Adaptive Control for Regulation of a Quadratic Function of the State

We propose an adaptive controller for a dynamical system, where we need a quadratic function of the state to track a given reference signal. This problem appears in the control of generators in weak-grid conditions for example. While the controller can measure the tracking error, the main difficulty arises from the fact that the parameters of the quadratic function itself are not known to the controller. Our approach consists of simultaneously estimating the quadratic function while tracking the reference signal, similar to the approach employed in adaptive control. The quadratic structure of the tracking function necessitates, however, a new adaptive law for estimating the parameters. Even though estimation and control are in general two contradicting requirements, using this new adaptive law and a multilevel controller we prove that the tracking error converges to zero in the absence of measurement noise. In the presence of bounded noise we show that the tracking error can be driven to a neighborhood around the origin.