Asymptotic Expansions, Root-Loci and the Global Stability of Nonlinear Feedback Systems

Abstract

Despite (or perhaps because of) the extreme complexity of the most general nonlinear control systems, there has been considerable recent interest in the control of certain nonlinear systems by a relatively straightforward application of techniques analogous to those used in linear control theory (e.g., [5]–[9], [15]). Among the most elegant of these recent methods is the theory of linearization via nonlinear feedback developed by Hunt-Su-Meyer [7] (see also [2], [8], [17]) and vastly extended in a spate of more recent contributions. Finding application, for example, in the design of helicopter autopilots [7] and the control of robot arms [18], linearization techniques owe their popularity to their advantage on the one hand of being conceptually appealing and on the other hand of allowing for the application of classical control intuition in a nonlinear context. It should be stressed that the apparent ubiquity of applications is not due to the genericity (even in ℝ2—see [1], [3]) of linearizable systems, but rather to the way nonlinear systems are actually designed. One can be sure that, in practice, nonlinear control systems are not designed so that certain distributions are involutive. Rather, systems are often designed so that there is an independent control for essentially everything in the system that moves.