1 Introduction

Switched systems, which are a class of special hybrid systems, include a group of continuous or discrete subsystems and a switching rule governing the activated subsystem [1]. Switched control has much more superiorities than the traditional single control in modeling complex systems. Thus, switched systems have an extensive range of practical applications in many fields of control, such as mechanical systems, chemical procedure systems, and power systems [2, 3]. Moreover, switched control provides more effective techniques than the traditional single control, which even can improve transient performance and control precision of systems better [4]. Generally speaking, the switched control has become one of the hottest topics in the control field.

With the increasing integration and complexity of the industry, switched systems play a more critical role than ever before; more and more results of switched systems have been obtained in recent years. Stability is a fundamental and essential issue when analyzing the performance of systems [5]. Compared with the non-switched systems, the switched systems have hybrid dynamic features, so it is more of complexity and challenge to analyze the stability of the switched systems. For a class of switched nonlinear systems whose subsystems are not supposed to be stable when the delay is present, Wang et al. provided sufficient conditions to ensure that the system is globally exponentially stable [6]. Wu et al. investigated the stability of switched systems with stochastic switching signals by using the probability analysis method [7]. By utilizing the nonlinear related techniques, the stability criteria for switched nonlinear sampled data systems are derived in [8]. Zhang et al. studied the stability of cascaded switched systems by using an input-to-state Lyapunov function method [9]. In addition to stability, there are many issues worthy of studying for switched nonlinear systems. Su et al. studied \(H_{\infty }\) control problem for cascaded switched nonlinear systems in [10]. By the virtue of multiple Lyapunov functional and average dwell-time approach, Park et al. constructed a fault estimator and the \(H_{\infty }\) fault estimation problem for a class of discrete-time switched nonlinear systems was solved [11]. Moreover, the dissipative control problem for wireless networked control systems was concerned in [12]. Zheng et al. proposed a performance index for a class of switched nonlinear systems in [13], which can be viewed as the mixed \(H_{\infty }\) performance and passive performance.

Many results show that the system may be unstable if the system switches so fast, even if every subsystem is stable. The simplest way to control the switching frequency is the activating time of each subsystem should not be too short; therefore, the concept of dwell time is proposed [5]. However, it is too restrictive to specify a dwell time. Thus, it is of interest to relax the condition of the dwell time concept, allowing the possibility of switching fast when necessary and then compensating for it by switching sufficiently slowly later. The concept of average dwell time serves this purpose. As a proven effective design method, the average dwell time technology has been adopted in [11,12,13,14]; what’s more, this technology was also used to study asynchronous control of switched systems [9].

The feedback controllers of switched systems are mostly continuous in the existing results. However, controllers are implemented on a digital platform in the actual operation. Therefore, an effective research method, event-triggered control, plays an important role in this situation. Event-triggered control means that the system performs control tasks after an external event occurs [15]. In detail, its signal samplings and controller operations are triggered by a specific event, rather than being executed regularly over time [16]. There is a lot of interaction between sensors, controllers, and actuators, which leads to information load problems and a waste of computational resources [17]. These problems can be effectively solved by applying the event-triggered control method. Owing to the excellent capability, the event-triggered strategy is used in the research of many control problems. Xiao et al. studied the event-triggered control of discrete-time and continuous-time switched linear systems in [18] and [19], respectively. Su et al. adopted the event-triggered mechanism to the study of sliding mode control of hybrid switched systems [20]. Li et al. applied the event-triggered method to study the \(H_{\infty }\) control problem for switched systems [21]. Qi and Cao introduced an event-triggered mechanism with a fixed threshold and the finite-time boundedness in [22], and the stabilization of switched linear systems is concerned.

Generally, if the event-triggered scheme is introduced into the system, then in the process of control, the actuator performs the operations in time-varying periods, besides, the intervals between events are also variable. That is, infinite events may appear in a limited time. Thence, Zeno behavior was concerned in [23]. It is a challenging task to exclude Zeno behavior. In the existing research results, Ni et al. proposed the positive lower bound for inter-event intervals for a class of linear systems [24]; Fei and Zhao extended this method to switched linear systems [25]. Li and Lian designed a Zeno-free event-triggered scheme for switched linear systems [26]. However, due to the complexity of switched nonlinear systems, there still are few results referring to this class of systems by virtue of the event-triggered scheme, not to mention the results of excluding the Zeno behavior. In [27], Li et al. introduced the event-triggered scheme to the output tracking control problem of switched nonlinear strict-feedback systems, while the Zeno behavior was not considered. Fortunately, this problem was concerned by Huo et al. where they studied the adaptive fuzzy output feedback control of MIMO switched nonlinear systems [28].

Most of the current-based event-triggered control theories refer to linear systems and are not fully applicable to nonlinear systems. In addition, switched systems are different from non-switched systems; even though each subsystem is stable, the switched system may be unstable if the switching signal is not selected properly. The study of event-triggered control for switched nonlinear systems is not the simple superposition of the properties of each subsystems. At the same time, cascade systems are an important class of systems [29]. This class of systems has special structure and facilitates the wide range of applications. Therefore, it is necessary and meaningful to study the event-triggered control of cascade switched nonlinear systems.

Motivated by the above discussion, we mainly studied the event-triggered control problem for switched nonlinear systems by an average dwell-time method. Our main contributions are in the following aspects:

  1. (1)

    For the cascaded switched nonlinear system, an event-triggered control scheme is first presented, under which the communication resources are effectively saved.

  2. (2)

    For the resulting closed-loop system, the sufficient condition for globally uniform boundedness is obtained by the virtue of an average dwell time method, and the exponential stability is guaranteed by the conditions we provided.

  3. (3)

    The case that infinite events may happen in a limited time interval is considered. The minimum interval of inter-event lower bound is calculated by applying the event-triggered sampling mechanism and the Zeno behavior is successfully excluded during the sampling procedure.

Notations: Unless otherwise stated, the notations used in this paper are standard. \(R^{n}\) denotes the Euclidean space with n dimensions. N represents the natural number. The superscript T denotes matrix transposition. \(P>0\) means that the matrix P is real symmetric and positive definite. I represents the identity matrix with appropriate dimension. \(\Vert \cdot \Vert \) denotes the Euclidean vector norm.

2 Preliminaries

Consider the following cascade switched nonlinear system

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{x}}_{1}(t)=A_{1\sigma (t)}x_{1}(t)+A_{2\sigma (t)}x_{2}(t)+B_{\sigma (t)}u_{\sigma (t)}(t)\\&{\dot{x}}_{2}(t)=f_{2\sigma (t)}(x_{2}(t))\\&y(t)=C_{\sigma (t)}x_{1}(t) \end{aligned}\right. \end{aligned}$$
(1)

where \(x=[x_{1},x_{2}]^{T}\) is the state vector and \(x_{1}(t)\in {R^{n-d}}, x_{2}(t)\in {R^{d}}\); \(y(t)\in {R^{q}}\) means the output of the system; \(\sigma (t):[0,\infty )\rightarrow {{\bar{M}}}=\{1,2,\ldots ,M\}\) is a piecewise constant value function with respect to time, called switching signal, where \({\bar{M}}\) is a finite index set, the ith subsystem is active when \(\sigma (t)=i\). \(u_{i}(t)\in {R^{m}}\) represents the control input of subsystem i. \(A_{1i}, A_{2i}, B_{i}, C_{i}\) are constant matrices with appropriate dimensions, \(i\in {{\bar{M}}}\). \(f_{2i}(\cdot )\) are known smooth vector fields with appropriate dimensions and \(f_{2i}(0)=0\)

According to the switching signal \(\sigma (t)\), we can get a group switching sequence presenting as follows:

$$\begin{aligned} {\{x_{t_{0}};(l_{0},t_{0}),(l_{1},t_{1}),\ldots ,(l_{i},t_{i}),\ldots |l_{i}\in {{\bar{M}}},i\in {N}}\} \end{aligned}$$
(2)

where \(l_{i}\) means the serial number of the subsystem and \(t_{i}\) is the switching instant. That is, the \(l_{i}\) subsystem is active when \(t\in [t_{i},t_{i+1})\). Supposing that there is no system state jump at the moment of switching.

Denoting instant with an event happens by \(\{{\hat{t}}_{k}\}_{k=0}^{\infty }\), \({\hat{t}}_{k}<{{\hat{t}}_{k+1}}\), and the error \(e(t)=x_{1}(t)-x_{1}({\hat{t}}_{k})\). The condition of event-triggered can be described as

$$\begin{aligned} {\Vert e(t)\Vert ^{2}\ge {\eta \Vert x_{1}(t)\Vert ^{2}+\varepsilon }} \end{aligned}$$
(3)

where \(\eta \) and \(\varepsilon \) are positive parameters. Sampling is performed immediately once the event is triggered. The value e(t) will be reduced to 0 if an event happens; besides, it begins to increase until the next event happens. Assume that when \({\hat{t}}_{0}=t_{0}\), the first event occurs. With the state \(x_{1}({\hat{t}}_{k})\) sampled at the time \({\hat{t}}_{k}\), we can describe the next sampling instant \({\hat{t}}_{k+1}\) by

$$\begin{aligned} {\hat{t}}_{k+1}=\inf \left\{ t>{{\hat{t}}_{k}} \mid \Vert e(t)\Vert ^{2}=\eta \Vert x_{1}(t)\Vert ^{2}+\varepsilon \right\} \end{aligned}$$
(4)

A controller is obtained under the above condition (3). Assume that there are n samplings occurred during \(t\in {[t_{i},t_{i+1})}\), and denote the first sampling instant by \({\hat{t}}_{k+1}\). Under the switching sequence (2), each interval corresponds to an active subsystem; then, the controller can be designed as follows:

$$\begin{aligned} u_{\sigma (t)}=u_{i}=\left\{ \begin{aligned}&K_{i}x_{1}({\hat{t}}_{k}),t\in {[t_{i},{\hat{t}}_{k+1})}\\&K_{i}x_{1}({\hat{t}}_{k+1}),t\in {[{\hat{t}}_{k+1},{\hat{t}}_{k+2})}\\&\qquad \vdots \\&K_{i}x_{1}({\hat{t}}_{k+n}),t\in {[{\hat{t}}_{k+n},t_{i+1})}\\ \end{aligned}\right. \end{aligned}$$
(5)

where \(K_{i}\) is the controller gain.

Definition 1

[13] For any \(\sigma \) and \(\forall {t}\ge {s}\ge {0}\), \(N_{\sigma }(s,t)\) means the number of switchings on the interval (s, t). If \(N_{\sigma }\le {1}\) and \(t-s\le {\tau _{d}}\), then \(\tau _{d}\) is called a dwell time; for \(\tau _{a}>{\tau _{d}}\) and \(N_{0}\ge {1}\), if there exists \(N_{\sigma }(s,t)\le {N_{0}+((t-s)/\tau _{a})}\), then \(\tau _{a}\) is called the average dwell time.

Definition 2

[14] For any constant \(\gamma >{0}\), if there is a switching signal \(\sigma \) and a constant \({\widetilde{\beta }}\), satisfying

$$\begin{aligned} {\Vert x(t_{0})\Vert<{\gamma }\Rightarrow {\Vert x(t)\Vert <{{\widetilde{\beta }}}},\forall {t}>{t_{0}}} \end{aligned}$$
(6)

where \({\widetilde{\beta }}={{\widetilde{\beta }}(\gamma )}<{\infty }\) is positive and independent from \(t_{0}\), then the considered switched nonlinear system is globally uniformly bounded.

Lemma 1

[14] For any u, v and positive definite matrix G,

$$\begin{aligned} {u^{T}v+v^{T}u\le {u^{T}Gu+v^{T}G^{-1}v}} \end{aligned}$$
(7)

holds, where u, v are real vectors and G has the appropriate dimension.

3 Main results

We will study the stability of the switched nonlinear system (1) in this section.

3.1 Event-triggered control

Substitute the controller (5) into the system (1) with the error \(e(t)=x_{1}(t)-x_{1}({\hat{t}}_{k+j}),(j=0,1,\ldots ,n)\) for \(\forall {t}\in {[t_{i},{\hat{t}}_{i+1}),[{\hat{t}}_{i+1},{\hat{t}}_{i+2}),\ldots ,[{\hat{t}}_{i+n},t_{i+1})}\), and the closed-loop system is presented as

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{x}}_{1}(t)=(A_{1i}+B_{i}K_{i})x_{1}(t)+A_{2i}x_{2}(t)-B_{i}K_{i}e(t)\\&{\dot{x}}_{2}(t)=f_{2i}(x_{2}(t))\\&y(t)=C_{i}x_{1}(t) \end{aligned}\right. \end{aligned}$$
(8)

Theorem 1

Consider the closed-loop system (8). If the system (8) satisfies the following conditions:

  1. (1)

    For the given scalars \(\eta>{0},\mu>{1},\varepsilon >{0}\) and \(N_{0}\ge {1}\), there exist matrices \(P_{i}>{0},P_{j}>{0}\) and \(K_{i}\) for \(\forall {i,j}\in {{{\bar{M}}}}\) satisfying

    $$\begin{aligned} \left[ \begin{array}{cc} A^{T}_{1i}P_{i}+P_{i}B_{i}K_{i}+P_{i}A_{1i}+K^{T}_{i}B^{T}_{i}P_{i}+(1+\eta )I &{}P_{i}B_{i}K_{i}\\ *&{}-I \end{array}\right] <{0}\end{aligned}$$
    (9)

    and

    $$\begin{aligned} {P_{i}\le {\mu {P_{j}}},P_{j}\le {\mu {P_{i}}}} \end{aligned}$$
    (10)
  2. (2)

    For the given constants \(\beta>{0},\alpha _{1}>{0},\alpha _{2}>{0}\), there exist functions \(W_{i}(x_{2})\) satisfying

    $$\begin{aligned} \frac{\partial W(x_{2})}{\partial x_{2}} f_{2i}(x_{2})\le {-\beta {\Vert x_{2}\Vert ^{2}}} \end{aligned}$$
    (11)

    and

    $$\begin{aligned} \alpha _{1}\Vert x_{2}\Vert ^{2}\le {W_{i}(x_{2})\le {\alpha _{2}\Vert x_{2}\Vert ^{2}}}, \end{aligned}$$
    (12)

    then for any switching signal \(\sigma \) satisfying \(\tau _{a}>{(\ln {\mu })/\gamma }\), the system (1) is globally uniform bounded, where \(\gamma >{0}\) can be calculated from valid matrix solutions (9) and (10); the state exponentially converges to the bounded region

    $$\begin{aligned}&B(\varepsilon )= \left\{ x(t) \mid \Vert x(t)\Vert \right. \nonumber \\&\quad \left. \le {\left[ \frac{\varepsilon {e}^{\gamma \tau _{a}}(({\widetilde{\mu }}-1)e^{\gamma \tau _{a}N_{0}}+1)- \varepsilon {\widetilde{\mu }}}{\varsigma \gamma (e^{\gamma \tau _{a}}-{\widetilde{\mu }})}\right] ^{\frac{1}{2}}} \right\} \end{aligned}$$
    (13)

    where \({\widetilde{\mu }}=\max {\left\{ \mu ,\frac{\alpha _{2}}{\alpha _{1}}\right\} }\), \(\varsigma \) is a scalar which will be determined later.

Proof

Construct a Lyapunov function as follows:

$$\begin{aligned} V(t)=V_{\sigma (t)}(x_{1},x_{2})=x^{T}_{1}(t)P_{\sigma (t)}x_{1}(t)+lW_{\sigma (t)}(x_{2}) \end{aligned}$$
(14)

when \(\sigma (t)=i\), by virtue of Lemma 1, the derivative of \(V_{i}\) is

$$\begin{aligned} \dot{V_{i}}(t)= & {} x^{T}_{1}\left[ (A_{1i}+B_{i}K_{i})^{T}P_{i}+P_{i}(A_{1i}+B_{i}K_{i})\right] x_{1} \nonumber \\&+x^{T}_{2}A^{T}_{2i}P_{i}x_{1}\nonumber \\&+x^{T}_{1}P_{i}A_{2i}x_{2}-e^{T}K^{T}_{i}B^{T}_{i}P_{i}x_{1} -x^{T}_{1}P_{i}B_{i}K_{i}e\nonumber \\&+\frac{\partial {W_{i}(x_{2})}}{\partial {x_{2}}}f_{2i} \nonumber \\\le & {} x^{T}_{1}\left[ (A_{1i}+B_{i}K_{i})^{T}P_{i}+P_{i}(A_{1i}+B_{i}K_{i})\right] x_{1}\nonumber \\&+2x^{T}_{1}P_{i}A_{2i}x_{2}\nonumber \\&+e^{T}e+x^{T}_{1}P_{i}B_{i}K_{i}K^{T}_{i}B^{T}_{i}P_{i}x_{1}-l\beta \Vert x_{2}\Vert ^{2}\nonumber \\ \end{aligned}$$
(15)

Notice that there exist two positive constants \(n_{i}\) and \(m_{i}\) satisfying

$$\begin{aligned} \Vert A_{2i}x_{2}\Vert \le {n_{i}\Vert x_{2}\Vert },\Vert x^{T}_{1}P_{i}\Vert \le {m_{i}\Vert x_{1}\Vert } \end{aligned}$$

Let \(p=\mathop {\max }\limits _{i\in {M}}\left\{ n_{i}m_{i}\right\} \), we have

$$\begin{aligned} \dot{V_{i}}(t)\le & {} x^{T}_{1}\left[ \left( A_{1i}+B_{i}K_{i}\right) ^{T}P_{i}+P_{i}\left( A_{1i}+B_{i}K_{i}\right) \right] \nonumber \\&x_{1}+e^{T}e+x^{T}_{1}P_{i}B_{i}K_{i}K^{T}_{i}B^{T}_{i}P_{i}x_{1}\nonumber \\&\quad +2p\Vert x_{1}\Vert \Vert x_{2}\Vert -l\beta \Vert x_{2}\Vert ^{2}\nonumber \\\le & {} x^{T}_{1}\left[ (A_{1i}+B_{i}K_{i})^{T}P_{i}+P_{i}(A_{1i}\right. \nonumber \\&\left. +B_{i}K_{i})+P_{i}B_{i}K_{i}K^{T}_{i}B^{T}_{i}P_{i}+I\right] x_{1}\nonumber \\&+e^{T}e+2P\Vert x_{1}\Vert \Vert x_{2}\Vert -l\beta \Vert x_{2}\Vert ^{2}-\Vert x_{1}\Vert ^{2}\nonumber \\\le & {} x^{T}_{1}\left[ (A_{1i}+B_{i}K_{i})^{T}P_{i}+P_{i}(A_{1i}+B_{i}K_{i})\right. \nonumber \\&\quad \left. +P_{i}B_{i}K_{i}K^{T}_{i}B^{T}_{i}P_{i}+I\right] x_{1}\nonumber \\&\quad +e^{T}e-\left( \Vert x_{1}\Vert ^{2}+2P\Vert x_{1}\Vert \Vert x_{2}\Vert +p^{2}\Vert x_{2}\Vert ^{2}\right) \nonumber \\&\quad +p^{2}\Vert x_{2}\Vert ^{2}-l\beta \Vert x_{2}\Vert ^{2}\nonumber \\&\quad \le x^{T}_{1}Q_{i}x_{1}+\varepsilon -\frac{\beta l}{2\alpha _{2}}W_{i}(x_{2})+\frac{\beta l}{2\alpha _{2}}W_{i}(x_{2})\nonumber \\&\quad -l\beta \Vert x_{2}\Vert ^{2}+p^{2}\Vert x_{2}\Vert ^{2}\nonumber \\\le & {} -\frac{\lambda _{min}(-Q_{i})}{\lambda _{max}(P_{i})}x^{T}_{1}P_{i}x_{1}+\varepsilon -\frac{\beta l}{2\alpha _{2}}W_{i}(x_{2})\nonumber \\&\quad -\left( \frac{1}{2}l\beta -p^{2}\right) \Vert x_{2}\Vert ^{2} \end{aligned}$$
(16)

where \(Q_{i}=\left( A_{1i}+B_{i}K_{i}\right) ^{T}P_{i}+P_{i}\left( A_{1i}+B_{i}K_{i}\right) +P_{i}B_{i}K_{i}K^{T}_{i}B^{T}_{i}P_{i}+\left( 1+\eta \right) I\).

Let \(\gamma _{i}=\min \left\{ \frac{\lambda _{\min }(-Q_{i})}{\lambda _{\max }(P_{i})},\frac{\beta l}{2\alpha _{2}}\right\} \), \(l\ge {\frac{2p^{2}}{\beta }}\). According to the linear matrix inequality (9), we can get \(Q_{i}<0\); therefore,

$$\begin{aligned} {\dot{V}}_{i}\le {-\gamma _{i}V_{i}}+\varepsilon \end{aligned}$$
(17)

Integrating (17) from \(t_{i}\) to t yields

$$\begin{aligned} V_{i}(t)\le e^{-\gamma _{i}(t-t_{i})}V_{i}(t_{i})+\varepsilon \int ^{t}_{t_{i}}e^{-\gamma _{i}(t-s)}ds \end{aligned}$$
(18)

According to the inequalities (10), (12), and (14), we have

$$\begin{aligned} V_{l_{i}}(t_{i})\le {\tilde{\mu }}V_{l_{i-1}}\left( t^{-}_{i}\right) , \forall l_{i},l_{i-1}\in {{{\bar{M}}}} \end{aligned}$$
(19)

where \({\tilde{\mu }}=\max \left\{ \mu ,\frac{\alpha _{2}}{\alpha _{1}}\right\} \). Let \(\gamma =\mathop {\min }\limits _{\forall l_{i}\in {\bar{M}}}\gamma _{l_{i}}>0\). Combining with the inequality (18) and \(N_{\sigma }\left( (t-t_{i})/\tau _{a}\right) +N_{0}\), we have

$$\begin{aligned}&V_{\sigma (t)}(t)=V_{l_{i}}(t)\nonumber \\&\quad \le e^{-\gamma (t-t_{i})}{\tilde{\mu }}V_{l_{i-1}}(t^{-}_{i})\nonumber \\&\qquad +\frac{\varepsilon }{\gamma }\left( 1-e^{-\gamma (t-t_{i})}\right) \nonumber \\&\quad \le e^{-\gamma (t-t_{i})}{\tilde{\mu }}\left( e^{-\gamma (t-t_{i})}V_{l_{i-1}}(t_{i-1})\right. \nonumber \\&\qquad \left. +\frac{\varepsilon }{\gamma }\left( 1-e^{-\gamma (t-t_{i})}\right) \right) +\frac{\varepsilon }{\gamma }\left( 1-e^{-\gamma (t-t_{i})}\right) \nonumber \\&\quad \le e^{-\gamma (t-t_{i})}{\tilde{\mu }}V_{l_{i-1}}(t_{i-1})\nonumber \\&\qquad +\frac{\varepsilon {\tilde{\mu }}}{\gamma }\left( e^{-\gamma (t-t_{i})}-e^{-\gamma (t-t_{i-1})}\right) \nonumber \\&\qquad +\frac{\varepsilon }{\gamma }\left( 1-e^{-\gamma (t-t_{i})}\right) \nonumber \\&\quad \le e^{-\gamma (t-t_{i})}{\tilde{\mu }}^{2}V_{l_{i-2}}(t^{-}_{i-1})\nonumber \\&\qquad +\frac{\varepsilon {\tilde{\mu }}}{\gamma }\left( e^{-\gamma (t-t_{i})}-e^{-\gamma (t-t_{i-1})}\right) \nonumber \\&\qquad +\frac{\varepsilon }{\gamma }\left( 1-e^{-\gamma (t-t_{i})}\right) \nonumber \\&\qquad \vdots \nonumber \\&\quad \le e^{-\gamma (t-t_{0})}{\tilde{\mu }}^{N_{\sigma }(t_{0},t)}V_{l_{0}}(t_{0})\nonumber \\&\qquad +\frac{\varepsilon {\tilde{\mu }}^{N_{\sigma }(t_{1},t)}}{\gamma }\left( e^{-\gamma (t-t_{2})}-e^{-\gamma (t-t_{1})}\right) \nonumber \\&\qquad +\frac{\varepsilon {\tilde{\mu }}^{N_{\sigma }(t_{2},t)}}{\gamma }\left( e^{-\gamma (t-t_{3})}-e^{-\gamma (t-t_{2})}\right) \nonumber \\&\qquad +\dots +\frac{\varepsilon {\tilde{\mu }}^{2}}{\gamma }\left( e^{-\gamma (t-t_{i-1})}-e^{-\gamma (t-t_{i-2})}\right) \nonumber \\&\qquad +\frac{\varepsilon {\tilde{\mu }}}{\gamma }\left( e^{-\gamma (t-t_{i})}-e^{-\gamma (t-t_{i-1})}\right) \nonumber \\&\qquad +\frac{\varepsilon }{\gamma }\left( 1-e^{-\gamma (t-t_{i})}\right) \nonumber \\&\quad \le e^{-\gamma (t-t_{0})}{\tilde{\mu }}^{N_{\sigma }(t_{0},t)}\left( V_{l_{0}}(t_{0})\right. \nonumber \\&\qquad \left. -\frac{\varepsilon }{\gamma {\tilde{\mu }}}\right) +\frac{\varepsilon ({\tilde{\mu }}-1)}{\gamma }\sum ^{N_{\sigma }(t_{2},t)}_{k=0}{\tilde{\mu }}^{k}e^{-\gamma (t-t_{i-k})}+\frac{\varepsilon }{\gamma }\nonumber \\&\quad \le e^{-(\gamma -(\ln {{\tilde{\mu }}}/\tau _{a}))(t-t_{0})}{\tilde{\mu }}^{N_{0}}\left( V_{l_{0}}(t_{0})\right. \nonumber \\&\qquad \left. -\frac{\varepsilon {\tilde{\mu }}}{\gamma }\right) + \frac{\varepsilon ({\tilde{\mu }}-1)}{\gamma }e^{\gamma \tau _{a}N_{0}}\sum ^{N_{\sigma }(t_{2},t)}_{k=0}e^{k(\ln {\tilde{\mu }}-\gamma \tau _{a})}\nonumber \\&\qquad +\frac{\varepsilon }{\gamma } \end{aligned}$$
(20)

From (14), we know that

$$\begin{aligned} V_{l_{i}}(t)= & {} x^{T}_{1}(t)P_{l_{i}}x_{1}(t)+lW_{l_{i}}(x_{2})\\\ge & {} \mathop {\min }\limits _{\forall l_{i}\in {M}}\left( \lambda \left( P_{l_{i}}\right) \right) \Vert x_{1}(t)\Vert ^{2}\\&+l\alpha _{1}\Vert x_{2}(t)\Vert ^{2}\ge \varsigma \Vert x(t)\Vert ^{2} \end{aligned}$$

and

$$\begin{aligned} V_{t_{0}}\le & {} \mathop {\max }\limits _{\forall l_{i}\in {M}}\left( \lambda \left( P_{l_{i}}\right) \right) \Vert x_{1}(t_{0})\Vert ^{2}\\&+l\alpha _{2}\Vert x_{2}(t_{0})\Vert ^{2}\le \xi \Vert x(t_{0})\Vert ^{2} \end{aligned}$$

where \(\varsigma =\min \left\{ \mathop {\min }\limits _{\forall k_{i}\in {\bar{M}}}\left( \lambda \left( P_{k_{i}}\right) \right) ,l\alpha _{1}\right\} ,\xi =\max \left\{ \mathop {\max }\limits _{\forall k_{i}\in {\bar{M}}}\left( \lambda \left( P_{k_{i}}\right) \right) ,l\alpha _{2}\right\} \).

Thus, according to the above inequalities, we can obtain that

$$\begin{aligned}&\Vert x(t)\Vert ^{2} \le \frac{1}{\varsigma }V_{l_{i}}(t)\nonumber \\&\quad \le \frac{\xi }{\varsigma }e^{-\left( \gamma -\left( \ln {\tilde{\mu }}/\tau _{a}\right) \right) \left( t-t_{0}\right) }{\tilde{\mu }}^{N_{0}}\left\{ \Vert x(t_{0})\Vert ^{2} -\frac{\varepsilon }{\gamma {\tilde{\mu }}\xi }\right\} \nonumber \\&\qquad +\frac{\varepsilon ({\tilde{\mu }}-1)}{\varsigma \gamma }e^{\gamma \tau _{a}N_{0}} \sum ^{N_{\sigma }(t_{2},t)}_{k=0}e^{k(ln{\tilde{\mu }}-\gamma \tau _{a})}+\frac{\varepsilon }{\varsigma \gamma } \end{aligned}$$
(21)

The condition \(\tau _{a}>(\ln {\tilde{\mu }})/\gamma \) means that \(\gamma -\left( (\ln {{\tilde{\mu }}})/\tau _{a}\right) >0\) and \(\ln {{\tilde{\mu }}}-\gamma \tau _{a}<0\). Then, according to inequality (21), we have

$$\begin{aligned} \Vert x(t)\Vert ^{2}\le & {} \frac{\xi }{\varsigma }e^{-(\gamma -(\ln {\tilde{\mu }}/\tau _{a}))(t-t_{0})}{\tilde{\mu }}^{N_{0}}\left\{ \Vert x(t_{0})\Vert ^{2} -\frac{\varepsilon }{\gamma {\tilde{\mu }}\xi }\right\} \nonumber \\&+\frac{\varepsilon ({\tilde{\mu }}-1)}{\varsigma \gamma }e^{\gamma \tau _{a}N_{0}} \sum ^{N_{\sigma }(t_{2},t)}_{k=0}e^{k(ln{\tilde{\mu }}-\gamma \tau _{a})}+\frac{\varepsilon }{\varsigma \gamma } \end{aligned}$$
(22)

which can guarantee the uniform boundedness of the system (1).

Remark 1

Let \(\varepsilon =0\), the inequality (22) can be rewritten as

$$\begin{aligned} \Vert x(t)|^{2}\le \frac{\xi }{\varsigma }e^{-(\gamma -(\ln {\tilde{\mu }}/\tau _{a}))(t-t_{0})}{\tilde{\mu }}^{N_{0}}\Vert x(t_{0})\Vert ^{2} \end{aligned}$$

Thus, when \(\varepsilon =0\), the switched nonlinear system (8) is exponentially stable.

Remark 2

According to Theorem 1, we assume the controller gain is \(K_i=B_i^TP_i\) , and the inequality (9) can be transformed the following LMIs:

$$\begin{aligned}&\left[ \begin{array}{ccc} X_iA^{T}_{1i}+2B_iB^{T}_{i}+A_{1i}X_i&{}B_iB^{T}_{i}&{}X_i\\ B^{T}_{i}B_i&{}-X_i&{}0\\ X_i&{}0&{}-(1+\eta )I \end{array}\right] \nonumber \\&\quad <{0}, X_i>I \end{aligned}$$
(23)

where \(X_i=P_i^{-1},\) \(i\in {{\bar{M}}}.\)

Remark 3

There is a lot of interaction between sensors, controllers, and actuators, which leads to information load problems and a waste of computational resources. Different from periodic sampling, under the event-triggered mechanism, data will only be transmitted when the system meets the set requirements, so that the amount of calculation can be saved.

3.2 Minimum inter-event interval

The Zeno behavior can be prevented in time-triggered sampling control by setting a positive time interval between two consecutive samples. However, this method cannot successfully be used in event-triggered sampling control. Therefore, a new method to exclude Zeno behavior is proposed in this section.

Theorem 2

Under the condition in inequality (3), the lower bound of the minimum inter-event interval is a positive scalar.

Proof

Firstly, we assume that there exist n samplings on \(\left[ t_{i},t_{i+1}\right) \), and the corresponding moments are \({\hat{t}}_{k+1},{\hat{t}}_{k+2},\ldots ,{\hat{t}}_{k+n}\). On any interval of \(\left[ t_{i},{\hat{t}}_{k+1}\right) ,\left[ {\hat{t}}_{k+1},{\hat{t}}_{k+2}\right) ,\ldots ,\left[ {\hat{t}}_{k+n},{\hat{t}}_{i+1}\right) \), the state \(x_{1}({\hat{t}}_{k+j}),j=0,1,\ldots ,n\) are always constants and \(e(t)=x_{1}(t)-x_{1}({\hat{t}}_{k+j})\). Thus, we can obtain that

$$\begin{aligned}&{\dot{e}}(t)={\dot{t}}_{1}(t)=A_{1i}x_{1}(t)+A_{2i}x_{2}(t)+B_{i}K_{i}x_{1}({\hat{t}}_{k+j})\nonumber \\&\quad = A_{1i}\left( e(t)+x_{1}({\hat{t}}_{k+l})\right) +A_{2i}x_{2}(t)+B_{i}K_{i}x_{1}({\hat{t}}_{k+j})\nonumber \\&\quad = A_{1i}e(t)+\left( A_{1i}+B_{i}K_{i}\right) x_{1}({\hat{t}}_{k+j})+A_{2i}x_{2}(t)\nonumber \\ \end{aligned}$$
(24)

Therefore,

$$\begin{aligned} e(t)= & {} e^{A_{1i}(t-{\hat{t}}_{k+j})}e({\hat{t}}_{k+j}) +\int ^{t}_{{\hat{t}}_{k+j}}e^{A_{1i}(t-s)}\left( (A_{1i}\right. \\&\left. +B_{i}K_{i})x_{1}({\hat{t}}_{k+j})+A_{2i}x_{2}(s)\right) ds \end{aligned}$$

Due to \(e({\hat{t}}_{k+j})=x_{1}({\hat{t}}_{k+j})-x_{1}({\hat{t}}_{k+j})=0\), we have

$$\begin{aligned} e(t)= & {} \int ^{t}_{{\hat{t}}_{k+j}}e^{A_{1i}(t-s)}\left( (A_{1i}+B_{i}K_{i})x_{1}({\hat{t}}_{k+j})\right. \\&\left. +A_{2i}x_{2}(s)\right) ds \end{aligned}$$

Thence,

$$\begin{aligned} \Vert e(t)\Vert\le & {} \int ^{t}_{{\hat{t}}_{k+j}}e^{\Vert A_{1i}\Vert (t-s)}\left( \Vert (A_{1i}+B_{i}K_{i})x_{1}({\hat{t}}_{k+j})\Vert \right. \\&\left. +\Vert A_{2i}x_{2}(s)\Vert \right) ds\\\le & {} \int ^{t}_{{\hat{t}}_{k+j}}e^{\Vert A_{1i}\Vert (t-s)}\Vert (A_{1i}+B_{i}K_{i})\Vert \Vert x_{1}({\hat{t}}_{k+j})\Vert ds\\&+\int ^{t}_{{\hat{t}}_{k+j}}e^{\Vert A_{1i}\Vert (t-s)}\Vert A_{2i}\Vert \Vert x_{2}(s)\Vert ds \end{aligned}$$

According to (22), we can find a positive constant \({\tilde{\beta }}\) such that

$$\begin{aligned} \Vert e(t)\Vert\le & {} \int ^{t}_{{\hat{t}}_{k+j}}e^{\Vert A_{1i}\Vert (t-s)}\Vert (A_{1i}+B_{i}K_{i})\Vert \Vert x_{1}({\hat{t}}_{k+j})\Vert ds\\&+\int ^{t}_{{\hat{t}}_{k+j}}e^{\Vert A_{1i}\Vert (t-s)}\Vert A_{2i}\Vert \sqrt{{\tilde{\beta }}}ds\\\le & {} \chi ({\hat{t}}_{k+j})\int ^{t}_{{\hat{t}}_{k+j}}e^{\Vert A_{1i}\Vert (t-s)}ds \end{aligned}$$

where \(\chi ({\hat{t}}_{k+j})=\Vert ((A_{1i}+B_{i}K_{i}))\Vert +\Vert A_{2i}\Vert \sqrt{{\tilde{\beta }}}\). If \(\Vert A_{1i}\Vert \ne {0}\), then

$$\begin{aligned} \Vert e(t)\Vert \le \frac{\chi ({\hat{t}}_{k+j})}{\Vert A_{1i}\Vert }\left( e^{\Vert A_{1i}\Vert (t-s)}-1\right) \end{aligned}$$

According to (4), when \(\Vert e(t)\Vert ^{2}=\eta \Vert x_{1}(t)\Vert _{2}+\varepsilon \), the next event will occur. Let \(T=t-{\hat{t}}_{k+j}\) denote the lower bound of inter-event interval, and T is determined by

$$\begin{aligned} \frac{\chi ({\hat{t}}_{k+j})}{\Vert A_{1i}\Vert }\left( e^{\Vert A_{1i}\Vert (t-s)}-1\right) =\sqrt{\eta \Vert x_{1}\Vert ^{2}+\varepsilon } \end{aligned}$$

Thus

$$\begin{aligned} T=\frac{1}{\Vert A_{1i}\Vert }\ln \left( \frac{\Vert A_{1i}\sqrt{\eta \Vert x_{1}\Vert ^{2}+\varepsilon }}{\chi ({\hat{t}}_{k+j})}+1\right) \end{aligned}$$

that means \(T>0\) for any given sampling instant.

4 Numerical example

In this section, we will prove the feasibility of the proposed methods by a numerical example.

Example

Consider system (1) with the subsystem, where

$$\begin{aligned} A_{11}= & {} \begin{bmatrix} -4.8 &{} 1 \\ 0 &{} -2.9 \end{bmatrix} ,\quad A_{21}=\begin{bmatrix} 1 \\ 0 \end{bmatrix} ,\quad B_{1}=\begin{bmatrix} 0.18 \\ 1 \end{bmatrix} ,\\ C_{1}= & {} \begin{bmatrix} 1&0 \end{bmatrix} ,\quad f_{21}=-x_{2}-x_{2}\sin ^{2}x_{2}; \\ A_{12}= & {} \begin{bmatrix} -5 &{} 1 \\ 0 &{} -3 \end{bmatrix} ,\quad A_{22}=\begin{bmatrix} 0 \\ 1 \end{bmatrix} ,\quad B_{2}=\begin{bmatrix} 0.2 \\ 1 \end{bmatrix} ,\\ C_{2}= & {} \begin{bmatrix} 0&1 \end{bmatrix} ,\quad f_{22}=-x_{2}-x_{2}\\cos^{2}x_{2}. \end{aligned}$$

Set \(\eta =1,\alpha _{1}=0.4,\alpha _{2}=0.5,\beta =1,\mu =1.1,\varepsilon =0, \kappa _{1}=0.9426,\kappa _{2}=1.3515\). Choose \(W_{1}=\frac{1}{2}x^{2}_{2}, W_{2}=\frac{2}{5}x^{2}_{2}\), and \(l=1\). The controller gains are chosen as

$$\begin{aligned} K_{1}=\begin{bmatrix} -0.2&-0.5 \end{bmatrix} , K_{2}=\begin{bmatrix} -0.4&-0.6 \end{bmatrix}. \end{aligned}$$

By solving inequalities (9) and (10) in Theorem 1, we get

$$\begin{aligned} P_{1}=\begin{bmatrix} 0.3952 &{} 0.0285 \\ 0.0285 &{} 0.5649 \end{bmatrix} , P_{2}=\begin{bmatrix} 0.3687 &{} 0.0115 \\ 0.0115 &{} 0.5150 \end{bmatrix}. \end{aligned}$$

\(\gamma =1.76,{\tilde{\mu }}=1.25\) and \(\tau ^{*}_{a}=(\ln {{\tilde{\mu }}})/\gamma =0.127\). Obviously, the conditions of Theorem 1 are satisfied; therefore, the considered system is exponentially stable under the switching signal \(\sigma \). Figures. 1, 2, 3, 4, 5, 6 show the results of the simulation, respectively. Figure 1 illustrates the state response of the system. From Fig. 1, we can see that the state of the system converges to zero, which indicates that the system is stable; Figure 2 gives the event-triggered condition; Figure 3 demonstrates the switching signal; Figure 4 displays the control input of the system. Figures 5 and 6 depict the time-triggered instants and event-triggered instants, respectively. One can obtain that, under the time-triggered scheme, there are 60 sampling instants in [0, 15s], by using the event-triggered scheme, only 45 sampling instants. In other words, \(75\%\) data information is used to stabilize the controlled system, which proves that the designed event-triggered scheme in (3) can save the communication resource effectively.

Fig. 1
figure 1

State responses

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Fig. 2
figure 2

Event-triggered condition

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Fig. 3
figure 3

Control input

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Fig. 4
figure 4

Switching signal

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Fig. 5
figure 5

Time-triggered instants

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Fig. 6
figure 6

Event-triggered instants

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5 Conclusions

In this paper, the problem of event-triggered control has been studied for the cascade switched nonlinear system. With the application of the average dwell time technique and the Lyapunov function method, we have obtained sufficient conditions to guarantee the system to be globally uniformly bounded and exponentially stable. What’s more, we have obtained a lower bound of the minimum inter-event interval to preclude the Zeno behavior in the procedure of event-triggered sampling. At present, the event-triggered control problems of the cascaded switched nonlinear systems still need further research. For example, noise signals are widely existing in the world [30] and Zeno behavior influencing the performance of systems. Therefore, considering the effect of noises in the framework of event-triggered systems and excluding the Zeno behavior in real time are our further research goals.