Homogeneity approach to high-order sliding mode design. (English) Zbl 1093.93003

Summary: It is shown that a general uncertain single-input-single-output regulation problem is solvable only by means of discontinuous control laws, giving rise to the so-called high-order sliding modes. The homogeneity properties of the corresponding controllers yield a number of practically important features. In particular, finite-time convergence is proved, and asymptotic accuracy is calculated in a very general way in the presence of input noises, discrete measurements and switching delays. A robust homogeneous differentiator is included in the control structure thus yielding robust output-feedback controllers with finite-time convergence. It is demonstrated that homogeneity features significantly simplify the design and investigation of a new family of high-order sliding-mode controllers. Finally, simulation results are presented.


93B12 Variable structure systems
93B50 Synthesis problems
Full Text: DOI


[1] Atassi, A.N.; Khalil, H.K., Separation results for the stabilization of nonlinear systems using different high-gain observer designs, Systems & control letters, 39, 183-191, (2000) · Zbl 0948.93007
[2] Bacciotti, A., & Rosier, L. (2001). Liapunov functions and stability in control theory. Lecture Notes in Control and Information Science, Vol. 267. London: Springer. · Zbl 0968.93004
[3] Bartolini, G.; Ferrara, A.; Punta, E., Multi-input second-order sliding-mode hybrid control of constrained manipulators, Dynamics and control, 10, 277-296, (2000) · Zbl 0980.93055
[4] Bartolini, G.; Ferrara, A.; Usai, E., Chattering avoidance by second-order sliding mode control, IEEE transactions on automatic control, 43, 2, 241-246, (1998) · Zbl 0904.93003
[5] Bartolini, G.; Pisano, A.; Punta, E.; Usai, E., A survey of applications of second-order sliding mode control to mechanical systems, International journal of control, 76, 9/10, 875-892, (2003) · Zbl 1070.93011
[6] Bartolini, G.; Pisano, A.; Usai, E., First and second derivative estimation by sliding mode technique, Journal of signal processing, 4, 2, 167-176, (2000)
[7] Edwards, C.; Spurgeon, S.K., Sliding mode controltheory and applications, (1998), Taylor & Francis London
[8] Emelyanov, S.V.; Korovin, S.K.; Levantovsky, L.V., Higher order sliding regimes in the binary control systems, Soviet physics, doklady, 31, 4, 291-293, (1986) · Zbl 0612.93035
[9] Ferrara, A.; Giacomini, L., Control of a class of mechanical systems with uncertainties via a constructive adaptive/second order VSC approach, Journal of dynamic systems-T ASME, 122, 1, 33-39, (2000)
[10] Filippov, A.F., Differential equations with discontinuous right-hand side, (1988), Kluwer Dordrecht, The Netherlands · Zbl 0664.34001
[11] Floquet, T.; Barbot, J.-P.; Perruquetti, W., Higher-order sliding mode stabilization for a class of nonholonomic perturbed systems, Automatica, 39, 1077-1083, (2003) · Zbl 1038.93063
[12] Fridman, L., An averaging approach to chattering, IEEE transactions on automatic control, 46, 1260-1265, (2001) · Zbl 1007.93010
[13] Fridman, L., Chattering analysis in sliding mode systems with inertial sensors, International journal of control, 76, 9/10, 906-912, (2003) · Zbl 1062.93011
[14] Furuta, K.; Pan, Y., Variable structure control with sliding sector, Automatica, 36, 211-228, (2000) · Zbl 0938.93011
[15] Isidori, A., Nonlinear control systems, (1989), Springer New York · Zbl 0714.93021
[16] Kobayashi, S., Suzuki, S., & Furuta, K. (2002). Adaptive vs. differentiator. Advances in variable structure systems. Proceedings of the seventh VSS Workshop, July 2002, Sarajevo.
[17] Krupp, D., Shkolnikov, I. A., & Shtessel, Y. B. (2000). 2-sliding mode control for nonlinear plants with parametric and dynamic uncertainties. Proceedings of AIAA guidance, navigation, and control conference, Denver, CO, AIAA paper No. 2000-3965.
[18] Levant, A. (Levantovsky, L. V.) (1993). Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58(6), 1247-1263. · Zbl 0789.93063
[19] Levant, A., Robust exact differentiation via sliding mode technique, Automatica, 34, 3, 379-384, (1998) · Zbl 0915.93013
[20] Levant, A., Universal SISO sliding-mode controllers with finite-time convergence, IEEE transactions on automatic control, 46, 9, 1447-1451, (2001) · Zbl 1001.93011
[21] Levant, A. (2002). Construction principles of output-feedback 2-sliding mode design. Proceedings of the 42nd IEEE conference on decision and control, Las-Vegas, Nevada, December 10-13, 2002.
[22] Levant, A., Higher-order sliding modes, differentiation and output-feedback control, International journal of control, 76, 9/10, 924-941, (2003) · Zbl 1049.93014
[23] Levant, A. (2003b). Quasi-continuous high-order sliding mode controllers. Proceedings of the 43rd IEEE conference on decision and control, Maui, Hawaii, December 9-12, 2003. · Zbl 1365.93072
[24] Levant, A.; Pridor, A.; Gitizadeh, R.; Yaesh, I.; Ben-Asher, J.Z., Aircraft pitch control via second order sliding technique, Journal of guidance, control and dynamics, 23, 4, 586-594, (2000)
[25] Man, Z.; Paplinski, A.P.; Wu, H.R., A robust MIMO terminal sliding mode control for rigid robotic manipulators, IEEE transactions on automatic control, 39, 12, 2464-2468, (1994) · Zbl 0825.93551
[26] Orlov, Y.; Aguilar, L.; Cadiou, J.C., Switched chattering control vs. backlash/friction phenomena in electrical servo-motors, International journal of control, 76, 9/10, 959-967, (2003) · Zbl 1040.93529
[27] Saks, S., Theory of the integral, (1964), Dover Publ. Inc New York
[28] Shkolnikov, I.A.; Shtessel, Y.B., Tracking in a class of nonminimum-phase systems with nonlinear internal dynamics via sliding mode control using method of system center, Automatica, 38, 5, 837-842, (2002) · Zbl 1001.93008
[29] Shkolnikov, I. A., Shtessel, Y. B., Lianos, D., & Thies, A. T. (2000). Robust missile autopilot design via high-order sliding mode control. Proceedings of AIAA guidance, navigation, and control conference, Denver, CO, AIAA paper No. 2000-3968.
[30] Shtessel, Y.B.; Shkolnikov, I.A., Aeronautical and space vehicle control in dynamic sliding manifolds, International journal of control, 76, 9/10, 1000-1017, (2003) · Zbl 1076.93523
[31] Sira-Ramírez, H., On the dynamical sliding mode control of nonlinear systems, International journal of control, 57, 5, 1039-1061, (1993) · Zbl 0772.93040
[32] Sira-Ramírez, H., Dynamic second-order sliding mode control of the hovercraft vessel, IEEE transactions on control systems technology, 10, 6, 860-865, (2002)
[33] Slotine, J.-J.E.; Li, W., Applied nonlinear control, (1991), Prentice-Hall, Inc London
[34] Spurgeon, S., Goh, K. B., & Jones, N. B. (2002). An application of higher order sliding modes to the control of a diesel generator set (genset), Advances in Variable Structure Systems. Proceedings of the seventh VSS Workshop, July 2002, Sarajevo.
[35] Utkin, V.I., Sliding modes in optimization and control problems, (1992), Springer New York · Zbl 0748.93044
[36] Yu, X.; Xu, J.X., An adaptive signal derivative estimator, Electronic letters, 32, 16, (1996)
[37] Zinober, A. S. I. (Ed.) (1994). Variable structure and Lyapunov control. Berlin: Springer. · Zbl 0782.00041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.