The Optimal Erection of the Inverted Pendulum

Matteo Massaro, Stefano Lovato, David J.N. Limebeer

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


The erection of the inverted pendulum is a classic control problem, which has appeared in several variants. One of the most challenging is the minimum-time erection of a pendulum that is mounted on a moving cart. The aim is to erect the pendulum from the ‘straight-down’ (stable equilibrium) to a ‘straight-up’ (unstable equilibrium) position in minimum time. The swing-up maneuver is usually addressed using a pre-defined control strategy, e.g., energy-based control or selecting the switching times in a bang-bang structure. The aim of this paper is to show that the minimum-time solution may have a singular arc, with the optimal control taking a bang-singular-bang form. The singular arc segment of the control law is a feedback law that is derived herein with the solution discussed. A sensitivity analysis of the solution structure is also performed by varying the model parameters. Finally, the time-optimal solution is compared with that obtained using an energy-based control strategy.

Original languageEnglish
Article number8112
JournalApplied Sciences (Switzerland)
Issue number16
Publication statusPublished - Aug 2022
Externally publishedYes


  • bang-bang
  • dynamics
  • inverted pendulum
  • multibody systems
  • optimal control
  • optimization
  • singular arc

ASJC Scopus subject areas

  • General Materials Science
  • Instrumentation
  • General Engineering
  • Process Chemistry and Technology
  • Computer Science Applications
  • Fluid Flow and Transfer Processes


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