The topic of the present dissertation is robustness and performance issues in nonlinear control systems. The control systems in our study are described by nominal models consisting of nonlinear deterministic or stochastic differential equations in a Euclidean state space. The nominal models are subject to perturbations which are completely unknown dynamic systems, except that they are known to possess certain properties of dissipation. A dissipation property restricts the dynamic behaviour of the perturbation to conform with a bounded resource; for instance energy. The main contribution of the dissertation is a number of sufficient conditions for robust performance of such systems. Since the perturbations in these uncertain models possess several dissipation properties simultaneously, we study fundamental properties of such multi-dissipative systems. These properties are related to convexity and topology on the space of supply rates. For instance, we give conditions under which the available storage is a continuous convex function of the supply rate. Dissipation theory in the existing literature applies only to deterministic systems. This is unfortunate since robust control applications typically also contain uncertainty which is better modelled in a probabilistic framework, such as measurement noise. This motivates an extension of the theory of dissipative dynamic systems to stochastic systems. This dissertation presents such an extension: We propose a definition and generalize fundamental results from deterministic dissipation theory to stochastic systems. Furthermore, we argue that stochastic dissipation is a natural fundament for a theory of robust performance of stochastic systems. To this end, we present a number of performance requirements to stochastic systems which can be formulated in terms of dissipation, after which we give sufficient conditions for these requirements to be robust towards multi-dissipative perturbations. A final contribution of the dissertation concerns the problem of simultaneous H-infinity control of a finite number of linear time invariant plants. This problem is a prototype of robust adaptive control problems. We show that the optimal (minimax) controller for this problem is finite dimensional but not based on certainty equivalence, and we discuss the heuristic certainty equivalence controller.
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Department of Mathematical Modelling, Technical University of Denmark, 1998