Tracking control for a class of switched non-linear systems with time-varying delay

Abstract

This paper investigates the problem of tracking control for a class of switched non-linear systems with time-varying delay. Firstly, based on the average dwell time method, a new exponential stability criterion for a class of switched non-linear systems with time-varying delay is derived; it is shown that this new criterion can provide fewer conservative results than those in the existing references. Then, tracking control for the systems is investigated, and a kind of state feedback control law and switching signal are proposed to satisfy the H_∞ model reference tracking performance. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

Keywords

Switched non-linear systems time-varying delay tracking control average dwell time

Introduction

Switched systems are an important class of hybrid dynamical systems that consist of several subsystems and a rule that orchestrates the switching among them. In recent years, switched systems have attracted considerable attention due to their significance both in theory development and practical applications (Agrachev and Liberzon, 2001; Engell et al., 2000; Horowitz and Varaiya, 2000; Liberzon and Morse, 1999; Xiang and Wang, 2009). A different switching rule would cause different behaviour of the system and hence lead to different system performances. Because of the complexity of the designing switched law for the systems, the stability analysis of switched systems becomes more difficult and attracts the interest of many scientists. To date, many works in the field of stability analysis and stabilization for switched systems have appeared (Cheng et al., 2005; Lin and Antsaklis, 2009; Mahmoud, 2010; Mahmoud and Elferik, 2010; Mahmoud and Xia, 2009, 2011; Sun, 2004; Sun and Ge, 2005). Meanwhile, when dealing with the stability analysis and stabilization for switched systems, common Lyapunov functions (Cheng, 2004), single Lyapunov functions (Wang and Zhao, 2010) and multiple Lyapunov functions (Branicky, 1998) are often considered as useful tools for switched systems.

It is well known that time delay is the inherent characteristic of many dynamic systems, which will lead to instability and poor performance. Thus, many researches have been done on the time delay systems (He et al., 2004; Liu and Su, 1998; Liu et al., 2008; Richard, 2003; Sun et al., 2006; Xia et al., 2010a, 2010b; Zhang and Yu, 2009). Zhang and Yu (2009) investigated the stability analysis for discrete-time switched time-delay systems. Xia et al. (2010b) considered the robust adaptive sliding mode control for uncertain discrete-time systems with time delay. In recent years, special attention has been paid to switched non-linear systems with time delay (Alwan and Liu, 2008; Liu and Yuan, 2011; Xiang and Chen, 2010; Xiang et al., 2011), and several useful results have been reported in the literature, such as the issues on stability analysis, robust control, $H_{\infty}$ filtering, and so on.

Tracking control is one of the most important issues currently under consideration by researchers in linear and non-linear systems. It has two categories: state tracking and output tracking. Tracking control is widely used in robot control (Fukao et al., 2000; Qu and Dorsey, 1991; Zhou et al., 1996). The main objective of tracking control is to make the output of the plant, via a controller, track the output of a given reference model as closely as possible. It has been universally known that tracking control design is more general and more difficult than stabilization, and in the past few decades, only a few results on tracking control for switched systems with time delay have been reported (Li et al., 2007, 2008, 2011). The issue of tracking control of switched linear systems with time delay is considered (Li et al., 2009), and a solution of robust tracking control of a class of switched non-linear systems is presented (Wang et al., 2009). However, to the best of our knowledge, the problem of tracking control of switched non-linear systems with time-varying delay has not been fully investigated, and this constitutes the main motivation of the present study.

In this paper, we investigate tracking control of a class of switched non-linear systems with time-varying delay. Based on the average dwell time method and Gronwall–Bellman inequality technique, a new exponential stability criterion for a class of switched non-linear systems with time-varying delay is derived; it is shown that this new criterion can provide fewer conservative results than those in the existing references. Then, we propose a non-linear state feedback control for exponential stability of a class of switched systems with Lipschitz non-linearity; the obtained control law can ensure the $H_{\infty}$ model reference tracking performance. The rest of the paper is organized as follows. In Section 2, problem formulation and some necessary lemmas are given. In Section 3, based on the average dwell time approach and matrix inequality analysis technique, $H_{\infty}$ tracking performance analysis and $H_{\infty}$ tracking controller have been considered. Two numerical examples illustrate the effectiveness of the proposed approach in Section 4. Concluding remarks are given in Section 5.

Notations

Throughout this paper, the superscript ‘T’ denotes the transpose, and the symmetric terms in matrices are denoted by *. The notation $X > 0 (X \geq, <, \leq 0)$ means that $X$ is a positive definite (positive semi-definite, negative definite, negative semi-definite, respectively). $R^{n}$ denotes the $n$ -dimensional Euclidean space. $‖ x (t) ‖$ denotes the Euclidean norm. $L_{2} [t_{0}, \infty)$ is the space of square integrable functions on $[t_{0}, \infty)$ and $t_{0}$ is the initial time. $λ_{max} (P)$ and $λ_{min} (P)$ denote the maximum and minimum eigenvalues of matrix $P$ , respectively. $I$ is an identity matrix with appropriate dimension. $diag {a_{i}}$ denotes a diagonal matrix with the diagonal elements $a_{i}, i = 1, 2, \dots, n$ .

Problem formulation and preliminaries

Consider the following switched non-linear systems with time-varying delay:

\overset{\cdot}{x} (t) = f_{σ} (x (t)) + A_{σ} x (t) + D_{σ} x (t - τ (t)) + B_{σ} u (t) + E_{σ (t)} w (t),

(1)

x (t) = φ (t), t \in [- τ, 0],

(2)

where $x (t) \in R^{n}$ is the state vector, $u (t) \in R^{l}$ is the control input, $f_{i} (\cdot) : R^{n} \to R^{n}$ is a non-linear function, $w (t) \in R^{p}$ is the noise input, which is assumed to belong to $L_{2} [t_{0}, \infty)$ , $τ (t)$ denotes the time-varying delay satisfying $0 \leq τ (t) \leq τ$ and $φ (t)$ is a continuous vector-valued initial function. The function $σ (t) : [t_{0}, \infty) \to \underline{N} = {1, 2, \dots, N}$ is the switching signal, which is deterministic, piecewise constant and right continuous, corresponding to it, the switching sequence $σ (t) : {(t_{0}, σ (t_{0})), (t_{1}, σ (t_{1})), \dots, (t_{k}, σ (t_{k})) \dots}, σ (t_{k}) \in \underline{N}, k = 0, 1, \dots$ , where $t_{0}$ is the initial time and $t_{k}$ denotes the $k$ th switching instant of the system. Moreover, $σ (t) = i$ means that the $i$ th subsystem is activated. $N$ denotes the number of subsystems. $A_{i}, B_{i}, D_{i}, E_{i}, i \in {1, 2, \dots, N}$ are known constant matrices with appropriate dimensions. Without loss of generality, we shall make the following assumptions for system (1) and (2).

Assumption 2.1 The functions $f_{i} (x (t))$ are Lipschitz for all $x (t) \in R^{n}$ and $\hat{x} (t) \in R^{n}$ , and satisfy

‖ f_{i} (x (t)) - f_{i} (\hat{x} (t)) ‖ \leq ‖ U_{i} (x (t) - \hat{x} (t)) ‖,

(3)

where $U_{i} \in R^{n \times n}$ are known real constant matrices.

Assumption 2.2 The matrices $B_{i}, i \in \underline{N}$ in (1) are full row rank.

The following lemma also will be used in the derivation of the main results.

Lemma 2.1 (Hale, 1977) (Gronwall–Bellman inequality) Let $x (t)$ , $y (t)$ be real-valued non-negative continuous functions with domain ${t | t \geq t_{0}}$ , where $a$ is a non-negative scalar, if the following inequality

x (t) \leq a + \int_{t_{0}}^{t} x (s) y (s) ds,

holds, for $t \geq t_{0}$ , then $x (t) \leq a \exp (\int_{t_{0}}^{t} y (s) ds)$ .

The reference model is given as

{\overset{\cdot}{x}}_{r} (t) = A_{r} x_{r} (t) + r (t),

(4)

where $x_{r} (t) \in R^{n}$ is the reference state and $r (t)$ is energy-bounded reference input. $A_{r}$ is a Hurwitz matrix with appropriate dimensions.

Combining (1) and (4), we obtain the augmented system:

[\begin{matrix} \overset{\cdot}{x} (t) \\ {\overset{\cdot}{x}}_{r} (t) \end{matrix}] = [\begin{matrix} f_{σ} (x (t)) + A_{σ} x (t) + D_{σ} x (t - τ (t)) + B_{σ} u (t) \\ A_{r} x_{r} (t) \end{matrix}] + [\begin{matrix} w (t) \\ r (t) \end{matrix}] .

(5)

Define the difference between the real state of the switched system (1) and the reference state as $e (t) = x (t) - x_{r} (t)$ , $\bar{w} (t) = {[w^{T} (t) E_{σ (t)}^{T} r^{T} (t) M^{T}]}^{T}$ .

Design the feedback control law:

u (t) = K_{1 σ} x (t) + K_{2 σ} x_{r} (t) - B_{σ}^{T} (B_{σ} B_{σ}^{T})^{- 1} f_{σ} (0),

(6)

where $K_{1 σ}, K_{2 σ}$ is the state feedback gain.

Remark 2.1 Traditionally, it is assumed that the function $f_{σ} (x)$ vanishes at $x = 0$ , and the state feedback controller has the form $u (t) = K_{1 σ} x (t) + K_{2 σ} x_{r} (t)$ . However, in this paper we modify this linear state feedback controller to deal with the systems that do not follow this characteristic. The control law (6) has the additional term $B_{σ}^{T} (B_{σ} B_{σ}^{T})^{- 1} f_{σ} (0)$ , which is essential in order to deal with systems having $f_{σ} (0) \neq 0$ . Then we obtain a sufficient condition for exponential stability and $H_{\infty}$ tracking performance of the switched non-linear system in Section 3.

Remark 2.2 In Assumption 2.2, matrix B needs to be full row rank, so the system has multi input. If the non-linear part in (1) satisfies $f_{σ} (0) = 0$ , then we do not require that Assumption 2.2 holds. Under this case, we can design control law $u (t) = K_{1 σ} x (t) + K_{2 σ} x_{r} (t)$ and the system can be single or multi input.

For a fixed switching signal $σ (t) = i$ , that is, the $i$ th subsystem is activated. The augmented systems (5) can be rewritten as

\overset{\cdot}{\bar{x}} (t) = {\bar{A}}_{i} \bar{x} (t) + {\bar{D}}_{i} \bar{x} (t - τ (t)) + {\bar{I}}_{i} [f_{i} (x (t)) - f_{i} (0)] + \bar{w} (t),

(7)

where $\bar{x} (t) = [\begin{matrix} x (t) \\ x_{r} (t) \end{matrix}]$ , ${\bar{A}}_{i} = [\begin{matrix} A_{i} + B_{i} K_{1 i} & B_{i} K_{2 i} \\ 0 & A_{r} \end{matrix}]$ , ${\bar{D}}_{i} = [\begin{matrix} D_{i} & 0 \\ 0 & 0 \end{matrix}]$ , ${\bar{I}}_{i} = [\begin{matrix} I \\ 0 \end{matrix}]$ .

By defining

{\hat{A}}_{σ} = [\begin{matrix} A_{σ} & 0 \\ 0 & A_{r} \end{matrix}], {\hat{B}}_{σ} = [\begin{matrix} B_{σ} \\ 0 \end{matrix}], K_{σ} = [\begin{matrix} K_{1 σ} & K_{2 σ} \end{matrix}],

we have ${\bar{A}}_{σ} = {\hat{A}}_{σ} + {\hat{B}}_{σ} K_{σ}$ , and the controller can be rewritten as

u (t) = K_{σ} \bar{x} (t) - B_{σ}^{T} (B_{σ} B_{σ}^{T})^{- 1} f_{σ} (0) .

(8)

Definition 2.1 Consider system (1), if there exist switching signal $σ = σ (t)$ and control input $u = u (t)$ , such that the trajectories of system (1) satisfy $‖ x (t) ‖ \leq α ‖ x (t_{0}) ‖_{cl} e^{- β (t - t_{0})}$ when $w (t) \equiv 0$ , where $‖ x (t_{0}) ‖_{cl} = {sup}_{- τ \leq θ \leq 0} {x (t + θ), \overset{\cdot}{x} (t + θ)}$ , $α \geq 1, β > 0, t \geq t_{0}$ , then system (1) is said to be exponentially stable with convergence rate $β$ .

Definition 2.2 For system (5), if there exist switching signal $σ = σ (t)$ and control input $u = u (t)$ , and the following conditions hold:

system (5) is exponentially stable when $\bar{w} (t) \equiv 0$ ;

system (5) satisfies $\int_{t_{0}}^{\infty} e^{- α (t - t_{0})} e^{T} (t) e (t) dt \leq γ^{2} \int_{t_{0}}^{\infty} {\bar{w}}^{T} (t) \bar{w} (t) dt$ when $\bar{w} (t) \neq 0$ under zero initial conditions, that is, $x (t) = 0$ , $x_{r} (0) = 0$ , $t \in [- τ, 0]$ ,

then system (1) is said to have weighted $H_{\infty}$ performance $γ$ , where $α, γ > 0$ are positive constants.

Definition 2.3 For any $T_{2} > T_{1} > 0$ , let $N_{σ} (T_{1}, T_{2})$ denote the switching number of $σ (t)$ on an interval $(T_{1}, T_{2})$ , if

N_{σ} (T_{1}, T_{2}) \leq N_{0} + (T_{2} - T_{1}) / τ_{a},

(9)

holds for given $N_{0} \geq 0$ , $τ_{a} > 0$ . Then, the constant $τ_{a}$ is called the average dwell time and $N_{0}$ is the chatter bound.

Main results

Stability and $H_{\infty}$ performance analysis

This section is concerned with the stability and the $H_{\infty}$ performance of the following switched non-linear systems with time-varying delay:

\overset{\cdot}{\bar{x}} (t) = {\bar{A}}_{σ} \bar{x} (t) + {\bar{D}}_{σ} \bar{x} (t - τ (t)) + {\bar{I}}_{σ} [f_{σ} (x (t)) - f_{σ} (0)] + \bar{w} (t) .

(10)

Theorem 3.1 Consider the augmented system (10) with $\bar{w} (t) = 0$ , if there exist positive definite matrices $P_{i}, S_{i} > 0$ and matrices $X_{i}, Y_{i}, Z_{i}$ , scalars $τ, α > 0$ , such that the following matrix inequality holds:

[\begin{matrix} φ_{i} & P_{i} {\bar{D}}_{i} + Y_{i}^{T} - X_{i} & P_{i} {\bar{I}}_{i} + Z_{i}^{T} & τ {\bar{A}}_{i}^{T} S_{i} & τ X_{i} & {\bar{U}}_{i}^{T} \\ * & - Y_{i} - Y_{i}^{T} & - Z_{i}^{T} & τ {\bar{D}}_{i}^{T} S_{i} & τ Y_{i} & 0 \\ * & * & - I & τ {\bar{I}}_{i}^{T} S_{i} & τ Z_{i} & 0 \\ * & * & * & - τ S_{i} & 0 & 0 \\ * & * & * & * & - τ S_{i} & 0 \\ * & * & * & * & * & - I \end{matrix}] < 0,

(11)

and the average dwell time satisfies

τ_{a} \geq τ_{a}^{*} = \frac{\ln μ}{α},

(12)

then the system is exponentially stable, where $φ_{i} = {\bar{A}}_{i}^{T} P_{i} + P_{i} {\bar{A}}_{i} + α P_{i} + X_{i}^{T} + X_{i}$ , ${\bar{U}}_{i} = [\begin{matrix} U_{i} & 0 \end{matrix}]$ , and $μ \geq 1$ satisfying

P_{i} \leq μ P_{j}, S_{i} \leq μ S_{j}, \forall i, j \in \underline{N} .

(13)

Proof Choose the following piecewise Lyapunov functional candidate for system (10):

V (t) = V_{σ} (\bar{x} (t)) = V_{1 σ} (\bar{x} (t)) + V_{2 σ} (\bar{x} (t))

The form of each $V_{i} (x)$ ( $i \in \underline{N}$ ) is given by

\begin{matrix} V_{1 i} (\bar{x} (t)) = {\bar{x}}^{T} (t) P_{i} \bar{x} (t) \\ V_{2 i} (\bar{x} (t)) = \int_{- τ}^{0} \int_{t + θ}^{t} {\overset{\cdot}{\bar{x}}}^{T} (s) S_{i} \overset{\cdot}{\bar{x}} (s) dsd θ \end{matrix},

(14)

which is positive definite since $P_{i}, S_{i}$ are positive definite matrices.

Along the trajectories of system (10), the time derivative of $V_{i} (\bar{x} (t))$ is given by

\begin{matrix} {\overset{\cdot}{V}}_{1 i} (\bar{x} (t)) = 2 {\overset{\cdot}{\bar{x}}}^{T} (t) P_{i} \bar{x} (t) \\ = {\bar{x}}^{T} (t) ({\bar{A}}_{i}^{T} P_{i} + P_{i} {\bar{A}}_{i}) \bar{x} (t) + {\bar{x}}^{T} (t) P_{i} \bar{w} (t) + {\bar{w}}^{T} (t) P_{i} \bar{x} (t) \\ + {\bar{x}}^{T} (t) P_{i} {\bar{D}}_{i} \bar{x} (t - τ (t)) + {\bar{x}}^{T} (t - τ (t)) {\bar{D}}_{i}^{T} P_{i} \bar{x} (t) + {\bar{x}}^{T} (t) P_{i} {\bar{I}}_{i} [f_{i} (x (t)) - f_{i} (0)] + {[f_{i} (x (t)) - f_{i} (0)]}^{T} {\bar{I}}_{i}^{T} P_{i} \bar{x} (t) . \end{matrix}

(15)

Inequality (3) can be written as

{[f_{i} (x (t)) - f_{i} (0)]}^{T} [f_{i} (x (t)) - f_{i} (0)] \leq x^{T} (t) U_{i}^{T} U_{i} x (t) = {\bar{x}}^{T} (t) {\bar{U}}_{i}^{T} {\bar{U}}_{i} \bar{x} (t) .

(16)

From (15) and (16), we have

\begin{matrix} {\overset{\cdot}{V}}_{1 i} (\bar{x} (t)) \leq {\bar{x}}^{T} (t) ({\bar{A}}_{i}^{T} P_{i} + P_{i} {\bar{A}}_{i} + {\bar{U}}_{i}^{T} {\bar{U}}_{i}) \bar{x} (t) \\ + {\bar{x}}^{T} (t) P_{i} {\bar{D}}_{i} \bar{x} (t - τ (t)) + {\bar{x}}^{T} (t - τ (t)) {\bar{D}}_{i}^{T} P_{i} \bar{x} (t) \\ + {\bar{x}}^{T} (t) P_{i} {\bar{I}}_{i} [f_{i} (x (t)) - f_{i} (0)] + {[f_{i} (x (t)) - f_{i} (0)]}^{T} {\bar{I}}_{i}^{T} P_{i} \bar{x} (t) \\ - {[f_{i} (x (t)) - f_{i} (0)]}^{T} [f_{i} (x (t)) - f_{i} (0)] + {\bar{x}}^{T} (t) P_{i} \bar{w} (t) + {\bar{w}}^{T} (t) P_{i} \bar{x} (t), \end{matrix}

(17)

{\overset{\cdot}{V}}_{2 i} (t) = τ {\overset{\cdot}{\bar{x}}}^{T} (t) S_{i} \overset{\cdot}{\bar{x}} (t) - \int_{t - τ}^{t} {\overset{\cdot}{\bar{x}}}^{T} (s) S_{i} \overset{\cdot}{\bar{x}} (s) ds \leq τ {\overset{\cdot}{\bar{x}}}^{T} (t) S_{i} \overset{\cdot}{\bar{x}} (t) - \int_{t - τ (t)}^{t} {\overset{\cdot}{\bar{x}}}^{T} (s) S_{i} \overset{\cdot}{\bar{x}} (s) ds .

(18)

On the other hand, the Newton–Leibniz formula gives

\bar{x} (t) - \bar{x} (t - τ (t)) = \int_{t - τ (t)}^{t} \overset{\cdot}{\bar{x}} (s) ds .

(19)

Then, for any appropriately dimensioned matrices $ψ_{i} = {[\begin{matrix} X_{i}^{T} & Y_{i}^{T} & Z_{i}^{T} \end{matrix}]}^{T}$ , we have

2 ξ^{T} (t) ψ_{i} [\bar{x} (t) - \bar{x} (t - τ (t)) - \int_{t - τ (t)}^{t} \overset{\cdot}{\bar{x}} (s) ds] = 0,

(20)

where $ξ (t) \overset{Δ}{=} {[\begin{matrix} {\bar{x}}^{T} (t) & {\bar{x}}^{T} (t - τ (t)) & {[f_{i} (x (t)) - f_{i} (0)]}^{T} \end{matrix}]}^{T}$ .

Considering (17), (18) and (20), when $\bar{w} (t) = 0$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{i} (\bar{x} (t)) + α V_{1 i} (\bar{x} (t)) \leq ξ^{T} (t) (Π_{i} + τ ψ_{i} S_{i}^{- 1} ψ_{i}^{T}) ξ (t) \\ - \int_{t - τ (t)}^{t} [ψ_{i}^{T} ξ (t) + S_{i} \overset{\cdot}{\bar{x}} (s)]^{T} S_{i}^{- 1} [ψ_{i}^{T} ξ (t) + S_{i} \overset{\cdot}{\bar{x}} (s)] ds \end{matrix},

(21)

where

Π_{i} = [\begin{matrix} ϕ_{11 i} & ϕ_{12 i} & P_{i} {\bar{I}}_{i} + τ {\bar{A}}_{i}^{T} S_{i} {\bar{I}}_{i} + Z_{i}^{T} \\ * & ϕ_{22 i} & τ {\bar{D}}_{i}^{T} S_{i} {\bar{I}}_{i} - Z_{i}^{T} \\ * & * & - I + τ {\bar{I}}_{i}^{T} S_{i} {\bar{I}}_{i} \end{matrix}],

with $ϕ_{11 i} = {\bar{A}}_{i}^{T} P_{i} + P_{i} {\bar{A}}_{i} + α P_{i} + {\bar{U}}_{i}^{T} {\bar{U}}_{i} + X_{i}^{T} + X_{i} + τ {\bar{A}}_{i}^{T} S_{i} {\bar{A}}_{i}$ , $ϕ_{12 i} = P_{i} {\bar{D}}_{i} + Y_{i}^{T} - X_{i} + τ {\bar{A}}_{i}^{T} S_{i} {\bar{D}}_{i}$ , $ϕ_{22 i} = - Y_{i} - Y_{i}^{T} + τ {\bar{D}}_{i}^{T} S_{i} {\bar{D}}_{i}$ .

Note that

\int_{t - τ (t)}^{t} [ψ_{i}^{T} ξ (t) + S_{i} \overset{\cdot}{\bar{x}} (s)]^{T} S_{i}^{- 1} [ψ_{i}^{T} ξ (t) + S_{i} \overset{\cdot}{\bar{x}} (s)] ds > 0 .

(22)

By the Schur complement, (11) implies

Π_{i} + τ ψ_{i} S_{i}^{- 1} ψ_{i}^{T} < 0 .

(23)

Thus, it follows from (21)–(23) that

{\overset{\cdot}{V}}_{i} (\bar{x} (t)) + α V_{1 i} (\bar{x} (t)) \leq 0, i \in \underline{N} .

(24)

Assume that the $i$ th subsystem is activated during $t \in [\begin{matrix} t_{k} & t_{k + 1} \end{matrix})$ , $k = 1, 2, \dots$ ; using (13) and (14), we have

V_{i} (\bar{x} (t_{k})) \leq μ V_{i} (\bar{x} (t_{k}^{-})),

(25)

V_{1 i} (\bar{x} (t_{k})) \leq μ V_{1 j} (\bar{x} (t_{k})), \forall i, j \in \underline{N} .

(26)

From (24)–(26), for any $t \in [t_{k}, t_{k + 1})$ , we have that

\begin{matrix} V_{1 i} (\bar{x} (t)) \leq V_{i} (\bar{x} (t)) = V_{i} (\bar{x} (t_{k})) + \int_{t_{k}}^{t} {\overset{\cdot}{V}}_{i} (\bar{x} (s)) ds \\ \leq μ V_{j} (\bar{x} (t_{k}^{-})) - α \int_{t_{k}}^{t} V_{1 i} (\bar{x} (s)) ds \\ \leq μ V_{j} (\bar{x} (t_{k - 1})) + μ α \int_{t_{k - 1}}^{t_{k}} {\overset{\cdot}{V}}_{j} (\bar{x} (s)) ds - α \int_{t_{k}}^{t} V_{1 i} (\bar{x} (s)) ds \\ \leq μ V_{j} (\bar{x} (t_{k - 1})) - α \int_{t_{k - 1}}^{t} V_{1 i} (\bar{x} (s)) ds \\ \leq \dots \\ \leq μ^{k} V_{j} (\bar{x} (t_{0})) - α \int_{t_{0}}^{t} V_{1 i} (\bar{x} (s)) ds . \end{matrix}

(27)

According to Definition 2.3 and Lemma 2.1, we obtain that

V_{1 i} (\bar{x} (t)) \leq μ^{N_{σ} (t_{0}, t)} V (\bar{x} (t_{0})) e^{- α (t - t_{0})} \leq μ^{N_{0}} e^{- (α - \frac{\ln μ}{τ_{a}}) (t - t_{0})} V (\bar{x} (t_{0})) .

(28)

Moreover, it can be obtained that

a ‖ \bar{x} (t) ‖^{2} \leq b μ^{N_{0}} ‖ \bar{x} (t_{0}) ‖_{cl}^{2} e^{- (α - \frac{\ln μ}{τ_{a}}) (t - t_{0})},

(29)

where $a = min_{i \in \underline{N}} λ_{min} (P_{i})$ , $b = max_{i \in \underline{N}} λ_{max} (P_{i}) + \frac{τ^{2}}{2} max_{i \in \underline{N}} λ_{max} (S_{i})$ .

Therefore

‖ \bar{x} (t) ‖ \leq \sqrt{\frac{b}{a}} μ^{\frac{N o}{2}} e^{- \frac{1}{2} (α - \frac{\ln μ}{τ_{a}}) (t - t_{0})} {‖ \bar{x} (t_{0}) ‖}_{c l} .

(30)

By Definition 2.1, we know that system (10) is exponentially stable. This completes the proof of Theorem 3.1.

The inequality (17) and (18) shows the relation between $V_{i} (\bar{x} (t))$ and $\bar{w} (t)$ , which is useful in the following theorem about $H_{\infty}$ tracking performance analysis.

Theorem 3.2 Consider the augmented system (10); if there exist positive definite matrices $P_{i}, S_{i} > 0$ and matrices $X_{i}, Y_{i}, Z_{i}$ , scalars $τ, α, γ > 0$ , such that the following matrix inequality holds:

[\begin{matrix} Ω_{i} & P_{i} {\bar{D}}_{i} + Y_{i}^{T} - X_{i} & P_{i} {\bar{I}}_{i} + Z_{i}^{T} & P_{i} & τ {\bar{A}}_{i}^{T} & τ X_{i} & {\bar{U}}_{i}^{T} \\ * & - Y_{i} - Y_{i}^{T} & - Z_{i}^{T} & 0 & τ {\bar{D}}_{i}^{T} & τ Y_{i} & 0 \\ * & * & - I & 0 & τ {\bar{I}}_{i}^{T} & τ Z_{i} & 0 \\ * & * & * & - γ^{2} I & τ I & 0 & 0 \\ * & * & * & * & - τ S_{i}^{- 1} & 0 & 0 \\ * & * & * & * & * & - τ S_{i} & 0 \\ * & * & * & * & * & * & - I \end{matrix}] < 0,

(31)

then the system is exponentially stable with an $H_{\infty}$ tracking performance $γ$ , where $Ω_{i} = {\bar{A}}_{i}^{T} P_{i} + P_{i} {\bar{A}}_{i} + \bar{Q} + α P_{i} + X_{i}^{T} + X_{i}$ , $\bar{Q} = [\begin{matrix} I & - I \\ - I & I \end{matrix}]$ .

Proof From Theorem 3.1, we know that system (10) is exponentially stable with $\bar{w} (t) = 0$ . In the sequel, we shall prove that the $H_{\infty}$ disturbance attenuation performance of system (10) is guaranteed for all non-zero $\bar{w} (t) \in L_{2} [t_{0}, \infty)$ under the zero initial condition.

By (14)–(20), and using the same method as in Theorem 3.1, we can obtain

{\overset{\cdot}{V}}_{i} (\bar{x} (t)) + α V_{1 i} (\bar{x} (t)) + e^{T} (t) e (t) - γ^{2} {\bar{w}}^{T} (t) \bar{w} (t) \leq 0 .

(32)

Let $Γ (s) = e^{T} (s) e (s) - γ^{2} {\bar{w}}^{T} (s) \bar{w} (s)$ .

From (9), (25), (26) and (32), we have

\begin{matrix} V_{1 i} (\bar{x} (t)) \leq V_{i} (\bar{x} (t)) = V_{i} (\bar{x} (t_{k})) + \int_{t_{k}}^{t} {\overset{\cdot}{V}}_{i} (\bar{x} (s)) ds \\ \leq μ V_{j} (\bar{x} (t_{k}^{-})) + μ \int_{t_{k}}^{t} V_{1 i} (\bar{x} (s)) ds - \int_{t_{k}}^{t} Γ (\bar{x} (s)) ds \\ \leq μ V_{j} (\bar{x} (t_{k - 1})) + μ \int_{t_{k - 1}}^{t_{k}} {\overset{\cdot}{V}}_{j} (\bar{x} (\bar{x} (s))) ds \\ - α \int_{t_{k}}^{t} V_{1 i} (\bar{x} (s)) ds - \int_{t_{k}}^{t} Γ (\bar{x} (s)) ds \\ \leq μ V_{j} (\bar{x} (t_{k - 1})) - μ α \int_{t_{k - 1}}^{t_{k}} V_{1 j} (\bar{x} (s)) ds \\ - α \int_{t_{k}}^{t} V_{1 i} (\bar{x} (s)) ds - μ \int_{t_{k - 1}}^{t_{k}} Γ (\bar{x} (s)) ds - \int_{t_{k}}^{t} Γ (\bar{x} (s)) ds \\ \leq μ V_{j} (\bar{x} (t_{k - 1})) - α \int_{t_{k - 1}}^{t} V_{1 i} (\bar{x} (s)) ds \\ - μ \int_{t_{k - 1}}^{t_{k}} Γ (\bar{x} (s)) ds - \int_{t_{k}}^{t} Γ (\bar{x} (s)) ds \\ \leq \dots \\ \leq μ^{k} V (\bar{x} (t_{0})) - α \int_{t_{0}}^{t} V_{1 i} (\bar{x} (s)) ds \\ - μ^{k} \int_{t_{0}}^{t_{1}} Γ (\bar{x} (s)) ds - \dots - μ \int_{t_{k - 1}}^{t_{k}} Γ (\bar{x} (s)) ds - \int_{t_{k}}^{t} Γ (\bar{x} (s)) ds \\ = μ^{N_{σ} (t_{0}, t)} V (\bar{x} (t_{0})) - α \int_{t_{0}}^{t} V_{1 i} (\bar{x} (s)) ds - \int_{t_{0}}^{t} μ^{N_{σ} (s, t)} Γ (\bar{x} (s)) ds . \end{matrix}

(33)

Under zero initial conditions, that is, $V (\bar{x} (t_{0})) = 0$ , and by Lemma 2.1, (33) becomes

0 \leq - \int_{t_{0}}^{t} e^{N_{σ} (s, t) \ln μ} Γ (s) ds .

(34)

Multiplying both sides of (34) by $e^{- N_{σ} (t_{0}, t) \ln μ}$ yields

\int_{t_{0}}^{t} e^{- N_{σ} (t_{0}, s) \ln μ} e^{T} (s) e (s) ds \leq \int_{t_{0}}^{t} e^{- N_{σ} (t_{0}, s) \ln μ} γ^{2} {\bar{w}}^{T} (s) \bar{w} (s) ds .

(35)

Note that $N_{σ} (t_{0}, s) \leq (s - t_{0}) / τ_{a}$ , and $τ_{a} > \ln μ / α$ ; we have

N_{σ} (t_{0}, s) \ln μ \leq α (s - t_{0}) .

(36)

Therefore, it follows from (35) and (36) that

\int_{t_{0}}^{t} e^{- α (s - t_{0})} e^{T} (s) e (s) ds \leq \int_{t_{0}}^{t} γ^{2} {\bar{w}}^{T} (s) \bar{w} (s) ds .

When $t \to \infty$ , it leads to

\int_{t_{0}}^{\infty} e^{- α (s - t_{0})} e^{T} (s) e (s) ds \leq γ^{2} \int_{t_{0}}^{\infty} {\bar{w}}^{T} (s) \bar{w} (s) ds .

(37)

This means that system (10) achieves $H_{\infty}$ tracking performance $γ$ . This completes the proof.

Remark 3.1 It should be noted that, in order to obtain the exponential stability criterion, Lyapunov functional $V_{i} (t) = x^{T} (t) P_{i} x (t) + \int_{t - τ}^{t} e^{- α (t - s)} x^{T} (s) Q_{i} x (s) ds$ , which has the exponential term $e^{- α (t - s)}$ , is often chosen. However, the term $e^{- α (t - s)}$ is avoided in this paper, and we can also obtain the exponential stability criterion, which can provide fewer conservative results than those in the existing references. The conservatism analysis will be given in Remark 3.3.

Remark 3.2 The exponential stability criterion of switched non-linear systems with time-varying delay is given in Theorem 3.1. When $f_{σ} (x (t)) = 0$ , system (10) degenerates into the following system, which was considered in Li et al. (2008):

\overset{\cdot}{\bar{x}} (t) = {\bar{A}}_{σ} \bar{x} (t) + {\bar{D}}_{σ} \bar{x} (t - τ (t)) + \bar{w} (t) .

(38)

Using the same method, we can get the exponential stability criteria as follows.

Corollary 3.1 Consider system (38), for given positive constants $α, γ, τ$ , if there exist positive definite matrices $P_{i}$ and $S_{i}$ , any matrices $Y_{i}$ and $T_{i}$ with appropriate dimensions, such that

Θ_{1 i} = [\begin{matrix} {\bar{A}}_{i}^{T} P_{i} + P_{i} {\bar{A}}_{i} + α P_{i} + Y_{i} + Y_{i}^{T} + \bar{Q} & P_{i} {\bar{D}}_{i} + T_{i}^{T} - Y_{i} & - Y_{i} & {\bar{A}}_{i}^{T} S_{i} & P_{i} \\ * & - T_{i}^{T} - T_{i} & - T_{i} & {\bar{D}}_{i}^{T} S_{i} & 0 \\ * & * & - τ^{- 1} S_{i} & 0 & 0 \\ * & * & * & - τ^{- 1} S_{i} & 0 \\ * & * & * & * & - γ^{2} I \end{matrix}] < 0,

(39)

holds, and the average dwell time satisfies (12), then the system is exponentially stable and the $H_{\infty}$ tracking performance $γ$ is achieved.

Remark 3.3 According to the result in Li et al. (2008), system (38) is exponentially stable and the $H_{\infty}$ tracking performance $γ$ is achieved when

Θ_{i} = [\begin{matrix} {\bar{A}}_{i}^{T} P_{i} + P_{i} {\bar{A}}_{i} + α P_{i} + Y_{i} + Y_{i}^{T} + \bar{Q} & P_{i} {\bar{D}}_{i} + T_{i}^{T} - Y_{i} & - Y_{i} & {\bar{A}}_{i}^{T} S_{i} & P_{i} \\ * & - T_{i}^{T} - T_{i} & - T_{i} & {\bar{D}}_{i}^{T} S_{i} & 0 \\ * & * & - τ^{- 1} e^{- α τ} S_{i} & 0 & 0 \\ * & * & * & - τ^{- 1} S_{i} & 0 \\ * & * & * & * & - γ^{2} I \end{matrix}] < 0,

(40)

where $Θ_{i} = Θ_{1 i} + Θ_{2 i}$ :

Θ_{2 i} = [\begin{matrix} 0 & 0 & 0 & 0 \\ * & 0 & 0 & 0 & 0 \\ * & * & τ^{- 1} (1 - e^{- α τ}) S_{i} & 0 & 0 \\ * & * & * & 0 & 0 \\ * & * & * & * & 0 \end{matrix}] .

(41)

For any $α, τ > 0$ and $S_{i} > 0$ , we have $Θ_{2 i} \geq 0$ , which means $Θ_{2 i}$ is semi-positive definite. Thus, if (40) is satisfied, then (39) holds. However, we would not obtain (40) from (39). This implies that the exponential stability criteria proposed in this paper is less conservative than the existing result (Li et al., 2008).

$H_{\infty}$ tracking controller design

In this section, we solve the $H_{\infty}$ tracking controller design problem based on Theorem 3.2. The controller design procedure is provided in the following theorem.

Theorem 3.3 Consider system (5); there exists a state feedback controller of the form (8), such that the augmented closed-loop system (7) achieves the $H_{\infty}$ tracking performance $γ$ , if there exist matrices $R_{i} > 0, S_{1 i} > 0, S_{2 i} > 0$ , matrices $L_{1 i}, L_{2 i}$ , ${\bar{X}}_{i} = [\begin{matrix} X_{11 i} & X_{12 i} \\ X_{21 i} & X_{22 i} \end{matrix}],$ ${\bar{Y}}_{i} = [\begin{matrix} Y_{11 i} & Y_{12 i} \\ Y_{21 i} & Y_{22 i} \end{matrix}],$ $\bar{Z} = [\begin{matrix} Z_{1 i} & Z_{2 i} \end{matrix}],$ and scalars $τ, α, γ > 0$ satisfying

[\begin{matrix} ψ_{11 i} & ψ_{12 i} & ψ_{13 i} & Y_{21 i}^{T} - X_{12 i} & I + Z_{1 i}^{T} & I & 0 & ψ_{18 i} & 0 & τ X_{11 i} & τ X_{12 i} & R_{i} U_{i}^{T} & R_{i} \\ * & ψ_{22 i} & ψ_{23 i} & Y_{22 i}^{T} - X_{22 i} & Z_{2 i}^{T} & 0 & I & τ L_{2 i}^{T} B_{i}^{T} & τ R_{i} A_{r}^{T} & τ X_{21 i} & τ X_{22 i} & 0 & - R_{i} \\ * & * & - Y_{11 i}^{T} - Y_{11 i} & - Y_{21 i}^{T} - Y_{12 i} & - Z_{1 i}^{T} & 0 & 0 & τ R_{i} D_{i}^{T} & 0 & τ Y_{11 i} & τ Y_{12 i} & 0 & 0 \\ * & * & * & - Y_{22 i}^{T} - Y_{22 i} & - Z_{2 i}^{T} & 0 & 0 & 0 & 0 & τ Y_{21 i} & τ Y_{22 i} & 0 & 0 \\ * & * & * & * & - I & 0 & 0 & τ I & 0 & τ Z_{1 i} & τ Z_{2 i} & 0 & 0 \\ * & * & * & * & * & - γ^{2} I & 0 & τ I & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & - γ^{2} I & 0 & τ I & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & - τ S_{1 i} & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & - τ S_{2 i} & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & * & ψ_{1010 i} & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & * & * & ψ_{1111 i} & 0 & 0 \\ * & * & * & * & * & * & * & * & * & * & * & - I & 0 \\ * & * & * & * & * & * & * & * & * & * & * & * & - I \end{matrix}],

(42)

where $ψ_{11 i} = A_{i} R_{i} + R_{i} A_{i}^{T} + B_{i} L_{1 i} + L_{1 i}^{T} B_{i} + α R_{i} + X_{11 i} + X_{11 i}^{T}$ , $ψ_{12 i} = B_{i} L_{2 i} + X_{12 i} + X_{21 i}^{T}$ , $ψ_{13 i} = D_{i} R_{i} + Y_{11 i}^{T} - X_{11 i}$ , $ψ_{18 i} = τ R_{i} A_{i}^{T} + τ L_{1 i}^{T} B_{i}^{T}$ , $ψ_{22 i} = A_{r} R_{i} + R_{i} A_{r}^{T} + α R_{i} + X_{22 i} + X_{22 i}^{T}$ , $ψ_{23 i} = Y_{12 i}^{T} - X_{21 i}$ , $ψ_{1010 i} = τ (S_{1 i} - 2 R_{i})$ , $ψ_{1111 i} = τ (S_{2 i} - 2 R_{i})$ and the average dwell time satisfies (12), $μ \geq 1$ , satisfying

R_{i} \leq μ R_{j}, S_{1 i} \leq μ S_{1 j}, S_{2 i} \leq μ S_{2 j} \forall i, j \in \underline{N} .

(43)

Moreover, the gain matrix of a desired controller of the form (8) is given by

K_{i} = [\begin{matrix} K_{1 i} & K_{2 i} \end{matrix}],

(44)

with $K_{1 i} = L_{1 i} R_{i}^{- 1}, K_{2 i} = L_{2 i} R_{i}^{- 1}$ .

Proof Performing a congruence transformation to (31) by $diag {P_{i}^{- 1}, P_{i}^{- 1}, I, I, I, P_{i}^{- 1}, I}$ , together with the change of matrix variables defined by

{\bar{P}}_{i} = P_{i}^{- 1}, {\bar{S}}_{i} = S_{i}^{- 1}, {\hat{Q}}_{i} = P_{i}^{- 1} \bar{Q} P_{i}^{- 1}, {\bar{X}}_{i} = P_{i}^{- 1} X_{i} P_{i}^{- 1}, {\bar{Y}}_{i} = P_{i}^{- 1} Y_{i} P_{i}^{- 1}, {\bar{Z}}_{i} = Z_{i} P_{i}^{- 1},

(45)

we obtain that (46) holds.

[\begin{matrix} Ψ_{i} & {\bar{D}}_{i} {\bar{P}}_{i} + {\bar{Y}}_{i}^{T} - {\bar{X}}_{i} & {\bar{I}}_{i} + {\bar{Z}}_{i}^{T} & I_{i} & τ {\bar{P}}_{i} {\bar{A}}_{i}^{T} & τ {\bar{X}}_{i} & {\bar{P}}_{i} {\bar{U}}_{i}^{T} \\ * & - {\bar{Y}}_{i} - {\bar{Y}}_{i}^{T} & - {\bar{Z}}_{i}^{T} & 0 & τ {\bar{P}}_{i} {\bar{D}}_{i}^{T} & τ {\bar{Y}}_{i} & 0 \\ * & * & - I & 0 & τ {\bar{I}}_{i}^{T} & τ {\bar{Z}}_{i} & 0 \\ * & * & * & - γ^{2} I & τ I & 0 & 0 \\ * & * & * & * & - τ {\bar{S}}_{i} & 0 & 0 \\ * & * & * & * & * & - τ {\bar{P}}_{i} {\bar{S}}_{i}^{- 1} {\bar{P}}_{i} & 0 \\ * & * & * & * & * & * & - I \end{matrix}] < 0,

(46)

where $Ψ_{i} = {\bar{A}}_{i} {\bar{P}}_{i} + {\bar{P}}_{i} {\bar{A}}_{i}^{T} + {\bar{X}}_{i} + {\bar{X}}_{i}^{T} + α {\bar{P}}_{i} + {\hat{Q}}_{i}$ .

Further, by noticing that ${\bar{S}}_{i}, {\bar{P}}_{i} > 0$ , we have $({\bar{S}}_{i} - {\bar{P}}_{i}) {\bar{S}}_{i}^{- 1} ({\bar{S}}_{i} - {\bar{P}}_{i}) \geq 0$ , which are equivalent to

- {\bar{P}}_{i} {\bar{S}}_{i}^{- 1} {\bar{P}}_{i} \leq {\bar{S}}_{i} - 2 {\bar{P}}_{i} .

(47)

From (46) and (47), (48) will hold:

[\begin{matrix} Ψ_{i} & {\bar{D}}_{i} {\bar{P}}_{i} + {\bar{Y}}_{i}^{T} - {\bar{X}}_{i} & {\bar{I}}_{i} + {\bar{Z}}_{i}^{T} & I_{i} & τ {\bar{P}}_{i} {\bar{A}}_{i}^{T} & τ {\bar{X}}_{i} & {\bar{P}}_{i} {\bar{U}}_{i}^{T} \\ * & - {\bar{Y}}_{i} - {\bar{Y}}_{i}^{T} & - {\bar{Z}}_{i}^{T} & 0 & τ {\bar{P}}_{i} {\bar{D}}_{i}^{T} & τ {\bar{Y}}_{i} & 0 \\ * & * & - I & 0 & τ {\bar{I}}_{i}^{T} & τ {\bar{Z}}_{i} & 0 \\ * & * & * & - γ^{2} I & τ I & 0 & 0 \\ * & * & * & * & - τ {\bar{S}}_{i} & 0 & 0 \\ * & * & * & * & * & τ ({\bar{S}}_{i} - 2 {\bar{P}}_{i}) & 0 \\ * & * & * & * & * & * & - I \end{matrix}] < 0 .

(48)

Define ${\bar{P}}_{i} = [\begin{matrix} R_{i} & 0 \\ * & R_{i} \end{matrix}], {\bar{S}}_{i} = [\begin{matrix} S_{1 i} & 0 \\ * & S_{2 i} \end{matrix}], {\bar{X}}_{i} = [\begin{matrix} X_{11 i} & X_{12 i} \\ X_{21 i} & X_{22 i} \end{matrix}], {\bar{Y}}_{i} = [\begin{matrix} Y_{11 i} & Y_{12 i} \\ Y_{21 i} & Y_{22 i} \end{matrix}], \bar{Z} = [\begin{matrix} Z_{1 i} & Z_{2 i} \end{matrix}]$ , then

{\hat{Q}}_{i} = [\begin{matrix} R_{i} R_{i} & - R_{i} R_{i} \\ * & R_{i} R_{i} \end{matrix}] .

(49)

Denoting $L_{1 i} = K_{1 i} R_{i}, L_{2 i} = K_{2 i} R_{i}$ , substituting ${\bar{A}}_{i} = {\hat{A}}_{i} + {\hat{B}}_{i} K_{i}$ and (49) into (48), and using the Schur complement, we readily get Theorem 3.3. The proof is completed.

Numerical examples

In this section we present two examples to illustrate the effectiveness of the proposed approach.

Example 1 Consider switched system (38) with the following parameters:

Subsystem 1

A_{1} = [\begin{matrix} - 1.45 & 0.25 \\ 0.15 & - 2.5 \end{matrix}], D_{1} = [\begin{matrix} 0.1 & 0.15 \\ 0.25 & 0 \end{matrix}],

Subsystem 2

A_{2} = [\begin{matrix} - 1.5 & 0.1 \\ 0.2 & - 2 \end{matrix}], D_{2} = [\begin{matrix} 0.1 & 0.1 \\ 0.2 & 0 \end{matrix}] .

and $γ = 1, α = 0.8$ , $τ (t) = 0.25 + 0.25 \sin t$ , so $τ = 0.5$ , solving the linear matrix inequality (LMI) (39), we can obtain that

P_{1} = [\begin{matrix} 0.9467 & - 0.3627 \\ - 0.3627 & 1.7683 \end{matrix}], S_{1} = [\begin{matrix} 0.0706 & 0.1882 \\ 0.1882 & 0.9491 \end{matrix}],

P_{2} = [\begin{matrix} 0.9721 & - 0.3410 \\ - 0.3410 & 1.4607 \end{matrix}], S_{2} = [\begin{matrix} 0.2165 & 0.2828 \\ 0.2828 & 0.6191 \end{matrix}],

Then, according to (13), we can get $μ = 4.7557$ ; thus, $τ_{a} \geq τ_{a}^{*}$ $= \frac{\ln μ}{α} = 1.9492$ , which means that the above switched time-varying system is exponentially stable. In order to illustrate that the exponential stability criterion proposed in this paper is less conservative than the existing results, we consider the switched time-varying system with same parameters as above, but we find that the LMIs (40) have no feasible solutions. This demonstrates that our result is less conservative.

Example 2 Consider switched system (1) and reference system (4) with parameters as follows:

A_{1} = [\begin{matrix} 3.5 & 1.5 \\ 1.5 & 4.5 \end{matrix}], B_{1} = [\begin{matrix} - 0.6 & - 0.2 \\ 0.2 & - 0.4 \end{matrix}], D_{1} = [\begin{matrix} 0.25 & 0.15 \\ 0.25 & - 0.15 \end{matrix}],

A_{2} = [\begin{matrix} 4.5 & 2 \\ - 1.5 & 3 \end{matrix}], B_{2} = [\begin{matrix} - 0.5 & 0.1 \\ - 0.1 & - 0.25 \end{matrix}], D_{2} = [\begin{matrix} - 0.1 & 0.1 \\ 0.2 & 0 \end{matrix}],

A_{r} = [\begin{matrix} - 5 & 0.5 \\ - 1.5 & - 4.5 \end{matrix}] .

It is assumed that $α = 2$ , $τ (t) = 0.15 + 0.15 \sin t$ , $γ = 0.5$ :

f_{1} (x (t)) = [\begin{matrix} 0.1 \cos (0.01 x_{1}) \\ 0.1 \cos (0.01 x_{2}) \end{matrix}], f_{2} (x (t)) = [\begin{matrix} 0.2 \cos (0.01 x_{1}) \\ 0.2 \cos (0.01 x_{2}) \end{matrix}],

where $f_{1} (0) = [\begin{matrix} 0.1 & 0.1 \end{matrix}]^{T} \neq 0, f_{2} (0) = [\begin{matrix} 0.2 & 0.2 \end{matrix}]^{T} \neq 0$ .

The Lipschitz matrices are given by

U_{1} = [\begin{matrix} 0.1 & 0 \\ 0 & 0.1 \end{matrix}], U_{2} = [\begin{matrix} 0.2 & 0 \\ 0 & 0.2 \end{matrix}] .

The aim is to design a state feedback controller such that the closed-loop system (7) is stable with $H_{\infty}$ tracking performance $γ$ .

Let $w (t) = r (t) = {\begin{array}{l} 2 \frac{\sin t}{t} & 5 \leq t \leq 17 \\ 0 & o t h e r s \end{array}$ , for $τ = 0.3$ ; by solving the matrix inequalities in Theorem 3.3, we obtain

R_{1} = [\begin{matrix} 1.3703 & 0.1296 \\ 0.1296 & 1.5330 \end{matrix}], R_{2} = [\begin{matrix} 1.3905 & 0.0967 \\ 0.0967 & 1.5533 \end{matrix}],

S_{11} = [\begin{matrix} 2.3211 & 0.2350 \\ 0.2350 & 2.6663 \end{matrix}], S_{21} = [\begin{matrix} 2.3172 & 0.2658 \\ 0.2658 & 2.5898 \end{matrix}],

S_{21} = [\begin{matrix} 2.2853 & 0.1690 \\ 0.1690 & 2.6023 \end{matrix}], S_{22} = [\begin{matrix} 2.2953 & 0.1972 \\ 0.1972 & 2.5275 \end{matrix}],

K_{1} = [\begin{matrix} 11.6445 & - 4.3938 & - 0.0431 & 0.0921 \\ 10.1561 & 21.3301 & - 0.4484 & - 0.6089 \end{matrix}]

K_{2} = [\begin{matrix} 16.3970 & 10.4759 & - 0.9619 & - 0.6835 \\ - 11.6764 & 31.8686 & - 0.2609 & - 3.0608 \end{matrix}]

Then, according to (43), we can get $μ = 22.6229$ , then $τ_{a}^{*} = 1.5595$ . Choosing $τ_{a} = 1.6$ , the simulation results are as shown in Figures 1 –4, where initial conditions $x (t) = 0, τ \in [- 0.3, 0),$ $x (0) = [0.4, - 0.3]^{T}$ and $x_{r} (0) = [- 0.2, 0.5]^{T}$ .

The switching signal $σ (t)$ , with average dwell time $τ_{a} = 1.6$ , is shown in Figure 1. Figures 2 and 3 indicate the state of the switched system and the given reference system, respectively. Figure 4 shows the error between the state of the switched system and the reference state.

From Figures 2 –4, it can be observed that the designed controller can guarantee the stability and $H_{\infty}$ tracking performance of the closed-loop system. This testifies the efficiency of the proposed method.

Conclusions

In this paper, we have studied the problem of tracking control for a class of switched non-linear systems with time-varying delay. A new exponential stability criterion for a class of switched non-linear systems with time-varying delay is derived; it is shown that this new criterion can provide fewer conservative results than the existing ones. Then, the $H_{\infty}$ tracking controller is designed by solving a set of LMIs. Two illustrative examples have demonstrated the effectiveness of the proposed approach.

Footnotes

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 60974027) and NUST Research Funding (2011YBXM26).

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Tracking control for a class of switched non-linear systems with time-varying delay

Abstract

Keywords

Introduction

Notations

Problem formulation and preliminaries

Main results

Stability and H ∞ performance analysis

H ∞ tracking controller design

Numerical examples

Conclusions

Footnotes

Funding

References

Stability and $H_{\infty}$ performance analysis

$H_{\infty}$ tracking controller design