A double integral quadratic cost and tolerance of feedback nonlinearities

A double integral quadratic cost with an associated integral constraint on state trajectories is shown to result in a stable feedback control law for linear time-varying differential systems. The gain matrix for this control is obtained by integrating a Riccati-type matrix differential equation over a finite time interval and is shown to allow for a large class of nonlinearities in the feedback loop without destroying its asymptotically stable property. The class of nonlinearities is larger than that which is generally permitted for the standard steady-state linear optimal regulator, and a phase margin of 90 ° can be approached by the closed-loop system in the case of time-invarlant systems.