Controllability of Linear and Nonlinear Control Systems related through Simulation Relations

Abstract

For nonlinear input-disturbance systems that are connected by a simulation relation, we examine to what extent they share certain controllability properties. Specifically our main objective is to determine the conditions under which the following holds: given two control systems $F$ and $\tilde{F}$ where $F$ is simulated by $\tilde{F}$ and $F$ is completely controllable, we have that $\tilde{F}$ is also completely controllable. To this end, we show that under some additional conditions the property of complete controllability is preserved for pointwise graph simulation relations and compact graph simulation relations. Next in an attempt to prove a similar result between a nonlinear system and an almost linear system, but with the simulation relation submanifold being a regular level set of a particular map instead of a graph, we achieve the result of the simulating system $\tilde{F}$ being at most completely controllable modulo the kernel of a linear map. We show through an example that $\tilde{F}$ may fail to be completely controllable if it does not fulfill a certain compactness condition. By imposing this compactness condition along with other somewhat restrictive assumptions, we are able to prove a similar result for nonlinear control systems connected through a simulation relation submanifold in the form of a regular level set of a smooth mapping. We then illustrate the features of our final main result with an example.