（School of Mechanical＆Electrical Engineering，Heilongjiang University，Harbin 150080，China）

Stabilization Control for Linear Switching Stochastic Systems Against Time-Delay in Communication Channel

Dongfang Lv?and Shen Cong

（School of Mechanical＆Electrical Engineering，Heilongjiang University，Harbin 150080，China）

The paper is concerned with stabilization problem for a class of stochastic switching systems with time-delay in the detection of switching signal.By using binomial model，Poisson process，and Wiener process to describe time-delay，switching signal，and exogenous disturbance，respectively，the system under investigation is entirely set in a stochastic framework.The influence of the random time-delay is combined into reconstructing the switching signal of overall closed-loop system and changes the distribution property of switching points.Therefore，based on the asymptotical behaviors of Poisson processes and Wiener processes，the almost surely exponential stability conditions are established.Furthermore，a design methodology is posed for solving the stabilization control.

switching systems；stochastic systems；random switching；almost sure stability；stabilization

1 Introduction

During the last two decades，the stability and stabilization problems of switching systems have attracted considerable interest.In this research area，one of the fundamental problems is to characterize the time evolution of switching signal and its effect on the dynamics of the overall system.In fact，this effect in strong way relies on the varying rate of switching signal. One way to capture the varying rate is to describe the distribution density of switching points in terms of average dwell-time.In this way，the basic idea behind proving stability is to guarantee the length of the interval between two successive switching points to be long enough so as to allow the overshoot caused by switching to be absorbed.Indeed，most of the results reported in the deterministic framework to address the stability problem of switching systems were obtained in this spirit［1-5］.

As for the control problem of switching systems，it is inevitable to encounter the time-delay in the detection of switching signal due to the aftereffect of the sensors in communication channel［6-8］.Indeed，the existence of such time-delay may constitute a great challenge to deal with since it can induce the unexpected dynamical performance and，at the same time，it is difficult to be accounted for［4，9］.Therefore，in order to guarantee the desired dynamical performance of the overall closed-loop system，the control strategy should accommodate such time-delay to certain degree. In Ref.［4］，it is observed that this aftereffect turns out to change the distribution density of switching points，which can be described by means of the technique of merging two switching signals to generate a new one.

In this paper，we extend the technique of merging switching signals to a stochastic framework so as to develop a systematical approach to achieve robust stabilization control against the time-delay in communication channel.To this end，we firstly set a stochastic framework to describe the switching signals and their varying rates.As a matter of fact，in parallel with the deterministic framework，it is a commonly used assumption that the switching signal evolves according to the law of continuous-time Markov chain［10-14］.This pure assumption provides us with a stochastic framework to deal with switching systems. However，it buries a fact that the occurrence of switching is a Poisson event［11，15］.This implicit characterization actually indicates the distribution density of switching points，which allows us to make a sense of the varying rate of switching signal.This observation motivates us to describe the switching signals by simply assuming their time evolution to obey Poisson distribution.Doing this indeed establishes a stochastic framework，which allows including another kind of Lev process，namely，Wiener processes，to account for the exogenous disturbance.In this way，wecan model the abrupt change and exogenous disturbance in terms of Poisson processes and Wiener processes，respectively.Moreover，within the stochastic framework，we are able to suppose the time-delay in the communication channel to obey Bernoulli distribution.

It is the main contribution of the paper that we set the system under investigation entirely in a stochastic framework and，moreover，we pose a systematical approach to deal with the feedback control problem so as to make the closed-loop system almost surely stable. Finally，a numerical example is worked out to demonstrate the theoretical results.

Notation：Throughout the paper，we use the following notations.Let｛Ω，F(xiàn)t，P｝be a complete probability space with the reference family｛Ft｝satisfying the usual conditions.We writeEfor the corresponding mathematical expectation operator.Letw（t）stand for the one-dimensional Wiener Process that is normalized and defined on the probability space. LetIA（t）denote the characteristic function of setA．

2 Problem Formulation and Preliminaries

Considering the controlled switching stochastic system as follows：

wherex（t）∈Rnandu（t）∈Rmare state vector and control input，respectively.The system（1）generated by the switching signalσ（t）to orchestrate among a family of subsystems as follows：

According to the evolution of switching signal over time，it can be expanded into the following sequential form：

which means that theσ（tk）th subsystem is activated during the interval［tk，tk+1）．Supposing the switching signal in Eq.（2）to obey Poisson distribution，for anyΔ＞0 we have

whereλ＞0 is referred to as Poisson exponent. Besides，we supposeσ（t）andw（t）to be independent of each other.Therefore，the switching pointst0＝0＜t1＜…＜tk＜…turn out to constitute a sequence of stopping times tending to infinity.Equivalently，it reads that

By means of Poisson exponentλ，one can make sense of the density of the switching points distributed within an interval of time and，moreover，categorize switching signals.Namely，denote bySλthe class of switching signals obeying Poisson distribution exactly with the exponentλ．For given switching signalσ，letx（t；x0，σ）denote the corresponding motion of system（1）at timetstarting fromx0at initial timet0＝0．

In order to achieve stabilization control，for each subsystem we construct the corresponding statefeedback controller as

Theoretically，it is a common assumption that the controllers are triggered into activation in synchronism with the change of subsystems.However，it is inevitable to encounter some aftereffect in detecting the change of subsystems and then changing controllers，which we refer to as the time-delay in communication channel.Such an aftereffect renders the state-feedback control the following form：

where the nonnegative scalarh（t）represents the timedelay in feedback channel to respond the change of subsystems.For modeling the aftereffect，we suppose it obey the Bernoulli distribution.Namely，h（t）is assumed in the binary set｛0，h｝according to the following probability：

whereη∈［0，1］．

Thus，the system under investigation is entirely set in a stochastic framework by modeling the exogenous disturbance，the abrupt changes，and the aftereffect in feedback channel as random processes.It is reasonable to assume thatw（t），σ（t），andh（t）are independent of each other.

3 Main Results

In this section，we shall establish conditions to guarantee system（1）to be almost surely exponentially stable with respect to a certain classesSλof switching signals.

Firstly，we interpret the influence of the aftereffect in feedback channel on the overall closed-loop system. Letσ′（t）＝σ（t-h（t））．If we represent the sets of the switching points belonging toσ（t）as follows：

and then it follows that

As shown in Fig.1，merging the switching signalsto generate a new one，which is denoted byThe switching pointsis the union of the switching points ofσ（t）and

Meanwhile，as opposed toσ（t）andσ′（t），which both are assumed in the index setis assumed in the product set｛1，2，…，N｝×｛1，2，…，N｝．With this construction，we can categorize the closed-loop subsystems into matched and mismatchedclasses.Precisely，the closed-loop subsystems are

Clearly，i＝jindicates that the closed-loop subsystem is matched；otherwise，i≠jcorresponds to a mismatched closed-loop subsystem.Thereby，the overall closed-loop system is written as follows：

Fig.1 An illustration for the construction of

Definition 1The closed-loop system（8）is said to be almost surely exponentially stable，if

wherex0∈Rn．

Remark 1It is worth noting thatno more constitute time-homogenous Poisson processes though they are generated fromσ（t）that itself is assumed to obey time-homogenous Poisson distribution. In other words，the existence of the time-delay in communication channel changes the statistic property of the switching signal.Indeed，if the switching signal remains a time-homogenous Poisson process，we would be able to formulate the infinitesimal generator，which plays a key role for capturing the dynamics of the overall system.For the sake of generality，we consider the switching stochastic differential system as follows：

where the switching signal obeys the Poisson distribution as in Eq.（4）.We suppose the drift coefficientsfi（x）and the diffusion coefficientsgi（x）to satisfy certain conditions so that we can guarantee there is a solution of system（9）to exist on［0，∞）.To each subsystemassign a functionVi（x），which has the partial derivatives at least to second order inx．LetLbe the infinitesimal generator corresponding to the stochastic differential system（9）. Forwe compute the action ofLon it as

It fails to capture the state-transition of the overall closed-loop system in terms of its infinitesimal generator due to the existence ofh（t），yet it is possible to describe the densities of the switching points belonging torespectively.Letbe the counter of the switching points ofthat are distributed within the intervalThen，by the interpretation forh（t），we know that

It brings about that

Furthermore，by the construction ofwe have

Theorem 1If there exist positive numbersα，β，and a family of positive-definite matricessuch that for allthe following inequalities hold：

and then the closed-loop system（8）is almost surely exponentially stable for all switching signals belonging to

Proof for system（1），we construct the Lyapunov function as

In particular，we notice that

By rewriting Young’s inequality as

The above equality is due to Eq.（15）.Letbe a sequence of the switching points belonging towe have

Hence，together with Eq.（21），it gives

Lettbe located between the successive switching pointsAccording to Eqs.（13）and（14），we get

where，to simplify typography，we denote the terminal pointtbysince there is no switching point within

Additionally，from Eq.（12）one can observe that

Since the randomly varying time-delayh（t），Poisson processσ（t），and Wiener processw（t）are assumed to be independent of each other，the reconstructed switching signalandw（t）are independent of each other.As a consequence，we rewriteM（t）as follows：

It constitutes a local martingale that vanishes att0．By using It? isometry，we can compute the quadratic variation ofM（t）as

Therefore，by the strong law of large numbers［16］，there is probability 1 thatAt the same time，by the law of large numbers for Poisson processes［17］，there is probability 1 that

This together with Eq.（17）yields that

The proof is thus completed.

Remark 2We now interpret the influence of the time-delay in communication channel by taking look at two special situations.The maximum tolerable timedelay is determined by Eq.（17）.Accordingly，it naturally requires that

When there is no time-delay at all，namely，η＝1，the condition reduces to

It falls into accordance with the stability conditions of switching systems.On the other hand，when there is a constant time-delay，namely，η＝0，it is seen that

In terms of merging switching signals，we can make sense of the condition in Eq.（25）.Actually，we haveand they are independent of each other. Clearly，σ（t-h）is dependent onσ（t），comparing Eq.（25）to Eq.（24）yet leads to the conclusion that the constant time-delay in communication channel makes the distribution density of switching points double. Nevertheless，one cannot assertbecause，actually，it may not be a timehomogenous Poisson process.

With the analysis result in hand，we now turn to consider the stabilization control design problem.The design procedure is divided into two steps.

Firstly，we solve the feedback controllers via the constraint conditions on the matched closed-loop subsystems.To proceed in this way，we need to specify positive numbersα，β，γ，andχthat are involved in the conditions of Theorem 1.Therefore，if there exist matricesWi，andQiisuch that the following matrix inequalities hold

where?represents symmetry element，then one can recover the inequalities in Eqs.（13），（14）and（16）for the matched subsystems withand

In other words，the state-feedback controlu（t）＝would make the closed-loop system（8）almost surely exponentially stable for all switching signals inSλif there were no time-delay in communication channel.

Secondly，we use the constraint conditions on the mismatched closed-loop subsystems to check the robustness of the controllers given in Eq.（29）. Precisely，if for all pairs（i，j），i≠jthere exist matricesPijsuch that the following inequalities are satisfied

and then the closed-loop system（8）staysexponentially stable provided

Remark 3Indeed，we consider the matched subsystems in the closed-loop and the mismatched ones separately.The former servers as the design constraint conditions as in Eqs.（26）-（28），while the later is left to test the robustness of the constructed controllers as in Eqs.（30）-（32）.In particular，the conditions in Eqs.（28）and（32）together guarantee Eq.（16）to be true， namely，

We now summarize the design procedure as follows.

Step 1Specify the parameterssubjected to Eq.（17）with

Step 2Test the existence of the controllers of subsystems via the conditions in Eqs.（26）-（28）for the matched subsystems and construct the controllers according to Eq.（29）；

Step 3Examine the robustness of the controllers by checking if the conditions in Eqs.（30）-（32）are feasible，if so，then the constructed controllers can stabilize the overall system；otherwise，then go back to Step 1 to reset the parameters.

4 Numerical Example

Considering the switched system composed of linear control subsystems with the following parameters：

With the Poisson exponentλ＝0.4 and Bernoulli coefficientη＝0.5，we specify the parametersχ1＝0.45，χ2＝2.25，α＝0.06，β＝1.18 and then find the matrix inequalities in Eqs.（26）-（28）feasible. Therefore，we get the feedback gain matrices as follows：

which，at the same time，make the matrix inequalities in Eqs.（26）-（28）feasible.Hence，by virtue of Eq.（17），we know that the overall closed-loop system stays exponentially stable almost surely provided the time-delay in communication channel does not exceed 0.345 6 s.

5 Conclusions

We investigate the robust stabilization control problem for linear switched systems against the timedelay in communication channel.A merging switching signal technique has been applied to reconstruct the switching signal to take into account this aftereffect.A robust design methodology has been posed，which guarantees the overall closed-loop system to be almost surely stable.Finally，the numerical example is included to demonstrate the theoretical results.

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TP72

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：1005-9113（2015）05-0110-06

10.11916／j.issn.1005-9113.2015.05.017

2015-03-12.

Sponsored by the Natural Science Foundation of Heilongjiang Province of China（Grant No.LC201428，F(xiàn)201429）.

?Corresponding author.E-mail：lvdfang＠hlju.edu.cn.