二维奇异系统扩展Roesser模型的H_∞控制和滤波
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摘要
本文主要研究了二维奇异系统扩展Roesser模型的H_∞控制、H_∞滤波以及正实控制和最小能量控制。这里提到的二维奇异系统扩展Roesser模型是二维奇异系统一般模型的特例,它包含二维奇异系统Roesser模型通过线性变换得到的所有系统。
首先,本文研究了二维奇异系统扩展Roesser模型的H_∞控制问题,给出了存在动态输出反馈控制器的充分条件。其次,本文研究了该系统的H_∞滤波问题,设计滤波器能够保证指定的H_∞噪声衰减性能指标,分别研究了基于观测器结构的二维滤波器和一般形式的二维滤波器,给出了基于LMI的充分条件。再次,本文研究了二维奇异系统扩展Roesser模型的正实控制,设计了一个动态控制器使得闭环系统渐近稳定且延展严格正实,给出了存在正实控制器的充分条件。接下来,本文研究了二维奇异系统扩展Roesser模型的最小能量控制,找到一控制序列,该控制序列将初始状态转移到期待的最终状态,且使得给定的性能指标最小,从而满足最小能量控制定义。另外,本文还研究了二维系统Roesser模型的鲁棒保代价控制。设计了一个静态状态反馈控制器使得闭环系统是渐近稳定的且闭环代价函数小于指定的上界,给出了存在控制器的充分条件。所有的这些结论都是基于LMIs的,因此非常方便计算机仿真。并且我们在每一部分都给出了相应的数值算例。
This paper studied H_∞control problems, H_∞filtering problems as well as positive real control, the minimum energy control of the 2-D extended singular Roesser model. 2-D extended singular Roesser model mentioned above is a particular case of the general singular model of 2-D systems. This class of systems includes all the systems resulted from the linear state transformations of 2-D singular Roesser models.
First of all, we studied H_∞control problems of the 2-D extended singular Roesser model and gave a sufficient condition of existence for dynamic output feedback controller. Secondly, we studied H_∞filtering problems of the 2-D extended singular Roesser model, designed a dynamical output feedback controller to achieve asymptotic stability and H_∞performance. The 2-D H_∞filtering problem is investigated for both the 2-D filters of an observer-based structure and a general form. We gave a sufficient condition based on the LMI. Third, we studied positive real control problems of the 2-D extended singular Roesser model. The purpose of this paper is to design a dynamic output feedback controller such that the resulting closed-loop system is asymptotically stable and the closed-loop system transfer function from the disturbance to the controlled output is extended strictly positive real. We obtained a sufficient condition for the existence of the desired output feedback controllers in terms of LMI. Fourth, we studied minimum energy control of the 2-D extended singular Roesser model, found a sequence of input u(i,j) which transfers extended singular Roesser model from the initial local state to the desired final local state. In addition, we considered the problem of the guaranteed cost control for a class of two-dimensional discrete systems described by the Roesser model. A linear matrix inequality (LMI)-based criterion for the existence of robust guaranteed cost controller is established. Such controller renders the closed-loop system asymptotically stable for all admissible uncertainties and guarantees an adequate level of performance. We gave a sufficient condition for the existence of the desired output feedback controllers. These conclusions are based on the LMI, therefore it very convenient for computer demonstration. Finally, we provide a numerical example to demonstrate the applicability of the proposed approach in each section.
首先,本文研究了二维奇异系统扩展Roesser模型的H_∞控制问题,给出了存在动态输出反馈控制器的充分条件。其次,本文研究了该系统的H_∞滤波问题,设计滤波器能够保证指定的H_∞噪声衰减性能指标,分别研究了基于观测器结构的二维滤波器和一般形式的二维滤波器,给出了基于LMI的充分条件。再次,本文研究了二维奇异系统扩展Roesser模型的正实控制,设计了一个动态控制器使得闭环系统渐近稳定且延展严格正实,给出了存在正实控制器的充分条件。接下来,本文研究了二维奇异系统扩展Roesser模型的最小能量控制,找到一控制序列,该控制序列将初始状态转移到期待的最终状态,且使得给定的性能指标最小,从而满足最小能量控制定义。另外,本文还研究了二维系统Roesser模型的鲁棒保代价控制。设计了一个静态状态反馈控制器使得闭环系统是渐近稳定的且闭环代价函数小于指定的上界,给出了存在控制器的充分条件。所有的这些结论都是基于LMIs的,因此非常方便计算机仿真。并且我们在每一部分都给出了相应的数值算例。
This paper studied H_∞control problems, H_∞filtering problems as well as positive real control, the minimum energy control of the 2-D extended singular Roesser model. 2-D extended singular Roesser model mentioned above is a particular case of the general singular model of 2-D systems. This class of systems includes all the systems resulted from the linear state transformations of 2-D singular Roesser models.
First of all, we studied H_∞control problems of the 2-D extended singular Roesser model and gave a sufficient condition of existence for dynamic output feedback controller. Secondly, we studied H_∞filtering problems of the 2-D extended singular Roesser model, designed a dynamical output feedback controller to achieve asymptotic stability and H_∞performance. The 2-D H_∞filtering problem is investigated for both the 2-D filters of an observer-based structure and a general form. We gave a sufficient condition based on the LMI. Third, we studied positive real control problems of the 2-D extended singular Roesser model. The purpose of this paper is to design a dynamic output feedback controller such that the resulting closed-loop system is asymptotically stable and the closed-loop system transfer function from the disturbance to the controlled output is extended strictly positive real. We obtained a sufficient condition for the existence of the desired output feedback controllers in terms of LMI. Fourth, we studied minimum energy control of the 2-D extended singular Roesser model, found a sequence of input u(i,j) which transfers extended singular Roesser model from the initial local state to the desired final local state. In addition, we considered the problem of the guaranteed cost control for a class of two-dimensional discrete systems described by the Roesser model. A linear matrix inequality (LMI)-based criterion for the existence of robust guaranteed cost controller is established. Such controller renders the closed-loop system asymptotically stable for all admissible uncertainties and guarantees an adequate level of performance. We gave a sufficient condition for the existence of the desired output feedback controllers. These conclusions are based on the LMI, therefore it very convenient for computer demonstration. Finally, we provide a numerical example to demonstrate the applicability of the proposed approach in each section.
引文
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[3]K.Galkowski,"State space realization of linear 2D systems with extension to the general n-D(n＞2)case",Springer Verlag,London 2001.
[4]T.Kaczorek,"Polynomial and rational matrices:Applications in dynamical systems theory",Springer,London 2006.
[5]T.Kaczorek,"Positive 1D and 2D systems",Springer Verlag,London 2002.
[6]Yun Zou,S.L.Campbell,"The jump behavior and stability analysis for 2-D singular systems",Multidimensional Systems and Signal Processing,11(2000),321-338.
[7]Chenxiao Cai,Weiqun Wang,Yun Zou,"A note on the internal stability for 2-D acceptable linear singular discrete systems",Multidimensional Systems & Signal Processing,15(3),197-204,2004.
[8]Yun Zou,Weiqun Wang,S.Xu,L.Xie,"Analysis and Control of the Jump Modes Behavior of 2-D Singular Systems Part Ⅱ:Regular Observer and Compensator Design," Systems and control letters,2006.
[9]Yun Zou,Weiqun Wang,S.Xu,L.Xie,"Analysis and control of the jump modes behavior of 2-D singular systems Part Ⅰ:structural stability",Systems and control letters,2006.
[10]Yun Zou,Chengwu Yang,"The realization of 2-D singular systems",Control Theory and Applications,16(1999),445-447.
[11]M.Sebek,“H_∞ Problem of 2-D Systems,"European Control Conference'93,1993,pp.1476-1479.
[12]C.Du,L.Xie,and C.Zhang,"H_∞ Control and Robust Stabilization of Two-Dimensional Systems in Roesser Models,"Automatica,37(2),2001,pp205-211.
[13]C.Du,L.Xie,and Y.C.Soh,"H_∞ Filtering of 2-D Discrete Systems",IEEE Trans.Signal Processing,vol.48,no.6,2000,pp.1760-1768.
[14]C.Du,L.Xie and C.Zhang,"Solutions for H_∞ Filtering of Two-Dimensional Systems," Multidimensional Systems and Signal Processing,vol.11,2000,pp.301 -320.
[15]L.Xie,C.Du,Y.C.Soh,and C.Zhang,"H_∞ and Robust Control of 2-D Systems in FM Second Model",Multidimensional Systems and Signal Processing,vol.13,2002,pp.265-287.
[16]Y.Ran,L.Xie,and C.Zhang,"Robust mixed H_2/ H_∞ control of 2-dimensional systems in Roesser model",2004 5th Asian Control Conference,v 2,2004 5th Asian Control Conference,2004,p1032-1040
[17]J.Xu,Y.Li,"H_∞ control of 2-D discrete state delay systems",International Journal of Control,Automation and Systems,v 4,n 4,August,2006,p 516-523.
[18]Tuan,H.D,Apkarian,P,Nguyen,T.Q.;Narikiyo T."Robust mixed H_2/ H_∞ filtering of 2-D systems",IEEE Transactions on Signal Processing,v 50,n 7,July,2002,p 1759-1771
[19]Gao,H.,Lain,J.,Wang,C.,and Xu,S.(2004)."Robust H∞ filtering for 2-D stochastic systems",Circuits,Systems and Signal Processing,23(6),479-505.
[20]Xu,H.,Lu,J.,and Zou,Y.(2006)."Observer-based H∞ filtering of 2-D singular system described by Roesser models",Acta Automatica Sinica,32,213-218.
[21]Lo Wu,Z.Wang,H.Gao and C,Wang,"Filtering for uncertain 2-D discrete systems with state delays",Signal Processing,v 87,n 9,September,2007,p 2213-2230
[22]Wen,J.T.,1988,"Time domain and frequency domain conditions for positive realness",IEEE Transactions on Automatic Control,33,988-992.
[23]Haddad,W.M.,and Bernstein,D.S.,1991,"Robust stabilization with positive real uncertainty:beyond the small gain theorem",Systems Control Letters,17,191-208.
[24]Haddad,W.M.,and Bernstein,Do S.,1994,"Explicit construction of quadratic Lyapunov functions for the small gain,positivity,circle,and Popov theorems and their application to robust stability.Part Ⅱ:discrete-time theory",International Journal of Robust and Nonlinear Control,4,249-265.
[25]Sun,Wo,Khargonekar,P.P.,and Shim,D.,"Solution to the positive real control problem for linear time-invariant systems",IEEE Transactions on Automatic Control,1994,39,2034-2046.
[26]Xie,L.,and Soh,Y.C.,1995,"Positive real control for uncertain linear time-invariant systems",Systems Control Letters,24,265-271.
[27]Vidyasagar,M.,"Nonlinear Systems Analysis",(Englewood Cliffs,NJ:Prentice-Hall)1993
[28]Mahmoud,M.S.,and Xie,L.,2000,"Positive real analysis and synthesis of uncertain discrete time systems",IEEE Transactions on Circuits Systems,I,47,403-406.
[29]Haddad,W.M.,and Bernstein,D.S.,1993,"Explicit construction of quadratic Lyapunov functions for the small gain,positivity,circle,and Popov theorems and their application to robust stability.Part I:continuous-time theory",International Journal of Robust and Nonlinear Control,3,313-339.
[30]S.Xu,J.Lam,Z.Lin and Krzysztof Galkowski."Positive Real Control for Uncertain Two-Dimensional Systems",IEEE Transactions on Circuits and Systems-I:Fundamental Theory and Applications,vol.49,No.11,November 2002.
[31]V.M.Popov,"Hyperstability of Control Systems",Berlin:Springer-Verlag,1973.
[32]K.S.Narendra and J.H.Taylor,"Frequency Domain Criteria for Absolute Stability",New York:Academic,1973.
[33]B.D.O.Anderson and S.Vongpanitlerd,"Network Analysis and Synthesis:A Modern Systems Theory Approach",Englewood Cliffs,NJ:Prentice-Hall,1973.
[34]M.G.Safonov and R.Y.Chiang,"Rea Vcomplex K,-synthesis without curve fitting,"in Control and Dynamic Systems,C.T.Leondes,Ed.New York:Academic,1993.
[35]P.Mohnder and J.C.Willems,"Synthesis of state feedback control laws with a specified gain and phase margin," IEEE Trans.Automat.Contr.,vol.AC-25,no.5,pp.928-931,Oct.1980.
[36]M.G.Safonov,E.A.Jonckheere,M.Verma,and D.J.N.Limebeer,"Synthesis of positive real multivariable feedback systems," Int.J.Contr.,vol.45,no.3,pp.817-842,1987.
[37]W.M.Haddad and D.S.Bemstein,"Parameter-dependent Lyapunov functions,constant real parameter uncertainty and the Popov criterion in robust analysis and synthesis," in Proc.30th IEEE Conf Decis.Contr,Brighton,UK,1991,pp.2274-2279,2632-2633.
[38]M.Araki,"Improved M-matrix condition for the existence of positivereal weighting,"Syst.Contr.Lett.,vol.6,pp.1-5,1985.
[39]Y.Jia,W.Gao and M.Cheng,"Robust strict positive real stabilization criteria for SISO uncertain systems," Syst.Contr.Lett.,vol.19,no.2,pp.111-117,1992.
[40]E.Noldus,"Design of robust state feedback laws," Int.J.Contr.,vol.35,no.6,pp.935-944,1982.
[41]Chang,S.S.L.,and Peng,T.K.C."Adaptive guaranteed cost control of systems with uncertain parameters".IEEE trans.Automat.Control,1972,AC- 17,474-483.
[42]Haddad,W.M.,& Bemstein,D.S."Robust reduced-order,nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds".IEEE 1rans.Automat.Control,1988,AC-33,591-595.
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