New results on state feedback control for a class of switched nonlinear systems

1 Introduction

During the past decades, the switched systems attracted more and more attention because of many physical or man-made systems possessing switching features [1]. A typical switched system is composed of a finite number of continuous-time or discrete-time subsystems and a switching signal which orchestrates the switching between these subsystems. Recently, the stability and stabilization problems of switched systems have been extensively studied [1–8]. Until now, two stability issues have been addressed, i.e., the stability under arbitrary switching and the stability under constrained switching. For the stability under arbitrary switching, the existence of a common Lyapunov function (CLF) for all subsystems provides a stability condition for the switched systems [1]. As for the stability under constrained switching, the multiple Lyapunov functions have been proven to be an useful method to study the stability of switched systems [2]. As one typical example of constrained switching, the average dwell time (ADT) logic is proposed in [4]. The ADT method is widely used to study the stability problem of switched systems [9–13]. The ADT switching can cover the dwell time switching [1], and the extreme case of the ADT switching is the arbitrary switching [10]. Therefore, it is important and theoretically significant for us to study the switched systems with ADT.

As we know, the T-S fuzzy model is proven to be an effective tool in approximating most complex nonlinear systems [14], which utilizes local linear system description for each rule. The last several decades have witnessed more and more applications of the T-S fuzzy model in the analysis and synthesis of dynamical systems [15–26]. Recently the T-S fuzzy model has been extended to describe each nonlinear subsystem of switched nonlinear systems. Many significant results have been obtained for switched T-S fuzzy systems [27–29].

Recently, based on different performance indexes, the control and filtering issues for continuous-time switched systems have been widely studied. To name a few, the weighted H_∞ control problem for switched linear systems has been studied in [9, 13]. The passivity-based control problem for switched systems was solved in [30, 31]. The dissipative control problem of switched systems has been investigated in [32–34]. It has been pointed out that by setting Q ≤ 0, the (Q, S, R)-dissipativity index can include the H_∞, positive realness and passivity as special cases [33, 34]. As we know, for the switched systems, the L₂ - L_∞ performance is another important performance index and has also received considerable attention [35, 36]. Unfortunately, the L₂ - L_∞ performance index cannot be included in the (Q, S, R)-dissipativity index. Naturally, an interesting and challenging problem arises: Can the dissipative and L₂ - L_∞ control problems of switched systems be solved in a unified framework? which motivates our study in this paper. This problem is well addressed in this paper.

The main contributions of our work are listed as follows: (I) A novel performance index, including the weighted H_∞, L₂ - L_∞, passivity and dissipativity performance indices as special cases, is proposed in this paper. (II) Based on this new performance index, a novel state feedback controller is designed for the continuous-time switched T-S fuzzy systems.

This paper studies the state feedback control problem for the continuous-time switched T-S fuzzy systems based on a novel performance index. First, the closed-loop switched T-S fuzzy systems with state feedback controllers are constructed. Then, a novel performance index which can be viewed as the extended dissipativity performance is proposed. Second, based on the multiple Lyapunov functions approach and the average dwell time technique, the state feedback controllers are designed. Finally, a numerical example is provided to illustrate the effectiveness of the obtained results. The remainder of this paper is organized as follows. Preliminaries are presented in Section 2. The main results of this paper are presented in Section 3. In Section 4, a numerical examples is provided. Finally, conclusions are shown in Section 5.

Notations: The notations used in this paper are fairly standard. The symbol “*” in a matrix stands for the transposed elements in the symmetric positions. The superscript “T” is the matrix transposition. N denotes the set of the natural numbers. R ⁿ denotes the n-dimensional Euclidean space. I represents the identity matrix in the block matrix with appropriate dimensions. The notation ∥ ·   ∥ refers to the Euclidean vector norm. L₂ [0, ∞) is the space of square-integrable. For v (t) ∈ L₂ [0, ∞), its norm is given by ∥ v ( t ) ∥ 2 = ∫ 0 ∞ v ( t ) T v ( t ) dt. A continuous function α: [0, ∞) → [0, ∞) is said to be of class K if it is strictly increasing and α (0) =0. If α is also unbounded, then it is said to be of class K ∞. A function β : [0, ∞) × [0, ∞) → [0, ∞) is said to be of class KL if β (· , t) is of class K for each fixed t ≥ 0 and β (s, t) decreases to 0 as t→ ∞ for each fixed s ≥ 0. We use P > 0  (≥0, < 0, ≤ 0) to denote a positive definite (semi-positive definite, negative definite, semi-negative definite) matrix P. If not explicitly stated, matrices are assumed to have compatible dimensions.

2 Preliminaries

In this paper, let us consider the following switched nonlinear systems: { x ̇ ( t ) = f σ ( t ) ( x ( t ) , u ( t ) , w ( t ) ) z ( t ) = g σ ( t ) ( x ( t ) , u ( t ) , w ( t ) )(1) where x (t) ∈ R ^{n _x} is the state vector, z (t) ∈ R ^{n _z} is the control output, and w (t) ∈ R ^{n _w} is the disturbance input that belongs to L₂ [0, ∞). A piecewise constant function of time σ ( t ) : [ 0 , + ∞ ) → S = { 1 , 2 , … , M } is called the switching signal, where M ∈ N is the number of subsystems. For a switching sequence 0 = t₀< t₁ < ⋯ < t _k  < t_k+1 < ⋯, σ (t) is continuous from right everywhere. When t ∈ [t _k , t_k+1), the σ (t _k ) subsystem is activated, and σ (t _k ) = i, i ∈ S.

The T-S fuzzy model which is described by fuzzy IF-THEN rules [14] is employed here to represent each subsystem of the switched nonlinear systems (1).

Rule m for the subsystem i: IF ν_i1 (t) is N_i1m and ⋯ and ν _ig  (t) is N _igm , THEN { x ̇ ( t ) = A im x ( t ) + B im u ( t ) + D i 1 m w ( t ) z ( t ) = E im x ( t ) + D i 2 m w ( t )(2) where ν _i  (t) = (ν_i1 (t) , ν_i2 (t) , ⋯ , ν _ig  (t)) are some measurable premise variables and N _ipm   (p = 1, 2, ⋯ , g) are fuzzy sets. The matrices A _im , B _im , D_i1m, D_i2m and E _im are system matrices with appropriate dimensions.

By using “fuzzy blending”, the final output of the ith subsystem is inferred as follows: { x ̇ ( t ) = ∑ m = 1 r h im ( t ) [ A im x ( t ) + B im u ( t ) + D i 1 m w ( t ) ] z ( t ) = ∑ m = 1 r h im ( t ) [ E im x ( t ) + D i 2 m w ( t ) ](3) where h im ( t ) = l im ( t ) / ∑ m = 1 r l im ( t ) , l im ( t ) = ∏ p = 1 g N ipm ( ν ip ( t ) ), r is the number of IF-THEN rules, and N _ipm  (ν _ip  (t)) is the grade of the membership function of ν _ip in N _ipm . It is assumed that l _im  (t) ≥0 for all t. Therefore, the normalized membership function h _im  (t) satisfies h im ( t ) ≥ 0 , ∑ m = 1 r h im ( t ) = 1 .(4)

For the ith subsystem, the state feedback controller has the following form u ( t ) = ∑ m = 1 r h im ( t ) K im x ( t ) .(5) By substituting (5) into (3), the following closed-loop switched T-S fuzzy systems are obtained { x ̇ ( t ) = A ¯ i ( t ) x ( t ) + D i 1 ( t ) w ( t ) z ( t ) = E i ( t ) x ( t ) + D i 2 ( t ) w ( t )(6) where A ¯ i ( t ) = A i ( t ) + B i ( t ) K i ( t ).

Now, the following assumptions and definitions are introduced, which are needed in the process of obtaining the main results of this paper.

Definition 1. [1] The switched T-S fuzzy system (6) with u (t) ≡0 and w (t) ≡0 is globally uniformly asymptotically stable (GUAS) if there exists a class KL function β such that for all initial conditions x (t₀) and all switching signals σ (t), the solutions of the systems (6) satisfy the following inequality ∥ x ( t ) ∥ ≤ β ( ∥ x ( t 0 ) ∥ , t ) , ∀ t ≥ t 0 .

Definition 2. [4] For a switching signal σ (t) and any T₂ ≥ T₁ ≥ 0, let N _σ  (T₁, T₂) denote the number of switching of σ (t) over (T₁, T₂). We say that σ (t) has an average dwell time τ _a if there exist two constants N₀ > 0 and τ _a  > 0 such that the following inequality holds: N σ ( T 2 , T 1 ) ≤ N 0 + T 2 - T 1 τ a .

Assumption 1. [37] ∀ i ∈ S, let matrices Φ, Ψ₁, Ψ₂ and Ψ₃ satisfy the following conditions:

Φ = Φ ^T , Ψ₁ = Ψ 1 T and Ψ₃ = Ψ 3 T.

Motivated by [37], and combined with the characteristics of switched systems, in this paper, a novel performance index is introduced in the Definition 3.

Definition 3. For given matrices Φ, Ψ₁, Ψ₂ and Ψ₃ satisfying the Assumption 1, the closed-loop switched T-S fuzzy system (6) is said to be extended dissipative if there exist scalars λ > 0 and ρ ≥ 0 such that the following inequality holds for any t > 0 and all w (t) ∈ L₂ [0, ∞): ∫ 0 t e - λ s J ( s ) ds - e - λ t z T ( t ) Φ z ( t ) ≥ ρ(7) where J (s) = z ^T  (s) Ψ₁z (s) +2z ^T  (s) Ψ₂w (s) + w ^T  (s) Ψ₃w (s). Since Ψ₃ ≥ 0 and λ > 0, from (7), the following inequality can be obtained ∫ 0 t J ¯ ( s ) ds - e - λ t z T ( t ) Φ z ( t ) ≥ ρ(8) where J ¯ ( s ) = e - λ s [ z T ( s ) Ψ 1 z ( s ) + 2 z T ( s ) Ψ 2 w ( s ) ] + w T ( s ) Ψ 3 w ( s ). Furthermore, since Φ ≥ 0, ρ ≥ 0 and λ > 0, from (7), the following inequality can be obtained ∫ 0 t J ( s ) ds - e - λ t z T ( t ) Φ z ( t ) ≥ ρ .(9)

It can be seen from the Definition 3 that the following performance indexes hold by choosing appropriate parameter values for Φ, Ψ₁, Ψ₂, Ψ₃ and ρ.

Choosing Φ = 0, Ψ₁ = - I, Ψ₂ = 0, Ψ₃ = γ²I and ρ = 0, the inequality (8) reduces to the weighted H_∞ performance index [9, 13].

Note from the Assumption 1 that Φ ≥ 0 and Ψ₁ ≤ 0. Thus, there always exist matrices Φ ¯ and Ψ ¯ 1 such that Φ = Φ ¯ T Φ ¯ , Ψ 1 = - Ψ ¯ 1 T Ψ ¯ 1 .(10)

Remark 1. It is necessary for us to give some explanations about the Assumption 1. The first item of the Assumption 1 guarantees that the inequality (7) is well defined. The second item enables one to derive a linear matrix inequality (LMI)-based condition for the investigation of the dissipativity analysis problem. Assumptions similar to 1), 2), and 5) of the Assumption 1 were used in [32–34]. On the other hand, it has been pointed out when considering the L₂ - L_∞ performance, the output of the considered system should not include disturbance inputs [38]. Therefore, it should be assumed that D_i2 = 0 when Φ ≠ 0, which justifies the need of the third item of Assumption 1. Finally, the fourth item of Assumption 1 is technically necessary for the development of our analysis and design methods.

Remark 2. It can be seen from the Definition 3 that if the inequality (7) holds, then the inequalities (8) and (9) hold. The inequality (8) can reduce to the weighted H_∞ performance index and L₂ - L_∞ performance index. The inequality (9) can reduce to the passivity performance index and the strict (Q, S, R)-dissipativity performance index. Therefore, to prove the continuous-time switched systems can achieve a unified performance index which includes the weighted H_∞, L₂ - L_∞, passivity and the strict (Q, S, R)-dissipativity performance indices, the inequality (7) only need to be proven to hold for the continuous-time switched systems.

Remark 3. It should be mentioned that the new performance index defined in the Definition 3 can be used to study the control synthesis problems for a variety of switched systems, for example networked control switched systems and so on.

Our aim in this paper is to design the state feedback controllers (5), such that the corresponding closed-loop switched T-S fuzzy system (6) is GUAS and satisfies the novel performance index defined in the Definition 3.

This section is concerned with the problem of the controller design for the closed-loop switched T-S fuzzy system (6). To begin with, the Lemma 1 is given, which is useful in the state feedback controller (5) design.

Lemma 1. Consider the closed-loop switched T-S fuzzy system (6) and let α > 0 and μ ≥ 1 be given constants. For given matrices Φ, Ψ₁, Ψ₂ and Ψ₃ satisfying the Assumption 1 and (10), ∀ ( i , j ) ∈ S × S , i ≠ j, if there exist matrices P _i  > 0, G _i  > 0 and L _i  > 0 satisfying the following LMIs P i < μ P j ,(11) [ - G i E i T ( t ) Φ ¯ T * - I ] < 0 ,(12) [ - L i L i * G i - 2 I ] < 0 ,(13) [ Δ i 1 Δ i 2 E i T ( t ) Ψ ¯ 1 T * Δ i 3 D i 2 T ( t ) Ψ ¯ 1 T * * - I ] < 0(14) where Δ i 1 = A ¯ i T ( t ) P i + P i A ¯ i ( t ) + α P i , Δ i 2 = P i D i 1 ( t ) - E i T ( t ) Ψ 2 , Δ i 3 = - D i 2 T ( t ) Ψ 2 - Ψ 2 D i 2 ( t ) - Ψ 3 , then the closed-loop switched T-S fuzzy system (6) is GUAS for any switching signal satisfying τ a ≥ τ a * = ln μ α(15) and satisfies the new performance index defined in the Definition 3.

Proof. Choose the following multiple Lyapunov functions for the system (6) V i ( t ) = x T ( t ) P i x ( t ) .(16) From the equality (16), the following equality can be obtained

V ̇ i ( t ) = x ̇ T ( t ) P i x ( t ) + x T ( t ) P i x ̇ ( t ) = x T ( t ) [ A ¯ i T ( t ) P i + P i A ¯ i ( t ) ] x ( t ) + w T ( t ) D i 1 T ( t ) P i x ( t ) + x ( t ) P i D i 1 ( t ) w ( t ) .(17) Assume that the subsystem is switching from the jth subsystem to the ith subsystem. Together (16) with (17), the following equality can be obtained V ̇ i ( t ) + α V i ( t ) + J ( t ) = η T ( t ) Π η ( t )(18) where η T ( t ) = [ x T ( t ) , w T ( t ) ] , Π = [ Π i 1 Δ i 2 - E i T ( t ) Ψ 1 D i 2 ( t ) * - D i 2 T ( t ) Ψ 1 D i 2 ( t ) + Δ i 3 ] , Π i 1 = Δ i 1 + E i T ( t ) Ψ ¯ 1 T Ψ ¯ 1 E i ( t ) . By Schur complement, the inequality (14) implies Π < 0, then V ̇ i ( t ) + α V i ( t ) - J ( t ) < 0 .(19) At the switching instant t _k , together (11) with (16), the following inequality can be obtained V i ( t k ) < μ V j ( t k - ) .(20) Together (19) with (20) and by induction, the following inequality can be obtained

V ( t ) ≤ e - α ( t - t k ) V ( t k ) + ∫ t k t e - α ( t - s ) J ( s ) ds ≤ μ e - α ( t - t k ) V ( t k - ) + ∫ t k t e - α ( t - s ) J ( s ) ds ≤ μ e - α ( t - t k ) [ e - α ( t k - t k - 1 ) V ( t k - 1 ) + ∫ t k - 1 t k e - α ( t k - s ) J ( s ) ds ] + ∫ t k t e - α ( t - s ) J ( s ) ds ≤ ⋯ ≤ μ N σ ( 0 , t ) e - α t V ( t 0 ) + ∫ 0 t μ N σ ( s , t ) e - α ( t - s ) J ( s ) ds = Ω ( t ) + ∫ 0 t μ N σ ( s , t ) e - α ( t - s ) J ( s ) ds(21) where Ω ( t ) = μ N σ ( 0 , t ) e - α t V ( t 0 ) .

From the Definition 2, the following inequality holds Ω ( t ) ≤ exp { ( ln μ τ a - α ) t } V ( t 0 ) . If supposing ln μ τ a - α ≤ 0 ,(22) a sufficient condition that guarantees the GUAS of the closed-loop switched nonlinear system is obtained. The inequality (22) can be rewritten as follows τ a ≥ τ a * = ln μ α . Then we can draw a conclusion that V (t) →0 as t→ ∞ if (15) is satisfied.

In the following, the performance index for the closed-loop switched T-S fuzzy system (6) is established. From (21), the following inequality can be obtained

∫ 0 t e - α ( t - s ) + N σ ( s , t ) ln μ J ( s ) ds ≥ V ( t ) - V ( t 0 ) .(23) Multiplying both sides of (23) by e^{-N _σ (0,t)ln μ}, the following inequality holds

∫ 0 t e - α ( t - s ) - N σ ( 0 , s ) ln μ J ( s ) ds ≥ e - N σ ( 0 , t ) ln μ [ V ( t ) - V ( t 0 ) ] .(24) Since -α (t - s) <0 and e^{-N _σ (0,t)ln μ} < 1, the following inequality can be obtained from (24)

∫ 0 t e - N σ ( 0 , s ) ln μ J ( s ) ds ≥ e - N σ ( 0 , t ) ln μ V ( t ) - V ( t 0 ) .(25) From the Definition 2, the following inequality holds - N σ ( 0 , s ) ≥ - s τ a .(26) Combining (15), (25) and (26), if let the inequality (27) hold, then the inequality (25) holds ∫ 0 t e - α s J ( s ) ds ≥ V ( t ) - V ( t 0 ) .(27) By setting the scalar ρ as ρ = - V (t₀), the following inequality can be obtained from (27) ∫ 0 t e - α s J ( s ) ds ≥ x T ( t ) P i x ( t ) + ρ .(28)

It can be obtained from ( G i - I ) G i - 1 ( G i - I ) ≥ 0 with G _i  > 0 that - G i - 1 ≤ G i - 2 I .(29) By letting L i = P i - 1, from (13) and (29), the following inequality holds P i > G i .(30) Together (16) with (30), the following inequality can be obtained V i ( t ) = x T ( t ) P i x ( t ) > x T ( t ) G i x ( t ) > 0 .(31) Combining (28) with (31), the following inequality can be obtained ∫ 0 t e - α s J ( s ) ds ≥ x T ( t ) G i x ( t ) + ρ .(32) According to the Definition 3, for any matrices Φ, Ψ₁, Ψ₂ and Ψ₃ satisfying the Assumption 1, the following inequality should be proven to be held ∫ 0 t e - α s J ( s ) ds - e - α t z T ( t ) Φ z ( t ) ≥ ρ .(33) The proof process is divided into two cases: (I) ∥Φ ∥ =0, (II) ∥Φ ∥ ≠0.

(I): ∥Φ ∥ =0

From (32), the following inequality can be obtained ∫ 0 t e - α s J ( s ) ds ≥ x T ( t ) G i x ( t ) + ρ ≥ ρ .(34) This implies that the (33) holds because of z ^T  (t) Φz (t) =0.

(II): ∥Φ ∥ ≠0

In this case, it can be concluded from the Assumption 1 that ∥Ψ₁ ∥ + ∥ Ψ₂ ∥ =0 and D_i2m = 0 which implies that Ψ₁ = 0, Ψ₂ = 0 and Ψ₃ > 0. By using Schur complement to (12) and with Φ = Φ ¯ T Φ ¯, it can be obtained that E i T ( t ) Φ E i ( t ) < G i. Then, the following inequality holds

∫ 0 t e - α s J ( s ) ds - e - α t z T ( t ) Φ z ( t ) ≥ ∫ 0 t e - α s J ( s ) ds - z T ( t ) Φ z ( t ) = ∫ 0 t e - α s J ( s ) ds - x T ( t ) E i T ( t ) Φ E i ( t ) x ( t ) ≥ ∫ 0 t e - α s J ( s ) ds - x T ( t ) G i x ( t ) ≥ ρ .(35) Based on the above two cases, it can be concluded that the closed-loop switched T-S fuzzy system (6) can achieve the new performance index defined in the Definition 3. The proof is ended.

Remark 4. Following the same procedure in the proof of Lemma 1, it can be concluded that the closed-loop switched T-S fuzzy system (6) with w (t) =0 is GUAS for any switching signal satisfying (15).

In Lemma 1, the controller parameter matrices are coupled with matrix variables P _i in (14). To decouple the variables in (14), a decoupling technique is introduced in the following lemma.

Lemma 2. Consider the closed-loop switched T-S fuzzy system (6) and let α > 0 and μ ≥ 1 be given constants. For given matrices Φ, Ψ₁, Ψ₂ and Ψ₃ satisfying the Assumption 1 and (10). ∀ ( i , j ) ∈ S × S , i ≠ j, if there exist matrices L _i  > 0, G _i  > 0 and W _i  (t) satisfying μ L i > L j ,(36) [ - L i L i * G i - 2 I ] < 0 ,(37) [ - G i E i T ( t ) Φ ¯ T * - I ] < 0 ,(38) Ψ i ( t ) < 0 ,(39) where

Ψ i ( t ) = [ Ξ i 1 Ξ i 2 L i E i T ( t ) Ψ ¯ 1 T * Ξ i 3 D i 2 T ( t ) Ψ ¯ 1 T * * - I ] , Ξ i 1 = A i ( t ) L i + B i ( t ) W i ( t ) + ( A i ( t ) L i + B i ( t ) W i ( t ) ) T + α L i , Ξ i 2 = D i 1 ( t ) - L i E i T ( t ) Ψ 2 , Ξ i 3 = - D i 2 T ( t ) Ψ 2 - Ψ 2 D i 2 ( t ) - Ψ 3 , then the closed-loop switched T-S fuzzy system (6) is GUAS for any switching signal satisfying (15) and satisfies the new performance index defined in the Definition 3. Moreover, if there exist feasible matrices L _i and W _i  (t), the switching state feedback controller gains are given as: K i ( t ) = W i ( t ) L i - 1 .(40)

Proof. Letting L i = P i - 1 and W i ( t ) = K i ( t ) P i - 1, and substituting them into the (39), the following inequality can be obtained [ Ξ ¯ i 1 Ξ ¯ i 2 P i - 1 E i T Ψ ¯ 1 T * Ξ ¯ i 3 D i 2 T Ψ ¯ 1 T * * - I ] < 0(41) where Ξ ¯ i 1 = A i ( t ) P i - 1 + B i ( t ) K i ( t ) P i - 1 + ( A i ( t ) P i - 1 + B i ( t ) K i ( t ) P i - 1 ) T + α P i - 1 , Ξ ¯ i 2 = D i 1 ( t ) - P i - 1 E i T ( t ) Ψ 2 , Ξ ¯ i 3 = - D i 2 T ( t ) Ψ 2 - Ψ 2 D i 2 ( t ) - Ψ 3 . Multiplying (41) from the left and right, respectively, by diag (P _i , I, I) and its transpose, the following inequality holds [ Ξ ˆ i 1 Ξ ˆ i 2 E i T ( t ) Ψ ¯ 1 T * Ξ ˆ i 3 D i 2 T ( t ) Ψ ¯ 1 T * * - I ] < 0(42) where Ξ ˆ i 1 = P i [ A i ( t ) + B i ( t ) K i ( t ) ] + [ A i ( t ) + B i ( t ) K i ( t ) ] T P i + α P i , Ξ ˆ i 2 = P i D i 1 ( t ) - E i T ( t ) Ψ 2 , Ξ ˆ i 3 = - D i 2 T ( t ) Ψ 2 - Ψ 2 D i 2 ( t ) - Ψ 3 . Since A ¯ i ( t ) = A i ( t ) + B i ( t ) K i ( t ), it can be seen that the condition (1) of Lemma 1 is satisfied. From (36) and L i = P i - 1, the following inequality holds P i < μ P j .(43) Thus, in addition to (37) and (38), all the conditions of Lemma 1 are satisfied. Therefore, the closed-loop switched T-S fuzzy system (6) is GUAS and satisfies the new performance index defined in the Definition 3. The proof is completed.

The following theorem provides the stabilization conditions with the new performance index defined in the Definition 3 for the closed-loop switched T-S fuzzy system (6), which are the main results of this paper.

Theorem 1. Consider the closed-loop switched T-S fuzzy system (6) and let α > 0 and μ ≥ 1 be given constants. For given matrices Φ, Ψ₁, Ψ₂ and Ψ₃ satisfying the Assumption 1 and (10). ∀ ( i , j ) ∈ S × S , i ≠ j, if there exist matrices L _i  > 0, G _i  > 0 and W _im satisfying μ L i > L j ,(44) [ - L i L i * G i - 2 I ] < 0 ,(45) [ - G i E im T Φ ¯ T * - I ] < 0 ,(46) Γ imm < 0 , 1 ≤ m ≤ r ,(47) Γ imn + Γ inm < 0 , 1 ≤ m < n ≤ r ,(48) where Γ imn = [ Λ i 1 Λ i 2 L i E im T Ψ ¯ 1 T * Λ i 3 D i 2 m T Ψ ¯ 1 T * * - I ] , Λ i 1 = A im L i + B im W in + ( A im L i + B im W in ) T + α L i , Λ i 2 = D i 1 m - L i E im T Ψ 2 , Λ i 3 = - D i 2 m T Ψ 2 - Ψ 2 D i 2 m - Ψ 3 , then the closed-loop switched T-S fuzzy system (6) is GUAS for any switching signal satisfying (15) and satisfies the new performance index defined in the Definition 3. Moreover, if there exist feasible matrices L _i and W _im , the switching state feedback controller gains are given as: K im = W im L i - 1 .(49)

Proof. From Lemma 2, for the switched T-S fuzzy system (6), the Ψ _i  (t) in (39) can be rewritten as

Ψ i ( t ) = ∑ m = 1 r ∑ n = 1 r h im ( t ) h in ( t ) Γ imn = ∑ m = 1 r h im 2 ( t ) Γ imm + ∑ m = 1 r ∑ n > m r h im ( t ) h in ( t ) [ Γ imn + Γ inm ] . If the conditions (47) and (48) of Theorem 1 are satisfied, it can be obtained Ψ _i  (t) <0. The inequality (38) can be obtained from the inequality (46). Together (44) with (45) and from the Lemmas 1 and 2, it can be concluded that the closed-loop switched T-S fuzzy system (6) is GUAS and satisfies the new performance index defined in the Definition 3. Finally, by (40), the switching state feedback controller gains can be given as (49). The proof is ended.

In this section, a numerical example is provided to illustrate the effectiveness of the state feedback control scheme developed in this paper. Based on the Definition 3, by setting ρ = 0, Φ = 0, Ψ₁ = - I, Ψ₂ = 0, and Ψ₃ = γ²I, the weighted H_∞ performance is considered for the closed-loop switched T-S fuzzy system (6). The effectiveness of other performance indices can also be illustrated following the same way, and they are omitted here.

To design the state feedback controller (5) for the switched T-S fuzzy system (3), let us consider the system (3) consisting of two subsystems (i = 1, 2), and each subsystem has two fuzzy rules (r = 2) where A 11 = [ - 0.4 - 0.08 0.4 0.3 ] , A 12 = [ - 0.4 0.08 0.6 0.2 ] , A 21 = [ 0.5 0.5 0.7 - 0.5 ] , A 22 = [ 0.75 - 0.5 - 0.25 - 0.5 ] , B 11 = [ - 0.35 - 0.05 0.11 0.15 ] , B 12 = [ - 0.3 - 0.1 0.15 0.1 ] , B 21 = [ 0.5 0.15 0.04 - 0.5 ] , B 22 = [ 0.5 - 0.15 0.04 0.5 ] , D 111 = [ 0.3 0.2 ] , D 112 = [ 0.3 - 0.2 ] , D 211 = [ 0.15 0.1 ] , D 212 = [ 0.1 - 0.2 ] , E 11 = [ 0.2 0.4 ] , E 12 = [ - 0.2 0.4 ] , E 21 = [ 0.1 0.2 ] , E 22 = [ 0.1 - 0.2 ] , D 121 = 0.15 , D 122 = 0.2 , D 221 = 0.2 , D 222 = 0.1 . The fuzzy membership functions are taken as { h 11 = sin 2 ( x 1 + 0.5 ) , h 12 = 1 - h 11 , h 22 = 1 [ 1 + exp ( - 3 x 2 0.5 - π / 2 ) ] × [ 1 + exp ( - 3 x 2 0.5 + π / 2 ) ] , h 21 = 1 - h 22 . First, the state response of the uncontrolled switched T-S fuzzy system is shown in Fig. 1. It is clearly shown in Fig. 1 that the uncontrolled system is unstable.

Here, a set of mode-dependent state feedback controllers will be designed such that the closed-loop switched T-S fuzzy system (6) is GUAS. The parameters used in Theorem 1 are given as α = 0.2, μ = 1.05 and γ = 1. By Theorem 1, the ADT is obtained as τ a * = 0.244. By using the LMI toolbox in MATLAB to solve the LMIs (44)–(48), it is found that the the LMIs (44)–(48) are feasible. By (49), the controller gains are obtained as follows: K 11 = [ 11.1494 4.0376 - 17.1771 - 18.3352 ] , K 12 = [ 14.2581 8.8791 - 20.1599 - 25.0950 ] , K 21 = [ - 7.8411 - 2.4199 1.4629 0.3276 ] , K 22 = [ - 7.6394 - 0.8211 1.2611 - 0.3344 ] . The initial conditions are assumed to be x (t₀) = [1, 2]  ^T , and the disturbance input is assumed to be w (t) =0.5 sin(2πt) exp(-0.2t). The state response for the closed-loop switched T-S fuzzy (6) is shown in Fig. 2. It is clearly shown in Fig. 2 that the unstable subsystems have been stabilized by the designed state feedback controllers, which demonstrates the effectiveness of the obtained results in this paper.

Next, by taking the weighted H_∞ performance index as an example, a comparison between our work and a related peer work [29] is given to show the advantage of our results. To complete the comparison, some parameters used in [29] are set as P _iu = P _i , α₁ = α₂ = α, μ₁ = μ₂ = μ, T _M  = 0. For different α and μ sets, by setting the same parameters for Theorem 1 of our work and Theorem 2 of [29], the minimum values γ min 2 for these two work are computed. The computation results are shown in Table 1. All the results of Table 1 reveal that under the same conditions Theorem 1 of our work has a better performance than Theorem 2 of [29].

5 Conclusion

The problem of state feedback controller design has been addressed for a class of continuous-time switched nonlinear systems in this paper. The T-S fuzzy model is used here to represent each nonlinear subsystem of the switched nonlinear systems. By proposing a novel performance index, the weighted H_∞, L₂ - L_∞, passive and dissipativity control problems for switched systems are solved in a unified framework. Through the use of the multiple Lyapunov functions approach and ADT technique, the existence conditions of the state feedback controller are obtained. Based on these results, the desired controllers are designed to ensure the switched T-S fuzzy system to be GUAS and satisfy the novel performance index. It is also remarked that the new performance index proposed in this paper can also be used to study the control synthesis problems for other switched systems. Finally, a numerical example is provided to illustrate the applicability of the obtained results. In our future work, based on this new performance index, the filtering problem for switched systems will be studied in a unified framework.