广义系统的状态估计与不定二次控制
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摘要
本文研究线性广义系统的两个基本问题:状态估计和线性二次最优控制.广义系统是动态系统的一般描述形式,是比正常状态空间系统更为一般的系统.这类模型在处理多目标,多层次,动态与静态相结合的系统时,具有重要的作用.但是,由于广义系统本身的特性,处理正常状态空间系统的许多方法和手段在广义系统中不能直接利用,因此需要发展和提出新的方法.
状态估计一直是控制理论与应用的基本问题之一.对于随机系统而言,线性最优状态估计通常指Kalman滤波,对于确定性系统而言,主要是指状态观测器设计.另外,线性二次最优控制,或称线性二次调节,也是控制理论的一个基本问题,一些新的热点问题,如H_∞滤波等都可以借助于讨论其对偶系统的线性二次最优控制进行研究.特别是最近引起广泛关注的不定线性二次最优控制问题,不但扩展了线性二次最优控制理论的研究范围,而且对于解决一些经济,生物等领域的实际工程控制问题也有很大的帮助.因此,尽管对这类的基本问题已有许多文献讨论,但有些问题仍有待于进一步研究,有些结果仍有待于改进.
本文将利用Krein空间理论,并结合一些经典的理论和方法,如:Riccati方程,线性矩阵不等式(LMI)等,对广义系统的这两类问题加以研究,改进某些已有结果或延伸出新的问题.具体来说,主要内容包括:
一.针对线性广义系统的线性最优状态估计(Kalman滤波),给出一种新的估计算法.其前提条件仅要求广义系统的正则性,其思路是利用Hilbert空间的投影定理而得到关于误差协方差阵的两个耦合的Riccati方程,通过求解这两个耦合的Riccati方程而得到与系统同维的状态估计器;
二.针对带有未知输入的线性广义状态时滞系统的观测器设计问题,利用矩阵变换和线性矩阵不等式方法,给出其观测器及输出反馈镇定控制器的设计算法.
三.针对线性广义系统以及线性广义多输入时滞系统的线性二次最优控制问题(LQR),延伸为不定的二次最优控制问题,并利用不定线性空间-Krein空间,给出了该不定二次最优控制问题的控制器存在唯一的充分必要条件,并得到与系统状态同维的控制器.
本文的创新点在于:
●对正则的线性广义系统,仅利用Hilbert空间的投影定理,通过求解两个耦合的Riccati方程而得到了其最优状态估计;
●对带有未知输入的线性广义状态时滞系统,结合矩阵理论和线性矩阵不等式方法,得到了其状态观测器及输出反馈镇定控制器的设计算法.
●提出确定性广义系统的不定二次最优控制问题,利用一种不定线性空间(Krein空间),得到了控制器存在的唯一性条件,及控制器的解析解形式.对线性广义多输入时滞系统,得到了与系统状态同维的最优控制器.
Two essential problems, including linear optimal state estimation and linear quadratic optimal control, of linear generalized systems are researched. Compared with standard state-space model, generalized systems are more general description of dynamic systems. Generalized systems are important because this model in many situations allows one to use all the physical information available and consequently generalized system models give more insight, convenience and generality for applications than traditional state-space models do. However, most of theories and methods dealing with regular systems (standard state-space models) are not directly applied for the characters of generalized systems, then new theories and techniques need to be developed.
Linear optimal state estimation is always one of the essential question of control theoretics and applications. It is mainly Kalman filtering for stochastic systems and observer design for determinate systems. Additionally, linear quadratic optimal control, which is also called linear quadratic regulation, is also one of the essential question of control theoretics. Some hot topics like H_∞filtering can be researched in virtue of discussion of linear quadratic optimal control of dual systems. It is worth pointed out that indefinite linear quadratic optimal control problem, which has attracted much more attention in recent years, not only extends the theory of linear quadratic optimal control, but also helps to solve many problems occurred in finance and biology fields. Then, the two classes of problem have always been active research areas though there are many literatures to discuss them.
Some new techniques and methods like Krein Space will be applied and some classical theories like Riccati equation and Linear Matrix Inequality(LMI) will be deeply discussed when the two problems for generalized systems are researched. Consequently, some results are improved or new problems are presented. This thesis mainly includes the following three aspects:
1. A new estimate arithmetic is presented for linear optimal state estimation of generalized systems. The assumption is only regularity of generalized systems. The filter is derived by solving two coupled Riccati-type difference equations. The approach is based on projection formula in Hilbert space.
2. Observer design of generalized linear systems with state time-delay and unknown inputs is researched. The corresponding design steps and observer-based feedback stabilizing controller are presented by using linear matrix inequality.
3. Linear quadratic optimal control problem for generalized linear systems and ones with multiple input delays is extended to indefinite LQ problem. Necessary and sufficient conditions guaranteeing existence of unique solution of the indefinite LQ problem are given by using Krein space. Meanwhile, optimal controller whose dimensions are same as those of the original systems is obtained.
Innovations of the thesis mainly include the following three aspects:
For filtering of regular generalized systems, the technique only used is projection theorem in Hilbert space and the method is to solve two coupled Riccati-type difference equations;
For generalized linear systems with state time-delay and unknown inputs, the algorithms of state observe design and observer-based feedback stabilizing controller design are presented by using matrix theory and linear matrix inequality.
For indefinite linear quadratic optimal control problem of determinate generalized system, necessary and sufficient conditions guaranteeing existence of unique solution and an explicit solution of optimal controller are obtained by using a kind of indefinite-metric space-Krein space. For that of generalized linear systems with multiple input delays, optimal controller whose dimensions are same as those of the original systems is obtained.
状态估计一直是控制理论与应用的基本问题之一.对于随机系统而言,线性最优状态估计通常指Kalman滤波,对于确定性系统而言,主要是指状态观测器设计.另外,线性二次最优控制,或称线性二次调节,也是控制理论的一个基本问题,一些新的热点问题,如H_∞滤波等都可以借助于讨论其对偶系统的线性二次最优控制进行研究.特别是最近引起广泛关注的不定线性二次最优控制问题,不但扩展了线性二次最优控制理论的研究范围,而且对于解决一些经济,生物等领域的实际工程控制问题也有很大的帮助.因此,尽管对这类的基本问题已有许多文献讨论,但有些问题仍有待于进一步研究,有些结果仍有待于改进.
本文将利用Krein空间理论,并结合一些经典的理论和方法,如:Riccati方程,线性矩阵不等式(LMI)等,对广义系统的这两类问题加以研究,改进某些已有结果或延伸出新的问题.具体来说,主要内容包括:
一.针对线性广义系统的线性最优状态估计(Kalman滤波),给出一种新的估计算法.其前提条件仅要求广义系统的正则性,其思路是利用Hilbert空间的投影定理而得到关于误差协方差阵的两个耦合的Riccati方程,通过求解这两个耦合的Riccati方程而得到与系统同维的状态估计器;
二.针对带有未知输入的线性广义状态时滞系统的观测器设计问题,利用矩阵变换和线性矩阵不等式方法,给出其观测器及输出反馈镇定控制器的设计算法.
三.针对线性广义系统以及线性广义多输入时滞系统的线性二次最优控制问题(LQR),延伸为不定的二次最优控制问题,并利用不定线性空间-Krein空间,给出了该不定二次最优控制问题的控制器存在唯一的充分必要条件,并得到与系统状态同维的控制器.
本文的创新点在于:
●对正则的线性广义系统,仅利用Hilbert空间的投影定理,通过求解两个耦合的Riccati方程而得到了其最优状态估计;
●对带有未知输入的线性广义状态时滞系统,结合矩阵理论和线性矩阵不等式方法,得到了其状态观测器及输出反馈镇定控制器的设计算法.
●提出确定性广义系统的不定二次最优控制问题,利用一种不定线性空间(Krein空间),得到了控制器存在的唯一性条件,及控制器的解析解形式.对线性广义多输入时滞系统,得到了与系统状态同维的最优控制器.
Two essential problems, including linear optimal state estimation and linear quadratic optimal control, of linear generalized systems are researched. Compared with standard state-space model, generalized systems are more general description of dynamic systems. Generalized systems are important because this model in many situations allows one to use all the physical information available and consequently generalized system models give more insight, convenience and generality for applications than traditional state-space models do. However, most of theories and methods dealing with regular systems (standard state-space models) are not directly applied for the characters of generalized systems, then new theories and techniques need to be developed.
Linear optimal state estimation is always one of the essential question of control theoretics and applications. It is mainly Kalman filtering for stochastic systems and observer design for determinate systems. Additionally, linear quadratic optimal control, which is also called linear quadratic regulation, is also one of the essential question of control theoretics. Some hot topics like H_∞filtering can be researched in virtue of discussion of linear quadratic optimal control of dual systems. It is worth pointed out that indefinite linear quadratic optimal control problem, which has attracted much more attention in recent years, not only extends the theory of linear quadratic optimal control, but also helps to solve many problems occurred in finance and biology fields. Then, the two classes of problem have always been active research areas though there are many literatures to discuss them.
Some new techniques and methods like Krein Space will be applied and some classical theories like Riccati equation and Linear Matrix Inequality(LMI) will be deeply discussed when the two problems for generalized systems are researched. Consequently, some results are improved or new problems are presented. This thesis mainly includes the following three aspects:
1. A new estimate arithmetic is presented for linear optimal state estimation of generalized systems. The assumption is only regularity of generalized systems. The filter is derived by solving two coupled Riccati-type difference equations. The approach is based on projection formula in Hilbert space.
2. Observer design of generalized linear systems with state time-delay and unknown inputs is researched. The corresponding design steps and observer-based feedback stabilizing controller are presented by using linear matrix inequality.
3. Linear quadratic optimal control problem for generalized linear systems and ones with multiple input delays is extended to indefinite LQ problem. Necessary and sufficient conditions guaranteeing existence of unique solution of the indefinite LQ problem are given by using Krein space. Meanwhile, optimal controller whose dimensions are same as those of the original systems is obtained.
Innovations of the thesis mainly include the following three aspects:
For filtering of regular generalized systems, the technique only used is projection theorem in Hilbert space and the method is to solve two coupled Riccati-type difference equations;
For generalized linear systems with state time-delay and unknown inputs, the algorithms of state observe design and observer-based feedback stabilizing controller design are presented by using matrix theory and linear matrix inequality.
For indefinite linear quadratic optimal control problem of determinate generalized system, necessary and sufficient conditions guaranteeing existence of unique solution and an explicit solution of optimal controller are obtained by using a kind of indefinite-metric space-Krein space. For that of generalized linear systems with multiple input delays, optimal controller whose dimensions are same as those of the original systems is obtained.
引文
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