Asynchronous Stabilization for Two Classes of Stochastic Switching Systems with Applications on Servo Motors

This paper addresses the asynchronous stabilization problem of two typical stochastic switching systems, i.e., dual switching systems and semi-Markov jump systems. By dual switching, it means that the systems contain both deterministic and stochastic switching dynamics. New stability criteria are firstly proposed for these two switched systems, which can well handle the asynchronous phenomenon. The conditional expectation of Lyapunov functions is allowed to increase during some unmatched interval to reduce the conservatism. Next, we present numerically testable asynchronous controller design methods for the dual switching systems. The proposed method is suitable for the situation where the asynchronous modes come from both inaccurate mode detection and time varying delay. Meanwhile, the transition probabilities are both uncertain and partly accessible. Finally, novel asynchronous controller design methods are proposed for the semi-Markov jump systems. The sojourn time of the semi-Markov jump systems can have both lower and upper bounds, which could be more practical than previous scenarios. Examples are utilized to demonstrate the effectiveness of the proposed methods.


Introduction
Switched systems, as a special class of hybrid systems, have received considerable attention in the past few years [1,2]. It usually contains a family of subsystems and a switching law that coordinates between them [3]. A variety of physical systems, such as mechanical systems [4,5] and network control systems [6][7][8][9], can be well modeled by the switched systems. Due to the abrupt and unpredictable phenomenons in real practice, a stochastic switching law is usually adopted for the switched systems. Hence, research on stochastic switching systems are of great practical importance [10,11]. A large number of excellent results have been obtained for various aspects of stochastic switching systems, such as stability analysis [12,13], controller design [14][15][16][17], state estimation [18,19], etc.
Markov jump systems are a typical class of stochastic switching systems [20][21][22][23]. The stochastic switching law is described by a Markov process. Numerous works have been conducted on the Markov jump systems. For instance, Ref. [24] considered the problem of state feedback stabilization for singular Markov jump systems by using the equivalent sets technique. In [25], a sliding mode controller was designed for a Markov jump system with digital data transmission. More recently, Ref. [10] considered the stabilization problem of a class of Markov jump systems with generally uncertain transition rates. Namely,

1.
New stability criteria are proposed for the considered stochastic switching systems, which can well handle the asynchronous phenomenon. It is noted that the Lyapunov function is allowed to increase during some unmatched interval to reduce the conservatism of controller design; 2.
Numerically testable asynchronous controller design methods are presented for the dual switching system. The proposed method is suitable for the situation where the asynchronous phenomenon can come from both inaccurate mode detection and time varying delay. Meanwhile, the transition probabilities are both uncertain and partly accessible; 3.
Novel asynchronous controller design methods are presented for the semi-Markov jump systems. The sojourn time of the semi-Markov jump systems can have both lower and upper bounds, which could be more practical than previous scenarios.
The organization is as follows: Section 2 formulates the problem. Section 3 proposes the stability and stabilization conditions for the considered two stochastic switching systems. Examples are presented in Section 4. Section 5 concludes the paper. All the proofs are put in the Appendix A.

Problem Formulation for Dual Switching Systems
Definition 1. Given the following dual switching systems.
x(k + 1) = A g(k),r(k) x(k) + B g(k),r(k) u(k) (1) where x(k) ∈ R n is the system state, u(k) ∈ R m is the control input. g(k) ∈ M 1 = {1, 2, . . . , M 1 } is a deterministic switching law. It is admissible with a average dwell time τ d [34]. Namely, it satisfies the following condition N g(k) (k 1 , k 2 ) ≤ N 0 + (k 2 − k 1 )/τ d (2) where N 0 ∈ N, τ d > 0. N g(k) (k 1 , k 2 ) denotes the switching numbers of g(k) over the time interval [k 1 , k 2 ). r(k) ∈ M 2 = {1, 2, . . . , M 2 } is a homogeneous Markov process defined in the probability space (Ω, F , Pr) where Ω is the sample space, F is a σ-field, and Pr is the probability measure. The evolution of r(k) is determined by the transition probability matrix defined as with ν ∈ M 1 , i, j ∈ M 2 . In practice, the transition probability could suffer from uncertainties and may not be fully accessible. Suppose that π νij ≤ π νij ≤ π νij .
Then, define the following set M 2νi = M K νi + M UK νi ; M K νi = ∅, i ∈ M 2 : Finally, assume that for each g(k) = ν and r(k) = i, A νi ∈ R n×n and B νi ∈ R m×n are known constant matrices.
In an ideal case, a mode-dependent state feedback controller can be considered for the above system, i.e., u ideal (k) = K g(k),r(k) x(k).
However, due to the asynchronous phenomenon, the mode of the dual switching system may not be detected exactly. In this case, we suppose that the actual control effort is expressed as: where d(k) ∈ N is an unknown time varying delay, such that 0 ≤ d(k) ≤ τ as ≤ τ d with known upper bound τ as ∈ N. φ(k) ∈ L = {1, 2, . . . , L} ⊆ M 2 , such that with ν ∈ M 1 , ϕ ∈ L, i ∈ M 2 . Figure 1, (4) can be interpreted as follows: first when the mode detector detects the mode of the dual switching system, due to the missing measurement, the detected mode may not be the exact current mode. Therefore, a stochastic variable φ(k) depending on (5) is presented to describe this pheromone. Second, when the detector transmits the mode information to the controller side, there exists a transmission delay τ as . Note that we assume the deterministic switching mode can be detected exactly. This may lie on that the deterministic g(k) has a larger dwell time τ d than the r(k). In fact, r(k) may change at every time instance, which implies that it switches much more frequently than g(k). Hence, r(k) is more difficult to be detected, and there may be some mismatch detection. Another reason is that since g(k) is deterministic. One can embed the switching instances of g(k) to the detector in advance. This can improve the accuracy of detection. Additionally, note that the proposed method can be extended to the case where g(k) is not detected exactly.

Remark 2.
Note that we have assumed that the stochastic switching r(k) represents the fast time-varying conditions, while the deterministic switching g(k) represents the slowly time-varying conditions. For example, consider the servo motor system, the motor may work in different situations, such as no load, external load, external inertia, etc. This can be represented by the deterministic switching. The fast time-varying parameters, which are from stochastic disturbance, abrupt failure, and noise, can be expressed as a stochastic switching sequence. Note that the proposed method can be easily extended to the case when g(k) switches more frequently than r(k). This can be performed by dividing the slowly switching modes into more modes for r(k) intentionally.
Based on the above analysis, we present our first problem.

Problem 1.
Propose a design method for the asynchronous controller (4), such that the dual switching system (1) is mean square stable.

Problem Formulation for Semi-Markov Jump Systems
Definition 2. Given the following semi-Markov jump systems where x(k) ∈ R n and u(k) ∈ R m are the same as Definition 1. r(k) ∈ M = {1, 2, . . . , M} is a semi-Markov process and the evolution of it is determined by a semi-Markov kernel (SMK), i.e., [Θ ij (τ)], ∀i, j ∈ M with where i, j ∈ M, R n represents the mode of system at n-th jump, S n is the sojourn time between (n − 1)th jump and nth jump. It is assumed that for the ith mode, its sojourn time S i n has a lower and upper bound like [11], i.e., τ i ≤ S i n ≤ τ i , ∀n ∈ N with τ i , τ i being known constants. π ij Pr{r(k + 1) = j|r(k) = i}, h ij (τ) Pr{S n = τ|R n = i, R n+1 = j} is the sojourn-time probability density function (PDF). Meanwhile, for mode i define function H i (τ) = Pr{S n ≤ τ|R n = i}.
Similar to Section 2.1, the asynchronous controller for the above system is given by where d(k) ∈ N is an uncertain time varying delay, such that 0 ≤ d(k) ≤ τ as ≤ τ i , i ∈ M with known upper bound τ as ∈ N.
Remark 3. Note that here we only consider the asynchronous phenomena caused by transmission delay. Meanwhile, we consider a small delay effect for the mode detection and a slowly switched law for the semi-Markov jump systems. Hence, compared with the time delay, the sojourn time of the Markov jump systems may be much larger. Please see Figure 2.
Then, the system is σ-error mean square stable where σ is defined as )| denoting the approximation error of the ith mode.
According to the above analysis, we present our second problem.

Problem 2.
Such that the semi-Markov jump systems (6) are σ-error mean square stable.
To handle the asynchronous phenomenon in (8), we first present the following stability criterion. Lemma 1. For the system (8), suppose there exists C 1 Lyapunov functions V g(k),r(k) (x(k)) : R n → R, g(k) ∈ M 1 , r(k) ∈ M 2 such that for anyν, ν ∈ M 1 , j, j 1 , j 2 , . . . , j τ as −1 , i ∈ M 2 , where λ > 1; K 1 (·) and K 2 (·) are two K ∞ (·) functions; k n with n ∈ N denotes switching instances for the signal g(k). r(k) is a vector of the previous system modes and given by with α > 1 and 0 < β < 1 being two positive constants. Then, the system (8) is mean square stable for any switching signal g(k) with average dwell time Based on the above lemma, we have the following theorem in terms of matrix inequalities.
(iii) There exist matrices T ννijφ 0, P σj 0 such that for anyν, . λ and χ νν are the same as statement (ii). π νjl , l ∈ M 2 is defined as: Remark 4. It is noted that statements (ii) and (iii) are equivalent and can be both used to check the stability of dual switching systems with asynchronous phenomenon and uncertain probability transition rates. However, statements (iii) are in strict LMI and can be solved efficiently.
Based on the above result, we can compute the control gain in (4).
Then, there exists a set of stabilizing controllers, such that (1) is mean square stable for dwell switching signal g(k) satisfying (12). The admissible controller can be given by:

Remark 5.
Compared with [25,37,38] the proposed stability criterion and controller design method can be used to simultaneously handle dual switching dynamics (switching sequence g(k), r(k)), mismatch mode detection (φ(k)), and mode transmission delay (τ as ). This is achieved by using a multiple Lyapunov function technique (see the proof of Lemma 1). It is noted that the conditional expectation of the Lyapunov function is allowed to increase from (9). This is more general than the existing studies [25,37,38]. Meanwhile, different from [25,37,38], the condition for the expectation of the Lyapunov function is not only dependent on the latest mode r(k − 1) but also on the latest τ as mode r(k − 1), . . . , r(k − τ as ). This will bring more difficulties to the controller design.

Asynchronous Controller Design for Semi-Markov Systems
Substituting (7) into (6), we obtain the closed loop system: where i, j ∈ M.
To handle the asynchronous phenomenon in (21), we first present the following stability criterion.

Remark 6.
Compared with [11,28], the condition for the expectation of the Lyapunov function is not only dependent on the latest mode r(k n ) but also on the r(k n−1 ). This will bring difficulties to the controller design.
Based on the above lemma, we will present the stability criterion in terms of matrix inequalities.

Lemma 4. Given matrices
Specifically, if d(k) = τ as , then If τ as ≤ τ i , i ∈ M, If τ as > τ i , i ∈ M, Remark 7. According to whether τ as ≤ τ i or not, I ij χ is defined separately. Meanwhile, as stated in Remark 3, we mainly consider a small delay effect and slowly switched law for the semi-Markov jump systems. Therefore, there is a high probability that τ as ≤ τ i , i ∈ M. Thus, (26)- (27) and (31)-(32) are applicable.
The above result can be converted into strict LMI form.

Lemma 5. Given matrices
Based on the above lemma, we can compute the control gain in (7).
Then, there exists a set of stabilizing controllers, such that (21) is σ-error mean square stable. The admissible controller can be given by:

Remark 9.
We have only considered the stabilization problem for semi-Markov jump systems. However, the proposed method can be extended to solve the controller design problem under performance constraints, such as mixed H ∞ and passivity [3,4]. Nevertheless, note that the state equation of the semi-Markov jump system needs to be iterated from time instant k to time k + τ i to describe the relationship between Lyapunov functions at time instant k to time k + τ i . In this case, the external disturbance terms in the system may bring some difficulties.

Examples
Example 1. Consider a DC motor system described in [37]. It is expressed by Definition 1.
The DC motor is driven by the traditional speed loop controller [6,40]. The state variables x 1 , x 2 represent the velocity and current of the DC motor, respectively. g(k) has two modes, which represent that the DC motors are working in two conditions with different loads. r(k) has three modes, which corresponds to (1)  We also assume that these two matrices are not known exactly. The accessible information about these two matrices are as follows: The switching signal g(k) and r(k) are shown in Figure 3. It can be seen that g(k) has an average dwell time τ d = 7 and r(k) is a stochastic process. Hence, the considered system contains both deterministic and stochastic dynamics.
In order to stabilize the DC motor system, the asynchronous controller (4) is utilized. Let α 1 = α 2 = α 3 = 1.1, β 1 = 0.92, β 2 = 0.90, β 3 = 0.95, λ 1 = 1.1, λ 2 = 1.2, λ 3 = 1.1, τ as = 2 in Theorem 1. It can be verified that for this parameters 6.5 = τ * d < τ d . Then, we can compute the mode-dependent control gain by solving the LMI in (19) and (20). The gray lines in Figures 4 and 5 represent the trajectories of x 1 (k), x 2 (k), x 3 (k), and u(k) for each simulation. The blue line is the average value of ||x(k)|| 2 and ||u(k)|| 2 for this 100 simulations which represent their expectation. It can be seen that both the expectation and trajectories converge to zero asymptotically. This verifies the validity of Theorem 2. We can also see that the controller has switched according to a different switching mode. The control effort is a piecewise signal and also finally converges to zero. This implies that the proposed controller can tolerate the mode transmission delay. In order to further show the effectiveness of the proposed method, we take the control gain from reference [11] where no delay and mismatch mode. In this case,    Example 2. Consider the system studied in [28], which has three distinct modes. This example can be used to represent some mechatronic system with possible failures in both structure and actuator. The three modes represent that the system suffers from different failures. The initial conditions are selected as x(0) = [1, 2] and r(0) is randomly generated from the set M. The switching signal r(k) and the sojourn time S n for each mode is shown in Figure 8. It can be seen that both the mode R n and the sojourn time S n are stochastic processes. This implies that r(k) is a semi-Markov jump process.
In order to stabilize the considered system, the asynchronous controller (7) is utilized. Note that Theorem 2 in fact provides a set of LMIs. By solving the LMIs, one can obtain the solution G j . Then, the controller can be determined by K j = K j G −1 j . We suppose that the controller suffers from a time varying mode transmission delay d(k). Let ρ 1 = ρ 2 = ρ 3 = 2, τ 1 = τ 2 = τ 3 = 3, τ 1 = τ 2 = τ 3 = 7, τ as = 2 in Theorem 2. Due to τ as ≤ τ i , (26) and (27)  The computed controller is utilized for the DC motor system. In total, 100 simulations have been conducted. As shown in Figure 9, all the trajectories have reached zero, including the state response x(k) for each simulation and E||x(k)|| 2 . This shows the effectiveness of the proposed method.  In order to stabilize the DC motor system, the asynchronous controller (7) is adopted. We suppose that the controller suffers from a time constant delay τ as = 2. Let ρ 1 = ρ 2 = ρ 3 = 2, τ 1 = τ 3 = 4, τ 1 = τ 3 = 7, τ 2 = 1, τ 2 = 5 in Theorem 2. Due to τ as > τ 2 , (33)- (35) are used for the set I

Conclusions
This paper focuses on the study of asynchronous stabilization of discrete time Markov jump systems. Two classes of typical Markov jump systems are considered, i.e., dual switching systems and semi-Markov jump systems. New stability criteria and numerically testable controller design methods are proposed for these two stochastic switching systems, which can well handle the asynchronous phenomenon. Future works may include extending the proposed results for more complex switched systems. Another interesting research line is considering the control of semi-Markov jump systems under cyber-attacks [41,42]. The additional attacks may further complex the structure of the controller, which is a challenging issue.
It follows that where x(0), r(0) are given constants. Therefore, (9) implies that where E[·] E[·]| x(0),r(0) . Then, during the time interval [k n , k n+1 ), we have where I s and I as denote the unions of time intervals that r(k) = r(k − d(k)) and r(k) = r(k − d(k)) respectively. |I s | and |I as | denote the total lengths of intervals I s and I as .
As shown in Figure 1, we first consider the case when r(k) = r(k − d(k)). Hence, we can compute the expectation in (9), Noting the asynchronous behavior described by (5), we have where P νj = ∑ l∈M π νjl P νl . According to whether or not the bounds of π νjl are known or not, (A2) can be expressed as Note that for l ∈ M K νi and l ∈ M UK νi , we have: Hence, we can conclude that It follows that (13)⇒(9) if r(k) = r(k − d(k)).
By the same reasoning, we have (13)⇒(9) if r(k) = r(k − d(k)). Meanwhile, it is obvious that (14)⇒(10). Hence, we show that if statement (i) is true, then the system (8) is mean square stable with dwell switching signal g(k) satisfying (12). This completes the proof.
(iii) ⇔ (ii). First note that (A3) is equivalent to the following matrix inequalities.
where ζ = k − k n . Meanwhile, let the system mode be k n−1 = i, k n = j, k n+1 = l with i, j, l ∈ M. The proof is divided into the following cases.