Abstract
In this chapter we further develop the ideas of Chap. 4 where second-order sliding modes were formulated. As we have seen, second-order sliding modes make the sliding variables vanish in finite time, when the relative degree of the variable equals two, and are able to solve the same problem by means of continuous control, if the relative degree is one. This helps to remove dangerous high-energy vibrations (the dangerous types of chattering). So-called higher-order sliding modes (HOSMs) solve these problems for arbitrary relative degrees. The realization of the scheme requires more information: usually one needs to calculate or measure a number of successive time derivatives of the sliding variables. However that problem is also solved within a similar framework. As a result, arbitrary-order exact robust differentiators are developed, having their own significance in terms of general observation theory. In particular, tracking problems are solved in finite time and with ideal accuracy, by means of smooth control, if the output relative degree is known. The accuracy remains high in the presence of small noises, switching inaccuracies and delays, etc.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
For details, see Sect. 2.1.1.
- 2.
For details see [82].
References
Atassi, A.N., Khalil, H.K.: Separation results for the stabilization of nonlinear systems using different high-gain observer designs. Syst. Control Lett. 39(3), 183–191 (2000)
Baccioti, A., Rosier, L.: Lyapunov Functions and Stability in Control Theory, 2nd edn. Springer, New York (2005)
Bartolini, G., Ferrara, A., Usai, E.: Chattering avoidance by second-order sliding mode control. IEEE Trans. Automat. Contr. 43(2), 241–246 (1998)
Bhat, S.P., Bernstein, D.S.: Finite time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)
DCCT: The Diabetes Control and Complications Trial Research Group, The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin-dependent diabetes mellitus. New Engl. J. Med. 329, 977–986 (1993)
Estrada, A., Fridman, L.: Integral HOSM semiglobal controller for finite-time exact compensation of unmatched perturbations. IEEE Trans. Automat. Contr. 55(11), 2644–2649 (2010)
Estrada, A., Fridman, L.: Quasi-continuous HOSM control for systems with unmatched perturbations. Automatica 46, 1916–1919 (2010)
Fisher, M.E.: A semi closed-loop algorithm for the control of blood glucose levels in diabetics. IEEE Trans. Biomed. Eng. 38(1), 57–61 (1991)
Isidori, A.: Nonlinear Control Systems. Springer, New York (1995)
Jaremco J., Rorstad, O.: Advances toward the implantable artificial pancreas for treatment of diabetes. Diabetes Care 21(3), 444–450 (1998)
Kaveh, P., Shtessel, Y.: Blood glucose regulation using higher order sliding mode control. Int. J. Robust. Nonlin. Special Issue on Advances in Higher Order Sliding Mode Control 18(4–5), 557–569 (2008)
Kochalummoottil, J., Shtessel, Y., Moreno, J.A., Fridman, L.: Adaptive twist sliding mode control: a Lyapunov design. In: Proceedings of the 50th Conference on Decision and Control, pp. 7623–7628, Orlando, FL (2011)
Kochalummoottil, J., Shtessel, Y., Moreno, J.A., Fridman, L.: Output feedback adaptive twisting control: a Lyapunov design. In: Proceedings of the American Control Conference, pp. 6172–6177, Montreal, Canada (2012)
Kolmogoroff, A. N.: On inequalities between upper bounds of consecutive derivatives of an arbitrary function defined on an infinite interval. Amer. Math. Soc. Transl. 2, 233–242 (1962)
Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379–384 (1998)
Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76(9/10), 924–941 (2003).
Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)
Levant, A.: Quasi-continuous high-order sliding-mode controllers. IEEE Trans. Automat. Contr. 50(11) 1812–1816 (2006)
Levant, A.: Construction principles of 2-sliding mode design. Automatica 43(4), 576–586 (2007)
Levant, A.: Finite differences in homogeneous discontinuous control. IEEE Trans. Automat. Contr. 527, 1208–1217 (2007)
Levant, A.: Chattering analysis. IEEE Trans. Automat. Contr. 55(6), 1380–1389 (2010)
Levant, A., Fridman, L.: Accuracy of Homogeneous Sliding Modes in the Presence of Fast Actuators. IEEE Trans. Automat. Contr. 55(3), 810–814 (2010)
Levant, A., Levantovsky, L.V.: Sliding order and sliding accuracy in sliding mode control. Int. J. Control 586, 1247–1263 (1993)
Levant, A., Michael, M.: Adjustment of high-order sliding-mode controllers. Int. J. Robust. Nonlin. 19(15), 1657–1672 (2009)
Neatpisarnvanit, C., Boston, J.R.: Estimation of plasma insulin from plasma glucose. IEEE Trans. Biomed. Eng. 49(11), 1253–1259 (2002)
Orlov, Y.: Finite time stability and robust control synthesis of uncertain switched systems. SIAM J. Cont. Optim. 43(4), 1253–1271 (2005)
Shtessel, Y., Taleb, M., Plestan, F.: A novel adaptive-gain super-twisting sliding mode controller: methodology and application. Automatica 48(5), 759–769 (2012)
Shtessel. Y., Kochalummoottil, J., Edwards, C., Spurgeon, S.: Continuous adaptive finite reaching time control and second order sliding modes. IMA J. Math. Control Inform. (2012). doi: 10.1093/imamci/dns013
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Shtessel, Y., Edwards, C., Fridman, L., Levant, A. (2014). Higher-Order Sliding Mode Controllers and Differentiators. In: Sliding Mode Control and Observation. Control Engineering. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4893-0_6
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4893-0_6
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-0-8176-4892-3
Online ISBN: 978-0-8176-4893-0
eBook Packages: EngineeringEngineering (R0)