In this paper, we study the stability and the stabilisation of 2D discrete linear systems with multiple state delays. All of the new results obtained are based on analysis of the Fornasini-Marchesini state space model with delays and the resulting conditions are given in terms of linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness of the overall approach.published_or_final_versio

Gałkowski, K

Lam, J

Owens, DH

Paszke, W

Rogers, E

Xu, S

HKU Scholars Hub

Title Stability and stabilisation of 2D discrete linear systems withmultiple delaysAuthor(s) Paszke, W; Gałkowski, K; Lam, J; Xu, S; Rogers, E; Owens, DHCitation Proceedings - Ieee International Symposium On Circuits AndSystems, 2003, v. 3, p. III670-III673Issued Date 2003URL http://hdl.handle.net/10722/46673Rights Creative Commons: Attribution 3.0 Hong Kong LicenseSTABILITY AND STABILISATION OF 2D DISCRETE LINEAR SYSTEMS WITH MULTIPLE DELAYS Wojciech Paszke', Krzysztof Gatkowski', James Lam2, Shengyuan Xu', Eric Rogers3, D. H.  Owens4 'Institute of Control and Computation Engineering. University of Zielona G6ra, Poland. 'Department of Mechanical Engineering, University of Hong Kong, China. 3Department of Electronics and Computer Science, University of Southampton, UK. 4Department of Automatic Control and Systems Engineering, University of Sheffield, UK. ABSTRACT In this paper, we study the stability and the stabilisation of 2 0  di- screte linear systems with multiple state delays. All of the new results obtained are based on analysis of the Fomasini-Marchesini state space model with delays and the resulting conditions are gi- ven in terms of linear matrix inequalities (LMls). A numerical example is given to illustrate the effectiveness of the overall ap- proach. 1. INTRODUCTION The analysis of time-delay systems is a very important part of (li- near) systems theory and has been a very active research area over the past few decades. The interest in time-delay systems stems from the fact that such delays wcur  often in, for example, electro- nic. mechanical, biological, metallurgical and chemical systems -see, for example, (IO, 111. . m e  existence of delays is frequently a source of instability. Much work has been reported on the problem of the stability of standard, termed I D  here, linear systems with delays [4. 121 but relatively little on the stability of 2 0  (nD) linear systems with delays. In this paper, we develop stability conditions for 2 0  linear systems with multiple state delays and then establish some con- nections between multidimensional delay and delay-free systems. Based on the these results conditions for the existence of stabili- sing controllers are developed. All of these conditions are formu- lated in terms'of linear matrix inequalities (LMIs) [Z. 61. where an advantage of using LMIs is the fact that there exist efficient nume- rical algorithms to solve them as demonstrated by the numerical example which concludes this paper. Throughout this paper, the null matrix and the identity matrix with appropriate dimensions are denoted by 0 and I respectively. M > 0 is used to denote the fact that M is a real symmetric posi- tive definite matrix. Also, delays h t , .  . . ~ h, are termed noncom- mensurate if 3 no integers 11, . . . ,1,  (not all of them zero) such that 1,hi = 0. The underlying delay differential system is ter- P .=I med commensurate if q = 1. We will also make extensive use of the following well known result. Lemma 1 fSchur complement) 121 For mafrices Cl, Ca. and Cp where C, > 0 and CJ = CT then (1) ~3 + C?C;'Ca < o if; and only if; 2. 2 0  LINEAR SYSTEMS WITH MULTIPLE DELAYS Consider a 2 0  linear system with multiple state delays which can be represented by the Fomasini-Marchesini state-space model [5 ]  with delays ~ ( i + l , j + l ) = A i ~ ( i + l , j )  + A a z ( i , j + l )  .* + AtrdZ(i + 1,j  - dtr) k=1 .. where z(i,j) E R", u ( i , j )  E Rm are the state and input vectors respectively, i, j E Z+, where Z+ denotes the set of nonnegative integers, A,, B, ( p  = 1,2) ,  A1kd.Btk.d k = 1,. ..,si. Azd. BZrd 1 = 1 , .  . . , s a  are known constant matrices with compatible dimensions, and SI  andsz denote the number of delay terms in eachdirection respectively. We also assume that0 < dll < dlz < 07803-7761-31031$17.00 02003 IEEE III-670 . . . < di , ,  and 0 < dzl < dzz < . . . < dzs2 and in this case the boundary conditions are defined as (4) { x ( i , j ) = v , j ] , V i t O  j = - d i s l , - d ~ 8 1 + l  ,..., 0 {x(i,j) = w , j } ,  V j  ? 0 i= -dza2 ,  -dzb2 + 1 , .  . . , O  where woo = WOO. WithX, = sup{llz(i,j)ll : i+j = r, i , j  € Z},asymptotic stability of the model (3) is defined as follows. Definition 1 [S, 51 The 2D linear sysrem wirh multiple srare de- lays (3) is said 10 be asymprorically stable if lim X, = 0 for zero input u ( i , j )  = 0 and for any bounded boundary conditions of the form (4). 7-m 2.1. Noncommensurate delays In  the case of noncommensurate delays, the following result cha- racterizes asymptotic stability of the class of systems considered in terms of an LMI condition. Theorem 1 The 2 0  delay sysrem (3) is asymptotically stable if 3 matrices P, Q > 0, U,, , . . . , U,,, > 0 and UZI, . . . , Uza2 > 0 suck that the following MI holds: where k=1 s* Proof: This is via a Lyapunov-Krasovskii approach. In particular, suppose that V ( { ,  E )  denotes a function that expresses the energy stored at x ( i  + {, j + E )  and consider the particular case when V(CIO = x r ( i + C , j + E ) W , , + ( i + C , 3 + E )  (7) where Wct > 0 is given and C ,  E E Z+, C 2 - d l 8 , ,  2 -dza.. Now introduce the following candidate Lyapunov functions for the (8) k=l @=--dit v d , ( C , c )  =.r(i+C,j+E)w(,x(i+C,j + E )  82  - 1  + xT(i+s,j+E)uzlx(i+s,j+S) 1=1 B=-d>z where W<c > 0 and u l k ,  UZI > 0 are given and C2 -dlS, ,  in the both sides of (3) consider the increment A V ( i , j )  where E Z, C 2 2 -dzs,. In order to determine the change of the energy A v ( i , j )  = v ~ , ~ ( ~ ~ ~ ) - ~ ~ ( ~ ~ ~ ) - ~ d ~ ( ~ ~ l )  (9) Now consider the result of substituting (7) and (8) into (9) and define the augmented state vector as i = [ zr ( i+l , j )  z r ( i , j+ l )  z r ( i + l , j - d i i )  z ' ( i+l , j -dlz)  . . .  x r ( i + l , j - d l s , )  z T ( i - d z l , j + l )  x T ( i - d z z , j + l )  . . . z T ( i - d Z d a 2 , j + l )  x? xr ] where x( includes all states from x(i + 1 , j  - 1 ) .  . . x ( i  + 1 , j  - dl,, + 1)  excluding those defined before and XI includes all states from z( i  - 1, j  + 1 ) .  . . x ( i  - dz.. + 1 ,  j + 1)  but also excluding those defined before. Then (9) can be rewritten as (using the same notation as in (6)) ~ v ( i , j ) = & ~ z  = iT(eTwlle - z) i  (10) where e =  [ Ai Az Aid AM 0 0 1 ,  5 =diag(Wm, Wor,Rl, Rz, RB. a,) Now, if AV(i , j )  < 0 for i # 0, then the 2D discrete linear system considered here is asymptotically stable. i n  order to gu- arantee that this stability condition holds, it is clear that n < 0 must hold. Also the last two rows and columns in this matrix (i.e. those which only consist of -R3 and -01) in (10) can be omitted because they only contribute terms that are guaranteed to be nega- live definite. By again making use of (7) and (8) it is easily seen that to guarantee the dissipative property ( A V ( i , j )  < 0) we can choose (11) e = ,  e=, Win  = P-Q-QI I - . .  . - Q i k - Q ~ i - .  . . - Q z I ,  Remark 1 I1 was shown in (91 (see also (31 and [ I 1  ) rhat rhere exist conneciions berween (linear) 2 0  delay-free systems and ID rime-delay sysrems. These arise because the delayed signal in the ID case can be viewed as a signal rmnsmirred through anorher dimension in rhe 2 0  framework. Theorem I here shows that the same result can be esrablishedfor 2 0  linearsysrems wirh multiple delays. Hence. asymproric srabiliry of a 2 0  linear system with ml and mz delayed rems in each direction respectively is equivalent lo asymptotic srabiliry of an mD linear system where m = ml + m z  + 2. 2.2. Commensurate Delays In what follows, we show that if all delays present in (5 )  are com- mensurate then investigation of the stability properties of 2D delay system can be equivalently treated as the stability investigation of a 4 0  delay free system. The key to establishing this fact is the Ele- mentary Operation Algorithm (EOA) developed by Gatkowski [71. In general case, the notation associated with this area is very cumbersome and hence for ease of presentation only we consider the particular case of a 2 0  linear system of the form (3) with two delays in each direction. i.e. we restrict attention to ml = mz = 2. In which case it is clear that the associated characteristic po- lynomial for stability is given by the determinant of the following 2D polynomial matrix I - Alz;' - Azz,! - - As~;P" - Asz;P2' - A4zLhZk (12) where k ,  1 E R+ and hl, hz, p l ,  p z  are natural numbers. Now introduce the new variables zf = 23. z: = ZI and then rewrite (12) as I - A&' -Azz; '  - A3zLh1 - Adz;h2 -A&'' - A ~ Z ; ~ ' .  Assume also that hl = 1, hz = 2, p l  = 1, p z  = 2 which yields I - AI;;'  - A z z ; ~  - A3zY1 -A&' (13) - - Asz;'. Application of the EOA to this last 4 0  polynomial matrix now gives z3-'A4 ] (14) Za-lA5 2;' z;' I - A1 2;' -A&' - A3zT1 - A,=;' which is equivalent to - - I - Alz;' - &;' - &z;' - Aaz;' (15) where 0 0  0 0 0 - A  0 -I AJ - I  0 As Here only elementary operations that preserve the matrix deter- minant are used and hence it is straightforward to see that (13) and (15) have the same determinant and hence the stability pro- perty for both system descriptions is the same. This result is easily extended to the partially commensurate case. In particular, as- sume that each delay dl , ,  v = 1,. . . , ml is a multiple of one of basic noncommensurate delays kl, kz,. . . , k,, and similarly for dlh, h = 1,. . . , mz of 1 1 , 1 2 , .  . . , I t , .  Then the previous method exploiting this fact requires the investigation of an nD linear sys- tem, where n = t l  + t z  + 2 whereas the metbod of Theorem 1 here requires the investigation of an m D  linear system with m = ml + m2 + 2. 3. STABILlZATlON ZD LINEAR SYSTEM WITH MULTIPLE DELAYS Consider a 2D linear system with multiple state and input delays described by (3) and assume that the state feedback control law u ( i , j )  = K z ( i , j ) .  (16) is used. The corresponding closed-loop system is z ( i+ l , j+ l )= (Ai+BiK)z ( i+ l , j )  + ( A z + B z K ) z ( i , j + l )  + C ( A i k d + B 1 k d K ) z ( i f l , j - d l * )  (17) dl k = I  82 +C(Az id+Bz idK)Z(Z-dz i ,  j +  1) ! = I  Definition 2 lfrhere exisrs K such rhar (17) is asymproricolly srn- ble, then rhe 2D delay sysrem (3)  is said ro be srabilisable. Theorem 2 The 2D delay sysrem (3)  is asymptotically srable if 3 marrices W, Z > 0. ZIL, .   . , ZI,, > 0, ZZI, . . . , ZZ., > 0 andany N such rhar rhe following LMI holds: -W A i W + B i N  W A T + N ~ B T  -r W A F + N ~ B , T  o 0 0 (18) A z W f B z N  r i d  T z d  0 - Z  0 -Zld 0 0 0 - Z Z d  where Ifrhis condirion holds, then the system is stabilised by feedback of K = N W - ' .  Proof: Using ( 5 )  and (16) and applying the Shur's complement (2). the closed-loop system is asymptotically stable if 3 matrices III-672 Now set P = W-'  and apply the congruence transformation de- fined by diag(W, w,. . . ,w). Then employ the notation (19) to obtain (18) 4. A NUMERICAL EXAMPLE We illustrate the results developed in this paper via one example where the computations involved have been undertaken using the LMI Control Toolbox [6]. Consider the following ZD system of the type (3) with 4 delays (for simplicity, we assume 2 delays in each direction) described by 0.2 0.1 0.4 0.5 0.5 0.5 ' I = [  0.4 0.91 3 0.4 0.31 I 0.3 0.81 I 0.5 0.2 0.7 0.4 0.7 0.6 "=[0.6 0.31 ' ""=[ 0.6 0.61 ' 0.1 0.11 ' 0.4 0.9 0 1.0 0.4 0.2 [ 0.3 0.71 10.7 0.41 , 10.6 0.81 A z l d = [  o 0.11 3 A z z d = [  0.9 0.71 1 0.3 0.41 ' l z d =  0.2 0.8 Bzid= 0.2 0.8 0.2 0.8 Note that this particular example is unstable. Also the following matrices solve the LMI condition (18) in this case -0.0398 3 9047 -0.03981 3 9436 ' = [ 1.7828 -0.34221 , z=[ -0.3422 2.1173 2.3288 0.0159 3.0958 0.0478 ''I=[ 0.0159 2.31331 ' '"=[ 0.0478 3.0491] 1 -0.6351 -1.4445 -0.3656 -0.7135 and N= Hence h e  system (4) is asymptotically stable independent of the delays under the control law (16) with -0.2784 -0.3820 1 ' -0.5028 -0.7635 K =  [ 5. CONCLUSIONS This paper has considered particular aspects of the stability of 2 0  linear systems with multiple state delays. All new results are expressed in terms of LMls and hence their actual implementation for numerical examples can, in principle, follow immediately. One potential problem with this approach, however, is the fact that the dimensions of the matrices involved in the LMI based conditions could well be very large and hence numerical difficulties could arise. This can occur, for example, when the system dimensiona- lity is large (n >> l) andlor many delays are present. This aspect is clearly one to which funher research effort could be applied. Another key feature of the stability tests in this paper is that they extend in a natural manner to the design of stabilizing control laws - a  feature which is not available for other stability tests in the many particular cases of processes with 2DlnD linear dynamics. 6. REFERENCES [I] P. Agathoklis and S. Foda. Stability and the matrix Lyapunov equation for delay differential systems. lnremrional Joumal of Conrml, 49(2):417432, 1989. [2] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan. Li- near Marrix Inequalilies In System And Conrml Theory, vo- lume IS of SlAM srudies in applied marhemorics. SIAM, Philadelphia, 1994. [3] I. N. Chiasson, S. D. Brierley. and E.B. Lee. A simplified de- rivation of the Zeheb-Walch 2-D stability test with applica- tions to time-delay systems. IEEE Trans. Auromar. Control, AC-30(4):411-414, 1985. [4] L. Dugard and E. 1. Vemest. Stabilify and Conrml of Eme- delay Systems, volume 228 of LNCIS. Springer-Verlag. Lon- don, 1998. [SI E. Fomasini and G. Marchesini. Doubly indexed dynami- cal systems ; state-space models and structural properities. Marhemarical Sysrem Theory, 1(1), 1978. [61 P. Gahinet. A. Nemirowski, A. I. Laub. and M. Chilali. MI Conrml Toolbox for use wirh MATLAB. The Mathworks Pm- ner Series. The Mathworks Inc., 1995. [7] K. Galkowski. Srare-Space Realisarions oJLinear 2 - 0  Sys- rems with Extensions ro rhe Geneml nD Case, volume 263 of LNCIS. Springer Verlag, London, 2001. [8] T. Kaczorek. Two-dimensional Lineor Systems. volume 68 of LNCIS. Springer-Verlag. Berlin, 1985. [91 E. W. Kamen. On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations. IEEE Trans. Automat. Conrml, AC- 23:983, 1980. [IO] M. S. Mahmoud. Robusr Conrml and Filrering For 'lime- Delay Systems, volume S of Conrml Engineering Series. 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Stability and stabilisation of 2D discrete linear systems with multiple delays

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Stability and stabilisation of 2D discrete linear systems with multiple delays

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