不确定广义系统鲁棒控制若干问题研究
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摘要
控制系统的设计都要以被控对象精确数学模型为依据,然而严格来说,对任一被控对象建模时都不可能做到完全精确,必然存在不确定性。这些不确定性包括参数不确定性、未建模动态和各种干扰等。鲁棒控制正是研究系统模型存在不确定性时如何设计控制器使闭环系统稳定且满足一定的动态性能。广义系统是一类形式更一般化的系统,是有着广泛应用背景的动力系统,许多实际系统用广义系统模型描述起来更为方便、自然。近年来广义系统的理论与应用问题已经吸引了国内外众多学者的关注,许多正常状态空间系统的结论被相继推广到广义系统中。
本文研究不确定广义系统的鲁棒控制问题。以线性矩阵不等式(LMI)为工具,讨论了不确定广义系统和广义时滞系统的鲁棒无源控制问题、鲁棒H_∞控制问题以及鲁棒变结构控制问题。本论文主要内容包括以下几部分:
(1).针对一类具有范数有界不确定线性广义系统,分别讨论了连续和离散情形下的鲁棒严格无源控制问题,给出线性广义系统容许且严格无源的充要条件。针对连续情形设计了鲁棒状态反馈控制器和无偏差滤波器;针对离散情形设计了鲁棒静态输出反馈控制器,且运用矩阵的正交补,把受限矩阵不等式转化为严格的线性矩阵不等式。
(2).针对一类具有范数有界不确定连续线性广义时滞系统,采用积分矩阵不等式方法给出该系统时滞相关鲁棒正则、无脉冲、稳定且严格无源的充分条件,结论表示为严格线性矩阵不等式形式,同其它方法所得的充分条件相比较,保守性低且不涉及矩阵的分解问题,同时基于该充分条件给出鲁棒严格无源控制器的设计方法。
(3).针对一类具有范数有界不确定离散线性广义时滞系统,采用基于二次型项的有限和不等式方法给出该系统时滞相关鲁棒正则、因果、稳定且具有H_∞性能指标的充分条件,结论表示为严格线性矩阵不等式形式,同其它方法所得的充分条件相比较,不涉及矩阵的分解问题,同时基于该充分条件,采用广义模型变换方法给出离散鲁棒H_∞控制器的设计方法,最终给出的控制器不涉及非线性问题。
(4).分别研究一类系统状态矩阵不确定性不满足匹配条件的广义系统变结构控制问题和一类系统状态矩阵与控制增益不确定性同时不满足匹配条件的正常状态空间时滞系统的变结构控制问题,给出相应切换面和变结构控制律的设计算法。
It is well known that the design of control system is established based on the exact mathematical mode of plant, it is difficult to obtain its exact model in general such that the system uncertainties which include parameter uncertainties, un-modeled dynamics, and disturbances should be taken into account for modeling. The robust control theory mainly focuses on the study of how to design an admissible controller for a given uncertain system such that the resulting closed-loop system is stable with some specified system performance measures. Singular system is a kind of dynamical systems of more general and has been widely applied. Many physical systems are described more convenient and natural by singular system. Recently, a lot of research efforts have been drawn on the study of singular systems, and many results of normal systems have been corresponding generalized to singular systems.
The robust control problem for uncertain singular systems is studied in this dissertation. Based on the linear matrix inequality (LMI) theory, the problems of robust passive control, robust H_∞control and robust variable structure control are studied. The main contents of this dissertation are outlined as follows.
(1) The robust strictly passive control of continuous- and discrete-time linear singular systems with norm-bounded uncertainties is developed. Sufficient and necessary conditions, which ensure that the system is admissible and strictly passive, are established. A robust state-feedback controller and an unbias filter are designed for continuous-time singular systems and a robust static output-feedback controller is developed for discrete-time singular systems and the constraint matrix is transformed into a strict linear matrix inequality by using the matrix orthogonal complement.
(2) Based on a integral inequality method, a new delay-dependent sufficient condition is presented for continuous singular system with variable time delay and norm-bounded uncertainties in terms of strict linear matrix inequalities, which guarantees that the system is robust regular, implusive-free, stable and strictly passive. Based on this sufficient condition, the robust strictly passive control problem is investigated. Compared with the previous methods, the proposed method is shown to be less conservative and involves no decomposition of system matrices.
(3) Based on a finite sum inequality for quadratic terms, a new delay-dependent sufficient condition is presented for discrete singular system with variable time delay in terms of strict linear matrix inequalities, which guarantees the system to be robust regular, causal, stable and with an H_∞performance. Contrasted to the previous method, the proposed method involves no decomposition of system matrices. Based on this sufficient condition, the discrete robust H_∞controller is correspondingly given by using a descriptor mode method, which involves no nonlinear problem.
(4) The variable structure control problem is studied for a class of singular systems with unmatched uncertainties in state matrix and for a class of normal time-delay state space systems with unmatched uncertainties both in state matrix and control gain matrix. And the corresponding switched surfaces and variable structure control laws are given respectively.
本文研究不确定广义系统的鲁棒控制问题。以线性矩阵不等式(LMI)为工具,讨论了不确定广义系统和广义时滞系统的鲁棒无源控制问题、鲁棒H_∞控制问题以及鲁棒变结构控制问题。本论文主要内容包括以下几部分:
(1).针对一类具有范数有界不确定线性广义系统,分别讨论了连续和离散情形下的鲁棒严格无源控制问题,给出线性广义系统容许且严格无源的充要条件。针对连续情形设计了鲁棒状态反馈控制器和无偏差滤波器;针对离散情形设计了鲁棒静态输出反馈控制器,且运用矩阵的正交补,把受限矩阵不等式转化为严格的线性矩阵不等式。
(2).针对一类具有范数有界不确定连续线性广义时滞系统,采用积分矩阵不等式方法给出该系统时滞相关鲁棒正则、无脉冲、稳定且严格无源的充分条件,结论表示为严格线性矩阵不等式形式,同其它方法所得的充分条件相比较,保守性低且不涉及矩阵的分解问题,同时基于该充分条件给出鲁棒严格无源控制器的设计方法。
(3).针对一类具有范数有界不确定离散线性广义时滞系统,采用基于二次型项的有限和不等式方法给出该系统时滞相关鲁棒正则、因果、稳定且具有H_∞性能指标的充分条件,结论表示为严格线性矩阵不等式形式,同其它方法所得的充分条件相比较,不涉及矩阵的分解问题,同时基于该充分条件,采用广义模型变换方法给出离散鲁棒H_∞控制器的设计方法,最终给出的控制器不涉及非线性问题。
(4).分别研究一类系统状态矩阵不确定性不满足匹配条件的广义系统变结构控制问题和一类系统状态矩阵与控制增益不确定性同时不满足匹配条件的正常状态空间时滞系统的变结构控制问题,给出相应切换面和变结构控制律的设计算法。
It is well known that the design of control system is established based on the exact mathematical mode of plant, it is difficult to obtain its exact model in general such that the system uncertainties which include parameter uncertainties, un-modeled dynamics, and disturbances should be taken into account for modeling. The robust control theory mainly focuses on the study of how to design an admissible controller for a given uncertain system such that the resulting closed-loop system is stable with some specified system performance measures. Singular system is a kind of dynamical systems of more general and has been widely applied. Many physical systems are described more convenient and natural by singular system. Recently, a lot of research efforts have been drawn on the study of singular systems, and many results of normal systems have been corresponding generalized to singular systems.
The robust control problem for uncertain singular systems is studied in this dissertation. Based on the linear matrix inequality (LMI) theory, the problems of robust passive control, robust H_∞control and robust variable structure control are studied. The main contents of this dissertation are outlined as follows.
(1) The robust strictly passive control of continuous- and discrete-time linear singular systems with norm-bounded uncertainties is developed. Sufficient and necessary conditions, which ensure that the system is admissible and strictly passive, are established. A robust state-feedback controller and an unbias filter are designed for continuous-time singular systems and a robust static output-feedback controller is developed for discrete-time singular systems and the constraint matrix is transformed into a strict linear matrix inequality by using the matrix orthogonal complement.
(2) Based on a integral inequality method, a new delay-dependent sufficient condition is presented for continuous singular system with variable time delay and norm-bounded uncertainties in terms of strict linear matrix inequalities, which guarantees that the system is robust regular, implusive-free, stable and strictly passive. Based on this sufficient condition, the robust strictly passive control problem is investigated. Compared with the previous methods, the proposed method is shown to be less conservative and involves no decomposition of system matrices.
(3) Based on a finite sum inequality for quadratic terms, a new delay-dependent sufficient condition is presented for discrete singular system with variable time delay in terms of strict linear matrix inequalities, which guarantees the system to be robust regular, causal, stable and with an H_∞performance. Contrasted to the previous method, the proposed method involves no decomposition of system matrices. Based on this sufficient condition, the discrete robust H_∞controller is correspondingly given by using a descriptor mode method, which involves no nonlinear problem.
(4) The variable structure control problem is studied for a class of singular systems with unmatched uncertainties in state matrix and for a class of normal time-delay state space systems with unmatched uncertainties both in state matrix and control gain matrix. And the corresponding switched surfaces and variable structure control laws are given respectively.
引文
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