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.
