Stability of linear time-periodic delay-differential equations via Chebyshev polynomials

2004 ◽  
Vol 59 (7) ◽  
pp. 895-922 ◽  
Author(s):  
Eric A. Butcher ◽  
Haitao Ma ◽  
Ed Bueler ◽  
Victoria Averina ◽  
Zsolt Szabo
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Delay Differential Equations ◽  
Linear Time ◽  
Periodic Delay ◽  
Delay Differential ◽  
Time Periodic

scholarly journals A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Gang Jin ◽  
Houjun Qi ◽  
Zhanjie Li ◽  
Jianxin Han ◽  
Hua Li
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Spindle Speed ◽  
Multiple Time ◽  
Variable Pitch ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the past literature. This indicates the effectiveness of our method in terms of time-periodic DDEs with multiple time-periodic delays. Moreover, for milling processes, the proposed method further provides a generalized algorithm, which possesses a good capability to predict the stability lobes for milling operations with variable pitch cutter or variable-spindle speed.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximate Model ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
Approximate System ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Least Squares Approach

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

Pole Placement for Delay Differential Equations with Time-Periodic Delays Using Galerkin Approximations (CND-20-1378)

2021 ◽  
Author(s):  
Shanti Swaroop Kandala ◽  
Thomas K. Uchida ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Delay Differential Equations ◽  
Second Order ◽  
Delay Systems ◽  
First Order ◽  
The Galerkin Method ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

Abstract Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems. In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory, and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

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