Optimal control of axial dispersion tubular reactors with recycle: Addressing state-delay through transport PDEs

Abstract

The optimal control of an axial tubular reactor with a recycle stream is addressed as a key type of setting for distributed parameter systems in chemical engineering. The intrinsic time delay from the recycle process, thus far overlooked in relevant literature, is modelled as a transport partial differential equation (PDE), resulting in a system of coupled parabolic and hyperbolic PDEs. Utilizing Danckwerts boundary conditions, the reactor is boundary-controlled with the control input at the inlet. A continuous-time optimal linear quadratic regulator is developed to stabilize the infinite-dimensional system, employing a late lumping approach in order to preserve the properties of the original infinite dimensional system in controller design. The full-state feedback regulator is designed by solving the Operator Riccati Equation (ORE), leveraging the system's Riesz-spectral properties. To address practical limitations of full-state feedback, a Luenberger observer is also proposed, enabling state reconstruction from boundary measurements. Numerical simulations are conducted to evaluate the proposed control strategies. The results demonstrate that the full-state feedback regulator effectively stabilizes the system. A comparison is made between two configurations where different numbers of eigenmodes were selected to design the controller. The observer-based regulator also stabilizes the system successfully using merely output measurements, effectively overcoming the challenge of limited state access.