Achieving Minimum Settling Time Subject to Undershoot Constraint in Systems With One or Two Real Right Half Plane Zeros

Abstract

This note concerns with the problem of achieving minimum settling time in linear systems with one or two real right half plane (RHP) zeros subject to the condition that undershoot does not exceed a given threshold. Such problem is of great practical significance, but it has not been formally addressed, to our knowledge. Time optimal control solutions for such systems are readily available based on the well known optimal control theory, but it does not address the practical consideration that the “wrong way response,â€‌ i.e., undershoot, must be limited. To be sure, the relationship between the minimum settling time and the undershoot constraint for systems with one or two real RHP zeros has already been given in the literature. How to find the control signal that achieves the minimum settling time, however, is still an open question. In this paper, such control signal is obtained constructively and, combined with feedback, is shown to be rather effective in controlling the system in the presence of model uncertainties and external disturbances, as shown in simulation.