An irreducible realization scheme for matrix fractions of multivariable systems

Systems & Control North-Holland

Letters

5 (1985)

383-388

May 1985

An irreducible realization scheme for matrix fractions of multivariable systems Shou-Yuan Depi.

of Elecmrical

ZHANG Engineering,

SUNY

ar Stony

Brook,

Srony Brook,

NY I 1794, USA

Received 26 November 1984 Revised 11 January 1985 We present an irreducible realization scheme for matrix fractions of multivariable systems under the Kalman module structure interpretation of dynamical linear systems. The realization is computationally and conceptually simple. It also provides an insight into the relation between the state space discription and the matrix fraction description of the linear systems. Some related problems are also discussed. Keyruords:

Irreducible

realization,

Matrix

fraction,

State space description,

Multivariable

systems,

Module

structure.

1. Introduction

Various irreducible realizations are availble at present time for matrix fractions of multivariable linear systems. For example, so-called controller-form and controllability-form realizations, in the dual case, observer-form and observability-form realizations as well as the Nerode realization are presented in details for matrix fractions of linear systems in [l-3]. We notice that Kalman gave a module structure exposition for the dynamic linear systems [4]. Under this interpretation, Fuhrmann developed a realization theory [S]. In this paper, we present and verify a realization scheme for the Kalman-Fuhrmann realization. Let the system G(s) be fractioned as D-‘(s)N(s). The realization introduced in this paper requires no computation under a reasonable assumption of D(s) row reduced with row degree coefficient matrix I. The realization is conceptually and computationally simple. Since all nontrivial elements in the realization come directly from the coefficients of the matrix fraction of the system, the realization provides an insight into the relation between the state space description and the matrix fraction description of the system. The relation can be shown on the resultant matrix of the system and can be used to explicitly explain the well known properties of the resultants [6-81.

2. The irreducible

realization

scheme

In this section, we discuss the Kalman-Fuhrmann realization, then we present the irreducible realization scheme. Let F be a field. F” denotes the vector space of n-tuples in F. F[s] is the ring of polynomials over the field F and F”[s] is the set of polynomials with coefficients in F”. Let F”(s) denote the rational functions with coefficients in F”, we define the map 17 such that IIF” is the strictly proper part of F”(s). For a given p x p nonsingular polynomial matrix D(s), K. is defined to be a vector space of elements f in Fp[s] such that D-If is strictly proper. We also define a map IT, for the polynomials by IT,e=Dfl(D-‘e),

eEFP[s],

i.e., for any e E P[s], 0167-6911/85/$3.30

II,e

8 1985, Elsevier

is the part of such that I7,e belongs to K,. Science Publishers

B.V. (North-Holland)

383

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5. Number

Consider

6

a

p x q

SYSTEMS

strictly

proper

& CONTROL

system with left coprime

LEPERS

B: A:

c:

realization

Fq-+K,; K,+K,; KD+FP;

1985

fraction

G = D-IN. Fuhrmann’s

May

(1)

{A,

B, C} is

UWNU, f-) n,(d),

(2)

f-(D-‘fh

where for H(s) = ZyDis-’ we define (H), = H,. It is shown that K, as a module over F[s] [5]. Since every module over F[s] is also a vector space over F, we can find a basis for the vector space of K,, and furthermore, express the basis in a constant vector form. Then, the basis is used in the realization. Consider the system shown in (1). Without loss of generality, we assume that D is row reduced with row degrees m,, i = 1, 2,. , p. Thus, the i-th row degree of f in FP[s] should be less than mi in order to ensure the strict properness of D-‘f, i.e., the degree of the i-th element of the vectors in K, should be less than mi. Now, we choose a natural basis for K, as follows:

To introduce D(s)

the irreducible = diag(s”‘0)

realization

+diag(s”‘*-‘}D,

N(s)=diag{s”o-‘}N,+diag{s”‘,-*}N,+

scheme for (2), we write + . . . +diag{s”‘,-“‘}D,,,,

(4

...

(5)

+diag{s”‘,-“‘)A!,,,

where m = Max mi is the observability index of G and the row degree coefficient matrix of D(s) is assumed to be I. Since mi < m for all i, there may exist some Di and Nj rows which are corresponding to the negative s-powers. We note that these rows are always zero and will be eliminated in the last step of the realization. Hence, we have expanded D and N by Dj, j = 1, 2,. . . , m, according to the basis (3). Thus, the realization {A, B, C) is presented as follows:

(6)

In the following, we verify the realization. The matrix B can degrees of N are less than the corresponding row degrees of D, we choose B according to the basis shown in (5). To find the K,, we define the shifting operator A. Consider the following sf=Dg+lI,sf, 384

be formed directly from (2). Since all row for every u in Fq, Nu belongs to K,. Thus, remainder l7,sf from sj, where f is from division: (7)

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1985

where g is a polynomial vector. If we denote the matrix representations of D and N to be D and N, which are formed in the same way as the last column block of A and B in (6) and so forth for the vectors in K,, let f’ = sf, then the division algorithm (7) can be accomplished by elementary column operations on the matrix [BJ,] to eliminate f[ of J’:

1 1 Z 0

-fi

=

1

Now, the following equation is evident: Af=n,sf.

Thus, the shifting function of A is verified. To show the validity fo C, we note lim sH=

s-m

lim

3’00 [

1

HI + g FZ,+~S-~ =H,. i=l

We choose the proper basis for f, then we have (D-‘f

)t = lim sD-‘f s+m

p-1

... = lim SD-1 s--m

s m,-1 -

‘.. ... I .. .

= lim D-’ s-+00

... .._ I

I..

.**

0 I I

.

J I

1i

. . .

= jm%D-‘diag{s”‘~}[ = [O

p,-1

0 I I d I

0

. .. ~s-‘I~/z~] j

0 I]J

where since D is row reduced with row degrees m,,

we

have

lim D-‘diag{ s’“a} = I. s-t* Thus, it follows that C=[O

-*.

0 I].

Finally, the A and B rows corresponding to the negative s-power bases as well as the same number columns in A and C should be deleted. We use the following example to illustrate the procedure. 385

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Example.

SYSTEMS

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May 1985

Let a strictly proper system G be coprime fractioned as s3+1

G=D-‘NC

[

s2

2-s

0

s-l

2

s

S-l

s2+s+l ,I

-’

1

s2+2s+1

4 s+2 [

s-t3

1

-1

.

We note that D and N are left coprime, D is row reduced with row degree coefficient matrix I. Following (6), we write A, B, and C as

A=

0 Q p 0 Q 0 -1 0 @- - f)--()-+-fj---@---&--* 8- I -(j-f)-+*--+-*---(j 1Qpobo 0 0 -(+-+J---fJ--+j u-*--s--c0 1 0 1 O t 0 1 0 : 0 -1 0 1 8 ii 8 0 : 3 ; -1 -1

8

0’ 1 -1 -1 -2 -1

-l-s -2 - s-‘-S-l I;- s2-l-s-

1 *--f

B=

6---00 2 Q---6-, 2 3

0 4

1 -1

-1

l-

C=

Hence, the irreducible realization is 000-l 100 A=OOO 0 1 000 I0 0

0 1

0 0 0 0 -1

0 0 0 1 l-l -1 -1 l-2 -1 -1

[ 000100 0000001 0 0

0

10.

1

3. Some remarks

In this section, we give some remarks on the relations between the realization presented in this paper and the Nerode realization and the observer-form realization [2]. We also discuss the relation between the state space description and the matrix fraction description of the systems by the realization of this paper. The shifting operator A plays a key role in the realization (6). Once a basis is chosen for Kc, say (3), we expand D and other matrices and vectors in the way of (4) and (5). Thus, the polynomial matrix problem is transformed to the constant matrix forms and the shifting operator A can be established by the coefficients of D. The shifting operator A for the basis (3) is shown in (9). We note that the Nerode realization [2] employs the same shifting property in defining A. It is of interest to note that although the Nerode realization is for the right fraction G = ND-‘, the row reducedness of D is still required. The technique shown in this paper can be used in the Nerode realization to find A. Since the nontrivial elements in the realization (6) come directly from the coefficients of the matrix fraction of the system, one can expect to have more insight into the relation between the state space description and the matrix fraction description of the system. We show this by a study of the resultant of the system. The resultant is formed by the coefficients of the matrix fraction of the system, say of (l), as 386

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follows:

Dn, 4,-,

4, K-1

S” =

1st block

. ..

nth

block

where n is the degree of the system. It is well known [6-81 that by searching the linear dependency of the columns of S,,, the controllability indices of the system can be found and the coprimeness of D and N can be identified. The controllability indices are defined from the controllability matrix [B AB . . * A”-iSI of the state space description (A, B, C} and the coprimeness of D and N is implied by the existence of a controllable and observable irreducible realization. In the following, we show that the realization (6) explicitly reveals the property of S, stated above. Note that the first N-block in S, is the matrix B in the realization (6). The second N-block can be taken as a one step shift of the first N-block _ Recall the equations (8) and (9), the remainder of the second N-block by elementary column operations to eliminate N, is equal to AB, and so forth for all other N-blocks. Thus, by elementary column operations and a column rearrangement, the resultant can be written as 4 4-1

0 4,

I I I : B

s,,u=

d, D2 . D,,, -----------------c-----------------Z 4 Z . Dlj 0 I

j I I I I

AB

...

A”-‘B

0

I

Thus, the row space of S,, can be partioned into two parts: one is an upper triangular matrix with the diagonal elements 1, which is always of full rank, the other is the controllability matrix [B AB - * * A”-‘B]. Hence, the results of [6-81 are self-explanatory. It is noted that after some interchange of rows of A and B and the corresponding columns of A and C, the observer-form realization of [2] can be identical with the realization introduced in this paper. In fact, if we choose the basis accordingly, the observer-form realization can be interpreted and verified by the Kalman theory, i.e., A acts as a shifting operator, B transforms the input vectors to the given module and C transforms the output of the operator A to the output vector. However, in this case, no neat forms of A, B, C like (6) can be found. We note that in (6), the matrices A, B, C are formed in a quasi-canonical form. It is not surprising that the realization presented in this paper has all properties of observer-form realization discussed in details in [2], but some properties possessedby this realization, for instance, those used to interpret the resultants, do not belong to the observer-form realization explicitly. 387

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References [l] [2] [3] [4] [S] [6] [7]

W.A. Wolovich, Linear Multioarioble Systen~s (Springer. New York, 1974). T. Kailath. Lb~eur Qsrents (Prentice-Hall, Englewood Cliffs, NJ, 1980). C.T. Chen, Linear SJcrem Theory and Design (Holt, Reinhart and Winston, New York, 1984). R.E. Kalman, P.L. Falb and M.A. Arbib, Topics irl Motltemaricul S)vlem Tlteq~ (McGraw-Hill. New York, 1969). P.A. Fuhrmarm, Algebraic system theory: An analyst’s poinl of view, J. Franklin Inst. 301 (1976) 521-540. A. Rowe, The generalized resultant matrix, J. Insf. Marh. Appl. 9 (1972) 390-396. S. Kung, T. Kailath and M. Morf, A generalized resultanl matrix for polynomial matrices, Proc. IEEE Con/ Decision mld Conrrol (1976) 892-895. [8] C.T. Chen, A contribution to the design of linear time-invariant multivariable systems, Proc. Amoicmr Conrrol Con/: (1982).