Direct Adaptive Control of Non-Minimum Phase Linear Infinite-Dimensional Systems in Hilbert Space Using a Zero Dynamics Estimator - Texas A&M University (TAMU) Scholar

Direct Adaptive Control of Non-Minimum Phase Linear Infinite-Dimensional Systems in Hilbert Space Using a Zero Dynamics Estimator
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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 recently developed 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 because a finite number of zeros are unstable. We previously developed a blending method to compensate for this issue where we modify or "blend" the output of the infinite dimensional plant, and then control this modified output rather than the original control output. In this paper we use a finite dimensional zero dynamics estimator based on a modified output but use the estimator to produce a fully minimum phase system. Then direct adaptive control for the infinite dimensional plant can focus on the original control output rather than the modified output. 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.