Stability analysis of continuous-time periodic systems via the harmonic analysis

Asymptotic stability of finite-dimensional linear continuous-time periodic (FDLCP) systems is studied by harmonic analysis. It is first shown that stability can be examined with what we call the harmonic Lyapunov equation. Another necessary and sufficient stability criterion is developed via this generalized Lyapunov equation, which reduces the stability test into that of an approximate FDLCP model whose transition matrix can be determined explicitly. By extending the Gerschgorin theorem to linear operators on the linear space l/sub 2/, yet another disc-group criterion is derived, which is only sufficient. Stability of the lossy Mathieu equation is analyzed as a numerical example to illustrate the results.

[1]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[2]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[3]  G. P. Szegö,et al.  Stability theory of dynamical systems , 1970 .

[4]  I. Gohberg,et al.  Classes of Linear Operators , 1990 .

[5]  Arch W. Naylor,et al.  Linear Operator Theory in Engineering and Science , 1971 .

[6]  Henk Nijmeijer,et al.  System identification in communication with chaotic systems , 2000 .

[7]  Pramod P. Khargonekar,et al.  Robust stability and performance analysis of sampled-data systems , 1993, IEEE Trans. Autom. Control..

[8]  Giuseppe De Nicolao,et al.  Zeros of Continuous-time Linear Periodic Systems , 1998, Autom..

[9]  Ruel V Churchill Prof.  Fourier Series and Boundary Value Problems , 1942, Nature.

[10]  Norman M. Wereley,et al.  Generalized Nyquist Stability Criterion for Linear Time Periodic Systems , 1990, 1990 American Control Conference.

[11]  Bruce A. Francis,et al.  Optimal Sampled-Data Control Systems , 1996, Communications and Control Engineering Series.

[12]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[13]  Jun Zhou,et al.  Trace Formulas for the H 2 Norm of Linear Continuous-Time Periodic Systems , 2001 .

[14]  Geir E. Dullerud,et al.  Robust stabilization of sampled-data systems to structured LTI perturbations , 1993, IEEE Trans. Autom. Control..

[15]  Miklós Farkas,et al.  Periodic Motions , 1994 .

[16]  Tomomichi Hagiwara,et al.  Stability analysis of continuous-time periodic systems via the harmonic analysis , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[17]  A. A. Coley.  The Theory of Dynamical Systems , 2003 .

[18]  Tomomichi Hagiwara,et al.  Absolute stability of sampled-data systems with a sector nonlinearity , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[19]  Jun Zhou,et al.  Existence Conditions and Properties of the Frequency Response Operators of Continuous-Time Periodic Systems , 2001, SIAM J. Control. Optim..

[20]  H. Antosiewicz,et al.  Differential Equations: Stability, Oscillations, Time Lags , 1967 .

[21]  Norman M. Wereley,et al.  Analysis and control of linear periodically time varying systems , 1990 .