On the equivalence of two nonlinear control approaches: Immersion and invariance and IDA-PBC

Authors

Abstract

In this paper we compare the two well-known nonlinear control design techniques Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) and Immersion and Invariance (I&I) at the example of the so-called Acrobot underactuated mechanical system. The immersion and matching equations in both approaches have a similar structure which is exploited to derive equivalent control laws, each of them providing a different perspective on the stabilization problem. In particular, the coordinate change which renders the potential energy matching PDE in IDA-PBC an ordinary differential equation is used to define the immersion map in I&I. It is shown that the energy shaping part of the IDA-PBC controller makes the closed-loop system an interconnection of two lower-dimensional port-Hamiltonian (pH) systems in the on- and off-manifold coordinates that appear in the I&I framework. The effect of damping injection output feedback can be identified with dissipation in the off-manifold part of the interconnected system. Dissipation is propagated to the on-manifold part which results in asymptotic stability of the system’s equilibrium. The particular choice of the I&I design parameters in the present example, including the unconventional definition of coordinates on the invariant manifold, provides an interesting re-interpretation of the IDA-PBC control law from the I&I perspective. Finally, a discussion on the equivalence of the two approaches is presented by examining the cases of linear mechanical systems with one unactuated pivot as well as of general linear mechanical systems.

Keywords

Underactuated mechanical systems; Port-Hamiltonian systems; Passivity based control; Immersion and invariance

Citation

Journal: European Journal of Control
Year: 2013
Volume: 19
Issue: 6
Pages: 445–453
Publisher: Elsevier BV
DOI: 10.1016/j.ejcon.2013.09.008
Note: Lagrangian and Hamiltonian Methods for Modelling and Control

BibTeX

@article{Kotyczka_2013,
  title={{On the equivalence of two nonlinear control approaches: Immersion and invariance and IDA-PBC}},
  volume={19},
  ISSN={0947-3580},
  DOI={10.1016/j.ejcon.2013.09.008},
  number={6},
  journal={European Journal of Control},
  publisher={Elsevier BV},
  author={Kotyczka, Paul and Sarras, Ioannis},
  year={2013},
  pages={445--453}
}

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References

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