# 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