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Exponential stabilization for nonlinear switched stochastic systems with interval timevarying delay under asynchronous switching
Advances in Difference Equations volume 2019, Article number: 295 (2019)
Abstract
The paper investigates meansquare exponential stabilization for a class of nonlinear switched stochastic systems with interval timevarying delay under asynchronous switching. Specifically, the delay occurs not only in the state equation, but also in the switching signal from the controller, which brings the difficulty of controller design to achieve meansquare exponential stabilization. Based on the Lyapunov stability theory, a new piecewise multiLyapunov–Krasovskii functional dependent on the size of time delay is constructed. By utilizing the matrix inequality technique and the average dwell time approach, delaydependent sufficient conditions are given to guarantee meansquare exponential stabilization for nonlinear switched stochastic systems under asynchronous switching. In accordance with the method, we also design state feedback controllers of the switched stochastic systems under asynchronous switching through special operations of matrices and Schur complement. Finally, a numerical example and a practical example of river pollution control are provided to show the effectiveness of the approach proposed in this paper.
Introduction
During the last two decades, hybrid systems have become increasingly important in contemporary society both in science and technology due mainly to the fact that hybrid systems have been extensively applied in many fields such as pattern recognition [1], network control [2], power systems [3], automotive systems [4], communication systems [5], neural networks [6], and so on. A switched system is one of the special dynamic hybrid systems that comprise a collection of subsystems equipped with a switching law orchestrating among these systems. Many of switched system models appear in the fields of industrial manufacturing, artificial intelligence, biochemical systems, actuator failures [7], and population dynamics [8,9,10].
The analysis and synthesis are important issues in the study of switched systems, and they have attracted extensive attention from domestic and foreign scientific research. So far, much progress has been made and many remarkable achievements for various types of switched systems have been studied. For example, behavior analysis [11, 12], property characterization [13], fault detection [14], and control synthesis [15,16,17]. These results show that stability is a crucial and fundamental problem for switched system with time delay. Essentially, time delay, perturbation, and stochastic term are common phenomena often encountered in real dynamical systems [18,19,20]. Moreover, the effect of time delays generally exists in system states and will be a source of control system instability, oscillation, and performance deterioration. As a result, the study of delay and stochastic term plays an important role in the stability analysis of switched system. With respect to those problems, we just mention here some representative work. In [21], a new method of uncertain matrix was proposed. Based on this approach, an exponential stabilization condition of nonlinear uncertain systems with timevarying delay was firstly established. By following this idea, [22] also studied the delaydependent stability analysis and relevant control problems for nonlinear switched with interval timevarying delay based on Lyapunov–Krasovskii functional method. Robust guaranteed cost control for a class of uncertain neutral system with timevarying delays was investigated in [23], delaydependent and delayindependent criteria were proposed for the stabilization of considered systems, state feedback control was considered to stabilize the uncertain neutral system, and upper bounds on the closedloop cost function were also given. [24] obtained sufficient conditions with delaydependent guaranteeing the exponential stability by a common Lyapunov functional (CLF). Recently analogous results have been found in [25], and [25] constructed a suitable Lyapunov–Krasovskii functional containing some novel triple integral terms with sufficient information about the actual sampling pattern. Based on the above discussion, the theory of timedelay systems can be divided into two classes: delayindependent control and delaydependent control. To the best of our knowledge, delaydependent stabilization condition gives less conservative result than the delayindependent one as it makes full use of information of the system. Specifically, systems with delay are of significant interest not only for their applicability in practice but also for their interesting theoretical properties. This is motivated by the need for systematic approach to investigate switched systems with delay.
On the other hand, the switching between the controller and the subsystem of switched systems is synchronous in the ideal case. In fact, the asynchronous phenomenon often occurs in practical industrial systems. For instance, when the system and the controller communicate via a communication channel and the current subsystem is switched to the next one, it is necessary to take some time to identify the active subsystem and then switch the controller from the current one to the corresponding subsystem, further causing asynchronous switching. With the great development of switched systems, the asynchronous control problem for switched systems, which is quite practical and energy efficient, has received increasing attention. In the past few years, it is noted that some valid results have appeared in studying nonlinear switched systems under asynchronous switching [26,27,28,29]. Specifically, [30] investigated the problem of output tracking control for switched systems with timevarying delay under asynchronous switching. Moreover, based on the dwell time approach, some sufficient conditions of exponential stabilization for a given switched system and a tracking error system were proposed in terms of linear matrix inequalities (LMIs). Due to the switching instants of the controllers lagging behind those of the subsystems, [31] dealt with the problem of stabilization for a class of switched delay systems with polytopic type uncertainties under asynchronous switching, and the running time was divided into two parts: matched periods \([t_{k}+\tau _{d},t_{k+1}),k=1,2,\ldots \) and mismatched periods \([t_{k},t_{k}+\tau _{d}),k=1,2,\ldots \) . In addition, by constructing the parameterdependent Lyapunov–Krasovskii functional and the average dwell time approach, the exponential stabilization problem for a class of nonlinear switched systems with mixed delays under asynchronous switching was investigated in [32]. In these papers mentioned above, the time delay is simple and no stochastic items are considered. Searching for delaydependent meansquare stability criteria for nonlinear switched stochastic systems with interval timevarying delays is obviously more preferable and challenging.
Furthermore, it is well known that few results have been devoted to the stability of nonlinear switched stochastic systems with interval timevarying delay under asynchronous switching based on the average dwell time approach. This paper considers interval timevarying delay. It is natural to look for an alternative view to derive a less conservative condition for exponential stabilization of nonlinear switched stochastic systems under asynchronous switching. Moreover, we can hardly use the existing methods to investigate a stochastic switched system due to the impact of stochastic factor. This has motivated our present study on the following questions.

Is it possible to find a delaydependent multiple Lyapunov–Krasovskii functional that studies the matched periods and the mismatched periods of the nonlinear switched stochastic systems, respectively?

Based on the average dwell time approach and Jense’s inequality, can we obtain a less conservative sufficient condition of meansquare exponential stabilization for nonlinear switched stochastic systems with interval timevarying delay under asynchronous switching?

Can we design a meansquare exponentially stable feedback controller for switched nonlinear systems under asynchronous switching by the matrix deformation technique and Schur compensation?
The core of this paper is the further development of switched stochastic nonlinear systems with interval timevarying delay under asynchronous switching. Moreover, we have proposed a detailed study and solutions on the above issues. Compared with the existing results on switched systems, the main contributions of this paper can be summarized as follows. (i) We consider the actual situation. In fact, the system needs to take some time to identify the active subsystem, and then switch the controller from the current subsystem to the corresponding subsystem, further causing asynchronous switching. (ii) According to Lyapunov stability theory, the Lyapunov–Krasovskii functional constructed in this paper is timedelaydependent and depends on the switching signal of the controller. At the same time, it is allowed to increase the running time of the active subsystem with mismatch controller. The established Lyapunov–Krasovskii functional facilitates the analysis of the proposed problem. (iii) By utilizing the matrix inequality technique and the average dwell time approach, delaydependent sufficient conditions are given to guarantee meansquare exponential stabilization for nonlinear switched stochastic systems under asynchronous switching. Moreover, state feedback controllers and switching signal of the switched stochastic systems are designed simultaneously under asynchronous switching through special operations of matrices and Schur complement without resorting to additional constraints on the switching signal.
The remainder of this paper is organized as follows. In Sect. 2, the problem description and preliminaries and some necessary lemmas are presented. Section 3 is devoted to deriving the results on exponential stabilization for switching signals by the average dwell time approach and delaydependent multiLyapunov–Krasovskii functional. Moreover, feedback controller of nonlinear switched stochastic systems with interval timevarying delay under asynchronous switching is designed, which is the main result of this paper. In Sect. 4, an example is given to illustrate the results. The paper is concluded in Sect. 5.
The notations used in this paper are fairly standard. \(R^{n}\) denotes the ndimensional Euclidean space. \(A^{T}\) denotes the transpose of A. The symbol ∗ is used to denote the corresponding transposed block matrix. Diag \(\{\cdots \}\) is a blockdiagonal matrix. I represents the identity matrix in the block matrix, and 0 represents a zero matrix with appropriate dimensions. The notation \(P>0\) indicates that P is a real symmetric and positive definite matrix, and \(\lambda _{\mathrm{min}}\) \((\lambda _{\mathrm{max}})\) is the minimum (maximum) eigenvalue of P.
Preliminaries
Consider the following switched stochastic nonlinear systems with interval timevarying delay:
where \(x(t)\in R^{n}\) and \(u(t)\in R^{m}\) are, respectively, the state vector and the control input of switched systems. \(\phi (s)\in R^{n}\) is the initial condition and \(f_{\sigma (t)}(\cdot )\) are nonlinear functions. \(\sigma (t):[0,\infty ] \rightarrow {M}=\{1,2,\ldots ,n \}\) is the switching signal. Specifically, denote \(\varSigma:\{(t_{0}, \sigma (t_{0})),\ldots ,(t_{k},\sigma (t_{k})),\ldots ,k=0,1,2,\ldots \}\), where \(0=t_{0}< t_{1}< t_{2}<\cdots <t_{k}<\cdots \) , in which \(t_{0}\) is the initial switching instant, \(t_{k}\) denotes the \(kth\) switching instant. For \(\sigma (t_{k})=i\), \(A_{1i},A_{2i},B_{i},C_{i}, D_{i}\) are constant matrices with appropriate dimensions. In this article, we assume that the delay function \(h(t)\) is interval timevarying and satisfies
\(\omega (t)\) is a onedimensional Brownian motion on a probability space \((\varOmega ,\mathcal{F},\mathcal{P})\) and it satisfies the following cases:
\(f_{i}(t,x(t),x(th(t)))\) are nonlinear perturbation functions, which satisfy the following condition:
where \(V_{i}\) and \(\varLambda _{i}\) are known constant matrices. Note that the assumption on the nonlinear perturbations is widely applicable in practice and considered by many researchers. When the controller synchronizes with the switching subsystem, the state feedback controller is often designed as
where \(K_{i}, i\in {M}\), denotes the feedback gain matrix.
In practice, since it inevitably takes some time to identify the system modes and apply the matched controllers, the switching instants of the controllers lag behind those of the subsystems. At this point, we consider the state feedback given by
where \(\tau _{d}\) is a known constant.
Remark 1
In this paper, \(\tau _{d}\) represents the period that the switching instants of the controller lag behind those of the system. Specifically, the running time of switched system is divided into two parts: matched periods \([t_{k}+\tau _{d},t_{k+1}),k=1,2,\ldots \) , and mismatched periods \([t_{k},t_{k}+\tau _{d}),k=1,2,\ldots \) . Correspondingly, we suppose that the jth subsystem is activated at the switching instant \(t_{k1}\), and the ith subsystem is activated at the switching instant \(t_{k}\), then the corresponding switching controllers are activated at the switching \(t_{k1}+\tau _{d}\) and \(t_{k}+\tau _{d}\), respectively.
The closed loop system of system (1) in the interval \([t_{k},t_{k+1})\) can be represented as:
where \(\bar{A}_{1ij}=A_{1i}+B_{i}K_{j},\bar{A}_{1i}=A_{1i}+B_{i}K_{i}\).
Definition 1
([28])
The equilibrium \(x^{\ast }=0\) of the closedloop system (7) is said to be meansquare exponentially stable under switching signal \(\sigma (t)\) if the solution \(x(t)\) of system satisfies
for constants \(k\geq 1, \alpha >0\).
Definition 2
([33])
For any \(T_{2}>T_{1}\geq 0\), let \(N_{\sigma }(T_{1},T _{2})\) denote the switching number of \(\sigma (t)\) on an interval \((T_{1},T_{2})\). If
holds for given \(N_{0}\geq 0, \tau _{\alpha }\geq 0\), then the constant \(\tau _{\alpha }\) is called the average dwell time and \(N_{0}\) is the chatter bound. Without loss of generality, we choose \(N_{0}=0\) in this paper.
Lemma 1
(Schur complement)
For a given matrix $\left(\begin{array}{cc}{S}_{11}& {S}_{12}\\ \ast & {S}_{22}\end{array}\right)$ with \(S_{11}=S_{11}^{T}, S_{22}=S_{22}^{T}\), then the following conditions are equivalent:

(1)
\(S<0\),

(2)
\(S_{11}<0, S_{22}S_{12}^{T}S_{11}^{1}S_{12}<0\),

(3)
\(S_{22}<0, S_{11}S_{12}S_{22}^{1}S_{12}^{T}<0\).
Lemma 2
(Jensen’s inequality)
For any symmetric and positive definite constant matrix \(G\in R^{l\times l}\), scalars α and β: \(\beta <\alpha \), vector function \(x:[\beta ,\alpha ]\rightarrow R ^{l}\) such that the integration concerned are well defined, then
Main results
In this section, based on the Lyapunov stability theory, a new piecewise multiLyapunov–Krasovskii functional dependent on the size of time delay is constructed. Moreover, we give sufficient conditions for the meansquare exponential stabilization of system (7) by the average dwell time approach and Jensen’s inequality. In addition, the state feedback controllers of nonlinear switched systems are designed under asynchronous switching.
Theorem 1
For given positive constants \(\alpha ,\beta ,h\), and \(\mu \geq 1\), if there exist symmetric and positive definite matrices \(P_{i},Q_{1i},Q _{2i},Q_{3i},R_{1i},R_{2i}\) such that the following matrix inequalities hold:
where
If the average dwell time of the switching signal \(\sigma (t)\) satisfies
then the closedloop system (7) is meansquare exponentially stabilizable under arbitrary switching signal for the feedback control (6).
Proof
When \(t\in [t_{k}+\tau _{d},t_{k+1})\), \(\sigma (t_{k})=i \in {M}\), switched systems run in matched periods. The closedloop system (7) is active within the ith subsystem, and the corresponding ith switching controller is also activated. We choose the Lyapunov–Krasovskii functional candidate as follows:
According to Itô’s differential formula, the stochastic differential is
with the infinitesimal operator
Inequality (4) can be written as follows:
By using Lemma 2, we get
From (16), (17), and (18) we have
Let
According to (19), we can obtain
We can get
Then, using (9) and (13), we have
When \(t\in [t_{k},t_{k}+\tau _{d})\), switched systems run in mismatched periods. The closedloop system (7) is active within the ith subsystem and the corresponding jth switching controller is also activated. We choose the Lyapunov–Krasovskii functional candidate as follows:
According to Itô’s differential formula, we have
Following the similar way, we have
According to (12), we have
Then
we can get
By recalling (2), we have
Therefore,
Considering the whole interval \([t_{0},t)\), the Lyapunov–Krasovskii functional \(V(t)\) is expressed as
When \(t\in [t_{k}+\tau _{d},t_{k+1})\), integrating both sides of (22) from \(t_{k}+\tau _{d}\) to t and taking expectation, we have
When \(t\in [t_{k},t_{k}+\tau _{d})\), integrating both sides of (28) and taking expectation, we obtain
Notice from (14) and (23) that
where
Finally, we can get
By Definition 1, we know that the closedloop system (7) is meansquare exponentially stabilizable. This completes the proof. □
Theorem 2
For given positive constants \(\alpha ,\beta ,h\), and \(\mu \geq 1\), if there exist symmetric and positive definite matrices \(X_{i},S_{1i},S _{2i},S_{3i},T_{1i},T_{2i}\) and any \(Y_{i}\) such that the following matrix inequalities hold:
where
then system (1) is meansquare exponentially stabilizable for arbitrary switching signal with the average dwell time satisfying (13). In addition, the feedback controller can be designed by the following formula:
Proof
According to \(S_{i}>0,T_{i}>0\), we have
Then the following inequality can be obtained:
Substituting (40) into (37) and multiplying both sides of (37) by \(\operatorname{diag}\{X_{i}^{1}, X_{i}^{1},X_{i}^{1},X_{i} ^{1},I, X_{i}^{1},X_{i}^{1},I,I,I,I,I,I,I,I\}\), we can get the following inequality:
where
Then set
Using Schur complement in (41), it can be concluded that (12) holds. By the same method, (38) implies (13). Correspondingly, controller gains are given by (39). The proof is completed. □
Remark 2
It is noticed that the Lyapunov–Krasovskii functional is delaydependent in this paper. On the one hand, the important information of \(h_{m}\) and \(h_{M}\) is taken into full consideration, which may overcome the conservatism of quadratic meansquare exponential stability conditions for nonlinear switched stochastic systems with interval timevarying delay under asynchronous switching. On the other hand, the delaydependent Lyapunov–Krasovskii functional is allowed to rise at both switching instants. However, the delaydependent Lyapunov–Krasovskii functional is decreasing on the entire interval and the meansquare exponential stabilization for nonlinear switched stochastic systems is also guaranteed.
Remark 3
[34] investigated the stabilization problem for a class of positive switched nonlinear systems under asynchronous switching. The author mainly focused on the study of positive switched systems in [34]. [35] addressed the problem of robust control for uncertain switched nonlinear systems with time delay under asynchronous switching. However, stochastic disturbance was not considered in [34] and [35]. The problem of robust reliable control for a class of uncertain stochastic switched nonlinear systems under asynchronous switching was investigated in [36], but the interval timevarying delay was not considered in [36] under asynchronous switching. In this paper, we consider stochastic disturbance and interval timevarying delay. Compared with [34,35,36], it is obvious that we have considered more external factors, and our switched systems model is more in line with engineering practice from the application level.
Remark 4
[18] obtained sufficient conditions with delaydependent guaranteeing the exponential stability by a common Lyapunov functional (CLF). In fact, we deeply realize that common Lyapunov functional (CLF) may not satisfy all subsystems and become conservative for switched systems. The multiLyapunov–Krasovskii functional (MLKF) and delaydependent method are better choices. They provide a powerful framework for analyzing the stability of switched nonlinear systems with interval timevarying delay.
Remark 5
We now summarize the controller design procedures as follows.

Step 1:
The desired convergence rates α and β are given. Choose a positive and appropriate parameter μ.

Step 2:
Define the variables \(X_{i},S_{1i},S_{2i},S_{3i},T_{1i},T_{2i},Y _{i}, i\in \{1,2\}\) to be solved.

Step 3:
Describe the block form to give a linear matrix inequalities (LMIs).

Step 4:
Complete LMIs model description.

Step 5:
Solve LMIs problems.

Step 6:
Solve (42).
Then the obtained feedback controller will make the desired performance indices be satisfied.
Numerical example
In this section, a numerical example is presented to confirm the effectiveness of the proposed approach.
Example 1
Consider system (1) composed of two subsystems with the following parameters:
Let \(\alpha =0.45, \beta =0.65, \mu =1.25, h_{m}=0.1, h_{M}=0.6, h=0.5, h(t)=0.5\sin (t)+0.1, \tau _{d}=0.2\). According to (13), we get the average dwell time
Choose
By solving (36), (37), and (38), we can get
Then the controller gains constructed by (39) are
By Theorem 2, the maximum value of interval timevarying delay \(h_{M}\) for the switched systems (1) is provided in Table 1 for different values of α and β.
In order to show the effectiveness of the proposed method, the responses of state trajectories of the openloop and closedloop systems and switching signals (system signals and controller signals) are given in Figs. 1, 2, and 3, respectively. It is clear that the openloop system with initial state \(x(0)=(2,2)^{T}\) is not stable form in Fig. 1. The closedloop system with initial state \(x(0)=(2,2)^{T}\) is meansquare exponentially stabilizable under the designed asynchronous switching and controllers form in Fig. 2 and 3. Therefore, the effectiveness of the designed asynchronous switching and controllers is fully illustrated.
Example 2
The problem of water pollution is an important issue facing every country, and its development is of great significance to social development. In this section, an example of applying this system to water pollution control systems will be demonstrated.
To facilitate the creation of models for water pollution control systems, we record \(p(t)\) and \(q(t)\) as the concentrations per unit volume of biochemical oxygen demand and dissolved oxygen, respectively. Simultaneously, let \(p^{*}\) and \(q^{*}\) indicate the desired steady values of \(p(t)\) and \(q(t)\) in a reach of a polluted river, respectively. Moreover, we take \(p^{*}\) and \(q^{*}\) as corresponding to some measure of water quality standards, given by the following definition:
As a result, the dynamic equation for \(x(t)\) can be expressed as
where
\(m_{i}(i=1,2,3), \varepsilon _{1}\) and \(\varepsilon _{2}\) are known constants, and \(\omega (t)\) is a onedimensional Brownian motion that satisfies condition (3). Moreover, \(u(t)= [u_{1}^{T}(t) u_{2} ^{T}(t)]^{T}\) is the control variable of river pollution system. We can learn the engineering significance of these parameters from [1]. This paper assumes that system actuators have good performance or failure, and according to the actual situation, we know that at least one actuator can ensure the normal operation of the river pollution system. In addition, for simulation of our purposes, we do not consider the nonlinear perturbation term, and the nonlinear perturbation term is not also considered in [26]. As a consequence, the river pollution system (43) can be modeled as a switched system consisting of two subsystems:
Next, we choose \(m_{1}=1.1, m_{2}=0.6, m_{3}=1.3, \varepsilon _{1}=0.5, \varepsilon _{2}=0.4\) and get that
Let \(\alpha =0.5, \beta =0.6, \mu =1.15, h_{m}=0.15, h_{M}=0.65, h=0.3, h(t)=0.3\sin (t)+0.2, \tau _{d}=0.1\).
By (36), (37), and (38), we have
Then the controller gains constructed by (39) are
Figure 4 describes state response of subsystem 1 with the initial condition \(x(0)=(1,0.5)^{T}\). Figure 5 describes state response of subsystem 2 with the initial condition \(x(0)=(0.2,0.5)^{T}\). Through the designed switching signal and our approach, we can get that system (43) with the initial condition \(x(0)=(2,2)^{T}\) is meansquare exponentially stabilizable for any switching signal under the feedback control form Fig. 6. As a consequence, this verifies the effectiveness of our results in the control of river pollution process.
Conclusions
The switching signal of the switched controller involves delay, which results in the asynchronous switching between the candidate controllers and subsystems. In the paper, we have investigated the problem of asynchronous switching for nonlinear switched stochastic systems with interval timevarying delay based on timedependent switching signal for the matched and mismatched sections, respectively. By constructing a new multiLyapunov–Krasovskii functional, which is related to the size of the time delay, using the matrix inequality technique and the average dwell time approach, the meansquare exponential stabilization criteria for nonlinear switched stochastic systems with interval timevarying delay are obtained under asynchronous switching. Then, the proposed approach is extended to design state feedback controller for switched stochastic systems by special operations of matrices. Finally, the numerical example illustrates the effectiveness of the theoretical results. Compared with the existing results, the new condition is less conservative. In order to better study the asynchronous switching issue, our future work will focus on extending the proposed method to a delaydependent robust dissipative problem for a class of nonlinear switched systems with mixed delays and the stabilization of stochastic switched nonlinear systems with Markov jumps.
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Acknowledgements
The author would like to thank the editor and the anonymous reviewers for their valuable comments and suggestions to improve this paper.
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Data sharing not applicable to this article as no datasets were generated or analysed during the current paper.
Funding
This work was supported by the National Natural Science Foundation for Young Scientists of China (Grant No. 71502050), the Key Scientific Research Projects of Colleges and Universities of Henan Province (No. 18A630001, 18A110008, and 18A110010).
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DOI
MSC
 93D20
 93E10
 93C10
 34D20
Keywords
 Switched stochastic system
 Asynchronous switching
 Interval timevarying delay
 Average dwell time
 Lyapunov Krasovskii functional