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This paper presents a stochastic model predictive control approach fornonlinear systems subject to time-invariant probabilistic uncertainties inmodel parameters and initial conditions. The stochastic optimal control problementails a cost function in terms of expected values and higher moments of thestates, and chance constraints that ensure probabilistic constraintsatisfaction. The generalized polynomial chaos framework is used to propagatethe time-invariant stochastic uncertainties through the nonlinear systemdynamics, and to efficiently sample from the probability densities of thestates to approximate the satisfaction probability of the chance constraints.To increase computational efficiency by avoiding excessive sampling, astatistical analysis is proposed to systematically determine a-priori the leastconservative constraint tightening required at a given sample size to guaranteea desired feasibility probability of the sample-approximated chance constraintoptimization problem. In addition, a method is presented for sample-basedapproximation of the analytic gradients of the chance constraints, whichincreases the optimization efficiency significantly. The proposed stochasticnonlinear model predictive control approach is applicable to a broad class ofnonlinear systems with the sufficient condition that each term is analytic withrespect to the states, and separable with respect to the inputs, states andparameters. The closed-loop performance of the proposed approach is evaluatedusing the Williams-Otto reactor with seven states, and ten uncertain parametersand initial conditions. The results demonstrate the efficiency of the approachfor real-time stochastic model predictive control and its capability tosystematically account for probabilistic uncertainties in contrast to anonlinear model predictive control approaches.

Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints