Error feedback and internal models on differentiable manifolds

Abstract

Structural aspects are studied of nonlinear control systems in a setting of differentiable manifolds. As a generalization of the setup commonly used in linear multivariable control, the regulator problem is defined as that of controlling a fixed plant to track (or reject) reference (or disturbance) signals generated by a fixed dynamic model called the exosystem. It is shown that, if the controller is error-driven, if perfect tracking is achieved in the limit <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t \rightarrow \infty</tex> , and if a suitable observability condition is present, then the controller necessarily incorporates a copy (internal model) of the exosystem dynamics. This result represents a counterpart in the nonlinear differentiable setting to various results already known for linear systems and for abstract automata.