Stability problem of systems with multiple delay channels

A new form of coupled differential-difference equations with one delay in each channel is proposed to model systems with multiple delays. This formulation has significant advantage over the traditional differential-difference equations or the prevailing form of coupled differential-difference equations with multiple delays. Fundamental solutions, general solutions, and the construction of the Lyapunov-Krasovskii functional are discussed. For systems with a large number of state variables with multiple low-dimensional delay elements, this formulation allows a drastic reduction of computational cost as compared to the traditional differential-difference equation formulation when the discretized Lyapunov-Krasovskii functional method is used.

[1]  Silviu-Iulian Niculescu,et al.  Oscillations in lossless propagation models: a Liapunov–Krasovskii approach , 2002 .

[2]  Robert K. Brayton,et al.  Small-signal stability criterion for electrical networks containing lossless transmission lines , 1968 .

[3]  V. Kharitonov,et al.  On delay-dependent stability conditions ☆ , 2000 .

[4]  Carlos J. Moreno,et al.  The zeros of exponential polynomials (I) , 1973 .

[5]  Huang Wenzhang,et al.  Generalization of Liapunov's theorem in a linear delay system , 1989 .

[6]  Kolmanovskii,et al.  Introduction to the Theory and Applications of Functional Differential Equations , 1999 .

[7]  Jack K. Hale,et al.  On the zeros of exponential polynomials , 1980 .

[8]  Antonis Papachristodoulou,et al.  Positive Forms and Stability of Linear Time-Delay Systems , 2006, CDC.

[9]  William R. Melvin.  Stability properties of functional difference equations , 1974 .

[10]  K. Gu,et al.  Lyapunov-Krasovskii functional for uniform stability of coupled differential-functional equations , 2009, Autom..

[11]  E. Fridman.  Stability of linear descriptor systems with delay: a Lyapunov-based approach , 2002 .

[12]  Keqin Gu.  Refined discretized Lyapunov functional method for systems with multiple delays , 2003 .

[13]  Pierdomenico Pepe,et al.  On the asymptotic stability of coupled delay differential and continuous time difference equations , 2005, Autom..

[14]  Vladimir L. Kharitonov,et al.  Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems , 2003, Autom..

[15]  Luis Carvalho.  On quadratic Liapunov functionals for linear difference equations , 1996 .

[16]  Zhong-Ping Jiang,et al.  Stability results for systems described by coupled retarded functional differential equations and functional difference equations , 2007 .

[17]  Sabine Mondié,et al.  Approximations of Lyapunov-Krasovskii functionals of complete type with given cross terms in the derivative for the stability of time delay systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[18]  Erik I. Verriest,et al.  On the stability of coupled delay differential and continuous time difference equations , 2003, IEEE Trans. Autom. Control..

[19]  Silviu-Iulian Niculescu,et al.  Additional dynamics in transformed time-delay systems , 2000, IEEE Trans. Autom. Control..

[20]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[21]  Jack K. Hale,et al.  Variation of constants for hybrid systems of functional differential equations , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[22]  G. Stein,et al.  Performance and robustness analysis for structured uncertainty , 1982, 1982 21st IEEE Conference on Decision and Control.

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

[24]  Pedro Martinez-Amores.  Periodic solutions for coupled systems of differential-difference and difference equations , 1979 .

[25]  V. Kharitonov,et al.  Lyapunov matrices for neutral type time delay systems , 2005, 2005 2nd International Conference on Electrical and Electronics Engineering.

[26]  Daniel B. Henry,et al.  Linear autonomous neutral functional differential equations , 1974 .

[27]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[28]  Andrew Packard,et al.  The complex structured singular value , 1993, Autom..

[29]  Elena Panteley,et al.  Advanced topics in control systems theory : lecture notes from FAP 2004 , 2005 .

[30]  Yi Liu,et al.  Lyapunov-Krasovskii functional for coupled differential-functional equations , 2007, 2007 46th IEEE Conference on Decision and Control.

[31]  A. Friedman.  Foundations of modern analysis , 1970 .

[32]  Zhong-Ping Jiang,et al.  A New Lyapunov-Krasovskii Methodology for Coupled Delay Differential Difference Equations , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[33]  Hongfei Li,et al.  Discretized Lyapunov-Krasovskii Functional for Systems With Multiple Delay Channels , 2008 .

[34]  S. Niculescu,et al.  Stability analysis of time-delay systems : A lyapunov approach , 2006 .