Optimal switching boundary control for two-dimensional piecewise linear dynamical systems

This paper deals with the optimal control design for a family of two-dimensional piecewise linear systems having a switching boundary defined by a straight line and vector fields with regular equilibrium points. Such a family is derived from discontinuous control systems characterized by exhibiting sliding motion and a single pseudo-equilibrium point at the switching boundary, which is the desired operating point of the system in the control design. The initial objective is to obtain conditions on the slope of the switching boundary so that the pseudoequilibrium is asymptotically stable, at least locally. The second and main objective is, based on the pseudo-equilibrium stability result, to find the optimal slope for the switching boundary in order to minimize the transient response time. A numerical method for obtaining the optimal slope is presented, and the effects caused by uncertainties in the parameters are discussed. The used approach explores an innovative way of determining the optimal slope of the switching boundary based on finite-time stability conditions. The results obtained are applied to a power electronic system that describes the dynamics of a Buck Converter under a sliding mode control strategy.