Matlab software to auto-tune a complex multi-input multi-output (MIMO) proportional-integral-derivative (PID) controller using Bayesian optimisation (BO)

Transfer functions for each of the nine input–output relationships are identified using MATLAB's System Identification Toolbox and subsequently validated against independent excitation data. The resulting 3×3 transfer-function matrix provides an accurate linear approximation of the nonlinear milling circuit about its nominal operating point.</p><p dir="ltr">The second stage designs a baseline diagonal PI controller using the SIMC tuning methodology and then performs a robust stability analysis to determine the permissible range of each tuning parameter.

Controller gains and integral time constants are represented as uncertain variables, after which μ-analysis is employed to determine the largest parameter variations that preserve closed-loop stability. The resulting robust stability margins define bounded search intervals for all six controller parameters, ensuring that any subsequent optimisation is confined to a region of guaranteed stable operation. These bounds also provide a theoretically justified search space for Bayesian optimisation rather than relying on arbitrary parameter limits.</p><p dir="ltr">The third stage applies Bayesian optimisation to the identified linear process model to determine the controller parameters that maximise closed-loop performance.

The optimisation searches over the six PI tuning parameters within the robustly stable parameter ranges using an expected-improvement acquisition function. Each candidate controller is evaluated by simulating the closed-loop linear system and computing an objective function based on tracking performance and control quality. Throughout the optimisation, all objective evaluations are logged, new best-performing solutions are identified, and the performance of the optimised controller is compared with the conventional SIMC controller using step responses, integrated squared errors and cumulative reward metrics.</p><p dir="ltr">The final stage transfers the optimal controller obtained from the linear model to the nonlinear milling circuit and performs a second Bayesian optimisation directly on the nonlinear process.

The best controller identified during the linear optimisation is used as the initial design, thereby significantly reducing the search effort required on the computationally expensive nonlinear model. Following optimisation, the nonlinear controller is validated against the nominal SIMC controller through comparative step-response analysis, interaction assessment, integrated squared error calculations, and reward evaluation.

The optimised controller parameters and optimisation history are saved for subsequent analysis, completing a hierarchical tuning framework in which inexpensive linear optimisation is first exploited to accelerate and improve optimisation on the full nonlinear process.</p>