Mixed incremental H∞ and incremental passivity analysis for Markov switched stochastic nonlinear systems

The current study addresses the mixed incremental H∞ and incremental passivity analysis for Markov switched stochastic (MSS) nonlinear systems. The multiple Lyapunov functions approach and the structure of Markov framework are utilized to establish some sufficient conditions for the MSS nonlinear systems, which will be used for the incrementally globally asymptotically stable in the mean(IGASiM) and performance index analysis. Then, the mixed incremental H∞ and incremental passivity performance issues are solved for two instances: in the first case, all subsystems are not IGASiM, while in the second one, both of IGASiM and unstable subsystems exist. Hence, it is shown that when none of the subsystems is IGASiM, the MSS nonlinear systems are IGASiM and possess the mixed incremental H∞ and incremental passivity performance metric in the presence of specified conditions. The mathematical induction is selected to guarantee the robust incremental stability of MSS systems with IGASiM and non-IGASiM subsystems and the performance index can be exhibited a prescribed decay rate. The effectiveness of the proposed results is demonstrated by two simulation examples.


I. INTRODUCTION
Stochastic hybrid systems include a group of dynamic systems consisting of continuous-time systems combined with discrete-time parts influencing by the measurement noise and discrete random events. These systems can be described by a variety of models, including stochastic switched systems [1], Markov jump systems [2]- [4], impulsive stochastic systems [5], [6]. MSS systems include a category of different active subsystems under actions governing a continuous-time Markov system, which can take values in defined state space. Based on the Markov switching principle, a subsystem activation along the system trajectory can be realized at defined samples. All subsystems will be endowed with continuous stochastic dynamics describing with a control-dependent stochastic differential equation. The Markov switching phenomenon may be due to switching events among various subsystems in dynamical systems caused by sudden changes in parameters or environmental features. Besides, by Markov process structural changes, the MSS systems are usually utilized to model some industrial hybrid systems with multiple or failure modes, including Hamiltonian systems, multiagent systems, manufacturing systems, and communication systems [7]- [11].
Incremental stability [12] deals with the mutual convergence of trajectories, rather than a certain equilibrium point or a definite path. More recently, incremental stability has been presented for various types of stochastic nonlinear systems, like stochastic switched systems [13], stochastic control systems [14], randomly switched stochastic systems [15], and their representation based on several notions of incremental multiple Lyapunov functions [16]. By resorting to the incremental multiple Lyapunov function method, the IGASiM for switched stochastic nonlinear systems [17] are investigated, and the incremental stability criteria was also presented for the feedback coupled switched stochastic nonlinear systems. Moreover, the contraction metric concept has been utilized to verify the incremental stability of stochastic nonlinear systems [18], [19].
In many practical control systems, there exist unstable subsystems, leading to various problems, including controller fault, sensor fault, and external random disturbance. Therefore, switched systems should be considered in both stable and unstable subsystems [20]- [23], where the active time for stable ones must be higher than a threshold to compensate unstable ones. Stabilization of such unstable systems may be very complicated and costly. Accordingly, the switching signal and the control protocols should be reconsidered to ensure the stability of the overall switched system. As an intuitive fact, if the signal rarely switches at stable subsystems, while often switches at unstable ones, the stability for the overall switched system can be preserved. It has been demonstrated that this idea could be reasonable using the dwell-time principle [20], [21], [24]. The use of incremental H ∞ performance issue is introduced in [24], where a performance metric has been obtained for unstable systems by resorting to a switching rule depending on states and the dwell time technique. It is popular that the stationary distribution of Markovian chains depends on their corresponding active time, which is very useful for confining the active time of stable and unstable subsystems for MSS [2]. Thus, the sojourn time for all subsystems should fulfill exponential and uniform distributions, which leads to the transition rates between different states are constants. Therefore, the Markov switching process and the mentioned sojourn times are essential in the stability analysis of such systems.
However, for MSS systems, the random noise and the Markov system framework have an essential influence on system stability. On the other hand, stochastic noise can tend the system to instability, while it can also be utilized to stabilize an unstable system. This can be considered as the main motivation of this study. Moreover, the Hessian term in Itô differentiation may lead to the complex dynamical treatment of the MSS nonlinear systems. Hence, a clear understanding how incremental H ∞ and incremental passivity performance connects to the general H ∞ and passivity theory remains largely unknown.
Regarding the mentioned considerations, the incremental Lyapunov function was employed to ensure that the switching law achieves the stabilization of the MSS nonlinear systems. The three main contributions of the current study are given below: (i) The mixed incremental H ∞ and incremental passivity disturbance attenuation features are developed for the MSS nonlinear systems. The stationary distribution of the Markov chain is utilized to verify the IGASiM for MSS systems.
(ii) The Markov switching system's framework indicates that the large sojourn time of non-IGASiM subsystem can be controlled by a lower probability of the switching procedure, which activates the corresponding subsystem.
(iii) It is demonstrated that even for non-IGASiM subsystems, the reasonable incremental H ∞ and incremental passivity performance can be attained for the MSS nonlinear system, if the whole activation time of non-IGASiM subsystems is lower than the corresponding one for IGASiM subsystems.

II. PRELIMINARIES AND PROBLEM STATEMENT
Let (Ω, F, {F t } t≥t0=0 , P) and Ω denote complete probability and sample spaces, respectively. F is a σ-field, {F t } t≥t0=0 indicates a filtration satisfying the usual conditions (i.e., it is increasing and right-continuous while F 0 contains all Pnull sets.), which P being a probability measure. E[·] denotes the expectation operator with respect to probability measure P. Let σ(t), t ≥ 0, be a right-continuous Markov chain on the probability space taking values in a finite state space Γ = {1, 2, · · · , M } with generator Q = (q ij ) M ×M given by where h > 0 and o(h) h → 0 as h → 0. Here, if i = j, then q ij ≥ 0 and q ii = − i =j q ij . The Markov chain σ(t) has a unique stationary distribution π = (π 1 , π 2 , · · · , π M ). Let {t k , k ∈ N} be the switching moments. Let N i (t 0 , t) be the occurrence number for the i-th subsystem over the interval [t 0 , t] and S i be the sojourn time at any state i ∈ Γ, with an exponential distribution, i.e., P( Assume a stochastic nonlinear system with Markovian switching describing by the following equations: where x(t) ∈ R n is the system state vector; ω σ (t) ∈ R m which belongs to L 2 [0, ∞), is either a disturbance input or a reference signal; y(t) ∈ R m indicates the controlled output; (t) indicates an r-dimensional Brownian motion satisfying E{d (t)} = 0 and E{d 2 (t)} = dt. Moreover, let (t) be independent of σ(t) in this paper.
In what follows, we impose some additional Lipschitz and linear growth conditions to ensure the existence and uniqueness of solutions. Then, we assume that these functions satisfy the following: This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
In addition, suppose that the external disturbance The following assumption for system (1) is adopted, which first appears in [2]. Assumption 2.1: The time series {t n+1 − t n , n ∈ N} consists of a set of independent random variables and independent of {σ(t n ), n ∈ N}.
The following key technical assumption will be proposed to handle mixed incremental performance index problem for system (1). Assumption 2.2: Consider that there exists some λ ∈ R + such that for each stochastic process σ, the probability of sojourning (remaining in a mode) within an infinitesimal time interval h is lower-bounded by the following, for any i ∈ Γ: In order to simultaneously analyze performance problem and IGASiM property of stochastic nonlinear systems, incremental stability theory is widely used, which has its roots in [25] for stochastic systems. Definition 2.1: [25] A system of the form (1) is said to be IGASiM, if there exists β(·, ·) ∈ KL such that for all t ≥ t 0 , The following key technical lemma will be given, it enables us to estimate an upper bound of switching number in probability. The estimation is based on the transition rate q ij . Lemma 2.1: [2] Let N σ (t 0 , t] be the switching number of σ(t) on the interval (t 0 , t], now, the following relation can be written whereq = max{q ij : i, j ∈ Γ},q = max{|q ii | : i ∈ Γ}. Now, for the system (1), let γ > 0, θ ∈ [0, 1] and σ(t) ∈ Γ, the mixed incremental H ∞ and incremental passivity performance issue can be addressed in this paper are formulated as follows: (i) In the case that ∆ω σ ≡ 0, the system (1) is IGASiM.

III. MIXED INCREMENTAL H∞ AND INCREMENTAL PASSIVITY PERFORMANCE ANALYSIS
This section aims at discussing two types of MSS nonlinear systems: (a) Each subsystem is non-IGASiM, (b)Simultaneous existence of both IGASiM and non-IGASiM subsystems. Just for the sake of notation, let where the convex function α 1 ∈ K ∞ , the concave function α 2 (·) ∈ K ∞ .

A. CASE (A): EACH SUBSYSTEM IS NON-IGASIM
Theorem 3.1: For the given system (1) and constants 0 < Proof: From (6) and (7), one obtains where (10), one gets Next, we aim at showing that for any k > 0 where . The above result can be obtained by the mathematical induction, the switching number and taking expectation about (II) Now, suppose that for the case of k − 1 the implication (III) Finally, the result of (12) is formulated for the case of Noting that (x(t k ),x(t k )) ∈ F t k for any k ≥ 0, applying the property of conditional expectation, we obtain (17) can be rewritten as follow Using (8) and (15), we can deduce that From (16), (18) and (19), it is not difficult to calculate that Now, for each t ≥ t 0 , by (9,20) and Lemma 2.1, one deduces that where c = e −λt0 .

Now, the system
Applying the expectation to (10), one leads to Next, we aim at showing that for any k > 0 The above result can be obtained by the mathematical induction, the switching number This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. (I) For t ∈ [t 0 , t 1 ) and N σ(t) (t 0 , t) = k = 0, by integrating from both sides of (23), we can get (II) Now, suppose that for the case of k − 1 the implication (24) holds, where t ≥ t 0 . Obviously, (18) and (26), the following relation could be obtained which implies that From (27) and (29), it follows that By (30), let t 0 = 0, one has where ψ(s) = θ∆y T (s)∆y(s) − 2(1 − θ)∆y T (s)∆ω i (s). From Lemma 2.1, (31) can be expressed as From (9), integrating from the inequality (32) from t = 0 to ∞ gives which is equivalent to From (5), one can get . Now, the system (1) has the performance index γ.

Remark 3.1: Condition (6) is a set of coupled incremental
Hamilton-Jacobi inequalities. It can be concluded from the condition λ i > 0 that the IGASiM condition is not necessarily true for the continuous dynamics of all subsystems. In fact, the corresponding proofs will not be affected if the condition (6) is replaced by H ∞ (V i (x,x)) = 0.. Condition (9) is introduced mainly to make sure that IGASiM of the whole system can be obtained by the framework of Markov switching process. According to inequality (9), the large sojourn time of non-IGASiM subsystems can be compensated for by VOLUME 4, 2016 a lower probability of the switching process, which activates the corresponding subsystem. Theorem 3.1 shows that if each subsystem is non-IGASiM, the IGASiM of all systems can be realized through the Markov switching process. Remark 3.2: Note that the switching time t k is random, so that the interval [t k , t k+1 ) is a random time interval. The method of iterating the time was used to estimate the EV σ(t) (t), which is not correct. Hence, we adopt the mathematical induction to estimate the EV σ(t) (t) and the performance index.

B. CASE(B): THE SIMULTANEOUS EXISTENCE OF BOTH IGASIM AND NON-IGASIM SUBSYSTEMS
Consider that all subsystems are not necessarily IGASiM. let Γ s describe an appropriate nonempty subset of Γ, which its complement with respect to Γ is indicated by Γ u , where Γ s = {i ∈ Γ : λ i ≥ 0} and Γ u = {i ∈ Γ : λ i < 0}.
Next, we show that the system (1) has a mixed incremental H ∞ and incremental passivity performance index γ 0 .
It can be easily seen from (40) that for ∀t ∈ [t k , t k+1 ), one can get From (37), by the mathematical induction, we can obtain Then, for any t ≥ t 0 , one has From (45), one can get

VOLUME 4, 2016
On the other hand, we show an inequality as follow By Lemma 2.1, from (49) and (54-56), we have that for every t ≥ t 0 , which implies that Note that Assumption 2.2 implies P(σ(s + h) = i|σ(s) = i) ≥ 1 − λh, s + h = t, λ ∈ R + , it follows that Hence, If both sides of (60) can be integrated from t = 0 to ∞, from (38), we have where λ s > λ, and thus, where Now, the system (1) has the performance index γ 0 .
Remark 3.3: Condition (38) represents the quantitative index of the unstable subsystem characterized by p, λ u ,q u ,q. The performance index in Theorem 3.2 is more extensive than [24], and when θ = 1, it can be degenerated into the incremental H ∞ performance index. Due to the sequence {t n+1 − t n , n ∈ N} is a collection of independent random variables, it is observed that the proof of the IGASiM results is more complicated than the corresponding proof in [24].
It is not too difficult to handle performance index problem by applying lower-bounded on the probability of sojourning time.

IV. EXAMPLE
In the current section, two examples of the MSS nonlinear systems are presented to evaluate the established results. Assume that a one-dimensional Brownian motion is described by (t). Consider a right-continuous Markov chain σ(t) with values in Γ = {1, 2} with generator. Example 4.1. Assume that the generator is described by Assume a MSS system represented by (1) with the following two subsystems The storage function of each subsystem is assumed as It can be simply verified that each subsystem is non-IGASiM for ∆ω i = 0, i = 1, 2. Let θ = 1, γ = √ 2, λ 1 = 6 5 , where x 1x1 ≥ 0. Let µ = 1 4 ,q = 2,λ = 6 5 ,q = 2, it can be checked that µq +λ −q = −0.3 < 0.

V. CONCLUSION
In the current research, the incremental Lyapunov functions and the stochastic analysis approaches are employed to solve the mixed incremental H ∞ and incremental passivity issue for MSS nonlinear systems. Based on the stationary distribution of Markovian switching procedure, some sufficient conditions represented by inequalities are established and the incremental H ∞ and incremental passivity performance problem to be solvable can be ensured. It is also suggested that via addressing mixed incremental H ∞ and incremental passivity performance problem of MSS nonlinear systems that the mixed performance index problem developed in this work is more powerful than the general incremental H ∞ control problem in those existing work. The presented theory is validated using a couple of examples and numerical simulations. YUSHI YANG an associated professor?in Mathematics at College of Science, Hebei Agricultural University. Mainly interested in Applications of Statistical, with current emphasis on incremental stability analysis and discrete-time systems. VOLUME 4, 2016