Exact model matching of 2-D systems

Exact model matching of 2-D systems

Citation for published version (APA):

Eising, R., & Emre, E. (1978). Exact model matching of 2-D systems. (Memorandum COSOR; Vol. 7815). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

PROBABILITY THEORY. STATISTICS AND OPERATIONS RESEARCH GROUP

Exact Model Matching of 2-D Systems

by

Rikus Eising and Erol Emre

Memorandum COSOR 78-15

Eindhoven, June 1978

Exact Model Matching of 2-D Systems Rikus Eising Eindhoven University of Technology Dept. of Math. Eindhoven the Netherlands Abstract by

and Erol Emre

Centre of Mathematical System Theory and Dept. of Electrical Engineering University of Florida Gainesville

Florida 32611

In this note the 2-D Model Matching Problem is considered. The purpose of this note is not to present a complete solution to this problem but

rather to indicate some possible solutions.

By using a Generalized Dynamic Cover (GDC)[SJ a first level realization [1J is constructed for the desired compensator.

2

-Introduction

In this note we consider the 2-D Model Matching Problem. Suppose there-fore that

Bl

(s,z) and

B

₂(S,z) are 2-D transfer matrices of causal 2-D systems (see [IJ). Now, completely analogous to the usual I-D case, an equation of the form

has to be solved for a 2-D transfer matrix G(s,z). As in the I-D case we can rewrite this equation and obtain

where B1(s,z), B

2(s,z) are given r x p and r x m polynomial matrices in z and G(s,z) is a proper rational matrix in z where the coefficients are

(not necessarily proper [3J) rational functions in s (see [IJ).

Main Results

For the solution of this problem we will use a theorem as can be found in [4J.

Suppose we are given the following problem:

where B1(Z), B

2(z) are r x p, r x m polynomial matrices with coefficients in a field

P,

then the solution of this problem (to find a proper rational matrix G(z» is equivalent to finding a Generalized Dynamic Cover (GDC) see also [4J.

Notation will be the same as in [4J, so {,} denotes a subspace generated by the columns of ~.

Theorem. Let

(3)

then to every reachable system ~

=

(A,B,C,D) with order n and transfer

matrix G(z) satisfying (2) there corresponds a subspace {~} with dimension ~ n satisfying:

3 -P{;j;} c {;j;} + {B 1} (4) {B 2} c {;j;} + {BI}

{*l

c

{~:jl

.

Here 0 I

o •

.

.

0 Bk r 0 0 I I Bk 2 r P .. Q

"

B .. I . I B '" 1/1 e: (k+ 1 )rxn 2 J F r BO 0

. . .

. .

.

• 0 I B O 2

Conversely to every subspace {;j;} of dimension n satisfying (4) there corresponds a system E of order n satisfying (2).

Proof. For a proof see [4J.

Furthermore we have

o

(5) Corollary (2) has a proper G(z) as a solution iff (4) has a solution {;j;}. The minimal order solutions of (2) correspond to the minimal dimensional

solutions of

(4).

The minimal possible order of the solutions of

(2)

is equal to the minimal possible dimension of the solution of (4).

0

Remark. For existence, properties and construction of a GDC see [5]. Now we choose a basis in {;j;} which will be denoted by ~. Then (4) gives rise to the following equation:

In [4J it is proved that every ;j; satisfying (6) gives rise to a system

E '" (A,B,C,D) such that (2) is satisfied. Here Get) is the transfet" matrix of E.

- 4

-We will now apply this result to (I) obtaining the following equation:

(7)

B(S~

•

D(S~

So we obtained again a matching problem of a very special kind. First of all we have freedom in the choice of {~(s)} such that (4) is satisfied and secondly, after having chosen a subspace, there is still freedom to choose a basis for that space.

Now the minimal dimension of A(s) and also the McMillan degree of

(8)

fA(s)

is(s)

B(SJ _{... A(s)}

D(s)

are important quantities for the construction of a second level realization (compare [IJ, [8J). Furthermore ~(s) has to be chosen such that a proper solution for (7) exists.

Now suppose we have a solution to (7) such that A(s) is a proper rational matrix, then (A(s). B(s), C(s), D(s» can be considered to be a first

level realization of the 2-D transfer matrixG(s,z) which can be used to construct a second level realization of the type that has been proposed in [IJ or a model of Roessers type (see C7J, [8J). For the latter case suppose that

(A,B,C,D)

is a minimal realization of (8) where

A,B,c,D

are partitioned according to (8) as:

then a 2-D state space model of Roessers type is:

l~+I'J ~I

_-J~

A2 ~k

_J

_~J

BI (9) Sh,k+l -

A3 A4

Shk +

B2

u_hk ,..,

I

C2{~j

+

..., Y hk at [el D uhk Shk

5

-with appropriate dimensions, zero initial conditions and h - 0,1, ••• , kill 0,1, ••••

Here ~k' Shk are local state variables.

Now to obtain a minimal 2-D state space realization, it is not clear how the space {~{s)} must be chosen to obtain the right balance between the dimension of A(s) and the McMillan degree of

fA{s)

~(s)

B{S"51 •

D(sJ

In general the choice of {~(s)} determines the dimension of A(s) but the choice of basis in {~(s)} will usually affect the McMillan degree of A(s). The requirement that {~(s)} has to be chosen such that a proper solution A(s) exists can be circumvented by allowing a more general class of causal 2-D state space models as is presented in [3J.

There is also a link between the technique proposed in this note and [6J.

For suppose we solve the exact model matching problem in the context of

[6J and obtain a state space description of G(s,z) in the form (9) where

is a modally controllable pair. Then it is proved in [2] that (A(s), B(s» is a reachable pair and therefore the GDC problem can be solved by theorem

(3).

Conclusion.

In this note we presented a GDC approach to the 2-D Exact Model Matching Problem. In (7) we stated the problem to be solved and pointed out the still existing freedom in that equation. The complete solution is far from being achieved. In the last part of the note contact has been made with [6] via [2].

6

-Acknowledgement. One of the authors (E. Emre) would like to thank the Dept. of Mathematics of Eindhoven University of Technology for their financial support and hospitality while this research was being done.

7

-References

[IJ R. Eising, Realization and Stabilization of 2-D Systems. To appear in IEEE trans. A.C. 1978.

[2J , Controllability and Observability of 2-D Systems. Submitted to IEEE trans. A.C. 1978.

[3J , Recurrence and Realization of 2-D Systems,

Memorandum COSOR 77-26, Eindhoven University of Technology 1977. [4J E. Emre , On the Exact Matching of Linear Systems by Dynamic

Compensation. Submitted to IEEE trans. A.C. 1977.

[5J , L.M. Silverman, K. Glover, Generalized Dynamic Covers for Linear Systems with Applications to Deterministic

Identification and Realization Problems. IEEE trans. A.C. vol. AC-22 Feb. 1977.

[6J S.Y. Kung, B. Levy, M. MOrf, T. Kailath, New Results in 2-D Systems Theory, part II, 2-D State Space Models-Realization and the Notions of Controllability, Observability and

Minimality, Proc. IEEE. Vol. 65, June 1977.

[7J R.P. Roesser, A Discrete State-Space model for Linear Image Processing.

IEEE trans. A.C. vol. AC-20 Jan. 1975.

[8J E.D. Sontag, On First-Order Equations for Multidimensional Filters. To appear in IEEE trans. ASSP. 1978.