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

This paper is concerned with a state observer-based sliding mode control design methodology for a class of continuous-time statedelayed switched systems with unmeasurable states and nonlinear uncertainties.The advantages of the proposed schememainly 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. Bymatrix 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.


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][15][16][17][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 Mathematical Problems in Engineering 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.

Problem Formulation and Preliminaries
where () ∈   denotes the vector of continuous-time state variables;  () () ∈   denotes the vector of control inputs; () ∈   denotes the system outputs;  () ((), ) ∈   denotes an unknown nonlinearity;  () ,  () , , and  are known real constant matrixes of corresponding dimension and the matrix  is of full column rank for arbitrary  ∈ I; Δ () () and Δ () () are unknown time-varying system parameter uncertainties; () : [0, ∞) → I {1, . . ., } is switching signal which is assumed to be a piecewise constant function of time ;  denotes the total number of switching modes.In this paper, the notations    in and    out are used to denote the time at which, for the th, the th subsystem is switched in and out, respectively, that is, ( +   in ) = ( −   out ) = .With these notations,  is used to replace () for    in ≤  <    out . is a known constant delay; () denotes a differentiable vector-valued initial function on [−, 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 where  ∈  × is a positive definite matrix that will be designed.
We know that an ideal sliding mode within the stateestimate space exists if there exists a finite time   , such that 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   Ṡ < 0 is satisfied.
Then, from we get The solution   () of ( 7), namely, sliding mode control law, can be broken down into two parts: equivalent control law  eq () and switching control term   (), where the equivalent control  eq () is designed for a class of certain systems without nonlinear disturbance and the switching control   () is robust control of nonlinear uncertain systems.
So,  eq () and   () are designed as the following functions (10) and (11), respectively: where,  1 ,  2 are real positive scalars to be specified and function (10) is the solution of the following equation: Remark 2. It should be noticed that the sliding surface function defined in (6) does not switch with switching single (so  not   and  not   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.

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.
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.

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 statedelayed system.Finally, we guarantee that the sliding modes within both the state estimate space and state estimation errors pace are attained, respectively.

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 where x() denotes the estimate of system state () and   ∈  × is the observer feedback matrix to be computed later.  is a robust control term used to eliminate impact of nonlinear   ((), ), given by The corresponding state-estimation error dynamics is given by According to the sliding mode control theory and ( 6),   ((), ) =   ().
In (6), it is assumed that matrix  satisfies We obtain This estimation error dynamics corresponds to a nonlinear uncertain state-delayed system, which is dependent on the observer feedback matrix   and state estimates x(), x(−).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   ((), ) =   () = (() − x()) = 0 in state-estimation error space and the sliding surface Ŝ(x(), ) =   x() = 0 in the state estimate space.Namely, as (7), the following function should be satisfied: Hence, as functions (10) and (11), equivalent control law and switching control term are designed as follows, respectively: where  1 and  2 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: where then the sliding mode control law (21)-( 22) guarantees that the combined closed-loop switched system is asymptotically stable for the switching signal satisfies Furthermore, an observer feedback matrix is given by   =  −1   .
Proof.Firstly, using the method of matrix transformation [23] function (15) is transformed, and the transformation matrix and the associated vector ẑ() are defined as where ẑ1 () ∈  − , ẑ2 () ∈   , B is an orthogonal complement of , which means B  = 0.So  is designed to a nonsingular matrix.It is easily known that  −1 = ( −1 B ) and ẑ2 () = (  ) −1 ŝ() due to the following function: Because of the sliding surface ŝ(, ) = 0, we obtain ẑ2 () = 0.So ( − ) reduced-order sliding mode dynamic system as follows: Similar to the previous method, function (17) is transformed as Then, the Lyapunov functional is chosen as Taking the time derivative along the state trajectories of ( 15) and ( 17), and considering ( 21)-( 22), it follows that where From (32), it is easily seen that V () < 0 if   () < 0 for  ̸ = 0. So, we just prove the inequality   () < 0 is satisfied.
Remark 7. As a result of the given Theorem 6, the observerbased SMC stable problem becomes a linear matrix inequality feasibility problem.Consider the linear equality condition    = , where ,  > 0 satisfies LMI (23), which can be equivalently converted to tr ((   − ) (   − )) = 0.
Introduce the condition and Lemma 4 gives Hence, it is now changed to a problem in which a global solution of the following minimization is found: min  subject to (23) and (40) .
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.

Accessibility Condition.
Finally, we prove the accessibility of sliding surfaces Ŝ(x(), ) = 0 in the state-estimate space and   ((), ) = 0 in estimation error space.So, Theorem 8 is given.
Remark 9.Only when the state trajectories enter the Ω  , could Ŝ() = 0 and   () = 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  1 ,  2 .seen that state response of the closed-loop switched system is clearly superior to the opened-loop switched system.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 Ŝ() is treated as reduced-order compensation of dynamic system.

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 () =  = arg{min(  ())}, 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][26][27] or arbitrary switching signal.