Approximate robust optimal control of nonlinear dynamic systems under process noise

Dynamic optimization techniques for nonlinear systems can provide the process industry with sustainable and efficient operating regimes. However, these regimes often lie close to the operating limits. It is therefore critical that these model based operating conditions are robust with respect to process noise, i.e, unmodeled time-varying random disturbances. Besides the effect of uncertainty in the satisfaction of constraints, also the effect of uncertainty on the objective function should be considered. Including uncertainty in an optimization problem typically leads to numerically challenging semi-infinite optimization problems. In this paper several computationally tractable methods are exploited to approximately solve robust optimal control problems. The presented approaches have the advantage that they allow the use of fast deterministic gradient based optimization techniques. The first method is based on a linearization approach while the second method exploits the unscented transformation to construct an estimate of the uncertainty propagation. Both methods yield an approximation of the variance-covariance matrix of the critical constraints and of the objective function. These variance-covariance matrices are employed in the optimization routine to obtain more robust control actions. The illustrative case study concerns a jacketed tubular reactor.