Mean square exponential stability for discrete-time stochastic fuzzy neural networks with mixed time-varying delay

Author's Accepted Manuscript

Mean Square Exponential Stability for Discrete-Time Stochastic Fuzzy Neural Networks with Mixed Time-varying Delay Di Liu, Lijie Wang, Yingnan Pan, Haoyi Ma

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PII: DOI: Reference:

S0925-2312(15)00891-7 http://dx.doi.org/10.1016/j.neucom.2015.06.045 NEUCOM15712

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Neurocomputing

Received date: 17 May 2015 Revised date: 18 June 2015 Accepted date: 25 June 2015 Cite this article as: Di Liu, Lijie Wang, Yingnan Pan, Haoyi Ma, Mean Square Exponential Stability for Discrete-Time Stochastic Fuzzy Neural Networks with Mixed Time-varying Delay, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.06.045 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 galley proof before it is published in its final citable 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.

1

Mean Square Exponential Stability for Discrete-Time Stochastic Fuzzy Neural Networks with Mixed Time-varying Delay Di Liu, Lijie Wang, Yingnan Pan and Haoyi Ma

Abstract This paper concerns with the passivity and mean square exponential stability problems for discretetime stochastic fuzzy neural network with parameter uncertainties based on interval type-2 (IT2) fuzzy model. For the nonlinear stochastic fuzzy neural network, novel sufficient conditions are presented by linear matrix inequalities (LMIs) to guarantee the passivity and mean square exponential stability of the resulting system, and the parameter uncertainties are handled via the IT2 fuzzy model approach. The main contribution of this paper is that we first propose the IT2 T-S discrete-time stochastic fuzzy neural network. Finally, a numerical example is provided to testify the effectiveness of the proposed scheme. Keywords: Interval type-2 (IT2) fuzzy systems; Neural networks; Discrete-time stochastic systems; Passivity analysis; Mean square exponential stability

I. I NTRODUCTION Neural network has received wide attention and research, mainly in dealing with the problems of timevarying delays and leakage delays [1]–[3], associative memory [4]–[8], optimization solution [9]–[13], and application of signal processing [14], etc. Many neural networks applications depend on the dynamic behaviors. Therefore, the stability of the neural networks has caused much attention of researchers. The results in [15], [16] showed the problem of stability for nonlinear neural networks with Markovian This work was partially supported by the National Natural Science Foundation of China (61304003, 61304002, 61403043), and the Program for Liaoning Excellent Talents in University (LJQ20141126). D. Liu, L. Wang and Y. Pan are with the College of Mathematics and Physics, Bohai University, Jinzhou 121013, Liaoning, China. (e-mail: [email protected], [email protected], [email protected] (Corresponding Author)). Haoyi Ma is with the Department of Control Science and Engineering, Harbin Institute of Technology Harbin 150080, China (e-mail: [email protected]). DRAFT

2

jumping parameters. Due to the effect of parameter uncertainty, the random interference often exists in the neural networks. The authors in [17] considered the exponential stability problem for discrete-time stochastic fuzzy uncertain neural networks. Recently, the type-1 fuzzy logic control approach has aroused more and more concern because it can solve a series of problems of complexity nonlinear systems [18]–[20]. Fuzzy logic control approach has been used in various industrial fields, such as the prediction of finance [21], vehicle speed estimation [22], automatic control, supervision, and fault diagnosis [23]. Based on a lot of fuzzy logic control approaches, Takagi-Sugeno (T-S) fuzzy model has becomed very popular, which offers an universal framework for studying nonlinear systems. The authors studied the stochastic stability of fuzzy Hopfield neural networks with time-varying delays on account of T-S fuzzy model in [24]. The authors in [25] designed an adaptive tracking controller for uncertain nonlinear stochastic fuzzy neural network-based adaptive control systems. In [26], the authors studied the mean square exponential stability for stochastic fuzzy Hopfield neural networks with time-varying delays by using T-S fuzzy model approach. Nevertheless, one can note that the above papers are on the strength of type-1 T-S fuzzy stochastic neural network-based control systems. Type-1 fuzzy sets can be employed to deal with the nonlinearities of the systems well but it can not be used when there exist parameter uncertainties in membership functions. Type-2 fuzzy sets [27]–[30] are expert in handling uncertainties due to its membership functions exist ambiguity of their own, which depict the uncertainties more precisely. Benefiting from the IT2 fuzzy sets, many considerable results have been given in [31]–[36]. The authors in [32] designed an IT2 fuzzy controller to make the nonlinear system to be stable. In [37], the authors proposed an IT2 fuzzy filter design approach for fuzzy systems with D stability constraints under a unified frame. The problem of fault detection for IT2 fuzzy systems was considered in [38]. The authors in [39] designed a sampleddata controller for IT2 fuzzy systems with actuator fault, which can ensure the closed-loop system is asymptotically stable with H∞ performance in the presence of actuator failure. However, it’s worth mentioning that there are few results on the passivity and mean square exponential stability for the IT2 T-S discrete-time stochastic fuzzy neural network. Hence, it is a very challenge to analyze the passivity and mean square exponential stability for discrete-time stochastic fuzzy neural network with constant delays based on IT2 fuzzy model. For all above discussion, the aim of this paper is to deal with the passivity and mean square exponential stability problems of IT2 discrete-time stochastic fuzzy neural network with constant delays. By means of the IT2 T-S fuzzy model approach, the uncertainties can be well captured by the lower and upper membership functions. First, the definitions of the stochastic passivity and mean square exponential DRAFT

3

stability are given (i.e.,in the sense of expectation) for discrete-time stochastic fuzzy neural network. Then, original sufficient conditions are obtained by developing some new techniques and using stochastic analysis tools to guarantee that the fuzzy system is passive and mean square exponentially stable. The obtained sufficient conditions are formed by linear matrix inequalities (LMIs), which can be easily resolved. The remainder of this paper is structured as follows. Section II provides the IT2 T-S fuzzy model and some pre-knowledge of this paper. Section III presents the passivity and mean square exponential stability conditions for the IT2 T-S discrete-time stochastic fuzzy neural network with constant delays. Section IV shows a simulation example to prove the availability of the proposed results. Finally, Section V concludes this paper. Notation: Throughout the paper, the notations used are fairly standard. “I ” denotes an identity matrix with appropriate dimension and and “0m×n ” denotes m × n zero matrix. “T ” represents the transpose. “P > 0” means that P is positive definite. Rn denotes n-dimension Euclidean space. diag{...} denotes a block diagonal matrix. “∗” represents symmetric terms in a block matrix. l2 [0,∞) means the space of square-integrable vector functions over [0,∞). II. P ROBLEM F ORMULATION Consider the following IT2 T-S discrete-time stochastic fuzzy neural network with ς -rules : Fuzzy Rule i : IF z1 (k) is Υi1 , · · · , and zα (k) is Υiα , · · · , and zι (k) is Υiι , THEN x (k + 1) = Ai x (k) + Bi g (x(k)) + u(k) + D

+∞ 

μi f (x(k − i))

i=1

+Ci x(k − δ) + [Ei x (k) + Fi g (x(k))]w (k) , y(k) = g(x(k)),

(1)

where Υiα stands for the fuzzy set of ith rule according to the measurable premise variable zα (k) for i = 1, 2, · · · , ς and α = 1, 2, · · · , ι ; u (k) ∈ Rq is the input vector; x (k) ∈ Rn denotes the system

state variable; δ is constant time delay; w (k) ∈ Rl is assumed to be a disturbance input belonging to l2 [0,∞); g (x(k)) = [g1 (x1 (k)), g2 (x2 (k)),..., gn (xn (k))]T ∈ Rn ; f (x(k)) = [f1 (x1 (k)), f2 (x2 (k)),

..., fn (xn (k))]T ∈ Rn ; gj (xj (k)) and fj (xj (k)) are jth neuron activation functions. Ai , Bi , Ci , Ei , Fi are known real constant matrices and D is the discretely delayed connection weight matrix; The following interval set stands for the emission intensity of the ith rule: Πi (x (k)) = [εi (x (k)) , εi (x (k))] , i = 1, 2, · · · , ς;

(2)

DRAFT

4

where εi (x (k)) = μZ i (z1 (k)) × μZ i (z2 (k)) · · · ×μZ i (zι (k)) , 1

2

ι

εi (x (k)) = μZ1i (z1 (k)) × μZ2i (z2 (k)) · · · ×μZιi (zι (k)) , 1 ≥ μZαi (zα (k)) ≥ 0, 1 ≥ μZ i (zα (k)) ≥ 0, α

μZαi (zα (k)) ≥ μZ i (zα (k)) ≥ 0, εi (x (k)) ≥ εi (x (k)) ≥ 0, α

εi (x (k)) denotes the lower grade of membership and εi (x (k)) denotes the upper grade of member-

ship. μZ i (zα (k)) and μZαi (zα (k)) represent the lower membership function and the upper membership α

function, respectively. Then system (1) can be written as: x (k + 1) =

ς 

εi (x (k)) [Ai x (k) + Bi g (x(k)) + u(k) + D

i=1

+∞ 

μi f (x(k − i))

i=1

+[Ei x (k) + Fi g (x(k))]w (k) + Ci x(k − δ)] , y(k) = g(x(k)),

(3)

where for i = 1, 2, · · · , ς , εi (x (k)) =

ς 

αi (x (k)) εi (x (k)) + αi (x (k)) εi (x (k))

s=1

(αs (x (k)) εs (x (k)) + αs (x (k)) εs (x (k)))

≥ 0,

ς 

εi (x (k)) = 1,

i=1

αi (x (k)) ∈ [0, 1] , αi (x (k)) ∈ [0, 1] , 1 = αi (x (k)) + αi (x (k)) , αi (x (k)) and αi (x (k)) depend on parameter uncertainties and are unnecessary to be known in this

paper. εi (x (k)) is the grade of the embedded membership functions. In discrete-time IT2 T-S stochastic fuzzy neural network (3), w (k) represents a Brownian motion with E{w (k)} = 0, E{wT (k) w (k)} = 1, E{wT (i) w (j)} = 0 (i = j),

(4)

Definition 1: [40] The discrete-time stochastic neural network (3) is called to be mean square exponentially stable, if there exist positive scalars θ > 0 and 0 < ϑ < 1 such that     E |x(k)|2 ≤ θϑk max E |x(j)|2 , k > 0. −δ≤j≤0

Definition 2: [41] The fuzzy system (3) is called as passive if there exists a scalar γ ≥ 0 such that 2

m  k=0

m      E y T (k)u(k) ≥ −γ E uT (k)u(k) . k=0

for all integers m ≥ 0.

DRAFT

5

Lemma 1: [42] For any constant positive definite matrix M and intergers h2 > h1 , vector function ϕ such that (h2 − h1 + 1)

h2 

ϕT (i)M ϕ(i) ≥ (

i−h1

h2 

ϕ(i))T M (

i−h1

h2 

ϕ(i)).

i−h1

Assumption 1: The neural activation functions fj (·) and gj (·) are continuous and bounded satisfies fj− ≤ gj− ≤

fj (κ1 ) − fj (κ2 ) ≤ fj+ , κ1 , κ2 ∈ R κ1 − κ2 gj (κ1 ) − gj (κ2 ) ≤ gj+ , κ1 , κ2 ∈ R κ1 − κ2

(5) (6)

for all κ1 = κ2 , where gj− , gj+ , fj− and fj+ are some contants, and Q1 = diag{f1− , f2− , ..., fn− }, Q2 = diag{f1+ , f2+ , ..., fn+ }, L1 = diag{g1− , g2− , ..., gn− }, L2 = diag{g1+ , g2+ , ..., gn+ }.

Assumption 2: The constants μi ≥ 0 satisfy the condition for convergence as follow: +∞ 

μi < .

i=1

Remark 1: From Assumption 1 and Assumption 2, we can be sure that and

+∞  i=1

μi

k−1  j=k−i

+∞  i=1

μi f (x(k − i)) is convergent

f T (x(j))Θf (x(j)) converges to a constant assumed to ˆ.

III. M AIN R ESULTS Theorem 1: The discrete-time IT2 T-S stochastic fuzzy neural network (3) is mean square exponentially stable, if there exist parameter dependency matrices P1 (ε) > 0, P2 (ε) > 0, Θ > 0, ε+ = (ε1 (x (k + 1)), ε2 (x (k + 1)), ..., ες (x (k + 1))), Kj = diag {k1 , k2 , ..., kn } and Mj = diag{m1 , m2 ,

..., mn } with appropriate dimensions satisfying the following LMI: ⎡ ⎤ Γ1ij Γ2ij ⎦ < 0, ∀i, j. Γij = ⎣ Γ3ij Γ4ij

(7)

DRAFT

6

where Γ1ij

⎡ = ⎣

⎡ Γ3ij

Γ11ij

Γ12ij

Γ13ij

Γ14ij

⎤

⎡

⎦ , Γ2ij = ⎣

DT P1j Ai DT P1j Bi

⎢ ⎢ = ⎢ CiT P1j Ai ⎣ 2Mj Q1

⎤

⎥ ⎥ CiT P1j Bi ⎥ , ⎦ 0

ATi P1j D

ATi P1j Ci 2QT2 Mj

⎤

⎦, BiT P1j D BiT P1j Ci 0 ⎡ DT P1j D − μ1¯ Θ DT P1j Ci 0 ⎢ ⎢ Γ4ij = ⎢ CiT P1j D CiT P1j Ci − P2i 0 ⎣ 0 0 μ ¯Θ − 2Mj

Γ11ij

= ATi P1j Ai + EiT P1j Ei − P1i + P2j − 2LT2 Kj L1 − 2QT2 Mj Q1 ,

Γ12ij

= ATi P1j Bi + EiT P1j Fi + 2LT2 Kj , Γ13ij = BiT P1j Ai + FiT P1j Ei + 2Kj L1 ,

Γ14ij

=

BiT P1j Bi

+

FiT P1j Fi

+∞ 

− 2Kj , μ ¯=

⎤ ⎥ ⎥ ⎥, ⎦

μi .

i=1

Firstly, we define P1 (ε) =

ς  i=1

εi P1i , P2 (ε) =

ς  i=1

εi P2i , P1 (ε+ ) =

we choose Lyapunov functions for system (3) as follows: V (k, x(k)) =

3 

ς  i=1

+ ε+ i P1i , P2 (ε ) =

ς  i=1

ε+ i P2i . Then,

Vi (k),

i=1

where V1 (k, x(k)) = xT (k) P1 (ε)x (k) , V2 (k, x(k)) =

+∞ 

μi

i=1

V3 (k, x(k)) =

k−1 

k−1 

f T (x(j))Θf (x(j)),

j=k−i

xT (s)P2 (ε)x(s).

s=k−δ

Then, for the difference of Vi (k, x(k)) along the trajectories of (3), taking the mathematical expectation, and using (3) we can obtain as: E{ΔV1 (k, x(k))} = E {V1 (k + 1)} − V1 (k) ⎧  ς  +∞ ς ⎨  2 + εi εj Ai x (k) + Bi g (x(k)) + u(k) + D μi f (x(k − i)) ≤ E ⎩ i=1 j=1

i=1

T

+[Ei x (k) + Fi g (x(k))]w (k) + Ci x(k − δ)] P1j [Ai x (k) + Bi g (x(k)) + u(k)  +∞  μi f (x(k − i)) + [Ei x (k) + Fi g (x(k))]w (k) + Ci x(k − δ) +D

=

i=1  T −ξ1 (k) P1i (ε)ξ1 (k) ς  ς  T ε2i ε+ j ξ1 (k)Γij ξ1 (k). i=1 j=1

(8)

DRAFT

7

where

⎡ Γij

= ⎣

⎡ Γ3ij

ξ1T (k)

Γ2ij

Γ3ij

Γ4ij

⎤

⎡

ATi P1j

ATi P1j D

ATi P1j Ci

⎤

⎦, BiT P1j D BiT P1j Ci ⎤ ATi P1j Ai + EiT P1j Ei − P1i ATi P1j Bi + EiT P1j Fi ⎦, BiT P1j Ai + FiT P1j Ei BiT P1j Bi + FiT P1j Fi ⎤ ⎡ P1j Ai P1j Bi P1j P1j D P1j Ci ⎥ ⎢ ⎥ ⎢ DT P1j Ai DT P1j Bi ⎥ , Γ4ij = ⎢ DT P1j DT P1j D DT P1j Ci ⎦ ⎣ T T Ci P1j Ai Ci P1j Bi CiT P1j CiT P1j D CiT P1j Ci  +∞  T T T T x (k) g (x(k)) u (k) μi f (x(k − i)) x (k − δ) .

= ⎣

⎡ Γ1ij

Γ1ij

⎢ ⎢ = ⎢ ⎣  =

⎦ , Γ2ij = ⎣

BiT P1j

⎤ ⎥ ⎥ ⎥ ⎦

i=1

E{ΔV2 (k, x(k))} = E {V2 (k + 1) − V2 (k)} ⎧ ⎫ k k−1 +∞ +∞ ⎨ ⎬    μi f T (x(j))Θf (x(j)) − μi f T (x(j))Θf (x(j)) = E ⎩ ⎭ i=1 i=1 j=k+1−i j=k−i  +∞  +∞   T T = E μi f (x(k))Θf (x(k)) − μi f (x(k − i))Θf (x(k − i))



i=1

i=1

 +∞  1 ≤ E μ ¯f T (x(k))Θf (x(k)) − ( μi f (x(k − i)))T Θ( μi f T (x(k − i))) μ ¯ i=1 i=1 ⎡ ⎤ −1Θ 0 ⎦ ξ2 (k). = ξ2T (k) ⎣ μ¯ (9) 0 μ ¯Θ  +∞   T T T μi f (x(k − i)) f (x(k)) . where ξ2 (k) = +∞ 

i=1

E{ΔV3 (k, x(k))} = E {V3 (k + 1) − V3 (k)} ⎧  k ς  ς ⎨  ε2i ε+ = E j ⎩ i=1 j=1

=

ς  ς  i=1 j=1

s=k+1−δ

xT (s)P2j x(s) −

k−1 

xT (s)P2i x(s)

s=k−δ

 T  T ε2i ε+ j x (k)P2j x(k) − x (k − δ)P2i x(k − δ) .

⎫ ⎬ ⎭ (10)

For another, from (6) and (5), we can obtain: (fi (xi (k)) − fi+ (xi (k)))(fi (xi (k)) − fi− (xi (k))) ≤ 0, (gi (xi (k)) − gi+ (xi (k)))(gi (xi (k)) − gi− (xi (k))) ≤ 0,

for i = 1, 2, ...n. DRAFT

8

Thus, we have −2

n 

mj (fi (xi (k)) − fi+ (xi (k)))(fi (xi (k)) − fi− (xi (k))) ≥ 0,

i=1 n 

−2

i=1

kj (gi (xi (k)) − gi+ (xi (k)))(gi (xi (k)) − gi− (xi (k))) ≥ 0,

that is −2(g(x(k)) − L2 x(k))T Kj (g(x(k)) − L1 x(k)) ≥ 0, −2(f (x(k)) − Q2 x(k))T Mj (f (x(k)) − Q1 x(k)) ≥ 0,

which are equivalent to

⎡

α(k) = ⎣

⎡ β(k) = ⎣

⎤T ⎡

x(k)

⎦ ⎣

g(x(k))

⎤T ⎡

x(k) f (x(k))

⎦ ⎣

−2LT2 Kj L1

2LT2 Kj

2Kj L1

−2Kj

⎦⎣

x(k)

⎤ ⎦ ≥ 0,

g(x(k)) ⎤⎡ ⎤ T 2Q2 Mj x(k) ⎦⎣ ⎦ ≥ 0. f (x(k)) −2Mj

−2QT2 Mj Q1 2Mj Q1

Then, when u(k) = 0 and ξ3T (k) = [ xT (k) g T (x(k)) we have

⎤⎡

+∞  i=1

(11)

(12)

μi f (x(k − i)) xT (k − δ) f T (x(k)) ]

3  E{ΔV (k)} ≤ E{ ΔVi (k) + α(k) + β(k)}

≤

i=1 ς ς  i=1 j=1

T ε2i ε+ j ξ3 (k)Γij ξ3 (k).

From (7), it follows that there exists a sufficiently small scalar σ > 0 such that   E{ΔV (k)} ≤ −σE{x(k)2 }. At present, we can establish the exponential stability of the system (3). k−1      E{V (k)} ≤ ρ1 E{x(k)2 } + ρ2 E{x(s)2 } + ˆ.

(13)

s=k−δ

where ρ1 = maxi {λmax (P1i )}, ρ2 = maxi {λmax (P2i )}.

DRAFT

9

For any constant τ > 1, we have τ k+1 E{V (k + 1)} − τ k E{V (k)} = τ k+1 E{ΔV (k)} + τ k (τ − 1)E{V (k)}

(14)

    ≤ −στ k+1 E{x(k)2 } + τ k (τ − 1)(ρ1 E{x(k)2 } + ρ2

k−1 

  E{x(s)2 } + ˆ)

s=k−δ

  = φ1 (τ )τ k E{x(k)2 } + φ2 (τ )

k−1 

  τ k E{x(s)2 } + τ k (τ − 1)ˆ .

(15)

s=k−δ

where φ1 (τ ) = −στ + (τ − 1)ρ1 , φ2 (τ ) = ρ2 (τ − 1). In addition, for any interger N ≥ δ + 1, we have N −1  

 τ k+1 E{V (k + 1)} − τ k E{V (k)}

k=0 N

= τ E{V (N )} − E{V (0)} ≤ φ1 (τ )

N −1 

N −1  k−1      τ k E{x(k)2 } + φ2 (τ ) τ k E{x(s)2 } + (τ N − 1)ˆ .

k=0

(16)

k=0 s=k−δ

By using the similar analysis method used in [43], we can obtain N

δ

τ E{V (N )} ≤ E{V (0)} + [φ1 (τ ) + δτ φ2 (τ )]

N −1 

  τ k E{x(k)2 }

k=0

  . +δτ φ2 (τ ) max E{x(i)2 } + (τ N − 1)ˆ δ

(17)

−δ≤i≤0

Define ρ0 = mini (λmax (P1 (ε))), ρˆ = max{ρ1, ρ2 , ˆ}. It’s easy to get that   E{V (N )} ≥ ρ0 E{x(N )2 }.

(18)

From (13), we can find the following inequality that   E{V (0)} ≤ ρˆ max E{x(i)2 }. −δ≤i≤0

Moreover, it can be confirmed that there is a τ0 > 1 such that φ1 (τ0 ) + δτ δ φ2 (τ0 ) = 0.

From (13) - (18), we can obtain   E{x(N )2 } ≤

1 E{V (N )} ρ0       1 x(i)2 } + δτ0δ φ2 (τ0 ) max E{x(i)2 } + (τ N − 1)ˆ ρ ˆ max E{ ≤ −δ≤i≤0 −δ≤i≤0 ρ0 τ0N   1 = 0 ( )N max E{x(i)2 }. τ0 −δ≤i≤0 DRAFT

10

where 0 =

1 ρ+ ρ0 [ˆ

δτ0δ φ2 (τ0 ) +

(τ N −1)ˆ  max−δ≤i≤0 E{|x(i)2 |} ]

> 0, 0 <

1 τ0

< 1.

From Definition 1, we can gain the discrete-time IT2 T-S stochastic fuzzy neural network (3) is mean square exponentially stable and the proof is completed. Theorem 2: System (3) is passive in a sense of Definition 2 if there exist parameter dependency matrices P (ε) > 0, ε+ = (ε1 (x (k + 1)), ε2 (x (k + 1)), ..., ες (x (k + 1))) and Kj = diag{k1 , k2 , ..., kn } and Mj = diag{m1 , m2 , ..., mn } with appropriate dimensions satisfying the following LMI: ⎡ ⎤ ˜ ˜ Γ Γ2ij ˜ ij = ⎣ 1ij ⎦ < 0, ∀i, j. Γ ˜ ˜ Γ3ij Γ4ij

where ˜ 1ij Γ

˜ 2ij Γ

⎡

˜ 11ij Γ

˜ 12ij Γ

⎤

⎡

DT P1j

DT P1j D − μ1¯ Θ

DT P1j Ci

0

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ˜ ⎥ ˜ ⎢ T ⎥ T T ˜ = ⎢ Γ , Γ = ⎢ Ci P1j ⎥, Γ14ij ⎥ Ci P1j D Ci P1j Ci − P2i 0 4ij ⎣ 13ij ⎦ ⎣ ⎦ 0 0 0 μ ¯Θ − 2Mj P1j Ai P1j Bi ⎡ ⎤ ⎡ ⎤ ATi P1j ATi P1j D ATi P1j Ci 2QT2 Mj DT P1j Ai DT P1j Bi ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ˜ ⎢ ⎥ = ⎢ BiT P1j − 2I BiT P1j D BiT P1j Ci ⎥ , Γ3ij = ⎢ CiT P1j Ai CiT P1j Bi ⎥ , 0 ⎣ ⎦ ⎣ ⎦ P1j − γI P1j D P1j Ci 0 2Mj Q1 0

˜ 11ij Γ

= ATi P1j Ai + EiT P1j Ei − P1i + P2j − 2LT2 Kj L1 − 2QT2 Mj Q1 ,

˜ 12ij Γ

˜ 13ij = BiT P1j Ai + FiT P1j Ei + 2Kj L1 , = ATi P1j Bi + EiT P1j Fi + 2LT2 Kj , Γ

˜ 14ij Γ

(19)

=

ξ T (k) =

BiT P1j Bi

+

FiT P1j Fi

− 2Kj , μ ¯=

+∞ 

μi ,

i=1

 xT (k) g T (x(k)) uT (k)

+∞  i=1



μi f (x(k − i)) xT (k − δ) f T (x(k))

.

From (8) - (12), we know E{ΔV (k, x(k)) − 2y T (k)u(k) − γuT (k)u(k)} ≤ E{ΔV (k, x(k)) − 2y T (k)u(k) − γuT (k)u(k) + α(k) + β(k)} ς  ς  ˜ ij ξ(k). = ε2 ε+ ξ T (k)Γ i=1 j=1

i j

From (19), we can know E{ΔV (k, x(k)) − 2y T (k)u(k) − γuT (k)u(k)} ≤ 0,

Therefore, we have 2

m  j=0

T

E{y (j)u(j)} ≥

m  j=0

E{ΔV (j, x(j))} −

m 

E{γuT (j)u(j)},

j=0 DRAFT

11

for all m ∈ N, for another, by using the definition of ΔV (k, x(k)), we can obtain m 

E{ΔV (j, x(j))} =

j=0

m 

E{V (m + 1, x(m + 1)) − V (0, x(0))} ≥ 0,

i=0

Then, we have 2

m 

T

E{y (j)u(j)} ≥ −

m 

j=0

E{γuT (j)u(j)}.

i=0

From Definition 2, we can gain the discrete-time IT2 T-S stochastic fuzzy neural network (3) is passive and the proof is completed. From Theorem 2, it is easy to know that if there is no Brownian motion, that is w(k) = 0 in the system (3), the following corollary can be easy to get. Corollary 1: The fuzzy system (3) with w(k) = 0 is passive in the sense of Definition 2 if there exist parameter dependency matrices P (ε) > 0, ε+ = (ε1 (x (k + 1)), ε2 (x (k + 1)), ..., ες (x (k + 1))), Kj = diag{k1 , k2 , ..., kn } and Mj = diag{m1 , m2 , ..., mn } with appropriate dimensions satisfying the following ⎡

LMI:

ˆ ij = ⎣ Γ

where ˆ 1ij Γ

˜ 2ij Γ

ˆ 14ij Γ

⎡

ˆ 11ij Γ

ˆ 12ij Γ

⎤

⎡

ˆ 1ij Γ

˜ 2ij Γ

˜ 3ij Γ

˜ 4ij Γ

DT P1j

⎤ ⎦ < 0, ∀i, j.

DT P1j D − μ1¯ Θ

(20)

DT P1j Ci

0

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ˆ ⎥ ⎥ ˜ ⎢ T TP D TP C − P ˆ = ⎢ Γ , Γ = ⎥, ⎥ ⎢ C P C C 0 Γ 4ij 14ij 2i i 1j i 1j i ⎣ 13ij ⎦ ⎦ ⎣ i 1j 0 0 0 μ ¯Θ − 2Mj P1j Ai P1j Bi ⎡ ⎤ ⎡ ⎤ T T T T T T Ai P1j Ai P1j D Ai P1j Ci 2Q2 Mj D P1j Ai D P1j Bi ⎢ ⎥ ⎢ ⎥ ⎢ T ⎥ ⎢ ⎥ ˜ T T = ⎢ Bi P1j − 2I BiT P1j D BiT P1j Ci , Γ = ⎥ ⎢ 0 C P A Ci P1j Bi ⎥ , 3ij ⎣ ⎦ ⎣ i 1j i ⎦ P1j − γI P1j D P1j Ci 0 2Mj Q1 0 = BiT P1j Bi − 2Kj , μ ¯=

+∞ 

μi ,

i=1

ˆ 11ij Γ

=

ˆ 12ij Γ

ˆ 13ij = BiT P1j Ai + 2Kj L1 . = ATi P1j Bi + 2LT2 Kj , Γ

ATi P1j Ai

− P1i + P2j − 2LT2 Kj L1 − 2QT2 Mj Q1 ,

The proof is similar to the method of Theorem 1 and is omitted to save space.

DRAFT

12

IV. S IMULATION E XAMPLES In this section, simulation result is provided to illustrate the effectiveness of the proposed method. Consider the 2-rule IT2 T-S discrete-time stochastic fuzzy neural network with parameters as follows : ⎡ ⎤ ⎡ ⎤ −0.0039 −0.0067 −0.0011 0.0053 ⎦ , A2 = ⎣ ⎦, A1 = ⎣ −0.0094 −0.0192 −0.0052 0.0006 ⎡ ⎤ ⎡ ⎤ 0.0107 0.0012 0.0024 −0.0010 ⎦ , B2 = ⎣ ⎦, B1 = ⎣ 0.0162 −0.0124 0.0140 0.0039 ⎡ ⎤ ⎡ ⎤ −0.0097 0.0040 −0.0167 −0.0103 ⎦ , C2 = ⎣ ⎦, C1 = ⎣ 0.0151 −0.0042 −0.0069 −0.0049 ⎡ ⎤ ⎡ ⎤ 0.0035 0.0016 0.0037 0.0033 ⎦ , E2 = ⎣ ⎦, E1 = ⎣ 0.0083 0.0008 0.0265 0.0014 ⎡ ⎤ ⎡ ⎤ 0.0158 −0.0067 −0.0036 0.0088 ⎦ , F2 = ⎣ ⎦, F1 = ⎣ 0.0009 0.0093 0.0020 0.0081 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −0.7 0 0.6 0 −0.1975 −0.0142 ⎦ , L2 = ⎣ ⎦, D = ⎣ ⎦, L1 = ⎣ 0 −0.35 0 0.35 −0.0153 −0.0952 ⎡ ⎤ ⎡ ⎤ −0.7 0 0.6 0 e−4 ⎦ , Q2 = ⎣ ⎦ , μi = e−4i , μ ¯= Q1 = ⎣ . (1 − e−4 ) 0 −0.35 0 0.35 Take the activation functions as f1 (x1 (k)) = −0.05 + 0.65 sin(x1 (k)), f2 (x2 (k)) = 0.35 sin(x2 (k)), g1 (x1 (k)) = tanh(−x1 (k)), g2 (x2 (k)) = tanh(0.3x2 (k)), f1− = −0.7, f2− = −0.35, g1− = −1, g2− = 0,

f1+ = 0.6, f2+ = 0.35,

g1+ = 0, g2+ = 0.3.

The constant delay δ = 4. The membership functions for Rules 1 and 2 of system (3) is ε1 (x1 ) = 1 − 1+exp(−x11+4+ν(x1 ))

and ε2 (x1 ) = 1 − ε1 (x1 ), which ν(x1 ) = 0.1 × sin(x1 ) ∈ [−0.1, 0.1]

stands

for the parameter uncertainties. We can obtain the parameter matrices as follows: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.2198 0 0.5127 0 0.0231 0 ⎦, M = ⎣ ⎦, Θ = ⎣ ⎦. K=⎣ 0 0.5282 0 0.5905 0 0.0219 

The state trajectories of the fuzzy system are depicted in Fig. 1 with the initial condition x0 = −0.2 0.5

T

. It is obvious that the discrete-time IT2 T-S stochastic fuzzy neural network is stable. DRAFT

13

And the discrete-time IT2 T-S stochastic fuzzy neural network (3) is passive with the parameter matrix as follows:

⎡

K=⎣

⎤

29.2048

0

0

89.3366

⎡

⎦, M = ⎣

80.1669

0

0

103.0725

⎤

⎡

⎦, Θ = ⎣

0.5

5.2817

0

0

4.0907

⎤ ⎦.

x 1 (t) x 2 (t)

0.4 0.3

x(t)

0.2 0.1 0 −0.1 −0.2 −0.3

0

100

200

Times

300

400

500

Fig. 1. State response of the closed-loop system.

V. C ONCLUSION In this paper, the problems of passivity and mean square exponential stability for the discrete-time IT2 T-S stochastic fuzzy neural network have been studied. The parameter uncertainties and stochastic disturbances have been taken into account to response more realistic behaviors of the IT2 fuzzy system. By applying the Lyapunov functional method and some advanced techniques, sufficient conditions have been obtained to guarantee the IT2 T-S fuzzy system to be passive and mean square exponentially stable in the sense of expectation. The results of simulation have illustrated the availability of the presented results. In future work, we will attempt to solve the dissipativity problem for discrete-time IT2 T-S fuzzy neural network with leakage delay. In future works, the problems of fault-tolerant control will be considered for the systems with actuator faults [44]–[49]. R EFERENCES [1] T. Li, L. Guo, C. Sun, and C. Lin, “Further results on delay-dependent stability criteria of neural networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 19, no. 4, pp. 726–730, 2008.

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14

[2] T. Li, L. Guo, and C. Sun, “Further result on asymptotic stability criterion of neural networks with time-varying delays,” Neurocomputing, vol. 71, no. 1, pp. 439–447, 2007. [3] R. Sakthivel, P. Vadivel, K. Mathiyalagan, A. Arunkumar, and M. Sivachitra, “Design of state estimator for bidirectional associative memory neural networks with leakage delays,” Information Sciences, vol. 296, pp. 263–274, 2015. [4] J. Hu, H. Tang, K. C. Tan, and S. B. Gee, “A spiking neural network model for associative memory using temporal codes,” in Proceedings of the 18th Asia Pacific Symposium on Intelligent and Evolutionary Systems, Volume 1.

Springer, 2015,

pp. 561–572. [5] G. A. Carpenter, “Neural network models for pattern recognition and associative memory,” Neural networks, vol. 2, no. 4, pp. 243–257, 1989. [6] G. Bao and Z. Zeng, “Analysis and design of associative memories based on recurrent neural network with discontinuous activation functions,” Neurocomputing, vol. 77, no. 1, pp. 101–107, 2012. [7] M. Xiao, W. X. Zheng, and J. Cao, “Hopf bifurcation of an-neuron bidirectional associative memory neural network model with delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 1, pp. 118–132, 2013. [8] R. Sakthivel, R. Raja, and S. M. Anthoni, “Exponential stability for delayed stochastic bidirectional associative memory neural networks with markovian jumping and impulses,” Journal of Optimization Theory and Applications, vol. 150, no. 1, pp. 166–187, 2011. [9] G. Li, Z. Yan, and J. Wang, “A one-layer recurrent neural network for constrained nonconvex optimization,” Neural Networks, vol. 61, pp. 10–21, 2015. [10] H.-X. Huang, J.-C. Li, and C.-L. Xiao, “A proposed iteration optimization approach integrating backpropagation neural network with genetic algorithm,” Expert Systems with Applications, vol. 42, no. 1, pp. 146–155, 2015. [11] Q. Liu, Z. Guo, and J. Wang, “A one-layer recurrent neural network for constrained pseudoconvex optimization and its application for dynamic portfolio optimization,” Neural Networks, vol. 26, pp. 99–109, 2012. [12] V. Latorre and D. Y. Gao, “Canonical dual solutions to nonconvex radial basis neural network optimization problem,” Neurocomputing, vol. 134, pp. 189–197, 2014. [13] R. Ge, L. Liu, and J. Wang, “A modified neural network for solving general singular convex optimization with bounded variables,” in Advances in Global Optimization.

Springer, 2015, pp. 363–374.

[14] D. Yu and L. Deng, “Fuse deep neural network and gaussian mixture model systems,” in Automatic Speech Recognition. Springer, 2015, pp. 177–191. [15] H. Li, B. Chen, Q. Zhou, and W. Qian, “Robust stability for uncertain delayed fuzzy hopfield neural networks with Markovian jumping parameters,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 39, no. 1, pp. 94–102, 2009. [16] A. Arunkumar, R. Sakthivel, K. Mathiyalagan, and J. H. Park, “Robust stochastic stability of discrete-time fuzzy markovian jump neural networks,” ISA Transactions, vol. 53, no. 4, pp. 1006–1014, 2014. [17] K. Mathiyalagan, R. Sakthivel, and S. M. Anthoni, “Exponential stability result for discrete-time stochastic fuzzy uncertain neural networks,” Physics Letters A, vol. 376, no. 8, pp. 901–912, 2012. [18] M. Sugeno and T. Yasukawa, “A fuzzy-logic-based approach to qualitative modeling,” IEEE Transactions on fuzzy systems, vol. 1, no. 1, pp. 7–31, 1993. [19] G.-C. Hwang and S.-C. Lin, “A stability approach to fuzzy control design for nonlinear systems,” Fuzzy Sets and Systems, vol. 48, no. 3, pp. 279–287, 1992.

DRAFT

15

[20] Y.-J. Liu and S. Tong, “Adaptive fuzzy control for a class of unknown nonlinear dynamical systems,” Fuzzy Sets and Systems, vol. 263, pp. 49–70, 2015. [21] J. W. Lin, M. I. Hwang, and J. D. Becker, “A fuzzy neural network for assessing the risk of fraudulent financial reporting,” Managerial Auditing Journal, vol. 18, no. 8, pp. 657–665, 2003. [22] K. Kobayashi, K. C. Cheok, and K. Watanabe, “Estimation of absolute vehicle speed using fuzzy logic rule-based kalman filter,” in American Control Conference, Proceedings of the 1995, vol. 5.

IEEE, 1995, pp. 3086–3090.

[23] R. Isermann, “On fuzzy logic applications for automatic control, supervision, and fault diagnosis,” IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, vol. 28, no. 2, pp. 221–235, 1998. [24] H. Huang, D. W. Ho, and J. Lam, “Stochastic stability analysis of fuzzy hopfield neural networks with time-varying delays,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 52, no. 5, pp. 251–255, 2005. [25] C. P. Chen, Y.-J. Liu, and G.-X. Wen, “Fuzzy neural network-based adaptive control for a class of uncertain nonlinear stochastic systems,” IEEE Transactions on Cybernetics, vol. 44, no. 5, pp. 583–593, 2014. [26] H. Li, B. Chen, C. Lin, and Q. Zhou, “Mean square exponential stability of stochastic fuzzy hopfield neural networks with discrete and distributed time-varying delays,” Neurocomputing, vol. 72, no. 7, pp. 2017–2023, 2009. [27] J. M. Mendel and R. B. John, “Type-2 fuzzy sets made simple,” IEEE Transactions on Fuzzy Systems, vol. 10, no. 2, pp. 117–127, 2002. [28] N. N. Karnik and J. M. Mendel, “Operations on type-2 fuzzy sets,” Fuzzy sets and systems, vol. 122, no. 2, pp. 327–348, 2001. [29] J. M. Mendel, “Type-2 fuzzy sets and systems: an overview,” IEEE Computational Intelligence Magazine, vol. 2, no. 1, pp. 20–29, 2007. [30] Q. Liang and J. M. Mendel, “Interval type-2 fuzzy logic systems: theory and design,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 5, pp. 535–550, 2000. [31] D. Wu and J. M. Mendel, “Designing practical interval type-2 fuzzy logic systems made simple,” IEEE International Conference on Fuzzy Systems, pp. 800–807, 2014. [32] H. Lam, H. Li, C. Deters, H. Wuerdemann, E. Secco, and K. Althoefer, “Control design for interval type-2 fuzzy systems under imperfect premise matching,” IEEE Transactions on Industrial Electronics, vol. 61, no. 2, pp. 956–968, 2014. [33] X. Luo, F. Liu, and F. Sun, “Attitude tracking control for hypersonic vehicles based on type-2 fuzzy dynamic characteristic modeling method,” IEEE International Conference on Fuzzy Systems, pp. 113–120, 2014. [34] H. Li, X. Sun, L. Wu, and H.-K. Lam, “State and output feedback control of a class of fuzzy systems with mismatched membership functions,” IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2014.2387876,2014. [35] H. Li, C. Wu, L. Wu, H.-K. Lam, and Y. Gao, “Filtering of interval type-2 fuzzy systems with intermittent measurements,” IEEE Transactions on Cybernetics, 2015. [36] H. Li, C. Wu, P. Shi, and Y. Gao, “Control of nonlinear networked systems with packet dropouts: Interval type-2 fuzzy model-based approach,” IEEE Transactions on Cybernetics, DOI: 10.1109/TCYB.2014.2371814,2014. [37] H. Li, Y. Pan, and Q. Zhou, “Filter design for interval type-2 fuzzy systems with d stability constraints under a unified frame,” IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2014.2315658. [38] Y. Pan, H. Li, and Q. Zhou, “Fault detection for interval type-2 fuzzy systems with sensor nonlinearities,” Neurocomputing, vol. 145, pp. 488–494, 2014. [39] H. Li, X. Sun, P. Shi, and H.-K. Lam, “Control design of interval type-2 fuzzy systems with actuator fault: Sampled-data control approach,” Information Sciences, DOI:10.1016/j.ins.2015.01.008.

DRAFT

16

[40] M. Hu, J. Cao, and A. Hu, “Mean square exponential stability for discrete-time stochastic switched static neural networks with randomly occurring nonlinearities and stochastic delay,” Neurocomputing, vol. 129, pp. 476–481, 2014. [41] J. Liang, Z. Wang, and X. Liu, “On passivity and passification of stochastic fuzzy systems with delays: the discrete-time case,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 40, no. 3, pp. 964–969, 2010. [42] S. Xu, J. Lam, B. Zhang, and Y. Zou, “A new result on the delay-dependent stability of discrete systems with time-varying delays,” International Journal of Robust and Nonlinear Control, vol. 24, no. 16, pp. 2512–2521, 2014. [43] Z. Wang, Y. Liu, M. Li, and X. Liu, “Stability analysis for stochastic cohen-grossberg neural networks with mixed time delays,” IEEE Transactions on Neural Networks, vol. 17, no. 3, pp. 814–820, 2006. [44] T. Li and W. X. Zheng, “Networked-based generalised h-infinity fault detection filtering for sensor faults,” INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE, vol. 46, no. 5, pp. 831–840, 2015. [45] T. Li, G. Li, and Q. Zhao, “Adaptive fault-tolerant stochastic shape control with application to particle distribution control,” Systems, Man, and Cybernetics: Systems, vol. DOI: 10.1109/TSMC.2015.2433896. [46] S. Yin, H. Luo, and S. X. Ding, “Real-time implementation of fault-tolerant control systems with performance optimization,” IEEE Transactions on Industrial Electronics, vol. 61, no. 5, pp. 2402–2411, 2013. [47] S. Yin, S. X. Ding, A. Haghani, H. Hao, and P. Zhang, “A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark tennessee eastman process,” Journal of Process Control, vol. 22, no. 9, pp. 1567–1581, 2012. [48] H. Li, Y. Gao, L. Wu, and H. Lam, “Fault detection for T-S fuzzy time-delay systems: Delta operator and input-output methods,” IEEE Transactions on Cybernetics, DOI: 10.1109/TCYB.2014.2323994, 2014. [49] H. Li, Y. Gao, L. Wu, and H. Lam, “Fault detection for T–S fuzzy time-delay systems: Delta operator and input-output methods,” IEEE Transactions on Cybernetics, vol. 45, no. 2, pp. 229–241, 2015.

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