Robust Stability Analysis of an Uncertain Aircraft Model with Scalar Parametric Uncertainty

This paper analyzes the robust stability of an uncertain aircraft model in which a key parameter, the location of the aircraft's center of gravity, is modeled as a real parametric uncertainty. A robust controller is specified and the stability bounds of the uncertain closed-loop system are determined using the small gain, circle, positive real, and Popov criteria. A graphical approach is employed in order to demonstrate the ease with which the above robustness tests can be carried out on a problem of practical interest. A significant improvement in stability bounds is observed as the analysis moves from the small gain test to the circle, positive real, and Popov tests. In particular, small gain analysis results in the most conservative robust stability bounds, while Popov analysis yields significantly less conservative bounds. This is because traditional small gain type tests allow the uncertainty to be arbitrarily time-varying, whereas Popov analysis restricts the uncertainty to be constant, real parametric uncertainty. Therefore, the results reported here indicate the conservatism associated with small gain analysis, and the effectiveness of Popov analysis, in gauging robust stability in the presence of constant, real parametric uncertainty.

I. Introduction

Mathematical models of physical systems often do not cater explicitly or exactly for all of the phenomena present in those systems. Although this simplification greatly facilitates controller design, and stability and performance analysis, it also introduces unmodeled dynamic uncertainty in the mathematical abstraction. Moreover, the parameters of any real system are seldom known precisely and may even vary with time, causing parametric variations and introducing further uncertainty in the associated model. Since practical control systems must ensure stability and acceptable performance in both nominal design conditions and in uncertain or perturbed conditions, the robustness of a control system to various parametric and unmodeled dynamic uncertainties is an important design requirement.