A Novel Frequency-Domain Approach for the Exact Range of Imaginary Spectra and the Stability Analysis of LTI Systems With Two Delays

This paper presents a novel frequency-domain approach to reveal the exact range of the imaginary spectra and the stability of linear time-invariant systems with two delays. First, an exact relation, i.e., the Rekasius substitution, is used to replace the exponential term caused by the delays in order to transform the transcendental characteristic equation to a quasi-polynomial. Second, this quasi-polynomial is uniquely tackled by our proposed Dixon resultant and discriminant theory, leading to the elimination of delay-related elements and the revelation of the exact range of the frequency spectra of the original system of interest. Then, by sweeping the frequency over this obtained range, the stability switching curves are declared exhaustively. Last, we deploy the cluster treatment of characteristic roots (CTCR) paradigm to reveal the exact and complete stability map. The proposed methodologies are tested and verified by a numerical method called Quasi-Polynomial mapping-based Root finder (QPmR) over an example case.

[1]  Deepak Kapur,et al.  Algebraic and geometric reasoning using Dixon resultants , 1994, ISSAC '94.

[2]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[3]  Nejat Olgaç,et al.  Optimal sign inverting control for time-delayed systems, a concept study with experiments , 2015, Int. J. Control.

[4]  F. S. Macaulay,et al.  The Algebraic Theory of Modular Systems , 1972 .

[5]  Nejat Olgaç,et al.  Critical Effects of the Polarity Change in Delayed States Within an LTI Dynamics With Multiple Delays , 2015, IEEE Transactions on Automatic Control.

[6]  Nejat Olgaç,et al.  Sign inverting and Delay Scheduling Control concepts with multiple rationally independent delays , 2014, 2014 American Control Conference.

[7]  Fanglai Zhu,et al.  Reduced-order switched UIO design for switched discrete-time descriptor systems , 2018, Nonlinear Analysis: Hybrid Systems.

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

[9]  Jun Huang,et al.  Further Result on Interval Observer Design for Discrete-Time Switched Systems and Application to Circuit Systems , 2020, IEEE Transactions on Circuits and Systems II: Express Briefs.

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

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

[12]  Qian Xiao,et al.  Time-delay stability switching boundary determination for DC microgrid clusters with the distributed control framework , 2018 .

[13]  Libor Pekař,et al.  Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays: A Literature Overview of Some Recent Results , 2018, IEEE Access.

[14]  Nejat Olgaç,et al.  Consensus analysis with large and multiple communication delays using spectral delay space concept , 2011, Int. J. Control.

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

[16]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[17]  Ismail Ilker Delice,et al.  Advanced Clustering With Frequency Sweeping Methodology for the Stability Analysis of Multiple Time-Delay Systems , 2011, IEEE Transactions on Automatic Control.

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

[19]  Seul Jung,et al.  Neural Network Control for Position Tracking of a Two-Axis Inverted Pendulum System: Experimental Studies , 2007, IEEE Transactions on Neural Networks.

[20]  Nejat Olgac,et al.  A test platform for cognitive delays: target tracking problem with multiple time-delayed feedback control , 2014 .

[21]  Nejat Olgaç,et al.  Stability analysis for LTI systems with multiple time delays using the bounds of its imaginary spectra , 2017, Syst. Control. Lett..

[22]  Michael Trott,et al.  The Mathematica GuideBook for Symbolics , 2006, The Mathematica Guidebook.

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

[24]  A. L. Dixon.  The Eliminant of Three Quantics in two Independent Variables , 1909 .

[25]  J. Hale,et al.  Stability in Linear Delay Equations. , 1985 .

[26]  Nejat Olgac,et al.  Investigation of Local Stability Transitions in the Spectral Delay Space and Delay Space , 2014 .

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

[28]  John F. Canny,et al.  An Efficient Algorithm for the Sparse Mixed Resultant , 1993, AAECC.

[29]  Jun Huang,et al.  Interval observer design method for asynchronous switched systems , 2020, IET Control Theory & Applications.

[30]  Nejat Olgac,et al.  Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS) , 2007 .

[31]  Huijun Gao,et al.  Delay-independent stability analysis of linear time-delay systems based on frequency discretization , 2016, Autom..

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

[33]  M. Stark,et al.  THE THEORY OF ELIMINATION , 1964 .

[34]  Tomás Vyhlídal,et al.  Mapping Based Algorithm for Large-Scale Computation of Quasi-Polynomial Zeros , 2009, IEEE Transactions on Automatic Control.

[35]  Nejat Olgaç,et al.  Combination of sign inverting and delay scheduling control concepts for multiple-delay dynamics , 2015, Syst. Control. Lett..