Abstract
We consider a class of stochastic nonlinear Markovian jump systems (MJSs) with partly unknown transition rate and time-varying delays. The system under consideration is subject to the mode uncertainties and nonlinear term and disturbance term which are unknown. The main mission is to design the observer-based sliding mode controller for such a complex system. An observer is first constructed, and then we design an integral sliding mode surface such that the MJSs satisfy the reaching condition. The sliding mode control law ensures the stochastic stability of the closed-loop system. Finally, an example is given to illustrate the proposed results.
1. Introduction
The sliding mode control (SMC) has attractive features to keep systems insensitive to the parameter uncertainties and external disturbances on the sliding mode plane [1]. Many practical dynamics, for example, manufacturing systems, chemical process systems, computer controlled systems and communication systems, solar receiver control, and power systems, experience abrupt random changes in their structure. These changes are usually caused by random failure or repairs of the components, changing in subsystems interconnections, sudden environmental changes, and so forth. Such systems can be modeled as hybrid systems. One special class of hybrid systems is named Markovian jump systems (MJSs), whose system mode is governed by a Markov process.
On the other hand, stability is one of the best topics in dynamic systems and a large number of results have appeared in the literature (e.g., see [2–28] and references therein). For MJSs, considerable attention has been recently paid to the problems of stability and stabilization [29–33], control and filtering [34–36], optimal tracking problem [37], etc. The robust stability and stabilization of uncertain stochastic systems with time-varying delays are investigated by using the linear matrix inequality approach (LMI) [38]. Huang and Mao [39] investigated the stabilization of stochastic linear systems by delayed state feedback controller. In [40], the method of Lyapunov functional is employed to study p-moment stability of nonlinear stochastic systems with impulsive and Markovian switching.
The SMC for stochastic systems has received an increasing attention. For example, see [41–51]. K. Chang and W. Chang [52] developd a SMC method to guarantee the robust state covariance assignment for perturbed stochastic multivariable systems via variable structure control. In [53], robust observer design for It’s stochastic time-delay systems was studied via sliding mode control and the sufficient conditions for the asymptotic stability (in probability) of the sliding motion were derived. Niu [54] dealed with the connection among the designed sliding surface corresponding to every mode for MJSs. However, there has been few results reported on the SMC of stochastic systems with Markovian switching. The existence of uncertainties, time-varying delays, Markovian switching with partly unknown transition probability will make the problem more complex and challenging.
Motivated by the above discussion, in this paper we investigate the problem of SMC problem for a class of stochastic nonlinear Markovian jump systems (MJSs) with partly unknown transition probabilities and time-varying delay. Firstly, an appropriate integral sliding mode surface is constructed. Secondly, the observer-based sliding mode controller is designed to guarantee the stochastic stability of the closed-loop system. Finally, a numerical example is provided to show the effectiveness of the proposed scheme.
The organization of this paper is given as follows. The system description and some preliminaries are given in Section 2. Section 3 presents the design of an integral sliding mode surface function and designs a sliding mode controller. Section 4 provides a numerical example to certify the feasibility of the mentioned method, and we conclude this paper in Section 5.
Notations. is a probability space, where denotes the sample space, represents the of subsets of the sample space and denotes the probability measure. means that is symmetric positive definite. denotes the identity matrix. refers to the Euclidean norm of a vector.
2. Preliminaries
In this section, we consider a class of stochastic nonlinear Markovian jump systems (MJSs) with partly unknown transition rates and time-varying delays on a complete probability space :
where represents the state vector, represents control input, is the nonlinear function, and is the system disturbance. denotes the measured output. , , , , and are the constant matrices and are the system uncertainties. is a standard Brownian motion defined on the complete probability spare . is time-varying delay and supposed to satisfy
where is known real constant scalar and is a continuous vector-valued initial function. represent the system mode, which take values in a given state space . Let represent the transition rate matrix. The mode transition probabilities can be denoted as follows:
where and , satisfies with , and for each mode . In addition, we assume that the probabilities of the Markovain jump system are partly unknown; for example, if there are four operation modes in system (1), the matrix of transition rates may be described as
where “?” denotes the unknown elements. We represent with and . We denote . We can rewrite MJSs for notational convenience when the system mode is :
where , , , , and .
Throughout this paper, we assume that the following conditions are satisfied.
Assumption 1. and satisfy the following form: where , , , and are the constant matrices with appropriate dimensions and , are unknown time-varying matrix function satisfying
Assumption 2. The external disturbance vector is an unknown function which satisfies where is an unknown scalar.
Assumption 3. has full column rank and the pair is controllable.
Assumption 4. The nonlinear function satisfies where is a constant scalar.
Obviously, under Assumptions 1–4, it follows from [55] that there exists a unique global solution for any initial data ,
For any given , we define an operator from to by
Next, we will gave a lemma.
Lemma 5 (see [43]). For any positive constant , , and , the following inequality holds:Finally, we give the definition of stochastic stability to end this section.
Definition 6 (see [56]). The stochastic system (1) is said to be stochastically stable when , if the condition satisfies where and is the initial mode.
3. Main Results
We first design a state observer to estimate unmeasured state components. Then we construct an integral sliding mode surface which is based on the state observer as the first step of SMC design. The state observer we synthesize is as follows:
where represents the estimation of , , is the observer gain matrix, and is chosen to reduce the effect of unknown nonlinear function and unknown external disturbance .
Remark 7. We can notice that the state observer not only contains state observation without time delays but also contains state observation with time delays; this can be a simplification of the controller.
The solution of the state observer can be expressed as follows:
Let be the estimation error:
and then we can obtain the following estimated dynamic:
Obviously, it follows from (16) that depends on . The integral sliding mode surface is constructed as follows:
where satisfies that is Hurwitz and is a constant matrix such that is nonsingular and . Combining (14) and (17), we have
According to SMC theory, when the system trajectory reaches the sliding surface, it follows that and .
Remark 8. The linear term is continuous in the integral sliding mode surface (17), the integral term is continuous, and then the integral sliding mode surface (17) is continuous when the MJS states switch.
Remark 9. It follows from (17) that the integral sliding mode surface depends on a specific set of matrices , , which means that switching effect from one state to another of the Markovian jump system is reflected on by the matrix . Thus, in the sliding mode control system accompanied by Markovian jump systems, the sliding mode surfaces establish contact with each other through .
By (18), we have
Based on the state observer , we design a new sliding mode control (SMC) law as follows:
where .
The control term is designed as
where .
Theorem 10. Assume that Assumptions 1–4 hold and the integral sliding mode controller is given by (18). Then, the reachabilty of the sliding mode surface can be satisfied under the action of the SMC law (20).
Proof. Taking then it follows from (2.14) thatSubstituting into , we get Thus, the reachabilty of the sliding mode surface can be satisfied.
The equivalent control law in the sliding mode can be obtained by solving as
Substituting into , then the state observer can be written as
The stability of system (1) can be determined by analyzing the stability of the closed-loop system which consists of (16) and (26).
The following theorem gives sufficient conditions for the stability of the closed-loop system, which is expressed in terms of linear matrix inequalities (LMIs).
Theorem 11. Considering system (16) and (26) with Assumptions 1–4, the sliding mode surface function is given by (18). If there exist positive definite symmetric matrix , matrices and , matrix , matrix , and positive scalars , , , and satisfy the following LMIs and the inequalities: where and then the overall closed-loop system composed of (16) and (26) is stochastically stable.
Proof. Take Then, According to (29), Using Assumption 1 and Lemma 5, we have where is the largest eigenvalue of .
Let , and then ,
where For , the term can be rewritten as follows: Then it follows from (28) and (32) that . For , , Then, combining (29), (32), and (33), we obtain .
Similarly, for , the term can be rewritten as follows: According to (28) and (34), . For , , Then, combining (29), (34), and (35), we get .
Hence, we obtain , , , and and and , which implies . Thus, we have which implies that the overall closed-loop system composed of (27) and (35) is stochastically stable.
Remark 12. Our result extends and improves those given in the previous literature. For example, if , then the system (1) is reduced to that in [36]; if , then the system (1) is reduced to that in [52]. Moreover, the transition rates in this paper are partially unknown, which is quite different from those in [36, 52].
Remark 13. When analyzing the stability of the stochastic system with Markovian switching and time-varying delay, Theorem 11 considers both state observer system and error system, which can conform the reality better and get better stability. When we construct the controller, because the system contains a Markovain stochastic term, we cannot directly apply the equivalent control method, so we construct the integral sliding surface and get our controller through the reachability of the integral sliding surface.
4. An Example
Let us consider system (1) above with the following matrix parameters:
The transition matrix is given as follows:
where “?” represents the unknown element. We select the matrices , , , and as follows:
Let , , and , The nonlinear function , external disturbance , and time-varying matrix function are given, respectively, as follows:
The initial conditions are given as and . Select the constant matrix . The adaptive laws are governed by (20); by solving the conditions (28)-(35), the observer is obtained as follows:
Then, the integral sliding mode surface in (18) for the 4 operating modes is synthesized as
Selecting , then we get
Finally, we use Figures 1–5 to show our results. In fact, Figure 1 shows the shows the trajectory of the Markov jumping signal with state 1,2,3. Figures 2–4 show state trajectory , state estimation trajectory , and state estimation error trajectory , respectively with state 1,2,3. Figure 5 shows the sliding mode surface function .
5. Conclusion
Because systems are inevitably influenced by stochastic factors, nonlinear factors, and time delay in the actual situation, discussing reachablity and stability of sliding mode control (SMC) in the context of stochastic system with time delay has more practical significance. This paper has investigated the SMC problem for stochastic nonlinear Markovian jump systems (MJSs) with partly unknown transition rates and time-varying delays. A SMC law has been synthesized such that the sliding mode surface can satisfy the reachable conditions. Sufficient conditions for the stability of the closed-loop system composed of the state observer system and error system have been proposed in terms of LMIs. A simulation result has been given to illustrate the feasibility of the results. Furture work will consider the SMC problem with Markov and impulse for nonlinear SDEs.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was jointly supported by the National Natural Science Foundation of China (61773217, 61374080), the Natural Science Foundation of Jiangsu Province (BK20161552), and the Construct Program of the Key Discipline in Hunan Province.