非线性不确定时滞系统的鲁棒性研究
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摘要
由于时滞和不确定性在实际工程中的广泛应用背景及分析与综合的复杂性,因此对不确定时滞系统的鲁棒稳定和鲁棒控制进行研究具有重要的理论价值和实际意义。但作为更一般的问题,也是更复杂的问题,非线性不确定时滞系统鲁棒控制的研究还没有得到充分的发展,目前非线性不确定时滞系统的鲁棒稳定性研究成为当前控制界研究的热点和难点之一。
基于Lyapunov稳定性理论、矩阵不等式技巧和非线性处理方法等,结合LMI技术,本文比较系统的研究了带非线性干扰和范数有界不确定参数的时滞系统的鲁棒稳定性和鲁棒控制,状态反馈H_∞鲁棒等问题。主要研究工作的内容包括:
(1)研究了一类带非线性扰动和范数有界不确定参数的状态滞后系统的时滞无关鲁棒稳定性判据和鲁棒镇定问题,首先给出了自由系统的鲁棒渐近稳定时滞无关的充分条件,然后通过构造无记忆状态反馈控制器,给出了满足系统镇定的时滞无关充分条件。结果表示为线性矩阵不等式(LMI)形式,容易数值处理,最后给出仿真算例,验证了所得结果的有效性。
(2)研究了一类带非线性扰动和范数有界不确定参数的状态滞后系统的时滞相关鲁棒稳定性判据和鲁棒镇定问题,提出了新的鲁棒可镇定判据和相应的鲁棒无记忆状态反馈控制器设计方法,导出的时滞相关结果以线性矩阵不等式(LMI)形式表示,最后以一个算例说明其用法,并与现有文献结果比较以显示其有效性。
(3)针对一类带非线性扰动和输入时滞的非线性不确定时滞系统,基于适当形式的Lyapunov泛函,利用线性矩阵不等式(LMI)方法,讨论了时滞无关的鲁棒H_∞状态反馈控制器设计问题,其中不确定项是范数有界的,非线性扰动满足线性约束,且控制器存在的充分条件由线性矩阵不等式的形式给出。最后给出一个具体算例说明了该方法的有效性。
(4)研究了一类带非线性扰动的多时变时滞不确定系统的时滞相关鲁棒H_∞控制问题,其中不确定项是范数有界的,非线性扰动满足线性约束。基于Lyapunov-Krasovskii泛函,根据矩阵不等式技术和非线性处理方法,讨论了时滞相关的鲁棒H_∞状态反馈控制器设计问题,且控制器存在的充分条件由线性矩阵不等式的形式给出。最后给出了两个示例,表明本文提出的方法与存在的文献结果相比,具有较少的保守性和良好的有效性。
Because of the abundant applied background of Uncertainty and time-delay in practice and the complexity of analysis and synthesis, the study of robust stability analysis and robust control of uncertain time-delay system has an important value in theory and practice. However, to be the general and complex issue, the research of uncertain nonlinear system with time-delay has not yet been fully developed. So it is one of the hotspots and also the difficulties in control domain.
In terms of Lyapunov stability theory, matrix inequality tactics, and the method for dealing with nonlinear term, some problems are systematically studied in this paper , such as robust stability analysis and robust control, state feedback H_∞robust control of time-delay systems with nonlinear perturbations and norm-bounded uncertain. Main studies include as following:
(1) Delay-dependent robust stability criterion and robust stabilization problem for a class of time-delay system with nonlinear perturbations and norm-bounded uncertain. First of all, the sufficient condition of freedom system' robust asymptotic stability of delay-independent is given, then constructing a state feedback controller with memoryless. Result is expressed as linear matrix inequality (LMI), it is easy for numerical processing. Finally, the simulation example is given to verify the validity of the result.
(2) Delay-dependent robust stability criterion and robust stabilization problem for a class of time-delay system with nonlinear perturbations and norm-bounded uncertain. New robust stability criterion and a sufficient condition for the existence of memoryless robust stabilizing control law are derived. The delay-dependent results are given in terms of linear matrix inequality (LMI). It is easy for numerical processing. Finally, the description of its usage is given in form of an example. The result's effectiveness is indicated by comparing with the existing literature.
(3) A delay-independent robust H_∞state feedback control is designed for a class of nonlinear uncertain time-delay system with nonlinear perturbations and the input delay. The linear matrix inequalities are used based on an appropriate Lyapunov function. The uncertainties are norm-bounded, and the nonlinear perturbations meet linear constraints. A sufficient condition independent on the delay of the state and input are presented for the existence of H_∞controller. The proposed controller stabilizes closed-loop uncertain systems and guarantees a prescribed H_∞norm bound of closed-loop transfer matrix from the disturbance to controlled output. By solving a linear matrix inequality, the robust H_∞controller is obtained. Examples are given to show the effectiveness of the proposed method.
(4)Delay-Dependent Robust H_∞Controller Design of uncertain time-varying or time-invariant delay system with nonlinear perturbation is studied. The uncertainties are norm-bounded, and the nonlinear perturbations meet linear constraints. Based on Lyapunov-Krasovskii functional and matrix inequality methods, sufficient conditions dependent on the delays of the state are presented for the existence of H_∞controller. By solving a linear matrix inequality, the robust H_∞controller is obtained. In the end, two examples are given to illustrate that the present method is less conservative and more effective than the existing ones.
基于Lyapunov稳定性理论、矩阵不等式技巧和非线性处理方法等,结合LMI技术,本文比较系统的研究了带非线性干扰和范数有界不确定参数的时滞系统的鲁棒稳定性和鲁棒控制,状态反馈H_∞鲁棒等问题。主要研究工作的内容包括:
(1)研究了一类带非线性扰动和范数有界不确定参数的状态滞后系统的时滞无关鲁棒稳定性判据和鲁棒镇定问题,首先给出了自由系统的鲁棒渐近稳定时滞无关的充分条件,然后通过构造无记忆状态反馈控制器,给出了满足系统镇定的时滞无关充分条件。结果表示为线性矩阵不等式(LMI)形式,容易数值处理,最后给出仿真算例,验证了所得结果的有效性。
(2)研究了一类带非线性扰动和范数有界不确定参数的状态滞后系统的时滞相关鲁棒稳定性判据和鲁棒镇定问题,提出了新的鲁棒可镇定判据和相应的鲁棒无记忆状态反馈控制器设计方法,导出的时滞相关结果以线性矩阵不等式(LMI)形式表示,最后以一个算例说明其用法,并与现有文献结果比较以显示其有效性。
(3)针对一类带非线性扰动和输入时滞的非线性不确定时滞系统,基于适当形式的Lyapunov泛函,利用线性矩阵不等式(LMI)方法,讨论了时滞无关的鲁棒H_∞状态反馈控制器设计问题,其中不确定项是范数有界的,非线性扰动满足线性约束,且控制器存在的充分条件由线性矩阵不等式的形式给出。最后给出一个具体算例说明了该方法的有效性。
(4)研究了一类带非线性扰动的多时变时滞不确定系统的时滞相关鲁棒H_∞控制问题,其中不确定项是范数有界的,非线性扰动满足线性约束。基于Lyapunov-Krasovskii泛函,根据矩阵不等式技术和非线性处理方法,讨论了时滞相关的鲁棒H_∞状态反馈控制器设计问题,且控制器存在的充分条件由线性矩阵不等式的形式给出。最后给出了两个示例,表明本文提出的方法与存在的文献结果相比,具有较少的保守性和良好的有效性。
Because of the abundant applied background of Uncertainty and time-delay in practice and the complexity of analysis and synthesis, the study of robust stability analysis and robust control of uncertain time-delay system has an important value in theory and practice. However, to be the general and complex issue, the research of uncertain nonlinear system with time-delay has not yet been fully developed. So it is one of the hotspots and also the difficulties in control domain.
In terms of Lyapunov stability theory, matrix inequality tactics, and the method for dealing with nonlinear term, some problems are systematically studied in this paper , such as robust stability analysis and robust control, state feedback H_∞robust control of time-delay systems with nonlinear perturbations and norm-bounded uncertain. Main studies include as following:
(1) Delay-dependent robust stability criterion and robust stabilization problem for a class of time-delay system with nonlinear perturbations and norm-bounded uncertain. First of all, the sufficient condition of freedom system' robust asymptotic stability of delay-independent is given, then constructing a state feedback controller with memoryless. Result is expressed as linear matrix inequality (LMI), it is easy for numerical processing. Finally, the simulation example is given to verify the validity of the result.
(2) Delay-dependent robust stability criterion and robust stabilization problem for a class of time-delay system with nonlinear perturbations and norm-bounded uncertain. New robust stability criterion and a sufficient condition for the existence of memoryless robust stabilizing control law are derived. The delay-dependent results are given in terms of linear matrix inequality (LMI). It is easy for numerical processing. Finally, the description of its usage is given in form of an example. The result's effectiveness is indicated by comparing with the existing literature.
(3) A delay-independent robust H_∞state feedback control is designed for a class of nonlinear uncertain time-delay system with nonlinear perturbations and the input delay. The linear matrix inequalities are used based on an appropriate Lyapunov function. The uncertainties are norm-bounded, and the nonlinear perturbations meet linear constraints. A sufficient condition independent on the delay of the state and input are presented for the existence of H_∞controller. The proposed controller stabilizes closed-loop uncertain systems and guarantees a prescribed H_∞norm bound of closed-loop transfer matrix from the disturbance to controlled output. By solving a linear matrix inequality, the robust H_∞controller is obtained. Examples are given to show the effectiveness of the proposed method.
(4)Delay-Dependent Robust H_∞Controller Design of uncertain time-varying or time-invariant delay system with nonlinear perturbation is studied. The uncertainties are norm-bounded, and the nonlinear perturbations meet linear constraints. Based on Lyapunov-Krasovskii functional and matrix inequality methods, sufficient conditions dependent on the delays of the state are presented for the existence of H_∞controller. By solving a linear matrix inequality, the robust H_∞controller is obtained. In the end, two examples are given to illustrate that the present method is less conservative and more effective than the existing ones.
引文
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[2]Zames.G.Functional analysis applied to nonlinear feedback systems.IEEE Tran s.Circuits and Systems,1963,Vol.cs-10:392-404
[3]Kalman.R.Z..When is a linear control system optimal?Trans.ASME,ser.D.Basic Engr.,1964,Vol.86:51-60
[4]Rosenbrock,H.H..The stability of multivariable systems.IEEE Trans.Automat.Co ntr.,1976,Vol.AC-17:105-107
[5]Youla,D C.,Jabr,H.A.and Bongiomo.J.J.Modem wiener Hopf design of optimal controllers,part Ⅱ:the multivariable case.IEEE Trans.Automat.Contr.,1976,Vol.A C-21:319-338
[6]Kucera,V.Discrete linear control:the polynomial equation approach.New York Wiley,1979.
[7]Safonobv M.G.Stability and robustness of multivariable feedback systems.MIT,press,1980.
[8]Zames.G.Feedback and opthnal sensitivity:model reference transformations,multi plicative seminorms and approximate inverses.IEEE Trans.Automat.Contr.,1981,Vol.AC-26:301-320
[9]周朝霞.不确定时滞非线性系统的鲁棒稳定性研究[D].南昌大学:2007.
[10]Hua C C,Guan X P,Shi P.Robust stabilization of a class of nonlinear time-delay systems[J].Applied Mathematics and Computation,2004,155(3):737-752
[11]Xi L,Souza C E.Criteria for robust stability and stabilization of uncertain linear systems with state delay[J].Automatica,1997,33(9):1657-1662.
[12]Luo JS,Van Denbosch P P J.Independent delay stability criteria for uncertain linear state space models[J].Automatica,1997,33(2):171-179.
[13]Trinh H,Aldeen M.On the stability of linear systems with delayed perturbations[J].IEEE Transaction on Automatic Control,1994,39(29):1948-1951.
[14]Xi L,Souza C E.Robust stabilization and H_∞ control of uncertain linear time delay systems Proceeding of IFAC 13th Triennial world Congress[C].San Francisco USA:1996.113-118.
[15]Xi L,Souza C.E.Delay-dependent robust stability and stabilization of uncertain linear delay systems:A linear matrix inequality approach[J].IEEE Transaction on Automatic Control.,1997,42(8):1144-1148.
[16]Vladimir L K,Caniel M A.On delay dependent stability conditions[J].Systems and Control Letters,2000,40(1):71-76.
[17]Wang J C,Shao H H.Delay-dependent robust and reliable H_∞ control for uncertain time-delayed systems with actuator[J].Journal of the Franklin Institute,2000,33(6):781-791.
[18]Yan J J,Tsai Jason S H.,Kung F C.A new result on the robust stability of uncertain systems with time-varying delay.IEEE Trans.Circ.and Syst.(I):Fundamental Theory and Application.,2001,48(7):914-916.
[19]Su T J,Lu C Y,Jong G.J.An LMI approach for robust stability of linear uncertain systems with time-varying multiple state delays.Proceedings of the IEEE Conference on Decision and Control,2000,2:1507-1508.
[20]Lee Y S,Moon Y S,Kwon W H,et al.Delay-dependent robust H_∞ control for uncertain systems with a state-delay[J].Automatica,2004,40:65-72.
[21]Xu S Y,Lam J,Zou Y.New results on delay-dependent robust H_∞ control for systems with time-varying delays[J].Automatica,2006,42:343-348.
[22]吴敏,张凌波,何勇.具有未知上界时滞状态扰动的非线性系统自适应鲁棒镇定[J].控制理论与应用,2005,22(1):114-117.
[23]A J Van der Schaft.Gain Analysis of Nonlinear Systems and Nonlinear State Feedback Control.IEEE Transactions on Automatic Control,1992,37(6):770-784.
[24]关新平,赵宇翔,刘奕昌.非线性系统的鲁棒控制.燕山大学学报,2001,25(1):1-8.
[25]Kwon O M,Park J H.Robust H_∞ Filtering for Uncertain Time-Delay Systems:Matrix Inequality Approach[J].Journal of Optimization Theory and Applications,2006,129(2):309-324.
[26]聂宏,赵军.一类非线性不确定时滞系统的混杂状态反馈H_∞鲁棒控制[J].控制理论与应用,2005,22(4):567-572.
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