Stability of continuous time dynamical systems with m-feedback nonlinearities.

In this paper a matrix version of the Kalman Yacubovich lemma and a modified Lur'e Lyapunov function introduced recently by the authors are applied to derive sufficient conditions for the absolute stability of continuous time dynamical systems with m-first and third quadrant, monotone increasing, and odd monotone increasing feedback nonlinearities. These frequency domain stability criteria provide a direct, systematic method for the generation of explicit Lyapunov functions. Numerous examples are presented to illustrate the application of the theoretical results. In particular, results obtained by N. N. Krasovskii for second-order systems with two nonlinearities are obtained in a straightforward manner using the frequency domain stability criteria derived in this paper.

[1]  Z. Rekasius,et al.  Construction of Liapunov functions for nonlinear feedback systems with several nonlinear elements , 1964 .

[2]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[3]  Z. Rekasius,et al.  A stability criterion for nonlinear feedback systems , 1964 .

[4]  K. Meyer.  On the Existence of Lyapunov Function for the Problem of Lur’e , 1965 .

[5]  K. Narendra,et al.  Stability of a Class of Differential Equations with a Single Monotone Nonlinearity , 1966 .

[6]  E. Jury,et al.  A stability inequality for a class of nonlinear feedback systems , 1966 .

[7]  Kumpati S. Narendra,et al.  Existence of quadratic type liapunov functions for a class of nonlinear systems , 1964 .

[8]  K. Narendra,et al.  A Geometrical Criterion for the Stability of Certain Nonlinear Nonautonomous Systems , 1964 .

[9]  L. Weinberg,et al.  Realizability Conditions on n-Port Networks , 1958 .

[10]  Z. Rekasius.  A stability criterion for feedback systems with one nonlinear element , 1964 .

[11]  K. Meyer,et al.  LIAPUNOV FUNCTIONS FOR THE PROBLEM OF LUR'E. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[12]  R. Kálmán.  LYAPUNOV FUNCTIONS FOR THE PROBLEM OF LUR'E IN AUTOMATIC CONTROL. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[13]  D. Youla,et al.  On the factorization of rational matrices , 1961, IRE Trans. Inf. Theory.

[14]  J. Hale,et al.  Stability of Motion. , 1964 .

[15]  J. Willems,et al.  Frequency domain stability criteria--Part II , 1965 .