Linear ultimate boundedness control of uncertain dynamical systems

Abstract

Given a state equation whose description includes uncertain parameters, it is often desirable to design a feedback controller which renders all solutions uniformly and ultimately bounded. When facing such a problem, one often has available or assumes an upper bound for the excursions of the uncertain parameters. Within this framework, the objective is to guarantee, no matter how ‘nature’ selects the uncertainty, that the state will eventually end up and remain within some pre-specified region. When this region is a small neighborhood about the origin, the concept of uniform ultimate boundedness is tantamount to ‘practical’ asymptotic stability. The objectives of this paper are twofold: first and foremost, our goal is to show that a class of ultimate boundedness problems which have been solved to date via nonlinear control, can in point of fact be solved via linear control. This is the main result and is given in Theorem 4.1. Secondly, we show that some of the so-called matching assumptions (which have been required in the earlier literature) can be weakened somewhat. This enables one to design controllers for a larger class of dynamical systems than previously considered.