A System Level Approach to LQR Control of the Diffusion Equation
This provides a method for imposing spatio-temporal constraints on controllers for distributed parameter systems while preserving convexity, which is incremental for control theory applications.
The paper tackles optimal controller design for distributed parameter systems by reformulating the Linear Quadratic Regulator (LQR) problem for the diffusion equation as an optimization over closed-loop mappings, analogous to System Level Synthesis, and solves it analytically to recover the LQR controller with internal stability.
The optimal controller design problem for a linear, first-order spatially-invariant distributed parameter system is considered. Through a case study of the Linear Quadratic Regulator (LQR) problem for the diffusion equation over the torus, it is illustrated that the optimal controller design problem can be equivalently formulated as an optimization problem over the system's closed-loop mappings, analogous to the System Level Synthesis framework. This reformulation is solved analytically to recover the LQR for the diffusion equation, and an internally stable implementation of this controller is recovered from the optimal closed-loop mappings. It is further demonstrated that a class of spatio-temporal constraints on the closed-loop maps can be imposed on this closed-loop formulation while preserving convexity.