复杂离散不确定系统的鲁棒滤波方法研究
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摘要
从最近十年的鲁棒滤波研究的发展来看，Krein空间方法已经逐渐被人认识并已成为鲁棒滤波方法研究的新领域。为了在现有的鲁棒滤波器研究基础上进一步降低设计复杂度，提高滤波精度，并进一步增强实用性，Krein空间滤波器设计方法已成为不可缺少的重要发展方向。因此，本文针对多种不确定线性及不确定非线性系统的Krein空间滤波方法进行研究，进一步完善Krein空间滤波器的设计方法并扩展Krein空间方法的研究领域。本文通过Matlab数值仿真实验，将Krein空间滤波器与其他的鲁棒滤波器进行对比，验证Krein空间滤波器的有效性。论文的主要工作有：
首先针对最常见的两种不确定性（即不确定参数和乘性噪声）同时存在于状态方程和量测方程的情况，在不引入任何非零因子的前提下，分别设计一种新颖的Krein空间形式系统，以消除非零因子所造成的人为误差隐患。为多种线性不确定系统提供新颖的Krein空间滤波器设计思路。
针对系统的外源随机扰动已知和未知两种情况，分别设计鲁棒Kalman滤波和鲁棒H_∞滤波。首先针对不确定系统系统中的能量和约束问题给出H2指标二次型，针对鲁棒H_∞问题给出H_∞指标的目标二次型；然后对二次型进行等价地矩阵变换运算，使得二次型的列向量包含原系统的所有结构，此时中心权重矩阵的逆即可作为Krein空间形式系统的误差方差阵；最后按照原始系统的状态空间结构和二次型的向量关系结构，设计出相应的Krein空间形式系统，并通过Matlab仿真实验进行验证。
然后，研究几类重要的线性时滞不确定系统的Krein空间滤波器设计。针对时变时滞不确定系统提出时滞无关的Krein空间滤波器设计方法，之后针对定常时滞的不确定系统设计时滞相关的Krein空间滤波器，以提高滤波精度。论文的研究涉及包括定常时滞、时变时滞、及多重时滞等多种状态时滞类型，并为多种线性时滞不确定系统提供新颖的Krein空间滤波器设计思路。
针对时变时滞与多重时滞同时存在于状态方程和量测方程的情况，论文设计一种时滞无关的滤波器；即将状态时滞项与不确定参数一起合并为Krein空间形式系统的噪声向量，并以此划分依据进行目标二次型的等价矩阵变换。
针对定常时滞系统，论文设计一种时滞相关的滤波器；即首先通过状态扩维方法将状态时滞纳入Krein空间形式系统的状态向量，然后通过给定初始条件的权重阵实现二次型中状态扩维处理，最后通过等价地矩阵变换确定Krein空间系统的形式噪声方差阵并给出Krein空间形式系统。Matlab仿真实验结果表明，论文提出的Krein空间滤波器设计方法具有良好的稳定性并且相比于其他方法精度有所提高。最后，依据之前章节中关于不确定性及时滞问题的处理方法，进一步研究非线性不确定系统的Krein空间滤波器设计。首先考查弱非线性且无状态时滞的系统，针对不确定
参数同时存在于状态方程和量测方程的情况，给出鲁棒EKF的设计方法；然后考查强非线性情况，提出了Krein空间鲁棒H_∞滤波；最后对于李普希兹式的非线性时滞不确定系统，提出一种时滞相关的鲁棒H_∞滤波器设计方法。最后，通过Matlab实验仿真验证滤波器的有效性。
针对非线性时滞不确定系统，特别提出一种无状态扩维的Krein空间滤波器设计方法，以减轻高维系统下的计算负担。系统中非线性函数的估计误差满足李普希兹条件，同时系统状态和时滞满足鲁棒能量约束。于是考查混合李普希兹非线性鲁棒二次型，并以此设计Krein空间形式系统。为了设计时滞相关滤波器，状态时滞和非线性函数都在Krein空间形式系统中分解为后验估计与后验估计误差。然后两个数值例子验证了该方法的有效性。
Reviewing the research of robust filtering design in the latest decade, Krein spaceestimation has been recognized and has been becoming into a novel researching field ofrobust filter design. Krein space approach is an essential method of filtering design to declinefilter design complexity, to enhance filter precision, and to fulfill practicality. Then this paperfocuses on Krein space approach to filter design for some classes of uncertain systems thatinvolve linear systems and nonlinear systems. Through this work, it is expected to improveKrein space method theory and to expand its research area. And the effectiveness of theproposed Krein space filter are supported through tests and comparations in numbericalexamples using Matlab. The main work of this paper is as follows.
Firstly, this paper studies novel design ideas of Krein space filtering for some classes oflinear uncertain systems. Uncertain parameter and multiplicative noise are respectivelyconsidered existing in both state and measurement equations. In premise of introducing nonon-zero factor into objective problem, novel Krein space formal state-space systems aredesigned. As a result, the hidden trouble that non-zero factor might leads to large error iseliminated.
Robust Kalman filter and robust H_∞filter are design respectively in the case of knownand unknown noise characteristics. Firstly,H2-indexed quadratic form and H_∞-indexedquadratic form are given respectively from the sum quadratic constraint(SQC) and robustH_∞problem. Secondly, the column of the objective quadratic form are expanded throughsome equivalent transformation to contain at least all vectors in the original system. In fact,the inverse of central weight matrix is the central of error Gramian in Krein space. Lastly,Krein space formal state-space system can be given according to the original system and thequadratic form. The effectiveness of the proposed filters are testified through Matlabsimulations.
Secondly, this paper proposes novel design ideas of Krein space filtering for someclasses of linear uncertain state-delay systems. Delay-independent filtering is developed totime-varying delay systems, while delay-dependent filtering is investigated to constant delaycases in order to improve filter precision. This paper researchs delay systems including casesof constant delay, time-varying delay, and multiple delays.
In the case that time-varying delay(whatever single or multiple) exists in both state and measurement equations, a novel method is proposed to design Krein space delay-independentfilter. The state-delay is regarded into the Krein space formal noise column, and accordingly,the quadratic form has to be equivalently transformed.
In the constant-delay case, a novel method is developed to design Krein spacedelay-dependent filter. The delayed state is regarded as a state vector in the augmented Kreinspace formal system. Meanwhile, state column in quadratic form is augmented throughappropriate initial conditions. Then Krein space formal system can be determined throughoriginal system and the quadratic form. The Matlab simulations show that the proposed Kreinspace filters are robustness and stable with enhancing precision.
Thirdly, this paper proposes novel design ideas of Krein space filtering for some classesof nonlinear uncertain systems. This paper investigates robust EKF for nonlinear uncertainsystems, then proposes robust H_∞filtering for the case that noise characteristics areunknown. At last, a novel method without augmenting state is attempted to designdelay-dependent filtering for nonlinear uncertain time-delay systems. The effectiveness of theproposed filters are all verified through Matlab simulations.
Considering the computational burden risk in the delay-dependent filter design for linearsystems, this paper tries to study a novel Krein space filter design without augmenting statesfor nonlinear uncertain time-delay systems. Both delayed state and nonlinear function aredevided into two parts(posteriori estimate and posteriori error), each part of which will behandle separately. Existance condition and Ricatti recursions of the proposed robust H_∞filtering are both given, and two numberical examples shows the effectiveness.
首先针对最常见的两种不确定性（即不确定参数和乘性噪声）同时存在于状态方程和量测方程的情况，在不引入任何非零因子的前提下，分别设计一种新颖的Krein空间形式系统，以消除非零因子所造成的人为误差隐患。为多种线性不确定系统提供新颖的Krein空间滤波器设计思路。
针对系统的外源随机扰动已知和未知两种情况，分别设计鲁棒Kalman滤波和鲁棒H_∞滤波。首先针对不确定系统系统中的能量和约束问题给出H2指标二次型，针对鲁棒H_∞问题给出H_∞指标的目标二次型；然后对二次型进行等价地矩阵变换运算，使得二次型的列向量包含原系统的所有结构，此时中心权重矩阵的逆即可作为Krein空间形式系统的误差方差阵；最后按照原始系统的状态空间结构和二次型的向量关系结构，设计出相应的Krein空间形式系统，并通过Matlab仿真实验进行验证。
然后，研究几类重要的线性时滞不确定系统的Krein空间滤波器设计。针对时变时滞不确定系统提出时滞无关的Krein空间滤波器设计方法，之后针对定常时滞的不确定系统设计时滞相关的Krein空间滤波器，以提高滤波精度。论文的研究涉及包括定常时滞、时变时滞、及多重时滞等多种状态时滞类型，并为多种线性时滞不确定系统提供新颖的Krein空间滤波器设计思路。
针对时变时滞与多重时滞同时存在于状态方程和量测方程的情况，论文设计一种时滞无关的滤波器；即将状态时滞项与不确定参数一起合并为Krein空间形式系统的噪声向量，并以此划分依据进行目标二次型的等价矩阵变换。
针对定常时滞系统，论文设计一种时滞相关的滤波器；即首先通过状态扩维方法将状态时滞纳入Krein空间形式系统的状态向量，然后通过给定初始条件的权重阵实现二次型中状态扩维处理，最后通过等价地矩阵变换确定Krein空间系统的形式噪声方差阵并给出Krein空间形式系统。Matlab仿真实验结果表明，论文提出的Krein空间滤波器设计方法具有良好的稳定性并且相比于其他方法精度有所提高。最后，依据之前章节中关于不确定性及时滞问题的处理方法，进一步研究非线性不确定系统的Krein空间滤波器设计。首先考查弱非线性且无状态时滞的系统，针对不确定
参数同时存在于状态方程和量测方程的情况，给出鲁棒EKF的设计方法；然后考查强非线性情况，提出了Krein空间鲁棒H_∞滤波；最后对于李普希兹式的非线性时滞不确定系统，提出一种时滞相关的鲁棒H_∞滤波器设计方法。最后，通过Matlab实验仿真验证滤波器的有效性。
针对非线性时滞不确定系统，特别提出一种无状态扩维的Krein空间滤波器设计方法，以减轻高维系统下的计算负担。系统中非线性函数的估计误差满足李普希兹条件，同时系统状态和时滞满足鲁棒能量约束。于是考查混合李普希兹非线性鲁棒二次型，并以此设计Krein空间形式系统。为了设计时滞相关滤波器，状态时滞和非线性函数都在Krein空间形式系统中分解为后验估计与后验估计误差。然后两个数值例子验证了该方法的有效性。
Reviewing the research of robust filtering design in the latest decade, Krein spaceestimation has been recognized and has been becoming into a novel researching field ofrobust filter design. Krein space approach is an essential method of filtering design to declinefilter design complexity, to enhance filter precision, and to fulfill practicality. Then this paperfocuses on Krein space approach to filter design for some classes of uncertain systems thatinvolve linear systems and nonlinear systems. Through this work, it is expected to improveKrein space method theory and to expand its research area. And the effectiveness of theproposed Krein space filter are supported through tests and comparations in numbericalexamples using Matlab. The main work of this paper is as follows.
Firstly, this paper studies novel design ideas of Krein space filtering for some classes oflinear uncertain systems. Uncertain parameter and multiplicative noise are respectivelyconsidered existing in both state and measurement equations. In premise of introducing nonon-zero factor into objective problem, novel Krein space formal state-space systems aredesigned. As a result, the hidden trouble that non-zero factor might leads to large error iseliminated.
Robust Kalman filter and robust H_∞filter are design respectively in the case of knownand unknown noise characteristics. Firstly,H2-indexed quadratic form and H_∞-indexedquadratic form are given respectively from the sum quadratic constraint(SQC) and robustH_∞problem. Secondly, the column of the objective quadratic form are expanded throughsome equivalent transformation to contain at least all vectors in the original system. In fact,the inverse of central weight matrix is the central of error Gramian in Krein space. Lastly,Krein space formal state-space system can be given according to the original system and thequadratic form. The effectiveness of the proposed filters are testified through Matlabsimulations.
Secondly, this paper proposes novel design ideas of Krein space filtering for someclasses of linear uncertain state-delay systems. Delay-independent filtering is developed totime-varying delay systems, while delay-dependent filtering is investigated to constant delaycases in order to improve filter precision. This paper researchs delay systems including casesof constant delay, time-varying delay, and multiple delays.
In the case that time-varying delay(whatever single or multiple) exists in both state and measurement equations, a novel method is proposed to design Krein space delay-independentfilter. The state-delay is regarded into the Krein space formal noise column, and accordingly,the quadratic form has to be equivalently transformed.
In the constant-delay case, a novel method is developed to design Krein spacedelay-dependent filter. The delayed state is regarded as a state vector in the augmented Kreinspace formal system. Meanwhile, state column in quadratic form is augmented throughappropriate initial conditions. Then Krein space formal system can be determined throughoriginal system and the quadratic form. The Matlab simulations show that the proposed Kreinspace filters are robustness and stable with enhancing precision.
Thirdly, this paper proposes novel design ideas of Krein space filtering for some classesof nonlinear uncertain systems. This paper investigates robust EKF for nonlinear uncertainsystems, then proposes robust H_∞filtering for the case that noise characteristics areunknown. At last, a novel method without augmenting state is attempted to designdelay-dependent filtering for nonlinear uncertain time-delay systems. The effectiveness of theproposed filters are all verified through Matlab simulations.
Considering the computational burden risk in the delay-dependent filter design for linearsystems, this paper tries to study a novel Krein space filter design without augmenting statesfor nonlinear uncertain time-delay systems. Both delayed state and nonlinear function aredevided into two parts(posteriori estimate and posteriori error), each part of which will behandle separately. Existance condition and Ricatti recursions of the proposed robust H_∞filtering are both given, and two numberical examples shows the effectiveness.
引文
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