Cancellation of unmatched biased sinusoidal disturbances for unknown LTI systems in the presence of state delay

In this paper, the problem of unmatched unknown sinusoidal disturbance cancellation in the presence of delay in the virtual control input of unknown LTI systems, is solved. Firstly, the disturbance is parameterized. Secondly, the delay is represented as a transport PDE. These two steps allow us to approach the problem as an adaptive control design problem for an uncertain coupled ODE-PDE system. An infinite-dimensional backstepping procedure is employed to compensate the state delay and a finite-dimensional backstepping procedure is used to overcome unmatched disturbance condition. Finally, an adaptive controller is designed and it is proven that the equilibrium of the closed-loop system is stable, and that the state of the closed-loop system converges to zero as t→∞ while perfect disturbance estimation is achieved. The effectiveness of the controller is illustrated with a simulation example of a third-order system.