Article | Proceedings of the 12th International Modelica Conference, Prague, Czech Republic, May 15-17, 2017 | Defining and Solving Hybrid Optimal Control Problems with Higher Index DAEs Linköping University Electronic Press Conference Proceedings
Göm menyn

Title:
Defining and Solving Hybrid Optimal Control Problems with Higher Index DAEs
Author:
Radoslaw Pytlak: Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland Damian Suski: Institute of Automatic Control and Robotics, Warsaw University of Technology, Poland Tomasz Tarnawski: Department of Quantitative Methods and Information Technology, Kozminski University, Poland Tomasz Zawadzki: Research and Academic Computer Network (NASK), Poland
DOI:
10.3384/ecp17132265
Download:
Full text (pdf)
Year:
2017
Conference:
Proceedings of the 12th International Modelica Conference, Prague, Czech Republic, May 15-17, 2017
Issue:
132
Article no.:
028
Pages:
265-273
No. of pages:
9
Publication type:
Abstract and Fulltext
Published:
2017-07-04
ISBN:
978-91-7685-575-1
Series:
Linköping Electronic Conference Proceedings
ISSN (print):
1650-3686
ISSN (online):
1650-3740
Publisher:
Linköping University Electronic Press, Linköpings universitet


Export in BibTex, RIS or text

The paper deals with optimal control problems defined for hybrid systems described by higher index DAEs. We present a prototype solution that supports the whole process from defining such problem to solving it and presenting results. Problem’s definition is done with Dynamic Optimization Modeling Language (DOML) which is based directly on Modelica. The proposed numerical procedure for solving the problems of interest has the following features: 1) it is based on the appropriately defined adjoint equations formulated for the discretized equations being the result of the numerical integration of system equations by an implicit Runge–Kutta method; 2) initialization for higher index DAEs is performed with the help of Pantelides’ algorithm; 3) it does not require the system to be transformed to ODEs (through differentiation of some algebraic equations).

The paper presents numerical examples related to hybrid systems described by index three DAEs, showing the validity of the proposed approach. All software components needed to carry out the computations, i.e. the code editor, compiler, numerical libraries and GUI for presenting results are prepared as parts of a combined platform: Interactive Dynamic Optimization Server (IDOS).



Keywords: hybrid systems, optimal control problems, higher index DAEs

Proceedings of the 12th International Modelica Conference, Prague, Czech Republic, May 15-17, 2017

Author:
Radoslaw Pytlak, Damian Suski, Tomasz Tarnawski, Tomasz Zawadzki
Title:
Defining and Solving Hybrid Optimal Control Problems with Higher Index DAEs
DOI:
http://dx.doi.org/10.3384/ecp17132265
References:

J. Åkesson. Tools and Languages for Optimization of Large-Scale Systems. PhD thesis, Department of Automatic Control, Lund University, Lund, Sweden, 2007.

J. Åkesson. Optimica–an extension of Modelica supporting dynamic optimization. In Proceedings of the 6th Modelica Conference, Bielefeld, Germany, March 2008. Modelica Association.

J. Åkesson, M. Gäfvert, and H. Tummescheit. JModelica—an open source platform for optimization of Modelica models. In Proceedings of MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling, Vienna, Austria, February 2009. TU Wien.

E. M. Gertz and S. J. Wright. Object-oriented software for quadratic programming. ACM Transactions on Mathematical Software, (29):58–81, 2003.

E. Hairer, Ch. Lubich, and M. Roche. The numerical solution of differential-algebraic equations by Runge–Kutta methods. Lecture Notes in Mathematics, 1409:56–225, 1989.

K. L. Hiebert and L. F. Shampine. Implicitly defined output points for solutions of ode-s. Technical report, United States Department of Energy, Sandia Laboratories, 1980.

J. Lygeros, K. H. Johansson, S. Sastry, and M. Egerstedt. On the existence of executions of hybrid automata. In Proceedings of the 38th IEEE CDC, pages 2249–2254, Phoenix, AZ; USA, December 1999. IEEE.

C. C. Pantelides. The consistent initialization of differentialalgebraic systems. SIAM Journal on Scientific and Statistical Computing, 9(2):213–231, 1988. doi: https://doi.org/10.1137/0909014.

R. Pytlak. Numerical Methods for Optimal Control Problems with State Constraints. Lecture Notes in Mathematics 1707. Springer Berlin Heidelberg, 1999. ISBN 9783540662143.

R. Pytlak. Numerical procedure for optimal control of higher index DAEs. Discrete Contin. Dyn. Syst., 29(2):647–670, 2011. ISSN 1078-0947; 0133-0189; 1553-5231/e. doi: https://doi.org/10.3934/dcds.2011.29.647.

R. Pytlak and D. Suski. On solving hybrid optimal control problems with higher index DAEs. Optimization Methods and Software, 2017. doi: https://doi.org/10.1080/10556788.2017.1288730.

R. Pytlak, J. Blaszczyk, A. Karbowski, K. Krawczyk, and T. Tarnawski. Solvers chaining in the IDOS server for dynamic optimization. In Proceedings of 52nd IEEE CDC, pages 7119–7124, Florence; Italy, 2013. IEEE. ISBN 978-1-4673-5714-2. URL http://dblp.uni-trier.de/db/conf/cdc/cdc2013.html#PytlakBKKT13.

R. Pytlak, T. Tarnawski, B. Fajdek, and M. Stachura. Interactive Dynamic Optimization Server – connecting one modelling language with many solvers. Optimization Methods and Software, 29(5):1118–1138, 2014. doi: http://doi.org/10.1080/10556788.2013.799159.

M. S. Shaikh. Optimal Control of Hybrid Systems: Theory and Algorithms. PhD thesis, McGill University, Montreal, Que., Canada, 2004. URL http://digitool.library.mcgill.ca/R/?func=dbin-jump-full&object_id=85095&local_base=GEN01-MCG02.AAINR06340.

D. Suski and R. Pytlak. The weak maximum principle for hybrid systems. In Proceedings of the of the 24th IEEE MED, pages 338–343, Athens; Greece, 2016. IEEE.

T. Tarnawski and R. Pytlak. DOML - a compiler environment for dynamic optimization supporting multiple solvers. In Proceedings of the 10th International Modelica Conference, pages 1095–1104, Lund; Sweden, 2014. Linköping University Electronic Press.

A. J. van der Schaft and J. M. Schumacher. An Introduction to Hybrid Dynamical Systems. Lecture Notes in Control and Information Sciences. Springer, 2000. ISBN 9781852332334.

A. Wachter and L. T. Biegler. On the implementation of an interior point line search flter algorithm for large scale nonlinear programming. Mathematical Programming, (106):25–57, 2006.

Proceedings of the 12th International Modelica Conference, Prague, Czech Republic, May 15-17, 2017

Author:
Radoslaw Pytlak, Damian Suski, Tomasz Tarnawski, Tomasz Zawadzki
Title:
Defining and Solving Hybrid Optimal Control Problems with Higher Index DAEs
DOI:
https://doi.org10.3384/ecp17132265
Note: the following are taken directly from CrossRef
Citations:
No citations available at the moment


Responsible for this page: Peter Berkesand
Last updated: 2019-11-06