Exponential Stability in Mean Square of Singular Markovian Jump System with Saturating Actuators and Time-Varying Delay

This paper investigates exponential stability in mean square of singular Markovian jump systems with saturating actuators and time-varying delay. The statistical property of the Markov process is fully used to derive the differential of the function. By using a delay decomposition method, a mode-dependent Lyapunov-Krasovskii function is established. A sufficient condition is proposed for exponential stability in mean square of the system designing the memoryless state feedback. A numerical example shows that the approach proposed is effective.


Introduction
Markovian jump systems are a special class of hybrid systems with both modes and state variables, which is described by a set of time-delayed linear systems with the transitions among models determined by a Markov chain in a finite mode set.In recent years, Markovian jump systems have received much attention, and many problems have been solved, including stability analysis, state feedback, and output feedback controller design [1,2].Both time-delay and saturating controls are commonly encountered in various engineering systems, which often lead to poor performance and instability of a control system.Therefore, much attention has been paid to the study on the stability problem of systems with saturating actuator.The problem of robust stabilization of uncertain time-delay systems with a saturating actuator was addressed by Niculescu et al. [3].Liu has obtained the condition of stability for a class of time-varying delay systems with saturating actuators according to the usage of the linear matrix inequality and Leibniz-Newton formula [4].For the systems with time-varying delays, the reported results are generally based on the assumption that the derivative of timevarying delays is less than one [3,4].Such restriction is very conservative.Furthermore, singular system model is a natural presentation of dynamic systems and can describe a larger class of systems than regular ones, such as large-scale systems, power systems, and constrained control systems, electrical circuits, power systems, and economics [5].For a singular time-delay system, it is important to develop conditions which guarantee that the given singular system is not only stable but also regular and impulse free.Wu et al. discussed the stability problem on uncertain singular Markovian jump time-delay systems [6,7].Ma et al. investigated the robust stochastic stability problem for discretetime uncertain singular Markov jump systems with actuator saturation [8].Boukas et al. established an LMI condition for the singular time-delay systems with Markovian jump to be regular, impulse free, and stochastically stable.However, the result of [9] is delay independent.Generally speaking, delay-independent conditions are more conservative than delay-dependent ones, especially for small time delays.To the best of our knowledge, the robust stability for singular Markov jump systems with actuator saturation and timevarying delay has not been investigated in the literature; this problem is important in both theory and practice.
In this paper, we are concerned with stability of singular Markovian jump system with saturating actuators and timevarying delay.A new Lyapunov function can be constructed by using a delay decomposition method and considering the statistical property of Markovian process.In terms of LMI approach, stability conditions are proposed to guarantee the considered systems to be regular, impulse free, and exponentially stable in mean square.Free weighting matrix and reduction method are not introduced in the derivation of the stability criterion.Thus, the results reduce conservation.
Assumption 1.The markovian jumping parameter   is rightcontinuous markovian process and takes values in finite set  = {1, 2, . . ., }, and is defined by where Δ > 0, lim Δ → ∞ (Δ)/Δ = 0,   is the transition rate from  to  and Definition 2. The infinitesimal generator of the solution to system (1) is defined as is said to be regular and impulse free, if the pairs (, (  )) are singular and impulse free for every   ∈ .
(2) The singular Markovian jump system with saturating actuators and time-varying delay ( 8) is said to be exponential stable in mean square if there exist scalars  > 0,  > 0, such that (3) The singular Markovian jump system with saturating actuators and time-varying delay ( 8) is said to be meansquare exponentially admissible, if it is regular, impulse free, and exponentially stable in mean square.
Lemma 5 (see [10]). is an  ×  diagonal matrix whose entries are either 1 or 0. Note that there are where  denotes convex hull.
Lemma 6 (see [11]).Given V > 0, for a vector () and a positive-definite matrix Ζ with appropriate dimensions, and the integrations concerned are well defined, then (13)

Main Results
Considering the system (1), for each region   =  ∈ , the system (1) becomes for every  ∈ , such that where Proof.According to the definition of the saturating function and Lemma 5, we have Substituting ( 19) into (16) gives Since rank  =  < , there exist nonsingular matrices  and Υ, such that Let According to (18a), it is easy to obtain that  2 = 0 for every  ∈ .By pre-and postmultiplying Σ 11 < 0 by Υ  and Υ, respectively, we have   4  4 +   4  4 < 0, which implies that  4 are nonsingular for every  ∈ .Thus, the pairs (,   ) are regular and impulse free for every  ∈ .Thus, by Definition 2, the system ( 8) is regular and impulse free for any time-varying delay () satisfying (2).

Choose a Lyapunov-Krasovskii function candidate as
where Moreover, the action of infinitesimal generator (7) on each term function (23) could be expressed as According to (20), (23), and (25), we have   (33) Similar to (33), by Lemma 6, it is easy to obtain where In conclusion, one has where Mathematical Problems in Engineering  () = [()  , ( − )  , ( −  ())  , ( − 2 ())  , . . ., ( − ( − 1)  ())  , ( −  () (37) using ( 18b) and (18c), it is easy to obtain Since  4 are nonsingular for every  ∈ , we set It is easy to get where Let Then, for every  ∈ , system (20) is a restricted system equivalent to To prove the exponential stability in mean square, we define a new function as where  > 0 and then by Dynkin's formula, we find that for every  ∈ ,  (  ,   , ) ≤  ( 0 ,  0 , 0) It can be seen from ( 23), (45), and (46) that if  is chosen small enough, a constant  > 0 can be found such that for  > 0, Hence, for any  > 0 where  = (min ∈ { min ( 1 )}) −1 .To study the exponential stability in mean square of  2 (), we apply the similar analysis method of [12] and define Then, according to (48), we have a constant  > 0; it can be found that when  > 0, We construct a function as By premultiplying the second equation of (44) with  2 ()    4 , we obtain that Adding ( 52) to (51) yields that where  1 is any positive scalar.By pre-and postmultiplying [ On the other hand, since  1 can be chosen arbitrarily,  1 is chosen small enough, such that  2 −  1 > 0.Then, we can always find a scalar  3 > 1, such that It follows from ( 51), (53), and (54) that It is clear that the above inequalities (55) and (56) imply that Thus, we get from (57) that Let() =  2 ()   4  2 (); from the above inequality and the fact (50) and (58), we have where We can find from (50) and the above inequality that the system (16) is exponentially stable in mean square.
Remark 8.It should be pointed out that sometimes too many free variables cannot reduce the conservatism of the obtained results.In the theorem, the system is exponentially stable in mean square without resorting to the free-weighting matrices method, which is in contrast with [6], where the free-weighting matrices method was used.Remark 9.In the proof of the theorem, the delay interval [0; ] is divided into  segments of equal length /, such that the information of delayed states ( − /),  = 1, 2, . . ., , are all taken into account.It is clear that the Lyapunov function defined in the theorem is more general than the ones in [7].

Numerical Examples
In this section, we provide a numerical example to verify the effectiveness of the proposed method.Consider systems (1) in  2 with two regines   ∈  = {1, 2}.The system parameters are described as follows:  (69) So we can conclude that the condition of the theorem is satisfied; the system (16) is exponentially stable in mean square.

Conclusions
In this paper, we have investigated the exponential stability in mean square of singular Markovian jump systems with singular Markovian jump systems with saturating actuators and time-varying delay.To guarantee the singular Markovian jump system to be regular, impulse free, and exponentially stable in mean square.Memoryless feedback control has been designed.By exploiting Lyapunov-Krasovskii theory, a new sufficient condition of exponentially stable in mean square is obtained.Finally, the numerical example shows that the approach proposed is effective.To reduce conservatism of the results, our future work focuses on reducing the constraint by an appropriate reduction method.
2  elements in .
= 1, by calculating we have