Planar Hybrid Systems

The design of nonlinear dynamical systems to realize the desired behavior of devices and processes is a fundamental problem that is often approached from the perspective of control theory. Control provides a large set of tools for tackling these design problems, but some classes of problems have remained resistant to these methods. This paper discusses a theory of hyb r id sys t ems that extends the concept of a smooth dynamical system to a context in which there is a mixture of continuous time and discrete time elements. This is particularly appropriate for the description of "switching systems" in which digital control is applied to change continuous control laws on a time scale much faster than the dynamical evolution of the device being controlled. Digital control makes it (relatively) easy to build devices in which the applied control is any computable function of system measurements. This enormously widens the scope for the system designs that can be implemented, but the additional design freedom brings with it many more design parameters. Consequently, we seek better theory to provide guiding principles that can help us cope with this new found freedom. This paper is a step in this direction. We consider hybrid systems with two dimensional phase spaces subject only to mild genericity requirements, and describe the types of long term dynamical phenomena displayed by these systems.