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Highlights: [•] Wavelet-based grid is applied to direct multiple shooting. [•] Adaptive multiple shooting is used to solve dynamic optimization (or optimal control) problems. [•] Adaptive multiple shooting, automatically generates a problem-dependent parameterization of the control profiles. [•] We demonstrate the superiority of adaptive grid refinement over an equidistant discretization for two examples. [ABSTRACT FROM AUTHOR]

Abstract: For the solution of optimal control problems, direct methods have been established in the process engineering community. If set up correctly they robustly provide more or less accurate approximations of the exact solution. In the usual engineering practice, neither the distance to the exact solution is reflected, nor the compliance with the continuous necessary conditions in form of Pontryagin''s Minimum Principle is checked. At the end, some approximate solution is available but its quality is at question. This tutorial addresses the problem of the verification of optimal controls computed by direct shooting methods. We focus on this popular transcription method though the results are also relevant for other solution strategies. We review known results spread in the mathematical literature on optimal control to show how the output of the nonlinear programs (NLPs) resulting from single shooting transcriptions of optimal control problems can be interpreted in the context of Pontryagin''s Minimum Principle. In particular, we show how to approximate continuous adjoint variables by means of the dual information provided by the NLP solver. Based on this adjoint approximation we use a multi-level setting to construct an estimate of the distance to a true extremal solution satisfying the continuous necessary conditions of optimality. A comprehensive case study illustrates the theoretical results. [Copyright &y& Elsevier]

Abstract: A new distributed model-predictive control method is introduced, which is based on a novel distributed optimization algorithm, relying on a sensitivity-based coordination mechanism. Coordination and therefore overall optimality is achieved by means of a linear approximation of the objective functions of neighboring controllers within the objective function of each local controller. As for most of the distributed optimization methods, an iterative solution of the distributed optimal control problems is required. An analysis of the method with respect to its convergence properties is provided. For illustration, the sensitivity-driven distributed model-predictive control (S-DMPC) method is applied to a simulated alkylation process. An almost optimal control sequence can be achieved after only one iteration in this case. [Copyright &y& Elsevier]

Abstract: Dynamic optimization allows for the determination of the upper bound of achievable performance in the operation of continuous chemical processes in transient operation which often results from load or grade changes. Performance assessment can rely on the computation of optimal trajectories for selected scenarios which properly reflect the operational envelope. Realistic industrial problems, however, involve very large-scale dynamic process models and consequently require highly-efficient and robust optimization algorithms. In this work we demonstrate the feasibility of operability assessment by means of dynamic optimization in an industrial case study involving a large-scale process model comprising about 12,000 differential-algebraic model equations. The numerical strategy employed relies on a single shooting method combined with adaptive control grid refinement to minimize the complexity of the numerical problem to the extent possible. This algorithm proves to be the key to success and saves about 95% of computational complexity in comparison to a conventional equidistant discretization. [Copyright &y& Elsevier]

One approach to solve optimal control problems by direct methods is the so-called sequential approach or single shooting. Only the control variables are discretized resulting in a nonlinear program (NLP) which can be solved with SQP or interior point methods. This paper presents a new methodology to efficiently provide the Hessian of the Lagrangian of that resulting NLP. The algorithm is based on the second-order adjoint method and introduces the novel concept of composite adjoints to reduce the computational effort of a Hessian evaluation. Though this contribution is for the sake of simplicity restricted to single shooting, the same methodology can also be easily applied to multiple shooting. [ABSTRACT FROM AUTHOR]

Abstract: In this contribution, a novel approach for the modeling and numerical optimal control of hybrid (discrete–continuous dynamic) systems based on a disjunctive problem formulation is proposed. It is shown that a disjunctive model representation, which constitutes an alternative to mixed-integer model formulations, provides a very flexible, intuitive and effective way to formulate hybrid (discrete–continuous dynamic) optimization problems. The structure and properties of the disjunctive process models can be exploited for an efficient and robust numerical solution by applying generalized disjunctive programming techniques. The proposed modeling and optimization approach will be illustrated by means of optimal control of hybrid systems embedding linear discrete–continuous dynamic models. [Copyright &y& Elsevier]

Abstract: In this paper we present a novel method for the numerical solution of dynamic optimization problems. After obtaining a first solution at a coarse resolution of the control profiles with a direct sequential approach, the structure of the control profiles is analyzed for possible switching times and arcs. Subsequently, the problem is reformulated automatically and solved as a multi-stage problem, with each stage corresponding to a potential arc. Order and resolution of the control parameterization are adapted to the type of the particular arc. By means of some case studies we show that accurate solutions with only few degrees of freedom can be obtained. [Copyright &y& Elsevier]

In this work we present an approach for the sensitivity analysis of linearly-implicit differential–algebraic equation systems. Solutions for both states and sensitivities are obtained by applying an extrapolated linearly implicit Euler discretization scheme. This approach is compared to the widely used sensitivity extensions of multi-step BDF methods by means of case studies. Especially, we point out the benefit of this method in the context of dynamic optimization using the sequential approach. [Copyright &y& Elsevier]

Abstract: Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that leads to a bi-level min–max optimization problem. We propose and investigate a numerical method to solve this min–max optimization problem exactly in the case that the underlying maximization problem always has its solution on the boundary of the uncertainty set. This is an adoption of the local reduction approach used to solve generalized semi-infinite programs. The approach formulates an equilibrium constraint employing first order derivatives of both the uncertainty set and the user defined constraints. We propose two different ways for computation of these derivatives, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show the equivalence of the proposed approach to a method based on geometric considerations that was recently developed by some of the authors. We show how to generalize the techniques to optimal control problems. The robust dynamic optimization of a batch distillation illustrates that both techniques are numerically efficient and able to overcome the inexactness of another recently proposed numerical approach to address uncertainty in optimal control problems. [Copyright &y& Elsevier]