Volume 12, Issue 4 (Journal of Control, V.12, N.4 Winter 2019)                   JoC 2019, 12(4): 1-14 | Back to browse issues page

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Shabani A, Fatehi A, Soltanian F, Jamilnia R. Design of nonlinear continuous time predictive controller by solving the differential-algebraic equations with boundary conditions using homotopy perturbation method. JoC. 2019; 12 (4) :1-14
URL: http://joc.kntu.ac.ir/article-1-553-en.html
1- Payame Noor University
2- K.N. Toosi University of Technology
Abstract:   (4376 Views)
In this paper, design of continuous time predictive controller and solving the resulting differential-algebraic equations are presented using the semi-analytical homotopy perturbation method. At any updating time of the continuous time predictive control algorithm, an optimal open loop control problem must be solved. In order to solve the predictive control problem in continuous time, the problem of optimal control is solved by an indirect method. For this purpose, the necessary and sufficient conditions for optimality are determined by applying the variational calculus and the Pontryagin's minimum principle. A system of differential-algebraic equations with boundary conditions is created. Homotopy perturbation method is proposed to semi-analytically solve this problem. By specifying the control and the state functions, we can obtain easily the control and the state values in every instance of the prediction horizon. The presented method can be used to design of continuous-time predictive controller of linear, nonlinear and time varying systems. To illustrate the reliability and efficiency of the proposed method, some numerical examples with simulation results are presented.
Type of Article: Research paper | Subject: Special
Received: 2018/01/2 | Accepted: 2018/08/18 | ePublished ahead of print: 2018/10/6 | Published: 2019/05/4

References
1. [1] Kirches C., Wirsching L., Bock J.P, 2012, "Efficient direct multiple shooting for nonlinear model predictive control on long horizon", Schloder Journal of Process Control 22, pp. 540- 550. [DOI:10.1016/j.jprocont.2012.01.008]
2. [2] Magni L., Scattolini R., 2004, "Model predictive control of continuouse time nonlinear systems with picewise constant control", IEEE transactions on automatic control, vol. 49, NO. 6, pp. 900 - 906. [DOI:10.1109/TAC.2004.829595]
3. [3] Magni L., Scattolini R., 2007, "Tracking of non-square nonlinear continuous time systems with piecewise constant model predictive control", Journal of Process Control 17, pp. 631-640. [DOI:10.1016/j.jprocont.2007.01.007]
4. [4] Wang L., 2001, "continuous time model predictive control design using orthonormal functions", INT.J.control, vol. 74, NO. 16, pp. 1588-1600. [DOI:10.1080/00207170110082218]
5. [5] Cizniar M., Fikar M., Latifi M.A., 2008, "Design of constrained nonlinear model predictive control based on global optimization", 18th European Symposium on Computer Aided Process Engineering - ESCAPE 18 Bertrand Braunschweig and Xavier Joulia (Editors). [DOI:10.1016/S1570-7946(08)80099-5]
6. [6] Findeisen R., Raff T., Allgower F., 2007, "Sampled-Data Model Predictive Control for Constrained Continuous Time Systems," Advanced Strategies in Control Systems with Input and Output Contraints, pp.207-235. [DOI:10.1007/978-3-540-37010-9_7]
7. [7] Li S.E., Xu SH., Ku D., 2016, "Efficient and accurate computation model predictive control using pseudospectral discretization", Neurocomputing, 177, pp. 363-372. [DOI:10.1016/j.neucom.2015.11.020]
8. [8] Chen W.H., 2004, "Predictive Control of General Nonlinear Systems Using Approximation", IEE proceedings: control theory and applications, 151 (2), pp. 137-144. [DOI:10.1049/ip-cta:20040042]
9. [9] Wang, Y., Boyd, S., (2008), "Fast model predictive control using online optimization", The International Federation of Automatic Control,Vol. 41, pp. 6974-6979. [DOI:10.3182/20080706-5-KR-1001.01182]
10. [10] Pannocchia, G., Rawlings, J. B., Mayne, D. Q., Marquardt, M., (2010)," On computing solutions to the continuous time constrained linear quadratic regulator", IEEE Transactions on Automatic Control, Vol. 55, pp. 2192-2198. [DOI:10.1109/TAC.2010.2053478]
11. [11] Daehlen, J. S., Otto Eikrem, G., (2014), "Nonlinear model predictive control using trust region derivative free optimization", Journal of Process Control, Vol. 24, pp. 1106-1120. [DOI:10.1016/j.jprocont.2014.04.011]
12. [12] D.E. Kirk, "Optimal control theory: an introduction", Dover Books on Electical Engineering Series, 2004.
13. [13] T.J. Böhme, B. Frank, "Direct Methods for Optimal Control", In: Hybrid Systems, Optimal Control and Hybrid Vehicles. Advances in Industrial Control. Springer, Cham, 2017. [DOI:10.1007/978-3-319-51317-1]
14. [14] Mangasarian O.L., 1996, "sufficient conditions for the optimal control of nonlinear systems", J. SIAM control, Vol. 4, No. 1. [DOI:10.1137/0304013]
15. [15] He J. H., 1999 "Homotopy perturbation technique", Computer Methods in Applied Mechanics and Engineering , Vol. 178 (3), pp. 257-262. [DOI:10.1016/S0045-7825(99)00018-3]
16. [16] Aslam Noor M., 2010, "Some iterative methods for solving nonlinear equations using homotopy perturbation method", international journal of computer mathematics, Vol. 87, No. 1, pp. 141-149. [DOI:10.1080/00207160801969513]
17. [17] Soltanian F., Dehghan M., Karbassi S.M., 2010, "Solution of the differential algebraic equations via homotopy perturbation method and their engineering applications", International Journal of Computer Mathematics, Vol. 87, No. 9, pp.1950-1974. [DOI:10.1080/00207160802545908]
18. [18] Roozi A., Alibeiki E., Hosseini S.S., Ebrahimi M., 2011, "Homotopy perturbation method for special nonlinear partial differential equations", Journal of King Saud University - Science, vol 23, issue 1, pp. 99-103. [DOI:10.1016/j.jksus.2010.06.014]
19. [19] Ayati Z., Biazar J., 2015, "On the convergence of Homotopy perturbation Method", Journal of the Egyptian Mathematical Society, vol.23 (2), pp. 424-428. [DOI:10.1016/j.joems.2014.06.015]
20. [20] Chen H., Allgower F., 1998, "A Quasi-Infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability", Automatica, Vol. 34, No. 10, pp. 1205-1217. [DOI:10.1016/S0005-1098(98)00073-9]
21. [21] Chen H., Allgower F., 1997, "A quasi infinite horizon nonlinear predictive control scheme for stable", IFAC Proceedings Volumes, Vol. 30, No. 9, pp. 529-534. [DOI:10.1016/S1474-6670(17)43203-4]
22. [22] Jadbabaie A., Hauser J., 2005, "On the stability of receding horizon control with a general terminal cost", IEEE Transactions on Automati control, Volume 50, issue 5, pp. 674 - 678. [DOI:10.1109/TAC.2005.846597]
23. [23] Camacho E.F., Bordons C., Model Peredictive Control, Advanced Texbooks in Control and Signal Processing, Springer, Cham, 2004.
24. [24] Fu H.S, Han B., 2006, A homotopy method for nonlinear inverse problems, Applied Mathematics and Computation, vol.183, pp. 1270-1279. [DOI:10.1016/j.amc.2006.05.139]
25. [25] طاهرسیما، حنیف."طراحی کنترل¬کننده هوشمند برای هلیکوپترآزمایشگاهی"، دانشگاه خواجه نصیرالدین طوسی، پایان نامه کارشناسی ارشد، 1385.
26. [26] قربانی، اصغر. "روش اختلال هموتوپی هی و کاربرد آن"، دانشگاه فردوسی مشهد، پایان نامه کارشناسی ارشد، 1386.