多阶分数线性系统的分析与综合

In recent decades，factional calculus theory is applied more and more in science and engineering, so that it has become a hot topic and attracted increasing interest. Fractional order control systems are the extension of traditional integer order control systems. Fractional order control systems have now been applied in mechanics, electrical, medical engineering, and so on. There are abundant research results in commensurate order fractional linear systems. The theory of commensurate order fractional linear systems has been gradually perfect. But the results of multi- order fractional linear systems are relatively less. The project takes multi-order fractional linear systems as the research objects. By using the Laplace transform, Mittag-Leffler function, piecewise orthogonal function and operational matrix , we investigate the related theory of analysis and synthesis in this kind of systems . Firstly, conditions of controllability and observability for multi-order fractional linear systems and the dual relationship of the systems will be studied. Secondly, we will explore the relationship between the multi-order fractional linear systems and the corresponding equivalent systems in performance. And the systems structure decomposition problem will be discussed. Thirdly, stability and stabilization conditions of multi-order fractional linear systems will be considered, as well as the robust stability and robust stabilization for multi-order fractional linear interval systems.The extension of analysis and synthesis of commensurate order fractional linear systems to multi-order case lay the theoretical foundation for the further applications of such systems.

近几十年来，分数阶微积分理论在实际科学与工程中的应用越来越多，已经成为学者们广泛关注的一个热门课题。分数阶控制系统是传统整数阶控制系统的扩展，目前已在力学、电学、医学工程等领域得到了应用。对于单阶分数线性系统，目前已经有比较丰富的研究成果，其理论已经日渐成熟，而对多阶分数线性系统的研究成果还相对较少。本项目以多阶分数线性系统为研究对象，以Laplace变换、Mittag-Leffler函数、分段正交函数和运算矩阵等为工具，探讨这类系统分析与综合的相关理论。首先，研究多阶分数线性系统能控与能观的条件，以及系统的对偶关系。其次，探讨多阶分数线性系统与其相应等价系统在性能上的关系，以及系统的结构分解问题。再次，研究多阶分数线性系统稳定性与能镇定的条件，以及多阶分数线性区间系统的鲁棒稳定性与鲁棒镇定问题。将单阶分数线性系统分析与综合的理论扩展到多阶的情形，为此类系统进一步的应用奠定理论基础。

该项目主要对多阶分数线性系统分析与综合方面的问题进行研究。围绕项目任务书拟定的研究内容开展研究，获得了以下几方面的研究成果和阶段性成果。第一，以Lapace变换为工具并利用扩展的Mitagge-Leffler函数，获得了多阶分数线性系统解的表达式，进而推导出系统能控与能观的充要条件。第二，通过定义一类多阶分数线性系统的对偶系统，推导出了多阶分数线性系统的对偶原理，阐述了系统能控性与其对偶系统能观性的等价关系。第三，利用多阶分数线性系统与其同阶系统的等价关系，获得了多阶分数线性系统稳定性与可镇定性的条件。第四，以整数阶奇异线性系统理论为基础，对具有奇异性的分数阶线性系统的能控性与能观性问题进行研究，推导出奇异分数阶线性系统能控与能观的条件。第五，对区间分数阶线性系统的鲁棒控制问题进行研究，首先考虑了整数阶线性系统鲁棒控制的强化学习方法，以最优控制理论为基础，我们提出了整数阶线性系统鲁棒控制的强化学习算法，获得了这个研究内容的阶段性成果。并且，目前完成分数阶线性系统鲁棒控制强化学习算法的理论推导。此外，项目组还对美式期权定价问题和几类矩阵特征值问题进行研究，获得了一些研究成果。