Robust Control Design with Real Parameter Uncertainty using Absolute Stability Theory

Jonathan P. How and Steven R. Hall


Abstract

The purpose of this thesis is to investigate an extension of u theory for robust control design by considering systems with linear and nonlinear real parameter uncertainties. In the process, explicit connections are made between mixed u and absolute stability theory. In particular, it is shown that the upper bounds for mixed u are a generalization of results from absolute stability theory. Both state space and frequency domain criteria are developed for several nonlinearities and stability multipliers using the wealth of literature on absolute stability theory and the concepts of supply rates and storage functions. The state space conditions are expressed in terms of Riccati equations and parameter -dependent Lyapunov functions. For controller synthesis, these stability conditions are used to form an overbound of the H2 performance objective. A geometric interpretation of the equivalent frequency domain criteria in terms of off-axis circles clarifies the important role of the multiplier and shows that both the magnitude and phase of the uncertainty are considered.

A numerical algorithm is developed to design robust controllers that minimize the bound on an H2 cost functionality and satisfy an analysis test based on the Popov stability multiplier. The controller and multiplier coefficients are optimized simultaneously, which avoids the iteration and curve-fitting procedures required by the D-K procedure of u synthesis. Several benchmark problems and experiments on the Middeck Active Control Experiment at M.I.T. demonstrate that these controllers achieve good robust performance and guaranteed stability bounds.


Serc Report #1-93, January 1993

Robust Control Design with Real Parameter Uncertainty using Absolute Stability Theory