Ralf Hannemann

Continuous and Discrete Composite Adjoints for the Hessian of the Lagrangian in Shooting Algorithms for Dynamic Optimization

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.

Discrete first- and second-order adjoints and automatic differentiation for the sensitivity analysis of dynamic models

We describe the use of first- and second-order tangent-linear and adjoint models of the residual of linear-implicit autonomous differential algebraic systems in the context of an extrapolated Euler scheme. The derivative code compiler dcc is applied to a C-implementation of the residual to get first derivative code. Second-(and higher-)order derivative models are obtained by reapplication of dcc to its own output. The resulting solver serves as a first proof of concept of a new platform for source-level manipulation of mathematical models that is currently under development at RWTH Aachen University.

FAST COMPUTATION OF THE HESSIAN OF THE LAGRANGIAN IN SHOOTING ALGORITHMS FOR DYNAMIC OPTIMIZATION

Abstract 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 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 sake of simplicity restricting to single shooting, the same methodology can also be easily applied to multiple shooting.

Model-based investment planning model for stepwise capacity expansions of chemical plants

Abstract In this contribution a novel investment planning model for the development of stepwise capacity expansion strategies for chemical plants is proposed. This method is implemented in a decision support tool that can be, used during the early stage of plant engineering — a phase which is concerned with the conversion of a chemical process into a highly profitable plant. Based on a previous work by Oldenburg et al. [1], who proposed a method for a quick economic comparison of possible stepwise plant expansion scenarios versus building a full capacity plant, the approach presented in this paper is capable of identifying the optimal process-specific investment strategy on the level of unit operations. A mixed-integer linear programming model dedicated for stepwise capacity expansion strategies for chemical process plants forms the core of the tool.

Integrated process and control design by the normal vector approach: Application to the Tennessee-Eastman process

Abstract This paper presents a large-scale application of the normal vector approach to demonstrate that the complexity of robust dynamic optimization with application to the integration of process and control design can be treated successfully for complex nonlinear systems. The case study further demonstrates that our approach can deal with a multi-dimensional uncertainty space. The normal vector approach is able to automatically identify the worst-case scenarios and find a solution that is optimal with respect to the cost function and robust with respect to path constraints on inputs and states in the presence of parameterized disturbances. The tedious analysis of a large number of different disturbance realizations is not required.

Input-constrained closed-loop systems with grazing bifurcations in optimal robust design

In this work, the normal vector method for robust design is considered to account for actuator saturation effects when unknown time-varying disturbances are present, and desired dynamic properties have to be guaranteed. The normal vector method ensures that desired dynamic properties hold despite uncertain parameters by maintaining a minimal distance between the operating point and so-called critical manifolds where the process behavior changes qualitatively. In this paper input saturation is considered for the first time in the normal vector framework. In order to solve the resulting optimization problem, first and second order derivatives of the flow of a dynamical system has to be computed efficiently. For this purpose, a new platform for source-level manipulation of mathematical models, currently under development at RWTH Aachen University, is proposed to solve the technical difficulties arising when the event of actuator saturation takes place.

Combining Direct and Indirect Methods for Optimal Control - a Case Study

Abstract Adaptive control vector parameterization for the solution of optimal control problems approximates the original infinite-dimensional optimal control problem by a set of finite-dimensional nonlinear programs (NLPs) whose control grids are iteratively refined. The refinement is stopped by a heuristic stopping criterion. The Hessians of the Lagrangian of these NLPs can be efficiently computed by the technique of composite adjoints as recently proposed by the authors. By means of a case study, namely the optimal control of the Williams-Otto semi-batch reactor, we show how to interpret composite adjoints as estimates for the continuous adjoints referred to by Pontryagins Minimum Principle. Thus, these composite adjoints can be used to (i) construct a novel and mathematical sound stopping criterion for the iterative refinement of the control grid and to (ii) setup an indirect multiple shooting method the solution of which verifies and improves the approximate solution to the exact one. Copyright

An Efficient Strategy for Real-Time Dynamic Optimization based on Parametric Sensitivities

Abstract The optimal operation of chemical processes is challenged by frequent transitions and by the influence of process or model uncertainties. Under uncertainties, it is necessary to quickly update the optimal trajectories in order to avoid the violation of constraints and the deterioration of the economic performance of the process. Although an economically optimal operation can be ensured by online dynamic optimization, the high computational load of dynamic optimization associated with nonlinear and complex models is often prohibitive in real-time applications. To reduce the computational time required for online computation of the optimal trajectories in the neighborhood of the optimal solution under uncertainty, different strategies have been explored recently. If the operation is affected by small perturbations, efficient techniques for updating the nominal trajectories based on parametric sensitivities are applied, which do not require the solution of the rigorous optimization problem. However for larger perturbations, the linear updates obtained by the neighboring extremal solutions are not sufficiently accurate, and the solution of the nonlinear optimization problem requires further iterations with updated sensitivities to give a feasible and optimal solution. In this work, the sensitivity-based approach of Kadam and Marquardt (2004) is extended with a fast computational method for second-order derivatives based on composite adjoints. The application of the method to a simulated semi-batch reactor demonstrates that fast and optimal trajectory updates can be obtained.