MPC performance monitoring and disturbance model identification

Although model predictive control (MPC) has been widely implemented in industry, no systematic method exists to assess if MPC controllers are performing optimally or to monitor their performance over time. We address this problem by proposing a benchmark called the key performance index (KPI), which is the expectation of the stage cost. For the linear, unconstrained case, the stage cost is a quadratic form of a normal variable and therefore has a generalized chi-squared distribution. The plant KPI is the time average of the stage cost and therefore has a normal distribution. We derive formulas for the mean and variance of the stage cost and plant KPI.;Calculation of the KPI (as well as estimator design) requires an accurate disturbance model. The standard autocovariance least squares (ALS) methods, which estimate the disturbance covariances from data, are not easily applicable to industrial systems with large-dimensional models, which often contain poorly observable states. The covariance estimates also may have a large variance, since the original ALS formulation weights the least squares problem with the identity matrix.;We overcome these challenges by reducing the model to contain only the necessary observable states and using a feasible generalized least squares technique to estimate the optimal weighting from data. Application of the improved ALS method to an industrial data set demonstrates that these improvements reduce the computational time and produce more reliable estimates as compared to the original ALS method.;As an alternative to the ALS method, a maximum likelihood estimation (MLE) method is proposed; this method eliminates the need to estimate the optimal weighting. Instead, the process and measurement noise covariances are estimated by maximizing the probability of observing the measured outputs. Thus this optimization problem has a more sound theoretical basis. Sufficient conditions for the existence of an MLE solution are given. The conditions for uniqueness are compared to those of the ALS method. Although the computational burden is large compared to the ALS method, the MLE method was applied to several small-scale examples and shown to maximize the likelihood compared to the ALS method.