A stability study on first-order neutral systems with three rationally independent time delays

First-order linear time invariant and time-delayed dynamics of neutral type is taken into account with three rationally independent delays. There are two main contributions of this study. (a) It is the first complete treatment in the literature on the stability analysis of systems with three delays. We use a recent procedure, the cluster treatment of characteristic roots (CTCR), for this purpose. This procedure results in an exact and exhaustive stability tableau in the domain of the three delays. (b) It provides a proof of a complex concept called the delay-stabilisability (also known as strong stability) as a by-product of CTCR. Furthermore, we deploy a numerical method (infinitesimal generator approach) to approximate the dominant characteristic roots of this class of systems, which concur with the stability outlook generated by CTCR.

[1]  R. Bellman,et al.  Differential-Difference Equations , 1967 .

[2]  Dimitri Breda,et al.  Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations , 2005, SIAM J. Sci. Comput..

[3]  Roger D. Nussbaum,et al.  Differential-delay equations with two time lags , 1978 .

[4]  Dimitri Breda,et al.  An adaptive algorithm for efficient computation of level curves of surfaces , 2009, Numerical Algorithms.

[5]  Lamberto Cesari,et al.  Dynamical systems : an international symposium , 1976 .

[6]  Min-Sen Chiu,et al.  Decoupling internal model control for multivariable systems with multiple time delays , 2002 .

[7]  G. Samaey,et al.  DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations , 2001 .

[8]  R. Vermiglio,et al.  Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions , 2006 .

[9]  Jie Chen.  On computing the maximal delay intervals for stability of linear delay systems , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[10]  Silviu-Iulian Niculescu.  On delay robustness analysis of a simple control algorithm in high-speed networks , 2001 .

[11]  F. du Burck,et al.  Stability and convergence speed of narrowband controller , 2005 .

[12]  Nejat Olgaç,et al.  A practical method for analyzing the stability of neutral type LTI-time delayed systems , 2004, Autom..

[13]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

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

[15]  J. G. Ziegler,et al.  Optimum Settings for Automatic Controllers , 1942, Journal of Fluids Engineering.

[16]  John L. Casti,et al.  Introduction to the theory and application of differential equations with deviating arguments , 1973 .

[17]  Onur Toker,et al.  Mathematics of Control , Signals , and Systems Complexity Issues in Robust Stability of Linear Delay-Differential Systems * , 2005 .

[18]  Sitian Qin,et al.  Global Exponential Stability and Global Convergence in Finite Time of Neural Networks with Discontinuous Activations , 2009, Neural Processing Letters.

[19]  Nejat Olgac,et al.  The Cluster Treatment of Characteristic Roots and the Neutral Type Time-Delayed Systems , 2005 .

[20]  Denis Dochain,et al.  Sensitivity to Infinitesimal Delays in Neutral Equations , 2001, SIAM J. Control. Optim..

[21]  Jack K. Hale,et al.  Effects of Small Delays on Stability and Control , 2001 .

[22]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[23]  Rifat Sipahi,et al.  Complete Stability Analysis of Neutral-Type First Order Two-Time-Delay Systems with Cross-Talking Delays , 2006, SIAM J. Control. Optim..

[24]  Nejat Olgac,et al.  Degenerate Cases in Using the Direct Method , 2003 .

[25]  Ismail Ilker Delice,et al.  Asymptotic stability of constant time headway driving strategy with multiple driver reaction delays , 2009, 2009 American Control Conference.

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

[27]  Rifat Sipahi,et al.  Stability Robustness of Retarded LTI Systems with Single Delay and Exhaustive Determination of Their Imaginary Spectra , 2006, SIAM J. Control. Optim..

[28]  Nejat Olgac,et al.  Direct method for analyzing the stability of neutral type LTI-time delayed systems , 2003 .

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

[30]  Chengjun Sun,et al.  Global existence of periodic solutions in a special neural network model with two delays , 2009 .

[31]  Brad Lehman,et al.  Setpoint PI controllers for systems with large normalized dead time , 1996, IEEE Trans. Control. Syst. Technol..

[32]  Carlos F. Daganzo,et al.  On the Stability of Supply Chains , 2002, Oper. Res..

[33]  Nejat Olgaç,et al.  An improved procedure in detecting the stability robustness of systems with uncertain delay , 2006, IEEE Transactions on Automatic Control.

[34]  Chuandong Li,et al.  Existence and Global Exponential Stability of Periodic Solution of Cellular Neural Networks with impulses and Leakage Delay , 2009, Int. J. Bifurc. Chaos.

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

[36]  Ismail Ilker Delice,et al.  Stability of inventory dynamics in supply chains with three delays , 2010 .

[37]  Silviu-Iulian Niculescu,et al.  On delay robustness analysis of a simple control algorithm in high-speed networks , 2002, Autom..

[38]  J. Hale,et al.  Global geometry of the stable regions for two delay differential equations , 1993 .

[39]  Z. Rekasius,et al.  A stability test for systems with delays , 1980 .

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

[41]  Nejat Olgaç,et al.  A unique methodology for the stability robustness of multiple time delay systems , 2006, Syst. Control. Lett..

[42]  Pavel Zítek,et al.  Quasipolynomial mapping based rootfinder for analysis of time delay systems , 2003 .

[43]  S. Niculescu,et al.  Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach , 2007 .

[44]  Nejat Olgaç,et al.  Complete stability robustness of third-order LTI multiple time-delay systems , 2005, Autom..

[45]  A. TUSTIN,et al.  Automatic Control Systems , 1950, Nature.