Maximal Open-Loop Operation Under Integral Error Constraints

The problem of controlling a linear time invariant system in open loop for the maximal time period is considered. The system is subjected to bounded parametric uncertainties and time varying disturbances. The open-loop operation is continued as long as the L ₂ norm of the states is bounded by a pre-specified threshold, for every possible value of the uncertainty and disturbance. It is shown that an optimal open-loop input that achieves the maximal open-loop operation, and a worst disturbance input exist. Moreover, both can be approximated for computational purposes by bang-bang functions with finite number of switches.