Active Unmatched Disturbance Cancellation and Estimation by State--Derivative Feedback for Plants Modeled as an LTI System

We design adaptive algorithms for both cancellation and estimation of unknown periodic disturbance, by feedback of state--derivatives ( i.e.,} without position information for mechanical systems) for the plants which are modeled as a linear time invariant system. We consider a series of unmatched unknown sinusoidal signals as the disturbance. The first step of the design consists of the parametrization of the disturbance model and the development of observer filters. The result obtained in this step allows us to use adaptive control techniques for the solution of the problem. In order to handle the unmatched condition, a backstepping technique is employed. Since the partial measurement of the virtual inputs is not available, we design a state observer and the estimates of these signals are used in the backstepping design. Finally, the stability of the equilibrium of the adaptive closed loop system with the convergence of states is proven. As a numerical example, a two-degree of freedom system is considered and the effectiveness of the algorithms are shown.