A Polynomial Method for Stability Analysis of LTI Systems Independent of Delays

A new method providing necessary and sufficient conditions to test delay-independent stability for general linear time-invariant systems with constant delays is proposed. The method is utilized for single delay and incommensurate multiple delay systems. The proposed method offers an approach to determine the exact boundaries of unknown parameters such as controller gains or system parameters ensuring delay-independent stability, in addition to exhibiting an efficient test for real parameters. The technique is based on nonexistence of unitary complex zeros of an auxiliary characteristic polynomial obtained via extended Kronecker summation. A special feature of the polynomial, i.e., the self-inversive property, is proved and utilized to check its unitary zeros to determine delay-independent stability by an efficient zero location test. The methodology is executed employing simple algebraic operations and inspection of the number of sign variations in the obtained sequence. For the single delay case, the pro...

[1]  Emilia Fridman,et al.  Tutorial on Lyapunov-based methods for time-delay systems , 2014, Eur. J. Control.

[2]  Rifat Sipahi,et al.  Controller design for delay-independent stability of linear time-invariant vibration systems with multiple delays ☆ , 2013 .

[3]  Jie Chen,et al.  Frequency sweeping tests for stability independent of delay , 1995, IEEE Trans. Autom. Control..

[4]  Keqin Gu,et al.  Stability and Stabilization of Systems with Time Delay , 2011, IEEE Control Systems.

[5]  Jie Chen,et al.  On sufficient conditions for stability independent of delay , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[6]  Yuval Bistritz,et al.  Zero location of polynomials with respect to the unit-circle unhampered by nonessential singularities , 2002 .

[7]  Rifat Sipahi,et al.  Analytical Boundaries of Controller Gains for Delay-Independent Stability of LTI Systems with Single Output Delay , 2012, TDS.

[8]  James Louisell,et al.  A matrix method for determining the imaginary axis eigenvalues of a delay system , 2001, IEEE Trans. Autom. Control..

[9]  Ali Fuat Ergenc,et al.  A New Method for Delay-Independent Stability of Time-Delayed Systems , 2010 .

[10]  Ismail Ilker Delice,et al.  Delay-Independent Stability Test for Systems With Multiple Time-Delays , 2012, IEEE Transactions on Automatic Control.

[11]  E. Jury,et al.  Stability independent and dependent of delay for delay differential systems , 1984 .

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

[13]  Joe Brewer,et al.  Kronecker products and matrix calculus in system theory , 1978 .

[14]  Yuval Bistritz.  A modified unit-circle zero location test , 1996 .

[15]  Yuval Bistritz,et al.  A circular stability test for general polynomials , 1986 .

[16]  Ismail Ilker Delice,et al.  Controller Design for Delay-Independent Stability of Multiple Time-Delay Systems via DéScirctes's Rule of Signs , 2010 .

[17]  A. Thowsen.  The Routh-Hurwitz method for stability determination of linear differential-difference systems† , 1981 .

[18]  S. Niculescu.  Stability and hyperbolicity of linear systems with delayed state: a matrix-pencil approach , 1998 .

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

[20]  V. Ionescu,et al.  On delay-independent stability criteria: a matrix-pencil approach , 1997 .

[21]  Pierre-Alexandre Bliman,et al.  Lyapunov equation for the stability of linear delay systems of retarded and neutral type , 2002, IEEE Trans. Autom. Control..

[22]  Arild Thowsen.  Delay-independent asymptotic stability of linear systems , 1982 .

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

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

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

[26]  S. Barnett,et al.  Matrix Differential Equations and Kronecker Products , 1973 .

[27]  Silviu-Iulian Niculescu,et al.  Survey on Recent Results in the Stability and Control of Time-Delay Systems* , 2003 .

[28]  E. W. Kamen,et al.  Linear systems with commensurate time delays: stability and stabilization independent of delay , 1982 .

[29]  Maurício C. de Oliveira,et al.  Stability independent of delay using rational functions , 2009, Autom..

[30]  W. Rogosinski,et al.  The Geometry of the Zeros of a Polynomial in a Complex Variable , 1950, The Mathematical Gazette.

[31]  Ali Fuat Ergenc,et al.  An Experimental Study for Delay-Independent State-Feedback Controller Design , 2014 .

[32]  Ping Wei,et al.  Easily testable necessary and sufficient algebraic criteria for delay-independent stability of a class of neutral differential systems , 2008, Syst. Control. Lett..

[33]  Haiyan Hu,et al.  Robust Hurwitz stability test for linear systems with uncertain commensurate time delays , 2004, IEEE Transactions on Automatic Control.

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

[35]  R. Datko.  A procedure for determination of the exponential stability of certain differential-difference equations , 1978 .

[36]  Nejat Olgaç,et al.  Extended Kronecker Summation for Cluster Treatment of LTI Systems with Multiple Delays , 2007, SIAM J. Control. Optim..

[37]  Ali Fuat Ergenc,et al.  A New Delay-Independent Stability Test for LTI Systems with Single Delay , 2015 .

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

[39]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[40]  T. SHEIL-SMALL,et al.  Complex Polynomials: Contents , 2002 .

[41]  Stability , 1973 .

[42]  Ismail Ilker Delice,et al.  On some features of core hypersurfaces related to stability switching of LTI systems with multiple delays , 2014, IMA J. Math. Control. Inf..

[43]  Bahram Shafai,et al.  On controller design for delay-independent stability of linear time-invariant systems with multiple delays , 2013, 2013 European Control Conference (ECC).

[44]  Y. Bistritz.  Zero location with respect to the unit circle of discrete-time linear system polynomials , 1984 .

[45]  Peilin Fu,et al.  Robust Stability of Quasi-Polynomials: Frequency-Sweeping Conditions and Vertex Tests , 2008, IEEE Transactions on Automatic Control.