不确定耦合PDE-ODE系统的自适应镇定

The adaptive stabilization is investigated for three classes of uncertain coupled PDE-ODE systems, whose actuator dynamics are described by heat equations or wave equations. Compared with the existing results, the presence of serious uncertainties/unknowns (e.g., there exist unknown parameters, states are unmeasurable and input is suffered from disturbance) and the coupling between PDE sub-systems and the ODE ones make the systems under investigation more general and representative, and hence more challenge to stabilize. The proposed academic problems will be solved by synthetically utilizing infinite-dimensional backstepping method, adaptive technique, the stability theory and the constructive methods of observers and controllers for distributed parameter systems. Specifically, an invertible infinite-dimensional backstepping/forwarding transformation will be searched for to change the original system into a target system whose stability in certain sense will imply the desirable stability of the original closed-loop system. From the target system, the control design and performance analysis will become quite convenient. Then, a stabilizing controller for the target system is designed, which can also guarantee the desirable stability of the original system. It is necessary to point out that, how to choose proper parameter updating laws so as to effectively compensate the uncertainties/unknowns and design reasonable state observers so as to reconstruct the unmeasurable states will be the key to the successful resolution of problems. This proposal will form new framework on control design and performance analysis of uncertain coupled PDE-ODE systems, extend the range of application of the existing results, and hence enrich and develop the control theory of distributed parameter systems.

计划研究执行器动态由热方程或波方程描述的三类典型不确定耦合PDE-ODE系统的自适应镇定问题。与已有结果相比，拟研究系统具有严重不确定性/未知性（如存在未知参数、状态不可测、输入端存在扰动）且PDE与ODE两子系统是耦合的，因而更具有一般性和代表性，实现其镇定更有挑战性。将综合利用无穷维反推方法、自适应技术、分布参数系统观测器和控制器构造方法及稳定性理论解决拟定科学问题。具体地，将寻找可逆无穷维反推/前推变换，把原系统转化为便于控制设计和性能分析的目标系统（该目标系统一定的稳定性意味着原闭环系统期望的稳定性），进而为目标系统设计控制器以实现反馈镇定。其中，选择适当的参数调节律实现不确定性/未知性的有效补偿、构造合理的状态观测器实现对不可测状态的重构是问题成功解决的关键。本研究将形成不确定耦合PDE-ODE系统控制设计与性能分析的新框架，拓宽现有结果适用范围，丰富和发展分布参数系统控制理论。

本项目主要研究了PDE子系统分别由抛物和双曲方程描述的不确定耦合和串联PDE-ODE系统的自适应镇定控制问题。实际工业生产过程中，许多动态过程可以建模成耦合或串联PDE-ODE系统,且系统不可避免的存在不确定性/未知性，如系统存在未知参数、状态不完全可量测或受扰动影响，这使得所研究系统更具有代表性且导致实现其镇定控制设计更具有挑战性。本项目综合利用无穷维反推方法、自适应动态补偿技术、PDE系统控制器和观测器构造性方法，解决了具有不同的耦合及串联形式，不同的不确定性/未知性刻画方式的多类不确定耦合和串联PDE-ODE系统的自适应状态反馈和输出反馈镇定控制问题，形成了不确定耦合和串联PDE-ODE系统的自适应镇定控制设计与性能分析的新框架。此外，还利用本项目发展的自适应控制设计框架解决了一类不确定线性化Ginburg-Landau方程的镇定问题。本项目成果的取得将丰富和发展分布参数系统控制理论，为实际工业生产提供指导。