基于时间加权H2指标的Markov跳变系统的模型降阶问题研究

It is well known that mathematical modelling of physical systems often involves high-order models, which bring difficulties in simulation, analysis, design of the systems concerned. This has motivated the investigation on a variety of model reduction problems. The prime purpose of model reduction is to find a lower-order model to approximate the original model, often subject to a given performance index. In this project, a new error measure named time-weighted H2 norm is defined based on the classical H2 norm for model reduction purpose. In this new error measure a heavy penalty is placed as time increases, hence the steady-state behavior of the error system will be greatly emphasized and the convergence rate of the error output is expected to be larger. In this project, we are concerned with the time-weighted H2 model reduction problem of Markov jump linear systems. First, the time-weighted H2 norm of discrete Markov jump system is defined and a computational method is constructed. One-order conditions are also given such that the corresponding time-weighted H2 model reduction problem can be solved.Second, efficient reduction approaches are proposed for Markov jump systems in both continuous and discrete time case. Finally, the definition of time-weighted H2 norm is improved by taking the amplitude of error output into account and corresponding reduction approaches for Markov jump linear systems are also given. The purpose of this project is to constuct the basic concepts and reduction methods of time-weighted H2 model reduction problem for Markov jump systems, which will bring much convenient for engineering practice,and promote the development of model reduction theory at the same time.

实际物理系统进行数学建模后往往具有较高的阶次，这给系统的分析、仿真和设计带来了很大的困难。模型降阶以简化模型为目的，在某种误差指标下寻求高阶系统的近似低阶模型。本项目拟在H2范数的基础上定义时间加权H2范数作为误差指标，并基于该指标研究Markov跳变系统的模型降阶问题。引入新型指标的目的是保证误差系统H2性能的同时，使得在相同输入下,降阶系统的输出更快地逼近原系统的输出。主要研究内容包括：1）定义并计算离散Markov跳变系统的时间加权H2范数，建立该降阶问题的可解条件；2）在连续及离散两种情况下，分别给出高效的Markov跳变系统的时间加权H2模型降阶算法；3）改进时间加权H2指标并设计相应的模型降阶方法，同时满足误差系统响应速度和超调量要求。本项目旨在建立Markov跳变系统时间加权H2模型降阶的基本概念和方法，不但有利于工程实践，对模型降阶理论的进一步完善和发展也将起到促进作用。

模型降阶问题根据控制系统性能的要求, 研究如何运用数学手段, 将原模型由高阶化成低阶, 由复杂变为简单, 同时确保简化后的模型（降阶模型）与原始模型在某种误差指标下达到一定的近似要求。本课题在前期工作的基础上，提出并研究了Markov跳变系统的时间加权H2模型降阶问题。主要取得的研究结果如下。 （1）将Markov跳变系统H2范数的定义推广为时间加权H2范数，并给出了计算公式。（2）以时间加权H2范数作为新的误差指标，提出了Markov跳变系统时间加权H2模型降阶问题；并给出了降阶模型参数的构造方法和算法。仿真算例表明，所给的时间加权H2模型降阶方法比普通的H2模型降阶误差减小得更快。（3）给出了降阶模型为最优时间加权H2降阶模型的必要条件，该条件为已有“Wilson条件”的一个推广。（4）针对一类正系统，定义了时间加权H1范数，同时给出了其时间加权H1控制器的设计方法。（5）提出正系统时间加权H∞ 控制问题，给出了正系统时间加权H∞ 控制器的设计方法。本课题探索并建立了基于时间加权H2 范数指标的Markov跳变系统模型降阶的基本概念和方法，为进一步丰富和完善模型降阶理论起到一定的推动和促进作用; 同时本课题的研究结果为其他控制问题提供了思路和可行的办法。