Sliding Mode Control Based on Observer for a Class of State-Delayed Switched Systems with Uncertain Perturbation

Abstract

This paper is concerned with a state observer-based sliding mode control design methodology for a class of continuous-time state-delayed switched systems with unmeasurable states and nonlinear uncertainties. The advantages of the proposed scheme mainly lie in which it eliminates the need for state variables to be full accessible and parameter uncertainties to be satisfied with the matching condition. Firstly, a state observer is constructed, and a sliding surface is designed. By matrix transformation techniques, combined with Lyapunov function and sliding surface function, a sufficient condition is given to ensure asymptotic stability of the overall closed-loop systems composed of the observer dynamics and the estimation error dynamics. Then, reachability of sliding surface is investigated. At last, an illustrative numerical example is presented to prove feasibility of the proposed approaches.

1. Introduction

A switched system consists of a finite number of subsystems described by a class of differential or difference equations and a logical law that is used to orchestrate switching between these subsystems [1]. As increasing demand, this theory is widely developed, which means that many working systems can be modeled as switched systems, such as automated highway systems [2].

It is well known that different switching strategies produce different systems stability and performance, and the systems states cannot be directly measured. Accordingly, choosing a suitable switching law that stabilizes switched systems and designing an observer become an important problem [3]. Various methods of observer design have been successfully proposed, such as algebraic transfer function and singular-value decomposition. However, designing state observer turns out to be much more difficult when the system is nonlinear [4] and uncertain [5]. In [6], this problem of state observer is considered for discrete time delay switched systems with Lipschitz’s nonlinearity and random switching law.

Since 1916, sliding mode control (SMC) has been proven to be an effective robust control strategy for hybrid or uncertain systems [7]. SMC belongs to variable structure control (VSC), which utilizes a discontinuous control to force the state trajectories of the system to some specific sliding surfaces on which the system acquires desired properties such as decay speed, disturbance rejection capability, and robustness. Developments on SMC involve uncertain systems [8, 9], time-delay systems [10, 11], fuzzy systems [12, 13], and Markovian jump systems [14–18].

Full state feedback is not always available and measurable in many practical systems. In Wu’s work [19] a new robust stability condition based on observer is proposed for a class of uncertain nonlinear neutral delay systems by using the sliding mode control theory combined with reaching law technique. In He’s paper [20], a sliding mode control strategy of uncertain switched linear systems based on observer is presented. The average dwell time is introduced and the matching condition of parameter uncertainties needs not to be satisfied. As the paper [19, 20] improved, this paper presents a new observer design for a class of uncertain nonlinear state-delayed switched systems by using the sliding mode control theory.

In the present paper, the motivation of our work is to introduce an SMC scheme based on observer for a class of switched systems that are uncertain nonlinear state-delayed systems with unmeasurable state, unknown nonlinear function, and mismatching parameter uncertainties. This is a new problem in SMC and switched systems research areas. In this work, the sliding mode observer for each subsystem is designed to estimate full state. By matrix transformation techniques, a sufficient condition is proposed to ensure Lyapunov asymptotic stability of the overall closed-loop switched system. In addition, the derived SMC law is provided to guarantee reachability of the designed sliding surface. Finally, an example is given to prove the feasibility of the proposed approaches.

2. Problem Formulation and Preliminaries

2.1. Switched Uncertain Nonlinear State-Delayed Systems

In this work, consider the following uncertain nonlinear state-delayed switched systems: 𝑥̇(𝑡)=𝐴_𝜎(𝑡)+Δ𝐴_𝜎(𝑡)(𝑡𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)})+Δ𝐴_{𝑑𝜎(𝑡)}(𝑡𝑥()𝑢)𝑡−𝑑+𝐵_𝜎(𝑡)(𝑡)+𝑓_𝜎(𝑡)(𝑥(𝑡),𝑦(𝑡)(𝑡),𝑥(𝑡)(𝑡)[],,𝑡)=𝐶𝑥=𝜑,𝑡∈−𝑑,0 (1) where 𝑥(𝑡)∈𝑅^𝑛 denotes the vector of continuous-time state variables; 𝑢_𝜎(𝑡)(𝑡)∈𝑅^𝑚 denotes the vector of control inputs; 𝑦(𝑡)∈𝑅^𝑝 denotes the system outputs; 𝑓_𝜎(𝑡)(𝑥(𝑡),𝑡)∈𝑅^𝑚 denotes an unknown nonlinearity; 𝐴_𝜎(𝑡), 𝐴_{𝑑𝜎(𝑡)}, 𝐵, and 𝐶 are known real constant matrixes of corresponding dimension and the matrix 𝐵 is of full column rank for arbitrary 𝑖∈𝐈; Δ𝐴_𝜎(𝑡)(𝑡) and Δ𝐴_{𝑑𝜎(𝑡)}(𝑡) are unknown time-varying system parameter uncertainties; 𝜎(𝑡)∶[0,∞)→𝐈  {1,…,𝑁} is switching signal which is assumed to be a piecewise constant function of time 𝑡; 𝑘 denotes the total number of switching modes. In this paper, the notations 𝑡_𝑖𝑟in and 𝑡_𝑖𝑟out are used to denote the time at which, for the 𝑟th, the 𝑖th subsystem is switched in and out, respectively, that is, 𝜎(𝑡⁺_𝑖𝑟in)=𝜎(𝑡⁻_𝑖𝑟out)=𝑖. With these notations, 𝑖 is used to replace 𝜎(𝑡) for 𝑡_𝑖𝑟in≤𝑡<𝑡_𝑖𝑟out. 𝑑 is a known constant delay; 𝜑(𝑡) denotes a differentiable vector-valued initial function on [−𝑑,0].

The following assumptions are useful for the development of our work.

Assumption 1. Each of the subsystems is completely observable.

Assumption 2. The uncertain parameters are of the form Δ𝐴_𝑖(𝑡)Δ𝐴_𝑑𝑖(𝑡)=𝐸_𝑖𝐹_𝑖(𝑡)𝐻_𝑖𝐻_𝑑𝑖. (2)Δ𝐴_𝑖(𝑡) and Δ𝐴_𝑑𝑖(𝑡) are composed by 𝐸_𝑖, 𝐻_𝑖, 𝐻_𝑑𝑖, and 𝐹_𝑖(𝑡), where 𝐸_𝑖, 𝐻_𝑖, and 𝐻_𝑑𝑖 are constant matrices and 𝐹_𝑖(𝑡) is a time-varying matrix function satisfying 𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼. (3)

Remark 1. Obviously, by comparison with Assumption 2, the matching condition that the following function must be satisfied is a stronger condition: Δ𝐴_𝑖(𝑡)Δ𝐴_𝑑𝑖(𝑡)=𝐵𝑀_𝑖(𝑡)𝑀_𝑑𝑖(𝑡). (4)

Assumption 3. There exists a known scalar function 𝜌(𝑡,𝑦) such that the nonlinearities 𝑓_𝑖(𝑥(𝑡),𝑡), for arbitrary 𝑖∈𝐈, satisfy 𝑓_𝑖(𝑥(𝑡),𝑡)≤𝜌_𝑖𝑡,𝑦. (5)

2.2. Sliding Mode Control

According to the sliding mode control theory, a sliding mode function is chosen as follows: 𝑆(𝑥,𝑡)=𝐵^𝑇𝑋𝑥(𝑡), (6) where 𝑋∈𝑅^𝑛×𝑛 is a positive definite matrix that will be designed.

We know that an ideal sliding mode within the state-estimate space exists if there exists a finite time 𝑡_𝑠, such that 𝑆(𝑡)=0,𝑆̇(𝑡)=0,𝑡≥𝑡_𝑠or𝑆^𝑇𝑆̇(𝑡)<0. (7)

The second function of (7) will be used to prove the accessibility condition. And in a conventional SMC design, a class of equivalent control laws and some switching control terms can be chosen to undertake that the accessibility condition 𝑆^𝑇𝑆̇<0 is satisfied.

Then, from 𝑆̇=𝜕𝑆𝜕𝑥𝜕𝑥𝜕𝑡=𝐵^𝑇𝑋𝑥̇(𝑡)=𝐵^𝑇𝑋𝐴_𝑖+Δ𝐴_𝑖(𝑡)𝑥(𝑡)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑡𝑥()𝑢)𝑡−𝑑+𝐵_𝑖(𝑡)+𝑓_𝑖(𝑥(𝑡),𝑡), (8) we get 𝐵^𝑇𝑋𝐴_𝑖+Δ𝐴_𝑖(𝑡)𝑥(𝑡)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑡𝑥()𝑢)𝑡−𝑑+𝐵_𝑖(𝑡)+𝑓_𝑖(𝑥(𝑡),𝑡)=0. (9)

The solution 𝑢_𝑖(𝑡) of (7), namely, sliding mode control law, can be broken down into two parts: equivalent control law 𝑢_eq𝑖(𝑡) and switching control term 𝑢_𝑐𝑖(𝑡), where the equivalent control 𝑢_eq𝑖(𝑡) is designed for a class of certain systems without nonlinear disturbance and the switching control 𝑢_𝑐𝑖(𝑡) is robust control of nonlinear uncertain systems.

So, 𝑢_eq𝑖(𝑡) and 𝑢_𝑐𝑖(𝑡) are designed as the following functions (10) and (11), respectively: 𝑢_eq𝑖(𝑡)=−𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥(),𝑢𝑡−𝑑_𝑐𝑖(𝑡)=−𝜌_𝑖𝑡,𝑦+𝛾_1𝑖+𝛾_2𝑖sgn(𝑆(𝑡)), (10)(11) where, 𝛾_1𝑖, 𝛾_2𝑖 are real positive scalars to be specified and function (10) is the solution of the following equation: 𝐵^𝑇𝑋𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥(𝑡−𝑑)+𝐵𝑢_eq𝑖(𝑡)=0. (12)

Remark 2. It should be noticed that the sliding surface function defined in (6) does not switch with switching single (so 𝐵 not 𝐵_𝑖 and 𝑋 not 𝑋_𝑖 are designed). It means that there is a unique nonswitched sliding surface in order to avoid repetitive jumps of the state trajectories between sliding surfaces leading to instability and chattering.

2.3. Some Lemmas

For further analysis, some lemmas are given that are useful for stability analysis of the sliding mode dynamics and the development of other theorems.

Lemma 3 (see [21]). Let 𝐸,  𝐹, and 𝐻 be real matrices of appropriate dimensions, with 𝐹^𝑇𝐹≤𝐼; then one has that for any scalar 𝜀>0, 𝐸𝐹𝐻+𝐻^𝑇𝐹^𝑇𝐸^𝑇≤𝜀⁻¹𝐸𝐸^𝑇+𝜀𝐻^𝑇𝐻. (13)

Lemma 4 (Schur complement). Let 𝑆=^𝑆11𝑆12_𝑆21𝑆22 be symmetrical matrix; then the following three functions are equivalent: 𝑆𝑆<0,₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂𝑆<0,₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0. (14)

Lemma 5 (see [22]). Given matrixes 𝐺∈𝑅^𝑛×𝑛 and 𝑈∈𝑅^𝑛×𝑚. Assume that  𝑈 has full rank 𝑚<𝑛 and 𝐺=𝐺^𝑇. Then, 𝛿𝑈𝑈^𝑇−𝐺>0 for some scalar 𝛿 if and only if   𝑈^𝑇𝐺𝑈<0 where 𝑈 is any matrix whose columns from basis of the null space of 𝑈^𝑇.

When the SMC is designed, we always suppose that all of the system states are available. But, this assumption is hardly satisfied on the practical viewpoint. Hence, in the next chapter, the sliding mode control-based observer will be designed.

3. Observer-Based Sliding Mode Control

In this section, a sliding mode observer is designed to provide the estimate of state vector, and then a sliding mode controller is synthesized based on state estimates. Furthermore, by applying the sliding mode control and the multiple Lyapunov function technique, a sufficient condition is given to ensure the asymptotic stability of the overall closed-loop state-delayed system. Finally, we guarantee that the sliding modes within both the state estimate space and state estimation errors pace are attained, respectively.

3.1. Sliding Mode Observer Design

The sliding mode observer to be designed, for state-delayed system (1)–(4), has the form of (6), which can be given by 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥(𝑡−𝑑)+𝐵𝑢_𝑖(𝑡)+𝐵𝑢_𝑒𝑖(𝑡)+𝐿_𝑖𝑦(𝑡)−𝐶𝑥(𝑡), (15) where 𝑥(𝑡) denotes the estimate of system state 𝑥(𝑡) and 𝐿_𝑖∈𝑅^𝑛×𝑝 is the observer feedback matrix to be computed later. 𝑢_𝑒𝑖 is a robust control term used to eliminate impact of nonlinear 𝑓_𝑖(𝑥(𝑡),𝑡), given by 𝑢_𝑒𝑖(𝑡)=𝜌_𝑖𝑡,𝑦+𝛾_1𝑖sgn𝑆_𝑒(𝑡). (16)

The corresponding state-estimation error dynamics is given by 𝑒̇(𝑡)=𝐴_𝑖−𝐿_𝑖𝐶+Δ𝐴_𝑖(𝑡)𝑒(𝑡)+𝐴_𝑑𝑖−Δ𝐴_𝑑𝑖(𝑡)𝑒(𝑡−𝑑)+Δ𝐴_𝑖(𝑡)𝑥(𝑡)+Δ𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝑑)−𝐵_𝑖𝑢_𝑒𝑖(𝑡)−𝑓_𝑖(𝑥(𝑡),𝑡). (17)

According to the sliding mode control theory and (6), 𝑆_𝑒(𝑒(𝑡),𝑡)=𝐵^𝑇𝑋𝑒(𝑡).

In (6), it is assumed that matrix 𝑋 satisfies 𝐵^𝑇𝑋=𝑁𝐶. (18) We obtain 𝑆_𝑒(𝑒(𝑡))(𝑡)(𝑥(𝑡)−𝑥(𝑡𝑦(𝑡)𝑥(𝑡).,𝑡=𝑁𝐶𝑒=𝑁𝐶))=𝑁−𝐶 (19)

This estimation error dynamics corresponds to a nonlinear uncertain state-delayed system, which is dependent on the observer feedback matrix 𝐿_𝑖 and state estimates 𝑥(𝑡), 𝑥(𝑡−𝑑). This means that stability analysis of the error dynamics (17) is dependent on the observer dynamics. So, when designing the sliding mode observer, the overall closed-loop system composed of (15) and (17) must be considered to guarantee system stability and accessibility of both the sliding surface 𝑆_𝑒(𝑒(𝑡),𝑡)=𝐵^𝑇𝑋𝑒(𝑡)=𝑁(𝑦(𝑡)−𝐶𝑥(𝑡))=0 in state-estimation error space and the sliding surface 𝑆(𝑥(𝑡),𝑡)=𝐵^𝑇𝑋𝑥(𝑡)=0 in the state estimate space. Namely, as (7), the following function should be satisfied: 𝑆(𝑡)=0,𝑆̇(𝑡)=0,𝑡≥𝑡_𝑠. (20)

Hence, as functions (10) and (11), equivalent control law and switching control term are designed as follows, respectively: 𝑢_eq𝑖(𝑡)=−𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥(),𝑢𝑡−𝑑_𝑐𝑖(𝑡)=−𝜌_𝑖𝑡,𝑦+𝛾_1𝑖+𝛾_2𝑖sgn𝑆(𝑡), (21) where 𝛾_1𝑖 and 𝛾_2𝑖 are positive design constants, which should be chosen suitably because approaching rate and chattering of sliding mode face can be influenced by it.

Hence, we obtain the following sliding mode controller: 𝑢_𝑖(𝑡)=𝑢_eq𝑖(𝑡)+𝑢_𝑐𝑖(𝑡). (22)

3.2. Stability of Closed-Loop System with Observer

Summing up, considering the design of sliding mode observer and synthesis of sliding mode controller, a sufficient condition for asymptotic stability of the overall closed-loop system can be given as follows.

Theorem 6. If there exist matrices 𝑌_𝑖,𝑁,𝑋>0, 𝑄_1𝑖>0, and 𝑄_2𝑖>0, and scalars 𝛿_1𝑖>0, and 𝛿_2𝑖>0 satisfyingΠ_1𝑖𝑋𝐴_𝑑𝑖+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑑𝑖𝐶^𝑇𝑌^𝑇_𝑖+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑖𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑑𝑖𝐴𝑋𝐸_𝑇𝑑𝑖𝑋+𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑖Π_2𝑖𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑖𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑑𝑖0𝑌_𝑖𝐶+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑖𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑑𝑖Π_3𝑖𝑋𝐴_𝑑𝑖+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑑𝑖0𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑖𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑑𝑖𝐴^𝑇_𝑑𝑖𝑋+𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑖Π_4𝑖0𝐸^𝑇𝑋000−𝛿_1𝑖𝐼≤0, (23)𝐵^𝑇𝑋=𝑁𝐶, (24) where Π_1𝑖=𝑋𝐴_𝑖+𝐴^𝑇_𝑖𝑋−𝑌_𝑖𝐶−𝐶^𝑇𝑌^𝑇_𝑖+𝑄_1𝑖+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑖−𝛿_2𝑖𝐵𝐵^𝑇,Π_2𝑖=−𝑄_1𝑖+𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑑𝑖−𝛿_2𝑖𝐵𝐵^𝑇,Π_3𝑖=𝐴^𝑇_𝑖𝑋+𝑋𝐴_𝑖+𝑄_2𝑖+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑖−𝛿_2𝑖𝐵𝐵^𝑇,Π_4𝑖=−𝑄_2𝑖+𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑑𝑖−𝛿_2𝑖𝐵𝐵^𝑇, (25) then the sliding mode control law (21)-(22) guarantees that the combined closed-loop switched system is asymptotically stable for the switching signal satisfies 𝜎(𝑡)=𝑖=argmin𝑉_𝑖(𝑡),𝑖∈𝐈. (26)
Furthermore, an observer feedback matrix is given by 𝐿_𝑖=𝑋⁻¹𝑌_𝑖.

Proof. Firstly, using the method of matrix transformation [23] function (15) is transformed, and the transformation matrix and the associated vector 𝑧(𝑡) are defined as 𝑧(𝑡)=𝑧₁(𝑡)𝑧₂(𝑡)=𝑊𝑥(𝑡), (27) where 𝑧₁(𝑡)∈𝑅^𝑛−𝑚, 𝑧₂(𝑡)∈𝑅^𝑚, 𝐵 is an orthogonal complement of 𝐵, which means 𝐵^𝑇𝐵=0. So 𝐵 is designed to a nonsingular matrix. It is easily known that 𝑊⁻¹=(𝑋⁻¹𝐵𝐵) and 𝑧₂(𝑡)=(𝐵^𝑇𝑋𝐵)⁻¹𝑠(̂𝑡) due to the following function: 𝐵^𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋𝑋⁻¹=𝐵𝐵𝐵_𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇𝑋⁻¹𝐵𝐵^𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇𝐵𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋𝑋⁻¹𝐵𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇=𝑋𝐵𝐼00𝐼=𝐼. (28)
Because of the sliding surface 𝑠(̂𝑥,𝑡)=0, we obtain 𝑧₂(𝑡)=0. So (𝑛−𝑚) reduced-order sliding mode dynamic system as follows: 𝑧̇₁=𝐵^𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇𝐴_𝑖−𝐿_𝑖𝐶𝑋⁻¹𝐵𝑧₁(𝑡)+𝐵^𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇𝐴_𝑑𝑖𝑋⁻¹𝐵𝑧₁()+𝐵𝑡−𝑑_𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇𝐿_𝑖𝐶𝑋⁻¹𝐵𝑧₁(𝑡). (29)
Similar to the previous method, function (17) is transformed as 𝑒̇_𝑧1(𝑡)=𝐵^𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇𝐴_𝑖−𝐿_𝑖𝐶+Δ𝐴_𝑖(𝑡)𝑋⁻¹𝐵𝑒_𝑧1(𝑡)+𝐵^𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇Δ𝐴_𝑖(𝑡)𝑋⁻¹𝐵𝑣₁(𝑡)+𝐵^𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑡)𝑋⁻¹𝐵𝑒_𝑧1()+𝐵𝑡−𝑑_𝑇𝑋⁻¹𝐵⁻¹𝐵^𝑇Δ𝐴_𝑑𝑖(𝑡)𝑋⁻¹𝐵𝑣₁(𝑡−𝑑). (30)
Then, the Lyapunov functional is chosen as 𝑉_𝑖(𝑡)=𝑒^𝑇_𝑧1(𝑡)𝐵^𝑇𝑋⁻¹𝐵𝑒_𝑧1(𝑡)+^𝑡_𝑡−𝑑𝑒^𝑇_𝑧1(𝜏)𝐵^𝑇𝑄_1𝑖𝐵𝑒_𝑧1(𝜏)+𝑧𝑑𝜏_𝑇1(𝑡)𝐵^𝑇𝑋⁻¹𝐵𝑧₁(𝑡)+^𝑡_𝑡−𝑑𝑧^𝑇₁(𝜏)𝐵^𝑇𝑄_2𝑖𝐵𝑧₁(𝜏)𝑉̇𝑑𝜏,_𝑖(𝑡)=𝑒̇^𝑇_𝑧1(𝑡)𝐵^𝑇𝑋⁻¹𝐵𝑒_𝑧1(𝑡)+𝑒^𝑇_𝑧1(𝑡)𝐵^𝑇𝑋⁻¹𝐵𝑒̇_𝑧1(𝑡)+𝑒^𝑇_𝑧1(𝑡)𝐵^𝑇𝑄_1𝑖𝐵𝑒_𝑧1(𝑡)^𝑡_𝑡−𝑑+𝑧̇^𝑇₁(𝑡)𝐵^𝑇𝑋⁻¹𝐵𝑧₁(𝑡)+𝑧^𝑇₁(𝑡)𝐵^𝑇𝑋⁻¹𝐵𝑧̇₁(𝑡)+𝑧^𝑇₁(𝑡)𝐵^𝑇𝑄_2𝑖𝐵𝑧₁(𝑡)^𝑡_𝑡−𝑑. (31) Taking the time derivative along the state trajectories of (15) and (17), and considering (21)-(22), it follows that 𝑉̇_𝑖(𝑡)=𝑍^𝑇(𝑡)𝐺_𝑖(𝑡)𝑍(𝑡), (32) where 𝐺_𝑖(𝑡)=𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵^𝑇𝐺_1𝑖+𝐺_2𝑖(𝑡)×𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵,𝑍(𝑡)=𝑒_𝑧1(𝑡)𝑒_𝑧1()𝑧𝑡−𝑑₁(𝑡)𝑧₁(),𝐺𝑡−𝑑_1𝑖=Π_1𝑖𝑋𝐴_𝑑𝑖𝐶^𝑇𝐿^𝑇_𝑖𝐴𝑋0_𝑇𝑑𝑖𝑋−𝑄_1𝑖00𝑋𝐿_𝑖𝐶0Π_2𝑖𝑋𝐴_𝑑𝑖00𝐴^𝑇_𝑑𝑖𝑋−𝑄_2𝑖,𝐺_2𝑖(𝑡)=000𝐹𝑋𝐸_𝑖(𝑡)𝐻_𝑖𝐻_𝑑𝑖𝐻_𝑖𝐻_𝑑𝑖+000𝐹𝑋𝐸_𝑖(𝑡)𝐻_𝑖𝐻_𝑑𝑖𝐻_𝑖𝐻_𝑑𝑖^𝑇,Π_1𝑖=𝐴_𝑖−𝐿_𝑖𝐶^𝑇𝑋+𝑋𝐴_𝑖−𝐿_𝑖𝐶+𝑄_1𝑖,Π_2𝑖=𝐴^𝑇_𝑖𝑋+𝑋𝐴_𝑖+𝑄_2𝑖,𝑄_1𝑖=𝑋𝑄_1𝑖𝑋,𝑄_2𝑖=𝑋𝑄_2𝑖𝑋. (33)
From (32), it is easily seen that 𝑉̇_𝑖(𝑡)<0 if 𝐺_𝑖(𝑡)<0 for 𝑍≠0. So, we just prove the inequality 𝐺_𝑖(𝑡)<0 is satisfied.
By Lemma 3, the previous matrix inequality holds for 𝐹_𝑖(𝑡) satisfying 𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼 if there exists a constant 𝛿_1𝑖>0 such that 𝐺_2𝑖(𝑡)≤𝐺_2𝑖=𝛿⁻¹000𝐸𝑋𝐸_𝑇𝐻𝑋000+𝛿^𝑇𝑖𝐻^𝑇_𝑑𝑖𝐻^𝑇_𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑖𝐻_𝑑𝑖𝐻_𝑖𝐻_𝑑𝑖. (34)
So, 𝐺_𝑖<0 only if 𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵^𝑇𝐺_1𝑖+𝐺_2𝑖×𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵<0. (35) By Lemma 4, inequality (35) is equivalent to the following inequality (36): 𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵^𝑇×𝐺_3𝑖𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵0000𝑋⁻¹𝐵<0, (36) where𝐺_3𝑖=Π_1𝑖𝑋𝐴_𝑑𝑖+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑑𝑖𝐶^𝑇𝑌^𝑇_𝑖+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑖𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑑𝑖𝐴𝑋𝐸_𝑇𝑑𝑖𝑋+𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑖Π_2𝑖𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑖𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑑𝑖0𝑌_𝑖𝐶+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑖𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑑𝑖Π_3𝑖𝑋𝐴_𝑑𝑖+𝛿_1𝑖𝐻^𝑇_𝑖𝐻_𝑑𝑖0𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑖𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑑𝑖𝐴^𝑇_𝑑𝑖𝑋+𝛿_1𝑖𝐻^𝑇_𝑑𝑖𝐻_𝑖Π_4𝑖0𝐸^𝑇𝑋000−𝛿_1𝑖𝐼. (37)
By Lemma 5 and given 𝐿_𝑖=𝑋⁻¹𝑌_𝑖, LMI (23) is obtained. That means 𝑉̇_𝑖(𝑡)<0 only if there exist matrices 𝑌_𝑖,𝑁,𝑋>0, 𝑄_1𝑖>0, and 𝑄_2𝑖>0 and scalars 𝛿_1𝑖>0, and 𝛿_2𝑖>0 satisfying LMI (23).

Remark 7. As a result of the given Theorem 6, the observer-based SMC stable problem becomes a linear matrix inequality feasibility problem. Consider the linear equality condition 𝐵^𝑇𝑋=𝑁𝐶, where 𝑁,𝑋>0 satisfies LMI (23), which can be equivalently converted to tr𝐵^𝑇𝑋−𝑁𝐶^𝑇𝐵^𝑇𝑋−𝑁𝐶=0. (38)
Introduce the condition 𝐵^𝑇𝑋−𝑁𝐶^𝑇𝐵^𝑇𝑋−𝑁𝐶≤𝛽𝐼, (39) and Lemma 4 gives −𝛽𝐼𝑋𝐵−𝐶^𝑇𝑁^𝑇𝐵^𝑇𝑋−𝑁𝐶−𝛽𝐼≤0. (40) Hence, it is now changed to a problem in which a global solution of the following minimization is found: min𝛽subjectto(23)and(40). (41)
So, it is a minimization problem relating linear objective and LMI constraints; it has an infimum and when infimum equals zero, the observed-based SMC problem is solvable.

3.3. Accessibility Condition

Finally, we prove the accessibility of sliding surfaces 𝑆(𝑥(𝑡),𝑡)=0 in the state-estimate space and 𝑆_𝑒(𝑒(𝑡),𝑡)=0 in estimation error space. So, Theorem 8 is given.

Theorem 8. If there exist matrices 𝑌_𝑖,𝑁,𝑋>0, 𝑄_1𝑖>0, 𝑄_2𝑖>0 and scalars 𝛿_1𝑖>0, and 𝛿_2𝑖>0 satisfying (23) with (24), and observer feedback matrix 𝐿_𝑖=𝑋⁻¹𝑌_𝑖, then the sliding mode control law (26) with (23) and (24) guarantees that the sliding motion is attained on the sliding surfaces 𝑆(𝑥(𝑡),𝑡)=0 and 𝑆_𝑒(𝑒(𝑡),𝑡)=0, respectively.

Proof. As (7), 𝑆(𝑥(𝑡),𝑡)=0 and 𝑆_𝑒(𝑒(𝑡),𝑡)=0 can be replaced by 𝑆^𝑇𝑆̇(𝑡)<0 and 𝑆^𝑇_𝑒𝑆̇_𝑒(𝑡)<0.
We know 𝑆^𝑇𝑆̇(𝑡)<0 is the same as 𝑆^𝑇(𝐵^𝑇𝑋𝐵)⁻¹𝑆̇(𝑡)<0, so 𝑆^𝑇𝑆̇(𝑡)<0 and 𝑆^𝑇_𝑒𝑆̇_𝑒(𝑡)<0 can be replaced by 𝑆^𝑇(𝐵^𝑇𝑋𝐵)⁻¹𝑆̇(𝑡)⁻¹<0 and 𝑆^𝑇_𝑒(𝐵^𝑇𝑋𝐵)⁻¹𝑆̇_𝑒(𝑡)<0 to make proving simpler: 𝑆^𝑇(𝑡)𝐵^𝑇𝑋𝐵⁻¹𝑆̇(𝑡)=𝑆^𝑇(𝑡)𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥(𝑡−𝑑)+𝐵𝑢_𝑖(𝑡)+𝐵𝑢_𝑒𝑖(𝑡)+𝐿_𝑖𝑦(𝑡)𝑥(𝑡)=𝑆−𝐶_𝑇(𝑡)𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋×𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥(𝑡−𝑑)+𝐿_𝑖𝐶𝑒(𝑡)+𝐵𝑢_eq𝑖−𝑆^𝑇(𝑡)𝑢_𝑐𝑖+𝑢_𝑒𝑖≤𝑆(𝑡)𝐵^𝑇𝑋𝐵⁻¹×𝐵−𝐼_𝑇𝑋𝐴_𝑖‖𝑥(𝑡)‖+𝐵^𝑇𝑋𝐴_𝑑𝑖𝑥(+𝐵‖𝑡−𝑑)‖_𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋𝐿_𝑖𝐶𝑒(𝑡−𝑆(𝑡𝜌‖)‖)_𝑖𝑡,𝑦+𝛾_1𝑖+𝛾_2𝑖+𝑆(𝑡)𝜌_𝑖𝑡,𝑦+𝛾_1𝑖≤𝑆(𝑡)−𝛾_2𝑖+𝛼_1𝑖𝛼_2𝑖‖𝑥(𝑡)‖+𝛼_1𝑖𝛼_3𝑖‖𝑥(𝑡−𝑑)‖+𝛼_4𝑖𝛼_5𝑖‖𝑒(𝑡,𝑆)‖_𝑇𝑒(𝑡)𝐵^𝑇𝑋𝐵⁻¹𝑆̇_𝑒(𝑡)=𝑆^𝑇_𝑒(𝑡)𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋𝐴_𝑖−𝐿_𝑖𝐶+Δ𝐴_𝑖(𝑡)𝑒(𝑡)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑡)𝑒()𝑢𝑡−𝑑−𝐵_𝑒𝑖(𝑡)−𝑓_𝑖(𝑥(𝑡),𝑡)+Δ𝐴_𝑖𝑥(𝑡)+Δ𝐴_𝑑𝑖𝑥(𝑡−𝑑)=−𝑆^𝑇_𝑒(𝑡)𝑢_𝑒𝑖(𝑡)−𝑓_𝑖(𝑥(𝑡),𝑡)+𝑆^𝑇_𝑒(𝑡)𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋𝐴_𝑖−𝐿_𝑖𝐶+Δ𝐴_𝑖(𝑡)𝑒(𝑡)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑡)𝑒(𝑡−𝑑)+Δ𝐴_𝑖𝑥(𝑡)+Δ𝐴_𝑑𝑖𝑥(≤𝑆𝑡−𝑑)_𝑒(𝑡)𝜌_𝑖𝑡,𝑦+𝑆_𝑒(𝑡)𝐵^𝑇𝑋𝐵⁻¹×𝐵^𝑇𝑋𝐴_𝑖−𝐿_𝑖𝐶‖𝑒(𝑡)‖+𝐵^𝑇𝑋𝐸𝐻_𝑖𝑒(𝑡+𝐵‖)‖_𝑇𝑋𝐴_𝑑𝑖‖𝑒(𝑡−𝑑)‖+𝐵^𝑇𝑋𝐸𝐻_𝑑𝑖𝑒(+𝐵‖𝑡−𝑑)‖_𝑇𝑋𝐸𝐻_𝑖‖𝑥(𝑡)‖+𝐵^𝑇𝑋𝐸𝐻_𝑑𝑖𝑥(≤𝑆‖𝑡−𝑑)‖_𝑒(𝑡)−𝛾_1𝑖+𝛼_4𝑖𝛽_1𝑖−𝛽_2𝑖‖𝑒(𝑡+𝛼)‖_3𝑖+𝛽_3𝑖‖𝑒(𝑡−𝑑)‖+𝛽_2𝑖‖𝑥(𝑡)‖+𝛽_3𝑖‖𝑥(𝑡−𝑑)‖ (42) with 𝛼_1𝑖=||(𝐵^𝑇𝑋𝐵)⁻¹−𝐼||, 𝛼_2𝑖=||𝐵^𝑇𝑋𝐴_𝑖||, 𝛼_3𝑖=||𝐵^𝑇𝑋𝐴_𝑑𝑖||, 𝛼_4𝑖=||(𝐵^𝑇𝑋𝐵)⁻¹||, 𝛼_5𝑖=||𝐵^𝑇𝑋𝐿_𝑖𝐶||, 𝛽_1𝑖=||𝐵^𝑇𝑋(𝐴_𝑖−𝐿_𝑖𝐶)||,  𝛽_2𝑖=||𝐵^𝑇𝑋𝐸||||𝐻_𝑖||,  𝛽_3𝑖=||𝐵^𝑇𝑋𝐸||||𝐻_𝑑𝑖||.
In the state space composed of state estimate vector and estimation error vector, we define the following domain: Ω_𝛿𝑖=𝑍∈𝑅^4𝑛∶𝛼_1𝑖𝛼_2𝑖‖𝑥(𝑡)‖+𝛼_1𝑖𝛼_3𝑖‖𝑥(𝑡−𝑑)‖+𝛼_4𝑖𝛼_5𝑖‖𝑒(𝑡)‖<𝛾_2𝑖𝛼−𝛿,_4𝑖𝛽_1𝑖−𝛽_2𝑖‖𝑒(𝑡)‖+𝛽_2𝑖‖𝑥(𝑡+𝛼)‖_3𝑖+𝛽_3𝑖‖𝑒(𝑡−𝑑)‖+𝛽_3𝑖‖𝑥(𝑡−𝑑)‖<𝛾_1𝑖−𝛿 (43) with constant 𝛿 satisfying 0<𝛿<min{𝛾_1𝑖,𝛾_2𝑖}, so from (42), in the domain Ω_𝛿𝑖, we have 𝑆^𝑇𝑆̇(𝑡)<0 and 𝑆^𝑇_𝑒𝑆̇_𝑒(𝑡)<0.
So, the sliding surfaces 𝑆(𝑥(𝑡),𝑡)=0 in the state-estimate space and 𝑆_𝑒(𝑒(𝑡),𝑡)=0 in estimation error space can be accessed.

Remark 9. Only when the state trajectories enter the Ω_𝛿𝑖, could 𝑆(𝑡)=0 and 𝑆_𝑒(𝑡)=0 happen. The region in which sliding motion takes place is usually referred to as sliding patch [24]. In the present scheme, it can be seen that size of the sliding patch depends on design constants 𝛾_1𝑖,𝛾_2𝑖.

4. Simulation

In this section, the following switch system will be used to illustrate the proposed sliding mode control scheme-based observer is feasible: 𝑥̇(𝑡)=𝐴_𝜎(𝑡)+Δ𝐴_𝜎(𝑡)(𝑡𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)})+Δ𝐴_{𝑑𝜎(𝑡)}(𝑡𝑥()𝑢)𝑡−𝑑+𝐵_𝜎(𝑡)(𝑡)+𝑓_𝜎(𝑡)(𝑥(𝑡),𝑦(𝑡)(𝑡),,𝑡)=𝐶𝑥 (44) where 𝐴₁=−240−7−1.5001−1,𝐴₂=,𝐴−1.20.902−2001.4−1.3_𝑑1=−0.30.10.10.10.1000.10.2,𝐴_𝑑2=,,𝐸0.10.20.30.120.09000.10.2𝐵=211211,𝐶=132220₁=𝐸₂=,𝐻0.10.20.200.20.1000.3₁=𝐻₂=,𝐻0.10.20.100.20.30.20.20.1_𝑑1=𝐻_𝑑2=,𝑓0.200.100.10.10.10.10.2₁(𝑥(𝑡))=𝑥,𝑡0.4sin₁(𝑡)0.2cos𝑥₂(𝑡),𝑓₂(𝑥(𝑡))=𝑥,𝑡0.3sin₂(𝑡)0.3cos𝑥₃(𝑡),𝐹₁(𝑡)=(𝑡)(𝑡)0(𝑡),𝐹sin000sin00cos₂(𝑡)=(𝑡)(𝑡)0,𝑑(𝑡)(𝑡)𝑥(𝑡)=cos000sin001=2sin+1,𝑑=3,1−10^𝑇[],𝑥(0)=,𝑡∈−2,01−10.5^𝑇. (45)

Then, the linear inequalities (23) and (24) with (41) are solved, and we obtain: ,,𝑌𝑋=2.8965−0.8585−5.1634−0.85851.79451.2445−5.16341.244510.6332𝑁=0.7754−0.50223.9794−3.9817₁=,𝑌−4.30620.7478−3.32640.1157−1.9617−1.3973₂=,𝐿−0.60940.41160.5758−0.6553−0.0320−0.1995₁=,𝐿−16.38870.3210−4.40450.2188−7.6272−0.0012₂=,𝑆(𝑡)−1.53030.62090.1156−0.2876−0.75960.3164=𝐵^𝑇𝑋𝑥(𝑡)=𝑥(𝑡),𝑢0.00910.00380.0017−0.00900.00600.0145_𝑖(𝑡)=𝑢_eq𝑖(𝑡)+𝑢_𝑐𝑖(𝑡),𝑢_𝑒𝑖(𝑡)=𝜌_𝑖𝑡,𝑦+𝛾_1𝑖sgn𝑆_𝑒(𝑡),𝑢_𝑐𝑖(𝑡)=−𝜌_𝑖𝑡,𝑦+𝛾_1𝑖+𝛾_2𝑖𝑆(𝑡),𝑢sgn_eq𝑖(𝑡)=−𝐵^𝑇𝑋𝐵⁻¹𝐵^𝑇𝑋𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥(),𝑢𝑡−𝑑_eq1(𝑡)=𝑥(𝑡)+𝑥(),𝑢2.000−3.0000−1.00001.00002.16671.0000−0.30000.00000.10000.1667−0.06670.1333𝑡−𝑑_eq2(𝑡)=𝑥(𝑡)+𝑥().1.20000.5000−1.3000−1.4667−0.13331.3000−0.1000−0.1000−0.10000.02670.0033−0.0000𝑡−𝑑 (46)

Let 𝛾₁₁=𝛾₁₂=60, 𝛾₂₁=𝛾₂₂=40, 𝑥(𝑡)=(000)^𝑇, 𝑡∈[−2,0], 𝑥(0)=(000)^𝑇. To restrain control signals from quiver, sgn(𝑆(𝑡)) is replaced by 𝑆(𝑡)/(||𝑆(𝑡)||+0.01), and 𝑆_𝑒(𝑡) is replaced by 𝑆_𝑒(𝑡)/(||𝑆_𝑒(𝑡)||+0.01). 𝜌_𝑖(𝑡,𝑦) is very tiny, compared with 𝛾_1𝑖, so it is ignored.

Figures 1–4 testify the simulation results. Figures 1 and 2 show the time evolution of states and confirm asymptotic stability of the respective switched system. According to Figure 1, we know that the opened-loop switched system is instability and oscillation. Nevertheless, Figure 2 shows that the closed-loop switched system is stability. So it can be seen that state response of the closed-loop switched system is clearly superior to the opened-loop switched system.

Figure 1 

State response of the opened-loop switched system.

Figure 2 

State response of the closed-loop switched system.

Figure 3 

State response of the observer system.

Figure 4 

Sliding mode variables.

And we check that, from Figure 3, the present observer based on sliding mode control scheme effectively eliminated effects of parameter uncertainties and nonlinearities and guaranteed asymptotic stability of the closed-loop system. Figure 4 shows that the sliding surface is accessibility but continue chattering. The chattering of the sliding mode variable 𝑆(𝑡) is treated as reduced-order compensation of dynamic system.

5. Conclusion

In this paper, an observer-based sliding mode control strategy is presented for a class of uncertain nonlinear state-delayed switched systems. Through this work, state-delayed switched system with immeasurable states and nonlinear uncertainties get better performance. Not only are the parameter uncertainties not needed to satisfy matching condition, but also the availabilities of all system states are no longer required. Meanwhile, a unique nonswitched sliding surface is designed in order to avoid repetitive jumps of the state trajectories between sliding surfaces leading to instability and chattering.

The weakness is that only the switching signal satisfies 𝜎(𝑡)=𝑖=arg{min(𝑉_𝑖(𝑡))}, can the switched system be asymptotically stable. It is expected that the results developed in the paper can be extended to common cases that the underlying systems are involved in Markovian switching signal [25–27] or arbitrary switching signal.

Acknowledgment

This work is partially supported by the Natural Science Foundation of Heilongjiang Province (no. F201135).