针对随机控制系统数值解的均方指数输入状态稳定性

祝乔; 崔家瑞; 胡广大

doi:10.3724/SP.J.1004.2013.01360

针对随机控制系统数值解的均方指数输入状态稳定性

doi: 10.3724/SP.J.1004.2013.01360

1.
北京科技大学自动化学院 100083

计量
- 文章访问数: 1910
- HTML全文浏览量: 82
- PDF下载量: 1469
- 被引次数: 0
出版历程
- 收稿日期: 2012-05-29
- 修回日期: 2012-12-20
- 刊出日期: 2013-08-20

Mean-Square Exponential Input-to-State Stability of Numerical Solutions for Stochastic Control Systems

1.
School of Automation & Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China

More Information

Corresponding author: ZHU Qiao

摘要

摘要: 分析了随机控制系统数值解的均方指数输入状态稳定性. 首先, 针对随机控制系统, 随机θ-方法满足有限时间强收敛条件. 然后, 我们证实, 在有限时间强收敛条件下, 随机控制系统是均方指数输入状态稳定的当且仅当随机θ-方法(充分小步长)是均方指数输入状态稳定的. 另外, 对一类满足单边Lipschitz条件的随机控制系统, 有两类隐式欧拉方法(对任意步长)能够继承原系统的均方指数输入状态稳定性. 最后, 一些数值实例证实了本文所获结论的正确性.
- 均方指数输入状态稳定性 /
- 随机控制系统 /
- 随机θ-方法 /
- 强收敛性
Abstract: This paper deals with the mean-square exponential input-to-state stability (exp-ISS) of numerical solutions for stochastic control systems (SCSs). Firstly, it is shown that a finite-time strong convergence condition holds for the stochastic θ-method on SCSs. Then, we can see that the mean-square exp-ISS of an SCS holds if and only if that of the stochastic θ-method (for suffciently small step sizes) is preserved under the finite-time strong convergence condition. Secondly, for a class of SCSs with a one-sided Lipschitz drift, it is proved that two implicit Euler methods (for any step sizes) can inherit the mean-square exp-ISS property of the SCSs. Finally, numerical examples confirm the correctness of the theorems presented in this study.
- Mean-square exponential input-to-state stability /
- stochastic control systems /
- stochastic θ-method /
- strong convergence

HTML全文

参考文献(20)

[1]	Sontag E D. Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control, 1989, 34(4): 435-443
[2]	Angeli D. An almost global notion of input-to-state stability. IEEE Transactions on Automatic Control, 2004, 49(6): 866-874
[3]	Sontag E D, Wang Y. On characterizations of the input-to-state stability property. Systems & Control Letters, 1995, 24(5): 351-359
[4]	Sontag E D, Wang Y. New characterizations of input-to-state stability. IEEE Transactions on Automatic Control, 1996, 41(9): 1283-1294
[5]	Xie W X, Wen C Y, Li Z G. Input-to-state stabilization of switched nonlinear systems. IEEE Transactions on Automatic Control, 2001, 46(7): 1111-1116
[6]	Yeganegar N, Pepe P, Dambrine M. Input-to-state stability of time-delay systems: a link with exponential stability. IEEE Transactions on Automatic Control, 2008, 53(6): 1526-1531
[7]	Sun F L, Guan Z H, Zhang X H, Chen J C. Exponential-weighted input-to-state stability of hybrid impulsive switched systems. IET Control Theory and Applications, 2012, 6(3): 430-436
[8]	Liu J, Liu X Z, Xie W C. Input-to-state stability of impulsive and switching hybrid systems with time-delay. Automatica, 2011, 47(5): 899-908
[9]	Zhao P, Feng W, Kang Y. Stochastic input-to-state stability of switched stochastic nonlinear systems. Automatica, 2012, 48(10): 2569-2576
[10]	Spiliotis J, Tsinias J. Notions of exponential robust stochastic stability, ISS and their Lyapunov characterizations. International Journal of Robust and Nonlinear Control, 2003, 13(2): 173-187
[11]	Tsinias J. The concept of exponential input to state stability for stochastic systems and applications to feedback stabilization. Systems & Control Letters, 1999, 36(3): 221-229
[12]	Tsinias J. Stochastic input-to-state stability and applications to global feedback stabilization. International Journal of Control, 1998, 71(5): 907-930
[13]	Higham D J, Mao X R, Stuart A M. Exponential mean square stability of numerical solutions to stochastic differential equations. LMS Journal of Computation and Mathematics, 2003, 6: 297-313
[14]	Higham D J. Mean-square and asymptotic stability of the stochastic theta method. SIAM Journal on Numerical Analysis, 2000, 38(3): 753-769
[15]	Mao X R. Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations. Journal of Computational and Applied Mathematics, 2007, 200(1): 297-316
[16]	Saito Y, Mitsui T. Stability analysis of numerical schemes for stochastic differential equations. SIAM Journal on Numerical Analysis, 1996, 33(6): 2254-2267
[17]	Hu G D, Liu M Z. Input-to-state stability of Runge-Kutta methods for nonlinear control systems. Journal of Computation and Applied Mathematics, 2007, 205(1): 633-639
[18]	Zhu Q, Hu G D, Zeng L. Mean-square exponential input-to-state stability of Euler-Maruyama method applied to stochastic control systems. Acta Automatica Sinica, 2010, 36(3): 406-411
[19]	Dekker K, Verwer J G. Stability of Runge-Kutta Methods for Stiff Nonlinear Equations. Amsterdam: North-Holland, 1984
[20]	Higham D J, Mao X R, Stuart A M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM Journal on Numerical Analysis, 2002, 40(3): 1041-1063