Robust stabilization of uncertain 2-D discrete-time delayed systems using sliding mode control.

*(English)*Zbl 1423.93338Summary: This paper aims to solve the problem of sliding mode control for an uncertain two-dimensional (2-D) systems with states having time-varying delays. The uncertainties in the system dynamics are constituted of mismatched uncertain parameters and the unknown nonlinear bounded function. The proposed problem utilizes the model transformation approach. By segregating the proper Lyapunov-Krasovskii functional in concert with the improved version of Wirtinger-based summation inequality, sufficient solvability conditions for the existence of linear switching surfaces have been put forward, which ensure the asymptotical stability of the reduced-order equivalent sliding mode dynamics. Then, we solve the controller synthesis problem by extending the recently proposed reaching law to 2-D systems, whose proportional part is appropriately scaled by the factor that does not depend on some constant terms but rather on current switching surface’s value, which in turn ensures the faster convergence and better robustness against uncertainties. Finally, the proposed results have been validated through an implementation to a suitable physical system.

##### MSC:

93D21 | Adaptive or robust stabilization |

93B12 | Variable structure systems |

93C55 | Discrete-time control/observation systems |

93C41 | Control/observation systems with incomplete information |

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\textit{I. Ghous} et al., J. Franklin Inst. 356, No. 16, 9407--9431 (2019; Zbl 1423.93338)

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##### References:

[1] | Koo, M. S.; Choi, H. L.; Lim, J. T., Stabilisation of feedback linearisable uncertain nonlinear systems with time delay using scaled sliding surface, IET Control Theory Appl., 2, 11, 974-979 (2008) |

[2] | Richard, J. P., Comments on time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 10, 1667-1694 (2003) · Zbl 1145.93302 |

[3] | Zhang, P.; Hu, J.; Zhang, H.; Chen, D., Robust \(H_∞\) control for delayed systems with randomly varying nonlinearities under uncertain occurrence probability via sliding mode method, Syst. Sci. Control Eng., 6, 1, 160-170 (2018) |

[4] | Yan, M.; Shi, Y., Robust discrete-time sliding mode control for uncertain systems with time-varying state delay, IET Control Theory Appl., 2, 8, 662-674 (2008) |

[5] | Wang, Y.; Shen, H.; Karimi, H. R.; Duan, D., Dissipativity-based fuzzy integral sliding mode control of continuous-time t-s fuzzy systems, IEEE Trans. Fuzzy Syst., 26, 3, 1164-1176 (2018) |

[6] | Wang, Y.; Gao, Y.; Karimi, H. R.; Shen, H.; Fang, Z., Sliding mode control of fuzzy singularly perturbed systems with application to electric circuit, IEEE Trans. Syst. Man Cybern., 48, 10, 1667-1675 (2018) |

[7] | Hou, H.; Zhang, Q., Novel sliding mode control for multi-input multi-output discrete-time system with disturbance, Int. J. Robust Nonlinear Control, 28, 8, 3033-3055 (2018) · Zbl 1391.93055 |

[8] | Zhang, B. L.; Han, Q. L.; Zhang, X. M.; Yu, X., Sliding mode control with mixed current and delayed states for offshore steel jacket platforms, IEEE Trans. Control Syst. Technol., 22, 5, 1769-1783 (2014) |

[9] | Zhang, X. M.; Han, Q. L., Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems, Automatica, 57, 199-202 (2015) · Zbl 1330.93213 |

[10] | Xiao, S.; Xu, L.; Zeng, H. B.; Teo, K. L., Improved stability criteria for discrete-time delay systems via novel summation inequalities, Int. J. Control Autom. Syst., 16, 4, 1592-1602 (2018) |

[11] | Zhang, B. L.; Han, Q. L.; Zhang, X. M., Recent advances in vibration control of offshore platforms, Nonlinear Dyn., 89, 2, 755-771 (2017) |

[12] | Choi, H. H., Variable structure control of dynamical systems with mismatched norm-bounded uncertainties: an LMI approach, Int. J. Control, 74, 13, 1324-1334 (2001) · Zbl 1017.93024 |

[13] | Bartoszewicz, A.; Lesniewski, P., New reaching law for quasi-sliding mode control of discrete-time systems, IEEE 52nd Annual Conference on Decision and Control (CDC), Florence, Italy, 2881-2887 (2013) |

[14] | Fei, J.; Lu, C., Adaptive sliding mode control of dynamic systems using double loop recurrent neural network structure, IEEE Trans. Neural Netw. Learn. Syst., 29, 4, 1275-1286 (2018) |

[15] | Li, H.; Wang, J.; Du, H.; Karimi, H. R., Adaptive sliding mode control for Takagi-Sugeno fuzzy systems and its applications, IEEE Trans. Fuzzy Syst., 26, 2, 531-542 (2018) |

[16] | Jiang, B.; Karimi, H. R.; Kao, Y.; Gao, C., Takagi-Sugeno modelbased sliding mode observer design for finite-time synthesis of semi-Markovian jump systems, IEEE Trans. Syst. Man Cybern. Syst. (2018) |

[17] | Perruquetti, W.; Barbot, J. P., Sliding Mode Control in Engineering (2002), Marcel Dekker: Marcel Dekker New York |

[18] | Efimov, D.; Polyakov, A.; Fridman, L.; Perruquetti, W.; Richard, J. P., Delayed sliding mode control, Automatica, 64, 37-43 (2016) · Zbl 1329.93043 |

[19] | Abidi, K.; Xu, J. X.; Xinghuo, Y., On the discrete-time integral sliding-mode control, IEEE Trans. Autom. Control, 52, 4, 709-715 (2007) · Zbl 1366.93091 |

[20] | Kaczorek, T., Two-Dimensional Linear Systems (1985), Springer-Verlag: Springer-Verlag Berlin, Germany · Zbl 0593.93031 |

[21] | Lu, W.; Antoniou, A., Two-dimensional Digital Filters (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0852.93001 |

[22] | Fei, Z.; Shi, S.; Zhao, C.; Wu, L., Asynchronous control for 2-d switched systems with mode-dependent average dwell time, Automatica, 79, 198-206 (2017) · Zbl 1371.93073 |

[23] | Duan, Z.; Xiang, Z., Finite frequency \(H_∞\) control of 2-d continuous systems in Roesser model, Multidim. Syst. Signal Process., 28, 4, 1481-1497 (2017) · Zbl 1381.93038 |

[24] | Duan, Z.; Ghous, I.; Wang, B.; Shen, J., Necessary and sufficient stability criterion and stabilization for positive 2-d continuous-time systems with multiple delays, Asian J. Control, 21, 4, 1-12 (2019) |

[25] | Ghous, I.; Xiang, Z., Robust state feedback \(H_∞\) control for uncertain 2-d continuous state delayed systems in the Roesser model, Multidimens. Syst. Signal Process., 27, 2, 297-319 (2016) · Zbl 1368.93150 |

[26] | Ghous, I.; Xiang, Z.; Karimi, H. R., \(H_∞\) control of 2-d continuous Markovian jump delayed systems with partially unknown transition probabilities, Inform. Sci., 382, 274-291 (2017) · Zbl 1432.93085 |

[27] | Zhang, D.; Shi, P.; Yu, L., Containment control of linear multiagent systems with aperiodic sampling and measurement size reduction, IEEE Trans. Neural Netw. Learn. Syst., 29, 10, 5020-5029 (2018) |

[28] | Zhang, D.; Liu, L.; Feng, G., Consensus of heterogeneous linear multiagent systems subject to aperiodic sampled-data and dos attack, IEEE Trans. Cybern., 1-11 (2018) |

[29] | Fallaha, C. J.; Saad, M.; Kanaan, H. Y.; Al-Haddad, K., Sliding-mode robot control with exponential reaching law, IEEE Trans. Ind. Electron., 58, 2, 600-610 (2011) |

[30] | Wu, L.; Gao, H., Sliding mode control of two-dimensional systems in Roesser model, IET Control Theory Appl., 2, 4, 352-364 (2008) |

[31] | Gao, W.; Wang, H.; Homaifa, A., Discrete-time variable structure control systems, IEEE Trans. Ind. Electron., 42, 2, 117-122 (1995) |

[32] | 13-15 December |

[33] | Choi, H. H., A new method for variable structure control system design: a linear matrix inequality approach, Automatica, 33, 11, 2089-2092 (1997) · Zbl 0911.93022 |

[34] | Adloo, H.; Karimaghaee, P.; Sarvestani, A. S., An extension of sliding mode control design for the 2-d systems in Roesser model, 48th IEEE Conference on Decision and Control (CDC) held Jointly with 28th Chinese Control Conference (CCC), Shanghai, China, 15-18, 7753-7758 (2009) |

[35] | 3584-3589 |

[36] | Bartoszewicz, A.; Lesniewski, P., New switching and nonswitching type reaching laws for SMC of discrete time systems, IEEE Trans. Control Syst. Technol., 24, 2, 670-677 (2016) |

[37] | Seuret, A.; Gouaisbaut, F.; Fridman, E., Stability of discrete-time systems with time-varying delays via a novel summation inequality, IEEE Trans. Autom. Control, 60, 10, 2740-2745 (2015) · Zbl 1360.93612 |

[38] | Zhang, C. K.; He, Y.; Jiang, L.; Wu, M., An improved summation inequality to discrete-time systems with time-varying delay, Automatica, 74, 10-15 (2016) · Zbl 1348.93185 |

[39] | Xie, L., Output feedback \(H_∞\) control of systems with parameter uncertainty, Int. J. Control, 63, 4, 741-750 (1996) · Zbl 0841.93014 |

[40] | Xu, B.; Liu, Y., An improved razumikhin-type theorem and its applications, IEEE Trans. Autom. Control, 39, 4, 839-841 (1994) · Zbl 0807.93056 |

[41] | Bartoszewicz, A., Remarks on discrete-time variable structure control systems, IEEE Trans. Ind. Electron., 43, 1, 235-238 (1996) |

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