Stability and stabilization for delayed fuzzy systems via reciprocally convex matrix inequality

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Stability and stabilization for delayed fuzzy systems via reciprocally convex matrix inequality Zhi Lian a,b , Yong He a,b,∗ , Min Wu a,b a School of Automation, China University of Geosciences, Wuhan, 430074, PR China b Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, Wuhan 430074, PR China

Received 7 March 2018; received in revised form 24 October 2019; accepted 20 December 2019

Abstract This paper focuses on the problems of stability and stabilization for time-delayed fuzzy systems. By establishing a novel Lyapunov-Krasovskii functional (LKF) and using an extended reciprocally convex matrix inequality with auxiliary function integral inequality, an improved stability condition is proposed. To further improve the results, a delay-product-type term is introduced into the LKF. Then, a delay-dependent stabilization condition is developed based on parallel distributed compensation (PDC) scheme by some linearization techniques. Finally, the efficiency and benefits of the stability criterion and controller design method are illustrated by several classical numerical examples. © 2019 Published by Elsevier B.V. Keywords: T-S fuzzy systems; Time-varying delay; Stabilization; Extended reciprocally convex matrix inequality

1. Introduction A majority of the real systems are nonlinear so that some traditional control theory is no longer applicable. For this reason, T-S fuzzy model has been widely studied in recent decades since it can effectively describe nonlinear systems by smoothly blending a set of local linear models with fuzzy membership functions. In consequence, the analysis and synthesis problems of nonlinear systems can be addressed by applying the mature linear control theory [1–5]. On the other hand, time delays, unavoidably existing in many practical systems such as aerospace systems, network transmissions, power systems, are the main factor of poor performance, oscillation, and instability of systems. Therefore, it is important to take time delays into account in practical analysis and synthesis problems [6–8]. At present, most delay-dependent stability and stabilization conditions are derived via the Lyapunov-Krasovskii functional (LKF) method with linear matrix inequality (LMI) method. The obtained results under this framework are sufficient and unnecessary, inevitably making the results conservative. For time-delayed systems, how to reduce the conservatism is the research hotspot, and therefore has received much concern of scholars. * Corresponding author at: School of Automation, China University of Geosciences, Wuhan, 430074, PR China.

E-mail address: [email protected] (Y. He). https://doi.org/10.1016/j.fss.2019.12.008 0165-0114/© 2019 Published by Elsevier B.V.

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Illustrated by a large number of existing work, the key to obtaining less conservative results is choosing appropriate LKF and applying suitable bounding techniques. In the aspect of establishing LKFs, augmented LKFs and delay-partitioning LKFs have enjoyed great popularity. Improved delay-dependent stability conditions were derived by equally or unequally separating the entire time delay interval into multiple segments and choosing different LKFs to different segments [9,10]. In recent years, the LKF establishing method of adding multiple integral terms has been widely applied to further reduce the conservatism of the obtained results. In respect of bounding techniques, the freeweighting matrix approach and integral inequality approach have been extensively applied to time-delayed systems. In [2], sufficient conditions were derived without requiring any time-delay derivative information for uncertain T-S fuzzy systems. In [11], by considering time-varying delay h(t), the allowable upper bound h2 and their difference, LMIbased delay-dependent stability criteria were proposed. In [1], the problem of robust H∞ control was investigated for fuzzy systems with interval time-varying delays. Note that the analysis and synthesis results of aforementioned work were derived by the free-weighting matrix approach [12]. Undoubtedly, less conservative results are obtained by introducing matrix variables. However, the computational complexity increases at the same time. For this reason, Jensen’s inequality [13] has been widely applied to estimate single integral terms directly without introducing any slack matrices [9,10,19–22]. After then, Wirtinger-based integral inequality was proposed [14]. The Wirtinger-based integral inequality encompasses Jensen’s one and therefore is capable of providing more accurate bounding results. In [23], less conservative stability criteria of uncertain fuzzy systems were obtained by combining delay-decomposition LKF with Wirtinger-based integral inequality. Similarly, via Wirtinger-based integral inequality, enhanced stability analysis results were presented in [24,25] of T-S fuzzy systems with time-varying delays. More recently, some other bounding techniques such as free matrix integral inequality [15] and auxiliary functions integral [16] inequality have been proposed and successfully applied to deal with various analysis and synthesis problems of time-delayed fuzzy systems. Since 1990s, the main method of stability problem for time-delayed systems is to add the quadratic dou0 t ˙ into the LKFs. Usually, traditional method estimates the derivative of ble integral term −h t+θ x˙ T (s)Z x(s)dsdθ 0 t t  t−h(t) T T ˙ as hx˙ (t)Z x(t) ˙ − t−h(t) x˙ T (s)Z x(s)ds ˙ directly by neglecting − t−h x˙ T (s)Z x(s)ds. ˙ −h t+θ x˙ (s)Z x(s)dsdθ Besides, the time-varying delay term h(t) with 0 ≤ h(t) ≤ h is used to be enlarged as h and another term h − h(t) is also used to be enlarged as h. Undoubtedly, these handling methods will result in much conservatism. To overcome the above issues, further improved free-weighting matrix approach [17] was proposed by considering the relationship that h = h(t) + h − h(t). After that, Jensen’s inequality has been proposed and widely used, which has the same conservatism as further improved free-weighting matrix approach does. However, when using Jensen’s inequality to estimate the derivative, the term h cannot be simply divided into h(t) and h − h(t), since h(t)-dependent terms will appear in the denominators of the result. Consequently, to deal with h(t)-dependent terms, reciprocally convex combination lemma (RCCL) [18] was proposed, becoming the most popular method when the integral inequality approach is employed. For time-varying delay systems, the main focus of obtaining less conservative stability and synthesis results is on developing tighter integral inequalities rather than improving reciprocally convex combination techniques. Currently, some extended reciprocally convex combination inequalities [27] have been put forward to handle time-varying delay terms, which are with more general forms and less conservatism compared with the common RCCL. Based on the research status discussed above, this paper is inspired to address the problem of stability and stabilization of T-S fuzzy systems with time-varying delays. This paper aims to develop less conservative analysis and design methods by the extended reciprocally convex combination techniques. First, by delay-partitioning approach and adding a delay-product-type term, a modified augmented LKF is established. Then, the extended reciprocally convex matrix inequality is utilized with auxiliary function integral inequality to estimate single integral quadratic terms. Subsequently, less conservative LMI-based stability conditions are obtained. Then, based on the parallel distributed compensation (PDC) scheme and Finsler’s lemma, delay-dependent stabilization conditions are derived. Note that the discussed time-varying delay has an interval, which exists commonly in many practical systems and has a strong application background. Finally, three numerical examples and a simulation example are presented to verify the merits and validity of the newly developed methods. The main contributions are summarized as: (1) Less conservative stability criteria are developed by introducing the delay-product-type term into LKF, utilizing the extended reciprocally convex matrix inequality and relaxing the conditions where every term of the constructed LKF is positive definite.

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(2) Improved and effective delay-dependent stabilization conditions are derived for time-delayed T-S fuzzy systems by PDC scheme. Notations. In this paper, 0 and I correspond to the zero matrix and identity matrix, respectively; Rm×n denotes the m × n matrix; the inverse and the transpose of matrix M are represented by M −1 and M T , respectively;  ·  refers to the Euclidean vector norm; Sym{M} = M + M T ; ∗ denotes the symmetric term; I and 0 stand for the identity matrix M 0 and zero matrix; diag{M, N } represents . 0 N 2. Problem statement Consider a time-delayed nonlinear systems described by the following T-S fuzzy model Plant Rule i: IF θ1 (t) is Mi1 and · · · and θp (t) is Mip , THEN  x(t) ˙ = Ai x(t) + Adi x(t − h(t)) + Bi u(t) x(t) = φ(t), t ∈ [−h2 , 0], h2 > 0

(1)

where i = 1, 2, . . . , r with r denoting the number of IF-THEN rules; θ1 (t), θ2 (t), . . . , θp (t) are premise variables; Mij (i = 1, 2, . . . , r, j = 1, 2, . . . , p) are fuzzy sets; x(t) ∈ Rn and u(t) ∈ Rm are the state vector and the control input vector; φ(t) is a given initial function; Ai , Adi and Bi are constant real matrices; h(t) is time-varying delay and satisfies  h1 ≤ h(t) ≤ h2 (2) ˙ ≤μ h(t) where h1 , h2 and μ are constants. By fuzzy blending, the global dynamics of T-S fuzzy system (1) can be inferred as x(t) ˙ = where

r

r 

ρi (θ (t)) (Ai x(t) + Adi x(t − h(t)) + Bi u(t))

i=1

i=1 ρi (θ (t)) = 1, ρi (θ (t)) =

rwi (θ(t)) i=1 wi (θ(t))

> 0, wi (θ (t)) =

(3) p

j =1 Mij (θj (t)) with Mij (θj (t)) representing the  ≥ 0, i = 1, 2, . . . , r, ri=1 grade of membership of θj (t) in Mij . Then, it can be seen that wi (θ i (θ (t)) ≥ 0 for (t)) w r r all t . For the simplicity of presentation, we define that A(θ (t)) = ρ (θ (t))A , A (θ (t)) = i i d i=1 i=1 ρi (θ (t))Adi ,  B(θ(t)) = ri=1 ρi (θ (t))Bi . Then, system (3) can be rewritten as

x(t) ˙ = A(θ (t))x(t) + Ad (θ (t))x(t − h(t)) + B(θ (t))u(t)

(4)

To stabilize system (4), a fuzzy controller is designed. Consider the following controller rules. Plant Rule j: IF θ1 (t) is Mj 1 and · · · and θp (t) is Mjp , THEN u(t) = Kj x(t),

j = 1, 2, . . . , r

(5)

where Kj , j = 1, 2, . . . , r are local control gains. The overall fuzzy controller is inferred as u(t) =

r 

ρj (θ (t))Kj x(t) = K(θ (t))x(t)

i=1

where K(θ (t)) = represented as

r

j =1 ρj (θ (t))Kj .

Under controller (6), the closed-loop system of T-S fuzzy system (4) can be

x(t) ˙ = [A(θ (t)) + B(θ (t))K(θ (t))]x(t) + Ad (θ (t))x(t − h(t)) r  r 

ρi (θ (t))ρj (θ (t)) (Ai + Bi Kj )x(t) + Adi x(t − h(t)) = i=1 j =1

(6)

(7)

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In this paper, the main focus is to analyze stability (under u(t) = 0) and stabilization problems so that the closedloop system(7) is asymptotically stable. the following lemmas are introduced. For simplicity, we   Before proceeding,  define that ri=1 ρi = ri=1 ρi (θ (t)), rj =1 ρj = rj =1 ρj (θ (t)). Lemma 1. ([14,16]) For a symmetric matrix R > 0, scalars a and b with a ≤ b, the following inequality holds b (b − a)

x˙ T (s)R x(s)ds ˙ ≥

3  (2i − 1)ωiT Rωi i=1

a

where ω1 = x(b) − x(a), ω2 = x(b) + x(a) − 6 ω3 = x(b) − x(a + b−a

b a

2 b−a

b x(s)ds a

12 x(s)ds − (b − a)2

b b x(u)duds a

s

Lemma 2. ([26]) For any constant matrix M > 0, scalars a, b and c with b ≥ a ≥ c ≥ 0, vector function x(s), s ∈ R, the following matrix inequality holds ⎛ −a t−c ⎞ ⎛ −a t−c ⎞ −a t−c     (b − a)(b + a − 2c) x T (s)Mx(s)dsdθ ≥ ⎝ x T (s)dsdθ ⎠ M ⎝ x(s)dsdθ ⎠ 2 −b t+θ

−b t+θ

−b t+θ

Lemma 3. ([27]) For a real scalar α ∈ (0, 1), symmetric matrices W1 > 0, W2 > 0, any matrices S1 and S2 , the following matrix inequality holds 1    0 W1 + (1 − α)T1 (1 − α)S1 + αS2 α W1 ≥ (8) 1 ∗ W2 + αT2 0 1−α W2 where T1 = W1 − S2 W2−1 S2T , T2 = W2 − S1T W1−1 S1 . 3. Main results 3.1. Stability analysis In this section, a newly augmented LKF is established, and an enhanced delay-dependent stability criterion is proposed by using auxiliary function integral inequality together with the extended reciprocally convex matrix inequality. Theorem 1. For given scalars h1 , h2 and μ, system (4) under u(t) = 0 is asymptotically stable if there exist symmetric matrices P and U , positive definite symmetric matrices Q1 , Q2 , Q3 , Q4 , R1 , R2 , T1 , T2 and any matrices S1 , S2 , such that the following conditions hold for i = 1, . . . , r: >0  i sum ∗  i sum ∗ where

1 −R˜ 2 2 −R˜ T

(9)

 <0

(10)

<0

(11)

h(t)=h1



h(t)=h2

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⎡

⎤T ⎡ ⎤   5 eˆ1 eˆ1  h − h(t) h(t) − h1 T T 2 i T  = P + ⎣ eˆ2 ⎦ U ⎣ eˆ2 ⎦ , sum =

k + i , 1 = W 1 S2 , 2 = W 2 S1 h12 h12 eˆ3 eˆ3 k=1

1 = Sym{ T1 P 2i },

2 = T3 Q1 3 − T4 Q1 4 + T5 Q2 5 − T6 Q2 6 + e3T Q3 e3 − (1 − μ)e4T Q3 e4 + e1T Q4 e1 − e5T Q4 e5 2h2 − h1 − h(t) T ˜ h2 + h(t) − 2h1 T ˜

3 = − T7 R1 7 − 3 T8 R1 8 − 5 T9 R1 9 − W1 RT W1 − W2 R2 W2 h12 h12    h(t) − h  h2 − h(t) 1 − Sym W1T S1 W2 − Sym W1T S2 W2 h12 h12 h212 T + ( 10 T2 10 + 3 T11 T2 11 + 5 T12 T2 12 ) 2  

4 = −h212 T13 T2 13 − h212 T14 T2 14 − T15 T1 15 , 5 = μ T16 U 16 + h(t)Sym T16 U 17i h21 T h4 T h4 T T R2 esi + 1 esi T1 esi + 12 esi T2 esi , esi R1 esi + h212 esi 4  64 4  h1 h1 1 = col e1 , e6 , e7 , (h(t) − h1 )e8 + (h2 − h(t))e9 2 2 2i = col {esi , e1 − e2 , e2 − e3 , e3 − e5 } , 3 = col {e1 , e2 } , 4 = col {e2 , e3 } , i =

5 = col {e6 , e7 } , 6 = col {e7 , e10 } 7 = e1 − e2 , 8 = e1 + e2 − 2e6 , 9 = e1 − e2 + 6e6 − 12e11 , 10 = e3 − e4 , 11 = e3 + e4 − 2e8 , 12 = e3 − e4 + 6e8 − 12e12

  h1 h1 h1 13 = e3 − (h(t) − h1 )e8 , 14 = e4 − (h2 − h(t))e9 , 15 = (e1 − e6 ), 16 = col e1 , e6 , e7 2 2 2 17i = col {esi , e1 − e2 , e2 − e3 } , W1 = col {e3 − e4 , e3 + e4 − 2e8 , e3 − e4 + 6e8 − 12e12 } h2 W2 = col {e4 − e5 , e4 + e5 − 2e9 , e4 − e5 + 6e9 − 12e13 } , R˜ T = diag {RT , 3RT , 5RT } , RT = R2 + 12 T2 2

R˜ 2 = diag {R2 , 3R2 , 5R2 } , h12 = h2 − h1 , eˆj = 0n×(j −1)n In×n 0n×(4−j )n , j = 1, 2, . . . , 4

ej = 0n×(j −1)n In×n 0n×(13−j )n , j = 1, 2, . . . , 13, esi = [Ai 0 0 Adi 0 0 0 0 0 0 0 0 0] Proof. Construct a LKF candidate as V (x(t)) = V1 (x(t)) + V2 (x(t)) + V3 (x(t)) + V4 (x(t)) + V5 (x(t))

(12)

where V1 (x(t)) = ε1T (t)P ε1 (t) t

t

V2 (x(t)) = t−

ε2T (s)Q1 ε2 (s)ds h1 2

h1 V3 (x(t)) = 2

t−

h1 2

x˙ (s)R1 x(s)dsdθ ˙ + h12

x˙ T (s)R2 x(s)dsdθ ˙

h2 x˙ (s)T1 x(s)dsdθ ˙ du + 12 2

−h1 −h1  t x˙ T (s)T2 x(s)dsdθ ˙ du

T

h1 2

t−h2

−h2 t+θ

t+θ

0  0  t −

x (s)Q3 x(s)ds +

−h1  t T

h1 2

+

t T

t−h(t)

0  t −

h2 V4 (x(t)) = 1 8

+

t−h  1

ε3T (s)Q2 ε3 (s)ds

u t+θ

V5 (x(t)) = h(t)ε4T (t)U ε4 (t)

−h2 u t+θ

x T (s)Q4 x(s)ds

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with

⎡

t

⎢ T ε1 (t) = ⎢ ⎣x (t)



t− T

⎢2 ε3 (s) = ⎢ ⎣ h1

2 h1

s−

h1 2

 h1 T x (s − ) 2 T

t−h2



s−

x T (θ )dθ

 ⎥ ⎥ x (s)ds ⎦ , ε2 (s) = x T (s) T

x (s)ds t−h1

s

⎤T

t−h  1 T

x (s)ds h t− 21

⎡

h1 2

h1 2

⎤T

⎡

t

⎥ ⎢ T ⎢ x T (θ )dθ ⎥ ⎦ , ε4 (t) = ⎣x (t)

s−h1



t−

h1 2

x T (s)ds t−

⎤T ⎥ x T (s)ds ⎥ ⎦

t−h1

h1 2

The time derivative of V1 (x(t)) along the trajectory of (4) is obtained as r 

V˙1 (x(t)) = 2ε1T (t)P ε˙ 1 (t) =

ρi ξ T (t) 1 ξ(t)

(13)

i=1

where

⎡

⎢ T ξ(t) = ⎢ ⎣x (t)

x T (t −

h1 ) 2

x T (t − h1 )

x T (t − h(t))

2 h1

x T (t − h2 )

t x T (s)ds t−

1 h(t) − h1

t−h  1

1 h2 − h(t)

T

x (s)ds t−h(t)

1 (h(t) − h1 )2

t−h  1 t−h  1

x T (u)duds t−h(t)

s

t−h(t) 

2 h1

T

x (s)ds t−h2

x (s)ds 3h1 2

⎤T

t−h(t)  t−h(t) 

2 h1

h1 2

x T (s)ds t−h1

t t

4 h21

T

t−

1 (h2 − h(t))2

h1 2

t−h  1



t−

x T (u)duds t−

h1 2

s

⎥ x T (u)duds ⎦

t−h2

s

Computing the time derivative of LKF V2 (x(t)) along the trajectory of the system yields h1 h1 h1 h1 V˙2 (x(t)) = ε2T (t)Q1 ε2 (t) − ε2T (t − )Q1 ε2 (t − ) + ε3T (t)Q2 ε3 (t) − ε3T (t − )Q2 ε3 (t − ) 2 2 2 2 T T T ˙ + x (t − h1 )Q3 x(t − h1 ) − (1 − h(t))x (t − h(t))Q3 x(t − h(t)) + x (t)Q4 x(t) − x T (t − h2 )Q4 x(t − h2 ) h1 h1 h1 h1 ≤ ε2T (t)Q1 ε2 (t) − ε2T (t − )Q1 ε2 (t − ) + ε3T (t)Q2 ε3 (t) − ε3T (t − )Q2 ε3 (t − ) 2 2 2 2 + x T (t − h1 )Q3 x(t − h1 ) − (1 − μ)x T (t − h(t))Q3 x(t − h(t)) + x T (t)Q4 x(t) − x T (t − h2 )Q4 x(t − h2 ) r  ρi ξ T (t) 2 ξ(t) =

(14)

i=1

Calculating the derivative of V3 (x(t)) and V4 (x(t)) yields V˙3 (x(t)) =

h21 T h1 ˙ − x˙ (t)R1 x(t) 4 2

t x˙ (s)R1 x(s)ds ˙ t−

 = ξ (t) T

r  i=1

 T ρi esi

t−h  1 T

h1 2

h21 R1 + h212 R2 4

+ h212 x˙ T (t)R2 x(t) ˙ − h12

x˙ T (s)R2 x(s)ds ˙

t−h2

 r  i=1

 ρi esi

h1 ξ(t) − 2

t x˙ T (s)R1 x(s)ds ˙ t−

h1 2

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t−h  1

− h12

x˙ T (s)R2 x(s)ds ˙

(15)

t−h2

V˙4 (x(t)) =

0  t

h41 T h2 ˙ − 1 x˙ (t)T1 x(t) 64 8

−

h2 − 12 2 =

0  t

= ξ (t)

−

h1 2

t+θ

−h(t) t+θ −h(t)  t−h(t) 

t−h  1

h2 (h2 − h(t)) x˙ (s)T2 x(s)dsdθ ˙ − 12 2

x˙ T (s)T2 x(s)dsdθ ˙

T

−h2 r 

t+θ

t−h(t)

 T ρi esi

i=1

h2 − 12 2

h412 T ˙ x˙ (t)T2 x(t) 4

x˙ T (s)T1 x(s)dsdθ ˙ +

−h1 t−h  1 x˙ T (s)T2 x(s)dsdθ ˙

 T

t+θ

−h2 t+θ

−

h2 − 12 2

h1 2

−h1 t−h  1 x˙ T (s)T2 x(s)dsdθ ˙

h2 h41 T ˙ − 1 x˙ (t)T1 x(t) 64 8 h2 − 12 2

h412 T ˙ x˙ (t)T2 x(t) 4

x˙ T (s)T1 x(s)dsdθ ˙ +

h4 h41 T1 + 12 T2 64 4



r 

 ρi esi

i=1

h2 ξ(t) − 1 8

0  t x˙ T (s)T1 x(s)dsdθ ˙ −

h1 2

t+θ

−h(t) −h1 t−h  1  t−h(t)  h212 T x˙ (s)T2 x(s)dsdθ ˙ − x˙ T (s)T2 x(s)dsdθ ˙ 2 −h2

−h(t) t+θ

h212 (h2 − h(t)) 2

t+θ

t−h  1

x˙ T (s)T2 x(s)dsdθ ˙

(16)

t−h(t)

Using Lemmas 1 and 3 to estimate the single integral terms appearing in (15) and (16) yields h1 − 2

t x˙ T (s)R1 x(s)ds ˙ t−

h1 2

T T T ≤ −ω11 R1 ω11 − 3ω12 R1 ω12 − 5ω13 R1 ω13 r  = ρi ξ T (t)(− T7 R1 7 − 3 T8 R1 8 − 5 T9 R1 9 )ξ(t)

(17)

i=1

with h1 h1 4 ω11 = x(t) − x(t − ), ω12 = x(t) + x(t − ) − 2 2 h1

t x(s)ds t−

h1 12 ω13 = x(t) − x(t − ) + 2 h1

t h t− 21

48 x(s)ds − 2 h1

h1 2

t t x(u)duds h t− 21

s

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and t−h  1

t−h  1

h2 (h2 − h(t)) x˙ (s)R2 x(s)ds ˙ − 12 2

−h12

x˙ T (s)T2 x(s)dsdθ ˙

T

t−h2

t−h(t)

t−h  1

= −h12

t−h(t) 

x˙ (s)R2 x(s)ds ˙ − h12

t−h(t)

t−h  1

h2 (h2 − h(t)) x˙ (s)R2 x(s)ds ˙ − 12 2

T

x˙ T (s)T2 x(s)dsdθ ˙

T

t−h2

t−h(t)

h12 h12 T T T T (ωT R2 ω21 + 3ω22 R2 ω22 + 5ω23 R2 ω23 ) − R2 ω32 + 5ω33 R2 ω33 ) (ωT R2 ω31 + 3ω32 h(t) − h1 21 h2 − h(t) 31  h2 (h2 − h(t))  T T T − 12 T2 ω22 + 5ω23 T2 ω23 ω21 T2 ω21 + 3ω22 2(h(t) − h1 ) h12 h12 T T T T =− (ωT RT ω21 + 3ω22 RT ω22 + 5ω23 RT ω23 ) − R2 ω32 + 5ω33 R2 ω33 ) (ωT R2 ω31 + 3ω32 h(t) − h1 21 h2 − h(t) 31 h2 T T T + 12 (ω21 T2 ω21 + 3ω22 T2 ω22 + 5ω23 T2 ω23 ) 2  r  h12 h12 ρi ξ T (t) − W1T R˜ T W1 − W T R˜ 2 W2 = h(t) − h1 h2 − h(t) 2 i=1  h212 T + ( 10 T2 10 + 3 T11 T2 11 + 5 T12 T2 12 ) ξ(t) 2    !  r h12 T  ˜ 0 W1 W1 T h(t)−h1 RT ρi ξ (t) − = h12 ˜2 W2 W 0 R 2 h2 −h(t) i=1  h2 + 12 ( T10 T2 10 + 3 T11 T2 11 + 5 T12 T2 12 ) ξ(t) 2       r  h2 − h(t) W1 T R˜ T − S2 R˜ 2−1 S2T S1 W1 T T ˜ T ˜ ρi ξ (t) −W1 RT W1 − W2 R2 W2 − ≤ W2 W2 ∗ 0 h12 i=1       h212 T h(t) − h1 W1 T 0 S2 W1 T T ( 10 T2 10 + 3 11 T2 11 + 5 12 T2 12 ) ξ(t) + − W2 ∗ R˜ 2 − S1T R˜ T−1 S1 W2 h12 2  r  2h2 − h1 − h(t) T ˜ h2 + h(t) − 2h1 T ˜ h2 − h(t) T = ρi ξ (t) − W1 RT W1 − W 2 R2 W 2 − Sym{W1T S1 W2 } h12 h12 h12 ≤−

i=1

h2 − h(t) T ˜ −1 T h(t) − h1 T T ˜ −1 h(t) − h1 Sym{W1T S2 W2 } + W1 S2 R2 S2 W1 + W2 S1 RT S1 W2 h12 h12 h12 ! h2 + 12 ( T10 T2 10 + 3 T11 T2 11 + 5 T12 T2 12 ) ξ(t) 2

−

with 2 ω21 = x(t − h1 ) − x(t − h(t)), ω22 = x(t − h1 ) + x(t − h(t)) − h(t) − h1 ω23 = x(t − h1 ) − x(t − h(t)) +

6 h(t) − h1

t−h  1

x(s)ds − t−h(t)

12 (h(t) − h1 )2

t−h  1

x(s)ds t−h(t)

t−h  1 t−h  1

x(u)duds t−h(t)

s

(18)

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2 ω31 = x(t − h(t)) − x(t − h2 ), ω32 = x(t − h(t)) + x(t − h2 ) − h2 − h(t) ω33 = x(t − h(t)) − x(t − h2 ) +

6 h2 − h(t)

t−h(t) 

x(s)ds − t−h2

12 (h2 − h(t))2

9 t−h(t) 

x(s)ds t−h2

t−h(t)  t−h(t) 

x(u)duds s

t−h2

Thus, from (17) to (18) we have t

h1 − 2 r  i=1

h2 (h2 − h(t)) x˙ (s)R2 x(s)ds ˙ − 12 2

x˙ (s)R1 x(s)ds ˙ − h12 t−

=

t−h  1 T

t−h2

h1 2

t−h  1

x˙ T (s)T2 x(s)dsdθ ˙

T

t−h(t)

  h2 − h(t) T ˜ −1 T h(t) − h1 T T ˜ −1 ρi ξ T (t) 3 + W 1 S 2 R 2 S2 W 1 + W2 S1 RT S1 W2 ξ(t) h12 h12

(19)

Then, by employing Lemma 2 to estimate the double integral terms in inequality (16), we have h2 − 12 2

−h(t) −h1 t−h  1  t−h(t)  0  t h212 h21 T T x˙ (s)T2 x(s)dsdθ ˙ − x˙ (s)T2 x(s)dsdθ ˙ − x˙ T (s)T1 x(s)dsdθ ˙ 2 8

−h(t) t+θ

⎛

h212 ⎜ ≤− ⎝ (h(t) − h1 )2

⎞

−

h212 (h2 ⎛

⎜ −⎜ ⎝

0  t

−

=

r 

− h(t))2

h1 2

⎜ ⎝

t+θ

⎞

⎛

−

h1 2

t+θ

−h1 t−h  1 −h1 t−h  1 ⎟ ⎜ ⎟ T x˙ (s)dsdθ ⎠ T2 ⎝ x(s)dsdθ ˙ ⎠

−h(t) t+θ

⎛

−h2

⎞

−h(t)  t−h  1

−h(t) t+θ

⎛

⎟ ⎜ x˙ T (s)dsdθ ⎠ T2 ⎝

−h2 t+θ

⎞

⎛

⎟ ⎜ ⎜ x˙ T (s)dsdθ ⎟ ⎠ T1 ⎝

t+θ

−h(t)  t−h  1

⎞

⎟ x(s)dsdθ ˙ ⎠

−h2 t+θ

0  t

−

h1 2

⎞

⎟ ⎟ x(s)dsdθ ˙ ⎠

t+θ

ρi ξ T (t) 4 ξ(t)

(20)

i=1

The derivative of V5 (x(t)) along the solution of system (4) is given by T ˙ V˙5 (x(t)) = h(t)ε 4 (t)U ε4 (t) + 2h(t)ε4 (t)U ε˙ 4 (t) ⎡ ⎡ ⎤T ⎡ ⎤ ⎤T ⎡ ⎤  t x(t)  t x(t)  t x(t) x(t) ˙ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ x(s)ds x(s)ds x(s)ds h h h h ⎥ U ⎢ t− 21 ⎥ +2h(t) ⎢ t− 21 ⎥ U ⎣ x(t) − x(t − 1 ) ⎦ ˙ ⎢ t− 21 = h(t) 2 ⎣ ⎣ ⎦ ⎣ ⎦ ⎦  t− h21  t− h21  t− h21 x(t − h21 ) − x(t − h1 ) t−h1 x(s)ds t−h1 x(s)ds t−h1 x(s)ds

≤

r 

$  % ρi ξ T (t) μ T16 U 16 + h(t)Sym T16 U 17i ξ(t)

i=1

=

r 

ρi ξ T (t) 5 ξ(t)

i=1

Therefore, from inequality (13) to inequality (21), we can easily obtain the upper bound of V˙ (x(t)):

(21)

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10

V˙ (x(t)) =

5 

V˙k (x(t)) ≤ ξ T (t)ξ(t)

(22)

k=1 r 

where  =

ρi

5 &

r 2 4 '  h(t)−h1 T T T ˜ −1 T )( h1 R +h2 R + h1 T + ˜ −1 T

k + h2 −h(t) ρi esi 12 2 h12 W1 S2 R2 S2 W1 + h12 W2 S1 RT S1 W2 +( 4 1 64 1

i=1 k=1 r  h412 ρi esi ). As for 4 T2 )( i=1

i=1

the P- and U-dependent terms of LKF (12), they can be rewritten as

ε1T (t)P ε1 (t) + h(t)ε4T (t)U ε4 (t) ⎛ ⎡ ⎤T ⎡ ⎤⎞ eˆ1 eˆ1 ⎜ ⎟ = ε1T (t) ⎝P + ⎣ eˆ2 ⎦ U ⎣ eˆ2 ⎦⎠ ε1 (t) = ε1T (t)ε1 (t) eˆ3 eˆ3 Thus, if  < 0 and  > 0 hold, system (4) is guaranteed to be the asymptotically stable under u(t) = 0.  < 0 is equivalent to inequalities (10) and (11) by applying Schur complement lemma and convex combination lemma. The proof is completed. 2 Remark 1. Inspired by the work of [26], the delay-partitioning method is used in establishing the LKF, which t  t− h21 x(s)ds appear in ξ(t). As a result, more informamakes terms such as x(t − h21 ), h1 x(s)ds and h1 t−

2

tion of time-delay is contained and less conservative results are obtained. Moreover, the triple integral term h212  −h1  −h1  t h2 −h2  −h  −h  t T ˙ du introduced into V4 (x(t)) is better than the term 2 2 1 −h21 u 1 t+θ x˙ T (s) × t+θ x˙ (s)T2 x(s)dsdθ 2 −h2 u  −h  −h  t ˙ du. Because the former triple integral term includes an extra term h12 −h21 u 1 t+θ x˙ T (s)T2 x(s)dsdθ ˙ du, T2 x(s)dsdθ which makes more information of lower bound h1 be used in the inner integral upper limit. Remark 2. In Theorem 1, the bounding techniques used to deal with quadratic single integral terms are auxiliary functions integral inequality and the extended reciprocally convex matrix inequality. Since auxiliary function integral inequality has a more general form, it is less conservative than Jensen inequality and Wirtinger-based inequality. Due to the utilization of auxiliary functions integral inequality, some techniques are expected to be utilized to handle h(t)-dependent terms in the denominators of the results. In this paper, the extended reciprocally convex matrix inequality is employed, which is less conservative than the popular RCCL. Note that less conservative analysis and control results are likely to be derived by combining the extended reciprocally convex matrix inequality with other bounding techniques such as generalized free-matrix-based integral inequality [34], generalized integral inequality [35], BesselLaguerre inequality [36], and other LKF constructing methods [3,4,37]. For example, for fuzzy-model-based control systems, further reduction of the conservatism can be achieved by introducing fuzzy-dependent information into the established LKFs. Remark 3. When computing the derivative of the triple integral term V4 (x(t)), a single integral term appears. And the following two single integral terms will be included in V˙ (x(t)), where the first term is introduced by the double integral term while the second term is introduced by the triple integral term. t−h  1

−h12 t−h2

h2 (h2 − h(t)) x˙ (s)R2 x(s)ds ˙ − 12 2

t−h  1

x˙ T (s)T2 x(s)dsdθ ˙

T

(23)

t−h(t)

In [29], a relaxed integral inequality proposed in [28] is applied to deal with single integral terms. However, this relaxed inequality is not applicable to handle (23), while the newly developed matrix integral inequality (8) can deal with (23) easily by combining with other inequalities. It further illustrates the advantages of the extended reciprocally convex matrix inequality. Remark 4. Based on the work of [3,4,30], a delay-product-type term V5 (x(t)) is introduced in the established LKF, nicely balancing the tradeoff between the conservatism and computational complexity. On the one hand, not only

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11

the form of V5 (x(t)) is simple but also the derivative of V5 (x(t)) contains much information of fuzzy systems and time-varying delays. On the other hand, V1 (x(t)) + V5 (x(t)) has a more general form than V1 (x(t)) does since the restriction of some locations is relaxed. Therefore, improved stability and synthesis results are derived by the introduction of the delay-product-type term V5 (x(t)). Remark 5. When using LKF approach to analyze the stability problems, we also need to estimate the established LKF itself and guarantee the LKF is positive definite. Usually, this condition is guaranteed by making every term in the LKF is positive definite, which causes much conservatism. For this reason, this paper considers all the nonintegral terms together. Thus, the conditions P > 0 and U > 0 can be relaxed to inequality (9), which is less conservative. Remark 6. Theorem 1 only considers the case when the time-varying delay h(t) is a differentiable function and ˙ ≤ μ. But sometimes the time-varying delay is not a differentiable function or the time delay is a fast satisfies h(t) time-varying delay. Thus, the case when the derivative of time-varying delay h(t) is unknown should also be discussed. Based on the above analysis, by setting Q3 = 0 and U = 0 in Theorem 1, the stability criteria when the derivative of time-varying delay is unknown can be easily obtained. 3.2. Controller design Based on Theorem 1, the stabilization condition for system (7) is presented. Theorem 2. For given scalars 0 ≤ h1 ≤ h2 , μ and δ, system (7) is asymptotically stable if there exist symmetric ˆ 1 , Qˆ 2 , Qˆ 3 , Qˆ 4 , Rˆ 1 , Rˆ 2 , Tˆ1 , Tˆ2 and any matrices Sˆ1 , Sˆ1 , X, Yj matrices Pˆ , Uˆ , positive definite symmetric matrices Q with appropriate dimensions, such that the following conditions hold for i = 1, . . . , r, j = 1, . . . , r: ˆ >0  ˆ˘ i,j 

sum

∗ ˆ˘ i,j  sum ∗

ˆ˘ 1 ˆ ˜ −R2 ˆ˘ 2 −Rˆ˜

T

(24)

! <0

(25)

<0

(26)

h(t)=h1

!

h(t)=h2

where ⎤T ⎡ ⎤ 5 eˆ1 eˆ1   ˆ˘ +  ˆ˘ + Sym L˘ ˘ i,j , ˆ˘ i,j = 

ˆ = Pˆ + ⎣ eˆ2 ⎦ Uˆ ⎣ eˆ2 ⎦ ,   k sum C eˆ3 eˆ3 k=1   h − h(t) 2 ˆ˘ = ˆ˘ = h(t) − h1 W˘ T Sˆ T W˘ 1T Sˆ2 , 1 2 2 1 h12 h12   ˆ˘ = Sym ˘ T1 Pˆ ˘2 ,

1 ⎡

ˆ˘ = ˆ 1 ˆ 1 ˆ 3 e˘3 − (1 − μ)e˘4T Qˆ 3 e˘4 + e˘1T Q ˆ 4 e˘1 − e˘T Q ˆ ˘ T3 Q ˘ T4 Q ˘ T5 Qˆ 2 ˘ T6 Qˆ 2 ˘3− ˘4+ ˘5− ˘ 6 + e˘3T Q

2 5 4 e˘5 2h − h1 − h(t) ˘ T ˆ˜ ˘ h2 + h(t) − 2h1 ˘ T ˆ˜ ˘ ˆ˘ = − ˘ T7 Rˆ 1 ˘ T8 Rˆ 1 ˘ T9 Rˆ 1 ˘ 7 − 3 ˘ 8 − 5 ˘9− 2 W1 RT W1 − W2 R2 W2

3 h12 h12    h(t) − h  h2 h2 − h(t) 1 ˘ T11 Tˆ2 ˘ T12 Tˆ2 ˘ T10 Tˆ2 ˘ 10 + 3 ˘ 11 + 5 ˘ 12 ) − Sym W˘ 1T Sˆ1 W˘ 2 − Sym W˘ 1T Sˆ2 W˘ 2 + 12 ( h12 h12 2   ˘ T Tˆ ˘ T Uˆ ˘ T Uˆ ˘ + h(t)Sym ˘ ˘ T Tˆ ˘ − h2 ˘ T Tˆ ˘ − ˘ ,

˘ˆ = μ ˘ˆ = −h2

4

12

13 2

13

12

14 2

14

15 1

15

5

2 4 4 ˆ˘ = h1 e˘T Rˆ e˘ + h2 e˘T Rˆ e˘ + h1 e˘T Tˆ e˘ + h12 e˘T Tˆ e˘ ,  2 14 14 1 14 12 14 2 14 14 1 14 4 64 4 14

16

16

16

17

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12

  h h ˘ 1 = col e˘1 , 1 e˘6 , 1 e˘7 , (h(t) − h1 )e˘8 + (h2 − h(t))e˘9 2 2 ) ( ˘ 3 = col {e˘1 , e˘2 } , ˘ ˘ 4 = col {e˘2 , e˘3 } , 2 = col eˇ14 , e˘1 − e˘2 , e˘2 − e˘3 , e˘3 − e˘5 , ˘ 6 = col {e˘7 , e˘10 } ˘ 5 = col {e˘6 , e˘7 } , ˘ 8 = e˘1 + e˘2 − 2e˘6 , ˘ 9 = e˘1 − e˘2 + 6e˘6 − 12e˘11 , ˘ 10 = e˘3 − e˘4 , ˘ 7 = e˘1 − e˘2 , ˘ 12 = e˘3 − e˘4 + 6e˘8 − 12e˘12 ˘ 11 = e˘3 + e˘4 − 2e˘8 , h ˘ 14 = e˘4 − (h2 − h(t))e˘9 , ˘ 15 = 1 (e˘1 − e˘6 ), ˘ 13 = e˘3 − (h(t) − h1 )e˘8 , 2   h1 h1 ˘ 17 = col {e˘14 , e˘1 − e˘2 , e˘2 − e˘3 } ˘ 16 = col e˘1 , e˘6 , e˘7 , 2 2 W˘ 1 = col {e˘3 − e˘4 ; e˘3 + e˘4 − 2e˘8 ; e˘3 − e˘4 + 6e˘8 − 12e˘12 } , W˘ 2 = col {e˘4 − e˘5 ; e˘4 + e˘5 − 2e˘9 ; e˘4 − e˘5 + 6e˘9 − 12e˘13 }  

h2 e˘j = 0n×(j −1)n In×n 0n×(14−j )n , j = 1, 2, . . . , 14, Rˆ˜ T = diag Rˆ T , 3Rˆ T , 5Rˆ T , Rˆ T = Rˆ 2 + 12 Tˆ2 2   i,j ˆ T T ˆ ˆ ˆ ˘ ˜ ˘ R2 = diag R2 , 3R2 , 5R2 , L = e˘1 + δ e˘14 , C = −X e˘14 + (Ai X + Bi Yj )e˘1 + (Adi X)e˘4 The corresponding controller gains are obtained as Kj = Yj X −1 . Proof. Choose LKF (12) and the asymptotic stability condition for system (7) are represented as  5   − h(t) ˘ T ˜ −1 T ˘ h h(t) − h 2 1 −1 T T T ˘ + ˘k + ξc (t) W1 S2 R2 S2 W1 + W˘ 2 S1 R˜ T S1 W˘ 2 ξc (t) ≤ 0,

h12 h12 k=1

s.t. 0 = CL (θ (t))ξc (t),

(27)

where

  ˘ T1 P ˘2 ˘ 1 = Sym

˘ T3 Q1 ˘ T4 Q1 ˘ T5 Q2 ˘ T6 Q2 ˘3− ˘4+ ˘5− ˘ 6 + e˘3T Q3 e˘3 − (1 − μ)e˘4T Q3 e˘4 + e˘1T Q4 e˘1 − e˘5T Q4 e˘5 ˘2=

2h − h1 − h(t) ˘ T ˜ ˘ h2 + h(t) − 2h1 ˘ T ˜ ˘ ˘ T7 R1 ˘ T8 R1 ˘ T9 R1 ˘ 3 = − ˘ 7 − 3 ˘ 8 − 5 ˘9− 2 W1 RT W1 − W2 R2 W2

h12 h12    h(t) − h  h2 − h(t) 1 − Sym W˘ 1T S1 W˘ 2 − Sym W˘ 1T S2 W˘ 2 h12 h12 2 h ˘ T11 T2 ˘ T12 T2 ˘ T10 T2 ˘ 10 + 3 ˘ 11 + 5 ˘ 12 ) + 12 ( 2   ˘ T15 T1 ˘ 5 = μ ˘ T16 U ˘ 16 + h(t)Sym ˘ T16 U ˘ 17 ˘ 4 = −h212 ˘ T13 T2 ˘ 13 − h212 ˘ T14 T2 ˘ 14 − ˘ 15 ,

$ %T h4 T h4 T h21 T T R2 e˘14 + 1 e˘14 T1 e˘14 + 12 e˘14 T2 e˘14 , ξc (t) = ξ T (t) x˙ T (t) e˘14 R1 e˘14 + h212 e˘14 4 64 4 CL (θ (t)) = [A(θ (t)) + B(θ (t))K(θ (t)) 0 0 Ad (θ (t)) 0 0 0 0 0 0 0 0 0 − I ]

˘ = 

According to system (7), the following zero equality holds for any matrices N1 and N2 . $ % 0 = 2 x T (t)N1 + x˙ T (t)N2 [−x(t) ˙ + (A(θ (t)) + B(θ (t))K(θ (t)))x(t) + Ad (θ (t))x(t − h(t))] =

r  r  i=1 j =1

where

  i,j ρi ρj ξcT (t)Sym L˘ ˘ C ξc (t)

(28)

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13

i,j T N2 , ˘ C = −e˘14 + (Ai + Bi Kj )e˘1 + Adi e˘4 L˘ = e˘1T N1 + e˘14

Therefore, from (27) and (28), we have  5 r r    h − h(t) ˘ T ˜ −1 T ˘ T ˘k+ 2 W1 S2 R2 S2 W1 ρi ρj ξc (t)

h12 i=1 j =1 k=1 * h(t) − h1 ˘ T T ˜ −1 ˘ i,j ˘ ˘ ˘ + W2 S1 RT S1 W2 +  + Sym(LC ) ξc (t) < 0 h12

(29)

And the asymptotic stability condition for system (7) can be  h2 − h(t) ˘ T ˜ −1 T ˘ h(t) − h1 ˘ T T ˜ −1 ˘ ˘ + Sym(L˘ ˘ i,j ) < 0 ˘k + W1 S2 R2 S2 W1 + W2 S1 RT S1 W2 +

C h12 h12 5

(30)

k=1

By choosing N2 = δN1 where δ is a given value to be tuned, the LMI-based stabilization conditions can be derived. h2

h4

h4

Since the term 41 R1 + h212 R2 + 41 T1 + 412 T2 − N2 − N2T in (30) must be negative definite and R1 , R2 , T1 , T2 are positive definite, thus N2 is positive definite, which means N1 , N2 are nonsingular. Thus, we can define X = (N1−1 )T . T and χ leads to By Schur complement lemma and pre- and post-multiplying both sides (30) by χ14 14 ⎡ ⎤ ˆ˘ ˆ˘ ˆ˘ i,j  1 2 ⎢ sum ⎥ ˆ˜ ⎢ ∗ ⎥<0 (31) − 0 R 2 ⎣ ⎦ ˆ ∗ ∗ −R˜ T where χm = diag{X, . . . , X } + ,- . m elements T ˆ 1 = χ2T Q1 χ2 , Qˆ 2 = χ2T Q2 χ2 , Q ˆ 3 = χ1T Q3 χ1 , Qˆ 4 = χ1T Q4 χ1 , Rˆ 1 = χ1T R1 χ1 Pˆ = χ4 P χ4 , Q Rˆ 2 = χ1T R2 χ1 , Tˆ1 = χ1T T1 χ1 , Tˆ2 = χ1T T2 χ1 , Uˆ = χ3T U χ3 , Sˆ1 = χ3T S1 χ3 , Sˆ2 = χ3T S2 χ3 The remaining process of deriving stability conditions (24)-(26) is very similar to the proof of Theorems 1. This completes the proof of Theorem 2. 2 4. Numerical examples This section presents four examples to verify the effectiveness of the proposed methods. Example 1. Consider the following time-delayed nonlinear systems: x˙1 (t) = 0.5(1 − sin2 (θ (t)))x2 (t) − x1 (t − h(t) − (1 + sin2 (θ (t)))x1 (t)  π (0.9 cos2 (θ (t)) − 1)x1 (t − h(t)) − x2 (t − h(t)) − (0.9 + 0.1 cos2 (θ (t)))x2 (t) x˙2 (t) = sgn |θ (t)| − 2 which can be described by the following two-rule fuzzy model π Rule1 : I F θ(t) is ± , T H EN x(t) ˙ = A1 x(t) + Ad1 x(t − h(t)) 2 Rule2 : I F θ(t) is 0, T H EN x(t) ˙ = A2 x(t) + Ad2 x(t − h(t)) with the following system parameters    −2 0 −1 A1 = , A2 = 0 −0.9 0

 0.5 , −1



−1 Ad1 = −1

 0 , −1



−1 Ad2 = 0.1

0 −1



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14

Table 1 MAUBs under different values h1 (Example 1). Methods

h1 0.4

0.8

1.2

Ukn μ

Tian et al. [2] Peng et al. [21] An et al. [9] Souza et al. [19] Peng et al. [10] Xia et al. [32] Theorem 1 (Q3 = 0, U = 0) Over [32]

0.8830 1.1818 1.3030 1.2836 1.3900 1.5274 1.7194 > 12.5%

1.0930 1.3130 1.3160 1.3394 1.4300 1.5361 2.0526 > 33.6%

1.3360 1.4333 1.4250 1.4815 1.5700 1.6340 2.4155 > 47.8%

μ = 0.1

Peng et al. [21] Theorem 1 (U = 0) Theorem 1 Over [21]

1.6500 1.9428 2.1161 > 28.2%

1.8000 2.1898 2.3116 > 28.4%

1.6900 2.5040 2.5818 > 52.7%

Table 2 MAUBS h2 under different values of h1 (Example 2). Methods Peng et al. [22] Tian et al. [1] Souza et al. [19] Lian et al. [31] Theorem 1 (Q3 = 0, U = 0) Over [31]

h1 0.4

0.6

0.8

0.9793 1.1500 1.1734 1.2376 1.5781 > 27.5%

1.0639 1.1720 1.1994 1.2775 1.7394 > 36.1%

1.1662 1.2090 1.2532 1.3372 1.9161 > 43.2%

The purpose of this example is to find the maximum allowable upper bounds (MAUBs) h2 , under which the system is stable. When μ = 0.1, Table 1 shows the obtained MAUBs under different values h1 , where μ is unknown (Ukn) are also given. The improvements of our methods can be seen clearly from Table 1. In case of μ = 0.1, compared with the results of Theorem 1 with U = 0, the results of Theorem 1 with U = 0 are less conservative, which strongly illustrates the merits of delay-product-type term. In order to further demonstrate the advantages of the proposed stability conditions, other numerical examples are given as follows. Example 2. Consider the nominal T-S fuzzy system with the following rules where        −2 0 −1.5 1 −1 0 −1 A1 = , A2 = , Ad1 = , Ad2 = 0 −0.9 0 −0.75 −1 −1 1

0 −0.85



In Table 2, the obtained MAUBs are tabulated for fast time-varying delay under different values of h1 . Compared with the results in [1,19,22,31], it can be seen that the results by Theorem 1 are obviously larger than those by other references. It illustrates that delay range by Theorem 1, where the system is asymptotically stable, is wider. That is, Theorem 1 is less conservative than the stability criteria of [1,19,22,31]. Example 3. Consider the nominal T-S fuzzy system with the following rules where        −2.1 0.1 −1.9 0 −1.1 0.1 −0.9 A1 = , A2 = , Ad1 = , Ad2 = −0.2 −0.9 −0.2 −1.1 −0.8 −0.9 −1.1

0 −1.2



Assuming h1 = 0, for various values of μ, the MAUBs of h(t) obtained from this paper and other existing works are tabulated in Table 3. As shown in Table 3, the criteria derived from this paper are less conservative. Furthermore,

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Table 3 MAUBS h2 under different values of μ (Example 3). Methods

μ

Liu et al. [11] An et al. [20] An et al. [9] (m = 10) Zeng et al. [23] (m = 3) Lian et al. [29] Kwon et al. [24] Theorem 1 Theorem 1 (Q3 = 0, U = 0) Over [24]

0

0.1

0.5

Ukn

3.30 3.70 4.41 4.37 4.42 5.5826 5.5973 > 0.2%

2.65 3.01 3.15 3.41 3.42 4.2044 4.2928 > 2.1%

1.50 1.65 1.87 1.95 2.02 2.0685 2.2571 > 9.1%

0.79 1.19 1.65 1.77 1.8791 1.9310 > 2.7%

not like the stability conditions derived from [24], our stability and stabilization criteria can handle time-varying delay with nonzero lower bound, which illustrates our results have more practical application value. Next, another simulation example is presented to verify the validity of Theorem 2. Example 4. Consider a truck-tailer system introduced in [33]. v t¯ v t¯ v t¯ x1 (t) − (1 − a) x1 (t − h(t)) + u(t) Lt0 Lt0 lt0 v t¯ v t¯ x1 (t) + (1 − a) x1 (t − h(t)) x˙2 = a Lt0 Lt0  * v t¯ v t¯ v t¯ x˙3 = sin x2 (t) + a x1 (t) + (1 − a) x1 (t − h(t)) t0 2L 2L x˙1 = −a

where x1 (t) is the angle difference between truck and trailer, x2 (t) is the angle of the trailer, x3 (t) is the vertical position of rear end of trailer. The model parameters are given as l = 2.8, L = 5.5, v = −1.0, t¯ = 2.0, t0 = 0.5. The constant a is the retarded coefficient, which satisfies the condition a ∈ [0, 1]. The limits 1 and 0 correspond to no delay term and to a complete delay term, respectively. In this example, we assume a = 0.7. Let θ (t) = x2 (t) + a(v t¯/2L)x1 (t) + (1 − a)(v t¯/2L)x1 (t − h(t)), and the system matrices modeling are: Rule1 : I F θ(t) is 0 rad, T H EN x(t) ˙ = A1 x(t) + Ad1 x(t − h(t)) + B1 u(t) Rule2 : I F θ(t) is − π rad or π rad, T H EN x(t) ˙ = A2 x(t) + Ad2 x(t − h(t)) + B2 u(t) where x(t) = [x1 (t) x2 (t) x3 (t)]T , and ⎤ ⎤ ⎡ ⎡ ⎡ ⎤ v t¯ v t¯ 0 0 0 0 −a Lt −(1 − a) Lt v t¯ 0 0 lt0 ⎥ ⎥ ⎢ v t¯ ⎢ ⎢ ⎥ ¯ v t ⎢ 0 0⎥ 0 0⎥ A1 = ⎢ ⎦ , B1 = ⎣ 0 ⎦ ⎦ , Ad1 = ⎣ (1 − a) Lt0 ⎣ a Lt2 02 v t¯ v t¯ v 2 t¯2 v t¯ 0 a 2Lt 0 0 (1 − a) 2Lt t0 0 t0 0 ⎤ ⎤ ⎡ ⎡ ⎡ ⎤ v t¯ v t¯ 0 0 0 0 −a Lt −(1 − a) Lt v t¯ 0 0 lt0 ⎥ ⎥ ⎢ v t¯ ⎢ ¯ ⎥ ⎢ vt ⎢ 0 0⎥ 0 0⎥ A2 = ⎢ ⎦ , Ad2 = ⎣ (1 − a) Lt0 ⎦ , B2 = ⎣ 0 ⎦ ⎣ a Lt0 ¯ 2 t¯2 gv ¯ t¯ ¯ 2 t¯2 v t¯ 0 a gv (1 − a) gv 0 0 2Lt0 t0 2Lt0 t0 Assume μ = 0.5 and h1 = 0, and then the maximum time delays under different adjusting parameters δ are tabulated in Table 4. From Table 4, it is observed that MAUB h2 varies with the change of value of δ, and the relationship between them is not a linear correlation. That is, improved results can be obtained in practical applications by turning parameters δ. Then, assuming h1 = 0, h2 = 10, δ = 1.4, the membership functions are defined as M1 (x1 (t)) = 1 −

1 , M2 (x1 (t)) = 1 − M1 (x1 (t)) 1 + ex1 (k)+2+sin(x1 (k))

(32)

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Table 4 MAUBS h2 under different values of δ (Example 4). Method Theorem 1

δ 0.2

0.6

1.0

1.4

1.8

243.5944

235.5743

266.0522

274.9969

262.4908

Fig. 1. State responses for T-S fuzzy system with time-varying delays without controllers.

Fig. 2. State responses for T-S fuzzy system with time-varying delays with controllers.

Under initial condition x(0) = [0.5π − 0.75π − 5]T , Fig. 1 shows the state responses of the open loop system. From Fig. 1, it is seen that the open-loop system is unstable. To stabilize this unstable system, utilizing Theorem 2 yields the following fuzzy controller gains K1 = [4.4640 − 10.8610 0.4312], K2 = [4.0864 − 9.9743 0.3946]

(33)

Under the same initial state condition, the state responses of the closed-loop system are presented in Fig. 2. One can clearly see that the designed fuzzy state-feedback controllers make the closed-loop states converge to zero as time goes by. The whole control procedure shows that the developed control method presented in Theorem 2 can stabilize system under time delays effectively. 5. Conclusion This paper proposes improved stability and control methods for a class of T-S fuzzy systems with interval timevarying delays. First, a novel augmented LKF is constructed by introducing a delay-product-type term and a triple integral term. Then, sufficient stability condition is derived by utilizing auxiliary function integral inequality with

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the extended reciprocally convex matrix inequality. Based on the PDC scheme, the corresponding controller design method is proposed. Finally, four classical examples are presented to verify the feasibility and improvements of the presented methods. Based on our heuristic research, correlative research can be conducted and expanded. By considering the information on membership functions, membership-dependent LKFs can be constructed. Moreover, the proposed controller design methods can be further generalized to type-2 fuzzy systems, where uncertainties are allowed to exist in membership functions. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant 61973284, by the Hubei Provincial Natural Science Foundation of China under Grant 2015CFA010, by the 111 Project under Grant B17040, by the China Scholarship Council, and by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan). References [1] E. Tian, D. Yue, Y. Zhang, Delay-dependent robust H∞ control for T-S fuzzy system with interval time-varying delay, Fuzzy Sets Syst. 160 (2009) 1708–1719. [2] E. Tian, C. Peng, Delay-dependent stability analysis and synthesis of uncertain T-S fuzzy systems with time-varying delay, Fuzzy Sets Syst. 157 (2006) 544–559. [3] Z. Lian, Y. He, C.K. Zhang, P. Shi, M. Wu, Robust H∞ control for T-S fuzzy systems with state and input time-varying delays via delayproduct-type functional method, IEEE Trans. Fuzzy Syst. 27 (10) (2019) 1917–1930. [4] Z. Lian, Y. He, C.K. Zhang, M. Wu, Stability and stabilization of T-S fuzzy systems with time-varying delays via delay-product-type functional method, IEEE Trans. Cybern. (2019), https://doi.org/10.1109/TCYB.2018.2890425. [5] R. Chai, A. Tsourdos, A. Savvaris, Y. Xia, S. Chai, Real-time reentry trajectory planning of hypersonic vehicles: a two-step strategy incorporating fuzzy multi-objective transcription and deep neural network, IEEE Trans. Ind. Electron. (2019), https://doi.org/10.1109/TIE.2019. 2939934. [6] X.J. Su, P. Shi, L. Wu, Y.D. Song, A novel approach to filter design for T-S fuzzy discrete-time systems with time-varying delay, IEEE Trans. Fuzzy Syst. 20 (6) (2012) 1114–1129. [7] M.J. Park, O.M. Kwon, Stability and stabilization of discrete-time T-S fuzzy systems with time-varying delay via Cauchy-Schwartz-based summation inequality, IEEE Trans. Fuzzy Syst. 25 (1) (2017) 128–140. [8] X.Z. Yang, L.G. Wu, H.K. Lam, X.J. Su, Stability and stabilization of discrete-time T-S fuzzy systems with stochastic perturbation and time-varying delay, IEEE Trans. Fuzzy Syst. 22 (1) (2014) 124–138. [9] J. An, G. Wen, Improved stability criteria for time-varying delayed T-S systems via delay partitioning approach, Fuzzy Sets Syst. 185 (2011) 83–94. [10] C. Peng, M. Fei, An improved result on the stability of uncertain T-S fuzzy systems with interval time-varying delay, Fuzzy Sets Syst. 212 (2013) 97–109. [11] F. Liu, M. Wu, Y. He, R. Yokoyamab, New delay-dependent stability criteria for T-S fuzzy systems with time-varying delay, Fuzzy Sets Syst. 161 (2010) 2033–2042. [12] M. Wu, Y. He, J.H. She, G.P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems, Automatica 40 (8) (2004) 1435–1439. [13] K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceeding of the 39th IEEE Conference on Decision and Control, Sydney, Australia, 2010, pp. 2805–2810. [14] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time-delay systems, Automatica 49 (9) (2013) 2860–2866. [15] H.B. Zeng, Y. He, M. Wu, J.H. She, Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE Trans. Autom. Control 60 (10) (2015) 2768–2772. [16] P. Park, W. Lee, S.Y. Lee, Auxillary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Franklin Inst. 352 (4) (2015) 1378–1396. [17] Y. He, G.P. Liu, D. Rees, M. Wu, Stability analysis for neural networks with time-varying interval delay, IEEE Trans. Neural Netw. 18 (6) (2007) 1850–1854. [18] P.G. Park, J.W. Ko, C.K. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica 47 (1) (2011) 235–238. [19] F.O. Souza, V.C.S. Campos, R.M. Palhares, On delay-dependent stability conditions for Takagi-Sugeno fuzzy systems, J. Franklin Inst. 351 (2014) 3707–3718. [20] J. An, T. Li, G. Wen, R. Li, New stability conditions for uncertain T-S fuzzy systems with interval time-varying delay, Int. J. Control. Autom. Syst. 10 (2012) 490–497. [21] C. Peng, Q.L. Han, Delay-range-dependent robust stabilization for uncertain T-S fuzzy systems with interval time-varying delay, Inf. Sci. 181 (19) (2011) 4287–4299. [22] C. Peng, L. Wen, J. Yang, On delay-dependent robust stability criteria for uncertain T-S fuzzy systems with interval time-varying delay, Int. J. Fuzzy Syst. 13 (2011) 35–44.

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