不确定随机系统的鲁棒滤波器设计
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摘要
滤波问题在控制和信号处理领域是较为关键的问题之一。自随机系统的最优滤波理论提出之后,随机系统的Kalman滤波理论被广泛应用于通讯、航天、航空、工业过程控制等领域,但Kalman滤波要求精确的系统模型和确切已知外部干扰信号的统计特性。而实际中的许多系统,或者其精确模型很难获取,或者外部干扰统计特性未知,甚至系统存在漂移现象等等,这些系统不确定性都将引起滤波发散。在这种情况下,研究改善的滤波算法,如鲁棒H∞滤波,鲁棒L2—L∞滤波等具有重要意义。
随机系统在实际中是广泛存在的,它含内部随机参数、外部随机干扰和观测噪声等随机变量。这些随机变量无法用已知的时间函数来描述,而只能了解其某些统计特性,其系统状态也无法通过观测来确定。很多实际系统的随机因素是不容忽视的,因此,研究不确定随机系统的滤波器设计问题是一个热点课题。
本文针对几类非线性不确定随机系统的鲁棒滤波器设计问题展开研究,其主要工作概括如下：
1)研究一类不确定随机T-S模糊系统的鲁棒L2—L∞滤波器设计问题。目标是设计一个依赖于模糊规则的滤波器,使其既能保证滤波误差系统的鲁棒稳定性,又能满足所要求的L2—L∞性能指标。基于线性矩阵不等式(LMI)方法,通过选取依赖于权重的模糊李亚普诺夫函数,得到问题可解性的充分条件。
2)研究一类非线性不确定随机系统的依赖于时滞的L2—L∞滤波器设计问题。系统模型中的非线性不确定项是向量有界的,这是一种比李普希兹条件更为宽泛的条件,因此该模型更具代表性和实用性。通过引入一个随机积分不等式,设计出一个随机L2—L∞滤波器,保证滤波误差系统对所有的容许不确定性是均方渐进稳定的,且满足所要求的L2—L∞性能指标。
3)研究一类非线性不确定随机中立时变时滞系统的鲁棒无源滤波器设计问题。所考虑的系统包含时变和范数有界的参数不确定性,随机干扰和向量有界非线性干扰。基于LMI方法和伊藤微分规则,通过构建恰当的李亚普诺夫函数,设计出一个全阶滤波器使其对所有可容许的不确定性,滤波误差系统是鲁棒稳定的并满足给定的无源性能指标。
Filtering problem has long been one of important fields in control and signal processing. Since the optimal filtering theory of stochastic system was proposed, the theory of Kalman filtering for stochastic systems is widely used in fields of telecommunications, aerospace, aviation, industrial process control and so on, but Kalman filtering requires that system model under consideration is accurately known and the external interference signals are with known statistical properties. However, in many actual systems, the exact system model or the statistical properties of the unknown external disturbances is difficult to obtain, or the system may drift, all of which may result in uncertainties. In such cases, research on improving the filtering algorithms, such as robust H∞, robust L2-L∞filtering, is of great significance.
Stochastic system is a class of systems that widely used in practice, which contains a series of stochastic variables, such as internal stochastic parameters, external stochastic disturbances and observation noise. These stochastic variables can not be described by known function of time, we can only understand some of its statistical properties, and also the system states can not be determined by observation. In practice, in many cases many stochastic factors of the system can not be ignored, so study on the filtering design problem for uncertain stochastic system is of great significance.
In this thesis, we will investigate the robust filter design problems for several classes of nonlinear uncertain stochastic systems, the main work are generalized as follows.
1) The robust L2-L∞filtering problem for a class of nonlinear uncertain stochastic delay systems-stochastic fuzzy delay systems is investigated. Attention is focused on the design of a fuzzy-rule-dependent filter that ensuring both the robust stability and a prescribed L2- L∞performance level of the filtering error system. An approach called weighting-dependent is adopted and delay-dependent sufficient conditions for the solvability of the problem are presented based on the linear matrix inequalities (LMIs) approach.
2) The problem of robust L2-L∞filtering for a class of nonlinear uncertain stochastic systems is addressed. The nonlinear uncertainties in the model are vector-bounded as an extension to the usual Lipschitz conditions, which makes the model more representative and practical. By proposing a stochastic integral inequality, we designed a stochastic L2-L∞filter that ensuring both the asymptotic mean-square stability and a prescribed L2-L∞performance level of the filtering error system. Sufficient conditions which can guarantee the existence of the desired filter are also obtained by using the Lyapunov stability theory.
3) The robust passive filtering for a class of uncertain stochastic neutral systems with nonlinear perturbations is investigated. The system under consideration contain time varying but norm bounded parameter uncertainties, stochastic disturbance, time-varying delay and vector-bounded nonlinearities. By combining ltd's differential rule with the stochastic Lyapunov stability theory, we design a full-order filter such that the dynamics of the filtering error system are guaranteed to be regular, robustly asymptotically stable in the mean square for all admissible uncertainties and nonlinearities, and the proposed passive performance is satisfied.
随机系统在实际中是广泛存在的,它含内部随机参数、外部随机干扰和观测噪声等随机变量。这些随机变量无法用已知的时间函数来描述,而只能了解其某些统计特性,其系统状态也无法通过观测来确定。很多实际系统的随机因素是不容忽视的,因此,研究不确定随机系统的滤波器设计问题是一个热点课题。
本文针对几类非线性不确定随机系统的鲁棒滤波器设计问题展开研究,其主要工作概括如下：
1)研究一类不确定随机T-S模糊系统的鲁棒L2—L∞滤波器设计问题。目标是设计一个依赖于模糊规则的滤波器,使其既能保证滤波误差系统的鲁棒稳定性,又能满足所要求的L2—L∞性能指标。基于线性矩阵不等式(LMI)方法,通过选取依赖于权重的模糊李亚普诺夫函数,得到问题可解性的充分条件。
2)研究一类非线性不确定随机系统的依赖于时滞的L2—L∞滤波器设计问题。系统模型中的非线性不确定项是向量有界的,这是一种比李普希兹条件更为宽泛的条件,因此该模型更具代表性和实用性。通过引入一个随机积分不等式,设计出一个随机L2—L∞滤波器,保证滤波误差系统对所有的容许不确定性是均方渐进稳定的,且满足所要求的L2—L∞性能指标。
3)研究一类非线性不确定随机中立时变时滞系统的鲁棒无源滤波器设计问题。所考虑的系统包含时变和范数有界的参数不确定性,随机干扰和向量有界非线性干扰。基于LMI方法和伊藤微分规则,通过构建恰当的李亚普诺夫函数,设计出一个全阶滤波器使其对所有可容许的不确定性,滤波误差系统是鲁棒稳定的并满足给定的无源性能指标。
Filtering problem has long been one of important fields in control and signal processing. Since the optimal filtering theory of stochastic system was proposed, the theory of Kalman filtering for stochastic systems is widely used in fields of telecommunications, aerospace, aviation, industrial process control and so on, but Kalman filtering requires that system model under consideration is accurately known and the external interference signals are with known statistical properties. However, in many actual systems, the exact system model or the statistical properties of the unknown external disturbances is difficult to obtain, or the system may drift, all of which may result in uncertainties. In such cases, research on improving the filtering algorithms, such as robust H∞, robust L2-L∞filtering, is of great significance.
Stochastic system is a class of systems that widely used in practice, which contains a series of stochastic variables, such as internal stochastic parameters, external stochastic disturbances and observation noise. These stochastic variables can not be described by known function of time, we can only understand some of its statistical properties, and also the system states can not be determined by observation. In practice, in many cases many stochastic factors of the system can not be ignored, so study on the filtering design problem for uncertain stochastic system is of great significance.
In this thesis, we will investigate the robust filter design problems for several classes of nonlinear uncertain stochastic systems, the main work are generalized as follows.
1) The robust L2-L∞filtering problem for a class of nonlinear uncertain stochastic delay systems-stochastic fuzzy delay systems is investigated. Attention is focused on the design of a fuzzy-rule-dependent filter that ensuring both the robust stability and a prescribed L2- L∞performance level of the filtering error system. An approach called weighting-dependent is adopted and delay-dependent sufficient conditions for the solvability of the problem are presented based on the linear matrix inequalities (LMIs) approach.
2) The problem of robust L2-L∞filtering for a class of nonlinear uncertain stochastic systems is addressed. The nonlinear uncertainties in the model are vector-bounded as an extension to the usual Lipschitz conditions, which makes the model more representative and practical. By proposing a stochastic integral inequality, we designed a stochastic L2-L∞filter that ensuring both the asymptotic mean-square stability and a prescribed L2-L∞performance level of the filtering error system. Sufficient conditions which can guarantee the existence of the desired filter are also obtained by using the Lyapunov stability theory.
3) The robust passive filtering for a class of uncertain stochastic neutral systems with nonlinear perturbations is investigated. The system under consideration contain time varying but norm bounded parameter uncertainties, stochastic disturbance, time-varying delay and vector-bounded nonlinearities. By combining ltd's differential rule with the stochastic Lyapunov stability theory, we design a full-order filter such that the dynamics of the filtering error system are guaranteed to be regular, robustly asymptotically stable in the mean square for all admissible uncertainties and nonlinearities, and the proposed passive performance is satisfied.
引文
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[2]Y. THEODOR, U. SHAKED, C. E. DE SOUZA. A Game Theory Approach to Robust Discrete-time H∞ Estimation. IEEE Trans. Signal Processing,1994,42:1486-1495
[3]M. FU, C. E. DE SOUZA, L. XIE. H∞ Estimation for Uncertain Systems. International Jour-nal of Robust and Nonlinear Control,1992,2:87-105
[4]D. A. WILSON. Convolution and Hankel Operator Norms for Linear Systems. IEEE Transac-tions on Automatic Control,1989,34(1):94-97
[5]RM. PALHARES, P. L. D. PERES. Robust Filtering with Guaranteed Energy-to-peak Performance-an LMI Approach. Automatica,2000,36(4):851-858
[6]H. J. GAO, C. H. WANG. New Approach to Robust l2-l∞ and H∞ Filtering for Uncertain Discrete Time Systems. Science in China (Series F),2003,46(5):355-358
[7]H. J. GAO, C. H. WANG. New Approach to Robust l2-l∞ Filter Design for Uncertain Continous-time Systems. Acta Automatica Sinica,2003,29(6):809-813
[8]H. J. GAO, C. H. WANG and Y. H. LI. Robust, l2-l∞ Filter Design for Uncertain Discrete-time State-delayed Systems. ACTA Automatica Sinica,2003,29(5):666-672
[9]C. LI, B. T. CUI. Delay-dependent Passive Control of Linear Systems with Nonlinear Perturba-tion. Journal of Systems Engineering and Electronics,2008,19(2):346-350
[10]Z. H. GUAN, H. ZHANG, S. H. YANG. Robust Passive Control for Internet-based Switching Systems with Time Delay. Chaos, Solitons and Fractals,2008,36(2):479-486
[11]C. LI, B.T. CUI. Robust Passive Control of Uncertain Linear Neutral Systems with Time-delays. Mathematica Application,2007,20(2):361-369
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