Local stability and stabilization of uncertain nonlinear systems with two additive time-varying delays

Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns

Research paper

Local stability and stabilization of uncertain nonlinear systems with two additive time-varying delays Tao Zhao∗, Chengsen Chen, Songyi Dian College of Electrical Engineering, Sichuan University, Chengdu 610065, China

a r t i c l e

i n f o

Article history: Received 5 August 2019 Revised 10 October 2019 Accepted 1 November 2019 Available online 3 November 2019 Keywords: Two additive delays Attraction domain Interval type-2 T-S fuzzy system Local stability

a b s t r a c t This paper investigates the local stability and stabilization problem of nonlinear time-delay systems subject to parameter uncertainties, which are represented by interval type-2(IT2) T-S fuzzy systems with two additive time-varying delays. Because T-S fuzzy models often represent primitive nonlinear systems well only in the region of validity, local stability and stabilization problems are considered. By constructing a novel Lyapunov–Krasovskii functional and using the matrix inequality scaling approach, a new delay-dependent local stability and stabilization criterion is derived in terms of linear matrix inequalities(LMIs). Using Lyapunov level set, the estimation of attraction domain is also developed. Finally, two numerical examples are employed to verify the effectiveness of proposed methods. © 2019 Elsevier B.V. All rights reserved.

1. Introduction During the past decades, fuzzy control is significant in dealing with the nonlinear systems. In particular, the T-S fuzzy model is well-known as great approximating ability to nonlinear systems [1]. Many problems about nonlinear systems, such as stability analysis and synthesis problems [2–6], filtering [7] and fault detection problems [8] are investigated based on the T-S fuzzy model. After the concept of type-2 fuzzy set introduced by Professor Zadeh [9], type-2 fuzzy systems attract more attention. IT2 fuzzy systems becomes one of the hot topics, since the IT2 fuzzy set can describe high uncertainties or parameter uncertainties. At the meantime, the computing cost is affordable [10]. Numerous experiments have showed that the IT2 fuzzy system is better than the type-1 fuzzy system in the applications with high uncertainties [10–13]. Stability is the most important performance of the systems. But due to the constraints of safe operation conditions, physical structure or other constraints, the system works inside a subpart of the state space [14]. Therefore, it is more reasonable to consider the local stability in this case. T-S fuzzy model can approximate the original nonlinear system in a state space region. Therefore, it is suitable to use the T-S fuzzy model technique to represent the nonlinear system in restricted region here. The restricted domain is called region of validity, which is chosen by physical constraints or other reasons [14]. In the T-S fuzzy model theme, the controller is designed based on the fuzzy model and then applied to the actual nonlinear system. Without considering the validity of region in the controller law, some trajectories of the actual closed-loop system may go outside of the region of the validity [14]. This may cause poor performance or even instability problem to the actual systems. A few results have already shown the effectiveness of this framework [15–18], but they are based on type-1 T-S fuzzy systems. To the best of our knowledge, no results are derived in the framework of IT2 fuzzy model. Comparing with type-1 T-S fuzzy model, the IT2 T-S fuzzy model has more powerful ability of representation, especially in highly uncertain situation. So it may improve the performance by approximating the nonlinear system subject to parameter ∗

Corresponding author. E-mail address: [email protected] (T. Zhao).

https://doi.org/10.1016/j.cnsns.2019.105097 1007-5704/© 2019 Elsevier B.V. All rights reserved.

2

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

uncertainties as a IT2 T-S fuzzy model instead. This motivates us to consider the local stability in the framework of IT2 T-S fuzzy model. Time delay is very common in the practical systems, and it has great influence on stability. Thus stability of systems with time delay attracts much attention last decades. The Lyapunov–Krasovskii approach is the most popular method and fruitful results have been derived based on it. Specially, delay-dependent criteria including more information, have less conservatism than delay-independent ones when the size of delay is small [19]. Thus, delay-dependent criteria is considered by more people. It seems credible that the less conservative criteria will derived if more delay information is included. However, much research only considers single delay [20–22]. Few research considers two additive delays [23–26] or multiple delays [27]. In the applications of remote control and network control systems, due to the difference of signal transmission between two network nodes, there may exist two time-delays with different properties [24]. Therefore, it is more reasonable to describe such delayed models as multiple delays. The model with two additive delays is the basic model of multi delays, which is proposed by James Lam for the first time [26]. The two additive delays in the model can be described as the delay between sensor and controller and the delay between controller and actuator. Some results have been obtained based on this model [20,23]. It can be found that few results are about IT2 fuzzy systems, which motivates our works on IT2 fuzzy systems with two additive time-varying delays. The Lyapunov method is popular, but the stability criteria based on it are sufficient but unnecessary. Conservatism can not be avoided in these results. Reducing the conservatism has been the topic of many studies [19]. Many methods have been proposed to reduce the conservatism: selection of appropriate Lyapunov functions [19,26,28], free weight matrix approach(FWM) [19,22], integral(summation) inequalities [2,20], reciprocally convex combination [25,29–32] and their combination. Specially, the FWM approach estimates the derivative of Lyaponov function by introducing zero-equality, which reduces the conservatism. Ignoring some items in the process of estimation brings the conservatism. And it has been proven that the FWM approach is equivalent to Jensen-based inequality and Wirtinger-based inequality by selecting the involved matrix [22]. FWM method will play an increasingly important role in future research, and thus it is used in this paper. In this paper, an IT2 T-S fuzzy model with two additive time-varying delays is employed to represent nonlinear delayed systems. The local stability is investigated in the framework of Lyapunov stability theory. The FWM approach and a novel Lyapunov functional are introduced to reduce to conservatism of stability criteria. The state-feedback control law is also considered. The effectiveness of the local stability criteria and designed controller is demonstrated through two numerical examples. The contribution of this paper can be summarized as follows: 1) The stability analysis and control design for nonlinear systems with two additive delays and parameter uncertainties are investigated via IT2 T-S fuzzy model. 2) The local stability analysis and synthesis of IT2 T-S fuzzy model with two additive delays are investigated. The problem of estimation of attraction domain is also considered. 3) A novel Lyapunov function is constructed and the General Free Weight Matrix (GFWM) approach is introduced to the derive less conservative local stability and stabilization conditions in the form of LMIs. This paper is organized as follows: In Section 2, some preliminary lemmas and the IT2 T-S systems with two additive delays are introduced. In Section 3, an augmented Lyapunov–Krasovskii functional is constructed. With the GFWM approach, a new stability criterion is proposed for discrete-time IT2 T-S fuzzy system with two additive delays. Then, based on the new stability criterion, a stabilization condition for such systems is derived in Section 4. Finally, the effectiveness of proposed methods is demonstrated by two numerical examples utilized in many previous works. Notation: Throughout this paper, the superscripts T and -1 mean the tranpose and the inverse of a matrix, respectively; Rn denotes the n-dimensional Euclidean space; Ban×an is the set of block diagonal matrices with an dimension, whose block elements are n × n real symmetric matrices; P > 0( ≥ ) means that P is real symmetric and positive definite(semidefinite); Sym(A) is defined as A + AT . 2. Preliminaries Consider the following IT2 T-S fuzzy delay model, which represents nonlinear systems with two additive time-varying delays: Rule i: IF α 1 (x(k)) is F1i and ... and α ψ (x(k)) is Fψi , THEN



x(k + 1 ) = Ai x(k )+ Adi x(k − d1 (k ) − d2 (k ) ) + Bi u(k ) x(k ) = φ (k ), k ∈ −d, 0

(1)

where Fpi is an IT2 fuzzy set, p = 1, . . . , ψ , i ∈ S  {1, 2, . . . , s}; x(k ) ∈ Rn is the state vector; u(k ) ∈ Rnu is the input vector; Ai ∈ Rn×n , Adi ∈ Rn×n and Bi ∈ Rn×n are known constant matrices; i is the number of IF-THEN rules; φ (k) is the initial condition; d1 (k) and d2 (k) are time delays and satisfy following conditions

d1 ≤ d1 (k )≤ d1 , d2 ≤ d2 (k ) ≤ d2

(2)

u1 ≤ d1 (k )≤ u1 ,

(3)

u 2 ≤ d 2 ( k ) ≤ u 2

where di , di , ui and ui , i = 1, 2 are constants. Let d (k ) = d1 (k ) + d2 (k ), d = d1 + d2 , d = d1 + d2 .

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

3

The firing strength of the ith rule is defined as follow:



i (x(k ) ) = w



ψ 

μF i (α p (x(k )) ), p

p=1

ψ 



  μ¯ Fpi (α p (x(k )) ) = wLi (x(k )), wUi (x(k ))

p=1







Note that μF i α p (x(k ) ) ∈ [0, 1] are the lower membership grades, and μ ¯ F i α p (x(k ) ) ∈ [0, 1] are the upper membership p

p

grades. The global model is described as follow: s 

x (k + 1 ) =

λi (x(k ))[Ai x(k ) + Adi x(k − d1 (k ) − d2 (k )) + Bi u(k )]

(4)

i=1

where

λi (x(k )) = wLi (x(k ))ν i (x(k )) + wUi (x(k ))ν i (x(k )) ≥ 0

(5)

with

ν i ( x ( k ) ) + ν i ( x ( k ) ) = 1,

s 

λi (x(k )) = 1.

(6)

i=1

where ν i (x(k ) ) ∈ [0, 1] and ν i (x(k)) ∈ [0, 1] are nonlinear functions. For the sake of simplicity, λi will be used to denote λi (x(k)) without confusion. In this paper, the membership functions of the state feedback controller are different from those of the IT2 fuzzy systems. The following control law is considered: Rule i: IF β 1 (σ (k)) is Gi1 and ... and β ψ (σ (k)) is Giψ THEN

u(k ) = Ki x(k )

(7)

where is an IT2 fuzzy set, p = 1, . . . , ψ , i = 1, . . . , s; Ki ∈ i ∈ S. The state-feedback controller is explained by Gip

u (k ) =

s 

Rnu ×n

denotes the control gain matrix at ith IF-THEN rules, for

hi (σ (k ) )Ki (x(k ) )

(8)

i=1

where

 hˆ i (σ (k ) ) =

ψ 

μGi (β p (σ (k ) ) ),

p=1

p

ψ 



  μ¯ Gip (β p (σ (k ) ) ) = hLi (σ (k ) ), hUi (σ (k ) )

p=1

hi (σ (k ) ) = hLi (σ (k ) )ϑ i (σ (k ) ) + hUi (σ (k ) )ϑ i (σ (k ) ) ≥ 0

ϑ i ( σ ( k ) ) + ϑ i ( σ ( k ) ) = 1,

s 

hi ( σ ( k ) ) = 1 .

i=1

ϑ i (σ (k ) ) ∈ [0, 1] and ϑi (σ (k)) ∈ [0, 1] are functions designed by author; hi (σ (k)) are the grades of membership of embedded membership functions. Similar to λi and λi (x(k)), hi will be used to denote hi (σ (k)). Under the control law (7), we investigate the local stabilization problem of discrete IT2 T-S fuzzy system with two additive delays (4) represented by x (k + 1 ) =

s  s 

λi hl [(Ai + Bi Kl )x(k ) + Adi x(k − d1 (k ) − d2 (k ))]

(9)

i=1 l=1

The following lemmas and method were introduced for developing the main results. Lemma 1. (Finsler’s lemma [20]) Let x ∈ Rn , AT = A ∈ Rn×n , C ∈ Rm×n such that rank{C} < n, and any matrix D ∈ Rn×m . The following statement are equivalent:

(a ) xT Ax < 0, ∀Cx = 0, x = 0; T

(b) C ⊥ AC ⊥ < 0; (c ) A + DC + C T DT < 0.

4

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

The GFWM Approach[22]: Estimation of summation term like −

k+α2 −1

of criterion. For simplifying description, let

φ (α , β ) = −

α −1 

xT (s )Rx(s ) = −

k+ α2 −1

s=β

s = k + α1

xT (s )Rx(s ) is the key during the deriving

xT (s )Rx(s )

s=k+α1

where x(s ) = x(s + 1 ) − x(s ), and α 1 , α 2 have the value like −d1 , −d2 , 0. GFWM approach estimate the summation terms aforementioned by introducing a new zero-value equality, see more details in [22].



φ (α , β ) = 1 (M, T , L, N, α , β ) + 2 (X, Z, α , β ) − 3 X , , T

≤ 1 (M, T , L, N, α , β ) + 2 (X, Z, α , β ) − 3 X , , T where

1 (M, T, L, N, α , β )

(10)



(11)

= xT (α )T x(α ) − xT (β )T x(β ) + 2 f T (k ) M[x(α ) − x(β )]



+ L[(α − β + 1 )σ (α , β ) − x(α )] + N [x(α ) + x(β ) − 2σ (α , β )]

(12)



 α−β −1 Z f (k ) 3 (α − β + 1 )   1 2 (X, Z, α , β ) = (α − β ) f T (k ) X + Z f (k ) ¯ 2 (X, Z, α , β ) = (α − β ) f T (k ) X + 

3

 −1

α X 3 X , , T = η3T (k, s ) T s=β

T

 η3 (k, s ).

with f(k) is a suitable vector, T is a symmetric matrix, X and Z are positive definite symmetric matrices, L, M, N, Y is any matrix.

 η1 (k, s ) σ (α , β ) = η3 (k, s ) = , s=β η2 ( s )       −α − β + 1 2s f (k ) x (s ) L M χ (s ) = + , η1 (k, s ) = , η2 ( s ) = , = . χ (s ) f (k ) x ( s ) 0 N α−β +1 α−β +1 

α





x (s ) , α−β +1



X X= YT

Y Z





0 ,T = T



T , T +R



3. Local stability analysis In this part, a new stability criterion for system (4) with u(k ) = 0 given by

x (k + 1 ) =

s 

λi [Ai x(k ) + Adi x(k − d1 (k ) − d2 (k ))]

(13)

i=1

will be derived. To simplify the proof in the latter part,the system in (4) is expressible in a compact form as

x(k + 1 ) = A(k )x(k ) + Ad (k )x(k − d1 (k ) − d2 (k ) ) s

(14)

s

where A(k ) = i=1 λi Ai , Ad (k ) = i=1 λi Adi . IT2 T-S fuzzy models are often used to approximate or equivalent nonlinear systems. Generally, this approximation is only valid in a restricted region of the state space. This paper assumes that J0 is a modeling domain and satisfies the following conditions:





x(k ) ≤ η J0 = x(k )H



(15)

 and η are matrix and vector given in advance, respectively. where H Since the state variables are only valid in J0 , the system cannot be globally stable. Therefore, the local stability analysis and control synthesis should be consider, that is, the domain of attraction should be estimated. Since two additive delays are considered, this paper divides the initial into two parts, similar to the point of view in [14]:

Jd = Jx0 × J−x0

(16)

Jx0 = {x(0 )

(17)

where

| x ( 0 ) ∈ Rn }

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

5

and

J−x0 = {x(0 ) with

| x(0 ) d < +∞ }

(18)



x(0 ) = x(−1 ), x(−2 ), . . . , x −d



, x(0 ) d =

sup −d≤i≤−1

x(i ) 22 .

In this section, we mainly discuss the local stability analysis, which determines the domain of attraction Jx0 ⊂ J0 and J−x0 , so that the state trajectory starting the domain of attraction is still in J0 and asymptotically converges to the equilibrium point. The related notations in Theorem 1 will be introduced to simplify the description of subsequent parts.





Xi YiT

Xi =





Yi L , i = i 0 Zi

U 3i−2 = U2i−1 ,





0 Mi , Ti = Ni TiT

Ti , i = 1, . . . , 9; Ti

x ( k ) = x ( k + 1 ) − x ( k )

U 3i−1 = U 3i = U2i , i = 1, 2, 3

h1d ( k ) = d ( k ) − d ,

h2d ( k ) = d − d ( k ) ,

h3d ( k ) = d1 ( k ) − d1 ,

h4d ( k ) = d1 − d1 ( k ) , h5d ( k ) = d2 ( k ) − d2 , k  x (s ) , d+1

ν1 ( k ) =

s=k−d (k )

s=k−d k 

ν4 ( k ) =

s=k−d1 k 

ν7 ( k ) =

s=k−d2

x (s ) , d1 + 1 x (s ) , d2 + 1

k−d 

ν2 ( k ) =

x (s ) , h1d ( k ) + 1

k−d1

ν5 ( k ) =



s=k−d1 (k ) k−d2

ν8 ( k ) =

h6d ( k ) = d2 − d2 ( k )



s=k−d2 (k )

k−d (k )



ν3 ( k ) =

s=k−d

x (s ) , h3d ( k ) + 1

x (s ) h2d ( k ) + 1

k−d1 (k )



ν6 ( k ) =

s=k−d1

x (s ) , h5d ( k ) + 1

k−d2 (k )



ν9 ( k ) =

s=k−d2

x (s ) h4d ( k ) + 1 x (s ) h6d ( k ) + 1

 ζ (k ) = xT (k ), xT (k − d ), xT (k − d (k )), xT (k − d ), xT (k − d1 ), xT (k − d1 (k )), ν1T (k ), ν2T (k ),

xT (k − d1 ), xT (k − d2 ), xT (k − d2 (k )), xT (k − d2 ),

ν3T (k ),

ν4T (k ),

ν5T (k ),



ν6T (k ),



ei = 0n×(i−1)n, In×n, 0n×(20−i )n , i = 1, 2, . . . , 20;

10d(k) = [h1d (k ) + 1]e12 + [h2d (k ) + 1]e13

ν7T (k ),

ν8T (k ),

ν9T (k ),

x T ( k )

T



T

e f = eT1 , eT3 , eT11 , eT12 , eT13 , eT14 , eT15 , eT16 , eT17 , eT18 , eT19

20d(k) = [h3d (k ) + 1]e15 + [h4d (k ) + 1]e16

30d(k) = [h5d (k ) + 1]e18 + [h6d (k ) + 1]e19

 11 =

 12 =

 13d(k) =

e1 + e20 (d + 1 )e11 − e2

 21 =



e1 , (d + 1 )e11 − e1

11



,



 , 23d (k ) =

 31 =

e1 , (d1 + 1 )e14 − e1

32 =



21





20d(k) − e6 − e7





, 33d (k ) =



e1 + e20 (d2 + 1 )e17 − e8 e1 (d2 + 1 )e17 − e1

31

  

30d(k) − e9 − e10   12 22 32 14d(k) = , 24d (k ) = , 34d (k ) = 10d(k) − e2 − e3 20d(k) − e5 − e6 30d(k) − e8 − e9



10d(k) − e3 − e4

 22 =



e1 + e20 , (d1 + 1 )e14 − e5



i j (d1 (k ), d2 (k ), d1 (k ), d2 (k )) = γ1,i j (d1 (k ), d2 (k ), d1 (k ), d2 (k )) + γ2 + γ31 + γ32 + γ33 (d1 (k ), d2 (k ))

6

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

γ1,i j (d1 (k ), d1 (k ), d2 (k ), d2 (k )) = (d (k ) + d (k ))T11 P1 j 11 − d (k )T12 P1i 12 + T13d(k) P2 13d(k) −T14d (k ) P2 14d (k ) + (d1 (k ) + d1 (k ))T21 P3 j 21 − d1 (k )T22 P3i 22 +T23d (k ) P4 23d (k ) − T24d (k ) P4 24d (k ) + (d2 (k ) + d2 (k ))T31 P5 j 31 −d2 (k )T32 P5i 32 + T33d (k ) P6 33d (k ) − T34d (k ) P6 34d (k )

γ2 = eT1 (Q1 + Q3 + Q5 )e1 − eT2 (Q1 − Q2 )e2 − eT5 (Q3 − Q4 )e5 − eT8 (Q5 − Q6 )e8 − eT4 Q2 e4 − eT7 Q4 e7 − eT10 Q6 e10 γ31 = WuT [dU1 + (d − d )U2 + d1U3 + (d1 − d1 )U4 + d2U5 + (d2 − d2 )U6 ]Wu 9 

γ32 =





eTi Ti ei − eTi+1 Ti ei+1 + Sym eTf (Mi (ei − ei+1 ) + Ni (ei + ei+1 ) − Li ei )

Wu = [e1 ; e20 ];



+ eT1 T4 e1 − eT4 T4 e4

i=1

+ eT1 T7 e1 − eT7 T7 e7 − 2

9 













Sym eTf Ni ei+10 + Sym (d + 1 )eTf L1 e11 + Sym d1 + 1 eTf L4 e14



i=1



+ Sym eTf (M4 (e1 − e4 ) + M7 (e1 − e7 ) + N4 (e1 − e4 ) + N7 (e1 − e7 ) − L4 (e1 − e4 ) − L7 (e1 − e7 ) ]











+ Sym d2 + 1 eTf L7 e17 + eTf dX1 + d1 X4 + d2 X7 e f



1 + eTf 3









d1 d1 − 1 d2 d2 − 1 d (d − 1 )

Z4 +

Z7 e f Z1 + (d + 1 ) d1 + 1 d2 + 1

γ33 (d1 (k ), d2 (k )) =

3  i=1





Sym h(2i−1)d (k ) + 1 eTf L3i−1 e3i+9 + (h2id (k ) + 1 )eTf L3i e3i+10









Z2 + h2d (k ) X3 + 3     Z6 + eTf h4d (k ) X6 + + h5d (k ) X8 + 3 + eTf h1d (k ) X2 +

i = (Ai − I )e1 + Adi e3 − e20 , i ∈ S,

Emi =



0n×(i−1)n

In





Z3 + h3d (k ) X5 + 3   Z8 + h6d (k ) X9 + 3 0n×(m−i)n



,





Z5 ef 3   Z9 ef 3

P i = P1 i + P2 i

P1 i = dE21 P1i E21 T + E31 P2 E31 T + d1 E21 P3i E21 T + E31 P4 E31 T + d2 E21 P5i E21 T + E31 P6 E31 T









P2 i = dE22U1 E22 T + d − d E22U2 E22 T + d1 E22U3 E22 T + d1 − d1 E22U4 E22 T





+ d2 E22U5 E22 T + d2 − d2 E22U6 E22 T + 12I P i = dE21 P1i E21 T + E31 P2 E31 T + d1 E21 P3i E21 T + E31 P4 E31 T + d2 E21 P5i E21 T + E31 P6 E31 T . Now, the Theorem 1 is expressed as follow: Theorem 1. For given scalars d1 , d1 , d2 , d2 , u1 , u1 , u2 , and u2 , system (13) with two additive time-varying delay satisfying (2) and (3) is asymptotically stale for any set of initial conditions Jd and the state trajectory starting the domain of attraction Jd is still in J0 , if there exist symmetric matrices Qi , Ui , T j , X j , Z j , i = 1, . . . , 6, j = 1, . . . , 9, block diagonal matrices P1i , P3i , P5i ∈ B2n×2n , i ∈ S, P2 , P4 , P6 ∈ B3n×3n , and any matrices L j , M j , N j,Y j , j = 1, . . . , 9, and matrix i , i ∈ S such that the following conditions hold

Qk > 0, Uk > 0,



0 > 0, 0

P3i 0

0 > 0, 0



(19)



P1i 0

P2 + a

P4 + b

k = 1, . . . , 6

a = d1 + d2 , d1 + d2

i∈S

(20)



b = d1 , d1

i∈S

(21)

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

 P6 + c

P5i 0

 k =

0 > 0, 0

Tk

i∈S

(22)



k

Xk

c = d2 , d2

≥ 0,

T k + Uk

k = 1, . . . , 9

(23)

  d ( k ) = d1 , d1  i j (d1 (k ), d2 (k ), d1 (k ), d2 (k ))  1 d2 ( k ) = d2 , d2   ⊥T i

−P i  H

7



d 1 ( k ) = u 1 , u 1 d 2 ( k ) = u 2 , u 2

T i

T H ≤ 0, −η2

< 0,

i, j ∈ S

i∈S

(24)

(25)

where the domain of attraction Jx0 is computed in (46) Proof. It is noted that d1 , d1 , d2 , d2 represents the lower bound and the upper bound of d1 (k) and d2 (k), u1 , u1 , u2 , u2 denotes lower bound and the upper bound of d1 (k) and d2 (k), respectively, which is defined in (2) and (3). A novel Lyapunov-Krasovskii functional is constructed as follow:

V (xk ) = V1 (xk ) + V2 (xk ) + V3 (xk )

(26)

where

V1 (xk ) = d (k )ξ1T (k )P1 (k )ξ1 (k ) + ξ2T (k )P2 ξ2 (k ) + d1 (k )ξ3T (k )P3 (k )ξ3 (k ) + ξ4T (k )P4 ξ4 (k ) +d2 (k )ξ5T (k )P5 (k )ξ5 (k ) + ξ6T (k )P6 ξ6 (k ) k−1 

V2 (xk ) =

xT ( s )Q1 x ( s ) +

s=k−d

k−d−1



s=k−d



−1  k−1 

η2T (s )U1 η2 (s ) +

+

η2T (s )U4 η2 (s ) +



k−1 

ξ1 ( k ) = x ( k ) , T

x (s )

k−1 

ξ3 ( k ) = x ( k ) , T

k−1 

ξ5 ( k ) = x ( k ) , T

η2T (s )U5 η2 (s ) +

 ,

ξ2 ( k ) = x ( k ) , T

T x (s ) T

s=k−d1



−1 k−1  

T T

ξ4 ( k ) = x ( k ) , T

T x (s ) T

,

k−1 

x (s ), T

−d2 −1 k−1  

k−1 

ξ6 ( k ) = x ( k ) , T

k−1 

k−d−1



η2T (s )U6 η2 (s )

T x (s ) T

s=k−d

T

k−d1 −1

x (s ), T

s=k−d1



s=k−d2

η2T (s )U3 η2 (s )

j=−d2 s=k+ j

s=k−d

 ,

(28)

j=−d1 s=k+ j

j=−d2 s=k+ j

s=k−d



−1 k−1  

η2T (s )U2 η2 (s ) +

j=−d s=k+ j

−d1 −1 k−1  

xT ( s )Q6 x ( s )

s=k−d2

−d−1 k−1  

j=−d1 s=k+ j

where



x T ( s ) Q 5 x ( s )+

s=k−d2

j=−d s=k+ j

xT ( s )Q3 x ( s ) k−d2 −1

k−1 

xT ( s )Q4 x ( s ) +

s=k−d1

V3 (xk ) =

k−1 

xT ( s )Q2 x ( s ) +

s=k−d1

k−d1 −1

+

(27)



x (s ) T

s=k−d1

T

k−d2 −1

x (s ), T

s=k−d2



x (s ) T

.

s=k−d2

Based on the convex combination approach,conditions (20) (21) and (22) lead that s 

  P1i λi P2 + (d1 (k ) + d2 (k ))

i=1 s  i=1 s  i=1

0

0 0

P3i 0

0 0

  P5i λi P6 + d2 (k )

0 0

 λi P4 + d1 (k )



0





> 0 ⇔ P2 + (d1 (k ) + d2 (k ) )





P (k ) > 0 ⇔ P4 + d1 (k ) 3 0



 > 0 ⇔ P6 + d2 (k )

P5 (k ) 0

P1 (k ) 0



0 >0 0



0 > 0. 0



0 >0 0

(29)

8

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

At the mean time, the matrices Qk , Uk , k = 1, . . . , 6 in functional V(xk ) are positive definite. Thus, V(xk ) ≥ ε x(k) 2 for a sufficiently small scalar ε > 0. Then, the conditions guaranteeing the negative of the forward difference of V(xk ), (V(xk )) is proven in the subpart. Firstly, calculating the forward difference of V1 (xk ), V2 (xk ) and V3 (xk ), respectively, leads that

V1 (xk ) = (d (k ) + d (k ))ξ1T (k + 1 )P1 (k + 1 )ξ1 (k + 1 ) − d (k )ξ1T (k )P1 (k )ξ1 (k ) + ξ2T (k + 1 )P2 ξ2 (k + 1 ) − ξ2T (k )P2 ξ2 (k ) + (d1 (k ) + d1 (k ))ξ3T (k + 1 )P3 (k + 1 )ξ3 (k + 1 ) − d1 (k )ξ3T (k )P3 (k )ξ3 (k ) + ξ4T P4 ξ4 (k + 1 ) − ξ4T (k )P4 ξ4 (k ) + (d2 (k ) + d2 (k ))ξ5T (k + 1 )P5 (k + 1 )ξ5 (k + 1 ) − d2 (k )ξ5T (k )P5 (k )ξ5 (k ) + ξ6T (k + 1 )P6 ξ6 (k + 1 ) − ξ6T (k )P6 ξ6 (k )   s  s 

λi λ+j γ1,i j (d1 (k ), d1 (k ), d2 (k ), d2 (k ) ζ (k )

(30)

V2 (xk ) = ζ (k )T [eT1 (Q1 + Q3 + Q5 )e1 − eT2 (Q1 − Q2 )e2 − eT5 (Q3 − Q4 )e5 −eT8 (Q5 − Q6 )e8 − eT4 Q2 e4 − eT7 Q4 e7 − eT10 Q6 e10 ]ζ (k ) T = ζ (k )γ2 ζ (k )

(31)

=

ζ T (k )

i=1 j=1

where λ+j = λ j (x(k + 1 ) ).

V3 (xk ) = dη2T (k )U1 η2 (k ) + (d − d )η2T (k )U2 η2 (k ) +d1 η2T (k )U3 η2 (k ) + (d1 − d1 )η2T (k )U3 η2 (k ) +d2 η2T (k )U5 η2 (k ) + (d2 − d2 )η2T (k )U6 η2 (k ) + Vb (xk ) =

ζ T (k )γ31 ζ (k ) + Vb (xk )

(32)

where

Vb (xk ) = −

k−1 

η2T (s )U1 η2 (s ) −

s=k−d k−d−1

−



η2T (s )U2 η2 (s ) −

k−d (k )−1

k−d1 (k )−1





(33)

k−d1 −1



η2T (s )U2 η2 (s ) −



η2T (s )U4 η2 (s )

(34)

s=k−d1 (k )

k−d2 −1

η2T (s )U6 η2 (s ) −

s=k−d2 (k )

s=k−d1

η2T (s )U5 η2 (s )

s=k−d2

s=k−d

η2T (s )U4 η2 (s ) −

k−1 

η2T (s )U3 η2 (s ) −

s=k−d1

s=k−d (k )

−

k−1 

k−d2 (k )−1



η2T (s )U6 η2 (s ).

(35)

s=k−d2

The GFWM approach aforementioned will used to estimate Vb (xk ). In detail, the equality (10) will be used to estimate the first three summation term of Vb (xk ) in (33), and the inequality (11) will be used to estimate the rest summation term of Vb (xk ) in (34) and (35). The vector f(k) in (12) can be component of some or all vectors in ζ (k). Here, let’s assume it as



T

f (k ) = xT (k ), xT (k − d (k )), ν1T (k ), ν2T (k ), ν3T (k ), ν4T (k ), ν5T (k ), ν6T (k ), ν7T (k ), ν8T (k ), ν9T (k ) Estimation of Vb (xk ) leads that

Vb (xk ) ≤ −

k−1  s=k−d

−

−

k−1 



η3T (k, s )2 η3 (k, s ) −

k−d (k )−1

s=k−d (k )

η3T (k, s )4 η3 (k, s ) −



s=k−d1 (k )

k−1 

k−d2 −1

η3T (k, s )7 η3 (k, s ) −





η3T (k, s )3 η3 (k, s )

s=k−d

k−d1 −1

s=k−d1

s=k−d2

+

k−d−1

η3T (k, s )1 η3 (k, s ) −

η3T (k, s )5 η3 (k, s ) −

k−d1 (k )−1



η3T (k, s )6 η3 (k, s )

s=k−d1

η3T (k, s )8 η3 (k, s ) −

s=k−d2 (k )

k−d2 (k )−1



η3T (k, s )9 η3 (k, s )

s=k−d2

ζ (k )[γ32 + γ33 (d (k ))]ζ (k ). T

(36)

If i ≥ 0, i = 1, . . . , 9 holds, the following can be obtained

V3 (xk ) ≤ ζ T (k )[γ31 + γ32 + γ33 (d1 (k ), d2 (k ))]ζ (k ) = ζ T (k )[γ3 (d1 (k ), d2 (k ))]ζ (k ).

(37)

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

9

Combining (30–32),(36) and condition (23) leads that

V (xk ) ≤ ζ T (k )[

s  s 

λi λ+j γ1,i j (d1 (k ), d2 (k ), d1 (k ), d2 (k )) + γ2 + γ3 (d1 (k ), d2 (k ))]ζ (k ).

i=1 j=1

=

s  s 

  λi λ+j ζ T (k ) γ1,i j (d1 (k ), d2 (k ), d1 (k ), d2 (k )) + γ2 +γ3 (d1 (k ), d2 (k )) ζ (k )

i=1 j=1

=

s  s 

λi λ+j ζ T (k )i j (d1 (k ), d1 (k ), d2 (k ), d2 (k ))ζ (k ).

(38)

i=1 j=1

According to inequality (38) and the Lyapunov stability theory, a stability condition for system (13) is s  s 

λi λ+j ζ T (k )i j (d1 (k ), d1 (k ), d2 (k ), d2 (k ))ζ (k ) < 0

(39)

i=1 j=1

Apparently, the following equality is true: s 

λi i ζ (k ) = 0, i = (Ai − I )e1 + Adi e3 − e20 .

(40)

i=1

According to Lemma 1, if condition (39) and (40) hold, the following relations are valid for any matrix D with appropriate dimension.

(39 ) and (40 ) ⇔

s  s 

λi λ+j [i j (d1 (k ), d1 (k ), d2 (k ), d2 (k )) + sym(Di )] < 0

i=1 j=1

⇐ i j (d1 (k ), d1 (k ), d2 (k ), d2 (k )) + sym(Di ) < 0 ⇔ i⊥ i j (d1 (k ), d2 (k ), d1 (k ), d2 (k ))i⊥ < 0. T

The relations above mean that if condition (24) holds, then we have V(k) < 0. Thus,V(k) < V(0). Note V (0 ) = V1 (0 ) + V2 (0 ) + V3 (0 ), we have

V1 (0 ) = d (0 )ε1T (0 )P1 (0 )ε1 (0 ) + ε2T (0 )P2 ε2 (0 ) + d1 (0 )ε3T (0 )P3 (0 )ε3 (0 ) + ε4T (0 )P4 ε4 (0 ) + d2 (0 )ε5T (0 )P5 (0 )ε5 (0 ) + ε6T (0 )P6 ε6 (0 )



 x(0 ) d + xT (0 )E31 P2 E31 T x(0 ) i=1  

s 2 + d1 xT (0 )E21 P3 (0 )E21 T x(0 ) + d1 max λmax E22 P3i E22 T x(0 ) d + xT (0 )E31 P4 E31 T x(0 ) i=1  

s 2 T T + d2 x (0 )E21 P5 (0 )E21 x(0 ) + d2 max λmax E22 P5i E22 T x(0 ) d + xT (0 )E31 P6 E31 T x(0 ) i=1





2 2 2 + d λmax E32 P2 E32 T x(0 ) d + d1 λmax E32 P4 E32 T x(0 ) d + d2 λmax E32 P6 E32 T x(0 ) d

2



2

+ d − d λmax E33 P2 E33 T x(0 ) d + d1 − d1 λmax E33 P4 E33 T x(0 ) d

2

+ d2 − d2 λmax E33 P6 E33 T x(0 ) d   s



≤ d xT (0 )E21 P1 (0 )E21 T x(0 ) + d max λmax E22 P1i E22 T 2

= xT ( 0 )

s 



λi P1i x(0 ) + ρ1 x(0 ) d

i=1

where







2

s ρ1 = dd2 max λmax E22 P1i E22 T + d2 λmax E32 P2 E32 T + d − d λmax E33 P2 E33 T i=1





2

s 2 2 +d1 d1 max λmax E22 P3i E22 T + d1 λmax E32 P4 E32 T + d1 − d1 λmax E33 P4 E33 T . i=1





2

s 2 2 +d2 d2 max λmax E22 P5i E22 T + d2 λmax E32 P6 E32 T + d2 − d2 λmax E33 P6 E33 T . i=1

V2 (0 ) ≤ dλmax (Q1 ) x(0 ) d + d1 λmax (Q3 ) x(0 ) d + d2 λmax (Q5 ) x(0 ) d



+ d−d =



ρ2 x(0 ) d





λmax (Q2 ) x(0 ) d + d1 − d1 λmax (Q4 ) x(0 ) d + d2 − d2 λmax (Q6 ) x(0 ) d

(41)

10

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

where







ρ2 = dλmax (Q1 ) + d − d λmax (Q2 ) + d1 λmax (Q3 ) + d1 − d1 λmax (Q4 ) + d2 λmax (Q5 ) + d2 − d2 λmax (Q6 ).





d−d d+d−1 d (d − 1 ) V3 (0 ) ≤ 5 λmax (U1 ) x(0 ) d + 5 λmax (U2 ) x(0 ) d 2

2

d1 d1 − 1 d1 − d1 d1 + d1 − 1 +5 λmax (U3 ) x(0 ) d + 5 λmax (U4 ) x(0 ) d 2



2

d2 d2 − 1 d2 − d2 d2 + d2 − 1 +5 λmax (U5 ) x(0 ) d + 5 λmax (U6 ) x(0 ) d 2  T 2



+ d x (0 )E22U1 E22 T x(0 ) + λmax E21U1 E21 T x(0 ) d + 2λmax E22U1 E22 T x(0 ) d + 2xT (0 )x(0 )



(42)





 x(0 ) d + λmax −E21U1 E22 T x(0 ) d + λmax −E22U1 E21 T x(0 ) d





+ d − d xT (0 )E22U2 E22 T x(0 ) + λmax E21U2 E21 T x(0 ) d + 2λmax E22U2 E22 T x(0 ) d + 2xT (0 )x(0 )





 + λmax E21U2 E22 T E22U2 E21 T x(0 ) d + λmax −E21U2 E22 T x(0 ) d + λmax −E22U2 E21 T x(0 ) d 



+ d1 xT (0 )E22U3 E22 T x(0 ) + λmax E21U3 E21 T x(0 ) d + 2λmax E22U3 E22 T x(0 ) d + 2xT (0 )x(0 )





 + λmax E21U3 E22 T E22U3 E21 T x(0 ) d + λmax −E21U3 E22 T x(0 ) d + λmax −E22U3 E21 T x(0 ) d





+ d1 − d1 xT (0 )E22U4 E22 T x(0 ) + λmax E21U4 E21 T x(0 ) d + 2λmax E22U4 E22 T x(0 ) d + 2xT (0 )x(0 )





 + λmax E21U4 E22 T E22U4 E21 T x(0 ) d + λmax −E21U4 E22 T x(0 ) d + λmax −E22U4 E21 T x(0 ) d 



+ d2 xT (0 )E22U5 E22 T x(0 ) + λmax E21U5 E21 T x(0 ) d + 2λmax E22U5 E22 T x(0 ) d + 2xT (0 )x(0 )





 + λmax E21U5 E22 T E22U5 E21 T x(0 ) d + λmax −E21U5 E22 T x(0 ) d + λmax −E22U5 E21 T x(0 ) d





+ d2 − d2 xT (0 )E22U6 E22 T x(0 ) + λmax E21U6 E21 T x(0 ) d + 2λmax E22U6 E22 T x(0 ) d + 2xT (0 )x(0 )





 + λmax E21U6 E22 T E22U6 E21 T x(0 ) d + λmax −E21U6 E22 T x(0 ) d + λmax −E22U6 E21 T x(0 ) d   + λmax E21U1 E22 T E22U1 E21 T

= xT ( 0 )

s 



λi P2i x(0 ) + ρ3 x(0 ) d

i=1

where



ρ3









d−d d+d−1 d1 d1 − 1 d (d − 1 ) =5 λmax (U1 ) + 5 λmax (U2 ) + 5 λmax (U3 ) 2 2 2









d1 − d1 d1 + d1 − 1 d2 d2 − 1 d2 − d2 d2 + d2 − 1 +5 λmax (U4 ) + 5 λmax (U5 ) + 5 λmax (U6 ) 2 2 2 





+ d λmax E21U1 E21 T + 2λmax E22U1 E22 T + λmax E21U1 E22 T E22U1 E21 T





 λmax −E21U1 E22 T + λmax −E22U1 E21 T







+ d − d λmax E21U2 E21 T + 2λmax E22U2 E22 T + λmax E21U2 E22 T E22U2 E21 T



 + λmax −E21U2 E22 T + λmax −E22U2 E21 T 





+ d1 λmax E21U3 E21 T + 2λmax E22U3 E22 T + λmax E21U3 E22 T E22U3 E21 T



 + λmax −E21U3 E22 T + λmax −E22U3 E21 T







+ d1 − d1 λmax E21U4 E21 T + 2λmax E22U4 E22 T + λmax E21U4 E22 T E22U4 E21 T



 + λmax −E21U4 E22 T + λmax −E22U4 E21 T 





+ d2 λmax E21U5 E21 T + 2λmax E22U5 E22 T + λmax E21U5 E22 T E22U5 E21 T



 +λmax −E21U5 E22 T + λmax −E22U5 E21 T







+ d2 − d2 λmax E21U6 E21 T + 2λmax E22U6 E22 T + λmax E21U6 E22 T E22U6 E21 T



 + λmax −E21U6 E22 T + λmax −E22U6 E21 T . +

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

Thus,



V ( 0 ) ≤ xT ( 0 )

s 

11

 λi Pi x(0 ) + ρ x(0 ) d

(43)

i=1

where

ρ = ρ1 + ρ2 + ρ3 .

(44)

On the other hand:

V (k ) ≥ dxT (k )E21 P1 (k )E21 T x(k ) + xT (k )E31 P2 E31 T x(k ) + d1 xT (k )E21 P3 (k )E21 T x(k ) + xT (k )E31 P4 E31 T x(k ) + d2 xT (k )E21 P5 (k )E21 T x(k ) + xT (k )E31 P6 E31 T x(k )



= x (k ) T

s 



λi Pi x(k ).

(45)

i=1

Now let

Jx0 =

s 



x(0 )xT (0 )P i x(0 ) ≤ 1 − ρ x(0 )

i=1

 d

.

(46)

Define

Jx(k ) =

s 





x(k )xT (k )P i x(k ) ≤ 1 .

(47)

i=1

From (43) and (45) we have

Jx0 ⊆ Jx(k ) .

(48)

To ensure Jx(k) ⊆J0 , we have

T η−2 H  − P ≤ 0. H i

(49)

From the above inequality, we have



−P i  H



T H ≤ 0. −η2

(50) 

This completes the proof.

Remark 1. Stability condition (39) is obtained by ignoring the positive summation terms in (36). Comparing to ignoring Vb (xk ) defined in (33)–(35), the GFWM approach brings less conservatism obviously. Applying the convex combination technique also avoid the enlargement of d1 (k ), d2 (k ), d (k ), d1 (k ) − d1 , d1 − d1 (k ), d2 (k ) − d2 , d2 − d2 (k ), d (k ) − d and d − d (k ). Remark 2. Theorem 1 is based on the analysis of IT2 fuzzy systems with two additive delays, which can be easily extended to the situation with multiple additive delays:



x ( k + 1 ) = A ( k )x ( k ) + Ad ( k )x k −

q i=1



di ( k )

with the constraint of di (k) and di (k ), i = 1, . . . , q

di ≤ di ( k ) ≤ di ,

u i ≤ d i ( k ) ≤ u i

Computational burden increases dynamically with the increment of the delay component. Reducing the computational burden will be investigated in our future research.  Remark 3. If x(0 ) = 0, then P i can be rewritten as P i = P1 i + P2 i where





 P2 i = dE22U1 E22 T + d − d E22U2 E22 T + d1 E22U3 E22 T + d1 − d1 E22U4 E22 T



+ d2 E22U5 E22 T + d2 − d2 E22U6 E22 T . Obviously, the domain of attraction can be further expanded.

12

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

4. Local control synthesis In the following part, a local stabilization condition for system (9) is derived based on the result of Theorem 1. Theorem 2. For given scalars d1 , d1 , d2 , d2 , u1 , u1 , u2 , u2 , and ϖ system (9) with two additive time-varying delay satisfying (2) and (3) is asymptotically stale for any set of initial conditions Jd and the state trajectory starting the domain of attraction Jd is still in J0 , if there exist scalars δii and δi2 , such that hi (σ (k ) )δi1 ≤ λi (x(k ) ) ≤ hi (σ (k ) )δi2 , symmetric matrices Qi , Ui , T j , X j , Z j, i = 1, . . . , 6, j = 1, . . . , 9, block diagonal matrices P1i , P3i , P5i ∈ B2n×2n , i ∈ S, P2 , P4 , P6 ∈ B3n×3n , any matrices L j , M j , N j,Y j , j = 1, . . . , 9 and matrix V ∈ Rn×n , ϒ ∈ Rm×n such that the following conditions hold

Qk > 0, Uk > 0,



k = 1, . . . , 6



P1i 0

0 > 0, 0

P P4 + b 3i 0

0 > 0, 0

P2 + a



 P6 + c

k =

a = d1 + d2 , d1 + d2

k

Tk

b = d1 , d1

i∈S

(53)

c = d2 , d2

i∈S

(54)

 ≥ 0,

T k + Uk

 d (k ) = d1 , d1 i ji  1 d2 ( k ) = d2 , d2

k = 1, . . . , 9

d 1 ( k ) = u 1 , u 1 < 0, d 2 ( k ) = u 2 , u 2

  d ( k ) = d1 , d1 δ i jl + δ l ji  1 d2 ( k ) = d2 , d2 

t l

−P i  H

(52)



0 > 0, 0

Xk

p i

i∈S



P5i 0



(51)

T H −η2

(55)

i, j ∈ S

d 1 ( k ) = u 1 , u 1 d 2 ( k ) = u 2 , u 2

(56)

< 0,

i, j, l ∈ S

i < l,

p, t = 1, 2

(57)

 ≤0

i∈S

(58)

where ijl denotes ijl (d1 (k), d2 (k), d1 (k), d2 (k)),





 i jl (d1 (k ), d2 (k ), d1 (k ), d2 (k ))=i j (d1 (k ), d2 (k ), d1 (k ), d2 (k )) + Sym eT1 + eT20  i + Bi ϒl e1

with i = (Ai − I )V e1 + AdiV e3 − V e20 , i ∈ S. i j (d1 (k ), d2 (k ), d1 (k ), d2 (k )), i, j ∈ S is defined in Theorem 1, and matrices Xk , k , Tk , Uk , k = 1, . . . , 9 are the same as Theorem 1. The domain of attraction Jx0 is computed in (46) and the gain matrices are defined as

Ki = ϒiV −1 .

(59)

Proof. To get the stabilization condition in the form of LMI for system (9), the following zero equality is introduced. The technique was firstly proposed in [33].

0=2



λi hl ζ T (k ) eT1 Z + eT20  Z ((Ai + Bi Kl − In )e1 + Adi e3 − e20 )ζ (k ) .   l=1

s  s  i=1

(60)

=0

where Z ∈ Rn×n , and ϖ is the given value. With the same Lyapunov functionals (26), combining the upper bound of V(xk ) obtained in (38) and the zero equality (60) aforementioned, a sufficient condition that guarantees the asymptotically stability for system (9) is presented as follow: s  s 

! λi λ i j (d1 (k ), d2 (k ), d1 (k ), d2 (k )) + sym + j

i=1 j=1

where i , i ∈ S is defined in (40).

s  s  i=1 l=1

λ



T i hl e 1

+

eT20



"

 (Z i + Z Bi Kl e1 ) < 0.

(61)

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

13

Because of the exitance of ZBi Kl , inequality (61) is not LMI. Let V = Z −1 ,and Kl = ϒl V −1 , and the congruent transformation of (61) for  20 leads to s  s 

!

λi λ  i j (d1 (k ), d2 (k ), d1 (k ), d2 (k ))20 + sym + j

T 20

i=1 j=1

=

s  s 

λ



T i hl e 1

+

eT20



 i + Bl ϒl e1



"

(62)

i=1 l=1

s  s  s 

 T





 λi λ+j hl 20 i j (d1 (k ), d2 (k ), d1 (k ), d2 (k ))20 + sym eT1 + eT20  i + Bi ϒl e1 <0

i=1 j=1 l=1

⎧ ⎨

where i =diag V , V , . . . , V

⎩

 i

(63)

⎫ ⎬ .

⎭

T  d ( k ) , d ( k ) , d ( k ) , d ( k )  As the symmetry and positive definiteness of matrix 20 ) 20 and  ij (d1 (k), d2 (k), 2 1 2 ij( 1 d1 (k), d2 (k)) is identical, which means inequality (63) is solvable if the following LMI is solvable: s  s  s 

λi λ+j hl i jl (d1 (k ), d2 (k ), d1 (k ), d2 (k )) < 0.

(64)

i=1 j=1 l=1

Let hi (σ (k ) )δ˜i (k ) = λi (x(k ) ). Using the similar idea of [34] to δ˜i (k ), the follow equality is obtained:

δ˜i (k )=δi1 ω˜ i(1) (k ) + δi2 ω˜ i(2) (k ) where

(1 )

ω˜ i (k ) 

minhi (σ (k ))=0

and

λi ( x ( k ) ) hi ( σ ( k ) )



(2 )

ω˜ i (k )

(65) are

nonlinear

and δi2 = maxhi (σ (k ))=0



functions

λi ( x ( k ) ) hi ( σ ( k ) )



satisfying

(1 )

(2 )

ω˜ i (k ) ∈ [0, 1], ω˜ i (k ) ∈ [0, 1]

and

δi1 =

. Combining (64) and (65) and using the same operation like

[12], the follow inequality is obtained s  s  s 

λi λ+j hl i jl =

i=1 j=1 l=1

s  s  2 

ω˜ i( p) (k )λ+j h2i δip i ji i=1 j=1 p=1

+

s−1  s  s  2  2 



t p hi hl λ+j ω ˜ l( ) (k )ω ˜ i( ) (k ) δip i jl + δlt l ji .

(66)

i=1 l=i+1 j=1 p=1 t=1

It can be seen that conditions (56) (57) guarantee that V(xk ) < 0. Using the similar proof procedure, We can obtain the same estimation of the attraction region as Theorem 1. The whole proof is completed.  Remark 4. Theorem 2 is stated as a convex feasibility problem. But in the simulation, we transform it to an optimization problem to make the estimation of the attraction region as big as possible. The idea is the same as [14]. Then, a symmetric matrix H is considered, satisfying the follow inequalities

H − P i ≥ 0,

i∈S

(67)

It turns to the follow convex optimization problem, which can be solved by LMI toolbox in Matlab.

min trace(H )

sub ject

to (51 ) − (58 )

and

(67 ).

(68)

Remark 5. In this paper, a new Lyapunov functional is constructed for nonlinear systems with two additive time-varying delays. More state information and delay information are considered in the proposed Lyapunov functional, so it has the potential to reduce conservativeness to some extent. It is noted that conservativeness is influenced by Lyapunov function, matrix inequality scaling technique, membership function information and so on. Therefore, to further reduce the conservativeness of the proposed method and to give a larger estimation of the attraction is the focus of future work. Remark 6. In the case of two additive delays, more state information and delay information are considered, so the number of decision variables in Theorem 1 and 2 is relatively large. Fortunately, Theorem 1 and 2 are solved in the framework of LMI and offline mode, so these complexities will not be too difficult. Compared with the existing literature [35–39], IT2 T-S fuzzy model, two additive delays and local stability are considered simultaneously, which solves the local control problem of nonlinear systems with two additive delays and parameter uncertainties for the first time. 5. Example studies Example 1. Consider the delayed system (1) with following parameters:

x (k + 1 ) =

2  i=1

λi (x(k ))[Ai x(k ) + Adi x(k − d1 (k ) − d2 (k ))]

(69)

14

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097 Table 1 Calculated AMUBs of d2 for various d2 . Method

d2

Theorem 1

3

5

8

10

12

22

30

8

12

18

23

28

47

59

Fig. 1. Trajectory of the open-loop system.

where

 A1 =



−0.291 0



1 −0.1 , A2 = 0.95 1







0 0.012 , Ad1 = −0.2 0



0.014 0.01 , Ad2 = 0.015 0.01



0 . 0.015

We calculate the admissible maximum upper bound of d2 for d2 ∈ {3, 5, 8, 10, 12, 22, 30}, d1 = 2, d1 = 1, −1 ≤ d1 (k ) ≤ 1 and −1 ≤ d2 (k ) ≤ 1, which is showed in Table 1. The admissible maximum upper bound of d2 is 59 when d2 is 30. Suppose that the time delay d1 (k) is the rand integer between 1 and 2, d2 (k) is the rand integer between 30 and 59, the initial state is x(0 ) = [0.25, 0.1], x(k ) = [0, 0], k = −d, −d + 1, . . . , −1 and the firing strength is defined as follow, in which the parameter γ ∈ [0, 1]. Fig. 1 shows that system (69) is asymptotically stable, which indicates the effectiveness of our proposed method.

λ1 (x2 (k ) ) =

0.2γ +0.8 − 0.8 sin (x2 (k ) ) , 2

λ2 (x2 (k ) ) = 1 − λ1 (x2 (k ) ).

(70)

Example 2. Consider the model car system in [40]. The mathematical equations of the system can be described as follows:

x1 ( k + 1 ) = x1 ( k ) + ud x1 ( k − d1 ( k ) − d2 ( k ) ) +

vTs l

tan (u(k ) )

(71)

x2 (k + 1 ) = x2 (k ) + vTs sin (x1 (k ) )

(72)

where x1 (k), x2 (k) denotes the angle of the car and the vertical position of the car, respectively. d1 (k) and d2 (k) are the delays. It is assumed that v ∈ [1, 2], ud = 0.12, Ts = 1(s ), and l = 2.8(m ). The objective is to control the car along the straight line(x2 (k ) = 0), while the states are in the region of valid. The model car system with uncertain parameter v is represented by the following IT2 T-S fuzzy model [12]: Rule i: IF x1 (k) is F1i , THEN



x(k + 1 ) = Ai x(k )+ Adi x(k − d1 (k ) − d2 (k ) ) + Bi u(k ) , x(k ) = φ (k ), k ∈ −d, 0

where



1 A1 = 0.01





0 1 , A2 = 1 2





0 0.12 , Ad1 = 1 0



i = 1, 2



0 0.12 , Ad2 = 0 0

(73)







F1i are IT2 fuzzy sets. The following upper and lower membership functions are considered:



uF 1 (x1 (k ) ) = 1

1−

sin (x1 (k ) ) , x1 ( k )

0,

x1 ( k ) = 0 x1 ( k ) = 0

uF 2 (x1 (k ) ) = 1

 sin (x

1 (k ) ) , 2x 1 ( k )

0.5,





0 0.5357 0.5357 , B1 = , B2 = . 0 0 0

x1 ( k ) = 0 . x1 ( k ) = 0

(74)

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

15

Fig. 2. Estimation of domain of attraction.

Fig. 3. Time-varying delay of Example 2.

 uF 1 ( x 1 ( k ) ) = 1

1−

sin (x1 (k ) ) , 2x 1 ( k )

0.5,

x1 ( k ) = 0 x1 ( k ) = 0

uF 2 (x1 (k ) ) =

 sin (x

1 (k ) ) , 2x 1 ( k )

1,

1

x1 ( k ) = 0 x1 ( k ) = 0

The IT2 fuzzy state feedback controller is described as follow: Rule i: IF x2 (k) is Gi1 , THEN u(k ) = Ki x(k ), i = 1, 2. In the case of controller, the lower and upper membership functions are chosen as:

hL1 (x2 (k ) ) = uG1 (x2 (k ) ) = 010.9 |x2 (k )| 1 hL2 (x2 (k ) ) = uG2 (x2 (k ) ) = 0.9 − 010.9 |x2 (k )| 1 ϑ 1 ( x2 ( k ) ) = ϑ 2 ( x2 ( k ) ) = 0.5.

hU1 (x2 (k ) ) = u¯ G1 (x2 (k ) ) = 010.9 |x2 (k )| + 0.1 1 hU2 (x2 (k ) ) = u¯ G2 (x2 (k ) ) = 1 − 010.9 |x2 (k )| 1

 = [1 0] and η = 1 The region of valid is modeled by (15) in this paper. The matrix and vector in (15) is set as H here. The estimation of domain of attraction is shown in Fig. 2. Three different points ( marks) in the domain of attraction and one point(∗ mark) outsides the region are selected to verify the effectiveness of our method. The initial condition is considered as follow: φ (k ) = 0, k = −d¯, . . . , −1. x(0 ) = [0.2890, 0.0076]T (a1 ); x(0 ) = [−0.3770, −0.0 0 03]T (a2 ); x(0 ) = [0.1567, −0.0084]T (a3 ); x(0 ) = [0.5482, 0.0302]T (a4 ). Time-variant delays are assumed d1 (k) ∈ [1 5], d2 (k) ∈ [5 10] and the delays satisfy −1 ≤ d1 (k ) ≤ 1, −1 ≤ d2 (k ) ≤ 1. The total delay d(k) is shown in Fig. 3. In these simulations, the nonlinear system (71) is employed in the close loop with control law (8). Solving the LMI of Theorem 2 with δ11 = 0, δ12 = 19.2994, δ21 = 0, δ22 = 20 and  = 1, the control gains can be obtained as follow



K1 = −2.0134



−0.0030



K2 = −2.0169



−0.0030 .

The state response trajectories are shown in Fig. 4. The trajectories starts from a1 , a2 , a3 converge asymptotically to origin and do not exceed the given region of valid. The trajectory starts from a4 is divergent. This demonstrates our method is effective.

16

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

Fig. 4. State responses of different initial conditions.

6. Conclusion In this paper, the local stability and stabilization problem is investigated in the case of discrete time IT2 T-S fuzzy system with two additive time-varying delay. By constructing an augmented Lyapunov–Krasovskii functional and applying the GFWM approach, a new delay-dependent stability criterion is derived in Theorem 1. Furthermore, a state-feedback controller that dose not share the same membership function as system model is obtained. The estimation of domain of attraction is also addressed by Lyapunov level sets. The simulation results show that the proposed method is effective. In the future work, more advanced Lyapunov functional and matrix inequality techniques should be further considered, so as to give a larger estimation of the domain of attraction. In the output feedback framework, the local stabilization methods for nonlinear systems with parameter uncertainties and time delays need to be further studied. Declaration of Competing Interest No conflict of interest exits in the submission of this manuscript. Acknowledgment This work is supported by the National Natural Science Foundation of China (61703291). References [1] Li XJ, Yang GH. Fault detection in finite frequency domain for Takagi–Sugeno fuzzy systems with sensor faults. IEEE Trans Cybern 2013;44(8):1446–58. [2] Ji WQ, Zhang HT, Qiu JB. Fuzzy affine model-based output feedback controller design for nonlinear impulsive systems. Commun Nonlinear Sci Numer Simul 2019;79:1–12. [3] Xiang W, Xiao J, Iqbal MN. H-Infinite control for switched fuzzy systems via dynamic output feedback: hybrid and switched approaches. Commun Nonlinear Sci Numer Simul 2013;18(6):1499–514. [4] Silva LFP, Leite VJS, Castelan EB, Lopes AND. Stability and controller design for t-s fuzzy discrete-time systems with time-varying delay in the state. In: 2018 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE); 2018. p. 1–7. [5] Xie X, Yue D, Zhang H, Peng C. Control synthesis of discrete-time t-s fuzzy systems: reducing the conservatism whilst alleviating the computational burden. IEEE Trans Cybern 2017;47(9):2480–91. [6] Su X, Xia F, Liu J, Wu L. Event-triggered fuzzy control of nonlinear systems with its application to inverted pendulum systems. Automatica 2018;94:236–48.

T. Zhao, C. Chen and S. Dian / Commun Nonlinear Sci Numer Simulat 83 (2020) 105097

17

[7] Chang X, Liu Q, Wang Y, Xiong J. Fuzzy peak-to-peak filtering for networked nonlinear systems with multipath data packet dropouts. IEEE Trans Fuzzy Syst 2018;27(3):436–46. [8] Li H, Gao Y, Shi P, Lam H. Observer-based fault detection for nonlinear systems with sensor fault and limited communication capacity. IEEE Trans Automat Contr 2016;61(9):2745–51. [9] Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning-i. Inf Sci 1975;8(3):199–249. [10] Zhao T, Liu J, Dian S. Finite-time control for interval type-2 fuzzy time-delay systems with norm-bounded uncertainties and limited communication capacity. Inf Sci 2019;483:153–73. [11] Zhao T, Dian S. Delay-dependent stabilization of discrete-time interval type-2 t-s fuzzy systems with time-varying delay. J Franklin Inst 2017;354(3):1542–67. [12] Zhao T, Dian S. State feedback control for interval type-2 fuzzy systems with time-varying delay and unreliable communication links. IEEE Trans Fuzzy Syst 2018;26(2):951–66. [13] Li H, Wu C, Shi P, Gao Y. Control of nonlinear networked systems with packet dropouts: interval type-2 fuzzy model-based approach. IEEE Trans Cybern 2015;45(11):2378–89. [14] Silva LFP, Leite VJS, Castelan EB, Klug M. Local stabilization of time-delay nonlinear discrete-time systems using Takagi–Sugeno models and convex optimization. Math Probl Eng 2014;2014. 587510,10 pages. [15] Wang L, Lam H. Local stabilization for continuous-time Takagi–Sugeno fuzzy systems with time delay. IEEE Trans Fuzzy Syst 2018;26(1):379–85. [16] Lee DH, Kim DW. Relaxed LMI conditions for local stability and local stabilization of continuous-time Takagi–Sugeno fuzzy systems. IEEE Trans Cybern 2014;44(3):394–405. [17] Lee DH, Joo YH. On the generalized local stability and local stabilization conditions for discrete-time Takagi–Sugeno fuzzy systems. IEEE Trans Fuzzy Syst 2014;22(6):1654–68. [18] Wang L, Hu L. Local stabilization for continuous-time fuzzy systems with time-varying delay. In: 2017 Chinese Automation Congress (CAC); 2017. p. 2328–32. [19] Zhang C, He Y, Jiang L, Wu QH, Wu M. Delay-dependent stability criteria for generalized neural networks with two delay components. IEEE Trans Neural Netw Learn Syst 2014;25(7):1263–76. [20] Park M, Kwon O. Stability and stabilization of discrete-time t-s fuzzy systems with time-varying delay via Cauchy–Schwartz-based summation inequality. IEEE Trans Fuzzy Syst 2017;25(1):128–40. [21] Kwon OM, Park MJ, Park JH, Lee SM, Cha EJ. New criteria on delay-dependent stability for discrete-time neural networks with time-varying delays. Neurocomputing 2013;121:185–94. [22] Zhang C, He Y, Jiang L, Wu M, Zeng H. Delay-variation-dependent stability of delayed discrete-time systems. IEEE Trans Automat Contr 2016;61(9):2663–9. [23] Shao H, Han Q. New delay-dependent stability criteria for neural networks with two additive time-varying delay components. IEEE Trans Neural Networks 2011;22(5):812–18. [24] Zhang X, Han Q, Wang Z, Zhang B. Neuronal state estimation for neural networks with two additive time-varying delay components. IEEE Trans Cybern 2017;47(10):3184–94. [25] Lin WJ, He Y, Zhang CK, Long F, Wu M. Dissipativity analysis for neural networks with two-delay components using an extended reciprocally convex matrix inequality. Inf Sci 2018;450:169–81. [26] Lam J, Gao H, Wang C. Stability analysis for continuous systems with two additive time-varying delay components. Syst Control Lett 2007;56(1):16–24. [27] Ko KS, Lee WI, Park P, Sung DK. Delays-dependent region partitioning approach for stability criterion of linear systems with multiple time-varying delays. Automatica 2018;87:389–94. [28] Wang L, Lam H. A new approach to stability and stabilization analysis for continuous-time Takagi–Sugeno fuzzy systems with time delay. IEEE Trans Fuzzy Syst 2018;26(4):2460–5. [29] Park P, Ko JW, Jeong C. Reciprocally convex approach to stability of systems with time-varying delays. Automatica 2011;47(1):235–8. [30] Chen B, Long F, Zhang C, He Y, Xiao H. Stability analysis of discrete t-s fuzzy systems with time-varying delay using an extended reciprocally convex matrix inequality. In: 2018 37th Chinese Control Conference (CCC); 2018. p. 1393–8. [31] Xu H, Zhang C, Jiang L, Smith J. Stability analysis of linear systems with two additive time-varying delays via delay-product-type lyapunov functional. Appl Math Model 2017;45:955–64. [32] Zhang C, He Y, Jiang L, Wu M, Wang Q. An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay. Automatica 2017;85:481–5. [33] Wu M, He Y, She J. New delay-dependent stability criteria and stabilizing method for neutral systems. IEEE Trans Automat Contr 2004;49(12):2266–71. [34] Lam HK, Leung FHF. Stability analysis of fuzzy control systems subject to uncertain grades of membership. IEEE Trans Syst Man CybernPart B (Cybernetics) 2005;35(6):1322–5. [35] Du P, Liang H, Zhao S, Ahn CK. Neural-based decentralized adaptive finite-time control for nonlinear large-scale systems with time-varying output constraints. IEEE Trans Syst Man Cybern: Syst 2019. doi:10.1109/TSMC.2019.2918351. [36] Zhang L, Liang H, Sun Y, Ahn CK. Adaptive event-triggered fault detection scheme for semi-Markovian jump systems with output quantization. IEEE Trans Syst Man Cybern: Syst 2019. doi:10.1109/TSMC.2019.2912846. [37] Zhang L, Lam H, Sun Y, Liang H. Fault detection for fuzzy semi-markov jump systems based on interval type-2 fuzzy approach. IEEE Trans Fuzzy Syst 2019. doi:10.1109/TFUZZ.2019.2936333. [38] Xie X, Yue D, Peng C. Relaxed real-time scheduling stabilization of discrete-time Takagi–Sugeno fuzzy systems via an alterable-weights-based ranking switching mechanism. IEEE Trans Fuzzy Syst 2018;26(6):3808–19. [39] Xie X, Yue D, Park JH. Observer-based state estimation of discrete-time fuzzy systems based on a joint switching mechanism for adjacent instants. IEEE Trans Cybern 2019. doi:10.1109/TCYB.2019.2917929. [40] Chang W, Sun C. Constrained fuzzy controller design of discrete Takagi–Sugeno fuzzy models. Fuzzy Sets Syst 2003;133(1):37–55.