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所设计结构的性能至关重要。利用近二十年来发展起来的适定线性系统理论，可以从理论上详细地分析控制和测量的某些选择。这种详细的分析随后导致更好的控制器设计为给定的控制和测量选择（例如动态控制器而不是静态控制器），并通过这来更好地设计结构的性能。不幸的是，似乎是最有效的控制和测量选择并不总是导致一个适定的线性系统。这些在工程文献中出现的控制和测量的有效选择确实总是导致分布式控制系统，这是首席研究员最近提出的概念。每个良定线性系统都是一个分布控制系统，上面提到的系统是分布控制系统的例子，而不是良定线性系统，提供了将现有的控制理论从良定线性系统扩展到更一般的分布控制系统的动机。为了测量所设计结构的性能，将使用一种被称为超可微半群的解析半群的推广。这是必要的，因为在结构动力学应用中，即使是最有效的控制和测量选择通常也不能产生解析闭环半群，而只能产生超可微半群。研究了当半群为解析半群时存在的平衡截断模型约简的有限误差界是否推广到超可微情况。静态反馈下闭环系统的这一特性对于开环系统的低阶动态控制器的设计至关重要。因此，在结构动力学应用的激励下，我们将研究开环系统是分布控制系统和静态反馈下闭环系统具有超可微半群的情况。这项研究的广泛影响将是双重的：1。目前，在数学上合理的柔性结构控制器设计受限于要求系统是适定的或反馈控制器是静态的。在这个项目的方法使数学分析更有效的控制器设计柔性结构。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.