A Stabilization of Fixed Gain Controlled Infinite Dimensional Systems by Augmentation With Direct Adaptive Control

Linear infinite dimensional systems are described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on a general Hilbert space of states and are controlled via a finite number of actuators and sensors. Many distributed applications are included in this formulation, such as large flexible aerospace structures, adaptive optics, diffusion reactions, smart electric power grids, and quantum information systems. We have developed the following stability result: an infinite dimensional linear system is Almost Strictly Dissipative (ASD) if and only if its high frequency gain CB is symmetric and positive definite and the open loop system is minimum phase, i.e. its transmission zeros are all exponentially stable. In this paper, we focus on infinite dimensional linear systems for which a fixed gain linear infinite or finite dimensional controller is already in place. It is usually true that fixed gain controllers are designed for particular applications but these controllers may not be able to stabilize the plant under all variations in the operating domain. Therefore we propose to augment this fixed gain controller with a relatively simple direct adaptive controller that will maintain stability of the full closed loop system over a much larger domain of operation. This can ensure that a flexible structure controller based on a reduced order model will still maintain closed-loop stability in the presence of unmodeled system dynamics. The augmentation approach is also valuable to reduce risk in loss of control situations. First we show that the transmission zeros of the augmented infinite dimensional system are the open loop plant transmission zeros and the eigenvalues (or poles) of the fixed gain controller. So when the open-loop plant transmission zeros are exponentially stable, the addition of any stable fixed gain controller does not alter the stability of the transmission zeros. Therefore the combined plant plus controller is ASD and the closed loop stability when the direct adaptive controller augments this combined system is retained. Consequently direct adaptive augmentation of controlled linear infinite dimensional systems can produce robust stabilization even when the fixed gain controller is based on approximation of the original system. These results are illustrated by application to a general infinite dimensional model described by nuclear operators with compact resolvent which are representative of distributed parameter models of mechanically flexible structures. with a reduced order model based controller and adaptive augmentation.