Equations with Advanced Arguments in Stick Balancing Models

  • Tamas Insperger
  • Richard Wohlfart
  • Janos Turi
  • Gabor Stepan
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 423)


A stick balancing problem is considered, where the output for the feedback controller is provided by an accelerometer attached to the stick. This output is a linear combination of the stick’s angular displacement and its angular acceleration. If the output is fed back in a PD controller with feedback delay, then the governing equation of motion is an advanced functional differential equation, since the third derivative of the angular displacement (the angular jerk) appears with a delayed argument through the derivative term. Equations with advanced arguments are typically non-causal and are unstable with infinitely many unstable poles. It is shown that the sampling of the controller may still stabilize the system in spite of its advanced nature. In the paper, different models for stick balancing are considered and discussed by analyzing the corresponding stability diagrams.


Angular Position Stability Boundary Functional Differential Equation Angular Acceleration Stability Diagram 
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Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  • Tamas Insperger
    • 1
  • Richard Wohlfart
    • 1
  • Janos Turi
    • 2
  • Gabor Stepan
    • 1
  1. 1.Department of Applied MechanicsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Programs in Mathematical SciencesUniversity of Texas at DallasRichardsonUSA

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