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Improved delay-dependent stability criteria for uncertain Lur’e systems with sector and slope restricted nonlinearities and time-varying delays. (English) Zbl 1172.34047
The paper investigates the absolute stability of uncertain time delay Lur’e systems that have time-varying delays and sector and slope restricted nonlinearities. In the mathematical model of this kind of Lur’e systems, it is assumed that the nonlinear function and the derivatives of the nonlinear function are bounded, respectively, the time-varying delays are bounded and the derivatives of the time-varying delays are less than one. By constructing Lyapunov-Krasovskii functionals, a new criterion for absolute stability of the Lur’e systems is presented in terms of linear matrix inequality (LMI). This criterion is dependent on the size of the time delays and the derivative of the time-varying delays, which is usually less conservative than delay-independent ones.
##### MSC:
 34K20 Stability theory of functional-differential equations 34K99 Functional-differential equations
##### Keywords:
Lur’e systems; time-varying delay; stability; LMIs
##### References:
 [1] Khalil, H. K.: Nonlinear systems, (1996) [2] Lur’e, A. I.; Postnikov, V. N.: On the theory of stability of control. Systems, Prikl. mat. Meh. 8 (1944) [3] Vidyasagar, M.: Nonlinear system analysis, (1993) · Zbl 0900.93132 [4] Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in system and control theory, (1994) [5] Gu, K.: Absolute stability of systems under block diagonal memoryless uncertainties, Automatica 31, 581-584 (1995) · Zbl 0822.93056 · doi:10.1016/0005-1098(95)98486-P [6] Lee, S. M.; Park, Ju H.: Robust stabilization of discrete-time nonlinear Lur’e systems with sector and slope restricted nonlinearities, Appl. math. Comput. 200, 450 (2008) · Zbl 1146.93017 · doi:10.1016/j.amc.2007.11.031 [7] Liu, D.; Molchanov, A.: Criteria for robust absolute stability of timevarying nonlinear continuous-time systems, Automatica 38, 627 (2002) · Zbl 1013.93044 · doi:10.1016/S0005-1098(01)00243-6 [8] Haddad, W. M.; Kapila, V.: Absolute stability criteria for multiple slope-restricted monotonic nonlinearities, IEEE trans. Automat. contr. 40, 361 (1995) · Zbl 0825.93618 · doi:10.1109/9.341811 [9] Suykens, J. A. K.; Vandewalle, J.; Moor, B. D.: An absolute stability criterion for the Lur’e problem with sector and slope restricted nonlinearities, IEEE trans. Circ. syst. 45, 1007 (1988) · Zbl 0965.93079 · doi:10.1109/81.721270 [10] Singh, V.: A stability inequality for nonlinear feedback systems with slope-restricted nonlinearity, IEEE trans. Automat. contr. 29, 743 (1984) · Zbl 0541.93058 · doi:10.1109/TAC.1984.1103622 [11] Park, P.: Stability criteria for sector- and slope-restricted Lur’e systems, IEEE trans. Automat. contr. 47, 308 (2002) [12] He, Y.; Wu, M.; She, J. H.; Liu, G. P.: Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Syst. contr. Lett. 51, 57 (2004) · Zbl 1157.93467 · doi:10.1016/S0167-6911(03)00207-X [13] Gan, Z.; Ge, W.: Lyapunov functional for multiple delay general Lur’e control systems with multiple non-linearities, J. math. Anal. appl. 259, 596 (2001) · Zbl 0995.93041 · doi:10.1006/jmaa.2001.7433 [14] Bilman, P.: Lyapunov – Krasovskiĭ functionals and frequency model: delay-independent absolute stability criteria for delay systems, Int. J. Robust nonl. Contr. 11, 771 (2001) · Zbl 0992.93069 · doi:10.1002/rnc.576 [15] Gan, Z.; Ge, W.: Lyapunov functional for multiple delay general Lur’e control systems with multiple non-linearities, J. math. Anal. appl. 259, 596 (2001) · Zbl 0995.93041 · doi:10.1006/jmaa.2001.7433 [16] He, Y.; Wu, M.: Absolute stability for multiple delay general Lur’e control systems with multiple nonlinearities, J. comput. Appl. math. 159, 241 (2003) · Zbl 1032.93062 · doi:10.1016/S0377-0427(03)00457-6 [17] L. Yu, Q.L. Han, S. Yu, J.F. Gao, Delay-dependent conditions for robust absolute stability of uncertain time-delay systems, in: Conference on Decision and Control, 2003, p. 6033. [18] He, Y.; Wu, M.; She, J. -H.; Liu, G. -P.: Robust stability for delay Lur’e control systems with multiple nonlinearities, J. comput. Appl. math. 176, 371 (2005) · Zbl 1076.93036 · doi:10.1016/j.cam.2004.07.025 [19] Han, Q. -L.: Absolute stability of time-delay systems with sector-bounded nonlinearity, Automatica 41, 2171 (2005) · Zbl 1100.93519 · doi:10.1016/j.automatica.2005.08.005 [20] Xu, S.; Feng, G.: Improved robust absolute stability criteria for uncertain time-delay systems, IET contr. Theory appl. 1, 1630 (2007) [21] Lee, S. M.; Park, J. H.; Kwon, O. M.: Delay-independent absolute stability for time-delay Lur’e systems with sector and slope restricted nonlinearities, Phys. lett. A 372, 4010 (2008) · Zbl 1220.93063 · doi:10.1016/j.physleta.2008.03.012 [22] Skelton, R. E.; Iwasaki, T.; Grigoradis, K. M.: A unified algebraic approach to linear control design, (1997) [23] Gu, K.; Kharitonov, V. L.; Chen, J.: Stability of time-delay systems, (2003) [24] Suplin, V.; Fridman, E.; Shaked, U.: H$\infty$ control of linear uncertain time-delay systems – a projection approach, IEEE trans. Automat. contr. 51, 680 (2006)