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https://hdl.handle.net/2142/22885
Description
Title
Sensitivity methods and slow adaptation
Author(s)
Rhode, Douglas Scott
Issue Date
1990
Doctoral Committee Chair(s)
Kokotovic, P.V.
Department of Study
Electrical and Computer Engineering
Discipline
Electrical and Computer Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, System Science
Language
eng
Abstract
In many existing model reference adaptive control (MRAC) schemes, the plant order and relative degree are assumed known. Then a full-order control parameterization is specified which allows for an exact transfer function match between the closed-loop plant and a reference model. Often for plants of modest complexity, this leads to a large number of adjustable parameters. Apart from the computational burden, an excessive parameterization imposes, a common assumption used to prove global stability requires the regressor to be Persistently Exciting (PE). This PE requirement has been tied to the spectral content of the system input, and is directly proportional to the number of parameters adjusted. Often this imposes unrealizable requirements upon the system input. This problem is especially pronounced for regulation problems where the reference and system input are zero.
In the approach presented in this thesis, the number of parameters and structure of the controller will not be tied to plant order. Instead, a reduced-order parameterization such as PID or lead-lag will be adapted. For many applications, slow adaptation offers an attractive means of increasing performance while retaining the simplicity of the underlying linear system.
Recent developments in integral manifolds and averaging are combined with sensitivity results from the 1960s to construct a pseudogradient approach to slow adaptation. Under slow adaptation, the parameters change much slower than the state of the underlying linear system. An integral manifold, the slow manifold, is used to separate the slow parameter dynamics from the fast linear states. This slow manifold can be approximated by a frozen parameter manifold, which is simply the steady-state response of the linear system with the parameters held constant. The signals from the frozen manifold are then substituted into the parameter update equation. This equation is then averaged to produce a set of nonlinear time invariant equations which approximate the dynamics of the parameters. Using sensitivity techniques, a pseudogradient algorithm will be constructed such that the averaged system will approximate a steepest descent optimization algorithm to reduce the mean square output error. This approach will encompass several existing algorithms such as those proposed by Narendra and Bodson (1,2). Sufficient conditions are given to guarantee the stability of these pseudogradient algorithms.
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