鲁棒滤波问题的若干研究
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摘要
状态滤波问题一直是控制理论界和工程应用中备受关注的领域之一。在许多实际应用中，不仅对外部干扰噪声信号的统计特性缺乏了解，而且系统模型本身存在一定范围的摄动，即外部干扰和系统有不确定性，因此鲁棒滤波问题一直是控制理论和应用研究的热点。在过去几十年里，很多学者对这一问题做了研究并给出大量有意义的结果，但是对关联时滞系统的鲁棒滤波问题，非线性系统的鲁棒滤波问题还没有得到全面的研究。
本文主要的工作是：
对鲁棒滤波理论的发展历程和研究现状给予了全面系统的介绍，重点论述鲁棒滤波理论的发展过程中各个阶段具有代表性的算法和结论，对鲁棒滤波理论今后面临的问题和发展进行了展望。
对一类时变时滞不确定线性系统的鲁棒滤波问题，提出一种改进算法，给出一个新的Lyapunov-Krasoviskii函数，利用有界实引理，设计出依赖时滞的H_∞滤波器。
对不确定中立时滞系统本文运用Riccati方程方法，通过解一对Riccati方程就可以得到满足条件的滤波器，而没有一个循环求解和判断的过程。考虑到线性矩阵不等式具有方便有效的计算方法，采用线性矩阵不等式(LMI)作为设计工具，给出了满足条件的H_∞滤波器存在的充分条件，同时给出了滤波器的具体表示形式。
对带有分布时滞的中立系统的鲁棒滤波的研究还不充分，因此对这一系统的研究还很有必要，本文运用不等式的放大技巧，降低了对Lyapunov函数求导数时进行放大所带来的保守性。对系统中的范数有界不确定性，本文将它做为外部扰动来处理，使得在证明有界实引理的过程中减少了对不确定性项的放大带来的保守性从而得到性能更好的H_∞滤波器，并且很好的解决了当测量方程中也含有分布时滞时的系统的滤波器设计问题。通过引入一种慢速类型的状态变换，将增广状态系统变为广义系统，在此系统上设计满足性能指标的H_∞滤波器，数值算例验证了设计方法的有效性和低保守性。
本文另一部分主要工作是对非线性系统的鲁棒滤波加以研究。本文主要是用Takagi-Sugeno(T-S)模糊模型来描述各类非线性系统，分别研究了连续T-S模糊时滞系统和离散不确定T-S模糊时滞系统的依赖时滞的H_∞滤波器存在的充分条件，因在设计过程中充分考虑了时滞而使我们的设计方法得到提高。然后基于广义T-S模糊系统研究了一类广义非线性时滞系统的鲁棒H_∞滤波，该模糊滤波器能够确保滤波误差系统的正则、无脉冲和渐近稳定性并能满足给定的H_∞性能指标。
State filtering has been attracting much attention as one of important fields in controltheory and engineering application. In many practical applications, however, the statisticalproperties of exogenous noise signals are rarely known and there are perturbations insystem model, that is, exogenous disturbances and uncertainties in system model, so therehas been rapidly growing interests in robust filtering. In the past decades, robust filteringwas studied and amount of meaningful results were obtained, but, the delay-dependentrobust filtering for time-delay systems and robust filtering for nonlinear systems have notbeen fully investigated.
The main content of this thesis can be briefly described as follows:
The history and research status of robust filtering are introduced systematically, theemphasis is focused on the representative algorithms and results in the development ofrobust filtering theory and we forecast the problem and development in the future.
We developed a robust filtering algorithm for a class of time-varying delayuncertainty linear systems, a new Lyapunov-Krasoviskii functional is given andbounded-real lemma is used in order to design a delay-dependent H_∞filter.
The Riccati equation method is used for uncertain neutral delay system, the suitablefilter can be obtained by solving a pair of Riccati equation, the cycle of solving andjudging is omitted. Considering the convenient and effective algorithm of LMI, we use itas a design tool. A sufficient condition for the existence of H_∞filter is given andformulated in terms of linear matrix inequalities.
The study of neutral delay systems with distributed delays is not sufficient, so it is stillessential to study this system. Some over-bounding techniques of inequality were proposedthat reduces the conservativeness coming from the derivative of Lyapunov function. Thenorm bounded uncertainty in system is regarded as exogenous disturbance, so theconservativeness is degraded in the proof of BRL and better H_∞index is obtained, what'more, the filtering problem is solved when there are distributed delays in measurementequation. An approach introduces a slow-type state transformation and is based onrepresenting the augmented system by a descriptor system and designing a H_∞filterwhich guarantees the performance level for the descriptor model. The numerical exampleis provided to demonstrate the effectiveness of the proposed method.
The other important content is about robust filtering for nonlinear system. T-S fuzzy model is used to describe kinds of nonlinear systems. The delay-dependent H_∞filter forcontinuous time delay T-S fuzzy model and discrete time delay uncertainty T-S fuzzymodel are studied respectively, the sufficient condition is given, and the method isdeveloped by considering time delays. We studied robust H_∞filtering for a descriptornonlinear time-delay system based on T-S fuzzy model. The fuzzy filter can insure theerror filtering system is regular, impulse-free and asymptotically stable.
本文主要的工作是：
对鲁棒滤波理论的发展历程和研究现状给予了全面系统的介绍，重点论述鲁棒滤波理论的发展过程中各个阶段具有代表性的算法和结论，对鲁棒滤波理论今后面临的问题和发展进行了展望。
对一类时变时滞不确定线性系统的鲁棒滤波问题，提出一种改进算法，给出一个新的Lyapunov-Krasoviskii函数，利用有界实引理，设计出依赖时滞的H_∞滤波器。
对不确定中立时滞系统本文运用Riccati方程方法，通过解一对Riccati方程就可以得到满足条件的滤波器，而没有一个循环求解和判断的过程。考虑到线性矩阵不等式具有方便有效的计算方法，采用线性矩阵不等式(LMI)作为设计工具，给出了满足条件的H_∞滤波器存在的充分条件，同时给出了滤波器的具体表示形式。
对带有分布时滞的中立系统的鲁棒滤波的研究还不充分，因此对这一系统的研究还很有必要，本文运用不等式的放大技巧，降低了对Lyapunov函数求导数时进行放大所带来的保守性。对系统中的范数有界不确定性，本文将它做为外部扰动来处理，使得在证明有界实引理的过程中减少了对不确定性项的放大带来的保守性从而得到性能更好的H_∞滤波器，并且很好的解决了当测量方程中也含有分布时滞时的系统的滤波器设计问题。通过引入一种慢速类型的状态变换，将增广状态系统变为广义系统，在此系统上设计满足性能指标的H_∞滤波器，数值算例验证了设计方法的有效性和低保守性。
本文另一部分主要工作是对非线性系统的鲁棒滤波加以研究。本文主要是用Takagi-Sugeno(T-S)模糊模型来描述各类非线性系统，分别研究了连续T-S模糊时滞系统和离散不确定T-S模糊时滞系统的依赖时滞的H_∞滤波器存在的充分条件，因在设计过程中充分考虑了时滞而使我们的设计方法得到提高。然后基于广义T-S模糊系统研究了一类广义非线性时滞系统的鲁棒H_∞滤波，该模糊滤波器能够确保滤波误差系统的正则、无脉冲和渐近稳定性并能满足给定的H_∞性能指标。
State filtering has been attracting much attention as one of important fields in controltheory and engineering application. In many practical applications, however, the statisticalproperties of exogenous noise signals are rarely known and there are perturbations insystem model, that is, exogenous disturbances and uncertainties in system model, so therehas been rapidly growing interests in robust filtering. In the past decades, robust filteringwas studied and amount of meaningful results were obtained, but, the delay-dependentrobust filtering for time-delay systems and robust filtering for nonlinear systems have notbeen fully investigated.
The main content of this thesis can be briefly described as follows:
The history and research status of robust filtering are introduced systematically, theemphasis is focused on the representative algorithms and results in the development ofrobust filtering theory and we forecast the problem and development in the future.
We developed a robust filtering algorithm for a class of time-varying delayuncertainty linear systems, a new Lyapunov-Krasoviskii functional is given andbounded-real lemma is used in order to design a delay-dependent H_∞filter.
The Riccati equation method is used for uncertain neutral delay system, the suitablefilter can be obtained by solving a pair of Riccati equation, the cycle of solving andjudging is omitted. Considering the convenient and effective algorithm of LMI, we use itas a design tool. A sufficient condition for the existence of H_∞filter is given andformulated in terms of linear matrix inequalities.
The study of neutral delay systems with distributed delays is not sufficient, so it is stillessential to study this system. Some over-bounding techniques of inequality were proposedthat reduces the conservativeness coming from the derivative of Lyapunov function. Thenorm bounded uncertainty in system is regarded as exogenous disturbance, so theconservativeness is degraded in the proof of BRL and better H_∞index is obtained, what'more, the filtering problem is solved when there are distributed delays in measurementequation. An approach introduces a slow-type state transformation and is based onrepresenting the augmented system by a descriptor system and designing a H_∞filterwhich guarantees the performance level for the descriptor model. The numerical exampleis provided to demonstrate the effectiveness of the proposed method.
The other important content is about robust filtering for nonlinear system. T-S fuzzy model is used to describe kinds of nonlinear systems. The delay-dependent H_∞filter forcontinuous time delay T-S fuzzy model and discrete time delay uncertainty T-S fuzzymodel are studied respectively, the sufficient condition is given, and the method isdeveloped by considering time delays. We studied robust H_∞filtering for a descriptornonlinear time-delay system based on T-S fuzzy model. The fuzzy filter can insure theerror filtering system is regular, impulse-free and asymptotically stable.
引文
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