Abstract
We address the problem of globally asymptotic stability for a class of stochastic nonlinear systems with time-varying delays. By the backstepping method and Lyapunov theory, we design a linear output feedback controller recursively based on the observable linearization for a class of stochastic nonlinear systems with time-varying delays to guarantee that the closed-loop system is globally asymptotically stable in probability. In particular, we extend the deterministic nonlinear system to stochastic nonlinear systems with time-varying delays. Finally, an example and its simulations are given to illustrate the theoretical results.
1. Introduction
In recent years, the stochastic nonlinear system has received much attention and has enjoyed a good development, which has been widely applied in many fields such as engineering and finance [1]. Particularly, owing to the reason that a system by the output feedback is more flexible to respond to the information of control systems, the stochastic nonlinear system via the output feedback control has been more widely studied. Marino and Tomei [2] and Battilotti [3] designed linear observer to study the output feedback control for nonlinear systems. As a result, a large number of researchers have been focused on the stability analysis of stochastic nonlinear systems and the design of controller.
Since the backstepping method has been introduced in the nonlinear system, the theory of stochastic nonlinear systems has achieved a remarkable development. According to the different Lyapunov function, there are two ways to solve the problem. One is that Pan and Basar [4–6] considered a quadratic Lyapunov function. By using the backstepping method to design the controller, they discussed the risk-sensitive optimal control problem. The other is that Deng and Krstić [7–10] used a quartic Lyapunov function to guarantee that the closed-loop system is globally asymptotically stable in probability. Based on their works, Liu et al. [11–13] employed the quartic Lyapunov function to design the output feedback control for stochastic nonlinear systems. M.-L. Liu and Y.-G. Liu [14] and Chen et al. [15] used such Lyapunov function to study the state feedback stability for stochastic nonlinear systems with time-varying delays. In [16], Du et al. discussed the global output feedback stability for a class of uncertain upper-triangular systems with the input delay. Finite-time stability for stochastic nonlinear systems in strict-feedback form was considered in [17]. More results can be found in [18–20].
In this paper, we are concerned with the globally asymptotic stability of stochastic nonlinear system with time-varying delays. We extend the results of the deterministic nonlinear systems in [21] to the stochastic nonlinear systems with time-varying delays by output feedback control and design a Lyapunov-Krasovskii functional to prove the globally asymptotic stability of the closed-system via the linear observer. It is obvious that the form of the linear controller is simpler than that used in [9]. And compared with the output feedback controller obtained by the homogeneous observers in [20], the controller designed by the linear observers in this paper is smooth.
The rest of this paper is organized as follows. In Section 2, we present some definitions and establish a new inequality, which plays an important role in the proof of our main results. In Section 3, we construct a Lyapunov-Krasovskii functional and design a linear output feedback control to prove the globally asymptotic stability based on the linear observers. In Section 4, we use an example to illustrate our theoretical results. Finally, we conclude this paper with some general remarks.
2. Preliminaries
In this section, we will give the following notations, definitions, and some preliminary lemmas. denotes the set of all nonnegative real numbers; denotes the real -dimensional space; denotes the trace for square matrix ; denotes the Euclidean norm of a vector . denotes the space of continuous -valued functions on endowed with the norm defined by for ; denotes the family of all -measurable bounded -valued random variables . denotes the set of all functions with continuous th partial derivatives; denotes the family of all nonnegative functions on which are in and in ; denotes the family of all functions which are in the first argument and in the second argument. denotes the set of all functions: , which are continuous and strictly increasing and vanish at zero; denotes the set of all functions which are of class and unbounded. is the set of all functions , which are of for each fixed and decrease to zero as for each fixed .
Consider the following stochastic time-varying delay system:with initial date , where is a Borel measurable function and is an -dimensional standard Wiener process defined on the complete probability space with being a sample space, being a filtration, and being a probability measure. , are assumed to be locally Lipschitz in uniformly in and satisfy , .
Definition 1 (see [22]). For any given associated with system (1), the differential operator is defined aswhere is called as the Hessian term of .
Definition 2 (see [11, 15]). For system (1), the equilibrium is said to be globally asymptotically stable in probability, if, for any , there exists a class of function such that and for any , one has
Definition 3 (see [22]). For fixed coordinates and real numbers , .
(1) The dilation is defined by for any ; are called the weights of the coordinates. For simplicity, we define dilation weight .
(2) A function is said to be homogeneous of degree if there is a real number such that for any , .
(3) A vector field is said to be homogeneous of degree if there is a real number such that for any , .
(4) A homogeneous -norm is defined as for any , where is a constant. For simplicity, in this paper, we choose and write for .
Lemma 4 (see [11, 15, 22]). For system (1), if there exist a function , two class functions , and a class function such thatthen there exists a unique solution on for each and the equilibrium is globally asymptotically stable in probability.
Lemma 5 (see [22]). Suppose that is a homogeneous function of degree with respect to the dilation weight ; then(i) is homogeneous of degree with being the homogeneous weight of ;(ii)there is a constant such that . Moreover, if is positive definite, then , where is a positive constant.
Lemma 6 (see [1]). Let be a continuous positive definite on a domain that contains the origin. Let for some . Then, there exist class functions and , defined on [0,r], such thatfor all . If , the functions and will be defined on and the foregoing inequality will hold for all . Moreover, if is radially unbounded, then and can be chosen to belong to class .
Lemma 7. For any constants and , one has that, for any ,where and .
Proof. We first prove (8). Let , where is a parameter. Then, we haveIt is clear that, for any , and , . With the sufficient condition of extreme value, is the maximum point of function . Therefore, we getWe now prove (9). Similarly, letting , where is a parameter, we haveObviously, for any , and , . With the sufficient condition of extreme value, is the maximum point of function . Therefore, we obtain
Lemma 8 (Cauchy-Schwartz’s inequality). For any vector and , one has
Lemma 9. For a sequence of numbers , one has
Lemma 10 (see [17]). For any given real numbers c, d and any real-valued functions , , the following inequality holds:where . Particularly when one takes , , and , then the inequality will become
3. The Output Feedback Model and Control Design
In this section, we will design a linear observer system for a class of stochastic nonlinear systems with time-varying delays. Using the backstepping method, a simple linear controller will be constructed to guarantee that the closed-loop stochastic system is globally asymptotically stable in probability.
Consider the following stochastic nonlinear system with time-varying delay:where the initial data , is a Borel measurable function and ; is an -dimensional standard Wiener process defined on the complete probability space . is the system state, is the time-delayed state vectors, is the system input, and is the system output. and are assumed to be locally Lipschitz in uniformly in with , , .
Remark 11. It should be pointed out that Zhai and Zha [23] considered the global adaptive output feedback control for system (18) without the diffusion terms. And Duan and Xie [24] discussed the globally asymptotic stability for system (18) without time-varying delay. Compared with these works, we focus on the output feedback stabilization of stochastic nonlinear systems with time-varying delay (see system (18)).
Assumption 12. There exist nonnegative constants such that
The linear observer system is designed aswhere is a constant to be determined and , , are coefficients of the Hurwitz polynomialThe observation error satisfieswhere is a Hurwitz matrix. Therefore, there is a positive-definite matrix such that
Theorem 13. Suppose Assumption 12 holds. Then, the equilibrium at origin of closed-loop stochastic nonlinear system (18) and (20) with the linear controller (50) below is globally asymptotically stable in probability and . Furthermore, the closed-loop system has a unique solution on for each .
Proof. Note , , and for simplicity. Consider the following Lyapunov function: . Then by Lemma 10, a direct computation yieldswhere denotes the minimum eigenvalue of the matrix and denotes the maximum eigenvalue of the matrix .
By Lemmas 8 and 9 and Assumption 12, one getsSubstituting (25) into (24), we havewhereTakethenSubstituting , , into (29), one getswhereNext, we will combine the backstepping design method with the mathematical induction to design the linear output feedback control.
Step 1. Construct the Lyapunov function . A direct calculation giveswhere is defined in Lemma 7.
Define with being a virtual control. Then, we haveChoose the virtual controllerwhere is independent of . Hence,Step . Suppose that, at Step , there exists a smooth Lyapunov function which is positive definite, radially unbounded, and twice continuously differentiable, satisfyingwith a set of virtual controllers , defined by , with being independent of the gain constant , where , .
From the above definition, we haveA direct computation yieldswhereObviously, are suitable real numbers and they are independent of the gain constant .
Let us consider the following Lyapunov function:Then, combining (36) and (38), we obtainIt follows from Lemma 7 thatIn particular, the last term of (41) yieldsHence,whereNow, we choose the following linear controller:with being independent of .
So it follows from (44) thatStep . Using the inductive argument step by step, at the ()th step, we getNoting the function , we haveWe now design the linear controllerwhere is a real number independent of the gain parameter . Thus,where .
Choosing the gain constant , it is obvious that is a positive-definite and proper function. Thus, we see that (6) is true.
Now, we prove (5). From the above process, we have constructed the Lyapunov functionwhere and are positive parameters andIt is easy to verify that is on . Suppose that is continuous, positive definite, and radially unbounded. Then by Lemmas 5 and 6, there exist positive constants and and two class functions and such thatIt follows from thatwhere , and are positive constants, and is a class function. Noting thatsoDefiningone getsTherefore, by Lemma 4, the equilibrium of closed-loop nonlinear stochastic system (18), (20), and (50) is globally asymptotically stable and the closed-loop system has a unique solution on for each .
Remark 14. Letting and in system (18), then the model will reduce to that introduced in [9]. Therefore, our result extends and improves those given in [9].
Remark 15. It should be pointed out that, compared with the output feedback controller in [20], we see that the linear controller (50) based on the observable linearization is a smooth one.
Remark 16. In this paper, combining the backstepping design with mathematical induction, we extend the method in [21] to the stochastic nonlinear system with time-varying delay. Although the globally asymptotic stability of stochastic nonlinear system has been discussed by Liu and Zhang in [12], they did not consider the term of time-varying delay. Therefore, our conclusion is proposed for the first time.
Remark 17. In [22], Liu and Xie considered the state feedback stability of stochastic feedforward nonlinear systems with time-varying delays. They introduced the homogeneous domination approach to construct a state feedback controller. It is generally known that in real lives the output feedback control is easier to achieve than the state feedback control and has great theory significance and utility value. Therefore, in this paper we combine the backstepping method with the mathematical induction to design a Lyapunov-Krasovskii functional and construct a linear output feedback controller for stochastic nonlinear systems with time-varying delay. Moreover, we impose different restricted conditions on functions and to ensure the global stability of stochastic nonlinear systems with time-varying delays.
4. An Example
In this section, we will use a simulation example to illustrate our main results.
Consider the following stochastic nonlinear system:where the time-varying delay . Obviously, we have , , which implies that Assumption 12 holds. Moreover, both of them are locally Lipschitz. The linear observer system is designed aswith a suitable choice of the parameter . The observation errorssatisfyFor the above matrix , there exists a positive-definite matrix satisfying , whereIt is easy to get the corresponding eigenvalues and .
Consider the Lyapunov-Krasovskii functionalThen, we take , , , , and the linear control . By Theorem 13, we see that closed-loop system (60), (61) and the linear controller are globally asymptotically stable in probability and there is a unique solution on . (The state response and control input with initial conditions , , , and are presented in Figures 1(a), 1(b), and 2, where and denote and .)
(a)
(b)
5. Conclusion
In this paper, we combine the backstepping method with the mathematical induction to prove that a class of stochastic nonlinear systems with time-varying delays by the output feedback control is globally asymptotically stable in probability. It should be pointed out that we not only extend the results in deterministic nonlinear systems to more complex stochastic case but also add time-varying delays to stochastic nonlinear systems. For the term of time-varying delays, we design a Lyapunov-Krasovskii functional and a linear output feedback controller to render the closed-loop system globally asymptotically stable.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was jointly supported by the Alexander von Humboldt Foundation of Germany (Fellowship CHN/1163390), the National Natural Science Foundation of China (61374080), the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Play of Nature Science Fundamental Research in Nanjing Xiao Zhuang University (2012NXY12).