Sliding Mode Control of a Class of Hybrid-Switched Systems with Disturbances

: This paper investigates the problem of sliding mode control for a class of hybrid switched systems with matching disturbances. Firstly, a sliding mode surface is designed, and the corresponding sliding mode equation for the switched system is derived. Then, we analyzed the stability of the sliding mode equation using the Lyapunov function and average dwell time. Moreover, a sliding mode control law is designed with the approach law method to drive the system state to a bounded sliding mode region and then maintain it there subsequently. Finally, an illustrative example is given to demonstrate the efficiency of the approach.


Introduction
With the continuous expansion of modern control system application field, switched systems become more and more important in practical application.Switched systems are composed of continuous or discrete-time subsystems, ang switching regulations coordinating the transitions between them.Switched system can simplify the mathematical model of the system, minimize the computational complexity and enhance the real-time efficiency of the system.It has been widely used in different fields, such as PI control of vehicle transmission [1], flight control system [2], power electronics system [3,4], network control system [5][6][7], and has attracted great attention from researchers.In the switched system, the switching signal indicates how each subsystem operates, and it is the core factor that determines the stable operation of the system.Because of the uncertainty and non-linearity in the switched process, it has become an important research focus to design effective control strategies to ensure the stability and performance.Using the event-triggered mechanism, Fei et al. [8] studied the dynamic output feedback control for switched systems with asynchronous switching.On the basis of the common Lyapunov function search method, Zhang et al. [9] proposed an integral sliding mode surface with good robustness to uncertainty, which is suitable for uncertain switched systems under arbitrary time-varying switched rules.Lian et al. [10] proposed a dynamic quantization strategy with known perturbation bounds for a class of switched linear systems with uncertain non-linearity.
As a robust control strategy, sliding mode control (SMC) is favored for its simple structure and rapid response.Sliding mode control technology regulates the dynamic characteristics of the system by predefined sliding mode surfaces, thereby guiding the system state to converge onto and subsequently slide along these surfaces.The design concept of SMC aims to ensure optimal performance and stability of the system, even in the presence of uncertainties such as external disturbances and inaccuracies in modeling.Due to these advantages of SMC, it has been an extensive application in diverse and intricate systems [11][12][13][14].In recent years, the SMC technology has seen a wide range of application and has made some key progress.It has been applied to Markov jump systems, switched systems, stochastic systems and fuzzy systems [15][16][17][18][19][20][21].Fei et al. [22] proposed a scheme combining fractional-order SMC and fuzzy neural network, which can effectively reduce the nonlinear influence caused by the combination of environmental fluctuations and unknown uncertainties, and this scheme has strong robustness.Meng et al. [23] researched the event-triggered SMC problem of switched systems with fractional order less than 1 based on the observer and constructed trigger rules to limit the updating of control signals.Zhao et al. [24] proposed an event-triggered observer SMC strategy for a class of Markov jump systems with time-varying delays.Huynh et al. [25] proposed a highly robust secondorder sliding mode control method for a class of multi-region steam-hydropower dynamics systems, which improves the reliability of MASHPS LFC and power systems.
Inspired by the above discussion, the aim of this paper is to provide an efficient SMC design method for discrete-time switched systems with matching perturbations, which differs from existing works [26,27].In this paper, considering the particularity of the system, a reduced-order sliding mode dynamical system of the switched system is constructed by state transformation and provides the sufficient conditions for the exponential stability of the related sliding mode dynamic systems.A new sliding mode controller design scheme based on the system model is proposed by using Lyapunov function method.This paper makes the following contributions: (1) We address the SMC problem for a category of switched systems with matching disturbances.By using a state transformation and proposing a sliding mode surface function, we construct the sliding mode equation for the switched system.(2) We present sufficient conditions for the exponential stability of switched sliding mode equation using Lyapunov function theory and ADT method.(3) Based on the characteristics of sliding mode control for switched systems, we devised a sliding mode controller for guaranteeing the convergence of the closed-loop system to the sliding mode region.Using an example, this conclusion is demonstrated to be valid.
The following is the structure of the rest of this article.In Section 2, the problem description is given, and the sliding mode surface function is designed.In Section 3, we analyze the exponential stability of the switching sliding mode dynamic equation is analyzed.In Section 4, we design an appropriate sliding mode controller to ensure the accessibility of the quasi-sliding modes.In Section 5, we provide a numerical example that verifies the effectiveness of the method proposed.In Section 6, we summarize our research work.
Notations: R and N + denote real numbers and positive integers, respectively; R n signals the real n-dimensional space; R m×n signals the real m × n dimensional matrix space.X T signals the transpose of matrix X.The notion P > 0 means that P is the positive definite symmetric matrix.I is used to represent an identity matrix of appropriate dimensions.' * ' represents an ellipsis of the symmetrically induced term.rank(•) represents the rank of the matrix, λ max (•) and λ min (•) represent the maximum and minimum eigenvalue of a symmetric matrix, respectively.

System Description and Problem Statement
Consider the following systems: where x(k) ∈ R n represent the state variables of the system, and u(k) ∈ R m is the control input.{A(σ(k)), F(σ(k)) : σ ∈ N } represent a family of known matrices depending on the index set N = {1, 2, • • • , N}. σ(k) : N + → N specifies the index of the active subsystem at the instant k.Suppose that the sequence of sub-modes of the switching signal σ(k) is unknown beforehand, but its instantaneous value is available in real-time.
For σ(k) = i, we denote the system associated with the i th subsystem by A(σ(k)) ≜ A(i), F(σ(k)) ≜ F(i).In this paper, f (x, k) represents a disturbance function satisfying , where η(i) > 0 is a known scalar.Therefore, system (1) can be expressed as Assumption 1. (A(i), D) is controllable for i ∈ N , and the matrix D is column full rank.
Based on the Assumption 1, there is an invertible matrix the system in (1) can be transformed into where where Then, system (3) can be represented as In this paper, we first analyze the stability of sliding mode dynamics and then design an SMC strategy to ensure that the system reaches the predetermined sliding mode surface in finite time.For system (4), we design the sliding mode surface function as follows: where M(i) = M(i) I and M(i) are matrices to be designed later.When the switched system states reaches the sliding mode surface (5), the sliding mode surface function satisfies s(k) = 0.Meanwhile, we have By substituting ( 6) into (4), we obtain where

Definition 1 ([28]
).The sliding trajectory of switched system (2) can be driven into a quasi-sliding mode band (QSMB) in finite time and if there are parameters δ > 0 and k > 0 that satisfy ∥s(k)∥ ⩽ δ.In this case, the constant δ is called a QSMB.

Controller Design and Stability Analysis
Theorem 1.Consider the systems (1) satisfying the Assumption 1.For given µ ⩾ 1, 0 < β < 1, if there exist matrices p(i) > 0 and M(i), satisfying the following inequalities ( for ∀i, j ∈ N , i ̸ = j ) where A(i) = A 11 (i) − A 12 (i)M(i), and the ADT T σ satisfying Then, the switched system (7) is exponentially stable, which is determined by with the parameters Proof.For the switched system (7), construct the following Lyapunov function: Thus, from ( 7) and (13), we have Then, where In view of Schur's complement, we can easily obtain We only have to obtain Π(i) < 0. It can be obtained by Equation (8).For any y 1 (k) ̸ = 0, we can figure out Therefore, Then, for k ∈ [k l , k l+1 ], it can be derived from ( 16) that From ( 9) and ( 13), we have According to (19), (20) and the relationship τ ≜ N σ (k 0 , k) ≤ k−k 0 T a , it holds that Considering ( 12), one has Combining ( 21) and ( 22), we have Therefore, the sliding mode ( 7) is exponentially stable, the proof is completed.

Sliding Mode Controller Design
In this section, we design a sliding mode controller to ensure that quasi-sliding modes are accessible.To accomplish control, an approach law method is applied: where T is sample time, ϖ and q are scalars that satisfying 0 < ϖ < 1, 1 − qT > 0. From ( 4), ( 5) and ( 24), we have From ( 25), the SMC law can be proposed as follows: where Since the presence of the uncertain term F i (k) in the control system (26), it is impractical for application.Therefore, the SMC law is constructed as follows: where In the following part, the following theorem is given to guarantee the reachability of the QSM.
Theorem 2. Consider the system (4) and the sliding surface function (5).If the SMC law is designed as (27), the system state trajectory can reach the quasi-sliding mode domain in a finite time.Moreover, once the system's state trajectory enters this region, it will not escape from the domain.
Proof.According to the system (4) and the sliding mode surface function ( 5), we have Next, construct the following Lyapunov function It follows that In view of (28), we have Combining ( 32)-(34), we have According to (29) and (35), there holds ∆V(k) ⩽ 0 when ∥s(k)∥ > ξ.Consequently, it is established that the system state trajectory can attain the quasi-sliding mode domain within a finite time.Once reached, the trajectory will persistently remain within the quasi-sliding mode domain without any deviation.Thus, the proof is concluded.

Simulation Examples
For the system (1) with (N = 2), the parameter matrices are as follows: For system (1), set µ = 1.2, β = 0.8 and select T * σ = 1.Suppose that the initial condition is x(0) = 0.5 −0.5 T .Our objective is to formulate the SMC law so that the sliding mode dynamics exponentially stable.First, we solve Equation (8) in Theorem 2 to obtain the sliding mode surface parameter of Equation ( 5) as follows Therefore, the sliding mode surface equation can be obtained as s(k) = −0.71741.0000 y(k), i = 1, −0.7222 1.0000 y(k), i = 2. (37) Suppose sampling time T = 0.1, select parameters ϖ = 0.5 and q = 2, according to ( 27) and ( 28), the SMC law can be designed as     Recently, the event-triggered control problem of discrete-time switching systems was studied in [7].In the case of ignoring disturbances, our system model corresponds to a special case in [7], that is, the uncertainty ∆A i = 0 in [7], and the matrix is B i consistent.However, when disturbances are present, it is possible to try to solve the problem using the method proposed in this paper.However, it could not be solved by the method in [7].In control systems, disturbances are generally considered as external uncertainties, which may adversely affect the stability and performance of the system.Therefore, developing effective control strategies to counteract these disturbances is very important.
To increase the readability of the simulation, our experimental background is provided here.This article is implemented under the Win10 system, with the help of Matlab R2021b (MathWorks Inc., Natick, MA, USA).

Conclusions
This paper has studied the issue of sliding mode control for switched systems in the presence of disturbances.Considering the particularity of the system, a reduced-order sliding mode dynamic system for the switched system has been constructed through state transformation, and sufficient conditions for the exponential stability of the related sliding mode dynamic systems have been proposed.A novel design scheme for sliding mode controllers based on the system model is presented using the Lyapunov function method.Finally, the effectiveness of this method has been demonstrated through a simulation example.Future research will be dedicated to extend this approach to switched systems with asynchronous switching.
Funding: This research was funded by the National Natural Science Foundation (NNSF) of China, grant number 62103289.

Figure 2 .
Figure 2. The trajectory of state x(k).

Figure 3 .
Figure 3.The trajectory of sliding surface s(k).

Figure 4 .
Figure 4.The trajectory of control input u(k).