Output Regulation of Switched Stochastic Systems with Sampled-Data Control

Dong, Xiaoxiao; Lan, Hui

doi:https://doi.org/10.1155/2023/9549909

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 9549909 | https://doi.org/10.1155/2023/9549909

Output Regulation of Switched Stochastic Systems with Sampled-Data Control

Xiaoxiao Dong¹and Hui Lan¹

Academic Editor: Ivan Giorgio

Received28 Oct 2022

Revised24 Feb 2023

Accepted27 Feb 2023

Published18 Mar 2023

Abstract

This paper studies the output regulation problem for a class of switched stochastic systems with sampled-data control. Solutions to the output regulation problem are given in two situations. On the one hand, the exogenous signal is assumed to be a constant. By designing a sampled-data state feedback controller, we obtain that the closed-loop system is mean-square exponentially stable and the regulation output tends to zero. On the other hand, the exogenous signal is assumed to be time-varying with bounded derivative. By constructing a class of Lyapunov-Krasovskii functional and a switching rule which satisfies the average dwell time, sufficient conditions for the solvability of practical output regulation problem are given for switched stochastic systems. Finally, numerical examples are given to illustrate the effectiveness of the method.

1. Introduction

Switched system is a typical class of hybrid system, which is composed of a series of subsystems and a switching rule which determines the active subsystem [1]. These systems have been applied to many practical engineering problems, such as flight control [2], communication engineering [3, 4], and network control [5, 6]. Therefore, switched systems and switching control are paid more and more attention [7–10]. As is well known, the random factors may result in the performance degeneration of practical systems. Consequently, it is very important to discuss the influence of random factors on the systems and there are many achievements about switched systems with random factors [11]. The related research results of switched stochastic systems have also been widely used in water quality engineering [12], industrial process control, biomedical engineering, and so on [13–18], although the switched system with random factors is more complex than the general switched system.

Output regulation problem (ORP) is aimed at designing the controllers to achieve the asymptotic tracking and disturbance rejection for reference signals. A majority of existing literature on output regulation focus on non-switched systems [19–21]. Nevertheless, the results on the ORP of switched systems are still few. Wang et al. have constructed a class of multiple linear Lyapunov functions and solved the ORP of positive switched systems by designing two kinds of controllers [22]. Reference [23] discusses the solvability of the ORP for discrete switched systems, and in order to utilize the communication network effectively, the designed feedback controller adopts the quantized output, and the solution to the problem is given by different coordinate transformations. There are still many problems of output regulation of switched systems to be studied.

With the development of digital control technology, many practical control systems adopt sampled-data control. The controller designed based on the time sampled-data is usually composed of time-triggered continuous step signal samplers and zero-order hold devices; thus, it can be directly implemented on the digital platform. The application of sampled-data control can improve the efficiency of system control and the utilization of the controller. In addition, the time-based sampling control will not trigger infinite times in a limited time interval. Therefore, Zeno phenomenon does not occur, and the controlled system under this kind of control strategy has the advantages of efficient execution and predictability, which meets the requirements of the practical engineering field [24]. In [25], the stability of the closed-loop system is analyzed directly, and the ORP of the linear system under aperiodic sampled-data control is solved. By extending the Halanay inequality to time-varying sampled-data systems, Zhang et al. obtained sufficient conditions for the stability of the closed-loop system [26]. Due to the wide application of switched systems, many scholars extend the relevant results of nonswitched systems under sampled-data control to switched systems. By constructing a new class of nonpositive definite functionals and giving a sampled-data feedback controller, Lian et al. obtained exponential stability of linear switched systems under asynchronous switching [27]. In [28], Fu et al. designed a sampled-data controller by using a fixed sampling period to ensure the stability of a class of switched neutral systems under asynchronous switching. The research results of ORP based on sampled-data have also been widely used in intelligent control fields such as mobile robots [29]. By virtue of the above analysis, the research on the ORP of switched stochastic systems under sampled-data control is essential. However, up to now, the results about this problem are not available.

This paper studies the ORP of a class of switched stochastic systems under a periodic sampling mechanism. For the exogenous signal generated by the exosystem, the constant signal and the time-varying signal with derivative bounded are considered, respectively. The main contributions are listed as follows: (1) the output regulation problem of stochastic switched systems is considered. In fact, even if the output regulation problem of each subsystem of the switched system is solvable, the output regulation problem of the switched system may not be solvable under any switching signal. Therefore, it is necessary to give the relationship between the average dwell time (i.e., the conditions to be satisfied by the switching signal) and the sampling period to obtain the solvable conditions of the output regulation problem of switched stochastic systems. This paper gives the relationship between the average dwell time and the sampling period. Meanwhile, this paper considers the asynchronous switching between the subsystem and the controller of the switched stochastic systems. (2) The sampled-data controller designed by time trigger is used to solve the output regulation problem of switched stochastic systems for the first time. The sampling mechanism applied in this paper will not trigger infinitely in a finite interval; that is, Zeno phenomenon will not occur. In addition, this paper also considers both output regulation and practical output regulation and gives sufficient conditions for the solvability of the problem.

2. Problem Formulation and Preliminaries

Consider the following switched stochastic system described by and the exosystem is described by where is the state vector of the system, is called the exogenous signal, represents the regulated output, is the switching signal, and it is a class of the piecewise right continuous constant value functions, and is control input. Brownian motion with -dimensional satisfies and . , denotes that the th subsystem is active. , , , , , and are constant matrices; the nonlinear coefficient satisfies the local Lipschitzness and the linear growth condition. Suppose that the matrix has nonnegative real parts and the pair is stable.

The sampled-data state feedback controllers are designed as follows: where and , , are constant matrices with suitable dimensions.

Then, the closed-loop system composed of the system (1) and the controller (3) is represented as

Let and ; the formula (4) can be rewritten as where , , , and .

The main results are based on the following assumptions:

Assumption 1. (see [25]).
There exists a matrix with appropriate dimension, satisfying the matrix equations According to equation (6), we have the coordinate transformation ; then, the closed-loop system (5) is transformed into the following form:

Assumption 2. (see [11]).
For nonlinear functions , there exist the constant matrices , such that

Definition 3. (see [30]).
Switched stochastic system (1) is said to be exponentially stable in mean square, if there exist and , such that

Definition 4. (see [27]).
If there exist and , such that where the notation denotes the number of switches in the interval , ; then, is called the average dwell time (ADT). The interval between any two switches is at least dwell time .
Next, the ORP of the switched stochastic system (1) based on sampled data is described as follows.
By designing an appropriate switching signal and sampled-data feedback controllers, such that (i)when , the regulated output of the closed-loop system (7) satisfies i.e., the ORP of the closed-loop system is solvable(ii)if there exists a constant and for any , when , the regulated output of the closed-loop system (7) satisfies i.e., the PORP of the closed-loop system is solvable

We considered the asynchronous switching phenomenon caused by the mismatch between the sampling instants and the switching instants; this phenomenon will last until the next sampling instant. denote the sequence of sampling instants; denote the sequence of switching instants. Assume that the sampling period is fixed. To ensure that there is at most one switch in each sampling interval, the period satisfies .

In the following sections, we will give the sufficient conditions for the solvability of the output regulation problems of the closed-loop system (7).

3. The Situation of

This section gives a solution to the ORP of the closed-loop system when the exogenous signal is constant.

Theorem 5. When , the system (1) which satisfies Assumptions 1 and 2 is considered. For given positive constants , , , , , and , if there exist positive definite matrices , , , , and and appropriate dimension matrices , , , and , , , such that where then the ORP of the closed-loop system (4) is solvable for any switching signal which satisfies ADT defined by where , , the matrix is composed of , , and the matrix is composed of , .

Proof. Since , thus, we get , , . And according to the previous coordinate transformation and , system (7) becomes where and .
Let and . When the th subsystem is active, the closed-loop system (18) can be rewritten as (i)Synchronous switching intervalAssume that the subsystem is active on the interval , and it is synchronous with the controller on the whole interval, i.e., . For system (19), construct the L-K functional candidates as follows: where , , and are positive definite symmetric matrices. From Itô’s differential formula, we get the stochastic differential with According to the Newton-Leibniz formula, the following equation is true: where and are positive definite matrices with suitable dimension. According to Assumption 2 and condition (15), there exist and constant matrices , such that By combining (22)–(24), we have Consider the arbitrary matrices which satisfy the following inequality:

By combining(23)–(27) and Schur complement lemma, we obtain where

For the inequality (13), we let , , , , , , and , . Multiply on both sides of inequality , and multiply both sides of inequality by . Then, formula (28) becomes

Thus, we have

Combining (21) and (31), we can derive that

Integrate both sides of (32) from to , and take expectation (ii)Asynchronous switching interval

For , suppose that a switch has occurred at instant , , and it satisfies . Then, the subsystem is active on the interval ; the subsystem is active on . When , we obtain the result from (i).

When , the closed-loop system has the following forms: where . For system (35), construct the L-K functional candidates as follows: where , , and are positive definite symmetric matrices. From Itô’s differential formula, we get with

According to the Newton-Leibniz formula, we have where and are positive definite matrices with suitable dimension. And there exist and constant matrices , such that

From (38) and (40), we get

Consider the arbitrary matrices which satisfies

By combining(41)–(43) and the Schur complement lemma, we have where

For the inequality (14), we let , , , , , and , . Multiply on both sides of inequality , and multiply both sides of inequality by . Then, formula (44) becomes

Thus, we have

Combining (21) and (47), we can derive that

Integrate both sides of (48) from to , and take expectation

For the entire interval , combining (15), (34), and (49), we have

According to the inequality (15), (50), and Definition 4, we can obtain where . According to the constructed L-K functional, we get where and . From (51) and (52), we have

Under condition (17), the above formula means that

Remark 6. When , the switched stochastic system (1) is simplified as Based on the result (53), we obtain i.e., the system (55) is mean-square exponentially stable.

4. The Situation of

For the case that the exogenous signal is time-varying with bounded derivative, this section gives the solution to the PORP of the closed-loop system. (i)Synchronous switching interval

Theorem 7. If there exists a constant , such that , , consider the closed-loop system (1) which satisfies Assumptions 1 and 2 and the following conditions: (1)For given positive constants , , , , , , , , and any , if there exist positive definite matrices , , , and , and , such that where (2)If there exists the matrix satisfying the matrix equations (6),then the PORP of the closed-loop system (4) is solvable for any switching signal which satisfies ADT defined by where and . And the matrix is composed of , ; the matrix is composed of ,

Proof. When , the closed-loop system has the following form: where . Let , . When the th subsystem is activated, (62) is equivalent to the following form:

Also, assume that the subsystem is activated on the interval . For the system (63), construct the L-K functional candidate as follows: where , , and are positive definite symmetric matrices. Besed on Itô’s differential formula and (64), we get the stochastic differential

According to the Newton-Leibniz formula, we have the following equations: where , , and are positive definite matrices with suitable dimension. By virtue of Assumption 2 and condition (59), there exist and constant matrices , such that

By combining (64)–(67), we have

Consider the arbitrary matrices which satisfies

By combining (68)–(70)) and the Schur complement lemma, we obtain where

For the inequality (57), we let , , , , , , , , , and . Multiply on both sides of inequality , and multiply both sides of inequality by . Then, formula (71) becomes

Since , thus, we have

Combining the above formula with (73), we have where . From (65) and (75), we can derive that

Integrate both sides of (76) from to , and take expectation (ii)Asynchronous switching interval

For , we also suppose that one switch has occurred at the instant and asynchronous switching occurs on the interval . When , we obtain the result from (i).

When , the closed-loop system has the following form: where , , and . For the system (79), construct the L-K functional candidate as follows: where , , and are positive definite symmetric matrices. Besed on Itô’s differential formula and (81), we get the stochastic differential

According to the Newton-Leibniz formula, we have where , , and are positive definite matrices with suitable dimension. And there exist and constant matrices , such that

From (81)–(84), we get

Consider the arbitrary matrices which satisfies

By combining(85)–(87) yields and the Schur complement lemma, we have where

For the inequality (58), we let , and . Multiply on both sides of inequality , and multiply both sides of inequality by . Then, formula (88) becomes

Since , formula (90) becomes where . Combining (82) and (91), we can derive that

Integrate both sides of (93) from to , and take expectation

For the entire interval and , combining (59), (78), and (93), we have where the notation represents the total time for asynchronous occurrence on the interval . And from (59) and (94), we get

Based on Definition 4, the following results hold: where and .

According to the constructed L-K functional, we obtain that where , , , and .

According to the ADT (61), we get . Thus, we have

Then, based on condition (97), for given any , letting from (97)–(99), we have

Then, the regulated output satisfies

Therefore, the PORP of the switched stochastic system can be solved.

5. Simulation

In this section, we will give two numerical examples corresponding to Theorem 5 and Theorem 7, which verify the effectiveness of the method.

Example 1. Consider the switched stochastic system (1) with two subsystems, where and and the matrix which satisfies Assumption 1 as follows:

Let , , , , and . From (17), we know that the minimum average dwell time is ; thus, we construct a switching strategy which satisfies . By solving the inequalities (13)–(15) in Theorem 5, we have

Under the initial state and , we obtain the state responses of the closed-loop system (18) shown in Figure 1, and the regulated output tends to zero which is shown in Figure 2. Figure 3 describes the time-triggered instants and shows that it only triggered 20 times in 10 seconds. Compared with continuous sampling, fixed-time sampling greatly reduces the waste of resources and greatly improves the efficiency. Based on the above simulation, we can draw the conclusion that the method to solve the ORP in Theorem 5 is effective.

Example 2. Consider the switched stochastic system (1) with two subsystems, where and and the matrices which satisfy Assumption 1 as follows:

Let , , , , , and . By (57), we can obtain the minimum average dwell time ; then, we construct a switching strategy with . By solving the inequalities (52)–(55) in Theorem 7, we can obtain that

We choose the initial state and ; then, the PORP with sampled-data control is solved according to Theorem 7. The state responses of the closed-loop system (58) is shown in Figure 4, Figure 5 means that the regulated output is bounded. Figure 6 describes the time-triggered instants, which shows that it only triggered 25 times in 10 seconds, proving that the scheme based on time sampling design can effectively save communication resources. Therefore, the simulation results show that the proposed method is effective.

6. Conclusion

We have solved the ORP of a class of switched stochastic systems via sampled-data control. We have discussed two cases of the controlled system under the constant exogenous signal and the time-varying exogenous signal with bounded derivative, respectively. For these two cases, the appropriate sampled-data state feedback controllers are designed and the stochastic signal is handled by utilizing Itô’s formula. Furthermore, by combining the ADT method and the free-weighting matrix method, the ORP of the closed-loop system can be solved. This problem will be further studied in the future. For example, the sampled-data control problem based on the design switching signal and the tracking control problem of switched stochastic systems based on the results of this paper will be studied.

Data Availability

Data are available on request from the authors.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61503254, 61673099) and the China Scholarship Council (202008210125).

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Copyright © 2023 Xiaoxiao Dong and Hui Lan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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