Zusammenfassung
Verteilte Optimierungsverfahren wie die duale Dekomposition oder die Alternating Direction Method of Multipliers (ADMM) erleben in den letzten Jahren ein erneutes steigendes Interesse in den unterschiedlichsten Anwendungen. Die zunehmende Vernetzung von Servern oder Mikrocontrollern weltweit sowie die Größe von heutigen Datensätzen liefern dabei die Grundlage für die Nachfrage nach iterativen, parallelisierbaren Optimierungsverfahren. In dieser Arbeit stellen wir verteilte Optimierungsalgorithmen und ihre Anwendungen bei der Berechnung von Zustandsrückführungen mithilfe der Modellprädiktiven Regelung vor. Wir konzentrieren uns auf die Systemdynamik sowie die Vernetzung der Systeme bei der Anwendbarkeit der Algorithmen. Darüber hinaus untersuchen wir die Algorithmen auf ihre Kommunikationsstruktur, den Austausch sensibler Daten, die Skalierbarkeit und die Flexibilität.
Abstract
Distributed optimization like dual decomposition or the alternating direction method of multipliers (ADMM), proposed centuries ago, experience an increased interest in various applications over the last years. Severs or microcontrollers connected all over the world and big data applications build the foundation and demand for iterative, parallelizable and distributed optimization algorithms. In this paper we present distributed optimization algorithms and their applications in the context of feedback design using model predictive control. We concentrate on the dynamics and the interconnection of the dynamical systems with respect to the applicability of the distributed optimization algorithms. Moreover, we focus on the communication structure in terms of the exchange of sensitive data, as well as the scalability and flexibility of the distributed optimization algorithms.
About the authors
Philipp Braun ist seit 2016 Senior Research Associate an der University of Newcastle, Australia, in der School of Electrical Engineering and Computing. Nach einem Mathematikstudium an der Technischen Universität Kaiserslautern (Dipl.-Math., 2012) promovierte er an der Universität Bayreuth (Dr. rer. nat., 2016). Als Wissenschaftler nach der Promotion war er als Akademischer Rat an der Universität Bayreuth und an der University of Newcastle tätig.Seine Forschungsinteressen liegen im Gebiet der Kontrolltheorie. Hierbei beschäftigt er sich insbesondere mit verteilten Optimierungsverfahren in der Modellprädiktiven Regelung sowie mit Lyapunov-Funktionen zur Stabilisierung dynamischer Systeme.
Lars Grüne ist seit 2002 Professor für Angewandte Mathematik an der Universität Bayreuth. Er wurde 1996 an der Universität Augsburg promoviert und habilitierte sich 2001 an der Goethe-Universität Frankfurt/Main, jeweils im Fach Mathematik. Er war als Gastwissenschafter an den Universitäten Rom ‘Sapienza’ (Italien), Padua (Italien), Melbourne (Australien), Paris IX – Dauphine (Frankreich) und Newcastle (Australien) tätig. Prof. Grüne ist Editor-in-Chief der Zeitschrift Mathematics of Control, Signals and Systems (MCSS) und Associate Editor bei verschiedenen weiteren Zeitschriften, z. B. beim Journal of Optimization Theory and Applications (JOTA), Mathematical Control and Related Fields (MCRF) und den IEEE Control Systems Letters (CSS-L). Seine Forschungsinteressen liegen im Gebiet der Mathematischen System- und Kontrolltheorie mit Schwerpunkt auf numerischen und optimierungsbasierten Methoden für nichtlineare Systeme.
Literatur
1. M. Annergren, A. Hansson and B. Wahlberg. An ADMM algorithm for solving ℓ1 regularized MPC. In Proc. 51st IEEE Conference on Decision and Control, S. 4486–4491, 2012.10.1109/CDC.2012.6426429Search in Google Scholar
2. I. Atzeni, L. G. Ordóñez, G. Scutari, D. P. Palomar and J. R. Fonollosa. Demand-side management via distributed energy generation and storage optimization. IEEE Transactions on Smart Grid, 4(2):866–876, 2013.10.1109/TSG.2012.2206060Search in Google Scholar
3. J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4(1):238–252, 1962.10.1007/BF01386316Search in Google Scholar
4. D. P. Bertsekas. Nonlinear Programming. Athena Scientific, 1999.Search in Google Scholar
5. D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Distributed Computation: Numerical Methods. Athena Scientific: Belmont, MA, USA, 1989.Search in Google Scholar
6. A. Bestler and K. Graichen. Distributed model predictive control for continuous-time nonlinear systems based on suboptimal ADMM. arXiv preprint, arXiv:1706.09599, 2017.10.1002/oca.2459Search in Google Scholar
7. S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3(1):1–122, 2011.10.1561/9781601984616Search in Google Scholar
8. P. Braun, T. Faulwasser, L. Grüne, C. M. Kellett, S. R. Weller and K. Worthmann. Hierarchical distributed ADMM for predictive control with applications in power networks. IFAC Journal of Systems and Control, 3:10–22, 2018.10.1016/j.ifacsc.2018.01.001Search in Google Scholar
9. P. Braun, L. Grüne, C. M. Kellett, S. R. Weller and K. Worthmann. A distributed optimization algorithm for the predictive control of smart grids. IEEE Transactions on Automatic Control, 61(12):3898–3911, 2016.10.1109/TAC.2016.2525808Search in Google Scholar
10. P. Braun, L. Grüne, C. M. Kellett, S. R. Weller and K. Worthmann. Towards price-based predictive control of a small-scale electricity network. International Journal of Control, 2017. https://doi.org/10.1080/00207179.2017.1339329.10.1080/00207179.2017.1339329Search in Google Scholar
11. T.-H. Chang, A. Nedić and A. Scaglione. Distributed constrained optimization by consensus-based primal-dual perturbation method. IEEE Transactions on Automatic Control, 59(6):1524–1538, 2014.10.1109/TAC.2014.2308612Search in Google Scholar
12. C. Conte, T. Summers, M. N. Zeilinger, M. Morari and C. N. Jones. Computational aspects of distributed optimization in model predictive control. In Proc. 51st IEEE Conference on Decision and Control, S. 6819–6824, 2012.10.1109/CDC.2012.6426138Search in Google Scholar
13. G. B. Dantzig and P. Wolfe. Decomposition principle for linear programs. Operations Research, 8(1):101–111, 1960.10.1287/opre.8.1.101Search in Google Scholar
14. M. Diehl, H. G. Bock and J. P. Schlöder. A real-time iteration scheme for nonlinear optimization in optimal feedback control. SIAM Journal on Control and Optimization, 43(5):1714–1736, 2005.10.1137/S0363012902400713Search in Google Scholar
15. J. C. Duchi, A. Agarwal and M. J. Wainwright. Dual averaging for distributed optimization: Convergence analysis and network scaling. IEEE Transactions on Automatic Control, 57(3):592–606, 2012.10.1109/TAC.2011.2161027Search in Google Scholar
16. H. Everett. Generalized Lagrange multiplier method for solving problems of optimum allocation of resources. Operations Research, 11(3):399–417, 1963.10.1287/opre.11.3.399Search in Google Scholar
17. C. Le Floch, F. Belletti, S. Saxena, A. M. Bayen and S. Moura. Distributed optimal charging of electric vehicles for demand response and load shaping. In Proc. 54th IEEE Conference on Decision and Control, S. 6570–6576, 2015.10.1109/CDC.2015.7403254Search in Google Scholar
18. M. Fortin and R. Glowinski. On decomposition-coordination methods using an augmented Lagrangian. In Augmented Lagrangian Methods: Applications to the numerical solution of Boundary-Value Problems. North-Holland: Amsterdam, 1983.10.1016/S0168-2024(08)70028-6Search in Google Scholar
19. D. Gabay. Applications of the method of multipliers to variational inequalities. In Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems. North-Holland: Amsterdam, 1983.10.1016/S0168-2024(08)70034-1Search in Google Scholar
20. D. Gabay and B. Mercier. A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Computers & Mathematics with Applications, 2(1):17–40, 1976.10.1016/0898-1221(76)90003-1Search in Google Scholar
21. P. Giselsson, M. D. Doan, T. Keviczky, B. De Schutter and A. Rantzer. Accelerated gradient methods and dual decomposition in distributed model predictive control. Automatica, 49(3):829–833, 2013.10.1016/j.automatica.2013.01.009Search in Google Scholar
22. P. Giselsson and A. Rantzer. Distributed model predictive control with suboptimality and stability guarantees. In Proc. 49th IEEE Conference on Decision and Control, S. 7272–7277, 2010.10.1109/CDC.2010.5717026Search in Google Scholar
23. K. Graichen and A. Kugi. Stability and incremental improvement of suboptimal MPC without terminal constraints. IEEE Transactions on Automatic Control, 55(11):2576–2580, Nov. 2010.10.1109/TAC.2010.2057912Search in Google Scholar
24. L. Grüne. Approximation properties of receding horizon optimal control. Jahresbericht der Deutschen Mathematiker-Vereinigung, 118(1):3–37, 2016.10.1365/s13291-016-0134-5Search in Google Scholar
25. L. Grüne and J. Pannek. Nonlinear Model Predictive Control. Theory and Algorithms. Springer, 2. Auflage, 2017.10.1007/978-3-319-46024-6Search in Google Scholar
26. X. Hou, Y. Xiao, J. Cai, J. Hu and J. E. Braun. Distributed model predictive control via proximal Jacobian ADMM for building control applications. In Proc. IEEE American Control Conference, S. 37–43, 2017.10.23919/ACC.2017.7962927Search in Google Scholar
27. B. Houska, J. Frasch and M. Diehl. An augmented Lagrangian based algorithm for distributed nonconvex optimization. SIAM Journal on Optimization, 26(2):1101–1127, 2016.10.1137/140975991Search in Google Scholar
28. G. K. H. Larsen, N. D. van Foreest and J. M. A. Scherpen. Distributed control of the power supply-demand balance. IEEE Transactions on Smart Grid, 4(2):828–836, 2013.10.1109/TSG.2013.2242907Search in Google Scholar
29. J. F. C. Mota, J. M. F. Xavier, P. M. Q. Aguiar and M. Püschel. Distributed optimization with local domains: Applications in mpc and network flows. IEEE Transactions on Automatic Control, 60(7):2004–2009, 2015.10.1109/TAC.2014.2365686Search in Google Scholar
30. I. Necoara and D. Clipici. Efficient parallel coordinate descent algorithm for convex optimization problems with separable constraints: Application to distributed MPC. Journal of Process Control, 23(3):243–253, 2013.10.1016/j.jprocont.2012.12.012Search in Google Scholar
31. I. Necoara and J. A. K. Suykens. Application of a smoothing technique to decomposition in convex optimization. IEEE Transactions on Automatic Control, 53(11):2674–2679, 2008.10.1109/TAC.2008.2007159Search in Google Scholar
32. A. Nedić and A. Ozdaglar. Cooperative distributed multi-agent optimization, S. 340–386. Cambridge University Press, 2009.10.1017/CBO9780511804458.011Search in Google Scholar
33. B. O’Donoghue, G. Stathopoulos and S. Boyd. A splitting method for optimal control. IEEE Transactions on Control Systems Technology, 21(6):2432–2442, 2013.10.1109/TCST.2012.2231960Search in Google Scholar
34. N. Parikh and S. P. Boyd. Proximal algorithms. Foundations and Trends in Optimization, 1(3):123–231, 2013.10.1561/9781601987174Search in Google Scholar
35. J. B. Rawlings, D. Q. Mayne and M. M. Diehl. Model Predictive Control: Theory, Computation, and Design. Nob Hill Publishing, 2. Auflage, 2017.Search in Google Scholar
36. H. Scheu and W. Marquardt. Sensitivity-based coordination in distributed model predictive control. Journal of Process Control, 21(5):715–728, 2011. Special Issue on Hierarchical and Distributed Model Predictive Control.10.1016/j.jprocont.2011.01.013Search in Google Scholar
37. B. T. Stewart, A. N. Venkat, J. B. Rawlings, S. J. Wright and G. Pannocchia. Cooperative distributed model predictive control. Systems & Control Letters, 59(8):460–469, 2010.10.1016/j.sysconle.2010.06.005Search in Google Scholar
38. A. N. Venkat, I. A. Hiskens, J. B. Rawlings and S. J. Wright. Distributed mpc strategies with application to power system automatic generation control. IEEE Transactions on Control Systems Technology, 16(6):1192–1206, 2008.10.1109/TCST.2008.919414Search in Google Scholar
39. Z. Wang and C. J. Ong. Distributed model predictive control of linear discrete-time systems with local and global constraints. Automatica, 81:184–195, 2017.10.1016/j.automatica.2017.03.027Search in Google Scholar
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