Nonlinear Model Predictive Control of Systems with Probabilistic Time-invariant Uncertainties*

Many model predictive control algorithms have been formulated with the objective of including chance constraints and the shaping of probability density functions (PDFs) of system states and outputs. Although many algorithms consider time-varying (TV) uncertainty, the consideration of time-invariant (TI) uncertainty is more challenging since the Markov property cannot be applied directly to the state-space system. Additional challenges associated with handling TI uncertainty include the incorporation of feedback, uncertainty propagation through the uncertain system, the handling of hard input and chance constraints, computational cost, and guaranteeing stability of the closed-loop system and feasibility of the control policy. Efforts to employ polynomial chaos theory (PCT) to mitigate some of these challenges are reviewed. Much of this literature performs stochastic analysis using PCT online in a receding-horizon framework. We discuss differences between the PDFs calculated online and the PDFs of the actual closed-loop system, and provide an example that proves that chance constraints, when implemented in the receding-horizon manner, are not necessarily satisfied by the closed-loop system. Opportunities for future research are suggested, which include the use of PCT for the online analysis of closed-loop systems and the modeling of TI uncertainty as TV uncertainty coupled with a low-pass filter.