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

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Delay Differential Equations ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
The Stability ◽  
The Galerkin Method ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

In this paper, we develop Galerkin approximations for determining the stability of delay differential equations (DDEs) with time periodic coefficients and time periodic delays. Using a transformation, we convert the DDE into a partial differential equation (PDE) along with a boundary condition (BC). The PDE and BC we obtain have time periodic coefficients. The PDE is discretized into a system of ordinary differential equations (ODEs) using the Galerkin method with Legendre polynomials as the basis functions. The BC is imposed using the tau method. The resulting ODEs are time periodic in nature; thus, we resort to Floquet theory to determine the stability of the ODEs. We show through several numerical examples that the stability charts obtained from the Galerkin method agree closely with those obtained from direct numerical simulations.

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.

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.

Spurious roots of delay differential equations using Galerkin approximations

2020 ◽  
Vol 26 (15-16) ◽  
pp. 1178-1184
Author(s):  
Shanti S Kandala ◽  
Surya Samukham ◽  
Thomas K Uchida ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Delay Differential Equations ◽  
Second Order ◽  
Partial Differential ◽  
First Order ◽  
Galerkin Approximations ◽  
Delay Differential

The dynamics of time-delay systems are governed by delay differential equations, which are infinite dimensional and can pose computational challenges. Several methods have been proposed for studying the stability characteristics of delay differential equations. One such method employs Galerkin approximations to convert delay differential equations into partial differential equations with boundary conditions; the partial differential equations are then converted into systems of ordinary differential equations, whereupon standard ordinary differential equation methods can be applied. The Galerkin approximation method can be applied to a second-order delay differential equation in two ways: either by converting into a second-order partial differential equation and then into a system of second-order ordinary differential equations (the “second-order Galerkin” method) or by first expressing as two first-order delay differential equations and converting into a system of first-order partial differential equations and then into a first-order ordinary differential equation system (the “first-order Galerkin” method). In this paper, we demonstrate that these subtly different formulation procedures lead to different roots of the characteristic polynomial. In particular, the second-order Galerkin method produces spurious roots near the origin, which must then be identified through substitution into the characteristic polynomial of the original delay differential equation. However, spurious roots do not arise if the first-order Galerkin method is used, which can reduce computation time and simplify stability analyses. We describe these two formulation strategies and present numerical examples to highlight their important differences.

Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Boundary Condition ◽  
Computational Complexity ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Kernel Functions ◽  
Distributed Delays ◽  
Dimensional Approximation ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic

Delay differential equations (DDEs) are infinite-dimensional systems, therefore analyzing their stability is a difficult task. The delays can be discrete or distributed in nature. DDEs with distributed delays are referred to as delay integro-differential equations (DIDEs) in the literature. In this work, we propose a method to convert the DIDEs into a system of ordinary differential equations (ODEs). The stability of the DIDEs can then be easily studied from the obtained system of ODEs. By using a space-time transformation, we convert the DIDEs into a partial differential equation (PDE) with a time-dependent boundary condition. Then, by using the Galerkin method, we obtain a finite-dimensional approximation to the PDE. The boundary condition is incorporated into the Galerkin approximation using the Tau method. The resulting system of ODEs will have time-periodic coefficients, provided the coefficients of the DIDEs are time periodic. Thus, we use Floquet theory to analyze the stability of the resulting ODE systems. We study several numerical examples of DIDEs with different kernel functions. We show that the results obtained using our method are in close agreement with those existing in the literature. The theory developed in this work can also be used for the integration of DIDEs. The computational complexity of our numerical integration method is O(t), whereas the direct brute-force integration of DIDE has a computational complexity of O(t2).

scholarly journals Stability of delay differential equations in the sense of Ulam on unbounded intervals

2019 ◽  
Vol 9 (2) ◽  
pp. 125-131 ◽  
Author(s):  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Stability Problem ◽  
The Stability ◽  
Delay Differential ◽  
Unbounded Intervals

In this paper, we consider the stability problem of delay differential equations in the sense of Hyers-Ulam-Rassias. Recently this problem has been solved for bounded intervals, our result extends and improve the literature by obtaining stability in unbounded intervals. An illustrative example is also given to compare these results and visualize the improvement.

Galerkin Approximations for Neutral Delay Differential Equations

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Lagrange Multipliers ◽  
Delay Differential Equations ◽  
Nonlinear Boundary ◽  
Hyperbolic Partial Differential Equations ◽  
Neutral Delay ◽  
Boundary Constraints ◽  
Galerkin Approximations ◽  
Neutral Delay Differential Equations ◽  
Delay Differential

In this work, Galerkin approximations are developed for a system of first order nonlinear neutral delay differential equations (NDDEs). The NDDEs are converted into an equivalent system of hyperbolic partial differential equations (PDEs) along with the nonlinear boundary constraints. Lagrange multipliers are introduced to enforce the boundary constraints. The explicit expressions for the Lagrange multipliers are derived by exploiting the equivalence of partial derivatives in space and time at a given point on the domain. To illustrate the method, comparisons are made between numerical solution of NDDEs and its Galerkin approximations for different NDDEs.

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