Pergamon 0005-1098(93)E0031-X Autornatica, Vol. 30. No. 9, pp. 1389-1397, 1994

Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0005-1098/94 $7.0(} + 0.00

The Design of Practically Stable nD Feedback Systems*

LI XU,t OSAMI SAITOt and KENICHI ABE~

A method for nD feedback "practical stabilization" is proposed which consists of 1D steps.

Key Words--Practical BIBO stability; multidimensional systems; feedback; stabilization; Bezout equation.

Abstract--This paper deals with, by using the matrix fractional description (MFD) approach, the problem of feedback practical stabilization of nD (multidimensional) discrete systems whose input and output signals are unbounded in, at most, one dimension. A constructive algorithm is first presented for solving the Bezout equation over the ring of practically stable rational functions. Then, a necessary and sufficient condition for an nD system to be practically stabilizable is derived and the parametrization of all nD practically stabilizing compensators is given. These results make it clear that the nD practical stabilization problem can be essentially solved by using 1D approaches.

1. INTRODUCTION

IN MANY practical situations of nD signal processing, such as seismic and image processing, the independent variables i~ . . . . . i, of an nD signal x( i l , . . . , in) are usually bounded spatial variables, except that perhaps one is the unbounded temporal variable. Taking this feature into account, Agathoklis and Bruton (1983) developed the concept of practical BIBO (bounded-input bounded-output) stability for nD discrete systems, and showed that the conventional BIBO stability conditions are too restrictive for many applications.

For designing practical BIBO stable nD digital filters, some results have been documented in the literature (see, for example, Reddy et al.,

*Received 12 October 1992; revised 19 April 1993; received in final form 5 November 1993. The original version of this paper was presented at the IFAC Workshop on System Structure and Control which was held in Prague, Czech Republic, during September 1992. The Published Proceedings of this IFAC Meeting may be ordered from Elsevier Science Limited, The Boulevard, Langford Lane, Kidlington, Oxford OX5 IGB, U.K. This paper was recommended for publication in revised form by Associate Editor Vladimir Ku~era under the direction of Editor Huibert Kwakernaak. Corresponding author Dr Li Xu. Fax: +81 532 47 5301; E-mail: xuli@mizar.tutkie.tut.ac.jp.

t Department of Knowledge-based Information Engineer- ing, Toyohashi University of Technology, Tempaku-cho, Toyohashi, 441 Japan.

Facutly of Engineering, Tohoku University, Sendai, 980 Japan.

1990). However, for feedback stabilization of nD systems in the practical sense of Agathoklis and Bruton (1983), a general effective method is not yet available. Since practical BIBO stability conditions are much weaker than the conven- tional ones, there exist systems that are practical BIBO stable but not conventional BIBO stable (Agathoklis and Bruton, 1983). This fact means that, to design such practical BIBO stable feedback systems, the existing methods de- veloped under the conventional BIBO stability concept cannot be applied. This is because the current design methods of an nD (n = 2) system (see, for example, Guiver and Bose, 1985) are mainly based on the idea of making the closed-loop system (denominator) separable (see, for example, Kaczorek, 1985) by using local state or output feedback, and obviously the separable structure precludes the practical BIBO stable situation mentioned above.

The objective of this paper is to consider the feedback stabilization problem of nD systems in the practical sense (Agathoklis and Bruton, 1983), that is, under the assumption that the input and output signals are unbounded in, at most, one dimension. From now on, an MIMO (multi-input multi-output) nD feedback system is said to be practically stable if every entry of its closed-loop transfer matrix corresponds to an SISO (single-input single-output) practical BIBO stable system. By practical stabilization, then, we mean the process to make a given system practically stable by a feedback scheme. In particular, this paper will solve the nD feedback practical stabilization problem by using a matrix fractional description (MFD) algebraic ap- proach. First, we briefly review the concept of practical BIBO stability for nD discrete systems and the necessary and sufficient conditions shown by Agathoklis and Bruton (1983). Then, a constructive algorithm is presented for solving

1389

1390 LI Xu et al.

the Bezout equation over the ring of practically stable rational functions. Based on these results, a necessary and sufficient condition for feedback practical stabilizability of nD systems is derived and the parametrization of all nD practically stabilizing compensators is given. Finally, ill- ustrative examples will be shown.

Some notations to be used are defined here. R: the field of real numbers; C: the field of complex numbers; R[z~ . . . . , z,]: commutative ring of nD polynomials in z~ . . . . . zn with coefficients in R; AT: transpose of matrix A; O = { v • C I lvl-< 1};

t-? n = {(za . . . . . zn) e C ~ Izkl--- 1, k = 1 . . . . . n}; U n = { ( z ~ , . . . , z . ) E C " I Z k [ < l , k = l . . . . . n}; r " = {(z, . . . . . z,,) • C" Izkl = 1, k : 1 . . . . . n}; Z~_ = {(i, . . . . . i,) l i, . . . . . i, e Z+, Z+: the set

of positive integers}; Z+ n = { ( i , , . . . , in) l i , , . . . , in • Z + , but not

more than one of them can be infinite simultaneously}.

2. P R A C T I C A L B I B O S T A B I L I T Y F O R n D D I S C R E T E S Y S T E M S

Consider the class of (SISO) linear shift invariant nD discrete systems characterized by the nD convolution sum:

i I in

y ( i , . . . . . in) = Z " ' " Z h(i~ - k , , . . . , in - k n ) k I =0 kn=O

)< U ( k l , • • • , kn), ( 1 )

where u ( i ~ , . . . , in), y ( i ~ , . . . , in) and h ( i ~ , . . . , in) are the input, output and impulse response, respectively. By using an nD z- transform we can obtain the transfer function of nD system (1):

n ( z , . . . . , z . ) H ( z , , . . . , Zn) d (Z l . . . . . z , ) " (2)

The conventional BIBO stability is defined as follows.

D e f i n i t i o n 1 (Huang, 1972; Jury, 1978). An nD system is BIBO stable if and only if, for all input signals u ( i j . . . . . i , ) such that

l u ( i , , . . . , i , )I-<M < ~ V(il . . . . , in) • Z+, (3)

where M is a finite real number, there exists a finite real number L such that, for the output y ( i ~ , . . . , i~) of the system, the relation

l y ( i l . . . . . in)l-< L < ~ (4)

holds.

On the other hand, the practical BIBO stability introduced by Agathoklis and Bruton (1983) is defined as follows.

D e f i n i t i o n 2 (Agathoklis and Bruton, 1983). An nD system is practical BIBO stable if and only if, for all input signals u(i~ . . . . . in) such that

lu( i l . . . . . i,)l <- M < ~ V(il . . . . . in) • Z~ / ' , (5)

where M is a finite real number, there exists a finite real number L such that, for the output of the system y ( i l , . . . , i,,), the relation

[y(i~ . . . . . i n ) l < - L < ~ (6)

is satisfied.

The difference between the two definitions is that, for the case of practical BIBO stability, the behaviour of the system at the points where more than one of the indeterminates take an infinite value are not considered (see Agathoklis and Bruton, 1983).

A well-known necessary and sufficient condi- tion for conventional BIBO stability is that the system's impulse response satisfies the following relation (Jury, 1978):

" ' " ~ Ih ( i l , i2 . . . . , in)l < c~. (7) /1=0 i2--0 in=()

In contrast, Agathoklis and Bruton (1983) have shown the following theorem, which reveals the relationship between the practical BIBO stability and the impulse response of an nD system.

T h e o r e m 1 (Agathoklis and Bruton, 1983). An nD discrete system is practical BIBO stable if and only if the n inequalities are satisfied:

NI N2 N k = ~

E E . - . E i l=O i2--0 ik--O

N.

• "" ~ Ih ( i , , i2 . . . . . i k , . . . , in )L< ~ , in = 0

k = 1 , 2 . . . . ,n , (8)

where N1, Ne . . . . , Nk ~, Nk+l . . . . . N, are any finite integers.

Further, the following theorem relates the practical BIBO stability to the singularities of an nD transfer function.

T h e o r e m 2 (Agathoklis and Bruton, 1983). nD system (2) is practical BIBO stable if and only if

d(0 . . . . . z k , . . . , 0 ) ~ 0 V z ~ e U,

k = l , 2 , . . . , n . (9)

As is well known, however, when an nD system given by transfer function (2) has no nonessential singularity of the second kind (Bose, 1982) on T 2 when n = 2 (Goodman, 1977), or on U n - U n when n > 2 (Swamy e t al. ,

Practically stable nD feedback systems 1391

1985), the system is BIBO stable in the conventional sense if and only if

d ( z , , . . . , z ~ ) ~ O V ( Z l , . . . , z ~ ) • ~r~. (10)

It is then obvious that the condition (9) for practical BIBO stability is in fact equivalent to the stabilities of n 1D systems, and this is much weaker than the condition (10) for conventional BIBO stability.

3. SOLUTION FOR BEZOUT EQUATION OVER THE RING OF PRACTICALLY STABLE RATIONAL

FUNCTIONS

In what follows, we call an nD rational function practically stable if its denominator satisfies the condition (9). Let G be the ring of nD causal rational function and H be the ring of nD practically stable rational function, that is,

G = {n/d [ n, d • R[z, . . . . , Zn], d(O . . . . . O) ~ 0},

H = { n / d • G I d(O . . . . . z , . . . . . o )

Yzk • U, k = 1, 2 . . . . , n},

and let

l = { h • H [ h - 1 • G},

J={h • H l h -t•H}.

Denote by M(*) the set of matrices with entries in set * (e.g. G, It). An element of M(H) is then said to be G-unimodular (respectively H- unimodular) if and only if it is square and its determinant belongs to | (J). If P • M(G) can be written as P = NpDp 1, where Dp, Np • M(H) and Dp is G-unimodular, we refer to such NpDp ~ as a right MFD of P (on {G, H, I, J}).

Definition 3. For right MFD NpDpl, we say that Np and Dp are right coprime on H and that NpDp I is a right coprime MFD on H if and only if there exist U, V • M(H) such that the Bezout equation

UDp + VNp = I (11)

holds.

The dual definitions on the left are given analogously. It is easy to see that for any P • M ( G ) , we can always find Np, D p • M(R[zl . . . . , z~]) c M(H) with det Dp • I such that P = N p D p 1, but Np and Dp are not in general right coprime on H. As a main result of this paper, the following theorem shows a necessary and sufficient condition for the existence of right coprime MFD of P on H. Suppose, without loss of generality, Np, Dp • M ( R [ z l , . . . , z~]) and det Dp • I. Let ~¢k denote the ideal generated by the all maximal order

minors of the matrix

Dp(0 . . . . ,Zk . . . . . 0)] (12) Np(0,. , z , , ,0)

and °//'(5~k) be the algebraic variety of 5~k, that is, the set of common zeros of the minors, where k = l , 2 . . . . ,n.

Theorem 3. For NpD; 1 where Np, D p • M(R[Zl . . . . , z,]) and det Dp • I, Dp and Np are right coprime on H if and only if

°V(~Ck) N /] = O, k = l , 2 . . . . . n. (13)

Proof. The solvability of (11) is obviously equivalent to the equation

XDp + YNp = (I), (14)

where X, Y, (I) E M(R[Zl . . . . , z,]) and det • • J. Then the necessity can be easily shown by

involving the Cauchy-Binet Theorem. Namely, if z ° • °7(~k), Z ° is also a zero of

det (I)(0 . . . . . zk . . . . . 0)

= det {[X(0 . . . . . Zk, . . . , O) Y(O . . . . , Zk , . . . , 0)]

IDa(0 . . . . . (15) X LNp(0 . . . . . zk . . . . 0)JJ"

It is then trivial to see that det (I) e .I, that is, (14) can never be satisfied if z ° • /2 .

The sufficiency is shown as follows. If the condition (13) is satisfied, the following 1D polynomial matrix equations can be solved by well-known 1D methods (Ku~ra , 1979):

YC (z,)D (O . . . . . z , . . . . . O)

+ Yk(Zk)Np(O . . . . , Z* . . . . ,0) = (bk(Zk), (16)

where X~(Zk), ~'k(Zk) • M(R[Zk]), and

d e t ~ k ( Z k ) # 0 ' C Z k • / ] , k = l , 2 . . . . . n. (17)

The general solution for (16) is given in the form

' ? , ( z , ) = . i ' , ( z , ) + (18)

L ( z , ) = - R , ( z , ) £ ) , ( z , ) .

where Rk(Zk) • M(R[zk]) is arbitrary, /)k(Zk), filk(Zk) satisfy the following relation and are left coprime on R[Zk]:

D ;'(Zk)filk(Zk) = Np(O . . . . . Zk, . . . , O)

× Dp'(O . . . . . Zk . . . . . 0). (19)

Suppose Ek(z , ) is the right greatest common factor of Dp(O . . . . . zk . . . . . 0) and Np(0 . . . . . Zk . . . . . 0), then it follows that

det Dp(O . . . . . zk . . . . . O)

= det Dk(Zk) det Ek(Zk). (20)

1392 LI Xu et al.

Since det Dp e I, that is, det Dp(0 . . . . . O) ~ O, we see that

det/)k(O) # O. (21)

So we can set:

Rk(Zk) = ~'k(0)D,'(0) =~Rk • M(R). (22)

Substituting Rk into (18) and using (16) and (17), we see that

det Xk(0) = det {~k(0)D~-'(0,.. •, 0)} # 0, (23)

l?k(O) = O. (24)

On the other hand, we can write Xk(Zk) as the s u m

2k(Zk) =_a 2,(Z~) + Xk(O), (25)

where Xk(O) corresponds to of S(Zk) and 2k(Zk) denotes which involve the variable zk.

the constant terms all the other terms Obviously,

Xk(0) =0. (26)

According to (23), S k l ( 0 ) exists. Substituting Xk(Zk), f'k(Zk) into (16) and then premultiplying by k~a(0) yields the result

z , . . . . . O)

z , . . . . . 0)

where

and

(27)

Xk(z~) = 2k-l(0)2k(Z~) = 2 k ~(O)2k(Zk) + I A t = Xk(z ) + I, (28)

Yk(Zk) ---- 2k~(0) f'k(Zk), (29)

~')k (Zk) ~- S k 1 (0)(~)k (Zk) (30)

X;(O) = O, (31)

Yk(O) = O, (32)

det ~k(Z*) # 0 VZk • /J. (33)

By the above results, then, the solution to (14) can be constructed as follows:

X ( z , . . . . . z , ) = Xk(zk) + I, (34) k=l

Y ( z , . . . . . z , ) = (351 k=l

• (Zl,. • •, z~) is obtained by the computation

'l'(Zl . . . . . z , ) = k zk + zl . . . . . z , '~k=l

-~- Yk(zk Zl . . . . , Z,). (36)

In view of (27)-(35), it follows the relation

a~(0 , . . . , zk . . . . . 0)

= { ~ X; (0 )+ X'k(Zk)+ 1} j=l j~k

X Dp(O . . . . . Zk . . . . . O)

+ { ~ l Yj(O) + Yk(Zk)}Np(O, . . . , . . . . . O)

j~k

= Xk(zk)Op(O . . . . , Zk . . . . . O)

+ Yk(zk)Np(O . . . . . Zk . . . . . O)

= ~,(Zk), (37)

which shows that

det ~(0 . . . . , Zk . . . . ,0) = det ~k(Zk) # 0 (38)

VZ~• / ) , k = l . . . . . n,

that is, det qb • J. Therefore, the solution to (11) directly reads as

u = (39)

V = ~ - l y . []

Using the right and left coprime MFDs on It , a doubly coprime MFD relation on I t can also be given.

Theorem 4. Suppose P • M(G), and let NpD~, ', /);1Np be any right and left coprime MFD of P on H, respectively. Then there exist U, V, (/, ~" • M(H) such that

r_'-'_ vlro, , I Np

Proof. The proof is similar to the 1D case (Desoer et al., 1980; Vidyasagar, 1985). []

4. O U T P U T F E E D B A C K P R A C T I C A L S T A B I L I Z A T I O N OF n D SYSTEMS

In this section, based on the results of the previous section, we show that the feedback stabilization problem of nD systems in the practical sense of Agathoklis and Bruton (1983) can be essentially solved by using 1D methods.

Consider the MIMO nD feedback system shown in Fig. 1, where NpD~ 1, with Np, Dp E M ( R [ z l , . . . , z,]), is a right coprime MFD on I t for the plant P E M(G), and D[~[Ncl No2], with Dc, Ncl, Nee • M(It), is a left coprime MFD on H for the controller C • M(G). Let

Y= Y2 u3

Practically stable nD feedback systems 1393

7 . . . . . . . . . . . . . . . . . . . . . . ~ F - ° - : F-----"~ r-'------~ I l+ , r---------a

"' ','1 ",, I-%'t o" ,>; h

, 12' I . . . . . . . . . . . . . . . . . . . . . . . , t . . . . . . . . . . . . .

F l o . 1. n D f e e d b a c k c o n t r o l s y s t e m s .

Then, as in Vidyasagar (1985), we can have

y = Oyuu ,

where

[ D.A-'Nc, - I + DvA-'D< Hy,, = LNpA-'N<, NpA-'D<

(42)

-Z ,,a-'Ucq - N p A - 1 N c 2 J

(43) and

A = Nc2Np + OcOp. (44)

If det A # 0 and Hy, • M(H), we say that the nD feedback system of Fig. 1 is practically stable. Further, if there exists C e M(G) such that Hy, ~ M(H), we say that P is practically stabilizable and that C is a practically stabilizing compensator for P.

Lemma 1. The nD feedback system of Fig. 1 is practically stable if and only if A is H- unimodular.

Proof The sufficiency is obvious. The necessity can be shown in the same way as Vidyasagar et al. (1982) and Vidyasagar (1985) under the conditions that Dp and Np are right coprime, Dc and [N~I N~2] are left coprime on H. []

Theorem 5. nD plant P = N p D p 1 e M(G) is practically stabilizable if and only if Dp and Np are right coprime on H.

Proof Suppose Dp and Np are right coprime on H. Then, as shown in the sufficiency proof of Theorem 3, the solution U, V e M ( H ) to equation (11) can be found. In view of equations (39), (34), (31) and (38), we see that

det U ( 0 , . . . , 0) # 0. (45)

Hence, C = U-~[Nc~ V] • M ( G ) for any N~I M(H). By the fact that I is H-unimodular and in view of Lemma 1, the sufficiency is established.

Conversely, if C = X - ~ Y = X-~[YI Y2] E M(G) is a practically stabilizing compensator of P, then by Lemma 1 we have

XOp + Y2Np = ¢}, (46)

where (I) is H-unimodular. Pre-multiplying equation (46) by (I)-~ gives

(dp-IX)Dp -b (dP-'Yz)N p = I, (47)

which shows that Dp and Np are right coprime on H. []

The following theorem gives the parametriza- tion of all nD practically stabilizing compen- sators.

Theorem 6. Suppose that NpDp I and/)TI.K/p are, respectively, any right and left coprime MFD on H for a given plant P e M(G), and that U, V E M(H) satisfy UDp + VNp = L Then the set of all practically stabilizing compensators of P is given by

C e {(U + SNp)-'[Q V - SDp] I

Q, S re(n), det (U + SAp) e !} (48)

and the set of all possible practically stable transfer matrices is in the form

OpQ Dp(U + SAp) - I -On(V - SDp)] (49)

NpQ Np(U + SAp) -Np(V SDp)_I" I

Proof The sufficiency is obvious. The necessity can be shown in the same way as Guiver and Bose (1985) and Desoer et al. (1980) by using the results of Lemma 1 and Theorem 4. []

5. E X A M P L E S

The design procedure based on the results of the previous sections and some properties of practically stable 2D systems are illustrated by examples.

Example 1 Consider a 2D MIMO system given by the

zL 2zl - 1 1 p(zl, z2)=122~i-5 6z, - 8z2- 7

5 z2 "

I_ 2z~ - 1 2Zl - 1

The right factor coprime MFD of P(z~, z2) over R[z~, z2] can be foun d by the algorithm of Mort et al. (1977) or Guiver and Bose (1982) as:

P(Zl, z2) = N.(z, . z2)O;'(z, , z2),

where

Op(z,, z2) [~(2z, - 1)(2z, - 5) - l ( 2 z 2 - 1)(2z, - 5)z2],, / 0 2(6z~ + 8ze - 7) J

NAz,, z2) = [~(2Zl - 1)z2 ½(8zl - 2z 3 + Z~ - 4) ]

/ ~ ( 2 Z l - 5 ) 2 -½(4Z~Z2 2Z, - 18Z2-3)z2J"

By making use of the results of Xu et al.

transfer function matrix

1394 L~ Xu et al.

(1991), it is easy to know that (z~,z2) = (½, ½)e /~2 is a common zero of all the 2 ×2 minors of the matrix T ID,,(Z,, Z2) N)I,(Zl, Z2)] T. Therefore, P is unstabilizable in the sense of conventional stability (see, for example. Guiver and Bose, 1985: Xu et al., 1991).

However, the right greatest common factor of Dp(z,, 0) and Np(z~, 0) is

G(z,) = L 0

and it is obvious that 7/'(~) ~ U = {~} ~ / J = ~. In addition, we can easily verify that the matrices Dp(O, z2) and Np(O, z:) are right coprime over R[z2], that is, ~(~2) = ~. In view of Theorems 5 and 3, P is practically stabilizable.

The practically stabilizing compensator for P can be constructed by the following procedure. First, we find the solutions to equation (16):

(2z'-5)0 7] 3(2z2- z~ + 4) 0

X2(z2) : z2 0 ' lO

~,2(Z2)= [4Z2(2z~-l) ~], ~)2(Z2)= [ ; --2 7].

where, obviously, det ~,(z) = (2z - 5) /2~0, det ~P2(z) = 1 ~ 0, Vz E/J.

Then/)~, N, and/)2, /V2 that satisfy equation (19) can be obtained:

D,(z,) : [ 3(6z'0- 7)

I 0 R, ( z , ) : -~(2z, - 5)

[ - (8z2 - 7) z):(z2) : L(8z~ - 7)z:

~:(z2) : [ , ~(8z23 - 7)z~2+ -~(8z2-7z2 25)

0 -~(2z, - 1)]'

3(2z,- 1) 1 0

_7], 'o].

Since det/)~(0) : - ~ ~ 0, det/)2(0) = -7 ~ 0, we can set Rk, k = 1,2, in equation (22) as

R I = Y l ( 0 ) ] ) l l ( 0 ) : I O - ~ ] ,

°

Substitution of these results in equation (18)

gives

2~(z~) = [4(ZJo 2) 0 _ ~(3z , + 2)['

[ 0 -4z , ] ? , ( z , ) : ~z, 0 '

[4(2Z3- Z~ + 4) 01 ] S 2 ( z 2 ) : k --3~Z2 i-4 '

[ 4z2(2z2 - 1) 72(z2) : L _4z~ ~ ]

The facts that det.,k'~(0)= 4~0, detX2(0)= 8 ~ 0 imply the existence of X~-~(0) and

)~'~(0). Hence, equation (27) admits the solution:

½(3z~ + 2) ' I° zl 2 ,

r , ( z , ) : -~z, 0 '

[ -~ (2z , - 5) o] Ol(Z0 = L 0 -14 '

X2(Z2)= [ ~(2z3-z~+4) ~] 8 2 _~Z2

[ ~z2(2z2 - 1) V2(z2): k 8z2 ~]'

O2(z2)=[~L0 -140]"

By the results of XI(O)=I, X2(O)=I equation (28), we get

and

[ - ! z ] t 2 1 0 x l ( z , ) = X , ( z , ) - z : o ~ z , '

, [~(2z2-1)z~ ~] X2(z2) = X2(z2) - t = 8 2

L sz2

with X;(0) = 0, X~(0) = 0. Now, by equations (34) and (35), the solution

X(zl , z2), Y(zl, z2) to equation (14) is con- structed as

X ( z , , z2) = X( ( z , ) + X~(z2) + I

[ - ¼ ( 2 Z l - 2 Z 3 + Z 2 - 4 ) 0 ] = 2

~z282 ~(3z, + 2) ' Y(Zl, Z2) = YI(Zl) + Y2(Z2)

= [ ~ z 2 ( 2 z 2 - l ) ½ o ' ] -1(9zi - 16z2)

Moreover, in light of equation (36), ~(z~, z2) is computed as follows:

[~, ,(z, , z:) o,2(z,, z2)] O(z~, z2) = t~21(Z,, Z2) O22(Zl, Z2) '

Practically stable nD feedback systems 1395

where

• ,,(z~, z~) 3 2 2 2

4ZEZ 1 -- 2Z3Z l -- 2 Z 2 Z 1 + Z~Zl -- 4Zl + 10

32

z 2 - 40z2 + 8) ztz2(4z 4 - 4z 3 + 2 ( 1 ) 1 2 ( Z l , Z 2 ) = - - 4 '

z~ z2(2Zl - 1)(32z2 - 45) ~zl(Zl, z2) = 160

~2(z,, z2) z~ z2(64z~ - 122z~ + 45z2 - 1120) + 280

20

and it is trivial to verify that ~(Zl, 0) -- qbl(Z~) and cI,(0, z2) = q'2(zz).

Finally, we can find U(z~, z2), V(z~, z2) by equation (39), and construct the class of all practically stabilizing compensators of P according to Theorem 6.

Example 2 Consider continuously the plant and the

designed closed feedback system in Example 1. If we set Q, the free parameter in the closed-loop transfer matrix of equation (49), as

Q(zt , z2) = q(zl , z2)-~I e M(I-I), (50)

where

q(zl , z2) = 3 + Zl + z2 - 1.08z~z2,

we have the transfer matrix from the input Ul to the output Y2

[h. hlq Hy~,,, = N p Q = q - a N p a__ th21 h2zl"

Since for brevity we will only show the simulation result for h12, we just give h~2 explicitly here, that is,

8 z l - 2 z 3 + z 2 - 4 h~(Zl, z2) =

2(3 + zl + z2 - 1.08z~z2)"

By the method, for example, of Huang (1972), one can confirm that q(zl, z2) has zero in 02, that is, it is not BIBO stable. By Theorem 2, however, it is easy to verify that q(z~, z2) is practical BIBO stable.

Figure 2 shows the simulation of the impulse response of h~2(zl, z2). From the result of Fig. 2, we can see a character common for practical BIBO stable impulse responses; namely, if oscillation occurs in such a response it must appear approximately along the diagonal of the coordinate plain. In fact, this is quite reasonable because, in view of the definition of practical BIBO stability, any practical BIBO stable (2D) impulse response has to obey the following properties: if i, is bounded by a fixed finite value while i2 is free, then the response must converge to zero as i2 increases sufficiently; the roles of il and i2 can be interchanged, but the response does not necessarily converge to zero when il and i2 approach infinity simultaneously. It should be apparent that these properties are not contradictory to the fact that a practical BIBO stable system may be unstable from the viewpoint of conventional BIBO stability.

Using the same denominator q(z, , z2) but a different numerator

b(z,, z2) = 1 + 2z~ + z~ + 2z2 + z~ + 4ztz2 + 2zlz 2

+ 2z~z2 + z~zl

1.5

1

0.5

o

-0.5 20

o lo

i~i2~ 15 20 0 25

FIG. 2. Impulse response of hl2(Zl, Z2).

1396 LI Xu et al.

0.5

0.4

0.3

0.2

0.i

0

20

15

0

i2 15 20 0 25

FIG. 3. Impulse response of b(zl, z2)/q(zl, Z2)"

we get another system b( z l , Z2)/q(zl, Z2), which has the impulse response shown in Fig. 3. By this result, we confirmed the existence of a practical BIBO stable system with rather satisfactory dynamic property, even though it does not satisfy the conventional BIBO stability. This can also be viewed as an example that proves the conclusion of Goodman (1976): there may exist 2D systems which are BIBO unstable but have a summable impulse response. Further, comparing the results of Figs 2 and 3, it is easy to see that zeros of the transfer function influence the dynamic property of the corresponding system greatly, which implies that to design satisfactory practically stable feedback systems the influence of zeros should also be investigated and considered carefully.

6. CONCLUDING REMARKS

Based on the practical BIBO stability concept introduced by Agathoklis and Bruton (1983), the feedback practical stabilization problem of nD systems has been considered. A constructive algorithm has been proposed for solving the Bezout equation over the ring of practically stable rational functions. Further, we derived a necessary and sufficient condition for an nD system to be practically stabilizable, and parametrized the class of all nD practically stabilizing compensators. Finally, the proposed method and some basic properties of practically stable systems have been illustrated by examples. The obtained results make it clear that the nD practical stabilization problem can be essentially solved by 1D methods.

It is well known that iterative learning control

systems and linear multipass processes can be described by a 2D system model (see, for example, Geng et al., 1990; Kurek and Zaremba, 1993; Boland and Owens, 1980; Rogers and Owens, 1992). A common feature of these systems is obviously that, while the iterations are not subjected to any boundary condition, the iterated passes are usually bounded on a finite discrete-time interval. It is therefore expected that the research results based on practical stability (see also Xu et al., 1992a, b, 1993) might supply a basis for establishing a unified design method for these systems.

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Huang, T. S. (1972). Stability of two-dimensional recursive filters. IEEE Trans. Audio Electroacoust., 20, 158-163.

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