Freezing method approach to an asymptotic stability of the discrete-time oscillator equation

The presented research work considers stability criteria of second-order differential equation. The second-order discrete-time oscillator equation is obtained from discretization of second order continuous-time equation using the forward difference operator. The stability criteria are drawn with freezing method and are presented in the terms of the equation coefficients. Finally, an illustrative example is shown.

[1]  A. Czornik,et al.  On the spectrum of discrete time-varying linear systems , 2013 .

[2]  Aleksander Nawrat,et al.  On new estimates for Lyapunov exponents of discrete time varying linear systems , 2010, Autom..

[3]  C. Di Bello,et al.  Asymptotic formulas in nearly sinusoidal nonlinear oscillators , 1996 .

[4]  A. Czornik,et al.  Lyapunov exponents for systems with unbounded coefficients , 2013 .

[5]  Ravi P. Agarwal,et al.  Discrete Oscillation Theory , 2005 .

[6]  A. Czornik,et al.  On the Lyapunov exponents of a class of second-order discrete time linear systems with bounded perturbations , 2013 .

[7]  Rigoberto Medina,et al.  Absolute Stability of Discrete-Time Systems with Delay , 2007 .

[8]  M. T. Qureshi,et al.  Lyapunov Matrix Equation in System Stability and Control , 2008 .

[9]  Jurang Yan,et al.  Oscillation and Stability of Linear Impulsive Delay Differential Equations , 1998 .

[10]  Dennis S. Bernstein,et al.  The Majorant Lyapunov Equation: A Nonnegative Matrix Equation for Robust Stability and Performance of Large Scale Systems , 1987, 1987 American Control Conference.

[11]  A. Yu. Kulikov.  Stability of a linear nonautonomous difference equation with bounded delays , 2010 .

[12]  Sui Sun Cheng,et al.  A stability criterion for a neutral difference equation with delay , 1994 .

[13]  R. A. Smith.  ASYMPTOTIC STABILITY OF x ″+ a ( t ) x ′+ x =0 , 1961 .

[14]  C. Desoer.  Slowly varying discrete system xi+1=Aixi , 1970 .

[15]  Artur Babiarz,et al.  On the number of upper Bohl exponents for diagonal discrete time-varying linear system , 2015 .

[16]  Jitsuro Sugie,et al.  Global Asymptotic Stability for Oscillators with Superlinear Damping , 2012 .

[17]  Jianshe Yu,et al.  Asymptotic stability for a linear difference equation with variable delay , 1998 .

[18]  P. Olver.  Nonlinear Systems , 2013 .

[19]  A. Czornik,et al.  On the exponential exponents of discrete linear systems , 2010 .

[20]  John R. Graef,et al.  On the asymptotic behavior of solutions of impulsively damped nonlinear oscillator equations , 1996 .

[21]  Aleksander Nawrat,et al.  On the stability of the discrete time jump linear system , 2006, ICINCO-SPSMC.

[22]  Hai Lin,et al.  Switched Linear Systems: Control and Design , 2006, IEEE Transactions on Automatic Control.

[23]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[24]  Aleksander Nawrat,et al.  On the Sigma Exponent of Discrete Linear Systems , 2010, IEEE Transactions on Automatic Control.

[25]  Guang Zhang,et al.  OSCILLATION CRITERIA FOR A NEUTRAL DIFFERENCE EQUATION WITH DELAY , 1995 .

[26]  Rigoberto Medina,et al.  The Freezing Method for Linear Difference Equations , 2002 .