zbMATH — the first resource for mathematics

Robust finite-time \(H_\infty\) control for one-sided Lipschitz nonlinear systems via state feedback and output feedback. (English) Zbl 1395.93128
Summary: Robust finite-time \(H_\infty\) control of a class of continuous-time nonlinear system with parameter uncertainties and disturbance input is discussed in this note. The nonlinear function is considered to satisfy the one-sided Lipschitz condition, which has less conservative than the well-known global Lipschitz nonlinear condition. By means of linear matrix inequality (LMI) techniques, both design algorithms of state-feedback controller and static output-feedback controller are developed. The designed controllers are proved to guarantee the corresponding closed-loop systems that are finite-time boundedness (FTB) with a desired \(H_\infty\) performance index. And the best scalars selection criterion is used to determine the finite-time scalars such that the LMI-based conditions with the best feasibility in the global field. Finally, a numerical example is included to verify the efficiency of the proposed methods.

93B07 Observability
93C10 Nonlinear systems in control theory
93B52 Feedback control
93C15 Control/observation systems governed by ordinary differential equations
93C55 Discrete-time control/observation systems
93B36 \(H^\infty\)-control
93C41 Control/observation systems with incomplete information
93C73 Perturbations in control/observation systems
Full Text: DOI
[1] M. Abbaszadeh, H. Marquez, Nonlinear observer design for one-sided Lipschitz systems, in: Proceedings of IEEE American Control Conference, USA, 2010.
[2] Amato, F.; Ariola, M.; Dorato, P., Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 37, 1459-1463, (2001) · Zbl 0983.93060
[3] Amato, F.; Cosentino, C.; Merola, A., Sufficient conditions for finite-time stability and stabilization of nonlinear quadratic systems, IEEE Trans. Autom. Control, 55, 430-434, (2010) · Zbl 1368.93524
[4] Amato, F.; De Tommasi, G.; Pironti, A., Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems, Automatica, 49, 2546-2550, (2013) · Zbl 1364.93737
[5] Benallouch, M.; Boutaveb, M.; Zasadzinski, M., Observer design for one-sided Lipschitz discrete-time systems, Syst. Control Lett., 61, 879-886, (2012) · Zbl 1270.93021
[6] Chen, B.; Niu, Y.; Zou, Y., Sliding mode control for stochastic Markovian jumping systems subject to successive packet losses, J. Frankl. Inst., 351, 4, 2169-2184, (2014) · Zbl 1372.93058
[7] Chen, M. S.; Chen, C. C., Robust nonlinear observer for Lipschitz nonlinear systems subject to disturbances, IEEE Trans. Autom. Control, 52, 2365-2369, (2007) · Zbl 1366.93459
[8] Chen, W.; Khan, A. Q.; Abid, M.; Ding, S. X., Integrated design of observer based fault detection for a class of uncertain nonlinear systems, Int. J. Appl. Math. Comput. Sci., 21, 423-430, (2011) · Zbl 1234.93036
[9] Dekker, K.; Verwer, J. G., Stability of Runge-Kutta methods for stiff nonlinear differential equations, (1984), North-Holland Amsterdam · Zbl 0571.65057
[10] P. Dorato, Short time stability in linear time-varying systems, in: Proceedings of the IRE International Convention Record, vol. 4, 1961, pp. 83-87.
[11] ElBsat, M. N.; Yaz, E. E., Robust and resilient finite-time bounded control of discrete-time uncertain nonlinear systems, Automatica, 49, 2292-2296, (2013) · Zbl 1364.93629
[12] Gahinet, P.; Nemirovski, A.; Laub, A. J.; Chilali, M., MATLAB LMI control tolbox, (1995), The MATH Works Inc. Natick
[13] Hairer, E.; Norsett, S. P.; Wanner, G., Solving ordinary differential equations II: stiff and DAE problems, (1993), Springer-Verlag New York
[14] He, S.; Liu, F., Finite-time \(H_\infty\) fuzzy control of nonlinear jump systems with time delays via dynamic observer-based state feedback, IEEE Trans. Fuzzy Syst., 20, 605-614, (2012)
[15] Hu, G., Observers for one-sided Lipschitz nonlinear systems, IMA J. Math. Control Inf., 23, 395-401, (2006) · Zbl 1113.93021
[16] Lin, X.; Li, X.; Zou, Y.; Li, S., Finite-time stabilization of switched linear systems with nonlinear saturating actuators, J. Frankl. Inst., 351, 1464-1482, (2014) · Zbl 1395.93277
[17] Liu, H.; Zhao, X., Finite-time \(H_\infty\) control of switched systems with mode-dependent average Dwell time, J. Frankl. Inst., 351, 1301-1315, (2014) · Zbl 1395.93278
[18] Niu, Y.; Jia, T.; Wang, X.; Yang, F., Output-feedback control design for NCSs subject to quantization and dropout, Inf. Sci., 179, 21, 3804-3813, (2009) · Zbl 1171.93328
[19] Rajamani, R., Observers for Lipschitz nonlinear systems, IEEE Trans. Autom. Control, 43, 397-401, (1998) · Zbl 0905.93009
[20] Song, J.; He, S., Observer-based finite-time passive control for a class of uncertain time-delayed Lipschitz nonlinear systems, Trans. Inst. Meas. Control, 36, 6, 797-804, (2014)
[21] Song, J.; He, S., Finite-time \(H_\infty\) control for quasi-one-sided Lipschitz nonlinear systems, Neurocomputing, 149, Part C, 1433-1439, (2015)
[22] Veluvolu, K. C.; Soh, Y. C., Multiple sliding mode observers and unknown input estimations for Lipschitz nonlinear systems, Int. J. Robust Nonlinear Control, 21, 1322-1340, (2011) · Zbl 1244.93037
[23] Wang, X.; Zhong, G.; Tang, K. S.; Man, K. F.; Liu, Z., Generating chaos in chua׳s circuit via time-delay feedback, IEEE Trans. Circuits Syst.—I: Fund. Theory Appl., 48, 1151-1156, (2001)
[24] Xu, S.; Lu, J.; Zhou, S.; Yang, C., Design of observers for a class of discrete-time uncertain nonlinear systems with time delay, J. Frankl. Inst., 341, 295-308, (2004) · Zbl 1073.93007
[25] Zemouche, A.; Boutayeb, M., Observer design for Lipschitz nonlinear systemsthe discrete-time case, IEEE Trans. Circuits Syst.—II: Express Briefs, 53, 777-781, (2006)
[26] Zemouche, A.; Boutayeb, M., Observer synthesis method for Lipschitz nonlinear discrete-time systems with time-delayan LMI approach, Appl. Math. Comput., 218, 419-429, (2011) · Zbl 1223.93018
[27] Zemouche, A.; Boutayeb, M., On LMI conditions to design observers for Lipschitz nonlinear systems, Automatica, 49, 585-591, (2013) · Zbl 1259.93031
[28] Zhang, W.; Su, H.; Liang, Y.; Han, Z., Non-linear observer design for one-sided Lipschitz systemsan linear matrix inequality approach, IET Control Theory Appl., 6, 9, 1297-1303, (2012)
[29] Zhang, W.; Su, H.; Wang, H.; Han, Z., Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations, Commun. Nonlinear Sci. Numer. Simul., 17, 4968-4977, (2012) · Zbl 1263.93051
[30] Zhang, W.; Su, H.; Zhu, F.; Yue, D., A note on observers for discrete-time Lipschitz nonlinear systems, IEEE Trans. Circuits Syst.—II: Express Briefs, 59, 123-127, (2012)
[31] Zhao, Y.; Tao, J.; Shi, N., A note on observer design for one-sided Lipschitz nonlinear systems, Syst. Control Lett., 59, 66-71, (2010) · Zbl 1186.93017
[32] Zhu, F.; Han, Z., A note on observers for Lipschitz nonlinear systems, IEEE Trans. Autom. Control, 47, 1751-1754, (2002) · Zbl 1364.93104
[33] Zong, G.; Wang, R.; Zheng, W.; Hou, L., Finite-time stabilization for a class of switched time-delay systems under asynchronous switching, Appl. Math. Comput., 219, 11, 5757-5771, (2013) · Zbl 1272.93099
[34] Zong, G.; Yang, D.; Hou, L.; Wang, Q., Robust finite-time \(H_\infty\) control for Markovian jump systems with partially known transition probabilities, J. Frankl. Inst., 350, 6, 1562-1578, (2013) · Zbl 1293.93773
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.