On the stability of a family of nonlinear time-varying systems

In the present paper we investigate several types of Lyapunov stability of an equilibrium x/sub e/ of a family of finite dimensional dynamical systems determined by ordinary differential (difference) equations. By utilizing the extreme systems of the family of systems, we establish sufficient conditions, as well as necessary conditions (converse theorems) for several robust stability types. Our results enable us to realize a significant reduction in the computational complexity of the algorithm of Brayton and Tong (1979) in the construction of computer generated Lyapunov functions. Furthermore, we demonstrate the applicability of the present results by analyzing robust stability properties of equilibria for Hopfield neural networks and by analyzing the Hurwitz and Schur stability of interval matrices.

[1]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

[2]  W. Walter.  Differential and Integral Inequalities , 1970 .

[3]  Robert K. Brayton,et al.  Stability of dynamical systems: A constructive approach , 1979 .

[4]  Robert K. Brayton,et al.  Constructive stability and asymptotic stability of dynamical systems , 1980 .

[5]  Stephen Grossberg,et al.  Absolute stability of global pattern formation and parallel memory storage by competitive neural networks , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[6]  V. Vittal,et al.  Computer generated Lyapunov functions for interconnected systems: Improved results with applications to power systems , 1983, The 22nd IEEE Conference on Decision and Control.

[7]  A. Michel,et al.  Stability analysis of fixed- point digital filters using computer generated Lyapunov functions- Part I: Direct form and coupled form filters , 1985 .

[8]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[9]  A. Michel,et al.  Stability analysis of fixed- point digital filters using computer generated Lyapunov functions- Part II: Wave digital filters and lattice digital filters , 1985 .

[10]  C. Jiang.  Sufficient condition for the asymptotic stability of interval matrices , 1987 .

[11]  Huang Lin,et al.  Root locations of an entire polytope of polynomials: It suffices to check the edges , 1987, 1987 American Control Conference.

[12]  Takehiro Mori,et al.  Convergence property of interval matrices and interval polynomials , 1987 .

[13]  B. Barmish.  A Generalization of Kharitonov's Four Polynomial Concept for Robust Stability Problems with Linearly Dependent Coefficient Perturbations , 1988, 1988 American Control Conference.

[14]  Wolfgang Porod,et al.  Qualitative analysis and synthesis of a class of neural networks , 1988 .

[15]  J. Farrell,et al.  Qualitative analysis of neural networks , 1989 .

[16]  S. Bhattacharyya,et al.  Robust stability under structured and unstructured perturbations , 1990 .

[17]  A.N. Michel,et al.  Associative memories via artificial neural networks , 1990, IEEE Control Systems Magazine.

[18]  Shankar P. Bhattacharyya,et al.  On robust stability of interval control systems , 1991 .

[19]  Yuguang Fang,et al.  Sufficient conditions for the stability of interval matrices , 1993 .

[20]  Hiromasa Haneda,et al.  Computer generated Lyapunov functions for a class of nonlinear systems , 1993 .

[21]  A. Michel,et al.  Robust stability: perturbed systems with perturbed equilibria , 1993 .

[22]  Anthony N. Michel,et al.  On sufficient conditions for the stability of interval matrices , 1993 .