Asynchronous $H_{\infty }$ Control of Switched Uncertain Discrete-Time Fuzzy Systems via Basis-Dependent Multiple Lyapunov Functions Approach

Abstract

This paper first investigates robust stability of open-loop switched uncertain discrete-time fuzzy systems (SUDFSs) under mode-dependent average dwell time (MDADT) switching. By a basis-dependent multiple Lyapunov functions (BLFs) approach, which has more flexibility than the multiple quadratic Lyapunov functions approach, computable robust stability conditions are presented in terms of linear matrix inequalities (LMIs). Then, the investigation is extended to robust $H_{\infty }$ control of closed-loop SUDFSs by using the same approach. The asynchronous state feedback $H_{\infty }$ controllers which can stabilize the SUDFSs and guarantee weighted $H_{\infty }$ performance are obtained by solving a set of LMIs. A numerical example and a practical example are provided to show the advantage of the proposed approach.

1 Introduction

Switched systems, which are made up of several continuous-time or discrete-time subsystems and the switching rules among them, have attracted a large number of researchers’ interest in recent years not only due to its excellent modeling ability for actual system, but also the applicability in relevant results of the existing control theory. It can be applied to many practical systems such as the flight control systems, the network control systems, the automobile engine systems [7, 16, 19, 20, 23]. Switched systems’ features, especially the stability and stabilization, are researched deeply in recent years, and many important results have been derived [1, 4, 7, 8].

Stability issue has always been important research content of switched systems. According to the property of switching signal, two stability issues have been addressed, i.e., the stability under arbitrary switching and the stability under constrained switching. For the arbitrary switching, we often try to construct a common Lyapunov function (CLF) for all subsystems [7]. But it is difficult to find this function in general. Due to this approach’s conservatism, some scholars put forward an improved approach in discrete-time domain by using the switched Lyapunov functions (SLFs) [4]. Despite the arbitrary switching’s stability issue, researchers are also interested in constrained switchings. The constrained switching is divided into time-constrained switching and state-constrained switching. We mainly discuss the time-constrained switching signals here. The average dwell time (ADT) and mode-dependent average dwell time (MDADT) logic are put forward to help research the time-constrained switching in [6, 28]. For these kinds of switching signals, many research results have been obtained by using the multiple Lyapunov-like functions (MLFs) approach [9, 12, 16, 23, 25, 26, 30,31,32]. Due to the switching signals’ particularity, if the declining rate of Lyapunov function in the system’s running time is high enough, then we can allow Lyapunov function rise to some extent at the switching moment. In this paper’s research, we also need to adopt this basic thought.

Nowadays, most research results focus on switched linear systems, which is mainly because most research results of non-switched linear systems have been obtained in the past. However, research results about nonlinear switched systems seem to be scattered. Due to the nonlinearity of the system, it is hard to analyze it directly. In this paper, we use the Takagi–Sugeno (T-S) fuzzy model to describe the nonlinear system. We can get the complex nonlinear switched systems’ local linear description for each fuzzy rule through T-S fuzzy model [21], and the whole nonlinear system can be expressed by using the “fuzzy blending,” and this can help us to investigate nonlinear systems’ stability issue by using linear systems’ research method. Some stability conditions and $H_{\infty }$ controllers design methods of non-switched nonlinear systems have been found, e.g., [2, 5, 10, 17, 22], but it seems that there exist few researches of switched nonlinear systems’ stability and controller design issues. A basis-dependent Lyapunov function [11, 29] is used to study the linear discrete-time systems’ stability and $H_{\infty }$ control issue. The basis-dependent Lyapunov functions depend on system’s fuzzy model, which lowers the difficulty in solving LMIs, while the single quadratic Lyapunov functions do not depend on it. Zhou et al. [29] had shown that the results of discrete-time nonlinear systems given by basis-dependent Lyapunov functions are less conservative than that given by the single quadratic Lyapunov functions which are basis-independent. Through combining switched systems’ switching information and fuzzy model, we apply basis-dependent multiple Lyapunov functions (BLFs) and multiple quadratic Lyapunov functions (QLFs) into SUDFSs’ stability analysis and $H_{\infty }$ control synthesis and compare the two approaches’ conservatism.

When dealing with practical problems, perturbation of systems’ characteristics or parameters is often unavoidable; therefore, robustness is needed to be considered for a control system. The $l_2$-gain also attracts more and more researchers’ attention due to its function of describing systems’ ability of constraining disturbing signal. Usually, in the case of guaranteeing systems’ stability we also hope it has good weighted $H_{\infty }$ performance. We need to find a controller which can guarantee that the $l_2$-gain from the disturbance to the estimation error is within a prescribed level. Some methodologies have been developed in recent years for the $H_{\infty }$ controller design, e.g., by using basis-dependent Lyapunov function, an $H_{\infty }$ control design approach is developed for a class of discrete-time fuzzy systems with uncertainty [29], two sufficient LMI conditions, which guarantee the existence of the $H_{\infty }$ controllers based on fuzzy observers for the T-S fuzzy systems have been proposed in [11]. However, most $H_{\infty }$ controllers are designed in the synchronous conditions. This circumstance is too ideal. In practice, it rarely holds since there is always a lag in the controller when the switching happens. Some time should be taken to identify the system modes and apply the matched controller. Asynchronous behaviors’ influence on the control of switched linear systems and nonlinear systems has been investigated in [12, 13, 30,31,32]. The research results have shown that developing $H_{\infty }$ controllers in asynchronous conditions is necessary. At present, some papers have investigated the design of discrete-time linear systems’ $H_{\infty }$ controllers in asynchronous conditions. For instance, the conditions of the existence of admissible asynchronous $H_{\infty }$ controllers are derived in [31]. $H_{\infty }$ controllers for a class of discrete-time switched linear parameter-varying systems with asynchronous switching are designed in [12]. Recently, there also exist some results on stability analysis and controller design of discrete-time switched nonlinear systems. For instance, stabilization of discrete-time switched nonlinear systems without stable subsystems is investigated in [14]. Robust stability and $H_{\infty }$ control of discrete-time switched T-S fuzzy systems with time-varying delays are investigated in [15]. However, in most of these papers, the controller is designed in synchronous conditions. The design of discrete-time nonlinear systems’ $H_{\infty }$ controller in asynchronous conditions remains unsolved.

The contributions of the paper focus on two points: first, SUDFSs’ stability condition and asynchronous state feedback $H_{\infty }$ controllers are obtained through BLFs approach. Second, we compare two different approaches’ performance to show the BLFs approach’s advantage. This paper is organized as follows: definitions and lemmas on stability and $l_2$-gain of switched systems are shown in Sect. 2. Main results are obtained in Sect. 3. Two numerical examples are presented to verify the feasibility and effectiveness of the proposed techniques in Sect. 4. We conclude the paper at last in Sect. 5.

1.1 Notations

The notation used in this paper is fairly standard. The superscript “T” stands for matrix transposition. The symbol “$*$” in a matrix stands for the transposed elements in the symmetric positions, and $diag\{\cdots \}$ stands for a block-diagonal matrix. $\mathbb R^n$ denotes the n-dimensional Euclidean space, $\mathbb N$ represents the set of nonnegative integers, and the notation $\parallel \cdot \parallel$ refers to the Euclidean vector norm. $l_2[ 0 ,\infty )$ is the space of square sum-able infinite sequence, and for $w={w(k)} \in l_2[ 0 , \infty )$, its norm is given by ${\parallel w \parallel }_2 = \sqrt{\sum \limits _{k=0}^{\infty }{\mid w(k)\mid }^2}$. $\mathbb C^1$ denotes the space of continuously differentiable functions, and a function $\alpha : [ 0 , \infty ) \rightarrow [ 0 , \infty )$ is said to be of class $\mathcal {K}_{\infty }$ if it is continuous, strictly increasing, unbounded, and $\alpha (0) = 0$. Also, a function $\beta : [ 0 , \infty ) \times [ 0 , \infty )$ is said to be of class $\mathcal {KL}$ if $\beta (\cdot ,t)$ is of class $\mathcal {K}$ for each fixed $t\ge 0$ and $\beta (s,t)$ decreases to 0 as $t \rightarrow \infty$ for each fixed $s\ge 0$.

2 System Descriptions and Preliminaries

Let us consider the following switched nonlinear systems described by

$$\begin{aligned} \left\{ \begin{array}{l} x(k+1) = {f_{\sigma (k)}}(x(k),u(k),w(k)) \\ y(k) = {s_{\sigma (t)}}(x(t),u(t),w(k)) \\ \end{array} \right. \end{aligned}$$

(1)

where $x(k) \in {\mathbb R}^{n_x}$ , $u(k) \in {\mathbb R}^{n_u}$ and $y(k) \in {\mathbb R}^{n_y}$ denote the state vector, input vector and output vector, respectively; $w(k) \in \mathbb R^{n_w}$ is the disturbance that belongs to $l_2[0,\infty )$; $f_{\sigma (k)}$ and $s_{\sigma (k)}$ are nonlinear functions; $\sigma (k)$ is defined as a switching signal, which is a piecewise constant function of time and takes its values in the finite set $\mathcal I=\{1,2,\ldots ,N\}$, where $N>1$ is the number of subsystems. For a switching sequence $0<k_0<k_1<\cdots<k_l<k_{l+1}<\cdots$, $\sigma (k)$ is continuous from right everywhere. $k_0$ is the initial time which can be any positive integer, and we assume that there exists no switching between any two adjacent switching times $k_l$ and $k_{l+1}$, $l\ge 0$. When $k \in [k_l,k_{l+1})$, $\sigma (k_l)=i$, $i \in \mathcal I$, we say the ith subsystem is activated.

The T-S fuzzy model which is described by fuzzy IF-THEN rules [21] is employed here to represent each subsystem of the switched nonlinear systems. The fuzzy model of ith subsystem is described as the following form:

Rule n: IF $v_{i1}(k)$ is $N _{i1m}$ and $\cdots$ and $v_{ig}(k)$ is $N _{igm}$, THEN

$$\begin{aligned} \left\{ \begin{array}{l} x(k+1) = \mathscr {A}_{im}x(k)+\mathscr {B}_{im}u(k)+E_{im}w(k) \\ y(k) = C_{im}x(k)+D_{im}u(k)+F_{im}w(k) \\ \end{array} \right. \end{aligned}$$

(2)

where $v_i(k)=(v_{i1}(k),v_{i2}(k),\ldots ,v_{ig}(k))$ are some measurable premise variables and $N _{ipm}(p = 1,2,\ldots ,g)$ are fuzzy sets. $\mathscr {A}_{im}$ , $\mathscr {B}_{im}$ ,$C_{im}$, $D_{im}$ , $E_{im}$ and $F_{im}$ are real matrices of the mth local model of the ith subsystem; among them matrix $\mathscr {A}_{im}$ and $\mathscr {B}_{im}$ have parametric uncertainties. $\mathscr {A}_{im}$ and $\mathscr {B}_{im}$ satisfy

$$\begin{aligned} \begin{bmatrix} \mathscr {A}_{im}&\mathscr {B}_{im} \end{bmatrix}=\begin{bmatrix} A_{im}&B_{im} \end{bmatrix} +M_{i}F(k)\begin{bmatrix} N_{1i}&N_{2i} \end{bmatrix} \end{aligned}$$

(3)

where $M_i$, $N_{1i}$ and $N_{2i}$ are known real constant matrices and F(k) is an unknown time-varying matrix function satisfying

$$\begin{aligned} F(k)^TF(k)\le I \quad \forall k \in \mathbb N \end{aligned}$$

(4)

Remark 1

Here we let $M_{i}$ and $N_{1i}$, $N_{2i}$ unchanged in different fuzzy rules; in other words, $M_{i}$ and $N_{1i}$, $N_{2i}$ are given linear matrixes which do not need to be fuzzified. This brings convenience to our research because the derivations will be very complex if $M_{i}$ and $N_{1i}$, $N_{2i}$ need to be fuzzified.

Through using “fuzzy blending,” the final output of the ith subsystem is inferred as follows:

$$\begin{aligned} \left\{ \begin{array}{l} x(k+1) = \sum \limits _{m=1}^{r_i}\sum \limits _{n=1}^{r_i}h_{im}h_{in}[\mathscr {A}_{imn}x(k)+E_{im}w(k)] \\ y(k) =\sum \limits _{m=1}^{r_i}\sum \limits _{n=1}^{r_i}h_{im}h_{in}[C_{imn}x(k)+F_{im}w(k)] \\ \end{array} \right. \end{aligned}$$

(5)

where $\mathscr {A}_{imn}=\mathscr {A}_{im}+\mathscr {B}_{im}K_{in}$ , $C_{imn}=C_{im}+D_{im}K_{in}$ , and

$$\begin{aligned} h_{im}=l_{im}/\sum \limits _{k=1}^{r_i}l_{im},\quad l_{im}=\prod \limits _{p=1}^{g}{ N }_{ipm}(v_{ip}(k)) \end{aligned}$$

(6)

in which $N _{ipm}(v_{ip}(k))$ is the grade of the membership function of $v_{ip}$ in $N _{ipm}$. It is assumed that $l_{im}\ge 0$ for all k and $m=1,2,\ldots ,r_i$. Therefore, the normalized membership function $h_{im}$ satisfies

$$\begin{aligned} h_{im}\ge 0 ,\sum \limits _{m=1}^{r_i}h_{im}=1 \quad \forall k \in \mathbb N. \end{aligned}$$

(7)

In view of the asynchronous behaviors, asynchronous control problem is considered. The controller u(k) is divided into two parts $\bar{u}(k)$ and $\hat{u}(k)$, where $\hat{u}(k)$ denotes the unmatched controller, and $\bar{u}(k)$ represents the matched controller. $\bar{u}(k)$ and $\hat{u}(k)$ have the following form:

$$\begin{aligned} \left\{ \begin{array}{l} \hat{u}(k) = \sum \limits _{n=1}^{r_i}h_{jn}K_{jn}x(k) , \quad k \in [k_l,k_l+T_i)\\ \bar{u}(k) = \sum \limits _{n=1}^{r_i}h_{in}K_{in}x(k) , \quad k \in [k_l+T_i,k_{l+1}) \\ \end{array} \right. \end{aligned}$$

(8)

where $k_l$ and $k_{l+1}$ are the beginning time and ending time of the activated subsystem $\sigma (k_l)$. When $k \in T[k_l,k_{l+1})$, $\forall l \in \mathbb {N}$, the subsystem $\sigma (k_l)=i$($i \in \mathcal I$) is activated. The energy function of the system when asynchronous switching occurs is shown in Fig. 1. It can be seen in Fig. 1 that $T[k_l,k_{l+1})$ is divided into $T[k_l,k_l+T_i)$ and $T[k_l+T_i,k_{l+1})$. During $T[k_l,k_l+T_i)$ period, the controller and the system are unmatched; then, the Lyapunov function may increase or decrease; $T_i$ represents the running time of the unmatched controller in $T[k_l,k_l+T_i)$, in other words, the ith subsystem’s asynchronous time in $T[k_l, k_l+T_i)$. During $T[k_l+T_i,k_{l+1})$ period, the controller and the system are matched. The Lyapunov function is strictly decreasing with the matched controller. $K_{in}$ and $K_{jn}$ are controller gain matrices which remain to be solved. Substituting (8) into (5), we can obtain the following closed-loop SUDFSs:

$$\begin{aligned} \left\{ \begin{array}{l} \left\{ \begin{array}{l} x(k+1) = \sum \limits _{m=1}^{r_i}\sum \limits _{n=1}^{r_i}h_{im}h_{jn}\hat{\mathscr {A}}_{imn}x(k)+\sum \limits _{m=1}^{r_i}h_{im}E_{im}w(k) \\ y(k)= \sum \limits _{m=1}^{r_i}\sum \limits _{n=1}^{r_i}h_{im}h_{jn}\hat{C}_{imn}x(k)+\sum \limits _{m=1}^{r_i}h_{im}F_{im}w(k) \\ \end{array} , \forall k \in [k_l,k_l+T_i)\right. \\ \left\{ \begin{array}{l} x(k+1) = \sum \limits _{m=1}^{r_i}\sum \limits _{n=1}^{r_i}h_{im}h_{in}\bar{\mathscr {A}}_{imn}x(k)+\sum \limits _{m=1}^{r_i}h_{im}E_{im}w(k) \\ y(k) = \sum \limits _{m=1}^{r_i}\sum \limits _{n=1}^{r_i}h_{im}h_{in}\bar{C}_{imn}x(k)+F_{im}w(k) \\ \end{array} , \forall k \in [k_l+T_i,k_{l+1})\right. \end{array} \right. \end{aligned}$$

(9)

Fig. 1

Energy function for the asynchronous switched system

where $\hat{\mathscr {A}}_{imn}=\mathscr {A}_{im}+\mathscr {B}_{im}K_{jn}$, $\hat{C}_{imn}=C_{im}+D_{im}K_{jn}$ and $\bar{\mathscr {A}}_{imn}=\mathscr {A}_{im}+\mathscr {B}_{im}K_{in}$, $\bar{C}_{imn}=C_{im}+D_{im}K_{in}$.

Our purpose in this paper is to design an asynchronous state feedback $H_{\infty }$ controller which can ensure weighted $H_{\infty }$ performance and GUAS of the SUDFS (9) under MDADT switching condition. The following definitions are first recalled.

Definition 1

[7] System (1) with $w(k)\equiv 0$ and $u(k)\equiv 0$ is globally uniformly asymptotically stable (GUAS) if there exists a class $\mathcal {KL}$ function $\beta$ such that for all switching signals $\sigma (k)$ and all initial condition $x(k_0)$, the solutions of (1) satisfy the following inequality

$$\begin{aligned} \Vert {x(k)}\Vert \le \beta (\Vert {x(k_0)}\Vert ,k), \quad \forall k \ge k_0. \end{aligned}$$

(10)

Definition 2

[28] For a switching signal $\sigma (k)$ and any $0\le k \le K$, let $N_{\sigma (k)i}(k,K)$ be the switching numbers of the ith subsystem activated during the interval [k, K], and $H_i(k,K)$ is the total running time of the ith subsystem in [k, K], $i \in \mathcal I$. We say that $\sigma (k)$ has a MDADT called $T_{oai}$, if there exist positive $N_{0i}$ (we call mode-dependent chattering bounds) and $T_{oai}$ such that

$$\begin{aligned} N_{\sigma (k)i}(k,K) \le N_{0i}+H_i(k,K)/T_{oai}, \quad 0\le k\le K. \end{aligned}$$

(11)

Remark 2

The MDADT $T_{oai}$ is mentioned here. If the switching signal of system (1) satisfies the MDADT property, its ith $(\forall i \in \mathcal I)$ subsystem’s average running time among the switching process should be larger than $T_{oai}$. In Fig. 1, we can see whether the subsystems’ average running time is long enough to offset the increment caused by the asynchronous phenomenon. The energy function of system (1) will converge to zero.

Definition 3

[33] For $0<\alpha <1$ and $\gamma >0$, system (1) is said to have a weighted $l_2$-gain no greater than $\gamma$, if under zero initial condition, the inequality $\sum \limits _{k=k_0}^{\infty }(1-\alpha )^ky_k^Ty_k\le \gamma ^2\sum \limits _{k=k_0}^{\infty }w_k^Tw_k$, $0<\alpha <1$ holds for all nonzero $w(k) \in l_2[ 0 , \infty )$.

Remark 3

The definition of non-weighted $l_2$-gain is different from the weighted one in Definition 3. For $\gamma >0$, system (1) is said to have a non-weighted $l_2$-gain no greater than $\gamma$, if the inequality $\sum \limits _{k=k_0}^{\infty }y_k^Ty_k\le \gamma ^2\sum \limits _{k=k_0}^{\infty }w_k^Tw_k$, holds for all nonzero $w(k) \in l_2[ 0 , \infty )$. The non-weighted one can be seen as a special case that the weighted index is 1. The non-weighted $l_2$-gain can describe the system’s input–output performance better than the weighted one. In [24], via a dwell time-dependent Lyapunov function approach, non-weighted $l_2$-gain results of uncertain discrete-time switched linear system under minimum dwell time switching are presented in terms of LMIs. However, this approach only applies to analysis of switched systems under minimum dwell time switching. There exist few results on non-weighted $l_2$-gain analysis of switched system under ADT switching and MDADT switching. In this paper, we mainly investigate the weighted $l_2$-gain of the SUDFSs under MDADT switching via the BLFs approach.

Lemma 1

[28] Consider SUDFS (9) with $w(k)\equiv 0$ and $u(k)\equiv 0$, and let $0<\alpha _{i}<1,\mu _{i}>1$, $i \in \mathcal I$ be given constants. If there exist positive definite $\mathbb C^1$ function $V_{\sigma (k_l)}:\mathbb R^n\rightarrow \mathbb R$, and class $\mathcal K_{\infty }$ functions $\kappa _{1i}$ and $\kappa _{2i}$, $i \in \mathcal I$ satisfying

$$\begin{aligned}&\kappa _{1i}(\Vert x_k\Vert )\le V_i(x_k) \le \kappa _{2i}(\Vert x_k\Vert ) \end{aligned}$$

(12)

$$\begin{aligned}&{V_i}({k_l}) \le {\mu _i}{V_j}({k_l} ) \end{aligned}$$

(13)

and

$$\begin{aligned} \Delta {V_i}(k)\le -\alpha _i {V_i}(k) ,\quad k \in [k_l,k_{l+1}). \end{aligned}$$

(14)

Then, SUDFS (9) is GUAS for any switching signal satisfying

$$\begin{aligned} T_{oai}\ge T_{oai}^*=\frac{-\ln {\mu _i}}{\ln (1-\alpha _i)} \end{aligned}$$

(15)

Remark 4

Lemma 1 is the stability result on discrete-time switched nonlinear system and has been proved in [28]. In [28], based on Lemma 1, some results on stability and stabilization of discrete-time switched linear systems under MDADT switching are given in terms of LMIs, and computable results of discrete-time switched nonlinear systems under MDADT switching are not given. In this paper, we utilize Lemma 1 to analyze stability of SUDFSs (9) and present some computable results in terms of LMIs.

Lemma 2

Given two matrices $A \in \mathbb R^{m\times n},B\in \mathbb R^{m\times n}$, and symmetric positive definite matrix $P \in \mathbb R^{m\times m}$, then

$$\begin{aligned} A^TPB+B^TPA \le A^TPA+B^TPB. \end{aligned}$$

Proof

From $P>0$, we have

$$\begin{aligned} (A-B)^TP(A-B)\ge 0, \end{aligned}$$

which is equal to

$$\begin{aligned} A^TPB+B^TPA \le A^TPA+B^TPB. \end{aligned}$$

Then, the proof is end.$\square$

Lemma 3

[29] Suppose $P_m > 0$, $P_n > 0$, and $P_u > 0$. If

$$\begin{aligned} A_m^TP_uA_m-P_m<0,\quad A_n^TP_uA_n-P_n<0, \end{aligned}$$

then

$$\begin{aligned} A_m^TP_uA_n+A_n^TP_uA_m-P_m-P_n<0. \end{aligned}$$

Lemma 4

[12] Consider SUDFS (9), and let $0<\alpha _i<1$, $\beta _i \ge 0$ and $\gamma _i>0$. $\forall i>0$ be given constants. Suppose that there exist positive $\mathbb C^1$ functions $V_{\sigma (k)}:R^n\rightarrow R,\sigma (k)\in \mathcal I$, with $V_{\sigma (k_0)}(x_{k_0})\equiv 0$ such that $\forall (i,j)\in \mathcal I\times \mathcal I ,i\ne j, V_i(x_{k_l})\le \mu _i V_j(x_{k_l})$ and $\forall i\in \mathcal I$, denoting $\varGamma (k)\triangleq y_k^Ty_k-\gamma _i^2\omega _k^T\omega _k$, if the following inequality is satisfied

$$\begin{aligned} \Delta V_i(x_k) \le \left\{ \begin{array}{l} -\alpha _iV_i(k)-\varGamma (k),\forall k \in [k_l,k_l+T_i)\\ \beta _iV_i(k)-\varGamma (k),\forall k \in [k_l+T_i,k_{l+1}) \end{array} \right. , \end{aligned}$$

(16)

then SUDFS (9) is GUAS for any switching signal satisfying

$$\begin{aligned} T_{oai}\ge T_{oai}^{*}=\frac{T_i(\ln {(1-\alpha _i)}-\ln {(1+\beta _i)})-\ln {\mu _i}}{\ln {(1-\alpha _i)}} \end{aligned}$$

(17)

and the $H_{\infty }$ performance no greater than $\hat{\gamma }=\sqrt{\prod \limits _{i \in \mathcal I}{}(\theta _i^{T_M}\mu _i)^{N_{0i}}\frac{\alpha _{\mathrm{max}}}{\alpha _{\mathrm{min}}}\theta _{\mathrm{max}}^{T_M-1}}\gamma$, where $T_i$ denotes the increasing interval of the ith subsystem, $T_M=\max \{T_i\}$, $\theta _{\mathrm{max}}=\max \{\theta _i\}=\max \{(1+\beta _i)/(1-\alpha _i)\}$, $\forall i \in \mathcal I$.

Remark 5

Lemma 4 has been proved in [12] and can apply to analysis of discrete-time switched nonlinear systems under asynchronous switching. In [12], based on Lemma 4, $H_{\infty }$ control of discrete-time switched linear parameter-varying systems with both MDADT and asynchronous switching is investigated. Some results are given in terms of LMIs. However, there exist no results on $H_{\infty }$ control of discrete-time nonlinear switched systems with both mode-dependent average dwell time (MDADT) and asynchronous switching. In this paper, we utilize Lemma 4 to analyze the problem of asynchronous $H_{\infty }$ control of SUDFSs (9) and fill the gap.

Lemma 5

[18] Given matrices P, M and N of appropriate dimensions and with P symmetrical, then

$$\begin{aligned} P+MF(k)N+N^TF^T(k)M^T<0 \end{aligned}$$

holds for all F(k) satisfying $F^T(k)F(k)\le I$ if and only if for some $\epsilon >0$

$$\begin{aligned} P+\epsilon ^{-1}MM^T+\epsilon N^TN<0. \end{aligned}$$

3 Main Results

3.1 Stability Analysis

In this section, we analyze the stability condition of open-loop SUBFSs (9) through BLFs and QLFs approach which have been mentioned before. The normal quadratic multiple Lyapunov function is like this form:

$$\begin{aligned} V_i(k)=x^T(k)P_ix(k), \end{aligned}$$

the basis-dependent multiple Lyapunov function assumes that

$$\begin{aligned} V_i(k)=x(k)^T\sum \limits _{u=1}^{r_i}h_{iu}P_{iu}x(k), \end{aligned}$$

this means:

$$\begin{aligned} P_i=\sum \limits _{u=1}^{r_i}h_{iu}P_{iu}. \end{aligned}$$

Theorem 1

Consider SUDFS (9) with $w(k)\equiv 0$ and $u(k)\equiv 0$ and let $0<\alpha _{i}<1$, $\mu _{i}>1$ be given constants, if there exist matrices $P_{im}>0$, for $\forall (i,j)\in \mathcal I\times \mathcal I$, $i\ne j$, $m=1,2,\ldots ,r_i$, $n=1,2,\ldots ,r_i$, $u=1,2,\ldots ,r_i$ satisfying

$$\begin{aligned}&P_{in}\le \mu _i P_{jm} \end{aligned}$$

(18)

$$\begin{aligned}&\mathscr {A}_{im}^TP_{iu}\mathscr {A}_{im}-(1-\alpha _i)P_{im}<0 \end{aligned}$$

(19)

then SUDFS (9) is GUAS for any switching signal satisfying (15).

Proof

We choose the BLFs which is described by $V_i(k)=x(k)^T\sum \limits _{u=1}^{r_i}h_{iu}P_{iu}x(k)$ to analysis open-loop SUDFS(9)’s stability conditions, due to $\sum \limits _{u=1}^{r_i}h_{iu}P_{iu}>0$, we surely can find $\kappa _{1i}$ and $\kappa _{2i}$ which satisfy (12). Make $P_i=\sum \limits _{u=1}^{r_i}h_{iu}P_{iu}$ and combine (9) we can get

from (21), we have

from Lemma 3, we can conclude that

$$\begin{aligned} \mathscr {A}_{im}^TP_{iu}\mathscr {A}_{in}+\mathscr {A}_{in}^TP_{iu}\mathscr {A}_{im}-(1-\alpha _i)P_{in}-(1-\alpha _i)P_{im}<0, \end{aligned}$$

then we can obtain (14). From (18), we can get $\sum \limits _{n=1}^{r_i}h_{in}P_{in}\le \mu _i \sum \limits _{m=1}^{r_i}h_{jm}P_{jm}$, which is equal to $x(k_l)^T\sum \limits _{n=1}^{r_i}h_{in}P_{in}x(k_l) \le \mu _i x(k_l)^T\sum \limits _{m=1}^{r_i}h_{jm}P_{jm}x(k_l)$, then (13) is derived. From Lemma 1, we can know the open-loop SUDFS (9) is GUAS for any switching signal satisfying (15). This completes the proof.$\square$

Theorem 2

Consider SUDFS (9) with $w(k)\equiv 0$ and $u(k)\equiv 0$ and let $0<\alpha _{i}<1$, $\mu _{i}>1$ be given constants, if there exist matrices $X_{im}>0$, matrices $\varOmega _i$, positive constant $\epsilon _i$ for $\forall (i,j)\in \mathcal I\times \mathcal I$, $i\ne j$, $m=1,2,\ldots ,r_i$, $n=1,2,\ldots ,r_i$, $u=1,2,\ldots ,r_i$ satisfying

$$\begin{aligned} X_{jm}\le \mu _iX_{in} \end{aligned}$$

(20)

$$\begin{aligned} \varXi _{imu}<0 \end{aligned}$$

(21)

where

$$\begin{aligned} \varXi _{imu}=\begin{bmatrix} \frac{X_{im}}{1-\alpha _i}-\varOmega _i^T-\varOmega _i&\varOmega _i^TA_{im}^T&0&\varOmega _i^TN_{1i}^T \\ *&-X_{iu}&\frac{M_i}{\epsilon _i}&0 \\ *&*&-\frac{I}{\epsilon _i}&0 \\ *&*&*&-\frac{I}{\epsilon _i} \end{bmatrix}. \end{aligned}$$

Then, SUDFS (9) is GUAS for any switching signal satisfying (15).

Proof

Consider the closed-loop SUDFS (9), applying the Schur complement to (21) yields:

$$\begin{aligned} \begin{bmatrix} \frac{X_{im}}{1-\alpha _i}-\varOmega _i^T-\varOmega _i&\varOmega _i^TA_{im}^T \\ *&-X_{iu} \end{bmatrix}+\epsilon _i\begin{bmatrix} 0 \\ \frac{M_i}{\epsilon _i} \end{bmatrix}\begin{bmatrix} 0 \\ \frac{M_i}{\epsilon _i} \end{bmatrix}^T+\epsilon _i\begin{bmatrix} \varOmega _i^TN_i^T \\ 0 \end{bmatrix}\begin{bmatrix} \varOmega _i^TN_i^T \\ 0 \end{bmatrix}^T<0, \end{aligned}$$

which is equal to

$$\begin{aligned} \begin{bmatrix} \frac{X_{im}}{1-\alpha _i}-\varOmega _i^T-\varOmega _i&\varOmega _i^TA_{im}^T \\ *&-X_{iu} \end{bmatrix}+\epsilon _{i}^{-1}\begin{bmatrix} 0 \\ M_i \end{bmatrix}\begin{bmatrix} 0 \\ M_i \end{bmatrix}^T+\epsilon _i\begin{bmatrix} \varOmega _i^TN_i^T \\ 0 \end{bmatrix}\begin{bmatrix} \varOmega _i^TN_i^T \\ 0 \end{bmatrix}^T<0. \end{aligned}$$

From Lemma 5, we can know:

$$\begin{aligned} \begin{bmatrix} \frac{X_{im}}{1-\alpha _i}-\varOmega _i^T-\varOmega _i&\varOmega _i^T\mathscr {A}_{im}^T \\ *&-X_{iu} \end{bmatrix}<0 \end{aligned}$$

which gives $0<\frac{X_{im}}{1-\alpha _i}<\varOmega _{i}^T + \varOmega _{i}$. This implies that $\varOmega _i$ is invertible. The inequalities $\left( \frac{X_{im}}{1-\alpha _i}-\varOmega _{i}\right) ^T{\left( \frac{X_{im}}{1-\alpha _i}\right) }^{-1}\left( \frac{X_{im}}{1-\alpha _i}-\varOmega _{i}\right) \ge 0$ imply that $\varOmega _{i}^T {(\frac{X_{im}}{1-\alpha _i})}^{-1}\varOmega _{i}\ge \varOmega _{i}^T + \varOmega _{j}-\frac{X_{im}}{1+\alpha _i}$, that is

$$\begin{aligned} \begin{bmatrix} -(1-\alpha _i)\varOmega _{i}^TX_{im}^{-1}\varOmega _{i}&\varOmega _i^T\mathscr {A}_{im}^T \\ *&-X_{iu} \end{bmatrix}<0. \end{aligned}$$

Note that $\varOmega _i$ is invertible. Pre-multiplying diag($\varOmega _i^{-T}$,I) and post-multiplying diag($\varOmega _i^{-1}$,I) to above inequality give

$$\begin{aligned} \begin{bmatrix} -(1-\alpha _i)X_{im}&\mathscr {A}_{im}^T \\ *&-X_{iu} \end{bmatrix}<0. \end{aligned}$$

Setting $X_{im}^{-1}= P_{im}$, we have

$$\begin{aligned} \begin{bmatrix} -(1-\alpha _i)P_{im}&\mathscr {A}_{im}^T \\ *&-P_{iu}^{-1} \end{bmatrix}<0. \end{aligned}$$

It follows from Schur complement equivalence that we can get (19), and (20) is equal to (18):

$$\begin{aligned} X_{jm}\le \mu _i X_{in} \Leftrightarrow X_{in}^{-1}\le \mu _i X_{jm}^{-1} \Leftrightarrow P_{in} \le \mu _i P_{jm}. \end{aligned}$$

From Theorem 1, we can know the open-loop SUDFS (9) is GUAS for any switching signal satisfying (15).

Remark 6

By setting $P_{im}=P_i$, $V_i(k)$ becomes $V_i(k)=x(k)^TP_ix(k)$ which is a multiple quadratic Lyapunov function which is widely used in related works. We get the following Corollary 1 where the QLFs are used. So the LMIs condition in Theorem 2 has less conservatism than those in Corollary 1.

Corollary 1

Consider SUDFS (9) with $w(k)\equiv 0$ and $u(k)\equiv 0$ and let $0<\alpha _{i}<1$, $\mu _{i}>1$ be given constants, if there exist matrices $X_{i}>0$, matrices $\varOmega _i$, positive constant $\epsilon _i$ for $\forall (i,j)\in \mathcal I\times \mathcal I$, $i\ne j$, $m=1,2,\ldots ,r_i$, $n=1,2,\ldots ,r_i$, satisfying

$$\begin{aligned} X_{jn}\le \mu _iX_{im} \end{aligned}$$

(22)

$$\begin{aligned} \varXi _{im}<0 \end{aligned}$$

(23)

where

$$\begin{aligned} \varXi _{im}=\begin{bmatrix} \frac{X_{i}}{1-\alpha _i}-\varOmega _i^T-\varOmega _i&\varOmega _i^TA_{im}^T&0&\varOmega _i^TN_{1i}^T \\ *&-X_{i}&\frac{M_i}{\epsilon _i}&0 \\ *&*&-\frac{I}{\epsilon _i}&0 \\ *&*&*&-\frac{I}{\epsilon _i} \end{bmatrix} \end{aligned}$$

Then, the open-loop SUDFS (9) is GUAS for any switching signal satisfying (15).

Proof

The proof is similar to the proof of Theorem 2 with the function $V_i(k)$ given by $V_i(k)=x(k)^TP_ix(k)$. It is omitted here.$\square$

3.2 Asynchronous $H_\infty$ Controller design

In this section, we are in a position to design the asynchronous $H_{\infty }$ controller for closed-loop SUDFSs (9). Through the BLFs approach and QLFs approach, we obtain the admissible controller’s existing conditions in the following theorem.

Theorem 3

Consider closed-loop SUDFS (9), and let $0<\alpha _i<1$, $\beta _i>0$, $\gamma >0$ and $\mu _i \ge 1$ be given constants. If there exist matrices $P_{im}>0$ for $\forall (i,j) \in \mathcal I\times \mathcal I$, $i\ne j$, $m=1,2,\ldots ,r_i$, $n=1,2,\ldots ,r_i$, $s=1,2,\ldots ,r_i$ such that

$$\begin{aligned}&P_{im} \le \mu _iP_{jn} \end{aligned}$$

(24)

$$\begin{aligned}&\Delta _{imn}^T \begin{bmatrix} P_{is}&0\\ 0&I \end{bmatrix} \Delta _{imn} -\begin{bmatrix} (1-\alpha _i)P_{im}&0 \\ 0&\gamma ^2 I \end{bmatrix} <0 \end{aligned}$$

(25)

$$\begin{aligned}&\varPsi _{imn}^T \begin{bmatrix} P_{is}&0\\ 0&I \end{bmatrix} \varPsi _{imn} -\begin{bmatrix} (1+\beta _i)P_{im}&0 \\ 0&\gamma ^2 I \end{bmatrix} <0 \end{aligned}$$

(26)

where

$$\begin{aligned} \Delta _{imn}=(\varPhi _{imn}+\varPhi _{inm})/2,\varPhi _{imn}=\begin{bmatrix} \bar{\mathscr {A}}_{imn}&E_{im}\\ \bar{C}_{imn}&F_{im} \end{bmatrix},\varPsi _{imn}=\begin{bmatrix} \hat{\mathscr {A}}_{imn}&E_{im}\\ \hat{C}_{imn}&F_{im} \end{bmatrix}, \end{aligned}$$

then closed-loop SUDFS (9) is GUAS for any switching signal satisfying (20) and has a weighted $H_{\infty }$ performance index no greater than $\hat{\gamma }=\sqrt{\prod \limits _{i \in \mathcal I}{}(\theta _i^{T_M}\mu _i)^{N_{0i}}\frac{\alpha _{\mathrm{max}}}{\alpha _{\mathrm{min}}}\theta _{\mathrm{max}}^{T_M-1}}\gamma$.

Proof

We choose the BLFs which has been defined in Theorem 1 to analyze stability conditions and $H_{\infty }$ performance of closed-loop SUDFS (9) here. Let $\xi _k=\begin{bmatrix}x_k^T \ \omega _k^T \end{bmatrix}^T$. First we consider the synchronous switching situation, considering $k \in T[k_l+T_i,k_{l+1})$, from SUDFS (9) we have that:

$$\begin{aligned}&\Delta V_k+\alpha _iV_k+y_k^Ty_k-\gamma ^2\omega _k^T\omega _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} h_{im}h_{in}h_{ip}h_{iq}\\&\quad \times \left( {\begin{bmatrix} \bar{\mathscr {A}}_{imn}^T \\ E_{im}^T \end{bmatrix}P_{is}\begin{bmatrix} \bar{\mathscr {A}}_{ipq}&E_{ip} \end{bmatrix} } \right. \left. {+ \begin{bmatrix} \bar{C}_{imn}^T \\ F_{im}^T \end{bmatrix}\begin{bmatrix} \bar{C}_{ipq}&F_{ip} \end{bmatrix}-\begin{bmatrix} (1-\alpha _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}} \right) \xi _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} h_{im}h_{in}h_{ip}h_{iq}\\&\quad \times \left( {\begin{bmatrix} \bar{\mathscr {A}}_{imn}^T&\bar{C}_{imn}^T \\ E_{im}^T&F_{im}^T \end{bmatrix}\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} \begin{bmatrix} \bar{\mathscr {A}}_{ipq}&E_{ip} \\ \bar{C}_{ipq}&F_{ip} \end{bmatrix} }\right. \left. { -\begin{bmatrix} (1-\alpha _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}} \right) \xi _k. \end{aligned}$$

Let $\begin{bmatrix} \bar{\mathscr {A}}_{imn}^T&\bar{C}_{imn}^T \\ E_{im}^T&F_{im}^T \end{bmatrix}=\varPhi _{imn}^T,\begin{bmatrix} \bar{\mathscr {A}}_{ipq}&E_{ip} \\ \bar{C}_{ipq}&F_{ip} \end{bmatrix}=\varPhi _{ipq}$, the above equation can be rewritten as

$$\begin{aligned}&\Delta V_k+\alpha _iV_k+y_k^Ty_k-\gamma ^2\omega _k^T\omega _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} h_{im}h_{in}h_{ip}h_{iq}\\&\quad \times \left( { \varPhi _{imn}^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} \varPhi _{ipq} }\right. \left. { -\begin{bmatrix} (1-\alpha _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}} \right) \xi _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} \frac{1}{4}h_{im}h_{in}h_{ip}h_{iq}\\&\quad \times \left( (\varPhi _{imn}+\varPhi _{inm})^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} (\varPhi _{ipq}+\varPhi _{iqp}) \right. \left. -4\begin{bmatrix} (1-\alpha _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix} \right) \xi _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} \frac{1}{8}h_{im}h_{in}h_{ip}h_{iq}\\&\quad \times \left( (\varPhi _{imn}+\varPhi _{inm})^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} (\varPhi _{ipq}+\varPhi _{iqp}) \right. \\&\quad \left. -8 \begin{bmatrix} (1-\alpha _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix} + (\varPhi _{ipq}+\varPhi _{iqp})^T \begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix}(\varPhi _{imn}+\varPhi _{inm}) \right) \xi _k. \end{aligned}$$

From Lemma 2, we can get the following inequality:

$$\begin{aligned}&\Delta V_k+\alpha _iV_k+y_k^Ty_k-\gamma ^2\omega _k^T\omega _k\\&\le \xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} \frac{1}{8}h_{im}h_{in}h_{ip}h_{iq}\\&\quad \times \left( (\varPhi _{imn}+\varPhi _{inm})^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} (\varPhi _{imn}+\varPhi _{inm}) \right. \\&\quad \left. -8 \begin{bmatrix} (1-\alpha _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix} + (\varPhi _{ipq}+\varPhi _{iqp})^T \begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix}(\varPhi _{ipq}+\varPhi _{iqp}) \right) \xi _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i} \frac{1}{4}h_{im}h_{in}\\&\quad \times \left( (\varPhi _{imn}+\varPhi _{inm})^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} (\varPhi _{imn}+\varPhi _{inm})- 4\begin{bmatrix}(1-\alpha _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}\right) \xi _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}h_{im}h_{in}\\&\quad \times \left( \Delta _{imn}^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix}\Delta _{inm}- \begin{bmatrix}(1-\alpha _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}\right) \xi _k. \end{aligned}$$

From (25), we can know

$$\begin{aligned} \Delta V_i(x_k) \le -\alpha _iV_i(k)-\varGamma (k),\quad \forall k \in T[k_l+T_i,k_{l+1}). \end{aligned}$$

(27)

Then, we consider the asynchronous switching situation. That is $k \in T[k_l,k_l+T_i)$, supposing the ith subsystem is activated and the former one is the jth subsystem, by the same procedure, from (9) we can know:

$$\begin{aligned}&\Delta V_k-\beta _iV_k+y_k^Ty_k-\gamma ^2\omega _k^T\omega _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} h_{im}h_{jn}h_{ip}h_{jq}\\&\quad \times \left( {\begin{bmatrix} \hat{\mathscr {A}}_{imn}^T \\ E_{im}^T \end{bmatrix}P_{is}\begin{bmatrix} \hat{\mathscr {A}}_{ipq}&E_{ip} \end{bmatrix} } \right. \left. {+ \begin{bmatrix} \hat{C}_{imn}^T \\ F_{im}^T \end{bmatrix}\begin{bmatrix} \hat{C}_{ipq}&F_{ip} \end{bmatrix}-\begin{bmatrix} (1+\beta _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}} \right) \xi _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} h_{im}h_{jn}h_{ip}h_{jq}\\&\quad \times \left( {\begin{bmatrix} \hat{\mathscr {A}}_{imn}^T&\hat{C}_{imn}^T \\ E_{im}^T&F_{im}^T \end{bmatrix}\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} \begin{bmatrix} \hat{\mathscr {A}}_{ipq}&E_{ip} \\ \hat{C}_{ipq}&F_{ip} \end{bmatrix} }\right. \left. { -\begin{bmatrix} (1+\beta _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}} \right) \xi _k. \end{aligned}$$

Let $\begin{bmatrix} \hat{\mathscr {A}}_{imn}^T&\hat{C}_{imn}^T \\ E_{im}^T&F_{im}^T \end{bmatrix}=\varPsi _{imn}^T,\begin{bmatrix} \hat{\mathscr {A}}_{ipq}&E_{ip} \\ \hat{C}_{ipq}&F_{ip} \end{bmatrix}=\varPsi _{ipq}$, the above equation can be rewritten as

$$\begin{aligned}&\Delta V_k-\beta _iV_k+y_k^Ty_k-\gamma ^2\omega _k^T\omega _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} h_{im}h_{jn}h_{ip}h_{jq}\\&\quad \times \left( { \varPsi _{imn}^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} \varPsi _{ipq} }\right. \left. { -\begin{bmatrix} (1+\beta _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}} \right) \xi _k\\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} \frac{1}{2}h_{im}h_{jn}h_{ip}h_{jq}\\&\quad \times \left( { \varPsi _{imn}^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} \varPsi _{ipq}+\varPsi _{ipq}^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} \varPsi _{imn} }\right. \left. { -2\begin{bmatrix} (1+\beta _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}} \right) \xi _k. \end{aligned}$$

From Lemma 2, we can get the following inequality:

$$\begin{aligned}&\Delta V_k-\beta _iV_k+y_k^Ty_k-\gamma ^2\omega _k^T\omega _k\\&\le \xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i}\sum \limits _{p=1}^{r_i}\sum \limits _{q=1}^{r_i} \frac{1}{2}h_{im}h_{jn}h_{ip}h_{jq}\\&\quad \times \left( { \varPsi _{imn}^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} \varPsi _{imn}+\varPsi _{ipq}^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} \varPsi _{ipq} }\right. \left. { -2\begin{bmatrix} (1+\beta _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}} \right) \xi _k \\&=\xi _k^T\sum \limits _{s=1}^{r_i}h_{is}\sum \limits _{n=1}^{r_i}\sum \limits _{m=1}^{r_i} h_{im}h_{jn}\\&\quad \times \left( { \varPsi _{imn}^T\begin{bmatrix} P_{is}&0 \\ 0&I \end{bmatrix} \varPsi _{imn} }\right. \left. { -\begin{bmatrix} (1+\beta _i)P_{im}&0 \\ 0&\gamma ^2I \end{bmatrix}} \right) \xi _k. \end{aligned}$$

From (26), we can know

$$\begin{aligned} \Delta V_i(x_k) \le \beta _iV_i(k)-\varGamma (k),\quad \forall k \in T[k_l,k_l+T_i). \end{aligned}$$

(28)

By combining (27) and (28), we have (16), and from (24), we can get

$$\begin{aligned} V_i(k_l)\le \mu _i V_j(k_l^-), \end{aligned}$$

(29)

then from Lemma 4, we can know SUDFS (9) is GUAS for any switching signal satisfying (17) and has a weighted $H_{\infty }$ performance index no greater than $\hat{\gamma }$. So SUDFS (9) is GUAS under asynchronous switching conditions. The proof is ended.

$\square$

Theorem 4

Consider closed-loop SUDFS (9), and let $0<\alpha _i<1$, $\beta _i>0$, $\mu _i\ge 1$ be the given constants. If there exist matrices $X_{im}>0$, matrices $\varOmega _{i}$ and $Y_{im}$, and positive constant $\epsilon _i$ such that for $\forall (i,j) \in \mathcal I\times \mathcal I, i\ne j$, $m=1,2,\ldots ,r_i$, $n=1,2,\ldots ,r_i$, $s=1,2,\ldots ,r_i$ satisfying

$$\begin{aligned}&X_{jn}\le \mu _i X_{im} \end{aligned}$$

(30)

$$\begin{aligned}&\varXi _{imns}<0 \end{aligned}$$

(31)

$$\begin{aligned}&\varXi _{imjns}<0 \end{aligned}$$

(32)

where

$$\begin{aligned}&\varXi _{imns}\triangleq \begin{bmatrix}\bar{Q}_{im}&0&\bar{M}_{imn}&\bar{N}_{imn}&0&\bar{P}_{imn} \\ *&-\gamma ^2I&\frac{E_{im}^T+E_{in}^T}{2}&\frac{F_{im}^T+F_{in}^T}{2}&0&0 \\ *&*&-X_{is}&0&\frac{M_i}{\epsilon _i}&0 \\ *&*&*&-I&0&0 \\ *&*&*&*&-\frac{I}{\epsilon _i}&0 \\ *&*&*&*&*&-\frac{I}{\epsilon _i} \end{bmatrix},\\&\varXi _{imjns}\triangleq \begin{bmatrix}\hat{Q}{imj}&0&\hat{M}_{imjn}&\hat{N}_{imjn}&0&\hat{P}_{inj} \\ *&-\gamma ^2I&E_{im}^T&F_{im}^T&0&0 \\ *&*&-X_{is}&0&\frac{M_i}{\epsilon _i}&0 \\ *&*&*&-I&0&0 \\ *&*&*&*&-\frac{I}{\epsilon _i}&0 \\ *&*&*&*&*&-\frac{I}{\epsilon _i} \end{bmatrix}, \end{aligned}$$

and

$$\begin{aligned}&\bar{Q}_{im}=\frac{X_{im}}{1-\alpha _i}-\varOmega _{i}^T - \varOmega _{i},\\&\bar{P}_{imn}=\frac{2\varOmega _{i}^TN_{1i}^T+(Y_{im}+Y_{in})^TN_{2i}^T}{2},\\&\bar{M}_{imn}=\frac{\varOmega _i^T(A_{im}+A_{in})^T+Y_{im}^TB_{in}^T+Y_{in}^TB_{im}^T}{2},\\&\bar{N}_{imn}=\frac{\varOmega _i^T(C_{im}+C_{in})^T+Y_{im}^TD_{in}^T+Y_{in}^TD_{im}^T}{2},\\&\hat{Q}_{imj}=\frac{X_{im}}{1+\beta _i}-\varOmega _{j}^T - \varOmega _{j},\\&\hat{P}_{inj}=\varOmega _{j}^TN_{1i}^T+Y_{jn}^TN_{2i}^T,\\&\hat{M}_{imjn}=\varOmega _j^TA_{im}^T+Y_{jn}^TB_{im}^T,\\&\hat{N}_{imjn}=\varOmega _j^TC_{im}^T+Y_{jn}^TD_{im}^T, \end{aligned}$$

then closed-loop SUDFS (9) is GUAS for any switching signal satisfying (17) and has a weighted $H_{\infty }$ performance index no greater than $\hat{\gamma }=\sqrt{\prod \limits _{i \in \mathcal I}{}(\theta _i^{T_M}\mu _i)^{N_{0i}}\frac{\alpha _{\mathrm{max}}}{\alpha _{\mathrm{min}}}\theta _{\mathrm{max}}^{T_M-1}}\gamma$ with the asynchronous state feedback controller gains given by

$$\begin{aligned} \left\{ \begin{array}{l} K_{jn}=Y_{jn}\varOmega _{j}^{-1} ,k \in [k_l,\bar{k}_l)\\ K_{in}=Y_{in}\varOmega _{i}^{-1} ,k \in [\bar{k}_l, k_{l+1}) \end{array} \right. \end{aligned}$$

(33)

Proof

Consider the closed-loop SUDFS(9), applying the Schur complement to (31) yields:

$$\begin{aligned} \begin{bmatrix}\bar{Q}_{im}&0&\bar{M}_{imn}&\bar{N}_{imn} \\ *&-\gamma ^2I&\frac{E_{im}^T+E_{in}^T}{2}&\frac{F_{im}^T+F_{in}^T}{2} \\ *&*&-X_{is}&0 \\ *&*&*&-I \end{bmatrix}+\epsilon _i\begin{bmatrix}0 \\ 0 \\ \frac{M_i}{\epsilon _i} \\ 0 \end{bmatrix}\begin{bmatrix}0\\ 0 \\ \frac{M_i}{\epsilon _i}\\ 0 \end{bmatrix}^T+\epsilon _i\begin{bmatrix} \bar{P}_{imn} \\ 0\\0\\0 \end{bmatrix}\begin{bmatrix} \bar{P}_{imn} \\ 0\\0\\0 \end{bmatrix}^T<0, \end{aligned}$$

which is equal to

$$\begin{aligned} \begin{bmatrix}\bar{Q}_{im}&0&\bar{M}_{imn}&\bar{N}_{imn} \\ *&-\gamma ^2I&\frac{E_{im}^T+E_{in}^T}{2}&\frac{F_{im}^T+F_{in}^T}{2} \\ *&*&-X_{is}&0 \\ *&*&*&-I \end{bmatrix}+\epsilon _i^{-1}\begin{bmatrix}0 \\ 0 \\ M_i \\ 0 \end{bmatrix}\begin{bmatrix}0\\ 0 \\ M_i\\ 0 \end{bmatrix}^T+\epsilon _i\begin{bmatrix} \bar{P}_{imn} \\ 0\\0\\0 \end{bmatrix}\begin{bmatrix} \bar{P}_{imn} \\ 0\\0\\0 \end{bmatrix}^T<0, \end{aligned}$$

from Lemma 5, we can know:

$$\begin{aligned} \begin{bmatrix}\frac{X_{im}}{1-\alpha _i}-\varOmega _{i}^T - \varOmega _{i}&0&\frac{\varOmega _i^T(\mathscr {A}_{im}+\mathscr {A}_{in})^T+Y_{im}^T\mathscr {B}_{in}^T+Y_{in}^T\mathscr {B}_{im}^T}{2}&\bar{N}_{imn} \\ *&-\gamma ^2I&\frac{E_{im}^T+E_{in}^T}{2}&\frac{F_{im}^T+F_{in}^T}{2} \\ *&*&-X_{is}&0 \\ *&*&*&-I \end{bmatrix}<0, \end{aligned}$$

which gives $0<\frac{X_{im}}{1-\alpha _i}<\varOmega _{i}^T + \varOmega _{i}$, which implies that $\varOmega _i$ is invertible. The inequalities $(\frac{X_{im}}{1-\alpha _i}-\varOmega _{i})^T{(\frac{X_{im}}{1-\alpha _i})}^{-1}(\frac{X_{im}}{1-\alpha _i}-\varOmega _{i})\ge 0$ imply that $\varOmega _{i}^T {(\frac{X_{im}}{1-\alpha _i})}^{-1}\varOmega _{i}\ge \varOmega _{i}^T + \varOmega _{j}-\frac{X_{im}}{1+\alpha _i}$, that is

$$\begin{aligned} \begin{bmatrix} -\varOmega _{i}^T {(\frac{X_{im}}{1-\alpha _i})}^{-1}\varOmega _{i}&0&\frac{\varOmega _i^T(\mathscr {A}_{im}+\mathscr {A}_{in})^T+Y_{im}^T\mathscr {B}_{in}^T+Y_{in}^T\mathscr {B}_{im}^T}{2}&\bar{N}_{imn} \\ *&-\gamma ^2I&\frac{E_{im}^T+E_{in}^T}{2}&\frac{F_{im}^T+F_{in}^T}{2} \\ *&*&-X_{is}&0 \\ *&*&*&-I \end{bmatrix}<0, \end{aligned}$$

Note that $\varOmega _i$ is invertible. Pre-multiplying diag($\varOmega _i^{-T}$,I,I,I) and post-multiplying diag($\varOmega _i^{-1}$,I,I,I) to above inequality give

$$\begin{aligned} \begin{bmatrix} - (1-\alpha _i)X_{im}^{-1}&0&\frac{\bar{\mathscr {A}}_{imn}+\bar{\mathscr {A}}_{inm}}{2}&\frac{\bar{C}_{imn}+\bar{C}_{inm}}{2} \\ *&-\gamma ^2I&\frac{E_{im}^T+E_{in}^T}{2}&\frac{F_{im}^T+F_{in}^T}{2} \\ *&*&-X_{is}&0 \\ *&*&*&-I \end{bmatrix}<0, \end{aligned}$$

Setting $X_{im}^{-1}= P_{im}$, we have

$$\begin{aligned} \begin{bmatrix} - (1-\alpha _i)P_{im}&0&\frac{\bar{\mathscr {A}}_{imn}+\bar{\mathscr {A}}_{inm}}{2}&\frac{\bar{C}_{imn}+\bar{C}_{inm}}{2} \\ *&-\gamma ^2I&\frac{E_{im}^T+E_{in}^T}{2}&\frac{F_{im}^T+F_{in}^T}{2} \\ *&*&-P_{is}^{-1}&0 \\ *&*&*&-I \end{bmatrix}<0, \end{aligned}$$

It follows from Schur complement equivalence that we can get (25); similarly, (26) can be obtained by (32), and (28) is equal to (24):

$$\begin{aligned} X_{jm}\le \mu _i X_{in} \Leftrightarrow X_{in}^{-1}\le \mu _i X_{jm}^{-1} \Leftrightarrow P_{in} \le \mu _i P_{jm}. \end{aligned}$$

Then, we can know the SUDFS (9) is GUAS with $\omega (k)=0$ and has a weighted $H_{\infty }$ performance index no greater than $\hat{\gamma }$ for any switching signals satisfying (17); the proof is ended.$\square$

Remark 7

Theorem 4 can also apply to relevant problems for SUDFSs (9) without asynchronous behaviors, i.e., $T_M=0$. For this case, if (40) in Theorem 4 is removed, we can easily conclude the SUDFS (9) is GUAS and has a weighted $H_{\infty }$ performance index no greater than $\bar{\gamma }=\sqrt{\prod \limits _{i \in \mathcal I}{}\mu _i^{N_{0i}}\frac{\alpha _{\mathrm{max}}}{\alpha _{\mathrm{min}}}}\gamma$ for any switching signal satisfying (15).

Remark 8

By setting $P_{im}=P_{i}$, we get the following Corollary 2 where the QLFs are used. So the LMIs condition in Theorem 4 has less conservatism than those in Corollary 2.

Corollary 2

Consider SUDFS (9), and let $0<\alpha _i<1$, $\beta _i>0$, $\mu _i\ge 1$ be the given constant. If there exist matrices $X_{i}>0$, matrices $\varOmega _{i}$ and $Y_{im}$ such that $\forall (i,j) \in \mathcal I\times \mathcal I, i\ne j$, $m=1,2,\ldots ,r_i$, $n=1,2,\ldots ,r_i$ satisfying

$$\begin{aligned}&X_{j}\le \mu _i X_{i}\\&\varXi _{imn}<0\\&\varXi _{imjn}<0 \end{aligned}$$

where

$$\begin{aligned}&\varXi _{imns}\triangleq \begin{bmatrix}\bar{Q}_{i}&0&\bar{M}_{imn}&\bar{N}_{imn}&0&\bar{P}_{imn} \\ *&-\gamma ^2I&\frac{E_{im}^T+E_{in}^T}{2}&\frac{F_{im}^T+F_{in}^T}{2}&0&0 \\ *&*&-X_{i}&0&\frac{M_i}{\epsilon _i}&0 \\ *&*&*&-I&0&0 \\ *&*&*&*&-\frac{I}{\epsilon _i}&0 \\ *&*&*&*&*&-\frac{I}{\epsilon _i} \end{bmatrix},\\&\varXi _{imjns}\triangleq \begin{bmatrix}\hat{Q}_{ij}&0&\hat{M}_{imjn}&\hat{N}_{imjn}&0&\hat{P}_{inj} \\ *&-\gamma ^2I&E_{im}^T&F_{im}^T&0&0 \\ *&*&-X_{i}&0&\frac{M_i}{\epsilon _i}&0 \\ *&*&*&-I&0&0 \\ *&*&*&*&-\frac{I}{\epsilon _i}&0 \\ *&*&*&*&*&-\frac{I}{\epsilon _i} \end{bmatrix}, \end{aligned}$$

and

$$\begin{aligned}&\bar{Q}_{i}=\frac{X_{i}}{1-\alpha _i}-\varOmega _{i}^T - \varOmega _{i},\\&\bar{P}_{imn}=\frac{2\varOmega _{i}^TN_{1i}^T+(Y_{im}+Y_{in})^TN_{2i}^T}{2},\\&\bar{M}_{imn}=\frac{\varOmega _i^T(A_{im}+A_{in})^T+Y_{im}^TB_{in}^T+Y_{in}^TB_{im}^T}{2},\\&\bar{N}_{imn}=\frac{\varOmega _i^T(C_{im}+C_{in})^T+Y_{im}^TD_{in}^T+Y_{in}^TD_{im}^T}{2},\\&\hat{Q}_{ij}=\frac{X_{i}}{1+\beta _i}-\varOmega _{j}^T - \varOmega _{j},\\&\hat{P}_{inj}=\varOmega _{j}^TN_{1i}^T+Y_{jn}^TN_{2i}^T,\\&\hat{M}_{imjn}=\varOmega _j^TA_{im}^T+Y_{jn}^TB_{im}^T,\\&\hat{N}_{imjn}=\varOmega _j^TC_{im}^T+Y_{jn}^TD_{im}^T. \end{aligned}$$

Then, the SUDFS (9) is GUAS for any switching signal satisfying (20) and has a weighted $H_{\infty }$ performance $\hat{\gamma }=\sqrt{\prod \limits _{i \in \mathcal I}{}(\theta _i^{T_M}\mu _i)^{N_{0i}}\frac{\alpha _{\mathrm{max}}}{\alpha _{\mathrm{min}}}\theta _{\mathrm{max}}^{T_M-1}}\gamma$ when the asynchronous state feedback controller gains are given by

$$\begin{aligned} \left\{ \begin{array}{l} K_{jn}=Y_{jn}\varOmega _{j}^{-1} ,k \in [k_l, k_l+T_i)\\ K_{in}=Y_{in}\varOmega _{i}^{-1} ,k \in [ k_l+T_i, k_{l+1}) \end{array} \right. \end{aligned}$$

(34)

Proof

The proof is similar to the proof of Theorem 3 and Theorem 4 with the function $V_i(k)$ given by $V_i(k)=x(k)^TP_ix(k)$. It is omitted here.$\square$

4 Numerical Examples

Example 1

First, a numerical example is presented to show the effectiveness of our results. Consider the following closed-loop SUDFS consisting of two subsystems, and each subsystem has two fuzzy rules, where

$$\begin{aligned}&A_{11}=\begin{bmatrix} 0.5&-2 \\ 1&3 \end{bmatrix},A_{12}=\begin{bmatrix} 0.3&-1.5 \\ 1&2 \end{bmatrix},A_{21}=\begin{bmatrix} 0.1&1.2 \\ 1&-0.05 \end{bmatrix},A_{22}=\begin{bmatrix} 0.5&1.3 \\ 1&-0.15 \end{bmatrix}, \\&B_{11}=\begin{bmatrix} 0.25&-2 \\ -1&3 \end{bmatrix},B_{12}=\begin{bmatrix} 0.2&-2 \\ -1&5\end{bmatrix},B_{21}=\begin{bmatrix} 0.05&-1.5 \\ -1&-0.5 \end{bmatrix},B_{22}=\begin{bmatrix} 0.06&-1.5 \\ -1&-0.4 \end{bmatrix}, \\&C_{11}=\begin{bmatrix} 0.3&0.5 \end{bmatrix},C_{12}=\begin{bmatrix} 0.2&0.2 \end{bmatrix},C_{21}=\begin{bmatrix} 0.2&0.2 \end{bmatrix},C_{22}=\begin{bmatrix} 0.2&0.2 \end{bmatrix},\\&D_{11}=\begin{bmatrix} 0.25&0.25 \end{bmatrix},D_{12}=\begin{bmatrix} 0.25&0.25 \end{bmatrix}, D_{21}=\begin{bmatrix} 0.2&0.2 \end{bmatrix},D_{22}=\begin{bmatrix} 0.18&0.22\end{bmatrix},\\&E_{11}=\begin{bmatrix} 0.35 \\ 0.22 \end{bmatrix},E_{12}=\begin{bmatrix} 0.1 \\ 0.1 \end{bmatrix}, E_{21}=\begin{bmatrix} 0.1 \\ 0.1 \end{bmatrix},E_{22}=\begin{bmatrix} 0.12 \\ 0.08 \end{bmatrix},\\&F_{11}=0.01,F_{12}=0.05,F_{21}=0.01,F22=0.01,\\&M_1=\begin{bmatrix} 0.01&0.08 \\ 0.02&0.03 \end{bmatrix},M_2=\begin{bmatrix} 0.02&0.03 \\ 0.5&0.01 \end{bmatrix},\\&N_{11}=\begin{bmatrix} 0.02&0.05 \\ 0.08&0.01 \end{bmatrix}, N_{21}=\begin{bmatrix} 0.03&0.4 \\ 0.1&0.015 \end{bmatrix}, N_{12}=\begin{bmatrix} 0.01&0.15 \\ 0.02&0.025 \end{bmatrix}, N_{22}=\begin{bmatrix} 0.01&0.2 \\ 0.02&0.01 \end{bmatrix}.\\ \end{aligned}$$

The fuzzy membership functions are taken as $h_{11}=\cos ^2(x_1+0.5)$, $h_{12}=1-h_{11}$, $h_{21}=\sin ^2(x_2+0.5)$, $h_{22}=1-h_{21}$. According to Lemma 4, the initial conditions are set to be $x(k_0)={\begin{bmatrix}0&0\end{bmatrix}}^T$, and the disturbance input is assumed to be $\omega (k)=e^{-0.15k}$. Our purpose here is to use our results to design a state feedback controller such that the closed-loop SUDFS is GUAS and the $H_{\infty }$ performance is achieved. We choose $N_{0i}=0$, $\mu _1=1.1$, $\mu _2=1.2$, $\alpha _1=0.18$, $\alpha _2=0.17$, $\beta _1=0.05$, $\beta _2=0.09$ and assume that the controlled signal is synchronously with the switched signal, which means $T_M=0$. From Theorem 4, we can get the synchronous state feedback controllers under the condition of minimizing $\gamma$

Fig. 2

State response of the closed-loop synchronous SUDFS with controller designed by synchronous conditions, $T_{oa1}=5$, $T_{oa2}=4$

At the same time, we can obtain $\bar{\gamma }_{\min }=0.8799$, $T_{oa1}^*= 0.4803$, $T_{oa2}^*= 0.9785$. We assume that $T_{oa1}=5>0.4803$, $T_{oa2}=4>0.9785$, $T_M=0$, the state response curve of the closed-loop synchronous switched system is shown in Fig. 2, from which we can know the state response converges to zero rapidly, and the weighted $H_{\infty }$ performance is 0.3481 which is below $\bar{\gamma }_{\min }$. This shows the synchronous controller works well in synchronous conditions.

But in reality, the synchronous condition is quite ideal. It inevitably takes some time to identify the system modes and apply the matched controller; thus, the asynchronous switching between the system and the controller generally exists. We assume that the controlled signal is asynchronous with the switched signal; subsystem 1’s asynchronous time is $T_1=3$, subsystem 2’s asynchronous time is $T_2=2$, obviously $T_M=3$. Through Theorem 4, we can obtain $\hat{\gamma }_{\min }=2.7896$, $T_{oa1}^*= 4.2178$, $T_{oa2}^*=3.9035$ and the asynchronous controller

We adopt two switching sequences which satisfy $T_{oa1}=5>4.2178$, $T_{oa2}=4>3.9035$ and $T_{oa1}=6>4.2178$, $T_{oa2}=5>3.9035$, respectively. First, we apply the synchronous controller to the given system. The state response curve under the two switching sequences is shown in Figs. 3 and 4, from which we can see in the case that $T_{oa1}=5>4.2178$, $T_{oa2}=4>3.9035$, the synchronous controller cannot stabilize the given system, and in the case that $T_{oa1}=6>4.2178$, $T_{oa2}=5>3.9035$, the synchronous controller can stabilize the given system but cannot guarantee the weighted $H_{\infty }$ performance.

Fig. 3

State response of the closed-loop asynchronous SUDFS with controller designed by synchronous conditions, $T_{oa1}=5$, $T_{oa2}=4$

Fig. 4

State response of the closed-loop asynchronous SUDFS with controller designed by synchronous conditions, $T_{oa1}=6$, $T_{oa2}=5$

Fig. 5

State response of the closed-loop asynchronous SUDFS with controller designed by asynchronous conditions, $T_{oa1}=5$, $T_{oa2}=4$

Fig. 6

State response of the closed-loop asynchronous SUDFS with controller designed by asynchronous conditions, $T_{oa1}=6$, $T_{oa2}=5$

Then, we adopt the asynchronous switching controller instead of the synchronous one, as is shown in Figs. 5 and 6 that the state response of the closed-loop asynchronous switched system which utilizes the asynchronous controller converges to zero. The weighted $H_{\infty }$ performance index is 0.3897 and 0.3840, which is much smaller than $\hat{\gamma }_{\min }=2.7896$. The simulated results show that the controller designed by asynchronous conditions can stabilize the given system and guarantee good weighted $H_{\infty }$ performance.

In Table 1, we change $\mu _1$, $\mu _2$, $\alpha _1$ and $\alpha _2$’s values and let other parameters remain the same. For different $T_{cai}^*$ sets, we compute the minimal value of $\hat{\gamma }_{\mathrm{min}}$ by BLFs approach and QLFs approach. In Table 1, we can see $\hat{\gamma }_{\mathrm{min}}$ is decreasing with $\mu _i$ increasing. This is because a smaller $\mu _i$ will bring a smaller $T_{oai}^*$ (faster switching), which will result in a worse system performance, i.e., a larger $\hat{\gamma }_{\mathrm{min}}$. However, it is also seen that although a smaller $\alpha _i$ brings a larger $T_{oai}^*$, $\hat{\gamma }_{\mathrm{min}}$ still may increase. This is because the change in $\alpha _i$ also influences the value of $\hat{\gamma }$’s weighted index $\sqrt{\frac{\alpha _{\mathrm{max}}}{\alpha _{\mathrm{min}}}\theta _{\mathrm{max}}^{T_M-1}}$ ($N_{0i}=0$), and the change in $\mu _i$ does not influence the value of weighted index. Although we can obtain a smaller $\gamma$ by letting $\alpha _i$ decrease, $\hat{\gamma }$ still may increase due to the change in weighted index. The same case will occur when $\beta _i$ changes; for brevity, the simulated results for different $\beta _i$ are omitted. We also found that the calculated $\hat{\gamma }_{\mathrm{min}}$ of BLFs method is smaller than the calculated $\hat{\gamma }_{\mathrm{min}}$ of QLFs method in the same condition. This proves that BLFs method has more flexibility than the QLFs method in designing $H_{\infty }$ controller and improves the switched system’s weighted $H_{\infty }$ performance.

Table 1 The minimal $\hat{\gamma }_{\mathrm{min}}$ of BLFs method and QLFs method for different $T_{cai}^*$ sets

Example 2

A practical example is also presented to show the effectiveness of our approach. Let us consider a continuous stirred tank reactor (CSTR) where an exothermic, irreversible reaction of the form $A\rightarrow B$ happens. There exist two different feeding streams to feed the reactor, and these two feeding streams are selected by a selector [27]. Source 1 feeds pure species A at the flow rate $F_1=50$ L/min, concentration $C_{A_1}=1.5$ mol/L and temperature $T_{A1}=350$ K, and source 2 feeds pure species A at the flow rate $F_2$ 200 L/min, concentration $C_{A_2}=0.75$ mol/L and temperature $T_{A_2}=350$ K, i.e., the reactor has two modes corresponding to the feeding stream. For each mode, the system model for the chemical process is given as follows [3, 27]:

$$\begin{aligned} \begin{array}{l} \dot{C_{A}}=\frac{F_{\sigma }}{V}(C_{A_{\sigma }}-C_{A})-k_0e^{-E/RT_{R}}C_{A}\\ \dot{T_{R}}=\frac{F_{\sigma }}{V}(T_{A_{\sigma }}-T_{R})+\frac{(-\Delta H)}{\rho c_p}k_0e^{-E/RT_{R}}C_{A}+\frac{Q_{\sigma }}{\rho c_p V} \end{array} \end{aligned}$$

(35)

It is clear that the system (35) is a switched nonlinear system, where $C_{A}$ denotes the concentration of the species A; $T_R$ denotes the temperature of the reactor; $Q_{\sigma }$ is the heat removed from the reactor; V is the volume of the reactor; $k_0$, E, $\Delta H$ are the pre-exponential constant, the activation energy and enthalpy of the reaction, respectively; $c_p$, $\rho$ are the heat capacity and fluid density in the reactor, and $\sigma (t)\in \{1,2\}$ is the switching signal which is a discrete variable. All process parameters have been given in [27]. By substituting all the process parameters into Eq. (35), we can get the following two subsystems:

Subsystem 1: ($\sigma =1$)

$$\begin{aligned}&\dot{C_{A}}=-0.0334C_{A}-1.2\times 10^{9}e^{-10000T_{R}}C_{A}+0.026386,\\&\dot{T_{R}}=-0.0334T_{R}+2.4\times 10^{11}e^{-10000T_{R}}C_{A}+11.77684+\frac{Q_{\sigma }}{23.9} \end{aligned}$$

Subsystem 2: ($\sigma =2$)

$$\begin{aligned}&\dot{C_{A}}=-0.0167C_{A}-1.2\times 10^{9}e^{-10000T_{R}}C_{A}+0.0167,\\&\dot{T_{R}}=-0.0167T_{R}+2.4\times 10^{11}e^{-10000T_{R}}C_{A}+5.177+\frac{Q_{\sigma }}{23.9} \end{aligned}$$

When $Q_{\sigma }=0$, the switched nonlinear systems’ two steady states: $(C_{A},T_{R})_1=(0.57,395.3)$ and $(C_{A},T_{R})_2=(0.57,395.3)$, can be obtained.

It has been shown in [27] that these subsystems are all unstable. Here we assume that the asynchronous phenomenon exists, and the asynchronous $H_{\infty }$ control problem of the system (35) is considered. Similar to [14], for the convenience of research, we discretize the system (35) by Euler approximation; then, the system (35) is transformed into a discrete-time switched nonlinear system. The discrete-time switched nonlinear system can be represented by the following T-S fuzzy models:

Subsystem $\sigma (t)$

Rule 1: IF the concentration of the species A is $M_{\sigma }(x_{1})$ (i.e., $x_1(t)$ is 0.57). THEN

$$\begin{aligned}&\delta x(k+1)=(A_{\sigma 1}+M_{\sigma }F(k)N_{1\sigma })\delta x(k) + B_{\sigma 1}\delta u(k)+ E_{\sigma 1} \omega (k),\\&\delta y(k)=C_{\sigma 1}\delta x(k)+ D_{\sigma 1}\delta u(k)+ F_{\sigma 1} \omega (k), \end{aligned}$$

Rule 2: IF the concentration of the species A is $M_{\sigma }(x_{1})$ (i.e., $x_1(t)$ is 0.738). THEN

$$\begin{aligned}&\delta x(k+1)=(A_{\sigma 2}+M_{\sigma }F(k)N_{1\sigma })\delta x(k)+ B_{\sigma 2}\delta u(k)+ E_{\sigma 2} \omega (k),\\&\delta y(k)=C_{\sigma 2}\delta x(k)+ D_{\sigma 2}\delta u(k)+ F_{\sigma 2} \omega (k), \end{aligned}$$

where $\sigma \in \{1,2\}$ is the subsystem subscript, $x(k)=[C_{A}\quad T_{R}]^T$, $\delta x(k)=x(k)-x_{d}$, $x_{d}$ is a stationary point of the discrete-time switched nonlinear system, $\delta u(k)$ is the asynchronous controller which need to be designed, and $\omega (k)$ is the disturbance input created by external disturbance. Due to change in environment, the system matrices have some uncertainties. The value of system matrices are given as follows:

$$\begin{aligned}&A_{11}=\begin{bmatrix} 0.9995&0.0000 \\ 0.0248&1.0000 \end{bmatrix},A_{12}=\begin{bmatrix} 0.9649&0.0000 \\ 6.9536&0.9999 \end{bmatrix},A_{21}=\begin{bmatrix} 0.9997&0.0000 \\ 0.0248&1.0000 \end{bmatrix},\\&A_{22}=\begin{bmatrix} 0.9650&0.0000 \\ 6.9366&0.9999 \end{bmatrix},M_1=\begin{bmatrix} 0.01&0.08 \\ 0.02&0.03 \end{bmatrix},N_{11}=\begin{bmatrix} 0.02&0.05 \\ 0.08&0.01 \end{bmatrix},\\&N_{12}=\begin{bmatrix} 0.01&0.15 \\ 0.02&0.025 \end{bmatrix},B_{11}=B_{12}=B_{21}=B_{22}=\begin{bmatrix} 0&0.003 \\ -0.05&0 \end{bmatrix},\\&E_{11}=E_{12}=E_{21}=E_{22}=\begin{bmatrix} 0.06 \\ 0.6 \end{bmatrix},C_{11}=C_{12}=C_{21}=C_{22}=\begin{bmatrix}0&1\end{bmatrix},\\&D_{11}=D_{12}=D_{21}=D_{21}=0,F_{11}=F_{12}=F_{21}=F_{22}=0, \end{aligned}$$

Fig. 7

State trajectory of $x_1(k)$

From above, we can see the temperature of the reactor $T_{R}$ can be obtained directly by system output y(k). The given system can be seen as a SUDFS with asynchronous controller. The membership functions for Rules 1 and 2 for each individual system are taken as those of [3].

We set $N_{0i}=0$, $\mu _1=1.2$, $\mu _2=1.2$, $\alpha _1=0.12$, $\alpha _2=0.12$, $\beta _1=0.14$, $\beta _2=0.14$ in Theorem 4, and assume that subsystem 1’s asynchronous time is $T_1$ = 3, and subsystem 2’s asynchronous time is $T_2= 2$; then, $T_M=3$. By using the MATLAB LMI toolbox and minimizing the value of $\gamma$, we can obtain $\hat{\gamma }_{\mathrm{min}}= 1.0163$, $T_{oa1}^*=5.4762$, $T_{oa2}^*=5.4762$, and the controller gain matrices are given as follows

We assume that the initial states are $x(k_0)=[0.14\quad 404.9]^T$ and the disturbance input is $\omega (k)=5cos(k)e^{-0.1k}$; the system’s switching signal satisfies $T_{oa1}=6>5.4762$, $T_{oa2}=6>5.4762$. Simulation results of the trajectory of the SUDFS are presented in Figs. 7 and 8, from which we can see although the asynchronous phenomenon and external disturbance exist, the states converge to the stationary point of the discrete-time switched nonlinear system gradually, which shows the designed asynchronous controller stabilizes the SUDFS and guarantees good $H_{\infty }$ performance.

Fig. 8

State trajectory of $x_2(k)$

5 Conclusion

In this paper, we use the BLFs approach to solve asynchronous SUDFSs’ stability issues and $H_\infty$ controller problems. We obtain open-loop SUDFSs’ robust stability conditions, and asynchronous state feedback $H_{\infty }$ controller which can ensure the stability of asynchronous closed-loop SUDFSs with a weighted $H_{\infty }$ performance is designed by solving LMIs. At last, both numerical and practical examples are given to show the importance of considering switched systems’ asynchronous behavior and demonstrate the effectiveness of BLFs approach. The research results show that the asynchronous behavior may influence the system’s performance or even cause the system’s instability. In future work, via the BLFs approach, filter design problem of asynchronous SUDFSs will be analyzed. Characteristics of asynchronous SUDFSs with impulsive dynamical behaviors and time-varying delays will also be investigated.