随机非线性系统的控制器设计和闭环性能分析

doi:10.7523/j.issn.2095-6134.2010.5.020

摘要/Abstract

摘要：

针对几类重要的随机非线性系统, 提出了一些新的概念,发展了一些基本分析工具, 研究了几类控制器的设计问题. 主要成果包括:(1) 针对一类部分动态不可量测的非线性随机系统,引入了随机输入状态稳定(SISS)的概念, 借助于分析概率理论,发展了随机系统改变能量函数方法, 成功地处理了随机微分中的伊藤项,给出了随机非线性串联系统SISS的小增益类条件. (2) 对一类具有SISS随机逆动态的大规模随机非线性系统,给出了分散自适应输出反馈镇定控制器的构造性设计方法. 既解决了实用镇定问题也解决了渐近镇定问题. 在分散控制框架内,给出了处理随机非线性逆动态的方法. (3) 对一类具有不稳定零动态的随机非线性系统,引入了随机输入状态可镇定的概念,给出了全局输出反馈镇定控制器构造性设计方法. (4) 对一类具有线性增长的不可量测状态的随机非线性系统,针对方差未知的噪声和一般随机输入,引入了广义随机输入状态稳定(GSISS)的概念,分别给出了随机干扰抑制和渐近镇定的输出反馈控制器的构造性设计方法.(5) 对一般的时滞随机非线性系统, 给出了解存在唯一的判定条件,引入了依概率全局(渐近)稳定的概念及相应的判定准则,丰富了随机时滞非线性系统的控制器设计理论. 对一类不确定随机时变时滞系统,构造性地设计出了自适应输出反馈镇定控制器.

关键词: 随机非线性系统, 零动态, 逆动态, 输出反馈, 积分反推法, 镇定控制器, 严格反馈, 随机输入状态稳定

Abstract:

For several classes of important stochastic nonlinear systems we develop new concepts and basic analysis tools, based on which some problems of controller design are investigated and solved. The main contents are the following. (1) For a class of stochastic nonlinear systems with unmeasurable dynamics, the concept of stochastic input-to-state stability (SISS) is introduced. By using the probability theory, a method of changing supply functions is developed in the stochastic framework, and a small-gain condition on the SISS is given for cascaded stochastic nonlinear systems. (2) For a class of large-scale stochastic nonlinear systems with SISS stochastic inverse dynamics, a decentralized adaptive stabilization controller based on output feedback is constructively designed. Both practical stabilization and asymptotical stabilization are solved. A method dealing with the stochastic nonlinear inverse dynamics is given in the framework of decentralized control. (3) For a class of stochastic nonlinear systems with unstable zero dynamics, the concept of stochastic input-to-state stabilization is introduced, and by using this concept a global stabilization controller based on output feedback is constructively designed. (4) For a class of stochastic nonlinear systems with linearly bounded unmeasurable states, the concept of generalized stochastic input-to-state stability is introduced with respect to the noises with unknown covariance and the general stochastic inputs, and by using this concept, output-feedback controllers are constructively designed for stochastic disturbance attenuation and asymptotically global stabilization, respectively. (5) For general stochastic nonlinear systems with time delays, some conditions on the existence and uniqueness of the solution are provided, global (asymptotical) stability in probability is introduced, and the corresponding criterion is obtained. Besides, for a class of uncertain stochastic nonlinear systems with time-varying time delays, an adaptive output-feedback stabilization controller is constructed.

Key words: stochastic nonlinear systems, zero dynamic, inverse dynamic, output feedback, integrator backstepping method, stabilization controller, strict-feedback, stochastic input-to-state stability

中图分类号:

O231
TP13

刘淑君, 张纪峰. 随机非线性系统的控制器设计和闭环性能分析[J]. 中国科学院大学学报, 2010, 27(5): 711-719.

LIU Shu-Jun, ZHANG Ji-Feng. Controller design for stochastic nonlinear systems and performance analysis of the closed-loop systems[J]. , 2010, 27(5): 711-719.

参考文献

[1] Isidori A. Nonlinear control systems
[M]. 3rd ed. London: Springer, 1995.

[2] Kanellakopoulos I, Kokotovic P V, More A S. Adaptive feedback linearization of nonlinear systems //Foundations of Adaptive Control. Berlin: Springer, 1991: 311-346.

[3] Frazzoli E, Dahleh M A, Feron E. Trajectory tracking control design for autonomous helicopters using a backstepping algorithm
[J]. Proceedings of the American Control Conference, 2000(6): 4102-4107.

[4] Roosta A R, Georges D, Hadj-Said N. Nonlinear control for power systems based on a backstepping method
[J]. Proceedings of the 40th IEEE Conference on Decision and Control, 2001(4): 3037-3042.

[5] Zhang Y P, Fidan B, Ioannou P A. Backstepping control of linear time-varying systems with known and unknown parameters
[J]. IEEE Transactions on Automatic Control, 2003,48(11): 1908- 1925.

[6] Mazenc F, Bowong S. Backstepping with bounded feedbacks for time varying systems
[J]. SIAM Journal on Control and Optimization, 2004,43(3): 856-871.

[7] Kushner H J. Stochasitic stability and control
[M]. New York: Academic, 1967.

[8] Deng H, Krstic M, Williams R J. Stabilization of stochastic nonlinear systems driven by noise of unknown covariance
[J]. IEEE Transactions on Automatic Control, 2001, 46(8): 1237-1252.

[9] Florchinger P. A university formula for the stabilization of control stochastic differential equations
[J]. Stochastic Analysis and Application, 1993, 11: 155-162.

[10] Florchinger P, Iggidr A, Sallet G. Stabilization of a class of nonlinear stochastic systems
[J]. Stochastic Processes and Application, 1994, 50: 235-243.

[11] Florchinger P. Lyapunov-like techniques for stochatic stability
[J]. SIAM Journal on Control and Optimization, 1995, 33(4): 1151-1169.

[12] Florchinger P. Feedback stabilization of affine in the control stochastic differential systems by the control Lyapunov function method
[J]. SIAM Journal on Control and Optimization, 1997, 35(2): 500-511.

[13] Pan Z G, Basar T. Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion
[J]. SIAM Journal on Control and Optimization, 1999, 37(3): 957-995.

[14] Liu Y G, Zhang J F. Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics
[J]. SIAM Journal on Control and Optimization, 2006, 45(3): 885-926.

[15] Deng H, Krstic M. Output-feedback stochastic nonlinear stabilization . IEEE Transactions on Automatic Control, 1999, 44(2): 328-333.

[16] Deng H, Krstic M. Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance
[J]. Systems and Control Letters, 2000, 39(1): 173-182.

[17] Pan Z G, Liu Y G, Shi S J. Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero dynamics
[J]. Science in China: Series F, 2001, 44(4): 292-308.

[18] Liu Y G, Pan Z G, Shi S J. Output feedback control design for strict feedback stochastic nonlinear systems under a risk-sensitive cost
[J]. IEEE Transactions on Automatic Control, 2003, 48(3): 509-513.

[19] Pan Z G. Canonical forms for stochastic nonlinear systems
[J]. Automatica, 2002, 38(7): 1163-1170.

[20] Krstic M, Kanellakopoulos I, Kokotovic P V. Nonlinear and Adaptive Control Design
[M]. New York: Wiley, 1995.

[21] Liu S J, Zhang J F, Jiang Z P. A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems
[J]. ACTA Mathematicae Applicatae Sinica: English Series, 2008, 24(1): 141-156.

[22] Liu S J, Zhang J F, Jiang Z P. Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems
[J]. Automatica, 2007, 43(2): 238-251.

[23] Liu S J, Jiang Z P, Zhang J F. Global output-feedback stabilization for a class of stochastic non-minimum phase nonlinear systems
[J]. Automatica, 2008, 44(8): 1944-1957.

[24] Liu S J, Zhang J F. Output-feedback control of a class of stochastic nonlinear systems with linearly bounded unmeasurable states
[J]. International Journal of Robust and Nonlinear Control, 2008, 18(6): 665-687.

[25] Liu S J, Ge S S, Zhang J F. Adaptive output-feedback control for a class of uncertain stochastic nonlinear systems with time delays
[J]. International Journal of Control, 2008, 81(8): 1210-1220.