Offset-free stochastic quadratic dynamic matrix control formulations using polynomial chaos expansions

Abstract

Probabilistic uncertainties in the model parameters result in distributional uncertainties in the model predictions. While such uncertainty descriptions have been incorporated into model predictive control (MPC) formulations using polynomial chaos theory (PCT), more care is required to ensure integral action than in traditional MPC. This article thoroughly examines offset-free formulations of PCT-based MPC for multiple-input, multiple-output linear time-invariant systems. We compile, prove, and validate features of multiple stochastic MPC formulations. Under mild assumptions, these features include (i) guarantees for the existence of a full column-rank integrator to eliminate offset in multiple performance indices; (ii) guarantees of nominal closed-loop stability for the unconstrained systems, and (iii) computationally efficient, spectrally accurate resolution of parametric uncertainty. Application of our stochastic MPC formulations to setpoint tracking and disturbance rejection in numerical case studies demonstrate the asymptotic removal of offset in all higher-order contributions to output variation due to parametric uncertainty.