Lyapunov-Krasovskii Functional Approach for Coupled Differential-Difference Equations with Multiple Delays

Coupled differential-difference equations (coupled DDEs) represent a very general class of time-delay systems. Indeed, traditional DDEs of retarded or neutral type, as well as singular systems, can all be considered as special cases of coupled DDEs. The coupled DDE formulation is especially effective when a system has a large number of state variables, but only a few of them involve time delays. In this chapter, the stability of such systems is studied by using a Lyapunov-Krasovskii functional method. For linear systems, a quadratic Lyapunov-Krasovskii functional is discretized to reduce the stability problem to a set of linear matrix inequalities for which effective numerical algorithms are available, and widely implemented in such software packages as MATLAB.

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

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

[3]  Jack K. Hale,et al.  Strong stabilization of neutral functional differential equations , 2002 .

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

[5]  R. Brayton.  Nonlinear oscillations in a distributed network , 1967 .

[6]  Rifat Sipahi,et al.  An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems , 2002, IEEE Trans. Autom. Control..

[7]  Nejat Olgaç,et al.  Stability Robustness Analysis of Multiple Time- Delayed Systems Using “Building Block” Concept , 2007, IEEE Transactions on Automatic Control.

[8]  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.

[9]  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.

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

[11]  K. Cooke,et al.  On zeroes of some transcendental equations , 1986 .

[12]  Zhong-Ping Jiang,et al.  Input-to-state stability for discrete-time nonlinear systems , 1999 .

[13]  Vladimir Răsvan.  FUNCTIONAL DIFFERENTIAL EQUATIONS OF LOSSLESS PROPAGATION AND ALMOST LINEAR BEHAVIOR , 2006 .

[14]  Zhong-Ping Jiang,et al.  Stability results for systems described by retarded functional differential equations , 2007, 2007 European Control Conference (ECC).

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

[16]  Keqin Gu,et al.  Large systems with multiple low-dimensional delay channels , 2008, 2008 Chinese Control and Decision Conference.

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

[18]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .

[19]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[20]  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.

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

[22]  K. Gu.  A further refinement of discretized Lyapunov functional method for the stability of time-delay systems , 2001 .

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

[24]  Jie Chen,et al.  On stability crossing curves for general systems with two delays , 2004 .

[25]  J. Hale,et al.  Existence, uniqueness and continuous dependence for hereditary systems , 1970 .

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

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

[28]  John C. Doyle.  Analysis of Feedback Systems with Structured Uncertainty , 1982 .

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

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

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

[32]  Kirk S. Walton,et al.  Direct method for TDS stability analysis , 1987 .

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

[34]  Sukhan Lee,et al.  Modeling, design, and evaluation of advanced teleoperator control systems with short time delay , 1993, IEEE Trans. Robotics Autom..

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

[36]  A. Teel.  Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem , 1998, IEEE Trans. Autom. Control..

[37]  Eduardo Sontag.  Further facts about input to state stabilization , 1990 .

[38]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

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

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

[41]  G. Stépán.  Retarded dynamical systems : stability and characteristic functions , 1989 .

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

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

[44]  Xinghuo Yu,et al.  A DISCRETIZED LYAPUNOV FUNCTIONAL APPROACH TO STABILITY OF LINEAR DELAY-DIFFERENTIAL SYSTEMS OF NEUTRAL TYPE , 2002 .

[45]  J. Hale,et al.  Stability of functional differential equations of neutral type , 1970 .

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

[47]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.