Near Optimal LQR Performance for Uncertain First Order Systems

In adaptive control, the objective is to provide stability and acceptable performance in the face of significant plant uncertainty. However, often there are large transients in the plant output and the control signal can become excessively large. Here we consider the first order case with the plant parameters restricted to a compact set; we show how to design a (linear time-varying) adaptive controller which provides near optimal LQR performance. This controller is periodic with each period split into two parts: during the Estimation Phase, an estimate of the optimal control signal is formed; during the Control Phase, a suitably scaled estimate of this signal is applied to the system. We demonstrate the technique with a simulation and discuss the benefits and limitations of the approach.