H. Georg Bock

Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations

Abstract Optimization problems in chemical engineering often involve complex systems of nonlinear DAE as the model equations. The direct multiple shooting method has been known for a while as a fast off-line method for optimization problems in ODE and later in DAE. Some factors crucial for its fast performance are briefly reviewed. The direct multiple shooting approach has been successfully adapted to the specific requirements of real-time optimization. Special strategies have been developed to effectively minimize the on-line computational effort, in which the progress of the optimization iterations is nested with the progress of the process. They use precalculated information as far as possible (e.g. Hessians, gradients and QP presolves for iterated reference trajectories) to minimize response time in case of perturbations. In typical real-time problems they have proven much faster than fast off-line strategies. Compared with an optimal feedback control computable upper bounds for the loss of optimality can be established that are small in practice. Numerical results for the Nonlinear Model Predictive Control (NMPC) of a high-purity distillation column subject to parameter disturbances are presented.

Introduction to Model Based Optimization of Chemical Processes on Moving Horizons

Dynamic optimization problems are typically quite challenging for large-scale applications. Even more challenging are on-line applications with demanding real-time constraints. This contribution provides a concise introduction into problem formulation and standard numerical techniques commonly found in the context of moving horizon optimization using nonlinear differential algebraic process models.

On the Role of Natural Level Functions to Achieve Global Convergence for Damped Newton Methods

The paper discusses a new view on globalization techniques for Newton’s method. In particular, strategies based on “natural level functions” are considered and their properties are investigated. A “restrictive mono-tonicity test” is introduced and theoretically motivated. Numerical results for a highly nonlinear optimal control problem from aerospace engineering and a parameter estimation for a chemical process are presented.

Newton-Type Methods for the Approximate Solution of Nonlinear Programming Problems in Real-Time

An efficient numerical method for the real-time solution of optimal control problems in optimal feedback control is presented, which is based on the direct multiple shooting method, and the contractivity of this real-time iteration scheme is proven.

Original article: Numerical computation of derivatives in systems of delay differential equations

This article deals with initial value problem solutions in systems of delay differential equations and their derivatives with respect to parameters, where the parameters may occur in the initial value, the initial function, the right-hand-side function, and the delay. Sufficient conditions for differentiability are given, and an efficient and reliable method for the numerical computation is presented. Emphasis is laid on the treatment of problems with a discontinuity at the initial time, for which it is shown that jumps occur in the derivative at the propagated discontinuity times. An explicit expression for the size of the jumps in the derivative is given. Features are discussed of the implementation of COLSOL-DDE, an experimental solver for initial value problems in delay differential equations that also computes the derivatives of the solution. The performance of the developed method is demonstrated by a comparison to standard techniques for derivative approximation.

Multiple Shooting Method for Initial Satellite Orbit Determination

After a satellite has been launched, the determination of its initial orbit is one of the primary tasks to be accomplished by mission control: e.g., in order to prepare for high-precision orbit determination or orbital transfers. For this purpose, tracking measurements are acquired by ground stations, which are the basis for the determination of the unknown orbit parameters. In this paper, the orbit determination problem is formulated as a least-squares parameter estimation problem, and a newly developed multiple shooting method is presented for its solution. For the generation of initial guesses, the method incorporates a so-called analytical projection, which is specifically tailored to the available measurement data. Original tracking data of the launch and early orbit phases of the Artemis satellite and the second pair of Cluster-II satellites are used for tests of the method. Both scenarios constitute challenging benchmark problems, because the satellites were launched into orbits significantly different from the expected ones. A comparison to the widely employed single shooting method reveals that the newly developed method significantly improves the convergence behavior.

Stable Walking and Running Robots Without Feedback

In this paper we give an overview of our previous work on open-loop stable walking and running robots. The concept of open-loop control implies that all actuators of a robot receive purely periodic pre-calculated torque or force inputs that are never altered by any feedback interference. Passive-dynamic robots can be considered as a special case with zero actuation for which a stabilization is much easier due to the possibility of time shifts. We have developed a numerical method that optimizes the open-loop stability of solutions of periodic optimal control problems with discontinuities. Using this method, we were able to find a number of open-loop stable one- and two-legged robot configurations and motions.

Real-Time Optimization and Nonlinear Model Predictive Control of Processes Governed by Differential-Algebraic Equations

Optimization problems in chemical engineering often involve complex systems of nonlinear DAE as the model equations. The direct multiple shooting method has been known for a while as a fast off-line method for optimization problems in ODE and later in DAE. Some factors crucial for its fast performance are briefly reviewed. The direct multiple shooting approach has been successfully adapted to the specific requirements of real-time optimization. Special strategies have been developed to effectively minimize the on-line computational effort, in which the progress of the optimization iterations is nested with the progress of the process. They use precalculated information as far as possible (e.g. Hessians, gradients and QP presolves for iterated reference trajectories) to minimize response time in case of perturbations. In typical real-time problems they have proven much faster than fast off-line strategies. Compared with an optimal feedback control computable upper bounds for the loss of optimality can be established that are small in practice. Numerical results for the Nonlinear Model Predictive Control (NMPC) of a high-purity distillation column subject to parameter disturbances are presented. # 2002 Published by Elsevier Science Ltd.

A nonlinear preconditioner for optimum experimental design problems

We show how to efficiently compute A-optimal experimental designs, which are formulated in terms of the minimization of the trace of the covariance matrix of the underlying regression process, using quasi-Newton sequential quadratic programming methods. In particular, we introduce a modification of the problem that leads to significantly faster convergence. To derive this modification, we model each iteration in terms of an initial experimental design that is to be improved, and show that the absolute condition number of the model problem grows without bounds as the quality of the initial design improves. As a remedy, we devise a preconditioner that ensures that the absolute condition number will instead stay uniformly bounded. Using numerical experiments, we study the effect of this reformulation on relevant cases of the general problem class, and find that it leads to significant improvements in both stability and convergence behavior.