Sensor Blending for Direct Adaptive Control of Non-Minimum Phase Linear Infinite-Dimensional Systems in Hilbert Space
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abstract
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. Using a new normal form for these 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 that are non-minimum phase and how we can make them minimum phase systems without complicated modifications. First we will show that the zeros are invariant under coordinate transformations and both static and dynamic feedback. So our only choice will be to modify the input-output operators B & C. This is called sensor (or actuator) blending. Our principal result will be a systematic way to do sensor blending on an infinite dimensional linear system with a finite number of unstable zeros. These results will be 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.
