Retarded, neutral and advanced differential equation models for balancing using an accelerometer

Balazs A. Kovacs, T. Insperger

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Stabilization of a pinned pendulum about its upright position via a reaction wheel is considered, where the pendulum’s angular position is measured by a single accelerometer attached directly to the pendulum. The control policy is modeled as a simple PD controller and different feedback mechanisms are investigated. It is shown that depending on the modeling concepts, the governing equations can be a retarded functional differential equation or neutral functional differential equation or even advanced functional differential equation. These types of equations have radically different stability properties. In the retarded and the neutral case the system can be stabilized, but the advanced equations are always unstable with infinitely many unstable characteristic roots. It is shown that slight modeling differences lead to significant qualitative change in the behavior of the system, which is demonstrated by means of the stability diagrams for the different models. It is concluded that digital effects, such as sampling, stabilizes the system independently on the modeling details.

Original languageEnglish
Pages (from-to)694-706
Number of pages13
JournalInternational Journal of Dynamics and Control
Volume6
Issue number2
DOIs
Publication statusPublished - Jun 1 2018

Fingerprint

Accelerometer
Pendulum
Pendulums
Accelerometers
Balancing
Differential equations
Differential equation
Unstable
Modeling
Retarded Functional Differential Equations
Characteristic Roots
Neutral Functional Differential Equation
Control Policy
Functional Differential Equations
Wheel
Governing equation
Wheels
Stabilization
Diagram
Model

Keywords

  • Accelerometer
  • D-subdivision method
  • Feedback delay
  • Functional differential equations
  • Semi-discretization
  • Stabilization

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Civil and Structural Engineering
  • Modelling and Simulation
  • Mechanical Engineering
  • Control and Optimization
  • Electrical and Electronic Engineering

Cite this

Retarded, neutral and advanced differential equation models for balancing using an accelerometer. / Kovacs, Balazs A.; Insperger, T.

In: International Journal of Dynamics and Control, Vol. 6, No. 2, 01.06.2018, p. 694-706.

Research output: Contribution to journalArticle

@article{d192fb1028a44273a7cc6e3cd15aec54,
title = "Retarded, neutral and advanced differential equation models for balancing using an accelerometer",
abstract = "Stabilization of a pinned pendulum about its upright position via a reaction wheel is considered, where the pendulum’s angular position is measured by a single accelerometer attached directly to the pendulum. The control policy is modeled as a simple PD controller and different feedback mechanisms are investigated. It is shown that depending on the modeling concepts, the governing equations can be a retarded functional differential equation or neutral functional differential equation or even advanced functional differential equation. These types of equations have radically different stability properties. In the retarded and the neutral case the system can be stabilized, but the advanced equations are always unstable with infinitely many unstable characteristic roots. It is shown that slight modeling differences lead to significant qualitative change in the behavior of the system, which is demonstrated by means of the stability diagrams for the different models. It is concluded that digital effects, such as sampling, stabilizes the system independently on the modeling details.",
keywords = "Accelerometer, D-subdivision method, Feedback delay, Functional differential equations, Semi-discretization, Stabilization",
author = "Kovacs, {Balazs A.} and T. Insperger",
year = "2018",
month = "6",
day = "1",
doi = "10.1007/s40435-017-0331-9",
language = "English",
volume = "6",
pages = "694--706",
journal = "International Journal of Dynamics and Control",
issn = "2195-268X",
publisher = "Springer International Publishing AG",
number = "2",

}

TY - JOUR

T1 - Retarded, neutral and advanced differential equation models for balancing using an accelerometer

AU - Kovacs, Balazs A.

AU - Insperger, T.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - Stabilization of a pinned pendulum about its upright position via a reaction wheel is considered, where the pendulum’s angular position is measured by a single accelerometer attached directly to the pendulum. The control policy is modeled as a simple PD controller and different feedback mechanisms are investigated. It is shown that depending on the modeling concepts, the governing equations can be a retarded functional differential equation or neutral functional differential equation or even advanced functional differential equation. These types of equations have radically different stability properties. In the retarded and the neutral case the system can be stabilized, but the advanced equations are always unstable with infinitely many unstable characteristic roots. It is shown that slight modeling differences lead to significant qualitative change in the behavior of the system, which is demonstrated by means of the stability diagrams for the different models. It is concluded that digital effects, such as sampling, stabilizes the system independently on the modeling details.

AB - Stabilization of a pinned pendulum about its upright position via a reaction wheel is considered, where the pendulum’s angular position is measured by a single accelerometer attached directly to the pendulum. The control policy is modeled as a simple PD controller and different feedback mechanisms are investigated. It is shown that depending on the modeling concepts, the governing equations can be a retarded functional differential equation or neutral functional differential equation or even advanced functional differential equation. These types of equations have radically different stability properties. In the retarded and the neutral case the system can be stabilized, but the advanced equations are always unstable with infinitely many unstable characteristic roots. It is shown that slight modeling differences lead to significant qualitative change in the behavior of the system, which is demonstrated by means of the stability diagrams for the different models. It is concluded that digital effects, such as sampling, stabilizes the system independently on the modeling details.

KW - Accelerometer

KW - D-subdivision method

KW - Feedback delay

KW - Functional differential equations

KW - Semi-discretization

KW - Stabilization

UR - http://www.scopus.com/inward/record.url?scp=85046812442&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046812442&partnerID=8YFLogxK

U2 - 10.1007/s40435-017-0331-9

DO - 10.1007/s40435-017-0331-9

M3 - Article

AN - SCOPUS:85046812442

VL - 6

SP - 694

EP - 706

JO - International Journal of Dynamics and Control

JF - International Journal of Dynamics and Control

SN - 2195-268X

IS - 2

ER -