Title

Robust Stabilization of Dynamic Systems with Uncertain Equilibrium States

Date of Award

Fall 2021

Project Type

Dissertation

Program or Major

Electrical and Computer Engineering

Degree Name

Doctor of Philosophy

First Advisor

Se Young Yoon

Second Advisor

Kent Chamberlin

Third Advisor

Shaad MD Mahmud

Abstract

In the study of dynamic systems, analysis and control techniques commonly rely on the knowledge of the equilibrium states. However, in many practical applications, the determination of the exact location of the equilibrium states is not trivial. This may be due to the complexity of the dynamic system or changes in the system's operating conditions. Stabilizing dynamic systems away from their equilibrium states requires a steady control effort to maintain the system at a non-equilibrium point. For practical systems with limited control bandwidth, stabilization away from system's true equilibrium state wastes valuable control resources. The objective of this study is to explore robust methods for stabilizing dynamic systems at their true equilibrium states even when their exact location is uncertain.

First, the derivative feedback controller is explored as a solution to stabilizing systems with uncertain equilibrium states. An advantage to this method is that knowledge of the equilibrium state is not required in its implementation. The robustness of the derivative feedback controllers to norm-bounded dynamic model uncertainty is investigated, and linear matrix inequality conditions are derived to guarantee the stability of the closed-loop system. Formulation of the control objectives as linear matrix inequalities facilitates the design of the robust derivative feedback controller. The proposed control solution is analytically shown to drive the closed-loop system exponentially to its unknown equilibrium states, and this observation is verified via simulation on the control of chaotic Rossler and Lorenz attractors. A practical example involving a magnetic levitation system, in which two disks are to be levitated at an unknown magnetic equilibrium, demonstrates the effectiveness of the output derivative feedback controller. Secondly, a structured linear adaptive mechanism which is designed to estimate the true equilibrium state of the controlled system using information from the feedback control signal is proposed. In particular, the proposed solutions overcome the odd-number condition, or parity condition, which appears as a recurring limitation in comparable solutions reported in the literature. Conditions for the existence of the adaptive solutions are presented, and the stability of the closed-loop system is demonstrated. The effectiveness of the proposed adaptation mechanism is verified through a numerical example based on a practical application in the active control of compressor surge instability. Finally, robustness is incorporated to the design of the linear adaptive compensator. A class of nonlinear uncertainties where the only information available is their norm upper bound is considered. Necessary conditions are derived with respect to the design parameters of the adaptive compensator to guarantee the robust stability of the closed-loop after the addition of the adjustment mechanism. A design procedure is then presented for the adaptive compensator in terms of the solutions to linear matrix inequality conditions. The control scheme is numerically validated on a chaotic Lorenz attractor system and experimentally on a magnetic levitation test bed, with an uncertain magnetic equilibrium.

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