关于分数阶系统的稳定性与反馈控制研究

发布时间：2018-05-12 06:35

  本文选题：分数阶系统 + 奇异系统　； 参考：《河北师范大学》2016年博士论文

【摘要】：分数阶系统是由微分阶次为任意实数甚至复数的微分方程所描述的动力学系统.分数阶控制系统是指被控系统为分数阶系统或者控制器为分数阶控制器的控制系统.本文以分数阶系统作为研究对象,主要从分数阶奇异系统、分数阶模糊系统、同分数阶非线性系统以及多分数阶非线性系统的稳定性与反馈控制等四个方面进行了研究.本文的主要研究内容包括:(1)研究了一类分数阶奇异不确定系统的稳定性与反馈控制问题.首先,根据分数阶奇异系统的稳定性理论,针对分数阶属于01的分数阶奇异不确定系统,给出了判断该类系统鲁棒渐近稳定的充分条件.其次,通过矩阵的奇异值分解和线性矩阵不等式(LMI)技术,讨论了分数阶奇异不确定系统的反馈控制问题,并设计了合适的状态反馈和输出反馈控制器,使得闭环系统是渐近稳定的.最后,通过三个数值仿真实例均验证了所得结论的正确性与设计思想的有效性.(2)研究了两类分数阶T-S模糊不确定系统的稳定性与反馈控制问题.首先,利用分数阶线性系统的稳定性理论,针对分数阶属于01以及1≤2两种不同情况下,分别给出了判断这两类分数阶T-S模糊不确定系统鲁棒渐近稳定的充分条件.其次,通过LMI技术,讨论了分数阶T-S模糊不确定系统的模糊反馈控制问题,并设计了合适的模糊输出反馈控制器,使得闭环系统对于所有可容许的不确定项是渐近稳定的.最后,通过两个数值仿真实例分别验证了所得结论的正确性与设计思想的有效性.(3)研究了一类同分数阶非线性不确定系统的稳定性与反馈控制问题.首先,根据分数阶系统Lyapunov稳定性理论,针对分数阶属于01的同分数阶非线性不确定系统,给出了判断该类系统鲁棒渐近稳定的充分条件.其次,通过LMI技术,讨论了同分数阶非线性不确定系统的反馈控制问题,并设计了合适的状态反馈控制器,使得闭环系统是渐近稳定的.最后,通过对同分数阶混沌Liu系统进行数值仿真验证了所得结论的正确性与设计思想的有效性.(4)研究了一类多分数阶非线性系统反馈控制问题.首先,利用多分数阶非线性系统的稳定性理论,给出了分数阶属于01的多分数阶非线性受控系统在不同平衡点处渐近稳定的充分条件.其次,利用受控系统在平衡点处Jacobian矩阵的特征值,设计了合适的状态反馈控制器,使得闭环系统在不同平衡点处是渐近稳定的.最后,通过对多分数阶的非混沌捕食-食饵系统和混沌Chen系统进行数值仿真分别验证了所得结论的正确性与设计思想的有效性.
[Abstract]:Fractional order system is a dynamic system described by differential equations with differential order being arbitrary real number or even complex number. Fractional control system refers to the control system which is a fractional system or a controller is a fractional controller. In this paper, fractional order systems are studied in four aspects: fractional singular systems, fractional fuzzy systems, nonlinear systems of the same fractional order and the stability and feedback control of multi-fractional nonlinear systems. In this paper, we study the stability and feedback control of a class of fractional singular uncertain systems. Firstly, according to the stability theory of fractional singular systems, a sufficient condition is given to judge the robust asymptotic stability of fractional singular uncertain systems with fractional order 01. Secondly, the feedback control problem of fractional singular uncertain systems is discussed by using singular value decomposition of matrices and LMI technique, and appropriate state feedback and output feedback controllers are designed. The closed loop system is asymptotically stable. Finally, three numerical simulation examples are given to verify the correctness of the conclusions and the validity of the design idea.) the stability and feedback control problems of two classes of fractional T-S fuzzy uncertain systems are studied. Firstly, by using the stability theory of fractional linear systems, sufficient conditions for judging the robust asymptotic stability of these two classes of fractional T-S fuzzy uncertain systems are given for the two different cases of fractional order 0 1 and 1 鈮，

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