Adding an integrator to backstepping: Output disturbances rejection for linear hyperbolic systems

We consider the output disturbance rejection problem for linear first-order hyperbolic systems with anti-collocated boundary input and output. We employ backstepping to construct the controller. We incorporate an integral action into the target system, yielding a Delay Differential Equation (DDE) and use a classical result to obtain a sufficient condition for its stability in L∞-norm. Then, we show that the full-state feedback control with the integral action rejects in-domain and boundary disturbances from the output. Besides, we show that when incorporating integral-action into the observer system, the resulting output feedback control rejects the disturbances too.