An Analysis of the Directional-Modifier Adaptation Algorithm Based on Optimal Experimental Design
Abstract
:1. Introduction
2. Optimal Experimental Design
2.1. Failure of Problem (2): An Example
2.2. Modified Optimality Loss Function
2.3. Problem Formulation
2.4. Second-Order Approximation of the Modified Optimality Loss Function
- 1.
- the noise e has a multivariate normal and centered distribution
- 2.
- for all , exists, is smooth, unique and satisfies the Second-Order Sufficient Condition (SOSC) condition of optimality.
- 3.
- the parameter estimation problem (20) has a unique solution satisfying SOSC for any e and is smooth and polynomially bounded in e
- 4.
- functions , and are all bounded by polynomials on their respective domains.
Algorithm 1: 2-run nominal optimal experimental design. |
Input : Current parameter estimation , covariance Σ.
|
2.5. Illustrative Example: Observability Problem
3. Link to the Modifier Approach and the DMA Approximation
3.1. The Modifier Approach
3.2. DMA as an Approximation of (41)
- the approximation for some
- a rank-one approximation of
3.3. Illustrative Example
4. Conclusions
Acknowledgments
Conflicts of Interest
Appendix
References
- Chen, C.; Joseph, B. On-line optimization using a two-phase approach: An application study. Ind. Eng. Chem. Res. 1987, 26, 1924–1930. [Google Scholar] [CrossRef]
- Jang, S.; Joseph, B. On-line optimization of constrained multivariable chemical processes. Ind. Eng. Chem. Res. 1987, 33, 26–35. [Google Scholar] [CrossRef]
- Forbes, J.; Marlin, T.; MacGgregor, J. Model adequacy requirements for optimizing plant operations. Comput. Chem. Eng. 1994, 18, 497–510. [Google Scholar] [CrossRef]
- Forbes, J.; Marlin, T. Design cost: A systematic approach to technology selection for model-based real-time optimization systems. Comput. Chem. Eng. 1996, 20, 717–734. [Google Scholar] [CrossRef]
- Agarwal, M. Feasibility of on-line reoptimization in batch processes. Chem. Eng. Commun. 1997, 158, 19–29. [Google Scholar] [CrossRef]
- Agarwal, M. Iterative set-point optimization of batch chromatography. Comput. Chem. Eng. 2005, 29, 1401–1409. [Google Scholar]
- Marchetti, A. Modifier-Adaptation Methodology for Real-Time Optimization. Ph.D. Thesis, EPFL, Lausanne, Switzerland, 2009. [Google Scholar]
- Roberts, P. An algorithm for steady-state system optimization and parameter estimation. Int. J. Syst. Sci. 1979, 10, 719–734. [Google Scholar] [CrossRef]
- Gao, W.; Engell, S. Comparison of iterative set-point optimization strategies under structural plant-model mismatch. IFAC Proc. Vol. 2005, 16, 401. [Google Scholar]
- Marchetti, A.; Chachuat, B.; Bonvin, D. Modifier-adaptation methodology for real-time optimization. Ind. Eng. Chem. Res. 2009, 48, 6022–6033. [Google Scholar] [CrossRef]
- Roberts, P. Coping with model-reality differences in industrial process optimisation—A review of integrated system optimization and parameter estimation (ISOPE). Comput. Ind. 1995, 26, 281–290. [Google Scholar] [CrossRef]
- Tatjewski, P. Iterative optimizing set-point control-the basic principle redesigned. IFAC Proc. Vol. 2002, 35, 49–54. [Google Scholar] [CrossRef]
- François, G.; Bonvin, D. Use of convex model approximations for real-time optimization via modifier adaptation. Ind. Eng. Chem. Res. 2014, 52, 11614–11625. [Google Scholar] [CrossRef]
- Bunin, G.; Wuillemin, Z.; François, G.; Nakajo, A.; Tsikonis, L.; Bonvin, D. Experimental real-time optimization of a solid oxide fuel cell stack via constraint adaptation. Energy 2012, 39, 54–62. [Google Scholar] [CrossRef]
- Serralunga, F.; Mussati, M.; Aguirre, P. Model adaptation for real-time optimization in energy systems. Ind. Eng. Chem. Res. 2013, 52, 16795–16810. [Google Scholar] [CrossRef]
- Navia, D.; Marti, R.; Sarabia, R.; Gutirrez, G.; Prada, C. Handling infeasibilities in dual modifier-adaptation methodology for real-time optimization. IFAC Proc. Vol. 2012, 45, 537–542. [Google Scholar] [CrossRef]
- Darby, M.; Nikolaou, M.; Jones, J.; Nicholson, D. RTO: An overview and assessment of current practice. J. Process Control 2011, 21, 874–884. [Google Scholar] [CrossRef]
- Costello, S.; François, G.; Bonvin, D.; Marchetti, A. Modifier adaptation for constrained closed-loop systems. IFAC Proc. Vol. 2014, 47, 11080–11086. [Google Scholar] [CrossRef]
- Faulwasser, T.; Bonvin, D. On the Use of Second-Order Modifiers for Real-Time Optimization. In Proceedings of the 19th IFAC World Congress, Cape Town, South Africa, 24–29 August 2014.
- Bunin, G.; François, G.; Bonvin, D. From discrete measurements to bounded gradient estimates: A look at some regularizing structures. Ind. Eng. Chem. Res. 2013, 52, 12500–12513. [Google Scholar] [CrossRef]
- Serralunga, F.; Aguirre, P.; Mussati, M. Including disjunctions in real-time optimization. Ind. Eng. Chem. Res. 2014, 53, 17200–17213. [Google Scholar] [CrossRef]
- Houska, B.; Telenb, D.; Logist, F.; Diehl, M.; Van Impe, J.F.M. An economic objective for the optimal experiment design of nonlinear dynamic processes. Automatica 2015, 51, 98–103. [Google Scholar] [CrossRef]
- Costello, S.; François, G.; Bonvin, D. Directional Real-Time Optimization Applied to a Kite-Control Simulation Benchmark. In Proceedings of the European Control Conference 2015, Linz, Austria, 15–17 July 2015.
- Costello, S.; François, G.; Bonvin, D. A directional modifier-adaptation algorithm for real-time optimization. J. Process Control 2016, 39, 64–76. [Google Scholar] [CrossRef]
- Marchetti, A.; Chachuat, B.; Bonvin, D. A dual modifier-adaptation approach for real-time optimization. J. Process Control 2010, 20, 1027–1037. [Google Scholar] [CrossRef]
- Oehlert, G. A note on the delta method. Am. Stat. 1992, 46, 27–29. [Google Scholar] [CrossRef]
- Withers, C. The moments of the multivariate normal. Bull. Aust. Math. Soc. 1985, 32. [Google Scholar] [CrossRef]
- Vignat, C. A generalized Isserlis theorem for location mixtures of Gaussian random vectors. Stat. Probab. Lett. 2012, 82, 67–71. [Google Scholar] [CrossRef]
© 2016 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Gros, S. An Analysis of the Directional-Modifier Adaptation Algorithm Based on Optimal Experimental Design. Processes 2017, 5, 1. https://doi.org/10.3390/pr5010001
Gros S. An Analysis of the Directional-Modifier Adaptation Algorithm Based on Optimal Experimental Design. Processes. 2017; 5(1):1. https://doi.org/10.3390/pr5010001
Chicago/Turabian StyleGros, Sébastien. 2017. "An Analysis of the Directional-Modifier Adaptation Algorithm Based on Optimal Experimental Design" Processes 5, no. 1: 1. https://doi.org/10.3390/pr5010001