On H-infinity Control and Large-Scale Systems

Carolina Bergeling

Abstract:  In this thesis, a class of linear time-invariant systems is identified for which a particular type of H-infinity optimal control problem can be solved explicitly. It follows that the synthesized controller can be given on a simple explicit form. More specifically, the controller can be written in terms of the matrices of the system’s state-space representation. The result has applications in the control of large-scale systems, as well as for the control of infinite-dimensional systems, with certain properties. For the large-scale applications considered, the controller is both globally optimal as well as possesses a structure compatible with the information-structure of the system. This decentralized property of the controller is obtained without any structural constraints or regularization techniques being part of the synthesis procedure. Instead, it is a result of its particular form. Examples of applications are electrical networks, temperature dynamics in buildings and water irrigation systems. In the infinite-dimensional case, the explicitly stated controller solves the infinite-dimensional H-infinity synthesis problem directly without the need of approximation techniques. An important application is diffusion equations. Moreover, the presented results can be used for evaluation and benchmarking of general purpose algorithms for H-infinity control. The systems considered in this thesis are shown to belong to a larger class of systems for which the H-infinity optimal control problem can be translated into an optimization problem at a single frequency. In certain cases, the optimization problem can be solved through a simple least-squares argument. This procedure is what renders the simple and explicit expression of the controller previously described. Moreover, the given approach is in contrast to conventional methods to the problem of H-infinity control, as they are in general performed numerically.