i , j ) E Zr, define s o~3, s ~3 1 and s~3 by SiOs : ~ 8 i j

10

87 := 18d +

82 :=

Generalized Quadratic Lyapunov Functions

8. - ~

18~d +

~

18~kl

- Y ~ 18jkl k~j

Then it is easy to see that s ~J I _> be verified that

169

)

Is~Jl and

Q

s~~ _>

Is~Jl due to

S E S. Moreover, it can

h--1 +

j=h+l

8 P = 8hh i=l

for all h = 1 , . . . , r. Therefore we conclude that S C S. We now prove Theorem 3. The second part (qSo C O(r for odd nonlinearity) directly follows from Lemmas 5 and 6. The first part can also be shown similarly. 10.4.2

Proof of Theorem

4

Consider the real parameters 5 and a such that

15I_~;, Io-1_~% I S - a [ < p

(10.36)

~

where % p ~ 0 are given scalars. We shall find a set O such that, for each O E O,

aI

aI

holds for any pair (5, a) satisfying condition (10.36). It should be clear that the set O is a QFM for the time-varying parametric uncertainty r = 5k( satisfying condition (10.32), for the case q = 1. The following lemma is useful for our development. L e m m a 7. Let scalars r, s, q and matrices R, S and Q be given. Suppose

R Then [rR sS]

sS' qQ

> O.

Proof. For any real vectors x and y, we have

where the last inequality holds due to the fact that A > 0 and B _> 0 imply t r ( A B ) > 0.

170

T e t s u y a Iwasaki

First, we can use t h e s t a n d a r d (D, G)-scaling for t h e gain b o u n d c o n s t r a i n t s 151 _< ~f and lal ~ ~/as follows:

m

D

~f2D2 0 G~ - D 1 2

0

G2 0

: Di=Di>0,_

Gi+G:=O,

(i=1,2)

.

-D2

N e x t we consider t h e "correlation" constraint [5 - a[ _< p. N o t e t h a t this constraint is equivalent to

O"--6

--

Hence, from L e m m a 7, we see t h a t

(a - ~)G'o

pD4

-

G'o D4

-

T h u s we have

~Pb =

La'o

pD4 -G'o -Go 0 o o

~

D3 Go

~

:

E(71o D4

> 0

-

"

Finally, we use all the three constraints. T h e key idea is t h a t these constraints imply

[

,./(,), _[_p ) - (~2

,.),2_5a

.y2 _ ~a

.y(.~ + p) _ a2

] _> 0.

This can be seen by noting t h a t .yp(2~/2 _ (~2 _ a2) _> 0

, ~ ~ ( ~ + p)2 _ ~(~ + p)(~2 + : ) >_ v~ _ 2 : ~ a (~(~ + p) - 5~)(~(~ + p) - ~2) > ( ~ _ ~ ) ~ and by using the Schur complement. Hence from L e m m a 7, for each 5 and a satisfying (10.36), we have

~'2D~

7('y + p)DTJ -

[D~

'

-

Thus,

~Pc =

I[

.).2 0

'6

7("/ + P)D7 0 0 0 -Ds-D6 0 -D'6 -D7

~~

:

[ D5 D6 LD6 D7

>0 -

"

10

Generalized Q u a d r a t i c L y a p u n o v Yhnctions

171

S u m m a r i z i n g t h e results, we have Q F M ~ : = ~Pa + ~Pb + ~Pc for t h e t i m e - v a r y i n g p a r a m e t r i c uncertainty. We now prove T h e o r e m 4. It is straightforward to verify t h a t ~' can be characterized as in T h e o r e m 4 by n o t i n g the c o r r e s p o n d e n c e

D = [

D;

D7 + D2

0

'

D4+"/Dr

[-Go

'

G2

0 D2

'

'

Q=

[ -Go/"/

DT+D2

T h e o r e m 4 t h e n follows i m m e d i a t e l y from t h e above d e v e l o p m e n t .

10.5

Conclusion

We have considered t h e class of n o n l i n e a r / u n c e r t a i n systems described by t h e feedback connection of a linear time-invariant s y s t e m and a n o n l i n e a r / u n c e r t a i n c o m p o n e n t . T h e generalized q u a d r a t i c L y a p u n o v f u n c t i o n is p r o p o s e d for stability analysis of such systems. We have shown t h a t a q u a d r a t i c - f o r m m o d e l of t h e n o n l i n e a r / u n c e r t a i n c o m p o n e n t can be effectively utilized to o b t a i n sufficient conditions for the existence of such L y a p u n o v functions t h a t proves g l o b a l / r e g i o n a l stability. T h e conditions are given in t e r m s of linear m a t r i x inequalities t h a t can be numerically verified in p o l y n o m i a l time. ACKNOWLEDGMENTS: T h e a u t h o r gratefully acknowledges helpful discussions w i t h C. Scherer and M. ~ .

A

Proof of Lemma

1

In the proof given below, vectors x E R n, u~ E a p, y~ C R m and x~ E R n (i = 0 , . . . , q - 1) are defined when x E R ; is given, as follows. Let x and ui be t h e p a r t i t i o n e d blocks of x such t h a t x' = [ x ' u~ . . . Uq-1 ]. Define x~ and y~ recursively by x~+l = A x i + B u i ,

y~ = C x i + Du~,

(10.37)

w i t h x0 := x. If in a d d i t i o n w E a p is given, t h e n vectors u C R p(q+l) and y E R re(q+1) are defined as follows.

y :--~

,

U :=

Yq- 1

LCAqx +

11,

1

Dw

In this case, it can be verified t h a t y = C~x + :Dyw,

u = C~x + :D~w

hold true. These identities are useful in t h e p r o o f below.

(10.38)

"

172

T e t s u y a Iwasaki

We now prove s t a t e m e n t (a). S u p p o s e x E T k ( r T h e n we have u~ ---- G(k, k + i, x), which in t u r n implies ui = r + i, yi) for all i = 0 , . . . , q - 1. Hence, for a given vector w, t h e condition Cx + 79w E G k ( r holds if and only if w = r

+ q, C.Aqx + Dw)

(10.39)

holds, where we n o t e d t h e identities in (10.38). N o t i n g t h a t c.Aqx = C [ A q A q - I B ... A B B ] x, we conclude t h a t t h e r e exists a u n i q u e v e c t o r w satisfying (10.39) due to t h e wellposedness assumption. This c o m p l e t e s t h e "if" p a r t of t h e proof. To show t h e converse, let x and w be such t h a t C x + T)w E G k ( r In v i e w of t h e identities in (10.38), we have u~ = r + i, y~) = r + i, Cx~ + Du~), or equivalently, ui = qo(k + i, xi). It t h e n follows t h a t

x~+l = A x i + Bqo(k + i, xi) = f ( k + i, xi),

xo = x

=r xi = F(k, k + i, x)

ui = qo(k + i , F ( k , k + i , x ) ) = G ( k , k + i , x ) . Hence we conclude t h a t x E T k ( r T h i s c o m p l e t e s t h e p r o o f of s t a t e m e n t (a). Next we prove s t a t e m e n t (b). F i x x and w such t h a t Cx + 79w E G k ( r Then, as shown above, we have ui = q o ( k + i , xi) for i = 0 , . . . , q 1. Let uq : = w, xq := A x q - 1 -q- B u q - 1 , and yq : = Cxq + Duq. T h e n it can be verified t h a t yq = C.Aqx+ D w holds. Hence, we see from t h e identities in (10.38) t h a t uq = r + q, yq) holds, implying t h a t ui = ~ ( k + i, xi) holds for i = q as well. Finally, n o t i n g t h a t l

A x + B w = [ x'l u l

,.

I]

. uq

!

and

ui ----99(k T i, F(k q- l, k q-i, x1)) = G ( k q - l , k W i ,

xl)

for i ---- 1 , . . . ,q, we conclude t h a t .Ax + Bw E T k + l ( r

S-procedure

B

T h e following is a version of the S-procedure [34] t h a t converts c o n s t r a i n e d q u a d r a t i c form inequalities to linear m a t r i x inequalities, and plays a key role in t h e proofs of our results. See [18,19] for lossless-ness results related to this version of t h e S-procedure. Lemma define

8. Given matrices Q -- Q' E R mxm, H E R l•

O:={O=O'ER~•

~'0~_>0, V~EG}.

Suppose Q > H'OH

for some

69 E 69.

Then ~'Q~>O,

V~o

such that

Hr

and a set G C R ~,

10

Proof. Fix

Generalized Quadratic Lyapunov Functions

6~ 6 ~9 such that Q >

~'Q~ > ~'H'OH~

H'~gH, and

= ~'0~ > O,

173

r ~ 0 such that H~ 6 G. Then

~ := H ~

where the last inequality holds due to ~ 6 G.

References 1. Barmish, B. R. (1985) Necessary and sufficient conditions for quadratic stabilizability of an uncertain linear system. J. Optimiz. Theory Appl., 46-4 2. Boyd, S., Yang, Q. (1989) Structured and simultaneous Lyapunov functions for system stability problems. Int. J. Contr., 49-6, 2215-2240 3. Boyd, S. P., E1 Ghaoui, L. et al. (1994) Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics 4. Chu, Y.-C., Glover, K. (1999) Bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities. IEEE Trans. Auto. Contr., 44-3, 471-483 5. D'Amato, F., Megretski, A. et al. (1999) Integral quadratic constraints for monotonic and slope-restricted diagonal operators. Proc. American Contr. Conf., 2375-2379 6. Dasgupta, S., Chockalingam, C., et al. (1994) Lyapunov functions for uncertain systems with applications to the stability of time varying systems. IEEE Trans. Circ. Syst., 41, 93-106 7. Doyle, J. C. (1982) Analysis of feedback systems with structured uncertainties. IEE Proc., 129, Part D(6), 242 250 8. Doyle, J. C., Packard, A., et al. (1991) Review of LFTs, LMIs, and #. Proc. IEEE Conf. Decision Contr., 1227-1232 9. Feron, E., Apkarian, P., et al. (1996) Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions. IEEE Trans. Automat. Contr., 41-7, 1041-1046 10. Gahinet, P., Apkarian, P., et al. (1996) Affine parameter-dependent Lyapunov functions and real parametric uncertainty. IEEE Trans. Auto. Contr., 41-3, 436-442 11. Geromel, J. C., Peres, P. L. D., et al. (1991) On a convex parameter space method for linear control design of uncertain systems. SIAM J. Contr. Opt., 29-2, 381-402 12. E1 Ghaoui, L., Niculescu, S.-I., editors. (2000) Advances in Linear Matrix Inequality Methods in Control. SIAM Advances in Design and Control 13. Haddad, W. M., Bernstein, D. S. (1991) Parameter-dependent Lyapunov functions, constant real parameter uncertainty, and the Popov criterion in robust analysis and synthesis. Proc. IEEE Conf. Decision Contr., 2274-2279, 26322633 14. Haddad, W. M., Bernstein, D. S. (1994) Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. Part II: Discrete-time theory. Int. J. Robust Nonlin. Contr., 4, 249-265 15. Hindi, H., Boyd, S. (1998) Analysis of linear systems with saturation using convex optimization. Proc. IEEE Conf. Decision Contr., 903--908

174

Tetsuya Iwasaki

16. Isidori, A. (1989) Nonlinear Control Systems. Springer-Verlag 17. Iwasaki, T., Hara, S. (1998) Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations. IEEE Trans. Auto. Contr., 43-5, 619-630 18. Iwasaki, T., Meinsma, G., et al. (2000) Generalized S-procedure and finite frequency KYP lemma. Mathematical Problems in Engineering, 6, 305-320 19. Iwasaki, T., Shibata, G. (1998) LPV system analysis via quadratic separator for uncertain implicit systems. Submitted for publication. 20. Iwasaki, T., Shibata, G. (1999) LPV system analysis via quadratic separator for uncertain implicit systems. Proc. IEEE Conf. Decision Contr., 287-292 21. Kokotovid, P. V. (1992) The joy of feedback: nonlinear and adaptive. IEEE Control Systems, 12, 7-17 22. Leitmann, G. (1979) Guaranteed asymptotic stability for some linear systems with bounded uncertainties. J. Dyn. Sys., Meas. Contr., 101, 202-216 23. Liu, D., Michel, A. (1994) Dynamical Systems with Saturation Nonlinearities: Analysis and Design. volume 195, Lecture Notes in Control and Information Sciences, Springer-Verlag 24. Lur'e, A. I. (1957) Some Nonlinear Problems in the Theory of Automatic Control. H. M. Stationery Off. 25. Megretski, A., Rantzer, A. (1997) System analysis via integral quadratic constraints. IEEE Trans. Auto. Contr., 42-6, 819-830 26. Nesterov, Yu, Nemirovsky, A. (1994) Interior-point Polynomial Methods in Convex Programming. SIAM Studies in Applied Mathematics 27. Packard, A., Doyle, J. (1993) The complex structured singular value. Automatica, 29-1, 71-109 28. Pittet, C., Tarbouriech, S., et al. (1997) Stability regions for linear systems with saturating controls via circle and Popov criteria. Proc. IEEE Conf. Decision Contr., 4518-4523 29. Rantzer, A. (1996) On the Kalman-Yakubovich-Popov lemma. Sys. Contr. Lett., 28-1, 7-10 30. Saberi, A., Lin, Z., et al. (1996) Control of linear systems with saturating actuators. IEEE Trans. Auto. Contr., 41-3, 368-378 31. Safonov, M. G., Athans, M. (1981) A multiloop generalization of the circle criterion for stability margin analysis. IEEE Trans. Auto. Contr., 26-2, 415422 32. Scherer, C., Gahinet, P., et al. (1997) Multiobjective output-feedback control via LMI optimization. IEEE Trans. Auto. Contr., 42-7, 896-911 33. Trofino, A., de Souza, C. E. (1999) Bi-quadratic stability of uncertain linear systems. Proc. IEEE Conf. Decision Contr. 34. Yakubovi6, V. A. (1971) S-procedure in nonlinear control theory. Vestnik Leningrad Univ., 1, 62-77 35. Zames, G. (1966) On the input-output stability of time-varying nonlinear feedback systems, Part I: Conditions using concepts of loop gain, conicity, and positivity. IEEE Trans. Auto. Contr., 11,228-238

11 Towards Online C o m p u t a t i o n of Information State Controllers M.R. James Department of Engineering, Australian National University, Canberra, A C T 0200, Australia, [email protected]

A b s t r a c t . Implementation of H ~ controllers for nonlinear systems requires the solution of a P D E online for the information state (which forms the controller state). This paper will describe two threads of research relating to online computation: (i) numerical techniques using max-plus expansions, and (ii) the "cheap sensor" problem which has dramatically less complexity t h a n the regular case.

11.1

Introduction

At the core of nonlinear H ~ control theory lie two partial differential equations (PDE). One is a first order evolution equation called the information state equation which plays the role of a generalized state estimator and in fact constitutes the dynamics of the controller. It must be implemented online. The second P D E is defined on the space of possible information states and determines the o u t p u t law for the controller. This can be implemented offline. For further details, the reader is referred to the books [2], [7], [12] and the many references cited therein. This paper discusses only the information state partial differential equation. The calculation of the information state is in general a computationally expensive task, requiring the solution of the information state equation in n dimensional space. The storage requirements and number of operations needed grow exponentially with n. This of course has negative practical implications, well-known as the "curse of dimensionality". Nevertheless, it is worthwhile developing effective computational methods for low dimensional problems, or for problems where the dimensionality can be reduced. In section 11.3 we exploit the underlying max-plus linearity of the information state evolution operator to obtain a numerical scheme implemented as max-plus matrixvector "multiplication", where as much of the computation is done offline (via LieTrotter type splitting). Then in section 11.4 we discuss briefly the so-called "cheap sensor" problem, where many states are assumed perfectly measured; this results in a reduced order information state evolution which in a wide range of applications is computationally tractable with current (2000) technology. In section 11.5 we make some additional comments. Before we begin, section 11.1 briefly reviews elements of the information state theory.

176

M.R. James

11.2

Nonlinear

H ~ Control and Information

States

We will now review the nonlinear H ~ controller synthesis theory developed in [9], [7] for the class of nonlinear systems or (regular) generalized plants:

= A(x) + Bl(x)w + B2(x)u

x(O) = xo

ill.l)

D12u

D~2012 = Im

(11.2)

y = Ca(x) + D21w

D21D~21 = Iq.

(11.3)

Z = el(x)

-~

In these equations x(t) C R '~ denotes the state of the system and y(t) E R q is the measurement signal. The output to be regulated is z(t) E R v. The control input is u(t) E R m, while w(t) C R l is an exogenous disturbance input. We assume that all the problem data are smooth functions of x with bounded first derivatives, that B1 and B2 are bounded, and that the origin is an equilibrium state: A(0) = 0, C1 (0) = 0 and Ca (0) = 0. In brief, the generalized plant is:

C1[ O

2

C2[D21 i)01

"

The (output feedback) controller K is assumed to be a causal mapping y C y H u E/4, where/g and y are the relevant signal spaces for outputs and inputs respectively, locally square integrable L2,loc. Such a controller is called admissible if the closed-loop equations associated with K and (11.1) are well defined in the sense that they have unique solutions in L2,loc. A controller K is said to solve the H ~ control problem provided the closed-loop system is 3,-dissipative and internally stable. The closed-loop system is 7-dissipative if

1/0T

1/0T

[z(s)[2ds <_ .y2

Iw(s)12ds +/3(x(0))

(11.4)

for some non-negative function /3 with /3(0) -- 0 and every w E L2[0, T], for all T >_ O. Internal stability means that if w C L2[0, oo) then u ( . ) , y ( . ) , z ( . ) , x ( . ) E L2[0, oo), and consequently x(t) ~ 0 as t -+ oo. To solve this problem the output feedback problem is transformed to a new state feedback problem using the information state pt(x) = p(x, t) defined by the equation

0 = -Vp(A+B1D'21(y ~-~p

C2) -{- B2u)

+7-2IVRBI(I-- D21D 2 1 ' ) B IpV'

(11.5) ,

(11.6)

21 1 - ' ~ ~[y - C2I 2 + ~[C1 + D12u[ 2.

(11.7)

In shorthand notation, we write pt(') = p(', t) and regard pt as the state of a new dynamical system with state equations

[9 ---- F ( p , u , y),

(11.8)

11

Towards Online Computation of Information State Controllers

177

where F(p,u, y) is the nonlinear differential operator defined in (11.7). The state space is an appropriate function space 9:' (e.g. the Banach space of continuous px functions with at most quadratic growth with the norm II P IIx----sup~eRn ~ ) . It is known (see [7, Section 3.1]) that if a controller K : y ~-~ u exists such that the closed-loop is 7-dissipative, then (11.7) has a solution. If the solution is not smooth, it can be interpreted in integrated form [7, Section 3.1, eq. (3.7)]), or perhaps in the viscosity sense, [1], [4]. The transformation afforded by the information state gives rise to a nonlinear PDE on an infinite dimensional (Banach) space 2d, viz.

inf

sup

{VW(p)[F(p,u,y)]}

-- 0

(11.9)

u E R m yE RP

where •W(p) is a linear operator (Frechet derivative). It is known that there exists a value function W(p) solving this equation in a suitable sense, [9], [10], [7]. In general it cannot be expected that W is smooth, and in fact at present there is no adequate theory for PDEs of the type (11.9); however, it can be interpreted in an integrated form, and various notions of smoothness and solution have been considered, [10], [7]. We assume smoothness to facilitate system-theoretic interpretation, and remark that smoothness issues do not arise in discrete time [9]. For full details, see [10], [7]. Here, it is sufficient to note that a smooth solution of (11.9) defines an output feedback controller K* via the optimal feedback function obtained by evaluating the infimum in (11.9):

u*(p) = VW(p)[-D~12C1 + B~2Vp'].

(11.10)

The optimal information state controller is given by

K* :

( Pu-- F(p,u,y), u*(p).

(11.11)

This controller feeds back the information state, suitably initialized, and produces a 7-dissipative closed loop. For precise statements and stability results, see

[7] In the sequel we focus on the online computation of the controller dynamics, the PDE (11.7).

11.3

Max-Plus A p p r o x i m a t i o n

Max-plus linearity refers to the commutative semi-field on R defined by aGb=max{a,b}

anda|

The transition or evolution operator (propagator, semigroup) for the information state is linear with respect to this semi-field. This can be used to advantage in formulating numerical schemes. Here we describe the approach of [5]; the results

178

M.R. J a m e s

q u o t e d are valid under the detailed hypotheses given there, and we refer t h e r e a d e r to this p a p e r for these details and proofs. Let ti ----iv for i E N . Henceforth in this section we work w i t h piecewise c o n s t a n t control u and discrete-time observations observations, so t h a t u(t) = u ~

y(t)=y,

for t ~ [ i v , ( i + l ) ~ ) .

Let /4 and y denote t h e spaces of such control processes and m e a s u r e m e n t sequences. In t h e sequel we assume T = NT. N o t e t h a t t h e n y is j u s t t h e space of sequences of length N taking values in R q. T h e values of u are in U C R m. T h e information state pt solves t h e P D E (11.7). In t e r m s of t h e e v o l u t i o n operator, the information state evolves as [pt~+l](x) ----St~'t~+l(pt,,u,y)(x) ~ sup {pt~(~)~-~ti,tiq_l(~,Z,U,y)}

(11.12)

~ER n

where

B t , t , + l ( e , x , u , y ) ~ sup { J ( t , t + T , ~ , u , w ) } wE kVx

where 142~ is t h e set of w C L2(ti,ti+l) for which

x(ti) = ~,

x(ti+l) = x and y = C2(x) + D21(x)w(ti).

Max-plus linearity m e a n s t h a t for p E X, c C R s ~''~'+1 (c | p) = c | s ~''~'+1 (p) and for pl, p2 E A'

St~,t~+, (pl @ p2) = S t~'t~+~ (pl) @ S t~'t~+* (p2). These expressions are to be interpreted relative to t h e p variable, i.e. p ~-~ Stl,ti+l (p, u, y). We assume there exists an equilibrium i n f o r m a t i o n state p~ such t h a t

p ~ ( . ) _< - c ~ , l . I ~ + c ~ where c~1 > 0 and c~2 >_ 0. Here the t e r m e q u i l i b r i u m m e a n s S t~'t~+l (p~, 0, 0) = p~. We will work on the space 7) = { p

: for a l l a > 0 , p ( x )

<_p~(x)+alxl 2+1},

invariant under t h e action of the evolution operator. Suppose we wish to a p p r o x i m a t e t h e p's in a c o m p a c t region Ix[ _< R d e t e r m i n e d by a radius R > 0. We assume there exists R1 > 0 such t h a t if Ix[ _< R1, t h e n t h e maximizers ~ in (11.12) satisfy [~[ ~ R3, for some R3 > 0, as described in [5]. So we will need to c o n s t r u c t our a p p r o x i m a t i o n s on a ball of radius R = R3. Let

pz~ ---- { ~ ) ? } j e j A ,

11

Towards Online Computation of Information State Controllers

179

be a finite collection of elements of 7), which we think of as "basis" vectors. Write ]JA I for the cardinality of j A . Define the max-plus span of 7)A by span~ A--{pET)

: p---- ~ jEJ

a?|162 A

where a ~ E R U { - c r This span is a finite dimensional max-plus linear subspace of 7). For any p E 7), a projection of p onto span 7~A will be denoted by r A p . Also note that pA E span7 ~a is characterized by the vector ( a ~ , . . . , a ~ A i ) E R IJ'~l (such vectors need not be unique). D e f i n i t i o n 1. A set T~A is called a monotone approximate basis with projection r a for a subset 7)0 C ~D provided: 1. For all p E :Do, then r A p E :Do and

7rAp(x) <_p(x) V x E R '~.

(11.13)

2. Given any ~ > 0, there exists ,5 > 0 such t h a t

p(x)--e<_rAp(x)

V I x I _ < R and all p E T ) 0 .

(11.14)

A natural choice for the projection operator is:

(rAp)(x) = m a x { a ? + r jEJ

,a

= ~ jEJ

a~|162 A

where

a? ---- m ax{r 2 (x) - p(x) }.

(11.15)

-

We note here that the geometric visualization is t h a t each a ~ is chosen so t h a t a ~ + CA ( x ) just "touches" p(x) at some point, and is no greater t h a n p(x) everywhere else. Indeed, we have: L e m m a 1. Let r A be the projection operator defined by (11.15) for p E 7). Then

1. p E 7) implies (11.13), and in fact r a p E 7). 2. If p E spanP A, then r a p = p. Hence ( r A ) 2 = r A. 3. The projection r A enjoys the following max-plus sublinearity properties: (a) For p E T) and c E R, 7rA ( c |

= c@rA(p).

(b) For pl,p2 E 7), r A ( p l ) O rA(p2) < r A ( p l Op2).

(c) For pl , p2 E 7),

rA(pl e p2)(x)- ~ < - A ( p l ) ( x ) 9 ~A(p2)(x),

Ixl _< R3.

180

M.R. James

Henceforth we use the projection defined by (11.15), and observe that it specifies a choice of "coordinates". We mention t h a t a monotone approximate basis can be constructed using quadratic functions. These quadratics span a subspace 7)0 of semiconcave functions. One of the key points of this approach is that the max-plus basis approximations of the information state reduce the propagation of the information state forward in time to max-plus matrix-vector multiplication. See [3] for more details, but the idea is as follows. We define our approximation pt~ to the information state pc, 0 < t < T as follows. Set po~ -- 7r,spo and for 0 < i < N ---- [T/T], ,5

Pti+l

:

G i r ,5 U I,P t ~ , i , y i + l ) ,

(11.16)

and P~ = Pt~ if t E [ti, t,+l). Here, the propagation operator G ~ is defined in terms of a monotone approximate basis p A by Gk,3(u,, Y~+I) | r

(11.17)

k E j z~

In terms of coefficients

p,=@

ati,5,j

|162

j E J z~

the operator G i acts by max-plus matrix-vector multiplication ,5 @ at~+l,k =

i ,5 Gk,j(u~,y~+l) | at~,j

~/k E

j,5

(11.18)

j E J "5

where the dimension of the vectors is J,5. The validity of this approach is somewhat dependent on an analysis of the errors it introduces. We now summarize an initial study of these errors. Lemma2.

For any u E l4, y E y , po E T) we have

,5

n

Pt~(x) < p t , ( x )

V x ER ,

(11.19)

and all i >_ O. Hence p ~ E 7). The next result requires careful choice of the discretization parameters ~- and ,5. Lemma3. Let po E 7), r > O, u E ld and y E Y with ]uil <_ R2, lYiI <- R2, 0 < i < N . Given ~ > 0 there exists z > 0 and Al > 0 such that ,5

and all 0 < i < N .

v Ixl < R3,

(11.20)

11

Towards Online Computation of Information State Controllers

181

Since the matrix Gi(ui, y~+l) depends on time via the signals u and y, it must be recomputed at each time step. Ways to reduce the complexity of this task are discussed in [5]. In particular, Lie-Trotter type splitting methods are used to separate homogeneous and inhomogeneous terms:

(11.21)

Sti ,t~+l (pt~, ui, yi+l ) ~" U2 (7",ui, yi+z)U1 (T)pti

The principal term U1 (T) can be calculated of[line, while the computationally simpler term U2(T, ui, Y~+I) must be calculated online. Figure 11.1 shows the result of a max-plus computation.

O -.M

-1~kO -2W -2M 4

4

-4

Fig. 11.1. Example information state computed using the max-plus method.

Finally, we remark that the controller output can be determined via

U*Z~t z~\

(11.22)

182

M.R. J a m e s

by (in principle) discretizing t h e control e q u a t i o n (11.9), or by some o t h e r app r o x i m a t e means.

11.4

T h e C h e a p Sensor P r o b l e m

T h e cheap sensor case has been studied for linear systems in [11] ( w i t h o u t using this terminology), where reduced-order H ~ c o m p e n s a t o r s were obtained. In this section we s t u d y a nonlinear analog of [11] in t h e c o n t e x t of t h e i n f o r m a t i o n s t a t e framework [7]. Specifically, we consider t h e non-regular generalized plant s t r u c t u r e

:

A

(/}1, B1)I B2

C1

I (0'0)ID12 00 0

(11.23) '

w i t h D12 full rank, where the s t a t e is d e c o m p o s e d as x ---- (x~, xt,), w i t h x~ measured perfectly and x~ imperfectly measured. T h e state feedback case corresponds to x -- x~, while in the full regular case (as in sections 11.2 and 11.3) x ---- x , . In s t a n d a r d notation,

D21 =

]

iu

C2u(x) .

(11.24)

T h u s the state space R n will have a d e c o m p o s i t i o n as a direct s u m of space X~ of directly measured states and X~ some subspace c o m p l e m e n t a r y to X~. Let P~ denote t h e projection onto X~ along Xt, , and let Pt, d e n o t e the p r o j e c t i o n o n t o Xu along X~. T h i n k of t h e state of G d e c o m p o s e d as x = (x~,xu), w i t h o u t p u t y -- (x~,y~) where y~ ---- x~ -- P~x are the perfectly m e a s u r e d states and y~, are imperfect m e a s u r e m e n t s of the plant state x. T h e d i s t u r b a n c e / c o m m a n d i n p u t is denoted (v, w), where the dimension of w equals t h a t of y~,. T h e plant G w i t h cheap sensor s t r u c t u r e (11.23) is given explicitly by

4= A(~)+Bl(~)w+[~(~)v+B2(~)u G:

z = C~(~)+ D~2([)u Y=

y.

(11.25)

[2,1 [o] c

~)

+

and we often write y as

y = C2(~) + D21 I v ] . In conventional n o t a t i o n t h e plant s t a t e e q u a t i o n is

(11.26)

11

Towards Online Computation of Information State Controllers

183

Here, ~(t) E R '~, y(t) E R p, z(t) e R r, u(t) E R m, while v(t) C R q, w(t) E R s. The input w includes sensor noise, while v does not (since v is not directly sensed). Information state methods can be applied to the cheap sensor case even though the matrix D21 is not full rank. Recall that the information state p is in general a function p : R" ~ R. In the cheap sensor case, this function can be replaced by the pair (y~,/5), where /5 : X~ --+ R. So the states of the reduced information state controller are the functions/5 : X , -o R with dynamics a cut down information state P D E of the form I5 ----/5(/5,u, y),

(11.27)

with suitable initialization. This is a P D E on the lower dimensional space X~,. In the case d i m X ~ _~ q the cut down information state P D E (equation (11.27) above) is as follows:

~

=

f

I - V . / S t P t , [ A + B l ( y . ( t ) - C2.) + B l v + B2u(t)] vEVt(xu)k sup

(11.28)

-~-1[C1 -~- Di2u(t)[ 2 - ~2[H2 + [C2~ - y~(t)12]} where 12t(xt,) is given by

V t ( x . ) - { v e (P~B1)-l[9~ - P~(A + Bl(yt,(t) - C2.) + B2u(t))]}.

(11.29)

Here (P~/~I) -1 is the set-valued inverse ((P~B1)-lb ---- {a : P~Bla = b}). Detailed analysis of the computational complexity of the P D E (11.29) P D E is given in [8], where a computational scheme is described in terms of the moving grid of Figure 11.2 and a finite difference scheme. The outcome is that there is a huge complexity reduction in the case dim X~ < < n because Gdim~x , < < < < G~. This is extremely important because often dim X~ < < dim R n ----n, and so computation of the information state becomes feasible in such cases. For example, for a plant with 6 states of which 4 are perfectly measured and 2 imperfectly that with Gx ---=50 grid points per dimension, the computational complexity of updateing the reduced information state P D E at each time step is 0 ( 6 • 502) for memory and O(107) Floating Point Operations (FLOPS). This is significantly less complex t h a n a direct solution of the full information state P D E in 6 dimensions, which would require 0 ( 6 • 506) memory elements and O(1012) FLOPS per time step, which is of course infeasible. The controller signal is given by u(t) = u* (y~(t),/st) where u* is determined in principle by a PDE of the type (11.9), or in practice using certainty equivalence. We remark that in the case of linear systems the control signal is given by

184

M.R. James

where H is a feedback matrix and xt, is the o u t p u t of a filter of the form

~ , = F.Y~t, + Gt.

Y. Y~

,

(11.30)

for suitable matrices Fu, G u.

Xtt

...............

(y~(t - a ) , ~ . ( t - a ) )

D. . . . . . . . . . . . . . .

, (y~(t - A), x . )

(~(t),x,)

9 Xtr

y~(t) + x .

y~(t - a ) + x .

F i g . 11.2. Location of grid G~. C Xu at times t and t - - A showing points (y~(t), xu) and (y~(t-A), x.) (solid dots), nearest neighbors .hf(xt, ) (empty dots), and (y~ ( t z~), ~ ( t - A ) )

11.5

Discussion

We would like to conclude that solving for the information state in real time is not as bad computationally as previously thought, at least for engineering problems where good knowledge of most, but not necessarily all, states is available. A choice of numerical methods is available, including the max-plus approximate basis technique described in section 11.3 which is well-matched to the mathematical structure of

11

Towards Online Computation of Information State Controllers

185

the information state dynamics, and versions of finite difference schemes as alluded to in section 11.4. Development of such numerical methods is the subject of current research. Another important issue not discussed in detail here is t h a t of determining the control output function u* (p). In the context of certainty equivalence, the receding horizon methods show some promise [6].

References 1. M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations", Birkhauser, Boston, 1997. 2. T. Basar and P. Bernhard, H a Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, Birkhuaser, Boston, 1995. 3. W.H. Fleming and W.M. McEneaney. A m a x - p l u s based algorithm for an HJB equation of nonlinear filtering. S I A M J. Control and Optim, 1998. 4. W.H. Fleming and H.M. Soner, "Controlled Markov Process and Viscosity Solutions", Springer-Vertag, New York, 1993. 5. E. Gallestey, M.R. James, and W.M. McEneaney, Max-Plus Approximation Methods in Partially Observed H ~ Control, in preparation, 2000. 6. E. Gallestey and M.R. James, The Receding horizon Approach to H ~ Control of Nonlinear Systems, submitted to Systems and Control Letters, August 2000. 7. J.W. Helton and M.R. James, "Extending H ~ Control to Nonlinear Systems: Control of Systems to Achieve Performance Objectives", Advances in Design and Control, SIAM, 1999. 8. J.W. Helton, M.R. James, and W.M. McEneaney, Measurement Feedback Nonlinear H ~ Control: The Cheap Sensor Case, s u b m i t t e d to IEEE TAC, 2000. 9. M.R. James and J.S. Baras, Robust H ~ Output Feedback Control for Nonlinear Systems, I E E E TAC 40, 1995, 1007-1017. 10. M.R. James and J.S. Baras, Partially Observed Differential Games, Infinite Dimensional HJB Equations, and Nonlinear H ~ Control, SIAM J. Control Optim., 34, 1996, 1342-1364. 11. A.A. Stoorvogel, A. Saberi and B.M. Chen, I E E E TAC,vol 39, Feb 1994, p p 355-360. 12. A. van der Schaft, "L~-Gain and Passivity Techniques in Nonlinear Control", Springer Verlag, New York, 1996.

12 T i m e D o m a i n I n t e g r a l s for Linear Sampled Data Control Systems R.H. Middleton 1, K. Law 1, and J.S. Freudenberg 2 1 Dept. of Electrical and Computer Engineering, The University of Newcastle, Callaghan 2308 NSW, A U S T R A L I A {rick,eekl}@ee.newcastle.edu.au 2 Dept. of Electrical and Computer Science, 4213 EECS Building, University of Michigan, 1301 B e l l Avenue, Ann Arbor, MI 48109-2122, USA [email protected] This author is supported by NSF Grant ECS-9810242

A b s t r a c t , It is well known t h a t linear time invariant control of plants with unstable poles results in time domain integral constraints on the closed loop system. This leads to the question of whether nonlinear or time varying control can be used to ameliorate these constraints. In this p a p e r a particular class of time varying controllers, namely sampled d a t a feedback controllers, is studied. A n integral constraint, analogous to one for continuous time systems, is derived for sampled d a t a systems with an unstable plant pole and an o u t p u t disturbance. Analysis of this constraint in several examples shows t h a t the performance limitation is often worse than in the continuous time case.

12.1

Introduction

It has long been recognised t h a t unstable plant poles and non minimum phase zeros, together with the normal requirement of internal closed loop stability, imply interpolation constraints which a feedback system must satisfy. Some of these interpolation constraints can be expressed as time domain integral constraints, see for example [5], [6], [7]. For example, consider the analog feedback control system shown in Figure 12.1 where r (t), e (t), y (t), u (t), d~ (t) and do (t) denote the reference signal, error signal, plant output, plant input, input disturbance and o u t p u t disturbance, respectively. The upper case is used for the Laplace transforms of signals. We then have the following result:

ld,(t) F i g . 12.1. Linear Analog Feedback Control Loop

.[do(t)

188

Middleton, Lau, Freudenberg

Proposition 1. (i) Suppose that the plant transfer function P (s) has a zero in the closed right half plane (CRHP) (also referred to as a non m i n i m u m phase ( N M P ) zero) at s : ~. Then for any internally stabilising linear time invariant (LTI) controller: W (r = 0 Y (r = Do (r E(r

= R(r

(12.1) - Do (r

U (~) -- C (r (R (r - Do (r (ii) Suppose that the plant transfer function has a pole in the C R H P at s = p. Then for any internally stabilising L T I controller: V (p) = 0 U (p) = - D i (p)

(12.2)

E (p) = - C -1 (p) D~ (p) Y (p) = R (p) + C -1 (p) Di (p) Proposition 1 gives rise to the following important corollary: C o r o l l a r y 1. (i) Consider any internally stable analog feedback loop as shown in Figure 12.1 with no input disturbance, no reference signal, and where the output disturbance is a unit step. Then for any plant N M P zero at s = ~ it follows that ~0 ~ e-~ty (t) dt = 1

(12.3)

(ii) Consider any internally stable analog feedback loop as shown in Figure 12.1 with no input disturbance, no reference signal, and where the output disturbance is a unit step. Then for any C R H P plant pole at s -- p it follows that o~ e-Pry (t) dt = 0

(12.4)

The results of Corollary 1 can be interpreted in a number of ways relating to the time domain performance of the feedback loop [5]. For example, it can be seen that if the CRHP p in (12.4) is real, then the response of y(t) to a unit step output disturbance will undershoot. 1 This is shown qualitatively in Figure 12.2. In fact, (12.4) can be interpreted as a weighted equal area criteria, and limits the possible output responses achievable whilst retaining stability. One question that arises when considering these performance limitations is to ask whether the constraints of (12.3) and (12.4) are inherent to the plant, or whether they depend, for example, on the type of feedback control law employed. In this context, we note t h a t the derivation of (12.3) does not depend in any way on linearity of the controller, instead, it depends entirely on the plant zero definition, closed loop stability, and the structure of the o u t p u t disturbance addition. On the other hand, 1 Note that "undershoot" here denotes t h a t there exists t such t h a t y(t) < 0. For the o u t p u t disturbance response, this is equivalent to "over correction" which for reference steps, gives "overshoot" as in [5].

12

Time Domain Integrals for Sampled Data Control

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 12.2. Sketch of Typical Continuous Time Disturbance Step Response.

(12.4) does indeed depend on linearity and time invariance of the controller. This latter fact can be seen by noting that to derive (12.4), we first use that for any stabilising control signal (independent of linearity), U (p) = 0. For the case of a LTI controller, E (s) = C -1 (s) U (s) and by the requirement for internal stability, we obtain Y (p) = 0. The fact that this derivation depends on linear time invariance of the controller has been used to advantage for example in [2], where improved performance is obtained for a linear integrating plant, using a switching control strategy. Further analysis and generalisation of these types of schemes is pursued in [1] and [4]. Given that, at least to some extent, nonlinear controls circumvent the LTI constraint (12.4), one natural question is to ask to what extent linear time varying controls remove this constraint. A particularly interesting and practically important class of these time varying controls is sampled data control systems [3]. Note however, that from a frequency domain viewpoint, sampled data control can be seen to inherit, and to some extent exacerbate, inherent limitations for LTI control [3]. In this paper we wish to consider the equivalent question for time domain performance limitations.

12.2

Sampled Data System Background

Following [3], consider a sampled data feedback control system as shown in Figure 12.3 where P (s) and F (s) represent the plant and anti-aliasing filter (AAF) transfer functions, respectively; and Cd ( es T) and H i s) represent the controller transfer function and hold response function2, respectively. If we consider only the sampled data response, then the system becomes linear time invariant, and the transfer function relating yk to uk with no disturbances present, is the discretised plant transfer function:

(FPH)d(e aT) = ~1 ~

F(s+jkcv~)P(s+jkws)H(s+jk~v~)

(12.5)

2 We include here the possibility of using holds other than a zero order hold.

190

Middleton, Lau, Freudenberg

~ r~

Hold (T)

,d,(t) J.

, do(t) J.

(t)

]~

Yk

Sampler (T)

AAF

F i g . 12.3. General Sampled D a t a Control block diagram.

where w~ = -~. The condition of internal stability, including the assumption of nonpathological sampling [3, Lemma 5] of the sampled d a t a system, implies t h a t if s ----p is an unstable plant pole, then z = e p T is also an unstable pole of the discretised plant, (FPH)d (z). Noting this, and t h a t the closed loop system is linear and time invariant when considered as a digital control problem, then we can establish t h a t with no input disturbance or reference, any stabilizing control gives: oo

Z . , e-kpTAyk = 0

(12.6)

k=O Whilst (12.6) resembles (12.4) it clearly ignores intersample behaviour. Our main aim in this paper is to consider the problem of finding integral expressions, analogous to (12.4) in the sampled d a t a case. In the development of these expressions, we will make use of the following definition: D e f i n i t i o n 1. The Discrete Time Sensitivity, Sd (Z) and the Fundamental Sensitivity function, Tf~,n (s) for the sampled d a t a system of Figure 12.3 are defined by:

sd (z) =

1

1 + Cd (z) (FPH)d (z)

T I ~ (s) = 1 p ( s ) H ( s ) C d ( e ~ T ) S d ( e ~ T ) F ( s )

(12.7) (12.8)

We are interested in the complete o u t p u t response, y(t), to an o u t p u t disturbance, do (t), as given in the following result: L e m m a 1. For the discrete time system of Figure 12.3, the output response with

zero reference and input disturbance is given in Laplace transforms by: Y (s) = Do (s) - Tf~,,~ (s) F -1 (s) (FD)u (e 8T) T

(12.9)

where (FD)d (e ST) = 1 ~--~k=-oo k=~ F (s + jkws) Do (s + jkws) is the frequency response of the sampled filtered disturbance.

12

Time Domain Integrals for Sampled Data Control

191

Proof. From [3, (23)] we have (with the appropriate notational changes)

Y (s)= Do (s) T = Do (s) xT

F (s + jkw~) Do (s + j k w , )

]

(12.9) follows directly from (12.10) and the definitions of T i ~

(12.10)

(12.8) and (FD)d.

With this background, we are now in a position to study sampled data time domain integrals.

12.3

Sampled Data Time Domain Integral

T h e o r e m 1. Consider any internally stable sampled data feedback loop as in Fig-

ure 12.3, where the reference and input disturbances are zero. Suppose that the plant has a right half plane pole at s = p. Then:

foo~ e-'ty (t)dt = Do (p) - TF -1 (p) (FD)a (e pT)

(12.11)

Proof. This result follows on substituting s = p in (12.9) and noting [3, Theorem 2] that Tf,,,~ (p) ----1. Note that (12.11) gives a time domain integral constraint on the output, which depends only on the disturbance and the anti-aliasing filter properties. In particular, note that the constraint is independent of the controller, save for the requirement that the controller be internally stabilising.

12.3.1

Interpretation and D i s c u s s i o n

Firstly, note that under fairly mild conditions on the anti-aliasing filter and output disturbance, the right hand side of (12.11) approaches zero as T --* 0 and therefore, as expected, we recover the analog control result (12.4). This can be seen by rewriting (12.11) as:

= F -1 (p)

dl (t) e-ptdt - T ~

d] (kT) e -pkT

(12.12)

k=0

where df (t) represents the signal formed by filtering, via the anti-aliasing filter F (s) the output disturbance, do (t). Provided this filtered disturbance has sufficient regularity such that the Riemann sum in (12.12) converges to the integral as T ~ 0, then Y (p) ~ 0.

192

Middleton, Lau, Freudenberg

We shall be frequently interested in the o u t p u t behaviour in response to a unit step output disturbance. Since we consider this disturbance to be an unknown disturbance, we cannot assume t h a t the disturbance arrives at a time which is synchronised with the sampling instants. In particular, we consider the o u t p u t disturbance to be the delayed unit step:

where without loss of generality the delay A is taken to be in the range (0, T). For a causal sampled d a t a feedback system, in response to the o u t p u t disturbance (12.13), the control signal must be zero until at least t ---- T. Therefore, the output behaviour in response to an o u t p u t disturbance should be qualitatively similar to t h a t illustrated in Figure 12.4. In particular, we have: y(t)=

{0:0
(12.14)

do(t)

I A

T

2

i~ r ~

I 4

" T

~

F i g . 12.4. Sketch of Typical Sampled D a t a O u t p u t Disturbance Step Response. Note that the constraint (12.14) is independent of the plant, and is purely dictated by the sampled d a t a control. Furthermore, this initial portion of the response does not affect arguments about overcorrection due to an open loop unstable pole. We shall therefore consider the following integral constraint: C o r o l l a r y 2. Consider any internally stable sampled data feedback loop as in Figure 12.3, where the reference and input disturbances are zero. Suppose that the plant has a right half plane pole at s = p, and that the output disturbance is a delayed step, as in (12.13). Then:

fT~e-Vty(t)dt=le-VT--TF-X(p)(FD)d(e

(12.15)

Proof. This result follows by considering: e-Pry (t) dt =

e-Pry (t) dt -

e-pry (t) dt

(12.16)

12

Time Domain Integrals for Sampled Data Control

193

and using the substitutions (12.11) and (12.14) together with the fact that Do (s) = e- s,~ 8

Note, for example, that for a real unstable open loop pole, if the right hand side of (12.15) is non-positive, then necessarily the output response to a delayed o u t p u t disturbance will overcorrect, in the sense that y (t) must be negative at some point. However, it is not clear at this point, what values the right hand side of (12.15) may take. If the right hand side is positive, then in some sense, it would appear that the use of sampled data control has ameliorated the time domain integral constraint which exists for a continuous time control system. On the other hand, if the right hand side is negative, then the use of sampled data control has exacerbated this time domain integral constraint. The interpretation of this situation is complicated by the presence of the anti-aliasing filter, and the possible delay, ,5. Although a general statement about the right hand side of (12.15) is not apparent, several simplified cases can be analysed.

12.3.2

N o Anti-Aliasing Filter

Consider the case where we have no anti-aliasing filter, that is where F (s) ---- 1. We then have the following corollary: C o r o l l a r y 3. Consider a system with conditions as in Corollary 2, and in addition, where F (s) = 1. Then: ~e_Pty

/T

(t) dt

(1 -- e -Tp -- T p ) p (e Tp - 1)

(12.17)

Furthermore, for p, T real and positive, f T e-PrY (t ) dt < O. Proof. Since the filter has transfer function equal to the identity, the filtered disturbance is precisely the disturbance. Therefore the transform of the sampled filtered disturbance is: (FD)d(e~T) =

1 e ~T -- 1

(12.18)

Evaluating (12.18) at s = p and substituting in (12.15) gives:

f

~ e - P ~ y (t) dt = l e - ~ P - T

1--L--e pT -- 1

p

(1 -- e - Tp T p ) p (e Tp -- 1) _ _

=: l f o ( p T ) (12.19) P as required. Note further that this function is negative f o r any positive real p, T . This can be seen from the inequality e -~ > 1 - x or f r o m the graph of fo which is shown in Figure 12.5. The result then follows. It therefore follows, that without an anti-aliasing filter, viewed from the perspective of the integral (12.17) over times during which the controller may take action, the sampled data controller has a 'worse' time domain integral constraint t h a n a purely analog controller. We next show that a similar result holds for an 'averager' (also called an 'integrate and reset') anti-aliasing filter.

194

Middleton, Lau, Freudenberg

0

!

k

-0"05t~

.... :........

I

i............ i

I \ -0,1

................................. :[-2

-0.15

-0.2

....

-025 0

0.5

1

pT

:. . . .

1.5

i ..............

2

2.5

Fig. 12.5. Plot of fo (pT) versus p T . 12.3.3

Integrate and Reset Anti-Aliasing Filter

Although not usually implemented in analog hardware, except perhaps implicitly if an integrating type analog to digital converter is used, an Integrate and Reset filter is a useful idealisation of an anti-aliasing filter. We define the behaviour of this filter by the time domain equation:

If/

y s (t) = T

-T

(12.20)

y (t) dt

or equivalently by the transfer function F(s)--

(1-e -sT ) sT

(12.21)

We then have the following Corollary: C o r o l l a r y 4. Consider a system with conditions as in Corollary 2, and in addition, where F (s) = (1--e-sT) Then: sT

e-'tY(t)dt

= p

e - ' T --

-~T----I~

Proof. Because of the nature of the filter, the filtered response to a step disturbance at t = A is given by:

d/(t) =

L ~ _ : t 9 (A, T + A) 1: t>T+A

(12.23)

12

Time Domain Integrals for Sampled D a t a Control

195

Therefore the transform of the sampled filtered disturbance is: (FD)d(Z)=Z_I(T-A)

T

( 1 ) 1 - z -1

+ z-2

_ z(T--A)+A z T ( z - 1)

(12.24)

Evaluating (12.24) at z = e pT and substituting in (12.15) gives: A

e-Pry (t) dt = l e - P T -- T p

pT 1

1 [

--

e-p T _

-

-

e -pT

e pT (1 -- ~-) + T e pT (e pT

-

-

1)

(pT)2(ePT(I---~)+-~)]

(12.25)

as required. Note that the function defined in (12.22) is not strictly speaking negative for all possible combinations of p, T and A. However, as illustrated in Figure 12.6, for practical values of p and T the integral is almost always negative. T h e only times where the integral is not negative is when A is very close to T (that is, when we are fortuitous enough to sample the output very shortly after a step disturbance has occurred) or when p T is large. Note however, that from [3, Corollary 6(a)] t h a t for p T > 2.5, we have a peak in [TI~,,~(jw)[ of greater than sinh2.5 2.42 or 7.68dB. Y--A Therefore we conclude that in general, and certainly in the case where A ----0 the time domain integral constraint with an integrate and reset anti-aliasing filter is more restrictive than its purely analog counterpart. -

12.3.4

First Order

Anti-Aliasing

-

Filter

Consider a first order anti-aliasing filter, which without loss of generality we take to have unity DC gain, with transfer function

F(s)-

a a s -b

(12.26)

We then have the following Corollary: C o r o l l a r y 5. Consider a system with conditions as in Corollary 2, and in addition,

where F (s) = 7-+-~'~ Then: ?e-pry =-

(t) dt

e -pT - p T

(12.27)

1+

196

Middleton, Lau, Freudenberg

:

0.1

~

i

i

f

A = I"T

-0.

. . . .

~~ ~-~-03. -0.4

"

-O.2

-0.6

.

.

.

-0.7

.

.

.

.

.

.

~

.

.

.

O. 5

.

.

.

.

.

.

.

.

.

- -

~

.

1

1.5

2

2.5

pT

Fig. 12.6. Plot of fz (pT, A / T ) versus p T for various A / T .

Proof. Because of the nature of the filter, the filtered response to a step disturbance at t -- A is given by: d/(t)=

{

0: t_ A

(12.28)

Therefore the Z-transform of the filtered disturbance is: 1 (FD)d (z) - z - 1

e -a(T-z~) z - e -~T

(12.29)

Evaluating (12.29) at z = e pT and substituting in (12.15) gives: ~ e-pty (t) dt 1 --pW ----

- e

p

.

T p + a [" 1 a ~ epT--- 1

= - e-vr-pT P

.

.

1+

=: ~ f2 (pT, aT, -~ )

.

~

.

e -a(T-zl) "~ e pT -- e--aT] -(~---~)(-~--e~-~

(12.30)

as required.

In this case, it is more difficult to plot this function due to the number of independent parameters. However, if we note that a T is precisely 7r times the ratio of the anti-aliasing filter cut-off to the Nyquist frequency, then we need only consider

12

02~ 0.3

-

Time Domain Integrals for Sampled Data Control

-

aT=l"u

:

:

. . . .

. . . . . .

i

: aT=0.5*~t

:

:

197

......

........... :

0.1

-0.1

0 ~-0.2

-0.3

-0.4

"

i . . . . . . . . .

~

.

.

.

~ L. -0.5

....

015

.

.

.

....

I 1

.

.

.

-

-

~

-

. . . .

A=0*T :

pT

'15

:

. . . . . . . . . . . . . .

I 2

2'.5

Fig. 12.7. Plot of fs (pT, aT, A/T) for a first order anti-aliasing filter.

a small number of values of aT. Adopting this philosophy, we obtain the graphs shown in Figure 12.7. Note once again that except for values of A very close to T the function is negative. We therefore conclude, once again, that sampled data control essentially always gives a more difficult time domain integral constraint compared to the purely analog feedback case.

12.4

Conclusion

We have seen that time domain integral constraints related to open loop unstable poles are often dictated by the use of LTI feedback control. Several authors have pointed out that non-linear controllers (e.g. switching controllers) can circumvent some of the inherent time domain limitations due to open loop unstable poles. We argued that it may also be of interest to consider whether time varying controllers may achieve a similar reduction in performance limitations. Here, we studied in detail a particular class of time varying controllers, namely, sampled data feedback controllers. In the case of sampled data feedback controllers, we showed that unstable plant poles do indeed generate time domain integral equality constraints which the closed loop response must satisfy. The right hand side (RHS) of these integral constraints is no longer exactly zero (as in the analog feedback case). In this sense, it could be then argued that sampled data control might alleviate or aggravate the performance limitations of analog control (depending on the sign of the RHS term). The RHS term in fact depends solely on the anti-aliasing filter used, and the type of output disturbance considered. In this paper, several classes of anti-aliasing filter with a unit step output disturbance were considered. The step arrival time was asyn-

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chronous with the sampling. It was found that the time domain integral constraints were nearly always worse than in the purely continuous case.

References 1. Chen, Q., Y. Chait and C. Hollot, "Stability and Asymptotic Performance Analysis of a Class of Reset Control Systems", Technical report http://www.ecs.umass.edu/mie/labs/dacs/pubs/bibo.pdf, also submitted for publication to IEEE transactions on Automatic Control 2. Feuer, A., G.C. Goodwin and M. Salgado, "Potential Benefits of Hybrid Control for Linear Time Invariant Plants", Proc. American Control Conference, 1999. 3. Freudenberg, J.S., R.H. Middleton and J.H. Braslavsky, "Inherent Design limitations for linear sampled-data feedback systems", International Journal of Control, V61,N6,pp1387-1421, 1995 4. Lau, K. and R.H. Middleton, "On the Use of Switching Control for Systems with Bounded Disturbances", Technical Report EE00012 http://eebrett.newcastle.edu.au/reports/reportsAndex.html, also, to appear, Proc. 39th IEEE Conference on Decision and Control, Sydney, December 2000 5. Middleton, R.H., "Trade offs in linear control system design", Automatica, V27, N2, pp281-292, 1991 6. Middleton, R.H. and G.C. Goodwin, "Digital Control and Estimation: A Unified Approach", Prentice Hall, 1990 7. Seron, M.M., J.H. Braslavsky and G.C. Goodwin, "Fundamental Limitations in Filtering and Control", Springer-Verlag, 1997.

13 A Linear T i m e - V a r y i n g A p p r o a c h to M o d e l Reference A d a p t i v e Control Daniel E. Miller Dept. of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

A b s t r a c t . In classical model reference adaptive control, the goal is to design a controller to make the closed loop system act like a prespecified reference model in the face of significant plant uncertainty. Typically the controller consists of an identifier (or tuner) which is used to adjust the parameters of an LTI compensator, and under suitable assumptions on the plant model uncertainty it is proven that asymptotic matching is achieved. However, the controller is highly nonlinear, and the closed loop system can exhibit undesirable behaviour, such as large transients or a large control signal, especially if the initial parameter estimates are poor. Here we propose an alternative approach, which yields a linear periodic controller. Rather t h a n estimating the plant or compensator parameters, instead we estimate what the control signal would be if the plant parameters were known; we are able to do so in a linear fashion. In this paper we consider the first order case, and prove that if the plant parameters lie in a compact set, then near exact model matching can be achieved. We explore the benefits and limitations of the approach and explain how it can be extended to the relative degree one case.

13.1

Introduction

Adaptive control is an approach used to deal with plant uncertainty. The basic idea is to have a controller which tunes itself to the plant being controlled; typically such controllers can be described by a nonlinear time-varying (NLTV) differential or difference equation. One of the most important problems in the area has been the model reference adaptive control problem (MRACP), where the goal is to have the output of the plant asymptotically track the output of a stable reference model in response to a piecewise continuous bounded input. It was shown around 1980 that this problem is solvable if (i) the plant is minimum phase, and if (ii) an upper bound on the plant order, (iii) the plant relative degree, and (iv) the sign of the high-frequency gain are known, e.g. see Morse [5], Narendra et al. [8], and Goodwin et al. [1]. Subsequent work has demonstrated that (iv) can be removed, e.g. see Mudgett and Morse [7], and that (iii) can be weakened to requiring that an upper b o u n d on the relative degree be known, e.g. see Tao and Ioannou [2]. Typically such an adaptive controller consists of an identifier (or tuner) which is used to adjust the parameters of an LTI compensator, and under the above assumptions on the plant model uncertainty it is proven that asymptotic matching

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Daniel E. Miller

is achieved. However, the controller is highly nonlinear, which makes the behaviour hard to predict, and the closed loop system can exhibit undesirable behaviour, such as large transients, especially if the initial parameter estimates are poor. In Miller and Davison [4] a high-gain adaptive controller is used to allow for increased plant uncertainty and to control the transient response; however, while the transient response can be made arbitrarily good, an undesirable side effect is that the control signal can become very large. Here we propose an alternative approach, which yields a linear periodic controller. Rather than estimating the plant or compensator parameters, instead we estimate what the control signal would be if the plant parameters were known; we are able to do so in a linear fashion. The essential features of the approach are embedded in the first order case, so we will use that to develop our controller and to prove our main result. We will then explain how this can be extended to the case where classical assumptions (i) and (ii) hold, (iii) is replaced by a relative degree one assumption, (iv) is omitted, and a crucial additional assumption is t h a t the plant parameters lie in a compact set. Given that all designs come with a pricetag, we also explore the benefits and limitations of the approach and compare it to the classical one.

13.2

Preliminary

Mathematics

Let Z denote the set of integers, Z + denote the set of non-negative integers, N denote the set of positive integers, R denote the set of real numbers, R + denote the set of non-negative real numbers, C denote the set of complex numbers, and C - denote the set of complex numbers with a real part less than zero. W i t h A a matrix, A T denotes its transpose. W i t h A E R '~x'~, sp(A) denotes the set of eigenvalues of A; its spectral radius is

p(A) := max{lAl : A c sp(A)}. With x E R n, the norm of x is defined by Ilxll := m a x { I x i l : i -- 1,...,n}. The norm of A E R nXm, denoted IIAll, is the corresponding induced norm. We let P C ( R "xm) denote the set of piecewise continuous functions from R + to R '~Xm. For f E PC(RnXm), define

Ilfllor := esssupt~R+ IIf(t)llLet P C ~ ( R n• denote the set of f E P C ( R ~• for which Ilftlo~ < ~ . Henceforth we drop the R n• and simply write PC and PC~. In this paper we will be dealing with linear time-varying systems, and it will be convenient to discuss the gain of such a system when starting with zero initial conditions at time zero. To this end, the gain of G : P C ~ ~ PC is defined by

"GII := sup { ''Gum''~ Ilumll~ We say that the system is

: um ~ PC~,

Ilumll~ r 0

} .

stable or bounded if IIGII < c~.

13

A Linear T i m e - V a r y i n g A p p r o a c h to the M R A C P

201

We say t h a t f : R + ---* R '~xm is of order T j, a n d write f = O ( T J ) , if t h e r e exist constants cl > 0 a n d 7"1 > 0 so t h a t [If(T)lr < ClT j, T e

(0, T1).

On occasion we have a function f which d e p e n d s n o t only on T > 0 b u t also on a pair (a, b) restricted to a compact set F C R2; we say t h a t f = O ( T ~) if there exist c o n s t a n t s cl > 0 a n d 7"1 > 0 so t h a t [If(T)l[ < cxT j, Z ~ (O, TO,

13.3

(a,b) E r.

The Problem

Our first order plant P is described by

y(t) = ay(t) + bu(t), y(O) = yo,

(13.1)

with y(t) C R the plant state (and m e a s u r e d o u t p u t ) a n d u(t) E R the p l a n t i n p u t ; we associate this plant with the pair (a, b). T h e associated transfer f u n c t i o n from u to y is given by

P(s) :=

b s--a"

We assume t h a t (a, b) is controllable, which m e a n s t h a t b # 0. Indeed, we a s s u m e that (a, b) lies in a compact set s satisfying E C {(a,b) e R 2 : b # 0 } . Our stable SISO reference model Pm is described by

ym(t) = amym(t) + bmum(t), ym(O) = Ymo,

(13.2)

with Ym (t) E R the reference model state a n d Um (t) E R the reference model i n p u t . The model is chosen to e m b o d y the desired b e h a v i o u r of the closed loop system; clearly we require it to be stable. Informally, our goal is to design a controller to make the plant o u t p u t track the reference model o u t p u t in the face of p l a n t uncertainty. To this end, we define the (tracking) error by c(t) := y~(t)

- y(t).

Our goal is to construct a single linear t i m e - v a r y i n g controller which n o t only provides closed loop stability b u t also provides near o p t i m a l performance for each possible model. It is our i n t e n t i o n to use a sampled d a t a controller, so it is n a t u r a l to use an anti-aliasing filter at the i n p u t . Hence, our control law has two parts: with a > 0 we have a n anti-aliasing filter at the i n p u t of the form ~m = --OLq~m"[- Ot~m, ~m(0) = Um0

(13.3)

whose i n p u t - o u t p u t m a p is labeled F~; this is followed by a s a m p l e d - d a t a controller of the form

z[k + 1] = F(k)z[k] + G(k)y(kh) + H(k)~m(kh), z[0] ----z0 E R z, ~z(kh + T) = g(k)z[k] + L(k)y(kh), v E [0, h)

(13.4)

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Daniel E. Miller

whose i n p u t - o u t p u t m a p is labeled K and whose gains F , G, H , J and L are periodic of period p E N ; t h e period of t h e controller is T :-- p h , and we associate this s y s t e m with t h e 7-tuple (F, G, H, J, L, h, p). O b s e r v e t h a t (13.4) can be i m p l e m e n t e d w i t h a sampler, a zero-order-hold, and an I th order periodically t i m e - v a r y i n g d i s c r e t e - t i m e system of period p . Closed loop stability clearly d e p e n d s solely on t h e s a m p l e d - d a t a c o m p o n e n t , which brings us to D e f i n i t i o n 1. T h e controller (13.4) stabilizes (13.1) if, for e v e r y set of initial conditions yo E R and z0 E R ~, w i t h ~m = 0 we have y(t) ~

o as t ~

cr

and z[k] ----~O as k ---* co.

Now suppose t h a t (13.4) stabilizes (13.1) and t h a t Y0 = 0 and z0 ---- 0; we let bY(P, K ) d e n o t e t h e closed loop m a p from fi,n ~ e. O u r goal is to design K so t h a t Ill-(P, K ) F ~ - PmlI is small for every possible P ; however, notice t h a t [ I . T ' ( P , K ) F ~ -- PmI[ _< [ [ . T ' ( P , K ) F ~ - P,~F,~[[ + HPm(F,~ - 1)H < II.~(P,K)F,-

21bml

P,~F~I I + lam +c~------~l"

Hence, we can first choose a sufficiently large ~ to m a k e the second t e r m small, and t h e n we can proceed to design K to m a k e t h e first t e r m small for all admissible P . To this end, we define ~,~ = am~m + bmfim,

~m(0) ----Ym0.

(13.5)

Notice t h a t for c~ > [am[, we have

,bml [~.~oleamt+la~+' ~lllumll~, [gm(t)--ym(t)l<_ lam+~

t_>O.

(13.6)

In t h e next section we will provide a high level description of t h e design approach.

13.4

The Approach

Here we explain the m o t i v a t i o n of t h e a p p r o a c h to t h e problem. O u r goal is to make t h e difference between y and ~m small, so let's form a differential e q u a t i o n describing this quantity: (~,~ - y ) = a,,~(~m - y ) + (bmftm - bu § ( a m - a ) y ) .

Since we would like t h e plant to act like t h e reference model, we m a y as well require t h a t t h e error caused by a m i s - m a t c h in initial condition go to zero like eamt, i.e. we m a y as well set 1 1 ( bm f t m + ( a m - a ) y ) . bm~tm - bu + (am - a ) y :- 0 ~ u -= -~

(13.7)

1 This is not essential - we can require the d y n a m i c s of t h e m i s - m a t c h in initial conditions to be as fast as we like; here we have m a d e t h e m o s t n a t u r a l choice.

13

A Linear T i m e - V a r y i n g A p p r o a c h to t h e M R A C P

203

Because of our c o m p a c t n e s s a s s u m p t i o n we can be assured t h a t 1/b is b o u n d e d above i n d e p e n d e n t of (a, b) E F . O f course, a and b are unknown, so some form of e s t i m a t i o n is required. Since we would like to end up with a linear controller, we would like to get rid of t h e 1/b term. It will t u r n out t h a t we will be able to deal w i t h polynomials. F r o m t h e compactness assumption, it is clear t h a t there exist positive b and b so t h a t

(a,b) E F = ~ 0 < b _< Ibl_< bF r o m t h e Stone-Weirstrass A p p r o x i m a t i o n T h e o r e m we know t h a t we can approxi m a t e 1/b arbitrarily well over t h e c o m p a c t set I - b , - b ] U [b, b] via a polynomial. While some proofs of this result are constructive, e.g. see [10], it is often difficult to i m p l e m e n t in practise. If we were only interested in [b, b] t h e n one could first form a Taylor series expansion a b o u t b; since it converges uniformly on every closed subinterval of (0, 2b), t h e n to get a good a p p r o x i m a t i o n one simply t r u n c a t e s t h e Taylor series. T h e p r o b l e m is not m u c h h a r d e r in our case. 1. The s u m m a t i o n

Proposition --~ ( - 1 ) i

b (b 2 - ~2)~

convenes unifo~ly to ~ on [-~, -hi u ~_,~]. Unfortunately, Taylor Series expansions, on which the above is based, t e n d to converge slowly. T h e r e have been investigations into o p t i m a l a p p r o x i m a t i o n d a t i n g back at least to t h e 1890's, e.g. see [11] (pp. 126 - 138) and the references c o n t a i n e d therein, which can yield alternative ways to o b t a i n an a p p r o x i m a t i o n . In any event, from Proposition 1 we see t h a t for every ~ > 0 we can choose a p o l y n o m i a l ]~(b) so that

I1 - bL(b)I <

~, b E I-b,-_b] LJ [b,b].

(13.8)

Now we consider our second a p p r o x i m a t i o n . R a t h e r t h a n a p p l y i n g t h e feedback control (13.7), instead we apply

= ]~(b)(bm~m + (am - a)y);

(139)

we label this 12 and the corresponding closed loop solution by ~; we set ~(0) = yo. In closed loop we end up with

(~m - ~) = [am + (a - am)(1 - bf~(b))](flm - fl) + [1 - bf~(b)]bmum - [1 - bf~(b)](a - am)rim. Define ::

Ym

--

Y, e0 : = Ym0 - yo.

P r o p o s i t i o n 2. Suppose that - ~ < am. Then there exist constants ~ > 0 and 7 > 0 so that for every ~ E (0,~), the closed loop system satisfies

I~(t)- e~

< ~/~(ly01 + lye0 I+ I ~ 0 I)e(~

+ ~l/~ll~lloo,

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Daniel E. Miller

So, as expected, as our a p p r o x i m a t i o n of ~ improves, our s t e a d y - s t a t e t r a c k i n g error improves. At this point we freeze e > 0 a n d choose a p o l y n o m i a l q

]e(b) = E c, b' i=l

so t h a t (13.8) holds; we assume t h a t e is sufficiently small t h a t the c o r r e s p o n d i n g control law (13.9) at least stabilizes every (a, b) E F . Now the goal is to design a s a m p l e d - d a t a control law of the form (13.4) so t h a t we can a p p r o x i m a t e l y imp l e m e n t (13.9) regardless of which admissible system t h a t we are controlling. W e use h small a n d with q the order of our p o l y n o m i a l a p p r o x i m a t i o n to ~, we choose p > 2 q + 1; recall t h a t the controller period is T ----ph. First we provide a c o n c e p t u a l description of the controller a n d a high-level description of why it should work. We do so in open loop a n d t h e n explain how to convert it to a controller of the form (13.4). To motivate our approach, first observe t h a t the s a m p l e d - d a t a control law ~(t) = {

0 p_(2Pq+l) ]~(b)[bm~tm(kT) + (am -a)~(kT)]

t E [kT + (2q + 1)h, (k + 1)T) t E [kT, kT + (2q + 1)h)

(13.10)

should be a good a p p r o x i m a t i o n to (13.9) if h a n d T ----ph are b o t h small. Here we will i m p l e m e n t s o m e t h i n g similar to this: every period [kT, (k + 1)T) is divided into two phases: 9 i d e n t i f i c a t i o n / e s t i m a t i o n phase: on the interval [kT, k T + (2q + 1)h) we e s t i m a t e

]e(b)[bm~m(kT) + (am - a)~l(kT)]. While we do n o t set fi(t) equal to zero here, we ensure t h a t the effect of t h e p r o b i n g used in the estimation yields only a second order effect. 9 control phase: on the interval [kT+ ( 2 q + 1)h, ( k + 1)T) we apply p--(2q-~l) P times the above estimate. Let us look at the first period [0, T). T h e first step is to form a n a p p r o x i m a t i o n of q i

-

a

c~ b [bmum(0) + ( m - a)y(0)]. =:(~i(o)

Suppose t h a t we initially set fi(t) ----0, t E [0, h). Since a is constrained to a compact set, it follows t h a t

y(h) = eahy(O) = [1 + ah + O(h2)]y(O). Hence, l [ y ( h ) - y(0)] = ay(O) + O(h)y(O).

13

A Linear Time-Varying Approach to the M R A C P

205

So at this point we have a good estimate of ay(0), with the quality of the estimate improving as h --~ 0. ~ Hence, we can form a good estimate of r

:----bmfim(0) + (a,~ - a)y(0),

namely r

1

----bmfim(0) + amy(O) - ~[y(h) - y ( 0 ) ]

= bm~tm(O) + (am - a)y(O) + O(h)y(O) = r

+

V(h)y(o).

To form estimates of r ----b~r we will carry out some experiments. W i t h p > 0 a scaling factor (we make this small so that it does not disturb the system very much), set

u(t) = pC0(0), t 9 [h, 2h). Then y(2h) = e2~hy(O) + (

e a~ dT")bu(h)

= [1 + 2ah + (.9(h2)]y(0) + [bh + O(h~)]p~So(O) ---- (1 + 2ah)y(O) + pbhr

+ (-9(h2)y(O) + O(h2)~tm(O).

(13.11)

Hence, r

= ~h[Y(2h) - 2y(h) + y(0)] = br

---- r

+ O(h)y(0) +

O(h)~m(O)

+ (9(h)y(O) + (9(h)~tm(O).

Of course, in carrying out this experiment we have given the state a boost - see (13.11). This can be largely undone by applying

u(t) = -pr

t 6 [2h, 3h),

for then y(3h) = (1 + 3ah)y(O) + O(h~)y(O) + O(h2)~m(0). To form a good estimate of r (0) we simply repeat the above procedure: we set

u(t) = pr

t 6 [3h, 4h)

2 The skeptics will be concerned that we are differentiating the plant output. We are indeed carrying out discrete-time approximation to differentiation, with the estimate improving but the noise tolerance degrading as h --+ 0. This is akin to the problem arising in PID design where one has to roll off the differentiator term at the appropriate frequency: the higher the rolloff frequency the better the approximation to a pure differentiator and the worse the noise behaviour.

206

Daniel E. Miller

and

u(t) = -pr

t e [4h, 5h);

it is easy to see t h a t y(4h) = (1 + 4ah)y(O) + pb2hr

+ O(h2)y(O) + O(h2)~tm(O)

and y(5h) = (1 + 5ah)y(O) + O(h2)y(O) + O(h2)~m(O). So a good estimate of r r

is

-- ~h [y(4h) - 4y(h) + 3y(0)] = r

+ O(h)y(O) + O(h)~tm(O).

This can be repeated q - 2 more times: for i = 2, ..., q - 1 we end u p with u(t)

S pq~i(0), t E [(2i + 1)h, (2i + 2)h) -pr t C [(2i + 2)h, (2i + 3)h);

we have y((2i + 2)h) = [1 + a(2i + 2)h]y(0) + pb~+lhr

O(h2)y(O) + O(h2)~tm(O) and y((2i + 3)h) -----[1 + a(2i + 3)h]y(0) + O(h2)y(O) + O(h~)~t,~(O), so t h a t $~+1(0) = ~h [y((2i + 2)h) - (2i + 2)y(h) + (2i + 1)y(0)] ---- r

+ O(h)y(O) + O(h)~tm(O).

At the end of the estimation phase, we are at t --- (2q + 1)h, a n d we have estimates of r162 to form our control signal to be applied d u r i n g t h e control phase. There is a lot of flexibility in how long one can make the control phase: the higher the percentage t h a t we carry out control the closer t h a t t h e m a g n i t u d e will be to the ideal one; however, if we make the percentage too large t h e n we will need a very small h to ensure t h a t T is small enough, which will exacerbate noise tolerance. In any event, at this point fix p>2q+l with a corresponding controller period of T ----ph. T h e n we set q

u(t) --

P E cir p - 2q - 1 i=1

t e [(2q + l)h, ph).

13

A Linear Time-Varying Approach to the M R A C P

207

It follows that q

y(ph) = y(T) -- [1 4-pah]y(O) 4- Tb ~ c,r

4- O(h2)y(O) 4- O(h2)fzm(O)

i~l

---- [1 4- aTlY(O) 4- Tb](b)[bmfzm(O) 4- (am - a)y(0)] 44-O(h2)y(0) 4- O(h2)fzm(O) = e~

+ Tb](b)[bm~m(0) + (am - a)y(0)] +

4-O(h2)y(0) 4- O(h2)f~m(O)

eamTy(O) 4-

/o

(13.12)

eam(T-'r)bmfZra(r)dT,

as desired. Of course, at time T = ph the procedure is repeated, but now using y(T) and 12re(T) instead of y(0) and fi,,~(0). F~arthermore, the above simply examines what happens between consecutive periods, whereas we wish to examine the interval [0, oc). Before doing so, let us first provide a closed form description of the proposed estimation and control scheme using a sampled-data system of the form (13.4). To this end, the state z has dimension 4: 9 9 9 9

Zl keeps track of y(jT) for later use. z2 keeps track of y(jT + h) for later use. z3 keeps track of the most current r z4 is used to construct the control signal to be applied during ~T + (2q 41)h, (j 4- 1)T).

We partition the periodically time-varying gains F, G, and H (with period p) accordingly as [ F1 (k) ]

IF=(k)l F(k) =/F3(k) / ,

[ G1 (k) ~

a(a)=

LF,(k) j

[ H1 (k) ]

la=(k) l /a3(k) / ,

iH=(a)I

H(k)= ira(a)/

La,(k) j

LH4(k) j

We would like zl(k) ----y(0), k = 1,'-" ,p so we set

(F1,G1,H1)(k)=

(0, 1,0) k = 0 ([1 0 0 0 ] , 0 ) k 1,-.. , p - 1 .

We would like

z2(k)----y(h), k = 2 , . . . , p , so we set (o,o,o)

(F2, G2, H2)(k) =

k = o

(0, 1, 0) k ([0 1 0 0 ] , 0 , 0 ) k

1 2,-..,p-1.

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Daniel E. Miller

We use za to keep track of the estimate of r

More precisely, we would like

$o(0) k=2 z3(k) =

k=2i+1,2i+2,

4i(0)

i=l,...,q.

To this end, we set

(F3,G3,H3)(k) = {

(0, am, bin) ( [ 1 / h 0 1 0 ] , - I / h , 0) (~ [2i-1 -2i 0 0],~,0) ([0 0 1 0 ] , 0 , 0 ) (0,0,0)

k = 0 k 1 k 2i ( i = l , . . . , q ) k 2i+1 (i= 1,.-.,q) k----2q+2,... ,p-1.

Our last state z4 is used to accumulate our control signal: we'd like q

z4(k) = ~ c&(0),

i = 2q + 2, ...,p,

i=l

so we set

(i,o,o) 0],0,0) k = 0,1,2 (o0cl 3 (F4, G4, I-Ia)(k) =

( ( ( (

0 0 0 0

0 0 0 0

0 ci 0 0

1] ,0,0) k 1],0,0) k 1],0,0) k lJ,0,0) k

4

2i+1(i=2,...,q)

2i+2(i=2, q) 2q+3,..-,p""l

Last of all, we need to form the control signal: we set

[iEo

0 ] --h~) 0 0] ,0) (J,L)(k)={([O 0 p 0 ] , 0 ) ! ([0 0

I ([0 0 2" o1,0)p-~q-1],O) qp-eq-1 (([0

__T__ 0 0 _p_~q_l],0)

k= 1 k= 2 k=2i+1 k=2i+2 k=2q+l

(i= 1,...,q-I) (i=l,...,q 1)

k=2q+2,...,p-1.

At this point, we need to prove that the proposed control law acts much like (13.9). To proceed, we label ~ to denote our control signal and ~) to denote our output; we set ~)(0) = yo. From (13.12) we see that 9[(k + 1)T] =

e'~T~)(kT)+ Tb.f(b)[bm~tm(kT) + (am - a)9(kT)] + +O(T2)fl(kT) + O(T2)f~m(kT).

(13.13)

Before proceeding, define : = m a x { sup

(a,b)er

[a+b]s(b)(am-a)],am};

since

a + bf~(b)(am - a) < 0 for every (a, b) C F by hypothesis, it follows from the compactness of F that A is negative.

13

A Linear T i m e - V a r y i n g A p p r o a c h to the M R A C P

209

P r o p o s i t i o n 3. For every A G (A, 0) there exists a T > O and a z/> 0 so that for every T 6 (0, T) we have

[9(t) - 9(t)l _< 2Te~*(lyol + I~mol) + ~,Tjlu,~[l~, t > 0. P r o o f ' . Fix A E (.~, 0). In closed loop we have

ij = (a + bf~(b)(am - a)] ~ + b]~(b)bm ftm. =:acl(a,b)

bcl(b)

It follows t h a t there exists a c o n s t a n t 3'1 > 0 so t h a t for every (a, b) 6 F , we have ]y(t)i _< 3'1eAt(lY0 ] + i~mol) + 3,,lluml[~,

t _> 0.

Now let us analyse the closed loop b e h a v i o u r at the sample points. We have

9[(k + 1)rl = ~ ' ( ~ ' ~ ) ~ ( k T ) +

~~176

+ T).

Now it is easy to show t h a t

~o T eact(a,b)(T ") b~z(b)ftm (kT + r) = Tbd(b)ftm(kT)+ O(T2)[Numl]o~ + e-~kTlumol], SO 9[(k + 1)T] = e~(~'b)Ty(kT) + Tbr O ( T 2) [[[Um I[oo + e--~kr[U~o 1]. F r o m (13.I3) we see t h a t ~)[(k + 1)7"] = e~Tfl(kT) + Tb]~(b)fbm~tm(kT) + (am - a)~)(kT)] +

+O(T2)fl(kT) + O(TZ)ftm(kT) = e~t(~'b)fl(kT) + Tb](b)bm~tm(kT) + O(T2)fl(kT) + O(T2)ftm(kT). Hence, if we set ~(t) = 9(t) - ~)(t), it follows t h a t (~[(k + 1)T] = [e~r

+ O(T2)]5[kT] + O(T2)fl(kT)+

O(T~)[ii~lioo + ~ - ~ l ~ m 0 1 ] . T h e r e exists T1 > 0 so t h a t [eacz(a'b)T d- O(T2)[ _~ e (X+A)T/2 < e AT, T 6 (0, T1),

210

Daniel E. Miller

SO

k

lS[kT]l <_ ~_e(X+~)(k-J)T/20(T2)[I~(jT)I + Ilu,~]loo + e-~3Tl~mot ] j=0 k

<- Ee(X+X>(k--3>T/20(T2)[71eXkT(]yoI + r~mol) § j=o

"Y~llumll~ + Ilumlloo + e-~Jrl~moll _< O(T)llumlloo + O(T)e)'kT(lyoI + I~m01), T E (0, T 0 .

(13.14)

Now let us look at what h a p p e n s in between the sample points. It follows from our b o u n d on ~ t h a t there exists a constant "1'2 so t h a t

lu(t)l -< "~e~t(lYol + lumo I) + "~2llumlloo 9 We also know t h a t

~t(t) = O(1)ftm(kT) + O(1)fl(kT), t E [kT, (k + 1)T). Hence, for t E [kT, (k + 1)T): ]t3(t) - ~3(kT)l _< (e '~*(~'b)T - 1)l~3(kT)l +

Te ~176

I + II~m II~] I~mo I) + ~ liUmlloo] + O ( T ) [ e - " ~ l ~ o I + II~mll~] = O(T)[eXkT(lYol + I~mol) + II~mll~] = O(T)bleXkT(lYoI

+

and 19(t) - ~(kT)l _ (el~l T _ 1)l~)(kT)l + TelalTIblO(1)[l~)(kT)l +

= O(T)[IY(kT)I + I~(kr)l + e-~kTJ~mol + IlUmll~] = o(r)[~/leXkT(lyo[ + If~mo]) + ~/lllUmlloo + O(T)lfumll~ + O(T)eXkT(lYo[ + [Umo[) + e--~kTlf~mo[ + [[Um[[oo] = o(r)[r

+ [~mo f) + Ilumll~].

Hence,

le(t) -5[kT]l <- O(T)[eXkT(lyol + [f~mol) + IlUmlloo], t e [kT, (k + 1)T). If we combine this with (13.14), it follows t h a t [~(t)[ _< O(T)[ext(lY0[ + [am0]) + [[Um[[oo], as required. []

13

13.5

The

Main

A Linear T i m e - V a r y i n g A p p r o a c h to the M R A C P

211

Result

Now we can leverage the results of the last section to prove t h a t we can o b t a i n near o p t i m a l performance in the face of significant plant uncertainty. T h e o r e m 1. For every 5 > 0 there exists a controller of the f o r m (13.3) and (13.4) so that for every (a, b) E F, the closed loop system satisfies

119(t) - ym(t)] - ~ ~ [y0-ymo]i_< 5~(a~+~)~(lY01 + lyre0 I + I ~ o t) + 51l~mlloc. P r o o f : First, choose a > - a m sufficiently large t h a t

Ibm______~l< 5/3. [a,~ + ol I It follows from (13.6) t h a t 19m(t) - ym(t)l <-- (5/3)1~m0 [eamt + (5/3)llumll~.

(13.15)

From P r o p o s i t i o n 2 there exist constants g > 0 and "7 > I so t h a t for every e E (0, g), the closed loop system satisfies t(Y - y m ) ( t )

-- e ~

1/2

~,c

-- Y m o ) l

--<

(lyol + lymol + I~mol)e (~m+='/=)~ +~/211u~ll~.

So now choose e C (0, ~) so t h a t ~ 1 / 2 < 5/3; notice t h a t :=max{

sup [ a + b ] ~ ( b ) ( a m - a ) ] , a m } ~_am+ (~,b)Er

1/2.

It follows t h a t

I(~- ~m)(t) --e ~ (Yo--Ym0)] _< (5/3)(lYol+lY.~ol+l~mol)e(~ (5/3)11u~11~.

(13.16)

Now set

)~:=am +5. From Proposition 3 there exists a T > 0 and ~ > 0 so t h a t for every T C (0, T) we have 19(t) - Y ( t ) l _< ~Te~(lY0i + I~mo I) + ~Tli~m I1~, t _> 0.

212

Daniel E. Miller

Choose T E (0, T) so that

~/T < 5/3; then

lY(t) - ~ ( t ) l _< (5/3)e(a"+~)t(lYo] + I~mol) + (5/3)llumll~,

t>0.

If we combine this with (13.15) and (13.16) we end up with

119(t) - ym(t)] - e ~

- ymo]l

<_ ig(t) - 9 ( t ) l + 119(t) - 9 r e ( t ) ] - e " m t ( y o - Y-,o)] +

Iflm(t) - ym(t)l

<_ (5/3)le(~'"~+'~)t([yol + I~,mo [) + (5/3)ltuml]~ + (5/3)(iyo[ + lymol + la-,o I)e (~

+ (5/3)Humll~ +

(5/3)e(,,m +~)t ifimo I + (5/3)I1~" II = _< 5(lyol + lymol + I~mol)e (~

+ ~11~',~11~.

Remark 1. This result demonstrates that we can control not only the steady-state behaviour but also the transient behaviour! Some of the features are: 9 the near perfect tracking performance begins immediately - there is no tuning phase; 9 any mismatch between y0 and Ym0 decays exponentially to zero approximately at the rate of e a'~t. Neither of these features are provided by a classical adaptive control law.

Remark 2. Since our linear time-varying control law computes u(t) every T seconds, one would expect that it would have the facility to handle slowly time-varying parameters. Although we will not prove this here, we will demonstrate this feature via simulation. Remark 3. From the proof of Theorem 1 it is clear that for the approach to yield extremely good tracking we need to choose a high order polynomial approximation to 1 (recall that q is the polynomial order) and we also need to choose the sampleddata controller period T to be small. This means that the sampling period h=T P

T q

must be quite small, which means that some of the controller gains will be quite 1 This may result in poor noise tolerance, large, since some are proportional to ~. and is probably the most important drawback of the design procedure.

13

13.6

A Linear T i m e - V a r y i n g A p p r o a c h to the M R A C P

to the Relative

Extensions

Degree

One

213

Case

It t u r n s out t h a t t h e m i n i m u m phase relative degree one case is not significantly more difficult t h a n t h e first order case. Here we assume t h a t our plant is d e s c r i b e d by = Ax + Bu, y = Cx

(13.17)

x(O) = xo

with (A, B ) controllable, (C, A) observable, and t h e s y s t e m m i n i m u m phase of relative degree one. O u r reference m o d e l is described by &m = A m x m Ym ~ Cmxm

-~ B m u m ,

(13.18)

xm(O) = X m 0

which is stable and of a r b i t r a r y relative degree. To proceed we borrow t h e following result from [6]: t h e m i n i m u m phase s y s t e m (13.17) of relative degree one a d m i t s a s t a t e space m o d e l of t h e form

y(t) = [0 1] [xl(t)] [x2(t) J

(13.19)

with a2, b2 C R and A1 stable w i t h eigenvalues at t h e zeros of C ( s I - A ) - I B . W i t h o u t loss o f generality, we will a s s u m e that o u r p l a n t model is already i n this form. W i t h am < 0, our goal is to choose u so t h a t

(ym - y) = am(ym -- y). Replacing t h e LHS by t h e a p p r o p r i a t e t e r m s we end up w i t h C m A m x m + C m B m u m - (a2y -4- b 2 c l x l + b2u) = a m ( y m - y);

solving for u yields 3 u = ~[a~(y~

- y) + (a2y + b 2 c l x l ) - C m A m x m

- CmBmum].

First of all, it is easy to form

am(Ym -- Y). Next, we can filter um as before to yield fi,~ and a s s u m i n g t h a t we can m e a s u r e xm, t h e n we can easily form Cm Amxm

-i- C m B m %tm.

3 Notice t h a t this control law decouples s u b - s y s t e m one from s u b - s y s t e m two: we have it1 = A l x l + bly;

since A1 is stable, x l is well-behaved as long as y is well-behaved.

214

Daniel E. Miller

Last of all, by setting u(t) = 0 for t E [0, h], we see that

y(h) = y(O) + h[a2y(O) + b2clxl(O)] + O(h2)x(0), which means that we can easily form a good approximation of

a2y(O) + b2clxl(O). Hence, the controller designed in this paper, with a mild modification, will work for our situation here. Implicit here is that there is an upper b o u n d on the plant order; following the proof of Theorem 1, we would first choose an appropriate value of which worked for all orders, and then we would choose an appropriate value of T which worked for all orders; the details are omitted.

13.7

Examples

Here we discuss two examples to illustrate the proposed design methodology.

13.7.1

Example

1

Here the set of plant uncertainty is F={(a,b)

ER2:

a E [-1,1],

b=-t-1}

and the reference model is X m = - - X r n -~ Urn.

Recall that there is no LTI controller to stabilize ( s ~ l , s~l }, so it follows that there does not exist a single LTI controller to stabilize every (a, b) C F, let alone provide good performance. We choose an anti-aliasing filter of the form ~rn ~- - - 5 0 ~ m -~- 50Urn,

which means that this degrades the performance by only 2%. Since our approach requires that the system state does not change very much during a period T, we will first choose T = 0.1, which is one-tenth of the magnitude of the largest pole. Here b has modest uncertainty: we can choose

]~(b) = b, ~ > 0, so that there is no error in the approximation. So we have the order q of ]s equal to one; we set p = ( 2 q + 1) + 1 = 4,

13 4/

!

21

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i ....

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i 8

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i .........

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L 4

llll,. ...........:,.,,,,,........ - ' ~ -2~

!

.,~,~,,L~ . . . . . . . .

_21 0

10

A Linear Time-Varying Approach to the MRACP

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s

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i 20

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u'so,,,,lt

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1

9

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''~

.........

-

, ....

i ........

-I

2

4

6

8

10

12

14

16

18

20

22

,.

, 4

i 6

i 8

i 10

12

14

16

i 18

9 20

22

2 1 0 -1 -2

2

F i g . 1 3 . 1 . E x a m p l e 1 w i t h a fixed b u t b c h a n g i n g sign.

so t h a t h =- T / q --- 0.025. A s i m u l a t i o n was c a r r i e d o u t (see F i g u r e 1) w i t h y(0) = 3, Ymo = umo --- 0, u m a s q u a r e wave of p e r i o d 10, a -- 1, a n d b a s q u a r e wave of p e r i o d 20. W e see t h a t t h e effect of t h e initial c o n d i t i o n s d e c a y s e x p o n e n t i a l l y t o zero, t h e t r a c k i n g is q u i t e good a n d t h e c o n t r o l signal is m o d e s t ; w h e n b c h a n g e s signs a t t -- 10 we see a s m a l l j u m p in y. Now we would like to d e m o n s t r a t e t h e c a p a b i l i t y of t h e c o n t r o l l e r t o t o l e r a t e t i m e v a r i a t i o n s : let U m a n d b b e as a b o v e b u t set

a(t) = c o s ( t ) Since a is m o v i n g q u i t e r a p i d l y we m u s t d e c r e a s e o u r s a m p l i n g p e r i o d - we c h o o s e T -- .02 so t h a t h ---- 0.005. W e see from F i g u r e 2 t h a t t h e c o n t r o l l e r d o e s a v e r y nice j o b of t r a c k i n g e v e n in t h e face of r a p i d l y c h a n g i n g p a r a m e t e r s . O f course, t h e faster t h a t t h e p a r a m e t e r s are c h a n g i n g t h e s m a l l e r t h e s a m p l i n g period; a t t h i s p o i n t we d o n o t h a v e a q u a n t i t a t i v e m e a s u r e of t h i s tradeoff. A s seen from Section 4 some of t h e c o n t r o l l e r gains are of t h e o r d e r of 1 / h ---- 200; t h i s is d u e to t h e d i f f e r e n t i a t o r - t i k e n a t u r e of p a r t of t h e controller. To i l l u s t r a t e t h e noise feature, we r e d o t h e a b o v e s i m u l a t i o n b u t w i t h a m e a s u r e m e n t noise of 0.005 * r a n d o m s e q u e n c e u n i f o r m l y d i s t r i b u t e d b e t w e e n =El a n d i l l u s t r a t e t h e s i m u l a t i o n in F i g u r e 3. W e see t h a t , as e x p e c t e d , t h e t r a c k i n g is d e g r a d e d . T h i s d e m o n s t r a t e s a key t r a d e o f f in t h i s a p p r o a c h : t h e larger t h e p a r a m eter u n c e r t a i n t y a n d t h e f a s t e r t h e t i m e - v a r i a t i o n of t h e p a r a m e t e r s , t h e s m a l l e r

216

Daniel E. Miller !

: . . . .

0

20

2

! :

'

.... :

0

2

21 r 1I~-

,

10

t

.......

8

12

14

i

i

i.

i

7

4

6

8

10

12

14

,

! :

! :

, :

, :

, :

!

,so,!~. L J

I "

....

10

..........................

'

i

-

ymaasne~t

16

18

;~

20

22

" u solid .-

'

'

r

0 -10 -20

- ....

- - - . ~ ' ~

o~ \

7

.....

~

~

0

I

2

4

, 6

.

J 10

i 8

.

,

\ .

I

18

I

I

~

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-1

-2

-

16

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i 12

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22

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i 16

20

i 18

!

i 20

22

F i g . 13.2. E x a m p l e 1 with b o t h a and b varying.

t h a t the sampling period must be and therefore the more susceptible to noise t h e closed loop system is. At this point we do not have a q u a n t i t a t i v e m e a s u r e of this tradeoff.

13.7.2

Example

2

Here the set of plant u n c e r t a i n t y is larger t h a n before: F={(a,b)

ER2:

aE[--1,1],

b2 E [1,21}.

We choose t h e same reference m o d e l as above: Xm

~-

-- Xm

"~- " t t m ~

as well as t h e s a m e anti-aliasing filter: ~,~ = - 5 0 ~ m + 50urn.

21/2].

Here we need a polynomial to a p p r o x i m a t e ~1 on the set [ - 2 i/2, - 1 ] U [1, While we could t r u n c a t e the series of P r o p o s i t i o n 1, it t u r n s out t h a t we would need a lot of terms. Instead we base our a p p r o a c h on t h e o p t i m a l a p p r o x i m a t i o n t e c h n i q u e discussed in [11]: we have ]0.0i (b) = 2 . 1 6 4 7 b - 1.5153b 3 + 0.3433b 5.

13

41

L

2~x.~

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!

!

:

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0

2

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4

6

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i ....... ! ......... i ........ i .....

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:

yso,d

......... ..... t

: ..... :........ :

o

-2

A Linear Time-Varying Approach to the M R A C P

10

12

i

14

16

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18

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20

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217

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! I asod IJ b dashed~ /

'...... t

18

20

22

F i g . 1 3 . 3 , E x a m p l e 1 w i t h a noisy m e a s u r e m e n t .

So q = 5 a n d we choose p = 22 = 2(2q + 1), i.e. t h e c o n t r o l p h a s e is 50% of t h e t i m e . W e set T = 0.048 (so t h a t h --- 0.002) a n d p = 1. A s i m u l a t i o n was c a r r i e d o u t (see F i g u r e 4) w i t h y0 = Ymo = urn0 = 0, Um a s q u a r e wave of p e r i o d 10,

a(t) = cos(t/2), and

b(t) ----1.2 + 0.2 c o s ( t / 2 ) . W e see t h a t t h e t r a c k i n g is q u i t e g o o d a n d t h e c o n t r o l signal is m o d e s t , e v e n in t h e face of significant t i m e variation.

13.8

Summary

and Conclusions

In classical m o d e l reference a d a p t i v e control, t h e goal is t o d e s i g n a c o n t r o l l e r t o m a k e t h e closed loop s y s t e m act like a p r e s p e c i f i e d r e f e r e n c e m o d e l in t h e face o f significant p l a n t u n c e r t a i n t y . T y p i c a l l y t h e c o n t r o l l e r c o n s i s t s of a n identifier (or

218

Daniel E. Miller 2[ 1

! 9

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0

2

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0

2

4

6

8

10

12

14

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1

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--1 I

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6

I

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8

10

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I

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14

16

~

18

20

22

F i g . 13.4. Example 2 with both a and b varying.

tuner) which is used to adjust the parameters of an LTI compensator, and under suitable assumptions on the plant model uncertainty it is proven that asymptotic matching is achieved. However, the controller is highly nonlinear, and the closed loop system can exhibit undesirable behaviour, such as large transients or a large control signal, especially if the initial parameter estimates are poor. Here we propose an alternative approach, which yields a linear periodic controller. Rather than estimating the plant or compensator parameters, instead we estimate what the control signal would be if the plant parameters were known; we are able to do so in a linear fashion. In this paper we prove everything for the first order case, but explain how it can be easily extended to the minimum phase relative degree one situation. The desirable features of the controller are 9 the effect of the plant and reference model initial conditions and the reference input are decoupled, 9 the linear periodic controller initial conditions do not play a role (the initial condition of the anti-aliasing filter plays a role but we can simply set it to zero), 9 the tracking of the reference input is immediate rather than asymptotic, and 9 the controller has the facility to allow for time-varying plant parameters, which we have demonstrated via simulation. An undesirable feature is that in order to obtain tight tracking performance, we may require a small sampling period and large gains; this may yield poor noise tolerance.

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Yhrther work is needed to extend this to the general m i n i m u m phase case of arbitrary relative degree. Yhrthermore, it is important to carefully characterize the ability to handle time varying parameters, and to compare these results with recent work on adaptive control of time-varying systems, e.g. [12], [9] and [3].

Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of Canada via a research grant. References 1. Goodwin G.C., Ramadge P.J., and Caines, P.E. (1980) Discrete Time Multivariable Control. IEEE Transactions on Automatic Control A C - 2 5 449 456. 2. Tao G., Ioannou P.A. (1989) Model Reference Adaptive Control for Plants with Unknown Relative Degree. Proceedings of the American Control Conference. 2297 - 2302. 3. Limanond S., Tsakalis K.S. (2000) Model Reference Adaptive and Nonadaptive Control of Linear Time-Varying Systems. IEEE Transactions on Automatic Control A C - 4 5 1290 - 1300. 4. Miller D.E., Davison E.J. (1991) An Adaptive Controller which Provides an Arbitrarily Good Transient and Steady-State Response. IEEE Transactions on Automatic Control A C - 3 6 68 - 81. 5. Morse A.S. (1980) Global Stability of Parameter-Adaptive Control Systems. IEEE Transactions on Automatic Control A C - 2 5 433 - 439. 6. Morse A.S. (1985) A Three-Dimensional Universal Controller for the Adaptive Stabilization of Any Strictly Proper Minimum-Phase System with Relative Degree Not Exceeding Two. IEEE Transactions on Automatic Control A C 30 1188-1191. 7. Mudgett D.R., Morse A.S. (1985) Adaptive Stabilization of Linear Systems with Unknown High-Frequency Gains. IEEE Transactions on Automatic Control A C - 3 0 549 - 554. 8. Narendra K.S., Lin Y.H., Valavani L.S. (1980) Stable Adaptive Controller Design, Part II: Proof of Stability. IEEE Transactions on Automatic Control A C - 2 5 440 - 448. 9. Narendra K.S., Balakrishnan J. (1997) Adaptive Control Using Multiple Models. IEEE Transactions on Automatic Control A C - 4 2 171 - 187. 10. Rudin W. (1976) Principles of Mathematical Analysis, 3rd ed. McGraw-Hill, New York. 11. Todd J. (editor) (1962) Survey of Numerical Methods. McGraw-Hill, New York. 12. Tsakalis K.S., Ioannou P.A. (1993) Linear Time-Varying Plants: Control and Adaptation. Prentice-Hall, Englewood Cliffs, NJ.

14 S a m p l e d - D a t a C o n t r o l of N o n l i n e a r S y s t e m s : an O v e r v i e w of R e c e n t R e s u l t s Dragan Nesi51 and Andrew R. Teel 2 1 Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, 3010, Victoria, Australia 2 CCEC, Electrical and Computer Engineering Department, University of California, Santa Barbara, CA, 93106-9560 A b s t r a c t . Some recent results on design of controllers for nonlinear sampled-data systems are surveyed.

14.1

Introduction

The prevalence of digitally implemented controllers and the fact that the plant dynamics a n d / o r the controller are often nonlinear strongly motivate the area of nonlinear sampled-data control systems. When designing a controller for a sampleddata nonlinear plant, one is usually faced with an important intrinsic difficulty: the exact discrete-time/sampled-data model of the plant can not be found. The absence of a good model of a sampled-data nonlinear plant for a digital controller design has lead to several methods that use different types of approximate models for this purpose. We single out two such methods that attracted lots of attention in the sampled-data control literature and that consist of the following steps: M e t h o d 1: continuous-time plant model continuous-time controller

M e t h o d 2: continuous-time plant model discretize plant model

discretize controller

discrete-time controller

implement the controller

implement the controller

Hence, in Method 1 (this method is often referred to as the controller emulation design) one first designs a continuous time controller based on a continuous time plant model. At this step the sampling is completely ignored. Then, the obtained continuous time controller is discretized and implemented using a sampler and hold device. On the other hand, in Method 2 one first finds a discrete time model of the plant, then designs a discrete-time controller based on this discrete-time model and finally implements the designed discrete-time controller using a sampler and hold device. In principle, Method 2 is more straightforward for linear systems t h a n for nonlinear systems. Indeed, for linear systems we can write down an explicit, exact

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discrete-time model while typically for nonlinear systems we cannot. Moreover, the exact discrete time model of a linear system is linear while the exact discrete-time model for a sampled-data nonlinear system does not usually preserve important structures of the underlying continuous time nonlinear system, like affine controls for example. Consequently, for nonlinear systems it is unusual to assume knowledge of the exact discrete-time model of the plant whereas this assumption is made in most of classical linear literature. However, often due to our inability to exactly compute the matrix exponential that generates the exact discrete-time model of a linear plant, we use its approximation and hence we actually use an approximate discrete-time plant model for controller design most of the time, even for linear systems. Consequently, we will always assume in the sequel that in Method 2 we use an approximate discrete-time model of the plant for controller design. The main question in Method 1 is whether the desired properties of the continuous time closed loop system that the designed controller yielded will be preserved and, if so, in what sense for the closed-loop sampled-data system. Similarly, the central question in Method 2 is whether or not the properties of the closed-loop system consisting of the exact discrete-time plant model and the discrete-time controller will have similar properties as the closed-loop system consisting of the approximate discrete-time plant model and the discrete-time controller. In this paper, we present some answers to these questions for both methods. In particular, we overview several recent results on Methods 1 and 2 for general nonlinear plants that appeared in [21,27-31,41]. These results do not contain algorithms for controller design. In the case of Method 1, controller design is the topic of the area of continuous-time nonlinear control. In the case of Method 2, controller design algorithms are yet to be developed and our results provide a unified framework for doing this. The paper is organized as follows. In Section 2 we present results on Method 1 and in Section 3 we present results on Method 3. We do not present any proofs and the reader may refer to original references for proofs of all results.

14.2

M e t h o d 1: E m u l a t i o n

Many results for nonlinear sampled-data systems uses controller emulation because of the simplicity of the method and the fact that a wide range of continuous-time controller design methods can be directly used for design of digital controllers, see [3,33,34,42,41]. An important drawback of the existing nonlinear sampled-data theory is that only the questions of stability (see [3,33,34,42]) and input-to-state stability (see [41]) were addressed within the emulation design framework. However, one may be interested in a range of other important system theoretic properties, such as passivity and Lp stability. It turns out that a rather general notion of dissipativity that we consider ((V,w)-dissipativity, see Definition 1) can be used to cover a range of most important system theoretic properties within a unified framework. Special cases of (V, w)-dissipativity are: stability, input-to-state stability, passivity, Lp stability, forward completeness, unboundedness observability, etc. Results in this section are taken from [30] and are concerned with preservation of (V, w)-dissipativity under sampling in the emulation design for the cases of static state feedback controllers and open loop controls. Results on emulation of dynamic state feedback controllers can be found in [21] and they are not presented here for

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223

space reasons. Hence, results in [21,30] provide a general and unified f r a m e w o r k for digital controller design using M e t h o d 1.

14.2.1

Preliminaries

In order to s t a t e our results precisely, we need to s t a t e some definitions and a s s u m p tions. Sets of real and n a t u r a l n u m b e r s are respectively d e n o t e d as R and N. T h e Euclidean n o r m of a vector x is d e n o t e d as lxt. G i v e n a set B C R ~, its e n e i g h b o r hood is d e n o t e d as N ( B , ~) :---- {x : infz~B Ix -- z I ~ e}. A function ~ : R>o --, R>o is of class-Af if it is continuous and non-decreasing. It is of classJC if it is continuous, zero at zero and strictly increasing; it is of c l a s s - ~ if it is of class-]C and is unbounded. A continuous function f~ : R>o • R_>o --~ R_>o is of class-IC/Z if f~(., T) is of class-]C for each T __~ 0 and f~(s, .) is decreasing to zero for each s > 0. For a given function d(-), we use the following n o t a t i o n d[tl,t2] :---- (d(t) : t E [tl,t2]}. If tl ~- kT, t2 = (k -}- 1)T, we use t h e shorter n o t a t i o n d[k], and t a k e t h e n o r m of d[k] to be t h e s u p r e m u m of d(-) over [kT, (k + 1)T], t h a t is IId[k]ll~ = e s s

sup Id(~)ITE[kT,(k~-I)T]

Consider t h e c o n t i n u o u s - t i m e nonlinear plant:

= f ( x , u , dc,ds), y = h(x, u, de, d~),

(14.1) (14.2)

where x E R '~, u E R T M and y E R p are respectively t h e state, control i n p u t and the o u t p u t of the system and dc C R TM and d~ E R '~ are d i s t u r b a n c e i n p u t s to the system. It is assumed t h a t f and h are locally Lipschitz, f ( 0 , 0, 0, 0) ---- 0 and h(0,0,0,0) =0. A s t a r t i n g point in t h e e m u l a t i o n design is to assume t h a t a c o n t i n u o u s - t i m e (open-loop or closed loop system w i t h an a p p r o p r i a t e l y designed c o n t i n u o u s - t i m e controller) possesses a certain property, such as stability. We will assume in t h e sequel t h a t the continuous s y s t e m satisfies the following dissipation p r o p e r t y : D e f i n i t i o n 1. T h e s y s t e m (14.1), (14.2) is said to be ( V , w ) - d i s s i p a t i v e if t h e r e exist a continuously differentiable function V, called t h e storage function, and a continuous function w : R ~ • R T M • R n~ • R "~ ---* R, called t h e dissipation rate, such t h a t for all x E R n, u C R m, d~ E R TM , ds C R '~ t h e following holds:

~

f(x,u,d~,d~) <_ w(x,u,d~,d~).

(14.3)

(V, w)-dissipation is a r a t h e r general p r o p e r t y whose special cases are stability, i n p u t - t o - s t a t e stability, passivity, etc. Besides t he continuous- t i m e m o d e l (14.1), (14.2 ), we consider t he e x a c t discret et i m e m o d e l for (14.1), (14.2) w h e n some of t h e variables in f u n c t i o n f of (14.1), (14.2) are sampled or assumed piecewise constant. More precisely, let T > 0 be a sampling period and suppose t h a t u and d8 in f in (14.1) are c o n s t a n t d u r i n g t h e sampling intervals, so t h a t u(t) = u(kT) ----:u(k) and d~ (t) = d~ (kT) =: d~ (k), Vt C [kT, (k + 1)T), Vk > 0, and y is m e a s u r e d only at sampling i n s t a n t s kT, k >_ O.

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D. Nesid and A.R. Teel

The exact discrete-time model for the system (14.1), (14.2) is obtained by integrating the initial value problem

&(t) = f(x(t), u(k), d~(t), d~(k)),

(14.4)

with given d~(k), d~[k], u(k) and x0 = x(k), over the sampling interval [kT, ( k + l ) T ] . Let x(t) denotes the solution of the initial value problem (14.4) with given ds(k), dc[k], u(k) and x0 -- x(k). Then, we can write the exact discrete-time model for (14.1), (14.2) as:

x(k + 1) = x(k) +JkT f](k+l)T f(x(T),u(k),d~(T),d~(k))dT :=F~(x(k),u(k),d~[k],d~(k)), y(k) = h(x(k),u(k),d~(k),d~(k)).

(14.5) (14.6)

The sampling period T is assumed to be a design parameter which can be arbitrarily assigned. In practice, the sampling period T is fixed and our results could be used to determine if it is suitably small. We emphasize that F~ in (14.5) is not known in most cases. We denote d~ := d~(0) and use it in the sequel.

14.2.2

Main results

We now present a series of results which provide a general framework for the emulation design method. In Theorems 1 and 2 we respectively consider the "weak" and the "strong" dissipation inequalities for the exact discrete-time model of the sampled-data system. Each of these dissipation inequalities is useful in certain situations, as illustrated in the last subsection, where they are applied to problems of input-to-state stability and passivity. Then several corollaries are stated for the closed-loop system consisting of (14.1), (14.2) and different kinds of static state feedback controllers. Examples are presented to illustrate different conditions in main results. For simplicity we do not present results on emulation of dynamic feedback controllers and these results can be found in [21]. T h e o r e m 1. (Weak form of dissipation) If the system (14.1), (14.2) is (V,w)-

dissipative, then given any 6-tuple of strictly positive real numbers (A~,A~,Ad~,A3c,Ad~,p), there exists T* > 0 such that for alI T C (0, T*) and all Ixl < Ax, lul < An, Idsl <_ Ad s and functions de(t) that are iipschitz and satisfy Ildc[0]ll~ < Adc and (~c[0] ~ < Ad~, the following holds for the exact discrete-time model (14.5) of the system (14.1), (14.2): AV

T

._ V(F~(x,u,

"--

dc[O],ds)) - V ( x )

T

< w(x,u, dc,ds) +v

-

.

(14.7)

Under slightly stronger conditions we can prove a stronger result that is useful in some situations:

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225

L e m m a 1. / f the system (14.1), (14.2) is (V,w)-dissipative, with ow being lo-

cally Lipschitz and oy Ox (0~] = O, then numbers (Ax, A~, Ad~ , A3c , Ads), K 1 , K 2 , K 3 , K 4 , K 5 such that for all [dsI ~_ Ads and functions de(t) that dc[0] ~_ A3~, we have:

given any quintuple of strictly positive real there exist T* > 0 and positive constants T 9 (O,T*) and all Ixl ~ A~, [u I ~ A~,, are Lipschitz and satisfy Ildc[0]l[~ ~ Age,

oo

AV < w ( x , u , dc,d~) + T -

T

K I I x l 2 + g2luJ 2 + K31d~I 2 + K4

IJdc[0llJ~ + K5 dc[0] ~

.

(14.8)

In the following theorem we use the strong form of dissipation inequality for the exact discrete-time model. This result is much more n a t u r a l to use in the situations when the disturbances d~ are not globally Lipschitz (see the ISS application in the next subsection and Example 1). 2. (Strong form of dissipation) If the system (14.1), (14.2) is (V,w)dissipative, then given any quintuple of strictly positive real numbers (/~x, An, /~dc,Ads, ll) there exists T* > 0 such that for all T 9 (O,T*) and all ]x I < A~, In[ < A , , Hdc[0]l[~ < J~dc, and ]d~] ~ Ads the following holds for the system (14.5):

T h e o r e m

A V < 1 .~T --~ _ ~ W(X, U, d~(T), d~)dT + u .

(14.9)

It is very important to state and prove Theorems 1 and 2 for the case when a feedback controller is used to achieve (V, w)-dissipativity of the closed loop system. To prove result on weak dissipation inequalities, we consider the situation when static state feedback controller of the form:

u = u(x,d~,d~)

(14.10)

is applied to the system (14.1), (14.2). It is assumed below t h a t the feedback (14.10) is bounded on compact sets. Note that this general form of feedback covers both, the full state (u = u(x)) and o u t p u t (u -- u(y)) static feedback. Note also, t h a t the dissipation rate for the closed-loop system (14.1), (14.2) and (14.10) in the definition of (V,w)-dissipativity can be taken as w = w(x,d~,ds). Direct consequences of Theorem 1 and Lemma 1 are the following corollaries: C o r o l l a r y 1. If the system (14.1), (14.2), (14.10) is (V, w)-dissipative, then given any quintuple of strictly positive real numbers (Ax, ~dc, /~dc' /~ds, V), there exists T* > 0 such that for alI T 9 (0, T*) and all Ixl ~_ A~, td~l ~ Ad~ and all functions

d~(t) that are Lipschitz and satisfy ]]d~[0]ll~ ~ A4~, (~[0] ~ _~ Ad~, the following holds for the discrete-time model of the closed-loop system (14.1), (14.2), (14.10): AV - - < w(x,d~,d~) + u . T

-

(14.11)

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D. Nesid a n d A.R. Teel

C o r o l l a r y 2. If the system (14.1), (14.2), (14.10)is (V,w)-dissipative, with ov Ox and u(x,d~,ds) in (14.10) being locally Lipschitz and ~Dx k ] = 0 and u ( 0 , 0 , 0 ) -- 0,

then given any quadT"uple of strictly positive real numbers ( /i~, ~ida,/IJU' A d s ) , there exists T* > 0 and positive constants K1, K2, g 3 , Ka such that for all T E (0, T*) and all Ix[ < A~, Id~] <_ /id~ and functions d~(t) that are Lipschitz and satisfy IId~[0]H~ -< /ia~ and ~l~[O] ~r _< /ida, the closed-loop discrete-time model for system (1~.1), (1~.2) and (1~.10) satisfies: ~IV < w(x, dc, d,) + T T

(

K1]xl2+K21dsI2+K3Hdc[Olll~+K4

dc[O]

2) B

It is interesting to note t h a t the condition on the derivative dc in T h e o r e m 1 is necessary to prove the weak dissipation inequality for the discrete-time system. T h e following example illustrates this.

Example 1. Consider the continuous t i m e system: = u(x) + dc = - x + de,

(14.12)

where x, dc E R. Using the storage function V ---- ~x 1 2 , the derivative of V is V ---_x2+xdc < 2 a n d (14.12) is ISS. We will show t h a t if d~(t) -- cos ( k ~ T ) _ - ~ x1 2 + 1dc, the claim of T h e o r e m 1 does not hold since d~ ~ =

-Tsin(~-)

~ =--T'I

(14.13)

which goes to infinity as T ---* 0. A s s u m e t h a t u(x) in (14.12) is piecewise cons t a n t for the s a m p l e d - d a t a system. So, the discrete-time model of the s a m p l e d - d a t a system is

x(k + 1) = (1 - T ) x ( k ) +

f (k-t-1)Tcos ( ~ _ )

dT,

(14.14)

JkT

a n d hence the exact discrete-time model is given by:

x(k+l)

= (1-T)x(k)+T[sin(k+3)-sin(k+2)],

Vk_>0.

(14.15)

Z~dc

Suppose t h a t for any given A s , a n d v, there exists T* > 0 such t h a t for all T E (0, T*) a n d k k 0 with Ix] < Ax a n d Hd~[0]llo~ < Adc we have AV < _lx2 + 1 2 T ~dc + u.

(14.16)

We show by contradiction t h a t the claim is not true for our case. Direct c o m p u t a tions yield: AV T

((1 - T ) x + T [sin(3) - sin(2)]) 2 - x 2 2T = - x 2 + x [sin(3) - sin(2)] + O(T).

(14.17)

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227

Let 9 = - 0 . 5 , (and hence A~ = 0.5, A ~ = 1). B y c o m b i n i n g (14.16) and (14.17) we conclude t h a t there should exist T* :> 0 such t h a t VT E (0, T*) we obtain:

_

1~2 + 2[sin(3) - sin(2)] - ~1 cos(2)2 _ v + O ( T ) < _ 0 ,

and since there exists we can rewrite (14.18) hold for u E (0, v*)). T* > 0, such t h a t VT

(14.18)

v* > 0 such t h a t - s 1x ~2 + ~ [sin(3) - sin(2)] - 89cos(2) 2 ~-- v* as v* - v + O ( T ) _~ 0 , which is a c o n t r a d i c t i o n (it does not Therefore, for A s ---- 0.5, Ad~ ~-- 1, b, < v*, t h e r e exists no E (0, T*) t h e condition (14.16) holds. N o t e t h a t t h e chosen

d~(t) does not satisfy the condition

d~ ~ ~ Ad~ for some fixed Ado > 0, which is

evident from (14.13). Hence, in this case we can not a p p l y T h e o r e m 1.

9

To s t a t e results on strong dissipation inequalities w i t h static s t a t e feedback controllers, we need to consider controllers of t h e following form:

u = u(x,d~),

(14.19)

in order to be able to s t a t e a general results on s t r o n g dissipation inequalities. T h i s is illustrated by the following example.

Example 2. Consider t h e system & = u, and u = - d ~ , where d~(0) = 0 and d~(t) = 1, Yt > 0. Using V(x) = x, such t h a t -ov[ ~ - ~ - d~ ~j = - d ~ and w(x,d~,d~) = -d~. Since u is sampled and d~(0) ----0, we have t h a t x(t) = O, Yt E [0, T] and so A V / T = O. O n the o t h e r hand f 7 w(d~(T))d~- = - T . Hence, if (14.20) was true, t h e n we would obtain o b t a i n 0 _~ - 1 + v, which is not true for small p.

9

3. If the system (14.1), (14.2), (14.19) is (V, w)-dissipative, then given any quadruple of strictly positive real numbers ( A s , Ado, Ads, u), there exists T* > 0 such that for aU T E (0, T*) and all Ix] _~ Ax, ][d~[0]][~ ~_ Adr and Ida] ~ Ad s the following holds for the closed-loop discrete-time model of the system (14.1), (14.2) and (14.19): Corollary

AV <

T -T

1/7

w(x,d~(r),d~)d~- + ~

(14.20)

4. If the system (14.1), (14.2), (14.19) is (V,w)-dissipative, with O V and u(x, ds) in (14.19) being locally Lipsehitz and ~O x \(o~! ---- 0, u ( 0 , 0 ) = O, then given any triple of strictly positive real numbers ( A~, Ado , Ad~), there exists T* > 0 and positive constants K 1 , K 2 , K 3 such that for all T E (0, T*) and all Ix] _~ A s , ]]dc[0]][~ ~ Age, and ]ds[ ~_ Ad s the closed-loop discrete-time model for the system (14.1), (14.2) and (14.19) satisfies: Corollary

AV < T -T

w(x, dc(T),ds)dT + T (K1]x]2 + K2[dsl2 + K3i[dr

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D. Nesid and A.R. Teel

14.2.3

Applications

The weak and strong discrete-time dissipation inequalities in Theorems 1 and 2 are tools that can be used to show t h a t the trajectories of the s a m p l e d - d a t a system with an emulated controller have a certain property. In order to illustrate what kind of properties can be proved for the s a m p l e d - d a t a system using the weak or strong inequalities, we apply our results to two i m p o r t a n t system properties: input-to-state stability and passivity. The results on input-to-state stability were proved in [30] and in [31]. The sketch of proof of the result on passivity can be found in [30]. Further applications of weak and strong inequalities to investigation of Lp stability, integral ISS, etc. are possible and axe interesting topics for further research. Input-to-state

stability:

Let us suppose that the system

&(t) = ](x(t), u(t), de(t))

(14.21)

can be rendered ISS using the locally Lipschitz static state feedback

u = u(x),

(14.22)

in the following sense: D e f i n i t i o n 2. The system & = f ( x , dc) is input-to-state stable if there exists/3 E ]C/: and ~/E ]C such t h a t the solutions of the system satisfy tx(t)l <_ fl(Ix(0)[ ,t) +

7(lld~lio~), Vx(O),d~

E

z:~,vt > o.

9

Suppose also that the feedback needs to be implemented using a sampler and zero order hold, t h a t is:

u(t) = u(x(k))

t e [kT, (k + 1)T), k > 0

(14.23)

The following result was first proved in [41] and an alternative proof was presented in [30]. The proof in [30] is based on the result on strong dissipation inequalities given in Corollary 3 and the results in [27]. In this case, the results on weak dissipation inequalities could not be used. This is because we do not want to impose the condition t h a t the disturbances are Lipschitz when proving the following result, and t h a t is a standing assumption in results on weak dissipation inequalities. C o r o l l a r y 5. If the continuous time system (14.21), (14.22) is ISS, then there exist fl E ]Cf~, ~f E ]C such that given any triple of strictly positive numbers (A~, Ado , u), there exists T* > 0 such that for all T E (0, T*), Ix(t0)l _< Ax, IId~ll~ _< nuc, the solutions of the sampled-data system (14.21), (14.23) satisfy:

Ix(t)l < ~ ( I x ( t o ) l , t vt > to > o.

- to) + 7(lld~ll~) + v,

(14.24) 9

Corollary 5 states t h a t if the continuous-time closed loop system is ISS, then the sampled-data system with the emulated controller will be semiglobally practically ISS, where the sampling period is the p a r a m e t e r t h a t we can adjust. Besides the above given property that is presented in an Lo~ setting, we can prove the following integral (or L2) version of the same result t h a t was proved in [31].

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Sampled-Data Control of Nonlinear Systems

229

C o r o l l a r y 6. If the system (14.21), (14.22) is ISS, then given any quadruple of

strictly positive real numbers (Ax, Aw, Vl, u2) there exists T* > 0 such that for all T E (0, T*), ]x(O)l ~_ A z and ]lwll~ ~_ A ~ , the following inequality holds for the

sampled-data system (14.21), (14.23) satisfy: t a3(Ix(s)l) ds ~ a2(Ix(O)l) +

7(Iw(s)l)ds + vlt + v2,

for all t > O. Passivity:

(14.25)

[] Consider the continuous time system with outputs

= f(x,u), y = h(x,u),

(14.26)

where x E R n, y, u E R m and assume that the system is passive, that is (V, w)dissipative, where V : R" --~ R_>0 and w = yTu. We can apply either results of Theorem 1 or 2 since u is a piecewise constant input, to obtain that the discretetime model satisfies the following: for any (A~, An, v) there exists T* > 0 such that for all T E (0, T*), Ixl ~ Ax, lul _~ Au we have:

,~V < yT u -~ v. T

(14.27)

-

In stability and ISS applications, adding v in the dissipation inequality deteriorated the property, but the deterioration was gradual. However, in (14.27) v acts as an infinite energy storage (finite power source) and hence it contradicts the definition of a passive system as one that can not generate power internally. As a result, conditions which guarantee that u in (14.27) can be set to zero are very important. These conditions are spelled out in the next corollary: C o r o l l a r y 7. Suppose that the system (14.26) is strictly input and state passive in

the following sense: the storage function has gradient oy that is locally Lipschitz and zero at zero and the dissipation rate can be taken as w(x, y, u) = yT u--r (x)--r where r and r are positive definite functions that are locally quadratic. Then given any pair of strictly positive numbers (A~, A~) there exists T* > 0 such that for all T ~ (0, T*), Ixl <_ ~ , lul <_ Z~ we have: AV < yTuT

14.3

-

1r

1r

(14.28)

-

Method 2: Approximate discrete-time model design

As we already indicated in the introduction, a majority of nonlinear and linear discrete-time control literature is based on the assumption, which is unrealistic in general, that the exact discrete-time model of the plant is known. In reality, even in the case of linear systems we use an approximate discrete-time model for the discrete-time controller design. We recognize this fact and whenever we refer

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D. Nesid and A.R. Teel

to Method 2, we always assume that only an approximate discrete-time model is available for the controller design. One may be tempted to believe that if a controller is designed for an approximate discrete-time model of the plant with a sufficiently small sampling period then the same controller will stabilize the exact discrete-time model. Note that if this was true then one could directly apply all the existing theory that assumes that the exact discrete-time model is known. However, this reasoning is wrong since no matter how small the sampling period is, we may always find a controller that stabilizes the approximate model for that sampling period b u t destabilizes the exact model for the same sampling period, as illustrated by the following example.

Example 3. Consider the triple integrator ~1 = x2 , 22 = x3 , 23 = u, its Euler approximate model xl(k + 1) = x~(k) + Tx2(k) x2(k + 1) = x2(k) + Tx3(k) x3(k + 1) = x3(k) + Tu(k) ,

(14.29)

and a minimum-time dead-beat controller for the Euler discrete-time model given by u(k)= (x~3k)

3x2(k)T 2

3x~k)) .

(14.30)

The closed loop system consisting of (14.29) and (14.30) has all poles equal to zero for all T > 0 and hence this discrete-time Euler-based closed loop system is asymptotically stable for all T > 0. On the other hand, the closed loop system consisting of the exact discrete-time model of the triple integrator and controller (14.30) has a pole at ~ -2.644 for all T > 0. Hence, the closed-loop sampleddata control system is unstable for all T > 0 (and, hence, also for arbitrarily small T)! So we see that, to design a stabilizing controller using Method 2, it is not sufficient to design a stabilizer for an approximate discrete-time model of the plant for sufficiently small T. Extra conditions are needed! Several control laws in the literature have been designed based on approximate discrete-time models of the plant, see [8,12,23]. These results are always concerned with a particular plant model and a particular approximate discrete-time model (usually the Euler approximation) and hence they are not very general. On the other hand, we present a rather general result for a large class of plants, a large class of approximate discrete-time models and the conditions we obtain are readily checkable. For different approximate discretization procedures see [25,38,39,37]. Results in this section are based mainly on [28] and they address the design of stabilizing static state feedback controllers based on approximate discrete-time plant models. A more general result was recently proved in [29] where conditions are presented for dynamic feedback controllers that are designed for approximate discrete-time models of sampled-data differential inclusions to prove stability with respect to arbitrary non-compact sets. 14.3.1

Main results

Consider the nonlinear continuous-time control system: = f ( x , u)

x ( 0 ) = ~o

(14.31)

14

Sampled-Data Control of Nonlinear Systems

231

where x E R ~, u E R TM are respectively the state and control vectors. The function f is assumed to be such that, for each initial condition and each constant control, there exists a unique solution defined on some (perhaps bounded) interval of the form [0, T). The control is taken to be a piecewise constant signal u(t) = u(kT) ----: u(k),Vt E [kT, (k + 1)T[, k E N where T > 0 is a sampling period. We assume that the state measurements x(k) := x(kT) are available at sampling instants kT, k C IN. The sampling period T is assumed to be a design parameter which can be arbitrarily assigned (in practice, the sampling period T is fixed and our results could be used to determine if it is suitably small). Suppose that we want to design a control law for the plant (14.31) using Method 2. The controller will be implemented digitally using a sampler and zero order hold element. As a first step in this direction we need to specify a discrete-time model of the plant (14.31), which describes the behavior of the system at sampling instants. We consider the difference equations corresponding to the exact plant model and its approximation respectively:

x(k + 1) = F~(x(k), u(k)) x(k + 1) = F~(x(k), u(k))

(14.32) (14.33)

which are parameterized with the sampling period T. We emphasize that F~ is not known in most cases. We will think of F~ and F~ as being defined globally for all small T even though the initial value problem (14.31) may exhibit finite escape times. We do this by defining F~ arbitrarily for pairs (x(k), u(k)) corresponding to finite escapes and noting that such points correspond only to points of arbitrarily large norm as T --* 0, at least when f is locally bounded. So, the behavior of F~ will reflect the behavior of (14.31) as long as (x(k),uT(x(k))) remains bounded with a bound that is allowed to grow as T -~ 0. This is consistent with our main results that guarantee practical asymptotic stability that is semiglobal in the sampling period, i.e., as T --* 0 the set of points from which convergence to an arbitrarily small ball is guaranteed to contain an arbitrarily large neighborhood of the origin. In general, one needs to use small sampling periods T since the approximate plant model is a good approximation of the exact model typically only for small T. It is clear then that we need to be able to obtain a controller UT(X) which is, in general, parameterized by T and which is defined for all small T. For simplicity, we consider only static state feedback controllers. For a fixed T > 0, consider systems (14.32), (14.33) with a given controller u(k) = UT(x(k)). We denote the state of the closed-loop system (14.32) (respectively (14.33)) with the given controller at time step k that starts from x(0) as xe(k, x(0)) or x~ (respectively xa(k, x(0)) or x~). We introduce the error: Ek(~,z) := xe(k,~) - x~(k,z),

(14.34)

and also use the notation ~k(~) :-- ek(~, ~) -- xe(k, ~) - xa(k, ~). In our results (see Theorems 3 and 4), we will make a stability assumption on the family of closed-loop approximate plant models and will draw a conclusion about stability of the family of closed-loop exact plant models by invoking assumptions about the closeness of solutions between the two families. O n e - s t e p c o n s i s t e n c y The first type of closeness we will use is characterized in the following definition. It guarantees that the error between solutions starting from

232

D. Nesid and A.R. Teel

the same initial condition is small, over one step, relative to the size of the step. The terminology we use is based on that used in the numerical analysis literature (see [39]). D e f i n i t i o n 3. The family (UT, F~) is said to be one-step consistent with (UT, F~) if, for each compact set 2( C R ~, there exist a function p E /Coo and T* > 0 such that, for all x E X and T E]0, T*[, we have [F~(x, UT(X)) -- F ~ ( x ,

UT(X)) I ~_~T p ( T ) .

(14.35)

A sufficient condition for one-step consistency is the following: Lemma

2.

If

1. (uT,F~.) is one-step consistent with (UT, F Euler) where FTEUIer(X,U) := X + T f ( x , u), 2. for each compact set .~ C R '~ there exist p E ]Coo, M > O, T* > 0 such that, for all T E]0, T*[ and all x , y E X , (a) II(y,u~(~))l < M, (b) I I ( y , u ~ ( ~ ) ) - f ( ~ , ~ ( ~ ) ) l < P(lY - ~l), then (UT , F~ ) is one-step consistent with (UT , F~ ). M u l t l - s t e p c o n s i s t e n c y The second type of closeness we will use is characterized in terms of the functions F~, F~ and UT(X) in the next definition. It will guarantee (see Lemma 3) that the error between solutions starting from the same initial condition is small over multiple steps corresponding "continuous-time" intervals with length of order one. D e f i n i t i o n 4. The family (UT, F~) is said to be multi-step consistent with (UT, F~) if, for each L > 0, ~ > 0 and each compact set X C R n, there exist a function a : R>o x R>0 --~ R>0 U {co} and T* > 0 such that, for all T El0, T* [ we have that {x,z E X , Ix - z I < 8} implies

]F~(x, UT(X)) -- F~(z, UT(Z))] < ~(5, T)

(14.36)

and k

k <_ L I T

==~

c~k(0,T) :----~(---c~(c~'(0,T ) , T ) . . .

,T) < ~ .

(14.37)

In terms of trajectory error over "continuous-time" intervals with length of order one, multi-step consistency gives the following: 3. If (UT, F~ ) is multi-step consistent with (UT, F~ ) then for each compact set X C R "~, L > 0 and ~? > 0 there exists T > 0 such that, if T and ~ satisfy

Lemma

Vk:kTE[O,L],

T E]0, T[ ,

x~(~) E X

I~k(~)I < ~

Vk: k T E [0, L] .

(14.38)

then (14.39)

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233

An interesting sufficient condition for multi-step consistency is given in the following: L e m m a 4 . If, for each compact set X C R n, there exist K > O, p E ICoo and T* > 0 such that for all T E]0, T*[ and all x , z E X we have

[F~(x, uT(x) ) -- F~(z, uT(z) ) I ~ (1 + K T ) Ix - z[ + T p ( T )

(14.40)

then (Ur , F~ ) is multi-step consistent with (UT , F~ ). Relative to the one-step consistency condition, the condition of L e m m a 4 is guaranteed by one-step consistency plus the following type of Lipschitz condition on either the family (UT, F~) or the family (uT, F~): for each compact set X C R ~ there exist K > 0 and T* > 0 such t h a t for all x , z E X and all T E]0, T*[,

IFT(x, UT(X)) -- FT(Z, UT(Z)) I < (1 + K T ) I x -- z I .

(14.41)

This condition is guaranteed for F~ when f ( x , u) and UT(X) are locally Lipschitz (uniformly in small T). Note that no continuity assumptions on UT(X) were made in Lemma 2 to guarantee one-step consistency. The condition given in L e m m a 4 for multi-step consistency is similar to conditions used in the numerical analysis literature (e.g., see conditions (i) and (iii) of Assumption 6.1.2 in [39, pg.429]). One-step and multi-step consistency do not imply each other and this is one motivation for developing different stability theorems that rely on one of these properties. Example 4 in Section 14.3.3 shows t h a t multi-step consistency may not hold when one-step consistency does hold. T h a t one-step consistency m a y not hold when multi-step consistency does hold can be seen from the plant ~ = x + u with Euler approximation x(k + 1) = x(k) + T ( x ( k ) + u(k)) = F~(x(k), u(k)) and controller uT(x) ----- - ( ~ + 1)x. The exact discrete-time model is x ( k + 1) = e T x ( k ) +

(e T -- i)u(k) and we have F~,(X, UT(X)) =-- 0 and T:~(X, UT(X)) = _(1 -- ~--~-2) X . Since, for x in a compact set, F~(x, uT(x)) is of order T we do not have one-step consistency. On the other hand, it follows from F~.(x, UT(X)) ==-0 and the fact t h a t F~(x, UT(X)) is of order T that we do have multi-step consistency. Indeed, for each compact set X C R and each ~ > 0 there exist strictly positive numbers K, T* such that, for all x , z E 2~, T El0, T*[, k > 0, e IF:~(x, UT(X))

F~(z, u:~(~))l a

-

e

= IF&(x, uT(x))[

_< K T

:= a(5, T) = ak(O,T) ~ ~ . 14.3.2

Stability

properties

We now give conditions on the family (UT, F~) t h a t guarantee asymptotic stability for the family (UT, F~). As we have already seen in Example 3, it is not enough to assume simply t h a t each member of the family (UT, F~) is asymptotically stable (at least for small T). Instead, we will impose partial uniformity of the stability property over all small T. For that, we make the following definitions: D e f i n i t i o n 5. Let/3 E/C/: and let N C R '~ be an open (not necessarily bounded) set containing the origin.

234

D. Nesi6 and A.R. Teel

1. The family (UT, FT) is said to be (f~, N)-stable if there exists T* > 0 such that for each T El0, T* [, the solutions of the system

x ( k + 1) = FT(x(k), UT(X(k)))

(14.42)

satisfy

Ix(k,x(O))l

<_/3(tx(0)l

, k T ) , Vx(0) E N, a >_ o.

(14.43)

2. The family (uT, FT) is said to be (8, N)-practically stable if for each R > 0 there exists T* > 0 such that for each T E]0, T*[ the solutions of (14.42) satisfy:

Ix(k,x(O))l _ O.

(14.44)

An equivalent Lyapunov formulation of (/3, R n)-stability is the following (local versions can also be formulated but are more tedious to state because of the need to keep track of basins of attraction): L e m m a 5. The following statements are equivalent:

1. There exists/3 E ]CL such that the family (uT,FT) is (~,R'~)-stable. 2. There exist T* > O, 41,42 E ]Cr 43 E ~ and for each T E]O,T*[,VT : R n --) R>_0 such that Vx E R~,VT E]0, T*[ we have: al(Ixl) _< VT(X) <_ a2(Ixl), VT( FT(x, UT(X) ) ) -- VT(X) < -T~3(lxl), 9

(14.45) (14.46)

In our first main result (Theorem 3) we will show that if the family (UT, F~) is (/3, N)-stable and multi-step consistent with (UT, F~) then the family (UT, F~.) is (/3, N)-practically stable. We will also show (in Theorem 4) that the multi-step consistency assumption can be changed to a one-step consistency assumption when (/3, N)-stability is formulated in terms of a family of Lyapunov functions satisfying (14.45),(14.46) and with an extra local Lipschitz condition that is uniform in small T. D e f i n i t i o n 6. The family (UT, FT) is said to be equi-globally asymptotically stable (EGAS) by equi-Lipschitz Lyapunov functions if the second statement of Lemma 5 holds and, moreover, for each compact set 2r C R'~\ {0} there exist M > 0 and T* > 0 such that, for all x , z E k' and all T E]0, T*[,

]VT(x) -- VT(Z)] < M i x - z] .

(14.47)

Our first result is expressed in terms of trajectory bounds for (uT, F~) and multi-step consistency: T h e o r e m 3. Let/3 E ]CL and let N be a bounded neighborhood of the origin. If the

family (UT,F~.) is: A: multi-step consistent with (UT , F~ ), and B: (fl, N)-stable,

then C: the family (UT, F~) is (/3, N)-practically stable.

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Sampled-Data Control of Nonlinear Systems

235

Our second result is expressed in terms of a family of Lyapunov functions for

(UT, F~,) and one-step consistency. It has some relations to the proof technique used to establish the main result of [7]. For simplicity we will only formulate the global result. Nonglobal results and results for stability of sets other t h a n the origin can also be established with the same proof technique.

T h e o r e m 4. If A I : (UT,F~) is one-step consistent with (itT, F~.), and B I : (UT, F~) is EGAS by equi-Lipschitz Lyapunov functions

then C I : there exists ~ C 1~s such that, for each bounded neighborhood N of the origin,

the family (UT, F~.) is (8, N)-practically stable. 14.3.3

Examples

Example 4 illustrates Theorem 4 by giving an example where multi-step consistency does not hold but one-step consistency does hold and there is a suitable family of Lyapunov functions. Example 5 shows situation where each element of the family (UT, F~) is globally exponentially stable with overshoots uniform in T but where the family (UT, F~) fails to be (8, N)-practically stable for any pair (/3, N). In reference to Theorem 3, we use the notation C for this situation.

Example 4. Consider the two-input linear system &l = xl + Ul

x2

(14.48)

~---U2

w h i c h has exact discretization

xl(k + 1) ----eTxl(k) + [eT - 1]ul(k) x2(k + 1) = x2(k) + Tu2(k)

(14.49)

and Euler approximate discretization

xl(k + 1) = [1 + T]xl(k) + Tul(k) x2(k + 1) ----x2(k) + Tu2(k) .

(14.50)

Consider the controller

u(x) =

[ :] (14.51) - 2 x l 1 otherwise. --X2 J

A 1 , A : It follows from Lemma 2 t h a t (uT,F~) is one-step consistent with (uT,F~). However, (uT, F~) is not multi-step consistent with (uT,F~). Indeed, consider the initial condition (~1, ~2) = (1, 0.1). It is easy to see that, in this case,

236

D. Nesid a n d A.R. Teel

(x1(k, ~), x~(k, ~)) = ( 1 - - T ) k ( 1 , 0.1), i.e., the positive ray x2 = 0.1xl > 0 is forward invariant for all T E (0, 1). O n the other hand, (x~(1, ~), x~(1, ~)) = ((2 - eT)l, (1 -T)0.1), i.e., for all small T > 0, x~(1,~) < 10x~(1,~) a n d x~(1,~) > 0.1x~(1,~) since e T > 1 + T. It follows t h a t , for k > 1, x(k, ~) will take values on the horizontal line given by x2 = (1 - T)0.1 moving in the direction of decreasing x l u n t i l it crosses the positive ray x2 = 10xl. Let fr denote the n u m b e r of steps required to cross t h e positive ray x2 = 10Xl. It is easy to p u t a n u p p e r a n d lower b o u n d on k T t h a t is i n d e p e n d e n t of T. T h e n since, for all k _< k, we have x~(k,~) = (1 - T)0.1 while x~(k, ~) ----(1 - T)k0.1 < e-kTo.1, it is clear t h a t t h e conclusion of L e m m a 3 is n o t satisfied. Hence (UT, F~) c a n n o t be multi-step consistent with (UT, F~). B I : We take VT(X) = IXll "~ IX21. We get, for T E (0, 1) a n d 0 < 0.1Xl < x2 < 10xl: VT( F~.(X, UT(X) ) ) -- Y(x) ----- T l x l [ < - T [ ] x l I + Ix21]

(14.52)

and, otherwise,

VT(F~(x, UT(X) ) ) -- V(x) = -T[IXl I + Ix2]] .

(14.53)

It follows t h a t the family (UT, F~) is E G A S by equi-Lipschitz L y a p u n o v functions. C I : We conclude from T h e o r e m 4 (and also using t h e homogeneity of VT(X) a n d F~(x, UT(X)) to pass from a semiglobal practical result to a global result a n d following the steps of the proof of T h e o r e m 4 to get a n exponential result) t h a t the f a m i l y (UT, F~,) is (13,R2)-stable with ~ ( s , t ) of t h e form ks exp(-At) with k > 0 a n d A > 0.

Example 5. (A, B, C) Consider the double integrator, its Euler a p p r o x i m a t i o n a n d its exact discrete-time model: double integrator:

Xl = X2

52 = u

(14.54)

approximate: Xl(k + 1) = xl(k) + Tx2(k)

x2(k + 1) -- x2(k) + Tu(k)

(14.55)

exact: Xl(k + 1) = xl(k) + Tx2(k) + 0.5T2u(k)

x2(k + 1) = x2(k) + Tu(k).

(14.56)

T h e following controller is designed for the Euler model:

u(x)-

xl T

2x2 T

(14.57)

T, A2 = --1,VT > 0 C : The eigenvalues of the exact closed-loop are At = 1 - 7 and thus the exact dosed-loop mode] is not (/3, N)-practically stable for any pair -

-

(~3, N). A : T h e eigenvalues of the Euler closed-loop system are A1 -- + x / T - T , A2 = - x / 1 - T. In a similar way as in the previous example we can show t h a t there exists b > 0 such t h a t for all T El0, 0.5[ we have: ]x(k)] < bexp(-O.hkT)Ix(0)[, Vx(0) e R 2, Hence, the a p p r o x i m a t e closed-loop system is (B, R 2)-stable with/3(s, t) : = b e x p ( - 0 . 5 t ) .

14

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237

B : It now follows from Theorem 3 t h a t (UT,F~) is not multi-step consistent with (UT, F~). In fact, (UT, F~) is not one-step consistent with (UT, F~) since

[el(x)[ -----[ T ~ / 2 ( - x l / T - 2x2/T)[ = 2T [xl q- x2[ ,Vx 9 R2,VT.

14.4

Conclusion

Several recent results on design of s a m p l e d - d a t a controllers that a p p e a r e d in [21,2731] were overviewed. These results are geared toward providing a unified framework for the digital controller design based either on the continuous-time plant model (Method 1) or on an approximate discrete-time plant model (Method 2). The conditions we presented are easily checkable and the results are applicable to a wide range of plants, controllers and system theoretic properties. Fklrther research is needed to provide control design algorithms based on approximate discrete-time models. Our results on Method 2 provide a unified framework for doing so.

References 1. J. P. Barbot, S. Monaco, D. Normand-Cyrot and N. Pantalos, Discretization schemes for nonlinear singularly perturbed systems, Proc. CDC'91, pp. 443-448. 2. C. I. Byrnes and W. Lin, "Losslessness, feedback equivalence and the global stabilization of discrete-time systems", IEEE Trans. Automat. Contr., 39 (1994), pp. 83-97. 3. B. Castillo, S. Di Gennaro, S. Monaco and D. Normand-Cyrot, On regulation under sampling, IEEE Trans. A u t o m a t . Contr., 42 (1997), pp. 864-868. 4. P. D. Christofides and A. R. Teel, Singular perturbations and input-to-state stability, IEEE Trans. Automat. Contr., 41 (1996), pp. 1645-1650. 5. T. Chen and B. A. Francis, Input-output stability of sampled-data systems, I E E E Trans. Automat. Contr., 36 (1991), pp. 50-58. 6. T. Chen and B. A. Francis, O p t i m a l sampled-data control systems. SpringerVerlag: London, 1995. 7. F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Trans. A u t o m a t . Contr., 42 (1997), pp. 1394-1407. 8. D. Dochain and G. Bastin, Adaptive identification and control algorithms for nonlinear bacterial growth systems, Automatica, 20 (1984), pp. 621-634. 9. G. F. Franklin, J. D. Powell and M. L. Workman, Digital control of dynamic systems. Addison-Wesley Pub. Co. Inc.: Reading, 1990. 10. B. A. Francis and T. T. Georgiou, Stability theory for linear time-invariant plants with periodic digital controllers, IEEE Trans. Automat. Contr., vol. 33 (1988), pp. 820-832. 11. S. T. Glad, Output dead-beat control for nonlinear systems with one zero at infinity, Systems and Control Letters, 9 (1987), pp. 249-255. 12. G. C. Goodwin, B. McInnis and R. S. Long, Adaptive control algorithm for waste water treatment ad pH neutralization, Optimal Contr. Applic. Meth., 3 (1982), pp. 443-459. 13. L. Grfine, Input-to-state stability of exponentially stabilized semi-linear control systems with inhomogeneous perturbations, preprint (1998).

238

D. Nesid and A.R. Teel

14. L. Hou, A. N. Michel and H. Ye, Some qualitative properties of sampled-data control systems, IEEE Trans. A u t o m a t . Contr., 42 (1997), pp. 1721-1725. 15. P. Iglesias, Input-output stability of sampled-data linear time-varying systems, IEEE Trans. Automat. Contr., 40 (1995), pp. 1646-1650. 16. A. Iserles and G. SSderlind, Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), pp. 97-112. 17. N. Kazatzis and C. Kravaris, System-theoretic properties of sampled-data representations of nonlinear systems obtained via Taylor-Lie series, Int. J. Control, 67 (1997), pp. 997-1020. 18. H. K. Khalil, Nonlinear systems. Prentice-Hall: New Jersey, 1996. 19. P.E. Kloeden and J. Lorenz. Stable attracting sets in dynamical systems and their one-step discretizations, SIAM J. Num. Anal., 23 (1986), pp. 986-995. 20. B. C. Kuo, Digital control systems. Saunders College Publishing: Ft. Worth, 1992. 21. D. S. Laila and D. Nevsi(~, A note on preservation of dissipation inequalities under sampling: the dynamic feedback case, s u b m i t t e d to Amer. Contr. Conf., 2001. 22. V. Lakshmikantham and S. Leela, Differential and integral inequalities, vol. 1. Academic Press: New York, 1969. 23. I. M. Y. Mareels, H. B. Penfold and R. J. Evans, Controlling nonlinear timevarying systems via Euler approximations, Automatica, 28 (1992), pp. 681-696. 24. S. Monaco and D. Normand-Cyrot, Zero dynamics of sampled nonlinear systerns, Syst. Contr. Lett., 11 (1988), pp. 229-234. 25. S. Monaco and D. Normand-Cyrot, Sampling of a linear analytic control system, Proc. CDC'85, pp. 1457-1462. 26. M. S. Mousa, R. K. Miller and A. N. Michel, Stability analysis of hybrid compos-

ite dynamical systems: descriptions involving operators and difference equations, IEEE Trans. Automat. Contr., 31 (1986), pp. 603-615. 27. D. Nevsid, A. R. Teel and E.D.Sontag, Formulas relating ~f~ stability estimates of discrete-time and sampled-data nonlinear systems, Sys. Contr. Lett., 38 (1999), pp. 49-60. 28. D. Nevsid, A. R. Teel and P. V Kokotovi5, Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations, Syst. Contr. Lett., 38 (1999), pp. 259-270. 29. D. Nevsi5 and A. R. Teel, Set stabilization of sampled-data differential inclusions via their approximate discrete-time models, to a p p e a r in Conf. Decis. Contr., Sydney, 2000. 30. D. NevsiS, D. S. Laila and A. R. Teel, On preservation of dissipation inequalities under sampling, to appear in Conf. Decis. Contr., Sydney, 2000. 31. D. Nevsi~ and P. Dower, Further results on preservation of input-to-state stability under sampling, to appear in ICARV, Singapore, 2000. 32. R. Ortega and D. Taoutaou, A globally stable discrete-time controller for current-fed induction motors, Systems and Control Letters, 28 (1996), pp. 123128. 33. D. H. Owens, Y. Zheng and S. A. Billings, Fast sampling and stability of nonlinear sampled-data systems: Part 1. Existence theorems, IMA J. Math. Contr. Informat., 7 (1990), pp. 1-11. 34. Z. Qu, Robust control of nonlinear uncertain systems. John Wiley &: Sons: New York, 1998.

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239

35. E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Contr., 34 (1989), 435-443. 36. E. D. Sontag, Remarks on stabilization and input-to-state stability, in Proc. CDC, Tampa, USA, 1989. 37. S. A. Svoronos, D. Papageorgiou and C. Tsiligiannis, Discretization of nonlinear control systems via the Carleman linearization, Chemical Eng. Science, 49 (1994), pp. 3263-3267. 38. H. J. Stetter, Analysis of discretization methods for ordinary differential equations. Springer-Verlag: New York, 1973. 39. A. M. Stuart and A. R. Humphries, Dynamical systems and numerical analysis. Cambridge University Press: New York, 1996. 40. A. R. Teel, Connection between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE Trans. Automat. Contr., to appear, 1998. 41. A. R. Teel, D. Nevsi5 and P. V. Kokotovid, A note on input-to-state stability of sampled-data nonlinear systems, In Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, Florida, 1998. 42. Y. Zheng, D. H. Owens and S. A. Billings, Fast sampling and stability of nonlinear sampled-data systems: Part 2: Sampling rate estimation, IMA J. Math. Contr. Informat., 7 (1990), pp. 13-33.

15 Stability Tests for Constrained Linear Systems Maurfcio C. de Oliveira and Robert E. Skelton University of California, San Diego Department of Mechanical and Aerospace Engineering, Mail Code 0411, 9500 Gilman Drive, La Jolla, California, 92093-0411, USA. A b s t r a c t . This paper is yet another demonstration of the fact that enlarging the design space allows simpler tools to be used for analysis. It shows that several problems in linear systems theory can be solved by combining Lyapunov stability theory with Finsler's Lemma. Using these results, the differential or difference equations that govern the behavior of the system can be seen as constraints. These dynamic constraints, which naturally involve the state derivative, are incorporated into the stability analysis conditions through the use of scalar or matrix Lagrange multipliers. No a priori use of the system equation is required to analyze stability. One practical consequence of these results is that they do not necessarily require a state space formulation. This has value in mechanical and electrical systems, where the inversion of the mass matrix introduces complicating nonlinearities in the parameters. The introduction of multipliers also simplify the derivation of robust stability tests, based on quadratic or parameter-dependent Lyapunov functions.

15.1

A Motivation from Lyapunov Stability

Consider the continuous-time linear time-invariant system described by the differential equation

5c(t) ----Ax(t),

x(O) = xo,

(15.1)

where x(t) : [0, co) ~ R '~ and A C R '~x~. Define the quadratic form V : R n --~ R as

V(x) := xT p x ,

(15.2)

where P C $n. The symbol (.)T denotes transposition and S n denotes the space of the square and symmetric real matrices of dimension n. If

V(x) > O, Vx # O, then matrix P is said to be positive definite. The symbol X ~- 0 (X -~ 0) is used to denote that the symmetric matrix X is positive (negative) definite. The equilibrium point x = 0 of the system (15.1) is said to be (globally) asymptotically stable if lim x(t) ----0,

t~oO

Vx(0) ----x0,

(15.3)

where x(t) denotes a solution to the differential equation (15.1). If (15.3) holds, then, by extension, the system (15.1) is said to be asymptotically stable. A necessary and

242

M . C . de Oliveira, R. E. Skelton

sufficient condition for the system (15.1) to be asymptotically stable is that matrix A be Hurwitz, that is, that all eigenvalues of A have negative real parts. According to Lyapunov stability theory, system (15.1) is asymptotically stable if there exists V(x(t)) > O, Vx(t) ~ 0 such that

~/(x(t)) < 0,

V~(t) ----Ax(t),

x(t) ~ O.

(15.4)

T h a t is, if there exists P ~ 0 such that the time derivative of the quadratic form (15.2) is negative along all trajectories of system (15.1). Conversely, it is well known that if the linear system (15.1) is asymptotically stable then there always exists P ~- 0 that renders (15.4) feasible. Notice that in (15.4), the time derivative f/(x(t)) is a function of the state x(t) only, which implicitly assumes that the dynamical constraint (15.1) has been previously substituted into (15.4). This yields the equivalent condition

fl(x(t)) = x(t) T (AT p + PA) x(t) < O, Vx(t) ~ 0. Hence, asymptotic stability of (15.1) can be checked by using the following lemma. L e m m a 1 ( L y a p u n o v ) . The time-invariant linear system is asymptotically stable

if, and only if, 3 P E S n : P ~ - O ,

ATp+PA-~O.

At this point, one might ask whether would it be possible to characterize the set defined by (15.4) without substituting (15.1) into (15.4)? The aim of this work is to provide an answer to this question. The recurrent idea is to analyze the feasibility of sets of inequalities subject to dynamic equality constraints, as (15.4), from the point of view of constrained optimization. By utilizing the well know Finsler's Lemma [9] it will be possible to characterize existence conditions for this class of problems without explicitly substituting the dynamic constraints. Equivalent conditions will be generated where the dynamic constraints appear weighted by multipliers, a standard expedient in the optimization literature. The method is conceptually simple, yet it seems that it has never been used with that purpose in the systems and control literature so far. The advantage of substituting the dynamic constraints in the stability test conditions is the reduced size of the space on which one must search for a solution. In the context of the problem of Lyapunov stability this reduced space is characterized by the state x(t). In contrast, the space composed of x(t) and ~(t) can be seen as an enlarged space. In this paper it will be shown that the use of larger search spaces for linear systems analysis provides better ways to explore the structure of the problems of interest. This will often lead to mathematically more tractable problems. Whereas working in a higher dimensional space requires the introduction of some extra variables to search for which one might think at first sight as being a disadvantage, - - this is frequently accompanied by substantial benefits. One example of a popular result that illustrates this is the use of the Schur complement, which is now widely employed in systems and control theory [2]. Consider the set defined by the quadratic form f : R n --* R

f(x)

X T ( ~ -- S T ~ - l s T) x,

f ( x ) > O,

Vx ~ O,

where Q E S n, $ E R n• and T~ E S m, T~ k- 0. Using Schur's complement, one can test the existence of feasible solutions to the above set looking for a solution to the

15

Stability Tests for Constrained Linear Systems

243

set defined by the enlarged quadratic form g : R n x R m --~ R

An advantage of the higher dimensional form g is the fact t h a t it is linear on the matrices Q, S and 7~, a property t h a t does not appear in the original f . The authors believe that the technique that will be introduced in this work has the potential to show new directions to be explored in a several areas, such as decentralized control [15], fixed order dynamic o u t p u t feedback control [18], integrating plant and controller design [12], singular descriptor systems [3]. In all these areas, the s t a n d a r d tests based on Lyapunov stability theory can be tough to manipulate. The introduction of a different perspective may reveal easier ways to deal with these difficult problems. Besides, several recent results can be given a broader and more consistent interpretation. For instance, the robust stability analysis results [11,7,4,14] and the extended controller and filter synthesis procedures [5,6,10] can be interpreted and generalized using these new tools.

15.2

Lyapunov Stability Conditions with Multipliers

Consider the set of inequalities with dynamic constraints (15.4) arising from Lyapunov stability analysis of the linear time-invariant system (15.1). Define the quadratic form V : 1~'~ x R '~ --~ 1~ as

?(x(t), ~c(t)) :----x(t)T p2(t) + 5c(t)Tpx(t),

(15.5)

which is the time derivative of the quadratic form (15.2) expressed as a function of

x(t) and 2(t). Do not explicitly substitute 2(t) in (15.5) using (15.1), and build the set

V(x(t),2(t)) < O, V2(t) -- Ax(t),

(x(t),2(t)) ~ O.

(15.6)

In the sequel, stability will be characterized by using (15.6) instead of (15.4). This replacement is possible even though (15.4) requires only that x(t) ~ 0 while (15.6) requires that (x(t), x(t)) ~ O. Utilizing an argument similar to the one found in [2], pp. 62-63, this equivalence between (15.4) and (15.6) can be proved by verifying that the set I / ( x ( t ) , 2 ( t ) ) < 0,

V~(t) = Ax(t),

x(t) = 0,

2(t) -~ 0

(15.7)

is empty. But from (15.5), it is not possible to make ~/(x(t),2(t)) < 0 with x(t) = 0, which shows that (15.7) is indeed empty. Moreover, ~/(x(t),2(t)) is never strictly negative for all (x(t), k(t)) ~ 0 without the presence of the dynamic equality constraint (15.1). The advantage of working with (15.6) instead of (15.4) is t h a t the set of feasible solutions of (15.6) can be characterized using the following lemma, which is originally a t t r i b u t e d to Finsler [9] (see also [19]). L e m m a 2 ( F i n s l e r ) . Let x

E ]Rn, Q E S n and B The following statements are equivalent:

E

~m•

such that rank (B)

< n.

244

M . C . de Oliveira, R. E. Skelton

i) x T ~ x < O, V B x : 0, ii) B "l"T ~B.L "~ O.

x ~ 0.

iii) 3 t~ E R : Q - ~BT B "~ O. iv) 3 X

E ~ n x m : ~_~. X B T B T x T

.~ O.

Although Lemma 2 has been proven many times, a brief proof is given in Appendix A for completeness. In Lemma 2, statement i) is a constrained quadratic form, where the vector x E R '~ is confined to lie in the null-space of B. In other words, vector x can be parametrized as x = B• y E R r, r :-- rank(B) < n, where B • denotes a basis for the null-space of B. Statement ii) corresponds to explicitly substituting that information back into i), which then provides an unconstrained quadratic form in R ~. Finally, items iii) and iv) give equivalent unconstrained quadratic forms in the original R '~, where the constraint is taken into account by introducing multipliers. In iii) the multiplier is the scalar # while in iv) it is the matrix X. In this sense, the quadratic forms given in iii) and iv) can be identified as Lagrangian functions. Reference [13] explicitly identifies # as a Lagrange multiplier and makes use of constrained optimization theory to prove a version of Lemma 2. Finsler's Lemma has been previously used in the control literature mainly with the purpose of eliminating design variables in matrix inequalities. In this context, Finsler's Lemma is usually referred to as Elimination Lemma. Most applications move from statement iv) to statement ii), thus eliminating the variable (multiplier) A'. Several versions of Lemma 2 are available under different assumptions. A special case of item iv) served as the basis for the entire book [17], which shows that at least 20 different control problems can be solved using Finsler's Lemma. Recalling that the requirement V(x(t)) > O, Vx(t) ~ 0 can be stated as P ~ 0, and rewriting (15.6) in the form

~5c(t),] < O,

V [A - I ]

= O,

\~(t)]

r O,

\~(t)]

it becomes clear that Lemma 2 can be applied to (15.6). T h e o r e m 1 ( L i n e a r S y s t e m S t a b i l i t y ) . The following statements are equiva-

lent: i) The linear time-invariant system (15.1) is asymptotically stable. ii) 3 P C S n : P ~ - O , ATp+PA-
iv) 3 P E S ' ~ , F , G C ~ •

LGA_Fr

A T G T - F -t- P]

+p

_G_G T

] <0.

Proof. Item i) can be stated as P >- 0 and (15.6). Lemma 2 can be used with x

~-

\~(t)]

' Q

~-

,

~-

, x

~-

Es] ,

~-

and (15.6) to generate the inequalities given in items ii) to iv).

9

15

Stability Tests for Constrained Linear Systems

245

It is a nice surprise that Finsler's Lemma has been able to generate item ii) of Theorem 1 which is exactly the standard Lyapunov stability condition given in Lemma 1. Items iii) and iv) are new stability conditions. Since A is a constant given matrix, all three conditions are LMI (Linear Matrix Inequalities) and the feasible sets of conditions ii), iii) and iv) are convex sets (see [2] for details). Notice that the first block of the second inequality in condition iii) is # A T A ~- O, which implies that # > 0 and A is nonsingular. This agrees with the fact that Lyapunov stability requires that no eigenvalues of matrix A should lie on the imaginary axis. The multipliers #, F and G represent extra degrees of freedom that can be used, for instance, for robust analysis or controller synthesis. In some cases, not all degrees of freedom introduced by the multipliers are really necessary, and it can be useful to constrain the multipliers. Notice that constraining a multiplier is usually less conservative than constraining the Lyapunov matrix (see [5]). Some constraints on the matrix multiplier can be enforced without loss of generality. For instance, the proof of Lemma 2 given in Appendix A shows that 2d can always be set to - ( # / 2 ) B T without loss of generality. Besides this "trivial" choice, some more elaborated options might be available. For example, choosing the variables in item iv) to be

F -~ F T -= P,

G = el,

introduces no conservativeness in the sense that there will always exist a sufficiently small e that will enable the proof of stability. This behavior is similar to the one exhibited by the stability condition developed in [11]. In fact, item iv) is a particular case of [11], which has been obtained as an application of the positive-real lemma. The introduction of extra variables, here identified as Lagrange multipliers, is the core of the recent works [11,7,4], which investigate robust stability conditions using parameter dependent Lyapunov functions. A link with these results is provided by considering that matrix A in system (15.1) is not precisely known but that all its possible values lie on a convex and bounded polyhedron .d. This polyhedron is described as the unknown convex combination of N given extreme matrices Ai E R n• i---- 1 , . . . , N , through A:=

A(():A(()=

A~(i,

~e~

,

i=l

where :=

~ = (~1,...

~i = 1,

, ~N) :

~ > 0, i = 1. . . . , N

.

(15.8)

i=l

If all matrices in ~4 are Hurwitz then system (15.1) is said to be robustly stable in ,4. The following theorem can be derived from Theorem 1 as an extension. T h e o r e m 2 ( R o b u s t S t a b i l i t y ) . I f at least one of the following s t a t e m e n t s is true:

i) 3 P E S ' ~ : P ~ - O , ii) 3 F , G E 1 R ~ • P~ ~- O,

ATp+PAi-~O, Vi=I,...,N, I .... ,N : [ ATF T+FA~ ATG T-F+Pi] [GA~ - F T -t- Pi - G - GT

j -~ O, Vi = 1 . . . . , N ,

246

M . C . de Oliveira, R. E. Skelton

then the linear time-invariant system (15.1) is robustly stable in ~4. Proof. Assume that i) holds. Evaluate the convex combination of the second inequality in i) to obtain P >- O,

A ( ~ ) T P + PA(~) -~ O, V~ C .~,

which imply robust stability in .A according to item ii) in Theorem 1. Now assume that ii) holds. The convex combination of the inequalities in ii) provide P(r

~- 0,

[ A ( ~ ) T F T "b FA(~) A ( ~ ) T G T - F + P(~)]

GA(~) - F T ~- P(r

-G - GT

J -~ O, V~ E ~ ,

where P(~) e S n is the affine (time-invariant) parameter dependent Lyapunov function N

P(r := ~ Pir >'- O. i~l

The above inequalities imply robust stability in A according to item iv) of Theorem 1. 9 Theorem 2 illustrates how the degrees of freedom obtained with the introduction of the Lagrange multipliers can be explored in order to generate less conservative robust stability tests. Notice that although the items ii) and iv) of Theorem 1 are equivalent statements, their robust stability versions provided in Theorem 2 have different properties. The Lyapunov function used in the robust stability condition i) is quadratic [1] while the one used in item ii) is parameter dependent [8]. Robust versions of all results presented in this paper can be derived using the same reasoning.

15.3

Discrete-time Lyapunov Stability

The methodology described so far can be adapted to cope with stability of discretetime linear time-invariant systems given by the difference equation

xk+l ----Axk,

xo given.

(15.9)

In this case, if the same quadratic Lyapunov function (15.2) is used, asymptotic stability is characterized as the existence of V ( x k ) > 0, Vxk ~ 0 such that

V ( x k + l ) - - V(xk) < 0,

Vxk+l = Axk,

xk ~ O.

(15.10)

As before, the above set is not appropriate for the application of Lemma 2. Instead, the enlarged set

V(xk+l) - V(xk) < 0,

Vxk+l = Axk,

(xk,xk+l) ~ O,

(15.11)

is considered. As for continuous-time systems, (15.10) and (15.11) can be shown to be equivalent since the set

V(xk+l) - V(xk) < 0,

Vxk+l -= Axk,

xk ----O,

xk+l ~ O.

(15.12)

15

Stability Tests for Constrained Linear Systems

247

is empty. Indeed, the first inequality in (15.12) is never satisfied with xk = 0 since V(xk+l) > 0 for all xk+l r 0. The following theorem is the discrete-time counterpart of Theorem 1. T h e o r e m 3 ( D i s c r e t e - t i m e L i n e a r S y s t e m S t a b i l i t y ) . The following state-

ments are equivalent: i) The linear time-invariant system (15.9) is asymptotically stable. ii) 3 P c S '~:P~-O, A T p A - P - ~ O . [-#ATA- p ItA r ] iii) 3 P E S "~,It E ~ : P >- O, [ ItA - I t I + Pj ~ O. iv) 3 P c S ~ , F ,

[ A T F T q- F A - P ATG T - F ] [ GA-FT p_G_GT] -<0.

GERnxn:p>-o,

Proof. This lemma follows as an application of Lemma 2 with X*--

, Qe--

[:o]

, B T * --

X*--

[:]

xkz~l

on (15.11).

9

As in the continuous-time case, it is possible to constrain the multipliers without introducing conservatism. For instance, the choice F----0,

G=GT=p,

in iv) produces

whose Schur complement is exactly ii). Indeed, this particular choice of multipliers recovers the stability condition given in [4]. In this form, stability and also H2 and H ~ norm minimization problems involving synthesis of linear controllers and filters can be handled as LMI using linearizing change-of-variables [5,6,10]. Finally, it is interesting to notice that, as expected, the discrete-time stability conditions do not require that A be nonsingular. Indeed, the first block of the second inequality in item iii) can now be satisfied with a singular matrix A.

15.4

Handling I n p u t / O u t p u t

Signals

At this point, a natural question is if the method introduced in this paper can be used to handle systems with inputs and outputs. For instance, consider the linear time-invariant system

x(t) = Ax(t) + Bw(t), z(t) = Cx(t) + Dw(t).

x(O) = O, (15.13)

In the presence of inputs, there is no sense in talking about stability of systern (15.13) without characterizing the input signal w(t). Thus, assume t h a t the signal w(t) : [0, oo) -+ R TM is a piecewise continuous function in /:2, that is,

IlwllL~ :=

(// w(~-)Tw(T)

d~-j

1/2

<
248

M . C . de Oliveira, R. E. Skelton

The system (15.13) will be said to be s stable if the output signal z(t) E R p is also i n / : 2 for all w(t) E f-.2. This condition can be checked, for instance, by evaluating t h e / : 2 to s gain

Hzll~ ~ ( ~ ) ~ II~IIL~ "

0'~ :---- sup

For a linear and time-invariant stable system (15.13) it can be shown that

IiH~(s)]l~ :-- sup []H~ijw)ll2 = ~'~, wEF.

where H ~ ( s ) is the transfer function from the input w(t) to the output z(t), and II(')112 denotes the maximum singular value of matrix (.). Now, define the same Lyapunov function Vizir)) > 0, Vx(t) ~ 0, considered in Section 15.2, and the modified Lyapunov stability conditions

V(x(t),ic(t)) < O, 2 w(t)Tw(t) <_ z(t)Tz(t), i15.14 ) V(xit), ~(t), w(t), z(t)) satisfying (15.13), (xit), xit), w(t), zit)) ~ O, defined for a given -/ > 0. These inequalities appear in the stability analysis of system (15.13) under the feedback

w(t) :----A(t)zit),

IIDll2

< %

V A ( t ) : IIA(t)]12 < ,~

-1

Following the same steps as in [2], pp. 62-63, the S-procedure can be used to generate the equivalent condition 1'2

I/(x(t), 2(t)) < V 2 w(t)Tw(t) -- z(t)Tz(t), V(x(t), 2(t), w(t), z(t)) satisfying (15.13),

(15.15)

ix(t), 2(t), wit), z(t)) ~ O.

Hence, when the above conditions are satisfied it is possible to conclude that

0 < V(x(t)) =

f/

V(X(T),JZ(T))dw <

/o

~/2W(T)Tw(v) -- Z(T)TZ(T)dT,

which is valid for all t > 0. In particular, taking t ~ co,

IIzll~ < ~72,~,,2s which implies that V > 7 ~ . In other words, feasibility of (15.15) yields an upperbound to I]Hwz(s)l]~. For the linear system (15.13), it is known that -y~ = inf 7 : (15.15). Therefore, if (15.15) is feasible for some 0 < V < oo then it is possible to conclude that the system (15.13) is E2 stable. Moreover, the conditions (15.15) also guarantee 1 In this particular case, the S-procedure produces a necessary and sufficient equivalent test. This result can also be seen as a version of Finsler's Lemma where the constraint is a quadratic form (see [2], pp. 23-24). 2 As in [2], p. 63, the function l/(x(t), 2(t)) is homogeneous in P so that the scalar introduced with the S-procedure can be set to 1 without loss of generality.

15

Stability Tests for Constrained Linear Systems

249

that (15.13) is internally asymptotically stable. When the state space realization of system (15.13) is minimal, i.e., controllable and observable, both notions of stability coincide. If minimality does not hold, then system (15.13) might be 122 stable but not internally asymptotically stable 3, in which case the set (15.15) is empty. A generalized version of (15.15) can be obtained by considering constraints on the input and on the output signals in the form

[?T where Q 9 S p, R 6 S m, S 6 yields the inequality

0 R pxm

9

After applying the S-procedure this constraint

~7(x(t),Jz(t)) < - - ( z ( t ) T w ( t ) T) [?T SR] \ w ( t ) ]~ V(x(t), 2(t), w(t), z(t)) satisfying (15.13),

(15.16)

(x(t), x(t), w(t), z(t)) # O,

The following theorem comes from using Finsler's Lemma on (15.16). Theorem 4 (Integral Quadratic Constraint).

The following statements are

equivalent: i) The set of solutions to (15.16) with P >- 0 is not empty. ii) 3 P 9 ~:P~-O,

ATp + PA + CTQC P B -~-c T s ~- c T Q D ] BT p + sTc + D T Q C R + ST D + DT s + DTQDJ ~- O, - # ( A T A + c T c ) p A T "4- P #A + P -#I #C 0 --IX (BTA

+ DTc)

IXBT

#C T

0 S+#D S T + IXDT R - # ( B T B + Q - M

iv) S P 6 S~,F1,G1 6 R nxn ,F2,G2 9 R pxp, J2 9 ~ m x p : p >._ O,

--#(ATB+CTD) IXB

[~nXp

]

I

-<0.

DTD)J

,H1 9 RPx~,J1 6 Tr}m z, X "~ , H

2

9

7"~ -~ ~ T _< O, where

[ FIA+F2C -FI |GIA + G2C + P -GI n := / H I A + H2C - H 1 L J 1 A + J2C -J1

F~B+F~D ] GI B + G2D | (1/2)Q - H2 H1B + H2D | " S T - J2 (1/2)R + J I B + J2DJ -F2 -G2

Proof. Assign

/x(t)

F2 G1 G2 '

\w(t)/

'

0 ST

LBT DrJ

H2

'

3"1 J2

and apply Lemma 2 on (15.16). 3 Some uncontrollable or unobservable mode of (15.13) may not be asymptotically stable.

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M . C . de Oliveira, R. E. Skelton

Several well known results can be generated as particular cases of Theorem 4. For instance, (15.16) reduces to (15.15) with the choice Q--I,

R=-~/2I,

S=0.

W i t h these matrices, as expected, item ii) of Theorem 4 reduces to the s t a n d a r d bounded-real lemma. The choice Q=0,

R=0,

S=-I,

produces the positive-real lemma. Items iii) and iv) can be seen as new equivalent statements of these well known results. It is interesting to notice that the introduction of the new signal z(t) brings an extra ' - I ' term into B. Thus preserving an identity full row rank block inside matrix B, that can be used to compute a straightforward B •

15.5

Analysis of Systems Described by Transfer Functions

So far Finsler's Lemma has been used to generate stability conditions for systems given in state space form. In this section, it will be used on linear time-invariant systems described by transfer functions. For simplicity, consider a second-order SISO system represented by the transfer function

b(s) b2s2+bls+bo Hwz(s)-- a(s---) -- s 2 + a l s + a o

(15.17)

The results to be presented can be generalized to cope with transfer functions of higher order. Asymptotic stability of this transfer function can be analyzed by considering the second order differential equation

!i(t) + al2(t) + aox(t) ----O,

(&(O), x(O)) --= (xo, xo).

(15.18)

The stability of this equation can be probed by the quadratic Lyapunov function

V(x(t)):=x(t)Tpx(t)'

P:=

[ p~l p02' 2] p 3

and the associated stability conditions

l/(x(t),~(t)) < O,

V(x(t),&(t),&(t)) • 0 satisfying (15.18).

(15.19)

Arguments similar to the ones used in Section 15.2 can be used to show t h a t the above conditions and P ~- 0 fully characterize the stability of (15.18) or, equivalently, of the transfer function (15.17). T h e o r e m 5 ( T r a n s f e r F u n c t i o n S t a b i l i t y ) . The following statements are equiv-

alent: i) The linear time-invariant system (15.17) is asymptotically stable.

15

ii) 3 P E S 2:P>-O, A := -ao

Stability Tests for Constrained Linear Systems

ATp+PA-40,

where

--al

iii) 3 P E S 2 , p C ~ : P > - 0 , U(P) :=

251

U ( P ) - l t a a T ~O, where

Pl 2p2 2 p3

,

iv) 3 P C S 2 , f G R 3 x l :P~-O,

a :=

,

U(P)+faT +afT-~o.

Proof. Set

/x(t)~ ~- [~(t)|,

a ~- u(P),

B T ~ - a,

x ~- f

\&(t)] and apply Lemma 2 on (15.19) with P ~- 0.

9

Item ii) of Theorem 5 recovers exactly the s t a n d a r d Lyapunov stability test that would have been obtained if item ii) of Theorem 1 had been applied to the companion state space realization (:~(t)~

5:(t)]

[ 0

1 ] [x(t)~

=-ao-al

(15.20)

\&(t)]'

On the other hand, items iii) and iv) are polynomial stability conditions. Notice that they differ from the state-space conditions iii) and iv) given by Theorem 1 for (15.20). The i n p u t / o u t p u t results of Section 15.4 can also be generalized to cope with transfer functions. Consider again the simple second-order SISO system (15.17), and define the dynamic constraints

&(t) + al&(t) + aox(t) = w(t), (&(0),x(0)) = (0,0), z(t) = b25~(t) + bl&(t) + box(t).

(15.21)

The analog of the integral quadratic performance conditions (15.16) can be shown to be given by

\~(t))

'

V(x(t), k(t), &(t), w(t), z(t) ) # 0 satisfying (15.21),

(15.22)

where q, s, r C R. The form of the dynamic constraint (15.21) deserves some commerits. First, it is based on the phase-variable canonical realization [16], where the transfer function (15.17) is implemented via

H~oz(s) -- Z(s)

w(s)'

Z(s) = b(s)~(s),

a(s)~(s) = W(s).

252

M . C . de Oliveira, R. E. Skelton

Second, in standard state space methods, the second equation (output equation) of (15.21) must have the term 5} substituted from the first equation. This yields the standard phase-variable canonical form

~(t) + al~(t) + aox(t) = w(t),

(~(0), x(0)) = (0, 0),

z(t) = b2w(t) + cl~(t) + cox(t).

(15.23)

where ci := (bi - b2ai), i = 0, 1. Finsler's Lemma can handle b o t h (15.21) and (15.23) without further ado. Theorem 6 (Transfer Function Integral Quadratic lowing statements are equivalent:

Constraint).

The fol-

i) The set of solutions to (15.22) with P ~ 0 is not empty. ii) 3 P E S a : P ~ - O , A T p + P A + q c T c P B + s C T + qb2C T] B T p + s C + qb2C r + 2sb2 + qb 2 J "~ O, where A :=

[o 1] .=[Ol] -ao -al

iii) 3 P E S 2 : P ~ O,

U(P):=

'

U ( P ) + raa T + s a b T + s b a T + q b b T -~ O, where

Pl 2p2 2 P3

,

a:=

,

b:=

bl , b2

iv) 3 P C S 2 , # E R : P > - 0 , "V(P)-#(aaT + b b T) # b #b T q- # /~a T

S

~a ] -~ 0, r -- #J

V) 3 P E S2, fl, f2 E R a X l , g l , g e , h l , h 2 E R : P ~- O, [ fU(P)+flbT+f~ar~ ] ~, + b f T -t- a f T ) g l b + g2a - fl h l b + h2a - f~ s--g2--hl I "~0. | glb T +g2a T -f~T q-2gl k h l b T d- h2a T - fT s -- g2 -- hi r -- 2h2 J Proof. Items ii) to v) have been generated applying Lemma 2 on (15.22) with P ~- 0, the dynamic constraint (15.21) and

[~(t)~

l~(t)l Q~- u(P) 01] , BT,-- [_~10] , x' ~-- [ fgll ~- l~(t)l, 0 q /z(t)/

\w(t)]

o

s

1

f~] g2 9 hi h2

15

Stability Tests for Constrained Linear Systems

253

Items ii) and iii) have been generated with item ii) of Lem m a 2 using

ii) : B

•

*--

[o0i)l -aO - a l

,

iii) : B • ~--

cl 0

. [aTJ

which are two possible choices for the null-space basis of B. It is interesting to notice that the same conditions obtained in Th eo r em 6 are generated if Lemma 2 is applied to (15.23) with

a0 al 1 0 In fact, it is straightforward to verify t h a t this matrix and matrix B used in the proof of Theorem 6 have the same range space, hence they share the same null space. Notice that there is also some freedom in the choice of B • This freedom has been used to generate items ii) and iii) of Theorem 6. While ii) is the standard integral quadratic constraint condition generated for the state space representation of (15.23), item iii) is a new condition where the coefficient vectors a and b are not involved in any product with the Lyapunov matrix P. Both Theorem 5 and 6 can be generalized to cope with higher order transfer functions by appropriately augmenting the vectors a, b and f. Extensions to general MIMO systems with m inputs and p outputs are also straightforward by considering

H~z(s) = Z ( s ) W ( s ) -1,

Z(s) = N(s)~(s),

D(s)~(s) = W ( s ) .

(15.24)

This factorization can be obtained as H ~ (s) ----N ( s ) D ( s ) -1, that is, by computing N ( s ) and D ( s ) as right coprime polynomial factors of H~z(s). From (15.24), one can compute matrices A and B so that

A x ( t ) = wit), z(t) -- B x ( t ) , where A E R m• B 9 R pxn and x(t) E R ~ is a vector containing the state ~(t) and the appropriate time derivatives. Another possible generalization of these results is for systems described by higher order vector differential equations as, for instance, vector second-order systems in the form

M&(t) + Dic(t) + K x ( t ) = B w ( t ) , z(t) = P ~ ( t ) + Q~(t) + R x ( t ) .

(15.25)

Robust versions of Theorems 5 and 6 would be able to provide stability conditions that enables one to take into account uncertainties on all matrices of (15.25), including the mass matrix M.

254

M . C . de Oliveira, R. E. Skelton

15.6

Some N o n - S t a n d a r d Applications

The ability to define extra signals and derive stability conditions directly involving these signals opens some new possibilities. For instance, consider an stability problem that is similar to the one discussed in [11,7]. Characterize the stability of the discrete-time linear time-invariant system given by xk+l = A B x k ,

xo given

(15.26)

where x E R '~ and A E ~nXm, B E ~rnXn, With the introduction of the auxiliary signal yk E ~m it is possible to rewrite this system in the equivalent form xk+l ---- Ayk,

xo given,

yk : B x k .

(15.27)

Following the same steps as in Section 15.3, asymptotic stability of this system can then be characterized in the enlarged space of (xk, xk+l, yk) as the existence of a quadratic Lyapunov function V ( x k ) > 0, Vxk ~ 0 such that Y ( x k + l ) -- Y ( x k ) <: 0,

V(xk, xk+l, Yk) ~ 0 satisfying (15.27)

(15.28)

The following theorem comes after applying Finsler's Lemma on (15.28). T h e o r e m 7 ( A B L i n e a r S y s t e m S t a b i l i t y ) . The following statements are equivalent: i) The linear time-invariant system (15.26) is asymptotically stable. ii) 3 P E S "~ : P >- 0, B T A T p A B - P -~ O. iii) 3 P C N n , # C R : P ~ O ,

-#I+P #A -~0. #A T L #B -#ATA - #I iv) 3 P E Sn, F~,G~ E RnXn,F2,G2 C RnXm,H~ C Rmx'~,H2 E R m x m : P >O, ?-l + 7-lT -< O, where [F2B - (1/2)P -F~ F~A - F2 ] Tl := [ G2B ( 1 / 2 ) P - G1 G1A O. G2 J -< L H2B -H1 H 1 A - H2

Proof. Define x~--

xk+x , Q~-\ yk /

P 0

,B T~-

,X*-T

and apply Lemma 2 on (15.28).

Gx G~ H1 H2

.

9

The stability result in [7] is a particular case of item iv) of Theorem 7 where the matrices A and B are assumed to be square and where the multipliers are set to a l G2 H i H2

=

(1/2)P L(1/2)AT p

,

G, H E N nxn.

15

Stability Tests for Constrained Linear Systems

255

Since this multiplier is a function of the system matrix A, the robustness analysis in [7] assumes that A is known. One advantage of dealing with (15.27) instead of (15.26) is that robust stability tests - - either using quadratic or parameter-dependent Lyapunov function - - for systems with multiplicative uncertainty in the form

xk+l = A(~)B(~)xk,

~ C ~,

where ~ is defined in (15.8), become readily available through item iv) of Theorem 7. Analogously, fractional or more involved uncertainty models can be taken care with no more effort. It is nice surprise that LMI robust stability conditions can be derived for uncertain models with complicated uncertainty structures such as

E(~)xk+l = A ( ~ ) C ( ~ ) - I B ( ~ ) z k ,

~ E ~,

by simply considering (15.28) and the yet linear dynamic constraint

Exk+1 = Ayk, Cyk = Bxk, where A E R '~• B E R mXn, C E R re• and E E R '~• Notice that the above system contains as a particular case the class of linear (nonsingular) descriptor systems.The subject of singular descriptor systems is slightly more involved and will be addressed in a separate paper. Counterparts of these results for continuous-time systems can be obtained as well. However, notice that in this case the dimension m should be necessarily greater or equal than n, since a singular dynamic matrix is never asymptotically stable in the continous-time sense.

15.7

Conclusion

In this paper Lyapunov stability theory has been combined with Finsler's Lemma providing new stability tests for linear time-invariant systems. In a new procedure, the dynamic differential or difference equations that characterize the system are seen as constraints, which are naturally incorporated into the stability conditions using Finsler's Lemma. In contrast with standard state space methods, where stability analysis is carried in the space of the state vector, the stability tests are generated in the enlarged space containing both the state and its time derivative. This accounts for the flexibility of the method, that does not necessarily rely on state space representations. Stability conditions involving the coefficients of transfer functions representing linear systems are derived using this technique. Systems with inputs and outputs can be treated as well. Alternative new formulations of stability analysis tests with integral quadratic constraints, which contain the bounded-real lemma and the positive-reM lemma as special cases, are provided for systems described by transfer functions or in state space. The philosophy behind the generation of these new stability tests can be summarized as follows: 1. Identify the Lyapunov stability inequalities (quadratic forms) in the enlarged space. 2. Identify the dynamic constraints in the enlarged space.

256

M . C . de Oliveira, R. E. Skelton

3. Apply Finsler's Lemma to incorporate the dynamic constraints into the stability conditions. The dynamic constraints are incorporated into the stability conditions via three main processes: a) evaluating the null space of the dynamic constraints, b) using a scalar Lagrange multiplier or c) using a matrix Lagrange multiplier. These multipliers bring extra degrees of freedom that can be explored to derive robust stability tests. Quadratic stability or parameter-dependent Lyapunov functions can be used to test robust stability.

Acknowledgements Maurlcio C. de Oliveira is supported by a grant from FAPESP, " F u n d a ~ o de Amparo ~ Pesquisa do Estado de S~o Paulo", Brazil.

A

Proof of Lemma 2 (Finsler's Lemma)

i) ~ ii): All x such that t3x ---- 0 can be written as x ----B• Consequently, i) =~ yTI3•177 < 0, for all y r 0 ~ B • • -~ 0. Conversely, assuming that the first part of ii) holds, multiply B• • on the right by any y r 0 and on the left by yT to obtain yTB• ~ B • < 0 ~ i). iii), iv) ~ ii): Multiply ii) or iii) on the right by B • and on the left by B •

so as

to obtain ii).

ii) ~ iii): Assume that ii) holds. Partition /3 in the full rank factors /3 = Bt/3)., define 7 ) : =

B~ (B~BT~)-I(B~Bz)1/2and apply the congruence transformation

L Since the second diagonal block is negative definite by assumption, a sufficiently large # exists so that the whole matrix is negative definite.

iii) ~ iv): Choose 2( -- - ( # / 2 ) • T.

9

References 1. B. R. Barmish. Necessary and sufficient conditions for quadratic stabilizability of an uncertain system. JOTA, 46:399-408, 1985. 2. S. P. Boyd, L. E1 Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, PA, 1994. 3. D. Cobb. Controllability, observability, and duality in descriptor systems. IEEE Transactions on Automatic Control, 29:1076-1082, 1984. 4. M. C. de Oliveira, J. Bernussou, and J. C. Geromel. A new discrete-time robust stability condition. Systems ~J Control Letters, 37(4):261-265, 1999.

15

Stability Tests for Constrained Linear Systems

257

5. M. C. de Oliveira, J. C. Geromel, and J. Bernussou. Extended //2 and Ho~ norm characterizations and controller parametrizations for discrete-time systems. Submitted paper. 6. M. C. de Oliveira, J. C. Geromel, and J. Bernussou. A n LMI optimization approach to multiobjective controller design for discrete-time systems. In Proceedings of the 38th IEEE Conference on Decision and Control, pages 3611-3616, Phoenix, AZ, 1999. 7. M. C. de Oliveira, J. C. Geromel, and L. Hsu. LMI characterization of structural and robust stability: the discrete-time case. Linear Algebra and Its Applications, 296(1-3):27-38, 1999. 8. E. Feron, P. Apkarian, and P. Gahinet. Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions. IEEE Transactions on Automatic Control, 41(7):1041-1046, 1996. 9. P. Finsler. (~ber das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formem. Commentarii Mathematici Helvetici, 9:188-192, 1937. 10. J. C. Geromel, M. C. de Oliveira, and J. Bernussou. Robust filtering of discretetime linear systems with p a r a m e t e r dependent Lyapunov functions. In Proceedings of the 38th IEEE Conference on Decision and Control, pages 570-575, Phoenix, AZ, 1999. 11. J. C. Geromel, M. C. de Oliveira, and L. Hsu. LMI characterization of structural and robust stability. Linear Algebra and Its Applications, 285(1-3):69-80, 1998. 12. K. M. Grigoriadis, G. Zhu, and R. E. Skelton. Optimal redesign of linear systems. Journal of Dynamic Systems Measurement and Control : transactions of the ASME, 118(3):598-605, 1996. 13. C. Hamburger. Two extensions to Finsler's recurring theorem. Applied Mathematics ~ Optimization, 40:183-190, 1999. 14. C. W. Scherer. Robust mixed control and linear parameter-varying control with full block scalings. In L. E. Gahoui and S.-L. Niculesco, editors, Advances in Linear Matrix Inequality Methods in Control, pages 187-207. SIAM, Philadelphia, PA, 2000. 15. D. D. Siljak. Decentralized Control of Complex Systems. Academic Press, London, UK, 1990. 16. R. E. Skelton. Dynamics Systems Control: linear systems analysis and synthesis. John Wiley &: Sons, Inc, New York, NY, 1988. 17. R. E. Skelton, T. Iwasaki, and K. Grigoriadis. A Unified Algebraic Approach to Control Design. Taylor ~z Francis, London, UK, 1997. 18. V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis. Static o u t p u t feedback - - a survey. Automatica, 33(2), 1997. 19. F. Uhlig. A recurring theorem about pairs of quadratic forms and extensions: a survey. Linear Algebra and its Applications, 25:219-237, 1979.

16 Equivalent Realizations for IQC Uncertain Systems* Ian R. Petersen School of Electrical Engineering, Australian Defence Force Academy, Canberra ACT, 2600, Australia, Phone +61 2 62688446, FAX +61 2 62688443, email: [email protected].

A b s t r a c t . This paper considers uncertain systems in terms of the corresponding system graph. The paper develops a necessary and sufficient condition for the graph of a given uncertain system to be contained in the graph of another uncertain system. This result also enables one to consider the equivalence between two uncertain systems. The uncertain systems under consideration are linear time-varying uncertain systems in which the uncertainty is described by a time domain integral quadratic constraint. K e y w o r d s : Uncertain Systems; Equivalent Realizations; Modelling; System Graph; Integral Quadratic Constraints.

16.1

Introduction

This paper is concerned characterizing equivalences between uncertain systems described by an integral quadratic constraint (IQC) uncertainty description. For a given linear time invariant system (with zero initial condition), only one inputoutput relation is possible. This i n p u t - o u t p u t relation defines the system graph which is a subspace of the input-output signal space; e.g., see [1]. For an uncertain system, a whole class of input-output relations are possible. However, we can still define the system graph as the set of possible i n p u t - o u t p u t pairs for this uncertain system. In this paper, we are concerned with the following question: Given two uncertain systems with an IQC uncertainty description, when is every possible i n p u t - o u t p u t pair of the first uncertain system also a possible i n p u t - o u t p u t pair for the second uncertain system. Our main result gives a necessary and sufficient condition for this condition to hold. This condition is given in terms of the solution to a certain singular optimal control problem. If we take a behavioural approach to uncertain system modelling (e.g., see [2]), we can regard an uncertain system as being characterized by its system graph. Then our results provide a method of determining if an uncertain system model can be replaced by an alternative (possibly simpler) uncertain system model whose system graph contains that of the original uncertain system. Our results also provide a way of determining whether two uncertain system models are equivalent from an input-output point of view. Thus, our results provide an important step in the * This work was supported by the Australian Research Council.

260

Ian R. Petersen

development of a systems theory for uncertain systems with an IQC uncertainty description; see also [3-5]. The class of uncertain systems considered in this paper are linear time-varying uncertain systems defined on a finite time interval. The uncertainty description involves a finite-horizon time-domain IQC; e.g., see [6-9]. This uncertainty description allows for a rich class of nonlinear dynamic time-varying uncertainties. Also, this uncertainty description has been found to yield tractable solutions to problems of minimax optimal control and state estimation; e.g., see [10,6]. It should be noted that a very complete theory of minimality and equivalence for a class of uncertain systems has been developed in the papers [11-14]. However, there are a number of important distinctions between the results presented in these papers and the results of this paper. The first distinction concerns the class of uncertain systems considered. This paper is concerned with a time domain IQC uncertainty description on a finite time horizon whereas the above papers are concerned with structured linear time varying (LTV) uncertainties defined on an infinite time horizon. Although both classes of uncertain system models have their own particular advantages, we will see that the results obtained in this paper are of a quite different form to those obtained in the above mentioned papers. This points to the fact that the system theory for these two classes of uncertain systems will be quite different. Another important difference between the approach presented in this paper and that developed in the papers [11-14] concerns the definition of equivalent uncertain systems. Our definition requires only the equality the system graphs whereas the definition given in papers [11-14] requires identical input-output relations be achieved with the same uncertainty operator. This is quite a significant restriction. The remainder of the paper proceeds as follows. In Section 16.2, we introduce the class of uncertain systems under consideration. We also recall a preliminary result from [7] which enables us to characterize the system graph for a given uncertain system. This section also includes definitions concerning the relationship between two uncertain systems and the equivalence of two uncertain systems. In Section 16.3, we present our main results characterizing when the system graph of a given uncertain system includes the system graph of another uncertain system. We also present some special cases in which such a relationship can be characterized in terms of the existence of a solution to a Riccati differential equation. In Section 16.4, we present some simple examples which illustrate our main results.

16.2

Definitions and P r e l i m i n a r y R e s u l t s

In this section, we introduce some definitions which will be required in order to present our main results. We also present a preliminary result on model validation for uncertain systems. This result is a key technical result required in the sequel. 16.2.1

Uncertain

System

Models

We consider a class of uncertain systems defined by state equations of the form:

it(t) = A ( t ) x ( t ) + D ( t ) w ( t ) + B(t)u(t);

x(O) ----O;

z(t) = K ( t ) x ( t ) + G(t)u(t); y(t) = C ( t ) x ( t ) + v(t)

(16.1)

16

Equivalent Realizations for U n c e r t a i n Systems

261

where x(t) E R n is the state, w(t) E R p a n d v(t) E R t are the uncertainty inputs, u(t) E R. m is the control input, z(t) E R q is the uncertainty output a n d y(t) E R z is the measured output, A(. ), D(. ), B(. ), K (. ), G(. ) a n d C(.) are b o u n d e d piecewise continuous m a t r i x functions. We will consider these state equations over a finite time interval [0, T].

System Uncertainty T h e u n c e r t a i n t y in the above system is required to satisfy t h e following Integral Q u a d r a t i c Constraint. Let d > 0 be a given constant, Q(-) - Q(.)~ a n d R(.) - R(.)' be given b o u n d e d piecewise c o n t i n u o u s m a t r i x weighting functions such t h a t there exists a c o n s t a n t 5 > 0 satisfying Q(t) >_ (iI a n d R(t) >_ 51 for all t. T h e u n c e r t a i n t y i n p u t s w(.) a n d v(.) are said to be admissible u n c e r t a i n t y i n p u t s if

/0 T(w(t)'Q(t)w(t) + v(t)'R(t)v(t))dt

~ d+

/0 F IIz(t)il2dt.

(16.2)

Here ]1' II denotes the s t a n d a r d E u c l i d e a n norm. Note t h a t the above u n c e r t a i n t y description allows for u n c e r t a i n t i e s in which the u n c e r t a i n t y i n p u t s w(.) a n d v(.) d e p e n d d y n a m i c a l l y on the u n c e r t a i n t y o u t p u t z(.). In this case, the c o n s t a n t d m a y be interpreted as a measure of t h e size of the initial conditions for the u n c e r t a i n t y dynamics. D e f i n i t i o n 1. Let d > 0 be given. Also, let uo(.) a n d y0(') be given vector functions defined over a given time interval [0, T]. T h e i n p u t - o u t p u t pair [u0('), y0(-)] is said to be realizable with p a r a m e t e r d if there exist [x(.), w(.), v(-)] satisfying c o n d i t i o n s (16.1), (16.2) with u(t) -- uo(t) a n d y(t) =--yo(t). We now present a result which gives a necessary a n d sufficient c o n d i t i o n for a given pair [u0(-), y0(-)] to be realizable. This result is a slight modification of t h e m a i n result of [7]. In order to a p p l y this condition, we assume t h a t the following Riccati differential e q u a t i o n (RDE) has a positive definite solution on (0, T]:

Po(t) ----A(t)Po(t) + D ( t ) Q ( t ) - l D(t) ' +Po(t)A(t)' + Po(t)[K(t)'K(t) - C(t)'R(t)C(t)]Po(t); g0(0) ----0.

(16.3)

In addition, we assume t h a t there exists a n g > 0 such t h a t the R D E

P~(t) = A(t)P~(t) + P~(t)A(t)' + D ( t ) Q ( t ) - l D ( t ) ' + P~(t)[K(t)' K(t) - C(t)' R(t)C(t)]P~(t); g~(0) ----r

(16.4)

has a positive definite solution on [0,T] for all 0 < r _< g. Note, it follows from the game theoretic i n t e r p r e t a t i o n of this R D E t h a t

P~(T) > Po(T)

V 0
e.g., see [15]. Also, it follows from t h e c o n t i n u i t y of solutions to the R D E with respect to the initial condition t h a t P~ (T) --* P0 (T) > 0 as r --~ 0.

262

Ian R. Petersen

In the sequel, it will be assumed that the uncertain system (16.1), (16.2) satisfies the above assumptions. It is shown in [7] that these assumptions are related to a property of strict verifiability. Our condition for a pair [u0(.), y0(')] to be realizable involves solving the following state equations:

~(t) = [A(t) + P ( t ) [ K ( t ) ' K ( t ) - C(t)'R(t)C(t)]] 2(t) + P ( t ) C ( t ) ' R ( t ) y o ( t ) + [ P ( t ) K ( t ) ' G ( t ) + B(t)]uo(t); 5(0) -= 0 (16.5) and forming the quantity

p[uo('), Yo(')] n =

/o

[ l l ( K ( t ) 2 ( t ) + G(t)uo(t))l[ 2 [ - ( C ( t ) ~ ( t ) - yo(t)) R(t)(C(t)2c(t) - yo(t))

]

(16.6)

dt.

L e m m a 1. Suppose that the uncertain system (16.1), (16.2) is such that conditions (16.3), (16.4) are satisfied and let uo(t) and yo(t) be given vector functions defined on [0, T]. Then, the pair [u0('),y0(-)] is realizable if and only if

p[uo(), yo(.)] > - d . P r o o f The proof of this lemma follows along the lines of the proof of Theorem 3.1 of [7]. Indeed, as in this proof, we consider the set of possible values of the state s YO(')I0, s d]. This set is defined as the set of state vectors XT such XT E Z s[u 0(')I0, that there exist uncertainty inputs w(-), v(-) leading to x ( T ) = XT and the given s s input-output pair u0 (t) and Y0(t). The set X ~[uo (-)10, y0 (')10, d] is characterized by points XT E R n such that J*(XT) ----

inf

w ( . ) E L 2 [0,T]

J ( x T -- x l ( T ) , w ( . ) ) < d

(16.7)

where

J ( ~ , ~(.)) = ~0 T

~w ( t ) ' Q ( t ) w ( t )

- II(K(t)[&(t) ~ xl(t)] + G(t)uo(t))lt 2 dt; +(yo(t) - C(t)[~(t) + xl(t)]) R(t)(yo(t) - C(t)[~c(t) + xl(t)]) ]

~l(t)=A(t)xl(t)+B2(t)uo(t);

xl(0)=0,

(16.8)

and 5:(t) = x(t) - xl(t). Hence, x(t) = A(t)~(t) + B l ( t ) w ( t ) ;

~(0) -= 0.

(16.9)

Also, :~(T) = ~CT = x'r -- x l ( T ) . To solve this fixed endpoint optimal control problem, we consider a sequence of free endpoint problems: J*(xT) =

inf

w ( - ) e L 2 [0,T]

J~(XT -- xl(T), w(.))

16

Equivalent Realizations for Uncertain Systems

where J~(xT -- x l ( T ) , w ( . ) ) = ~ll~(0)ll ~ + that for all 0 <: e _< g, the RDE

J(~T,W(')).

263

Now it follows from (16.4)

-.~(t) = X(t)A(t) + A(t)'X(t) + X(t)D(t)Q(t)-In(t)'X(t) + K ( t ) ' K ( t ) - C(t)' n ( t ) C ( t ) ; X(0) ---- 1 I

(16.10)

E

has a positive definite solution on [0, T] such t h a t X~(t) ---- Pe(t) -~. Furthermore, Pe(T) >_ Po(T) for all 0 < e < g. Hence, X ~ ( T ) = P~(T) -~ < Po(T) -~ for all 0 < e < g. Also,

X ~ ( T ) --* Po(T) -1 as e --+ O.

(16.11)

Now as in [7], inf

w(') EL2 [0,T]

:

( X T --

J~(&T,W(')) Sce(T))'X(T)(XT - &e(T)) - p~[uo(.), y0(')]

where &~(t) :

[A(t) + P ( t ) [ g ( t ) ' K ( t ) - C(t)' R(t)C(t)]] &~(t) + P ( t ) C ( t ) ' R ( t ) y o ( t ) + [ P ( t ) K ( t ) ' G ( t ) + B(t)]uo(t);

~(0) = 0 (16.12)

and

+ G(t)uo(t))ll 2 ] p~[~0(.), yo(.)] =~ ~o T [f -II(g(t)&~(t) ( C ( t ) & ~ ( t ) - yo(t)) R(t)(C(t)&~(t) - yo(t)) dt; (16.13) see also [16,17]. Therefore, it follows as in the proof of Theorem II.4.2 of [18] t h a t J*(xT) in (16.7) satisfies

J*(XT) : lira

inf

e ~ O w(.)eL2 [0,T ]

ffe(XT,W(')).

Hence, taking the limit as e --~ 0 in (16.11), (16.12) and (16.13), we obtain

J* (XT ) ---- (XT -- ~(T) )' Po(T) -1 (XT -- :~(T) ) -- p[uo('), Yo(')] where 2(T) is defined by (16.5) and p[uo(.), yo(')] is defined by (16.6). Therefore, X~ [uo (')Io, Yo(')Io, d]

= {XT: (XT - - ~ c ( T ) ) ' P o ( T ) - i ( x T -

&(T)) < d + p[uo('),yo(')]}.

This set is non-empty if and only if d+p[uo(.), y0(')] >_ 0. Hence, the pair [uo(-), yo(')] is admissible if and only if p[u0(.), y0(-)] _> - d . [] R e m a r k s The above lemma requires t h a t assumption (16.3) be satisfied. However, for assumption (16.3) to be satisfied, it is necessary t h a t the pair (A, D) be controllable. If the pair (A, D) is not controllable, assumption (16.3) can be relaxed as follows (in the time invariant case): First decompose the system in controllable and uncontrollable subsystems. Then assumption (16.3) can be relaxed to the requirement t h a t Po(t) >_ 0 for t E [0, T] and x ' P o ( t ) x = 0 only for x in the uncontrollable subspace of the system. To establish this result, the above proof can be modified to consider only the set of possible states for the controllable subsystem.

264

Ian R. Petersen

16.2.2

Realization

Relationships

In this section, we consider relationships between uncertain systems defined as above in terms of their system graphs. We can think of the uncertain system (16.1), (16.2) as being described by its system graph B which is the set of realizable inputoutput pairs [u0 (.), Y0(')]. Alternatively, we can think of the uncertain system (16.1), (16.2) as being a realization of the corresponding system graph B. We now consider a pair of uncertain systems of the form (16.1), (16.2):

(,F,1) :

&(t) = A l ( t ) x ( t ) + D l ( t ) w ( t ) + Bl(t)u(t);

x(O) = 0

z(t) = K1 (t)x(t) + G1 (t)u(t); y(t) = C l ( t ) x ( t ) + v(t);

fo r ( w ( t ) ' Q l ( t ) w ( t ) (5:~) :

+ v(t)'Rl(t)v(t))dt

<_ dl +

fo r IIz(t)ll2dt;

2(t) = A2(t)x(t) + D2(t)w(t) + B~(t)u(t); z(t) = K~(t)x(t) + G2(t)u(t);

x(O) = 0

y(t) = C2(t)x(t) + v(t);

(w(t)'Q2(t)w(t) + v ( t ) ' R 2 ( t ) v ( t ) ) d t <_ d2 +

]o F IIz(t)ll~dt.

Note, it is assumed that the respective dimensions of u and y are the same for these two systems. However, any of the other dimensions may be different. D e f i n i t i o n 2. The uncertain system ( E l ) is said to be a subrealization of (~72) if for any d~ > 0, there exists a dl > 0 such that any realizable i n p u t - o u t p u t pair for (~1) is a realizable input-output pair for (~2). To indicate this relationship, we will use the notation (~1) C (5:2). If (~2) C (~U1) and (5:1) C (,U2), then we say the realizations (~U1) and (5:2) are equivalent. Roughly speaking, if (~U1) C (,U2), then any i n p u t - o u t p u t property we can establish for the uncertain system (~2) will also hold for uncertain system (~71). For example, if we can find an output feedback controller for the uncertain system (,U2) which guarantees some i n p u t - o u t p u t property for the closed loop system, then the same controller will guaranteed the same i n p u t - o u t p u t property when applied to the system (~U1).

16.3

The Main Result

We now develop the main result of this paper which gives a necessary and sufficient condition for the uncertain system (~1) to be a subrealization of the uncertain

16 Equivalent Realizations for Uncertain Systems

265

system (Z2). This result involves the following system formed by augmenting the uncertain system (~1) with the system (16.5) corresponding to (,U2): A1 A2 + P2 [K~K2 -

'

, t P2K2G2 + B2 P2C2R2 0

;

x(0) = 0; ~?(0) = 0.

(16.14)

In particular, this system was obtained by using the signals u(-) and y(-) for (El) as the signals u0(.) and y0(') in (16.5) corresponding to the system (2~2). Here P2(t) is the solution to the Riccati differential equation (16.3) for the system (I;'2). Note, that in order to form this augmented system, we assume that the uncertain system (E2) satisfies assumptions (16.3),(16.4). Associated with this augmented system is the quantity: p[u(-), v(.), w(.)] fo r [IfK2~ + G2u]J2

(62~

[x' ~'] [C~R2C,

IT

+2 [x' e']

c1~

-

v) !

R~ ( C ~ -

Clx

v)] dt

K~K2 - CIR2C~

K;G2 C;R2

+ [u' v' w']

at.

(16.15)

-R2 0

The inputs in the above augmented system are required to satisfy the IQC:

fo r {~'Q,~ +

vI R l v

- (Klx + Glu)'(KlX + G ~ ) } dt _< d~.

This is equivalent to the condition:

~0T

+2[X'

&']

+ [~' v' ~']

OK~al 00

'u

dt <_ dl.

(16.16)

R1 0 0

Q~

We will think of (16.14), (16.16) as defining an uncertain system with an IQC uncertainty description and uncertainty inputs [u(.), v(.), w(-)]. In order to prove our main result, we will use the following lemma.

266

Ian R. Petersen

L e m m a 2. (El) is a subrealization of (E2) if and only if the following condition holds: For all d2 > O, there exists a d l > 0 such that for all inputs [u(.), v(.), w(.)] E L2[0, T] for the augmented system (16.14) satisfying (16.16), then (16.17)

p[u(-), v(.), w(.)] ~ -d2.

Proof First note that it follows from Lemma 1 and the construction of the augmented uncertain system (16.14), (16.16) that the following conclusion holds: Any input-output pair [so('), y0(-)] which is realizable for the uncertain system (El) will be realizable for the uncertain system (~2) if and only if the augmented uncertain system (16.14), (16.16) is such that condition (16.17) holds for all admissible uncertainty inputs Is(.), v(-), w(.)]. From this, the lemma now follows immediately using Definition 2. [] In order to develop a convenient test for the condition (2~1) C (~2), we will use a certain S-procedure result for two quadratic forms. To state this result, we first note that the augmented uncertain system (16.14), (16.16) together with the condition (16.17), can be re-written in the form: ~(t) ----71(t)2(t) +/~(t)fi(t); fo T r p[~(.)] =

2(0) ----0

(16.18)

< dl;

(16.18)

O(t)'N(t)O(t)dt >_ -d2

(16.18)

where

[i]

;

f/-=

;

_~=

At - C~R2C2] P~C~R2CI 0A2 + P2 [K2K: ;

[~=

0 D1 ] . B1 P:K~G2 + B2 P2C2R2 0 J '

N-=

I -C1R2C1 C~ R2 C1 0 -R2C1 0

C[R2C2 K~K2 - C ~ R 2 6 2 G': K2 R 2C2 0

[oK~K1 0 00 -GIG1 /V/= [ OG~K1 0 0 o

0 K~G2 G'2 G2 0 0

1"

- 1R2 0 C~R2 0 0 - R2 0

c 000] ;

0 0 R~ o QlJ

(16.19)

Hence, we consider the constrained optimal control problem: J* =

inf

fi(.)EL2 [0,T]

~(t)'IV(t)~(t)dt

(16.20)

16

Equivalent Realizations for Uncertain Systems

267

subject to

~(t) = / l ( t ) 2 ( t ) +/~(t)~(t);

2(0) = 0

and

f0 r

~(t)' l~I (t)~?(t)dt

_
We will solve this constrained optimal control problem using a Lagrange-Multiplier/Sprocedure approach. This involves considering a corresponding unconstrained optimal control problem: J* =

inf ~(.)eL2[0,T]

O(t)'[Nr(t) + rM(t)]O(t)dt

(16.21)

subject to

~(t) = f~(t)~2(t) +/~(t)~(t);

2(0) ----0.

L e m m a 3. The constrained optimal value J* defined in (16.20) is finite if and

only if there exists a ~- >_ 0 such that J* defined in (16.21) is finite. Furthermore, if J* is finite, then J* = max { J ; - Tdl}.

(16.22)

~->0

Proof First suppose J* is finite. We will use the standard S-procedure result for two quadratic forms; e.g., see [19,9]. To apply this result, we first define some quadratic functions on the vector space B = L~[0, T]. Let 9r(fi(.)) : B --, R be defined by O(t)'N(t)O(t)dt- J*

~-(u(.)) =

where #(t) corresponds to the solution to (16.18) with input ~(-). Also, let G(~(-)) : B ~ R be defined by G(~(')) = dl -

O(t)'M(t)~(t)dt.

Clearly both ~-(~(.)) and G(~(')) are quadratic functionals on B. Furthermore, ~(.) -- 0 implies 0(') ----0 and hence G(0) = d l > 0 . Now using the definition of these functionals and the definition of J*, we have ~(~(.)) > 0 ~

~

/o

O(t)'M(t)~(t)dt <