广义大系统的稳定性与分散控制
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摘要
广义大系统的稳定性与镇定问题是广义大系统理论的基本问题之一，对其稳定性与镇定的研究要比状态空间系统要复杂得多，因为广义大系统不仅需要考虑稳定性，而且还要考虑正则性和因果性(离散广义系统)及脉冲自由(连续广义系统)。又因为大系统规模庞大，因素众多以及计算机的计算能力等原因，所以对大系统的控制是很困难的。由于子系统之间交换信息是不可避免的，故很难采用集中控制方法，而采用分散控制方法。因此，分散控制是处理大系统的一个有效方法。
本论文基于常系数线性广义大系统的模型，特别是常系数线性广义大系统的研究现状，较系统地研究了常系数线性广义大系统和参数不确定广义大系统的稳定性和分散镇定控制以及保性能控制问题。本文的主要内容包括以下几个方面：
一、在所有孤立子系统都是正则的且脉冲自由的条件下，利用Lyapunov方程，应用Lyapunov函数方法，研究了广义连续线性大系统和广义连续非线性大系统的稳定性和不稳定性问题，给出了连续广义大系统的稳定性判定定理和不稳定性判定定理，得到了连续广义大系统的关联稳定参数域和不稳定域。
二、在所有孤立子系统都是正则的且具有因果关系的条件下，利用Lyapunov方程，应用Lyapunov函数方法，研究了广义离散线性大系统和广义离散非线性大系统的稳定性和不稳定性问题，给出了离散广义大系统的稳定性判定定理和不稳定性判定定理，得到了离散广义大系统的关联稳定参数域和不稳定域。
三、在所有孤立子系统都是正则的、具有因果性(离散广义系统)及脉冲自由(连续广义系统)且R-能控的条件下，利用广义Lyapunov方程，应用广义Lyapunov函数方法，研究了广义离散(连续)线性大系统的镇定问题，给出了广义大系统渐近稳定的判定定理：设计了适当的反馈律，以实现广义大系统的镇定，方法简单、直观。并给出了例子说明该方法的可行性。
四、应用线性矩阵不等式(LMI)方法研究一类具有数值界的参数时不变不确定性连续广义大系统的鲁棒稳定性和分散鲁棒镇定问题。目的是要设计一个状态反馈控制器，使得对所有不确定项，闭环系统是正则的、脉冲自由且渐近稳定。得到了其可分散状态反馈镇定的一组严格矩阵不等式(Linear Matrix Inequality)的充分条件，提出了该类不确定广义大系统的分散鲁棒控制器的参数化设计方法。最后举例说明了该方法的应用。
五、应用线性矩阵不等式(LMI)方法研究一类具有数值界的参数时不变不确定性离散广义大系统的鲁棒稳定性和分散鲁棒镇定问题。目的是要设计一个状态反馈控
制器，使得对所有不确定项，闭环系统是正则的、具有因果关系且稳定。得到了其可
分散状态反馈镇定的一组严格矩阵不等式(Linear Matrix Inequality)的充分条件，提
出了该类不确定广义大系统的分散鲁棒控制器的参数化设计方法。最后举例说明了该
方法的应用。
六、对一类不确定项具有数值界的参数不确定广义大系统和一个二次型性能指标，
研究了其保性能状态反馈控制律的设计问题.基于不确定项的表达形式，应用线性矩
阵不等式(LMI)方法和LyaPunov函数法，导出了存在保性能分散控制器的LMI条件，
LM工方法求解简单，最后用例子说明该方法的应用。
对参数时不变不确定关联离散广义大系统，还考虑了其保性能分散最优控制问
题，在给出保性能控制器存在的基础上，建立了最优控制器满足的凸优化问题并求解。
并给出了例子说明该方法的可行性。
关键词:广义系统;大系统;广义大系统;参数域;镇定;线性矩阵不等式;
Lyap。n。v方程;Lyapunov函数;鲁棒控制;保性能控制.
The asymptotic stability and stabilization problem is one of the fundamental problems in the theory of singular large-scale system. The study of them is much more complicated than that of state-space systems because the singular large-scale system requires considering not only stability, but also regularity and causality (for discrete singular systems) or impulse immunity (for continuous singular systems) . The large-scale systems are difficult to control due to large in scale, numerous factors and lack of centralized computing capability, etc. Some of the difficulties associated with a centralized control scheme can be alleviated via a decentralized control structure in which information transfer between subsystems is unavoidable. Therefore, decentralized control is considered as an effective method to deal with large-scale systems.In the light of the recent work on singular large-scale systems models, especially in linear singular large-scale systems models, the dissertation provides a systematic study on the asymptotic stability, decentralized stabilization and guaranteed cost control of linear singular large-scale systems models and singular large-scale systems with parameter uncertainty. The main results obtained in this dissertation are as follows:i) The problems of asymptotic stability and unsteadiness of continuous singular linear large-scale system and continuous singular non-linear large-scale system are investigated by means of Lyapunov equation and Lyapunov function under the conditions that all their isolated subsystems are of regularity and impulse immunity. The theorems of asymptotic stability and unsteadiness of continuous singular large-scale systems are presented. The interconnecting parameter regions of asymptotic stability and unsteadiness for them are obtained.ii) The problems of asymptotic stability and unsteadiness of discrete singular linear large-scale system and discrete singular non-linear large-scale system are investigated by means of Lyapunov equation and Lyapunov function under the conditions that all their isolated subsystems are of regularity and causality. The theorems of discrete singular large-scale systems asymptotic stability and unsteady are presented. The interconnecting parameter regions of asymptotic stability and unsteadiness for them are obtained.iii) The Lyapunov s method and Lyapunov equation are employed to study the asymptotic stability and stabilization problem on discrete (continuous) singular large scale
systems under the conditions that all their isolated subsystems are regularity and impulse immunity (for continuous singular systems) or causality (for discrete singular systems) and R-controllable. The theorem of asymptotic stability is obtained. The controller is designed for the stabilization of singular large-scale systems. The utilized method is simple, intuitional and easily understood. An example shows that the theorem is feasible.iv) This paper addresses the problems of robust stable and robust stabilization for uncertain continuous-time singular large-scale systems with parameter uncertainties via linear matrix inequality (LMI) method. The parameter uncertainties are assumed to be time invariant, but norm-bounded. The purpose of the underlying robust stabilization problem discussed in this paper is to design state feedback controllers so that, for all admissible uncertainties, the closed-loop system is of regularity and impulse immunity. In terms of strict LMIs, sufficient conditions for the solvability of above problems are presented, and the parameterizations of the desired state feedback controllers are also given. A numerical example is given to demonstrate the applications of the proposed design.v) This paper addresses the problems of robust stable and robust stabilization for uncertain discrete-time singular large-scale systems with parameter uncertainties by LMI method. The parameter uncertainties are assumed to be time invariant, but norm-bounded. The purpose of the underlying robust stabilization problem discussed in this paper is to design state fee
本论文基于常系数线性广义大系统的模型，特别是常系数线性广义大系统的研究现状，较系统地研究了常系数线性广义大系统和参数不确定广义大系统的稳定性和分散镇定控制以及保性能控制问题。本文的主要内容包括以下几个方面：
一、在所有孤立子系统都是正则的且脉冲自由的条件下，利用Lyapunov方程，应用Lyapunov函数方法，研究了广义连续线性大系统和广义连续非线性大系统的稳定性和不稳定性问题，给出了连续广义大系统的稳定性判定定理和不稳定性判定定理，得到了连续广义大系统的关联稳定参数域和不稳定域。
二、在所有孤立子系统都是正则的且具有因果关系的条件下，利用Lyapunov方程，应用Lyapunov函数方法，研究了广义离散线性大系统和广义离散非线性大系统的稳定性和不稳定性问题，给出了离散广义大系统的稳定性判定定理和不稳定性判定定理，得到了离散广义大系统的关联稳定参数域和不稳定域。
三、在所有孤立子系统都是正则的、具有因果性(离散广义系统)及脉冲自由(连续广义系统)且R-能控的条件下，利用广义Lyapunov方程，应用广义Lyapunov函数方法，研究了广义离散(连续)线性大系统的镇定问题，给出了广义大系统渐近稳定的判定定理：设计了适当的反馈律，以实现广义大系统的镇定，方法简单、直观。并给出了例子说明该方法的可行性。
四、应用线性矩阵不等式(LMI)方法研究一类具有数值界的参数时不变不确定性连续广义大系统的鲁棒稳定性和分散鲁棒镇定问题。目的是要设计一个状态反馈控制器，使得对所有不确定项，闭环系统是正则的、脉冲自由且渐近稳定。得到了其可分散状态反馈镇定的一组严格矩阵不等式(Linear Matrix Inequality)的充分条件，提出了该类不确定广义大系统的分散鲁棒控制器的参数化设计方法。最后举例说明了该方法的应用。
五、应用线性矩阵不等式(LMI)方法研究一类具有数值界的参数时不变不确定性离散广义大系统的鲁棒稳定性和分散鲁棒镇定问题。目的是要设计一个状态反馈控
制器，使得对所有不确定项，闭环系统是正则的、具有因果关系且稳定。得到了其可
分散状态反馈镇定的一组严格矩阵不等式(Linear Matrix Inequality)的充分条件，提
出了该类不确定广义大系统的分散鲁棒控制器的参数化设计方法。最后举例说明了该
方法的应用。
六、对一类不确定项具有数值界的参数不确定广义大系统和一个二次型性能指标，
研究了其保性能状态反馈控制律的设计问题.基于不确定项的表达形式，应用线性矩
阵不等式(LMI)方法和LyaPunov函数法，导出了存在保性能分散控制器的LMI条件，
LM工方法求解简单，最后用例子说明该方法的应用。
对参数时不变不确定关联离散广义大系统，还考虑了其保性能分散最优控制问
题，在给出保性能控制器存在的基础上，建立了最优控制器满足的凸优化问题并求解。
并给出了例子说明该方法的可行性。
关键词:广义系统;大系统;广义大系统;参数域;镇定;线性矩阵不等式;
Lyap。n。v方程;Lyapunov函数;鲁棒控制;保性能控制.
The asymptotic stability and stabilization problem is one of the fundamental problems in the theory of singular large-scale system. The study of them is much more complicated than that of state-space systems because the singular large-scale system requires considering not only stability, but also regularity and causality (for discrete singular systems) or impulse immunity (for continuous singular systems) . The large-scale systems are difficult to control due to large in scale, numerous factors and lack of centralized computing capability, etc. Some of the difficulties associated with a centralized control scheme can be alleviated via a decentralized control structure in which information transfer between subsystems is unavoidable. Therefore, decentralized control is considered as an effective method to deal with large-scale systems.In the light of the recent work on singular large-scale systems models, especially in linear singular large-scale systems models, the dissertation provides a systematic study on the asymptotic stability, decentralized stabilization and guaranteed cost control of linear singular large-scale systems models and singular large-scale systems with parameter uncertainty. The main results obtained in this dissertation are as follows:i) The problems of asymptotic stability and unsteadiness of continuous singular linear large-scale system and continuous singular non-linear large-scale system are investigated by means of Lyapunov equation and Lyapunov function under the conditions that all their isolated subsystems are of regularity and impulse immunity. The theorems of asymptotic stability and unsteadiness of continuous singular large-scale systems are presented. The interconnecting parameter regions of asymptotic stability and unsteadiness for them are obtained.ii) The problems of asymptotic stability and unsteadiness of discrete singular linear large-scale system and discrete singular non-linear large-scale system are investigated by means of Lyapunov equation and Lyapunov function under the conditions that all their isolated subsystems are of regularity and causality. The theorems of discrete singular large-scale systems asymptotic stability and unsteady are presented. The interconnecting parameter regions of asymptotic stability and unsteadiness for them are obtained.iii) The Lyapunov s method and Lyapunov equation are employed to study the asymptotic stability and stabilization problem on discrete (continuous) singular large scale
systems under the conditions that all their isolated subsystems are regularity and impulse immunity (for continuous singular systems) or causality (for discrete singular systems) and R-controllable. The theorem of asymptotic stability is obtained. The controller is designed for the stabilization of singular large-scale systems. The utilized method is simple, intuitional and easily understood. An example shows that the theorem is feasible.iv) This paper addresses the problems of robust stable and robust stabilization for uncertain continuous-time singular large-scale systems with parameter uncertainties via linear matrix inequality (LMI) method. The parameter uncertainties are assumed to be time invariant, but norm-bounded. The purpose of the underlying robust stabilization problem discussed in this paper is to design state feedback controllers so that, for all admissible uncertainties, the closed-loop system is of regularity and impulse immunity. In terms of strict LMIs, sufficient conditions for the solvability of above problems are presented, and the parameterizations of the desired state feedback controllers are also given. A numerical example is given to demonstrate the applications of the proposed design.v) This paper addresses the problems of robust stable and robust stabilization for uncertain discrete-time singular large-scale systems with parameter uncertainties by LMI method. The parameter uncertainties are assumed to be time invariant, but norm-bounded. The purpose of the underlying robust stabilization problem discussed in this paper is to design state fee
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