Filtering for Takagi-Sugeno Fuzzy Neural Networks

L2-L∞ Filtering for Takagi-Sugeno Fuzzy Neural Networks Based on Wirtinger-Type Inequalities Hyun Duck Choi1 , Choon Ki Ahn∗1 , Peng Shi2,3 , Myo Taeg Lim1 , and Moon Kyou Song4 1

School of Electrical Engineering, Korea University, Anam-Dong 5-Ga, Sungbuk-Gu, Seoul 136-701, Korea

2

College of Automation, Harbin Engineering University, Heilongjiang Province, Harbin 150001, China

3

School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia 4

Department of Electronics Convergence Engineering, Wonkwang University 344-2, Shinyong-dong, Iksan 570-749, Korea

Abstract

This paper deals with the L2 -L∞ filtering problem for continuous-time TakagiSugeno fuzzy delayed Hopfield neural networks based on Wirtinger-type inequalities. A new set of delay-dependent conditions is established to estimate the state variables of fuzzy neural networks through the observed input and output variables. This ensures that the state estimation error system is asymptotically stable with a guaranteed L2 -L∞ performance. The presented criterion is formulated in terms of linear matrix inequalities (LMIs). An example with simulation results is given to illustrate the effectiveness of the proposed fuzzy neural state estimator. Key Words: L2 -L∞ filtering, Takagi-Sugeno fuzzy Hopfield neural network, linear matrix inequality (LMI), Wirtinger-type inequality ∗

Corresponding author

1

1

Introduction Several types of neural networks have been investigated over the past decade because they

have practical applications in many areas such as signal processing, combinatorial optimization, and communication [1, 2, 3, 4, 5, 6]. These applications mainly depend on the dynamic characteristics of the neural networks. Hopfield neural networks [7, 8] are the most popular and widely used among the many neural networks, and they have been extensively researched and successfully utilized in several fields such as combinatorial optimization, computer vision, intelligent control, and pattern recognition [9]. Recently, Takagi-Sugeno (T-S) fuzzy systems have attracted much interest. The linguistic description of T-S fuzzy systems can provide an efficient way to describe many nonlinear systems using local linear systems [10, 11, 12]. T-S fuzzy systems can be used to represent complex nonlinear systems by having a set of Hopfield neural networks as their consequent parts. This nice approximation property of the T-S fuzzy models has resulted in the recent recognition of T-S fuzzy Hopfield neural networks [13, 14, 15, 16] as an important method for approximating complex nonlinear systems. Some stability analysis results for T-S fuzzy Hopfield neural networks have been researched in [13, 14, 15, 16]. Recently, Ahn proposed several new results on system identification, weight learning, and robust stability for T-S fuzzy delayed neural networks in [17, 18, 19, 20]. In large-scale neural networks, only partial information of the state variables of neural networks can be measured in the output variables. Therefore, estimation of the state variable of the neural networks through available input and output measurements becomes desirable in order to use the neural networks in several applications. The state estimation problem of neural networks has been extensively researched in [4, 5, 6, 21]. Wang et al. obtained a criterion for state estimation of delayed neural networks in [22]. The authors in [23] studied the state estimator design method of neural networks and presented a condition for the asymptotic stability of the estimation error system. Recently, the authors in [24] researched the state estimator for neural networks with discrete and distributed delays. State estimation methods based on passivity and an H∞ approach for T-S fuzzy Hopfield neural networks were also proposed in [25, 26, 27, 28]. On the other hand, external disturbances always exist in physical systems and can generate 2

instability and poor performance. For this reason, L2 -L∞ control has received much attention. The L2 -L∞ approach [29, 30, 31, 32, 33, 34] is now recognized as a handy tool for dealing with external disturbances or model uncertainties. For example, the design of an L2 -L∞ filter was introduced in [35, 36] for stochastic systems with time-varying delay. Huang and Feng proposed an L2 -L∞ state estimator for delayed neural networks [37] in order to attenuate the effect of external disturbances in the state estimation problem for neural networks. This raises the following question: Is there an L2 -L∞ state estimator for fuzzy delayed neural networks? This paper answers this question. To the best of our knowledge, no result for the L2 -L∞ state estimation of fuzzy delayed neural networks has yet been presented in the literature. Thus, this situation leads to our present research. In this paper, we investigate the L2 -L∞ filtering problem for continuous-time T-S fuzzy delayed Hopfield neural networks. A new set of delay-dependent conditions is established for estimation of the state variables of fuzzy neural networks through input and output measurements, which gives a state estimation error system that is asymptotically stable with a guaranteed L2 -L∞ performance. We improve the conservatism of the L2 -L∞ filtering problem for T-S fuzzy delayed Hopfield neural networks by employing a Lyapunov-Krasovskii functional with triple integral term, Wirtinger-type inequalities [38, 39] and the Leibniz-Newton formula. The criterion is formulated in terms of LMIs [40], which can be checked efficiently by the use of some standard numerical algorithms [41]. Compared to [37], the main contributions of this paper are that it initially formulates the L2 -L∞ filtering problem for fuzzy delayed neural networks and it develops a delay-dependent and LMI-based approach to solve it. This paper is organized as follows. In Section 2, we formulate the problem. In Section 3, an LMI problem for the L2 -L∞ filtering of T-S fuzzy delayed Hopfield neural networks is proposed. In Section 4, a numerical example is given, and finally, conclusions are presented in Section 5.

2

Problem Formulation Consider the following delayed Hopfield neural network: x(t) ˙ = Ax(t) + W ϕ(x(t − τ )) + J(t) + Gw(t), 3

(1)

y(t) = Cx(t) + Dx(t − τ ) + Ew(t),

(2)

z(t) = Hx(t),

(3)

where x(t) = [x1 (t) ... xn (t)]T ∈ Rn is the state vector, z(t) = [z1 (t) ... zp (t)]T ∈ Rp , which needs to be estimated, is a linear combination of the states, y(t) = [y1 (t) ... ym (t)]T ∈ Rm is the output vector, w(t) = [w1 (t) ... wk (t)]T ∈ Rk is the disturbance vector, τ ≥ 0 is the time-delay, A = diag{−a1 , . . . , −an } ∈ Rn×n (ak > 0, k = 1, . . . , n) is the self-feedback matrix, W ∈ Rn×n is the delayed connection weight matrix, ϕ(x(t)) = [ϕ1 (x(t)) ... ϕn (x(t))]T : Rn → Rn is the nonlinear function vector satisfying the global Lipschitz condition with Lipschitz constant Lϕ > 0, G ∈ Rn×k , C ∈ Rm×n , D ∈ Rm×n , E ∈ Rm×k , and H ∈ Rp×n are known constant matrices, and J(t) ∈ Rn is an external input vector. According to [17], Hopfield neural networks have become an efficient approximator for complex nonlinear systems. The T-S fuzzy model concept is used as the basis to construct the T-S fuzzy delayed Hopfield neural network by a set of Hopfield neural networks that are smoothly connected by a fuzzy membership functions as follows:

Fuzzy Rule i: IF ω1 is µi1 and . . . ωs is µis THEN x(t) ˙ = Ai x(t) + Wi ϕ(x(t − τ )) + Ji (t) + Gi w(t),

(4)

y(t) = Ci x(t) + Di x(t − τ ) + Ei w(t),

(5)

z(t) = Hx(t),

(6)

where ωj (j = 1, . . . , s) is the premise variable, µij (i = 1, . . . , r, j = 1, . . . , s) is the fuzzy set that is characterized by a membership function, r is the number of the IF-THEN rules, and s is the number of the premise variables. Using a standard fuzzy inference method, the system (4)-(5) is inferred as follows:

x(t) ˙ =

r ∑

hi (ω)[Ai x(t) + Wi ϕ(x(t − τ )) + Ji (t) + Gi w(t)],

i=1

4

(7)

y(t) =

r ∑

hi (ω)[Ci x(t) + Di x(t − τ ) + Ei w(t)],

(8)

i=1

z(t) = Hx(t),

(9)

where ω = [ω1 , . . . , ωs ], hi (ω) = wi (ω)/

∑r

j=1 wj (ω),

wi : Rs → [0, 1] (i = 1, . . . , r) is the

membership function of the system with respect to the fuzzy rule i. hi can be regarded as the normalized weight of each IF-THEN rule and it satisfies r ∑

hi (ω) ≥ 0,

hi (ω) = 1.

(10)

i=1

Now, we propose the following fuzzy neural state estimator:

Fuzzy Rule i: IF ω1 is µi1 and . . . ωs is µis THEN x ˆ˙ (t) = Ai x ˆ(t) + Wi ϕ(ˆ x(t − τ )) + Ji (t) + L(y(t) − yˆ(t)),

(11)

yˆ(t) = Ci x ˆ(t) + Di x ˆ(t − τ ),

(12)

zˆ(t) = H x ˆ(t),

(13)

where x ˆ(t) = [ˆ x1 (t) ... x ˆn (t)]T ∈ Rn is the state vector of the state estimator, yˆ(t) = [ˆ y1 (t) ... yˆm (t)]T ∈ Rm is the output vector of the state estimator, zˆ(t) = [ˆ z1 (t) ... zˆp (t)]T ∈ Rp , which is the estimate value of z(t), is a linear combination of the states of the state estimator, and L ∈ Rn×m is the gain matrix of the state estimator to be designed. Using a standard fuzzy inference method, the fuzzy neural state estimator (11)-(13) is inferred as follows: x ˆ˙ (t) =

r ∑

hi (ω)[Ai x ˆ(t) + Wi ϕ(ˆ x(t − τ )) + Ji (t) + L(y(t) − yˆ(t))],

(14)

hi (ω)[Ci x ˆ(t) + Di x ˆ(t − τ )],

(15)

i=1

yˆ(t) =

r ∑ i=1

zˆ(t) = H x ˆ(t).

(16)

5

Define the state estimation errors e(t) = x(t)− x ˆ(t) and z˜(t) = z(t)− zˆ(t). The state estimation error system is then given by

e(t) ˙ =

r ∑

{ ˜ hi (ω) (Ai − LCi )e(t) − LDi e(t − τ ) + Wi ϕ(x(t − τ ))

i=1

} + (Gi − LEi )w(t) ,

(17)

z˜(t) = He(t),

(18)

˜ where ϕ(x(t − τ )) = ϕ(x(t − τ )) − ϕ(ˆ x(t − τ )). In this paper, given a prescribed level of disturbance attenuation γ > 0, we design a fuzzy neural state estimator such that the state estimation error system (17)-(18) with w(t) = 0 is asymptotically stable (limt→∞ e(t) = 0) and ∫ sup{˜ z T (t)˜ z (t)} < γ 2 t≥0

∞

wT (t)w(t)dt

(19)

0

under the zero-initial conditions for all nonzero w(t) ∈ L2 [0, ∞), where L2 [0, ∞) is the space of the square integrable vector functions over [0, ∞).

3

L2 -L∞ Filtering for T-S Fuzzy Hopfield Neural Networks In this section, we design an L2 -L∞ filter for the T-S fuzzy delayed Hopfield neural network

(4)-(6), such that the state estimation error system is asymptotically stable with a guaranteed L2 -L∞ performance. A new set of delay-dependent conditions for the existence of the desired fuzzy neural state estimator is presented for the T-S fuzzy delayed Hopfield neural network (4)-(6) in the following theorem:

Theorem 1. Let η and ϵ be given positive scalars. For a given level γ > 0, assume that there exist a scalar δ > 0 and common matrices P = P T > 0, Q = QT > 0, R = RT > 0, [ ]T U1 = U1T > 0, U3 = U3T > 0, S = S T > 0, T = T T > 0, N = N1T N2T N3T N4T N5T N6T N7T ,

6

and M such that  Ξi11 Ξi12 Ξi13   ∗ Ξi22 Ξi23   ∗ ∗ Ξi33   ∗ ∗  ∗   ∗ ∗ ∗   ∗ ∗ ∗    ∗ ∗ ∗   ∗ ∗  ∗   ∗ ∗ ∗ ∗ ∗ ∗

√ √ τ N1 τ (P Ai − M Ci )T √ √ Ξi24 Ξi25 Ξi26 Ξi27 τ N2 − τ DiT M T √ τ N3 0 Ξi34 Ξi35 0 0 √ Ξi44 Ξi45 Ξi46 Ξi47 τ N4 0 √ τ N5 0 ∗ Ξi55 0 0 √ √ ∗ ∗ Ξi66 0 τ N6 τ WiT P √ √ ∗ ∗ ∗ Ξi77 τ N7 τ (P Gi − M Ei )T Ξi14 Ξi15 Ξi16 Ξi17

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

−S ∗ ∗

√τ 2

(P Ai − M Ci )T



    0   0    0  < 0, (20)  √τ W T P  i 2  T τ √ (P Gi − M Ei )   2  0 0   2  −2ηP + η S 0 2 ∗ −2ϵP + ϵ T     T  U1 U2   P H    > 0,   > 0, (21) U2T U3 H γ2I − √τ2 DiT M T

for i = 1, . . . , r, where Ξi11 = P Ai + ATi P − M Ci − CiT M T + τ Q + R + N1 + N1T , Ξi12 = −M Di − N1 + N2T , Ξi13 = U1 + τ U2T + N3T , Ξi14 = −N1 + N4T , Ξi15 = U2 + τ U3 + N5T , Ξi16 = P Wi + N6T , Ξi17 = P Gi − M Ei + N7T , 2 2 Ξi22 = δL2Q I − R − S − N2 − N2T , Ξi23 = −U1 + 2 S − N3T , τ τ Ξi24 = −N2 − N4T , Ξi25 = −U2 − N5T , Ξi26 = −N6T , Ξi27 = −N7T , 2 2 2 1 Ξi33 = − Q − 3 S − 2 T − U2 − U2T , Ξi34 = −N3 , Ξi35 = 3 T − U3 , τ τ τ τ Ξi44 = −N4 − N4T , Ξi45 = −N5T , Ξi46 = −N6T , Ξi47 = −N7T , Ξi55 = −

2 T, Ξi66 = −δI, Ξi77 = −I. τ4

(22)

The L2 -L∞ filtering for T-S fuzzy delayed Hopfield neural networks with the disturbance attenuation level γ is then achieved. Moreover, the gain matrix of the fuzzy neural state estimator (14)-(15) is given by L = P −1 M .

7

Proof. Consider the following Lyapunov-Krasovskii functional with triple integral term: ∫

∫

0

∫

t

T

V (t) = e (t)P e(t) +

 + ∫

∫t

∫

0



T 

∫

∫

t

0

∫

0∫ t

T

+

e˙ T (σ)T e(σ)dσdθdµ. ˙

e˙ (σ)S e(σ)dσdθ ˙ + −τ



∫t

U2    t−τ e(σ)dσ   ∫ ∫ 0 t e(σ)dσdθ U3 −τ t+θ

  U1   U2T e(σ)dσdθ t+θ

∫t

0 −τ

−τ

t+θ

e(σ)dσ

t−τ

eT (t + σ)Re(t + σ)dσ

e (σ)Qe(σ)dσdθ + −τ



0

T

−τ

t+θ

µ

(23)

t+θ

Its time derivative along the trajectory of the state estimation error system (17) is ∫

t

V˙ (e(t)) = e˙ T (t)P e(t) + eT (t)P e(t) ˙ + τ eT (t)Qe(t) −

eT (σ)Qe(σ)dσ t−τ

[∫

+ eT (t)Re(t) − eT (t − τ )Re(t − τ ) + [e(t) − e(t − τ )]T U1

]

t

e(σ)dσ t−τ

[∫

]T

t

+

[∫ T

U1 [e(t) − e(t − τ )] + [e(t) − e(t − τ )] U2

e(σ)dσ ]T

t

+

[

∫

U2 τ e(t) −

e(σ)dσ

]

t

[∫

[ ∫ × [e(t) − e(t − τ )] + τ e(t) −

]T

t

e(σ)dσ

[∫

[

t

e(σ)dσ

× U3 τ e(t) −

t

t−τ

+

U3

t−τ

∫

τ2 T e˙ (σ)T e(σ) ˙ − 2

[∫ ]

0

e(σ)dσ t−τ

∫

]

t

t+θ

∫

e(σ)dσ + τ e˙ T (t)S e(t) ˙ − ∫

0

∫

]

t

[∫

0

∫

]T

t

e(σ)dσdθ

e(σ)dσdθ + −τ

U2T

t+θ

U2T

t−τ

]T

t+θ

e(σ)dσdθ −τ

t−τ

]

t

]T

t

e(σ)dσ +

t−τ

[ ∫ + τ e(t) −

∫

0

∫

e(σ)dσdθ −τ

t−τ

[∫

0

−τ

t+θ

t

e˙ T (σ)S e(σ)dσ ˙ t−τ

t

e˙ T (σ)T e(σ)dσdθ. ˙ −τ

(24)

t+θ

Since the function ϕ(x(t)) satisfies the global Lipschitz condition, it is clear that ˜ ϕ˜T (x(t − τ ))ϕ(x(t − τ )) = [ϕ(x(t − τ )) − ϕ(ˆ x(t − τ ))]T [ϕ(x(t − τ )) − ϕ(ˆ x(t − τ ))] ≤ L2ϕ (x(t − τ ) − x ˆ(t − τ ))T (x(t − τ ) − x ˆ(t − τ )) = L2ϕ eT (t − τ )e(t − τ ).

8

(25)

Then, for a positive scalar δ, ˜ δ[L2ϕ eT (t − τ )e(t − τ ) − ϕ˜T (x(t − τ ))ϕ(x(t − τ ))] ≥ 0.

(26)

Using (26), we obtain V˙ (t) ≤

r ∑

{ hi (ω) eT (t)[(Ai − LCi )T P + P (Ai − LCi )]e(t) − eT (t)P LDi e(t − τ )

i=1

˜ − τ )) + ϕ˜T (x(t − τ ))WiT P e(t) − eT (t − τ )DiT LT P e(t) + eT (t)P Wi ϕ(x(t } + e(t)T P (Gi − LEi )w(t) + wT (t)(Gi − LEi )T P e(t) + τ eT (t)Qe(t) ∫ −

t

eT (σ)Qe(σ)dσ + eT (t)Re(t) − eT (t − τ )Re(t − τ )

t−τ

[∫ T

+ [e(t) − e(t − τ )] U1

]

t

[∫

e(σ)dσ +

[∫

0

[e(t) − e(t − τ )] U2 [

∫

× τ e(t) −

∫

t−τ

+ τ e(t) −

]

[∫

×

0

∫

∫

0

−τ

]T

t

e(σ)dσ

U2T [e(t) − e(t − τ )]

t+θ

[∫

t

U2T

]

[

∫

]

[∫

0

∫

e(σ)dσ

∫

]T

t

[

∫

τ2 e˙ (σ)S e(σ)dσ ˙ + e˙ T (σ)T e(σ) + τ e˙ (t)S e(t) ˙ − ˙ − 2 t−τ

∫

0

]

t

U3 τ e(t) −

t+θ

t

e(σ)dσ ∫

t−τ t

T

T

U3

t−τ

e(σ)dσdθ −τ

]T

t

e(σ)dσ + τ e(t) −

t−τ

t

t+θ

U2

]T

t

e(σ)dσdθ + −τ

e(σ)dσ

e(σ)dσdθ

t−τ

[∫

]T

t t−τ

e(σ)dσ + ∫

[∫

e(σ)dσdθ + t+θ

t−τ

[

]

t

−τ

t

U1 [e(t) − e(t − τ )]

e(σ)dσ

t−τ T

]T

t

e˙ T (σ)T e(σ)dσdθ ˙ −τ

t+θ

˜ + δ[L2ϕ eT (t − τ )e(t − τ ) − ϕ˜T (x(t − τ ))ϕ(x(t − τ ))].

Using the Jensen’s inequality [42] and Wirtinger-type inequality [39]: ∫

[∫

t

1 − e(σ) Qe(σ)dσ ≤ − τ t−τ

]T

t

T

e(σ)dσ

[∫ Q

t−τ

]

t

e(σ)dσ t−τ

and ∫ −

0

∫

t

2 e˙ (σ)T e(σ)dσdθ ˙ ≤− 4 τ t+θ

[∫

0

∫

−τ

]T

t

T

e(σ)dσdθ −τ

9

t+θ

[∫

0

∫

]

t

T

e(σ)dσdθ −τ

t+θ

[∫

2 − 2 τ

[∫

]T

t

e(σ)dσ

e(σ)dσ

T t−τ

t−τ

[∫

4 + 3 τ

∫

0

]

t

]T

t

e(σ)dσdθ −τ

[∫

]

t

T

e(σ)dσ ,

t+θ

(27)

t−τ

we have V˙ (t) ≤

r ∑

{ hi (ω) eT (t)[(Ai − LCi )T P + P (Ai − LCi )]e(t) − eT (t)P LDi e(t − τ )

i=1

˜ − eT (t − τ )DiT LT P e(t) + eT (t)P Wi ϕ(x(t − τ )) + ϕ˜T (x(t − τ ))WiT P e(t) } + e(t)T P (Gi − LEi )w(t) + wT (t)(Gi − LEi )T P e(t) + τ eT (t)Qe(t) 1 − τ

[∫

]T

t

e(σ)dσ

[∫

]

t

e(σ)dσ + eT (t)Re(t) − eT (t − τ )Re(t − τ )

Q

t−τ

t−τ

[∫ T

+ [e(t) − e(t − τ )] U1

]

t

[∫

e(σ)dσ + t−τ

[∫ T

+ [e(t) − e(t − τ )] U2 [

∫

∫

]

t

∫

+ τ e(t) −

]

[∫

× U3

0

∫

0

−τ

]T

[∫

[ ∫ e(σ)dσ + τ e(t) −

t−τ

]

t

t+θ

U2T [e(t) − e(t − τ )]

]

t

[∫

0

∫

∫

]T

t

e(σ)dσdθ −τ

]T

t

e(σ)dσ

t−τ

e(σ)dσdθ + −τ

]T

t

t+θ

U2T

e(σ)dσ

∫

e(σ)dσ

e(σ)dσdθ

t−τ

[∫

]T

t

t−τ

e(σ)dσ +

t

[∫

e(σ)dσdθ + t+θ

t−τ

[

U1 [e(t) − e(t − τ )]

e(σ)dσ t−τ

−τ

t

× U2 τ e(t) −

0

]T

t

[ U3 τ e(t) −

t+θ

∫

]

t

e(σ)dσ t−τ

t

τ2 T e˙ (σ)T e(σ) ˙ 2 t−τ [∫ 0 ∫ t ]T [∫ 0 ∫ t ] [∫ t ]T 2 2 − 4 e(σ)dσdθ T e(σ)dσdθ − 2 e(σ)dσ τ τ −τ t+θ −τ t+θ t−τ [∫ t ] [∫ 0 ∫ t ]T [∫ t ] 4 ×T e(σ)dσ + 3 e(σ)dσdθ T e(σ)dσ τ t−τ −τ t+θ t−τ + τ e˙ (t)S e(t) ˙ − T

e˙ T (σ)S e(σ)dσ ˙ +

˜ + δ[L2ϕ eT (t − τ )e(t − τ ) − ϕ˜T (x(t − τ ))ϕ(x(t − τ ))].

(28)

Using the Leibniz-Newton formula [43], for any appropriate dimensional matrix N , the follow-

10

ing equation is true: [

∫

2ξ (t)N e(t) − e(t − τ ) −

]

t

T

e(σ)dσ ˙ = 0,

(29)

t−τ

where [

∫

ξ(t) = e (t), e (t − τ ), T

∫

t

T

0

∫

t

T

e (σ)dσ, t−τ

∫

t

T

eT (σ)dσdθ,

e˙ (σ)dσ, −τ

t−τ

]T ϕ˜T (x(t − τ )), wT (t) .

t+θ

(30)

Adding the terms in the right-hand side of (29) to the upper bound of V˙ (t) in (28) yields V˙ (t) ≤

r ∑ i=1

∫ −

{ [ ] } τ2 T T −1 T T T hi (w) ξ (t) Φi + τ N S N + τ Ψi SΨi + Ψi T Ψi ξ(t) + w (t)w(t) 2 [

t

] ξ T (t)N + e˙ T (σ)S]S −1 [N T ξ(t) + S e(σ) ˙ dσ

t−τ

=

r ∑ i=1

{ [ ] } τ2 T T −1 T T T hi (w) ξ (t) Φi + τ N S N + τ Ψi SΨi + Ψi T Ψi ξ(t) + w (t)w(t) 2

− τ ξ (t)N S T

−1

∫ N ξ(t) − 2ξ(t) N T

t

T

∫ e(σ)dσ ˙ −

t

e˙ T (σ)S e(σ)dσ, ˙ t−τ

t−τ

where Ψi = [(Ai − LCi ) , −LDi , 0, 0, 0, Wi , (Gi − LEi )] ,  Φ  i11   ∗    ∗    Φi =  ∗    ∗     ∗  ∗

 Φi12

Φi13

Φi14

Φi15

Φi16

Φi22

Φi23

Φi24

Φi25

Φi26

∗

Φi33

0

Φi35

0

∗

∗

0

0

0

∗

∗

∗

Φi55

0

∗

∗

∗

∗

Φi66

∗

∗

∗

∗

∗

Φi17

  Φi27    0     , 0    0     0   Φi77

Φi11 = (Ai − LCi )T P + P (Ai − LCi ) + τ Q + R + N1 + N1T ,

11

(31)

Φi12 = −M Di − N1 + N2T , Φi13 = U1 + τ U2 + N3T , Φi14 = N4T , Φi15 = U2 + τ U3 + N5T , Φi16 = P Wi + N6T , Φi17 = P Gi − M Ei + N7T , Φi22 = δL2Q I − R − N2 − N2T , Φi23 = −U1 − N3T , Φi24 = −N4T , Φi25 = −U2 − N5T , Φi26 = −N6T , Φi27 = −N7T , 1 2 2 Φi33 = − Q − 2 T − U2 − U2T , Φi35 = 3 T − U3 , τ τ τ Φi55 = −

2 T, Φi66 = −δI, Φi77 = −I. τ4

(32)

Using the Wirtinger-type inequality [38]: ∫

t

2 − e˙ (σ)S e(σ)dσ ˙ 0, −τ ξ T (t)N S −1 N T ξ(t) ≤ 0. So, if the following inequality is satisfied: Ξi + τ N S −1 N T + τ ΨTi SΨi +

τ2 T Ψ T Ψi < 0, 2 i

(34)

for i = 1, . . . , r, we have V˙ (t)
0. γ2

Noting (18),(23) and (37) imply that

z˜T (t)˜ z (t) = eT (t)H T He(t) < γ 2 eT (t)P e(t) ≤ γ 2 V (t) 13

(37)

∫ 0 leads to (19). From the Schur complement, the matrix inequality (34) is equivalent to Γi < 0 for i = 1, . . . , r, where           Γi =          

√ √ τ N1 τ (Ai − LCi )T S √ √ Ξi24 Ξi25 Ξi26 Ξi27 τ N2 − τ DiT LT S √ τ N3 0 Ξi34 Ξi35 0 0 √ Ξi44 Ξi45 Ξi46 Ξi47 τ N4 0 √ ∗ Ξi55 0 0 τ N5 0 √ √ ∗ ∗ Ξi66 0 τ N6 τ WiT S √ √ ∗ ∗ ∗ Ξi77 τ N7 τ (Gi − LEi )T S

Ξi11 Ξi12 Ξi13 Ξi14 Ξi15 Ξi16 Ξi17 ∗ Ξi22 Ξi23 ∗ ∗ ∗ ∗

∗ Ξi33 ∗ ∗ ∗ ∗ ∗ ∗

∗

∗

∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

−S ∗ ∗

0 −S ∗



√τ (Ai − LCi )T T 2  − √τ2 DiT LT T  

      .    τ T √ (Gi − LEi ) T   2  0    0 −T 0 0 0 √τ W T T 2 i

(39)

Pre- and post-multiplying diag{I, I, I, I, I, I, I, I, P S −1 , P T −1 } and diag{I, I, I, I, I, I, I, I, S −1 P, T −1 P } by Γi < 0, respectively, and applying a change of variable such that M = P L, we can obtain                    

√ √ τ N1 τ (P Ai − M Ci )T √ √ Ξi24 Ξi25 Ξi26 Ξi27 τ N2 − τ DiT M T √ Ξi34 Ξi35 0 0 τ N3 0 √ Ξi44 Ξi45 Ξi46 Ξi47 τ N4 0 √ ∗ Ξi55 0 0 τ N5 0 √ √ ∗ ∗ Ξi66 0 τ N6 τ WiT P √ √ ∗ ∗ ∗ Ξi77 τ N7 τ (P Gi − M Ei )T

Ξi11 Ξi12 Ξi13 Ξi14 Ξi15 Ξi16 Ξi17 ∗ Ξi22 Ξi23 ∗ ∗ ∗ ∗

∗ Ξi33 ∗ ∗ ∗ ∗ ∗ ∗

∗

∗

∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

−S ∗ ∗

0 −P S −1 P ∗

√τ 2

(P Ai − M Ci )T



    0   0    0 