Switching Systems: Controllability and Control Design

We consider two classes of switches switches-on-time and switches-on-state. Switching-on-time is the simplest of the two and it can be considered as an intrinsic or endogenous switching scheme in the sense that it involves only changes in the tangent space switching from one element to another one of the family of vector fields without need to check what happens on the flow of the solution. In endogenous switching one assumes to have complete control over the time of switch, i.e., the time instant in which the switching occurs, and over the switching pattern, i.e., which of the systems is activated selected in the switching event. This type of switching is called open-loop switching. Switching-on-state is more complicated it requires a check on the integral curve of the system in order to decide when to pass from a dynamic mode to another one exogenous switching. It is an important observation that any switching-on-state path has a unique corresponding switching-on-time path, however these switching times and switching patterns depend on the state. Controllability of switching systems has been investigated mostly for the linear case, i.e., when the dynamics in the given modes are linear time invariant LTI and the case when arbitrary switching is possible open--loop switching. By using geometric methods and imbedding linear switching systems in the class of the linear parameter varying systems LPV we have obtained controllability results. In contrast, bimodal systems are special classes of switching systems, where the switch from one mode to the other one depends on the state closed-loop switching. In the simplest case the switching condition is described by a hypersurface in the state space.