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Economic NMPC for averaged infinite horizon problems with periodic approximations. (English) Zbl 1442.93013

Summary: Feedback control for averaged infinite horizon problems is a challenging task because the optimal way to operate such systems often exhibits a time-varying behavior, i.e. it cannot be represented as a steady state. We present a controller that is based on the excellent approximation properties of periodic solutions to such systems. The controller is capable of autonomously adapting both the optimal periodic trajectory as well as the optimal period itself in case the system parameters change. The amount of necessary a priori information for the controller setup is reduced compared to other economic model predictive control (EMPC) schemes and the resulting economic performance of the closed-loop is superior to schemes that use a fixed period. Complementary to the standard stability-theory for EMPC, our approach does not require dissipativity conditions and is based solely on assumptions on controllability of the dynamical system, existence of optimal periodic trajectories with descent directions for suboptimal periodic orbits and regularity of the EMPC subproblems. Based on these assumptions we show that the resulting closed-loop economically performs equally well as the optimal periodic trajectory. We illustrate the results with a numerical simulation of a highly nonlinear, unstable air-borne powerkite under varying wind conditions.

MSC:

93B45 Model predictive control
93B52 Feedback control
93C10 Nonlinear systems in control theory

Software:

ACADO; Ipopt
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Full Text: DOI

References:

[1] Amrit, R.; Rawlings, J. B.; Biegler, L. T., Optimizing process economics online using model predictive control, Computers and Chemical Engineering, 58, 334-343 (2013)
[2] Angeli, D.; Amrit, R.; Rawlings, J. B., Receding horizon cost optimization for overly constrained nonlinear plants, (Proceedings of the 48th IEEE conference on decision and control (CDC) held jointly with 2009 28th Chinese control conference (2009), IEEE)
[3] Angeli, D.; Amrit, R.; Rawlings, J. B., On average performance and stability of economic model predictive control, IEEE Transactions on Automatic Control, 57, 7, 1615-1626 (2012) · Zbl 1369.93209
[4] Bellman, R., On the computational solution of differential-difference equations, Journal of Mathematical Analysis and Applications, 2, 1, 108-110 (1961) · Zbl 0098.31603
[5] Bittanti, S.; Fronza, G.; Guardabassi, G., Optimal steady-state versus periodic operation in discrete systems, Journal of Optimization Theory and Applications, 18, 4, 521-536 (1976) · Zbl 0304.49014
[6] Bock, H. G.; Plitt, K. J., A multiple shooting algorithm for direct solution of optimal control problems, IFAC Proceedings Volumes, 17, 2, 1603-1608 (1984)
[7] Canale, M.; Fagiano, L.; Milanese, M., Power kites for wind energy generation [Applications of control], IEEE Control Systems, 27, 6, 25-38 (2007) · Zbl 1395.93142
[8] Clarke, F.; Ledyaev, Y.; Stern, R., Asymptotic stability and smooth Lyapunov functions, Journal of Differential Equations, 149, 1, 69-114 (1998) · Zbl 0907.34013
[9] Cotrell, J.; Stehly, T.; Johnson, J.; Roberts, J. O.; Parker, Z.; Scott, G., Analysis of transportation and logistics challenges affecting the deployment of larger wind turbines: Summary of resultsTechnical report (2014), Office of Scientific and Technical Information (OSTI)
[10] Diehl, M., Real-time optimization for large scale nonlinear processes (2001), Heidelberg University, (Ph.D. thesis) · Zbl 0991.49023
[11] Dong, Z.; Angeli, D., Analysis of economic model predictive control with terminal penalty functions on generalized optimal regimes of operation, International Journal of Robust and Nonlinear Control, 28, 16, 4790-4815 (2018) · Zbl 1402.93214
[12] Ellis, M.; Durand, H.; Christofides, P. D., A tutorial review of economic model predictive control methods, Journal of Process Control, 24, 8, 1156-1178 (2014)
[13] Engell, S., Feedback control for optimal process operation, Journal of Process Control, 17, 3, 203-219 (2007)
[14] Fagiano, L.; Teel, A. R., Generalized terminal state constraint for model predictive control, Automatica, 49, 9, 2622-2631 (2013) · Zbl 1364.93240
[15] Faulwasser, T.; Bonvin, D., Exact turnpike properties and economic NMPC, European Journal of Control, 35, 34-41 (2017) · Zbl 1367.93227
[16] Faulwasser, T.; Korda, M.; Jones, C. N.; Bonvin, D., On turnpike and dissipativity properties of continuous-time optimal control problems, Automatica, 81, 297-304 (2017) · Zbl 1373.49026
[17] Frasch, J. V.; Wirsching, L.; Sager, S.; Bock, H. G., Mixed-level iteration schemes for nonlinear model predictive control, IFAC Proceedings Volumes, 45, 17, 138-144 (2012)
[18] Grammel, G., Estimate of periodic suboptimal controls, Journal of Optimization Theory and Applications, 99, 3, 681-689 (1998) · Zbl 1085.93501
[19] Grüne, L., Economic receding horizon control without terminal constraints, Automatica, 49, 3, 725-734 (2013) · Zbl 1267.93052
[20] Guardabassi, G.; Locatelli, A.; Rinaldi, S., Status of periodic optimization of dynamical systems, Journal of Optimization Theory and Applications, 14, 1, 1-20 (1974) · Zbl 0264.49018
[21] Houska, B., Enforcing asymptotic orbital stability of economic model predictive control, Automatica, 57, 45-50 (2015) · Zbl 1330.93199
[22] Houska, B.; Diehl, M., Optimal control for power generating kites, (2007 European control conference (ECC) (2007), IEEE) · Zbl 1131.93333
[23] Houska, B.; Diehl, M., Robustness and stability optimization of power generating kite systems in a periodic pumping mode, (2010 IEEE international conference on control applications (2010), IEEE)
[24] Houska, B.; Ferreau, H. J.; Diehl, M., ACADO toolkit-an open-source framework for automatic control and dynamic optimization, Optimal Control Applications & Methods, 32, 3, 298-312 (2010) · Zbl 1218.49002
[25] Ilzhöfer, A.; Houska, B.; Diehl, M., Nonlinear MPC of kites under varying wind conditions for a new class of large-scale wind power generators, International Journal of Robust and Nonlinear Control, 17, 17, 1590-1599 (2007) · Zbl 1131.93333
[26] Jiang, Z.-P.; Wang, Y., A converse Lyapunov theorem for discrete-time systems with disturbances, Systems & Control Letters, 45, 1, 49-58 (2002) · Zbl 0987.93072
[27] Jose, Z.; Michael, D.; Patrick, G.; Ananthan, S.; Lantz, E.; Cotrell, J., Enabling wind power nationwideTechnical report (2015), Office of Scientific and Technical Information (OSTI)
[28] Kellett, C. M., A compendium of comparison function results, Mathematics of Control, Signals, and Systems, 26, 3, 339-374 (2014) · Zbl 1294.93071
[29] Limon, D.; Pereira, M.; de la Peña, D. M.; Alamo, T.; Grosso, J., Single-layer economic model predictive control for periodic operation, Journal of Process Control, 24, 8, 1207-1224 (2014)
[30] Manwell, J. F., Wind energy explained : Theory, design and application (2009), Wiley: Wiley Chichester, U.K
[31] Müller, M. A.; Angeli, D.; Allgöwer, F., On necessity and robustness of dissipativity in economic model predictive control, IEEE Transactions on Automatic Control, 60, 6, 1671-1676 (2015) · Zbl 1360.93397
[32] Müller, M. A.; Grüne, L., Economic model predictive control without terminal constraints for optimal periodic behavior, Automatica, 70, 128-139 (2016) · Zbl 1339.93051
[33] Müller, M. A.; Grüne, L.; Allgöwer, F., On the role of dissipativity in economic model predictive control, IFAC-PapersOnLine, 48, 23, 110-116 (2015)
[34] Polak, E., Optimization (1997), Springer New York · Zbl 0886.90140
[35] Rawlings, J., Model predictive control : Theory, computation, and design (2017), Nob Hill Publishing: Nob Hill Publishing Madison, Wisconsin
[36] Wächter, A.; Biegler, L. T., On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106, 1, 25-57 (2005) · Zbl 1134.90542
[37] Wirsching, L., Multi-level iteration schemes with adaptive level choice for nonlinear model predictive control (2018), Heidelberg University Library, (Ph.D. thesis)
[38] Zanon, M.; Grüne, L.; Diehl, M., Periodic optimal control, dissipativity and MPC, IEEE Transactions on Automatic Control, 62, 6, 2943-2949 (2017) · Zbl 1369.93356
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