Saturating and time-optimal feedback controls

The open-loop solution of the soft-constrained time-optimal control problem can be efficiently computed in terms of the controllability Grammian matrix, but a closed-loop implementation was found to be cumbersome. This control was observed to have a saturation property strongly reminiscent of the hard-constrained time-optimal control problem. In this paper, we present a theoretical justification for the observed saturation and propose a modification of the problem statement that gives a suboptimal solution and results in a drastically simpler implementation of the feedback time-optimal soft-constrained control. Moreover, we generalize the proposed approach to generate a family of saturating control laws occupying a middle ground between linear state-feedback and hard-constrained time-optimal controls. For illustration, we consider the simultaneous slewing and vibration suppression of an undamped flexible beam that is reducible to a marginally stable linear system. As an example, we design a simple and elegant feedback control law where the regulation time and control amplitude saturate like the square root of the norm of the state vector.