Least Squares and Marginal Log-Likelihood Model Predictive Control using Normalizing Flows

TL;DR

This paper introduces a novel stochastic process modeling approach using conditional normalizing flows for model predictive control, improving accuracy and stability over traditional methods in chemical reactor applications.

Contribution

It develops a new MPC framework employing normalizing flows to explicitly model stochastic dynamics and derives a marginal log-likelihood objective for enhanced stability and performance.

Findings

Normalizing flow MPC halves setpoint error compared to nominal control.

Chance constraints reduce constraint violations.

MLL objective offers more stability with small scenario sets.

Abstract

Real-world (bio)chemical processes often exhibit stochastic dynamics with non-trivial correlations and state-dependent fluctuations. Model predictive control (MPC) often must consider these fluctuations to achieve reliable performance. However, most process models simply add stationary noise terms to a deterministic prediction. This work proposes using conditional normalizing flows as discrete-time models to learn stochastic dynamics. Normalizing flows learn the probability density function (PDF) of the states explicitly, given prior states and control inputs. In addition to standard least squares (LSQ) objectives, this work derives a marginal log-likelihood (MLL) objective based on the explicit PDF and Markov chain simulations. In a reactor study, the normalizing flow MPC reduces the setpoint error in open and closed-loop cases to half that of a nominal controller. Furthermore, the chance constraints lead to fewer constraint violations than the nominal controller. The MLL objective yields slightly more stable results than the LSQ, particularly for small scenario sets.