Deep learning adaptive Model Predictive Control of Fed-Batch Cultivations

TL;DR

Bioprocess control faces nonlinear, uncertain dynamics that hinder real-time MPC. The authors develop a DC-TMPC approach that learns a DC dynamics representation via input-convex neural networks and bounds linearisation errors with simplex tubes, enabling convex subproblems at each step. They add online parameter estimation through set-membership and LMS to support adaptive robust control, with the nominal predictions updated using parameters within the estimated set and a horizon of $N=25$ over $T=100$ hours. The method is demonstrated on a penicillin-producing fed-batch bioreactor, achieving robust constraint satisfaction and reduced conservatism compared with traditional TMPC, suggesting practical impact for parallel bioreactor development.

Abstract

Bioprocesses are often characterised by nonlinear and uncertain dynamics, posing particular challenges for model predictive control (MPC) algorithms due to their computational demands when applied to nonlinear systems. Recent advances in optimal control theory have demonstrated that concepts from convex optimisation, tube MPC, and differences of convex functions (DC) enable efficient, robust online process control. Our approach is based on DC decompositions of nonlinear dynamics and successive linearisations around predicted trajectories. By convexity, the linearisation errors have tight bounds and can be treated as bounded disturbances within a robust tube MPC framework. We describe a systematic, data-driven method for computing DC model representations using deep learning neural networks with a special convex structure, and explain how the resulting MPC optimisation can be solved using convex programming. For the problem of maximising product formation in a cultivation with uncertain model parameters, we design a controller that ensures robust constraint satisfaction and allows online estimation of unknown model parameters. Our results indicate that this method is a promising solution for computationally tractable, robust MPC of bioprocesses.

Figures (4)

• Figure 1: DC decomposition: the actual model evaluated at $30\times 30$ sample points (blue dots), the convex functions $f_1$ and $f_2$ (orange and green surfaces), and the DC decomposition $f=f_1-f_2$ (blue surfaces) approximating $\dot{X}$ (left), and $\dot{S}$ (right), as functions of the concentrations $X$ and $S$, with product concentration $P=$ 0.4gL, volume $V=$ 122L, and feed rate $u =$ 0.07Lh.
• Figure 2: Predicted state and control trajectories at initial time, $t =0$. Solid lines: nominal predicted trajectories $\{x^\circ_k,u^\circ_k\}_{k=0:N}$, dashed/dash-dotted lines: lower/upper bounds on predicted states within $\{\mathbb{S}_k\}_{k=0:N}$.
• Figure 3: Closed loop state and control trajectories (left) and parameter estimates (right).
• Figure 4: Closed loop state and control trajectories with noisy inputs (left) and parameter estimates (right).