An optimal strategy for a saturating sampled-data system

Consider the usual sampled-data control system in which the sampler is followed by a zero-order hold and the transfer function is G(s) = 1/s(s+a). Saturation is represented by the fact that the forcing function applied to G(s) may not be larger than 1 in absolute value. The problem is to determine a saturating zero-order hold forcing function which forces the system from an arbitrary initial state to equilibrium in the least number of sampling periods. Such a forcing function is defined as an optimal strategy. The state plane is divided into boundary states and interior states. To each boundary state corresponds a unique optimal strategy. To each interior state correspond infinitely many optimal strategies. From the system parameters a polygonal curve, called the critical curve, is defined in the state plane. An optimal strategy is then proposed in which the required forcing function is simply obtained by computing the distance of the representative point in state plane to the critical curve. A simple computer is proposed to implement this optimal strategy. Finally, the proposed optimal strategy is shown to reduce in the limit as T→0 to that of the corresponding continuous system.