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Journal of Applied Mathematics
Volume 2014, Article ID 341716, 6 pages
Research Article

Partitioned Quasi-Newton Approximation for Direct Collocation Methods and Its Application to the Fuel-Optimal Control of a Diesel Engine

1Institute for Dynamic Systems and Control, ETH Zurich, Sonneggstrasse 3, 8092 Zurich, Switzerland
2FPT Motorenforschung AG, Schlossgasse 2, 9320 Arbon, Switzerland

Received 26 November 2013; Revised 17 February 2014; Accepted 18 February 2014; Published 25 March 2014

Academic Editor: M. Montaz Ali

Copyright © 2014 Jonas Asprion et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The numerical solution of optimal control problems by direct collocation is a widely used approach. Quasi-Newton approximations of the Hessian of the Lagrangian of the resulting nonlinear program are also common practice. We illustrate that the transcribed problem is separable with respect to the primal variables and propose the application of dense quasi-Newton updates to the small diagonal blocks of the Hessian. This approach resolves memory limitations, preserves the correct sparsity pattern, and generates more accurate curvature information. The effectiveness of this improvement when applied to engineering problems is demonstrated. As an example, the fuel-optimal and emission-constrained control of a turbocharged diesel engine is considered. First results indicate a significantly faster convergence of the nonlinear program solver when the method proposed is used instead of the standard quasi-Newton approximation.