类分数阶微分系统的能控和能观性  

Controllability and Observability for Fractional-Like Differential System

作　　者：郭小春[1] GUO Xiao-chun(School of Information Science and Technology/Taishan University,Tai'an 271000,China)

机构地区：[1]泰山学院信息科学技术学院,山东泰安271000

出　　处：《山东农业大学学报（自然科学版）》2024年第6期943-949,共7页Journal of Shandong Agricultural University：Natural Science Edition

摘　　要：动态微分系统的能控性和能观性是现代控制理论的基本问题,紧密联系着微分系统的极点配置、最优控制和观测器设计等。能控性是评估系统是否具有通过控制作用支配系统状态的能力,而能观性则研究系统测量输出是否具备推断系统内部状态的能力。本文研究了类分数阶线性微分系统和矩阵李雅普诺夫微分系统的能控和能观性问题。首先,应用常数变易法给出类分数阶线性系统的解析解,基于解析解导出类分数阶线性微分系统能控能观的充分必要条件;其次,应用相似方法提出了类分数阶矩阵李雅普诺夫微分系统能控能观性的判定准则,同时研究其稳定性问题;最终,通过两个具体的数值案例,本文提出的理论结果的有效性得到了充分的验证,从而不仅证实了我们的分析的正确性,而且展示了所提出方法在实际应用中的潜力和价值。The controllability and observability of dynamical differential system are fundamental issues in modern control theory,closely related to pole assignment,optimal control,and observer design in differential systems.Controllability assesses the ability of the system to control the state of the system through control action,while observability examines the ability of a system's measurement outputs to infer its internal states.In this paper,the controllability and observability of the fractional order linear differential systems and matrix Lyapunov differential system was studied.Firstly,we derive the explicit solutions via the variation of constant approach for the fractional-like linear differential system,based on the analytical solution,the necessary and sufficient conditions for the controllability and observability of fractional-order linear differential systems are derived.Secondly,a similar approach is employed to propose criteria for the controllability and observability of fractional-order matrix Lyapunov differential systems,while also examining their stability.Ultimately,the theoretical results validity proposed in this paper is thoroughly validated through two specific numerical examples,not only confirming the correctness of our analysis but also demonstrating the potential and value of the proposed methods in practical applications.

关 键 词：类分数阶线性微分系统 类分数阶矩阵李雅普诺夫微分系统 能控性 能观性 

分 类 号：O231.1[理学—运筹学与控制论]