Noncollocated robust output regulation of an unstable heat equation with recycle

In this article, we study the output regulation of an unstable heat equation with recycle, where the right boundary heat flux is fed back to the other end through a recycle loop, and the reference signal and the disturbances in all channels are generated by an exosystem with known frequencies. By designing a static feedback with the noncollocated observation and choosing the proper control gain, we stabilize the heat system without disturbances. Then we solve the regulator equation and construct an observer based on the tracking error to obtain the robust output feedback regulator. The closed-loop system is proved to be well-posed and uniformly bounded, and the regulated output tracks the reference signal exponentially. The numerical simulations are carried out to demonstrate the effectiveness of the proposed control.