System-theoretic Analysis and Controller Design for PDEs

The analysis and control of systems is advantageously based on mathematical models of the underlying (physical) processes. Dynamical processes can be described by so-called differential equations, where ordinary and partial differential equations are distinguished. The analysis of the latter system class is a challenging field of research in systems and control theory, and many results that are known for ordinary differential equations since decades could not yet be transferred to the more complex class of partial differential equations. In this project we want to contribute here. With mathematical methods we want to perform system analysis, i.e. characterize formal properties of the dynamical systems, and furthermore design control laws to impose a desired behaviour on the processes. In this research project differential geometric methods shall be employed, but we strive for an interdisciplinary project, and therefore we are also interested in a symbiosis of several mathematical disciplines. Starting from a formal approach based on the mentioned geometric methods, we plan to use, amongst others, also functional analytic methods. In the literature there already exist approaches that are purely formal (i.e. differential geometric), as well as methods that use function theory. However, up to the knowledge of the applicant, a holistic/integrated approach for the analysis of partial differential equations from a system and control theoretic viewpoint has not yet been considered. With the current state of the art, a systematic controller design for partial differential equations is only possible for very simple systems, and one of the main goals of the present project is to develop a systematic design method that is based on energy flows. For this purpose, we use a system representation that is linked to the energy relations, and we try to manipulate the energy flows in such a way that the system shows a desired behaviour. For the examination of energy balances, the formal methods based on differential geometric concepts are perfectly suited. However, the mathematical proof that the system then exhibits the desired behaviour requires functional analytic methods.

Dynamic processes can be described using so-called differential equations (DEs), whereby a distinction is made here between ordinary and partial differential equations. The analysis for the latter system class is a demanding area within the framework of systems and control theory. In this project, mainly differential-geometric methods were used to characterize system properties on the one hand and to design controllers on the other hand. The controller designs analyzed in this project are based on the idea of influencing the energy flows in such a way that the system shows a desired behavior. This has been systematically investigated for mechanical systems such as beams or plates, whereby the actuators intervene either on the boundary (forces or moments) or in the domain (e.g. through force densities generated by piezoelectric actuators). However, these control concepts also require information about the current state of the system. Therefore, as part of this energy-based approach, virtual sensors (observers) were also designed to accomplish this. For systems that are described by partial differential equations, this is highly non-trivial - in particular the proof that these virtual sensors actually provide correct estimates of the system variables. In addition to these controller and observer designs, more fundamental questions of a system-theoretical nature were also considered. This involved the analysis of system classes (here partial differential equations) with regard to the question of the extent to which controllers, observers or controls can in principle be designed for such systems. For this purpose, different mathematical concepts were used, such as classical and generalized symmetry groups, but also functional-theoretical concepts. The latter are indispensable, e.g. in particular for proving the exact controllability of a system. Also very useful in this context is the concept of flatness, which was used to design a feed-forward control for an under-actuated beam. For partial differential equations, however, the concept of flatness is mainly restricted to the linear case. However, partial differential equations can always be converted to difference equations by discretization with regard to space and time. For this system class, a systematic test for the property of flatness was developed as part of this project, which is computationally much easier than all the methods previously known in the literature on this subject.