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An EP algorithm for stability analysis of interval neutral delay-differential systems
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Expert Systems with Applications Expert Systems with Applications 34 (2008) 920–924 www.elsevier.com/locate/eswa
An EP algorithm for stability analysis of interval neutral delay-differential systems Jun-Juh Yan a b
a,*
, Meei-Ling Hung
b,c
, Teh-Lu Liao
b
Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan, ROC Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan, ROC c Department of Electrical Engineering, Far-East College, Tainan 744, Taiwan, ROC
Abstract As well-known, evolutionary programming (EP) algorithms have been considered as promising techniques for global optimal search. The main objective of this paper is to develop a novel EP algorithm for the robust stability analysis of interval neutral delay-differential systems. Two main results are obtained in this paper. First a delay-dependent criterion is derived for ensuring the stability of degenerate neutral time-delay systems, and then by solving some optimization problems, which will be defined later, the robust stability of interval neutral delay-differential systems can be guaranteed. Two numerical examples are given to illustrate the results. 2006 Elsevier Ltd. All rights reserved. Keywords: Evolutionary programming approach; Stability; Interval neutral systems; Delay-differential systems
1. Introduction As we know, evolutionary programming (EP) algorithms have been considered as available and promising techniques for global optimization of complex functions, and also have been applied to solve difficult problems in control engineering (Cao, 1997; Chen & Huang, 2003; Fogel, 1995; Huang, Tzeng, & Ong, 2006; Kim, Min, & Han, 2006). Generally, the EP algorithm for global optimization contains four parts: initialization, mutation, competition, and reproduction. Furthermore, in the work of Cao (1997) a quasi-random sequence (QRS) is used to generate the initial population for EP to avoid causing clustering around an arbitrary local optimal. On the other hand, systems with delayed states are frequently encountered in various engineering systems, such as the turbojet engine, microwave oscillator, nuclear reactor, chemical process, manual control and long transmission lines in pneumatic, hydraulic systems. Such time*
Corresponding author. Tel.: +886 7 6158000x4806; fax: +886 7 6158000x4899. E-mail address: [email protected] (J.-J. Yan). 0957-4174/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2006.10.025
delays can be a source of instability or of degradation on performance. As a result, stability-testing methods for time-delay systems were proposed in the literature (Oucheriah, 1995; Trinh & Aldeen, 1995; Liu & Su, 1988; Luo, van den Bosch, Weiland, & Goldenberge, 1998; Yan, 2001). Furthermore, in the real world, systems will contain some information about past derivatives of a state (Hale & Verduyn Lunel, 1993; Hui & Hu, 1997; Lien, 1999; Xu, Lam, Ho, & Zou, 2005). Therefore, the stability of neutral delaydifferential systems has also received some attention (Hui & Hu, 1997, 1996; Lien, 1999; Li, 1988). However, up to our knowledge, there have been few results of an investigation for the stability of interval neutral delay systems in the literature. Therefore, the goal of this study is to fill the gap and develop a new EP algorithm to guarantee the robust stability of interval neutral delay-differential systems. In this paper, the EP algorithm is used for deriving a less conservative condition for ensuring the stability of interval neutral delay-differential systems. By using the Lyapunov method, we first derive a delay-dependent stability criterion for a class of degenerate neutral systems. Three optimization problems are then well defined and an EP algorithm is presented to solve the above three optimization problems
J.-J. Yan et al. / Expert Systems with Applications 34 (2008) 920–924
such that the robust stability of interval system is guaranteed. Two numerical examples are given to verify our results and to show the superiority of our results. The notations are used throughout this paper. k(W) denotes any eigenvalue of the matrix W. kmax(W) represents the maximum value of ki(W), i = 1, . . ., n. kWk and kxk represent the standard Euclidean norm of matrix W and vector x, respectively; kWk = [kmax(WTW)]1/2 and kxk = [xTx]1/2; WT denote the transpose of matrix W. l(W) represents the matrix measure of matrix W; l(W) = [kmax (WT + W)/2]. In is the identity matrix of n · n.
921
Lemma 1 Lien (1999). For matrices Bd,Cd 2 Rn·n, if s < qd ¼ 1kBkCd kd k, then the operator D:C ! Rn with Dðxt Þ ¼ xðtÞþ Rt Bd ts xðsÞds C d xðt sÞ is stable. Lemma 2 Doyle, Glover, Khargonekar, and Francis (1989). Define the 2n · 2n degenerate (real) Hamiltonian matrix as 2 3 b 1 A kC d kI n þ sBd BTd d 6 7 H d ¼: 4 5; b 1 2 b 1 ÞT ð s þ kC d kÞ A d þ e I n ð A d ð6Þ
2. Main results Consider the interval neutral delay-differential systems SI described by S I : x_ ðtÞ C 1 x_ ð1 sÞ ¼ A1 xðtÞ þ B1 xðt sÞ; xðtÞ ¼ /ðtÞ; t 2 ½s; 0;
ð1Þ
where x(t) 2 Rn is the state, s 2 R+ is the delay, and /(t) is a continuous vector-valued initial function. The n · n interval matrices AI, BI and CI are sets of degenerate (real) matrices whose elements vary in prescribed ranges defined as AI ¼ aIij ;
BI ¼ bIij
and
C I ¼ cIij ;
ð2Þ
where a1ij 6 aIij 6 a2ij ; b1ij 6 bIij 6 b2ij and c1ij 6 cIij 6 c2ij , i, j = 1, . . ., n. The degenerate neutral delay-differential system Sd of Eq. (1) is defined as Sd :
x_ ðtÞ C d x_ ðt sÞ ¼ Ad xðtÞ þ Bd xðt sÞ
ð3Þ
where Ad 2 AI, Bd 2 BI and Cd 2 CI are real matrices. In the following, we present the main results of this paper. It consists of two main parts. The first part we consider the degenerate systems Sd as described by Eq. (3) and derive a sufficient condition to guarantee the stability of system. In the second part, we develop an algorithm, based on the evolutionary programming approach, to ensure the stability for all possible Sd 2 SI such that the stability of the interval system SI can be guaranteed. 3. Stability analysis of the degenerate neutral delay-differential system By adopting the following relation: Z t d Bd xðsÞds ¼ Bd ðxðtÞ xðt sÞÞ: dt ts We can rewrite the degenerate system (3) as Z t d b d xðtÞ; xðtÞ þ Bd xðsÞds C d xðt sÞ ¼ A dt ts
where e is sufficiently small but greater than zero. Assume b 1 is a stable matrix, and (ii) Hd has no eigenvalues that (i) A d on the imaginary axis. Then the algebraic Riccati equation (ARE) b 1 ÞT P þ P A b 1 þ P kC d kI n þ sBd BT P ðA d d d b 1 2 ð7Þ þ ðs þ kC d kÞ A d þ e I n ¼ 0; has a positive definite solution P. Proof. The proof is an immediate consequence of Lemma 4 in the work (Doyle et al., 1989) and hence is omitted. Now, we present a sufficient condition that guarantees the stability of the degenerate system (3). h Theorem 1. Consider the degenerate neutral delay-differential system (3). If the following conditions, b 1 is a stable matrix, (1) A d (2) H as (6) has no eigenvalues on the imaginary axis, (3) s < qd ¼ 1kBkCd kd k, are satisfied, then system (3) is asymptotically stable in the large. Proof. Let the legitimate Lyapunov finctional candidate be constructed as Z 0Z t T V ðxt Þ ¼ Dðxt Þ PDðxt Þ þ xT ðhÞxðhÞdh ds þ kC d k
Z
s
tþs
t
xT ðhÞxðhÞdh;
ð8Þ
ts
ð4Þ
where D(xt)is defined as in Lemma 1. Then the time derivative of V(xt) along the trajectory of system (5) satisfies
ð5Þ
b 1 ÞT P þ P A b 1 ^xðtÞ V_ ðxt Þ ¼ b x T ðtÞ½ð A d d Z t T xðsÞds C d xðt sÞ þ sxT ðtÞxðtÞ þ 2½P^xðtÞ Bd
b d ¼ Ad þ Bd . where A The following lemmas will be used to prove the main results.
ts
Z
t
xT ðsÞxðsÞds þ kC d k½xT ðtÞxðtÞ xT ðt sÞxðt sÞ ts
ð9Þ
922
J.-J. Yan et al. / Expert Systems with Applications 34 (2008) 920–924
b 1 ÞT P þ P A b 1 ^xðtÞ 6 ^xT ðtÞ½ð A d d Z t T ^x ðtÞPBd kxðsÞkds þ 2kC d k ^xT ðtÞP kxðt sÞk þ2 ts Z t T xT ðsÞxðsÞds þ kC d k½xT ðtÞxðtÞ xT ðt sÞxðt sÞ þ sx ðtÞxðtÞ ts
ð10Þ
b 1 ðzÞÞÞ ðP1Þ : max max Reðki ð A
b 1^xðtÞ has introduced. where xðtÞ ¼ A d
z2S
1 From the fact that 2ab 6 a2 þ cb2 8a;b P 0; and c > 0 c ð11Þ one has Z
t
ts
2^xT ðtÞPBd kxðsÞkds 6
Z
2
for z 2 S, where S ¼ fz 2 R3n j0 6 zi 6 1g: The optimization problems involve finding z 2 S such that the maximum b 1 ðzÞ is maximized or the real part of all eigenvalues of A absolute value of the real parts of all eigenvalues of H(z) is minimized or q(z) is minimized. More accurately, the optimization problems are stated mathematically as 16i6n
for the maximization problem: ðP2Þ : min min absðReðki ðH ðzÞÞÞÞ
ð15Þ
for the minimization problem: ðP3Þ : min qðzÞ
ð16Þ
for the minimization problem:
ð17Þ
z2S
16i62n
z2S
t
½^xT ðtÞPBd BTd P^xðtÞ þ xT ðsÞxðsÞds; ts Z t 6 s^xT ðtÞPBd BTd P^xðtÞ þ xT ðsÞxðsÞds; ts T
2kC d k^xT ðtÞP kxðt sÞk 6 kC d k½^xT ðtÞPP^xðtÞ þ x ðt sÞxðt sÞ:
ð12Þ ð13Þ
b 1^xðtÞk 6 k A b 1 kk^xðtÞk and By using the fact kxðtÞk ¼ k A d d substituting Eqs. (12) and (13) into Eq. (10) yields n b 1 ÞT P þ P A b 1 þ P kC d kI n þ sBd BT P V_ ðxt Þ 6 ^xT ðtÞ ð A d d d o b 1 k2 I n ^xðtÞ: þ ðs þ kC d kÞk A ð14Þ d b 1 is a stable matrix and Hd has no eigenSuppose A d values on the imaginary axis, ARE Eq. (7) has a positive definite solution P by Lemma 2 and V_ ðxt Þ < ek^xðtÞk2 . dk , this implies that the operator Note that s < 1kC RkBt d k Dðxt Þ ¼ xðtÞ þ Bd ts xðsÞds C d xðt sÞ is stable by Lemma 1. Thus according to Theorem 9.8.1 (Hale & Verduyn Lunel, 1993), we conclude that the systems given in Eq. (5) are asymptotically stable in the large. This implies that system (3) is also asymptotically stable in the large and the proof is complete. h Until now, we have proposed Theorem 1 to guarantee the stability of the degenerate neutral delay-differential systems Sd. However, to guarantee the robust stability of the interval neutral systems SI, there still exists a problem to be solved. That is how to ensure that for all possible degenerate systems Sd 2 SI are asymptotical stable. In the following part, an evolutionary programming (EP) algorithm is introduced to solve this problem.
Since Ad, Bd and Cd are enclosed by AI ¼ aIij ; BI ¼ and C I ¼ cIij , respectively, where a1ij 6 aIij 6 a2ij ; b1ij 6 6 b2ij and c1ij 6 cIij 6 c2ij . Now, we introduce variables, with zaij ; zbij ; zcij 2 ½0; 1 such that one has bIij bIij
aIij ¼ a1ij þ zaij ða2ij a1ij Þ; bIij ¼ b1ij þ zbij ðb2ij b1ij Þ and
Since EP algorithm have been considered as available and promising techniques for global optimization of complex functions, we introduce the EP algorithm to solve above problems. 3.2. Problem formulation b 1 ðzÞ 2 Rnn ; H ðzÞ 2 R2n2n and qðzÞ 2 Rþ be the Let A continuously differentiable matrix-valued function defined
ð18Þ
2
and z ¼ ½zaij ; zbij ; zcij 2 R3n , then all the possible degenerate b 1 ; H d and qd can be denoted as A b 1 ðzÞ; matrices A d H ðzÞ and qðzÞ, respectively. The problems, to ensure b 1 is stable and for any that for any degenerate matrix A d degenerate Hamiltonian matrix Hd has no eigenvalues on the imaginary axis and the delay time s is always small than all possible q, can be well solved by solving the optimizab 1 ðzÞ; H ðzÞ and qðzÞ as described in tion problems of A (P1)–(P3). Proposition 1. If z* is the global optimum of (P1) and the optimal objective value b 1 ðz ÞÞÞ < 0; f ðz Þ ¼ max Reðki ð A 16i6n
ð19Þ
b 1 2 A b 1 ðzÞ are stable. then all the degenerate matrices A d Proposition 2. If z* is the global optimum of (P2) and the optimal objective value f ðz Þ ¼ min absðReðki ðH ðz ÞÞÞÞ > 0; 16i62n
3.1. An evolutionary programming (EP) algorithm to solve the optimization problem
cIij ¼ c1ij þ zcij ðc2ij c1ij Þ
ð20Þ
all the degenerate Hamiltonian matrices Hd 2 H(z) have no eigenvalues on the imaginary axis. Proposition 3. If z* is the global optimum of (P3) and the optimal objective value s < qðz Þ;
ð21Þ
the delay time s is always small than all possible qd 2 q(z). Based on the results of Cao (1997), the extended EP algorithm for solving the above optimization problems is described as follows:
J.-J. Yan et al. / Expert Systems with Applications 34 (2008) 920–924
Step I. Generate an initial population P0 = [p1, p2, . . ., pN] of size N by randomly initializing each 3n2dimensional solution vector pi 2 S, i = 1, . . ., N, based on QRS. Step II. Calculate the fitness score (objective function) fi = f (pi) for every pi, i = 1, . . ., N, where b 1 ðpi ÞÞÞ; f ðpi Þ ¼ max Reðki ð A 16i6n
for the maximization problem of ðP1Þ; ð22Þ and f ðpi Þ ¼ min absðReðki ðH ðpi ÞÞÞ; 16i62n
for the minimization problem of ðP2Þ; ð23Þ and f ðpi Þ ¼ qðpi Þ; for the minimization problem ofðP3Þ:
ð24Þ
Step III. Mutate every pi, i = 1, . . . ,N, based on the statistics to double the population size from N to 2N, and generate pi+N in the following way: ! fi piþN ;j ¼ pi;j þ N 0; b P ; 8j ¼ 1; . . . ; 3n2 ; f ð25Þ where pi,j denotes the ! jth element of the ith indifi vidual; N 0; b fP
represents a Gaussian ran-
dom variable with a mean zero and variance fi b fP ; fP is the sum of every fitness scores, b is fi a parameter to scale fP .
Step IV. Calculate the fitness score fi+N for every pi + N, i = 1, . . ., N by Eq. (22) for (P1) and Eq. (23) for (P2) and Eq. (24) for (P3), respectively. By the stochastic competition process, pi, i = 1, . . ., N, compete with pj, j = N + 1, . . ., 2N, randomly. If fi > fj for (P1) or fi < fj for (P2) and (P3), pi is the winner and survives; otherwise, pj is the winner and pi is replaced by pj. After the competition process, we select the N winners for the next generation and let the individual with the minimum objective function in the winners be p1. Step V. If the value fP converges to a minimum value, then z* = p1,j, j = 1, . . ., 3n2 is the global optimum and f(z*) is the global objective value. Otherwise return to Step III.
4. An illustrative example This section presents two examples to demonstrate the validity of the proposed results. The first example involves the stability of neutral-type systems with constant delays, while the second example involves the stability of interval neutral delay-differential systems.
923
Example 1. Consider nominal neutral system, which is a special case of system (1), described as x_ ðtÞ C x_ ðtÞ ¼ AxðtÞ þ Bxðt sÞ;
ð26Þ
where
3 2 ; A¼ 1 0 0:1 0 C¼ ; 0 0:1
0 B¼a 1
1 ; 0
s ¼ 0:01
If a = 1.24, we can easily construct a Hamiltonian matrix H with e = 0.0001 as 2 3 0 0:4464 0:1154 0 6 1:3158 1:7622 0 0:1154 7 6 7 H :¼ 6 7: 4 0:5464 0 0 1:3158 5 0:5464
0
0:4464 1:7622
It is also easy to check H has no eigenvalues on the imagb 1 is stable. Furthermore, s < 1 ¼ 0:8065. inary axis and A kBk Thus according to Theorem 1, system (26) is asymptotical stable. However, from the result of report (Park & Won, 1999), the stability of system (26) can be guaranteed when jaj < 0.4. Furthermore, l(A) = 0.0811 > 0 and l(A1) = 0.0406 > 0, thus the criteria of Hui & Hu (1997) & Li (1988) are also not applicable. Example 2. Consider the interval neutral delay-differential system described as x_ ðtÞ C I x_ ðt sÞ ¼ AI xðtÞ þ BI xðt sÞ xðtÞ ¼ /ðtÞ; t 2 ½s; 0; where
½ 7:2 6:5 ½ 1 2 ½ 0:35 1 ½ 1 0 AI ¼ ; BI ¼ ; ½ 2 2:5 ½ 7 6 ½ 1:6 0:5 ½ 0:4 0:6 ½ 0:1 0 ½ 0:1 0:2 ; s ¼ 0:1 CI ¼ ½ 0:2 0 ½ 0:2 0:1
Now, introduce e = 0.001 and introduce variables z ¼ zaij zbij zcij ; i; j ¼ 1; 2 where zaij ¼ ½ za11 zcij ¼ ½ zc11
za12
za21
za22 ;
zc12
zc21
zc22 :
zbij ¼ zb11
zb12
One has " # 7:2 þ 0:7za11 1 þ za12 Aðzaij Þ ¼ ; 2 þ 0:5za21 7 þ za22 " # 0:35 þ 0:65zb11 1 þ zb12 b ; Bðzij Þ ¼ 1:6 þ 1:1zb21 0:4 þ zb22 " # 0:1 þ 0:1zc11 0:1 þ 0:1zc12 c Cðzij Þ ¼ 0:2 þ 0:2zc21 0:2 þ 0:1zc22
zb21
zb22 ;
924
J.-J. Yan et al. / Expert Systems with Applications 34 (2008) 920–924
b 1 ðzÞ ¼ ½Aðza Þ þ Bðzb Þ1 and Thereby, A ij ij H ðzÞ 2
3 c ½Aðzaij Þ þ Bðzbij Þ1 Cðzij ÞI n þ sBðzbij ÞBT ðzbij Þ 6 7 7; 2 ¼6 4 5 c a b 1 a b 1 T ðs þ Cðzij ÞÞ½Aðzij Þ þ Bðzij Þ þ 0:001 I n f½Aðzij Þ þ Bðzij Þ g
and qðzÞ ¼
1kCðzcij Þk
: kBðzbij Þk b 1 ðzÞ, H(z) and q(z), we solve the optiFor the above A mization problems (P1)–(P3), respectively, with N = 300 and b = 0.1. It converges after 250, 621 and 499 interactions for (P1)–(P3), respectively. And the global objectives are f(z*) = 0.1093 for (P1) and f(z*) = 0.0345 for (P2) and f(z*) = 0.3262 for (P3). According to Propositions 1–3, all b 1 ðzÞ are stable and all b 1 2 A the degenerate matrices A d the degenerate Hamiltonian matrices Hd 2 H(z) have no eigenvalues on the imaginary axis and the delay time s is always small than all possible qd 2 q(z). Thus by Theorem 1, this interval neutral delay-differential system is robust stable. 5. Conclusions In this paper, an EP approach for the robust stability analysis of interval neutral delay-differential systems is newly derived. The following results are obtained: (1) Less conservative criterion than the ones derived in the literature is obtained. (2) An evolutionary programming algorithm has been developed to analyze the robust stability of interval neutral delay-differential systems. References Cao, Y. J. (1997). Eigenvalue optimisation problems via evolutionary programming. Electronics Letters, 33, 642–643. Chen, M. C., & Huang, S. H. (2003). Credit scoring and rejected instances reassigning through evolutionary computation techniques. Expert Systems with Applications, 24(4), 433–441.
Doyle, J. C., Glover, K., Khargonekar, P. P., & Francis, B. A. (1989). State-space solutions to standard H2 and H1 control problem. IEEE Transactions on Automatic Control, 34, 831–846. Fogel, D. B. (1995). Evolutionary computation: Toward a new philosophy of machine intelligence. IEEE Press. Hale, J. K., & Verduyn Lunel, S. M. (1993). Introduction to functional differential equations. New York: Springer. Huang, J. J., Tzeng, G. H., & Ong, C. S. (2006). Optimal fuzzy multicriteria expansion of competence sets using multi-objectives evolutionary algorithms. Expert Systems with Applications, 30(4), 739–745. Hui, G. D., & Hu, G. D. (1996). Some simple stability criteria of neutral delay-differential systems. Applied Mathematics and Computation, 80, 257–271. Hui, G. D., & Hu, G. D. (1997). Simple criteria for stability of neutral systems with multiple delays. International Journal of Systems Science, 28, 1325–1328. Kim, M. J., Min, S. H., & Han, I. (2006). An evolutionary approach to the combination of multiple classifiers to predict a stock price index. Expert Systems with Applications, 31(2), 241–247. Li, L. M. (1988). Stability of linear neutral delay-differential systems. Bulletin of the Australian Mathematical Society, 38, 339–344. Lien, C. H. (1999). Asymptotic criterion for neutral systems with multiple time delays. Electronics Letters, 35, 850–852. Liu, P. L., & Su, T. J. (1988). Robust stability of interval time-delay systems with delay-dependence. Systems and Control Letters, 33, 231–239. Luo, J. S., van den Bosch, P. P. J., Weiland, S., & Goldenberge, A. (1998). Design of performance robustness for uncertain linear systems with state and control delays. IEEE Transactions on Automatic Control, 43, 1593–1596. Oucheriah, S. (1995). Measures of robustness for uncertain time-delay linear systems. ASME Journal of Dynamic Systems, Measurement, and Control, 117, 633–635. Park, J. H., & Won, S. (1999). A note on stability of neutral delaydifferential systems. Journal of Franklin Institute, 336, 543–548. Trinh, H., & Aldeen, M. (1995). Stability robustness bounds for linear systems with delayed perturbations. IEE Proceeding-Control Theory Applications, 142, 345–350. Xu, S., Lam, J., Ho, W. C., & Zou, Y. (2005). Delay-dependent exponential stability for a class of neural networks with time delays. Journal of Computational and Applied Mathematics, 183, 16–28. Yan, J. J. (2001). Robust stability analysis of uncertain time delay systems with delay-dependence. Electronics Letters, 37, 135–137.
Expert Systems with Applications Expert Systems with Applications 34 (2008) 920–924 www.elsevier.com/locate/eswa
An EP algorithm for stability analysis of interval neutral delay-differential systems Jun-Juh Yan a b
a,*
, Meei-Ling Hung
b,c
, Teh-Lu Liao
b
Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan, ROC Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan, ROC c Department of Electrical Engineering, Far-East College, Tainan 744, Taiwan, ROC
Abstract As well-known, evolutionary programming (EP) algorithms have been considered as promising techniques for global optimal search. The main objective of this paper is to develop a novel EP algorithm for the robust stability analysis of interval neutral delay-differential systems. Two main results are obtained in this paper. First a delay-dependent criterion is derived for ensuring the stability of degenerate neutral time-delay systems, and then by solving some optimization problems, which will be defined later, the robust stability of interval neutral delay-differential systems can be guaranteed. Two numerical examples are given to illustrate the results. 2006 Elsevier Ltd. All rights reserved. Keywords: Evolutionary programming approach; Stability; Interval neutral systems; Delay-differential systems
1. Introduction As we know, evolutionary programming (EP) algorithms have been considered as available and promising techniques for global optimization of complex functions, and also have been applied to solve difficult problems in control engineering (Cao, 1997; Chen & Huang, 2003; Fogel, 1995; Huang, Tzeng, & Ong, 2006; Kim, Min, & Han, 2006). Generally, the EP algorithm for global optimization contains four parts: initialization, mutation, competition, and reproduction. Furthermore, in the work of Cao (1997) a quasi-random sequence (QRS) is used to generate the initial population for EP to avoid causing clustering around an arbitrary local optimal. On the other hand, systems with delayed states are frequently encountered in various engineering systems, such as the turbojet engine, microwave oscillator, nuclear reactor, chemical process, manual control and long transmission lines in pneumatic, hydraulic systems. Such time*
Corresponding author. Tel.: +886 7 6158000x4806; fax: +886 7 6158000x4899. E-mail address: [email protected] (J.-J. Yan). 0957-4174/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2006.10.025
delays can be a source of instability or of degradation on performance. As a result, stability-testing methods for time-delay systems were proposed in the literature (Oucheriah, 1995; Trinh & Aldeen, 1995; Liu & Su, 1988; Luo, van den Bosch, Weiland, & Goldenberge, 1998; Yan, 2001). Furthermore, in the real world, systems will contain some information about past derivatives of a state (Hale & Verduyn Lunel, 1993; Hui & Hu, 1997; Lien, 1999; Xu, Lam, Ho, & Zou, 2005). Therefore, the stability of neutral delaydifferential systems has also received some attention (Hui & Hu, 1997, 1996; Lien, 1999; Li, 1988). However, up to our knowledge, there have been few results of an investigation for the stability of interval neutral delay systems in the literature. Therefore, the goal of this study is to fill the gap and develop a new EP algorithm to guarantee the robust stability of interval neutral delay-differential systems. In this paper, the EP algorithm is used for deriving a less conservative condition for ensuring the stability of interval neutral delay-differential systems. By using the Lyapunov method, we first derive a delay-dependent stability criterion for a class of degenerate neutral systems. Three optimization problems are then well defined and an EP algorithm is presented to solve the above three optimization problems
J.-J. Yan et al. / Expert Systems with Applications 34 (2008) 920–924
such that the robust stability of interval system is guaranteed. Two numerical examples are given to verify our results and to show the superiority of our results. The notations are used throughout this paper. k(W) denotes any eigenvalue of the matrix W. kmax(W) represents the maximum value of ki(W), i = 1, . . ., n. kWk and kxk represent the standard Euclidean norm of matrix W and vector x, respectively; kWk = [kmax(WTW)]1/2 and kxk = [xTx]1/2; WT denote the transpose of matrix W. l(W) represents the matrix measure of matrix W; l(W) = [kmax (WT + W)/2]. In is the identity matrix of n · n.
921
Lemma 1 Lien (1999). For matrices Bd,Cd 2 Rn·n, if s < qd ¼ 1kBkCd kd k, then the operator D:C ! Rn with Dðxt Þ ¼ xðtÞþ Rt Bd ts xðsÞds C d xðt sÞ is stable. Lemma 2 Doyle, Glover, Khargonekar, and Francis (1989). Define the 2n · 2n degenerate (real) Hamiltonian matrix as 2 3 b 1 A kC d kI n þ sBd BTd d 6 7 H d ¼: 4 5; b 1 2 b 1 ÞT ð s þ kC d kÞ A d þ e I n ð A d ð6Þ
2. Main results Consider the interval neutral delay-differential systems SI described by S I : x_ ðtÞ C 1 x_ ð1 sÞ ¼ A1 xðtÞ þ B1 xðt sÞ; xðtÞ ¼ /ðtÞ; t 2 ½s; 0;
ð1Þ
where x(t) 2 Rn is the state, s 2 R+ is the delay, and /(t) is a continuous vector-valued initial function. The n · n interval matrices AI, BI and CI are sets of degenerate (real) matrices whose elements vary in prescribed ranges defined as AI ¼ aIij ;
BI ¼ bIij
and
C I ¼ cIij ;
ð2Þ
where a1ij 6 aIij 6 a2ij ; b1ij 6 bIij 6 b2ij and c1ij 6 cIij 6 c2ij , i, j = 1, . . ., n. The degenerate neutral delay-differential system Sd of Eq. (1) is defined as Sd :
x_ ðtÞ C d x_ ðt sÞ ¼ Ad xðtÞ þ Bd xðt sÞ
ð3Þ
where Ad 2 AI, Bd 2 BI and Cd 2 CI are real matrices. In the following, we present the main results of this paper. It consists of two main parts. The first part we consider the degenerate systems Sd as described by Eq. (3) and derive a sufficient condition to guarantee the stability of system. In the second part, we develop an algorithm, based on the evolutionary programming approach, to ensure the stability for all possible Sd 2 SI such that the stability of the interval system SI can be guaranteed. 3. Stability analysis of the degenerate neutral delay-differential system By adopting the following relation: Z t d Bd xðsÞds ¼ Bd ðxðtÞ xðt sÞÞ: dt ts We can rewrite the degenerate system (3) as Z t d b d xðtÞ; xðtÞ þ Bd xðsÞds C d xðt sÞ ¼ A dt ts
where e is sufficiently small but greater than zero. Assume b 1 is a stable matrix, and (ii) Hd has no eigenvalues that (i) A d on the imaginary axis. Then the algebraic Riccati equation (ARE) b 1 ÞT P þ P A b 1 þ P kC d kI n þ sBd BT P ðA d d d b 1 2 ð7Þ þ ðs þ kC d kÞ A d þ e I n ¼ 0; has a positive definite solution P. Proof. The proof is an immediate consequence of Lemma 4 in the work (Doyle et al., 1989) and hence is omitted. Now, we present a sufficient condition that guarantees the stability of the degenerate system (3). h Theorem 1. Consider the degenerate neutral delay-differential system (3). If the following conditions, b 1 is a stable matrix, (1) A d (2) H as (6) has no eigenvalues on the imaginary axis, (3) s < qd ¼ 1kBkCd kd k, are satisfied, then system (3) is asymptotically stable in the large. Proof. Let the legitimate Lyapunov finctional candidate be constructed as Z 0Z t T V ðxt Þ ¼ Dðxt Þ PDðxt Þ þ xT ðhÞxðhÞdh ds þ kC d k
Z
s
tþs
t
xT ðhÞxðhÞdh;
ð8Þ
ts
ð4Þ
where D(xt)is defined as in Lemma 1. Then the time derivative of V(xt) along the trajectory of system (5) satisfies
ð5Þ
b 1 ÞT P þ P A b 1 ^xðtÞ V_ ðxt Þ ¼ b x T ðtÞ½ð A d d Z t T xðsÞds C d xðt sÞ þ sxT ðtÞxðtÞ þ 2½P^xðtÞ Bd
b d ¼ Ad þ Bd . where A The following lemmas will be used to prove the main results.
ts
Z
t
xT ðsÞxðsÞds þ kC d k½xT ðtÞxðtÞ xT ðt sÞxðt sÞ ts
ð9Þ
922
J.-J. Yan et al. / Expert Systems with Applications 34 (2008) 920–924
b 1 ÞT P þ P A b 1 ^xðtÞ 6 ^xT ðtÞ½ð A d d Z t T ^x ðtÞPBd kxðsÞkds þ 2kC d k ^xT ðtÞP kxðt sÞk þ2 ts Z t T xT ðsÞxðsÞds þ kC d k½xT ðtÞxðtÞ xT ðt sÞxðt sÞ þ sx ðtÞxðtÞ ts
ð10Þ
b 1 ðzÞÞÞ ðP1Þ : max max Reðki ð A
b 1^xðtÞ has introduced. where xðtÞ ¼ A d
z2S
1 From the fact that 2ab 6 a2 þ cb2 8a;b P 0; and c > 0 c ð11Þ one has Z
t
ts
2^xT ðtÞPBd kxðsÞkds 6
Z
2
for z 2 S, where S ¼ fz 2 R3n j0 6 zi 6 1g: The optimization problems involve finding z 2 S such that the maximum b 1 ðzÞ is maximized or the real part of all eigenvalues of A absolute value of the real parts of all eigenvalues of H(z) is minimized or q(z) is minimized. More accurately, the optimization problems are stated mathematically as 16i6n
for the maximization problem: ðP2Þ : min min absðReðki ðH ðzÞÞÞÞ
ð15Þ
for the minimization problem: ðP3Þ : min qðzÞ
ð16Þ
for the minimization problem:
ð17Þ
z2S
16i62n
z2S
t
½^xT ðtÞPBd BTd P^xðtÞ þ xT ðsÞxðsÞds; ts Z t 6 s^xT ðtÞPBd BTd P^xðtÞ þ xT ðsÞxðsÞds; ts T
2kC d k^xT ðtÞP kxðt sÞk 6 kC d k½^xT ðtÞPP^xðtÞ þ x ðt sÞxðt sÞ:
ð12Þ ð13Þ
b 1^xðtÞk 6 k A b 1 kk^xðtÞk and By using the fact kxðtÞk ¼ k A d d substituting Eqs. (12) and (13) into Eq. (10) yields n b 1 ÞT P þ P A b 1 þ P kC d kI n þ sBd BT P V_ ðxt Þ 6 ^xT ðtÞ ð A d d d o b 1 k2 I n ^xðtÞ: þ ðs þ kC d kÞk A ð14Þ d b 1 is a stable matrix and Hd has no eigenSuppose A d values on the imaginary axis, ARE Eq. (7) has a positive definite solution P by Lemma 2 and V_ ðxt Þ < ek^xðtÞk2 . dk , this implies that the operator Note that s < 1kC RkBt d k Dðxt Þ ¼ xðtÞ þ Bd ts xðsÞds C d xðt sÞ is stable by Lemma 1. Thus according to Theorem 9.8.1 (Hale & Verduyn Lunel, 1993), we conclude that the systems given in Eq. (5) are asymptotically stable in the large. This implies that system (3) is also asymptotically stable in the large and the proof is complete. h Until now, we have proposed Theorem 1 to guarantee the stability of the degenerate neutral delay-differential systems Sd. However, to guarantee the robust stability of the interval neutral systems SI, there still exists a problem to be solved. That is how to ensure that for all possible degenerate systems Sd 2 SI are asymptotical stable. In the following part, an evolutionary programming (EP) algorithm is introduced to solve this problem.
Since Ad, Bd and Cd are enclosed by AI ¼ aIij ; BI ¼ and C I ¼ cIij , respectively, where a1ij 6 aIij 6 a2ij ; b1ij 6 6 b2ij and c1ij 6 cIij 6 c2ij . Now, we introduce variables, with zaij ; zbij ; zcij 2 ½0; 1 such that one has bIij bIij
aIij ¼ a1ij þ zaij ða2ij a1ij Þ; bIij ¼ b1ij þ zbij ðb2ij b1ij Þ and
Since EP algorithm have been considered as available and promising techniques for global optimization of complex functions, we introduce the EP algorithm to solve above problems. 3.2. Problem formulation b 1 ðzÞ 2 Rnn ; H ðzÞ 2 R2n2n and qðzÞ 2 Rþ be the Let A continuously differentiable matrix-valued function defined
ð18Þ
2
and z ¼ ½zaij ; zbij ; zcij 2 R3n , then all the possible degenerate b 1 ; H d and qd can be denoted as A b 1 ðzÞ; matrices A d H ðzÞ and qðzÞ, respectively. The problems, to ensure b 1 is stable and for any that for any degenerate matrix A d degenerate Hamiltonian matrix Hd has no eigenvalues on the imaginary axis and the delay time s is always small than all possible q, can be well solved by solving the optimizab 1 ðzÞ; H ðzÞ and qðzÞ as described in tion problems of A (P1)–(P3). Proposition 1. If z* is the global optimum of (P1) and the optimal objective value b 1 ðz ÞÞÞ < 0; f ðz Þ ¼ max Reðki ð A 16i6n
ð19Þ
b 1 2 A b 1 ðzÞ are stable. then all the degenerate matrices A d Proposition 2. If z* is the global optimum of (P2) and the optimal objective value f ðz Þ ¼ min absðReðki ðH ðz ÞÞÞÞ > 0; 16i62n
3.1. An evolutionary programming (EP) algorithm to solve the optimization problem
cIij ¼ c1ij þ zcij ðc2ij c1ij Þ
ð20Þ
all the degenerate Hamiltonian matrices Hd 2 H(z) have no eigenvalues on the imaginary axis. Proposition 3. If z* is the global optimum of (P3) and the optimal objective value s < qðz Þ;
ð21Þ
the delay time s is always small than all possible qd 2 q(z). Based on the results of Cao (1997), the extended EP algorithm for solving the above optimization problems is described as follows:
J.-J. Yan et al. / Expert Systems with Applications 34 (2008) 920–924
Step I. Generate an initial population P0 = [p1, p2, . . ., pN] of size N by randomly initializing each 3n2dimensional solution vector pi 2 S, i = 1, . . ., N, based on QRS. Step II. Calculate the fitness score (objective function) fi = f (pi) for every pi, i = 1, . . ., N, where b 1 ðpi ÞÞÞ; f ðpi Þ ¼ max Reðki ð A 16i6n
for the maximization problem of ðP1Þ; ð22Þ and f ðpi Þ ¼ min absðReðki ðH ðpi ÞÞÞ; 16i62n
for the minimization problem of ðP2Þ; ð23Þ and f ðpi Þ ¼ qðpi Þ; for the minimization problem ofðP3Þ:
ð24Þ
Step III. Mutate every pi, i = 1, . . . ,N, based on the statistics to double the population size from N to 2N, and generate pi+N in the following way: ! fi piþN ;j ¼ pi;j þ N 0; b P ; 8j ¼ 1; . . . ; 3n2 ; f ð25Þ where pi,j denotes the ! jth element of the ith indifi vidual; N 0; b fP
represents a Gaussian ran-
dom variable with a mean zero and variance fi b fP ; fP is the sum of every fitness scores, b is fi a parameter to scale fP .
Step IV. Calculate the fitness score fi+N for every pi + N, i = 1, . . ., N by Eq. (22) for (P1) and Eq. (23) for (P2) and Eq. (24) for (P3), respectively. By the stochastic competition process, pi, i = 1, . . ., N, compete with pj, j = N + 1, . . ., 2N, randomly. If fi > fj for (P1) or fi < fj for (P2) and (P3), pi is the winner and survives; otherwise, pj is the winner and pi is replaced by pj. After the competition process, we select the N winners for the next generation and let the individual with the minimum objective function in the winners be p1. Step V. If the value fP converges to a minimum value, then z* = p1,j, j = 1, . . ., 3n2 is the global optimum and f(z*) is the global objective value. Otherwise return to Step III.
4. An illustrative example This section presents two examples to demonstrate the validity of the proposed results. The first example involves the stability of neutral-type systems with constant delays, while the second example involves the stability of interval neutral delay-differential systems.
923
Example 1. Consider nominal neutral system, which is a special case of system (1), described as x_ ðtÞ C x_ ðtÞ ¼ AxðtÞ þ Bxðt sÞ;
ð26Þ
where
3 2 ; A¼ 1 0 0:1 0 C¼ ; 0 0:1
0 B¼a 1
1 ; 0
s ¼ 0:01
If a = 1.24, we can easily construct a Hamiltonian matrix H with e = 0.0001 as 2 3 0 0:4464 0:1154 0 6 1:3158 1:7622 0 0:1154 7 6 7 H :¼ 6 7: 4 0:5464 0 0 1:3158 5 0:5464
0
0:4464 1:7622
It is also easy to check H has no eigenvalues on the imagb 1 is stable. Furthermore, s < 1 ¼ 0:8065. inary axis and A kBk Thus according to Theorem 1, system (26) is asymptotical stable. However, from the result of report (Park & Won, 1999), the stability of system (26) can be guaranteed when jaj < 0.4. Furthermore, l(A) = 0.0811 > 0 and l(A1) = 0.0406 > 0, thus the criteria of Hui & Hu (1997) & Li (1988) are also not applicable. Example 2. Consider the interval neutral delay-differential system described as x_ ðtÞ C I x_ ðt sÞ ¼ AI xðtÞ þ BI xðt sÞ xðtÞ ¼ /ðtÞ; t 2 ½s; 0; where
½ 7:2 6:5 ½ 1 2 ½ 0:35 1 ½ 1 0 AI ¼ ; BI ¼ ; ½ 2 2:5 ½ 7 6 ½ 1:6 0:5 ½ 0:4 0:6 ½ 0:1 0 ½ 0:1 0:2 ; s ¼ 0:1 CI ¼ ½ 0:2 0 ½ 0:2 0:1
Now, introduce e = 0.001 and introduce variables z ¼ zaij zbij zcij ; i; j ¼ 1; 2 where zaij ¼ ½ za11 zcij ¼ ½ zc11
za12
za21
za22 ;
zc12
zc21
zc22 :
zbij ¼ zb11
zb12
One has " # 7:2 þ 0:7za11 1 þ za12 Aðzaij Þ ¼ ; 2 þ 0:5za21 7 þ za22 " # 0:35 þ 0:65zb11 1 þ zb12 b ; Bðzij Þ ¼ 1:6 þ 1:1zb21 0:4 þ zb22 " # 0:1 þ 0:1zc11 0:1 þ 0:1zc12 c Cðzij Þ ¼ 0:2 þ 0:2zc21 0:2 þ 0:1zc22
zb21
zb22 ;
924
J.-J. Yan et al. / Expert Systems with Applications 34 (2008) 920–924
b 1 ðzÞ ¼ ½Aðza Þ þ Bðzb Þ1 and Thereby, A ij ij H ðzÞ 2
3 c ½Aðzaij Þ þ Bðzbij Þ1 Cðzij ÞI n þ sBðzbij ÞBT ðzbij Þ 6 7 7; 2 ¼6 4 5 c a b 1 a b 1 T ðs þ Cðzij ÞÞ½Aðzij Þ þ Bðzij Þ þ 0:001 I n f½Aðzij Þ þ Bðzij Þ g
and qðzÞ ¼
1kCðzcij Þk
: kBðzbij Þk b 1 ðzÞ, H(z) and q(z), we solve the optiFor the above A mization problems (P1)–(P3), respectively, with N = 300 and b = 0.1. It converges after 250, 621 and 499 interactions for (P1)–(P3), respectively. And the global objectives are f(z*) = 0.1093 for (P1) and f(z*) = 0.0345 for (P2) and f(z*) = 0.3262 for (P3). According to Propositions 1–3, all b 1 ðzÞ are stable and all b 1 2 A the degenerate matrices A d the degenerate Hamiltonian matrices Hd 2 H(z) have no eigenvalues on the imaginary axis and the delay time s is always small than all possible qd 2 q(z). Thus by Theorem 1, this interval neutral delay-differential system is robust stable. 5. Conclusions In this paper, an EP approach for the robust stability analysis of interval neutral delay-differential systems is newly derived. The following results are obtained: (1) Less conservative criterion than the ones derived in the literature is obtained. (2) An evolutionary programming algorithm has been developed to analyze the robust stability of interval neutral delay-differential systems. References Cao, Y. J. (1997). Eigenvalue optimisation problems via evolutionary programming. Electronics Letters, 33, 642–643. Chen, M. C., & Huang, S. H. (2003). Credit scoring and rejected instances reassigning through evolutionary computation techniques. Expert Systems with Applications, 24(4), 433–441.
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