An efficient method for stochastic optimal control with joint chance constraints for nonlinear systems

Abstract

Stochastic model predictive control hinges on the online solution of a stochastic optimal control problem. This paper presents a computationally efficient solution method for stochastic optimal control for nonlinear systems subject to (time-varying) stochastic disturbances and (time-invariant) probabilistic model uncertainty in initial conditions and parameters. To this end, new methods are presented for joint propagation of time-varying and time-invariant probabilistic uncertainty and the nonconservative approximation of joint chance constraint (JCC) on the system state. The proposed uncertainty propagation method relies on generalized polynomial chaos and conditional probability rules to obtain tractable expressions for the state mean and covariance matrix. A moment-based surrogate is presented for JCC approximation to circumvent construction of the full probability distribution of the state or the use of integer variables as required when using the sample average approximation. The proposed solution method for stochastic optimal control is illustrated on a nonlinear semibatch reactor case study in the presence of probabilistic model uncertainty and stochastic disturbances. It is shown that the proposed solution method is significantly superior to a standard random sampling method for stochastic optimal control in terms of computational requirements. Furthermore, the moment-based surrogate for the JCC is shown to be substantially less conservative than the widely used distributionally robust Cantelli-Chebyshev inequality for chance constraint approximation. KEYWORDS generalized polynomial chaos, joint chance constraint, joint propagation of probabilistic model uncertainty and stochastic disturbances, stochastic optimal control Int J Robust Nonlinear Control. 2019;29: 5017-5037.wileyonlinelibrary.com/journal/rnc

A data-driven robust optimization approach to scenario-based stochastic model predictive control

“…It is worth mentioning that, the proposed method cannot be applied to the case of multiplicative uncertainty, while classic SSMPC is able to handle multiplicative uncertainty. For predictive control of general nonlinear systems with both multiplicative and additive uncertainty, one can utilize the polynomial chaos expansion-based and moment-based formulations of SMPC [41,42].…”

Stochastic predictive control

“…Computationally efficient solution methods is the subject of the third contribution . The authors focus on stochastic optimal control for nonlinear systems subject to possibly time‐varying stochastic disturbances and uncertainty in initial conditions and parameters.…”

Stochastic Nonlinear Model Predictive Control Using Gaussian Processes

“…Note to adjust the back-offs we use the ecdf, which does account for the true shape of the underlying probability distribution. This avoids the problem that is often faced in stochastic optimization utilizing Chebyshev's inequality to robustly approximate chance constraints, which is often excessively conservative [56].…”