control systems; differential equations; time-variance; observability; asymptotic stability; circle criterion
Abstract :
[en] This paper presents some new criteria for uniform and non uniform asymtotic stability of equilibria for time-variant differential equations and this within a Liapunov approach. The stability criteria are formulated in terms of certain observability conditions with the output derived from the Liapunov function. For some classes of systems, this system theoretic interpretation proves to be fruitful since - after establishing the invariance of observability under output injection - this enables us to check the stability criteria on a simpler system. This procedure is illustrated for some classical examples.
Disciplines :
Electrical & electronics engineering
Author, co-author :
Aeyels, D.; Universiteit Gent - Ugent > Department of Systems Dynamics
Sepulchre, Rodolphe ; Université Catholique de Louvain - UCL > Center for Systems Engineering and Applied Mechanics
Peuteman, J.; Universiteit Gent - Ugent > Department of Systems Dynamics
Language :
English
Title :
Asymptotic stability conditions for time-variant systems and observability : uniform and non-uniform criteria
Aeyels, D., Stability of nonautonomous systems by Liapunov's direct method, in Geometry in Nonlinear Control and Differential Inclusions, B. Jakubczyk, W. Respondek, and T. Rzezuchowski, eds., Polish Academy of Sciences, Institute of Mathematics, Banach Center Publications, vol. 32, 1995, pp. 9-17.
Aeyels, D., A new criterion for asymptotic stability of nonautonomous differential equations, Systems & Control Letters, vol. 25, no. 4, 1995, pp. 273-280.
Aeyels, D., Sepulchre, R., On the convergence of a time-variant linear differential equation arising in identification, Kybernetica, vol. 30, no. 6, 1994, pp. 715-723.
Anderson, B. D. O., Exponential stability of linear equations arising in adaptive identification, IEEE Transactions on Automatic Control, vol. 22, 1977, pp. 84-88.
Anderson, B. D. O., Moore, J. B., New results in linear system stability, SIAM Journal on Control, vol. 7, no. 3, 1969, pp. 398-414.
Artstein, Z., Uniform asymptotic stability via the limiting equations, Journal of Differential Equations, vol. 27, 1978, pp. 172-189.
Artstein, Z., Stability, observability and invariance, Journal of Differential Equations, vol. 44, 1982, pp. 224-248.
D'Anna, A., Maio, A., Moauro, V., Global stability properties by means of limiting equations, Nonlinear Analysis, Theory, Methods and Applications, vol. 4, no. 2, 1980, pp. 407-410.
Dieudonné, J., Foundations of Modern Analysis, Academic Press, New York, 1960.
Hermes, H., Discontinuous vector fields and feedback control, in Differential Equations and Dynamical Systems, J. K. Hale and J. P. LaSalle, eds., Academic Press, New York, 1967, pp. 155-165.
Kalman, R. E., Contributions to the theory of optimal control, Boletín de la Sociedad Matemática Mexicana, vol. 5, 1960, pp. 102-119.
Khalil, H. K., Nonlinear Systems, Macmillan, New York, 1992.
Miller, R. K., Michel, A. N., Asymptotic stability of systems: results involving the system topology, SIAM Journal on Control and Optimization, vol. 18, no. 2, 1980, pp. 181-190.
Morgan, A. P., Narendra, K. S., On the uniform asymptotic stability of certain linear nonautonomous differential equations, SIAM Journal on Control and Optimization, vol. 15, no. 1, 1977, pp. 5-24.
Narendra, K. S., Annaswamy, A. M., Persistent excitation in adaptive systems, International Journal of Control, vol. 45, no. 1, 1987, pp. 127-160.
Rouche, N., Habets, P., Laloy, M., Stability Theory by Liapunov's Direct Method, Springer-Verlag, New York, 1977.
Sepulchre, R., Contributions to nonlinear control systems analysis by means of the direct method of Lyapunov, Ph.D. Dissertation, Université Catholique de Louvain, September 1994.
Willems, J. L., Stability Theory of Dynamical Systems, Nelson, London, 1970.
