Adaptive cancelation of matched unknown sinusoidal disturbances for unknown LTI systems by state derivative feedback

Solutions already exist for the problem of canceling sinusoidal disturbances by measurement of the state or by measurement of an output for unknown linear and nonlinear systems. In this paper, we design an adaptive controller to cancel matched sinusoidal disturbances forcing a linear time-invariant system with unknown system parameters in controllable canonical form by using only measurement of state-derivatives. Our design is based on three steps, 1) parametrization of the sinusoidal disturbance as the output of a known feedback system with an unknown output vector that depends on both unknown disturbance parameters and unknown plant parameters, 2) design of an adaptive disturbance observer and, 3) design of an adaptive controller. We prove that the equilibrium of the closed loop adaptive system is stable and state of the considered system goes to zero as t → ∞ with perfect disturbance estimation. The effectiveness of the controller is illustrated with a simulation example of a second order system.