| Title:
|
Optimal control processes associated with a class of discontinuous control systems: Applications to sliding mode dynamics (English) |
| Author:
|
Gil García, Arturo Enrique |
| Author:
|
Azhmyakov, Vadim |
| Author:
|
Basin, Michael V. |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
50 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
5-18 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper presents a theoretical approach to optimal control problems (OCPs) governed by a class of control systems with discontinuous right-hand sides. A possible application of the framework developed in this paper is constituted by the conventional sliding mode dynamic processes. The general theory of constrained OCPs is used as an analytic background for designing numerically tractable schemes and computational methods for their solutions. The proposed analytic method guarantees consistency of the resulting approximations related to the original infinite-dimensional optimization problem and leads to specific implementable algorithms. (English) |
| Keyword:
|
sliding mode |
| Keyword:
|
nonlinear systems |
| Keyword:
|
absolute continuous approximations |
| MSC:
|
49J15 |
| MSC:
|
49J30 |
| MSC:
|
62A10 |
| MSC:
|
93B12 |
| MSC:
|
93E12 |
| idZBL:
|
Zbl 1302.93222 |
| idMR:
|
MR3195001 |
| DOI:
|
10.14736/kyb-2014-1-0005 |
| . |
| Date available:
|
2014-05-02T06:42:08Z |
| Last updated:
|
2016-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143760 |
| . |
| Reference:
|
[1] Aliprantis, C. D., Border, K. C.: Infinite Dimensional Analysis..Springer, Berlin 1999. Zbl 1156.46001, MR 1717083 |
| Reference:
|
[2] Atkinson, K., Han, W.: Theoretical Numerical Analysis..Springer, New York 2005. Zbl 1181.47078, MR 2153422 |
| Reference:
|
[3] Gómez-Gutiérrez, D., Čelikovský, S., Ramírez-Trevino, A., Ruiz-León, J., Gennaro, S. Di: Robust regulation via sliding modes of a rotary inverted pendulum..In: Preprints 3rd IFAC Symposium on Robust Control Design, ÚTIA AV ČR, Praha 2000. |
| Reference:
|
[4] Attia, S. A., Azhmyakov, V., Raisch, J.: On an optimization problem for a class of impulsive hybrid systems..In: Discrete Event Dynamical Systems, 2009. Zbl 1197.49038, MR 2609675 |
| Reference:
|
[5] Azhmyakov, V., Raisch, J.: Convex control systems and convex optimal control problems with constraints..IEEE Trans. Automat. Control 53 (2008), 993-998. MR 2419445, 10.1109/TAC.2008.919848 |
| Reference:
|
[6] Azhmyakov, V., Boltyanski, V. G., Poznyak, A.: Optimal control of impulsive hybrid systems..Nonlinear Anal.: Hybrid Systems 2 (2008), 1089-1097. Zbl 1163.49038, MR 2478392 |
| Reference:
|
[7] Azhmyakov, V., Galvan-Guerra, R., Egerstedt, M.: Hybrid LQ-optimization using dynamic programming..In: Proc. 2009 American Control Conference, St. Louis 2009, pp. 3617-3623. |
| Reference:
|
[8] Azhmyakov, V., Egerstedt, M., Fridman, L., Poznyak, A.: Continuity properties of nonlinear affine control systems: applications to hybrid and sliding mode dynamics..In: Proc. 2009 IFAC Conference on Analysis and Design of Hybrid Systems, Zaragoza 2009, pp. 204-209. |
| Reference:
|
[9] Azhmyakov, V., Boltyanski, V. G., Poznyak., A.: The dynamic programming approach to multi-model robust optimization..Nonlinear Anal.: Theory, Methods Applications 72 (2010), 1110-1119. MR 2579373 |
| Reference:
|
[10] Azhmyakov, V.: Optimal control of sliding mode processes: A general approach..In: Proc. 11th International Workshop on Variable Structure Systems, Mexico City 2010, pp. 504-509. |
| Reference:
|
[11] Basin, V. Azhmyakov M., García, A. E. Gil: A general approach to optimal control processes associated with a class of discontinuous control systems: Applications to the sliding mode dynamics..In: Proc. 2012 IEEE International Conference on Control Applications, Dubrovnik 2012, pp. 1154-1159. |
| Reference:
|
[12] Bartolini, G., Fridman, L., Pisano, A., (eds.), E. Usai: Modern Sliding Mode Control Theory..Lecture Notes in Control and Inform. Sci. 375, Springer, Berlin 2008. Zbl 1140.93005, MR 2435347 |
| Reference:
|
[13] Berkovitz, L. D.: Optimal Control Theory..Springer, New York 1974. Zbl 1257.49001, MR 0372707 |
| Reference:
|
[14] Boiko, I.: Discontinuous Control Systems Frequency-Domain Analysis and Design..Birkhauser, New York 2009. Zbl 1165.93002, MR 2683127 |
| Reference:
|
[15] Boiko, I., Fridman, L., Pisano, A., Usai, E.: On the transfer properties of the ``generalized sub-optimal" second-order sliding mode control algorithm..IEEE Trans. Automat. Control 54 (2009), 399-403. MR 2491973, 10.1109/TAC.2008.2008361 |
| Reference:
|
[16] Čelikovský, S.: Numerical algorithm for nonsmooth stabilization of triangular form systems..Kybernetika 32 (1996), 261-274. Zbl 0873.93074, MR 1438219 |
| Reference:
|
[17] Branicky, M. S., Borkar, V. S., Mitter, S. K.: A unifed framework for hybrid control: model and optimal control theory..IEEE Trans. Automat. Control 43 (1998), 31-45. MR 1604250, 10.1109/9.654885 |
| Reference:
|
[18] Caines, P., Egerstedt, M., Malhame, R., Schoellig, A.: A hybrid Bellman equation for bimodal systems..Lecture Notes in Computer Sci. 4416, Springer, Berlin 2007, pp. 656-659. Zbl 1221.49054, 10.1007/978-3-540-71493-4_54 |
| Reference:
|
[19] Deimling, K.: Multivalued Differential Equations..de Gruyter, Berlin 1992. Zbl 0820.34009, MR 1189795 |
| Reference:
|
[20] Edwards, R. E.: Functional Analysis..Dover, New York 1995. Zbl 0354.06008, MR 1320261 |
| Reference:
|
[21] Egerstedt, M., Wardi, Y., Axelsson, H.: Transition-time optimization for switched-mode dynamical systems..IEEE Trans. Automat. Control 51 (2006), 110-115. MR 2192797, 10.1109/TAC.2005.861711 |
| Reference:
|
[22] Fattorini, H. O.: Infinite-Dimensional Optimization and Control Theory..Cambridge University Press, Cambridge 1999. Zbl 1200.49001, MR 1669395 |
| Reference:
|
[23] Ferrara, A., Rubbagoti, M.: A sub-optimal second order sliding mode controller for systems with saturating actuators..IEEE Trans. Automat. Control 54 (2009), 1082-1087. MR 2518127, 10.1109/TAC.2008.2010992 |
| Reference:
|
[24] Filippov, A. F.: Differential Equations with Discontinuous Right-Hand Sides..Kluwer, Dordrecht 1988. Zbl 1098.34006 |
| Reference:
|
[25] Gallardo-Hernández, A. G., Fridman, L., Islas-Andrade, S., Shtessel, Y.: Quasi-continuous high order sliding modes controllers applied to glucose-insulin regulatory system models..In: Proc. 47th IEEE Conference on Decision and Control, Cancun 2008, pp. 2208-2213. |
| Reference:
|
[26] Gamkrelidze, R.: Principles of Optimal Control Theory..Plenum Press, London 1978. Zbl 0401.49001, MR 0686793 |
| Reference:
|
[27] Gómez-Gutiérrez, D., Čelikovský, S., Ramírez-Trevino, A., Ruiz-León, J., Gennaro, S. Di: Sliding mode observer for switched linear systems..In: Proc. 2011 IEEE Conference on Automation Science and Engineering, IEEE Conference on Automation Science and Engineering, Trieste 2011. |
| Reference:
|
[28] Haddad, W. M., Chellaboina, V.: Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach..Princeton University Press, New Jersey 2008. Zbl 1142.34001, MR 2381711 |
| Reference:
|
[29] Hale, J.: Ordinary Differential Equations..J. Wiley, New York 1969. Zbl 0433.34003, MR 0419901 |
| Reference:
|
[30] Himmelberg, C. J.: Measurable relations..Fund. Math. 87 (1975), 53-72. Zbl 0465.28002, MR 0367142 |
| Reference:
|
[31] Isidori, A.: Nonlinear Control Systems..Springer, New York 1989. Zbl 0931.93005, MR 1015932 |
| Reference:
|
[32] Jahn, J.: Introduction to the Theory of Nonlinear Optimization..Springer, Berlin 2007. Zbl 1115.49001, MR 2292173 |
| Reference:
|
[33] Khalil, H. K.: Nonlinear Systems..Prentice Hall, New Jersey 2001. Zbl 1194.93083 |
| Reference:
|
[34] Levant, A.: Universal SISO sliding-mode controllers with finite-time convergence..IEEE Trans. Automat. Control 46 (2001), 1447-1451. MR 1853689, 10.1109/9.948475 |
| Reference:
|
[35] Liberzon, D.: Switching in Systems and Control..Birkhäuser, Boston 2003. Zbl 1036.93001, MR 1987806 |
| Reference:
|
[36] Orlov, Y.: Discontinuous Systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions..Springer, New York 2008. Zbl 1180.37004, MR 2731468 |
| Reference:
|
[37] Paden, B. E., Sastry, S. S.: Calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulator..IEEE Trans. Circuits Systems 34 (1987), 73-82. MR 0871547, 10.1109/TCS.1987.1086038 |
| Reference:
|
[38] Poznyak, A.: Advanced Mathematical Tools for Automatic Control Engineers..Elsevier, Amsterdam 2008. |
| Reference:
|
[39] Boltyanski, V. G., Poznyak, A.: The Robust Maximum Principle..Birkhäuser, Boston 2011. Zbl 1239.49002, MR 3024943 |
| Reference:
|
[40] Pytlak, R.: Numerical Methods for Optimal Control Problems with State Constraints..Springer-Verlag, Berlin 1999. Zbl 0928.49002, MR 1713434 |
| Reference:
|
[41] Rockafellar, R. T.: Monotone operators and the proximal point algorithm..SIAM J. Control Optim. 14 (1976), 877-898. Zbl 0358.90053, MR 0410483, 10.1137/0314056 |
| Reference:
|
[42] Ramos-Velasco, L. E., Ruiz-León, J. J., Čelikovský, S.: Rotary inverted pendulum: Trajectory tracking via nonlinear control techniques..Kybernetika 38 (2002), 217-232. Zbl 1265.93138, MR 1916453 |
| Reference:
|
[43] Roubíček, T.: Approximation theory for generalized young measures..Numer. Funct. Anal. Optim. 16 (1995), 1233-1253. Zbl 0854.65051, MR 1374974, 10.1080/01630569508816671 |
| Reference:
|
[44] Shaikh, M. S., Caines, P. E.: On the hybrid optimal control problem: theory and algorithms..IEEE Trans. Automat. Control 52 (2007), 1587-1603. MR 2352436, 10.1109/TAC.2007.904451 |
| Reference:
|
[45] Utkin, V.: Sliding Modes in Control and Optimization..Springer, Berlin 1992. Zbl 0748.93044, MR 1295845 |
| . |
