Approximate robust optimization of nonlinear systems under parametric uncertainty and process noise

Abstract

a b s t r a c tDynamic optimization techniques for complex nonlinear systems can provide the process industry with sustainable and efficient operating regimes. The problem with these regimes is that they usually lie close to the limits of the process. It is therefore paramount that these operating conditions are robust with respect to the parameter uncertainties and to the process noise such that critical constraints are not violated. Besides the uncertainty in the constraints, also the uncertainty in the objective function needs to be taken into account. However, including robustness in an optimization problem typically leads to semi-infinite optimization problems that are challenging to solve in practice. In the current manuscript several computationally tractable methods are exploited to approximately solve the robust dynamic optimization problem. These methods allow the use of fast deterministic gradient based optimization techniques. The first type of methods are based on a linearization approach while the second method exploits the unscented transformation to construct an estimation of the uncertainty propagation. Both types provide the user with an approximation of the variance-covariance matrix of the critical constraints and of the objective function. This allows the user to easily take them into account in the dynamic optimization routine in a stochastic setting without the need of using computationally expensive Monte Carlo simulations in the optimization procedure. Moreover, an iterative scheme is mentioned to evaluate the approximate results and to improve them if necessary. Two illustrative case studies are discussed, a jacketed tubular reactor and the Williams-Otto reactor.

Interactive Multi‐objective Dynamic Optimization of Bioreactors under Parametric Uncertainty

“…where E [ c i ] is the expected value on the constraint function c i , Var [ c i ] is the variance on the constraint function c i , and is a backoff parameter for c i . This backoff parameter can be chosen based on probabilistic inequalities as, e.g., Cantelli‐Chebyshev's inequality or following the procedure presented in .…”

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

“…As discussed in Section 1, the linearization and the UT methods are the most commonly used deterministic approaches to uncertainty propagation in stochastic optimal control. 12,15 Linearization is usually computationally inexpensive but requires the existence and calculation of the Jacobians, which can be cumbersome for complex model functions. On the other hand, the UT method propagates a set of heuristically chosen sigma points through the system dynamics to circumvent the inaccuracies associated with linearization.…”

“…However, the UT method can be expensive for online optimization, as the sigma points must be adapted at each optimization step for any candidate input profile. 15 Furthermore, error bounds and convergence results are not readily available for the UT method. Alternatively, random sampling methods look to replace the probabilistic operators E[·], Var(·), and P(·) with their corresponding sample approximations.…”

“…Remark 3. The number of terms in the gPC expansions scales exponentially with respect to the number of uncertainty sources (see (15)). This can become a computational limitation of gPC.…”

“…However, its accuracy is largely dependent on the choice of the sigma points, which are typically defined in terms of the predicted state covariance and, therefore, must be adapted in every step of the optimization as a function of the input profile. 15 Alternatively, random sampling approaches to uncertainty propagation such as Monte Carlo-based (MC-based) methods have been applied for solving the stochastic OCP. 16,17 A key advantage of MC-based uncertainty propagation methods is that the approximation error is independent of the number of uncertainty sources.…”