Accepted Manuscript Robust stability of stochastic fuzzy delayed neural networks with impulsive time window Xin Wang, Junzhi Yu, Chuandong Li, Hui Wang, Tingwen Huang, Junjian Huang PII: DOI: Reference:

S0893-6080(15)00066-0 http://dx.doi.org/10.1016/j.neunet.2015.03.010 NN 3466

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Received date: 2 November 2014 Revised date: 16 March 2015 Accepted date: 19 March 2015 Please cite this article as: Wang, X., Yu, J., Li, C., Wang, H., Huang, T., & Huang, J. Robust stability of stochastic fuzzy delayed neural networks with impulsive time window. Neural Networks (2015), http://dx.doi.org/10.1016/j.neunet.2015.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Robust Stability of Stochastic Fuzzy Delayed Neural Networks with Impulsive Time Window Xin Wang, Junzhi Yu, Chuandong Li, Hui Wang, Tingwen Huang, and Junjian Huang

Abstract—The urgent problem of impulsive moments can’t be determined in advance brings new challenges beyond the conventional impulsive systems theory. In order to solve this problem, the novel concept of impulsive time window is proposed in this paper. And the stability problem of stochastic fuzzy uncertain delayed neural networks with impulsive time window is investigated. By combing the discretized Lyapunov function approach with mathematical induction method, several novel and easy-to-check sufficient conditions concerning the impulsive time window are derived to ensure that the model considered here is exponentially stable in mean square. Numerical simulations are presented to further demonstrate the effectiveness of the proposed stability criterion. Index Terms—Robust stability, Stochastic fuzzy Neural networks, impulsive time window.

I. I NTRODUCTION Since mathematical modeling of physical systems and processes in many areas of engineering often leads to complex nonlinear systems, which brings several difficulties to analysis and synthesis, researchers have been seeking effective methods for the control of nonlinear system. It is well known that there has been a turning point in one of the most effective methods in accordance with the advent of the fuzzy model[1], which is among all of modes to solve the control of complex nonlinear system. Recently, an army of results have been advanced for the fuzzy model which has received increasing attention research because it can provide an effective solution to the control of plants that are mathematically ill-defined, uncertain and nonlinear[1]-[10]. The main feature of the fuzzy model is to express the local dynamics of each fuzzy rule by a linear system model and to express the overall system by fuzzy ”blending” of the local linear system models. To date, fuzzy model has been suggested as an alternative approach to conventional control techniques for complex control systems. For many applications which have been found in various fields, neural networks have been extensively studied and developed[11]-[20]. In the real world, neural networks are often subjected to external disturbances. Generally speaking, there are two kinds of disturbances considered: parameter X. Wang, C. Li are with College of Electronic and Information Engineering, Southwest University, Chongqing 400044, PR China (emails: [email protected], [email protected]). J. Yu is with Institute of Automation, Chinese Academy of Sciences Beijing 100190, P.R. China (email:[email protected]). H. Wang is with college of Mathematics Science, Chongqing Normal University, Chongqing 401331 PR China T. Huang is with Texas A&M University at Qatar, Doha, P.O.Box 23874, Qatar. (email:[email protected]). J. Huang is with Department of Computer Science, Chongqing University of Education, Chongqing 400067, China(emails: [email protected])

uncertainties and stochastic perturbations. Therefore, it is necessary to both consider parameter uncertainties and stochastic effects on the stability of neural networks[21]-[25]. On the other hand, the states of electronic networks and biological networks are often subjected to instantaneous disturbances and experienced abrupt changes at certain instants, which may be caused by switching phenomenon, frequency changes, or other sudden noise, i. e., they exhibit impulsive effects[26]-[36]. Moreover, time delays are often encountered in the real world due to finite switching speed of amplifiers. Hence, it is of great importance to both investigate delay and impulsive effects on the stochastic stability of neural networks. It is widely recognized that there are two kinds of impulsive effects, i.e., stabilizing impulses and destabilizing impulses[26]. Recently, in[22], a novel strategy named mixed impulses has been proposed. Hence, the previous results which only concerned the lower bound or upper bound of the impulsive sequences become trivial. Moreover, in many actual control systems, the impulsive moments almost can not be specified in advance. Therefore, it becomes desirable to discuss the impulsive systems with impulsive time window. To the best of the authors’ knowledge, the problem of stochastic stability for fuzzy uncertain delayed neural networks with impulsive time window is still an open issue. It is, therefore, the motivation behind our efforts to bridge this gap by studying stochastic stability of fuzzy neural networks with impulsive time window. Motivated by the shortcoming of the aforementioned research in this area, in this paper, the problem of stochastic stability for uncertain delayed fuzzy neural networks with impulsive time window is investigated. Based on the discretized Lyapunov function method and mathematic induction method, several stability criterion are derived under which stochastic uncertain delayed fuzzy neural networks with impulsive time window are exponentially stable in mean square. The main contributions of this paper can be listed as follows: 1)the stochastic uncertain delayed neural networks both considered the T-S model and impulsive time window are firstly constructed; 2) a unified framework is established to handle stochastic, parameter uncertainty, impulsive time window and fuzzy rule; 3)some approximation algorithms are proposed to compute the lower and the upper bounds of the impulsive time window, respectively. The rest of this paper is arranged as follows. In the next section, the problem to be considered and some needed preliminaries are presented. The main results are derived in Section III. In Section IV, we present several simulation examples to verify the effectiveness of our theoretical results. Finally, the conclusions are drawn in Section

2

V. II. M ODEL DESCRIPTION AND SOME PRELIMINARIES In this section, some preliminaries are given including model formulation, lemmas, and definitions. Consider the following stochastic uncertain delayed neural networks:  dx(t) = [−(C + ∆C)x(t) + (A + ∆A)f (x(t))     + (D + ∆D)f (x(t − τ (t)))]dt (1)  + [∆W x(t) + ∆W x(t − τ (t))]dW (t) 0 1    x(t) = ϕ(t), t ∈ [−τ, 0]

has a unique global solution on t ≥ 0 with the initial value ϕ(t) ∈ L([−τ, 0], Rn ). By the singleton fuzzifier, the product inference engine and the center average defuzzifier, the final output of the fuzzy system (3) is inferred as follows:  dx(t) =    ∑r    i=1 hi (z(t)){[−(Ci + ∆Ci )x(t)   +(A + ∆A )f (x(t)) i i (4)  +(D + ∆D i i )f (x(t − τ (t)))]dt     +[∆Wi0 x(t) + ∆Wi1 x(t − τ (t))]dW (t)}, t ̸= tk   ∑r  + x(tk ) = i=1 hi (z(t))Bi x(t− tk ∈ Dk k ),

where x(t) = [x1 (t), x2 (t), ..., xn (t)]T ∈ Rn is the state where p vector associated with the neurons; C = diag(c1 , c2 , ..., cn ) > ∏ wi (z(t)) 0 is a positive diagonal matrix, A = (aij )n×n and hi = ∑r , wi (z(t)) = Mij (zj (t)) i=1 wi (z(t)) j=1 D = (dij )n×n are the connection weight matrices; ∆C, ∆A, ∆D, ∆W0 and ∆W1 are time-varying matrices on and Bi = (Bi1 , Bi2 , ..., Bik )T , Mij (zj (t)) denotes the grade Rn×n that denote the parameter uncertainties. f (x(t)) = of membership of zj (t) in Mij . Note that (f1 (x(t)), f2 (x(t)), ..., fn (x(t)))T denotes the neuron activar ∑ tion function vector ; τ (t) is the transmission delay satisfies hi (z(t)) = 1, hi (z(t)) ≥ 0, i = 1, 2, ..., r 0 ≤ τ (t) ≤ τ , where τ is a positive scalar. W (t) = i=1 T [ω1 (t), ω2 (t), ..., ωn (t)] is an n-dimension Brown motion. Remark 1: From the second equation of (4), it is obvious Taking impulsive time window effects into account, we have that a set of control matrices Bi and impulsive time window the following model:  Dk to be designed to guarantee the stochastic exponential  dx(t) = [−(C + ∆C)x(t) + (A + ∆A)f (x(t))  stability of model (4) in mean square. The impulsive control     + (D + ∆D)f (x(t − τ (t)))]dt strategy considered here has some favorable features which  described as follow: 1)The impulse effects here are dependent + [∆W0 x(t) + ∆W1 x(t − τ (t))]dW (t), t ̸= tk   + −  x(tk ) = Bk x(tk ), tk ∈ Dk on the fuzzy rules, namely, both stabilizing and destabilizing    impulses are considered. 2)The impulses occur in a random x(t) = ϕ(t), t ∈ [−τ, 0] manner in the impulsive time window. 3)The impulsive effects (2) can be distinct at different impulsive instants. Moreover, it is where Dk are the time window of impulsive times tk , i. noted that impulsive control strategy considered here can be e., Dk = [dkmin + tk−1 , dkmax + tk−1 ), where dkmin and directly used to realize the state-dependent impulsive control dkmax denote the minimum and maximum residence time, strategy. Remark 2:In the following sequel, we will illustrate that respectively. Bk are impulsive gain at impulsive instants tk ; − the impulses considered here encompass the impulses in the + − x(t+ ) = lim x(t + σ), x(t ) = lim x(t + σ). k k σ→0 σ→0 k k In this paper, a general class of stochastic fuzzy uncertain previous results. To be more specific, if we assume that the delayed neural networks with impulsive time window, are lower and upper bounds of the impulsive time window are discussed. As in [1], the model of stochastic fuzzy uncertain identical, i. e., dkmin = dkmax = d > 0, where d is a constant, delayed neural networks with impulsive time window are then the impulsive time window turns into the fixed impulses considered in [26]-[31]. If r = n, i.e., i = 1, ..., n, it turns into composed of r plant rules that can be described as follows: the heterogeneous impulses considered in [33]. If r < n, i.e., Plant Rule i: i = 1, 2, .., r, it will turns into the pinning impulsive strategy IF z1 (t) is Mi1 and· · · and zp (t) is Mip considered in [37]. Moreover, the heterogeneous impulses can THEN  encompass the time-varying impulsive considered in [36]. In dx(t) =    summary, the impulses based on fuzzy rules and impulsive   [−(Ci + ∆Ci )x(t) + (Ai + ∆Ai )f (x(t))   time window are general enough to include most of the   +(Di + ∆Di )f (x(t − τ (t)))]dt existing impulses as special cases [26]-[37]. 0 1  In the following, we will establish theoretical results to +[∆W x(t) + ∆W x(t − τ (t))]dW (t), on t = ̸ t k  i i   + −  characterize that the stochastic uncertain delayed fuzzy neural  x(tk ) = Bik x(tk ), on tk ∈ Dk    networks with impulsive time window are exponentially stable x(t) = ϕ(t), on t ∈ [−τ, 0]. in mean square. The following definitions and assumptions are (3) necessary for getting our main results. where i = 1, 2, ..., r, Mij (j = 1, ..., p) are the fuzzy sets, Definition 1. For the given impulsive time sequences tk ∈ z(t) = (z1 (t), z2 (t), ..., zp (t))T is the premise variable vector, Dk , the stochastic uncertain delayed fuzzy neural networks (4) r is the number of fuzzy IF-THEN rules. It is known that (3) are said to be global exponentially stable in mean square if

3

there exist positive scalar λ, such that for every ϱ > 0, there is a positive scalar ϱ1 such that E∥ϕ∥ ≤ ϱ1 implying E∥x(t)∥ ≤ ϱe−λ(t−t0 )

(5)

holds for all t > t0 . For the parameter uncertainty ∆Ci , ∆Ai , ∆Di , ∆Wi0 , ∆Wi1 and the nonlinear function f (·), we make the following assumptions. Assumption 1. Assume the parameter uncertainties ∆Ci , ∆Ai , ∆Di , ∆Wi0 and ∆Wi1 are satisfied the following condition: [∆Ci , ∆Ai , ∆Di , ∆Wi0 , ∆Wi1 ] =

M F (t)[Φ1i , Φ2i , Φ3i , Φ4i , Φ5i ]

(6)

where M, Φji , i = 1, ..., r, j = 1, 2, ..., 5 are known real constant matrices with appropriate dimensions, and F (t) is the time varying uncertain matrix that satisfies F T (t)F (t) ≤ I

(7)

where I denotes an identity matrix with appropriate dimensions. Assumption 2. There exist scalars li+ , li− such that for any x, y ∈ R, x ̸= y li− ≤

fi (x) − fi (y) ≤ li+ x−y

(8)

Assumption 3. There exist scalars dmin , dmax such that for any k ∈ N+ tk ∈ [tk−1 + dmin , tk−1 + dmax )

(9)

where dmin = mink∈N+ {dkmin }, dmax = maxk∈N+ {dkmax } III. M AIN RESULTS In this section, the exponential stability of stochastic fuzzy uncertain delayed neural networks with impulsive time window is investigated by using the discretized Lyapunov function and mathematical induction method. For convenience, denoting l1− + l1+ l2− + l2+ l− + ln+ , , ..., n }, 2 2 2 + − + − + − L2 = diag{l1 l1 , l2 l2 , ..., ln ln }.

L1 = diag{

Now, our main results are given in the following theorem, which show that the stochastic uncertain delayed fuzzy neural networks are exponentially stable in the mean square for all admissible parameter uncertainties if the following matrix inequalities are feasible. Theorem 1: Consider the stochastic uncertain fuzzy delayed neural networks (4) with impulsive time window. Suppose that Assumptions 1-3 hold. If for prescribed positive scalar µ ∈ (0, 1), there exist a set of matrices Pq > 0, q = 0, 1, ..., L, ,

positive scalars λ1 , σ, β, small enough constant ϵ0 and ϵ0 ∈ (0, 1 − µ) such that the following inequalities hold: (10) (11)

Pq ≤ λ1 In , − ln(µ + ϵ1 )/dmax > 0

M T Pq M − σI ≤ 0, q = 1, 2, ..., L, − µP0 + 

Ξ(j)

  = 

BT i= i PL Bi < 0, (j) Γ11 Pj Ai + K1 L1 (j) ∗ Γ22

∗ 0 j = q, q + 1, L (j)

where Γ11

=

0 0

(12)

1, 2, ..., r P j Di 0 (j) Γ33 ∗

0 0 0 (j) Γ44



   < 0, 

(13)

(14)

β µ Pj + Ψq − (j) 2 T 2 = −K1 + σ(Φi ) Φi , Γ33 = −βPj − K2 L2 + 2σ(Φ5i )T Φ5i .

(ϵ0 + 3)Pj − 2Pj Ci +

(j) K1 L2 + 2σ(Φ4i )T Φ4i , Γ22 (j) −K2 + σ(Φ3i )T Φ3i , Γ44 =

K1 = diag{k11 , k12 , ..., k1n } > 0 and K2 = diag{k21 , k22 , ..., k2n } > 0, Ψq = L(Pq+1 − Pq )/dmin , dmin and dmax are the lower and upper bounds of the impulsive time window, respectively. ∗ is used to denote the term that is induced by symmetry. Then, the stochastic uncertain fuzzy delayed uncertain neural networks with impulsive time window (4) are exponentially stable in mean square. Proof: Dividing the proof procedures into the following three steps: Step 1: Dividing the interval [tk , tk +dmin ) into L segments described as Nk,q = [tk + θq , tk + θq+1 ), q = 1, 2, ..., L − dmin 1 of equal length h = dmin L , and then θq = qh = d L , q = 0, 1, ..., L − 1. Choosing the continuous matrix function P (t), t ∈ [tk , tk+1 ) which is linear within each segment Nk,q , q = 0, 1, ..., L − 1. Letting Pq = P (tk + θq ), then the matrix function P (t) is piecewise linear with in the interval [tk , tk + dmin ), it can be expressed in terms of the values at dividing points using linear interpolation formula, i. e., for 0 ≤ α ≤ 1, q = 0, 1, ..., L − 1 and t ∈ Nk,q P (t) = P (tk + θq + αh) = (1 − α)Pq + αPq+1 = Pq (α)

(15)

where α = (t − tk − θq )/h. Then the continuous matrix function P (t) is completely determined by Pq , q = 0, 1, ..., L in [tk , tk + dmin ). Afterwards, in the interval [tk + dmin , tk+1 ), matrix function P (t) is fixed as a constant matrix P (t) = PL . Hence, the discretized matrix function P (t) is described as follows: { Pq (α), t ∈ Nk,q , q = 0, 1, .., L − 1 P (t) = (16) PL , t ∈ [tk + dmin , tk+1 ) Step 2: In view (18), constructing the following discretized Lyapunov function : { xT (t)Pq (α)x(t), t ∈ Nk,q , q = 1, 2, ..., L − 1 V (t) = xT (t)PL x(t), t ∈ [tk + dmin , tk+1 ) (17)

4

Step 3: In the following sequel, we will prove that EV (t) ≤ λ0 ϱ2 e−ϵ0 (t−t0 ) ,

t ∈ [t0 − τ, +∞)

(18)

where λ0 = min{λmin (Pq (α)), λmin (PL )}, q = 0, 1, ...., L − 1. For any given scalar ϱ > 0, choose ϱ1 > 0 such that λ1 ϱ21 < µλ0 ϱ2 . Set W (t) = eϵ0 (t−t0 ) V (t), µ1 = − ln(µ + ϵ1 )/dmax . In order to prove (20), it is equivalent to proving EW (t) < λ0 ϱ ,

t ∈ [t0 − τ, +∞)

2

(19)

In order to do so, the mathematical induction method is used. From (12), one can easily observes that for t ∈ [t0 − τ, t0 ) EW (t) = Eeϵ0 (t−t0 ) xT P (t)x(t)

≤ λ1 E∥x(t)∥2 ≤ λ1 ϱ21 < µλ0 ϱ2

(20)

We now claim that EW (t) < λ0 ϱ2 ,

t ∈ [tk , tk+1 )

(21)

In order to prove this claim, we firstly show that it is true for k=0 EW (t) < λ0 ϱ2 ,

t ∈ [t0 , t1 )

(22)

If (24) is not true, then there exist time t¯ satisfying t¯ = inf{t ∈ [t0 , t1 ) : EW (t) ≥ λ0 ϱ2 }

(23)

Obviously, t¯ > t0 , and

EW (t) < λ0 ϱ , t0 ≤ t < t¯

Let t∗ = sup{t ∈ [t0 , t¯] : EW (t) ≤ µλ0 ϱ2 }. This implies that t∗ < t¯ and EW (t) > µλ0 ϱ2 , t ∈ (t∗ , t¯]. For θ ∈ [−τ, 0], then, we have (24)

˜ (t) = D+ W (t) + β( 1 W (t) − w(t − τ (t))) − µ1 W (t). Set W µ It is obvious from [41] that when t ∈ [tk , tk+1 ), D+ EW (t) = ELW (t), where L is the Itˆo operator [42] and the upperright Dini derivative D+ W (t) is defined as D+ W (t) = limh→0+ (W (t + h) − W (t))/h. Then, for t ∈ [t∗ , t¯], from ˜ (t), we have the definition of W

(25)

(26)

Since α = (t − t0 − θq )/h, we have α˙ = 1/h, where h = dmin /L. Thus, we have P˙ (t) = (Pq+1 − Pq )/h = L(Pq+1 − Pq )/dmin = Ψq

hi (z(t))E{ϵ0 xT (t)Pq (α)x(t)

i=1

− 2xT (t)Pq (α)Ci x(t) + 2xT (t)Pq (α)Ai f (x(t))

+ 2xT Pq (α)Di f (x(t − τ (t))) − 2xT (t)Pq (α)∆Ci x(t) + 2xT (t)Pq (α)∆Ai f (x(t))

+ 2xT (t)Pq (α)∆Di f (x(t − τ (t))) + xT (t)(∆Wi0 )T Pq (α)∆Wi0 x(t)

+ xT (t − τ (t))(∆Wi1 )T Pq (α)∆Wi1 x(t − τ (t))

+ 2xT (t)(∆Wi0 )T Pq (α)∆Wi1 x(t − τ (t)) β + xT (t)Ψq x(t) + xT Pq (α)x(t) µ − βe−ϵ0 τ xT (x(t − τ (t)))Pq (α)x(t − τ (t)) − µ1 xT (t)Pq (α)x(t)}

(28)

From Assumption 1 and fi (0) = 0(i = 1, 2, ..., n), (li+ xi (t) − fi (xi (t))(fi (xi (t)) − li− xi (t)) ≥ 0 for i = 1, 2, ..., n, this together with K1 = diag(k11 , k12 , ..., k1n ) > 0, by the same method used in[30] lead to 0≤

n ∑ i=1

k1i (li+ xi (t) − fi (xi (t)))(fi (xi (t)) − li− xi (t))) (29)

Similarly as (31), it yields 0≤

n ∑ i=1

k2i (li+ xi (t − τ (t)) − fi (xi (t − τ (t)))

× (fi (xi (t − τ (t))) − li− xi (t − τ (t))))

= 2xT (t − τ (t))K2 L1 f (x(t − τ (t)))

− f T (x(t − τ (t)))K2 f (x(t − τ (t))) − xT (t − τ (t))K2 L2 x(t − τ (t))

(30)

where K2 = diag(k21 , k22 , ..., k2n ) > 0. Combining Lemma 1 with Assumption 1, from (13), one observes that the following inequalities hold: 2xT (t)P q (α)∆Ci x(t)

In each discretized segment N0,q , the following equation can be derived that for t ∈ N0,q P˙ (t) = P˙ (t0 + θq + αh) = P˙q (α) = (Pq+1 − Pq )α˙

r ∑

− xT (t)K1 L2 x(t)

2

˜ (t) = eϵ0 (t−t0 ) (ϵ0 EV (t) + LEV (t)) EW β + EW (t) − βEW (t − τ (t)) − µ1 EW (t) µ

˜ (t) ≤ eϵ0 (t−t0 ) EW

= 2xT (t)K1 L1 f (x(t)) − f T (x(t))K1 f (x(t))

EW (t¯) ≥ λ0 ϱ2

EW (t) ≥ µλ0 ϱ2 ≥ µEW (t + θ), t ∈ [t∗ , t¯]

Moreover, by the linear interpolation relationship of Pq (α), one has

= 2xT P q (α)M F (t)Φ1i x(t) ≤ xT (t)P q (α)x(t)

+ xT (t)(Φ1i )T F T (t)M T P q (α)M F (t)Φ1i x(t)

≤ xT (t)[P q (α) + σ(Φ1i )T Φ1i ]x(t)

(31)

2xT (t)P q (α)∆Ai f (x(t)) = 2xT P q (α)M F (t)Φ2i f (x(t)) ≤ xT (t)P q (α)x(t)

+ f T (x(t))(Φ2i )T F T (t)M T P q (α)M F (t)Φ2i f (x(t))

(27)

≤ xT (t)P q (α)x(t) + f T (x(t))[σ(Φ2i )T Φ2i ]f (x(t))

(32)

5

Thus, we have, for t ∈ [t∗ , t¯] ,

and

1 D+ EW (t) − µ1 EW (t) + βE[ W (t) − W (t − τ (t))] < 0 µ (41)

2xT (t)P q (α)∆Di f (x(t)) = 2x P T

q

(α)M F (t)Φ3i f (x(t q

≤ x (t)P (α)x(t) T

− τ (t)))

+ f T (x(t − τ (t)))(Φ3i )T F T (t)M T P q (α)×

(33)

M F (t)Φ3i f (x(t − τ (t))) q

+ f (x(t − T

τ (t)))[σ(Φ3i )T Φ3i ]f (x(t

− τ (t)))

(34)

In view of Assumption 1, it can be checked that

(35)

≤ x (t − T

Defining

− τ (t))

Pq (α)Ai + K1 L1 Γq22 (α) 0 0

Pq (α)Di 0 Γq33 (α) ∗

q=0,1,...,L−1

r ∑ i=1

(44)

(45)

r ∑

− hi (z(t))[xT (t− r )P0 x(tr )]

i=1

0  0   0 q Γ44 (α)

β µ Pq (α)

(46)

Hence, from (47), we have EW (tr ) ≤ µλ0 ϱ2 . If (47) is not true, then there exists a natural number t˜ satisfying



t˜ = min{t ∈ [tr , tr+1 ) : EW (t) ≥ λ0 ϱ2 }

(47)

t∗∗ = max{t ∈ [tr , t˜) : EW (t) ≤ λ0 ϱ2 }

(48)

EV (t) ≤ λ0 ϱ2 e−ϵ0 (t−t0 )

(49)



(50)

Obviously, t˜ > tr and EW (t˜) > EW (tr ). Let + + =

(38)

(39)

where η(t) = [xT (t), f T (x(t)), f T (x(t − τ (t))), xT (t − τ (t))]T . On the other hand, since P (t) = PL , when t ∈ [t0 + dmin , t1 ). Based on (14), this yields ˜ (t) EW ≤ eϵ0 (t−t0 )

t ∈ [tr , tr+1 )

≤ µW (t− r )

i=1

N0,q = [t0 , t0 + dmin )

≤µ

(37)

Thus, from (14), we see r ∑ ˜ (t) ≤ eϵ0 (t−t0 ) EW hi (z(t))E[η T (t)Ξq (α)η(t)] < 0 t∈

t ∈ [tk , tk+1 ), k = 1, 2, ..., r − 1

+ + = eϵ0 (tr −t0 ) xT (t+ r )P (tr )x(tr ) r ∑ T − ≤ µeϵ0 (tr −t0 ) hi (z(t))[xT (t− r )Bi PL Bi x(tr )]

− τ (t))

(q) Ψq − K1 L2 + σ(Φ1i )T Φ1i + 2σ(Φ4i )T Φ4i , Γ22 (α) = −K1 (q) σ(Φ2i )T Φ2i , Γ33 (α) = −K2 + σ(Φ3i )T Φ3i , Γq44 (α) −ϵ0 τ −βe Pq (α) − K2 L2 + 2σ(Φ5i )T Φ5i . We have



(43)

i=1

where Γq11 (α) = (ϵ0 + 3)Pq (α) − 2Pq (α)Ci +

Ξq (α) = (1 − α)Ξq + αΞq+1

EW (t) < λ0 ϱ2 ,

(36)

τ (t))(Φ5i )T F T (t)M T P q (α)M F (t)Φ5i x(t τ (t))[σ(Φ5i )T Φ5i ]x(t − τ (t))

Ξq (α) =  q Γ11 (α)  ∗   ∗ 0

≤ µλ0 ϱ2 eµ1 dmax ≤ λ0 ϱ2

W (tr ) = eϵ0 (tr −t0 ) V (t+ r )

T

≤ x (t −

)

From (15), we have

+ xT (t − τ (t))(∆Wi1 )T P q (α)∆Wi1 x(t − τ (t))

xT (t − τ (t))(∆Wi1 )T P q (α)(∆Wi1 )x(t − τ (t))



EW (t) < λ0 ϱ2 ,

≤ xT (∆Wi0 )T P q (α)∆Wi0 x(t)

T

¯

In this sequel, we will show (24) holds for k = r.

2xT (t)(∆Wi0 )T P q (α)(∆Wi1 )x(t − τ (t))

≤ x (t)[σ(Φ4i )T Φ4i ]x(t) + xT (t − τ (t))[σ(Φ5i )T Φ5i ]x(t

W (t¯) ≤ W (t∗ )eµ1 (t−t

(42)

This is a contradiction. Hence, (24) is true. Now, we assume that the claim (24) also holds for k = 1, 2, ..., r − 1(r > 2)

x (t)(∆Wi0 )T P q (α)(∆Wi0 )x(t) ≤ xT (t)[M F (t)Φ4i ]T P q (α)[M F (t)Φ4i ]x(t) ≤ xT (t)[σ(Φ4i )T Φ4i ]x(t) T

and

D+ EW (t) ≤ µ1 EW (t)

It leads to

≤ x (t)[P (α)]x(t) T

From (26), we have

hi (z(t))E[η T (t)ΞL η(t)], t ∈ [t0 + dmin , t1 ) (40)

Then, (47) can be obtained similar as the proof of (24) easily. Therefore (47) holds. Namely, the claim (23) holds for k = r + 1. By mathematical induction method, the claim holds for all k ∈ N. From the definition of W (t), we have

Hence,

E∥x(t)∥ ≤ ϱ

λ0 −ϵ0 (t−t0 ) e 2 λ1

The proof is thus completed. Remark 3: The discretized Lyapunov function method which was proposed in[38], was used to analyze the linear neural system. And in [40], it was used to analyze the switched systems with unstable modes. Different from the previous results, in this paper, it is applied to investigate the stochastic stability problem of the impulsive systems with time window. Here it should be noted that the number of discretized matrices Pq , q = 0, 1, ..., L are L + 1 which have to be prescribed in advance and different choices of L could lead to different

6

analysis results. Roughly speaking, the larger L is chosen, the denser the division of the interval [tk , tk + dmin ) is therefore produced and, intuitively, a less conservative results can be obtained. Remark 4: In [4][33], the novel time-dependent Lyapunov function method was used to investigate the stability and synchronization problems of nonlinear dynamic systems, respectively. Motivated by the above interesting results, the discretized Lyapunov function method is applied in this paper. Obviously, the method here is more general and the results are less conservative than above literatures[4][33]. To be more specific, if q = 1, PL = P2 , the discretized method is reduced into the time-dependent Lyapunov function method. Remak 5: The following properties for the impulsive time window {dmin , dmax } hold with fixed constants 0 < u < 1, it is easy to see that if Theorem 1 holds for some d∗max , then it holds for any dmax < d∗max , since (13) can be satisfied for any dmax < d∗max . Thus, if LMIs(12),(14)-(16)are feasible with given u and dmin , the corresponding maximal admissible impulse interval d∗max can be computed by the following optimization algorithm: d∗max = maxdmax d∗min . To see this, assume that conditions (12),(14)-(16)of Theorem 1 are satisfied for given d∗min , we can still choose [tk , tk +d∗min ) to be divided into the same segments Nk,q , q = 0, 1, ..., L−1, ∪ then (12),(14)-(16)can ensure D+ EW (t) ≤ µ1 EW (t), t ∈ q=0,1,...,L−1 Nk,q = [tk , tk + d∗min ). And also has D+ EW (t) ∪ ≤ µ1 EW (t) ∀t ∈ [tk + d∗min , tk+1 ) = [tk + d∗min , tk + dmin ) [tk + dmin , tk+1 ). Thus, the dmin > d∗min can also be obtained with given µ1 , µ, σ, and dmax satisfying condition (13) can be estimated by d∗min = mindmin 0, q = 0, 1, ..., L, , positive scalars λ1 ,β, small enough constant ϵ0

and ϵ0 ∈ (0, 1 − µ) such that the following inequalities hold:

(54) (55)

Pq ≤ λ1 In , − ln(µ + ϵ1 )/dmax > 0

− µP0 + BT i = 1, 2, ..., r i PL Bi < 0,  (j) Γ11 Pj Ai + K1 L1 Pj Di 0  (j) Γ22 0 0  ∗ (j) Ξ = (j)  ∗ 0 Γ33 0 (j) 0 0 ∗ Γ44 j = q, q + 1, L (j)

where Γ11 (j) Γ22

=

ϵ0 Pj − 2Pj Ci + (j) Γ33

β µ Pj + (j) Γ44



(56)

   < 0, (57)  (58)

Ψq − K1 L2 ,

= −K1 , = −K2 , = −βPj − K2 L2 . K1 = diag{k11 , k12 , ..., k1n } > 0 and K2 = diag{k21 , k22 , ..., k2n } > 0, Ψq = L(Pq+1 − Pq )/dmin , dmin and dmax are the lower and upper bounds of the impulse time window, respectively. ∗ is used to denote the term that is induced by symmetry. Then, the stochastic fuzzy uncertain delayed neural networks with impulsive time window (53) are exponentially stable in mean square. If there is no state delay in model (4), then model (4) reduces to the following model:  ∑r  dx(t) = i=1 hi (z(t))[−(Ci + ∆Ci )x(t) + (Ai + ∆Ai )f (x(t))]dt + [∆Wi0 x(t)]dW (t), t ̸= tk  ∑r  + x(tk ) = i=1 hi (z(t))Bi x(t− tk ∈ Dk k ), (59) then the following corollary can be obtained easily. Corollary 2: Consider the stochastic delay uncertain neural networks (6) with impulse time window based on T-S model. Suppose that Assumption 1 and 2 hold. If for prescribed positive scalar µ ∈ (0, 1), there exist a set of matrices Pq > 0, q = 0, 1, ..., L, , positive scalars λ1 , σ, β, small enough constant ϵ0 and ϵ0 ∈ (0, 1 − µ) such that the following inequalities hold: (60)

Pq ≤ λ1 In ,

− ln(µ + ϵ1 )/dmax > 0

(61)

− µP0 + BT PL Bi < 0, i = 1, 2, ..., r (62) [ i ] (j) Γ11 Pj Ai + K1 L1 Ξ(j) = < 0, j = q, q + 1, L (63) (j) ∗ Γ22 (j)

where Γ11

β (j) + Ψq µP 2 T 2 σ(Φi ) Φi , K1

= (ϵ0 + 2)Pj − 2Pj Ci +

σ(Φ4i )T Φ4i ,

(j) Γ22



K1 L2 + = −K1 + = diag{k11 , k12 , ..., k1n } > 0 and Ψq = L(Pq+1 − Pq )/dmin , dmin and dmax are the lower and upper bounds of the impulsive time window, respectively. ∗ is used to denote the term that is induced by symmetry. Then, the stochastic delayed fuzzy uncertain delayed neural networks with impulsive time window (59) are exponentially stable in mean square. IV. N UMERICAL EXAMPLES In order to show the usefulness of the theoretical results derived in the preceding section, we present three numerical examples in this section.

7

We let

[

=

Φ42

−1 C1 = 0 [ 0.4 B1 = 0.6 [ 0.4 D1 = 0.2

=

0.5I, Φ51

=

Φ52

= 0.3I, F (t) = 0.5I

] [ 0 −0.9 , C2 = −1 0 ] [ 0.3 0.2 , B2 = 0.28 0.2 ] [ 0.5 0.5 , D2 = 0.9 0.1

1.5 1

] 0 , −0.9 ] 0.4 , 0.4 ] 0.3 , 0.3

Then, we can get L1 = diag{0.5, 0.5}, L2 = 0, τ = 1. Let B1k = 0.4, B2k = 0.6. Then, by (12)-(15), if we fix L = 1,µ = 0.9, the feasible solution can be obtained(It is omitted here for saving space). Moreover, we have 0.1324 ≤ tk − tk−1 ≤ 0.3258 . Let tk − tk−1 = 0.3, the corresponding trajectories of the stochastic uncertain delayed neural networks without and with impulsive time window effects in (66) are presented in Figs. 1 and 2, respectively. One can easily see that, when there are no impulse effects, the above network is not exponentially stable in mean square. However, Fig.2 shows that the it is exponentially stable in mean square with the impulsive interval tk − tk−1 = 0.3. The simulation further confirm the previous results derived well. Example 2:In this example, we will show that if the impulsive effects here is destabilizing impulse. Consider the neural networks model with [ ] [ ] −0.6 0 −0.3 0 C1 = , C2 = , 0 −0.6 0 −0.3 [ ] [ ] 0.4 −0.2 0.6 −0.4 B1 = , B2 = , −0.3 0.3 0.6 −0.2 [ ] [ ] 0.2 −0.7 0.4 0.6 D1 = , D2 = , 0.4 0.4 −0.5 0.6

and B1k = B2k = 1.1,tk − tk−1 = 0.48, and other parameters are given in Example 1. According to Theorem 1, it can be conclude that the neural networks (66) can be exponentially stable in mean square. Fig.3 depict state trajectories of ei1 and ei2 . The simulation results confirm that the stochastic

0.5 0 −0.5 −1

=

1 1 fi (xi ) = tanh(xi ), τ (t) = sin(t) + , M = diag(0.1, 0.5), 2 2 Φ11 = Φ12 = 0.3I, Φ21 = Φ22 = 0.2I, Φ31 = Φ32 = 0.4I Φ41

2

−1.5

0

5

10 t

15

20

Fig. 1: The state trajectories of ei1 of subsystem 1 of Example 1 without impulsive effects .

1 0.8 0.6 0.4 0.2 xi1,xi2

for i = 1, 2, where Mip is a fuzzy set, z(t) [z1 (t), ..., zp (t)]T is the premise variable vector and

2.5

xi1,xi2

Example 1: Let r = 2. Consider the following plant rules of a stochastic uncertain delayed neural networks with impulsive time window: Plant Rule i IF z1 (t) is Mi1 and· · · zp (t) is Mip THEN  ∑r   dx(t) = i=1 hi (z(t))[−(Ci + ∆Ci )x(t)    + (Bi + ∆Bi )f (x(t))  + (Di + ∆Di )f (x(t − τ (t)))]dt    + [∆Wi0 x(t) + ∆Wi1 x(t − τ (t))]dW (t), t ̸= tk    x(t+ ) = ∑r h (z(t))B x(t− ), tk ∈ Dk ik i=1 i k k (64)

0 −0.2 −0.4 −0.6 −0.8 −1

0

1

2

3 t

4

5

Fig. 2: The state trajectories of ei1 of subsystem 1 of Example 1 with impulsive effects. uncertain delayed neural networks is exponentially stable in mean square. Example 3: In this example, we will show that the results derived are also valid for mixed impulse sequence. Consider the neural networks (66) with B1k = 0.6, B2k = 1.1 and other parameters are the same as those in Example 2. Fig.4 depicts the impulsive sequence, and the corresponding state trajectories are plotted in Fig.5. The simulation results show that our results are also valid for mixed impulsive sequences. V. C ONCLUSION In this paper, we have investigated the stochastic stability problem for a class of fuzzy delayed neural networks with impulsive time window. The impulses considered here are more

6

8

5

2

4

1.5

3 1

0.5

1

xi1,xi2

xi1,xi2

2

0

0

−1 −0.5 −2 −1

−3 −4

0

5

10

15 t

20

25

0

2

4

6

8

Fig. 5: The state trajectories of ei1 of subsystem 1 of Example 3 with mixed impulsive effects. supported by Natural Science Foundations of China (grant no: 61374078 and grant no:61302180).

1.4

1.2

R EFERENCES

1 impulsive strength

−1.5

t

Fig. 3: The state trajectories of ei1 of subsystem 1 of Example 2 with impulsive effects .

0.8

0.6

0.4

0.2

0

30

0

2

4

6

8

t

Fig. 4: Impulsive sequence of Example 3. general than most existing results. Based on the discretized Lyapunov approach and mathematical induction method, several novel stability criteria are obtained such that the proposed fuzzy neural networks with impulsive time window can be exponentially stable in mean square. And then, some effective optimization algorithms are presented to computing the lower and upper bounds of the impulsive time window, respectively. Finally, the simulation examples are given to illustrate the validation of the proposed results. ACKNOWLEDGMENTS This publication was made possible by NPRP grant # NPRP 4-1162-1-181 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. This work was also

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Robust stability of stochastic fuzzy delayed neural networks with impulsive time window.

The urgent problem of impulsive moments which cannot be determined in advance brings new challenges beyond the conventional impulsive systems theory. ...
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