Nonlinear Feedback System Stability via Coprime Factorization Analysis

In this paper right coprime factorization results are derived for a general class of nonlinear plants and stabilizing feedback controllers. Both input-output descriptions and state space realizations of the plant and controller are used. It is first shown that if there exist stable right coprime factorizations for the plant and controller, and if a certain matrix of nonlinear operators has a stable inverse then the feedback system is well-posed and internally stable. The links between the right and left coprime factorizations for a stable plant controller pair will be explored for this purpose. A generalization of the notion of linear fractional maps is explored as a means of characterizing the class of plants stabilized by this controller, and dually classes of controllers which stabilize the plant. A state-space approach is then taken to show that such factorizations exist for some nonlinear plants. It is shown that if there exists a stabilizing state feedback for a plant in the class of interest, then there exists a right coprime factorization for the plant. Additionally if there exists a stabilizing output injection, then there will exist a stabilizing controller with a right coprime factorization. An important assumption in this work is to assume that the plant and controller have the same initial conditions.