非常数扩散率的分数阶分布参数系统的控制

Control of a Fractional Distributed Parameter System with Non-constant Diffusivity

研究了具有混合边界条件和非常数扩散率的分数阶分布参数系统(原系统)的边界反馈控制问题.该问题可以视为具有常数扩散率的分数阶分布参数系统边界反馈镇定问题的推广.具体而言,通过变量变换将原系统转化为更具一般性的分数阶分布参数系统(新系统).利用反步法、积分变换设计新系统的Dirichlet边界反馈控制器,再根据给定的变量变换得到原系统的边界反馈控制器.进一步而言,基于Mittag-Leffler稳定性理论(分数阶李亚普诺夫稳定性理论)获得了在边界反馈控制器作用下的原系统Mittag-Leffler稳定的充分条件.最后给出了具体的数值仿真算例,从而说明了本文所提方法的有效性.

We consider the boundary feedback control of a fractional distributed parameter system( original system)with mixed boundary conditions and non-constant diffusivity. This situation can be viewed as a generalization of the boundary feedback stabilization problem of a fractional distributed parameter system with constant diffusivity. Specifically,we convert the original system into a general fractional distributed parameter system( new system) by a change of variables. We utilize the backsteping method and integral transformation to design a Dirichlet boundary feedback controller for the new system. Then,we can obtain a boundary feedback controller for the original system via the change of variables. Moreover,based on the Mittag-Leffler stability theory( fractional Lyapunov stability theory),we obtain sufficient conditions for Mittag-Leffler stability of the original system by the boundary feedback controller. Lastly,we present a specific numerical simulation example to verify the effectiveness of our proposed method.