Run-indexed time-varying Bayesian optimization with positional encoding for auto-tuning of controllers: Application to a plasma-assisted deposition process with run-to-run drifts

Abstract

Bayesian optimization (BO) has emerged as a useful paradigm for automatic calibration (aka auto-tuning) of advanced optimization- and learning-based controllers whose closed-loop performance depends on the choice of several tuning parameters in highly nonlinear and nonconvex ways. However, BO approaches to controller auto-tuning commonly rely on the assumption that system dynamics remain constant, which does not hold for systems with time-varying dynamics, for example, due to gradual aging or persistent environmental drifts. This challenge can be further compounded when gradual and persistent system drifts occur over a series of process runs. Existing time-varying BO (TVBO) approaches with spatio-temporal kernels fall short of effectively handling an integer run index, which is imperative for capturing run-to-run changes in the system behavior. To this end, this paper presents a run-indexed TVBO (RI-TVBO) approach that can systematically account for run-to-run process drifts as the system is queried over sequential process runs. The proposed approach relies on adapting the non-stationary Wiener process kernel to accommodate an integer run index, instead of time. This is done via positional encoding that incorporates the integer run index and, thus, enables describing run-to-run variations in system dynamics. The positional embedding vector associated with each run index is then mapped onto a scalar value to leverage the relationships between different process runs within the probabilistic surrogate model of the objective function in RI-TVBO. The performance of RI-TVBO is evaluated for auto-tuning of an offset-free model predictive controller for a low-temperature plasma-assisted process for thin film deposition. Simulation results demonstrate the superior performance of RI-TVBO over standard BO and TVBO under different scenarios of run-to-run process drifts encountered in plasma-assisted deposition processes in semiconductor manufacturing.