Abstract
This paper deals with the problem of mixed / control for Itô-type stochastic time-delay systems. First, the / control problem for stochastic time-delay systems is presented, which considers the mean square stability, control performance index, and the ability of disturbance attenuation of the closed-loop systems. Second, by choosing an appropriate Lyapunov–Krasoviskii functional and using matrix inequality technique, some sufficient conditions for the existence of state feedback / controller for stochastic time-delay systems are obtained in the form of linear matrix inequalities. Third, two convex optimization problems with linear matrix inequality constraints are formulated to design the optimal mixed / controller which minimizes the guaranteed cost of the closed-loop systems with known and unknown initial functions, and the corresponding algorithm is given to optimize / performance index. Finally, a numerical example is employed to show the effectiveness and feasibility of the proposed method.
1. Introduction
Over the past decades, there has been a rapid increase of interest in the study of stochastic systems due to the importance of stochastic models in science and engineering, such as finance systems [1] and power systems [2]. And a lot of excellent results have been obtained. For example, Zhu et al. [3] investigated the tracking control issue of stochastic systems subject to time-varying full state constraints and input saturation. In [4], the stability of a class of discrete-time stochastic nonlinear systems with external disturbances was considered. The finite-time tracking control of a class of stochastic quantized nonlinear systems was studied in [5]. Furthermore, since stochasticity and time delay are the main sources resulting in the complexities of systems in reality, considerable interests have been focused on a general model of stochastic time-delay systems. For example, the problem of guaranteed cost robust stable control was considered via state feedback for a class of uncertain stochastic systems with time-varying delay in [6]. In [7], the mean square exponential stability of neutral-type linear stochastic time-delay systems with three different delays by using the Lyapunov–Krasovskii functionals was studied. In [8], the finite-time dissipative control for stochastic interval systems with time delay and Markovian switching was investigated. Some other nice results can be referred to [9–17] and the references therein.
At present, control has been receiving increased attention because it can suppress external interference, and many efforts have been devoted to extending the results for control over the last few decades. For instance, Ma and Liu [18] investigated the finite-time control problem for singular Markovian jump system with actuator fault through the sliding mode control approach. In [19], the problem of nonfragile observer-based control for stochastic time-delay systems was considered. The problems of robust stabilization and robust control with maximal decay rate were investigated for discrete-time stochastic systems with time-varying norm-bounded parameter uncertainties in [20]. Some other nice results can be referred to [21–26]. On the contrary, control is an effective way to attenuate the disturbance, while control can guarantee quadratic performance cost. By combining control and control theory, the mixed / control theory is obtained. Owing to the fact that the mixed control can minimize a desired control performance and eliminate the effect of disturbance, it is more attractive than the sole control in engineering practice. For example, Gao et al. [12] investigated the problem of / control for nonlinear stochastic systems with time-delay and state-dependent noise. / control problem of stochastic systems with random jumps was solved in [27]. Sathananthan et al. [28] studied guaranteed cost control of linear stochastic Markovian switching systems. Although the problem of / control has been investigated, there are few literature studies on Itô-type stochastic time-delay systems.
Motivated by the abovementioned discussions, in this work, we aim to investigate the mixed / control for Itô-type stochastic time-delay systems. It is difficult to design state feedback / controller because of the complicated structure of the system. The main contributions of this paper are as follows. (i) The definition of / control for Itô-type stochastic time-delay systems is presented, which considers stability, control performance index, and control performance index. (ii) The new sufficient conditions for the existence of state feedback / controller are provided in the form of linear matrix inequalities. (iii) An algorithm is given to optimize / performance index.
The organization of this paper is as follows. Section 2 is devoted to the problem statement, preliminaries, and lemmas. Section 3 provides the sufficient conditions for the existence of state feedback / controller for Itô-type stochastic time-delay systems. Section 4 gives an algorithm to solve the theorems. Section 5 presents a numerical example to demonstrate the effectiveness of the proposed method. Section 6 is our conclusions.
Notations: denotes the transpose of matrix ; indicates the trace of matrix ; indicates that is a positive definite (positive semidefinite) matrix; represents a -dimensional identity matrix; shows -dimensional Euclidean space; represents the mathematical expectation of random process; and the asterisk “∗” in the matrix indicates symmetry term.
2. Preliminaries
Consider the following Itô-type stochastic time-delay system described bywhere is the state of the system, is the control input, is the control output, is the initial state function, and is a one-dimensional standard Wiener process defined on probability space (, , , P). stands for the smallest -algebra generated by , , i.e., . is the time delay. , , , , , , , and are constant matrices with appropriate dimensions.
Next, a new definition of the mean square stability for system (1) is given.
Definition 1. System (1) ( and ) is said to be mean square stable, ifThen, some lemmas for obtaining the main results are introduced.
Lemma 1 (see [29]). Let be a scalar function, and , for the following stochastic system:The Itô formula of is given as follows:where
Lemma 2 (see [30]). For given , , , and , then we have
Lemma 3 (see [6]). For some real matrices the following three conditions are equivalent:
3. Mixed / Control for Stochastic Time-Delay Systems
In this section, a state feedback / controller will be designed.
We consider a state feedback controller for system (1) iswhere is the state feedback gain to be determined.
The closed-loop system can be obtained by substituting (2) into (1):
Associated with system (1), the cost function is provided as follows:where and are the given positive scalars or given weighting matrices.
By substituting (2) into (4), we can obtain
Based on the above analysis, the problem of / control for stochastic time-delay systems is provided as follows.
Definition 2. For a given scalar , if there exist a positive scalar and a state feedback controller (2) such that(i)The closed-loop system (3) is asymptotically stable in mean square sense.(ii) cost function (5) satisfies under the condition of .(iii)For any nonzero disturbance , the control output satisfies the following inequality with zero initial condition:then (2) is said to be a state feedback / controller for system (1).
The sufficient conditions for the existence of the state feedback / controller (2) are given below. For this reason, an important lemma is first given.
Lemma 4. For a given scalar and two symmetric positive definite matrices and , if there are two symmetric positive definite matrices and such thathold, where , then (2) is a mixed / controller of system (3), and the corresponding guaranteed cost for system (3) is .
Proof. The following proof is divided into three parts. First, it is proved that the closed-loop system (3) is mean square stable.
According to Lemma 3, condition (7) impliesDue to , , and , we can obtain , , and ; then, (8) implieswhere .
Let a quadratic function and the differential generation operator of system (3) be with ; then,that is,where .
In the light of (9), we can derive that , that is, the closed-loop system (3) is asymptotically stable in mean square sense.
Secondly, we prove that the control output satisfies index for any nonzero disturbance under zero initial condition.
According to (7), , and , we can obtainwhere .
Notice thatwhere is the infinitesimal operator of system (3) for any nonzero disturbance , andThen, we can see thatwhere .
Based on (12), we can see , that is, (12) implies that . Therefore, system (3) satisfies index.
Thirdly, we prove that system (3) satisfies index under the condition of .
Based on (13) and (14), , and , we obtain thatholds, where .
Due towhere .
In view of (22), we obtainAccording to (23) and (24), we can seeThe proof is completed here.
In order to solve the complex problem to seek the solution caused by the nonlinear terms in Lemma 4, we give the following Lemma 5.
Lemma 5. For a given scalar and two symmetric positive definite matrices and , if there are two symmetric positive definite matrices and and a matrix such thathold, where , , and ; then, (8) is a mixed controller of system (9), and the corresponding guaranteed cost for system (9) is . In this case, .
Proof. According to Lemma 2 and (13), if the following inequalityhold, where , then (13) holds.
Using diag to premultiply and postmultiply inequality (27), we havehold, where . Let , , , , and ; by Lemma 3, we obtain (26) from (28).
Summarizing the process, the proof is completed.
Next, in order to get the least upper bound for cost function among all the possible solutions to inequality (26), the convex optimization problem is provided as follows.
Theorem 1. For system (9), if the following optimization problemsubject to (26) andhas a solution , , , , and , then controller is an optimal state feedback / controller which ensures the minimization of guaranteed cost for system (9), where .
Proof. From Lemma 5, the controller is a guaranteed cost control law of system (9). (30) is equivalent to ; (31) is equivalent to .
Therefore, we can obtainThus, we can obtain .
Therefore, the minimization of implies the minimization of guaranteed cost for system (9).
The proof is completed here.
Remark 1. It is an ideal case that the initial function is known. However, in general, the initial function of system (1) is not known, but the guaranteed cost depends on it. In order to avoid the dependence, we assume that the initial function is a white noise process with zero expectation function and unit covariance function.
When the initial function is not known, we haveTherefore, we have the following optimization problem:which subjects to (26) and
Theorem 2. If there exist solution to (26), (34)–(36) then controller is an optimal state feedback / controller which ensures the minimization of guaranteed cost (18) for system (1).
Proof. From Lemma 5, the controller is a / controller of system (9). We can see (34) is equivalent to and (35) is equivalent to from Lemma 3. Therefore, the minimization of implies the minimization of the guaranteed cost for system (1).
The proof is complete.
4. Numerical Algorithms
In this section, an algorithm is presented in order to find the minimum value of in Theorem 1. The similar algorithm can also be applied to Theorem 2.
By analyzing (26), (30), (31) in Theorem 1, we find that if (26), (30), (31) have no feasible solutions when takes the initial value, then (26), (30), (31) will have no feasible solutions for all . Next, we search for from the initial value that makes (26), (30), (31) have feasible solutions to optimize by using linear search algorithm. The specific algorithm is as follows.
5. Numerical Examples
The coefficient matrices of system (1) are given as follows:
First case: when the initial function is known and , . In order to find the minimum value of , we obtain the relationship between and by Algorithm 1, which is shown in Figure 1.
|
As can be seen from Figure 1, decreases with the increase of , and when , and when .
Take , according to Theorem 1, we obtain that
Therefore, the optimal state feedback / controller is , and the guaranteed cost of closed-loop system is .
Take external disturbance , then we can obtain the curves of and and in Figure 2. From Figure 2, we can see that and , that is, closed-loop system (9) is mean square stable.
Second case: when the initial function is a white noise process with zero expectation function and unit covariance function, in order to find the minimum value of , we obtain the relationship between and by Algorithm 1, which is shown in Figure 3.
As can be seen from Figure 3, decreases with the increase of , and when , when .
Take , according to Theorem 2, we obtain that
Therefore, the optimal state feedback / controller is , and the guaranteed cost of closed-loop system is .
6. Conclusion
In this paper, the mixed / control problem for Itô-type stochastic time-delay systems is presented, and the description of / control problem for stochastic time-delay systems is given. On the basis of matrix transformation and convex optimization method, state feedback / controller is obtained to make the system satisfy performance index and performance index. Moreover, an algorithm is given to solve state feedback controller and optimize / performance index. Finally, a numerical example is used to show the feasibility of the results. In the future work, we will investigate mixed control for the more complex systems, such as, stochastic Markov jump systems with time delay.
Data Availability
The data used to support the findings of this study are available from the corresponding upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 61877062 and 61977043), China Postdoctoral Science Foundation (Grant no. 2017M610425), and Open Foundation of Key Laboratory of Pulp and Paper Science and Technology of Ministry of Education of China (Grant no. KF201419).