Controllability and observability of 2-D systems

Controllability and observability of 2-D systems

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Eising, R. (1978). Controllability and observability of 2-D systems. (Memorandum COSOR; Vol. 7808). Technische Hogeschool Eindhoven.

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

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

PROBABILITY THEORY J STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 78-08

Controllability and Observability of 2-D Systems by F. Eising Eindhoven, April 1978 The Netherlands

Controllability and Observability of 2-D Systems

by

F. Eising

Abstract

In this paper a necessary and sufficient condition for modal controllability (modal observability) of a 2-D system, as defined in [4J is obtained in terms of controllability (observability) of a system as is derived in [lJ. Further-more it is shown that modal controllability (modal observability) is a

_ 2 _

1. Introduction

1.1.

The state-space model for a 2-D system will be the model as has been proposed by Roesser in [5]

[:+l.k

h/k+d

l .

Y _hk

=

[C . ₁ n m p r where x hk E lR , ahk E lR , uhk E JR , Yhk E lR •

The matrices have appropriate dimensions. We will take the usual D-matrix to be zero.

In [4J some state-space models for 2-D systems are compared and the authors arque in favor of Roessers model. For this model the concept of modal contro-llability (modal observability) is defined in terms of left (right) coprime-ness of two-variable polynomial matrices in the following way.

1.2.

Definition. The system (1.1) is modally controllable iff

are left coprime with respect to C[s,zJ. The system (1.1) is modally observable iff

are right coprime with respect to e[s, For the definition of coprimeness see [4J.

In [3J the following t~eorem is obtained.

1.3. Theorem.

The matrices

are Zeft coprime w.r.t.

~[s,zJ

iff they are left coprime w.r.t.

~(s)[z] and

- 3

-It can easily be seen that in the case of real matrices we may take~(s)[ andm(z)[s] in stead of ~(s}[zJ and ~(z)[sJ respectively. An analogous result can be obtained for modal observability.

In the following necessary and sufficient conditions for modal controllabili-ty (modal observabilicontrollabili-ty) are obtained in terms of rational matrices. Also a very simple sufficient condition will be derived •

. 2. Some definitions and concepts

mrs] denotes the set of polynomials in the variable s with real co~fficients. m[s,z] denotes the set of polynomials in the variables sand z with real

coef-ficients.

mrxp[s,z] denotes the set of r x p matrices with entries in~[s,z]. m(s) denotes the set of real rational functions 1n s.

m(s)rz] denotes the set of polynomials 1n z with coefficients in~(s). mrxp(s) denotes the set of r x p matrices with entries inm(s).

m(s,z) denotes the set of real rational functions on sand z.

mrxp(s,z) denotes the set of r x p matrices with entries in~(s,z).

The elements ofm[s,z] can also be considered as polynomials in z with coef-ficients inm[s], thusm[s,z]

=:m.[

[zJ.

Analogously I lRrxp[ s, .z] == ~[s Jrxp[ zJ •

A polynomial q E: lR[s,z] seen as an element of ~[s][zJ will be notated as q.

Analogously for P and

Ii

where P c ~rxp[s/zJ and

P

IS lR[s]rxp[zJ. Let

T E lRrxp(s,z), T can be written in the form p/q ==

P/q

where P,

P,

q, q are as above. We will also use some of the above sets wherelR is replaced by ~

(the complex numbers) •

2.1. Definition. An element T IS lRrxp(s,z) will be called a 2-D transfer matrix:

T is called a 2-D transfer function if r :::: p = 1.

- - Y'xp

2.2. Definition. T

=

p/q € JR~ (s,z) is called proper if the degree in z of

q(z) is not less than the degree in z of P(z) •

A proper trans fer matrix T is called causal if the degree in s of the coeffi-cient of the highest power in z of

q(z)

is not less than the degree in s of all other coefficients of q(zj and the entries of P(z) •

Remark. If z is replaced by s in definition (2;2) and interchanged also

in

the next,a completely parallel theory can be obtained.

4

-3. The results

In r1J it is shown that Roessers model can be obtained by realizing'the so-called first level realization (A(s),B(s),C(s) ,D(s» of a causal 2-D transfer

nXn . nXp rXn

matrix T(s,z). Here we have .A(s) E lR . (s) ,B{s) E lR (s), C(s) € JR (s), D(s) E lRrxp(s). In fact these matrices are proper rational matrices

themsel-ves. For an interpretation of the first level realization see [1J. The above matrices play the role of the usual A,B,C,D-matrices in 1-0 system theory. Now suppose that T(s,z) is a causal 2-D transfer matrix and that (1.1) is a state space realization. Then we have:

T{s,z)

=

[C 1

and if (ACs) ,B(s) ,C(s),D(s» is a first level realization then we have: T(s,z)

=

C(S)[ZI - A(S)]-1B (s) + D(s) •

A possible first level realization is (see [lJ):

A (s) =: Al + A2 [sl A₄

J-

1 A3 B (5) _{BA + A2} [51 - A

J-

1 _B2 I .I. 4

c

(s) C 1 + C2 [sI - A

J-1 4 A3 D (5)

By interchanging the role of sand Z we can construct another first level realization. A possible first level realization is then:

-

_A

_J-

1 A(z) _{= A4 + A3} [z1 _A2 1 B (z) _{B2 + A3 [ZI - A}

J-

1 1 BI

-

_A

_J-

1 C(z}

₌

C 2 + CI [z1 - 1 A2 D (z) C 1 [z1 - A

J-1 1 Bl

5

-. ~n ~p

Now suppose A(s) Em (s), B(s) Em (s).

3.1. Definition. (A(s) ,8(s» is said to be controllable w.r.t. m(s) iff

n-l

rank [8 (s)

I

A ( s) 8 (s)

I • • • I

A ( s) 8 (5)]

=

n where the rank is considered over the fieldm(s).

Remark. It can easily be seen that the controllability condition is satisfied if the rank condition holds for some complex number s. (see [2J).

We can now state:

~3.2.

Theorem.

The system

(l.l) is

modaZZy

-1 1) (A l + A2[SI - A4] A3, 81 + A2[51

aontroZZabZe

iff

-1 - A4J 8 2) is

aontroUable

w.r.

t.

m(s) and Proof. [ ZI - Al -A 3 -A 2 sl - A4

::]

-

[:

-A2

J [

zI - A(S) 51 - A -[sl - A ]-lA 4 4 3 -1 -1 where A(s) ;:: Al + A

2[SI - A4J A3, 8(5)

=

81 + A2[SI - A4] B2•

o

I

If zI - A(5) and B(s) have a nonunimodular left common factor then i t is clear by the above factorization that also

are not left coprime w.r.t. ~(s)[z] and therefore (1.1) is not modally con-trollable. Failure of 2) gives the same result in an analogous way and there-fore necessity has been proven. 1) together with 2) is also sufficient.

that 1) is true. Then [zI - A(S) jB(s)] is right invertible. Therefore there exist matrices Land Q with entries is m(s) [z] such that:

6

-Now i t is straightforward to verify that

l

ZI - A1 -A

3

This implies left coprimeness of

[

ZI - Ai

-A

3

w.r.t. ~(s)[z]. In the same way 2) implies left coprimeness w.r.t. ~(z)[s]

and therefore modal controllability of (2.1) has been proven.

3.3. Theorem.

If

(A

1,B1) is a controllable pair then

-1 -1

(A(s) ,B(s» = (A

1 + A2[S! - A4] A3, B1 + A2[S! - A4] B2)

is a controllable pair w.r.t. ]R(s).

o

Proof. From the right invertibility of [BlIA1Bll •••

IA~-lB1]

follows the right invertibilityof [B(s) IA(s)B(s) 1 • • • \A(s)n-1 B (s)]. 0

3.4. Theorem. Modal controllability is a generic property.

Proof. Suppose (1.1) is not modally controllable then by (3.2) and (3.3) (A B) or (A B) is not a controllable pair, from which i t follows that

l' 1 4' 2

the set of points where modal controllabilit:;.j fails is contained in a Zaris-(n+m) (n+m+p)

ki closed set in the parameter space ~ •

Remark. By duality all the results of this paper are also valid for modal observability.

For example the observability counterpart of (3.2) becomes:

3.5. The~. The system (1.1) is modaUy obser'JabZe

-iff

o

-1 -1 .

+ C

2[sl - A4] A3, Al + A2[SI - A4~1 A3) 'l-S obser'Oable W.I', t. :IR(s) and

+ C

1[ZI - A11A2, A4 + A3[ZI - A1] A2) is observable w.r.t. :IR(z).

0

Here "observability w.r.t.:IR(s) (lR(z»" is as usual defined as dual to "con-trollability w.r.t.:IR(s) (lR(z»".

7

-4. Conclusions

In this paper an equivalent characterization of modal controllability has been obtained in terms of a so-called first level realization. It has been shown that modal controllability is a generic property. Also a very simple sufficient condition for modal controllability of (1.1) has been obtained, namely U(Al,B

I) and (A4,B2) both are a controllable pair". By duality analo-gous results hold for modal observability.

8

-5. References

[1] F. Eising: "Realization and Stabilization of 2-D Systems". COSOR Memorandum 77-16.

[2J F. Eising: "Realization of NSHP,...Filters". COSOR Memorandum 78-04.

[3 ] M. Morf, B. Levy, S. Y. Kung: "New Results in 2-D Systems Theory, Part

I: 2-D Polynomial Matrices, Factorization and Coprimeness". Proc. IEEE, June 1977.

[4J S.Y. Kung, B, Levy, M. ~1orf, T. Kailath: "New Results in 2-D Systems Theory I Part II: 2-D State-Space ~!odels - Realization and the

no-tions of Controllability, Observability and Minimality". Proc.

IEEE, June 1977.

[5J R.P. Roesser: itA Discrete State-Space nodel for Linear Image Processing".