Generalized bang–bang control for feedforward constrained regulation

In the behavioral framework for continuous-time linear scalar systems, simple sufficient conditions for the solution of the minimum-time rest-to-rest feedforward constrained control problem are provided. The investigation of the time-optimal input–output pair reveals that the input or the output saturates on the assigned constraints at all times except for a set of zero measure. The resulting optimal input is composed of sequences of bang–bang functions and linear combinations of the modes associated to the zero dynamics. This signal behavior constitutes a generalized bang–bang control that can be fruitfully exploited for feedforward constrained regulation. Using discretization, an arbitrarily good approximation of the optimal generalized bang–bang control is found by solving a sequence of linear programming problems. Numerical examples are included.