Previous |  Up |  Next


Popov criterion; stability; time-varying nonlinearity; Lur’e type nonlinearity; interval transfer function
Systems with time-varying non-linearity confined to a given sector (Luré type) and a linear part with uncertainty formulated by an interval transfer function, are considered. Sufficient conditions satisfying the Popov criterion for stability, which are computationally tractable, are derived. The problem of checking the Popov criterion for an infinite set of systems, is reduced to that of checking the Popov criterion for a finite number of fixed coefficient systems, each in a prescribed frequency interval. Illustrative numerical examples are provided.
[1] Brockett R. W., Lee H. B.: Frequency–domain instability criteria for time–varying and nonlinear systems. Proc. IEEE 55 (1967), 604–618
[2] Fruchter G., Srebro U., Zeheb E.: An analytic method for design of uncertain discrete nonlinear control systems. In: Proc. of the 15th Conference of IEEE in Israel, Tel-Aviv 1987
[3] Fruchter G.: Generalized zero sets location and absolute robust stabilization of continuous nonlinear control systems. Automatica 27 (1991), 501–512 DOI 10.1016/0005-1098(91)90107-D | MR 1104510 | Zbl 0754.93027
[4] Fruchter G., Srebro U., Zeheb E.: Stability of discrete nonlinear control systems. IMA J. Math. Control Inform. 14 (1997), 333–351 DOI 10.1093/imamci/14.4.333 | MR 1492837 | Zbl 0896.93027
[5] Kerbelev A. M.: A circle criterion for robust stability and instability of nonstationary nonlinear systems. Automat. Remote Control 54 (1993), 12, 1820–1823 MR 1261540 | Zbl 0825.93595
[6] Kharitonov V. L.: Asymptotic stability of an equilibrium of a family of system of linear differential equations. Differential Equations 14 (1979), 1483–1485
[7] Levkovich A., Zeheb E., Cohen N.: Frequency response envelopes of a family of uncertain continuous–time systems. IEEE Trans. Circuits and Systems 42 (1995), 156–165 DOI 10.1109/81.376874 | MR 1332412 | Zbl 0820.93044
[8] Popov V. M.: Absolute stability of nonlinear systems of automatic control. Translated from Automatika i Telemekhanika 22 (1961), 961–979 MR 0133563 | Zbl 0107.29601
[9] Rozenvasser E. N.: The absolute stability of non–linear systems. Translated from Automatika i Telemekhanika 24 (1963), 304–313 MR 0160996
[10] Siljak D.: Parameter analysis of absolute stability. Automatica 5 (1969), 385–387 DOI 10.1016/0005-1098(69)90079-X | Zbl 0179.41203
[11] Siljak D.: Polytopes of nonnegative polynomials. In: Proc. of the American Control Conf. (ACC), Pittsburgh 1989, pp. 193-199
[12] Zeheb E.: Robust stability of non–linear time varying systems. In: Proc. of the 5th IEEE Mediterranean Conf. on Control and Systems, Paphos 1997
Partner of
EuDML logo