A novel method for the design of switching surfaces for discretized MIMO nonlinear systems

José Darío Luis-Delgado ; Basil Mohammed Al-Hadithi ; Agustín Jiménez

International Journal of Applied Mathematics and Computer Science, Tome 27 (2017), p. 5-17 / Harvested from The Polish Digital Mathematics Library

Résumé

Designing variable structure control with sliding mode (VSC-SM) control schemes needs a switching function or a sliding surface which guarantees the global stability of the closed-loop system. Despite the fact that a wide range of design approaches has been proposed for solving this mathematical problem, the number of proposed methodologies for nonlinear systems is not very extensive, especially for discrete time nonlinear MIMO systems, and most of them require some coordinate system transformation. Therefore, it is not an easy task to find a design scheme that can be applied to discrete time nonlinear MIMO systems. The proposed methodology introduces a mathematical tool: a switching surface equation for a class of MIMO nonlinear systems through an explicit equation without any coordinate transformation. This equation makes use of an implicit linearizing process via the Taylor expansion that allows the use of linear procedures for the design of switching surfaces and the forward Euler method to obtain a discrete time dynamics representation. An illustrative example is included to show the advantages of the proposed design methodology.

Détail

Publié le : 2017-01-01
EUDML-ID : urn:eudml:doc:288100

Bibliographie

[000] Ababneh, M., Salah, M. and Alwidyanm, K. (2011). Linearization of nonlinear dynamical systems: A comparative study, Jordan Journal of Mechanical and Industrial Engineering 5(6): 567-571.

[001] Ackermann, J. and Utkin, V. (1994). Sliding mode control design based on Ackermann's formula, Proceedings of the 33rd IEEE Conference on Decision and Control, Lake Buena Vista, FL, USA, Vol. 4, pp. 3622-3627.

[002] Åström, K.J. and Wittenmark, B. (1997). Computer-Controlled Systems. Theory and Design, 3rd Edn., Prentice-Hall, Upper Saddle River, NJ.

[003] Bartoszewicz, A. (1998). Discrete-time quasi-sliding-mode control strategies, IEEE Transactions on Industrial Electronics 45(4): 633-637.

[004] Bartoszewicz, A. and Leśniewski, P. (2014). An optimal sliding mode congestion controller for connection-oriented communication networks with lossy links, International Journal of Applied Mathematics and Computer Science 24(1): 87-97, DOI: 10.2478/amcs-2014-0007. | Zbl 1292.93045

[005] Camacho, O. and Smith, C.A. (2000). Sliding mode control: An approach to regulate nonlinear chemical processes, ISA Transactions 39(2): 205-218.

[006] Choi, H.H. (2003). An LMI-based switching surface design method for a class of mismatched uncertain systems, IEEE Transactions on Automatic Control 48(9): 1634-1638.

[007] DeCarlo, R., Zak, S. and Matthews, G. (1988). Variable structure control of nonlinear multivariable systems: A tutorial, Proceedings of the IEEE 76(3): 212-232.

[008] Dorling, C. and Zinober, A. (1986). Two approaches to hyperplane design in multivariable structure control systems, International Journal of Control 44(1): 65-82. | Zbl 0596.93036

[009] Draženović, B., Milosavljević, Č. and Veselić, B. (2013). Comprehensive approach to sliding mode design and analysis in linear systems, in B. Bandyopdhyay et al. (Eds.), Advances in Sliding Mode Control: Concept, Theory and Implementation, Springer, Heidelberg, pp. 1-19. | Zbl 1291.93067

[010] Edwards, C. and Spurgeon, S. (2003). Linear matrix inequality methods for designing sliding mode output feedback controllers, IEE Proceedings: Control Theory and Applications 150(5): 539-545.

[011] Furuta, K. (1990). Sliding mode control of a discrete system, Systems and Control Letters 14(2): 145-152. | Zbl 0692.93043

[012] Furuta, K. and Pan, Y. (1995). A new approach to design a sliding sector for VSS controller, Proceedings of the 1995 American Control Conference, Seatle, WA, USA, Vol. 2, pp. 1304-1308.

[013] Furuta, K. and Pan, Y. (2002). Discrete-time variable structure control, in X. Yu and J.-X. Xu (Eds.), Variable Structure Systems: Towards the 21st Century, Springer, Berlin, pp. 57-81. | Zbl 1023.93016

[014] Gao, W. and Hung, J. (1993). Variable structure control of nonlinear systems: A new approach, IEEE Transactions on Industrial Electronics 40(1): 45-55.

[015] Gao, W., Wang, Y. and Homaifa, A. (1995). Discrete-time variable structure control systems, IEEE Transactions on Industrial Electronics 42(2): 117-122.

[016] Gatzke, E.P., Meadows, E.S., Wang, C. and Doyle, F.J. (2000). Model based control of a four-tank system, Computers & Chemical Engineering 24(2): 1503-1509.

[017] Ghaffari, A. and Yazdanpanah, M. (2008). Nonlinear sliding surfaces: Computing and existence of solution, International Conference on Control, Automation and Systems, ICCAS 2008, Seoul, Korea, pp. 1610-1615.

[018] Hung, J.Y., Gao, W. and Hung, J.C. (1993). Variable structure control: A survey, IEEE Transactions on Industrial Electronics 40(1): 2-22.

[019] Johansson, K. and Nunes, J. (1998). A multivariable laboratory process with an adjustable zero, Proceedings of the 1998 American Control Conference, Philadelphia, PA, USA, Vol. 4, pp. 2045-2049.

[020] Kim, K.-S., Park, Y. and Oh, S.-H. (2000). Designing robust sliding hyperplanes for parametric uncertain systems: A Riccati approach, Automatica 36(7): 1041-1048. | Zbl 0955.93005

[021] Lin, Y., Shi, Y. and Burton, R. (2013). Modeling and robust discrete-time sliding-mode control design for a fluid power electrohydraulic actuator (EHA) system, IEEE/ASME Transactions on Mechatronics 18(1): 1-10. (1985). General conditions for the existence

[022] Milosavljević, Č. of a quasi-sliding mode on the switching hyperplane in discrete variable structure systems, Automation and Remote Control 46(3): 307-314.

[023] Mondal, S., Gokul, T. and Mahanta, C. (2012). Chattering free sliding mode controller for mismatched uncertain system, 7th IEEE International Conference on Industrial and Information Systems (ICIIS), Chennal, India, pp. 1-6.

[024] Nadzinski, G., Vladev, G. and Zheng, Y. (2012). A design of discrete-time SMC for nonlinear systems based on fuzzy T-S model, 6th IEEE International Conference on Intelligent Systems (IS), Sofia, Bulgaria, pp. 317-324.

[025] Pai, M.-C. (2008). Discrete-time output feedback sliding mode control for uncertain systems, Journal of Marine Science and Technology 16(4): 295-300.

[026] Perruquetti, W. and Barbot, J.-P. (2002). Sliding Mode Control in Engineering, Marcel Dekker, New York, NY.

[027] Potts, R. and Yu, X. (1991). Discrete variable structure system with pseudo-sliding mode, Journal of the Australian Mathematical Society B: Applied Mathematics 32(04): 365-376. | Zbl 0733.34021

[028] Qu, S., Xia, X. and Zhang, J. (2014). Dynamics of discrete-time sliding-mode-control uncertain systems with a disturbance compensator, IEEE Transactions on Industrial Electronics 61(7): 3502-3510.

[029] Rui, D. and Dong-wei, S. (2011). Optimal sliding mode design for nonlinear discrete-time systems, 30th Chinese Control Conference (CCC), Yantai, China, pp. 738-742.

[030] Sira-Ramírez, H. (1986). Variable structure control of nonlinear systems through simplified uncertain models, 25th IEEE Conference on Decision and Control, Athens, Greece, pp. 2037-2041.

[031] Sira-Ramírez, H. (1991). Non-linear discrete variable structure systems in quasi-sliding mode, International Journal of Control 54(5): 1171-1187. | Zbl 0738.93049

[032] Soroush, M. and Kravaris, C. (1992). Discrete-time nonlinear controller synthesis by input/output linearization, AIChE Journal 38(12): 1923-1945.

[033] Spurgeon, S. and Davies, R. (1993). A nonlinear design approach for sliding mode control systems, Proceedings of the 32nd IEEE Conference on Decision and Control, San Antonio, TX, USA, Vol. 2, pp. 1440-1445.

[034] Spurgeon, S. and Pugh, A. (1991). On output deadbeat control of discrete-time multivariable systems, IEEE Transactions on Automatic Control 36(7): 894-896.

[035] Su, W.C., Drakunov, S.V. and Özgüner, Ü. (1994). Constructing discontinuity planes for variable structure systems - A Lyapunov approach, American Control Conference, Baltimore, MD, USA, Vol. 1, pp. 1169-1173.

[036] Su, W.-C., Drakunov, S.V. and Özgüner, Ü. (1996). Constructing discontinuity surfaces for variable structure systems: A Lyapunov approach, Automatica 32(6): 925-928. | Zbl 0854.93020

[037] Tapia, A., Márquez, R., Bernal, M. and Cortez, J. (2014). Sliding subspace design based on linear matrix inequalities, Kybernetika 50(3): 436-449. | Zbl 1298.93110

[038] Utkin, V. (1977). Variable structure systems with sliding modes, IEEE Transactions on Automatic Control 22(2): 212-222. | Zbl 0382.93036

[039] Utkin, V., Guldner, J. and Shi, J. (1999). Sliding Mode Control in Electro-mechanical Systems, CRC Press, Boca Raton, FL.

[040] Utkin, V. and Yang, K. (1978). Methods for construction of discontinuity planes in multidimensional variable structure systems, Automation and Remote Control 39(6): 1466-1470. | Zbl 0419.93045

[041] Wang, Y., Xia, Z., Jiang, Z. and Xie, G. (2011). A quasi-sliding mode variable structure control algorithm for discrete-time and time-delay systems, Chinese Control and Decision Conference, Mianyang, China, pp. 107-110.

[042] Yadav, N. K. and Singh, R. (2012). Robust discrete-time nonlinear sliding mode controller with plant uncertainties, International Journal of Engineering, Science and Technology 4(1): 38-45.

[043] Yu, S., Yu, X. and Qian, W. (2000). Time delayed discrete variable structure control with quasi-sliding modes, in X. Yu and J.-X. Xu (Eds.), Advances in Variable Structure Systems: Analysis, Integration and Applications, World Scientific, Singapore, pp. 84-92.

[044] Zhang, X., Wang, P., Yan, M. and Ju, Y. (2010). Discrete-time sliding mode control of nonlinear time-delay systems based on T-S fuzzy model, International Conference on Intelligent Control and Information Processing (ICICIP), Dalian, China, pp. 304-309.