Maximum Likelihood Identification of Linear Models with Integrating Disturbances for Offset-Free Control

TL;DR

This paper develops a maximum likelihood identification method for linear models with integrating disturbances, enhancing offset-free model predictive control by ensuring stable eigenvalues through LMI constraints and applying the approach to real-world systems.

Contribution

It introduces a novel eigenvalue constraint approach using barrier functions and a Cholesky-based approximation for nonlinear semidefinite programming in model identification.

Findings

Effective disturbance estimation improves control performance.

Eigenvalue constraints prevent unstable or oscillatory filter behavior.

Method successfully applied to temperature control and chemical reactor data.

Abstract

This report addresses the maximum likelihood identification of models for offset-free model predictive control, where linear time-invariant models are augmented with (fictitious) uncontrollable integrating modes, called integrating disturbances. The states and disturbances are typically estimated with a Kalman filter. The disturbance estimates effectively provide integral control, so the quality of the disturbance model (and resulting filter) directly influences the control performance. We implement eigenvalue constraints to protect against undesirable filter behavior (unstable or marginally stable modes, high-frequency oscillations). Specifically, we consider the class of linear matrix inequality (LMI) regions for eigenvalue constraints. These LMI regions are open sets by default, so we introduce a barrier function method to create tightened, but closed, eigenvalue constraints. To solve the resulting nonlinear semidefinite program, we approximate it as a nonlinear program using a Cholesky factorization method that exploits known sparsity structures of semidefinite optimization variables and matrix inequalities. The algorithm is applied to real-world data taken from two physical systems: a low-cost benchmark temperature microcontroller suitable for classroom laboratories, and an industrial-scale chemical reactor at Eastman Chemical's plant in Kingsport, TN.