Workshop on Operator Theory and Complex Analysis: Sapporo, Japan, June 1991 (Operator Theory: Advances and Applications)

C".(z - A".)-lx - W(z)
(1.1)

(iii) for every

= D+C(z-A)-l B for W(z)j this is discussed

in [BR5]. For a more complete discussion, see [BGK] or [BGR4]. To discuss null or zero structure in the nonregular case, we need to introduce a definition. We define a real non-archimedean valuation

Irlz=.\

1•

={

Iz=.\ of'R by putting 0, e,-'I

if r = 0 if r '" 0

where 1/ is such that r(z) = (z-A)'If(z) with f(z) analytic and nonzero at A. If x 'Rn, where 'Rn can be identified with 'Rnx1 or 'R1X ,., let

= (Xl. X2,··· , xn) E

('Rn, II . IIz=.\) is a non-Archimedean nQrmed vector space over the real valued field ('R,I . Iz='\). Subspaces A and

n of ('Rn, 1I·llz=.\) are said to be orthogonal (see [M]) if

(1.2) for all x E A and y

= n.

We say that subspaces A and

holds for all x E A, yEn, A E 0'.

if (2.2) n We say that an element x of'R is orthogonal to a subspace of n

if'Rx is orthogonal to n (resp. to 'Ry) on 0'. of subspaces n and A of'Rn has a straightforward cHaracterization

of ~n (resp. an element y E 'Rn) on The orthogonality on

0'

n of 'Rn are orthogonal on 0' C ct

0'

in terms of linear algebra. Let n(A) (resp. A(A)) denote the subspace of ctn formed by the values at A of those functions in orthogonal on

0'

n which are analytic at A.

By Proposition 2.3 in [BR2],

if and only if n(A) n A(A)

= (0)

n and A are

102

Ban and Rakowski

for each A E (J. Let WE

n mxn , and let wo l

denote the left kernel of W, that is

w ol = {ip E nIxm : ipW = O}. A controllable pair of matrices (A" Be) of sizes

n, x n, n, x and

m, respectively, is said to be a

left null pair for W over (J (see [BR2]) if

(i) (J(Ae) C (J, (ii) for each x E (£IXnc the function x(z - Ae)-I B, is orthogonal to

wol on (J(Ae),

(iii) the function (z-Ae)-I B(W(z)h(z) is analytic on (J whenever hE nnXI((J) and W(z)h(z) is analytic on (J, (iv) the size of A, is maximal subject to the above conditions. A left null pair can be related to a canonical set of left null chains or left null functions for W(z) over (J (see [BR2] , [BR3] for the nonregular case, [BGK], [BGR4] for the regular case) and to a piece of a realization WX(z) = Dt - DtC(z - A + BDtC)-1 BDt for a generalized inverse WX(z) for W(z) (see [BCRR], [R] for the nonregular case and [BGK], [BGR4] for the regular case). Complete information regarding left null-pole structure of WE nmxn over (J is given by the null-pole subspace

An indication of this statement is that the null-pole subspace is a complete invariant for right equivalence over W; more precisely, two rational m x n matrix functions (assumed for simplicity to have trivial right annhilators) WI and W 2 are related by WI = W 2 Q where Q is analytic with nonsingular values on (J if and only if Su(Wt)

= Su(W2 ).

Let (C"., A".) be a right pole pair and let (A(, Be) be a left null pair for a function W E nffixn. Then there exists a unique matrix

(1.3) S .. (W)

= (WnnXI) n {C".(z -

r

(see [BR3]) such that

A".)-IX + h(z): x E (£n.xt,h E nmXI((J) and

L Resz=zo(z -

Ae)-I B(h(z) = rx}

zoEO'

The matrix

r

is called the coup1ing matrix associated with the right pole pair (C"., A".) and a left

null pair (A" Be) for W over (J. The triple

(1.4) is called a left D1!1I-pole triple for W over (J or a left a-spectral triple of W. More detailed motivation for these concepts and connections with more established notions in systems theory can be found in [BGR1] and [BGR4] for the regular case and [BRS] for the nonregular case.

103

Ball and Rakowski

Let W E 'R,.mxn. Any matrix polynomial P,. whose rows form a minimal polynomial basis for

wol

(see [F]) is said to be a left kernel polynomial for W. The following proposition

characterizes matrix polynomials P,. and data sets w arising as a left kernel polynomial and a left u-spectral triple for some rational matrix function. PROPOSITION

1.1. (cf. Proposition 4.1 in [BR2] and Theorem 3.1 in [BR3]) Let W E 'R,.mxn be a

function with a left kernel polynomial P,. and left u-spectral triple w as in (1.4). Then wand P,. satisfy the following conditions: (NPDSi) the pair (C"., A".) is observable and u(A".) C u,

Cd is controllable and u(Ad C u, A,r = B,c".,

(NPDSii) the pair (A"

(NPDSiii) rA". (NPDSiv) P,. has no zeros in CC and the leading coefficients of the rows of P,. are HnearIy independent, (NPDSv) the function P,.(z)C".(z - A".)-l is analytic on ct, (NPDSvi) if u(Ac)

= {AI! .x 2 , · · · ,.xr }, where the points .x}, .x 2 , · · · ,.xr are distinct, A, (

.xlI

.x2I

then the pair

1 [p,.(i

l) ) , P,.(.x2)

...

..

.xrI

P,.(.xr)

is controllable. Conversely, if {C.", A.", A" B" r} is a collection of matrices and P" is a matrix polynomial such that (NPDSi)-(NPDSvi) hold, then there exists a rational matrix function W with a left uspectral triple w as in (1.4) and with a left kernel polynomial equal to P". The construction of such a function W is indicated in [BR2] (see also Section 6 in [BRa]). Given a matrix polynomial P" and a triple w

r), where C.", A"., A" B"

r

= « c.", A.,,), (A"

B,), are matrices of appropriate sizes, we may associate with wand P,. a

linear space

(1.5) StreW, P,,)

= {h E 'R,.mXI : P"h = O} n {C,,(z -

A.,,)-lx + h(z}: x E ctn.xt, hE 'R,.mXI(u) and

E Resz=zo(z - Ad- B,h(z) = rx}. l

ZoEtr

It follows from formulas (1.3) and (1.5) that if w is a left u-spectral triple and P" is a left kernel polynomial for a function W E 'R,.mxn, then StreW, P,,} = StreW}. We will use the notation StreW, p .. ) and StreW} interchangeably. We shall refer to the collection (w, P,.) = (C.", A", A" B" a complete (left) nun-pole data set for W over u.

r, P,,(z}) as

104

Ball and Rakowski

2. A homogeneous interpolation problem. In the previous section we saw that a complete null-pole data set (w, P,.) over

q

for

a rational matrix function W is a useful set of invariants for describing the null-pole subspace S... (W)

= W1lnXl(q) associated with W.

However, for the 4-block interpolation problem, a more

general submodule of 1lm arises in a natural way. Specifically, assume that W in 1lmxn has a block row decomposition W 1l(q)-submodule of 1lmx1

= [WI

W 2 ] with WI E '7l mx (n-lc) and W 2 E '7lmxlc, and consider the

(2.1) We refer to S... (Wh W2 ) as the extended null-pole sub§Pace associated with the block row matrix function [WI W 2 ] over q. For our applications [WI W 2] has trivial right annhilator in 1lnxl, i.e., the columns of [WI W2] are linearly independent over 1l. Hence we make the assumption on W

= [WI

W 2] that the decomposition in (2.1) is direct. Note that we can recover W 21l1cx1 from

S...(W1, W2) as W21llcX1

=

nrS...

(Wh W2)

rE'R

The submodule W 11l(n-lc)xl(q) is not as uniquely deterlnined by S.. (Wh W 2); all one can say is that it is a direct sum complement to

n rS..(Wh W2) inside the 1l(q)- submodule S.. (Wh W 2).

rE'R

In any case the direct sum decomposition of S.. (W1 , W2 ) can be viewed as an affine version of the Wold decomposition for an isometric operator on a Hilbert space (see [NFl). From another point of view, note that S.. (Wf, W~)

= S.. (Wh W 2) if and only if [Wf

W~] = [WI W2] [ ~ ~] = [WIF + W 2 G

W2H]

where F and F-l are in '7l(n-lc)x(n-Ic)(q), G E '7l Icx (n-lc) and H and H-l are in '7l1cxlc . Indeed, to see this, note that both

map 1l(n-lc)xl(q) E91l1cx1 onto itself, and conversely, any multiplier in 1l"xn with this property necessarily must be ofthe form

[~ ~]

with p±l E '7l(n-lc) x(n-Ic)( q), G E '7l Icx (n-Ic), H±l E '7l1cxlc .

In particular we may always choose H so that the columns of W~ basis for their span, namely,

= W2H form a lninima.l polynolnia.l

n rS..(WI, W2). Note that the zeros and poles of Wf = W1F + W. 2 G re'R

can be quite different from those of WI; the invariant is the left zero-pole structure in a subspace

Ball and Rakowski

105

complementary to the 1l-span of the columns of W2 • We may always choose G E 1lA:x(n-/c) so that the span of the columns of W{ = WI + W 2G is orthogonal over the set of zeros and poles of W{ to the span of the columns of W2 (or, equivalently, of W~). Let us call any u-spectral triple

w = «C"., A".), (A" Bd,r) for W{ obtained from the pair (WI ,W2 ) in this way a W2-normalized left u-spectral triple for WI. The 1l-span of S,,(WI. W 2), U rS,,(WI. W 2 ) = [WI W 2 ]1lnxI , can rE'R

be specified via a left kernel polynomial P,. for the function [WI W 2 ]. Given a block row matrix function W

= [WI

W 2 ], we have now introduced (1) a W2-normalized left u-spectral triple w for WI. (2) a left kernel polynomial P,. for [WI W 2] and (3) a polynomial matrix Q E 1lmxL whose columns form a minimal polynomial basis for the 1l-span of the columns of W2 • Let us call the whole collection (w, P,., Q) a complete extended null-pole data set for W

= [ WI W2 ].

The analysis

above shows that two block row matrix functions [WI W 2] and [W{ W~] for which the associated extended null-pole subspaces are the same (Su(W{,

W~)

= Su(WI. W 2)) share the same complete

extended null-pole data sets. Moreover, we can recover the extended null-pole subspace Su(WI. W2) from the data set via the formula

({f E 1lmxI : P,.! = O}n {C".(z - A".)-I x + h(z): x E o:n.xI and h E 1lmXI(u) such that

E Resz=zo(z- Ac)-IB,h(z) = rx}) + Q1lA:xI

(2.2)

zoE"

For our application to interpolation problems in Section 4, we need to understand the inverse problem, namely: which data sets (w, P,., Q) arise as the extended complete null-pole data set over

u of a block row W

= [WI

W2] E 1lmxn. We view this problem as a homogeneous interpolation

problem. The answer is given by the following result.

THEOREM

= «C"., A".), (A"Bc), r» is the extended complete W = [WI W 2] E 1lmxn (where WI E 1lmx(n-/c) and

2.1. Suppose (w,P,.,Q) (where w

null-pole data set for a block row function

W 2 E 1lmx/c). Then the data set (w, P,., Q) satisfies the following conditions: (NPDSi) the pair (C".,A".) is observable and u(A".) C u, (NPDSii) (NPDSiii) (NPDSiv) (NPDSv)

the pair (4" Bd is controllable and u(Ad C u, rA". - A,~= B,C".,

P,. has no zeros in 0: and the leading coefficients of the rows of P,. are linearly independent, the function P,.(z)C".(z - A ... )-l is analytic on 0:,

(NPDSvi) if u(Ad

= {Ah ~2"

•• , ~r}'

where the points ~h ~2' ••• ~r are distinct, then the pair

106

Ball and Rakowski

is controllable. (NPDSvii) Q has no zeros in 0: and the leading coefIicients of the columns ofQ are linearly independent

(NPDSviii) the function (z - Ad- l B,Q(z) is analytic on 0: (NPDSix) if u(A,..)

= {wt, W2,··· , w

(Ie.

6 },

then the pair

Q(w,) Q(",,) ... Q(w.)l.

[A.

is observable;

J)

=

(NPDSx) P,.Q o. Conversely, if {C,.., A,.., A"B" r} is a collection of matrices and P,. and Q are matrix polynomials such that conditions (NPDSi) - (NPDSx) hold, then there is a block row matrix function W

= [WI W2] E 'Rmxn such that (w, P,., Q) is an extended complete null-pole data set for

W overu. REMARK.

Note that there is some redundancy in the conditions (NPDSi) - (NPDSx). In particular

(NPDSvi) implies the first part of (NPDSii) and (NPDSix) implies the first part of (NPDSi). Nevertheless, we list them in this form to make the connections with the I-block case (NPDSi) (NPDSiii) and the 2-block case (NPDSi) - (NPDSvi) clear. PROOF:

To prove the first statement, we note that condition (NPDSvii) follows from Proposition

3.18 in [BR2], and condition (NPDSix) follows from the fact that Q is analytic on 0: and the column spans of WI and Q are orthogonal on u. We give now the constructive proof of the second statement. The construction is a modification of the construction in [BR2] (see also Section 6 in

[BRa]). Let {C,.., A ... , A" B" r} be a collection of matrices and let P,. and Q be matrix polynomials such that conditions (NPDSi) - (NPDSx) hold. Step 1 Using the results from [GK], find a regular rational matrix function H with the u-spectral

«C. .

=

= ,A,..),(A"Bc),r). Find a Smith-McMillan factorization EDF of H and set WI ED. Step 2 Let v be the largest geometric multiplicity of a pole of H in u, let p. be the largest geometric multiplicity of a zero of H in u, and let '7 be the largest sum of the geometric multiplicity of a pole

triple w

107

Ball and Rakowski

and the geometric multiplicty of a zero of H at any single point of 0'. Let di denote the ith diagonal entry of D. For i = 7] - P. + 1,7] - P. + 2, ... , v, let P; be the minimal degree monic polynomial such that if dm-'I+i has a zero point

~

E

of order k then Pidi has a zero at

0'

~

of order k. Define an

m X 7] matrix polynomial Q = [%1 by

I, ifi=j$7]-p. { Pi, if7]-p.m-p. 0, otherwise, and put

Step 3 For i = 1, 2, ... , p., let

a point

~

E

0'

4>; be the (m -

functions for H

at~.

i - I )th row of E-l, so that if H has a zero at

{4>1, 4>2, ... ,4>,.} is a canonical set of left null Note that the left null pair constructed from the functions 4>t. ... ,4>/J is

of the geometric multiplicity

then

K.

left-similar to the pair (A" Be). We may, in fact, assume that both pairs are equal. Modify the functions

4>10 4>2, ... ,4>/J as follows. Suppose that O'(A,..) U O'(Ad = {At. ~2,··· , ~.} and the largest

multiplicity of a zero of H in

0'

multiplicity of a zero of H at

~;

is p., and consider the point be

K..

~i

(i = 1, 2, ... , s). Let the geometric

By condition (NPDSvi), the matrix

Ai

=

[~~~:~l 4>/J(~i)

P"(~i)

has full row rank. By conditions (NPDSviii) and (NPDSx), geometric multiplicity of a zero of H in

0',

there is a point

~

AiQ(~;)

=

o.

Since p. is the largest

such that the matrix

A= 4>/J(~) P,.(~)

has full row rank and AQ( ~) = O. Hence we can add to

4>; (j = K. + 1, K. + 2, ... , p.) a function

p(z)c,

where c is a vector and p(z) is a scalar polynomial vanishing at ~t. ~2' .•• ,~. to the order p., so that the modified 4>K+l, 4>,,+2, ••• ,4>/J (which are again called 4>,,+1, ••• ,4>/J) are such that the matrix

Ball and Rakowski

108

Hi=

4>"(.~i)

P,,(Ai)

has full row rank and HiQ( Ai)

= O.

In this manner we obtain functions 4>1,4>2,· .• ,4>,. whose span

is orthogonal to the row span of P" on q( A".) U q( A,) and such that the function

+(z)

=

[~1;ll

Q(z)

4>,.(z) vanishes on q(A".) U q(Ad. Now it follows from the construction and condition (NPDSvili) that if

4>i is a left null function for H at AI. A2,··· ,A. of orders h, 12, ... ,I., then 4>i(Z)Q(Z) vanishes at Ai to the order at least kj where kj

~

max{l,li}. Suppose ki

4>i(Z)Q(Z)

= (.Ii (z 3=1

< 00. By condition (NPDSvii),

Ai)k;)Vi(Z)Q(Z)

= ~i(Z)Q(Z). where Vi is analytic, and does not vanish, at AI, A2, ... ,A•.

Let,pi

= 4>i -

~i (i

= 1,2,··· ,1-').

Then the set {,p1o,p2,··· ,,p,.} contains a canonical set ofleft

null functions for H at each zero of H in span of P" on q(A".) U q(Ad, and ,piQ

q,

the span of {,pI, ,p2, ... ,,p,.} is orthogonal to the row

= 0 (i = 1,2,··· ,1-').

Extend the span of {,p1o ,p2, ... ,,p,.} to an orthogonal complement B of the row span of

P" in (Qol, q(A".) U q(A,», and project each column of W 2 onto an orthogonal complement of the row span of Q in

(P~T,

q( A".) U q( A,» along BOT

+ {column span of Q}

to get W3 •

Step 4 Multiply W3 on the right by a regular rational matrix function without poles or zeros in

q(A".) U q(Ad, so that the resulting function W4 has no zeros nor poles in q \ (q(A".) U q(Ad). Step 5 Find a minimal polynomial basis column span of [W4 Q] in

(P~T,

{UIo U2, .••

,uT } for an orthogonal complement of the

q(A".) U q(Ad). Set

The extended null-pole subspace S,,(w, p .. , Q) given in (2.2) has a special relationship with the null-pole subspace Sew, p .. ) studied in [BRa] as the following result shows.

109

Ball and Rakowski

2.2 .. Suppose (w,PIC(z),Q(z))

= (C".,A".,A"BC,r,PIC(z),Q(z»

is a a-admissible extended null-pole data set and let StreW, PIC) given by (1.5) and StreW, PIC' Q) given by (2.2) be the associated "R( 0' )-modules. Then THEOREM

PROOF:

The containment C is trivial from the definition. Conversely, suppose that j E StreW, PIC' Q)

is analytic on

0'.

By (2,2)

j(z)

(2.3) where x E

rx.

ctnw,h E "RmXl(a), r

E

= c".(z -

A". )-l x + h(z) + Q(z)r(z)

"Rkxl are such that PICj = 0 and

In general, if 9 E "Rmx1 has partial fraction decomposition 9

(elements of "Rmx1 with all poles inside ~c(9)

= 9_. Since C".(z -

0'

E

Resz=z.(z-Ad-1B,h(z) =

z.E"

= 9_ + 9+ where 9_ E "RO'Xl(a c )

and vanishing at infinity) and 9+ E "RmXl(a), we define

A".)-l x E "RO'Xl(aC) and h E "RmXl(a) in (2.2), we have

(2.4) But a consequence of (NPDSix) is that (2.4) can happen only if

From (NPDSi) and Lemma 12.2.2 in [BGR4), we conclude that x

= o.

Hence (2.3) becomes

j(z) = h(z) + Q(z)r(z) where PICj

= 0 and E Res..=z. (z -

A". )-1 BCh(z)

= O.

Since Qr is analytic on

(J'

and by (NPDSvii)

zaEr

Q has no zeros on 0', we must have that r is analytic on 0'. Furthermore, from (NPDSviii) we see that

E Resz=z.(z-Ad-1 BcQ(z)r(z) = o. Then j = h+Qr in fact is an element of StreW, PIC)n"RmXI(a) z.E"

as asserted. 3. Interpolation Problem.

Let

0'

be

&. subset

of the complex plane

ct. We look for a rational matrix function W

which is analytic on/O" and satisfies the following five interpolation conditions. Let

Z1,Z2,··· ,ZM

be given (not necessarily distinct) points in

scribed 1 X m and 1 x n vector functions analytic at zi (j be positive integers. We require that

0'.

Let

Xi

and

Yi

be pre-

= 1,2,··· ,M), and let k 1 , k 2 ,··· , kM

Ball and Rakowski

110

(3.1) for i

= 1,2,··· , kj and j

= 1,2,··· , M.

Let WI, W2, ..• , WN be given (not necessarily distinct) points on

0'.

Let Uj and Vj be

prescribed n x 1 and m x 1 vector functions analytic at Wj (j = 1,2,··· , N) and let It, l2,··· , IN be positive integers. The second condition is that

(3.2) for i

= 1,2,··· ,Ij

and j = 1,2,··· ,N.

For each pair of points Zi and Wj such that Zi

Wj,

we are given numbers /fg(1

1,2,··· , ki and 9 = 1,2,··· , Ij) so that df +1- 1

(3.3)

(J)

-:-(1-=-+-g------:1)-:"! -:"dz--:f:-:-+-g--:-1 (xi

(g)

( z) W ( Z )uj

I

( z)) Z=Zi

= / f g,

where M'1)(z) is a polynomial obtained by discarding all but the first 1/ coefficients in the Taylor expansion of a function h at Zi; that is, if h(z) = L:~1 h{i}(z - Zi)j-l in a neighborhood of Zi, M'1)(z) =

L:']=1 h{i}(z -

Zi)j-l.

The fourth interpolation condition is as follows. We are given 1 x m and 1 x n rational vector functions Pj+ and pj_ (j = 1,2,··· , K) analytic on

0'

and require that

(3.4) for all positive integers i and for at least one (and hence for all) A EO'. Finally, we are given n x 1 and m analytic on

0'

X

1 rational vector functions qj_ and qj+ (j = 1,2,··· , L)

and demand that i 1

dd i-I (W(z)qj_(Z))

(3.5)

Z

I

= ddZ i-I qj+(Z) I Z=A Z=A i 1

for all positive integers i and for at least one (and hence all) points A E 0'. Below, we reformulate the conditions (3.1) - (3.5) and show when they are consistent. The combination (3.1) - (3.4) was considered in detail in [BR5]; we summarize this case (the so-called 2-block case) first. We note that the problem of finding a rational matrix function which is analytic on

0'

and satisfies conditions (3.1) - (3.3) is called the two-sided Lagra.nge-Sylyester interpolation

problem. Its complete solution is presented in Chapter 16 of [BGR4J. The problem of finding a rational matrix function which is analytic on in [BRS].

0'

and satisfies conditions (3.1) - (3.4) has been solved

Ball and Rakowski

111

3.1 2-block Interpolation Problem.

In [BR5] the following general interpolation problem was considered. We are given a subset u of the complex plane a; and an interpolation data set

(3.6) consisting of matrices C+ E a;mxn·,C_ E a;nxn., A". E a;n.xn·,A( E a;n(xn(,B+ E a;n(xm,

B_ E a;n( xn,

r

E a;n( xn. and matrix polynomials P+(z) E nKxm and P_(z) E nKxn which

satisfy the admissibility requirements (IDSi) the pair (C _, A".) is observable and u( A".) C Uj (IDSii) the pair (A(, B+) is controllable and u(Ad C Uj (IDSiii)

r A". - A(r = B+C+ + B_C_

(IDSiv) p_CRnXl) C p+(nmXl), P+(z) has no zeros in

U

and the rows of [P+(z) P_(z)] form a

minimal polynomial basisj (IDSv) the function

is analytic on a;. (IDSvi) if 0"( A".) = {At. A2, ... , Ar }, the pair

j[

P+(Al) B+ ) P+(A2)

ArI

'

P+~Ar)

is controllable. A collection of data w in (3.1.1) satisfying (IDSi) - (IDSvi) is said to be a uadmissible interpolation data set. The problem then is to describe all rational matrix functions with W(z) analytic on u which satisfy the interpolation conditions

(3.1') zoEO'

(3.2') zoEa'

(3.3')

L zoECI

and

(3.4')

Resz=zo(z -

Ad- 1 B+W(z)C_(z - A".)-l

=

r

112

Ball and Rakowski

We mention that the problem (3.1') - (3.4') is simply a more compact way of writing conditions (3.1) - (3.4). Indeed (3.1') is equivalent to (3.1) if we let, for each j

(3.7)

A(j

['

= {

1

Zj

,8;+ =

1

= 1,2, .. · ,M, {l}

[:!:){,) 1

Yj

{2}

,8;_ =-

z{l.j}

Zj

yJI.i}

J

where f{i} denotes the

ith

Yj

coefficient in the Taylor expansion of a function fat

Zj,

and set

(3.8)

Conversely, if matrices A(, B+, B_ of appropriate sizes are given, there exists a nonsingular matrix

S such that SA,S-l is a block diagonal matrix with the diagonal blocks in lower Jordan form. After replacing A( by SA,S-I, B+ by SB+ and B_ by SB_, we can convert the interpolation condition (3.1) to the equivalent condition (3.1'). Similarly, if for each j

= 1,2,··· , N

we let

(3.9)

Wj

A"'j =

[

1 Wj

and then set

A".

= [ A""

A"..

1

. A".N

then (3.2') collapses to (3.2). Conversely, if the matrices C+, C_, A". are given, we can find a nonsingular matrix S such that S-lA".S is in Jordan form. After replacing A ... by S-lA".S, C_ by

C_S, and C+ by C+S, we can reformulate the interpolation condition {3.2'} in the more detailed form {3.2}.

Ball and Rakowski

113

A similar analysis gives an equivalence between (3.4) and (3.4'). Suppose first that and let rii and

Wj

If Zi

f=

Z;

= wi

= hlg] where 'Y!i! = 1,2, .. ·ki and 9 = 1,2,,,, ,Ii) are numbers associated with Zi

as in (3.3). Then the interpolation conditions (3.3) hold if and only if

Wj

let rij be the unique solution of the Sylvester equation in X

where A".j and A" are as in (3.7) and (3.8) (see [BGR4], Appendix A.l) and set r = [fij] with 1 ::::; i ::::; M and 1 ::::; j ::::; N. Then the set of interpolation conditions (3.3) is equivalent to the single

block interpolation condition (3.3'). We also mention that validity of the Sylvester equation

is a necessary and sufficient condition for the consistency of (3.1') - (3.3') (see [BGR4] or [BR5]). Conditions (3.3) can be recovered from (3.3') by multiplying both sides of the equality by nonsingular matrices Sand T such that SA,S-l and T-l A,T are in lower and upper Jordan forms, respectively, and reading the numbers "(Ig from the appropriate blocks ofthe matrix SrT. Finally, if P+ and P_ are rational matrix functions whose jt" rows (j

= 1""

, K) are

equal to Pj+ and -Pj_ respectively, then it is easy to see that conditions (3.4) are equivalent to (3.4'). Demanding that P+ and P_ are polynomials is with no loss of generality. The admissibility conditions (IDSi) - (IDSvi) guarantee consistency and minimize redundancies in the interpolation conditions. For complete details we refer to [BR5]. 3.2 The general 4-block case.

Let Q_ and Q+ be rational matrix functions whose

P"

columns (j

= 1,2"" , L) are

equal to qj_ and qj+, respectively. Conditions (3.5) are equivalent to

(3.5') Similarly as with P+ and P_, we will assume first Q_ and Q+ are matrix polynomials and the columns of

(3.10) form a minimal basis.

Ball and Rakowski

114

We consider now the consistency of condition (3.5'). It follows immediately from (3.5') that

Also, since W is analytic on u, Q_ has no zeros in u. Indeed, if Q_ had a zero in u, then Q+ = W Q_ would have a zero in u, and so the rank of the matrix polynomial (3.10) would drop at some point of u, contradicting the fact that the columns of (3.10) form a minimal polynomial basis. Condition (3.1) says that the first kj Taylor coefficients at Zj of Xj(z)W(z) coincide with the first kj Taylor coefficients at Zj of Yj(z). Therefore, by (3.5'), the first kj Taylor coefficients at

zi of Xj(z)Q+(z) coincide with the first kj Taylor coefficients at Zj of y;(z)Q_(z). Consequently, the function

(z - AC;)-l[Bj+ Bj-l

[3~~~~n

is analytic on CJ: where AC;' Bj+ and Bj_ are as in (3.7). Hence the function

is analytic on CJ:. Suppose that a vector Uj(Wj) is in the column span of Q_(Wj), where functions in (3.2). Then Uj(Wj)

Uj

is one of the

= Q_{Wj)Uo, and it follows from (3.2) and (3.5') that

So in this case conditions (3.2) and (3.5) are either redundant or contradictory. To exclude such situations we require that the matrix

[Ui,(Wj,) uj,(wj,) .. . ui. (wi.) Q-{Wi'>] have full column rank whenever Wj,

= wi. = ... = Wj..

In terms of the data A"., C+, C_, this

assumption is equivalent to the controllability of the pair

(3.11)

where {wi"wi.,··· ,Wi.} is the set of distinct points among {WtoW2,··· ,WN}. We will call this assumption the consistency of right generic and right lumped interpolation conditions. Finally, in view of (3.4') and (3.5'),

Ball and Rakowski

115

P+Q+ = P+WQ_ = -P-Q-. So conditions (3.4') and (3.5') are consistent if

We summarize various consistency and minimality properties our interpolation data must have in the following definition. We will call the data

(3.12) a q-admissible interpolation JiiWl.ill if C+ E a: mxn • , C_ E a: nxn ., A". E a:n• xn., A( E a:ne xne, B+ E

a:nexm,B_ E a:nexn,r E a:nexn·,p+(z) E nKxm, P_(z) E nKxn,Q_ E nmxL,and Q+ E nnxL are such that (IDSi) the pair (C_,A".) is observable and O'(A".) C 0'; (IDSii) the pair (A(, B+) is controllable and O'(A".) C 0'; (IDSiii) r A". - A,r = B+C+ (IDSiv) P_ nnxI

c

+ B_C_;

p+nmxI, P+(z) has no zeros in 0' and the rows of [P+(z) P_(z)] form a

minimal polynomial basis; (IDSv) the function

is analytic on a:. (IDSvi) if 0'( A".)

= {AI, A2, ... , AT}, the pair P+(At) B+ ) P+(A2)

P+(AT) is controllable; (IDSvii) n1xmQ+(z)

c nlxnQ_(z), Q_(z) has

form a minimal polynomial basis;

no zeros in 0', and the columns of

Ball and Rakowski

116

(IDSviii) the function

is analytic on 0:. (IDSix) if u(A".)

= {w}, W2,··· , w.}, then the pair

J)

is observable. (IDSx) [P+ P_]

[~~] = o.

Given a u-admissible interpolation data set, the interpolation problem is to find which m x n rational matrix functions W analytic on u satisfy conditions (3.1') - (3.5'). 4. Parametrization of solutions.

Theorems 4.1 and 4.2 below present our results on parameterization of solutions of the interpolation problem (3.1') - (3.5') (respectively with or without an additional norm constraint). We first need the following observation. Ifw

= (C+, C_, A"., A"

B+, B_, r, P+(z), P_(z), Q_(z),Q+(z»

is a u-admissible interpolation data set, then the collection of matrices

(4.1) together with the matrix polynomials PI<

= [P+

P_] and Q

= [~~]

satisfy conditions (NPDSi)

- (NPDSx) in Theorem 2.1. Therefore there exists a rational block row matrix function 9

8 = [8

: 9 1S ] in n,(m+n)x(m-K+n) (where 9 1 E n,(m+n)x(m-K+n-L), 9 2 E : 92S n,(m+n)xL and 9 1 = [9 11 8 12 ] with 9 11 E n,mx(m-K) 9 s = [9 1S ] with 8 1s E n,mXL) which 9 21 922 ' 9 2S [91

s]

11

=

8 21

has

9 12 9 22

(w, [P+(z) P"':(z)], [~~~~~]) as an extended complete null-pole data set over u. The 2-block

version of the following appears as Theorem 4.1 in [BR5]. THEOREM

4.1. Let u E 0: and let was in (3.12) be a u-admissible interpolation data set. Then there

exist functions W E n,mxn(u) which satisfy the interpolation conditions (3.1') - (3.5'). Moreover, if

117

Ball and Rakowski

is a block row rational matrix function with extended complete nun-pole data set over 0' equal to

(w, [P+(z) P_(z)], [g~~~~]), then a function WE 1lmxn is analytic on 0' and satisfies interpola-

tion conditions (3.1') - (3.5') if and only if W

= [011Q1 + 0 12 Q2 0 13] [021 Ql + 022Q2 0 23r 1

where Ql E 1l(m-K)x(n-L)(0') and Q2 E 1l(n-L)x(n-L) are such that

[021Ql

+ 0 22 Q2 0 23] (1l(n-L)Xl(0') EB 1lLXl)

= [021022023] (1l(m-K+n-L)X1(0')EB1l LX1 ).

(4.2)

We note that condition (4.2) holds for a generic pair (Q1, Q2) of rational matrix functions

lr IT::::::::~: ~::~ri::'':::ho:',::':~:M~;: :-:::~u:::,: function [0 21 0 22 0 23 ], an 1l-subspace inside 1l(m-K+n)xl of dimension m - K. For the Hoo-control problem, one takes 0' to be either the right half plane n+

Re z > O} or the unit disk V

= {z : Izl <

= {z

:

I} and asks that W, in addition to meeting a set

of interpolation conditions on 0', satisfy IIWlloo := rms1LPzEuIlW(z)1I

<

'Y for some prescribed

tolerance level 'Y. Without loss of generality we assume that 'Y has been normalized to 'Y = 1. The following refinement of Theorem 4.1 gives the solution ofthis generalized Nevanlinna Pick problem (Le. interpolation problem with norm constraint). The 2-block version of the following result is Theorem 4.2 in [BRS]. THEOREM 4.2.

Let 0' be either the right half plane n+ or the unit disk V and let w as in (3.12) be

a O'-admissible interpolation data set. Suppose that there exists a rational (m + n) x (m - K

+ n)

block row matrix function 0(z) = [0 11 (Z) 0 21 (Z)

0 12 (z) 0 22 (Z)

(with 0 11 E 1lmx (m-K), 0 12 E 1lmx (n-L») such that

: 0 13 (Z)] : 0 23 (Z)

(a) ([ g~] ,A", A(, [B+ B_], r, [p+(z) P_(z)],

[g~~~~])

is an extended complete

nuJI-pole data set for 0 over 0' and

/

(b) e is analytic on 80' (including at infinity if 0' = n+) and satisfies 0(z)* J0(z) j EB -h,z E 80' where J Then the fonowing are equivalent.

= 1m EB -In,j = Im-K EB -In-L.

=

118

Ball and Rakowski

(i) There exist functions W E

'Rmxn(O')

(3.5') and in addition satisfy

which satisfy the interpolation conditions (3.1') -

IIWlloo < 1.

(ii) 9 satisfies 9(z)* J9(z) $ j Ell O(n-L)x(n-L) at all points z of analyticity in Moreover, if (i) and (ii) hold, function W E 'Rmxn is analytic on lation conditions (3.1') - (3.5') and has

W= [9

(4.3)

11 H

satisfies the interpo-

IIWlloo < 1 if and only if

+ 9 12 9 13]

for an H E 'R(m-K)x(n-L)(O') with

0',

0'.

[9

21 H

+ 9 22

9 23 ] -1 =: ge[H)

IIHlloo < 1.

In fact the existence of a matrix function 9 satisfying conditions (a), (b) in Theorem 4.2

REMARK.

is also necessary for interpolants

W with IIWlloo < 1 to exist, but we do not prove this here.

To make Theorem 4.2 useful we need a systematic way of computing the rational matrix function

e

appearing in Theorem 4.2. For practical purposes we need only compute a rational

matrix function 9 of the form

9=e[Jl

J2

~J

where 11,12, and 13 are any rational matrix functions (of the appropriate sizes) with 13 invertible and

e is as in Theorem 4.2, since in such a case 9

map ge

= ge

and

e induce the same linear fractional

(see [BHV)). One can obtain such a matrix function 9 from a preliminary

constructed to meet the specifications in Theorem 4.1 as follows. IT by W'j(z) = 0li(Z)*01'(Z) 1<'J<3 Partition W as a block 2 X 2-matrix function by

define W(z)

= [W.j(Z)]

W= where Wo

O2.(z)*02j(z).

[~! ~:]

= [W'jh~'.j~2 and set V equal to the Schur complement

Finally find a square outer matrix function Ro(z) such that V(z)

= Ro(z)*jRo(z).

Then

_= 9-9

[R00 0] I 1

e=

[011 021

012 022

e

013] 023 '

Ball and Rakowski

119

is the desired matrix function such that the associated linear fractional map

ge parametrizes

interpolants with norm less than 1 as in Theorem 4.2. Hence, once one has constructed a Theorem 4.1, one arrives at an appropriate

e as in

e via one j-spectral factorization V = R*jR. For more

complete details we refer to Theorem 4.3 in [BHV]. Alternatively, once one has a matrix function

e meeting the specifications in Theorem 4.1,

one could transform to a plant P such that the transform P of P to the generalized chain formalism is equal to

e (see Section 5). Then the interpolation problem with norm constraint IIWlloo < 1 is

equivalent to the Boo-problem for the plant P (with tolerance level 1); one can then use any of the known solutions of the Boo-problem (e.g. [DGKF], [G], [PAJ]) to produce the parameterizer

e of

contractive (or "bounded real" in engineering parlance) interpolants. Of course, it would be desirable to have the criterion for existence of interpolants W with IIWlloo < 1 and the construction of the linear fractional map

ge expressed more directly in terms

of the interpolation data w, as has been done for the I-block case in [BGR4]i this together with a more explicit realization formula for in this area. As was remarked above, pole data set whenever w

e in terms of the data set u, is the remaining open problem

(w, [P+(z) P_(z)], [~~~;~]) is a u-admissible extended null-

= ( C+,C_, A"., A"

B+, B_, r, P+(z), P_(z), Q_(z), Q+(z») is au-admis-

sible interpolation data set. The converse statement is not true in general; the following theorem, which will be used in our characterization of stabilizable linear feedback systems in the next section, characterizes which u-admissible extended null-pole data sets

(w, [P+(z) P_(z)], [~~~;~]) arise

from u-admissible interpolation data sets in terms of the associated extended null-pole subspace S(w,[p+ P_],

[~~]); see Theorem 4.3 [BR5] for the 2-block case and Theorem 13.2.3 in [BGR4]

for the I-block case. THEOREM

4.3. Let

(w, P, Q) be a u-admissible extended null-pole data set of the form

and set w

= (c+, C_, A"., A"

B+,B_,r, P+(z), P_(z),Q_(z),Q+(z»).

Then the following are equivalent. (i) w is a u-admissible interpolation data set. (ii) The extended null-pole subspace S

= S,,(w, P, Q) c

following properties: (a) S n [n.mxl ED 0] c n.mXl(u) Ell 0,

n.(m-K+n)xl (see (2.2)) has the

120

Ball and Rakowski

(b) POE9R ftX1(Snn.<m+n)Xl(O'») =

oEll n.nXl(O') WherePOE9R

ft . . ( [ : ] )

[~].

=

PROOF: Suppose first that w is a O'-admissible interpolation data set. By definition, any function

f

E S has the form

= [h(Z)] = [C+] (z _ A".)-lz + [h+(Z)] + [Q+(z)] r(z) f-(z) C_ L(z) Q_(z)

f(z)

where z E o:n., h+ E n.mXl(O'), h_ E n.nXl(O'), r E n.LXl. Suppose that f-(z)

h_(z)

+ Q_(z)r(z)

= O. Let P~c denote the projection map P~c : k

partial fraction decomposition k = k_

+ k+

where k_ has all poles in

=0

(i.e. k_ E n.~Xl(O'C» and k+ E n.nXl(O'). Then from fC_(z - A".)-lz E n.~Xl(O'c) and h_ E n.nXl(O'), we see that

0'

->

= C_(z -

A". )-lZ +

k_ if k E n.nxl has

and vanishes at infinity

we get that P~c(f-)

= OJ

since

(4.4) But now by condition (IDSix), (4.4) forces

(4.5) By Lemma 12.2.2 in [BGR41, the first condition in (4.5) forces z = O. Also, since by (IDSvii) Q_ has no zeros in

0',

the second condition in (4.5) forces r to be analytic on

h(z) = h+(z) + Q+(z)r(z) is analytic on

0'.

[}~]

But then

Thus S satisfies conditions (ii-a) in Theorem 4.3.

To verify (ii-b) we must show that for any given so that

0'.

f-

E n.mXl(O') we can find hE n.mXl(O')

E S. But by Theorem 2.2, S.,.(w,p)nn.<m+n)xl(O') = S.,.(w,p,Q)nn.<m+n)xl(O').

Hence this result follows in the same way as for the 2-block case (see Theorem 4.3 in [BR5]). Conversely, suppose which S

= S.,.(w,[p+

P-l,

(w, [P+ P-l, [8~]) is a O'-admissible extended null-pole data set for

[8~])

satisfies (ii-a) and (ii-b). Thus (w,[p+ P-l,

[8~])

satisfies

(NPDSi)-(NPDSx) and we must verify that w satisfies (IDSi) - (IDSx). First note that (IDSiii), (IDSv), (IDSviii) and (IDSx) follow from their counterparts (NPDSiii), (NPDSv), (NPDSviii) and (NPDSx) respectively without use of any assumptions on S.,.

(w, [P+ P-l, [8~]). Next, one can

use Theorem 2.3, assumption (ii-b) and the same arguments as those for the 2-block case (see Theorem 4.3 in [BR5]) to deduce (IDSii), (IDSiv) and (IDSvi) from their counterparts (NPDSii), (NPDSiv) and (NPDSvi). It remains now only to verify (IDSi), (IDSvii) and (IDSix). Suppose now that (C_, A ... ) is not observable. Then by Lemma 12.2.2. in [BGR4l there is an z

e 0:".

not zero with C_(z - A... )-lz

we can solve for h+

e n.mXl(O') so that

= 0 identically in z.

From Theorem 4.5 (i) in [BR3I,

121

Ball and Rakowski

is in S,,(W, P) C S,,(w, P, Q) (here we use that (IDSvi) has already been verified). Furthermore, since ([ g~]

,A,,)

is observable by (NPDSi), and since C_(z -

A" )-lx = 0, by

Lemma 12.2.2. in [BGR4j again necessarily C+(z - A" )-l x is not identically zero. Then fin (4.6) is an element of S,,(w,P,Q) in violation of (ii-a). Hence we must have (C_,A".) observable, i.e. (IDSi) holds. Next suppose (IDSvii) is not true. Then we can find r E 1lLX1 such that Q+r ~ 1lmXl(0') but Q_r E 1lnXl(0"). Again by an argument based on Theorem 4.5 (i) in [BR3j, we can then produce an element h+ E 1lmXl(0") so that

[~~r]

E S,,(w, P)

c S,,(w, P, Q). The n

is an element of S,,(W, P, Q) in violation of (ii-a). Hence (IDSvii) must hold. Finally, suppose (IDSix) is violated. Then we can find x =F 0 in o;n. and r =F 0 in 1lLX1 so that L(z)

= C_(z -

A,,)-lx

+ Q_(z)r(z)

can produce h+ E 1lmXl(0') so that

[~~]

E

E

1lnXl(0'). From Theorem 4.5 (iii) in [BR3j we

S,,(w,P) C S,,(w,P,Q) where h_ = (I -

Also by Theorem 4.5 (i) from [BR3j there is 9 E 1lnx1 (0') so that

[g~] (z -

P~c)(Q_r).

A" )-1 X +

[g~)]

E

S,,(w, P) C S,,(w, P, Q) (here we use that (IDSvi) has already been verified). But then

+ Q_(z)r(z) is analytic on 0' and (NPDSix) is in force, we + Q+(z)r(z) is not analytic on 0". Hence f is an element of

However, since x =F 0, C_(z - A,,)-l x are guaranteed that C+(z - A,,)-I X

S,,(w, P, Q) in violation of (ii - a). Thus, (IDSix) must hold and Theorem 4.3 follows. The proof of Theorem 4.1 requires the following lemma, an adaptation of one of the main ideas in [Be]. LEMMA 4.4. Let 0' C 0;, let w as in (3.12) be a a-admissible interpolation data set, and let

122

Ball and Rakowski

be a block row rational matrix function with extended complete null-pole data set over u equal to

(w, [P+ P-1, [~~])

where

wis as in (4.1). Then a function W E nmxn is analytic on u and

satisfies the interpolation conditions (3.1 ')-(3.5') if and only if

[Wjz)] [021 02 2 0 231(n(m-K+n-L)X1(u) EB n LX1 )

C 0 ( n(m-K+n-L)x1(u) EB nLX1).

(4.7) PROOF:

As in the proof of Theorem 4.3, one can show that

[0 21 0 22 0 231( n(m-K+n-L)x1(u) EB n LX1 )

={C_(z -

A"y 1x: x E C n.} + nmX1(u) + Q_nLX1

S,,( (C_,A,.),(0,0),0),0,Q_).

=

Suppose W is analytic on u and satisfies the interpolation conditons (3.1 ')-(3.5'). By Lemma 4.4 in [BR51,

[lj] ({C_(Z-A,,)-1X:XEG:n.}+nmX1(u»)

C

S,,(w, P) C S,,(w, P, Q)

=

0(

n(m-K+n-L)x1(u) EB nLX1).

But by condition (3.5'),

and hence the inclusion (4.7) follows. Conversely, suppose that the inclusion (4.7) holds. Then

[lj] Q_nLX1

C

nr0(n(m-K+n-L)Xl(u)EBnLXl) ~ER

= 0(0 EB n LX1 ) = [8~] n LX1 and hence W satisfies (3.5'). Moreover, by an argument in the proof of Theorem 4.3 we know that

123

Ball and Rakowski

Using also that Q_r = 0 implies that Q+r = 0 (a consequence of (IDSvii», we see that the inclusion (4.7) implies that

[lj] c

S"

({C_(Z-A1r)-lx:x E o:n.}+1lnXl

(u»)

(w, [P+ p-1) = fm(m-K+n)xl(u).

Now Lemma 4.4 from [BR51 implies that W is analytic on u and satisfies the remaining interpolation conditions (3.1 ') - (3.4'). PROOF OF THEOREM

4.1. By Lemma 4.4 it suffices to show that W E 1lmxn satisfies

[lj]

[9 21 9 22 9 231( 1l(m-K+n-L)Xl(u)EIl1lLXl )

C 9 ( 1l(m-K+n-L)xl (u) EIl1lLXl)

(4.8)

if and only if W has a representation of the form

(4.9) where

Ql

E 1l(m-K)x(n-L)(u), Q2 E 1l(n-L)x(n-L)(u) are such that

[9 2l Ql

(4.10)

+ 9 22 Q2 9 231( 1l(n-L)xl(u) EIl1lLXl)

= [9 21 9 22 9 231(1l(m-K+n-L)Xl(u) Ell 1lLXl).

Suppose first the (4.9) and (4.10) hold. Rewrite (4.9) in the form

[lj]

[9 2l Ql

+ 9 22Q2

9 2319

[~~ ~].

Apply both sides to 1l(n-L)xl(u) EIl1l LXl and use (4.10) to get

[If] [9

21

9 22 9 231(1l(m-K+n-L)Xl(U) Ell 1lLXl)

= 9 [~~ ~]

(1l(n-L)Xl(U) EIl1lLXl).

Ball and Rakowski

124

Since Ql and Q2 are analytic on u, we have

and (4.8) follows. Conversely, suppose (4.8) holds. Note that r~r 'RLXl)

=

[lj]

9 23] ( 'R(m-K+n-L) Xl (u)E9

823'RLxl is a 'R-subspace of 'Rnxl of dimension L which (by (4.8)) is contained

n

r9('R(m-K+n-L)Xl(u) E9 'RLXl) == rER dimension count,

in

[lj] [821 822

[lj]

[~13] 'R LXl , also an 'R-subspace of dimension 8 23

823'RLXl ==

L. By

[~~:] 'RLXl.

Next note that the quotient module

[lj] [821 8 22 8 23]

('R(m-K+n-L)Xl(U) E9 'RLXl) /

[lj]

823'RLxl

is a free 'R(u)-submodule of

[qn-L,+ ] ql,+ ] with n - L generators; we may choose n - L generators of the form 8 [ qti- , .. ·8 qn-oL,- and

then set

Q+ == [ql,+ ... qn-L,+] E'R(m-K)x(n-L)(u), Q_ == [ql,- .. . qn-L,-] E'R(n-L)x(n-L)(u).

Then the left side of (4.8) has the representation

[lj] [92l 9 22 923] == 8

[~~ ~]

== [8 UQ1 9 21 Ql

('R(m-K+n-L)Xl(u) E9 'RLXl)

('R(n-L)Xl(U)E9'R LX1 )

l2 Q2 813] + + 9922Q2 9 23

('R(n-L)Xl(U) E9 'RLXl)

From equality of the bottom components in (4.11) we deduce that the pair (QhQ2) satisfies (4.10) and then (4.11) can be rewritten as

Ball and Rakowski

125

[lj] [921Ql + 9 22 Q2

923] ( n.(n-L)xl( u) Ell n. LX1 )

= [99 2111Q1 + 9 12 Q2 9 13 ] Ql + 8 22Q2 9 23

(4.12)

(n.(n-L)Xl(U) Ell n.LX1 )

Also from (4.10) we see that the square matrix function [9 21 Ql +9 22 Q2 9 23 ] cannot have determinant vanishing indenticaIly, and hence [9 21 Ql +9 22 Q2 9 23 ]-1 exists. Now (4.12) leads immediately to the representation (4.9) for W. The proof of Theorem 4.2 requires the following lemma. LEMMA 4.5. Suppose u,w and 9 =

[~11 ~12 ~13] 021 022 023

are as in the hypothesis of Theorem 4.2.

Then 9 satisfies

9(z)* J9(z) :5 i Ell 0, z E u (where J

= 1m Ell -In

and

i = Im-K Ell In-L)

if and only if [In-L 0][922 923r1921 has analytic

continuation to all of u. PROOF:

By assumption 9(z)*J9(z)

= i Ell -h for z E ou. In particular

[~t~] [912 9 13]- [~~~] [922 923] = -In' Hence

[9* ][922 9 23] 9~~

=

hn +

Z

E OU.

[9* ][912 9 13] for z E ou. As [922 9 23] is square, we conclude 9i~

that [922(Z) 9 23 (Z)] is invertible for Z E

ou.

Then for each fixed z we can rewrite the system of

equations

(4.13) (for z E ou) in the equivalent form

(4.14) where

= [912 9 13][922 9 23r1 U12 = 9 11 - [912 9 13][922 923r1921 g:~] = [822 823r1 U11

[

[g:] = -[822 823]-1821 .

126

Ball and Rakowski

Note that 0(z)* J0(z)

= j Ell -h on au is equivalent to

or equivalently, to

whenever z E au and (u,y,yO,v,w) satisfies (4.13). Since (4.13) and (4.14) are equivalent, we see that 0(z)* J0(z)

=j

EIl-h on

U12] au is equivalent to U(z) = [UU U2I U22 (z) being isometric on au.

Similarly, the validity of 0(z)* J0(z)

:s; j

hI

U32

Ell 0 on u is equivalent to

or otherwise stated, to

whenever z E u and (u,y,yO,v,w) satisfy (4.13). Since (4.13) and (4.14) are equivalent, we read off that this in turn is equivalent to [Im+n-L O]U(z) being contractive on u. In particular

must be contractive at all points of analyticity of u, and hence has analytic continuation to all of the set u. Conversely, to show that 0(z)* J0(z)

:s; jEllO on u is equivalent to showing that [Im+n-L 0]

U(z) has contractive values at all points of analyticity in u, or, equivalently by the maximum modulus theorem, that

have analytic continuation to all of u. By assumption we know only that

has such an analytic continuation. The fact that this forces Un, U12, and Un to have such an analytic continuation as well relies on the special structure of the extended null-pole subspace

e(1l(m-K+n-L)Xl(U) Ell 1lLXl) = s. . (w,[p+,P_J, [8~]) given by Theorem 4.3. Details for the I-block case (the case where K

= 0 and L = 0) are given in Theorem 13.2.3 in [BGR4Ji we leave

it to the reader to check that the general case here can be verified by analogous arguments.

Ball and Rakowski

PROOF OF THEOREM

127 4.2. Let a,w and 8 be as in the statement of Theorem 4.2. Suppose first

that 8(z)* J8(z) ::::: JED O(n-L)x(n-L) at all points z E a where 8(z) is analytic and that H E 1l(m-K)x(n-L)(a) with

IIHlloo < 1.

Then

(4.15) By Lemma 4.5 we know that [In-L 0][8 22 8 23 ]-18 21 is analytic on a and, from the proof of Lemma 4.5, that 11[8 21 823J-1821(z)1I invertible for all

Z

< 1 for z E Ba. Hence t[In-L 0][8 22 8 23 ]-18 21 H + In-L is

E Ba and for all t with 0 ::::: t ::::: 1. Hence wno det([In_L 0][8 22 823r1821H

+

In-L) = wno det In-L = 0, where wno X is the change of the argument of X(z) along Ba.

As we observed above that [In-L 0][8 22 823r1821H det([In_L 0][822 823r1821H

+ In-L)

+ In-L

is analytic on a, we conclude that

has no zero in a. From this we see that

From (4.15) we conclude that

[8 21 H

+ 8 22 8 23] (1l(n-L)Xl(a) ED1lL)

= [8 22 8 23 ] ( 1l(n-L)xl( a) ED1lL ).

(4.16) On the other hand,

[8 21 8 22 8 23 ] ( 1l(m-K+n-L)X1(a) ED1lL) ( 4.17)

= [8 22 8 23 ][[8 22 823J-1821 In] (1l(m-K+n-L)X1(a)ED1l L ).

But by Lemma 4.5 [In-L 0][8 22 823]-1821 is analytic on a, so

[[822823]-1821 In] ( 1l(m-K+n-L)x1 (a) ED1lLX1) (4.18)

= 1l(n-L)Xl(a) ED1lLX1.

Combining (4.17) and (4.18), we obtain

Ball and Rakowski

128

[9 21 9 22 9 231 ( R(m-K+n-L)xl(U) E9 RLX1) = [9 22 9 231 ( R(n-L)xl(U) E9 RLX1).

(4.19)

Combine (4.19) and (4.16) to see that the pair (H,I) is an admissible parameter pair in the sense of Lemma 4.4. Hence, by Lemma 4.4 we know that W = [9 11 H + 9 12 9 13 )[9 21 H + 9 22 923r 1 is analytic on u and satisfies the interpolation conditions (3.1') - (3.5'). Moreover, for z E 8u,

where 'I/J

= [9 21 H + 9 22

923r 1. Hence IIW(z)1I

< 1 for z E

8u. By the maximum modulus

principle, IIW(z)1I < 1 for z E u. Conversely, suppose that there exists a function W E Rmxn which is analytic on u, satisfies the interpolation conditions (3.1') - (3.5') and has IIWlloo < 1. Then by Theorem 4.1 we know that there are Ql E R(m-K)x(n-L)(u) and Q2 E R(n-L)x(n-L)(u) such that

[9 21 Ql (4.20)

+ 9 22 Q2

9 231( R(n-L)xl(u) E9 RLX1)

= [9 21 9 22 9 231(R(m-K+n-L)Xl(u) E9 RLX1),

from which we recover W as

(4.21) An alternative way to write (4.21) is

= [9 21 Ql + 9 22 Q2

9 231. Since IIWlloo < 1 and on 8u by assumption, we have, for z E u, where 'I/J

e satisfies condition (b) in Theorem 4.2

129

Ball and Rakowski

( 4.22) Thus Q2(Z) cannot have any null space for z E

0',

and hence, as it is square, Q2(Z)-1 exists for

z E au. Also (4.22) then implies that H(z) = Ql(Z)Q2(Z)-1 iswell defined and satisfies IIH(z)1I < 1 for z E

au. Next note from (4.20) the inclusion

[0 22 0 23 ]( 'R("-L)Xl(u)ED'R LX1 )

c [0 21 Ql + 0 22 Q2 023] ( 'R(,,-L)Xl(u) ED 'RLXl).

(4.23)

We also know from the proof of Lemma 4.5 that [0 22 0 23 ] is invertible on

au.

Then (4.23) can be

written in the form

'R(n-L)Xl(U) ED'RLX1 (4.24)

C ([0 22 023r1021[Ql 0]+

[~2 ~])('R(n-L)Xl(u)ED'RLXl).

A

= [1,,-L

0][022 023r1021Ql + Q2 has size (n -

has size L X (n - L). In this notation (4.24) assumes the form 'R(n-L)Xl(U) ED 'R LX1 C This in turn requires

(4.25) Note that

JJ

[~2 ~] has the form [~ where L) X (n - L) and B = [01£][022 023r1021Ql

Note that the nxn matrix function [022 023r 1021[Ql 0]+

[~ ~]

('R(n-L)Xl(u) ED 'RLXl).

130

Ball and Rakowski

A = ([In-L 0][0 22 023r 10 21 H proof of Lemma 4.5) and where IIH(z)1I

+ I)Q2

where 11[0 22 0 23r 1 0 21 11 < Ion au (from the

< 1 and Q2(Z) is invertible on au from observations made

above. Hence A(z) is invertible on az and wno det A(z) is well defined. The inclusion (4.25) gives us that

On the other hand,

(4.27) As was observed above, II[In-L 0][0 22 023r1021HII

argument wno det ([In-L 0][0 22 023r1021H

+ I)

<

1 on au. Then by a standard homotopy

= 0, so

wno det A = wno det Q2.

(4.28)

Since Q2 is analytic on 0', by the argument principle we deduce

wno det Q2

(4.29)

~

o.

Combining (4.26), (4.28) and (4.29) leads to

(4.30)

wno det A

= 0 = wno det Q2.

Condition (4.30) has several consequences. First of all, det Q2 has no zeros on 0' so Q;1 and

H := QIQ;1 are analytic on u. Secondly, we deduce that equality must hold in (4.24) and hence in (4.23). This has a consequence that

[0 22 9 23 ] (1l(n-L)Xl(U)

or equivalently

EIl1lLX1)

= [021

0 22 0 23] ( 1l(m-K+n-L) xl (0') Ell 1lLX1) ,

131

Ball and Rakowski

This condition in turn is equivalent to

By Lemma 4.5 this last condition is equivalent to 0(z)* J0(z) :::; j ffi 0 for z E u. Thus we have verified that condition (ii) in Theorem 4.2 is necessary for interpolants W with

IIWlloo < 1 to exist.

It remains only to verify that W has the form (4.3) for an H E n(m-K)x(n-L)(u) with

IIWlloo < 1.

We have already verified that H := Q1Q;1 is analytic on

<

the maximum modulus theorem it follows that supIIH(z)1I

0'

with norm < 1 on Bu. By

1. Moreover, from (4.21) we deduce

zED"

that

= ([0 n H + 0 12 0 13] [~2 ~] )([0 21 H + 0 22 0 13] [~2 =[0 n H + 012 0 13 ][0 21 H + 0 22 023r1

W

as needed. 5. Interpolation and internally stable feedback systems.

In this section we establish the connections between the interpolation theory presented in the previous sections and the problem of designing a compensator to stabilize a given plant in a standard feedback configuration which has been studied in the control literature (see [Fr], [DGKFl). We emphasize that the connection between internal stability and interpolation has been a recurring theme in the systems theory literature (see e. g. [YBL], [V], [Kil). The matrix version of the result is usually derived via a coprime factorization of the plant and the Youla parametrization of stabilizing compensators (see [YJB]). Our contribution here is to relate the interpolation conditions directly to the original plant P; the connection is given in terms of the extended complete null-pole data set of a related block row rational matrix function

P.

We also solve the inverse problem of

describing which plants P go with a prescribed set of interpolation conditions

5.1 Preliminaries on feedback systems. Suppose we are given a rational block matrix function P

= [~~~ ~~:] , where Pn, P12,

P 21 , P22 have respective sizes n z x n w , n z x n u , nil

X n w , ny

complex plane 0: 00 • (For discrete time systems

is usually taken to be closed unit disk while for

0'

x n", and

0'

is a subset of the extended

continuous time systems 0' is usually taken to be the closed right half plane including infinity.) The problem is to design a rational n,. x nil matrix function K (the compensator) so that the closed loop system depicted in Figure 5.1 is internally au.bk, a notion which we shall make precise in a moment. (Here we assume that all input-output maps are causal, linear, time invariant and finite dimensional, and that the Laplace transform has already been implemented so all input-output maps are represented as multiplication by rational matrix functions.)

Ball and Rakowski

132

w

~

u

z

f

,~

--"'-

Y

K

'" .....

'

Figure 5.1 In Figure 5.1,

w,z,'U,y

are functions with values in CC,,·xt,ccn.xt,ccn.xt and CC n• xt , re-

spectively, which are analytic in some right half plane (for the continuous time case) or in some disk centered at the origin (for the discrete time case); in our discussion here we also assume that all the functions are rational, although this assumption is not necessary. The configuration depicted

in Figure 5.1 is equivalent to the system of algebraic equations

PllW

+ P t 2'U =

P2tW

+ P 22'U = Y Ky =

(5.1) In the control theory context, the function

aipal, z is the mm: ~ and y is the

K which computes the control signal system E(P,K):

W

~

W

z

'U.

is the disturbance or

measurement~.

'U

reference~, 'U

is the J:mltml

The idea is to design a compensator

based on the measurement y so as to make the overall

z perform better. The standard BOO-control problem is to design K which

minimizes the largest error z (in the sense of L2- norm) over all disturbances w of L2- norm at most 1, subject to the additional constraint that K stabilizes the system: (5.2)

K

min 6tr>bilizing

max IIz1l2. IIwll.$1

Here the L2- norm is over the imaginary line for continuous time systems and over the unit circle for discrete time systems. In this section we put aside the norm constraint (5.2) and analyze only the connection

between the stability of the system in Fig. 5.1 and interpolation theory. To define precisely internal stability and the related notion of well-posedness for the system in Figure 5.1, following [Fr] we introduce auxiliary signals tit E 1l".xt and 112 E 1l".xt as in Figure 5.2.

Ball and Rakowski

133

w u

,.....

v,

z

..... ~

fP

...

Y,

K

.;

.....

..... ,.

_

' I....

./

I'

u,

Y

....

vi

Figure 5.2 The diagram in Figure 5.2 is equivalent to the system of equations

+ 'P12U = 'P21W + 'P22U = 'Pn w

(5.3)

Z

Y - V2

KY=U-Vl.

Well-posed ness means that the system (5.3) can be solved for (z, u, y) in terms of (w, Vb V2) and that the resulting map

[~]

-+ [ ; ]

is given by multiplication by a proper rational matrix function 1t

(for the continuous time case) or by a rational matrix function 1t analytic at zero (for the discrete time case). Internal stabi1ity means that in addition this function 1t is analytic on all of u. Thus internal stability amounts to the assertion that the output signal z and the internal signals

U

and

yare uniquely determined and stable for any choice of stable disturbance signals w, Vb v2; for a more complete discussion we refer to [Fr] and [V]. By elementary algebra one can show that His given by the explicit formula

(5.4) where

~

= 1- 'P22K.

In particular the closed-loop transfer function Tzw from disturbance (and/or

reference) signal w to error z is given by

Tzw = 'Pn + 'P12K(I - 'P22K)-1'P21 (5.5)

=: Fp[K].

134

Ball and Rakowski

A standard assumption in the literature is that P 1 2 is injective (as a multiplication operator from 'Rn• x1 into 'Rn.x1) and that P21 is surjective (as a multiplication operator from 'Rnw x1 into 'Rn• x1 )j in particular, n" ~ n z and nil ~ nw. An equivalent assumption is that the linear fractional map K

--+

Fp[K] defined by (5.5) is injective.

IT P21 in fact is square and invertible, one can solve the system of equations

Puw + P12U = Z P21 w + P 22U = Y for (z, w) in terms of (u, y)j the result is

=z 1'21u + 1'22Y = w

1'uu + 1'12Y

where

l' = [P12 - Pu P;,?P22 PuP;,? ] -P;,?P22

In the language of circuit theory the transform P

--+

P;,?

l' amounts to the transform from the scattering

to the chain formalism (see [B]), and was the basis for the analysis in [BRS]. To remove the assumption that P21 be invertible, we use here the generalized transform to the chain formalism introduced in [BHV]. Since P 21 by assumption is surjective, we may add extra rows P~1 to P21 so that the augmented matrix function

[~~~]

is square and invertible. Define

P~2 of a compatible size arbitrarily

and set

£= [

~ :'~2l

:'1.1 ... ..

P21

: P22

P~1 : P~2 Here yO is to be considered as a physically meaningless fictitious signal created for mathematical convenience. Since

[~~~]

is invertible, we may solve the system of equations

Puw + P12U = Z P21 W + P22 U = Y P~lW + P~2U

= yO.

135

Ball and Rakowski

for (z,w) in terms of (u,y,yO). The result is

(5.6)

where

P=

[rn 'P21

r

P12

13 ]

P22

'P23

is given by

(5.6a) (5.6b) (5.6c) and

(5.6d) The system of equations (5.1) associated with the feedback configuration in Figure 5.1 can be expressed in terms of P as

(5.7)

0 [ -J

o

0 0 J

o ] [ VI W] + o

-K

-V2

[-J 0 0

In particular the closed-loop transfer function T zw from w to z works out to be

(5.8)

Tzw = [Pn K + P12, P13] [P21K + P22,P23]-1

We also remark that the assumptions on 'P imply that

P is

[P22 P23]

is square and invertible and that

injective as a multiplication operator from n.(n.+n w )xl into n.(n.+nv)xl, and that one can

always backsolve for 'P from

P. For more complete details we refer to [BHV].

Below, in this section we shall say that a square rational matrix function W is hjprOller if W is analytic and invertible at infinity and we are in the continuous time case, or if W is analytic

and invertible at zero and we are in the discrete time case. In addition we say that the square rational matrix function R is u-outer if both R and R-l are analytic on u (where u is the closed right half plane in the continuous time case and the closed unit disk in the discrete time case).

136

Ball and Rakowski

5.2 Stability of feedback systems and interpolation In this section we delineate the connections between internal stability of a feedback system

in Figure 5.1 and the theory of null-pole structure and interpolation developed in the previous sections. We assume that we are given a rational matrix function P

5.1 with P2I surjective and P I2 injective, and then define P

= [~~~ ~~~]

= [~ll ~12 P2I

P22

as in (5.6). We consider P as a block row matrix function (With PI

[~~:])

as in Section

: ~13] : P23

= [~~~ ~~~]

and P2

=

and apply the notions of extended null-pole structure for block row matrix functions

explained in Section 2. Define integers m, n, K, L by m = n.. ,n = nw,K = n .. - n .. ,L = nw - ny,

so that P has size (m + n)

X

(m - K

+ n) with Pll

of size m

X

(m - K), P12 of size m

X

(n - L),

etc. Let u be the closed right half plane (continuous time case) or the closed unit disk (discrete time case), and let (w,P,Q) = (C",A",A"B"r,p,Q) be an extended complete null-pole data set over u for the block row matrix function

P in

the sense explained in Section 2. Then in fact

(w, P, Q) has the finer block structure (w, P, Q) = ([

g~] ,A,., A"

[B+ B-1, r, [P+ P-1,

[~~])

where C+, C_, A", A" B+, B_, r are matrices of respective sizes mxn", nxn,., n". xn"., n,xn" n,x

m,n, xn, and n, xn,., and where P+, P_,Q+, Q_ are matrix polynOlnials of respective sizes K X m, K X n, m X L, n X L. The following is the main result of this section. THEOREM

5.1. Let P

= [~~~ ~~~]

with PI2 injective and P2I surjective be a rational matrix

function describing the plant in Figure 5.1, and define the rational block row matrix function P

= [~ll ~12

P2I equivalent:

P 22

: ~13] :

of size (m

+ n)

P23

(i) Pis stabilizable

X(m -

K

+ n)

as in (5.6). Then the following are

p(1l(m-K+n-L)xI(u) E91lLXI ) ofP satisfies: 0) 1lmXI(u) 0

(ii) The extended null-pole subspace S :=

(a) S n (1lmXI E9

C

E9

and

(b) PIlE91la .. (sn1l(m+n)XI(u») =OE91lnXl(u) wherePOED1la .. ([:]) = (iii) If (w, P, Q)

= ([ g~] ,A,., A"

[B+ B-1, r, [P+ P-1,

[g~])

is

&D. extended

of null-pole data over u for the bloclc row matrix functioll P, thell

[~].

complete set

Ball and Rakowski

137

w = (C+,C_,A",A"B+,B_,r,p+,p_,Q+,Q_) is a O'-admissible interpolation data set (see (3.12)). Moreover, ifP is stabilizable, the following holds true:

(1) A rational matrix function K stabilizes P if and only if K has a O'-coprime factorization K=NKDi/ with NK E 'R(m-K)x(n-L)(O') and with DK E 'R(n-L)x(n-L)(O') O'-biproper such that

(2) An m

X

n rational matrix function W is the closed loop transfer function T zw associated

with a stabilizing compensator K for P if and only if W is analytic on

0'

and satisfies the

interpolation conditions (3.1') - (3.5') associated with the O'-admissible interpolation data set w, and [0 In-L 01P-L

[lj] 111