Sliding Mode Control for Markovian Switching Singular Systems with Time-Varying Delays and Nonlinear Perturbations

This paper is devoted to investigating sliding mode control (SMC) for Markovian switching singular systems with time-varying delays and nonlinear perturbations. The sliding mode controller is designed to guarantee that the nonlinear singular system is stochastically admissible and its trajectory can reach the sliding surface in finite time. By using Lyapunov functional method, some criteria on stochastically admissible are established in the form of linear matrix inequalities (LMIs). A numerical example is presented to illustrate the effectiveness and efficiency of the obtained results.


Introduction
The sliding mode control (SMC) theory has made rapid progress since it was proposed by Utkin [1].As an effective robust control strategy, SMC has been successfully applied to a wide variety of practical engineering systems such as robot manipulators, aircrafts, underwater vehicles, spacecrafts, flexible space structures, electrical motors, power systems, and automotive engines [2].The SMC system has various attractive features such as fast response, good transient performance, and insensitiveness to the uncertainties on the sliding surface [2].These advantages provide more freedom in designing the controllers for the system models which can be easily modified by introducing virtual disturbances to satisfy some requirements.The SMC strategy has been successfully applied to many kinds of systems, such as uncertain time-delay systems and Markovian jump system [3][4][5][6][7][8][9][10][11][12][13][14].
Singular systems, also referred to as descriptor systems, generalized state-space systems, differential-algebraic systems, or semistate systems, are more appropriate to describe the behavior of some practical systems, such as economic systems, power systems, and circuits systems, because singular systems mix up dynamic equations and static equations.Basic control theory for singular systems has been widely studied, such as stability and stabilization [14][15][16][17][18],  ∞ control problem [19][20][21][22], and optimal control [23] and filtering problem [24][25][26].Xu et al. have designed an integral sliding mode controller for singular stochastic hybrid systems [27].They put up with new sufficient conditions in terms of strict LMIs, which guarantees stochastic stability of the sliding mode dynamics.
In practice, many physical systems may happen to have abrupt variations in their structure, due to random failures or repair of components, sudden environmental disturbances, changing subsystem interconnections, and abrupt variations in the operating points of a nonlinear plant.Therefore, Markovian jump systems have received increasing attention, see [28][29][30][31][32][33] and the references therein.Wu et al. [28] have probed sliding mode control with bounded  2 gain performance of Markovian jump singular time-delay systems.Kao et al. [29] have investigated delay-dependent robust exponential stability of Markovian jumping reactiondiffusion Cohen-Grossberg neural networks with mixed delays.Zhang and Boukas have discussed mode-dependent  ∞ filtering for discrete-time Markovian jump linear systems with partly unknown transition probability.To the authors' best knowledge, sliding mode control for a class of Markovian switching singular systems with time-varying delays and nonlinear perturbations has not been properly investigated.
Especially few consider exponential stabilization for this kind of nonlinear singular systems by sliding mode control.
Motivated by the above discussion, we consider exponential stabilization for Markovian switching nonlinear singular systems via sliding mode control.First, we develop two lemmas.Based on these lemmas, delay-dependent sufficient condition on exponential stabilization for singular timevarying delay systems is given in terms of nonstrict LMIs.Some specified matrices are introduced and the non-strict LMIs are translated into strict LMIs which are easy to check by MATLAB LMI toolbox.Second, a sliding surface is derived using an equivalent control approach.A sliding mode controller is developed to drive the systems to the sliding surface in finite time and maintain a sliding motion thereafter.Finally, a numerical example is provided to show the effectiveness of the proposed result.
Notations.(Ξ, F, {F  } ≥0 , P) is a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions.  F 0 is the family of all F 0 -measurable ([−, 0];   ) valued random variables  = () : − ≤  ≤ 0 such that sup −≤≤0 E‖()‖ 2 2 < ∞, where E{⋅} stands for the mathematical expectation operator with respect to the given probability measure P.   and  × denote, respectively, the -dimensional Euclidean space and the set of  ×  real matrices.The superscript  denotes the transpose, and the notation ,  (resp.,  > ) where  and  are symmetric matrices means that  −  is positive semi-definite (resp., positive definite). 2 stands for the space of square integral vector functions.‖ ⋅ ‖ will refer to the Euclidean vector norm, and * represents the symmetric form of matrix.
The system ( 5) is said to be stochastically admissible if it is regular, impulse free, and stochastically stable.
We will assume the followings to be valid.
Lemma 3 (see [34]).Let  ∈  × be symmetric such that      > 0 and  ∈  (−)×(−) nonsingular.Then,  +     is nonsingular and its inverse is expressed as where X is symmetric and T is a singular matrix with where  and  are any matrices with full row rank and satisfy  = 0 and  = 0, respectively;  is decomposed as  =      with   ∈  × and   ∈  × are of full column rank.
and this means Therefore by Definition 2, the Markovian jump singular system ( 5) is stochastically stable.This completes the proof.

Sliding Motion Stability Analysis
where   ∈ R × is real matrix to be designed and   ∈ R × is designed to satisfy that     is nonsingular.According to SMC theory, when the system trajectories reach onto the sliding surface, it follows that () = 0 and Ṡ () = 0. Therefore, by Ṡ () = 0, we get the equivalent control as Substituting  eq (r() = (())() − ((), , ()) into (1), we obtain the following sliding mode dynamics:

Sliding Mode Dynamics Analysis.
In this section, we pay attention to establishing  conditions to check the stochastical admissibility of the system (29).Based on Lemmas 4, it is easy to get the sufficient condition provided in the following theorem.
Theorem 9. Given scalars  and , for any delays () satisfying (2), the system (29) is stochastically admissible if there exist nonsingular matrices   and symmetric positivedefinite matrix  and  such that [ where Remark 10.Note that the conditions in Theorem 9 are not strict  conditions due to matrix equality constraint of (30).According to Lemma 3 and Theorem 9, the strict  conditions are given as follows.
Theorem 11.Given scalars  and , for any delays () satisfying (2), the system (29) is stochastically admissible if there exist symmetric positive-definite matrices X, ,   , , , and symmetric matrix T ∈ R (−)×(−) , matrix L  ∈  × , and any matrices with full row rank   ,   satisfying     = 0,     = 0, respectively, such that [ [ where Proof.Let   ≜   +       in Theorem 9. We can get where Using Lemma 3, we get By pre-and postmultiplying (37) by diag[   , , ] and diag[  , , ], we get where In light of Lemma 4, there exists symmetric matrix   such that It is easy to know that Using Shur's complement, the following inequality can ensure (41) as where There exists a matrix   =  > 0 such that if and only if The above inequality is equivalent to the following inequality: And the following inequality can guarantee that (48) is true: By Shur's complement, the right of ( 46) is equivalent to where   to design a slide mode controller to guarantee the existence of a sliding mode.Now, we design an SMC law, by which the trajectories of singular system (1) can be driven onto the designed sliding surface () = 0 in a finite time.
Theorem 13.With the constant matrix   mentioned in Theorem 11 and the integral sliding surface given by (29), the trajectory of the closed-loop system (1) can be driven onto the sliding surface in finite time with the control (51) as where   is a positive constant.Proof.Choose   under the condition of     is nonsingular.Consider the following Lyapunov function: Due to (29), we have Differentiating   () where   >   √ min (          ).Then the state trajectory converges to the surface and is restricted to the surface for all subsequent time.This completes the proof.

Numerical Example
In this section, a numerical example is presented to illustrate the effectiveness of the main results in this paper.

Conclusion
In this paper, the stochastically admissible using sliding mode control for singular system with time-varying delay and nonlinear perturbations is studied by LMI method.The sliding mode control is designed to ensure that the closedloop system is stochastically admissible.A numerical example demonstrates the effectiveness of the method mentioned above.

Figure 1 :
Figure 1: State trajectories of the open-loop system.

Figure 3 :
Figure 3: State trajectories of the closed-loop system.