SATURATING AND TIME-OPTIMAL FEEDBACK CONTROLS

The open-loop solution of the soft-constrained time-optimal control pr oblem can be efficiently computed in terms of the controllability Gram mian matrix, but a closed-loop implementation was found to be cumberso me. This control was observed to have a saturation property strongly r eminiscent of the hard-constrained time-optimal control problem. In th is paper, we present a theoretical justification for the observed satu ration and propose a modification of the problem statement that gives a suboptimal solution and results in a drastically simpler implementat ion of the feedback time-optimal soft-constrained control. Moreover, w e generalize the proposed approach to generate a family of saturating control laws occupying a middle ground between linear state-feedback a nd hard-constrained time-optimal controls. For illustration, we consid er the simultaneous slewing and vibration suppression of an undamped f lexible beam that is reducible to a marginally stable linear system. A s an example, we design a simple and elegant feedback control law wher e the regulation time and control amplitude saturate like the square r oot of the norm of the state vector.