线性奇异摄动系统极限性能分析与综合
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摘要
通过对奇异摄动系统状态解极限性质的深入研究，本论文探讨了广义系统理论与方法在奇异摄动系统分析与综合中的适用性问题。奇异摄动系统的控制理论已得到了较为系统的研究。当忽略奇异摄动系统模型中的小参数时，可得极限系统模型；该极限系统模型具有广义系统形式。目前，广义系统理论已取得了非常系统深入的结果，为奇异摄动系统的综合控制提供了一个很有前景的新的方法途径。然而，由于研究的对象不同，广义系统控制器的分析与设计方法一般不能直接应用于原来的系统。针对上述问题，本论文将探讨当小参数趋于零时奇异摄动系统的极限特性。在此基础上，基于极限系统模型利用广义系统方法对原系统进行性能分析和综合设计。本论文完成了以下工作。
1研究了当小参数趋于零时奇异摄动系统状态解的收敛性问题。得到了奇异摄动系统状态解在广义函数意义下收敛的条件，并且得到了状态解在广义函数意义下的极限解，以及极限解与极限系统广义状态解的关系。
2基于极限系统模型，利用广义系统方法研究了奇异摄动系统H_∞性能分析和控制器设计问题。通过引入状态解收敛约束，给出了奇异摄动系统H_∞性能与其对应的极限系统H_∞性能之间的关系。在此基础上，进一步利用广义系统方法设计奇异摄动系统输出动态反馈H_∞控制器和状态反馈H_∞控制器。
3基于极限系统模型，利用广义系统方法讨论了奇异摄动系统鲁棒稳定性分析以及鲁棒镇定控制器设计问题。通过对奇异摄动系统鲁棒稳定性与其对应的极限系统鲁棒稳定性之间关系的探讨，在一定条件下给出了设计奇异摄动系统鲁棒镇定控制器的一类方法。
4基于极限系统模型，利用广义系统方法探讨了奇异摄动系统正实性问题。得到了奇异摄动系统正实性与极限系统正实性之间的关系，进而在一定的条件下提出并证明了判断奇异摄动系统正实性的判据。
5分析和研究了奇异摄动系统的H_∞模型降阶问题，得到了利用快、慢子系统的H_∞降阶模型构造原系统H_∞降阶模型的算法。
另外，我们通过数值算例说明了本论文得到方法的有效性。
In this. dissertation, by means of the further investigation of the limitingbehavior of linear time invariant (LTI) singularly perturbed systems we discuss theproblem of the applicability of the singular system theory and synthesisapproaches in the-control design of singularly perturbed systems. This limitingmodel of a singularly perturbed system is of the form of the singular systems,while nowadays, the theory of singular systems is well developed and thus pavesthe way to develop a possible new synthesis approach to singularly perturbedsystems. However, because of the controlled plant models is different, singularsystem theory cannot be applied into singularly perturbed systems in general.Hence, via survey of the algebraic condition that the limiting solution is availableto approach that of the singularly perturbed systems, the limiting models areemployed to design controller for their original singularly perturbed systems. Thedesired performance is thus achieved by feedback controller that is designed basedon the limiting system model. The major works in this dissertation are as follows.
Firstly, the limiting behavior of the solution of singularly perturbed controlsystems is discussed in distributional sense. Some improved conditions areproposed based on the existing results, under which, the solution of linear timeinvariant (LTI) singularly perturbed systems converges to that of the limitingsystem in distributional sense.
Secondly, the singular system models are employed to analyze the H_∞propertyof singularly perturbed systems. The conventional design techniques of the outputdynamic feedback and state feedback H_∞controller for singular systems areimproved to suitable for singularly perturbed systems. It is shown that when suchan improved controller is applied to corresponding singularly perturbed systems,the stability and H_∞performance of the closed loop can be guaranteed.
Thirdly, the singular system models are implemented to analyze the robuststability property of singularly perturbed systems. The conventional designtechniques of the state feedback robust stabilization controller for singularsystems are further improved to stabilize the original singularly perturbed system.
Fourthly, the singular system models are applied to analyze the strictly positive realness property of singularly perturbed systems. The relationship between thepositive realness of singularly perturbed system and that of its limiting system isobtained.
Finally, the problem of H_∞model reduction is investigated for singularlyperturbed control systems. An algorithm is proposed for the construction ofreduced model based on the slow-fast decomposition of original system.
1研究了当小参数趋于零时奇异摄动系统状态解的收敛性问题。得到了奇异摄动系统状态解在广义函数意义下收敛的条件，并且得到了状态解在广义函数意义下的极限解，以及极限解与极限系统广义状态解的关系。
2基于极限系统模型，利用广义系统方法研究了奇异摄动系统H_∞性能分析和控制器设计问题。通过引入状态解收敛约束，给出了奇异摄动系统H_∞性能与其对应的极限系统H_∞性能之间的关系。在此基础上，进一步利用广义系统方法设计奇异摄动系统输出动态反馈H_∞控制器和状态反馈H_∞控制器。
3基于极限系统模型，利用广义系统方法讨论了奇异摄动系统鲁棒稳定性分析以及鲁棒镇定控制器设计问题。通过对奇异摄动系统鲁棒稳定性与其对应的极限系统鲁棒稳定性之间关系的探讨，在一定条件下给出了设计奇异摄动系统鲁棒镇定控制器的一类方法。
4基于极限系统模型，利用广义系统方法探讨了奇异摄动系统正实性问题。得到了奇异摄动系统正实性与极限系统正实性之间的关系，进而在一定的条件下提出并证明了判断奇异摄动系统正实性的判据。
5分析和研究了奇异摄动系统的H_∞模型降阶问题，得到了利用快、慢子系统的H_∞降阶模型构造原系统H_∞降阶模型的算法。
另外，我们通过数值算例说明了本论文得到方法的有效性。
In this. dissertation, by means of the further investigation of the limitingbehavior of linear time invariant (LTI) singularly perturbed systems we discuss theproblem of the applicability of the singular system theory and synthesisapproaches in the-control design of singularly perturbed systems. This limitingmodel of a singularly perturbed system is of the form of the singular systems,while nowadays, the theory of singular systems is well developed and thus pavesthe way to develop a possible new synthesis approach to singularly perturbedsystems. However, because of the controlled plant models is different, singularsystem theory cannot be applied into singularly perturbed systems in general.Hence, via survey of the algebraic condition that the limiting solution is availableto approach that of the singularly perturbed systems, the limiting models areemployed to design controller for their original singularly perturbed systems. Thedesired performance is thus achieved by feedback controller that is designed basedon the limiting system model. The major works in this dissertation are as follows.
Firstly, the limiting behavior of the solution of singularly perturbed controlsystems is discussed in distributional sense. Some improved conditions areproposed based on the existing results, under which, the solution of linear timeinvariant (LTI) singularly perturbed systems converges to that of the limitingsystem in distributional sense.
Secondly, the singular system models are employed to analyze the H_∞propertyof singularly perturbed systems. The conventional design techniques of the outputdynamic feedback and state feedback H_∞controller for singular systems areimproved to suitable for singularly perturbed systems. It is shown that when suchan improved controller is applied to corresponding singularly perturbed systems,the stability and H_∞performance of the closed loop can be guaranteed.
Thirdly, the singular system models are implemented to analyze the robuststability property of singularly perturbed systems. The conventional designtechniques of the state feedback robust stabilization controller for singularsystems are further improved to stabilize the original singularly perturbed system.
Fourthly, the singular system models are applied to analyze the strictly positive realness property of singularly perturbed systems. The relationship between thepositive realness of singularly perturbed system and that of its limiting system isobtained.
Finally, the problem of H_∞model reduction is investigated for singularlyperturbed control systems. An algorithm is proposed for the construction ofreduced model based on the slow-fast decomposition of original system.
引文
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