Abstract
In this chapter, we propose an explicit discretization scheme for class of disturbed systems controlled by homogeneous quasi-continuous high-order sliding-mode controllers which are equipped with a homogeneous Lyapunov function. Such a Lyapunov function is used to construct the discretization scheme that preserves important features from the original continuous-time system: asymptotic stability, finite-time convergence, and the Lyapunov function itself.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Some parts of Lemma 2, Theorem 3, Theorem 5, and their proofs have been reproduced with permission from [Sanchez T, Polyakov A, Efimov D. Lyapunov-based consistent discretization of stable homogeneous systems. Int J Robust Nonlinear Control. 2021;31:3587–3605. https://doi.org/10.1002/rnc.5308] ©2020 John Wiley & Sons Ltd., and with permission from the IFAC License Agreement IFAC 2020#1150 of [T. Sanchez, A. Polyakov, D. Efimov, A Consistent Discretisation method for Stable Homogeneous Systems based on Lyapunov Function, IFAC-PapersOnLine 53(2), 5099–5104 (2020). DOI https://doi.org/10.1016/j.ifacol.2020.12.1141.].
- 2.
Under the assumption of \(d\in \mathscr {D}\), it can be seen that the right-hand side of (6) is locally Lipschitz in x on \(\mathbb {R}^n\setminus \{0\}\).
- 3.
- 4.
This equation is known as Euler’s theorem for weighted homogeneous functions, see, e.g., [2, Proposition 5.4].
- 5.
In this chapter, we mean by numerical solution a sequence \(\{z_k\}_{k\in \mathbb {Z}_{+}}\) such that \(z_0=z(0)\), and for some \(h\in \mathbb {R}_{+}^*\), \(z_k\) approximates z(kh).
- 6.
- 7.
The local truncation error is the one-step error computed by assuming that \(E(t)=0\), i.e., \(x(t)=x_k\).
- 8.
Since \([a_1,a_2]\subset \mathbb {R}\) is compact and \(a_1>0\), the function \(g:[a_1,a_2]\subset \mathbb {R}_{+}^* \rightarrow \mathbb {R}\) given by \(g(V)=V^{\frac{r_i}{m}}\) is Lipschitz continuous. Also, \(z_k\) and z belong to a compact subset of \(\mathbb {R}^n\) on which V is Lipschitz continuous.
References
Acary, V., Brogliato, B., Orlov, Y.V.: Chattering-free digital sliding-mode control with state observer and disturbance rejection. IEEE Trans. Autom. Control 57(5), 1087–1101 (2012). https://doi.org/10.1109/TAC.2011.2174676
Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory, 2nd edn. Communications and Control Engineering. Springer, Berlin (2005). https://doi.org/10.1007/b139028
Bernuau, E., Efimov, D., Perruquetti, W., Polyakov, A.: On homogeneity and its application in sliding mode control. J. Frankl. Inst. 351(4), 1866–1901 (2014). https://doi.org/10.1016/j.jfranklin.2014.01.007
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004). https://doi.org/10.1017/CBO9780511804441
Brogliato, B., Polyakov, A., Efimov, D.: The implicit discretization of the supertwisting sliding-mode control algorithm. IEEE Trans. Autom. Control 65(8), 3707–3713 (2020). https://doi.org/10.1109/TAC.2019.2953091
Crouzeix, J.P., Ferland, J.A.: Criteria for quasi-convexity and pseudo-convexity: relationships and comparisons. Math. Program. 23(1) (1982). https://doi.org/10.1007/BF01583788
Cruz-Zavala, E., Moreno, J.A.: Homogeneous high order sliding mode design: a Lyapunov approach. Automatica 80, 232–238 (2017). https://doi.org/10.1016/j.automatica.2017.02.039
Drakunov, S.V., Utkin, V.I.: On discrete-time sliding modes. IFAC Proc. Vol. 22(3), 273–278 (1989). https://doi.org/10.1016/S1474-6670(17)53647-2
Efimov, D., Polyakov, A., Levant, A., Perruquetti, W.: Realization and discretization of asymptotically stable homogeneous systems. IEEE Trans. Autom. Control 62(11), 5962–5969 (2017). https://doi.org/10.1109/TAC.2017.2699284
Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht, The Netherlands (1988). https://doi.org/10.1007/978-94-015-7793-9
Goodwin, G.C., Agüero, J.C., Garrido, M.E.C., Salgado, M.E., Yuz, J.I.: Sampling and sampled-data models: the interface between the continuous world and digital algorithms. IEEE Control Syst. Mag. 33(5), 34–53 (2013). https://doi.org/10.1109/MCS.2013.2270403
Grimm, V., Quispel, G.R.W.: Geometric integration methods that preserve Lyapunov functions. BIT Numer. Math. 45(4), 709–723 (2005). https://doi.org/10.1007/s10543-005-0034-z
Haimo, V.T.: Finite time controllers. SIAM J. Control Optim. 24(4), 760–770 (1986)
Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Berlin, Heidelberg (1993). https://doi.org/10.1007/978-3-540-78862-1
Hong, Y., Huang, J., Xu, Y.: On an output feedback finite-time stabilisation problem. In: Proceedings of the 38th IEEE Conference on Decision and Control, vol. 2, pp. 1302–1307 (1999). https://doi.org/10.1109/CDC.1999.830117
Huber, O., Acary, V., Brogliato, B.: Lyapunov stability and performance analysis of the implicit discrete sliding mode control. IEEE Trans. Autom. Control 61(10), 3016–3030 (2016). https://doi.org/10.1109/TAC.2015.2506991
Kawski, M.: Stability and nilpotent approximations. In: Proceedings of the 27th IEEE Conference on Decision and Control, vol. 2, pp. 1244–1248 (1988). https://doi.org/10.1109/CDC.1988.194520
Koch, S., Reichhartinger, M.: Discrete-time equivalents of the super-twisting algorithm. Automatica 107, 190–199 (2019). https://doi.org/10.1016/j.automatica.2019.05.040, http://www.sciencedirect.com/science/article/pii/S0005109819302596
Koch, S., Reichhartinger, M., Horn, M., Fridman, L.: Discrete-time implementation of homogeneous differentiators. IEEE Trans. Autom. Control 65(2), 757–762 (2020). https://doi.org/10.1109/TAC.2019.2919237
Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley, New York (1991)
Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76(6), 924–941 (2003). https://doi.org/10.1080/0020717031000099029
Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005). https://doi.org/10.1016/j.automatica.2004.11.029
Levant, A.: Quasi-continuous high-order sliding-mode controllers. IEEE Trans. Autom. Control 50(11), 1812–1816 (2005). https://doi.org/10.1109/TAC.2005.858646
Levant, A.: On fixed and finite time stability in sliding mode control. In: 52nd IEEE Conference on Decision and Control (2013)
Mangasarian, O.L.: Nonlinear Programming. SIAM (1994)
Nakamura, H., Yamashita, Y., Nishitani, H.: Smooth Lyapunov functions for homogeneous differential inclusions. In: Proceedings of the 41st SICE Annual Conference, pp. 1974–1979 (2002). https://doi.org/10.1109/SICE.2002.1196633
Nesić, D., Teel, A.R.: Perspectives in Robust Control, chap. Sampled-Data Control of Nonlinear Systems: An Overview of Recent Results. Springer, London (2001)
Polyakov, A.: Generalized Homogeneity in Systems and Control. Springer, Cham, Switzerland (2020). https://doi.org/10.1007/978-3-030-38449-4
Polyakov, A., Efimov, D., Brogliato, B.: Consistent discretization of finite-time and fixed-time stable systems. SIAM J. Control Optim. 57(1), 78–103 (2019). https://doi.org/10.1137/18M1197345
Polyakov, A., Fridman, L.: Stability notions and Lyapunov functions for sliding mode control systems. J. Frankl. Inst. 351(4), 1831–1865 (2014). https://doi.org/10.1016/j.jfranklin.2014.01.002
Rosier, L.: Homogeneous Lyapunov function for homogeneous continuous vector field. Syst. Control Lett. 19(6), 467–473 (1992). https://doi.org/10.1016/0167-6911(92)90078-7
Sanchez, T., Efimov, D., Polyakov, A., Moreno, J.A.: Homogeneous discrete-time approximation. IFAC-PapersOnLine 52(16), 19–24 (2019). https://doi.org/10.1016/j.ifacol.2019.11.749. 11th IFAC Symposium on Nonlinear Control Systems NOLCOS 2019
Sanchez, T., Polyakov, A., Efimov, D.: A consistent discretisation method for stable homogeneous systems based on lyapunov function. IFAC-PapersOnLine 53(2), 5099–5104 (2020). https://doi.org/10.1016/j.ifacol.2020.12.1141. 21st IFAC World Congress
Sanchez, T., Polyakov, A., Efimov, D.: Lyapunov-based consistent discretization of stable homogeneous systems. Int. J. Robust Nonlinear Control 31(9), 3587–3605 (2021). https://doi.org/10.1002/rnc.5308
Sepulchre, R., Aeyels, D.: Homogeneous Lyapunov functions and necessary conditions for stabilization. Math. Control Signals Syst. 9(1), 34–58 (1996). https://doi.org/10.1007/BF01211517
Utkin, V.I.: Variable Structure and Lyapunov Control, chap. Sliding mode control in discrete-time and difference systems. Springer, Berlin, Heidelberg (1994)
Walter, J.: Proof of Peano’s existence theorem without using the notion of the definite integral. J. Math. Anal. Appl. 59(3), 587–595 (1977). https://doi.org/10.1016/0022-247X(77)90083-X
Acknowledgements
The authors acknowledge the support of: the project ANR DIGITSLID (ANR 18-CE40-0008); CONACYT CVU-371652.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Sanchez, T., Polyakov, A., Efimov, D. (2023). Lyapunov-Based Consistent Discretization of Quasi-continuous High-Order Sliding Modes. In: Oliveira, T.R., Fridman, L., Hsu, L. (eds) Sliding-Mode Control and Variable-Structure Systems. Studies in Systems, Decision and Control, vol 490. Springer, Cham. https://doi.org/10.1007/978-3-031-37089-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-031-37089-2_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-37088-5
Online ISBN: 978-3-031-37089-2
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)