Time-Variant Systems and Interpolation

Editor: I. Gohberg Tel Aviv University Ramat Aviv, Israel
Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel
Editorial Board:
A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) L. de Branges (West Lafayette) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. G. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P. A. Fillmore (Halifax) C. Foias (Bloomington) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) J. A. Helton (La Jolla)
Honorary and Advisory Editorial Board:
P. R. Halmos (Santa Clara) T. Kato (Berkeley) P. D. Lax (New York)
Birkhauser Verlag Basel· Boston· Berlin
M. A. Kaashoek (Amsterdam) T. Kailath (Stanford) H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) E. Meister (Darmstadt) B. Mityagin (Columbus) J. D. Pincus (Stony Brook) M. Rosenblum (Charlottesville) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville)
M. S. Livsic (Beer Sheva) R. Phillips (Stanford) B. Sz.-Nagy (Szeged)
Time-Variant Systems and Interpolation
Editors' address:
1. Gohberg Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel Aviv University 69978 Tel Aviv, Israel
Deutsche Bibliothek CataJoging-in-Publication Data
Time-variant systems and interpolation / ed. by 1. Gohberg. - Basel ; Boston ; Berlin : Birkhuser, 1992
(Operator Thcory ; VoI. 56) ISBN 978-3-0348-9701-3 ISBN 978-3-0348-8615-4 (eBook) DOI 10.1007/978-3-0348-8615-4
NE: Gochberg, Izrail' [Hrsg.]; GT
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© 1992 Springer Basel AG Originally published by Birkhuser Verlag Basel in 1992 Softcover reprint of the hardcover 1 st edition 1992
ISBN 978-3-0348-9701-3
J.A.Ball, 1. Gohberg and M.A.Kaashoek Nevanlinna-Pick interpolation for time-varying input-output maps: The discrete case 1
O. Introduction . . . . . . 1 1. Preliminaries . . . . . . 4 2. J-Unitary operators on £2 17 3. Time-varying Nevanlinna-Pick interpolation 27 4. Solution of the time-varying tangential Nevanlinna-Pick interpolation problem 34 5. An illustrative example 41
References . . . . . . . . . 50
J.A.Ball, 1. Gohberg, M.A.Kaashoek Nevanlinna-Pick interpolation for time-varying input-output maps: The continuous time case . . 52
O. Introduction . . . . . . . 52 1. Generalized point evaluation 55 2. Bounded input-output maps 62 3. Residue calculus and diagonal expansion 65 4. J-unitary and J-inner operators 68 5. Time-varying Nevanlinna-Pick interpolation 76 6. An example 85
References . . . . 88
A.Ben-Artzi, 1. Gohberg Dichotomy of systems and invertibility of linear ordinary differential operators . . . 90
1. Introduction . . . . . . . . . . . . . . . . . 90 2. Preliminaries . . . . . . . . . . . . . . . . . 94 3. Invertibility of differential operators on the real line 95 4. Relations between operators on the full line and half line 102 5. Fredholm properties of differential operators on a half line 106 6. Fredholm properties of differential operators on a full line 110 7. Exponentially dichotomous operators 113 8. References . . . . . . . . . . . . 118
A.Ben-Artzi and 1. Gohberg Inertia theorems for block weighted shifts and applications 120
1. Introduction . . . . . . . . . . . . . . . . . 120 2. One sided block weighted shifts . . . . . . . . . 121 3. Dichotomies for left systems and two sided systems 131 4. Two sided block weighted shifts. . . . . . . . . 139
VI
P.Dewilde, H.Dym Interpolation for upper triangular operators
1. Introduction . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . 3. Colligations & characteristic functions 4. Towards interpolation . . . . . . . 5. Explicit formulas for e ..... . 6. Admissibility and more on general interpolation. 7. Nevanlinna-Pick Interpolation 8. Caratheodory-Fejer interpolation 9. Mixed interpolation problems. .
10. Examples ......... . 11. Block Toeplitz & some implications 12. Varying coordinate spaces 13. References . . . . . . . . .
1. Gohberg, M.A.K aashoek, L.Lerer Minimality and realization of discrete time-varying systems
Introduction . . . . . . . . 1. Preliminaries . . . . . . . . . . 2. Observability and reachability 3. Minimality for time-varying systems 4. Proofs of the minimality theorems 5. Realizations of infinite lower triangular matrices 6. The class of systems with constant state space dimension 7. Minimality and realization for periodical systems
References . . . . . . . . . . . . . . . . . . . . .
147 152
153 154 164 168 177 193 203 210 215 224 226 245 251 259
261 261 264 268 271 274 278 285 292 295
VII
EDITORIAL INTRODUCTION
This volume consists of six papers dealing with the theory of linear time
varying systems and time-varying analogues of interpolation problems. All papers are
dedicated to generalizations to the time-variant setting of results and theorems from oper
ator theory, complex analysis and system theory, well-known for the time-invariant case.
Often this is connected with a complicated transition from functions to infinite dimensional
operators, from shifts to weighted shifts and from Toeplitz to non-Toeplitz operators (in
the discrete or continuous form). The present volume contains a cross-section of recent
progress in this area.
The first paper, "Nevanlinna-Pick interpolation for time-varying input
output maps: The discrete case" of J .A. Ball, I. Gohberg and M.A. Kaashoek, general
izes for time-varying input-output maps the results for the Nevanlinna-Pick interpolation
problem for strictly contractive rational matrix functions. This paper is based on a sys
tem theoretic point of view. The time-variant version of the homogeneous interpolation
problem developed in the same paper, plays an important role.
The second paper, also of J.A. Ball, I. Gohberg and M.A. Kaashoek, is enti
tled "Nevanlinna-Pick interpolation for time-varying input-output maps: The continuous
time case". The previous paper contains a time-varying analogue of the Nevanlinna-Pick
interpolation for the disk. This paper contains the time-varying analogue for the half
plane, and hence the latter results may be viewed as appropriate continuous analogues
of the results of the first paper. Here, as well as in the previous paper, all solutions are
described via a linear fractional formula.
In the third paper, "Dichotomy of systems and invertibility of linear ordinary
differential operators" of A. Ben-Artzi and I. Gohberg, are considered linear ordinary
differential operators of first order with bounded matrix coefficients on the half line and on
VIII
the full line. Conditions are found when these operators are invertible or Fredholm on the
half line. The main theorems are stated in terms of dichotomy. In the case of invertibility,
the main operator is a direct sum of two generators of semigroups, one is supported on the
negative half line and the other on the positive half line.
The fourth paper, "Inertia theorems for block weighted shifts and applica
tions" of A. Ben-Artzi and 1. Gohberg, contains time-variant versions of the well-known
inertia theorem from linear algebra. These theorems are connected with linear time depen
dent dynamical systems and are stated in terms of dichotomy and Fredholm characteristics
of weighted block shifts.
The fifth paper, "Interpolation for upper triangular operators" of P. deWilde
and H. Dym, treats for the time-varying case the tangential problems of Nevanlinna-Pick
and Caraththeodory-Fejer, as well as more complicated ones for operator-valued functions.
Here both the cantractive and the strictly contractive cases are considered. The description
of all solutions in a linear fractional form is given. The general case of varying coordinate
spaces is analysed. The main method is based on an appropriate generalization of the
theory of reproducing kernel spaces.
The sixth paper, "Minimality and realization of discrete time-varying sys
tems" of I. Gohberg, M.A. Kaashoek and L. Lerer, analyses time-varying finite dimensional
linear systems with time-varying state space. A theory which is an analogue of the classi
cal minimality and realization theory for time independent systems, is developed. Special
attention is paid to periodical systems.
1. Gohberg
Operator Theory: Advances and Applications, Vol. 56 © 1992 Birkhiiuser Verlag Basel
NEVANLINNA-PICK INTERPOLATION FOR TIME-VARYING
J.A. Ball*, 1. Gohberg and M.A. Kaashoek
1
This paper presents the conditions of solvability and describes all solutions of the matrix version of the Nevanlinna-Pick interpolation problem for time-varying input output maps. The system theoretical point of view is employed systematically. The tech nique of solution generalizes the method for finding rational solutions of the time-invariant version of the problem which is based on reduction to a homogeneous interpolation prob lem.
O. INTRODUCTION
The simplest interpolation problem of Nevanlinna-Pick type reads as follows.
Given N different points Zl, ..• ,ZN in the open unit disc D of the complex plane and
arbitrary complex numbers Wl,"" WN, determine a function f, analytic in D, such that
(i) f(zj)=wj, j =l, ... ,N,
(ii) sUPI%I<l If(z)1 < 1.
This problem can be restated as a problem involving double infinite Toeplitz matrices
acting on .e2 (Z), namely find a lower triangular Toeplitz matrix T,
(0.1)
* The first author thanks the Netherlands organization for scientific research (NWO) for supporting his research.
2
(ii)' IITII < l.
Ball, Gohberg and Kaashoek
Here Zj = ZjS*, where S is the forward shift on £2(Z), and Wj = wjI, j = i, ... , N. In
this operator form the problem has a natural non-Toeplitz version, in which the operators
T, Zj, and Wj appearing in (i)' and (ii)' are replaced by
1-1,-1 0 0
!I.-I !I,o !I. 1
(0.3) A d' ( (j) (1) (j) ) ~j = lag ""Z_I'ZO ,ZI , ..• ,
d W · h d' al . W d' ( (1) (1) (j)) N an j IS t e lagon matnx j = lag ""W_l'WO ,WI , ••• , j = 1, ... , . More
precisely, the non-Toeplitz version of the prohlem reads as follows. One has given diagonal
matrices 6.j, j = 1, ... , N, as in (0.3) such that
(0.4) 1· ( I (j) (j) (j) I) t 1 . - 1 N 1m sup zi zi+l'" zi+k-l <, J - , ... , , k ..... oo iEI
and
(0.5) (1 + ~ z(i) ... z(i) z-(j) ... z-(j)) N > ~IN k E Z ~ k k+lI-l k+lI-l k - ~, , 11=1 iJ=1
where € is a positive number independent of k and IN is the N x N identity matrix. In the
classical case condition (0.4) means that the interpolation points are inside the unit disc
and the positive definiteness in (0.5) replaces the condition that the points are different.
Now the problem is to find a lower triangular double infinite matrix T as in (0.2) such that
T acts as a bounded linear operator on £2(Z), IITII < 1, and for j = 1, ... , N the following
interpolation requirements are fulfilled:
(0.6) 00
.f +" (j) (j) f - (j) k Z Jk,k L..-, zk ... zk+,,_1 k+",k - w k ' E . u=1
3
It turns out that this generalized Nevanlinna-Pick problem is solvable whenever for some
e > 0 we have
(0.7)
( 1 - w(i)w(j) + ~ z(i) ••• z<i) (1 - w(i) w(j) )z(j) ... z(j)) N > eIN k E Z k k L..-, k k+,,-1 k+" k+" k+,,-1 k -, •
,,=1 i,j=1
The left hand side of (0.6) is the generalized point evaluation which appears
(for the scalar case) in [1] in the role of W-transform and has been developed further in
[2] for triangular operators with matrix or operator coefficients, in connection with time
variant lossless inverse scattering problems. The non-Toeplitz Nevanlinna-Pick problem
mentioned above with the entries iij being matrices and no tangential restrictions has been
treated in [8]. The solution of the most general version of the problem, i.e., with operator
entries ii; and tangential interpolation conditions appears in [9] (in the present volume).
In these papers the main tools come from an appropriate generalization of the reproducing
kernel space method. The starting point for the present research was H. Dym's lecture at
Oberwolfach in October 1989 in which he discussed Nevanlinna-Pick interpolation in the
general context of upper triangular operators.
In the Toeplitz-version of the problem there is a special interest in solutions
i that are rational. The latter means that the operator T in (0.1) is the input-output map
of a causal time-invariant discrete system. In that case the relation
(0.8)
is given in the following way:
{ Xn+1 = AXn + BUn, n = 0, ±1, ±2, ... , Yn = CXn + DUn,
4 Ball, Gohberg and Kaashoek
where A, B, C and D are matrices of appropriate sizes. In the non-Toeplitz version the
requirement that the solution is rational is replaced by the condition that the operator T
is the input-output map of a time-variant system, i.e., the action of Tin (0.8) is given by
{ Xn+1 = Anxn + Bnun, n = 0, ±1, ±2, ... , Yn = Cnxn + Dnun,
where now the input, output and state matrices may vary in time.
In the description of the solutions of the non-Toeplitz version of the tangential
Nevanlinna-Pick interpolation problem we are mainly interested in input-output maps of
time-variant finite dimensional systems. This restriction allowed us to use for the technique
of solution a modification of the method for finding rational solutions for the time-invariant
version of the problem as developed in [6] (see also the book [7]). In this way we derive
an algorithm to compute the solutions explicitly. Also the conditions of solvability are
obtained. The results are illustrated by computing in detail the solutions for one example,
namely when N = 1 and ~1 = ( ... , 0, 0, w, 0, 0, ... ). Throughout the paper our aim is to
keep the exposition and techniques elementary at the level of linear algebra and standard
matrix computations. This means that some proofs would be shorter if one uses known
results concerning the geometry of Krein spaces (see [3], [10]).
The main theorem of this paper with an application to sensitivity minimiza
tion of time-variant systems was presented at MTNS-91 (see [4]).
The classical Nevanlinne-Pick interpolation problem has a half plane version.
In the system theoretical approach to the latter problem the role of discrete time systems
is taken over by continuous time systems (see [7]). This led us to a time-variant version
of the half plane interpolation problem which we treat in another paper [5] (also in this
volume).
A few words about notation. An identity operator is denoted by Ij from the
context it should be clear on which space it acts. For an invertible matrix or Hilbert space
operator A we let A -* denote the adjoint of A -1.
Acknowledgement: The authors are grateful to J. Kos for his careful reading
of the manuscript and his useful remarks.
Ball, Gohberg and Kaashoek 5
1. PRELIMINARIES
sider a discrete time-varying linear system
(1.1) r: {Xk+1 = AkXk + Bkuk, Yk = GkXk + DkUk.
The input sequence (Uk) is assumed to take values in the input space U = Cr , the state
vector sequence (Xk) takes values in the state space Xk = Cnk whose dimension we allow to
depend on the time k and the output sequece (Yk) takes values in the output space Y = em. Thus A k, B k, Gk, Dk are matrices of respective sizes nk+1 x nk, nk+1 X r, m x nk and
m x r. Sometimes it is convenient to view U, X k and Y in a more coordinate-free form as
simply finite dimensional linear spaces; then Ak, Bk, Gk, Dk are linear transformations
acting between finite dimensional spaces and become matrices only after some particular
choice of basis.
We shall assume that the system r: in (1.1) is at rest until some time instant
1\;; thus we take Xk = ° for k :::; I\; and Uk = ° for k < 1\;. Then the equations in (1.1)
give Yk = ° for k < 1\;. By.e+ we denote the space of all (input) sequences (Uk)~oo
with values in e r such that Uk = ° for all k < I\; for some integer I\; depending on the
sequence. Such sequences are said to have finite negative support. To handle the sequence
of state vectors, denote by .e<.;.) the space of all sequences (Xk)~oo such that Xk E en. for k = ... ,-1,0,1, ... and Xk = 0 for all k < K for some integer K (depending on the
sequence). Given a i1 = (Uk)~oo E.e+ with Uk = ° for k < 1\;, if the system (1.1) is initialized
with XI( = 0, the equations (1.1) generate a state vector sequence x = (Xk)~oo E .e~k and an
output sequence if= (Yk)~oo E.e+ such that Yk = ° and Xk = ° for all k E I\; depending on
the sequence (i.e., the bound I\; on the negative support is the same for all three sequences
U, X, iJ).
Let S : .e~n.) -+ .e!(n.) be the (bilateral) forward shift operator on state vector
sequences x defined by (Sxh = Xk-b where x = (Xk)~oo' and let S-1 : .e<';k) -+ .e!-l(n.) be the (bilateral) backward shift. Note S-1 is the inverse of the forward shift from .e!-l(nk) to .e<';.) which we also denote by S. Then the system (1.1) can be expressed as a system
of equations on the sequences U E .e+, x E .e<';.) and if E .e+ in the form:
{ S-1£ =A£ + Bu, (1.2)
Here A, B, C, 'D are block diagonal operators,
given by
Note that action of the operator I - SA : i~·) -+ i~·) may be described by
block matrix multiplication in the following way:
II] Ao I
Here the unspecified entries are zero. Thus as a block matrix I - SA is block lower
triangular with block main diagonal entries equal to I. It follows that I - SA is invertible
in the algebra of linear operators acting on i~·) with inverse given by
00
... E i(n.) x +.
Since Xk = 0 for all k < K for some integer K, the sequence (SA)jX' is equal to the zero
sequence for j sufficiently large. Therefore the infinite series in (1.3) degenerates to a finite
sum and hence is well-defined. Thus one can solve the first equation in (1.2) for X'to get
and plug this value into the second equation to get
This gives us a closed form expression for the input-output map TI: : i1 r-+ Y from i+ into
i+ generated by the linear time-varying system E given by (1.1).
We note that Tr. is causal in the sense that
(1.4) k = 0, ±1, ±2, ... ,
where v = QkW is defined by
Vj = { Wj, if j ~ k, 0, if j > k.
Here v = (Vk )~oo and W = (Wk )~CXl. In more physical language, causality means that the
output up to time k is independent of inputs after time k.
A linear transformation T : £+ ~ £+, which is of the form T = Tr. for some
linear finite dimensional time-varying system I; as in (1.1), will be called a causal, rational
(linear) transformation. The terminology is inspired by the time-invariant case where a
block Toeplitz matrix T = (ai-j )0=-CXl is causal and rational in the above sense if and only
if T is lower triangular (i.e., ak = 0 for k < 0) and r(z) = L:;:'o akzk is a rational matrix
function. The realization theorems in [12], Section 5, give the necessary and sufficient
conditions in order that a linear transformation T : £+ ~ £+ is causal and rational, and
they also provide an algorithm to construct a realization of T (i.e, a time-varying system
I; as in (1.1) such Tr. = T) when these conditions are fulfilled.
1.2 Anticausal time-varying systems. When working in the algebra of
linear operators mapping f+ into £+, the matrix associated with any such operator is
necessarily of the form S-kT, where T is lower triangular and S is the forward shift.
Therefore to generate more examples we consider also input-output maps for anticausal
systems.
By an anticausal discrete time-varying system we mean a system of the form
(1.5) (j { Xk = AkXk+1 + BkUk,
Yk = CkXk+1 + DkU k ,
where the state evolves in backwards time. Here at each point in time Xk is an element of
the state space Xk = en., the vector Uk is an element of the input space U = er and Yk
is an element of the output space Y = em. Furthermore, Ak : X k+1 ~ Xk, Bk : U ~ Xk,
Ck : Xk+1 ~ Y and Dk : U ~ Yare linear transformations.
If the system (j in (1.5) is initialized at some time instant k = '" by Xk+1 = 0
and then inputs Uk, Uk-I, ... are fed in in decreasing time, then the equations (1.5) generate
a well-defined sequence ( ... 0,0, x .. , x .. -1, ••• ) of state vectors and a well-defined sequence
( ... 0,0, Y .. , Y .. -1, ••• ) of output vectors. The associated input-output map T = T.,. is
naturally defined as a map from ~ into i~ where i~ is the set of sequences iii = (Wk)~oo with values in CV such that Wk = ° for all k > K for some K, where K depends on the
sequence iii. Any such input-output map T.,. is anticausal in the sense that
(1.6)
for all integers k where Pk = 1- Qk-l is the projection operator defined on biinfinite
sequences iii = (w j )~oo by Pk (iii) = v with v = (v j )~oo given by
v' = {Wj for j ~ k, J ° for j < k.
The theory developed in Section 1.1 for causal linear time-varying systems
and causal maps has an analogue for anticausallinear time-varying systems and anticausal
maps. In particular (J' is an anticausal linear time-varying system as in (1.5), then the
associated input-ou~put map T.,. : i~ -+ i~ can be expressed as the upper-triangular
matrix
where A, S, C, V are the block diagonal matrices
acting between the spaces
and, as before, S is the forward bilateral shift.
1.3 Input-output mapS' defined on i 2 • In this section we consider a
system E as in (1.1) assuming additionally that
(1.10) sup IIAkll < 00, sup IIBkll < 00, sup liCk II < 00, sup IIDkll < 00. k k k k
Here IIMII denotes the spectral norm of the matrix M, i.e., IIMII is the largest singular
value of M. The first inequality in (1.10) implies that
[Q] Ao J o
defines a bounded linear operator on £~n.), the Hilbert space of all doubly infinite norm
square summable sequences with entry x" at time k in Cn •• We shall also assume that
p(SA), the spectral radius of SA, is strictly less than one or, more explicitly, that
(1.11) limsup(sup IIAj+II-1 ... Ail!):- < 1. JI-+OO j
Then I - SA is invertible as an operator on £~n.) with inverse given by
00
(1.12) (I - SA)-l = ~)SA)II. 11=0
Because of condition (1.11), the series in the right hand side converges in the operator
norm for operators on £~ n.) .
From (1.12) and the boundedness conditions on B", 0" and D" in (1.10)
it follows that the input-output map TE, initially defined only on sequences in £2 having
finite negative support, extends uniquely by continuity to a bounded linear operator from
£2 into £-q' which we also denote by TE. Note that any such TE defined on £2 also has the
causality property (1.4), and hence is given by a lower triangular matrix. The expansion
00
(1.13) TE = V+ :LC(SA)"SB 11=0
gives an expansion of TE in terms of its diagonals (the main diagonal V together with all
diagonals below the main diagonal). Exactly which linear discrete time-varying systems E
have input-output maps TE which acts as bounded operators from £2 into £-q' we leave as
a topic for future work.
To get bounded upper-triangular input-output maps from f2 into fr; we have
to consider the anticausal systems from Section 1.2. Let (J' be the anticausal discrete time
varying system (1.5), and assume that the boundedness conditions in (1.10) are fulfilled.
Condition (1.11) is now replaced by
(1.14) lim sup(sup IIAi Ai+l ... Ai+v-lll)': < 1. V-'oo i
which is equivalent to the requirements that p(AS-l), the spectral radius of AS-l , IS
strictly less than one. Under the conditions (1.10) and (1.14) the associated input-output
map Tu given by (1.7) defines a bounded operator from f2 into fr; which has an upper
triangular matrix representation. We shall say that a linear bounded operator from f2 into fr; is a time-varying rational operator if it can be written as the sum of a bounded
input-output map of a causal linear time-varying system and a bounded input-output map
of an anticausal linear time-varying system.
1.4 A time-varying calculus for lower triangular matrices. Denote by
f 2 (Z) the'space of doubly infinite square summable complex-valued sequences (Xk)r:-oo
(indexed by the set of all integers Z). For m a positive integer, fr;(Z) denotes the set of
block sequences (Xk)r:_oo where each entry Xk is in the space em of complex m tuples
(viewed as column vectors). The space of all bounded linear operators on f2 we denote by
Xj note that each element F in X has a biinfinite matrix representation
such that if = Fx is given by
F-l,-l FO,-l Fl,-l
F_l,o
Fl,l
if x = (x j )~-oo E f 2(Z) and if = (Yj )~-oo E f2 (Z). For m and r positive integers, xm x r
denotes the bounded linear operators from f2 into fr;. These similarly can be identified
with doubly infinite block matrices F = (Fij)i,j=_oo where each block has size m x r. We
call (Fij)r,'j=_oo the (standard) block matrix representation of F or just the matrix of F.
Denote by C the subclass of X consisting of all operators F in X which leave invariant
each of the subspaces
for all k E Z. This class C coincides with the set of F's in X having a lower triangular
matrix representation, i.e., F = (Fij)r,'j=_oo with Fij = 0 if i < j.
Analogously, we define transposed versions of the.se spaces, U, umxr which
consist of upper triangular matrices, for example, umxr consits of all operators F in xmxr
which leave invariant the subspace
i;«-oo,kJ):= {(xi)~oo E i;: Xi = 0 for j > k}
for all k E Z. The intersection cmxr n umxr consists of diagonal matrices; this class we
denote by vmxr.
A key operator for the setup which we now describe is the bilateral forward
shift operator
I 0 I [Q]
I 0 J consisting of the identity matrix (of a size determined by the context) on the diagonal
below the main diagonal. We use the same symbol S to denote the shift operator in c mxm
for any positive integer m. If G = (Gij)r,'j=-oo is in xmxr, then SG E xmxr and for the
(i,j)-th entry in the block matrix representation of SG we have
i,j = 0, ±1, ±2, ...
Thus the block matrix representation of SG is equal to the one of G with each row shifted
one down. Note that
o I
and thus the block matrix representation of S-lG is the one of G with each row shifted
one up.
For G E xm)(r and k an integer we let G[kl be the block diagonal operator
in vm)(r of which the main diagonal entries are given by (G[k])jj = Gj+k,j for each j E Z.
We have (cf., [2], Lemma 2.7)
IIG[k]1I = s~p II Gj+k.i II :5I1GII· J
Note that
It follows that G admits the series expansion:
00
(1.15) G = E SkG[k)l k=-oo
where the convergence is entrywise. The right hand side of (1.15) amounts to breaking G up
along diagonals parallel to the main diagonal. The series expansion in (1.15) corresponds
to the FOUl'ier series expansion of a bounded measurable function on the unit circle T.
Indeed, if 9 is such a function with Fourier series
00
g(z) = E gjzj, ;=-00
then for the Laurent operator G on i 2 (Z) with symbol 9 we have
00
G = (9i-j)i'J=-00 = E Sk(gkJ) k=-oo
where gkJ is the diagonal matrix with constant value gk along the main diagonal.
Now we introduce the generalized point evaluation map for operators in
cmxr studied by Alpay-Dewilde-Dym [2]. Let F E cmxr and ~ E vmxm. We suppose
that ~S-l (whose block matrix representation has all nonzero entries on the diagonal
immediately above the main diagonal) has spectral radius p(~S-l) strictly less than 1 as
an operator on ir(Z), i.e.,
(1.16) p(~S-l) := lim sup II(~S-l)jlllfj < 1. j-+oo
Then S - A is an invertible operator on .er(Z), and its inverse on .e~(Z) is given by
00
(S - A)-l = S-l(1 - AS-l )-l = E S-l(AS-l )-i i=O
with convergence in the operator nonn. We define F(A) to be the block diagonal operator
in 1)mxr with main diagonal equal to the main diagonal of S(S - A)-l F. The map
A f-+ F(A) is the natural generalization of the point evaluation map for analytic functions
on the unit disc. The analogy appears if we consider the diagonal expansion
00
of FE .cmxr. In terms of (1.17) we have
00
(1.18) F(A) = E(AS-l )iSiF(,1 , i=O
where the series concerges in the operator norm. If F[jJ = 1;1 and A = >'1 with 1>'1 < 1
are all constant diagonals, then the shift operators in the right hand side of (1.18) cancel
and (1.18) collapses to
(1.19) F(>.J) = E >.i 1;1 = 1(>.)1. i=O
From another point of view, for any G E xm x r define the total residue n( G)
of G to be the coefficient of S-l in its diagonal expansion (1.15), i.e., put n(G) = G[-lJ.
Then F(A) can alternatively be defined as
(1.20)
because (SG)[k) = G[k-l) for any G E xmxr and k E Z. In the time invariant case where
F = (ji-i)'ti=-oo and A = >.I whith 1>'1 < 1, it is readily checked that
n{(s - A)-l F} = n{(z - >.)-1/(>')}1 = 1(>')1,
where 1(>') = E;'o >.i!; and n in the second tenn denotes the sum of the residues inside
the unit disk in the usual complex variables sense. We shall see more striking parallels
with this more general noncomrnutative time-varying calculus and the standard calculus
for analytic functions as we proceed.
We now present some basic properties of the time varying point evaluation
F 1-+ F( L\) which we shall need in the sequel. These results appear in [2]; we include
proofs for the sake of completeness.
PROPOSITION 1.1. Let F E £mxr and L\ E vmxm be given, wbere
p(L\S-l) < 1. Tben F(L\) is tbe unique element ofvmxr sucb tbat (S-L\)-l(F-F(L\)) E
PROOF. We first settle the uniqueness issue. If D1 and D2 were two diagonal
operators in vmxr such that (S - L\)-l(F - D j ) E £mxr for j = 1,2, then by linearity
(S-L\)-l E E £mxr where E = D 1-D2 E vmxr. However (S-L\)-l = 2:~0 S-l(L\S-l)i
is strictly upper triangular, and hence remains so when multiplied by a diagonal E. This
forces (S - L\)-1 E = 0, and hence E = D1 - D2 = O. This establishes uniqueness.
In general, if E E vmxr is diagonal, then (1.18) shows that E(L\) = E. By
linearity of the map H 1-+ H(L\), we see that whenever F E £mxr, then G =
F -F(L\) E £mxr has G(L\) = o. The lower triangularity is thus settled if we show that
(5 - ..6.)-lG E £mxr whenever G E £mxr and 8(..6.) = o. To do this we must show that the l-th diagonal above the main diagonal of
(S - L\)-lG is zero for l = 1,2, .... Write
00
W:= (S - L\)-lG = L SkW[kj. k=-oo
For (S - L\) -1 and G we have the diagonal expansions
00 00
(S - L\)-1 = L S-(Hl)(Si(L\S-l)i), G= LS"G[lj. i=O 11=0
It follows that for k = -1, -2, ...
00
00
because of (1.18). Since G(~) = 0, we see that W[kJ = 0 for k = -1, -2, ... , and hence
(S - ~) -1 G is lower triangular. D
PROPOSITION 1.2. H F E cmxr , G E Crxp and ~ E vmxm with
p(~S-l) < 1, then
PROOF. Since (S - ~)-l(FG - Fc(~» E Cmxp , we see from Proposition
1.1 that it suffices to show that
We calculate
since H = (S - ~)-l(F - F(~» E cmxr by Proposition 1.1 and G E Crxp by assumption.
D
It is convenient also to consider certain classes of Hilbert-Schmidt matrices.
By x 2mxr we denote the class of all doubly infinite block matrices
F = (Fij )i,'j=-oo =
where each block has size m x r such that
(1.21) L l!Fiill~ < 00.
I;'o,~ I 1,0
where IIFii II~ is the sum of the squares of the moduli of the matrix entries of Fij (i.e., the
square ofthe Hilbert-Schmidt norm of Fij). Since l!Fijll ~ IIFiill2 for all i andj, such an F
defines a bounded linear operator from £2 into £r which we also denote by F. In fact, F is
a Hilbert-Schmidt operator and the quantity in (1.21) is the square of the Hilbert-Schmidt
norm of F. In particular
x 2mxr = {F E xmxr I F is Hilbert - Schmidt}.
By C';,xr we denote the subclasses of Fin x;,xr which are lower triangular
(Fii = 0 for i < j). Similarly, u;,xr consists of the upper triangular matrices in X2mxr.
Our primary focus will be on block column lower triangular matrices of this type (,C;X1 or
,C~X1). Note that ,C;X1 is a Hilbert space in the natural inner product
IT X E :V1xr, 6 E :v1x1 such that p(6S-1) < 1 and D E :V~X1, then (Xf)I\(6) E :V~X1
(c.f., Lemma 7.4 in [2]) and
f 1-+ tr (D*(Xft(6)) = (Xf)I\(6),D)1)1Xl 2
is a bounded linear functional on the Hilbert space ,C;X1, hence by the Riesz representation
theorem there must be an element k = k(X, 6, D) in ,C;X1 for which
The following proposition identifies this element k(X, 6, D); in the time-invariant case,
the element k(X, 6, D) is associated with the kernel function k(z, w) = (1- ZW)-1 for the
Hardy space H2 . A form of this result also appears in [2J.
PROPOSITION 1.3. Let X E :v1xr, 6 E :v1X1 with p(6S-1) < 1 and
D E :V~X1 be given. Then X*(I - S6*)-1 DE ,C;X1 and satisfies the identity
for all f E ,C;X1.
PROOF. Indeed,
(j, X*(I - S6 *)-1 D) eX1 = tr D*(I - 6S-1 )-1 X f 2
= L Di;[(I - 6S-1 )-1 X fJii = L Dii[(S - 6)-1 X f]i+1,i i i
= L Dii[(X f)1\(6)]ii = tr (D*(X 1)1\(6» i
= (Xft(6),D)1)lXl. D 2
If F E em x r, then the operator L f : I f-+ F I of multiplication by F on the
left is a bounded operator on e;X1. The adjoint L'F,
requires the orthogonal projection PCXl from X;X1 onto e;X1 following left multiplication 2
by F*. It is useful to know that (LF)* can be computed explicitly on the "kernel function"
elements X*(I - SL':l.*)-1 D.
PROPOSITION 1.4. Let X E V 1xm , L':l. E V1X1 with p(L':l.S-1) < 1 and
D E V~X1 be given. Suppose F E e mxr and Y = (XF)I\(L':l.) E v1xr. Denote by LF the
operator on e;X1 given by multiplication on the left by F. Then
(LF)* X*(I - SL':l.*)-1 D = Y*(I - SL':l. *)-1 D
PROOF. Take I in e;X1. We compute
(J, (LF)* X*(I - SL':l. *)-1 D) crX1 = (FI, X*(I - SL':l. *)-1 D) CmX1 2 2
= ((X F J)1\(L':l.), D)V'X', 2
where we used Proposition 1.3. But then we can apply Proposition 1.2 to show that
= (J,Y*(I-SL':l.*)-1D)C rXl , 2
2. J-UNITARY OPERATORS ON £2
2.1 J-unitary and J-contractive operators. Suppose 1t+ and 1-L are
two separable Hilbert spaces and J is the self-adjoint and unitary operator on the or
thogonal direct sum space 1t = 1t+ Ef) 1t_ defined by J = hf.+ - hc. Here h; denotes
the identity operator on the space K.. A bounded linear operator 8 on 1t is said to be
J-isometric if 8* J8 = J, or equivalently, 8 preserves the indefinite inner product induced
by J:
(J8h,8h) = (Jh, h), hE 1t.
We say that 6 is J-unitaryif both 6 and 6* are J-isometric; equivalently, 6 is J-isometric
and 1m 6 = 1i. We say that 6 is J-contractive (or 6 is a J-contraction) if 6* J6 ~ J, or
equivalently
(J6h, 6h) ~ (Jh, h), hE 1i.
If both 6 and 6* are J-contractive, then 6 is called J-bicontractive. The Hilbert space
1i with the Hilbert space inner product
replaced by the indefinite inner product
is known as a Krein space, and J-isometric, J-unitary, J-contractive and J-bicontractive
operators correspond to isometric, unitary, contractive and bicontractive operators in the
Krein space sense. A systematic study of the operator theory and geometry associated with
such operators is given in [3]; a useful summary can be found in [10]. Here we set down
only a few basic properties, well known among specialists but not so well known in general,
which we shall need. In the next subsection, we specialize to the setting where 1i+ = £r and 1i_ = £2. Although these results can be gleaned from the more general results of [3]
and [10], we include simple, direct proofs to keep the exposition self-contained.
We begin with a result concerning J-bicontractions.
THEOREM 2.1. Suppose 6 = (~ll ~12) is a J -contraction. Then 8 is 021 022
a J-bicontraction if and only if 822 is invertible. In this case, 1182"l821 11 < 1.
PROOF. Suppose first that 6 is a J-bicontraction. Then the relations
8*J8 ~ J, 8J6* ~ J
yield
(2.1)
and
(2.2)
(2.3)
(2.4)
From (2.4) we see that 8 22 is onto while (2.3) gives that 8 22 is one-to-one. Hence 8 22 is
invertible on 'H- as asserted. Moreover (2.4) yields
This implies 11822182111 < 1.
Next, suppose that 8 is a J-contraction and 8 22 is invertible. Then we can
solve the system of equations
(2.5) 8 n u + 8 12y = z,
for (z,y) in terms of (w,u). The result is
(2.6)
where
(2.7)
Moreover, the J-contractive property of 8 implies
whenever u, y, z, win 'H+, 'H-, 'H+, 'H_ respectively satisfy (2.5). We conclude that
whenever u, y, Z, W satisfy (2.6). This gives us that U defined by (2.7) is an ordinary
Hilbert space contraction. For the case J = I there is no distiction between contractions
and J-contractions; hence also
(2.8)
whenever
(2.9)
But since U2l = e22* is invertible, we can solve (2.9) for (ZbWl) in tenns of (Ul,Yl);
~ (e* e*) indeed, note that U has the same fonn as U but with e* = er~ e;~ in place of e. The result is
(2.10)
Since (2.10) and (2.9) are equivalent systems of equations, from (2.8) we see that
whenever (Ul' Yl, Zb wd satisfy (2.10), i.e., e* is J-contractive. This verifies all assertions
in Theorem 2.1. o 2.2 J-unitary and J-inner operators on i~+r. In this subsection we
specialize the results of the previous section to the case where 1i+ = i!f.' and 1i- = i 2. The signature operator J then is given by J = It'2 EB -It';. on i2' EB i2 ~ i~+r. We
abuse notation and denote also by J the signature operator J = Im EB - Ir on the finite
dimensional space cm +r ; this should cause no confusion as the meaning will be clear
from the context. By a J-unitary map on i~+r we therefore mean a bounded linear
operator e : e;+r ~ i~+r such that e* Je = J and eJe· = J. If we write e in
block form as 9 = (~11 ~12 ), Theorem 2.1 guarantees that 9 22 is invertible on 12 and 021 022
92ls21 : lr -+ 12 has 11921921 11 < 1 if 9 is J-unitary.
In this section we wish to derive some basic properties of J-unitary maps e which have an additional property with respect to the time structure of l~+r. If 9 is a
J-unitary map on l~+r we say that 9 is J-inner if 9 also satisfies
(2.11)
for all integer k. Recall that Qk is the projection operator defined by
{ Uj ifj:5 k,
(Qki1)j = 0 if j > k.
If 9 is also lower triangular (i.e., in the terminology of subsection 1.4 we have 8 E
.c(m+r»«m+r»), then Qk9 = Qk9Qk and (2.11) can be simplified to
(2.12)
The following gives an equivalent formulation of the J -inner property in terms of the
projections Pk = 1- Qk-l rather than Qk.
PROPOSITION 2.2. Suppose 9 is a lower triangular J -unitary map on
l~+r. Then 9 is J -inner if and only if
k = ... ,-1,0,1, ....
PROOF. Suppose that 9 is lower triangular J-unitary. After reindexing in
(2.12) we see that 9 is J-inner if and only if
(2.13)
for k = ... , -1, 0,1, .... Substitute Qk-l = 1- Pk and use that 9* J9 = J to get that
(2.13) is equivalent to
(2.14). 9* JPk9 ~ JPk
Multiply on the left by eJ and on the right by Je- and we use that eJe- = J to get
JPk ;::: eJPke- as required. D
The following is a useful characterization of the J -inner property for lower
triangular J-unitary maps.
eel2 ) is a lower triangular J -unitary - 22
map on .e~+r; in particular, e 22 is invertible on .e2. Then e is J -inner if and only if e;.l is lower triangular.
PROOF. Note that e 22 is invertible by Theorem 2.1. Suppose that e is
lower triangular and J-unitary. We consider [elk = el.e~+r([k, 00)) as a mapping on
.e~+r([k, 00)). Since e is lower triangular and J-unitary, [elk is (Jllm Pk)-isometric
([elkt JPk[elk = e- Jellm Pk = Jllm Pk.
In particular, [elk is (Jllm Pk)-contractive. The content of Proposition 2.2 is that a lower
triangular J-unitary e is J-inner if and only if ([eDk)- is (Jllm Pk)-contractive, i.e., if and
only if [elk is a (Jllm Pk)-bicontraction for every k = ... , -1,0,1, .... On the other hand,
by Theorem 2.1 such a e has the property that [elk is a (Jllm Pk)-bicontraction if and
only if [622]k = 6221.e2([k, 00)) is invertible for every k. This last condition is equivalent
to e-;:21 mapping .e2([k, 00)) into itself, i.e., to e-;:l being lower triangular. D
2.3 Realization of J-unitary and J-inner maps. In this section we
define a class of systems which yield lower-triangular J-unitary or J-inner operators on
.e~+r as its input-output maps. By a (causal) stable J -unitary time-varying system we
mean a linear time-varying system
(2.15) = AkXk + BkUk = GkXk +DkUk
with state space X k = X = eno independent of k which satisfies the sufficient conditions
(1.10) and (1.11) to generate a bounded lower triangular input-output map on .e~+r and
for which there exist invertible Hermitian linear transformations H k on X k = X = eno
such that
~) ,k E Z.
We say that the stable J-unitary system E is J-inner if in addition the Hermitian matrix
H" is positive definite for all k. The following result will be a basic tool in our solution of
the time-varying version of the tangential Nevanlinna-Pick interpolation problem.
THEOREM 2.4. The input-output map a = TI: of a stable J -unitary system
E is a J -unitary map on i~+r . .FUrthermore, E is J -inner as a system if and only if a = TI:
is J -inner as an operator on i;,+r.
PROOF. Suppose E as in (2.15) is a stable J-unitary system, and suppose
a = ( ... ,O,O,U",U,,-l. ... ) is an input sequence in i++r n i~+r. Then the J-unitary
property of E implies the equality
or equivalently
(2.18)
at each point k in time. Here, in general, pw(w) = (Ww,w) is the Hermitian form induced
by the Hermitian matrix W acting on the vector w. Summing from k = It to k = j in
(2.18) and using x" = 0 gives
j j
(2.19) PHi+t(XjH) = L pAUle) - L PJ(YIe). "=-00 "=-00
Since the system E is stable by assumption, the sequence ii = (x")~oo = (1 - SA)-lSBa
is in i~Oj hence, in particular, lim" ..... oox" = O. By assumption (2.16) it follows that
lim" ..... ooPH.(x,,) = 0 as well. From (2.19) we conclude that
00 00 (2.20) L PJ(Y,,) = L PJ(u,,)
"=-00 "=-00
whenever ii = aa and a E i++r n i~+r. By an approximation argument it follows that
(2.20) continues to hold for all U E i;,+r. We conclude that a is J-isometric.
To show that e is J -unitary, we must show that e* is also a J -isometry. If
e is the input-output map of the causal stable system (2.15), it is straightforward to see
that e* is the input-output map of the anticausal antistable system
(2.21)
( Ak is a right inverse for BA;
also a left inverse. Hence,
(2.22)
= AkXk+1 + CA;Uk, = BA;Xk+1 + DA;uk.
~) ~i ). Since all these matrices are square, a right inverse is
0) (Ak J BA; ~)
for all k. Since E satisfies (1.11), E* satisfies (1.14). Now one can proceed to show that
e" is J-isometric in the same way that e was shown to be J-isometric above, with the
minor modification that the direction of time should be reversed.
Now suppose in addition that E is J-inner as a system, i.e., Hk > ° for all
k. Then (2.19) gives
(2.23) :E PJ(Yk):5 :E PJ(Uk). k=-oo k=-oo
for all j whenever y = eit. Rewriting (2.23) in operater form gives
e*QjJe ~ QjJ, j E Z,
and hence e is J-inner.
To establish the reverse implication we first show that a stable J-unitary
system is completely reachable, that is (see [12], Section 2), for any time l + 1 and x E c no
there is a sequence of inputs
(2.24) it = ( ... ,0, 0, U",U,,+l, ... , Ut-l,Ut)
so that with x Ie = 0 the resulting state Xt+1 at time f + 1 is equal to x. To prove this, note
that (2.22) implies that
It follows that for k < f i-I
Hi/I - At ... AkH;;I AZ ... Ai = Bt J B; + L At ... AII+1BJ B:A:+I ... Ai· ,,=k
Since (1.11) holds, there exist 0 < f3 < 1 and an integer fo < f such that
Recall that the sequences (IIBkll)~oo and (IIH;;III)~oo are bounded. So we may conclude
that
t-I
(2.25) Hi/I = BtJB; + L At··· A"+1 B"JB:A:+1 ... Ai, 11=-00
where the convergence is in the operator norm for operators on Xl+ I = eno . The left
hand side of (2.25) is invertible, and hence we may use the fact that the set of invertible
operators is open to conclude that for some integer II: < f the operator
i-I
BiJB; + LAt·"A ... +1B ... JB:A:+I ,,·Ai 11="
is also invertible. So, given x E eno , there exist vectors z", ... ,Z,- in eno such that
i-I
X = BtJB;zt + L At", A ... +1B ... JB:A:+I ". Aiz .... V=K.
Now, put Uj = JBjAi+I ". AiZj for j = 11:,,,. ,f-l and Ut = JBiz/. Then
t-I
X = Btut + L At· .. A"'+1 B ... u ... , 11=1(
which implies that for this choice of u", ... , Ut the input sequence ( ... ,0,0, u", ... , Ut) has
the desired property.
Now, assume that a is J-inner as an operator on i~+r. Fix an integer i and
let x be an arbitrary vector in Xl+! = enG. By the result of the previous paragraph, we
can find an input sequence il as in (2.24) so that with x,. = 0 the resulting state Xl+! at
time i + 1 is equal to x. We consider il as an element of i~+r by setting u" = 0 for k > i.
Put fi = ail. Since a is J-inner, we have
l l
But from the identity (2.19) this in turn yields
(2.26)
By assumption, Hi+! is invertible. Since x is an arbitrary element of the state space cna ,
formula (2.26) implies that Hi+! is positive definite for alIi. o Note that J-unitary maps on i~+r arising as the input-output map of a stable
J-unitary system are necessarily lower triangular. We can produce upper triangular J
unitary maps on i~+r by considering input-output maps of systems evolving in backwards
time; we have already seen that e" is such a map whenever e is the input-output map of
a causal, stable, J-unitary system. In general, let
(2.27) q { x" = A"X"+l + B"u" y" = C"x,,+! + D"u"
be a linear time-varying system evolving in backwards time. We shall say that q is an
(anticausal) anti-stable J-unitary system if q satisfies the sufficient conditions (1.10) and
(1.14) to induce an input-output map T tr which is bounded on i~+r and, in addition, there
exist a sequence (H,,)~_oo of invertible Hermitian linear transformations (or matrices) on
the state space cno such that
(2.28)
(2.29) ~)
for all k E Z. If, in addition, H k is positive definite for all k, we shall say that 17 IS
anti-J-inner. At the input-output level, we call a J-unitary map e on £!;+r anti-J-inner
if
k E Z,
where Pk = 1- Qk-I. The following result is the analogue of Theorem 2.4 for systems
evolving in backwards time; as the proof is also completely analogous, it is omitted.
THEOREM 2.5. The input-output map e = TiT of an antistable J -unitary
system 17 is a J -unitary map on C':;+r. Furthermore, 17 is anti-J -inner as a system if and
only if e is anti-J -inner as an operator.
Proposition 2.2 shows that there is a simple connection between lower trian
gular J-inner maps and upper triangular anti-J-inner maps (independent of any realiza
tions as input-output maps); we state the result explicitly in the next proposition.
PROPOSITION 2.6. A lower triangular J-unitary map e is J-inner if and
only if e* is an upper triangular anti-J -inner map.
PROOF. Apply Theorem 2.3. 0
3. TIME-VARYING NEVANLINNA-PICK INTERPOLATION
In this section we consider the time-varying (tangential) N evanlinna-Pick
interpolation problem. We are given 2N row diagonal matrices Xj E vIxm and Yj E
VI Xr (j = 1, ... , N) and N scalar diagonal matrices D.j E VI Xl for which the spectral
radius of D.jS-1 is strictly less than one (j = 1, ... ,N). Put
and consider the operator
:;:; = (X AS-IX (AS-I)2X ) . ffioonm _, nN - ~ ~ • •• • IJ70 {.2 ~ {.2 ,
where E9g"£2' stands for the Hilbert space of square summable sequences (xo, Xl, X2"")
with entries in £2'. Since p( D.S- I ) is strictly less than one,:=: is a well-defined bounded
operator. In what follows we also require that the pair (X,~) is exactly controllable in
the sense that
(3.1) 33"' >0
as an operator on tf. Note that 33"' may be written as an N x N operator matrix of
which the entries are bounded linear operators on t2' In fact, in terms of the original data
we have
The time-varying (tangential) Nevanlinna-Pick interpolation (TVNPI) prob
lem is: Find necessary and sufficient conditions for the existence of a lower triangular
matrix F E £mxr such that
(3.2) IIFII < 1
When these conditions are satisfied describe all such F.
The norm in (3.2) is the induced operator norm of F as an operator from
t2 into ir. We shall refer to {Xi, y;, ~i : i = 1, ... ,N} as an admissible TVNPI data set
provided (3.1) is satisfied.
The following theorem settles the existence problem
THEOREM 3.1. Let {Xi, Y;, ~i : i = 1, ... , N} be an admissible TVNPI
data set. Then the associated TVNPI problem has a solution if and only if the Hermitian
matrix
A( {Xi, y;, ~i}) = ({(XiX; - Y;Yj"')(I - S~;)-1 Y(~i))~ . • ,)=1
is positive definite on if. PROOF OF NECES~ITY. Suppose that F E £mxr is a solution of the
TVNPI problem. Then by Proposition 1.4 and the interpolation conditions (3.3) we see
that
for j = 1, ... , N, where LF is the operator from .c;Xl into .c~Xl given by LF(f) = F f and D 1 , .•• , D N belong to V~ XI. It is straightforward to check that the induced operator
norm IILFII of LF as an operator from .c~Xl to .c~Xl is the same as the induced operator
norm IIFII of F as an operator from e2 to er. By condition (3.2) it follows that IIL'FII =
IILFII = IIFII < 1. Hence, if Dl, ... ,DN are any diagonal matrices in V~X\ then
N N N II LX;(I _S~j)-1 Djll~-II(LF)* LX;(I _S~j)-1 Djll~ 262 11 LX;(I _S~j)-1 Djll~
j=1 j=1 j=1
for some 6 > O. Writing the squares of norms in terms of inner products, expanding and
using (3.4) gives
N N
L L (X; (I - S~;)-1 Dj, X;(I - S~n-l Di) CmX1 2
i=1 j=1 N N
- L L(Yj*(I - S~;)-1 Dj , i=1 j=1
N N
;=1 j=1 N N
262 L L ({XiX; (I - S~;)-1 }"(~;)Dj, D i ) C~Xl ;=1 j=1
Since Dl"'" D N are arbitrary diagonal matrices in V~ Xl, and we are assuming that
{(Xj,~j): j = 1, ... ,N} satisfies the exact controllability assumption (3.1), it follows
from (3.5) that A = A( {X;, Yi,~; : i = I}) is positive definite as asserted. 0
The proof of sufficiency will be postponed until the next section, where more
over a linear fractional description for the set of all solutions will be constructed.
Let us remark here that condition (3.1) is automatically fulfilled if the gen
eralized Pick matrix A({X;,Yi,~;: i = 1, ... ,N}) is positive definite. This follows from
the equality
33* - A( {X;, Yi,~; : i = 1, ... , N}) = (f(~iS-l )IIYiY/(S~;)II) N . 11=0 1,)=1
and the fact that the latter operator is positive semi-definite.
In the rest of this section we set down some general principles concerning
the connections between linear fractional maps and solutions of the TVNPI problem.
Following the approach of [7], we first show how the interpolation condi
tions (3.3) can be reduced to a homogeneous interpolation problem for a matrix 9 E
.c(m+r)x(m+r) which can be used to parametrize the set of all solutions. Indeed, if F
satisfies (3.3), then (~) E .c(m+r)x(m+r) satisfies the following set of homogeneous in
terpolation conditions:
(3.6) {(Xj -Yj) (~)}"(~j) = 0, j =1, ... ,N.
Moreover, by the property for the time-varying calculus given by Proposition 1.2, (3.6)
implies that
(3.7) j = 1, ... ,N.
for all H E .c(m+r)xl. If we construct a lower triangular matrix 9 in .c(m+r)x(m+r) such
that
(3.8) e.c( .... +r)xl = {H E .c( .... +r)xl : {( Xj -Yj ) H}A(.6.j) = 0 for j = 1,2, ... , N},
then we see that
(3.9)
for some (g~) E .c(m+r)xr. This is the first step to obtaining a parametrization of the
set of all solutions F E .cmxr of (3.3). More precisely we have the following result.
THEOREM 3.2. Let {Xj,Yj,~j: j = 1, ... ,N} be an admissible TVNPI
data set, and suppose that 9 E .c(m+r)x(m+r) satisfies (3.8). Then an operator Fin .cmxr
satisfies the interpolation conditions (3.3) if and only if F has a representation of the form
for some pair of lower triangular matrices G1 E .cmxr and G2 E .crxr such that
9 21 G1 + 9 22G2 E .crxr is invertible with inverse again in .crxr .
PROOF. Suppose that F E e mxr satisfies (3.3). Write (~) in the fonn
(/l,h, ... ,Ir) where each li E e(m+r)xl. Then (3.3) implies {(Xj -Yj )li}"(~j) = 0
for j = 1, ... , N and for each i = 1, ... , r. Hence by property (3.8) of 9, each Ii E
ge(m+r)xl for each i = 1, ... , r. We conclude that (~) itself ~as a factorization (3.9)
with (g~) E e(m+r)xr. In particular, G1 E e mxr , G2 E e rxr and 9 21 G1 + 9 22G2 = I
has a lower triangular inverse. We conclude that F has a representation as asserted.
Conversely, suppose that G1 E e mxr , G2 E e rxr and 9 21 G1 + 9 22 G2 is
invertible with inverse in e mxr, and
Then F is the product of lower triangular matrices, so itself is in em x r. Moreover
Since ( g~) (921 G1 +922 G2 )-1 E e(m+r)xr, each column is in e(m+r)xl. By the defining
characteristic (3.8) of 9, we see that each column it of ( ~) satisfies
j = 1, ... ,N.
From this we see that (~) satisfies the homogeneous form (3.6) of the interpolation
conditions, and hence F satisfies (3.3). 0
The next step is to adapt Theorem 3.2 to handle the norm constraint (3.2).
THEOREM 3.3. Let {Xi> Yj, l::!.j : j = 1, ... , N} be an admissible TVNPI
data set. Suppose that the lower triangular matrix
in addition to (3.8), is J-inner. Then there exist solutions F E e mxr of the TVNPI
problem. Moreover any solution F is given by
where G is any lower triangular matrix in .cmxr with IIGII < 1.
PROOF. Suppose first that F E .cmxr is a solution of the TVNPI prob
lem. In particular F satisfies (3.3). Since 8 satisfies (3.8), it follows that (~) has a
factorization (3.9) with (~~) E .cmxr .
We argue next that G2 is invertible with G"2 1 also lower triangular. Indeed,
since F satisfies (3.2) and 8 is a J-isometry we have
So we have proved that
(3.10)
= ( Gi Gn 8* J8 ( ~~ )
= ( Gi G2 )J ( ~~ ) = G~Gl - G~G2.
If x E £2 and G 2 x = 0, then (3.10) forces G 1 x = o. But then we can use (3.9) to show that
and we conclude that G2 has a trivial kernel. What's more, since 8 is J-unitary, 8 has a
bounded inverse 8-1 on e~+r. Hence
(Gi G2) (~~) = (F* I) 8-*8-1 ( ~) ;::: m(F*F+I);::: mI
for some m > O. But from (3.10) we have
2G* G ( G* G2* ) (GG12) . 2 2> 1
We conclude that G2G2 > tmI, and so G2 has closed range. To show that G2 is invertible
with inverse lower triangular, it remains only to show that G2£2([k, 00» is a dense subset
of l2( [k, 00» for every integer k. Therefore, suppose x E l~ ([ k, 00» is orthogonal to
G2 f2([k , 00» for some k. Set
Then for all w E f~([k,oo»,
(J (~) w, (~~)) = (J0 (g~) w,0 (~))
= (J (~~) w, (~)) = -(G2 w, x) = o.
In particular, this holds with w = Y2. Hence
IIYl - FY2112 = (J{ (~~) - (~) Y2}, (~~) - (~) Y2)
= (J (~~), (~~)) + (J (~) Y2, (~) Y2)
= (J (~), (~)) = -llxll 2 ~ 0,
where the inequality stems from the norm constraint (3.2). We conclude that Yl = FY2.
Thus
from which we get x = G2Y2. As x was chosen to be orthogonal to G2f2([k, 00», we
conclude that x = 0 as needed. We now have shown that G:;l E crxr•
Since G:;l E crxr , we see that G := GlG:;l E cmxr• Moreover (3.10)
implies that IIGII < 1. Finally from the identity (3.9) we read off
and
Hence F has a representation F = (0u G + 0 12 )(021G + 0 22 )-1 of the required form.
Conversely, suppose that 8 is a J-inner matrix satisfying (3.8), and let G be
any lower triangular matrix in cmxr with IIGII < 1. Define (~~) E c(m+r)xn by
(3.11)
From Theorem 2.3 we know that 8 2l E crxr , and according to Theorem 2.1 we have
11821821 11 < 1. Since 8;}821 G E cmxr with 118221821 GII < 1, by the Neumann series
expansion we see that 8 2l 8 21 G + I has a lower triangular inverse. Hence
has a lower triangular inverse Fi1 E crxr• Hence F := F1 F2- 1 E cmxr and (3.11) give
Since 8 satisfies (3.8), we see from Theorem 3.2 that F satisfies the interpolation conditions
(3.3). Moreover, since 8 is a J-isometry,
F"F-I=(F" I)J(~) =F2-"(G" I)8"J8(~)F2-1
=F2-*(G" I)J(~)F2-1 =F2-*(G* I)J(~)F2-1 = F2-*(G"G - I)F2- 1 < 0,
and so IIFII < 1. 0
4. SOLUTION OF THE TIME-VARYING TANGENTIAL NEVANLINNA
PICK INTERPOLATION PROBLEM
In this section we deal with the construction of a J-unitary 8 meeting the
hypotheses of Theorem 3.3 and go on to complete the proof of Theorem 3.1, including a
parametrization of the set of all solutions.
Throughout this section {Xj, Yj,!::J.j : j = 1, ... , N} is an admissible TVNPI
data set as defined in Section 3. Introduce the following operators
(4.1) ,,~») ,
(4.2)
Here xY), Yj(k) and ~~k) denote the k-th diagonal entries of X;, Y; and ~i' respectively.
The next theorem provides a useful sufficient condition for the existence of
a 8 e .c(m+r)x(m+r) satisfying at least (3.8).
THEOREM 4.1. Let {X;,Y;'~i: j = 1, ... ,N} be an admissible TVNPI
data set, and suppose 8 e .c(m+r)x(m+r) has inverse 8-1 e u(m+r)x(m+r) which admits
a representation of the form
00
(4.3) 8-1 = V + C(S - A)-18 = V + 1: CS-1(AS-1)" 8, ,,=0
where A and 8 are as in (4.1) and (4.2) respectively, C e v(m+r)xN is such that
( 4.4)
and V is any element of v(m+r) x(m+r) . Then 8 satisfies (3.8).
PROOF. Let H e .c(m+r)xl be given. Then H e 8.c(m+r)xl if and only if
e-1 H e .c(m+r)xl. Since H is lower triangular, H has a series expansion
00
H = 1:SiH(j] j=O
where H[ll e v(m+r)xl. From the expansion (4.3) for e-I, we see that the j-th diagonal
of e-1 H above the main diagonal is given by
00
S-j(8- 1 H)[_j) = 1: CS-1(AS- 1 )l+j-18St H[t) t=o
= CS-1(AS-1 )i-1 (~(AS-l)t8StH[l)).
Thus 8-1 H e .c(m+r)xl if and only if
00
(4.5) CS- 1(AS- 1 )i-1 .1:(AS-1 )i8Si H[l) = 0, j = 1,2, ... i=O
From (4.4) we see that (4.5) in turn is equivalent to
00
(4.6) L(AS-llBSlH[l] = O. l=O
Recalling now the definitions of A and B in (4.1) and (4.2), we see that (4.6) is the same
as 00
L(.6.jS-l)l(Xj -Yj )SlH[l] = 0, j = 1, ... ,N. l=O
In other words
j = 1, ... ,N,
as required. 0
Next we obtain a sufficient condition for the existence of a J-inner E> satisfy
ing condition (3.8); happily this sufficient condition coincides with the necessary condition
for existence of solutions of the TVNPI problem already established (necessity in Theorem
3.1). In what follows we write 1{ for the operator A( {Xi, Yi,.6.;}) introduced in Theorem
3.1. Thus
(4.7) 1{ = ({XiX; - Yilj")(I - S.6.;)-1 Y(.6.i») N. 1,)=1
Note that 1{ act on .e!j and is also given by
00
(4.8) 1{ = 2)AS-1)kBJB"(SA")\ k=O
where A and B are as in (4.1) and (4.2), respectively. Indeed, in terms of A and B the
right hand side of (4.7) can be written as {BJB*(I - SA*)-l yeA). Now, we use that
and apply the definition in (1.18) of the point evaluation map to get (4.8). From (4.8) we
see that 1{ E VNxN.
THEOREM 4.2. Let {Xj, Yj,.6. j : j = 1, ... , N} be an admissible TVNPI
data set. Define block diagonal operators A E V NxN and BE vNx(m+r) as in (4.1) and
(4.2), and let 1-{ E V NxN be the Hermitian operator defined by (4.7). Assume that 1-{ is
invertible on £!j and that the diagonal entries H k of 1-{ have signature independent of k.
Then there exists a J-unitary lower triangular matrix e E .c(m+r)x(m+r) which satisfies
(3.8). Moreover e is J -inner if and only if, in addition, the operator 1-{ given by (4.7) is
positive definite.
The following corollary presents a recipe for the construction of a realization
for the matrix e in Theorem 4.1.
COROLLARY 4.3. Let {Xj,Yj,Aj : j = 1, .. . ,N},A,8 and 1-{ be as in
Theorem 4.2, and assume that 1-{ is invertible with diagonal entries Hk having constant
signature. Then 11 = diag (Hk)'r::-oo satisfies the following time-varying Stein equation:
( 4.9) k E l,
where Ak and Bk are as in (4.1) and (4.2), respectively. Furthermore, one can find matrices
(Ck Dk) such that (Ck Dk) are bounded in norm uniformly with respect to k and
(4.10)
(4.11) CiJ ) JDZJ '
( 4.12)
= rkXk + bkUk.
( 4.13) e = Tr, = JV* J + J8*(J - SA*)-l SC* J,
Dk).
where C = diag (Ck)'r::-oo' V = diag (Dk)'r::_oo' and e is a lower triangular J-unitary
operator satisfying (3.8).
PROOF OF THEOREM 4.2 AND COROLLARY 4.3. We seek a lower
triangular J-unitary (or even J-inner) e satisfying (3.8). Condition (3.8) is a condition on
an anticausal realization for \If = e-1 = Je" J. Certainly \If is necessarily upper triangular
J-unitary and, by Proposition 2.6, is anti-J-inner if and only if e is J-inner. Theorem 4.1
suggests that we seek an anticausal realization for \If = Je" J of the form
(4.14) { Xk = AkXk+1 + BkUk q Yk = Ck Xk+1 + DkUk.
where (Ak,Bk) are given by the data as in (4.1) and (4.2), and Ck, Dk are to be determined.
Note that Ak acts on eN, and hence the state spaces in (4.14) do not depend on time. In
order for the input-output map of q to be J-unitary, by Theorem 2.5 we are led to seek a
sequence of invertible Hermitian matrices (H;;1 )f::-oo so that
(4.15)
Since Ak, Bk are given and Ck, Dk are to be found, it is more convenient to work with the
equivalent formulation
(4.16) 0) (A" C") (Hk 0) J B: DZ = 0 J .
Equality of the (l,l)-blocks in (4.14) leads to the time-varying Stein equation (4.9).
Now, let 11. be as in (4.7). Since 11. also arunits the representation (4.8), we
have
(4.17) 11. = (AS-1 )1l(SA") + BJB".
By comparing the diagonal entries of the left and right hand side in (4.17) we see that the
diagonal entries of 1l satisfy (4.9).
The next step is to find matrices (Ck Dk) satisfying (4.10) and such that
(Ck Dk) are bounded in norm uniformly with respect to k. This problem may be viewed
as the time-variant analogue of the embedding problem solved in [11]. To find the matrices
(Ck Dk) set
Ilk = (~~) H;;I(Ak Bk ) (H~+1 ~). Note that (4.9) can be rewritten as
(4.18) Hk = (Ak Bk) (H~+1 ~) (~~) ,
and hence we can use this identity to show that rIA, is a projection operator acting on the
space cN +m +r • Since Hk is invertible, formula (4.18) also implies that
and hence rank Ilk = N. Therefore the Hermitian matrix
(4.19) ~) has rank m + r. Via the functional calculus for Hermitian matrices, one sees that Ak has
a factorization
(4.20)
for some (m + r) x (m + r) signature matrix j. In fact, one may choose (Ck D k ) to have
the form
(4.21)
where f(t) = Itl' and Uk is a partial isometry. Note that our assumptions on A k , Bk
and Hk guarantee that Ak in (4.19) is bounded in norm uniformly with respect to k. But
then we see from (4.21) that the matrices (Cle Dk) in (4.20) also may be chosen to be
uniformly bounded relative to k.
Next, let us prove that in (4.20) we may take j = J. Note that (4.18) implies
that
(4.22)
From (4.20) we know that 1m Ak = 1m (gi), and hence the equality (4.22) yields
(4.23)
On the other hand from (1 - 11,,)2 = 1 - II" we get
( Ck) . (C D) (Hk+l D* J " Ii: 0 "
Since (~i) is injective and ( C" D,,) ( H ~+l ~) is surjective, this gives
(4.24)
In particular, the right hand side of (4.23) is invertible. The latter implies that the first
and third term in the left hand side of (4.24) are invertible, and thus the matrices
0) (Hk 0) J ' 0 j
have the same signature. According to our hypotheses, the signatures of H"+l and H" are
equal. It follows that J and j have the same signatures. Therefore in (4.20) we may take
j = J.
From (4.23) and (4.24) we get the identity (4.16), and hence the system a
in (4.14) is (anticausal) anti-stable J-unitary. Furthermore, for the input-output map we
have
Now put 8 = JT;J. Then 8 is the J-unitary input-output map of the system ~ in (4.12)
and 8 admits the representation (4.13). Since 8-1 = J8J = Tu , also (4.3) holds.
Next, let us check that condition (4.4) is fulfilled. The identity (4.15), which
is equivalent to (4.16), yields
1{-1 - (SA*)1{-1(AS-1) = SC* JCS-1,
and hence 00
1{-1 = ~)SA*)jSC* JCS-1(AS-1)j. j=O
So, if x is a vector in the space defined by the left hand side of (4.4), then 1{-l x must be
zero, and therefore x = O. So condition (4.4) is fulfilled.
Thus, by Theorem 4.1, e satisfies (3.8). Note that, by definition, the system
u is anti-J-inner exactly when 11. > O. Hence, by Theorem 2.5, the same holds true for the
associated input-output map TO'. It follows (cf. Proposition 2.6) that e is J-inner if and
only if 11. > O. So e meets all the requirements in Theorem 4.2. 0
Putting together the pieces we have the following more detailed form of
Theorem 3.1.
THEOREM 4.4. Let {Xj, Yj,~j : j = 1, ... ,N} be an admissible TVNPI
data set, and let 11. be the block diagonal matrix in V NxN given ·by (4.7). Then solutions
of the TVNPI problem exist if and only if?-i is positive definite. In this case any solution
F of the TVNPI problem (3.2) and (3.3) is given by
where G is any lower triangular matrix in .cmxr with IIGII < 1 and where e constructed as in Corallary 4.3.
PROOF. Necessity of the condition 11. > 0 has already been noted. Con-
versely, suppose 11. > O. Then we may construct e as in Corollary 4.3 satisfying all the
conditions of Theorem 4.2. Now Theorem 3.3 gives that solutions F of the TVNPI prob
lem exist and that the set of all such solutions is given by the linear fractional formula as
described above. 0
5. AN ILLUSTRATIVE EXAMPLE
As an example which one may compute by hand, we consider the special case
of one (N = 1) interpolation condition
(5.1)
where the unknown F E .c1X1 as a scalar entries, and Xl
diag (Yl(k»f:_oo and ~l = diag (~~k»f:_oo are given by
(5.2a) X (k) -1 1 - , k E l,
(5.2b) k E l,
with (Yk)k::-oo a bounded sequence of complex numbers, and
(5.2c) ~~k) = { 0 ~f k ;6 0, W If k = O.
In this case (~IS-l)i = 0 of eachj ~ 2. Hence, for F = 2:;0 SiF[i] E C1XI, the diagonal
matrix (XIF)I\(~l) is given by
Thus the interpolation condition (5.1) can be given explicitly in terms of the entries
(i ~ j) of F as follows:
(5.3) { Fii = Yi 0 ;6 i E 1,
Foo + wF10 = Yo·
We also want the interpolant FE Clxt in (5.1) to be a strict contraction.
For a lower triangular matrix F = (Fii)iJ=-oo to be a strict contraction, it
is certainly necessary that all diagonal entries Fii (i E 1) have modulus less than 1. We
see from (5.3) immediately therefore that a necessary condition for (5.3) to be satisfied by
a strict contraction Fin C1X1 is that IYil < 1 for all i ;6 O. If Yo also has IYol < 1, clearly
we Ulay set F = Y = diag (Yk )k::-<Xl to get a solution. It is not obvious froUl a casual
glance what the precise necessary and sufficient condition for strictly contractive solutions
to exist should be. Such a condition is easily computed by using the theory developed in
the preceding sections. Note that in this case condition (3.1) is fulfilled.
PROPOSITION 5.1. There exists F = (Fii)iJ=-<Xl E C1xl with I!PII < 1
satisfying the interpolation conditions (5.3) if and only if the following two conditions are
fulfilled:
(i) for some c: > 0, we have ly;I ~ 1 - c: for all 0 ;6 i E 1,
(ii) IYol2 < 1 + Iw1 2(1 - IYlI2).
PROOF. By Theorem 4.4, solutions exist if and only if 1t = diag (Hk)k::-oo
is positive definite on £2, where H k (k E 1) is determined as the solution of the time-varying
Stein equation
For our case, AI: = 0 for k =F 0, Ao = W, BI: = (1 - YI:) for all k. Thus (5.4) becomes
Solving gives
(5.5) k =0.
The condition HI: ~ e for some e > 0 then leads to (i) and (ii) in the theorem. 0
Theorem 4.4 of course provides not only a necessary and sufficient condition
for existence of solutions of the TVNPI problem, but also a linear fractional parametriza
tion for the set of all solutions. For the specific data set (5.2) which we are discussing
here, implementation of the algorithm for the construction of the linear fractional map 8
involves decisions at various steps as to whether a certain quantity is positive, negative or
zero. The explicit formula for 8 as a result breaks out into five special cases. Here we
present two of these cases explicitly for purposes of illustration.
PROPOSITION 5.2. Suppose that the TVNPI data set given by (5.2) satis
fies the following additional condition:
(j) for some e > 0 we have IYil :5 1 - e for all i E Z.
Set
(5.6) bi = 1 - ly;/2 (i E Z),
Then the interpolation problem (5.3) has a solution FE C1x1 with I!PII < 1 and the block
lower triangular matrix
8 = (811 812 ) E C2X2 8 22 8 22
which parametrizes the set of all solutions is given by 8 = (8(i,j»iJ=_oo, where
o =F i E Z,
and 9(i,j) = 0 for all other pairs (i,j). In particular, the central interpolant F = 9 12 9;-21
is the diagonal matrix F = diag (Yk)f:,-oo.
PROOF. Note that condition (j) implies that conditions (i) and (ii) in The
orem 5.1 are fulfilled. Hence, the interpolation problem (5.3) has a solution F E .c1X1 with
IIFI! < 1. By Theorem 4.4 combined with Theorem 4.2 and Corollary 4.3, the desired 9 is
given by
(5.7) 9 = JV* J + JB*(1 - SA*)-1 se* J
where
with
B = diag (Bk)f:,-oo
V = diag (Dk)f:,-oo
where Ck ~s a 2 x 1 matrix and Dk is a 2 x 2 matrix, bounded in norm uniformly with
respect to k, such that
with J = (~ ~1). For our case here, Hk is given by (5.5). Thus
(5.8) ~k) , -1
k i= O.
By inspection we observe the factorization
Since lik > 0 by assumption, we conclude that a viable choice of (~:) for k i= 0 is
(5.9) k i= O.
For k = 0 a straigtforward computation gives us
(5.10)
To factor Ao we perform a sequence of row and symmetric column operations. The result
IS
Yo -00
~) , E,~ G "GJ0;101 n 1 1 0 0
Note that in the definition of E2 we have already used that 00 = 1 -IYoI2 :/: O. Almost by
inspection, where we now use the assumption that 00 > 0, we get the factorization
(5.12)
Putting the pieces together, we conclude that a viable choice for (Co Do) is
(5.13)
Taking adjoints and multiplying by J as appropriate, in summary we have
(5.14)
k:/:O
We are now ready to plug (5.14) into (5.7) to get 0. To do this observe that
(I - SA*)-1S is given by
Hence
(5.15a)
(5.15b)
(5.15c)
and
(5.15d)
W 1 0
0(1,-1) = wJB~1C~1J
0 1 0 0 1 0
This leads to the formula for 0 stated in the theorem. From this formula for 0 we read
off that 0 12 is diagonal with its diagonal entries given by
while 0 22 is also diagonal with diagonal entries
i = 0,
{
8j , i i- O.
From this we read off that the central solution F = 0120~21 is the diagonal solution
F = diag (Yk)~-oo which one can see by inspection (for the case where the strong sufficient
condition (j) holds) without applying the theory. 0
We next present another special case where the structure of the solution is
somewhat different.
PROPOSITION 5.3. Suppose that the TVNPI data set given by (5.2) satis-
fies the following condition:
(i) for some e > 0 we have IYil < 1 - e for all 0 f. i E Z,
(ii) IYo 12 < 1 + Iw1 2(1 - IYI1 2),
(iii) IYol > 1.
5i = 1 - IYi 12 (i E Z),
Then the interpolation problem (5.3) has a solution F E .cIXl with I!PII < 1 and the block
lower triangular matrix
8 = (811 8 11
which parametrizes the set of all solutions is given by 8 = (8(i,i»;J=_oo, where
with e(i,i) = 0 for all other pairs (i,j). In particular, the central solution F = e128;-l is
given by
Y-2 Y-I
Y2
where unspecified entries are all zero and where x = -w-l (1-lyoI2)yo l.
PROOF. Since (i), (ii) hold, Theorem 5.1 implies that the associated TVNPI
problem has a solution. To get the parametrization of all solutions we proceed as in the
proof of Proposition 5.2. The first step is to find uniformly bounded matrices (Ck Dk)
which solve the factorization problem
where Ak is again given by (5.8) and (5.10). For k i:. 0 the situation is exactly the same as
in the proof of Proposition 5.2; a viable choice for ( Ck Dk) is given by (5.9) for k i:. O.
For k = 0, all the details are the same up to formula (5.11). Since in our present setting
60 < 0, we factorize
We conclude that a viable choice of ( Co Do ) is
(Co Do) = (H;iI6~li6;i = (H;'16~li6;i
Next, taking asjoints and multiplying by J as appropoiate, in summary we have
(5.16)
), k=O.
As before the entries e(i,j) are given by (5.15a) - (5.15d). Use of (5.16) then leads to the
formulas for e(i,i) as stated in Proposition 5.3. In this case e 12 and e 22 are not diagonal.
Indeed, we have
c_I -i-Ll
9 22 = l-wlcol-tclYoH;t I _1 1 _ 1 c- t -YIHO '5'lcol'5'c1 '5' 1
-t c2
-.!.. Y2 C2 2
Since 9 22 has only one nonzero off diagonal entry, its inverse is easily computed; the result
is
1 8'5' -2
1 c"i 1
1 c'5' 2
Multiplying out now gives that F = 91292"l as is specified in the theorem. D
Of course it is also possible to verify directly that F as in Proposition 5.3
is a solution of the TVNPI problem associated with the data set (5.2). The interpolation
conditions (5.3) are clearly fulfilled when i =I o. When i = 0, we have
as required. To show that F has norm less than 1, since IYkl $ 1 - c < 1 for all k f= 0 for
some c > 0, it suffices to show that
By assumption we know Iyoll < 1, IYII < 1. In general one can show that
II (y~1 ~I) II < 1 if and only if
With x = -w-I (1-IYoI2)Yol, the condition to be verified is
(5.17)
Multiply both sides by Iwl21yol2 and divide by lyol2 - 1 to convert (5.17) to
This condition in turn is exactly equivalent to Ho = ,so + Iw1 2,s1 > O. Thus, given that
Iyol > 1, F is a strict contraction exactly when the necessary and sufficient condition for
solutions of the TVNPI problem holds.
A similar analysis can be done for the case Iyo I = 1. In this case the explicit
formula for e breaks into three cases depending on whether Iwl > 1, Iwl = 1 or Iwl < 1.
We invite the interested reader to explore the details of this case for his or her self. For
more complicated examples, of course, one would want to automate the algorithm on a
computer.
[2J
REFERENCES
D. Alpay and P. Dewilde, Time-varying signal approximation and estima tion, in: Signal processing, scattering and operator theory, and numerical methods, Proceedings of the international symposium MTNS-89, Volume III (Eds. M.A. Kaashoek, J.H. van Schuppen and A.C.M. Ran), Birkhauser Verlag, Boston, 1990. D. Alpay, P. Dewilde and H. Dym, Lossless inverse scattering and reproduc ing kernels for upper triangular operators, in: Extension and interpolation of linear operators and matrix functions, OT 47 (Ed. 1. Gohberg), Birkhauser Verlag, Basel, 1990, pp. 61-135.
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
J.A. Ball
T. Azizov and loS. Yokhvidov, Foundations of the theory of linear operators in spaces with an indefinite metric, Wiley, New York, 1989. J.A. Ball, I. Gohberg and M.A. Kaashoek, Time-varying systems: Nevanlinna-Pick interpolation and sensitivity minimization, Proceedings MTNS-91, submitted. J.A. Ball, I. Gohberg and M.A. Kaashoek, Nevanlinna-Pick interpolation for time-varying input-output maps: the continuous time case, In this Volume. J .A. Ball, I. Gohberg and L. Rodman, Realization and interpolation of ratio nal matrix functions, in: Topics in interpolation theory of rational matrix valued functions, OT 33 (Ed. I. Gohberg), Birkhauser Verlag, Basel, 1988, pp. 1-72. J .A. Ball, I. Gohberg and L. Rodman, Interpolation of rational matrix func tions, OT 45 , Birkhauser Verlag, Basel, 1990 P. Dewilde, A course on the algebraic Schur and Nevanlinna-Pick interpo lation problems, in Algorithms and Parallel VLSI Architectures, Volume A: Tutorials (Eds. E.F. Deprettere and A.-J. van der Veen), Elsevier, Amster dam, 1991. P. Dewilde and H. Dym, Interpolation for upper triangular operators, in this Volume. M.A. Drischel and J. Rovnyak, Extension theorem for contraction operators on Krein spaces, in: Extension and interpolation of linear operators and matrix functions, OT 47 (Ed. I. Gohberg), Birkhauser Verlag, Basel, 1990, pp. 221-305. Y. Genin, P. Van Dooren, T. Kailath, M. Delosme and M. Morl, On ~ lossless transfer functions and related questions, Linear Algebra Appl. 50 (1983), 251-275. I. Gohberg, M.A. Kaashoek and L.Lerer, Minimality and realization of dis crete time-varying systems, in this Volume.
Department of Mathematics, Virginia Tech
Blacksburg, VA 24061, U.S.A.
School of Mathematical Sciences, Tel-Aviv University
Ramat-Aviv, Israel.
M.A. Kaashoek
Amsterdam, The Netherlands
52 Operator Theory: Advances and Applications, Vol. 56 © 1992 Birkhiiuser Verlag Basel
NEVANLINNA-PICK INTERPOLATION FOR TIME-VARYING INPUT-OUTPUT MAPS: THE CONTINUOUS TIME CASE
J .A. Ball, I. Gohberg and M.A. Kaashoek
In this paper the tangential Nevanlinna-Pick interpolation problem for time varying continuous time input-output maps is introduced and solved. The conditions of solvability are derived and all solutions are described via a linear fractional representa tion.
o. INTRODUCTION
For functions analytic on the open right half plane C+ the simplest N evanlinna Pick interpolation problem reads as follows. Given N different points Zl, • .. , Z N in C+ and arbitrary complex numbers Yl, ... , YN, determine a function F, analytic on C+, such that
(i) F(zj) = Yj, j = 1, ... ,N,
(ii) sup{IF(>')11 >. E C+} < 1.
Let us assume that we look for solutions F of the form
(0.1)
where d is a complex number and J is in Ll(lR) with suPPJ c [0,00). Then (i) can be rewri t ten as
(0.2) j =l, ... ,N,
and the above interpolation problem can be restated as a problem involving operators on L2(1R). To see this, note that for the function F in (0.1) the operator of multiplication by F on L2(ilR) is unitarily equivalent via the bilateral Laplace transform to the convolution operator T on L2(1R) given by
(0.3) (Tcp)(t) = dcp(t) + J~oo J(t - s)cp(s)ds, tER
The number Zj we view as the operator of multiplication by Zj on L2(1R). Since Zj E C+,
the maximal operator on L2(1R) 8.'3sociated with the differential expression 1t - Zj 18
t E JR.
Thus eft - Zj) -IT is an integral operator on L2(JR) with kernel function
kj(t,s)=_deZj(t-s)_ t'" eZj(t-a)f(a-s)da, lmax(t,s)
and the interpolation condition (0.2) is equivalent to the requirement that
j = 1, ... ,N.
53
The interpolation problem (i), (ii) mentioned above can now be reformulated as a problem involving operators on L2(JR), namely find a lower triangular Wiener-Hopf operator T on L2(JR) of the form (0.3) such that
(i)' the kernel function ofthe integral operator -(ft - Zj)-IT evaluated at t = s is equal to Y j, j = 1, ... , N,
(ii)' IITII < 1.
In this form the problem can be extended in a natural way to an interpolation problem for operators that are not of Wiener-Hopf type and in which the interpolation data ZI,··· , ZN and YI,·· ., YN are Loo-functions on JR.
More precisely, in the present paper we study the following problem. Let ZI, ... ,ZN and Yb ... , YN be in Loo(JR). Assume that the maximal operator on L2(JR) associated with the differential expression ft - Zj(t) is invertible, and let its inverse be the upper triangular integral operator
(0.4) t E JR.
(0.5) t E JR,
where c is a positive number independent of t and IN is the N X N identity matrix. In the classical case condition (0.4) means that the points where the interpolation takes place are in C+ and (0.5) is equivalent to the requirement that these points are different. Now the problem is to find a lower triangular integral operator T on L2(JR) of the form
(0.6) (Tcp)(t) = d(t)cp(t) + [too f(t,s)cp(s)ds,
such that IITII < 1 and for j fulfilled:
1, ... ,N the following interpolation requirements are
(0.7) d(t) + 1000 Zj(t, t + a)f(t + a, t)da = Yj(t), t E IR.
The function d(·) in (0.6) is required to be an Loo-function on JR., and the kernel fundion f in (0.6) has to be measurable on JR. x JR. and such that
The class of operators T as in (0.6) for which d and f satisfy these conditions can be viewed as the time-varying analogue of the Wiener algebra on the line. We shall proVf~ that this generalized continuous-time Nevanlinna-Pick interpolation problem is solvable if and only if for some € > 0
(O.S) t E IR.
In what follows we also treat the matrix version of this problem.
In the classical Nevanlinna-Pick interpolation problem there is a special inter est in solutions (0.1) that are rational, i.e., a ratio of polynomials. Using the realization theorem from systems theory the latter can be interpreted to mean that the operator T in (0.2) is the input-output map of a causal stable time-invariant system, i.e., the relation T<.p = g is given in the following way:
{ x'(t) = Ax(t) + B<.p(t), get) = Gx(t) + D<.p(t),
t E JR.,
where A, B, G and D are matrices of appropriate sizes. In the general problem the requirement that the solution is rational is replaced by the condition that the operator T in (0.6) is the input-output map of a time-variant system. In other words the action of the operator T in (0.6) is given by
{ x'(t) = A(t)x(t) + B(t)<.p(t),
get) = G(t)x(t) + D(t)<.p(t),
t E JR.,
where now the state, input, output and feedthrough matrices A(t), B(t), G(t) and D(t), respectively, may vary in time.
The left hand side of (0.7) will be denoted by T(zj)(t). The map Z f--+ T(z), which assigns to an Loo-function z the function T(z), is the continuous analogue of the generalized point evaluation map for lower triangular doubly infinite matrices appearing in [Dew], [ADeDy], [DeDy]' and [BallGK]. The generalized Nevanlinna-Pick interpolation
problem introduced above is the natural continuous version of the interpolation problem appearing in [DewDy] and [BallGK].
In the present paper we develop the continuous analogue of the method of soluti

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