Dissipative control of switched nonlinear systems with sliding mode: an event-triggered control scheme

This paper introduces the event-triggered scheme to study the problems of dissipative control and dissipativity-based sliding mode control (SMC) for switched nonlinear systems. Firstly, based on a designed event-triggered scheme, a piecewise continuous state feedback controller is constructed; sufficient conditions for Q,S,R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( Q,S,R\right) $$\end{document}-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-dissipativity of the resulting system are derived by the approaches of a switched Lyapunov function and an average dwell time. Secondly, these two methods are extended to the study of H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\infty $$\end{document} performance and passive performance; the corresponding sufficient conditions are provided. Thirdly, the dissipative control problem is solved by introducing an event-triggered scheme into SMC method. Sufficient conditions for dissipativity of the resulting system are derived, and an SMC law is designed to guarantee the reachability. At the end of this paper, the correctness of the proposed methods is proved by two numerical examples.


Introduction
Switched systems are a special kind of hybrid dynamical systems [1]. Many practical processes, such as mechanical systems and electrical systems, are often described by switched systems. Due to its superiority in application, the switched systems have become one of the main directions of scholars' research in recent years. However, the existing results on switched systems are mainly for linear systems. The switched nonlinear systems have more complex application backgrounds; the research on switched nonlinear systems is more challenging and meaningful. As the basic control problems, stability analysis and stabilization are issues that most of scholars are interested in, and many methods introduced in this process have been proved to be effective [2][3][4][5][6], for example, the average dwell time method.
Recently, the sampling control is paid more and more attention. In fact, there exists a lot of redundant information in most practical systems, which has a great negative impact on the improvement of control efficiency and the reduction of energy loss [7]. As an effective control strategy, event-triggered control theory is proposed to solve this problem, with which, the control behavior is executed when it is needed instead of being executed regularly or periodically [8][9][10]. Event-triggered control is becoming a very hot research topic in control theory. In [11][12][13], scholars studied the control synthesis problems for switched lin-ear systems via the event-triggered strategy; however, these results have not been extended to switched nonlinear systems. Lian and Li studied the tracking control problem for switched nonlinear systems with the eventtriggered approach, but not referring to the dissipativity and SMC problems [14]. By the event-triggered approach, a design of sliding mode observer was investigated for nonlinear systems [15]. For nonlinear networked control systems, a kind of event-triggered communication scheme was investigated by Gu et al. [16]. Unfortunately, the above two event-triggered schemes have not been introduced to switched nonlinear systems.
The dissipativity theory has always been a reliable tool for system research and design, especially for power systems and network control systems [17]. Dissipativity means the energy stored does not exceed the energy supplied, which are usually expressed in terms of storage function and supply rate [18]. Dissipativity provides us with novel ideas and methods for analyzing and designing the controlled systems from the perspective of energy, which are mainly reflected in the following three points: Firstly, the Lyapunov function of the controlled system can be directly replaced with the storage function; secondly, dissipativity theory can be used to study and solve many important problems of controlled systems, such as stabilization, optimal control, robust control, and H ∞ control [19,20]. Finally, under a general structure, when we select some special forms of the supply rate, we will obtain dissipativity forms with special significance such as (Q, S, R)-dissipativity, passive performance, and H ∞ performance. Dissipativity also has a significant effectiveness on analysis and designing of switched systems. Using the relationship between energy storage and supply in switched systems, Zhao et.al constructed a dissipativity theory framework and proposed a dissipativity concept for switched systems [21][22][23]. For switched nonlinear systems, Pang and Zhao investigated incremental (Q, S, R)-dissipativity without assuming that subsystems are incremental (Q, S, R)-dissipative [24]. For switched stochastic systems, Liu et al. studied the (Q, S, R)-α-dissipative control problems in [25]. For discrete-time switched systems, McCourt and Antsaklis established a notion of (Q, S, R)-dissipativity through the multiple supply rates method [26]. However, up to now, there are few results of dissipative control for switched nonlinear systems based on the event-triggered method. Therefore, it is worth trying to extend these methods to the research of switched nonlinear systems.
Additionally, SMC is an effective analysis and synthesis tool for complex dynamical systems. In sliding mode control, the most important works can be divided into two points: selecting a sliding mode surface function and designing a controller. Recently, researchers are trying to introduce the event-triggered scheme into SMC [27][28][29]. Event-triggered SMC problems for nonlinear systems have been studied in [30,31]. However, how to extend these results to switched systems is a challenging issue. In [32,33], dissipative control problems based event-triggered scheme and SMC method have been studied for linear switched systems and TS fuzzy systems, respectively. But there are almost no results of introducing event-triggered scheme to SMC for switched nonlinear systems.
Under the motivation of the above discussions, an event-triggered scheme is introduced to dissipative control and SMC for switched nonlinear systems. The main contributions can be reflected in three points: (1) Based on a designed event-triggered scheme, the dissipative control strategy for switched cascade nonlinear systems is firstly proposed by applying a switched Lyapunov function method and the average dwell time technique. Correspondingly, an event-triggered-based piecewise continuous controller is developed. (2) Sufficient conditions are obtained to guarantee the (Q, S, R)-α-dissipativity and exponential stability for the considered switched nonlinear system, based on that sufficient conditions of H ∞ performance and passive performance can also be obtained. (3) An event-triggered scheme is introduced to SMC in the analysis of dissipativity for switched nonlinear systems. On the one hand, both the linear part and the nonlinear part are concerned when the observers are designed, and the dissipativity of sliding dynamics is discussed. On the other hand, an SMC law is provided to ensure the trajectories can be maintained onto a predefined sliding manifold within a limited time.
The main content of this paper consists of two parts. In the first part, the dissipative control for the switched nonlinear system is concerned; sufficient conditions for (Q, S, R)-α-dissipativity, H ∞ performance, and pas-sive performance are provided. In the second part, the event-triggered SMC with dissipativity is concerned.

Preliminaries
Consider a switched nonlinear system described by the following dynamical equations: where is a piecewise constant value function of time, called switching signal; σ (t) = i means the ith subsystem is active. v(t) ∈ R p represents the disturbance input that belongs to L 2 [0, +∞). f 2i (·) are known smooth vector fields with appropriate dimensions and f 2i (0) = 0.
Corresponding to σ (t), we can obtain the following switching sequence: where the pair (s i , t i ) means the s i th subsystem is active on t i , t i+1 ), and the switching occurs at the moment t i . To improve computing efficiency for the considered system, we design an event-triggered scheme. Let x 1 (t) denote the current state and x 1 (t k ) denote the sampled state. The error is e(t) = x 1 (t) − x 1 (t k ). For a given positive scalar η, the event-triggered condition is: Remark 1 The event will be triggered once inequality (3) is satisfied; then the data are transmitted to the controller. When an event occurs, the value of e(t) reduces to 0; otherwise, it begins to increase until a new event happens. When the s i th subsystem is activated, n i + 1(n i > 0) samplings will be triggered on the interval t i , t i+1 ), that is, t i , t i+1 ) = t i ,t k+1 ∪ t k+1 ,t k+2 ∪ · · · ∪ t k+n i , t i+1 ,t 0 = t 0 represents the first trigger moment.
Definition 1 [34] For any switching signal and any t > s ≥ 0, N σ (s, t) means the number of switchings on the interval (s, t). For τ a > 0 and N 0 ≥ 0, if there exists N σ (s, t) ≤ N 0 + t−s τ a , then the constant τ a is called the average dwell time and N 0 is the chatter bound. Without loss of generality, we let N 0 = 0. Definition 2 [35] If there are two constants α > 0, λ > 0, satisfying Then, system (1) is globally exponentially stable under the switching signal σ (t).

Event-triggered dissipative control
In this section, the dissipativity of the considered system in (1) will be analyzed. Main results will be presented in the form of theorems, and the necessary proofs will be also provided.
Firstly, combining the event-triggered condition in (3), for the s i th subsystem, the corresponding controller is constructed as follows: . . .
where K i are controller gains with proper dimensions.
A system is dissipative when the energy dissipation occurs. In a dissipative system, energy can only be consumed but not produced; in other words, the energy stored cannot be more than the energy supplied. The theories of dissipativity have been introduced in many different kinds of systems, for the considered system, the energy supply function is usually described as: where Q, S and R are constant matrices of appropriate Definition 3 [36] For the given energy supply function in (6), the system is dissipative if there is a nonnegative function (i.e., the energy storage function) From Definition 4, we can deduce the H ∞ performance and the passivity performance. In general, Definition 4 The passivity performance is adopted as Based on the above discussions, we will give sufficient conditions for the considered cascaded nonlinear system to satisfy the dissipativity under event-triggered scheme (3).
and the switching signal satisfies Proof Design the following Lyapunov function: For the s i th activated subsystem, according to the derivative of V (t), it can be deduced thaṫ There are two constants ς i > 0, l i > 0, satisfying Setting l = max {ς i l i |i ∈ M} and using the eventtriggered condition in (3), we havė From (7), we havė Let From (8) and (12), there is whereδ = max δ, κ 2 κ 1 . By (15), we can get Then we can calculate that Multiplying both sides of (17) by e −N σ (0,t) lnδ , we have The proof of Theorem 1 is completed. (17), we can yield that the following inequality holds with v(t) = 0.

Remark 2 By the inequality in
For the Lyapunov function in (12), we can easily find two constants ε 1 and ε 2 , satisfying Then, according to (18) and (19), it is not difficult to obtain the inequality Thus, for any t ≥ 0, This proves that the considered system is stable.
The following are details: For any t ∈ t k ,t k+1 there exists β > 0 and ε > 0 such that Obviously, the value of ℘ (t) evolves from 0 to √ η.

Thus, we can obtain the positive lower bound of the inter-execution intervals by the solution ℘ (t) = √ η.
This completes the proof.
In the next content, we will discuss two special cases of the dissipativity-H ∞ performance and passive performance.
Next, sufficient conditions will be obtained by referencing to the above methods.

Then, system (5) is exponentially stable with H ∞ performance.
Proof Design the following Lyapunov function: For the s i th activated subsystem, according to the derivative of V (t), we easily geṫ According to the proof process of Theorem 1, it is obvious that the following inequality holds.
This completes the proof.
Proof Choose the function: This proof can be completed by referring to the techniques used in the proof of Theorem 1.

Event-triggered sliding mode control with dissipativity
Next, we will investigate the dissipativity of switched nonlinear systems via the event-triggered SMC method. As shown in Fig. 1, an observer is constructed between the sensor and controller to estimate unmeasured state variables in system. As we mentioned in the previous content, there is a lot of redundant data between the sensor and the controller. To solve this problem, the event-triggered strategy is applied again, which can improve the calculation efficiency. We also set up the event detectors on two different channels, respectively. Consider the following system: where z(t) ∈ R q represents the controlled output, y(t) ∈ R n y represents the measurable output, D i ∈ R n y ×(n−d) , i ∈M are constant matrices. Let y(t) denote the current output, while y(t k ) is the sampled output att k . The error is e y (t) = y(t) − y(t k ). Then the event-triggered condition between the sensor and observer can be described as the following inequality.

e T y (t)e y (t) ≥ ηy T (t)y(t)
where η is the given positive scalar. Letx 1 (t) denote the current estimated state, whilê . Then the event-triggered condition between the observer and controller can be described as the following inequality.
where η is the given positive scalar. For the unavailable state variables of the subsystem s i , we construct an observer as follows: wherex 1 (t) ∈ R n−d ,ŷ(t) ∈ R n y represent the observed state and output, respectively. E i ∈ R (n−d)×n y , i ∈M are the matrices to be given.
We design an sliding surface function for the considered switched nonlinear system as follows: (24) where G i , K i are real matrices with proper dimensions and G i B 1i is nonsingular and positive definite.
With the event-triggered condition in (22), equation (25) can be rewritten as System (27) can be rewritten as: wherē t)) , z ∈ R nz is the output of system (28).
In the next content, we will study the stability and dissipativity for system (28).
Proof Construct the following Lyapunov function: When the s i th subsystem is activated, we havė There are two constants j i > 0, q i > 0, satisfying (21) and (22), we havė Let l 1 ≥ 2 a 2 λ 0 +β 1 , l 2 ≥ 2 b 2 λ 0 +β 2 , according to (29), then we can geṫ From (38) we havė By (30) and (36), we have Integrating both sides of (39) yields Then we can calculate that When v = 0, we easily get the exponential stability of the switched system. Next, by multiplying both sides of (41) by e −N σ (0,t)lnδ , we have This completes the proof of Theorem 4.
In the following content, we will study the reachability of the system. An SMC law is constructed and the system trajectories will be forced onto the specified sliding manifold within a limited time.
Theorem 5 Under the conditions of Theorem 4, consider designed sliding surface function (24) and observer (23), via event-triggered condition (22), the trajectories of system (20) are forced onto the specified sliding manifold within a limited time by the following proposed controller where ϕ satisfies the following condition Proof Choose the following Lyapunov function: by (24), (42), and (43), it can be deduced thaṫ < 0 ( 4 4 ) From (44) we can get thatV (t) < 0 always holds, which implies that the trajectories can be driven onto the predefined manifold S(t) = 0 by control law (42) within a limited time.

Numerical example
This section provides two numerical examples to verify the correctness of the proposed method.

Example 2 Consider switched nonlinear systems (20) with
By solving matrix inequality (29) and inequality (30) Figure 7 shows the state responses of the  Fig. 9 shows the switching signal; Fig. 10 gives the errors of the observer. Figures 7, 8, 9 and 10 depict that the system is stable. Figure 11 shows the SMC control input. Figure 12 shows the sliding surface function. These figures show the effectiveness of Theorems 4 and 5.

Conclusions
This paper introduces the event-triggered scheme into the dissipativity and SMC for switched nonlinear systems; the sufficient conditions are obtained to ensure the (Q, S, R)-α-dissipativity, H ∞ performance and passive performance. Meanwhile, for estimating the unavailable state of the system, we construct an observer, based on which we design the sliding surface function. Event detectors also are set up on two different channels, which can greatly reduce the computation. By virtue of the event-triggered scheme and SMC method, the dissipative conditions of the switched nonlinear system are proposed. For the event-triggered SMC for switched systems, there are still many problems to be researched. In the further, we will discuss the sliding domain for the sliding mode dynamics.