Stability criteria for largescale timedelay systems: the LMI approach and the Genetic Algorithms


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1 Control and Cybernetics vol 35 (2006) No 2 Stability criteria for largescale timedelay systems: the LMI approach and the Genetic Algorithms by JenqDer Chen Department of Electronic Engineering National Kinmen Institute of Technology Jinning, Kinmen, Taiwan, 892, ROC Abstract: This paper addresses the asymptotic stability analysis problem for a class of linear largescale systems with time delay in the state of each subsystem as well as in the interconnections Based on the Lyapunov stability theory, a delaydependent criterion for stability analysis of the systems is derived in terms of a linear matri inequality (LMI) Finally, a numerical eample is given to demonstrate the validity of the proposed result Keywords: delaydependent criterion, largescale systems, linear matri inequality, genetic algorithms 1 Introduction Time delay naturally arises in the processing of information transmission between subsystems and its eistence is frequently the main source of instability and oscillation in many important systems Many practical control applications are encountered, involving the phenomena in question, including electrical power systems, nuclear reactors, chemical process control systems, transportation systems, computer communication, economic systems, etc Hence, the stability analysis problem for timedelay systems has received considerable attention (Kim, 2001; Kolmanovskii and Myshkis, 1992; Kwon and Park, 2004; Li and De Souza, 1997; Niculescu, 2001; Park, 1999; Richard, 2003; Su et al, 2001) In the recent years, the problem of stability analysis for largescale systems with or without time delay has been etensively studied by a number of authors Moreover, depending on whether the stability criterion itself contains the delay argument as a parameter, stability criteria for systems can be usually classified into two categories, namely the delayindependent criteria (Hmamed, 1986; Huang et al, 1995; Lyou et al, 1984; Michel and Miller, 1977; Schoen and Geering, 1995; Siljak, 1978; Wang et al, 1991) and the delaydependent criterion (Tsay et al, 1996) In general, the latter ones are less conservative than
2 292 JD CHEN the former ones, but the former ones are also important when the effect of time delay is small However, there are few papers to derive the delaydependent and delayindependent stability criteria for a class of largescale systems with time delays, to my knowledge This has motivated my study In this paper, delayindependent and delaydependent criteria for such systems can be derived to guarantee the asymptotic stability for largescale systems with timedelays in the state of each subsystem as well as in the interconnections Appropriate model transformation of original systems is useful for the stability analysis of such systems, and some tuning parameters, which satisfy the constraint on an LMI can be easily obtained by the GAs technique A numerical eample is given to illustrate that the proposed result is useful The notation used throughout this paper is as follows We denote the set of all nonnegative real numbers by R, the set of all continuous functions from [ H, 0 to R n by C 0, the transpose of matri A (respectively, vector ) by A T (respectively T ), the symmetric positive (respectively, negative) definite by A>0 (respectively A<0) We denote identity matri by I and the set {1, 2,, N} by N 2 Main results Consider the following largescale timedelay systems, which is composed of N interconnected subsystems S i, i N Each subsystem S i, i N is described as ẋ i (t) =A i i (t) A ij j (t h ij ), t 0, (1a) where i (t) is the state of each subsystem S i, i N, then, A i, A ij, i,j N are known constant matrices of appropriate dimensions, and h ij, i, j N, are nonnegative time delays in the state of each subsystems as well as in the interconnections The initial condition for each subsystem is given by i (t) =θ i (t), t [ H, 0, H=ma {h ij}, i,j N, (1b) i,j where θ i (t), i N is a continuous function on [ H, 0 A model transformation is constructed for the largescale timedelay systems of the form: d [ [ i (t) B ij j (s)ds = A i i (t) B ij j (t) dt (A ij B ij ) j (t h ij ), (2) where B ij R ni nj, i, j N are some matrices that can be chosen by GAs such that the matri Āi = A i B ij, i, j H is Hurwitz
3 Stability criteria for largescale timedelay systems 293 Now, we present a delaydependent criterion for asymptotic stability of system (1) Theorem 21 The system (1) is asymptotically stable for any constant time delays h ij satisfying 0 h ij h ij, i, j N under the condition that Ā i are Hurwitz and there eist symmetric positive definite matrices P i, R ij, T ij, V ij, and W ijk, i, j, k N such that the following LMI condition holds: Ξ 11 Ξ12 Ξ13 Ξ14 Ξ15 Ξ T 12 Ξ Ξ = Ξ T 13 0 Ξ < 0 (3) Ξ T Ξ 44 0 Ξ T Ξ 55 where Ψ 1 B21 T P 2 P 1 B 12 BN1 T P N P 1 B 1N P 2 B 21 B12P T 1 Ψ2 BN2 T Ξ 11 = P N P 2 B 2N P M B N1 B1N T P 1 P N B N2 B2N T P 2 ΨN Ψ i = ĀT i P i P i Ā i [ hji R ji V ij h ki (T kji W kji ), ˇΞ 12 = diag(γ 1,, Γ N ), Γ i =[ hi1 A T i P i B i1 hin A T i P i B in,, ˇΞ 22 = diag(ˆγ 1,, ˆΓ N ), ˆΓi = diag(r i1,,r in ), ˇΞ 13 = diag(π 1,, Π N ), Π i =[Π i1 Π in, Π ki =[ hi1 Bik T P ib i1 hin Bik T P ib in, ˇΞ 33 = diag(ˆπ 1,, ˆΠ N ), ˆΠi = diag(ˆπ i1 ˆΠ in ), ˆΠ ki = diag(t ik1,,t ikn ), ˇΞ 14 = diag(λ 1,, Λ N ), Λ i =[(A 1i B 1i ) T P 1 (A Ni B Ni ) T P N, ˇΞ 44 = diag(ˆλ 1,, ˆΛ N ), ˆΛi = diag(v 1i,,V Ni ), ˇΞ 15 = diag(ω 1,, Ω N ), Ω i =[Ω i1 Ω in, Ω ki =[ hi1 (A ik B ik ) T P i B i1 hin (A ik B ik ) T P i B in, ˇΞ 55 = diag(ˆω 1,, ˆΩ N ), ˆΩi = diag(ˆω i1 ˆΩ in ), ˆΩ ki = diag(w ik1,,w ikn )
4 294 JD CHEN Proof From the Schur Complements of Boyd et al (1994), the condition (3) is equivalent to the following inequality: Ξ( h, P i,r ij,t ijk,v ij,w ijk )= φ 1 ( h) B21 T P 2 P 1 B 12 BN1 T P N P 1 B 1N P 2 B 21 B12P T 1 φ 2 ( h) BN2 T P N P 2 B 2N P M B N1 B1N T P 1 P N B N2 B2N T P 2 φ N ( h) < 0 (4) where h denotes { h ij,i,j N}, and φ i ( h) =ĀT i P i P i Ā i [ hij A T i P ib ij Rij 1 BT ij P ia i h ji R ji (A ji B ji ) T P j V 1 ji P j (A ji B ji )V ij [ hjk Bjk T P jb jk T 1 jik BT jk P jb ji h ki T kji h jk (A ji B ji ) T P j B jk W 1 jik BT jk P j(a ji B ji ) h ki W kji The Lyapunov functional is given by V ( t )=V 1 ( t )V 2 ( t )V 3 ( t ), (5a) where V 1 ( t )= V 2 ( t )= L i ( t ) T P L i ( t ),L i ( t ) i=1 = i (t) B ij j (s)ds, i N (A ij B ij ) T P i (V 1 ij V 3 ( t )= T j (s)[(s t h ij) R ij (5b) h ik B ik W 1 ijk BT ik)p i (A ij B ij ) j (s)ds, (5c) t h ik (s t h ik ) T k (s)(t ijk W ijk ) k (s)ds, (5d)
5 Stability criteria for largescale timedelay systems 295 are the legitimate Lyapunov functional candidates The time derivatives of V i ( t ), i = 3, along the trajectories of the system (2) are given by V 1 ( t )= T i (t)(at i P i P i A i ) i (t) i=1 2 T i (t)a T i P i B ij j (s)ds 2 T j (t)bijp T i i (t) 2 T j (t)bt ij P ib ik k (s)ds t h ik 2 T j (t h ij )(A ij B ij ) T P i i (t) 2 T j (t h ij)(a ij B ij ) T P i B ik k (s)ds It is known fact that for any, y R n and Z R n n 2 T y T Z y T Z 1 y being true, we have > 0, the inequality V 1 ( t ) T i (t)(at i P i P i A i ) i (t) i=1 [ hij T i (t)a T i P i B ij R 1 ij BT ijp i A i i (t) [ T i (t)(bii T P i P i B ii ) i (t)2 i=1,j i T i (s)r ij j (s)ds T j (t)bt ij P i i (t) [ t hik T j (t)bijp T i B ik T 1 ijk BT ikp i B ij j (t) T k (s)t ijk k (s)ds t h ik [ T j ()(A ij B ij ) T P i V 1 P i (A ij B ij ) j ( ) T i (t)v ij i (t) ij [ hik T j ( )(A ij B ij ) T P i B ik W 1 ijk BT ikp i (A ij B ij ) j ( ) t h ik T k (s)w ijk k (s)ds
6 296 JD CHEN V 2 ( t )= V 3 ( t )= { T j (t) [ h ij R ij (A ij B ij ) T P i (V 1 ij h ik B ik W 1 ijk BT ik)p i (A ij B ij ) j (t) T j (s)r ij j (s)ds T j (t h ij)[(a ij B ij ) T P i (V 1 ij h ik B ik W 1 ijk BT ik )P i(a ij B ij ) j (t h ij ) }, [ hik T k (t)(t ijk W ijk ) k (t) t h jk T k (s)(t ijk W ijk ) k (s)ds Now, considering that T j (t) [ h ij R ij (A ij B ij ) T P i V 1 ij P i (A ij B ij ) h ik B T ij P ib ik T 1 ijk BT ik P ib ij h ik (A ij B ij ) T P i B ik W 1 ijk BT ikp i (A ij B ij ) j (t) = T j (t)[h ji R ji (A ji B ji ) T P j V 1 ji P j (A ji B ji ) h jk B T jip j B jk T 1 jik BT jkp j B ji h jk (A ji B ji ) T P j B jk W 1 jik BT jkp j (A ji B ji ) i (t) H ik T k (t)(t ijk W ijk ) k (t) = h ki T i (t)(t kji W kji ) i (t)
7 Stability criteria for largescale timedelay systems 297 Then, the derivative of V ( t ) satisfies V ( t )= V 1 ( t ) V 2 ( t ) V 3 ( t ) [ T i (t)φ i(h) i (t)2 i=1,j i T j (t)bt ij P i i (t) X T (t)ξ( h, P i,r ij,t ijk,v ij,w ijk )X(t), (6) where X(t) =[ T 1 (t) T N (t)t Hence, by Theorem 981 of Hale and Verduyn Lunel (1993) with (4)(6), we conclude that systems (1) and (2) are both asymptotically stable for any constant time delays h ij satisfying 0 h ij h ij, i, j N Remark 21 Notice that for fied h ij,andb ij, i, j N, theentriesof Ξ in (3) are affine in P i, R ij, T ijk, V ij,andw ijk, i, j, k N, so that the asymptotic stability problem for the largescale timedelay systems can be converted into a strictly feasible LMI problem Remark 22 Notice that for any chosen matri B ij =0, i, j N it means that the delay terms B ij j (s)ds, i, j N have not been converted to the left side of the system (2) By the above Theorem 21, the corresponding matrices R ij, T ijk, U ijk,andw ijk, i, j, k N could be chosen as zero matrices, and the LMI condition in (3) is reduced by it s corresponding elements Letting B ij =0, i, j N in Theorem 21, we achieve the following result that does not depend on delay arguments Corollary 21 The system (1) is asymptotically stable with h ij R, i,j N, provided that A i is Hurwitz, and there eist P i, and V ij, i, j N, such that the following LMI condition holds: ˆΞ11 ˆΞ12 < 0 (7) ˆΞ T 11 ˆΞ 22 where Ψ Ψ2 0 ˆΞ 11 =, Ψi = A T i P i P i A i V ij, 0 0 ΨN Ψ 12 = diag(λ 1,, Λ N ), Λ i =[A T 1iP 1 A T NiP N, Ψ 22 = diag(ˆλ 1,, ˆΛ N ), ˆΛi = diag(v 1i,,V Ni )
8 298 JD CHEN Remark 23 Suppose that for any chosen matri B ij = A ij, i, j N we have that the delay term A ij j (s)ds, i, j N, have been converted to the left side of the system (2) By the proof of Theorem 21, the corresponding matrices V ij,andw ijk, i, j, k N, could be chosen as zero matrices, and the LMI condition in (3) is reduced by its corresponding elements Choosing B ij = A ij, i, j N in Theorem 21 we achieve the following result that depends on delay arguments Corollary 22 The system (1) is asymptotically stable for any constant time delays h ij satisfying 0 h ij h ij, i, j N, provided that Â i = A i A ii is Hurwitz and there eist symmetric positive definite matrices P i, R ij,andt ij, i, j N satisfying the following LMI condition: Ξ 11 Ξ12 Ξ13 Ξ T 12 Ξ 22 0 Ξ T 13 0 Ξ 33 < 0 (8) where Ψ 1 A T 21 P 2 P 1 A 12 A T N1 P N P 1 A 1N P 2 A 21 A T 12 Ξ 11 = P 1 Ψ2 A T N2 P N P 2 A 2N, P N A N1 A T 1N P 1 P N A N2 A T 2N P 2 ΨN Ψ 1 = ÂT i P i P i Â i ( h ji R ji h ki T kji ), Ξ 12 = diag(γ 1,, Γ N ), Γ i =[ hi1 A T i P i A i1 hin A T i P i A in, Ξ 22 = diag(ˆγ 1,, ˆΓ N ), ˆΓi = diag(r i1,,r in ), Ξ 13 = diag(π 1,, Π N ), Π i =[Π i1 Π in, Π ki =[ hi1 A T ik P ia i1 hin A T ik P ia in, Ξ 33 = diag(ˆπ 1,, ˆΠ N ), ˆΠi = diag(ˆπ i1 ˆΠ in ), ˆΠ ki = diag(t ik1,,t ikn ) Now, we provide a procedure for testing the asymptotic stability of the system (1) Step 1 Choosing all B ij =0, i, j N the delayindependent criterion in Corollary 21 is applied to test the stability of the system (1) Step 2 If the condition in Corollary 21 cannot be satisfied, we choose all B ij = A ij, i, j N and the delaydependent criterion in Corollary 22 is applied to test the stability of the system (1)
9 Stability criteria for largescale timedelay systems 299 Step 3 By employing the method of GAs (Chen, 1998; Eshelman and Schaffer, 1993; Michalewicz, 1994; Gen and Cheng, 1997), the matri parameters B ij, i, j N that maimize the fitness function h ij, i, j N and Theorem 21 will be search in this algorithm Continue this algorithm until finding optimal matri parameters B ij =0, i, j N such that the system (1) and (2) are both asymptotically stable for any constant timedelays h ij satisfying 0 h ij h ij, i, j N 3 Eample Eample 31 Largescale systems with time delays: Subsystem 1: ẋ 1 (t) = [ Subsystem 2: ẋ 2 (t) = [ (t) 2 (t) (t h 11 ) (t h 12 ), (t h 21 ) (t h 22 ) (9) Comparison of system (1) with system [ (9), we have N =2 By Theorem and GAs with h11 =21143, B 11 =,andb = B 21 = B 22 =, the LMI (3) is satisfied with P 00 1 =, P = , R =, V =, V 21 =, V =, V =, [ T 111 =, W =, W = Hence, we conclude that system (9) is asymptotically stable for 0 h , andh 12,h 21,h 22 R The delayindependent criteria in Hmamed (1986), Huang et al (1995), Lyou et al (1984), Schoen and Geering (1995), Wang et al (1991) cannot be satisfied The delaydependent stability criteria of Tsay et al (1996) cannot be applied for sufficiently large time delays, h 12,h 21,h 22 R 4 Conclusions In this paper, asymptotic stability analysis problem for a class of largescale neutral timedelay systems has been considered A model transformation and
10 300 JD CHEN the Lyapunov stability theorem have been applied to obtain some stability criteria for such systems Furthermore, the computational programming LMI with GAs has been used to improve the proposed results Finally, it has been shown using a numerical eample that the result shown in this paper is fleible and sharp Acknowledgement The author would like to thank anonymous reviewers for their helpful comments The research reported here was supported by the National Science Council of Taiwan, ROC under grant NSC E References Boyd,S,Ghaoui,LEl,Feron,Eand Balakrishnan, V (1994) Linear Matri Inequalities In System And Control Theory SIAM, Philadelphia Chen, ZM, Lin, HS and Hung, CM (1998) The Design and Application of a GeneticBased Fuzzy Gray Prediction Controller The Journal of Grey System 1, Eshelman, LJ and Schaffer, JD (1993) RealCoded Genetic Algorithms and Interval Schemata In: LD Whitley, ed, Foundation of Genetic Algorithms2, Morgan Kaufmann Publishers Gen, M and Cheng, RW (1997) Genetic Algorithms and Engineering Design John Wiley and Sons, New York Hale, JK and Verduyn Lunel, SM (1993) Introduction to functional differential equations SpringerVerlag, New York Hmamed, A (1986) Note on the stability of large scale systems with delays International Journal of Systems Science 17, Huang, SN, Shao, HH and Zhang, ZJ (1995) Stability analysis of largescale systems with delays Systems & Control Letters 25, Kim, J H (2001) Delay and its timederivative dependent robust stability of timedelayed linear systems with uncertainty IEEE Transaction Automatic Control 46, Kolmanovskii, VB, and Myshkis, A (1992) Applied Theory of Functional Differential Equations Kluwer Academic Publishers, Dordrecht, Netherlands Kwon, OM, and Park, JH (2004) On improved delaydependent robust control for uncertain timedelay systems IEEE Transaction Automatic Control 49, Li, X, and De Souza, CE (1997) Delaydependent robust stability and stabilization of uncertain linear delay systems: a linear matri inequality approach IEEE Transaction Automatic Control 42,
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