Time-optimal control and high-gain linear state feedback

Abstract

It is known that the fastest possible response of a linear system under input constraints is typically achieved using extreme control efforts at all times, or bang-bang control. However, closed-loop minimum-time control is very difficult to implement except for a few simple second-order systems, since a closed-form solution for the time-optimal switching hypersurface usually cannot be found for high-order systems. In this paper the connection between linear state feedback and time-optimal control is explored. It is shown that for a class of systems an equivalent time-optimal, linear, switching hypersurface exists corresponding to an initial condition. Based on these switching planes, a high-gain linear state feedback law can be used to achieve minimum-time control on systems of order higher than two, provided that the feedback coefficients are adjusted according to the initial conditions at the start of each operation. Three different cases are investigated in detail: a second-order DC servo drive, a third-order servo system including actuator dynamics, and a fourth-order flexible mechanism. Since the proposed algorithm resembles in form a sliding-mode controller, the similarities and differences between the two control algorithms are also discussed.