Stability crossing set for systems with three scalar delay channels

The characteristic function of a system with three scalar delay channels contains cross terms of different delays. This article studies the parameterization and geometric structure of the stability crossing set (the set of delay combinations with at least one characteristic root on the imaginary axis) for such systems. Understanding the structure of this set is crucial to the identification of stable regions in the delay parameter space using the D-subdivision method. The presence of the cross terms significantly complicates the analysis, and requires a quite different method than the case without these cross terms, and it involves more numerical computation.

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

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

[3]  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..

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

[5]  Tomás Vyhlídal,et al.  A New Perspective in the Stability Assessment of Neutral Systems with Multiple and Cross-Talking Delays , 2008, SIAM J. Control. Optim..

[6]  S. Ruan,et al.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays , 2003 .

[7]  Dirk Roose,et al.  Limitations of a class of stabilization methods for delay systems , 2001, IEEE Trans. Autom. Control..

[8]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[9]  Ismail Ilker Delice,et al.  Extraction of 3D stability switching hypersurfaces of a time delay system with multiple fixed delays , 2009, Autom..

[10]  Boris Polyak,et al.  D-decomposition technique state-of-the-art , 2008 .

[11]  V. Van Assche,et al.  Some problems arising in the implementation of distributed-delay control laws , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[12]  N. Olgac,et al.  Direct method implementation for the stability analysis of multiple time delayed systems , 2003, Proceedings of 2003 IEEE Conference on Control Applications, 2003. CCA 2003..

[13]  A. Hatley.  Mathematics in Science and Engineering , Volume 6: Differential- Difference Equations. Richard Bellman and Kenneth L. Cooke. Academic Press, New York and London. 462 pp. 114s. 6d. , 1963, The Journal of the Royal Aeronautical Society.

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

[15]  Gjerrit Meinsma,et al.  Robustness of the stability of feedback systems with respect to small time delays , 1999 .

[16]  Keqin Gu,et al.  A Review of Some Subtleties of Practical Relevance , 2012 .

[17]  Mohammad Naghnaeian,et al.  Stability Crossing Set for Systems With Three Delays , 2011, IEEE Transactions on Automatic Control.

[18]  Elias Jarlebring,et al.  Critical delays and polynomial eigenvalue problems , 2007, 0706.1634.

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

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

[21]  V. Kolmanovskii,et al.  Stability of Functional Differential Equations , 1986 .

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

[23]  Hongfei Li,et al.  Discretized Lyapunov-Krasovskii functional for coupled differential-difference equations with multiple delay channels , 2010, Autom..

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

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

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

[27]  Keqin Gu,et al.  Stability problem of systems with multiple delay channels , 2010, Autom..

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

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

[30]  Zalman J. Palmor,et al.  Stability properties of Smith dead-time compensator controllers , 1980 .

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