On the Youla-Kucera parametrization for nonlinear systems

Abstract

We consider the problem of stabilizing a nonlinear plant through the use of a, possibly unstable, pre-compensator and a stable feedback-compensator, and parameterizing a class of such stabilizers in terms of a BIBO stable map Q. In the linear case, by means of superposition the pre-and feedback-compensator can be combined in such a way that the BIBO map Q occurs in just one feedback loop within the controller, feeding back output residuals (prediction errors) to the control inputs. The main contribution of this paper is to develop a nonlinear generalization of this property. Building on earlier work in forming the Youla-Kucera parametrization for nonlinear systems, we show the equivalence of the class of all (bounded-input) stabilizing nonlinear pre- and feedback-compensators to a class of possibly unstable feedback controllers in which a map Q s is present in only the one feedback loop. We then show that necessary and sufficient condition to achieve stability of the system is that Q s be BIBO stable. One advantage of the new formulation is that differential boundedness assumptions do not involve the parametrization Q s in any way. Just as the linear versions of our results have applications in the areas of optimal control and adaptive control of linear systems, it is conjectured that the present results will underlie more general results for adaptive control and nonlinear systems.