Restricted Stabilization of Markovian Jump Systems Based on a Period and Random Switching Controller

This paper considers the stabilization of continuous-time Markovian jump systems (MJSs) via a restricted controller. It is actually a period and random switching controller. It also contains some existing controllers as special ones. Sufficient conditions for existence of such a controller are established by studying a discrete-time MJS, which are presented in terms of LMIs and depend on its period and probability. Moreover, an extension about a similar but aperiodic controller is considered. Finally, a numerical example is used to demonstrate the effectiveness and superiority of the proposed methods.

In the above problems, stabilization is one of most important problems and could get better performance. By investigating the references about MJSs, it is found that most of them are mainly classified as three cases. The first kind of controller is a usual one and always referred to be mode-dependent. Because of operation mode available online and synchronous, it is the least conservative. This is also its drawback since the above assumption about operation mode is hard to be satisfied in applications. In order to The associate editor coordinating the review of this manuscript and approving it for publication was Usama Mir . remove this assumption, another mode-independent methods [30], [31] were proposed and has nothing to do with mode. Because it ignored operation mode totally even it is available sometimes, it is said to be an absolute approach. Recently, a kind of partially mode-dependent method was presented in [32] and bridged the above two cases, where a Bernoulli variable was introduced. By applying the polytopic uncertainty method to a controller, the fault-tolerant control of MJSs was considered in [34]. Though the above methods can be applied to nonmode-dependent cases, it is seen that the switchings of Bernoulli variable and polytopic uncertainty are fast even instantaneous. It is said that such a fast switching will lead to a higher cost even a damage to an equipment. In this case, it is natural to design a controller for an MJS which could sustain a period. A typical example is semi-Markov jump systems. Because of its sojourn time being any distribution, the corresponding switching will be slower than one of traditional MJSs. Very recently, the stability and stabilization of discrete-time semi-Markov jump linear systems subject to exponentially modulated periodic probability density function of sojourn time was considered in [35] and very important to make further research about semi-Markov jump systems. Particularly, necessary and sufficient VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ criterion about mean square stability was first developed. However, there are still many problems to be further studied which of them will be different in essence. For example, when the considered system is continuous-time, the stability criterion of discrete-time case based on the Lyapunov function approach will be disabled because of the switching signal right-continuous and belonging to any distribution instead of only exponential distribution. Moreover, when some general kinds of controllers such as ones mentioned above are considered, how to obtain the easily solvable conditions will be not easy but necessary and important. Thus, in order to overcome such as problems and difficulties, some new techniques for analyzing its stability and giving LMI conditions of such a generally stabilizing controller are necessary to be developed. By summarizing a large number of literatures, it is found that very few of them were considered about such a controller. All the aforementioned facts and observations motivate the current research.
In this paper, the stabilization problem of continuous-time MJSs is studied via a restricted controller. The main contributions of this paper are summarized as follows: 1) A kind of restricted controller in terms of period and random switching is proposed, which contains some existing controllers as special ones; 2) By studying a discrete-time MJS indirectly, sufficient linear matrix inequality conditions for the controller are presented, in which both period and conditional switching probabilities are included. It is shown that our results are less conservative in terms of having larger application scope; 3) Compared with some traditional controllers, the modes of original system and proposed controller are not necessary synchronous. Moreover, its switching is not so fast and has less damage to equipments; 4) More extension about an aperiodic controller is further considered to remove the assumption of constant period.
Notation: R n denotes the n-dimensional Euclidean space, R q×n is the set of all q × n real matrices. ( , F, P) is a complete probability space equipped with a filtration {F t : t ∈ R + } satisfying the usual hypotheses, that is a right-continuous filtration augmented by all null sets in the P-completion of F. Here, is the sample space, F is the σ -algebras of subsets of the sample space and P is the probability measure on F. E [·] denotes the expectation operator. · refers to the Euclidean vector norm or spectral matrix norm. N represents the set of natural number. λ min (M ) and λ max (M ) denote the smallest and largest eigenvalues of a symmetric matrix M . In symmetric block matrices, we use '' * as an ellipsis for the terms induced by symmetry, diag {· · ·} for a block-diagonal matrix, and (M ) M + M T .

II. PROBLEM FORMULATION
Consider a class of continuous-time MJSs defined on a complete probability space ( , F, P), it is described as followṡ where x(t) ∈ R n is the system state vector, and u(t) ∈ R m is the control input vector. Here, A(r t ) and B(r t ) are known matrices of compatible dimensions. Process {r t , t ≥ 0} is a homogeneous Markov process taking values in a finite set S {1, 2, . . . , N }. It is adapted to the filtration {F t : t ∈ R + } and is F t -measurable. It is a right-continuous trajectory and represents the switching among different modes. The evolution of Markov process {r t , t ≥ 0} with transition rate matrix = (λ ij ) ∈ R N ×N is governed by where λ ij denotes the transition rate from state i to state j, and λ ij ≥ 0, if i = j, and λ ii = − j =i λ ij , for all i, j ∈ S. As for system synthesis problems such as stabilization, the common controllers designed for system (1) could be summarized as follows: Mode-dependent controller [7], [8]: where K r t is the control gain and depends on operation mode r t all the time; Mode-independent controller [30], [31]: where K is the control gain and totally ignores r t ; Partially mode-dependent controller [32]: where both K r t and K are control gains and similar to the above ones, and α(t) is the Bernoulli variable and denotes the current operation mode available or not. Different from the above controllers, the state feedback controller in this paper is proposed to be where K [r t ] is the control gain to be determined but different for the above ones. The detailed construction of controller (6) is clearly stated. Fig. 1. Here, operation moder t is another switching signal and defined aŝ where τ is a constant and denotes the periodic dwell time ofr t . Its random switching on jump points is related to r t and defined as where operation mode r kτ is the value of r t at instant kτ such as r kτ = r t=kτ . Then it is said that the implementation of controller (6) has some restrictions such as (7) and (8). Firstly, switching signalr t is restricted to be a piecewise constant function, whose dwell time is not very small or instantaneous but τ . In this case, the fast switching among controllers could be avoided and lead to fewer damages to equipment of a controller. However, it is also mentioned that such a dwell time is constant and will be with some limitation. A more general assumption about τ is time-varying, which will be our further work. Secondly, it is defined that the sojourn time and switching number of operation mode r t = i on interval [kτ, (k + 1)τ ) are τ i and n i respectively. Because n i and τ i are very closed to r t = i, ∀t ∈ [kτ, (k + 1)τ ), it is reasonable that they are stochastic variables and with finite expectations. In this case, it is naturally assumed that N j=1 n j ≤ n max and τ i ∈ [τ min , τ max ], ∀i ∈ S, hold for any interval [kτ, (k + 1)τ ). Here, parameters n max , τ min and τ max are given in advance, but the preassumption are without loss of generality. Particularly, n max is a natural number, while τ min and τ max are positive real constants. This assumption is also reasonable in practice, since the switching of any equipment or system among modes should be finite. So, the accumulated sojourn time of each mode should also be with upper and lower bounds. For simple description, the variables such as x(kτ ) and r kτ are simply denoted as x(k) and r(k) respectively. Then, system (1) is equal to Remark 1: Compared with the above existing controllers, controller (6) is better in terms of having less constriction on current mode r t but doesn't neglect it at all. In other words, when the operation mode of controller (3) such as [7], [8], [12] is not accessible on time, it will be disabled while controller (6) is an effective choice. Moreover, more information about the correlation between modes r t andr t are further considered and will be less conservative than controller (4) referred in [30], [31]. Thirdly, but not the last, in contrast to controllers (3) and (5) having fast switchings even instantaneously in [32], [33], the switching of (6) is more slower though r t and α(t) are fast switchings. Such a slower switching will lead to less damage to equipment or system and have a wider application scope. More importantly, controller (6) could be specialized to be (3) and (4) respectively. However, it is worth mentioning that there are still some disadvantages of controller (6). One of them is that the jump points are periodic, and the dwell times are equal or constant. This assumption is ideal and will make the application with some limitations. Thus, more effort will be applied to deal with this problem.
Definition 1: System (9) or system (1) closed by controller (6)is said to be asymptotically mean square stable, if for initial conditions x 0 ∈ R n and r 0 ∈ S, there is Lemma 1: [36] For any real matrix A ∈ R n×n and positive-definite matrix P ∈ R n×n , if an arbitrary scalar ς is selected to be ς ≥ λ max (P) λ min (P) , it is always obtained that Particularly, by defining two pairs of conditions such as (a) λ min (P) could be removed.

Theorem 1: Suppose that there exists a scalar r [ ]
i . There is a controller (6) such that the closed-loop system (9) is asymptotically mean square stable, if for given scalars, satisfying either of the following conditions and The feedback gain of controller (3) is computed by Proof: Based on system (9) with condition (8), for any t ∈ [kτ, (k + 1)τ ) with any given r(k) = h, ∀h ∈ S, a discrete-time MJS is constructed to be According to the Kolmogorov differential equation, the transition probability matrix = (π hj ) ∈ R N ×N could be obtained that Its detail is described to be Then, a stochastic Lyapunov function of system (17) is constructed as follow Then, for any given where t 1 , t 2 , . . . , t q , are switching instants of r t on interval [kτ, (k + 1)τ ), and q is the switching number of r t on the same interval. For any real matrixĀ [ ] There, parameters α [ ] i and β [ ] i are computed by where Then, formula (21) is further obtained that By condition (14), it is known that S [ ] is nonsingular. Then, it is concluded from formula (16) (13), it is easy to get that which is further obtained that Then, inequality (26) could be implied by condition (15). As a result, one concludes that lim k→∞ E x(k) 2 |x 0 , r 0 = 0.
At the same time, for any t ∈ [kτ, (k + 1)τ ), it is known that where } is a value with limited bound. It is further obtained that Then, the closed-loop system (1) is asymptotically mean square stable. This completes the proof. Remark 2: For continuous-time MJSs, it is said that the time-scheduled Lyapunov function method proposed for LTI control systems [38] is not suitable. The main reason is that nonlinear term eĀ [ ] i t has NM modes and is very closed to stabilizing controller, which cannot be handled by the above method. In other words, when stabilization problems are considered, solvable conditions with easy computation forms such as LMI conditions are not easily obtained. All these facts will make the analysis and synthesis of system (9) difficult, and novel methods should be developed. To the contrary, based on the proposed methods, convex conditions for the existence of controller (6) are obtained and computed easily. Moreover, our results will include the deterministic case as a special one. Thus, the obtained results can be viewed as extension results on stabilization by a controller with failures from deterministic systems to stochastic systems.

Remark 3:
As for the conditions in this theorem, some additional explanations are necessary given in the following. Firstly, all the conditions are given in terms of LMIs and could be solved directly and easily. However, the complexity of computation will be larger, especially M and N becomes very large. There will be (N + 1)M + N variables to be computed, while N 2 + (3M + 1)N inequalities are needed to solved; Secondly, more information about probability (8), parameters n max , τ min and τ max are taken into account and could further demonstrate theirs effect on system analysis and synthesis. On the other hand, there is also an unavoidable problem that parameter n max related to max i∈S { } plays a large negative effect in terms of making condition (15) having smaller region of solvable solution. This phenomenon results from the switching property of r t on interval [kτ, (k + 1)τ ) and is inevitable. Fortunately, one could reduce this effect by selecting suitable values of r [ ] j , τ min and τ max . However, how to further reduce this negative effect and obtain less conservative results with smaller computation complexity are not easy and will be our further work.
Since the jump points of controller (6) are periodic, another type of jump system is considered and described aṡ where {η t , t ≥ 0} is a jump process and takes values in a finite set S {1, 2, . . . , N }, x(t) ∈ R n is the system state vector, u(t) ∈ R m is the control input vector, and A(η t ) and B(η t ) are known matrices of compatible dimensions. Different from Markovian switching r t , η t jumps randomly. As a result, the instant of switching is random, whose dwell time of a given mode is also random and may be not an exponential distribution. Without loss of generality, the jump points are denoted as 0 = t 0 < t 1 < · · · < t k < t k+1 < · · · , k ∈ N. Here, τ (k) denotes the dwell time of a given switching η t and is defined as τ (k) t k+1 − t k . It is stated above that η t is different from r t and not a Markov signal. In detail, dwell time τ (k) is time-varying and random but without any statistical property. The switching among modes is different from (2) and described as where θ hj denotes the switching probability from state h to state j, and θ hj ∈ [0, 1], if h = j, and θ hh ≡ 0. Moreover, it should also be satisfied that N j=1 θ hj = 1, for all h, j ∈ S. In other words, the jump happening in the next time should be changed to another different mode. Without loss of generality, for any given interval [t k , t k+1 ), its operation mode is assumed to be ∀η t = h ∈ S, ∀t ∈ [t k , t k+1 ). Thus, system (32) with above description ∀η t = h ∈ S, ∀t ∈ [t k , t k+1 ) becomes to Similarly, the state feedback controller in this section is also proposed to be where K [η t ] is the control gain to be determined. Andη t is another switching signal and defined aŝ Its value is related to η t and defined as Since τ (k) is time-varying and random, it is naturally assumed that τ (k) ∈ [τ min , τ max ], where 0 < τ min < τ max .
Similarly, variables such as x(kτ ) and η kτ are simply denoted as x(k) and η(k) respectively. Remark 4: By investigating models (1) and (32), it is found that the main difference is about the property of switching signal. In the former one, r t is a traditional Markov process, and its dwell time belongs to an exponential distribution. The latter η t is only a switching one, whose dwell time is arbitrary. Moreover, the current operation mode must change to another different one at jump points. It may be seen as a semi-Markovian process. The reason considering such a system is to remove periodic interval [kτ, (k + 1)τ ), ∀k ∈ N. In this case, the switching of controller (35) is naturally not instantaneous, and only conditional probability (37) is needed considered. However, it is not said that model (1) could be removed. The reason is either of them cannot be totally included or specialized as another one, though some aspects in one model are more general than ones in the other one.  (11) or (12), (13), (14), and Then, the gain of controller (35) could be obtained by (16).
Proof: Based on system (34) closed by controller (35), for any t ∈ [t k , t k+1 ) with any given η(k) = h, ∀h ∈ S, a discrete-time jump system is constructed to be . A stochastic Lyapunov function of system (39) is constructed to be Then, it is computed that Based on inequality (22) implied by conditions (13) and (14), it is further computed that which is guaranteed by (38). The next steps are similar to the ones in Theorem 1, which are omitted here. This completes the proof.
Remark 5: Compared with conditions in Theorems 1 and 2, only conditions (15) and (38) are different. Such a difference is totally determined by the considered different systems. It seems that the latter one could lead to a higher probability about solvable solutions. This doesn't say that Theorem 2 is better than Theorem 1. The reason is that the considered problems between them are different, and no any additional jumps in interval [t k , t k+1 ) happen in the latter. Thus, condition (38) could easily obtain solvable solutions. How to obtain more general conditions containing them both is not easy and will be our future research topics.
Based on the above analysis, an algorithm for Markovian jump systems with a period and random switching controller is presented to solve this problem.
Computation Algorithm: Step 1: For system (9) with given r i , i ∈ S, ∈ T, and determine setsN + ,N − and maximum iteration time k max .
Step 4: If there are solvable solutions, compute control gains by (16), and exit; Otherwise go to Step 5, and set k = k + 1; i , while increasing ς or not depends on which one of conditions (11) and (12) is used, and go to Step 3; Otherwise exit. It means that there is no solvable solution to controller (6) for system (9) with given r i , i ∈ S, ∈ T. In order to obtain solutions, one could select much more smaller values of r i such as r i < 0, i ∈ S, ∈ T. Then, repeat this process from Step 1.

Example 1:
Consider an VTOL helicopter model partly cited from [39]. Its form is described as (1), whose parameters are given to be By the traditional methods for designing a mode-dependent controller such as [7], [8], [12], one could get the corresponding gains as For this kind of controller, it is said that it is ideal sometimes, since its operation mode should be available online. In other word, when its mode experiences general case such as (7), the desired controller may be disabled. Without loss of generality, it is assumed here that stochastic processr(t) has three modes such asr(t) ∈ M {1, 2, 3}. Meanwhile, the other parameters are given as τ = 20, τ min = 10 and n max = 2. The is computed to  (7), one could get the corresponding simulation of the closed-loop system given in Fig. 2, while Fig. 3 is the simulation  of operation modes r t andr t . Based on these simulations, it is seen that the closed-loop system becomes unstable when controller experiences condition (7). In other words, general condition (7) plays a negative effect in terms of reducing system performance even making the stable system unstable. At the same time, one could design controller (6)   Under the same conditions and after applying the above controller, one could get the simulation of closed-loop system given in Fig. 4. It is obvious that it is stable and demonstrates the utility of the proposed method. On the other hand, one could also design controller (35) with conditions (36) and (37) for a jump system described by (33) and (34). Jump parameters η t andη t take values in sets S and M respectively. The transition probability of (θ hj ) ∈ R 3×3 is given as   Under the same initial condition, one could also obtain the simulations of the resulting closed-loop system. Particularly, Fig. 5 is the simulation of operations modes satisfying (33), (36) and (37). From Fig. 6, it is seen that the designed controller is still effective since the states of closed-loop system are stable. Based on such simulations and comparisons, it is said that our methods are less conservative that they can be used to more cases in terms of general operation mode of controller.

V. CONCLUSIONS
In this paper, the stabilization problem of continuous-time Markovian jump systems has been investigated by a restrict controller. Different from the existing ones, the main restriction about the controller is that the dwell time of each controller is period, whose switching signal is a piecewise continuous function. Moreover, the switching of controllers at jump points is random and conditionally dependent on the original mode at such jump points. By studying a discrete-time MJS indirectly, the existence conditions have been with LMI forms and related to period and conditional probability. Then, the proposed model and method have been applied to propose an aperiodic controller. The utility and advantage of the established results have been proved by a numerical example. Finally, it is said that there are still many problems to be considered. Firstly, because of lots of LMIs and variables to be solved, how to further reduce the complexity is another important problem; Secondly, it is seen that in order to obtain solvable conditions, some enlarged inequalities have been introduced, which also bring some conservatism. How to further reduce the conservatism is necessary to be considered; Thirdly, some extensions about such models could be applied to describe other problems directly, such as filtering, observer design, and fault detection.
Fourthly, but not the last, it is worth mentioning that there are still some disadvantages of the proposed controller. One of them is that the jump points are periodic, and the dwell times are equal or constant. This assumption is ideal and will make the application with some limitations. Thus, more effort will be applied to deal with this problem. In a word, all the observations are necessary studied, and some of them may be not easy.