Control por modos deslizantes de orden superior basado en funciones de Lyapunov
DOI:
https://doi.org/10.4995/riai.2022.17013Palabras clave:
Control integral, Modos deslizantes, Control de estructura variable, Métodos de Lyapunov, Observadores no linealesResumen
En este trabajo se presenta una panorámica del desarrollo de los métodos básicos de análisis y diseño de controladores y observadores por modos deslizantes de orden superior. Inicialmente se describen los controladores por retroalimentación de estados con una ley de control discontinua, que generan un modo deslizante de cualquier orden. Posteriormente se presenta una nueva clase de algoritmos por modos deslizantes de orden superior, que consisten en una retroalimentación de estados continua y una acción de control integral discontinua. Se describen también observadores por modos deslizantes, que estiman los estados del sistema en tiempo finito, y que permiten obtener un controlador por retroalimentación de la salida. Todos los diseños presentados se basan en el uso de funciones de Lyapunov (explícitas), que constituyen una contribución importante del grupo de trabajo de los autores en la Universidad Nacional Autónoma de México. La presentación es tutorial y solo se dan los resultados, dejando a un lado la formalización rigurosa y las pruebas matemáticas. Para ello se refiere al lector a la literatura pertinente. Se ilustran los resultados mediante simulaciones y la validación experimental en un sistema de levitación magnética.
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Andrieu, V., Praly, L., Astolfi, A., 2008. Homogeneous approximation, recur- sive observer design and output feedback. SIAM J. Control Optim. 47 (4), 1814-1850. https://doi.org/10.1137/060675861
Bacciotti, A., Rosier, L., 2005. Liapunov functions and stability in control theory, 2nd Edition. Springer-Verlag, New York. https://doi.org/10.1007/b139028
Bartolini, G., Ferrara, A., Usai, E., 1998. Chattering avoidance by second-order sliding mode control. IEEE Transactions on Automatic Control 43 (2), 241- 246. https://doi.org/10.1109/9.661074
Bartolini, G., Pisano, A., Usai, E., 2000. First and second derivative estimation by sliding mode technique. Journal of Signal Processing 4 (2), 167 - 176.
Bejarano, F., Fridman, L., 2010. High order sliding mode observer for linear systems with unbounded unknown inputs. International Journal of Control 83 (9), 1920 - 1929. https://doi.org/10.1080/00207179.2010.501386
Bernuau, E., Efimov, D., Perruquetti, W., Polyakov, A., Jul. 2013a. On an extension of homogeneity notion for differential inclusions. In: European Control Conference. Zurich, Switzerland. https://doi.org/10.23919/ECC.2013.6669525
Bernuau, E., Efimov, D., Perruquetti, W., Polyakov, A., 2014. On homogeneity and its application in sliding mode control. Journal of the Franklin Institute 351 (4), 1816-1901. https://doi.org/10.1016/j.jfranklin.2014.01.007
Bernuau, E., Efimov, D., Perruquetti, W., Polyakov, A., 2016. Homogeneity of differential inclusions. Control, Robotics & Sensors. Institution of Engineering and Technology, pp. 103-118.
Bernuau, E., Polyakov, A., Efimov, D., Perruquetti, W., 2013b. Verification of ISS, iISS and IOSS properties applying weighted homogeneity. Systems & Control Letters 62 (12), 1159-1167. https://doi.org/10.1016/j.sysconle.2013.09.004
Bhat, S., Bernstein, D., 2005. Geometric homogeneity with applications to finite-time stability. Mathematics of Control, Signals, and Systems 17 (2), 101-127. https://doi.org/10.1007/s00498-005-0151-x
Boiko, I., 2009. Discontinuous control systems: frequency-domain analysis and design. Birkha ̈user, Boston.
Cruz-Zavala, E., Moreno, J., 2014a. Improved convergence rate of discontinuous finite-time controllers. Vol. 19. pp. 8636-8641. https://doi.org/10.3182/20140824-6-ZA-1003.02496
Cruz-Zavala, E., Moreno, J. A., July 2014b. A new class of fast finite-time discontinuous controllers. In: 13th IEEE Workshop on Variable Structure Systems (VSS14). Nantes, France, pp. 1-6. https://doi.org/10.1109/VSS.2014.6881097
Cruz-Zavala, E., Moreno, J., 2017. Homogeneous high order sliding mode design: A Lyapunov approach. Automatica 80, 232-238. https://doi.org/10.1016/j.automatica.2017.02.039
Cruz-Zavala, E., Moreno, J., Fridman, L., 2010. Uniform robust exact differen-tiator. In: Proceedings of the IEEE Conference on Decision and Control. pp.102-107. https://doi.org/10.1109/CDC.2010.5717345
Cruz-Zavala, E., Moreno, J., Fridman, L., 2011a. Adaptive gains super-twistingalgorithm for systems with growing perturbations. In: IFAC Proceedings Vo-lumes (IFAC-PapersOnline). Vol. 18. pp. 3039-3044. https://doi.org/10.3182/20110828-6-IT-1002.03529
Cruz-Zavala, E., Moreno, J., Fridman, L., 2011b. Second-order uniform exactsliding mode control with uniform sliding surface. In: Proceedings of theIEEE Conference on Decision and Control. pp. 4616-4621. https://doi.org/10.1109/CDC.2011.6160493
Cruz-Zavala, E., Moreno, J., Fridman, L., 2012. Uniform sliding mode controllers and uniform sliding surfaces. IMA Journal of Mathematical Control andInformation 29 (4),491-505. https://doi.org/10.1093/imamci/dns005
Cruz-Zavala, E., Moreno, J., Fridman, L., 2013. Fast second-order sliding mo-de control design based on Lyapunov function. In: Proceedings of the IEEEConference on Decision and Control. pp. 2858-2863. https://doi.org/10.1109/CDC.2013.6760317
Cruz-Zavala, E., Moreno, J. A., 2016a. Lyapunov approach to Higher-OrderSliding Mode design. In: Fridman, L., Barbot, J.-P., Plestan, F. (Eds.), Re-cent Trends in Sliding Mode Control. The Institution of Engineering andTechnology (IET), London, Ch. 1.1, pp. 3-28.
Cruz-Zavala, E., Moreno, J. A., 2016b. Lyapunov functions for continuous and discontinuous differentiators. IFAC-PapersOnLine 49 (18), 660-665, 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016. https://doi.org/10.1016/j.ifacol.2016.10.241
Cruz-Zavala, E., Moreno, J. A., Jul. 2019. Levant's arbitrary order exact differentiator: a Lyapunov approach. IEEE Transactions on Automatic Control 64 (7), 3034-3039. https://doi.org/10.1109/TAC.2018.2874721
Davila, A., Moreno, J., Fridman, L., 2010. Variable gains super-twisting al-gorithm: A Lyapunov based design. In: Proceedings of the 2010 AmericanControl Conference, ACC 2010. pp. 968-973. https://doi.org/10.1109/ACC.2010.5530461
Davila, A., Moreno, J. A., Fridman, L., 2009. Optimal Lyapunov function selection for reaching time estimation of super twisting algorithm. In: Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference. pp. 8405-8410. https://doi.org/10.1109/CDC.2009.5400466
Davila, J., Fridman, L., Levant, A., 2005. Second-order sliding-mode observer for mechanical systems. IEEE transactions on automatic control 50 (11), 1785-1789. https://doi.org/10.1109/TAC.2005.858636
Deimling, K., 1992. Multivalued Differential Equations. Walter de gruyter. Berlin. https://doi.org/10.1515/9783110874228
Ding, S., Levant, A., Li, S., Dec. 2015. New families of high-order sliding-mode controllers. In: 54th IEEE Conference on Decision and Control. Osaka, Japan, pp. 4752-4757. https://doi.org/10.1109/CDC.2015.7402960
Ding, S., Levant, A., Li, S., 2016. Simple homogeneous sliding-mode controller. Automatica 67 (5), 22-32. https://doi.org/10.1016/j.automatica.2016.01.017
Dorel, L., Levant, A., Dec 2008. On chattering-free sliding-mode control. In: 2008 47th IEEE Conference on Decision and Control. pp. 2196-2201. https://doi.org/10.1109/CDC.2008.4739126
Efimov, D., Fridman, L., 2011. A hybrid robust non-homogeneous finite-time differentiator. IEEE Trans. on Aut. Control 56, 1213-1219. https://doi.org/10.1109/TAC.2011.2108590
Filippov, A., 1988. Differential equations with discontinuous righthand side. Kluwer. Dordrecht, The Netherlands. https://doi.org/10.1007/978-94-015-7793-9
Floquet, T., Barbot, J. P., 2007. Super twisting algorithm-based step-by-step sliding mode observers for nonlinear systems with unknown inputs. International Journal of Systems Science 38 (10), 803-815. https://doi.org/10.1080/00207720701409330
Freeman, R., Kokotovic, P., 1996. Robust Nonlinear control Design: State space and Lyapunov Techniques. Modern Birkha ̈user Classics, Boston.
Fridman, L., Levant, A., 2002. Higher-Order Sliding Modes. In: Perruqueti, W.,Barbot, J. (Eds.), Sliding Mode Control in Engineering. Marceel Dekker,Inc., New York, Ch. 3, pp. 53-102. https://doi.org/10.1201/9780203910856.ch3
Fridman, L., Moreno, J. A., Bandyopadhyay, B., Kamal, S., Chalanga, A., 2015. Continuous nested algorithms: The fifth generation of sliding mode controllers. In:Yu,X.,O ̈nder Efe, M.(Eds.), Recent Advances in Sliding Modes: From Control to Intelligent Mechatronics. Vol. 24. Springer International Publishing, Cham, pp. 5-35. https://doi.org/10.1007/978-3-319-18290-2_2
Hahn, W., 1967. Stability of Motion. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-50085-5
Hermes, H., 1991. Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, in, Differential Equations, Stability and Control (S. Elaydi, ed.). Vol. 127 of Lecture Notes in Pure and Applied Math. Marcel Dekker, Inc., NY, pp. 249-260.
Hestenes, M. R., 1966. Calculus of variations and optimal control theory. John Wiley & Sons, New York.
Isidori, A., 1995. Nonlinear control systems. Springer Verlag, Berlin. https://doi.org/10.1007/978-1-84628-615-5
Isidori, A., 1999. Nonlinear control systems II. Springer Verlag, London. https://doi.org/10.1007/978-1-4471-0549-7
Kamal, S., Chalanga, A., Moreno, J. A., Fridman, L., Bandyopadhyay, B., June 2014. Higher order super-twisting algorithm. In: 2014 13th International Workshop on Variable Structure Systems (VSS). pp. 1-5. https://doi.org/10.1109/VSS.2014.6881129
Kamal, S., Moreno, J., Chalanga, A., Bandyopadhyay, B., Fridman, L., 2016. Continuous terminal sliding-mode controller. Automatica 69, 308-314. https://doi.org/10.1016/j.automatica.2016.02.001
Khalil, H., 2002. Nonlinear Systems, 3rd Edition. Prentice-Hall, New Jersey.
Kobayashi, S., Suzuki, S., Furuta, K., 2007. Frequency characteristics of Levant's differentiator and adaptive sliding mode differentiator. International Journal of Systems Science 38 (10), 825 - 832. https://doi.org/10.1080/00207720701631198
Kochalummoottil, J., Shtessel, Y., Moreno, J., Fridman, L., 2011. Adaptive twist sliding mode control: A lyapunov design. pp. 7623-7628. https://doi.org/10.1109/CDC.2011.6160982
Laghrouche, S., Harmouche, M., Chitour, Y., July 2017. Higher order super-twisting for perturbed chains of integrators. IEEE Transactions on Automatic Control 62 (7), 3588-3593. https://doi.org/10.1109/TAC.2017.2670918
Laghrouche, S., Harmouche, M., Chitour, Y., Obeid, H., Fridman, L. M., 2021. Barrier function-based adaptive higher order sliding mode controllers. Automatica 123, 109355. https://doi.org/10.1016/j.automatica.2020.109355
Levant, A., 1993. Sliding order and sliding accuracy in Sliding Mode Control. International Journal of Control 58 (6), 1247-1263. https://doi.org/10.1080/00207179308923053
Levant, A., 1998. Robust exact differentiation via sliding mode technique. Automatica 34 (3), 379-384. https://doi.org/10.1016/S0005-1098(97)00209-4
Levant, A., 2001. Universal single-input single-output (siso) sliding-mode controllers with finite-time. IEEE Trans. Autom. Control 46 (9), 1447-1451. https://doi.org/10.1109/9.948475
Levant, A., 2003. High-order sliding modes: differentiation and output-feedback control. Int. J. Control 76 (9), 924-941. https://doi.org/10.1080/0020717031000099029
Levant, A., 2005a. Homogeneity approach to high-order sliding mode design. Automatica 41, 823-830. https://doi.org/10.1016/j.automatica.2004.11.029
Levant, A., 2005b. Quasi-continuous high-order sliding-mode controllers. IEEE Trans. Autom. Control 50 (11), 1812-1816. https://doi.org/10.1109/TAC.2005.858646
Levant, A., April 2007. Principles of 2-Sliding Mode design. Automatica 43 (4), 576-586. https://doi.org/10.1016/j.automatica.2006.10.008
Levant, A., Alelishvili, L., July 2007. Integral high-order sliding modes. IEEE Transactions on Automatic Control 52 (7), 1278-1282. https://doi.org/10.1109/TAC.2007.900830
Levant, A., Livne, M., 2016. Weighted homogeneity and robustness of sliding mode control. Automatica 72, 186 - 193. https://doi.org/10.1016/j.automatica.2016.06.014
Levant, A., Livne, M., 2018. Globally convergent differentiators with variable gains. Int. J. of Control 91 (9), 1994-2008. https://doi.org/10.1080/00207179.2018.1448115
Levant, A., Michael, A., 2009. Adjustment of high-order sliding-mode contro-llers. International Journal of Robust and Nonlinear Control 19 (15), 1657-1672. https://doi.org/10.1002/rnc.1397
Mendoza-Avila, J., Moreno, J. A., Fridman, L. An idea for Lyapunov functiondesign for arbitrary order continuous twisting algorithms. In: 2017 IEEE56th Annual Conference on Decision and Control (CDC). Dec 2017. pp.5426-5431. https://doi.org/10.1109/CDC.2017.8264462
Mendoza-Avila, J., Moreno, J. A., Fridman, L. M., 2020. Continuous twisting algorithm for third-order systems. IEEE Transactions on Automatic Control 65 (7), 2814-2825. https://doi.org/10.1109/TAC.2019.2932690
Mercado-Uribe, A., Moreno, J., 2018a. Full and partial state discontinuous integral control. IFAC-PapersOnLine 51 (13), 573-578, 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems MICNON 2018. https://doi.org/10.1016/j.ifacol.2018.07.341
Mercado-Uribe, A., Moreno, J., July 2018b. Output feedback discontinuous integral controller for siso nonlinear systems. In: 2018 15th International Workshop on Variable Structure Systems (VSS). Vol. 2018-July. pp. 114- 119. https://doi.org/10.1109/VSS.2018.8460305
Mercado-Uribe, A., Moreno, J. A., 2020a. Homogeneous integral controllers for a magnetic suspension system. Control Engineering Practice 97, 104325. https://doi.org/10.1016/j.conengprac.2020.104325
Mercado-Uribe, J. A., Moreno, J. A., 2020b. Discontinuous integral action for arbitrary relative degree in sliding-mode control. Automatica 118, 109018. https://doi.org/10.1016/j.automatica.2020.109018
Moreno, J. A., 2009. A linear framework for the robust stability analysis of ageneralized super-twisting algorithm. In: 2009 6th International Conferenceon Electrical Engineering, Computing Science and Automatic Control, CCE2009. pp. 12-17. https://doi.org/10.1109/ICEEE.2009.5393477
Moreno, J. A., 2010. Lyapunov analysis of non homogeneous Super-Twistingalgorithms. In: Proceedings of the 2010 11th International Workshop on Va-riable Structure Systems, VSS 2010. pp. 534-539. https://doi.org/10.1109/VSS.2010.5544672
Moreno, J. A., 2011. Lyapunov approach for analysis and design of second order sliding mode algorithms. In: Fridman, L., Moreno, J., Iriarte, R. (Eds.), Sliding Modes after the first decade of the 21st Century. LNCIS, 412. Springer-Verlag, Berlin - Heidelberg, pp. 113-150. https://doi.org/10.1007/978-3-642-22164-4_4
Moreno, J. A., 2013. On discontinuous observers for second order systems: Pro-perties, analysis and design. In: Bandyopadhyay, B., Janardhanan, S., Spur-geon, S. K. (Eds.), Advances in Sliding Mode Control - Concepts, Theoryand Implementation. LNCIS, 440. Springer-Verlag, Berlin - Heidelberg, pp.243-265. https://doi.org/10.1007/978-3-642-36986-5_12
Moreno, J. A., June 2016. Discontinuous integral control for mechanical sys-tems. In: 2016 14th International Workshop on Variable Structure Systems(VSS). 2016. pp. 142-147. https://doi.org/10.1109/VSS.2016.7506906
Moreno, J. A., 2018a. Discontinuous integral control for systems with relativedegree two. In: Clempner, J., (Eds.), W. Y. (Eds.), New Perspectives and Ap-plications of Modern Control Theory: In Honor of A. S. Poznyak. SpringerInternational Publishing, pp. 187-218. https://doi.org/10.1007/978-3-319-62464-8_8
Moreno, J. A., 2018b. Exact differentiator with varying gains. InternationalJournal of Control 91 (9), 1983-1993. https://doi.org/10.1080/00207179.2017.1390262
Moreno, J. A., 2018c. Lyapunov-based design of homogeneous high-order sli-ding modes. In: Li, S., Yu, X., Fridman, L., Man, Z., Wang, X. (Eds.), Ad-vances in Variable Structure Systems and Sliding Mode Control-Theory andApplications. Vol. 115. Springer International Publishing, Cham, pp. 3-38. https://doi.org/10.1007/978-3-319-62896-7_1
Moreno, J. A., 2020. Asymptotic tracking and disturbance rejection of time-varying signals with a discontinuous PID controller. Journal of Process Control 87, 79-90. https://doi.org/10.1016/j.jprocont.2020.01.006
Moreno,J.A., Cruz-Zavala, E., Mercado-Uribe, A., 2020. Discontinuous Integral Control for Systems with Arbitrary Relative Degree. Springer International Publishing, Cham, pp. 29-69. https://doi.org/10.1007/978-3-030-36621-6_2
Moreno, J. A., Osorio, M., Dec. 2008. A Lyapunov approach to second-ordersliding mode controllers and observers. In: 47th IEEE Conference on Deci-sion and Control. Cancun, Mexico, pp. 2856-2861. https://doi.org/10.1109/CDC.2008.4739356
Moreno, J., Osorio, M., 2012. Strict Lyapunov functions for the Super-Twisting algorithm. IEEE Transactions on Automatic Control 57 (4), 1035-1040. https://doi.org/10.1109/TAC.2012.2186179
Moulay, E., Perruquetti, W., 2006. Finite time stability and stabilization of a class of continuous systems. Journal of Mathematical Analysis and Applications 323 (2), 1430 - 1443. https://doi.org/10.1016/j.jmaa.2005.11.046
Nakamura, H., Yamashita, Y., Nishitani, H., 2002. Smooth Lyapunov functions for Homogeneous Differential Inclusions. In: Proc. 41st SICE Annual Conference. Vol. 3. pp. 1974-1979.
Nakamura, N., Nakamura, H., Yamashita, Y., Nishitani, H., 2009. Homogeneous stabilization for input affine homogeneous systems. IEEE Trans. Autom. Control. 54 (9), 2271-2275. https://doi.org/10.1109/TAC.2009.2026865
Orlov, Y., Dec 2003. Finite time stability of homogeneous switched systems. In: Decision and Control, 2003. Proceedings. 42nd IEEE Conference on. Vol. 4. pp. 4271-4276.
Orlov, Y. V., 2009. Discontinuous systems: Lyapunov analysis and robust synthesis under uncertainty conditions. Springer-Verlag, Berlin-Heidelberg.
Perez-Ventura, U., Fridman, L., 2019a. Design of super-twisting control gains: A describing function based methodology. Automatica 99, 175 - 180. https://doi.org/10.1016/j.automatica.2018.10.023
Perez-Ventura, U., Fridman, L., 2019b. When is it reasonable to implement the discontinuous sliding-mode controllers instead of the continuous ones? frequency domain criteria. International Journal of Robust and Nonlinear Control 29 (3), 810-828. https://doi.org/10.1002/rnc.4347
Polyakov, A., Efimov, D., Perruquetti, W., 2015. Finite-time and fixed-time stabilization: Implicit Lyapunov function method. Automatica 51 (1), 332-340. https://doi.org/10.1016/j.automatica.2014.10.082
Polyakov, A., Efimov, D., Perruquetti, W., 2016. Robust stabilization of MIMO systems in finite/fixed time. Int. J. Robust Nonlinear Control 26 (1), 69-90. https://doi.org/10.1002/rnc.3297
Polyakov, A., Poznyak, A., 2009a. Lyapunov function design for finite-time convergence analysis: "twisting" controller for second-order sliding mode realization. Automatica 45 (2), 444-448. https://doi.org/10.1016/j.automatica.2008.07.013
Polyakov, A., Poznyak, A., 2009b. Reaching time estimation for "super-twisting" second-order sliding mode controller via Lyapunov function designing. IEEE Trans. Autom. Control 54 (8), 1951-1955. https://doi.org/10.1109/TAC.2009.2023781
Polyakov, A., Poznyak, A., 2012. Unified Lyapunov function for a finite-time stability analysis of relay second-order sliding mode control systems. IMA J. Math. Control & Information 29 (4), 529-550. https://doi.org/10.1093/imamci/dns007
Rosier, L., 1992. Homogeneous Lyapunov function for homogeneous continuous vector field. Systems & Control Letters 19, 467-473. https://doi.org/10.1016/0167-6911(92)90078-7
Sanchez, T., Moreno, J. Construction of Lyapunov functions for a class of hig-her order sliding modes algorithms. In: Proceedings of the IEEE Conferenceon Decision and Control. pp. 6454-6459
Sanchez, T., Moreno, J., 2013. On a sign controller for the triple integrator. In: Proceedings of the IEEE Conference on Decision and Control. pp. 3566-3571. https://doi.org/10.1109/CDC.2013.6760431
Sanchez, T., Moreno, J., 2014. Lyapunov functions for Twisting and Terminal controllers. In: 13th IEEE Workshop on Variable Structure Systems (VSS14). https://doi.org/10.1109/VSS.2014.6881136
Sanchez, T., Moreno, J. A., 2019. Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach. International Journal of Robust and Nonlinear Control 29 (3), 661-681. https://doi.org/10.1002/rnc.4274
Santiesteban, R., Fridman, L., Moreno, J., 2010. Finite-time convergence analy-sis for "twisting" controller via a strict Lyapunov function. In: Proceedingsof the 2010 11th International Workshop on Variable Structure Systems,VSS 2010. pp. 1-6. https://doi.org/10.1109/VSS.2010.5545144
Shtessel, Y., Edwards, C., Fridman, L., Levant, A., 2014. Sliding Mode Control and Observation. Birkha ̈user, Springer, New York. https://doi.org/10.1007/978-0-8176-4893-0
Shtessel, Y. B., Shkolnikov, I. A., 2003. Aeronautical and space vehicle control in dynamic sliding manifolds. International Journal of Control 76 (9/10), 1000-1017. https://doi.org/10.1080/0020717031000099065
Torres-Gonzalez, V., Fridman, L., Moreno, J. Continuous twisting algorithm.In: 2015 54th IEEE Conference on Decision and Control (CDC). pp. 5397-5401. https://doi.org/10.1109/CDC.2015.7403064
Torres-Gonzalez, V., Sanchez, T., Fridman, L. M., Moreno, J. A., 2017. Design of continuous twisting algorithm. Automatica 80, 119 - 126. https://doi.org/10.1016/j.automatica.2017.02.035
Utkin, V., Guldner, J., Shi, J., 2009. Sliding Mode Control in Electro-Mechanical Systems, 2nd Edition. CRC Press, Taylor & Francis, London, UK.
Vasiljevic, L. K., Khalil, H. K., 2008. Error bounds in differentiation of noisy signals by high-gain observers. Systems and Control Letters 57, 856-862. https://doi.org/10.1016/j.sysconle.2008.03.018
Zamora, C., Moreno, J. A., Kamal, S., Oct 2013. Control integral discontinuo para sistemas mecanicos. In: 2013 Congreso Nacional de Control Automatico (CNCA AMCA). Asociacion de Mexico de Control Automatico (AM-CA), Ensenada, Baja California, Mexico, pp. 11-16.
Zubov, V. I., 1964. Methods of A. M. Lyapunov and their applications. Groningen: P. Noordhoff Limited.
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