Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs

Abstract

We solve in this article the problem of robust output feedback regulation for a system composed of two hyperbolic equations with collocated input and output in presence of a general class of disturbances and noise. Importantly, the robustness of the controller is considered with respect to delays in the actuation and in the measurements but also with respect to uncertainties on parameters, most importantly transport velocities. The proposed control law introduces three degrees of freedom (by means of tuning parameters) on which we give general conditions to guarantee the existence of robustness margins. We show that to tune these degrees of freedom and allow potential robustness trade-offs, it is necessary to consider all the different types of uncertainties simultaneously as it is the only way to ensure the existence of non-zero robustness margins. Provided that these conditions are satisfied, these tuning parameters enable a trade-off between performance and robustness, between disturbance rejection and sensitivity to noise. The existence of robustness margins and the Input To State Stability of the system are proved combining backstepping transformations and classical complex analysis techniques.