Stabilizing Partially Observable Uncertain Linear Systems

Abstract

We consider linear systems which involve no stochastic element, but whose state equations depend on time-varying unknown parameters. This parameter uncertainty is not complete because we know that the parameters are constrained to lie within known bounded intervals. The objective is to choose a dynamic observer and a feedback on the state vector of this observer guaranteing uniform asymptotic stability for all admissible variations of the parameters of this control law.

After having defined fixed and mobile pairs of matrices and presented some of their properties, we will show that if a set of two fixed pairs are conveniently oriented, then we can construct such an observer and a feedback control law that we obtain the desired stabilization. This representation with fixed pairs makes the introduction of homotopy (or: continuation) methods very natural. Theses methods are very convenient computationnally (usable on a pocket computer) and construct possible paths from a known situation to the actual uncertain one.