基于多项式参数依赖Lyapunov函数的鲁棒滤波器设计
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摘要
滤波是以测量信号为基础对系统内部不可测量的信号进行估计,系统模型存在不确定情况下的滤波问题即鲁棒滤波问题。本论文研究了一类凸多面体不确定系统的鲁棒滤波器设计问题,主要包括线性系统、时滞系统及多维系统,并讨论了鲁棒滤波降低滤波保守性的问题。
首先针对标准的Kalman滤波理论无法应用于不确定系统这一问题,提出了不确定系统的鲁棒H2滤波器设计方法。在此基础上,研究了时变时滞不确定连续和离散时间系统的鲁棒H∞滤波问题,所提出的滤波器设计方法与系统中时滞的信息相关。其次,基于不确定FM模型研究了在信号处理、热能工程等领域具有较强应用背景的2D离散系统的鲁棒H∞滤波问题。基于参数依赖Lyapunov稳定思想,将鲁棒滤波器存在的条件转化为一组线性矩阵不等式的可行解问题。由于存在统一的设计规范、并存在有效的数值解法、容易对参数进行优化等优点,线性矩阵不等式方法逐渐成为处理鲁棒滤波问题的主流方法。任何鲁棒性的设计都是以牺牲系统的性能为代价的。基于二次稳定概念的鲁棒滤波结果是针对“最坏情况”设计而得到,即对所有的不确定参数都要保持系统的稳定性及相应的滤波性能,这通常具有较高的保守性。针对凸多面体不确定系统参数依赖的处理方法在一定程度上降低了设计的保守性,是近几年来鲁棒滤波领域使用最多的技巧,但并不是保守性最小的结果,进一步降低设计的保守性仍是摆在国内外学者面前的一个重要课题。
本文创造性地提出了不确定动态系统鲁棒滤波的多项式参数依赖Lya-punov函数方法,以一个统一的框架解决了凸多面体不确定系统在多种性能指标下的滤波问题,就设计的保守性而言明显优于目前该领域的最新研究成果。在航空、航天以及工业生产过程等领域,被控对象的动态特性一般都难以用精确的数学模型来描述,使得用于描述被控对象的数学模型和实际对象之间总是不可避免地存在误差,因此研究存在模型不确定性及噪声输入情况下的滤波算法具有重大的理论和实际意义,是控制和信号处理领域的前沿研究课题。
The filtering problem is one of producing an estimate of the system state usingmeasurements, while robust filtering is to estimate an unknown hidden state whenthe system model exists uncertainty. This dissertation is concerned with the problemof robust filter design for a class of convex polyhedron uncertain systems, includ-ing linear systems, time-delay systems, multi-dimensional systems, and conservatismreduction issues are discussed.
Robust filter design methods for uncertain continuous and discrete-time systemshave been proposed in Chapter 2. Based on this, Chapter 3 studies the problem of de-signing robust H∞filters for uncertain time-varying delay systems in both continuous-and discrete-time cases. Moreover, Chapter 4 considers the robust H∞filtering prob-lem for 2D discrete systems which have strong application backgrounds in signalprocessing and heat energy engineering etc. Linear matrix inequality (LMI) condi-tions are obtained for the existence of admissible filters and based on polynomiallyparameter-dependent matrices of arbitrary degree, the filter design problems are castinto convex optimizations, which can be readily solved via standard numerical soft-ware. Due to its utilization of a common Lyapunov function for the entire uncertaintydomain, quadratic stability has been well recognized to be conservative, which pre-vents robust control theory from further development and limits its applications.
This thesis, based on previous works of others, systematically and deeply investi-gates the problems of filter design for polytopic uncertain systems in both continuous-and discrete-time cases based on polynomially parameter-dependent Lyapunov func-tions, and presents analysis and synthesis methodologies for uncertain dynamic sys-tems in the unified polynomially parameter-dependent framework. In aerospace, as-tronautics and industrial process, it is usually difficult to characterize the dynamics ofthe controlled object exactly by a mathematical model, therefore the problem investi-gated in this paper is of great theoretical and practical significance.
首先针对标准的Kalman滤波理论无法应用于不确定系统这一问题,提出了不确定系统的鲁棒H2滤波器设计方法。在此基础上,研究了时变时滞不确定连续和离散时间系统的鲁棒H∞滤波问题,所提出的滤波器设计方法与系统中时滞的信息相关。其次,基于不确定FM模型研究了在信号处理、热能工程等领域具有较强应用背景的2D离散系统的鲁棒H∞滤波问题。基于参数依赖Lyapunov稳定思想,将鲁棒滤波器存在的条件转化为一组线性矩阵不等式的可行解问题。由于存在统一的设计规范、并存在有效的数值解法、容易对参数进行优化等优点,线性矩阵不等式方法逐渐成为处理鲁棒滤波问题的主流方法。任何鲁棒性的设计都是以牺牲系统的性能为代价的。基于二次稳定概念的鲁棒滤波结果是针对“最坏情况”设计而得到,即对所有的不确定参数都要保持系统的稳定性及相应的滤波性能,这通常具有较高的保守性。针对凸多面体不确定系统参数依赖的处理方法在一定程度上降低了设计的保守性,是近几年来鲁棒滤波领域使用最多的技巧,但并不是保守性最小的结果,进一步降低设计的保守性仍是摆在国内外学者面前的一个重要课题。
本文创造性地提出了不确定动态系统鲁棒滤波的多项式参数依赖Lya-punov函数方法,以一个统一的框架解决了凸多面体不确定系统在多种性能指标下的滤波问题,就设计的保守性而言明显优于目前该领域的最新研究成果。在航空、航天以及工业生产过程等领域,被控对象的动态特性一般都难以用精确的数学模型来描述,使得用于描述被控对象的数学模型和实际对象之间总是不可避免地存在误差,因此研究存在模型不确定性及噪声输入情况下的滤波算法具有重大的理论和实际意义,是控制和信号处理领域的前沿研究课题。
The filtering problem is one of producing an estimate of the system state usingmeasurements, while robust filtering is to estimate an unknown hidden state whenthe system model exists uncertainty. This dissertation is concerned with the problemof robust filter design for a class of convex polyhedron uncertain systems, includ-ing linear systems, time-delay systems, multi-dimensional systems, and conservatismreduction issues are discussed.
Robust filter design methods for uncertain continuous and discrete-time systemshave been proposed in Chapter 2. Based on this, Chapter 3 studies the problem of de-signing robust H∞filters for uncertain time-varying delay systems in both continuous-and discrete-time cases. Moreover, Chapter 4 considers the robust H∞filtering prob-lem for 2D discrete systems which have strong application backgrounds in signalprocessing and heat energy engineering etc. Linear matrix inequality (LMI) condi-tions are obtained for the existence of admissible filters and based on polynomiallyparameter-dependent matrices of arbitrary degree, the filter design problems are castinto convex optimizations, which can be readily solved via standard numerical soft-ware. Due to its utilization of a common Lyapunov function for the entire uncertaintydomain, quadratic stability has been well recognized to be conservative, which pre-vents robust control theory from further development and limits its applications.
This thesis, based on previous works of others, systematically and deeply investi-gates the problems of filter design for polytopic uncertain systems in both continuous-and discrete-time cases based on polynomially parameter-dependent Lyapunov func-tions, and presents analysis and synthesis methodologies for uncertain dynamic sys-tems in the unified polynomially parameter-dependent framework. In aerospace, as-tronautics and industrial process, it is usually difficult to characterize the dynamics ofthe controlled object exactly by a mathematical model, therefore the problem investi-gated in this paper is of great theoretical and practical significance.
引文
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9 R. C. L. F. Oliveira, P. L. D. Peres. LMI Conditions for Robust Stability AnalysisBased on Polynomially Parameter-dependent Lyapunov Functions. Systems &Control Letters. 2006, 55(1):52–61
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2 D. Peaucelle, D. Arzelier, O. Bachelier, et al. A New Robust D-stability Condi-tion for Real Convex Polytopic Uncertainty. Systems & Control Letters. 2000,40(1):21–30
3 J. C. Geromel, M. C. De Oliveira, J. Bernussou. Robust Filtering of Discrete-time Linear Systems with Parameter Dependent Lyapunov Functions. SIAM J.Control Optim. 2002, 41(3):700–711
4 H. D. Tuan, P. Apkarian, T. Q. Nguyen. Robust Filtering for Uncertain Nonlin-early Parameterized Plants. IEEE Trans. Signal Processing. 2003, 51(7):1806–1815
5 Y. Ebihara, T. Hagiwara. A Dilated LMI Approach to Continuous-time Gain-scheduled Controller Synthesis with Parameter-dependent Lyapunov Variables.Proceedings of the 4th IFAC Symposium on Robust Control Design. Milan, Italy,2003:FA01–4
6 Y. Ebihara, T. Hagiwara. New Dilated LMI Characterizations for Continuous-time Multiobjective Controller Synthesis. Automatica. 2004, 40(11):2003–2009
7 Y. Ebihara, T. Hagiwara. A Dilated LMI Approach to Robust PerformanceAnalysis of Linear Time-invariant Uncertain Systems. Automatica. 2005,41(11):1933–1941
8 K. A. Barbosa, C. E. de Souza, A. Trofino. Robust H2 Filtering for Uncertain Lin-ear Systems: LMI Based Methods with Parametric Lyapunov Functions. Systems& Control Letters. 2005, 54(3):251–262
9 R. C. L. F. Oliveira, P. L. D. Peres. LMI Conditions for Robust Stability AnalysisBased on Polynomially Parameter-dependent Lyapunov Functions. Systems &Control Letters. 2006, 55(1):52–61
10 C. W. Scherer. LMI Relaxations in Robust Control. European J. Contr. 2006,12(1):3–29
11 U. Shaked. H∞-minimum Error State Estimation of Linear Stationary Processes.IEEE Trans. Automat. Control. 1990, 35(5):554–558
12 K. M. Nagpal, P. P. Khargonekar. Filtering and Smoothing in an H∞Setting.IEEE Trans. Automat. Control. 1991, 36(2):152–166
13 I. Yaesh, U. Shaked. Game Theory Approach to Optimal Linear State Estimationand its Relation to the Minimum H∞-norm Estimation. IEEE Trans. Automat.Control. 1992, 37(6):828–831
14 C. E. de Souza, M. D. Fragoso. H∞Filtering for Markovian Jump Linear Systems.Int. J. Systems Sci. 2002, 33(11):905–915
15 M. Fu, C. E. de Souza, L. Xie. H∞Estimation for Uncertain Systems. Int. J.Robust & Nonlinear Control. 1992, 2(2):87–105
16 C. E. de Souza, L. Xie, Y. Wang. H∞Filtering for a Class of Uncertain NonlinearSystems. Systems & Control Letters. 1993, 20(6):419–426
17 E. Fridman, U. Shaked, L. Xie. Robust H∞Filtering of Linear Systems withTime-varying Delay. IEEE Trans. Automat. Control. 2003, 48(1):159–165
18 L. Xie, L. Lu, D. Zhang, et al. Improved Robust H2 and H∞Filtering for Uncer-tain Discrete-time Systems. Automatica. 2004, 40(5):873–880
19 Z. Duan, J. Zhang, C. Zhang, et al. Robust H2 and H∞Filtering for UncertainLinear Systems. Automatica. 2006, 42(11):1919–1926
20 H. Gao, J. Lam, P. Shi, et al. Parameter-dependent Filter Design with GuaranteedH∞Performance. IEE Proc. Part D: Control Theory Appl. 2005, 152(5):531–537
21 H. Gao, J. Lam, C. Wang. Mixed H2/H∞Filtering for Continuous-time Poly-topic Systems: A Parameter-dependent Approach. Circuits, Systems and SignalProcessing. 2005, 24(6):689–702
22 W. Zhang, L. Yu, X. Jiang. Delay-dependent Generalized H2 Filtering for Uncer-tain Systems with Multiple Time-varying State Delays. Signal Processing. 2007,87(4):709–724
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