Optimal Parameter Tuning of Feedback Controllers with Application to Biomolecular Antithetic Integral Control

We consider the deterministic setting of a general nonlinear plant in feedback with a nonlinear controller that is parameterized by a finite number of unknown constants to be tuned. This setting is particularly useful in applications where the architecture of the feedback controller is constrained to have a specific structure, but the controller parameters can be tuned to optimize a given performance measure (e.g. biomolecular controllers, PID controllers, etc.). We first cast the tuning problem as a dynamically constrained optimization problem, then we convert the latter to an unconstrained one by introducing a suitable nonlinear operator. It is shown that the necessary conditions of optimality can be written as a parameter-dependent Two-Point Boundary Value Problem (TPBVP) that is difficult to solve analytically. Hence, we derive and compare two (first order) numerical methods to solve the optimization problem based on the Gradient Descent (GD) and the Conjugate Gradient Descent (CGD) algorithms. Finally, we apply the developed algorithms to tune a biomolecular antithetic integral controller. Tuning this controller has the advantages of shaping the dynamic response of the plant and minimizing the effect of dilution of the controller species.