Christof Büskens

Sensitivity Analysis and Real-Time Optimization of Parametric Nonlinear Programming Problems

Basic results for sensitivity analysis of parametric nonlinear programming problems [11] are revisited. Emphasis is placed on those conditions that ensure the differentiability of the optimal solution vector with respect to the parameters involved in the problem. We study the explicit formulae for the sensitivity derivatives of the solution vector and the associated Lagrange multipliers. Conceptually, these formulae are tailored to solution algorithm calculations. However, we indicate numerical obstacles that prevent these expressions from being a direct byproduct of current solution algorithms. We investigate post-optimal evaluations of sensitivity differentials and discuss their numerical implementation. The main purpose of this paper is to describe an important application of sensitivity analysis: the development of real-time approximations of the perturbed solutions using Taylor expansions. Two elementary examples illustrate the basic ideas.

Sensitivity Analysis and Real-Time Control of Parametric Optimal Control Problems Using Nonlinear Programming Methods

We discuss nonlinear programming (NLP) methods for solving optimal control problems with control and state inequality constraints. Suitable discretizations of control and state variables are used to transform the optimal control into a finite dimensional NLP problem. In [8] we have proposed numerical methods for the post-optimal calculations of parameter sensitivity derivatives of optimal solutions to NLP problems. The purpose of this paper is to extend the methods of post-optimal sensitivity analysis and real-time optimization to discretized control problems. The dimension of the discretized control problem should be kept small to obtain accurate sensitivity results. This can be achieved by taking only the discretized control variables as optimization variables whereas the state variables are computed recursively through an appropriate integration routine. We discuss the implications of this approach for the calculations of parameter sensitivity derivatives with respect to optimal control, state and adjoint functions. The efficiency of the proposed methods are illustrated by two numerical examples.

The ESA NLP Solver WORHP

We Optimize Really Huge Problems (WORHP) is a solver for large-scale, sparse, nonlinear optimization problems with millions of variables and constraints. Convexity is not required, but some smoothness and regularity assumptions are necessary for the underlying theory and the algorithms based on it. WORHP has been designed from its core foundations as a sparse sequential quadratic programming (SQP) / interior-point (IP) method; it includes efficient routines for computing sparse derivatives by applying graph-coloring methods to finite differences, structure-preserving sparse named after Broyden, Fletcher, Goldfarb and Shanno (BFGS) update techniques for Hessian approximations, and sparse linear algebra. Furthermore it is based on reverse communication, which offers an unprecedented level of interaction between user and nonlinear programming (NLP) solver. It was chosen by ESA as the European NLP solver on the basis of its high robustness and its application-driven design and development philosophy. Two large-scale optimization problems from space applications that demonstrate the robustness of the solver complement the cursory description of general NLP methods and some WORHP implementation details.

Approximation of reachable sets by direct solution methods for optimal control problems

A numerical method for the approximation of reachable sets of linear control systems is discussed. The method is based on the formulation of suitable optimal control problems with varying objective function, whose discretization by Runge–Kutta methods leads to finite-dimensional convex optimization problems. It turns out that the order of approximation for the reachable set depends on the particular choice of the Runge–Kutta method in combination with the selection strategy used for control approximation. For an inappropriate combination, the expected order of convergence cannot be achieved in general. The method is illustrated by two test examples using different Runge–Kutta methods and selection strategies, in which the run times are analysed, the order of convergence is estimated numerically and compared with theoretical results in similar areas.

Towards optimization of probe placement for radio-frequency ablation

We present a model for the optimal placement of mono- and bipolar probes in radio-frequency (RF) ablation. The model is based on a numerical computation of the probes electric potential and of the steady state of the heat distribution during RF ablation. The optimization is performed by minimizing a temperature based objective functional under these constraining equations. The paper discusses the discretization and implementation of the approach. Finally, applications of the optimization to artificial data and a comparison to a real RF ablation are presented.

Real-Time Solutions of Bang-Bang and Singular Optimal Control Problems

In many applications of optimal control some or all of the control variables appear linearly in the objective function and the dynamical equations. Therefore, the optimal solutions may exhibit both bang-bang and singular subarcs. Unfortunately, the theory for linear problems of that type is not as well developed as for regular problems, in particular with respect to second order sufficiency conditions. This results in serious problems in developing real-time capable methods to approximate optimal solutions in the presence of data perturbations. In this paper, two discretization methods are presented by which linear optimal control problems can be transcribed into nonlinear programming problems. Based on a stability and sensitivity analysis of the resulting nonlinear programming problems it is possible to compute sensitivity differentials for the discretized problems, by means of which near-optimal solutions can now be computed in real-time for linear problems, too. The performance of one of these methods is demonstrated for the optimal control of a batch reactor.

Sensitivity Analysis and Real-Time Control of Nonlinear Optimal Control Systems via Nonlinear Programming Methods

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Based on recent stability results we propose a robust nonlinear programming method to compute the sensitivity derivatives of optimal solutions. Realtime control approximations of perturbed optimal solutions are obtained by evaluating a first order Taylor expansion of the perturbed solution. The numerical methods are illustrated by two examples. We consider the Rayleigh problem from electrical engineering and the maximum range flight of a hang glider.

Enhanced method for the evaluation of the thermal impact of dry machining processes

From today’s point of view the modelling of machining operations is a promising tool to extend the productivity and the precision of future industrial manufacturing. The importance of predictive simulation and compensation of thermally induced workpiece deformation during machining is especially important in dry machining because of the absence of cooling lubricants. Since the simulation results mainly depend on the boundary conditions of the model, a detailed knowledge of them is necessary. In this case the most important boundary condition is the intensity and possibly the distribution of the surface heat flux representing the heat flow into the workpiece resulting from the chip formation. The surface heat flux cannot be measured directly. One possible way to determine surface heat fluxes is to employ a thermal model of the machining process and match simulated and measured time and space dependent temperature fields. This procedure is time-consuming and is in most cases subjective because the congruency of temperature fields is rated manually, e.g. by the position of single isotherms. Therefore an enhanced method for the determination of surface heat fluxes is proposed in this paper. The method is based on nonlinear optimisation techniques and a simple finite difference scheme for numerical solution of the heat equation (WORHP-FDM). The procedure is objective between repeat measurements and works in a fully automated manner. The implementation is validated by the comparison to an analytical solution of the moving heat source based on the model of Carslaw and Jaeger and then applied to measured thermal images from milling experiments.

Optimal Guidance and Control of Lunar Landers with Non-throttable Main Engine

Autonomous soft, safe and precise landing on celestial bodies like the Moon, planets and asteroids is still a challenging task for future exploration missions. To achieve a maximum of payload mass landed on the target body the trajectories of landing vehicles need to be (fuel) optimized. In order to allow an adjustability of the trajectory and a compensation of disturbances for all vehicles so far a thrust modulation is required. The drawback is that currently no main engine is available which allows the needed thrust modulation for an efficient, robust and safe landing on a celestial body like the Moon. The technology of the Apollo missions is not available anymore.Most planned lunar missions rely on the modulation capability of multiple engines where in some cases the thrust of the auxiliary engines for modulation is in the order of main engine thrust. This approach requires a large number of smaller engines to achieve the needed thrust modulation adding complexity and increasing the probability of failure.

Multidisciplinary design optimization models and algorithms for space launch vehicles

The paper presents in details the engineering models and optimization algorithms for a Multidisciplinary Design Optimization research framework developed within ESA’s PRESTIGE PhD program. The application focuses on the conceptual design of classical unmanned Expendable Launch Vehicles, with future extensions to the early preliminary detail level and to more complex systems such as manned and reusable vehicles. Results are presented from the validation of the disciplinary models and optimization algorithms. Besides, sensitivity analyses and Multidisciplinary Analysis and Optimization runs on European test cases (Ariane-5 ECA and VEGA) show how relatively simple models and a mixed global/local optimization approach allow to obtain reasonable results for conceptual level design (10 to 20% errors on global performance figures) with very limited computational effort.