Spectral Approach to the Servomechanism Problem for a Sturm–Liouville Boundary Control System

Abstract This work focuses on addressing the servomechanism problem within a Sturm–Liouville boundary control system, employing a spectral approach within an infinite-dimensional framework. The primary goal is to achieve tracking of a reference trajectory despite disturbances that are generated by a distributed parameter exosystem. First, a state feedback stabilizing regulator is devised, leveraging a series expansion of the eigenvectors of the SL generator, to steer the system output toward the reference trajectory. Subsequently, a servomechanism system is developed utilizing the tracking error as input, which offers practical and theoretical benefits. It reduces the need for full-state or full-output measurements, simplifies controller design, and aligns naturally with the internal model principle by directly focusing on eliminating the output deviation. Moreover, it is shown that the closed-loop plant exhibits exponential stability, with the tracking error asymptotically approaching zero. The effectiveness of the designed servomechanism is illustrated through its application to a tubular reactor.