Co-design method for H ∞ control of quantized T-S fuzzy system over the networked system

1 Introduction

To appropriately model the physical systems it is appealing to describe both dynamic and static behaviors simultaneously. Fuzzy systems attained sufficient attention from the researchers in the past decades. In contrast to conventional state-space representation fuzzy systems describes, which comes from [1,2, 1,2], the practical demonstration of real-world applications which includes large-scale systems, electric power systems [3], tracking control [4] and economic systems. Fuzzy systems whose dynamic part is represented by differential equations and relationship between different sections of the systems are described by algebraic equations [5–8].

It is known that the quantization phenomenon is commonly observed in real-world applications and usually deteriorates the stability of a control system [9, 10] {and observer-based controller in [11, 12]. Under the circumstances of a communication channel, in the closed-loop circuit, a real-valued data of the controller and ones of the model are always projected into some piecewise-constant signals before broadcast. In some kinds of literature, some authors have been investigated in this direction, for example [13, 14]. In [14], for T-S fuzzy systems, the state-feedback controller was designed. Therefore, in the practical and realistic example, the state-feedback consists of some critical limitations that the states of the system can not be attained on account of method or are always difficult to get [15–17]. Besides, authors have been designed as an effective algorithm to overcomes the nonlinearities via an adaptive fuzzy controller in [18–20]. Furthermore, the measurements of the sensor before the transmission of signal and the output of the fuzzy controller require new quantized techniques. Therefore, motivated by the above-mentioned work, we want to design a fuzzy feedback controller with network control systems.

Recently Network control systems are getting too much attention in modern research [21, 22]. On the other hand, communication resource saving is also a significant issue for networked systems because the bandwidth resource becomes more and more limited as the complexity of the network increase. In NCS the data from the sensor to a controller and from the controller to the actuator is transferred through a common network [23, 24]. Such type of mechanism is preferred in distributed control of modern industries including large power systems [25]. Comparing with traditional control NCS is unpredictable due to time-varying delays in network and data dropouts. Therefore match premise requirement is not fulfilled in NCS [26]. As mentioned above in Network controlled applications time-varying delays occur due to limited capacities of data analysis and transmission between different sections of plants over a common network. So delay is considered as a key source of instability [27–30]. Currently, T-S fuzzy systems got an extension to handle the problem synchronization for networked control systems [31, 32]. The stability of such type systems is more challenging than normal regular systems because of networked induced delay. In this regard, a few result is obtained for details sees [35–37]. But still is open research for T-S fuzzy systems which is one of the motivation.

In this research work, the H_∞ fuzzy controller design under the stochastic nonlinearity of networked systems with energy constrain is investigated. The main aim of this article is to manage the network burden of a network-based controlled plant. To accomplish the goal, we introduced two methods to increase the network rate and guarantee communication reliability. Firstly, a new time-event triggered mechanism is established in such a way that the signal packages are communicated according to the law of event trigger and time trigger toggling to each other. Also, the co-design method can not only eradicate a huge number of irrelevant data packages but also keep some of the common information, which keeps the stability of the system and tackles the channel burden. After this, we prosed the logarithmic quantizer to further decrease the magnitude of the signal package. Based on the Lyapunov function method with stochastic analysis methodology and fuzzy controller gains are obtained by solving the linear matrix inequalities. In addition, this paper considers the state feedback H_∞ control design problem for the T-S fuzzy system by using the co-design method. The main contributions of this paper can be highlighted as follows:

Different from the existing literature [38, 39], a new hybrid-time-event triggered communication mechanism is constructed such that quantized data is chosen for communication. Compared with the existing theory, co-design method not only eliminates the communication burden efficiently but also can increase the performance of the system.

The rest part of the paper is organized as follows. In Section 2, we describe the problem formulation. Section 3 presents the sufficient conditions for co-design method design scheme for the underlying system. Section 4 shows the fuzzy controller designing. Numerical examples and simulation results are provided in Section 5. The conclusion of this paper is made in Section 6.

Notations: Throughout this paper, a real symmetric matrix X > 0 (X ≥ 0) means that X is positive definite (respectively, positive semi-definite) matrix; X ^T represents transpose of the matrix X; R ^{n _x} and R^{n _x ×m _y} stand for n _x -dimensional Euclidean space and set of all n _x  × m _y real matrices; In denotes the n-dimensional identity matrix; diag {·· ·} stands for a block diagonal matrix. The notation ★ is used as an ellipsis for terms that are induced by symmetry; L 2 [ 0 , ∞ ) is the space of square-integrable vector function over [0, ∞); ∣· ∣ denotes the Euclidean norm for vectors. Matrix dimensions, if not explicitly stated are assumed to be compatible for algebraic operations.

2 System description and problem formulation

Consider a continues class of T-S fuzzy system with network control system described as: Plant Rule i: IF ϑ₁ is M 1 i and...and ϑ _g is M g i , THEN { x ̇ ( t ) = A i x ( t ) + B i u ( t ) + g ( t , x ( t ) ) + D i ω ( t ) y ( t ) = C i x ( t ) , i = 1 , 2 , · · · , r(1) Where r is the number of IF-THEN rules, ϑ₁ (t) , ϑ₂ (t) , ·· · , ϑ _g  (t) are the premise variables, x (t) ∈ R ^{n _x} , y (t) ∈ R ^{n _y} and u (t) ∈ R ^{n _u} represents the state variable, output vector and control input respectively and ω (t) ∈ R ^{n _ω} represents the disturbance signal that belongs to L 2 [ 0 , ∞ ] , g (t, x (t)) is nonlinear function. A i , B i , C i and D i are the system matrices with proper dimensions. By using center-average defuzzifier, product interference and singleton fuzzifier, the obtained fuzzy system (1) is inferred as follows: { x ̇ ( t ) = A ( μ ) x ( t ) + B ( μ ) u ( t ) + g ( t , x ( t ) ) + D ( μ ) ω ( t ) y ( t ) = C ( μ ) x ( t )(2) where A ( μ ) = Σ i = 1 r h i A i , B ( μ ) = Σ i = 1 r h i B i , C ( μ ) = Σ i = 1 r h i C i and D ( μ ) = Σ i = 1 r h i D i . Abbreviation of h _i stands for h _i  (ϑ (t)). h i ( ϑ ( t ) ) = ξ i ( ϑ ( t ) ) ∑ i = 1 r ξ i ( ϑ ( t ) ) , ξ i ( ϑ ( t ) ) = ∏ j = 1 g M j i ( ϑ j ( t ) ) shows the grade membership value of ϑ _j  (t) in M j i and h _i  (ϑ (t)) fulfills h _i  (ϑ (t)) >0, ∑ i = 1 r h i ( ϑ ( t ) ) = 1 . For our convince and simplicity, we present h _i instead of h _i  (ϑ (t)).

In Fig. 1, we constructed the networked control systems, which contain the continuous T-S fuzzy plant, fuzzy controller with the quantization effect, hybrid-triggered communication mechanism and an actuator. This framework presents a system in which the actuation is cheap for a heavy transmission but transmission of sensor measurements to the controller is expensive. To encounter such a problem in a networked environment, it is compulsory that signal must measure before the transmission from sensor to controller. Main theme of this paper is to eliminate the communication burden as well as to increase the system performance. In the following, we concentrate on designing a hybrid–triggered state feedback H_∞ fuzzy controller for Eq. (1) with the following form: Controller Rule i: IF ϑ₁ is M 1 i and...and ϑ _g is M g i , THEN u ( t ) = K i x ( t ) , i = 1 , 2 , · · · , r(3) where K i is the controller gain, which is determined to be later. Then the overall fuzzy state feedback controller can be expressed as: u ( t ) = ∑ i = 1 r h i K i x ( t )(4) In the next sections, the main mathematical methods of the networked systems are given step by step. In Section 2.1, we considering the stochastic nonlinearities, while hybrid–triggered mechanism presents in Section 2.2 and quantized measurement formulated in Section 2.3.

OPEN IN VIEWER

Fig. 1

The structure of hybrid-triggered NCSs with quantization.

2.2 Hybrid-triggered communication mechanism

In NCSs, energy constraint is still changeling for the researchers. Before transmission, the local measurements are scheduled according to the need and is one of the solutions to this problem. Event-trigger control problems have been investigated to sent the next transmission execution time for saving sensor data transmission energy in [41]. On the other hand, event-triggered mechanism neglects discards some important data packages, which cause the poor performance of the system. In the cop of efficient bandwidth utilization and increase the system performance, we introduced the co-design method which can be called event-triggered and time-triggered network toggle randomly according to their probability distribution. A descriptive example of discrete co-design method is shown in Fig. 2.

OPEN IN VIEWER

Fig. 2

Timing diagram of hybrid-triggered NCSs.

Assumption 1. In the communication network the sensor is time triggered with sampling rate h, and the sampled state is determined by set {0, h, 2h, ·· · , kh, ·· ·}, where k ∈ N. Whereas the controller and actuators are hybrid-triggered. The successful transmitted instants are represented by {0, i₁h, i₂h, ·· · , i _k h, ·· ·}, where k ∈ N.

Assumption 2. The event triggered communication is proposed to be implemented from sensor to controller. The network induced delay is taken into consideration with some upper bound. τ i k ∈ [ d m , d ¯ ] , where 0 ≤ d m < d ¯ .

Assumption 3. The data packets are to be time stamped and assumed to be transmitted without any packet loss.

In order to explain the behavior of switch effect of co-design method, first we have to introduce a stochastic variable sequence β (t) with Bernoulli distribution, which defined as follows: β ( t ) = { 1 , if event - triggered happened 0 , if time - triggered happened(10) The expectation of β (t) can be given as: E { β ( t ) } = prob { β ( t ) = 1 } = β(11) The value of 0 ≤ β ≤ 1 is constant.

As we have already discussed in the literature that we are considering a common communication channel for the measured output. Due to the common communication channel, we have a limitation of bandwidth. To deal with this issue an event-triggered communication-based transmitter is inserted between the plant and the controller. In order to enhance the communication performance. In addition, to deciding the transmission of the current measured data y (t) to the observer from the ETC transmitter is based on logic function [25]. Network-induced delays are considered between the sensor and controller in our proposed scheme. The ETC is designed to handle the event generated by the instant (i _k h). The current transmitted sampled data is denoted by i _k h. The error between the current sampled data and the latest transmitted data can be expressed as: e k ( t ) = y ( z k h ) - y ( i k h )(12) Where z _k h = i _k h + mh, m ∈ ℕ and z _k h denotes the sampling instant between two simultaneous instants. The future transmission instants are based on ETC scheme which is expressed as follows. i k + 1 h = i k h + min { kh ∣ e k ( t ) T O e k ( t ) } > ρ y T ( i k h ) O y ( i k h )(13) O > 0 and 0 < ρ < 1 represents the conditions for triggering will be determined later. From the above mentioned conditions, it is clear that the next transmitting instants are experienced by two important aspects. Which are triggering parameters and output y (i _k h). From (13) it is seen that the transmitted instant {i _k h ∣ i _k  ∈ N} yielding to sampled instants denoted as {  _k h ∣   _k  ∈ N} . i₀h = 0 represents the initial transmitting instant. In this paper we consider the network induced transmission delays, d _{i k} and d _{i k+1} at transmitting instants i _k h and i_k+1h respectively. Then i _k  + d _{i k} represents the instant when the transmitted signal arrives at zero-order-holder (ZOH).

It is examined that y ˜ ( t ) retains the value of y (i _k h) with interval [i _k h + d _{i k} , i_k+1h + d _{i k+1} ) influenced by ZOH. y ˜ ( t ) = y ( i k h ) , t ∈ [ i k h + d i k , i k + 1 h + d i k + 1 )(14) The subset used to call the holding zone Ξ of ZOH [25]: Ξ = [ i k h + d i k , i k + 1 h + d i k + 1 ) = ⋃ p = 0 o l Ξ p(15) Where Ξ _p  = [i _k h + d _{i k} , i_k+1h + d _{i k+1} ), p = 0, 1, 2, ·· · , o _l , o _l  = i_k+1 - i _k  - 1.

The network delay could be defined as d (t) = t - i _k h, satisfying d m ≤ d i k ≤ h + d ¯ ≐ d M . Based on above mentioned analysis the original input of the system could be expressed as: y ˇ ( t ) = y ( i k h ) = y ( t - d ( t ) ) - e k ( t )(16) For above summary, we implement the co-design method with two channels (time trigger and event trigger), the output y (i _k h) expression can be defined as: y ˇ ( t ) = β ( t ) [ y ( t - d ( t ) ) - e k ( t ) ] + [ 1 - β ( t ) ] y ( t - d ( t ) ) = y ( t - d ( t ) ) - β ( t ) e k ( t )(17)

Remark 1. The time-triggered communication scheme be able to be considered as a special illustration of event-triggered communication scheme, therefore the constraint (13) at all times holds owing to e ^k  (t) =0 under time-triggered scheme.

Remark 2. Certainty, the co-design method is a random event-triggered scheme, in which the state output data can choose a channel according to a specified probability. Sometime there is less information to be transmitted. In such environment if we transmit the data through event-triggered scheme, may degrades the network resource management. By considering co-design method in network control systems, efficient utilization of bandwidth could be ensured along with prevention from significant data loss.

The arrangement of this part is as follows. In Section 3.1, a process of establishing the state feedback fuzzy controller (4) is developed such that the closed loop system (25) is stochastically stable. Then, the fuzzy controller design for two special cases is discussed in Section 4.

Theorem 1. For the given scalars which satisfying d _m  < d (t) < d _M , α, β and γ > 0, augmented system (25) is stochastically stable with the H_∞ performance index under co-design method with quantization, if there exist matrices P > 0, Q_ℓ > 0, R_ℓ > 0, ℕ , O > 0 , M and S , ℓ = 1, 2, such that the following inequalities hold [ Ψ ij 11 + ϒ + ϒ T ★ ★ Ψ ij 21 Ψ ij 22 ★ Ψ ij 31 s 0 - R 2 ] < 0 , s = I , II(28) where Ψ ij 11 = [ ( 1 , 1 ) = ( S 1 A i + A i T S 1 T + ∑ g = 1 2 ( Q g ) ) - R 1 ] ( 2 , 1 ) = R 1 , ( 2 , 2 ) = - Q 1 - R 1 ( 3 , 1 ) = S 2 A i + ( I + ∇ q j ) T K j T B i T S 1 T ( 3 , 3 ) = ( I + ∇ q j ) T K j T B i T S 2 T + S 2 B i K j ( I + ∇ q j ) + H T ρ C i T O C i H , ( 4 , 4 ) = - Q 2 ( 5 , 1 ) = β ( I + ∇ q j ) T K j T B i T S 1 T + S 3 A i ( 5 , 3 ) = β ( I + ∇ q j ) T K j T B i T S 2 T + S 3 B i K j ( I + ∇ q j ) ( 5 , 5 ) = β ( I + ∇ q j ) T K j T B i T S 3 T + β S 3 B i K j ( I + ∇ q j ) - O ( 6 , 1 ) = α S 1 T I , ( 6 , 3 ) = α S 2 T I ( 6 , 5 ) = α S 3 T I , ( 6 , 6 ) = - α P ( 7 , 1 ) = ( 1 - α ) S 1 T I , ( 7 , 3 ) = ( 1 - α ) S 2 T I ( 7 , 5 ) = ( 1 - α ) S 3 T I , ( 7 , 7 ) = - ( 1 - α ) P ( 8 , 1 ) = P - S 1 T I + S 4 A i ( 8 , 3 ) = - S 2 T I + S 4 B i K j ( I + ∇ q j ) ( 8 , 5 ) = - S 3 T I + β S 4 B i K j ( I + ∇ q j ) , ( 8 , 6 ) = α S 4 I ( 8 , 7 ) = ( 1 - α ) S 4 I ( 8 , 8 ) = ( d m 2 R 1 + ( d M - d m ) R 2 ) - S 4 I - S 4 T I ( 9 , 1 ) = D i T S 1 T , ( 9 , 3 ) = D i T S 2 T ( 9 , 5 ) = D i T S 3 T , ( 9 , 8 ) = D i T S 4 T , ( 9 , 9 ) = - γ 2 I ϒ = [ 0 ℕ - ℕ + M - M 0 0 0 0 0 ] Ψ ij 22 = diag { - P , - P , - I } , Ψ ij 31 I = ( d M - d m ) ℕ T Ψ ij 31 II = ( d M - d m ) M T Ψ ij 21 = [ α P G 0 0 0 0 0 0 0 0 ( 1 - α ) P H 0 0 0 0 0 0 0 0 C i 0 0 0 0 0 0 0 0 ]

Proof. First, we implement the Lyapunov-Krasovskii function candidate for system (25)

V ( x t , i , t ) = ∑ l = 1 3 V l ( t )(29) where

V 1 ( x t , i , t ) = x ( t ) T P x ( t ) V 2 ( x t , i , t ) = ∫ t - d m t x ( s ) T Q 1 x ( s ) ds + ∫ t - d M t x ( s ) T Q 2 x ( s ) ds V 3 ( x t , i , t ) = d m ∫ t - d m t ∫ s t x ̇ ( v ) T R 1 x ̇ ( v ) dvds + ∫ t - d M t - d m ∫ s t x ̇ ( v ) T R 2 x ̇ ( v ) dvds P > 0, Q_ℓ > 0, R_ℓ > 0 (ℓ =1, 2). Then, by implementing similar techniques to those in [53], we have to expectation for each V _p  (t) (p = 1, 2, 3) E { V ̇ 1 ( x t , i , t ) } = 2 E { x ̇ ( t ) T } P x ( t )(30) E { V ̇ 2 ( x t , i , t ) } = x ( t ) T ∑ ℓ = 1 2 Q ℓ x ( t ) - x T ( t - d m ) Q 1 x ( t - d m ) - x T ( t - d M ) Q 1 x ( t - d M )(31) E { V ̇ 3 ( x t , i , t ) } = d m ∫ t - d m t x ̇ ( v ) T R 1 x ̇ ( v ) dv + ∫ t - d M t - d m x ̇ ( v ) T R 2 x ̇ ( v ) dv + E { d m 2 x ̇ ( t ) T R 1 x ̇ ( t ) + ( d M - d m ) x ̇ ( t ) T R 2 x ̇ ( t ) }(32) Taking the expectation of (25), we can get:

A i x ( t ) + B i K j ( I + ∇ q j ) [ x ( t - d ( t ) ) - β e k ( t ) ] + α g 1 ( t , x ( t ) ) + ( 1 - α ) g 2 ( t , x ( t ) ) + D i ω ( t ) ) - E { x ̇ T ( t ) } = 0(33) Now, characterize the augmented matrix

Ξ ( t ) = col [ x ( t ) , x ( t - d m ) , x ( t - d ( t ) ) , x ( t - d M ) , e k ( t ) , g 1 ( t , x ( t ) ) , g 2 ( t , x ( t ) ) , E { x ̇ ( t ) } , ω ( t ) ]

Γ = [ A i 0 B i K j ( I + ∇ q j ) 0 - β B i K j ( I + ∇ q j ) α I ( 1 - α ) I - I D i ]

So, rewrite the eq. (33) as:

Γ Ξ ( t ) = 0(34) According to the (34):

Ξ T ( t ) ( S Γ + Γ T S T ) Ξ ( t ) = 0(35) where S T = [ S 1 T 0 S 2 T 0 S 3 T 0 0 S 4 T 0 ] Integrating the (30)–(35) with introducing the slack matrices under H_∞ control performance index, we get:

E { V ̇ ( x t , i , t ) } + y T ( t ) y ( t ) - γ 2 ω T ( t ) ω ( t ) ≤ 2 E { x ̇ ( t ) T } P x ( t ) + x ( t ) T ( ∑ ℓ = 1 2 Q ℓ ) x ( t ) - x T ( t - d m ) Q 1 x ( t - d m ) - x T ( t - d M ) Q 1 x ( t - d M ) + d m ∫ t - d m t x ̇ ( v ) T R 1 x ̇ ( v ) dv + ∫ t - d M t - d m x ̇ ( v ) T R 2 x ̇ ( v ) dv + d m 2 E { x ̇ ( t ) T R 1 x ̇ ( t ) } + ( d M - d m ) E { x ̇ ( t ) T R 2 x ̇ ( t ) } + 2 Ξ T ( t ) ℕ [ x ( t - d m ) - x ( t - d ( t ) ) - ∫ t - d ( t ) t - d m x ̇ ( s ) ds ] + 2 Ξ T ( t ) M [ x ( t - d ( t ) ) - x ( t - d M ) - ∫ t - d m t - d ( t ) x ̇ ( s ) ds ] + Ξ T ( t ) ( S Γ + Γ T S T ) Ξ ( t ) + y T ( t ) y ( t ) - γ 2 ω T ( t ) ω ( t ) + e k ( t ) T O e k ( t ) - e k ( t ) T O e k ( t )(36)

where ℕ T = [ 0 ℕ 1 T ℕ 2 T 0 0 0 0 0 0 ] M T = [ 0 0 M 1 T M 2 T 0 0 0 0 0 ] At that point, we use the definition in (7) and (8) ( 7 ) ⇒ α x T ( t ) G T ( t ) P G ( t ) - α g 1 T ( t , x ( t ) ) P g 1 ( t , x ( t ) ) ≥ 0(37) ( 8 ) ⇒ ( 1 - α ) x T ( t ) H T ( t ) P H ( t ) - ( 1 - α ) g 2 T ( t , x ( t ) ) P g 2 ( t , x ( t ) ) ≥ 0(38) Recalling the triggering condition (6),  t ∈ [i _k h + d _{i k} , i_k+1h + d _{i k+1} ) we have: e k ( t ) T O e k ( t ) ≤ ϱ y ( t - d ( t ) ) T O y ( t - d ( t ) )(39) which is equivalent to

y ( t - d ( t ) ) T O y ( t - d ( t ) ) = x ( t - d ( t ) ) T H T C i T O C i H x ( t - d ( t ) )(40) Recalling the Lemma of [46], which satisfy the following inequalities

- d m ∫ t - d m t x ̇ ( v ) T R 1 x ̇ ( v ) dv ≤ [ x ( t ) x ( t - d m ) ] T [ - R 1 R 1 R 1 - R 1 ] [ x ( t ) x ( t - d m ) ](41) Further more:

- 2 Ξ ( t ) ℕ ∫ t - d ( t ) t - d m x ̇ ( s ) ds ≤ ( d ( t ) - d m ) Ξ T ( t ) ℕ R 2 - 1 ℕ T Ξ ( t ) + ∫ t - d ( t ) t - d m x ̇ ( s ) R 2 x ( s ) ds(42)

- 2 Ξ ( t ) M ∫ t - d M t - d ( t ) x ̇ ( s ) ds ≤ ( d m - d ( t ) ) Ξ T ( t ) M R 2 - 1 M T Ξ ( t ) + ∫ t - d m t - d ( t ) x ̇ ( s ) R 2 x ( s ) ds(43) Combing the (36)–(43):

E { V ̇ ( x t , i , t ) } + y T ( t ) y ( t ) - γ 2 ω T ( t ) ω ( t ) ≤ 2 E { x ̇ ( t ) T } P x ( t ) + x ( t ) T ( ∑ ℓ = 1 2 Q ℓ ) x ( t ) - x T ( t - d m ) Q 1 x ( t - d m ) - x T ( t - d M ) Q 1 x ( t - d M ) + [ x ( t ) x ( t - d m ) ] T [ - R 1 R 1 R 1 - R 1 ] [ x ( t ) x ( t - d m ) ] + 2 Ξ ( t ) ℕ [ x ( t - d m ) - x ( t - d ( t ) ) ] + 2 Ξ ( t ) M [ x ( t - d ( t ) ) - x ( t - d M ) ] + d m 2 E { x ̇ ( t ) T R 1 x ̇ ( t ) } + ( d M - d m ) E { x ̇ ( t ) T R 2 x ̇ ( t ) } + ( d ( t ) - d m ) Ξ T ( t ) ℕ R 2 - 1 ℕ T Ξ ( t ) + ( d M - d ( t ) ) Ξ T ( t ) M R 2 - 1 M T Ξ ( t ) + α x T ( t ) G T ( t ) P G ( t ) - α g 1 T ( t , x ( t ) ) P g 1 ( t , x ( t ) ) + ( 1 - α ) x T ( t ) H T ( t ) P H ( t ) - ( 1 - α ) g 2 T ( t , x ( t ) ) P g 2 ( t , x ( t ) ) + Ξ T ( t ) ( S Γ + Γ T S T ) Ξ ( t ) + y T ( t ) y ( t ) + x ( t - d ( t ) ) T H T ρ C i T O C i H x ( t - d ( t ) ) - e k ( t ) T O e k ( t ) - γ 2 ω T ( t ) ω ( t )(44) By implementing the well–known Schur complement with Lemma 1 of [46], one can easily attained: E { V ̇ ( x t , i , t ) } + y T ( t ) y ( t ) - γ 2 ω T ( t ) ω ( t ) < 0(45) According to the zero-initial condition with integrate the (45) from 0 → t, and let t→ ∞, we can easily get: ∫ 0 + ∞ y T ( t ) y ( t ) dt ≤ ∫ 0 + ∞ γ 2 ω T ( t ) ω ( t ) dt < 0(46) which yields ∥ y ( t ) ∥ 2 ≤ ∥ ω ( t ) ∥ 2 It is well known, the remaining part of the proof is similar to those in [48, 49] and here, we omitted for avoid the complication.

Now, we are going to prove the stochastically stability of the system (25), with condition ω (t) =0. For this, first we have to define new function:

M ( t ) = e λ t V ( t )(47) Taking the derivative

M ̇ ( t ) = λ e λ t V ( t ) + e λ t V ̇ ( t )(48) From the above equation, we can conclude that:

E { M ( t ) } - E { M ( 0 ) } = ∫ 0 t λ e λ s E { V ( s ) } ds + ∫ 0 t e λ s E { V ̇ ( s ) ds }(49) We can see that there exit a positive number ɛ, for such that t ≥ 0

E { V ( t ) } ≤ ɛ sup ︸ d m < s < d M { ∥ φ ( s ) ∥ 2 } e - λ t(50) Therefore, V (t) ≥ λ _min  (P) x ^T  (t) x (t), for t ≥ 0, the equality appeared in above can be shown as:

E { V ( t ) } ≤ ζ sup ︸ d m < s < d M { ∥ φ ( s ) ∥ 2 } e - λ t(51) where definition of ζ = ɛ λ min ( P ) . So from the Definition 1, the system (25) is stochastically stable. The proof is complete. □

Remark 3. Sufficient condition is derived in Theorem 1. It is worth mentioning that in inequalities, there is some nonlinear term like K j ∇ q j from Eq. (28) and we cannot be handled these LMIs directly. For this, we have to use some common method to tackle the K j ∇ q j . Based upon Theorem 1, the next theorem will proceed.

For system (25) the main results for the solvability of T-S fuzzy based controller is presented as bellows:

Theorem 2. For the given scalars which satisfying d _m  < d (t) < d _M , α, β and γ > 0, augmented system (25) is stochastically stable with the H_∞ performance index under co-design method with quantization, if there exist matrices P > 0, Q_ℓ > 0, R_ℓ > 0, O > 0 , ℕ , M and S , ℓ=1, 2, and ɛ > 0 such that the following inequalities hold [ Ψ ¯ ij 11 + ϒ + ϒ T ★ ★ ★ Ψ ij 21 Ψ ij 22 ★ ★ Ψ ij 31 s 0 - R 2 ★ Ψ ij 41 0 0 Ψ ij 44 ] < 0 , s = I , II(52) where mostly parameters are same like Ψ ij 21 , Ψ ij 22 , Ψ ij 31 s , ϒ, (1, 1), (8, 1), (8, 8) etc. as mentioned in Theorem 1. Note:(We also mentioned change parameters in this theorem.) Ψ ¯ ij 11 = ( 3 , 1 ) = S 2 A i + K j T B i T S 1 T ( 3 , 3 ) = K j T B i T S 2 T + S 2 B i K j + H T ρ C i T O C i H ( 5 , 1 ) = β K j T B i T S 1 T + S 3 A i ( 5 , 3 ) = β K j T B i T S 2 T + S 3 B i K j ( 5 , 5 ) = β K j T B i T S 3 T + S 3 B i K j - O ( 8 , 3 ) = - S 2 T I + S 4 B i K j ( 8 , 5 ) = - S 3 T I + S 4 B i K j , Ψ ij 41 = [ U B T ɛ U K T ] T Ψ ij 44 = diag { - ɛ ρ g - 2 I 1 , - ɛ I 1 } U B = [ B i T S 1 T 0 B i T S 2 T 0 B i T S 3 T 0 0 B i T S 4 T 0 ] U K = [ 0 0 K i 0 β K i 0 0 0 0 ] Proof. The matrix Ψ ij 11 appeared in Eq. (28) can be redraft as: Ψ ij 11 = Ψ ¯ ij 11 + γ K T ∇ q j γ B T + γ B T ∇ q j γ K T(53) Using Lemma 3 of [47], for any ɛ > 0. Then the following inequalities can be attained Ψ ij 11 = Ψ ¯ ij 11 + ɛ - 1 γ B T ∇ q j 2 γ B T + ɛ γ K T γ K(54) Therefore, in ∇ q j is a diagonal matrix, which means ∇ q j ( y ˇ ( t ) ) ≤ ρ g . Suppose ρ g = max { ρ g i } , then proceeding inequalities can easily get: ∇ q j ≤ ρ g 2 I(55) Apply the schur complement and integrating the Eq. (28), (54) and (55), the LMIs of Eq. (52) can be obtained. The proof is complete. □