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Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems
Commun Nonlinear Sci Numer Simulat xxx (2013) xxx–xxx
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Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems Song Zhu ⇑, Kai Zhong, Yufeng Zhang College of Sciences, China University of Mining and Technology, Xuzhou 221116, China
a r t i c l e
i n f o
Article history: Received 23 September 2012 Received in revised form 12 May 2013 Accepted 8 June 2013 Available online xxxx Keywords: Stochastic delayed systems Global exponential stability Robustness
a b s t r a c t In this paper, we analyze the robustness of global exponential stable stochastic delayed systems subject to the uncertainty in parameter matrices. Given a globally exponentially stable systems, the problem to be addressed here is how much uncertainty in parameter matrices the systems can withstand to be globally exponentially stable. The upper bounds of the parameter uncertainty intensity are characterized by using transcendental equation for the systems to sustain global exponential stability. Moreover, we prove theoretically that, the globally exponentially stable systems, if additive uncertainties in parameter matrices are smaller than the upper bounds arrived at here, then the perturbed systems are guaranteed to also be globally exponentially stable. Two numerical examples are provided here to illustrate the theoretical results. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Time delays are often encountered in various practical systems such as chemical processes, neural networks and long transmission lines in pneumatic systems [1,2]. It has been shown that the existence of time-delays may lead to oscillation, divergence, instability, greatly increasing the difficulty of stability analysis and control design. Many researchers in the field of control theory and engineering study the robust control of time-delay systems. The main methods of stability analysis can be classified into two types: frequency-domain and time-domain. The former use the sum of squares technique. As to the time-domain approach, Lyapunov functional is a powerful tool, which can deal with time varying delays. For most successful applications of the systems, the stability is usually a prerequisite. The stability of the systems depends mainly on their parametrical configuration. Moreover, in the applications of the systems, external random disturbances and time delays are common and hardly avoided. It is known that random disturbances and time delays in the systems may result in oscillation or instability of the nonlinear systems. The stability analysis of the delayed systems and the systems with external random disturbances has been widely investigated in recent years (see, e.g., [3–8], and the references cited therein). In practice, when we estimate systems parameter matrices, there are always some uncertainty and errors. For the parameter matrices uncertainty, two types are studied widely: time varying structured uncertainty [9–12] and polytopic-type uncertainty [13–15]. If the uncertainty is too large, the stable systems may becomes instable, the intensity of parameter matrices uncertainty is often the sources of instability and they can destabilize stable delay systems if it exceeds its limits. The instability depends on the intensity of parameter matrices uncertainty. Many people analyze the robust stability of parameter uncertainty systems for given structured uncertainty or polytopic-type uncertainty. For a stable delay system, if the intensity of parameter matrices uncertainty is low, the perturbed delay system may still be stable. Therefore, it is interesting to determine how much parameter matrices uncertainty of stable delay systems can withstand without losing global ⇑ Corresponding author. E-mail addresses: [email protected] (S. Zhu), [email protected] (Y. Zhang). 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.06.022
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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S. Zhu et al. / Commun Nonlinear Sci Numer Simulat xxx (2013) xxx–xxx
exponential stability. Shen et al. [10] investigated the robustness of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances. Although the various stability properties of stable delay systems have been extensively investigated using the Lyapunov and the linear matrix inequality methods, the robustness of the global stability for parameter matrices of systems is rarely analyzed directly by estimating the upper bounds of parameter matrices uncertainty level. Motivated by the above discussion, our aim in this paper is directly to quantify the parameter uncertainty level for stable systems only use the definition of stability. That is, we characterize the robustness for parameter matrices in general form of systems by deriving the upper bounds of parameter matrices uncertainty for global exponential stability. We prove theoretically that, for globally exponentially stable systems, if additive parameter matrices uncertainty is smaller than the derived upper bounds herein, then the systems are guaranteed to be globally exponentially stable. The remainder of this paper is organized as follows. Section 2 provides problem formulation. Section 3 discusses the impact of uncertainty in parameter matrices for the global exponential stability of systems. In Section 4, two numerical examples are given to illustrate the theoretical results. Finally, concluding remarks are given in Section 5. 2. Problem formulation Throughout this paper, unless otherwise specified, Rn and Rnm denote, respectively, the n-dimensional Euclidean space and the set of n m real matrices. Let ðX; F ; fF t gtP0 ; PÞ be a complete probability space with a filtration fF t gtP0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). xðtÞ be a scalar Brownian motion defined on the probability space. If A is a matrix, its operator norm is denoted by kAk = sup fjAxj : jxj ¼ 1g, where j j is ; 0; Rn Þ as the family of all F 0 measurable Cð½s ; 0; Rn Þ valued random variables the Euclidean norm. Denote L2F 0 ð½s 2 6 h 6 0g such that sups6h60 EjwðhÞj < 1 where Efg stands for the mathematical expectation operator with w ¼ fwðhÞ : s respect to the given probability measure P. Consider a stochastic delayed systems
dxðtÞ ¼ ½AxðtÞ þ Bxðt sðtÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdxðtÞ; xðtÞ ¼ wðt t 0 Þ 2
L2F 0 ð½t0 T
n
; t 0 ; R Þ; s
t > t0
6 t 6 t0 t0 s
n
ð1Þ n
nn
where xðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞ 2 R is the state vector of the system, t 0 2 Rþ and w 2 R are the initial values, A 2 R ; B 2 Rnn are parameter matrices, W 1 ; W 2 are Rnn matrices, which stand for noise function. We assume that origin is an equilibrium ; s0 ðtÞ 6 l < 1; w ¼ fwðsÞ : s 6 s 6 0g 2 point of (1), sðtÞ is a time varying delay that satisfies sðtÞ : ½t0 ; þ1Þ ! ½0; s ; 0; Rn Þ; s is the maximum of delay. Cð½s Now we define the global exponential stability of the state of systems (1). ; 0; Rn Þ, Definition 1 [16]. Systems (1) is said to be almost sure globally exponentially stable if for any t 0 2 Rþ ; w 2 L2F 0 ð½s there exist a > 0 and b > 0 such that 8t P t0 ; jxðt; t 0 ; wÞj 6 akwk expðbðt t0 ÞÞ hold almost surely; i.e., the Lyapunov exponent lim supt!1 ðln jxðt; t0 ; wÞj=tÞ < 0 almost surely, where xðt; t 0 ; wÞ is the state of system (1). Systems (1) is said to be ; 0; Rn Þ, there exist a > 0 and b > 0 such that mean square globally exponentially stable if for any t0 2 Rþ ; w 2 L2F 0 ð½s 8t P t0 ; Ejxðt; t0 ; wÞj2 6 akwk expðbðt t0 ÞÞ hold; i.e., the Lyapunov exponent lim supt!1 ðln ðEjxðt; t 0 ; wÞj2 Þ=tÞ < 0, where xðt; t 0 ; wÞ is the state of systems (1). From the definition, it is clear that the almost sure global exponential stability of systems (1) implies the mean square global exponential stability of systems (1) [16], but not vice versa. However, as (1) is a linear system, we have the following lemma [16, Theorem 6.2, pp. 175].
Lemma 1. The global exponential stability in sense of mean square of systems (1) implies the almost sure global exponential stability of systems (1). Numerous criteria for ascertaining the global exponential stability of systems (1) have been developed; e.g., [10,11,15] and the references therein.
3. Main results Now, the question is given a globally exponentially stable stochastic system, how much the parameter uncertainty intensity of parameter matrices the systems can bear? We first consider the parameter uncertainty intensity adding to parameter matrix A, the perturbed systems changes as
dyðtÞ ¼ ½ðA þ DAÞyðtÞ þ Byðt sðtÞÞdt þ ½W 1 yðtÞ þ W 2 yðt sðtÞÞdxðtÞ; ; t 0 ; Rn Þ; yðtÞ ¼ wðt t 0 Þ 2 L2F 0 ð½t 0 s
t > t0
6 t 6 t0 t0 s
ð2Þ
where the notations of (2) are the same as in Section 2, DA stands for parameter matrix uncertainty intensity. Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
S. Zhu et al. / Commun Nonlinear Sci Numer Simulat xxx (2013) xxx–xxx
3
It is easily known that systems (2) have a unique state for any initial value t 0 ; w and the origin point is the equilibrium point. We will characterize how much the parameter matrix uncertainty intensity that system (2) can bear to still be globally exponentially stable. Theorem 1. Let systems (1) be mean square globally exponentially stable, parameter uncertainty systems (2) is mean square globally exponentially stable and almost sure globally exponentially stable if kDAk < r, where r is the positive solution of the transcend equation for kDAk
c1 Þ þ 2a2 expð2bs Þ ¼ 1 2c2 expð3s
ð3Þ
where
kAk2 þ 2kDAk2 þ 3kBk2 þ 4kW 1 k2 þ 12kW 2 k2 þ 108s kBk2 þ 24kW 2 k2 c1 ¼ 18s " # ! kW 2 k2 kBk2 4s 2 2 2 2 2 kAk þ kDAk þ kW 1 k þ þ 4s 4s ð1 lÞ ð1 lÞ 2 3 kBk2 þ 2s 2 kW 2 k2 2s kBk2 þ 24kW 2 k2 4s ð1 lÞ1 þ þs c2 ¼ 108s ð1 lÞ " # ! # kW 2 k2 kBk2 2s 2 2 kAk2 þ kDAk2 þ kW 1 k2 a2 =b þ þ 2s þ 2s ð1 lÞ ð1 lÞ h i 2 2 2 kDAk þ 54s kBk þ 12kW 2 k ð1 þ expð2bs ÞÞ a2 =b: þ 18s 6 s 6 0g; for simplicity, we write xðt; t 0 ; wÞ; yðt; t0 ; wÞ as xðtÞ; yðtÞ respectively. From (1) and (2), Proof. Fix t 0 ; w ¼ fwðsÞ : s we have
xðtÞ yðtÞ ¼
Z
t
½AðxðsÞ yðsÞÞ DAyðsÞ þ Bðxðs sðsÞÞ yðs sðsÞÞÞds
t0
þ
Z
t
½W 1 ðxðsÞ yðsÞÞ þ W 2 ðxðs sðsÞÞ yðs sðsÞÞÞdwðsÞ:
ð4Þ
t0 2
2
, from the fundamental inequality ða þ b þ cÞ 6 3ða2 þ b þ c2 Þ and Holder inequality [16], we have When t 6 t0 þ 3s
Z t 2 EjxðtÞ yðtÞj2 6 2E ½AðxðsÞ yðsÞÞ DAyðsÞ þ Bðxðs sðsÞÞ yðs sðsÞÞÞds t0
Z t 2 þ 2E ½W 1 ðxðsÞ yðsÞÞ þ W 2 ðxðs sðsÞÞ yðs sðsÞÞÞdwðsÞ t0 8 9 Z t 2 Z t 2 = <Z t 2 6 6E ½AðxðsÞ yðsÞÞdsj þ DAyðsÞds þ Bðxðs sðsÞÞ yðs sðsÞÞÞ ds ; : t0
þ 2E
Z
t
t0
j½W 1 ðxðsÞ yðsÞÞ þ W 2 ðxðs sðsÞÞ yðs sðsÞÞÞj2 ds
t0
6 18s
t0
Z th
i kAk2 EjxðsÞ yðsÞj2 þ kBk2 Ejxðs sðsÞÞ yðs sðsÞÞj2 ds
t0
kDAk2 þ 18s þ 4kW 2 k2
Z
Z
t
EjyðsÞ xðsÞ þ xðsÞj2 ds þ 4kW 1 k2
t0 t
Z
t
EjxðsÞ yðsÞj2 ds
t0
Ejxðs sðsÞÞ yðs sðsÞÞj2 ds
t0
h iZ t ðkAk2 þ 2kDAk2 þ 3kBk2 Þ þ 4kW 1 k2 þ 12kW 2 k2 EjxðsÞ yðsÞj2 ds 6 18s Z t kBk2 þ 12kW 2 k2 EjyðsÞ yðs sðsÞÞj2 ds þ 54s
t0
t0
iZ t kDAk2 þ ð108s kBk2 þ 24kW 2 k2 Þð1 þ expð2bs ÞÞ EjxðsÞj2 ds: þ 36s h
ð5Þ
t0
, from (2), In addition, when t P t 0 þ s
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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Z
t
Z
EjyðsÞ yðs sðsÞÞj2 ds 6
t 0 þs
t
ds
Z
t 0 þs
s
nh
i o kAk2 þ kDAk2 þ 4kW 1 k2 EjyðrÞj2 þ 4s kBk2 þ 4kW 2 k2 Ejyðr sðrÞÞj2 dr: 8s
ss
ð6Þ By reversing the order of integral, we have
Z
t
ds
t 0 þs
Z
s
EjyðrÞj2 dr ¼
ss
Z
t
dr
Z
t0
;rÞ minðrþs ;tÞ ðt 0 þs
EjyðrÞj2 ds 6 s
Z
max
t
EjyðrÞj2 dr:
ð7Þ
t0
For the same reason, we have
Z
t
ds
t 0 þs
Z
s
Z
Ejyðr sðrÞÞj2 dr ¼
ss
t
dr
Z
t0
;rÞ minðrþs ;tÞ ðt 0 þs
max
s2
6
Ejyðr sðrÞÞj2 ds 6 !
sup EjyðsÞj
ð1 lÞ
2
þ
6s6t 0 t 0 s
Z
s ð1 lÞ
t
Z
s ð1 lÞ
t
EjyðuÞj2 du
t 0 s
EjyðuÞj2 du:
ð8Þ
t0
, by substituting (7) and (8) into (6), we have So, when t P t 0 þ s
Z
t
t 0 þs
EjyðsÞ yðs sðsÞÞj2 ds 3 kBk2 þ 4s 2 kW 2 k2 4s
6
! sup EjyðsÞj2
ð1 lÞ 6s6t 0 t 0 s " # )Z t kW 2 k2 kBk2 4s 2 2 2 2 þ EjyðsÞj2 ds: 4s 2 kAk þ kDAk þ þ 4skW 1 k ð1 lÞ ð1 lÞ t0
( þ
ð9Þ
, we have Substituting (9) into (5), when t P t 0 þ s
h iZ t ðkAk2 þ 2kDAk2 þ 3kBk2 Þ þ 4kW 1 k2 þ 12kW 2 k2 EjxðtÞ yðtÞj2 6 18s EjxðsÞ yðsÞj2 ds
2
2
kBk þ 12kW 2 k þ 54s
Z
t 0 þs
þ
Z
t
EjyðsÞ yðs sðsÞÞj2 ds
t0
t 0 þs
t0
i Z t kDAk þ 108s kBk þ 24kW 2 k2 ð1 þ expð2bs ÞÞ þ 36s a2 kwk2 expð2bðs t0 ÞÞds h
h
2
2
2
2
kAk þ 2kDAk þ 3kBk 6 18s þs ð1 lÞ1 ½s
sup 6s6t 0 þs t 0 s
iZ
t0
kBk2 þ 24kW 2 k2 EjxðsÞ yðsÞj2 ds þ 108s þ 4kW 1 k þ 12kW 2 k t0 ! kBk2 þ 12kW 2 k2 EjyðsÞj2 þ 54s
2
8 < 4s 3 kBk2 þ 4s 2 kW 2 k2
2
2
!
t
# ! kW 2 k2 kBk2 4s 2 kW 1 k þ sup EjyðsÞj þ 4s 2ðkAk þ kDAk Þ þ þ 4s : ð1 lÞ ð1 lÞ ð1 lÞ 6s6t 0 t 0 s ! h Z t i kDAk2 þ 54s kBk2 þ 12kW 2 k2 ð1 þ expð2bs ÞÞ a2 =b EjyðsÞ xðsÞ þ xðsÞj2 ds þ 18s sup EjyðsÞj2 : 2
"
2
2
2
6s6t0 t 0 s
t0
ð10Þ
From (10), we further have
n kAk2 þ 2kDAk2 þ 3kBk2 þ 4kW 1 k2 þ 12kW 2 k2 EjxðtÞ yðtÞj2 6 18s " # !) Z t kW 2 k2 kBk2 4s 2 2 2 2 2 2 þ þ 108skBk þ 24kW 2 k EjxðsÞ yðsÞj2 ds þ 4skW 1 k 4s 2 kAk þ kDAk þ ð1 lÞ ð1 lÞ t0 8 2 < 3 kBk2 þ 2s 2 kW 2 k2 2s 2 2 4 1 kBk þ 24kW 2 k s þ sð1 lÞ þ þ 108s : ð1 lÞ " # ! # 2 2 kW 2 k kBk 2s 2 2 2 2 2 2 kAk þ kDAk þ kW 1 k a =b þ þ 2s þ 2s ð1 lÞ ð1 lÞ ! h i o 2 2 2 2 2 kDAk þ 54s kBk þ 12kW 2 k ð1 þ expð2bs ÞÞ a =b sup EjyðsÞj þ 18s ¼: c1
Z
t
t0
! EjxðsÞ yðsÞj2 ds þ c2
sup 6s6t 0 þs t 0 s
EjyðsÞj2 :
6s6t 0 þs t 0 s
ð11Þ
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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6 t 6 t0 þ 3s , applying the Gronwall inequality [16], we obtain When t 0 þ s
! c1 Þ EjxðtÞ yðtÞj2 6 c2 expð3s
EjyðsÞj2 :
sup
ð12Þ
6s6t 0 þs t 0 s
Therefore
! c1 Þ þ 2a2 expð2bðt t 0 ÞÞ EjyðtÞj2 6 2EjxðtÞ yðtÞj2 þ 2EjxðtÞj2 6 ½2c2 expð3s
EjyðsÞj2 :
sup 6s6t0 þs t 0 s
ð13Þ
6 t 6 t 0 þ 3s Thus, when t 0 þ s
c1 Þ þ 2a2 expð2bs Þ EjyðtÞj2 6 2c2 expð3s
! sup
EjyðsÞj2
! ¼: ^c
6s6t 0 þs t 0 s
EjyðsÞj2 :
sup
ð14Þ
6s6t 0 þs t 0 s
c1 Þ þ 2a2 expð2bs Þ. From (3), when kDAk < r, then ^c < 1. Select c ¼ ln ^c=s , we can conclude where ^c ¼ 2c2 expð3s
! 2
sup 6t6t 0 þ3s t 0 þs
Þ Ejyðt; t 0 ; wÞj 6 expðcs
sup 6t6t 0 þs t 0 s
2
Ejyðt; t 0 ; wÞj
ð15Þ
:
Then, for any positive integer m ¼ 1; 2; . . ., denote
ðt 0 þ ð2m 1Þs ; t 0 ; wÞ :¼ fyðt 0 þ ð2m 1Þs þ s; t0 ; wÞ : s 6 s 6 0g 2 Cð½s ; 0; Rn Þ: y From the existence and uniqueness of the state of system (2), we have
; y ðt0 þ ð2m 1Þs ; t 0 ; wÞÞ: yðt; t0 ; wÞ ¼ yðt; t 0 þ ð2m 1Þs , from (15) When t P t 0 þ ð2m 1Þs
! sup
2
Þ Ejyðt; t 0 ; wÞj 6 expðcs
6t6t 0 þð2mþ1Þs t 0 þð2m1Þs
sup
Ejyðt; t 0 ; wÞj
6t6t 0 þð2m1Þs t 0 þð2m3Þs
Þ 6 expðcms
sup 6t6t 0 þs t 0 s
Ejyðt; t 0 ; wÞj2
2
! Þ; ¼ c0 expðcms
, there must have a positive integer m such that where c0 ¼ supt0 s6t6t0 þs Ejyðt; t 0 ; wÞj2 . So for any t > t0 þ ð2m 1Þs 6 t 6 t 0 þ ð2m þ 1Þs , we have t 0 þ ð2m 1Þs
Ejyðt; t 0 ; wÞj2 6 c0 exp
cs
2
c exp ðt t 0 Þ : 2
ð16Þ
6 t 6 t0 þ s . So systems (2) is mean square globally exponentially stable. According to Condition (16) is also true when t0 s Lemma 1, systems (2) is also almost sure globally exponentially stable. h Remark 1. Different from the robust stability of time varying structured uncertainty and polytopic-type uncertainty, the parameter uncertainty are given. Theorem 1 gives a more general result for parameter uncertainty norm condition of robust stability. Remark 2. Different from the traditional Lyapunov stability theory or linear matrix inequality method. We directly obtain the stability results only from the definition of stochastic delayed systems. Let us consider another condition. If we add parameter uncertainty to matrix B, we have a new systems, where DB stands for parameter uncertainty intensity.
dyðtÞ ¼ ½AyðtÞ þ ðB þ DBÞyðt sðtÞÞdt þ ½W 1 yðtÞ þ W 2 yðt sðtÞÞdxðtÞ; t > t 0 ; t 0 ; Rn Þ; t0 s 6 t 6 t0 yðtÞ ¼ wðt t 0 Þ 2 L2F 0 ð½t 0 s
ð17Þ
we have the following theorem. Theorem 2. Let systems (1) be mean square globally exponentially stable, parameter uncertainty systems (17) is mean square globally exponentially stable and almost sure globally exponentially stable if kDBk < k, where k is the positive solution of the transcend equation for kDBk
c3 Þ þ 2a2 expð2bs Þ ¼ 1 2c4 expð3s
ð18Þ
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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S. Zhu et al. / Commun Nonlinear Sci Numer Simulat xxx (2013) xxx–xxx
where
kAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 c3 ¼ 18s " # ! h i kW 2 k2 2kBk2 þ 2kDBk2 4s 2 2 2 2 2 2 ðkBk þ kDBk Þ þ 24kW 2 k kAk þ kW 1 k þ þ 108s þ 4s 4s ð1 lÞ ð1 lÞ 2 3 kBk2 þ kDBk2 þ 2s 2 kW 2 k2 h i 4s kBk2 þ kDBk2 þ 24kW 2 k2 4s ð1 lÞ1 þ þs c4 ¼ 108s ð1 lÞ " # ! # kW 2 k2 2kBk2 þ 2kDBk2 2s 2 kAk2 þ kW 1 k2 a2 =b þ 2s þ 2s þ ð1 lÞ ð1 lÞ h i kDBk2 þ ð54s kBk2 þ 12kW 2 k2 Þð1 þ expð2bs ÞÞ a2 =b: þ 27s The proof of Theorem 2 is similar to Theorem 1, so we omit it here. Now, we consider this condition, if the matrices A; B all have uncertainties, that is
dyðtÞ ¼ ½ðA þ DAÞyðtÞ þ ðB þ DBÞyðt sðtÞÞdt þ ½W 1 yðtÞ þ W 2 yðt sðtÞÞdxðtÞ; t > t0 ; t 0 ; Rn Þ; t0 s 6 t 6 t0 : yðtÞ ¼ wðt t 0 Þ 2 L2F 0 ð½t 0 s
ð19Þ
For this conditions, we have following theorem. Theorem 3. Let systems (1) be mean square globally exponentially stable, parameter uncertainty systems (19) is mean square globally exponentially stable and almost sure globally exponentially stable if ðkDAk; kDBkÞ are in the inner of the transcend equation curve
c5 Þ þ 2a2 expð2bs Þ ¼ 1 2c6 expð3s
ð20Þ
where
kAk2 þ 2kDAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 c5 ¼ 24s " # ! h i kW 2 k2 kBk2 þ kDBk2 4s 2 2 2 2 2 2 2 þ 144s kBk þ kDBk þ 24kW 2 k 8s kAk þ kDAk þ þ þ 4skW 1 k ð1 lÞ ð1 lÞ 2 3 kBk2 þ kDBk2 þ 2s 2 kW 2 k2 h i 4s 2 2 2 4 1 kBk þ kDBk þ 24kW 2 k s þ sð1 lÞ þ c6 ¼ 144s ð1 lÞ " # ! # kW 2 k2 kBk2 þ kDBk2 2s 2 kAk2 þ kDAk2 þ kW 1 k2 a2 =b þ þ 4s þ 2s ð1 lÞ ð1 lÞ h i 2 2 2 kDAk þ 36s kDBk þ 72s kBk þ 12kW 2 k2 ð1 þ expð2bs ÞÞ a2 =b: þ ½24s Proof. Since the proof of Theorem 3 is similar to those of Theorems 1 and 2, we give only an outline of the proof of 6 s 6 0g; for simplicity, we write xðt; t 0 ; wÞ; yðt; t 0 ; wÞ as xðtÞ; yðtÞ respectively. From (1) Theorem 3. Fix t0 ; w ¼ fwðsÞ : s , from the Holder inequality [16], we have and (19), and when t 6 t 0 þ 3s
h i Z t kAk2 þ 2kDAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 EjxðtÞ yðtÞj2 6 24s EjxðsÞ yðsÞj2 ds i Z t ðkBk2 þ kDBk2 Þ þ 12kW 2 k2 þ 72s EjyðsÞ yðs sðsÞÞj2 ds
t0
h
t0
iZ t kDAk þ 72s kDBk þ 144s kBk2 þ 24kW 2 k2 ð1 þ expð2bs ÞÞ EjxðsÞj2 ds: þ 48s h
2
2
ð21Þ
t0
, from (19), In addition, when t P t 0 þ s
Z
Z t Z s nh i kAk2 þ kDAk2 þ 4kW 1 k2 EjyðrÞj2 EjyðsÞ yðs sðsÞÞj2 ds 6 ds 8s t 0 þs t 0 þs ss h i o 2 2 2 kBk þ kDBk þ 4kW 2 k Ejyðr sðrÞÞj2 dr: þ 8s t
ð22Þ
, by substituting (7) and (8) into (22), we have So, when t P t 0 þ s Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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Z
h
t
2
EjyðsÞ yðs sðsÞÞj ds 6
i 3 kBk2 þ kDBk2 þ 4s 2 kW 2 k2 8s
! sup EjyðsÞj2
ð1 lÞ
t 0 þs
(
6s6t 0 t 0 s
"
# )Z t kW 2 k2 kBk þ kDBk2 4s 2 þ EjyðsÞj2 ds: þ 4skW 1 k ð1 lÞ ð1 lÞ t0 2
2 kAk2 þ kDAk2 þ 8s
þ
ð23Þ
, we have Substituting (23) into (22), when t P t 0 þ s
h iZ t kAk2 þ 2kDAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 EjxðtÞ yðtÞj2 6 24s EjxðsÞ yðsÞj2 ds h
2
2
kBk þ kDBk þ 72s
þ 12kW 2 k
2
iZ
t 0 þs
þ
Z
t0
EjyðsÞ yðs sðsÞÞj2 ds
t
t 0 þs
t0
i Z t kDAk2 þ 72s kDBk2 þ 144s kBk2 þ 24kW 2 k2 ð1 þ expð2bs ÞÞ a2 kwk2 expð2bðs t0 ÞÞds þ 48s t0 n h i kAk2 þ 2kDAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 þ 144s kBk2 þ kDBk2 þ 24kW 2 k2 6 24s " # !) Z t kW 2 k2 kBk2 þ kDBk2 4s 2 2 2 2 kAk þ kDAk þ kW 1 k þ EjxðsÞ yðsÞj2 ds þ 4s 8s ð1 lÞ ð1 lÞ t0 8 2
¼: c5
Z
t
6s6t 0 þs t 0 s
! EjxðsÞ yðsÞj2 ds þ c6
t0
EjyðsÞj2 :
sup
ð24Þ
6s6t 0 þs t 0 s
The rest of the proof can be completed similarly to the proof of Theorem 1. h Remark 3. Theorems 1–3 show that when the system is globally exponentially stable, the uncertainty in parameter matrices systems can be also mean square globally exponentially stable and almost sure globally exponentially stable, provided that the parameter uncertainty is smaller than given upper bounds. Remark 4. From the proofs of Theorems 1–3, we can see that the upper bounds of parameter uncertainty intensity are derived via subtle inequalities and can be estimated by solving transcendental equation. As transcendental equation can be solved by using some software such as MATLAB, the derived conditions in these theorems can be verified easily.
4. Illustrative examples
Example 1. Consider a bi-state systems
dxðtÞ ¼ ½AxðtÞ þ Bxðt sðtÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdwðtÞ:
ð25Þ
The parameters are as follows:
A¼
1
0
0
1
;
B¼
0:1
0
0
0:1
;
W1 ¼
0:1
0
0
0:1
;
W2 ¼
0 0:1 0 0:1
;
T
sðtÞ ¼ 0:005sint; xð0Þ ¼ ½0:6; 0:3 . Hence, according to Theorem 4.2 in [15], systems (25) is globally exponentially stable with a ¼ 0:5; b ¼ 0:5. In the presence of uncertainty in parameter matrices, the systems become new systems: dyðtÞ ¼ ½ðA þ DAÞyðtÞ þ ðB þ DBÞyðt sðtÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdwðtÞ;
ð26Þ
where sðtÞ is the time-varying delay, DA; DB is the uncertainty intensity in parameter matrices. According to Theorem 3, let l ¼ 0, and by solving (20)
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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n 0:2472 þ 0:72kDBk2 kDAk2 þ kDBk2 þ 203:01 104 þ 0:12kDAk2 o n þ0:88kDBk2 þ 0:24 exp 0:015 0:2836 þ 0:24kDAk2 þ 0:36kDBk2 h ih i o þ 0:2472 þ 0:72kDBk2 4:2 þ 0:2kDAk2 þ 0:2kDBk2 105 þ 0:4998 ¼ 1:
ð27Þ
Using the software of MATLAB, we can obtain its stability region for ðkDAk; kDBkÞ. Fig. 1 shows the maximum stability region for ðkDAk; kDBkÞ in (26) if systems (25) is mean square globally exponentially stable as the coefficient a ¼ 0:5; b ¼ 0:5. Fig.2 depicts the transient states of perturbed systems (26) with ðDA; DBÞ ¼ ð1:4I; 0:1IÞ as the uncertainty intensify in parameter matrices. From Theorem 3, as it located in the maximum stability region for systems (26), it shows that systems (26) is means square globally exponentially stable and also almost sure globally exponentially stable, as the parameter ðkDAk; kDBkÞ in the inner of transcend equation of (27). Figs. 3–5 show that the transient states of perturbed systems (26) with ðDA; DBÞ as ð1:8I; 0Þ; ð0; 2:2IÞ; ð1:8I; 0:6IÞ. From Theorems 1–3, it shows that when the conditions in Theorems 1–3 do not hold, the systems (26) may become unstable.
0.7 0.6
||Δ B||
0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
||Δ A||
1.5
Fig. 1. The maximum stability region with ðkDAk; kDBkÞ in perturbed systems (26).
0.6
y1 y2
0.5 0.4 0.3
y
0.2 0.1 0 −0.1 −0.2 −0.3
0
1
2
t
3
4
5
Fig. 2. The transient state of perturbed systems (26) with ðDA; DBÞ ¼ ð1:4I; 0:1IÞ and sðtÞ ¼ 0:005.
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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Example 2. Consider a stochastic delayed recurrent neural networks (SDRNN) model
dxðtÞ ¼ ½AxðtÞ þ Bf ðxðtÞÞ þ Df ðxðt sðtÞÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdwðtÞ:
ð28Þ
The parameters are as follows:
A¼
1 0 0 1
;
B¼
0
2
2
0
;
D¼
2 0 0 2
;
W1 ¼
0:1
0
0
0:1
;
W2 ¼
0 0:01
0 0:01
T
f ðxj Þ ¼ tanhðxj Þ; ðj ¼ 1; 2Þ; sðtÞ ¼ 0:00025ð1 þ sintÞ; xð0Þ ¼ ½0:6; 0:3 . Hence, according to Theorem 3.1 in [16, Theorem 3.1, pp. 367]]. SDRNN (28) is globally exponentially stable with a ¼ 0:5; b ¼ 0:5. In the presence of parameter uncertainty in connection weight matrices, the SDRNN becomes a new perturbed SDRNN:
dyðtÞ ¼ ½AyðtÞ þ ð1 þ rÞBf ðyðtÞÞ þ ð1 þ kÞDf ðyðt sðtÞÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdwðtÞ;
1
ð29Þ
y1 y2
0
−1
y −2 −3
−4
−5
0
1
2
t
3
4
5
Fig. 3. The transient state of perturbed systems (26) with ðDA; DBÞ ¼ ð1:8I; 0Þ and sðtÞ ¼ 0:005.
5
y1 y2
0
−5
y −10 −15
−20
−25
0
1
2
t
3
4
5
Fig. 4. The transient state of perturbed system (26) with ðDA; DBÞ ¼ ð0; 2:2IÞ and sðtÞ ¼ 0:005.
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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10
y1 y2
0 −10
y
−20 −30 −40 −50 −60
0
1
2
t
3
4
5
Fig. 5. The transient state of perturbed system (26) with ðDA; DBÞ ¼ ð1:8I; 0:6IÞ and sðtÞ ¼ 0:005.
1.5
1
λ
0.5
0
−0.5
−1
−1.5 −1.5
−1
−0.5
0 σ
0.5
1
1.5
Fig. 6. The maximum stability region with (r,k) in SDRNN (29).
where sðtÞ is the time-varying delay, r; k is the parameter uncertainty intensity in connection weight matrices. Similarly to Theorem 3, let l ¼ 0, and by solving (20)
n o ð0:3624 þ 0:36k2 Þð1:5r2 þ 1:5003k2 þ 3r þ 3:0006k þ 204:385Þ 105 þ 0:06r2 þ 0:09k2 þ 0:3624 n o exp 0:015 0:2962 þ 0:12r2 þ 0:18k2 þ 0:36k þ ½0:3624 þ 0:36k2 ½3:37 þ 0:6r2 þ 0:6k2 þ 1:2r þ 1:2k 105 þ 0:4998 ¼ 1:
ð30Þ
Fig. 6 shows the maximum stability region for ðr; kÞ in SDRNN (29).
5. Conclusion In this paper, the robustness of stochastic delayed systems with uncertainty in parameter matrices is analyzed. The upper bounds of parameter uncertainty are derived for stochastic delayed systems to remain global exponential stability. The Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
S. Zhu et al. / Commun Nonlinear Sci Numer Simulat xxx (2013) xxx–xxx
11
results herein provide a theoretic basis for the design and application of stochastic delayed systems in the presence of uncertainty in parameter matrices. Further investigations may be aimed at the improvement of the upper bounds to allow bigger stability margins for withstanding parameter uncertainty. Acknowledgments The authors thank the Editor Albert Luo and the anonymous reviewers for their insightful comments and valuable suggestions, which have helped us in finalizing the paper. This work was supported by the National Natural Science Foundation of China with Grant No. 61203055 and supported by the Fundamental Research Funds for the Central Universities of 2012QNA48. References [1] Hale J. Functional Differential Equations. New York: Springer Verlag; 1977. [2] Richard JP. Time-delay systems: an overview of some recent advances and open problems. Automatica 2008;39:1667–94. [3] Yuan K, Cao J, Li H. Robust stability of switched Cohen–Grossberg neural networks with mixed time-varying delays. IEEE Trans Syst Man Cybern B, Cybern 2006;36:1356–63. [4] Shen Y, Wang J. Almost sure exponential stability of recurrent neural networks with Markovian switching. IEEE Trans Neural Netw 2009;20:840–55. [5] Xu S, aLam J. Improved delay-dependent stability criteria for time-delay systems. IEEE Trans Autom Control 2005;50:384–7. [6] Meng X, Lam J, Du B, Gao H. A delay-partitioning approach to the stability analysis of discrete-time systems. Automatica 2010;46:610–4. [7] Jiang F, Shen Y, Liu L. Taylor approximation of the solutions of stochastic differential delay equations with Poisson jump. Commun Nonlinear Sci Numer Simulat 2011;16:798–804. [8] Jiang F, Shen Y, Hu J. Stability of the split-step backward Euler scheme for stochastic delay integro–differential equations with Markovian switching. Commun Nonlinear Sci Numer Simulat 2011;16:814–21. [9] Liu P. Robust exponential stability for uncertain time-varying delay systems with delay dependence. J Franklin I 2009;346:958–68. [10] Shen Y, Wang J. Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances. IEEE Trans Neural Netw Learn Syst 2012;23:87–96. [11] Zhu S, Shen Y, Chen G. Exponential passivity of neural networks with time-varying delay and uncertainty. Phys Lett A 2010;375:136–42. [12] Zhu S, Shen Y, Liu L. Exponential stability of uncertain stochastic neural networks with markovian switching. Neural Process Lett 2010;32:293–309. [13] He Y, Wu M, She J, Liu G. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties. IEEE Trans Autom Control 2004;49:828–32. [14] Gao H, Shi P, Wang J, Bittanti, Cuzzola FA. Parameter-dependent robust stability of uncertain time-delay systems. J Comput Appl Math 2007;206:366–73. [15] Mao X. Stability and stabilization of stochastic differential delay equations. IET Control Theory Appl 2007;1:1551–66. [16] Mao X. Stochastic differential equations and applications. second ed.. Chichester: Harwood; 2007.
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Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems Song Zhu ⇑, Kai Zhong, Yufeng Zhang College of Sciences, China University of Mining and Technology, Xuzhou 221116, China
a r t i c l e
i n f o
Article history: Received 23 September 2012 Received in revised form 12 May 2013 Accepted 8 June 2013 Available online xxxx Keywords: Stochastic delayed systems Global exponential stability Robustness
a b s t r a c t In this paper, we analyze the robustness of global exponential stable stochastic delayed systems subject to the uncertainty in parameter matrices. Given a globally exponentially stable systems, the problem to be addressed here is how much uncertainty in parameter matrices the systems can withstand to be globally exponentially stable. The upper bounds of the parameter uncertainty intensity are characterized by using transcendental equation for the systems to sustain global exponential stability. Moreover, we prove theoretically that, the globally exponentially stable systems, if additive uncertainties in parameter matrices are smaller than the upper bounds arrived at here, then the perturbed systems are guaranteed to also be globally exponentially stable. Two numerical examples are provided here to illustrate the theoretical results. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Time delays are often encountered in various practical systems such as chemical processes, neural networks and long transmission lines in pneumatic systems [1,2]. It has been shown that the existence of time-delays may lead to oscillation, divergence, instability, greatly increasing the difficulty of stability analysis and control design. Many researchers in the field of control theory and engineering study the robust control of time-delay systems. The main methods of stability analysis can be classified into two types: frequency-domain and time-domain. The former use the sum of squares technique. As to the time-domain approach, Lyapunov functional is a powerful tool, which can deal with time varying delays. For most successful applications of the systems, the stability is usually a prerequisite. The stability of the systems depends mainly on their parametrical configuration. Moreover, in the applications of the systems, external random disturbances and time delays are common and hardly avoided. It is known that random disturbances and time delays in the systems may result in oscillation or instability of the nonlinear systems. The stability analysis of the delayed systems and the systems with external random disturbances has been widely investigated in recent years (see, e.g., [3–8], and the references cited therein). In practice, when we estimate systems parameter matrices, there are always some uncertainty and errors. For the parameter matrices uncertainty, two types are studied widely: time varying structured uncertainty [9–12] and polytopic-type uncertainty [13–15]. If the uncertainty is too large, the stable systems may becomes instable, the intensity of parameter matrices uncertainty is often the sources of instability and they can destabilize stable delay systems if it exceeds its limits. The instability depends on the intensity of parameter matrices uncertainty. Many people analyze the robust stability of parameter uncertainty systems for given structured uncertainty or polytopic-type uncertainty. For a stable delay system, if the intensity of parameter matrices uncertainty is low, the perturbed delay system may still be stable. Therefore, it is interesting to determine how much parameter matrices uncertainty of stable delay systems can withstand without losing global ⇑ Corresponding author. E-mail addresses: [email protected] (S. Zhu), [email protected] (Y. Zhang). 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.06.022
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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exponential stability. Shen et al. [10] investigated the robustness of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances. Although the various stability properties of stable delay systems have been extensively investigated using the Lyapunov and the linear matrix inequality methods, the robustness of the global stability for parameter matrices of systems is rarely analyzed directly by estimating the upper bounds of parameter matrices uncertainty level. Motivated by the above discussion, our aim in this paper is directly to quantify the parameter uncertainty level for stable systems only use the definition of stability. That is, we characterize the robustness for parameter matrices in general form of systems by deriving the upper bounds of parameter matrices uncertainty for global exponential stability. We prove theoretically that, for globally exponentially stable systems, if additive parameter matrices uncertainty is smaller than the derived upper bounds herein, then the systems are guaranteed to be globally exponentially stable. The remainder of this paper is organized as follows. Section 2 provides problem formulation. Section 3 discusses the impact of uncertainty in parameter matrices for the global exponential stability of systems. In Section 4, two numerical examples are given to illustrate the theoretical results. Finally, concluding remarks are given in Section 5. 2. Problem formulation Throughout this paper, unless otherwise specified, Rn and Rnm denote, respectively, the n-dimensional Euclidean space and the set of n m real matrices. Let ðX; F ; fF t gtP0 ; PÞ be a complete probability space with a filtration fF t gtP0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). xðtÞ be a scalar Brownian motion defined on the probability space. If A is a matrix, its operator norm is denoted by kAk = sup fjAxj : jxj ¼ 1g, where j j is ; 0; Rn Þ as the family of all F 0 measurable Cð½s ; 0; Rn Þ valued random variables the Euclidean norm. Denote L2F 0 ð½s 2 6 h 6 0g such that sups6h60 EjwðhÞj < 1 where Efg stands for the mathematical expectation operator with w ¼ fwðhÞ : s respect to the given probability measure P. Consider a stochastic delayed systems
dxðtÞ ¼ ½AxðtÞ þ Bxðt sðtÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdxðtÞ; xðtÞ ¼ wðt t 0 Þ 2
L2F 0 ð½t0 T
n
; t 0 ; R Þ; s
t > t0
6 t 6 t0 t0 s
n
ð1Þ n
nn
where xðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞ 2 R is the state vector of the system, t 0 2 Rþ and w 2 R are the initial values, A 2 R ; B 2 Rnn are parameter matrices, W 1 ; W 2 are Rnn matrices, which stand for noise function. We assume that origin is an equilibrium ; s0 ðtÞ 6 l < 1; w ¼ fwðsÞ : s 6 s 6 0g 2 point of (1), sðtÞ is a time varying delay that satisfies sðtÞ : ½t0 ; þ1Þ ! ½0; s ; 0; Rn Þ; s is the maximum of delay. Cð½s Now we define the global exponential stability of the state of systems (1). ; 0; Rn Þ, Definition 1 [16]. Systems (1) is said to be almost sure globally exponentially stable if for any t 0 2 Rþ ; w 2 L2F 0 ð½s there exist a > 0 and b > 0 such that 8t P t0 ; jxðt; t 0 ; wÞj 6 akwk expðbðt t0 ÞÞ hold almost surely; i.e., the Lyapunov exponent lim supt!1 ðln jxðt; t0 ; wÞj=tÞ < 0 almost surely, where xðt; t 0 ; wÞ is the state of system (1). Systems (1) is said to be ; 0; Rn Þ, there exist a > 0 and b > 0 such that mean square globally exponentially stable if for any t0 2 Rþ ; w 2 L2F 0 ð½s 8t P t0 ; Ejxðt; t0 ; wÞj2 6 akwk expðbðt t0 ÞÞ hold; i.e., the Lyapunov exponent lim supt!1 ðln ðEjxðt; t 0 ; wÞj2 Þ=tÞ < 0, where xðt; t 0 ; wÞ is the state of systems (1). From the definition, it is clear that the almost sure global exponential stability of systems (1) implies the mean square global exponential stability of systems (1) [16], but not vice versa. However, as (1) is a linear system, we have the following lemma [16, Theorem 6.2, pp. 175].
Lemma 1. The global exponential stability in sense of mean square of systems (1) implies the almost sure global exponential stability of systems (1). Numerous criteria for ascertaining the global exponential stability of systems (1) have been developed; e.g., [10,11,15] and the references therein.
3. Main results Now, the question is given a globally exponentially stable stochastic system, how much the parameter uncertainty intensity of parameter matrices the systems can bear? We first consider the parameter uncertainty intensity adding to parameter matrix A, the perturbed systems changes as
dyðtÞ ¼ ½ðA þ DAÞyðtÞ þ Byðt sðtÞÞdt þ ½W 1 yðtÞ þ W 2 yðt sðtÞÞdxðtÞ; ; t 0 ; Rn Þ; yðtÞ ¼ wðt t 0 Þ 2 L2F 0 ð½t 0 s
t > t0
6 t 6 t0 t0 s
ð2Þ
where the notations of (2) are the same as in Section 2, DA stands for parameter matrix uncertainty intensity. Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
S. Zhu et al. / Commun Nonlinear Sci Numer Simulat xxx (2013) xxx–xxx
3
It is easily known that systems (2) have a unique state for any initial value t 0 ; w and the origin point is the equilibrium point. We will characterize how much the parameter matrix uncertainty intensity that system (2) can bear to still be globally exponentially stable. Theorem 1. Let systems (1) be mean square globally exponentially stable, parameter uncertainty systems (2) is mean square globally exponentially stable and almost sure globally exponentially stable if kDAk < r, where r is the positive solution of the transcend equation for kDAk
c1 Þ þ 2a2 expð2bs Þ ¼ 1 2c2 expð3s
ð3Þ
where
kAk2 þ 2kDAk2 þ 3kBk2 þ 4kW 1 k2 þ 12kW 2 k2 þ 108s kBk2 þ 24kW 2 k2 c1 ¼ 18s " # ! kW 2 k2 kBk2 4s 2 2 2 2 2 kAk þ kDAk þ kW 1 k þ þ 4s 4s ð1 lÞ ð1 lÞ 2 3 kBk2 þ 2s 2 kW 2 k2 2s kBk2 þ 24kW 2 k2 4s ð1 lÞ1 þ þs c2 ¼ 108s ð1 lÞ " # ! # kW 2 k2 kBk2 2s 2 2 kAk2 þ kDAk2 þ kW 1 k2 a2 =b þ þ 2s þ 2s ð1 lÞ ð1 lÞ h i 2 2 2 kDAk þ 54s kBk þ 12kW 2 k ð1 þ expð2bs ÞÞ a2 =b: þ 18s 6 s 6 0g; for simplicity, we write xðt; t 0 ; wÞ; yðt; t0 ; wÞ as xðtÞ; yðtÞ respectively. From (1) and (2), Proof. Fix t 0 ; w ¼ fwðsÞ : s we have
xðtÞ yðtÞ ¼
Z
t
½AðxðsÞ yðsÞÞ DAyðsÞ þ Bðxðs sðsÞÞ yðs sðsÞÞÞds
t0
þ
Z
t
½W 1 ðxðsÞ yðsÞÞ þ W 2 ðxðs sðsÞÞ yðs sðsÞÞÞdwðsÞ:
ð4Þ
t0 2
2
, from the fundamental inequality ða þ b þ cÞ 6 3ða2 þ b þ c2 Þ and Holder inequality [16], we have When t 6 t0 þ 3s
Z t 2 EjxðtÞ yðtÞj2 6 2E ½AðxðsÞ yðsÞÞ DAyðsÞ þ Bðxðs sðsÞÞ yðs sðsÞÞÞds t0
Z t 2 þ 2E ½W 1 ðxðsÞ yðsÞÞ þ W 2 ðxðs sðsÞÞ yðs sðsÞÞÞdwðsÞ t0 8 9 Z t 2 Z t 2 = <Z t 2 6 6E ½AðxðsÞ yðsÞÞdsj þ DAyðsÞds þ Bðxðs sðsÞÞ yðs sðsÞÞÞ ds ; : t0
þ 2E
Z
t
t0
j½W 1 ðxðsÞ yðsÞÞ þ W 2 ðxðs sðsÞÞ yðs sðsÞÞÞj2 ds
t0
6 18s
t0
Z th
i kAk2 EjxðsÞ yðsÞj2 þ kBk2 Ejxðs sðsÞÞ yðs sðsÞÞj2 ds
t0
kDAk2 þ 18s þ 4kW 2 k2
Z
Z
t
EjyðsÞ xðsÞ þ xðsÞj2 ds þ 4kW 1 k2
t0 t
Z
t
EjxðsÞ yðsÞj2 ds
t0
Ejxðs sðsÞÞ yðs sðsÞÞj2 ds
t0
h iZ t ðkAk2 þ 2kDAk2 þ 3kBk2 Þ þ 4kW 1 k2 þ 12kW 2 k2 EjxðsÞ yðsÞj2 ds 6 18s Z t kBk2 þ 12kW 2 k2 EjyðsÞ yðs sðsÞÞj2 ds þ 54s
t0
t0
iZ t kDAk2 þ ð108s kBk2 þ 24kW 2 k2 Þð1 þ expð2bs ÞÞ EjxðsÞj2 ds: þ 36s h
ð5Þ
t0
, from (2), In addition, when t P t 0 þ s
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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Z
t
Z
EjyðsÞ yðs sðsÞÞj2 ds 6
t 0 þs
t
ds
Z
t 0 þs
s
nh
i o kAk2 þ kDAk2 þ 4kW 1 k2 EjyðrÞj2 þ 4s kBk2 þ 4kW 2 k2 Ejyðr sðrÞÞj2 dr: 8s
ss
ð6Þ By reversing the order of integral, we have
Z
t
ds
t 0 þs
Z
s
EjyðrÞj2 dr ¼
ss
Z
t
dr
Z
t0
;rÞ minðrþs ;tÞ ðt 0 þs
EjyðrÞj2 ds 6 s
Z
max
t
EjyðrÞj2 dr:
ð7Þ
t0
For the same reason, we have
Z
t
ds
t 0 þs
Z
s
Z
Ejyðr sðrÞÞj2 dr ¼
ss
t
dr
Z
t0
;rÞ minðrþs ;tÞ ðt 0 þs
max
s2
6
Ejyðr sðrÞÞj2 ds 6 !
sup EjyðsÞj
ð1 lÞ
2
þ
6s6t 0 t 0 s
Z
s ð1 lÞ
t
Z
s ð1 lÞ
t
EjyðuÞj2 du
t 0 s
EjyðuÞj2 du:
ð8Þ
t0
, by substituting (7) and (8) into (6), we have So, when t P t 0 þ s
Z
t
t 0 þs
EjyðsÞ yðs sðsÞÞj2 ds 3 kBk2 þ 4s 2 kW 2 k2 4s
6
! sup EjyðsÞj2
ð1 lÞ 6s6t 0 t 0 s " # )Z t kW 2 k2 kBk2 4s 2 2 2 2 þ EjyðsÞj2 ds: 4s 2 kAk þ kDAk þ þ 4skW 1 k ð1 lÞ ð1 lÞ t0
( þ
ð9Þ
, we have Substituting (9) into (5), when t P t 0 þ s
h iZ t ðkAk2 þ 2kDAk2 þ 3kBk2 Þ þ 4kW 1 k2 þ 12kW 2 k2 EjxðtÞ yðtÞj2 6 18s EjxðsÞ yðsÞj2 ds
2
2
kBk þ 12kW 2 k þ 54s
Z
t 0 þs
þ
Z
t
EjyðsÞ yðs sðsÞÞj2 ds
t0
t 0 þs
t0
i Z t kDAk þ 108s kBk þ 24kW 2 k2 ð1 þ expð2bs ÞÞ þ 36s a2 kwk2 expð2bðs t0 ÞÞds h
h
2
2
2
2
kAk þ 2kDAk þ 3kBk 6 18s þs ð1 lÞ1 ½s
sup 6s6t 0 þs t 0 s
iZ
t0
kBk2 þ 24kW 2 k2 EjxðsÞ yðsÞj2 ds þ 108s þ 4kW 1 k þ 12kW 2 k t0 ! kBk2 þ 12kW 2 k2 EjyðsÞj2 þ 54s
2
8 < 4s 3 kBk2 þ 4s 2 kW 2 k2
2
2
!
t
# ! kW 2 k2 kBk2 4s 2 kW 1 k þ sup EjyðsÞj þ 4s 2ðkAk þ kDAk Þ þ þ 4s : ð1 lÞ ð1 lÞ ð1 lÞ 6s6t 0 t 0 s ! h Z t i kDAk2 þ 54s kBk2 þ 12kW 2 k2 ð1 þ expð2bs ÞÞ a2 =b EjyðsÞ xðsÞ þ xðsÞj2 ds þ 18s sup EjyðsÞj2 : 2
"
2
2
2
6s6t0 t 0 s
t0
ð10Þ
From (10), we further have
n kAk2 þ 2kDAk2 þ 3kBk2 þ 4kW 1 k2 þ 12kW 2 k2 EjxðtÞ yðtÞj2 6 18s " # !) Z t kW 2 k2 kBk2 4s 2 2 2 2 2 2 þ þ 108skBk þ 24kW 2 k EjxðsÞ yðsÞj2 ds þ 4skW 1 k 4s 2 kAk þ kDAk þ ð1 lÞ ð1 lÞ t0 8 2 < 3 kBk2 þ 2s 2 kW 2 k2 2s 2 2 4 1 kBk þ 24kW 2 k s þ sð1 lÞ þ þ 108s : ð1 lÞ " # ! # 2 2 kW 2 k kBk 2s 2 2 2 2 2 2 kAk þ kDAk þ kW 1 k a =b þ þ 2s þ 2s ð1 lÞ ð1 lÞ ! h i o 2 2 2 2 2 kDAk þ 54s kBk þ 12kW 2 k ð1 þ expð2bs ÞÞ a =b sup EjyðsÞj þ 18s ¼: c1
Z
t
t0
! EjxðsÞ yðsÞj2 ds þ c2
sup 6s6t 0 þs t 0 s
EjyðsÞj2 :
6s6t 0 þs t 0 s
ð11Þ
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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6 t 6 t0 þ 3s , applying the Gronwall inequality [16], we obtain When t 0 þ s
! c1 Þ EjxðtÞ yðtÞj2 6 c2 expð3s
EjyðsÞj2 :
sup
ð12Þ
6s6t 0 þs t 0 s
Therefore
! c1 Þ þ 2a2 expð2bðt t 0 ÞÞ EjyðtÞj2 6 2EjxðtÞ yðtÞj2 þ 2EjxðtÞj2 6 ½2c2 expð3s
EjyðsÞj2 :
sup 6s6t0 þs t 0 s
ð13Þ
6 t 6 t 0 þ 3s Thus, when t 0 þ s
c1 Þ þ 2a2 expð2bs Þ EjyðtÞj2 6 2c2 expð3s
! sup
EjyðsÞj2
! ¼: ^c
6s6t 0 þs t 0 s
EjyðsÞj2 :
sup
ð14Þ
6s6t 0 þs t 0 s
c1 Þ þ 2a2 expð2bs Þ. From (3), when kDAk < r, then ^c < 1. Select c ¼ ln ^c=s , we can conclude where ^c ¼ 2c2 expð3s
! 2
sup 6t6t 0 þ3s t 0 þs
Þ Ejyðt; t 0 ; wÞj 6 expðcs
sup 6t6t 0 þs t 0 s
2
Ejyðt; t 0 ; wÞj
ð15Þ
:
Then, for any positive integer m ¼ 1; 2; . . ., denote
ðt 0 þ ð2m 1Þs ; t 0 ; wÞ :¼ fyðt 0 þ ð2m 1Þs þ s; t0 ; wÞ : s 6 s 6 0g 2 Cð½s ; 0; Rn Þ: y From the existence and uniqueness of the state of system (2), we have
; y ðt0 þ ð2m 1Þs ; t 0 ; wÞÞ: yðt; t0 ; wÞ ¼ yðt; t 0 þ ð2m 1Þs , from (15) When t P t 0 þ ð2m 1Þs
! sup
2
Þ Ejyðt; t 0 ; wÞj 6 expðcs
6t6t 0 þð2mþ1Þs t 0 þð2m1Þs
sup
Ejyðt; t 0 ; wÞj
6t6t 0 þð2m1Þs t 0 þð2m3Þs
Þ 6 expðcms
sup 6t6t 0 þs t 0 s
Ejyðt; t 0 ; wÞj2
2
! Þ; ¼ c0 expðcms
, there must have a positive integer m such that where c0 ¼ supt0 s6t6t0 þs Ejyðt; t 0 ; wÞj2 . So for any t > t0 þ ð2m 1Þs 6 t 6 t 0 þ ð2m þ 1Þs , we have t 0 þ ð2m 1Þs
Ejyðt; t 0 ; wÞj2 6 c0 exp
cs
2
c exp ðt t 0 Þ : 2
ð16Þ
6 t 6 t0 þ s . So systems (2) is mean square globally exponentially stable. According to Condition (16) is also true when t0 s Lemma 1, systems (2) is also almost sure globally exponentially stable. h Remark 1. Different from the robust stability of time varying structured uncertainty and polytopic-type uncertainty, the parameter uncertainty are given. Theorem 1 gives a more general result for parameter uncertainty norm condition of robust stability. Remark 2. Different from the traditional Lyapunov stability theory or linear matrix inequality method. We directly obtain the stability results only from the definition of stochastic delayed systems. Let us consider another condition. If we add parameter uncertainty to matrix B, we have a new systems, where DB stands for parameter uncertainty intensity.
dyðtÞ ¼ ½AyðtÞ þ ðB þ DBÞyðt sðtÞÞdt þ ½W 1 yðtÞ þ W 2 yðt sðtÞÞdxðtÞ; t > t 0 ; t 0 ; Rn Þ; t0 s 6 t 6 t0 yðtÞ ¼ wðt t 0 Þ 2 L2F 0 ð½t 0 s
ð17Þ
we have the following theorem. Theorem 2. Let systems (1) be mean square globally exponentially stable, parameter uncertainty systems (17) is mean square globally exponentially stable and almost sure globally exponentially stable if kDBk < k, where k is the positive solution of the transcend equation for kDBk
c3 Þ þ 2a2 expð2bs Þ ¼ 1 2c4 expð3s
ð18Þ
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where
kAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 c3 ¼ 18s " # ! h i kW 2 k2 2kBk2 þ 2kDBk2 4s 2 2 2 2 2 2 ðkBk þ kDBk Þ þ 24kW 2 k kAk þ kW 1 k þ þ 108s þ 4s 4s ð1 lÞ ð1 lÞ 2 3 kBk2 þ kDBk2 þ 2s 2 kW 2 k2 h i 4s kBk2 þ kDBk2 þ 24kW 2 k2 4s ð1 lÞ1 þ þs c4 ¼ 108s ð1 lÞ " # ! # kW 2 k2 2kBk2 þ 2kDBk2 2s 2 kAk2 þ kW 1 k2 a2 =b þ 2s þ 2s þ ð1 lÞ ð1 lÞ h i kDBk2 þ ð54s kBk2 þ 12kW 2 k2 Þð1 þ expð2bs ÞÞ a2 =b: þ 27s The proof of Theorem 2 is similar to Theorem 1, so we omit it here. Now, we consider this condition, if the matrices A; B all have uncertainties, that is
dyðtÞ ¼ ½ðA þ DAÞyðtÞ þ ðB þ DBÞyðt sðtÞÞdt þ ½W 1 yðtÞ þ W 2 yðt sðtÞÞdxðtÞ; t > t0 ; t 0 ; Rn Þ; t0 s 6 t 6 t0 : yðtÞ ¼ wðt t 0 Þ 2 L2F 0 ð½t 0 s
ð19Þ
For this conditions, we have following theorem. Theorem 3. Let systems (1) be mean square globally exponentially stable, parameter uncertainty systems (19) is mean square globally exponentially stable and almost sure globally exponentially stable if ðkDAk; kDBkÞ are in the inner of the transcend equation curve
c5 Þ þ 2a2 expð2bs Þ ¼ 1 2c6 expð3s
ð20Þ
where
kAk2 þ 2kDAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 c5 ¼ 24s " # ! h i kW 2 k2 kBk2 þ kDBk2 4s 2 2 2 2 2 2 2 þ 144s kBk þ kDBk þ 24kW 2 k 8s kAk þ kDAk þ þ þ 4skW 1 k ð1 lÞ ð1 lÞ 2 3 kBk2 þ kDBk2 þ 2s 2 kW 2 k2 h i 4s 2 2 2 4 1 kBk þ kDBk þ 24kW 2 k s þ sð1 lÞ þ c6 ¼ 144s ð1 lÞ " # ! # kW 2 k2 kBk2 þ kDBk2 2s 2 kAk2 þ kDAk2 þ kW 1 k2 a2 =b þ þ 4s þ 2s ð1 lÞ ð1 lÞ h i 2 2 2 kDAk þ 36s kDBk þ 72s kBk þ 12kW 2 k2 ð1 þ expð2bs ÞÞ a2 =b: þ ½24s Proof. Since the proof of Theorem 3 is similar to those of Theorems 1 and 2, we give only an outline of the proof of 6 s 6 0g; for simplicity, we write xðt; t 0 ; wÞ; yðt; t 0 ; wÞ as xðtÞ; yðtÞ respectively. From (1) Theorem 3. Fix t0 ; w ¼ fwðsÞ : s , from the Holder inequality [16], we have and (19), and when t 6 t 0 þ 3s
h i Z t kAk2 þ 2kDAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 EjxðtÞ yðtÞj2 6 24s EjxðsÞ yðsÞj2 ds i Z t ðkBk2 þ kDBk2 Þ þ 12kW 2 k2 þ 72s EjyðsÞ yðs sðsÞÞj2 ds
t0
h
t0
iZ t kDAk þ 72s kDBk þ 144s kBk2 þ 24kW 2 k2 ð1 þ expð2bs ÞÞ EjxðsÞj2 ds: þ 48s h
2
2
ð21Þ
t0
, from (19), In addition, when t P t 0 þ s
Z
Z t Z s nh i kAk2 þ kDAk2 þ 4kW 1 k2 EjyðrÞj2 EjyðsÞ yðs sðsÞÞj2 ds 6 ds 8s t 0 þs t 0 þs ss h i o 2 2 2 kBk þ kDBk þ 4kW 2 k Ejyðr sðrÞÞj2 dr: þ 8s t
ð22Þ
, by substituting (7) and (8) into (22), we have So, when t P t 0 þ s Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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Z
h
t
2
EjyðsÞ yðs sðsÞÞj ds 6
i 3 kBk2 þ kDBk2 þ 4s 2 kW 2 k2 8s
! sup EjyðsÞj2
ð1 lÞ
t 0 þs
(
6s6t 0 t 0 s
"
# )Z t kW 2 k2 kBk þ kDBk2 4s 2 þ EjyðsÞj2 ds: þ 4skW 1 k ð1 lÞ ð1 lÞ t0 2
2 kAk2 þ kDAk2 þ 8s
þ
ð23Þ
, we have Substituting (23) into (22), when t P t 0 þ s
h iZ t kAk2 þ 2kDAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 EjxðtÞ yðtÞj2 6 24s EjxðsÞ yðsÞj2 ds h
2
2
kBk þ kDBk þ 72s
þ 12kW 2 k
2
iZ
t 0 þs
þ
Z
t0
EjyðsÞ yðs sðsÞÞj2 ds
t
t 0 þs
t0
i Z t kDAk2 þ 72s kDBk2 þ 144s kBk2 þ 24kW 2 k2 ð1 þ expð2bs ÞÞ a2 kwk2 expð2bðs t0 ÞÞds þ 48s t0 n h i kAk2 þ 2kDAk2 þ 3kBk2 þ 3kDBk2 þ 4kW 1 k2 þ 12kW 2 k2 þ 144s kBk2 þ kDBk2 þ 24kW 2 k2 6 24s " # !) Z t kW 2 k2 kBk2 þ kDBk2 4s 2 2 2 2 kAk þ kDAk þ kW 1 k þ EjxðsÞ yðsÞj2 ds þ 4s 8s ð1 lÞ ð1 lÞ t0 8 2
¼: c5
Z
t
6s6t 0 þs t 0 s
! EjxðsÞ yðsÞj2 ds þ c6
t0
EjyðsÞj2 :
sup
ð24Þ
6s6t 0 þs t 0 s
The rest of the proof can be completed similarly to the proof of Theorem 1. h Remark 3. Theorems 1–3 show that when the system is globally exponentially stable, the uncertainty in parameter matrices systems can be also mean square globally exponentially stable and almost sure globally exponentially stable, provided that the parameter uncertainty is smaller than given upper bounds. Remark 4. From the proofs of Theorems 1–3, we can see that the upper bounds of parameter uncertainty intensity are derived via subtle inequalities and can be estimated by solving transcendental equation. As transcendental equation can be solved by using some software such as MATLAB, the derived conditions in these theorems can be verified easily.
4. Illustrative examples
Example 1. Consider a bi-state systems
dxðtÞ ¼ ½AxðtÞ þ Bxðt sðtÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdwðtÞ:
ð25Þ
The parameters are as follows:
A¼
1
0
0
1
;
B¼
0:1
0
0
0:1
;
W1 ¼
0:1
0
0
0:1
;
W2 ¼
0 0:1 0 0:1
;
T
sðtÞ ¼ 0:005sint; xð0Þ ¼ ½0:6; 0:3 . Hence, according to Theorem 4.2 in [15], systems (25) is globally exponentially stable with a ¼ 0:5; b ¼ 0:5. In the presence of uncertainty in parameter matrices, the systems become new systems: dyðtÞ ¼ ½ðA þ DAÞyðtÞ þ ðB þ DBÞyðt sðtÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdwðtÞ;
ð26Þ
where sðtÞ is the time-varying delay, DA; DB is the uncertainty intensity in parameter matrices. According to Theorem 3, let l ¼ 0, and by solving (20)
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n 0:2472 þ 0:72kDBk2 kDAk2 þ kDBk2 þ 203:01 104 þ 0:12kDAk2 o n þ0:88kDBk2 þ 0:24 exp 0:015 0:2836 þ 0:24kDAk2 þ 0:36kDBk2 h ih i o þ 0:2472 þ 0:72kDBk2 4:2 þ 0:2kDAk2 þ 0:2kDBk2 105 þ 0:4998 ¼ 1:
ð27Þ
Using the software of MATLAB, we can obtain its stability region for ðkDAk; kDBkÞ. Fig. 1 shows the maximum stability region for ðkDAk; kDBkÞ in (26) if systems (25) is mean square globally exponentially stable as the coefficient a ¼ 0:5; b ¼ 0:5. Fig.2 depicts the transient states of perturbed systems (26) with ðDA; DBÞ ¼ ð1:4I; 0:1IÞ as the uncertainty intensify in parameter matrices. From Theorem 3, as it located in the maximum stability region for systems (26), it shows that systems (26) is means square globally exponentially stable and also almost sure globally exponentially stable, as the parameter ðkDAk; kDBkÞ in the inner of transcend equation of (27). Figs. 3–5 show that the transient states of perturbed systems (26) with ðDA; DBÞ as ð1:8I; 0Þ; ð0; 2:2IÞ; ð1:8I; 0:6IÞ. From Theorems 1–3, it shows that when the conditions in Theorems 1–3 do not hold, the systems (26) may become unstable.
0.7 0.6
||Δ B||
0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
||Δ A||
1.5
Fig. 1. The maximum stability region with ðkDAk; kDBkÞ in perturbed systems (26).
0.6
y1 y2
0.5 0.4 0.3
y
0.2 0.1 0 −0.1 −0.2 −0.3
0
1
2
t
3
4
5
Fig. 2. The transient state of perturbed systems (26) with ðDA; DBÞ ¼ ð1:4I; 0:1IÞ and sðtÞ ¼ 0:005.
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Example 2. Consider a stochastic delayed recurrent neural networks (SDRNN) model
dxðtÞ ¼ ½AxðtÞ þ Bf ðxðtÞÞ þ Df ðxðt sðtÞÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdwðtÞ:
ð28Þ
The parameters are as follows:
A¼
1 0 0 1
;
B¼
0
2
2
0
;
D¼
2 0 0 2
;
W1 ¼
0:1
0
0
0:1
;
W2 ¼
0 0:01
0 0:01
T
f ðxj Þ ¼ tanhðxj Þ; ðj ¼ 1; 2Þ; sðtÞ ¼ 0:00025ð1 þ sintÞ; xð0Þ ¼ ½0:6; 0:3 . Hence, according to Theorem 3.1 in [16, Theorem 3.1, pp. 367]]. SDRNN (28) is globally exponentially stable with a ¼ 0:5; b ¼ 0:5. In the presence of parameter uncertainty in connection weight matrices, the SDRNN becomes a new perturbed SDRNN:
dyðtÞ ¼ ½AyðtÞ þ ð1 þ rÞBf ðyðtÞÞ þ ð1 þ kÞDf ðyðt sðtÞÞÞdt þ ½W 1 xðtÞ þ W 2 xðt sðtÞÞdwðtÞ;
1
ð29Þ
y1 y2
0
−1
y −2 −3
−4
−5
0
1
2
t
3
4
5
Fig. 3. The transient state of perturbed systems (26) with ðDA; DBÞ ¼ ð1:8I; 0Þ and sðtÞ ¼ 0:005.
5
y1 y2
0
−5
y −10 −15
−20
−25
0
1
2
t
3
4
5
Fig. 4. The transient state of perturbed system (26) with ðDA; DBÞ ¼ ð0; 2:2IÞ and sðtÞ ¼ 0:005.
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10
y1 y2
0 −10
y
−20 −30 −40 −50 −60
0
1
2
t
3
4
5
Fig. 5. The transient state of perturbed system (26) with ðDA; DBÞ ¼ ð1:8I; 0:6IÞ and sðtÞ ¼ 0:005.
1.5
1
λ
0.5
0
−0.5
−1
−1.5 −1.5
−1
−0.5
0 σ
0.5
1
1.5
Fig. 6. The maximum stability region with (r,k) in SDRNN (29).
where sðtÞ is the time-varying delay, r; k is the parameter uncertainty intensity in connection weight matrices. Similarly to Theorem 3, let l ¼ 0, and by solving (20)
n o ð0:3624 þ 0:36k2 Þð1:5r2 þ 1:5003k2 þ 3r þ 3:0006k þ 204:385Þ 105 þ 0:06r2 þ 0:09k2 þ 0:3624 n o exp 0:015 0:2962 þ 0:12r2 þ 0:18k2 þ 0:36k þ ½0:3624 þ 0:36k2 ½3:37 þ 0:6r2 þ 0:6k2 þ 1:2r þ 1:2k 105 þ 0:4998 ¼ 1:
ð30Þ
Fig. 6 shows the maximum stability region for ðr; kÞ in SDRNN (29).
5. Conclusion In this paper, the robustness of stochastic delayed systems with uncertainty in parameter matrices is analyzed. The upper bounds of parameter uncertainty are derived for stochastic delayed systems to remain global exponential stability. The Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022
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results herein provide a theoretic basis for the design and application of stochastic delayed systems in the presence of uncertainty in parameter matrices. Further investigations may be aimed at the improvement of the upper bounds to allow bigger stability margins for withstanding parameter uncertainty. Acknowledgments The authors thank the Editor Albert Luo and the anonymous reviewers for their insightful comments and valuable suggestions, which have helped us in finalizing the paper. This work was supported by the National Natural Science Foundation of China with Grant No. 61203055 and supported by the Fundamental Research Funds for the Central Universities of 2012QNA48. References [1] Hale J. Functional Differential Equations. New York: Springer Verlag; 1977. [2] Richard JP. Time-delay systems: an overview of some recent advances and open problems. Automatica 2008;39:1667–94. [3] Yuan K, Cao J, Li H. Robust stability of switched Cohen–Grossberg neural networks with mixed time-varying delays. IEEE Trans Syst Man Cybern B, Cybern 2006;36:1356–63. [4] Shen Y, Wang J. Almost sure exponential stability of recurrent neural networks with Markovian switching. IEEE Trans Neural Netw 2009;20:840–55. [5] Xu S, aLam J. Improved delay-dependent stability criteria for time-delay systems. IEEE Trans Autom Control 2005;50:384–7. [6] Meng X, Lam J, Du B, Gao H. A delay-partitioning approach to the stability analysis of discrete-time systems. Automatica 2010;46:610–4. [7] Jiang F, Shen Y, Liu L. Taylor approximation of the solutions of stochastic differential delay equations with Poisson jump. Commun Nonlinear Sci Numer Simulat 2011;16:798–804. [8] Jiang F, Shen Y, Hu J. Stability of the split-step backward Euler scheme for stochastic delay integro–differential equations with Markovian switching. Commun Nonlinear Sci Numer Simulat 2011;16:814–21. [9] Liu P. Robust exponential stability for uncertain time-varying delay systems with delay dependence. J Franklin I 2009;346:958–68. [10] Shen Y, Wang J. Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances. IEEE Trans Neural Netw Learn Syst 2012;23:87–96. [11] Zhu S, Shen Y, Chen G. Exponential passivity of neural networks with time-varying delay and uncertainty. Phys Lett A 2010;375:136–42. [12] Zhu S, Shen Y, Liu L. Exponential stability of uncertain stochastic neural networks with markovian switching. Neural Process Lett 2010;32:293–309. [13] He Y, Wu M, She J, Liu G. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties. IEEE Trans Autom Control 2004;49:828–32. [14] Gao H, Shi P, Wang J, Bittanti, Cuzzola FA. Parameter-dependent robust stability of uncertain time-delay systems. J Comput Appl Math 2007;206:366–73. [15] Mao X. Stability and stabilization of stochastic differential delay equations. IET Control Theory Appl 2007;1:1551–66. [16] Mao X. Stochastic differential equations and applications. second ed.. Chichester: Harwood; 2007.
Please cite this article in press as: Zhu S et al. Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems. Commun Nonlinear Sci Numer Simulat (2013), http://dx.doi.org/10.1016/j.cnsns.2013.06.022