Stabilizability preserving quotients for non-linear systems

Abstract

The design of feedback stabilizing controllers is an essential component of control engineering theory and practice. A large part of the modern literature in control theory is devoted to coming up with new methods of designing feedback stabilizing controllers. This sustained interest in answering the question \how to design a feedback stabilizing controller" has not been accompanied by equal interest in the more existential and fundamental problem of \when is a system stabilizable by feedback". As such the theory of control Lyapunov functions still remains the most general framework that characterizes stabilizability as a system property. This approach however simply replaces one elusive and di cult concept (i.e stabilizability) with an equally diffcult concept, the existence of control Lyapunov functions. In this thesis we analyse control system stabilizability from the perspective of control system quotients which are generalized control system reductions, the focus being on the propagation of the stabilizability property from the lower order quotient system to the original system. For the case where the quotient system is a linear controllable system we prove that propagation of the stabilizability property to the original system is possible if the zero dynamics of the original system are stable. A novel way of constructing the zero dynamics which does not involve the solution of a system of partial differential equations is devised. More generally for analytic non-linear systems given a stabilizable quotient system, we develop a new method of constructing a control Lyapunov function for the original system, this construction involves the solution of a system of partial di erential equations. By studying the integrability conditions of this associated system of partial differential equations we are able to characterize obstructions to our proposed method of constructing control Lyapunov functions in terms of the structure of the original control system.