Performance output regulation of coupled heat equations with recycle

In this article, we study the output regulation of two coupled heat equations with recycle, where the disturbances are distributed in all channels, the control is implemented at the left boundary $x=0$ and the point temperature at $x=2$ is the only noncollocated measurement. The output control is divided into two parts: one is to stabilize the unstable heat system by the noncollocated static feedback, and the other is to achieve output regulation by solving the regulator equation and estimating the disturbances, where an observer is constructed to estimate the state of heat system and exosystem synchronously. The closed-loop system is proved to be well-posed and uniformly bounded using the Riesz basis approach, and the integral output tracks successfully the reference signal exponentially. The numerical simulations are carried out to demonstrate the proposed controller is effective and the tracking error converges to zero.