Modelling and Control of Infinite-Dimensional Mechanical Systems: A Port-Hamiltonian Approach

Authors

Abstract

We consider a port-Hamiltonian representation for infinite-dimensional systems described by partial differential equations. Then the control by interconnection method is applied, by using a finite-dimensional controller system interacting via an energy port at the boundary of the infinite-dimensional system. This will be demonstrated by means of a heavy chain system, modelled as a partial differential equation. Furthermore, we sketch the stability proof in the infinite-dimensional setting. To motivate for the presented ideas we recapitulate the well-known concepts for finite-dimensional systems as well, but mainly as a starting point for the discussion of the infinite setting.

Keywords

Interconnected System; Hamiltonian Density; Suspension Point; Casimir Function; Hamiltonian Representation

Citation

ISBN: 9783709112885
Publisher: Springer Vienna
DOI: 10.1007/978-3-7091-1289-2_5

BibTeX

@inbook{Sch_berl_2012,
  title={{Modelling and Control of Infinite-Dimensional Mechanical Systems: A Port-Hamiltonian Approach}},
  ISBN={9783709112892},
  DOI={10.1007/978-3-7091-1289-2_5},
  booktitle={{Multibody System Dynamics, Robotics and Control}},
  publisher={Springer Vienna},
  author={Schöberl, Markus and Siuka, Andreas},
  year={2012},
  pages={75--93}
}

Download the bib file

References

van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
H Khalil, Nonlinear systems (1996)
Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Port Hamiltonian formulation of infinite dimensional systems I. Modeling. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3762-3767 Vol.4 (2004) doi:10.1109/cdc.2004.1429324 – 10.1109/cdc.2004.1429324
Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Port Hamiltonian formulation of infinite dimensional systems II. Boundary control by interconnection. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3768-3773 Vol.4 (2004) doi:10.1109/cdc.2004.1429325 – 10.1109/cdc.2004.1429325
Maschke, B. & van der Schaft, A. 4 Compositional Modelling of Distributed-Parameter Systems. Lecture Notes in Control and Information Sciences 115–154 (2005) doi:10.1007/11334774_4 – 10.1007/11334774_4
van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42, 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
Schlacher, K. Mathematical modeling for nonlinear control: a Hamiltonian approach. Mathematics and Computers in Simulation 79, 829–849 (2008) – 10.1016/j.matcom.2008.02.011
Schöberl, M., Ennsbrunner, H. & Schlacher, K. Modelling of piezoelectric structures–a Hamiltonian approach. Mathematical and Computer Modelling of Dynamical Systems 14, 179–193 (2008) – 10.1080/13873950701844824
Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1986). doi:10.1007/978-1-4684-0274-2 – 10.1007/978-1-4684-0274-2
Schoberl, M. & Siuka, A. On Casimir functionals for field theories in Port-Hamiltonian description for control purposes. IEEE Conference on Decision and Control and European Control Conference 7759–7764 (2011) doi:10.1109/cdc.2011.6160430 – 10.1109/cdc.2011.6160430
Siuka, A., Schöberl, M. & Schlacher, K. Port-Hamiltonian modelling and energy-based control of the Timoshenko beam. Acta Mech 222, 69–89 (2011) – 10.1007/s00707-011-0510-2
Z Liu, Semigroups associated with dissipative systems (1999)
Luo, Z.-H., Guo, B.-Z. & Morgul, O. Stability and Stabilization of Infinite Dimensional Systems with Applications. Communications and Control Engineering (Springer London, 1999). doi:10.1007/978-1-4471-0419-3 – 10.1007/978-1-4471-0419-3
Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38, 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
Curtain, R. F. & Zwart, H. An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics (Springer New York, 1995). doi:10.1007/978-1-4612-4224-6 – 10.1007/978-1-4612-4224-6
Morgul, O. Stabilization and disturbance rejection for the wave equation. IEEE Trans. Automat. Contr. 43, 89–95 (1998) – 10.1109/9.654893
D Thull, Tracking control of mechanical distributed parameter systems with applications (2010)