Stabilization of Fixed Gain Controlled Nonminimum Phase Infinite Dimensional Systems with Actuator Dynamics by Augmentation with Direct Adaptive Control

摘要

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. Almost Strictly Dissipative (ASD) is sufficient to guarantee global asymptotic stability of the adaptively augmented system and bounded adaptive gains. In many cases a direct adaptive control must be implemented on a nonminimum phase infinite dimensional plant. By nonminimum phase we mean that the plant has some transmission zeros that are not exponentially stable. One way to deal with this situation is to add a feedthrough or D term in the output which can also be seen as an output feedback term for the adaptive controller. In this paper, we focus on such nonminimum phase infinite dimensional linear systems for which a fixed gain linear infinite or finite dimensional controller is already in place and the system is operated through actuator dynamics. We augment this controller with a direct adaptive controller that will maintain stability of the full closed loop system. We prove that the transmission zeros of the combined system are the open loop transmission zeros of the infinite dimensional plant with an added feedthrough or D term, the actuator zeros, and the point spectrum of the controller alone. Therefore the combined plant plus controller is ASD if and only if modified open loop system is minimum phase, the actuator dynamics are minimum phase, and the fixed gain controller alone is exponentially stable. This result is true whether the fixed gain controller or actuator dynamics are finite or infinite dimensional subsystems. These results are illustrated by application to direct adaptive control of general linear systems on a Hilbert space that are described by self-adjoint operators with compact resolvent.