Discrete-time sliding mode control of linear systems with input saturation

Veselić, Boban ; Milosavljević, Čedomir ; Peruničić-Draženović, Branislava ; Huseinbegović, Senad ; Petronijević, Milutin

International Journal of Applied Mathematics and Computer Science, Tome 30 (2020) no. 3, pp. 517-528.

Résumé

The paper proposes a discrete-time sliding mode controller for single input linear dynamical systems, under requirements of the fast response without overshoot and strong robustness to matched disturbances. The system input saturation is imposed during the design due to inevitable limitations of most actuators. The system disturbances are compensated by employing nonlinear estimation by integrating the signum of the sliding variable. Hence, the proposed control structure may be regarded as a super-twisting-like algorithm. The designed system stability is analyzed as well as the sliding manifold convergence conditions are derived using a discrete-time model of the system in the δ-domain. The results obtained theoretically have been verified by computer simulations.

Détail

Keywords: discrete time sliding mode control, super twisting controller, input saturation, disturbance compensation
Mots-clés : sterowanie ślizgowe, nasycenie wejściowe, kompensacja zakłóceń

Bibliographie

[1] Abidi, K., Xu, J. and Yu, X. (2007). On the discrete-time integral sliding-mode control, IEEE Transactions on Automatic Control 52(4): 709–715.

[2] Ackermann, J. and Utkin, V. (1998). Sliding mode control design based on Ackermann’s formula, IEEE Transactions on Automatic Control 43(2): 234–237.

[3] Bartolini, G., Ferrara, A., Pisano, A. and Usai, E. (1998). Adaptive reduction of the control effort in chattering-free sliding-mode control of uncertain nonlinear systems, International Journal of Applied Mathematics and Computer Science 8(1): 51–71.

[4] Bartolini, G., Ferrara, A. and Utkin, V.I. (1995). Adaptive sliding mode control in discrete-time systems, Automatica 31(5): 769–773.

[5] Bartolini, G., Pisano, A. and Usai, E. (2001). Digital second-order sliding mode control for uncertain nonlinear systems, Automatica 37(9): 1371–1377.

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

[7] Bartoszewicz, A. and Adamiak, K. (2019). A reference trajectory based discrete time sliding mode control strategy, International Journal of Applied Mathematics and Computer Science 29(3): 517–525, DOI: 10.2478/amcs-2019-0038.

[8] Bartoszewicz, A. and Latosinski, P. (2017). Reaching law for DSMC systems with relative degree 2 switching variable, International Journal of Control 90(8): 1626–1638.

[9] 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.

[10] Bondarev, A.G., Bondarev, S.A., Kostyleva, N.E. and Utkin, V.I. (1985). Sliding modes in systems with asymptotic state observers, Avtomatika i Telemekhanika 46(6): 5–11.

[11] Castillo, I., Steinberger, M., Fridman, L., Moreno, J.A. and Horn, M. (2016). Saturated super-twisting algorithm: Lyapunov based approach, IEEE 55th Conference on Decision and Control (CDC), Las Vegas, NV, USA, pp. 269–273.

[12] Chakrabarty, S., Bandyopadhyay, B. and Bartoszewicz, A. (2017). Discrete-time sliding mode control with outputs of relative degree more than one, in A. Bartoszewicz (Ed.), Recent Developments in Sliding Mode Control Theory and Applications, InTech, London, pp. 21–44.

[13] Corradini, M.L., Cristofaro, A. and Orlando, G. (2014). Sliding-mode control of discrete time linear plants with input saturation: Application to a twin-rotor system, International Journal of Control 87(8): 1523–1535.

[14] Draženović, B. (1969). The invariance conditions in variable structure systems, Automatica 5(3): 287–295.

[15] Draženović, B., Milosavljević, Č. and Veselić, B. (2013). Comprehensive approach to sliding mode design and analysis in linear systems, in B. Bandyopadhyay et al. (Eds), Advances in Sliding Mode Control: Concept, Theory and Implementation, Springer, Berlin/Heidelberg, pp. 1–19.

[16] Drakunov, S.V. and Utkin, V.I. (1989). On discrete-time sliding mode, Proceedings of IFAC Symposium on Nonlinear Control Systems Design, Capri, Italy, pp. 484–489.

[17] Emelyanov, S.V. (1957). A method to obtain complex regulation laws using only the error signal or the regulated coordinate and its first derivatives, Avtomatika i Telemekhanika 18(10): 873–885.

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

[19] Ghane, H. and Menhaj, M. B. (2015). Eigenstructure-based analysis for non-linear autonomous systems, IMA Journal of Mathematical Control and Information 32(1): 21–40.

[20] Golkani, M.A., Koch, S., Reichhartinger, M. and Horn, M. (2018). A novel saturated super-twisting algorithm, Systems and Control Letters 119: 52–56.

[21] Golo, G. and Milosavljević, Č. (2000). Robust discrete-time chattering free sliding mode control, Systems and Control Letters 41(1): 19–28.

[22] Golo, G., Schaft, A. and Milosavljević, Č. (2000). Discretization of control law for a class of variable structure control systems, Technical Report 1551, University of Twente, Enschede.

[23] Huber, O., Brogliato, B., Acary, V., Boubakir, A., Plestan, F. and Wang, B. (2016). Experimental results on implicit and explicit time-discretization of equivalent-control-based sliding-mode control, in L. Fridman et al. (Eds), Recent Trends in Sliding Mode Control, IET, London, pp. 207–235.

[24] Koch, S. and Reichhartinger, M. (2019). Discrete-time equivalents of the super-twisting algorithm, Automatica 107: 190–199.

[25] Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58(6): 1247–1263.

[26] Lješnjanin, M., Peruničić, B., Milosavljević, Č. and Veselić, B. (2011). Disturbance compensation in digital sliding mode, 2011 IEEE EUROCON, International Conference on Computer as a Tool, Lisboa, Portugal, pp. 1–4.

[27] Milosavljević, Č. (1985). General conditions for the existence of quasi-sliding mode on the switching hyper-plane in discrete variable structure systems, Automatic and Remote Control 46(3): 307–314.

[28] Milosavljević, Č., Peruničić-Draženović, B., Veselić, B. and Mitić, D. (2007). A new design of servomechanisms with digital sliding mode, Electrical Engineering 89(3): 233–244.

[29] Milosavljević, Č., Petronijević, M., Veselić, B., Peruničić-Draženović, B. and Huseinbegović, S. (2019). Robust discrete-time quasi-sliding mode based nonlinear PI controller design for control of plants with input saturation, Journal of Control Engineering and Applied Informatics 21(3): 31–41.

[30] Salgado, I., Kamal, S., Bandyopadhyay, B., Chairez, I. and Fridman, L. (2016). Control of discrete time systems based on recurrent super-twisting-like algorithm, ISA Transactions 64: 47–55.

[31] Salgado, I., Kamal, S., Chairez, I., Bandyopadhyay, B. and Fridman, L. (2011). Super-twisting-like algorithm in discrete time nonlinear systems, Proceedings of the 2011 International Conference on Advanced Mechatronic Systems, Zhengzhou, China, pp. 497–502.

[32] Shtessel, Y., Taleb, M. and Plestan, F. (2012). A novel adaptive-gain supertwisting sliding mode controller: methodology and application, Automatica 48(5): 759–769.

[33] Slotine, J.J.E. (1984). Sliding controller design for non-linear systems, International Journal of Control 40(2): 421–434.

[34] Su, W.C., Drakunov, S.V. and Ozguner, U. (2000). An O(T 2) boundary layer in sliding mode for sampled-data systems, IEEE Transactions on Automatic Control 45(3): 482–485.

[35] Utkin, V. (2016). Discussion aspects of higher order sliding mode control, IEEE Transactions on Automatic Control 61(3): 829–833.

[36] Utkin, V.I. (1992). Sliding Modes in Control and Optimization, Springer, Heidelberg.

[37] Yan, Y., Yu, X. and Sun, C. (2015). Discretization behaviors of a super-twisting algorithm based sliding mode control system, 2015 International Workshop on Recent Advances in Sliding Modes (RASM), Istanbul, Turkey, pp. 1–5.