The Application of Shifted Legendre Polynomials to Time-Delay Systems and Parameter Identification

1985 ◽  
Vol 107 (1) ◽  
pp. 79-85 ◽  
Author(s):  
Rong-Yeu Chang ◽  
Maw-Ling Wang
Keyword(s):  
Time Delay ◽  
Parameter Identification ◽  
Legendre Polynomials ◽  
Linear Time ◽  
Delay System ◽  
Delay Systems ◽  
Time Interval ◽  
Algebraic Equations ◽  
Shifted Legendre Polynomials ◽  
Expansion Coefficients

A linear time-delay state equation is solved by the proposed shifted Legendre polynomials method. The parameter identification of such a system with time delay is also studied. The system is partitioned into several time intervals. Within a certain time interval, the state and control functions are assumed to be expressed by the shifted Legendre polynomials series. Time-delay differential equations are transformed into a series of algebraic equations of expansion coefficients. An effective algorithm is proposed to solve the time-delay system problem and to estimate the system parameters. Only a small number of leading terms of expansion coefficients is enough to get accurate results. By using such an effective computational algorithm, the calculation procedures are greatly simplified. Thus much computer time is saved.

Parameterization approach to stability and feedback stabilization of linear time-delay systems

2009 ◽  
Vol 223 (7) ◽  
pp. 929-939
Author(s):  
M S Mahmoud ◽  
A Ismail ◽  
F M Al-Sunni
Keyword(s):  
Time Delay ◽  
Linear Time ◽  
Delay System ◽  
Feedback Stabilization ◽  
Delay Systems ◽  
Linear Matrix ◽  
Parameter Uncertainties ◽  
Systematic Analysis ◽  
Feedback Scheme ◽  
Delay Dependent

This paper develops a new parameterized approach to the problems of delay-dependent analysis and feedback stabilization for a class of linear continuous-time systems with time-varying delays. An appropriate Lyapunov-Krasovskii functional is constructed to exhibit the delay-dependent dynamics. The construction guarantees avoiding bounding methods and effectively deploying injecting parametrized variables to facilitate systematic analysis. Delay-dependent stability provides a characterization of linear matrix inequalities (LMIs)-based conditions under which the linear time-delay system is asymptotically stable with a γ-level £2 gain. By delay-dependent stabilization, a state-feedback scheme is designed to guarantee that the closed-loop switched system enjoys the delay-dependent asymptotic stability with a prescribed γ-level £2 gain. It is established that the methodology provides the least conservatism in comparison with other published methods. Extension to systems with convex-bounded parameter uncertainties in all system matrices is also provided. All the developed results are tested on representative examples.

scholarly journals Stochastic stability of linear time-delay system with Markovian jumping parameters

1997 ◽  
Vol 3 (3) ◽  
pp. 187-201 ◽  
Author(s):  
K. Benjelloun ◽  
E. K. Boukas
Keyword(s):  
Time Delay ◽  
Stochastic Stability ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Delay System ◽  
Delay Systems ◽  
Necessary Condition ◽  
Markovian Jumping Parameters ◽  
Markovian Jumping ◽  
The Stability

This paper deals with the class of linear time-delay systems with Markovian jumping parameters (LTDSMJP). We mainly extend the stability results of the deterministic class of linear systems with time-delay to this class of systems. A delay-independent necessary condition and sufficient conditions for checking the stochastic stability are established. A sufficient condition is also given. Some numerical examples are provided to show the usefulness of the proposed theoretical results.

Rapid parameter identification of linear time-delay system from noisy frequency domain data

2020 ◽  
Vol 83 ◽  
pp. 736-753
Author(s):  
Guang Liu ◽  
Min-li Yu ◽  
Li Wang ◽  
Zhi-yi Yin ◽  
Ji-ke Liu ◽  
...  
Keyword(s):  
Time Delay ◽  
Parameter Identification ◽  
Frequency Domain ◽  
Linear Time ◽  
Delay System ◽  
Time Delay System

scholarly journals Dynamic Environment Coupling Induced Synchronized States in Coupled Time-Delayed Electronic Circuits

2014 ◽  
Vol 24 (05) ◽  
pp. 1450067 ◽  
Author(s):  
R. Suresh ◽  
K. Srinivasan ◽  
D. V. Senthilkumar ◽  
K. Murali ◽  
M. Lakshmanan ◽  
...  
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Phase Synchronization ◽  
Linear Time ◽  
Piecewise Linear ◽  
Dynamic Environment ◽  
Delay System ◽  
Delay Systems ◽  
Electronic Circuits ◽  
Low Dimensional

We experimentally demonstrate the effect of dynamic environment coupling in a system of coupled piecewise linear time-delay electronic circuits with mutual and subsystem coupling configurations. Time-delay systems are essentially infinite-dimensional systems with complex phase-space properties. Dynamic environmental coupling with mutual coupling configuration has been recently theoretically shown to induce complete (CS) and inverse synchronizations (IS) [Resmi et al., 2010] in low-dimensional dynamical systems described by ordinary differential equations (ODEs), for which no experimental confirmation exists. In this paper, we investigate the effect of dynamic environment for the first time in mutual as well as subsystem coupling configurations in coupled time-delay differential equations theoretically and experimentally. Depending upon the coupling strength and the nature of feedback, we observe a transition from asynchronization to CS via phase synchronization and from asynchronization to IS via inverse-phase synchronization in both coupling configurations. The results are corroborated by snapshots of the time evolution, phase projection plots and localized sets as observed from the oscilloscope. Further, the synchronization is also confirmed numerically from the largest Lyapunov exponents, correlation of probability of recurrence and correlation coefficient of the coupled time-delay system. We also present a linear stability analysis and obtain conditions for different synchronized states.

scholarly journals Asymptotic Analysis of Spectrum and Stability for One Class of Singularly Perturbed Neutral-Type Time-Delay Systems

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 325
Author(s):  
Valery Y. Glizer
Keyword(s):  
Time Delay ◽  
Asymptotic Analysis ◽  
Linear Time ◽  
Neutral Type ◽  
Original System ◽  
Delay System ◽  
Singularly Perturbed ◽  
Delay Systems ◽  
Time Delay System ◽  
Small Positive Parameter

In this study, a singularly perturbed linear time-delay system of neutral type is considered. It is assumed that the delay is small of order of a small positive parameter multiplying a part of the derivatives in the system. This system is decomposed asymptotically into two much simpler parameter-free subsystems, the slow and fast ones. Using this decomposition, an asymptotic analysis of the spectrum of the considered system is carried out. Based on this spectrum analysis, parameter-free conditions guaranteeing the exponential stability of the original system for all sufficiently small values of the parameter are derived. Illustrative examples are presented.

scholarly journals Stability and Performance of First-Order Linear Time-Delay Feedback Systems: An Eigenvalue Approach

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Shu-An He ◽  
I-Kong Fong
Keyword(s):  
Time Delay ◽  
Linear Time ◽  
Delay System ◽  
Feedback Stabilization ◽  
Delay Systems ◽  
Lambert Function ◽  
Time Delay Systems ◽  
Feedback Systems ◽  
Order Linear ◽  
First Order

Linear time-delay systems with transcendental characteristic equations have infinitely many eigenvalues which are generally hard to compute completely. However, the spectrum of first-order linear time-delay systems can be analyzed with the Lambert function. This paper studies the stability and state feedback stabilization of first-order linear time-delay system in detail via the Lambert function. The main issues concerned are the rightmost eigenvalue locations, stability robustness with respect to delay time, and the response performance of the closed-loop system. Examples and simulations are presented to illustrate the analysis results.

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