Delay-dependent stability analysis by using delay-independent integral evaluation

For the stability analysis of time-delay systems, the available methods usually require the exact evaluation of some quantities. The definite integral stability method, originated from the Argument Principle or the Cauchy Theorem, is effective because it only requires a rough estimation of the testing integral over a finite interval to judge stability. However, no general rule is given in the literature for properly choosing the upper limit of the testing integral. In this paper, two simple algorithms are presented for finding the parameter-dependent critical upper limit and a parameter-independent upper limit without any restriction on the number of time delays. These results improve and complete the definite integral stability method. As illustrated by the numerical examples, the proposed algorithms work effectively and accurately.

[1]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

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

[3]  Haiyan Hu,et al.  Symbolic computation of normal form for Hopf bifurcation in a neutral delay differential equation and an application to a controlled crane , 2012 .

[4]  Gábor Stépán,et al.  Traffic jams: dynamics and control , 2010, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  Magnus Egerstedt,et al.  Autonomous driving in urban environments: approaches, lessons and challenges , 2010, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  S.-I. Niculescu.  A model transformation class for delay-dependent stability analysis , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[7]  C. Abdallah,et al.  Delayed Positive Feedback Can Stabilize Oscillatory Systems , 1993, 1993 American Control Conference.

[8]  Yanjun Shen,et al.  On delay-dependent stability for linear neutral systems , 2005, IEEE International Conference Mechatronics and Automation, 2005.

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

[10]  Gbor Stpn.  Modelling nonlinear regenerative effects in metal cutting , 2001 .

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

[12]  Ali H. Nayfeh,et al.  Sway Reduction on Container Cranes Using Delayed Feedback Controller , 2003 .

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

[14]  Minyue Fu,et al.  The edge theorem and graphical tests for robust stability of neutral time-delay systems , 1989, Autom..

[15]  Mike McDonald,et al.  Car-following: a historical review , 1999 .

[16]  Balakumar Balachandran,et al.  Stability analysis for milling process , 2007 .

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

[18]  Wim Michiels.  Stability and stabilization of time-delay systems , 2002 .

[19]  T. Erneux.  Applied Delay Differential Equations , 2009 .

[20]  B. Hassard,et al.  Counting Roots of the Characteristic Equation for Linear Delay-Differential Systems , 1997 .

[21]  Qi Xu,et al.  Exact stability test of neutral delay differential equations via a rough estimation of the testing integral , 2014 .

[22]  Silviu-Iulian Niculescu,et al.  Additional dynamics in transformed time-delay systems , 2000, IEEE Trans. Autom. Control..

[23]  Jean-Michel Dion,et al.  On delay-dependent stability for linear neutral systems , 2003, Autom..

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

[25]  Wim Michiels,et al.  Stability and Stabilization of Time-Delay Systems (Advances in Design & Control) (Advances in Design and Control) , 2007 .

[26]  A. Olbrot,et al.  Robust stability for time-delay systems: the edge theorem and graphical tests , 1989 .

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

[28]  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..

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