Observer Based Adaptive Estimation/Cancellation of Unmatched Sinusoidal Disturbances in Known LTI Systems by State Derivative Measurement

In this paper, an adaptive observer and backstepping controller are designed to cancel and estimate sinusoidal disturbances forcing a linear time-invariant by using only the measurements of the state-derivatives. The parametrization of the sinusoidal disturbance as the output of a known feedback system with an unknown output vector that depends on unknown disturbance parameters with the necessary filter designs enables to approach the problem as an adaptive control problem. An observer is designed for the unmeasured virtual input to apply a backstepping procedure which handles the unmatched disturbance and input condition. Firstly, it is shown that the disturbance and the unmeasured actuator state are observed perfectly in the open loop case. Secondly, the closed loop case is considered and it is proven that the equilibrium of the closed-loop adaptive system is stable and the state of the considered original system converge to zero as t → ∞ with perfect disturbance estimation. The effectiveness of the controller and the observers are illustrated with a simulation example of a third order system.