Brief paperGeneralized Lyapunov criteria on finite-time stability of stochastic nonlinear systems☆
Introduction
Stochastic stability has become a wide and important research field for controlled systems modeled by stochastic differential equations in the past few decades, and we here mention (Arnold, 1974, Deng et al., 2001, Ito and Nishimura, 2015, Khas’minskii, 2012, Krstic and Deng, 1998, Kushner, 1967, Mao, 1994, Mao, 2007, Zhao and Deng, 2016) among others.
In classical stochastic stability theory, generally speaking,asymptotic stability in probability, -order moment asymptotic stability, and almost sure asymptotic stability are often considered. These three types of stability describe the asymptotic behavior of the trajectories of a stochastic system as time goes to infinity. In many applications, however, it is desirable that a stochastic system possesses the property that its trajectories can converge to a Lyapunov stable equilibrium state in finite time rather than merely asymptotically. This kind of stability is referred as stochastic finite-time stability.
To develop the theory of finite-time stability of deterministic systems (Bhat & Bernstein, 2000) to the stochastic case, the notions and Lyapunov criteria of stochastic finite-time stability were introduced separately in Chen and Jiao (2010) and Yin, Khoo, Man, and Yu (2011). Based on stochastic finite-time stability theory, finite-time stabilization of stochastic nonlinear systems by state or output feedback are considered in Khoo, Yin, Man, and Yu (2013), Yin and Khoo (2015), Wang and Zhu (2015) and Zha, Zhai, Fei, and Wang (2014) for example. Properties of finite-time stable stochastic systems are further discussed in Yin, Ding, Liu, and Khoo (2015). Recently, finite-time stability of homogeneous stochastic nonlinear systems is studied in Yin, Khoo, and Man (2017).
In those existing papers, stochastic finite-time stability is required that the differential operator satisfies with and . So far, to the best of our knowledge, there are not any papers that could demonstrate whether stochastic finite-time stability still holds, if this condition is not fulfilled.
In this paper, our target is to establish Lyapunov criteria of stochastic finite-time stability under more general conditions. The main contributions of this paper are as follows: A general Lyapunov theorem on finite-time stability of stochastic nonlinear systems is proved, and an important corollary follows directly. By comparing the Lyapunov theorem with the previous results of stochastic finite-time stability, it is shown that this generalized criterion not only relaxes the constraint on the differential operator in some degree, but also reveals an active role of Brownian noise in finite-time stabilizing a system. In addition, a multiple Lyapunov function approach is proposed for finite-time stability analysis of stochastic systems, which further relaxes the constraint of the differential operator . In particular, by using this method, it shows from an illustrative example that an unstable deterministic system can even be finite-time stabilized by Brownian noise. Some examples are constructed to show the significant features of our results.
The rest of the paper is organized as follows. The mathematic preliminaries are given in Section 2. A general Lyapunov theorem of stochastic finite-time stability is presented in Section 3. In Section 4, we derive an important corollary and discuss the comparisons with the existing results. In Section 5, multiple Lyapunov functions-based criteria of stochastic finite-time stability are presented. Section 6 gives some simulation results to illustrate the theoretical results. Finally, Section 7 gives some conclusions and future work.
Notations: stands for the set of all nonnegative real numbers, is the -dimensional Euclidean space, is the space of real -matrices. denotes the transpose of matrix . is its trace when is a square matrix. is the usual Euclidean norm of a vector . denotes the set of all functions that are continuously twice differentiable in . A function means that this function is continuously twice differentiable in its domain of definition. A random variable means that . is a class function means that it is continuous, strictly increasing, and . .
Section snippets
Preliminary results
In this paper, we consider a stochastic nonlinear system modeled by the following stochastic differential equation: where is the system state; is an -dimensional standard Brownian motion defined on a complete probability space ; and are continuous in and satisfying and , which implies that (1) has a trivial zero solution.
As discussed in Yin et al. (2017), in general, we are interested in having a unique solution
Generalized stochastic finite-time stability theorem
In this section, we first review and refine the definition of stochastic finite-time stability introduced in Yin and Khoo (2015) and Yin et al. (2011). Then, a new Lyapunov theorem on finite-time stability of stochastic nonlinear systems will be given.
Definition 1 The trivial zero solution of (1) is said to be stochastically finite-time stable, if the stochastic system admits a solution (either in the strong sense or in the weak sense) for any initial data , and the following properties hold: (i)
Comparisons with the existing results
Let us first recall the existing results on the stochastic finite-time stability (Yin and Khoo, 2015, Yin et al., 2011), and take the classical result in Yin et al. (2011) as a theorem.
Theorem 2 For system (1) , If there exists a function , class functions and , positive real numbers and , such that for all , then the trivial solution of (1) is stochastically finite-time stable. To see the important contributions of this paper, letYin et al., 2011
Criteria of stochastic finite-time stability via multiple Lyapunov functions
Criteria via multiple Lyapunov functions have an advantage over those via single Lyapunov function in stochastic asymptotic stability (Mao, 2001). Similarly, in this section, we shall develop Theorem 1 to establish some criteria using multiple Lyapunov functions, by which we can further relax the constraint of the differential operator .
Theorem 3 For system (1) , assume that there exists a positive definite and radially unbounded function such that Furthermore, if there exists
Simulations
In this section, Example 1, Example 2, Example 3, Example 4 are considered again, and their simulation results are given to verify the theoretical results.
In the three cases of Example 1, the initial state is set to be . In Case 1, we choose parameters , , , , and . In Case 2, we choose parameters , , , , , and . In Case 3, we choose parameters , , , , , and . Fig. 1 shows the
Conclusions
In this paper, some general Lyapunov criteria of stochastic finite-time stability via single Lyapunov function and multiple Lyapunov functions are given. Compared with the existing results about stochastic finite-time stability, these criteria not only relax the constraint on the differential operator , but also reveal the importance of Brownian noise in stochastic finite-time stability. Some examples are constructed to show the significant features of the proposed results, including an
Acknowledgments
The authors would like to thank the Associate Editor and anonymous referees for their helpful comments and suggestions which have greatly improved this paper.
Xin Yu received his Ph.D. degree in Control Theory and Control Engineering from the Southeast University, China, in 2012. During 2012–2015, he was a Lecturer in the Department of Automation, Jiangsu University, China, where he is currently an Associate Professor. His research interests include stochastic systems, stability theory, control of stochastic nonlinear systems and applications.
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Xin Yu received his Ph.D. degree in Control Theory and Control Engineering from the Southeast University, China, in 2012. During 2012–2015, he was a Lecturer in the Department of Automation, Jiangsu University, China, where he is currently an Associate Professor. His research interests include stochastic systems, stability theory, control of stochastic nonlinear systems and applications.
Juliang Yin received his Ph.D. degree in Probability and Statistics from the University of Zhongshan, China, in 2003 and was then a Post-Doctoral Research Fellow at the University of Nankai, China from 2003 to 2005. From 2000 to 2017, he was with the Department of Statistics, Jinan University, China, where he became an Associate Professor and Professor of Probability and Statistics in 2004 and 2009, respectively. In 2011, he was appointed as an Honorary Professor in the School of Engineering, Deakin University, Australia. In 2017, he joined the School of Economics and Statistics, Guangzhou University, China, where he was a Distinguished Professor of Probability and Statistics. He has authored 2 books and over 50 research papers. His main research interests are in stochastic control, stochastic analysis, mathematical finance and statistics of stochastic processes.
Suiyang Khoo received the B.Eng. degree in Electronics and Communications Engineering from Tasmania University and the Ph.D. degree in Computer Engineering from Nanyang Technological University, in 2005 and 2008, respectively. In 2008, he was a Research Fellow in the Department of Electrical and Electronics Engineering, Nanyang Technological University, Singapore. Since 2009, he has been with the School of Engineering, Deakin University, Victoria, Australia, where he is now Senior Lecturer in Electrical & Electronics Engineering. His research interests include stochastic differential equations, variable structure control, co-operative control, robotics, adaptive signal processing, time-varying systems, and neural networks.
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This paper was supported in part by the National Natural Science Foundation of China under Grants 61573006 and 61304073. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Debasish Chatterjee under the direction of Editor Daniel Liberzon.