Finite-time stabilization of weak solutions for a class of non-local Lipschitzian stochastic nonlinear systems with inverse dynamics☆
Introduction
Background of finite-time control A standard problem in system theory is to develop controllers which drive a system to a given position as fast as possible (Haimo, 1986). Finite-time control can make the controlled system to reach the given position in finite time. And it is shown that the closed-loop system with the finite-time control law has a better disturbance rejection performance compared with the infinite-time control case by theoretical analysis and simulation study in Ding, Li, and Li (2009). Moreover, finite-time control has been applied into some practical systems, such as robotic manipulators (Feng, Yu, & Man, 2002) and spacecrafts (Du, Li, & Qian, 2011). Due to the usefulness of finite-time control, the problem of finite-time control has attracted many researchers’ interest (Du et al. (2011), Fu et al. (2015), Hong and Jiang (2006), Huang et al. (2005), Liu (2014), Menard et al. (2010), Polyakov (2012), Song et al. (2017) and reference therein). Recently, based on Theorem 3.1 in Yin, Khoo, Man, and Yu (2011), the finite-time stabilization for some stochastic nonlinear systems has been studied (Khoo et al. (2013), Lan et al. (2015), Wang and Zhu (2015), Yin and Khoo (2015), Zha et al. (2015) and reference therein).
Existing work on inverse dynamics For nonlinear systems with input-to-state stable inverse dynamics, an adaptive partial-state feedback finite-time controller is designed by the methods of changing supply functions and adding a power integrator in Hong and Jiang (2006). Since the inverse dynamic is unobservable, the controller design for nonlinear systems with inverse dynamics is generally under some stability of inverse dynamics (see e.g., Hong and Jiang (2006), Jiang (1999), Liu et al. (2015), Liu et al. (2007), Wu et al. (2007), Xie et al. (2011)). However, to the best of the authors’ knowledge, there is no work about the finite-time stabilization of stochastic nonlinear systems with inverse dynamics.
Difficulty of analyzing the existence of strong solutions instochastic finite-time control Since Theorem 3.1 in Yin et al. (2011) is about a pathwise unique strong solution, the existing results about finite-time stabilization of the stochastic systems are on strong solutions. And yet, the existence of a strong solution is hard to be guaranteed without local Lipschitzian condition. On the one hand, local Lipschitzian condition cannot be used in the finite-time stabilization, because under the local Lipschitzian condition, almost all the sample paths of any solution starting from non-zero state will never reach the origin (Mao, 1997); on the other hand, the local Lipschitzian condition can guarantee pathwise uniqueness of the solution, while the pathwise uniqueness is very closed to the existence of a strong solution (Ikeda & Watanabe, 1989). Thus, the existence analysis about the strong solutions is an obstacle for the further development of the finite-time control of stochastic nonlinear systems. Also, we can say that the main reason of this obstacle is the lack of pathwise uniqueness.
The existing work about finite-time control of stochastic systems can be classified into two classes according to the analysis methods of the existence of the solution: (i) the existence of a local strong solution is assumed by imposing some growth conditions on the drift and diffusion terms besides the continuity; (ii) the existence of a local solution is guaranteed by Theorem 5.2 in Skorokhod (1965), and then a global solution is constructed by introducing the sequence of stopping time. However, we should point out that (a) in the first case, it is very hard to give such a system which satisfies the growth conditions but without local Lipschitzian condition (The finite-time stability is impossible under local Lipschitzian condition); (b) in the second case, Theorem 5.2 in Skorokhod (1965) (almost all related work uses this theorem even though weak solutions are not referred) is actually about the existence of a weak solution on interval ( in this paper). Thus, it is hard to use Theorem 5.2 in Skorokhod (1965) to construct the global solution of system by on , . Since, without the local Lipschitzian condition (thus pathwise uniqueness of is hard to be guaranteed), for different , the definition of is different and thus is not well-defined. Here is the state, is an -dimensional standard Brownian motion, and are Borel measurable and satisfy , ; , is a weak solution on the interval to with and having the similar definition. Besides these two classes of work, authors in Lan et al. (2015) refer to the weak solution but no finite-time stability theorems are supplied and the global existence of weak solution is not investigated rigorously.
Weak solutions Fortunately, some stochastic differential equations have no strong solutions but have weak solutions. A strong solution is clearly a weak one, but the converse is not true in general (Mao, 1997). It is shown that stochastic differential equations with continuous coefficients have local weak solutions (Ikeda & Watanabe, 1989) but may have no strong solutions. Such an example is given in Barlow (1982). Now, referring to the definitions of a weak solution in Ikeda and Watanabe (1989) and Li and Liu (2014), we give the following definition.
Definition 1 If there exist a continuous adapted process on a probability space (, , ) with a filtration satisfying the usual conditions, and an -dimensional -adapted Brownian motion with , such that has the given distribution, and for all , then the tuple (, , , , , ), or simply is called a local weak solution of system (1), where is the explosion time of the local weak solution with and . Moreover, if , then is called a global weak solution of system (1).
Different from a strong solution, which is a functional of the given Brownian motion and the initial value (Ikeda & Watanabe, 1989), a weak solution includes a probability space, a filtration and a driven Brownian motion, which are not given in advance. Simply speaking, the construction of the solution is totally different: a strong solution is constructed on a given probability space with respect to a given filtration and a given Brownian motion, while the weak solution is constructed simultaneously with the construction (the proof of the existence) of a probability space.
Our work In this paper, we investigate the finite-time stabilization of non-local Lipschitzian stochastic nonlinear systems with stochastic inverse dynamics under the framework of weak solutions. This work extends the deterministic work in Hong and Jiang (2006) to the stochastic case. Our work mainly involves two points: (1) We reestablish the framework of finite-time control in the sense of weak solutions for larger classes of stochastic nonlinear systems including rigorous analyzing about the existence of the global weak solutions and establishing of finite-time stability theorem for weak solutions. It is worth mentioning that the existence of the global weak solutions is established for the analysis of asymptotical stability and asymptotical stabilization of stochastic systems in Li and Liu (2014), and we refer to work in Li and Liu (2014) to analyze the existence of the global weak solutions. (2) A Lyapunov function for the whole system including controllable dynamics and stochastic inverse dynamics (unmodeled dynamics) is designed by the method of changing supply functions, and a controller is constructively designed such that the closed-loop system has a global weak solution and the trivial weak solution is proved to be finite-time stable in probability.
Our work is remarkably different from the existing work: (i) Theorem of finite-time stability for weak solution is given for the first time in this paper. Different from the existing theorems about finite-time stability of stochastic nonlinear systems (e.g., Theorem 3.1 in Yin et al. (2011)), Picard iteration cannot be used to prove the finite-time attractiveness in this paper, because the convergence of the approximation generated by Picard iteration cannot be guaranteed only by the continuity of coefficients. Thus, we construct the approximation in Lemma 2 to show the finite-time attractiveness. (ii) The problem of finite-time control for stochastic nonlinear systems with inverse dynamics is studied first in this paper. And, the existing methods of stability analysis (including the method of stability analysis for asymptotical stabilization of weak solutions) cannot be used to analyze the finite-time stability of the closed-loop system in this paper. The reasons are given as follows. (a) (Compared with the case without inverse dynamics) In the control design and finite-time stability analysis, the interconnection of the controlled dynamics and inverse dynamics should be considered. Thus, the existing methods of analyzing finite-time stability for stochastic closed-loop systems (e.g., Khoo et al., 2013; Wang & Zhu, 2015) cannot be used in this paper; (b) (Compared with the asymptotical stabilization) Because of the especial rate of change for finite-time Lyapunov function, the judgement theorems about asymptotical stability cannot be used here. Thus, the methods of analyzing infinite-time stability of stochastic systems with inverse dynamics cannot be used here (e.g., Liu et al. (2015), Liu et al. (2007)). (c) (Compared with the deterministic case) Due to the diffusion term in the stochastic system, stochastic analysis is needed, and so, it is hard to directly use the deterministic method of finite-time stabilization in Hong and Jiang (2006) to analyze the finite-time stability of the stochastic closed-loop system in this paper. Thus, we develop a finite-time stability theorem for the interconnected stochastic nonlinear systems (Theorem 1) to analyze the finite-time stability of the closed-loop system.
The limitations of this work lie in two aspects: (i) it is hard to extend the results (about stochastic time-invariant systems) of this paper to stochastic time-varying systems, because of the difficulty of constructing a local weak solution for the time-varying case; (ii) Stochastic inverse dynamics studied here is required to satisfy some specifical structure (see Assumption 3 and Lemma 3).
Organization of the Paper Theorems of finite-time stability for weak solutions are presented in Section 2. The controller design problem is described in Section 3. The finite-time control design procedure is given in Section 4. The finite-time stability analysis of the closed-loop system is provided in Section 5. One numerical simulation example is presented in Section 6. Conclusions are given in Section 7.
Section snippets
Finite-time stability of weak solutions
Definition 2 For any given , associated with stochastic differential equation (1), the differential operator is defined as Mao, 1997
Now, to present the definition of global finite-time stability in probability of weak solutions, we give the following definition, which is essentially the same as the existing finite-time stability notions of stochastic systems (e.g., Definition 3.1 in Yin et al. (2011)).
Definition 3 The trivial weak solution of system (1) is said to be
Problem formulation for finite-time control
Consider where is the control input, is the state vector, is the state of stochastic inverse dynamics or unmodeled dynamics, ; Borel measurable functions are continuous and satisfy ; Borel measurable functions are continuous and satisfy
Finite-time controller design
We now consider the controller design for the -subsystem of (38). In the procedure, we use the method of adding a power integrator to construct a Lyapunov function for the -subsystem of (38).
Step 1: Take being odd in both denominator and numerator. Let and . By Assumption 1, Lemma 5, we have that for any given positive constant , there exists a function such that , where with
Finite-time stability analysis of the closed-loop system
To analyze the finite-time stability in probability of the closed-loop system composed of (38), (48), we consider the following assumption and lemma.
Assumption 3 There exist known continuous functions and satisfying and .
Lemma 3 If and then, there exists a nondecreasing positive function defined on with such that
Simulation example
Different from the conference version in Li, Zhao, and Liu (2016), where the diffusion coefficient of the subsystem is a zero function, here we consider a more general mathematical example Choose . For the -subsystem of (58), choosing , then, Assumption 2, Assumption 3 are satisfied with , , ,
Conclusions
We have established the framework of finite-time control of stochastic nonlinear systems under the sense of weak solutions. The finite-time control for a class of stochastic nonlinear systems with stochastic input-to-state stable inverse dynamics is investigated for the first time. The Lyapunov function for the whole system including the controllable dynamics and stochastic inverse dynamics is constructively designed by the method of changing supply functions and the closed-loop system is
Gui-Hua Zhao received her B.S. degree and Ph.D. degree in mathematics from Harbin Institute of Technology, China, in 2004 and 2009, respectively. Since 2009, she has been with Jiangsu University of Science and Technology, Zhenjiang, China. Now, she is a post-doctor fellow in School of Mathematics of Southeast University, Nanjing, China. Her research interest includes numerical analysis of stochastic systems, and finite-time control of stochastic nonlinear systems.
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Gui-Hua Zhao received her B.S. degree and Ph.D. degree in mathematics from Harbin Institute of Technology, China, in 2004 and 2009, respectively. Since 2009, she has been with Jiangsu University of Science and Technology, Zhenjiang, China. Now, she is a post-doctor fellow in School of Mathematics of Southeast University, Nanjing, China. Her research interest includes numerical analysis of stochastic systems, and finite-time control of stochastic nonlinear systems.
Jian-Chao Li received the B.S. degree and the M.S. degree in mathematics from Nantong University and Southeast University, China, in 2013 and 2016, respectively. Now, he is a data analyst in Shanghai Fudata Technology Limited Company.
Shu-Jun Liu received the B.S. degree in Mathematics from Sichuan University, Chengdu, China, in 1999, the M.S. degree in Operational Research and Cybernetics from the same university, in 2002, and the Ph.D. degree in Operational Research and Cybernetics from Institute of Systems Science (ISS), Chinese Academy of Sciences (CAS), Beijing, China, in 2007. From 2008 to 2009, she held a postdoctoral position in the Department of Mechanical and Aerospace Engineering, University of California San Diego. From August 2002 to August 2015, she was with Department of Mathematics of Southeast University, Nanjing, China. Since September 2015, she has been with College of Mathematics of Sichuan University, where she is now a professor. Her current research interests include stochastic control and optimization, adaptive control, and stochastic extremum seeking.
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The research was supported by National Natural Science Foundation of China (No. 61673284, No. 61807017). The material in this paper was partially presented at the 28th Chinese Control and Decision Conference, May 28. This paper was recommended for publication in revised form by Associate Editor Hiroshi Ito under the direction of Editor Daniel Liberzon.