Design of piecewise-linear switching functions for relay control systems

The design of piecewise-linear switching functions is investigated for linear, constant, lumped systems with a single ideal relay controller with respect to a transient response (minimum-time) performance criterion. A design method is presented which consists of a series of steps leading finally to the optimization of parameters by a gradient or other search technique suitable for computer implementation. The preliminary steps are devoted to reducing the dimensionality of the parameter space which must be searched, defining the criterion function so as to yield a surface which is reasonably smooth and free of relative minima, and insuring that the search is started with a feasible parameter set. It is concluded that easily implemented piecewise-linear switching functions can be designed so as to yield improved performance, both in terms of settling times and region of stability in the state space, over that attainable with linear switching functions. The complexities of the performance surface are such, however, that it is felt that frequently the determination of a switching function by the heuristic guides described, constitutes an adequate design without the refinement of parameter optimization.