基于LMI技术的线性系统模型降阶与静态输出反馈控制器设计
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摘要
随着现代社会信息化、系统化的发展,人类面临的各种控制系统的规模越来越大,由此导致系统模型以及控制器的阶数也越来越高,相应的对系统分析计算和综合实现的复杂度也越来越大。为此,模型降阶和降阶控制器的设计一直都是控制理论中的热门研究领域,并在过去的几十年里取得了长足的发展和广泛的应用。然而,其中的一些问题通过现有的方法中仍不能得到很好的解决,如在模型降阶的研究中,针对系统工作频率范围已知的情况,现有的如基于加权矩阵的平衡截断等方法会带来一定的不准确性和不可靠性,而且无法给出降阶模型和高阶模型在已知频率范围内的逼近性能。此外现有的一些基于线性矩阵不等式的模型降阶和降阶控制器(如静态输出反馈控制器)的设计条件存在着一定的保守性,如何给出保守性更少的设计条件也是一个非常重要的问题。
本论文在前人工作的基础上,给出了新的基于线性矩阵不等式(LMI)技术的模型降阶和静态输出反馈控制器设计方法。针对已知输入信号频率范围情况下的模型降阶问题,通过结合广义KYP引理给出了可以准确刻画有限频逼近误差的设计条件,解决了现有方法如频率加权法等带来的不准确性问题。对全频范围考虑的模型降阶问题以及离散时间系统的静态输出反馈问题,均给出了相对于现有结果保守性更少的设计条件。另外,针对实际系统中存在不确定性和时滞的情况,研究了系统中含有多胞不确定性和时不变状态时滞情况下的静态输出反馈控制问题。本文的一些结果用到了对RLC电路系统的模型降阶中,数值算例与仿真验证了本文提出方法的优越性和有效性。
第一、二章系统地分析和总结了模型降阶与静态输出反馈控制这两个控制理论中的热门研究领域的发展现状及研究方法,并给出了与本文相关的一些预备知识。
第三、四章分别就连续线性系统和离散线性系统的H_∞、H_2模型降阶问题给出了新的基于线性矩阵不等式的设计条件。在考虑H_∞模型降阶时,通过结合新提出的广义KYP引理,根据输入信号不同的频率范围分别给出在低频,中频,高频和全频时的H_∞模型降阶结果,这样就避免了过去方法处理有限频模型降阶时的不确定性和不可靠性。此外埘全频H_∞模型降阶问题,本章的方法也比现有文献中的一些同类方法具有更少的保守性。数值算例和仿真进一步说明了本章提出方法的有效性和优越性。
第五章研究了离散时间系统的静态输出反馈控制问题。基于LMI技术,分别给出了一组针对镇定控制,H_∞控制,以及正实控制的静态输出反馈控制设计条件。和现有文献中的同类方法相比,本章的设计方法结合了鲁棒控制领域中的参数依赖Lyapunov函数方法,引入了更多的辅助变量,进而具有更少的保守性。此外,本章中的设计条件都借助于Finsler引理在一个统一的框架得出,可以清楚的说明现有结果之间以及本章结果与现有结果之间的区别与联系。数值算例进一步说明了本章方法的有效性与优越性。
第六章考虑了具有多胞不确定性的离散线性系统的静态输出反馈控制问题。首先给出了基于参数依赖Lyapunov函数方法的鲁棒正实性分析结果,并从理论上证明了这些结果与现有文献中的同类结果之间的关系。然后根据第五章中的设计方法,并结合不确定系统中的一些放缩技巧,分别就H_∞控制,正实控制问题给出了一组不确定系统的鲁棒静态输出反馈控制器设计方法。数值例子说明了方法的有效性。
第七章研究了带有时不变状态时滞的线性离散时间系统的静念输出反馈控制问题。根据第五章中的设计方法,并结合处理时滞项的Jensen不等式方法,给出了时滞依赖的H_∞静念输出反馈控制器设计方法。最后通过数值例子进一步表明本章方法的有效性。
最后对全文所做的工作进行了总结,并指出了下一步研究的方向。
Due to the increasing development of informationization,systematization of the modern society,the dimensions of various control systems are becoming larger and larger, and the resulting complexity for system analysis and synthesis are also increased because the increasing order of the system model and the corresponding controller.Therefore, the reduction theory(i.e,model reduction and reduced-order controller design) is always a burgeoning research area.Great developments and wide applications have been made during the last several decades.However,there are still some problems that cannot be properly solved via the existing methods.For example,to some extent there exists inaccuracy and unreliability while using the existing method to cope with the known operating frequency information of the system,and there exists no approximation performance information over the known fiequency interval.Besides,how to reduce the conservatism of the existing LMI-based design methods for model reduction and static output feedback control is also an important problem.
This thesis,based on previous works of others,presents new methods for model reduction and static output feedback control problems via LMI-based approach.For the model reduction problem that with known fiequency information about the input signal, the design conditions are developed with the aid of the generalized KYP lemma,which can deal with the approximation error over finite frequency directly.Therefore,the inaccuracy resulted by the existing methods such as frequency-weighted method can be avoided.For model reduction problems over entire frequency interval and static output feedback control problems for discrete-time systems,design methods with less conserv-ativeness compared with the counterpart ones in the literatures are developed.Besides, static output feedback controller design methods for systems with polytopic uncertainties and time-invariant delay are also presented respectively.Parts of the developed methods are applied to the model reduction of RLC circuit systems.Numerical examples and simulations illustrate the advantages and effectiveness of our approaches.
Chapters 1-2 summarize the development and main research methods in the burgeoning research areas:model reduction and static output feedback control.Preliminaries about the considered problems are also given.
Chapters 3-4 present new LMI-based design methods for H_∞and H_2 model reduction problems for linear continuous-time systems and discrete-time systems,respectively. Based on the recently developed generalized KYP lemma,design methods of H_∞model reduction are developed under low-frequency,middle frequency,high frequency,and entire frequency interval considerations according to the frequency information about input signal.Consequently,the uncertainty and unreliability of the existing methods for finite frequency model reduction problems are avoided.For the entire frequency H_∞model reduction problems,it is also pointed out that the conservativeness of the proposed methods in this chapter is less than the existing ones.Numerical examples and simulations illustrate the effectiveness and advantages of the proposed approach.
Chapter 5 investigates the static output feedback control problem for linear discretetime systems.Stabilization,H_∞and positive real static output feedback control design methods are presented based on LMI technique respectively.By utilizing the parameter-dependent Lyapunov function method which originated in the research area of robust control and introducing more auxiliary variables,the conservativeness of the proposed methods is further reduced compared with the existing ones.Besides,the differences and relationships between the proposed methods and the existing methods can be clearly demonstrated due to those methods are presented in a unified framework in terms of the Finsler lemma.Numerical examples illustrate the effectiveness and advantages of the proposed approach.
Chapter 6 focuses on the static output feedback control problem for linear discretetime systems with polytopic uncertainties.Firstly,robust positive realness analysis results are given based on the parameter-dependent Lyapunov function method and the relationship between the proposed result and the existing one is clarified theoretically.Combining some relaxation techniques,H_∞and positive realness static output feedback controller design methods are presented.Numerical examples illustrate the effectiveness and advantages of the proposed approach.
Chapter 7 investigates the static output feedback control problem for linear discretetime systems with time-invariant state delay.Combining the Jensen inequality approach that dealing with the delay items,H_∞static output feedback controller design methods are presented.Numerical examples illustrate the effectiveness and advantages of the proposed approach.
Finally,the results of the dissertation are summarized and further research topics are pointed out.
本论文在前人工作的基础上,给出了新的基于线性矩阵不等式(LMI)技术的模型降阶和静态输出反馈控制器设计方法。针对已知输入信号频率范围情况下的模型降阶问题,通过结合广义KYP引理给出了可以准确刻画有限频逼近误差的设计条件,解决了现有方法如频率加权法等带来的不准确性问题。对全频范围考虑的模型降阶问题以及离散时间系统的静态输出反馈问题,均给出了相对于现有结果保守性更少的设计条件。另外,针对实际系统中存在不确定性和时滞的情况,研究了系统中含有多胞不确定性和时不变状态时滞情况下的静态输出反馈控制问题。本文的一些结果用到了对RLC电路系统的模型降阶中,数值算例与仿真验证了本文提出方法的优越性和有效性。
第一、二章系统地分析和总结了模型降阶与静态输出反馈控制这两个控制理论中的热门研究领域的发展现状及研究方法,并给出了与本文相关的一些预备知识。
第三、四章分别就连续线性系统和离散线性系统的H_∞、H_2模型降阶问题给出了新的基于线性矩阵不等式的设计条件。在考虑H_∞模型降阶时,通过结合新提出的广义KYP引理,根据输入信号不同的频率范围分别给出在低频,中频,高频和全频时的H_∞模型降阶结果,这样就避免了过去方法处理有限频模型降阶时的不确定性和不可靠性。此外埘全频H_∞模型降阶问题,本章的方法也比现有文献中的一些同类方法具有更少的保守性。数值算例和仿真进一步说明了本章提出方法的有效性和优越性。
第五章研究了离散时间系统的静态输出反馈控制问题。基于LMI技术,分别给出了一组针对镇定控制,H_∞控制,以及正实控制的静态输出反馈控制设计条件。和现有文献中的同类方法相比,本章的设计方法结合了鲁棒控制领域中的参数依赖Lyapunov函数方法,引入了更多的辅助变量,进而具有更少的保守性。此外,本章中的设计条件都借助于Finsler引理在一个统一的框架得出,可以清楚的说明现有结果之间以及本章结果与现有结果之间的区别与联系。数值算例进一步说明了本章方法的有效性与优越性。
第六章考虑了具有多胞不确定性的离散线性系统的静态输出反馈控制问题。首先给出了基于参数依赖Lyapunov函数方法的鲁棒正实性分析结果,并从理论上证明了这些结果与现有文献中的同类结果之间的关系。然后根据第五章中的设计方法,并结合不确定系统中的一些放缩技巧,分别就H_∞控制,正实控制问题给出了一组不确定系统的鲁棒静态输出反馈控制器设计方法。数值例子说明了方法的有效性。
第七章研究了带有时不变状态时滞的线性离散时间系统的静念输出反馈控制问题。根据第五章中的设计方法,并结合处理时滞项的Jensen不等式方法,给出了时滞依赖的H_∞静念输出反馈控制器设计方法。最后通过数值例子进一步表明本章方法的有效性。
最后对全文所做的工作进行了总结,并指出了下一步研究的方向。
Due to the increasing development of informationization,systematization of the modern society,the dimensions of various control systems are becoming larger and larger, and the resulting complexity for system analysis and synthesis are also increased because the increasing order of the system model and the corresponding controller.Therefore, the reduction theory(i.e,model reduction and reduced-order controller design) is always a burgeoning research area.Great developments and wide applications have been made during the last several decades.However,there are still some problems that cannot be properly solved via the existing methods.For example,to some extent there exists inaccuracy and unreliability while using the existing method to cope with the known operating frequency information of the system,and there exists no approximation performance information over the known fiequency interval.Besides,how to reduce the conservatism of the existing LMI-based design methods for model reduction and static output feedback control is also an important problem.
This thesis,based on previous works of others,presents new methods for model reduction and static output feedback control problems via LMI-based approach.For the model reduction problem that with known fiequency information about the input signal, the design conditions are developed with the aid of the generalized KYP lemma,which can deal with the approximation error over finite frequency directly.Therefore,the inaccuracy resulted by the existing methods such as frequency-weighted method can be avoided.For model reduction problems over entire frequency interval and static output feedback control problems for discrete-time systems,design methods with less conserv-ativeness compared with the counterpart ones in the literatures are developed.Besides, static output feedback controller design methods for systems with polytopic uncertainties and time-invariant delay are also presented respectively.Parts of the developed methods are applied to the model reduction of RLC circuit systems.Numerical examples and simulations illustrate the advantages and effectiveness of our approaches.
Chapters 1-2 summarize the development and main research methods in the burgeoning research areas:model reduction and static output feedback control.Preliminaries about the considered problems are also given.
Chapters 3-4 present new LMI-based design methods for H_∞and H_2 model reduction problems for linear continuous-time systems and discrete-time systems,respectively. Based on the recently developed generalized KYP lemma,design methods of H_∞model reduction are developed under low-frequency,middle frequency,high frequency,and entire frequency interval considerations according to the frequency information about input signal.Consequently,the uncertainty and unreliability of the existing methods for finite frequency model reduction problems are avoided.For the entire frequency H_∞model reduction problems,it is also pointed out that the conservativeness of the proposed methods in this chapter is less than the existing ones.Numerical examples and simulations illustrate the effectiveness and advantages of the proposed approach.
Chapter 5 investigates the static output feedback control problem for linear discretetime systems.Stabilization,H_∞and positive real static output feedback control design methods are presented based on LMI technique respectively.By utilizing the parameter-dependent Lyapunov function method which originated in the research area of robust control and introducing more auxiliary variables,the conservativeness of the proposed methods is further reduced compared with the existing ones.Besides,the differences and relationships between the proposed methods and the existing methods can be clearly demonstrated due to those methods are presented in a unified framework in terms of the Finsler lemma.Numerical examples illustrate the effectiveness and advantages of the proposed approach.
Chapter 6 focuses on the static output feedback control problem for linear discretetime systems with polytopic uncertainties.Firstly,robust positive realness analysis results are given based on the parameter-dependent Lyapunov function method and the relationship between the proposed result and the existing one is clarified theoretically.Combining some relaxation techniques,H_∞and positive realness static output feedback controller design methods are presented.Numerical examples illustrate the effectiveness and advantages of the proposed approach.
Chapter 7 investigates the static output feedback control problem for linear discretetime systems with time-invariant state delay.Combining the Jensen inequality approach that dealing with the delay items,H_∞static output feedback controller design methods are presented.Numerical examples illustrate the effectiveness and advantages of the proposed approach.
Finally,the results of the dissertation are summarized and further research topics are pointed out.
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