Symbolic Regression-Based Robust Control of Batch Crystallization: Design and PLC Implementation

This work presents symbolic regression as a route to compress nonlinear model predictive control (NMPC) into an explicit, deployable control law for resource-limited industrial hardware. The key barrier is the limited memory and processing power of programmable logic controllers (PLCs), which make on-board real-time optimization impractical. Unseeded batch cooling crystallization of paracetamol in ethanol was used as a case study. First, a detailed population balance model (PBM) was embedded in an NMPC to generate closed-loop trajectories for the representative operating scenarios. Symbolic regression was then applied to the resulting data to identify an algebraic mapping from temperature, supersaturation, and crystal size and mass tracking errors to the next temperature set point. The learned expression was compact, interpretable, and PID-like, with gain scheduling and only basic arithmetic operations. Parameter uncertainty was quantified via confidence regions obtained from a statistical analysis of the regression. Simulations over parameters within these regions showed that the symbolic regression controller (SRC) maintained satisfactory closed-loop performance across all statistically plausible settings, thereby inheriting robustness from the underlying NMPC in a data-driven manner. The SRC was implemented on an Allen-Bradley Micro 870 PLC at typical scan-cycle rates as a Structured Text function block and evaluated in a hardware-in-the-loop setup, where the PBM ran on a host computer and exchanged measurements and set points with the PLC via Modbus/TCP. Across multiple scenarios, the PLC-based SRC drove crystal size and mass close to their set points using smooth temperature trajectories that respected actuator constraints. The approach delivered NMPC-level performance while avoiding the online optimization.