Maximum Likelihood Estimation of Linear Disturbance Models for Offset-free Model Predictive Control

The performance of industrially successful model predictive control (MPC) and offset-free MPC is reliant on identifying an adequate linear state-space model using plant data. While the models for MPC can be identified using one of many subspace identification methods, there are no methods for identifying the linear disturbance models used in offset-free MPC. Here we formulate a series of maximum likelihood estimation (MLE) problems for identifying linear disturbance models. To formulate the first problem, the state is estimated as a linear combination of past inputs and outputs, and the state-space model is then written as a linear estimation problem. The second problem is formulated as a linear estimation problem relating the long-range prediction error sequence to the disturbance and noise sequences. The last problem is simply a covariance estimation problem for the noises in the linear disturbance model. Each MLE problem has a closed-form solution. While size of the second MLE problem makes it computationally demanding, it can be simplified considerably in the case where the system has no integrators. Hardware experiments (TCLab, an Arduino-based heat transport laboratory) demonstrate that the proposed method generates offset-free performance under realistic conditions on systems without integrators. Numerical simulation experiments demonstrate that the results also generalize to systems with integrators.