Infinite-Dimensional Control Systems: A Combined Distributional/Ultradifferentiable Approach

在结构动力学（例如机器人或空间中的柔性结构）中，控制和测量的选择对于所设计结构的性能至关重要。某些控制和测量的选择可以使用适定线性系统理论进行详细的理论分析，该理论是在过去二十年中发展起来的。这种详细的分析随后导致针对给定的控制和测量选择的更好的控制器设计（例如，动态控制器而不是静态控制器），并且通过此导致所设计的结构的更好的性能。不幸的是，似乎是最有效的控制和测量的选择并不总是导致一个适定的线性系统。然而，这些出现在工程文献中的控制和测量的有效选择总是导致分布式控制系统，这是最近由主要研究者引入的概念。每个适定的线性系统都是分布式控制系统，并且上面提到的是分布式控制系统但不是适定的线性系统的系统的例子提供了将现有的控制理论从适定的线性系统扩展到这类更一般的分布式控制系统的动机。将使用被称为超可微半群的解析半群的推广。这是必要的，因为即使是最有效的选择，控制和测量结构动力学应用往往不能产生一个分析闭环半群，但只有一个超可微的。它将调查是否有限的误差界模型减少平衡截断，已知存在时，半群是解析的，推广到超可微的情况下。静态反馈下闭环系统的这一特性对于开环系统低阶动态控制器的设计是至关重要的。 因此，受结构动力学应用的启发，本文将研究开环系统是分布控制系统，而闭环系统在静态反馈下具有超可微半群的情况。目前，数学上合理的柔性结构控制器的设计是有限的，要求系统是适定的或反馈控制器是静态的。在这个项目中的方法，使数学分析更有效的控制器设计的柔性结构。 2.在研究中获得的理论结果和见解将有可能应用于结构动力学以外的其他领域，如声学系统，热力学系统和复合材料系统的研究。

In structural dynamics (for example flexible structures for robots or in space) the choice of control and measurement is of fundamental importance for the performance of the designed structure. Certain choices of control and measurement can be theoretically analyzed in great detail using the theory of well-posed linear systems, which was developed during the last two decades. This detailed analysis subsequently leads to better controller designs for the given choice of control and measurement (for example dynamic controllers instead of static controllers) and through this to better performance of the designed structure. Unfortunately, what seem to be the most effective choices for control and measurement do not always lead to a well-posed linear system. These effective choices of control and measurement that appear in the engineering literature do however always lead to a distributional control system, a notion recently introduced by the principal investigator. Every well-posed linear system is a distributional control system and the above mentioned examples of systems that are distributional control systems, but not well-posed linear systems, provides the motivation for extending the existing control theory from well-posed linear systems to this more general class of distributional control systems.To measure the performance of the designed structure, a generalization of analytic semigroups known as ultradifferentiable semigroups, will be used. This is needed since even the most effective choices for control and measurement in structural dynamics applications often cannot produce an analytic closed-loop semigroup, but only an ultradifferentiable one. It will be investigated whether finite error-bounds for model reduction by balanced truncation, known to exist when the semigroup is analytic, generalize to the ultradifferentiable case. This property of the closed-loop system under static feedback is crucial in designing low-order dynamic controllers for the open-loop system. Thus, motivated by structural dynamics applications, the situation where the open-loop system is a distributional control system and the closed-loop system under static feedback has an ultradifferentiable semigroup will be investigated.The broader impact of the research will be twofold: 1. Presently, mathematically justified designs of controllers for flexible structures are limited by requiring either that the system be well-posed or that the feedback controller is static. The approach in this project enables mathematical analysis of more effective controller designs for flexible structures. 2. The theoretical results and insights obtained in the study will have the potential to be applied to other areas than structural dynamics such as the study of acoustic systems, thermodynamic systems and systems comprised of composite materials.