不确定线性时滞系统鲁棒控制问题的研究
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摘要
线性矩阵不等式(Linear Matrix Inequality—LMI)以其易于同系统的性能指标或约束条件相结合的特点，已成为解决许多鲁棒控制问题的一个重要方法。本文基于Lyapunov稳定性原理，采用LMI方法研究了状态空间描述下的不确定线性时滞系统的鲁棒性能分析与鲁棒控制器设计问题，主要内容如下：
(1)对于不确定线性时变时滞系统，不需对系统模型进行变换，直接采用新提出的二次型积分不等式，导出了基于LMI的不确定线性时变时滞系统的鲁棒稳定性判据，并据此给出该系统的状态反馈鲁棒控制律和动态输出反馈鲁棒控制律设计新方法；以及系统鲁棒可靠控制的状态反馈控制律设计新方法。
(2)研究了一类不确定线性时滞系统对于给定的二次型性能目标函数的状态反馈保性能控制问题，给出了具有状态时滞信息的保性能控制律的设计方法；研究了当系统的状态不能被观测或被严重污染时基于观测器的保性能控制问题，给出了最优动态输出反馈控制律的设计方法。
(3)对于不确定线性时变时滞系统的γ次优H_∞的控制问题，提出了采用LMI方法、针对不具有状态时滞信息和具有状态时滞信息的两类γ次优H_∞状态反馈控制律设计的新方法；通过γ的极小化，给出了系统最优H_∞控制律的设计方法。
(4)针对具有圆盘极点约束的鲁棒D稳定控制问题，给出了鲁棒D稳定控制律设计的新方法；当系统的状态信息不可测或者被严重干扰时，通过设计动态输出反馈控制器，给出了该系统的动态输出鲁棒D稳定控制律的设计新方法。
(5)将新提出的二次型积分不等式结合LMI方法推广到具有非线性关联的Lurie系统中，给出了该系统鲁棒绝对稳定的新判据，据此给出了系统对于不具有状态时滞信息和具有状态时滞信息两类静态状态反馈鲁棒绝对稳定控制律设计的新方法。
对于本文提出的新方法进行了计算机仿真研究，验证了本文方法的有效性。
Linear Matrix Inequality (LMI) has become a very important method to solve many kinds of robust control problems by combining with performance index or constraints of the systems easly. This master thesis focuses on some robust performance analysis and robust controller' design with LMI approaches together with Lyapunov stability theory, and the main works are summarized as the following five aspects:
(1) A robust stability criterion for uncertain linear time-varying systems based on LMI is proposed adopting directly the new quadratic term integral inequality without model transformation. Further, a robust state-feedback controller and a dynamic-output-feedback controller are derived by the criterion, respectively, a robust reliable state-feedback controller's design method is provided also.
(2) The problem of state-feedback guaranteed cost control for uncertain linear time delay systems is investigated for given quadratic object function, and a design method of guaranteed cost controller with state delay information is derived. When system's state can not be observed or disturbed badly by noise, the problem of guaranteed cost control based on state observer for the system is considered and an optimal dynamic-output-feedback controller's design method is given.
(3) After the research of γ suboptimal H_∞ control for uncertain linear time-varying
delay, two type of γ suboptimal H_∞ controller's designing methods are proposed with state delay or without state delay information, father, an optimal H_∞ controller's designing method is derived by the minimization of parameter γ.
(4) A type of robust D stabilization controller's designing method is proposed after the research of D stability with disk pole control for uncertain linear systems. When system's states can not be observed or be disturbed badly, a robust dynamic-output-feedback D stabilization controller is proposed by dynamoic-output-feedback controller.
(5) The combination of new proposed quadratic term integral inequality and LMI technology are extended to nonlinear-interconned Lurie system, a new robustly absolute stability criterion is proposed. Two types of robustly absolute stability criteria with state delay or without state delay information are proposed based on the new obtained stability criterion.
Computer simulations have been done for the proposed methods of this thesis, and obtained results show the effectiveness of methods.
(1)对于不确定线性时变时滞系统，不需对系统模型进行变换，直接采用新提出的二次型积分不等式，导出了基于LMI的不确定线性时变时滞系统的鲁棒稳定性判据，并据此给出该系统的状态反馈鲁棒控制律和动态输出反馈鲁棒控制律设计新方法；以及系统鲁棒可靠控制的状态反馈控制律设计新方法。
(2)研究了一类不确定线性时滞系统对于给定的二次型性能目标函数的状态反馈保性能控制问题，给出了具有状态时滞信息的保性能控制律的设计方法；研究了当系统的状态不能被观测或被严重污染时基于观测器的保性能控制问题，给出了最优动态输出反馈控制律的设计方法。
(3)对于不确定线性时变时滞系统的γ次优H_∞的控制问题，提出了采用LMI方法、针对不具有状态时滞信息和具有状态时滞信息的两类γ次优H_∞状态反馈控制律设计的新方法；通过γ的极小化，给出了系统最优H_∞控制律的设计方法。
(4)针对具有圆盘极点约束的鲁棒D稳定控制问题，给出了鲁棒D稳定控制律设计的新方法；当系统的状态信息不可测或者被严重干扰时，通过设计动态输出反馈控制器，给出了该系统的动态输出鲁棒D稳定控制律的设计新方法。
(5)将新提出的二次型积分不等式结合LMI方法推广到具有非线性关联的Lurie系统中，给出了该系统鲁棒绝对稳定的新判据，据此给出了系统对于不具有状态时滞信息和具有状态时滞信息两类静态状态反馈鲁棒绝对稳定控制律设计的新方法。
对于本文提出的新方法进行了计算机仿真研究，验证了本文方法的有效性。
Linear Matrix Inequality (LMI) has become a very important method to solve many kinds of robust control problems by combining with performance index or constraints of the systems easly. This master thesis focuses on some robust performance analysis and robust controller' design with LMI approaches together with Lyapunov stability theory, and the main works are summarized as the following five aspects:
(1) A robust stability criterion for uncertain linear time-varying systems based on LMI is proposed adopting directly the new quadratic term integral inequality without model transformation. Further, a robust state-feedback controller and a dynamic-output-feedback controller are derived by the criterion, respectively, a robust reliable state-feedback controller's design method is provided also.
(2) The problem of state-feedback guaranteed cost control for uncertain linear time delay systems is investigated for given quadratic object function, and a design method of guaranteed cost controller with state delay information is derived. When system's state can not be observed or disturbed badly by noise, the problem of guaranteed cost control based on state observer for the system is considered and an optimal dynamic-output-feedback controller's design method is given.
(3) After the research of γ suboptimal H_∞ control for uncertain linear time-varying
delay, two type of γ suboptimal H_∞ controller's designing methods are proposed with state delay or without state delay information, father, an optimal H_∞ controller's designing method is derived by the minimization of parameter γ.
(4) A type of robust D stabilization controller's designing method is proposed after the research of D stability with disk pole control for uncertain linear systems. When system's states can not be observed or be disturbed badly, a robust dynamic-output-feedback D stabilization controller is proposed by dynamoic-output-feedback controller.
(5) The combination of new proposed quadratic term integral inequality and LMI technology are extended to nonlinear-interconned Lurie system, a new robustly absolute stability criterion is proposed. Two types of robustly absolute stability criteria with state delay or without state delay information are proposed based on the new obtained stability criterion.
Computer simulations have been done for the proposed methods of this thesis, and obtained results show the effectiveness of methods.
引文
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[2] Black H S. Stabilized feedback amplifiers. U.S.: Patent.1927.
[3] Nyquist H. Regeneration theory. Bell Syst. Tech. J., 1932. 11: 126-147.
[4] Bode H W. Network analysis and feedback amplifier design. Princeton,NJ: Van Nostrand.1945.
[5] Horowitz. Systhesis of feedback systems. New York: Academic Press.1945.
[6] 梅生伟，申铁龙，刘康志．现代鲁棒控制理论与应用．北京：清华大学出版社．2003．
[7] Davison E J. The output of linear time invariant multivariable systems with unmeasurable arbitarary disturbances. IEEE Trans. Automat. Contr., 1971.17: 621-630.
[8] Pearson J B,Staats P W Jr. Robust controllers for linear regulators. IEEE Trans. Automat. Contr., 1974. 19:231-234.
[9] Zames G. Functional analysis applied to nonlinear feedback systems. IEEE Trans. on Circuit Theory, 1963. 10: 392-404.
[10] Kaiman R E. When is a linear control system optimaL Trans. ASME, 1964. 86: 51-60.
[11] Doyle J C. Analysis of feedback systems with structured uncertainties, in IEE Proc. 1982:242-251.
[12] Boyd S. Linear matrix inequalities in systems and control. PA: SIAM. 1994.
I13] Qu Z. Robust control of nonlinear uncertain systems. New York: John Wiley & Sons. 1998.
[14] Willems J C. Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans., 1971. 16(1): 621-634.
[15] Nesterov Y. Nemirovskii A. A general approach to polynomial-time algorithms design for convex programming, in Centr. Econ & Math. 1988. Moscow.
[16] Laurent El Ghaoui. Francois Qustry, Mustapha Aitrami. A cone complementarity linearization algorithm for static output-feedback and related problems, in Proceeding of the 1996 IEEE international symposium on computer-aided control systems design. 1996. Dearborn.
[17] Hale J.K. Theory of functional differential equations. New York: Spring-Verlag. 1977.
I18] Yu Y. On stabilizing uncertain linear delay systems. J. Optim. Theory Appl., 1983.41(3): 503-508.
[19] Thowsen A. Uniform ultimate boundedness of the solutions of uncertain dynamoic delay systems with state-dependent and memoryless feedback controL Int. J. Control, 1983.37:1135-1143.
[20] 俞立．不确定线性时滞系统的稳定化控制器设计．控制理论与应用，1991．8(1)：68-73．
[21] Shen J.C.,Kung EC. Memoryless stabilization of uncertain dynamic delay systems: Riccati equation approach. IEEE Trans. Automat. Contr., 1991.36(5): 638-640.
[22] Petersen I.R. Hollot C.V., A Riccati equation approach to the stabilization of uncertain linear systems. Automatica, 1986. 22(4): 95-102.
[23] Hale J. Theory of function Differential Equations. New York: Springer- Verlag.1997.
[24] Xi Li. Carlos E,De Souza, Criteria for robust stability and stabilization of uncertain linear systems with state delay. Automatica, 1997. 33(9): 1657-1662.
[25] 张先明，吴敏，何勇．不确定线性多时变时滞系统的时滞相关鲁棒控制．控制与决策，2004．19(5)： 496-500．
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