Global and semi-global stabilizability in certain cascade nonlinear systems

This paper addresses the issue of global and semi-global stabilizability of an important class of nonlinear systems, namely, a cascade of a linear, controllable system followed by an asymptotically (even exponentially) stable nonlinear system. Such structure may arise from the normal form of “minimum phase” nonlinear systems that can be rendered input-output linear by feedback. These systems are known to be stabilizable in a local sense. And, in some cases, global stabilizability results have also been obtained. It is also known, however, that when the linear “connection” to the nonlinear system is nonminimum phase, i.e,, it has zeros with positive real part, then global or semi-global stabilizability may be impossible. Indeed, it has been shown that for any given nonminimum phase linear subsystem, there exists an asymptotically stable nonlinear subsystem for which the cascade cannot be globally stabilized. We expand on the understanding of this area by establishing, for a broader class of systems, conditions under which global or semiglobal stabilization is impossible for linear and nonlinear feedback.