1 Introduction

A switched system belongs to an important class of hybrid systems. Typically, a switched system consists of a number of subsystems and a switching signal, which defines a specific subsystem being activated during a certain time interval. Switched systems arised in many practical models in manufacturing, communication networks, automotive engine control, chemical processes etc.[1–3] In the last two decades, there has been increasing interest in stability analysis and control design of switched linear systems[4–7]. The main approach for stability analysis relies on the use of Lyapunov-Krasovskii functional and linear matrix inequalities (LMIs).

For nonlinear systems, there has been no systematic tool capable of finding necessary and sufficient conditions for stability till now. This makes the control problem of nonlinear systems a challenging task. The problem becomes more challenging when the system is constituted by several subsystems, and only one of them is activated at a given time because the determination of the switching rule between subsystems is a fundamental task in the stability analysis.

Fuzzy control represents an important approach to handle complex and ill-determined systems. Compared with conventional modeling techniques which use a single model to describe the global behavior of a system, fuzzy modeling is essentially a multi-model approach in which simple submodels are fuzzily combined to describe the global behavior of a system. Numerous control problems have been studied based on this T-S fuzzy model[8–11]. The fuzzy logic approach is used to study the stability analysis of switched systems[8, 12–14].

As is well known, the introduction of time delay systems render the stability analysis more complicated. Furthermore, time delay systems are of great importance since they are frequently encountered in various engineering systems, such as chemical processes, mechanical systems, systems in economy, transmission lines, etc. Linear time delay switched systems have been extensively studied in the literatures[15–20]. For nonlinear switched time delay systems there are only few results[21–24]. Benzaouia[14] considered discrete time systems while Yong et al.[25] investigated continuous time systems with constant delay.

In this paper, we address the problem of stability of uncertain switched nonlinear systems with time delay using T-S fuzzy model. Note here that the transient process of a system can be better described if its decay rate is determined. Hence, the problem of exponential stability is of great importance. To the best of our knowledge, there is no paper dealing with exponential stability of nonlinear time delay switched systems. This point represents our objective in this paper. Delay dependent exponential stability conditions are presented by using T-S fuzzy model. Our method is based on Lyapunov-Krasovskii functional approach, and the results are expressed in terms of LMIs. With our results, we can compute simultaneously the two bounds that characterize the exponential stability. These bounds are the stability factor and decay rate of the solution. Two numerical examples are given for illustration.

2 Problem formulation

The switched nonlinear systems with time delay can be described by a differential equation as

$$\left\{ {\begin{array}{*{20}{l}} {\dot x(t) = {f_\sigma }(x(t),x(t - h),t)}&{t > 0} \\ {x(t) = \phi (t)}&{t \in [ - \bar h,0]} \end{array}} \right.$$
(1)

where x ∈ Rn is the state vector, φ ∈ (\([ - \bar h,0]\),Rn) is the initial function, with norm \(\left\| \phi \right\| = {\sup _{\theta \in [ - \bar h,0]}}\left\| {\phi (\theta )} \right\|,\), where \(\theta \in [\bar h,0],{f_\sigma }\) are sufficiently regular functions from Rn to Rn that are parameterized by the index set I := {1,2, ⋯, N}, i.e., σ ∈ I. σ(x): Rn → I is the switching rule which is a piece-wise constant function depending on the state x(t) in each time, i.e., f σ(x) (x(t),x(t − h,t) switches between f 1 (x(t),x(t − h),t), f 2(x(t),x(t − h),t), ⋯, and f N (x(t),x(t−h),t). h is a constant time delay which can take its values in \([ - \bar h,0]\), i.e.,

$$0 \leqslant h \leqslant \bar h.$$
(2)

Moreover σ(x) = j implies that the j-th subsystem is activated, and we have the following subsystem:

$$\left\{ {\begin{array}{*{20}{l}} {\dot x(t) = {f_j}(x(t),x(t - h),t),}&{t > 0} \\ {x(t) = \phi (t),}&{t \in [ - \bar h,0].} \end{array}} \right.$$
(3)

Furthermore, following the idea of [26], we consider a fuzzy switched time delay system described by T-S fuzzy model for switched nonlinear systems (1). The i-th rule of the T-S fuzzy model is described by the following IF-THEN form: Plan rule i: R i: If z 1(t) is W 1 i and ⋯ and z s (t) is \(W_s^i\), then

$$\left\{ {\begin{array}{*{20}{c}} {\dot x(t) = ({A_{i\sigma }} + \Delta {A_{i\sigma }}(t))x(t) + ({B_{i\sigma }}(t) + \Delta {B_{i\sigma }}(t) + \Delta {B_{i\sigma }})x(t - h)} \\ {x(t) = \phi (t),\,\,\,\,\,\,\,t \in [ - \bar h,0],\,\,\,\,i = 1,...,r\,\,\,t > 0} \end{array}} \right.$$
(4)

where z 1 (t), z 2 (t), ⋯ , z s (t) are the premise variables, \(W_l^i\), l = 1, ⋯, s are the fuzzy sets, r is the number of IF-THEN rules, [A iσ B iσ ∆ A iσ(t) ∆B iσ(t)] ∈ {[A ij , B ij ∆ A ij (t) ∆Bij(t)],j = 1, ⋯, N}, Aij and Bij are constant matrices, ∆ A ij (t) and ∆B ij (t) are unknown perturbation matrices. The matrix A iσ switches between matrices A i 1 ,A i 2, ⋯, A iN , the matrix B iσ switches between matrices B i 1, B i2, ⋯, B iN. The perturbation matrices ∆A iσ(t) and ∆B iσ(t) switch between matrices ∆ A i 1(t) A i2(t), ⋯,A iN (t) and ∆B i 1 (t), ∆B i2 (t), ⋯, ∆B iN (t), respectively. Hence, system (4) can be represented as

Plan rule i for individual subsystem j: Ri: If z 1 (t) is W 1 i and ⋯ and z s (t) is \(W_s^i\), then

$$\left\{ {\begin{array}{*{20}{c}} {\dot x(t) = ({A_{ij}} + \Delta {A_{ij}}(t))x(t) + ({B_{ij}}(t) + \Delta {B_{ij}}(t) + \Delta {B_{ij}})x(t - h)} \\ {x(t) = \phi (t),\,\,\,\,\,\,\,t \in [ - \bar h,0],\,\,\,\,i = 1,...,r\,\,\,t > 0.} \end{array}} \right.$$
(5)

The perturbation matrices are assumed to be norm bounded and given by

$$\begin{array}{*{20}{c}} {{A_{ij}} = {E_{aij}}{F_{aij}}(t){H_{aij}}} \\ {\Delta {B_{ij}} = {E_{bij}}{F_{bij}}(t){H_{bij}}} \end{array}$$
(6)

where E aij , E bij , H aij and H bij are known constant matrices with appropriate dimensions, F aij (t) and F bij (t) are unknown time-varying matrices with lebesgue measurable elements bounded by

$$\begin{array}{*{20}{c}} {F_{aij}^T(t){F_{aij}}(t) \leqslant I} \\ {F_{bij}^T(t){F_{bij}}(t) \leqslant I.} \end{array}$$
(7)

By using the center-average defuzzifier, product inference and singleton fuzzifier, dynamics of T-S fuzzy system (5) can be expressed as

$$\dot x(t) = \sum\limits_{i = 1}^r {{\mu _i}(z(t))[({A_{ij}} + \Delta {A_{ij}}(t))x(t) + ({B_{ij}} + \Delta {B_{ij}}(t))x(t - h)]} $$
(8)

where \({\mu _i}(z(t)) = \frac{{{w_i}(z(t))}}{{\sum {_{i = 1}^r{w_i}(z(t))} }}\) , \({w_i}(z(t)) = \prod {_{j = 1}^sw_j^i({z_i}(t))} \), \(W_j^i\) ( z j (t)) is the membership value of z j (t) in \(W_j^i\), and it is assumed that μ i (z(t)) ⩾ 0, \(\sum {_{i = 1}^r{\mu _i}(z(t)) = 1} \) for all t.

The system (8) may be written as

$$\dot x(t) = {\tilde A_j}x(t) + {\tilde B_j}x(t - h)$$
(9)

where

$$\begin{array}{*{20}{l}} {{{\tilde A}_j} = \sum\limits_{i = 1}^r {{\mu _i}(z(t))({A_{ij}}(t) + \Delta {A_{ij}}(t))} } \\ {{{\tilde B}_j} = \sum\limits_{i = 1}^r {{\mu _i}(z(t))({B_{ij}}(t) + \Delta {B_{ij}}(t)).} } \end{array}$$

Lemma 1. For any x,y ∈ Rn, and M > 0, we have

(10)

Definition 1. Given α >0, system (9) is robustly α- exponentially stable, if there exists a switching rule σ and a constant β > 1 such that every solution x(t, φ) of the system satisfies

$$\left\| {x(t,\phi )} \right\|\beta {\operatorname{e} ^{ - \alpha t}}\left\| \phi \right\|,\quad t \geqslant 0.$$

3 Main results

Let the matrices R, Q and \(\tilde Q\) be given by

(11)

And let

(12)
(13)
(14)

where * denotes the symmetric part in a symmetric matrix, and

The switching rule ([S R ]) is constructed as follows:

Step 1. Let x(t) = φ(t).

Step 2. .

Step 3. σ(x) = i as long as

Step 4. If x(t) hits the boundary of Ω i , go to Step 2 to etermine the next mode.

Theorem 1. System (9) is α-exponentially stable if here exist symmetric positive definite matrices Z, P, S, Q 22, Q 11, R 11, R 22, S 0ij , S 2ij, matrices R 12, Q 12, F 0, F 1, F 2, F 3, S 1ij , j = 1, ⋯, N and i = 1, ⋯, r satisfying:

  1. 1)

    There exists 0 < ρ j < 1, such that \(\sum {_{j = 1}^N{\rho _j} = 1} \) and

    $$\sum\limits_{j = 1}^N {{\rho _j}{L_{ij}} < 0.} $$
    (15)
  2. 2)
    (16)

The switching rule is given by [S R ], and the solution x(t, φ) of the system satisfiesThe switching rule is given by [S R ], and the solution x(t, φ) of the system satisfies

$$\left\| {x(t,\phi )} \right\| \leqslant \beta {\operatorname{e} ^{ - \alpha t}}\left\| \phi \right\|,\quad \beta = \sqrt {\frac{{{\beta _2}}}{{{\beta _1}}}} ,\quad t \geqslant 0.$$

Proof. Let x t := {x(t + s), s ∈ \([ - 2 \bar h,0]\)} and consider the following Lyapunov-Krasovskii functional: V(x t ) = V 1(x t ) + V2(x t ) + V3(x t ), where

It is easy to verify that

where \({\bar F_{ij}} = \operatorname{diag} \left\{ {{F_{aij}},{F_{bij}}} \right\}\). So we have

$${\beta _1}{\left\| {x(t)} \right\|^2} \leqslant V({x_t}) \leqslant {\beta _2}{\left\| {{x_t}} \right\|^2}$$
(17)

where β 1 and β 2 are defined by (12) and (13). Computing the first time derivative of V (x t ), we obtain

Let \(\chi (t) = \left( {\begin{array}{*{20}{c}} {x(t)} \\ {x(t - h)} \\ {\int_{t - h}^t {x(s)\operatorname{d} s} } \\ {\dot x(t)} \end{array}} \right)\), then

where

Let θ ij = T 1 + T 2 PT 2 T + T 4, we get

(18)

Now let \({B_j} = [{\tilde A_j}\quad {\tilde B_j}\quad 0\quad - I]\) and \(F = \left[ {\begin{array}{*{20}{c}} {{F_0}} \\ {{F_1}} \\ {{F_2}} \\ {{F_3}} \end{array}} \right]\) , we can easily verify that \({B_j}\chi = 0,\;\forall \chi \ne 0\), and

(19)

Adding (19) to the right hand side of (18), and adding and subtracting the term x T(t)Zx(t) with the positive definite matrix Z, we get

(20)

where ∆π ij is given at the top of the next page. Let \(\tilde F = \left[ {\begin{array}{*{20}{c}} {{F_{aij}}(t)}&0 \\ 0&{{F_{bij}}(t)} \end{array}} \right]\) and \({\tilde H_{ij}} = \left[ {\begin{array}{*{20}{c}} {{H_{aij}}}&0&0&0 \\ 0&{{H_{bij}}}&0&0 \end{array}} \right]\). ∆π ij can be written as

Applying Lemma 1, we have

(21)

where θ ij can be written as

where M ij is given at the top of the next page.

where \(\lambda (h) = - \frac{{{\operatorname{e} ^{ - 2\alpha h}}}}{h}\). Since λ(h) is an increasing function of h we have \(\lambda (h) \leqslant \lambda (\bar \lambda )\) so that

Since condition (16) holds, by schur complement, for all h satisfying \(0 < h \leqslant \bar h\), we have

where . It follows that

(22)

From condition (15), we have \(\sum {_{j = 1}^N{\rho _j}{L_{ij}}} < 0\), where 0 < ρ j < 1, j = 1, ⋯, N, and \(\sum {_{j = 1}^N{\rho _j}} = 1\). Consequently, we can write

By choosing the switching rule

we have

This implies that \(V({x_t}) \leqslant V(\phi ){\operatorname{e} ^{ - 2\alpha t}},t \geqslant 0\). Taking account of (17), we obtain \({\beta _1}{\left\| {x(t)} \right\|^2} \leqslant V({x_t}) \leqslant V(\phi ){\operatorname{e} ^{ - 2\alpha t}} \leqslant {\beta _2}{\operatorname{e} ^{ - 2\alpha t}}{\left\| \phi \right\|^2}\). And then, \(\left\| {x(t)} \right\| \leqslant \beta {\operatorname{e} ^{ - \alpha t}}\left\| \phi \right\| \geqslant \;t \geqslant 0\) which concludes the proof. D

Remark 1. For the nominal system of (9), i.e., ∆A ij (t) = ∆B ij (t) = 0, from Theorem 1 we can obtain the α-exponential stability if the condition (15) holds and the following LMI is satisfied.

$${\psi _{ij}} + {\pi _{ij}} < 0$$
(23)

Remark 2. If we set α = 0 in the Lyapunov functional, Theorem 1 reduces to a robust stability criterion. In [27], the free weighting matrix method was used to deal with the cross terms. It has been shown that the weighting matrix approach is less restrictive than model transformation method used in [28]. Peng et al.[29] employed Jensen’s integral inequality, and it is shown by examples that it gives improved results. The work of [30] is based on Finsler’s lemma and leads to improved results over the previous references. We emphasize here that the results of [27–30] can not be applied to switched systems, because the stability of the switched system depends on the switching rule. As is well known, all the subsystems may be stable, but the overall system may be unstable if the switching rule is not adequately chosen and the converse is true. In this paper, we are motivated by Finsler’s lemma although this lemma is not used. In manipulating the derivative of the Lyapunov functional, we used the state equation. As shown in example 1, our approach gives improved results.

Remark 3. For the case N = 1, without switching, and without perturbations, system (9) reduces to the following T-S fuzzy time delay system:

$$\dot x(t) = \sum\limits_{i = 1}^r {{\mu _i}[{A_i}x(t) + {B_i}x(t - h)].} $$
(24)

In this case, from (20), since there is no switching rule, we can insert L ij in the first element of the matrix θ ij , which leads to the result stated in the next corollary. First, let ψ i = ψ ij and π i = π ij , where in the expressions of ψ ij and π ij , we replace A ij and B ij by A i and B i , respectively.

Corollary 1. The system (24) is α-exponentially stable if there exist symmetric positive definite matrices P, Q 22, Q 11, R 11, R 22, Z 1, Z 2, matrices R 12, Q 12, F 0, F 1, F 2, F 3 satisfying

$${\Omega ^i} + {\pi _i} < 0.$$
(25)

The solution x(t, φ) of the system satisfies

$$\begin{array}{*{20}{c}} {\left\| {x(t,\phi )} \right\| \leqslant \beta {\operatorname{e} ^{ - \alpha t}}\left\| \phi \right\|} \\ {\beta = \sqrt {\frac{{{\beta _2}}}{{{\beta _1}}}} } \end{array}$$
(26)

where the elements \(\Omega _{k1}^i = \psi _{k1}^i\) for k,l = 1, ⋯, 4,(k,l) = (1,1), , and we omit j in the calculation of β.

Note that in general, the results of fuzzy T-S switched systems when applied to only fuzzy systems give large time delay upper bounds because of the absence of the switching modes.

4 Numerical examples

In this section, we will present two numerical examples to illustrate the proposed results.

Example 1[29]. Let us consider a non switched system with the following rules

Rule 1: If x 1(t) is W 1, then

$$\dot x = {A_1}x(t) + {B_1}x(t - h).$$
(27)

Rule 2: If x 1(t) is W 2, then

$$\dot x = {A_2}x(t) + {B_2}x(t - h).$$
(28)

And the membership functions for Rules 1 and 2 are

$$\begin{array}{*{20}{c}} {{\mu _1}({x_1}(t)) = \frac{1}{{1 + {\operatorname{e} ^{ - 2{x_1}}}(t)}}} \\ {{\mu _2}({x_2}(t)) = 1 - {\mu _1}({x_1}(t))} \end{array}$$

where A i and B i (i = 1,2) are given by

$$\begin{array}{*{20}{l}} {{A_1} = \left[ {\begin{array}{*{20}{c}} { - 2}&0 \\ 0&{ - 0.9} \end{array}} \right],\quad {A_2} = \left[ {\begin{array}{*{20}{c}} { - 1}&{0.5} \\ 0&{ - 1} \end{array}} \right],} \\ {{B_1} = \left[ {\begin{array}{*{20}{c}} { - 1}&0 \\ { - 1}&{ - 1} \end{array}} \right],\quad \quad {B_2} = \left[ {\begin{array}{*{20}{c}} { - 1}&0 \\ {0.1}&{ - 1} \end{array}} \right].} \end{array}$$

Taking α = 0 and applying Corollary 1, we obtain the maximum allowable delay bound \(\bar h = 1.9110\).

To compare with the literature results, the upper bounds on the time delay obtained are listed in Table 1. It is clear that the obtained upper h is significantly larger than those in [29-35].

Table 1 The maximal allowable time delay bound
Full size table

Example 2. Consider continuous switched time-delay system (1) with two subsystems.

For the subsystem 1,

Rule 1: If x 1(t) is W 1, then

$$\dot x = ({A_{11}} + \Delta {A_{11}}(t))x(t) + ({B_{11}} + \Delta {B_{11}}(t))x(t - h).$$

Rule 2: If x 1(t) is W 2, then

$$\dot t = ({A_{21}} + \Delta {A_{21}}(t))x(t) + ({B_{21}} + \Delta {B_{21}}(t))x(t - h).$$

For the subsystem 2,

Rule 1: If x 1(t) is W 1, then

$$\dot x(t) = ({A_{12}} + \Delta {A_{12}}(t))x(t) + ({B_{12}} + \Delta {B_{12}}(t))x(t - h).$$

Rule 2: If x 1(t) is W 2, then

$$\dot x(t) = (({A_{22}} + \Delta {A_{22}}(t))x(t) + ({B_{22}} + \Delta {B_{22}}(t))x(t - h).$$

And the membership functions are taken the same as in Example 1. The system matrices are given by

$$\begin{array}{*{20}{l}} {{A_{11}} = \left( {\begin{array}{*{20}{c}} { - 6}&{0.2} \\ { - 1.05}&1 \end{array}} \right),\;{A_{12}} = \left( {\begin{array}{*{20}{c}} { - 0.2}&{0.5} \\ { - 1}&{ - 5} \end{array}} \right),} \\ {{A_{21}} = \left( {\begin{array}{*{20}{c}} { - 3.6}&{ - 0.3} \\ { - 0.2}&{0.3} \end{array}} \right),\;{A_{22}} = \left( {\begin{array}{*{20}{c}} { - 0.2}&{ - 0.5} \\ {0.31}&{ - 4.9} \end{array}} \right),} \\ {{B_{11}} = \left( {\begin{array}{*{20}{c}} { - 3}&{0.5} \\ { - 1}&{1.5} \end{array}} \right),\;{B_{12}} = \left( {\begin{array}{*{20}{c}} { - 0.4}&{0.5} \\ { - 0.1}&{ - 1} \end{array}} \right),} \\ {{B_{21}} = \left( {\begin{array}{*{20}{c}} { - 0.1}&{0.15} \\ { - 1}&{0.1} \end{array}} \right),\;{B_{22}} = \left( {\begin{array}{*{20}{c}} { - 0.12}&{ - 0.5} \\ { - 0.3}&{0.5} \end{array}} \right),} \\ {{H_{a11}} = \left( {\begin{array}{*{20}{c}} { - 0.01}&{0.0002} \\ { - 0}&{0.00003} \end{array}} \right),\;{H_{a21}} = \left( {\begin{array}{*{20}{c}} { - 0.01}&0 \\ {0.002}&0 \end{array}} \right),} \\ {{H_{a12}} = \left( {\begin{array}{*{20}{c}} {0.01}&{0.0002} \\ 0&{ - 0.00003} \end{array}} \right),\;{H_{a22}} = \left( {\begin{array}{*{20}{c}} {0.0001}&0 \\ {0.0002}&0 \end{array}} \right),} \\ {{H_{b11}} = \left( {\begin{array}{*{20}{c}} {0.01}&0 \\ 0&{0.5} \end{array}} \right),\;{H_{a21}} = \left( {\begin{array}{*{20}{c}} {0.01}&{0.02} \\ {0.25}&{0.3} \end{array}} \right),} \\ {{H_{b12}} = \left( {\begin{array}{*{20}{c}} {0.01}&0 \\ {0.15}&{0.5} \end{array}} \right),\;{H_{b22}} = \left( {\begin{array}{*{20}{c}} {0.01}&{0.02} \\ {0.25}&{0.3} \end{array}} \right),} \\ {{E_{a11}} = \left( {\begin{array}{*{20}{c}} 0&1 \\ 0&{ - 2} \end{array}} \right),\;{E_{a21}} = \left( {\begin{array}{*{20}{c}} 2&0 \\ 0&1 \end{array}} \right),} \\ {{E_{a12}} = \left( {\begin{array}{*{20}{c}} 0&1 \\ 0&{ - 2} \end{array}} \right),\;{E_{a22}} = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&2 \end{array}} \right),} \\ {{E_{b11}} = \left( {\begin{array}{*{20}{c}} {0.3}&0 \\ {2.2}&{0.3} \end{array}} \right),\;{E_{b21}} = \left( {\begin{array}{*{20}{c}} {0.1}&{0.2} \\ {0.3}&{0.4} \end{array}} \right),} \\ {{E_{b12}} = \left( {\begin{array}{*{20}{c}} {0.2}&0 \\ {2.2}&{0.3} \end{array}} \right),\;{E_{b22}} = \left( {\begin{array}{*{20}{c}} {0.1}&2 \\ {0.3}&{0.4} \end{array}} \right),} \\ {{F_{a11}}(t) = {F_{a21}}(t) = {F_{a12}}(t) = {F_{a22}}(t) = \left( {\begin{array}{*{20}{c}} {\cos t}&0 \\ 0&{\sin t} \end{array}} \right).} \end{array}$$

Letting α = 0.5 and applying Theorem 1, the upper bound \(\bar h = 0.3863\) of h is computed.

5 Conclusions

This paper has proposed a switching design for the exponential stability of uncertain linear switching fuzzy time-delay systems. Delay-dependent stability conditions are established by using Lyapunov approach. The perturbations considered are norm-bounded and the results are expressed in terms of LMIs. The numerical examples have been provided to demonstrate the effectiveness and applicability of the proposed method.