混杂系统稳定性及其在电力系统中的应用研究
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摘要
混杂系统是包含离散事件系统和连续变量动态系统、两者又相互作用的动态系统。自八十年代以来，混杂系统的研究得到了很大的关注。由于混杂系统的复杂性和特殊性，即使是线性切换型混杂系统也具有非常复杂的非线性动态行为，传统的研究方法不能直接应用，稳定性研究很困难。而稳定性是对于控制系统的一个基本要求，许多实际系统从本质上说又都是混杂的。故研究混杂系统稳定性分析和控制方法在理论和应用两个方面都具有重要意义。
本文以李雅普诺夫函数法理论为基础，采用线性矩阵不等式方法对带有时变时滞摄动的混杂系统的稳定性进行研究。针对微分代数混杂系统的稳定性分析进行深入研究，并将其应用于电力系统电压稳定性的分析和控制。
重点从自动控制的角度综述了混杂系统的研究现状，在对现有混杂系统模型分析的基础上，指出混杂系统的稳定性研究是现阶段混杂系统的研究重点，基于李雅普诺夫函数法得出了许多重要的结论。对混杂系统在电力系统中的应用作了重点分析。
针对带有时变时滞摄动的混杂系统，基于单李雅普诺夫函数法和多李雅普诺夫函数法给出了能够使整个系统渐近稳定的切换律的设计办法，并把单李雅普诺夫函数法的切换律设计方法的结果以线性矩阵不等式形式给出，能够基于线性矩阵不等式算法来确定使系统渐近稳定的稳定边界。
建立了含有微分代数子系统的混杂系统的数学模型，并提出了包括该类混杂系统稳定性和(大范围)渐近稳定性定义在内的理论框架。通过单李雅普诺夫函数和多李雅普诺夫函数方法对此类混杂系统进行稳定性研究，得出了混杂系统在任意切换及慢切换条件下的稳定性及渐近稳定性结果。
给出了以微分代数混杂系统描述的电力系统电压稳定性的数学模型，分别利用单李雅普诺夫函数法和多李雅普诺夫函数法得到了具体实现慢切换条件下的微分代数混杂系统稳定性判定的充分条件，并进行应用，可找到切换律来使整个混杂系统达到稳定状态，进而分析电力系统电压稳定性。由于多李雅普诺夫函数法对于每个子系统的运行区域都提出一个李雅普诺夫函数，而不是所有的子系统都采用同一个李雅普诺夫函数，故比单李雅普诺夫函数法有更多优势。因此把此
浙江大学博士学位论文
混杂系统稳定性及其在电力系统中的应用研究
方法用于电力系统电压稳定性分析的时候，此优势不仅带来了稳定性分析上的简
便，而且可给出系统的稳定域并可用来分析系统参数变化对电力系统电压稳定性
的影响。该方法在电压稳定性分析中的实际应用，验证了切换律设计方法的有效
性。
Hybrid systems consist of continuous time and/or discrete time dynamic systems and discrete event systems, which interact on each other. There has been an intensified interest among control engineers in discrete event and hybrid systems during the last decade. For the complex dynamic and particularity of hybrid system, even though the linear hybrid system has very complex nonlinear character. Stability is one of the central properties in system theory and engineering. Many practical systems are hybrid naturally. So it is import for the theory and application to research on the stability and control method of hybrid system.
Based on the Lyapunov stability theory of hybrid systems and using Linear Matrix Inequality (LMI) method, this dissertation investigates the stability of hybrid system with uncertain time-varying time-delay perturbation. And the further research is carried out for stability of differential algebraic hybrid system. And the stability theory put forward in the dissertation is used in the analysis and control of voltage stability of power system.
The research of hybrid system is overviewed form the view of automatically control. Based on the analysis of hybrid system model in existence, it is pointed out that stability of hybrid system is the research emphasis on the stage. Many significant results are derived from Lyapunov direct method. Further more, it is discussed in detail for the application of hybrid system in power system.
Using Single Lyapunov function and Multiple Lyapunov function, the switching laws are designed to analyze the stability of the whole hybrid system with uncertain time-varying time-delay perturbation. And the design based on Single Lyapunov function is put forward in the form of LMI. And the stability bounds can also be determined.
Differential-Algebraic Hybrid System (DAHS) is modeled whose subsystems are formulated by differential-algebraic equations besides pure differential equations. A framework of stability theory including the definition of stability and (globally) asymptotic stability was put forward based on the model of DAHS. And the sufficient conditions to test for stability and asymptotic stability were achieved in the case of arbitrary switching and slow switching by Single Lyapunov Function and Multiple Lyapunov Functions.
The model of voltage stability is analyzed and abstracted to the model of Differential-Algebraic Hybrid System. So the judgment of voltage stability can be come down to the stability research of Differential-Algebraic Hybrid System. Using Single Lyapunov Function and Multiple Lyapunov Function, the stability problem of Differential-Algebraic Hybrid System is researched in the case of slow switching scenario. The sufficient conditions are derived to stabilized whole hybrid system by the switching laws put forward in this dissertation and used in the research of voltage stability. Multiple Lyapunov Function is better than Single Lyapunov Function in the analysis of hybrid system for the former is that each Lyapunov Function is active in an area rather than in the whole state space. The advantage simplifies the stability analysis. And the method can be used to analyze the influence of parameter varying
on the whole system. Therefore, when the method is used in the analysis of power system's voltage stability, it not only accounts for the validity of the switching law designed by the stability criterion put forward in this dissertation but also gives the stability margin of the system and judges the influence of the parameter varying on the whole system's stability.
本文以李雅普诺夫函数法理论为基础，采用线性矩阵不等式方法对带有时变时滞摄动的混杂系统的稳定性进行研究。针对微分代数混杂系统的稳定性分析进行深入研究，并将其应用于电力系统电压稳定性的分析和控制。
重点从自动控制的角度综述了混杂系统的研究现状，在对现有混杂系统模型分析的基础上，指出混杂系统的稳定性研究是现阶段混杂系统的研究重点，基于李雅普诺夫函数法得出了许多重要的结论。对混杂系统在电力系统中的应用作了重点分析。
针对带有时变时滞摄动的混杂系统，基于单李雅普诺夫函数法和多李雅普诺夫函数法给出了能够使整个系统渐近稳定的切换律的设计办法，并把单李雅普诺夫函数法的切换律设计方法的结果以线性矩阵不等式形式给出，能够基于线性矩阵不等式算法来确定使系统渐近稳定的稳定边界。
建立了含有微分代数子系统的混杂系统的数学模型，并提出了包括该类混杂系统稳定性和(大范围)渐近稳定性定义在内的理论框架。通过单李雅普诺夫函数和多李雅普诺夫函数方法对此类混杂系统进行稳定性研究，得出了混杂系统在任意切换及慢切换条件下的稳定性及渐近稳定性结果。
给出了以微分代数混杂系统描述的电力系统电压稳定性的数学模型，分别利用单李雅普诺夫函数法和多李雅普诺夫函数法得到了具体实现慢切换条件下的微分代数混杂系统稳定性判定的充分条件，并进行应用，可找到切换律来使整个混杂系统达到稳定状态，进而分析电力系统电压稳定性。由于多李雅普诺夫函数法对于每个子系统的运行区域都提出一个李雅普诺夫函数，而不是所有的子系统都采用同一个李雅普诺夫函数，故比单李雅普诺夫函数法有更多优势。因此把此
浙江大学博士学位论文
混杂系统稳定性及其在电力系统中的应用研究
方法用于电力系统电压稳定性分析的时候，此优势不仅带来了稳定性分析上的简
便，而且可给出系统的稳定域并可用来分析系统参数变化对电力系统电压稳定性
的影响。该方法在电压稳定性分析中的实际应用，验证了切换律设计方法的有效
性。
Hybrid systems consist of continuous time and/or discrete time dynamic systems and discrete event systems, which interact on each other. There has been an intensified interest among control engineers in discrete event and hybrid systems during the last decade. For the complex dynamic and particularity of hybrid system, even though the linear hybrid system has very complex nonlinear character. Stability is one of the central properties in system theory and engineering. Many practical systems are hybrid naturally. So it is import for the theory and application to research on the stability and control method of hybrid system.
Based on the Lyapunov stability theory of hybrid systems and using Linear Matrix Inequality (LMI) method, this dissertation investigates the stability of hybrid system with uncertain time-varying time-delay perturbation. And the further research is carried out for stability of differential algebraic hybrid system. And the stability theory put forward in the dissertation is used in the analysis and control of voltage stability of power system.
The research of hybrid system is overviewed form the view of automatically control. Based on the analysis of hybrid system model in existence, it is pointed out that stability of hybrid system is the research emphasis on the stage. Many significant results are derived from Lyapunov direct method. Further more, it is discussed in detail for the application of hybrid system in power system.
Using Single Lyapunov function and Multiple Lyapunov function, the switching laws are designed to analyze the stability of the whole hybrid system with uncertain time-varying time-delay perturbation. And the design based on Single Lyapunov function is put forward in the form of LMI. And the stability bounds can also be determined.
Differential-Algebraic Hybrid System (DAHS) is modeled whose subsystems are formulated by differential-algebraic equations besides pure differential equations. A framework of stability theory including the definition of stability and (globally) asymptotic stability was put forward based on the model of DAHS. And the sufficient conditions to test for stability and asymptotic stability were achieved in the case of arbitrary switching and slow switching by Single Lyapunov Function and Multiple Lyapunov Functions.
The model of voltage stability is analyzed and abstracted to the model of Differential-Algebraic Hybrid System. So the judgment of voltage stability can be come down to the stability research of Differential-Algebraic Hybrid System. Using Single Lyapunov Function and Multiple Lyapunov Function, the stability problem of Differential-Algebraic Hybrid System is researched in the case of slow switching scenario. The sufficient conditions are derived to stabilized whole hybrid system by the switching laws put forward in this dissertation and used in the research of voltage stability. Multiple Lyapunov Function is better than Single Lyapunov Function in the analysis of hybrid system for the former is that each Lyapunov Function is active in an area rather than in the whole state space. The advantage simplifies the stability analysis. And the method can be used to analyze the influence of parameter varying
on the whole system. Therefore, when the method is used in the analysis of power system's voltage stability, it not only accounts for the validity of the switching law designed by the stability criterion put forward in this dissertation but also gives the stability margin of the system and judges the influence of the parameter varying on the whole system's stability.
引文
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