Beyond the Construction of Optimal Switching Surfaces for Autonomous Hybrid Systems

Abstract

This paper presents the first steps to design a new method to determine optimal switching surfaces for hybrid systems with autonomous modes. The idea is to characterize the variations in the optimal switching times corresponding to variations in the initial conditions, and to apply this formula for transverse shifts in the initial conditions, according to the considerations. The new result which identifies those situations where an optimal switching surface independent of initial condition is well defined, allows pursuing the program, based on the investigation of the effect of transverse variations in the initial condition on the switching states. Existing techniques design the optimal switching surfaces based on a singular sample evolution of the system; hence providing a solution dependent on the initial conditions. On the other hand, the optimal switching times can be found, providing an open loop control to the system, but those also are dependent on the initial conditions. The technique presented relies on a variational approach, giving the derivative of the switching times with respect to the initial conditions, thus providing a tool to design programs/algorithms generating switching surfaces that are optimal for any possible execution of the system.