Interconnections of input-output Hamiltonian systems with dissipation

Authors

Abstract

Negative imaginary and counter-clockwise systems have attracted attention as an interesting class of systems, which is well-motivated by applications. In this paper first the formulation and extension of negative imaginary and counter-clockwise systems as (nonlinear) input-output Hamiltonian systems with dissipation is summarized. Next it is shown how by considering the time-derivative of the outputs a port-Hamiltonian system is obtained, and how this leads to the consideration of alternate passive outputs for port-Hamiltonian systems. Furthermore, a converse result to positive feedback interconnection of input-output Hamiltonian systems with dissipation is obtained, stating that the positive feedback interconnection of two linear systems is an input-output Hamiltonian system with dissipation if and only if the systems themselves are input-output Hamiltonian systems with dissipation. This implies that the Poisson and resistive structure matrices can be redefined in such a way that the interaction between the two systems only takes place via the coupling term in the Hamiltonian of the interconnected system. Subsequently, it is shown how the positive feedback interconnection of two nonlinear input-output Hamiltonian systems with dissipation can be extended to the network interconnection of such systems, and how this leads to a stability analysis of the interconnected system in terms of the Hamiltonians and output mappings of the component systems associated to the vertices, as well as of the network topology.

Citation

Journal: 2016 IEEE 55th Conference on Decision and Control (CDC)
Year: 2016
Volume:
Issue:
Pages: 4686–4691
Publisher: IEEE
DOI: 10.1109/cdc.2016.7798983

BibTeX

@inproceedings{van_der_Schaft_2016,
  title={{Interconnections of input-output Hamiltonian systems with dissipation}},
  DOI={10.1109/cdc.2016.7798983},
  booktitle={{2016 IEEE 55th Conference on Decision and Control (CDC)}},
  publisher={IEEE},
  author={van der Schaft, Arjan},
  year={2016},
  pages={4686--4691}
}

Download the bib file

References

Xiong J, Petersen IR, Lanzon A (2010) A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems. IEEE Trans Automat Contr 55(10):2342–2347. https://doi.org/10.1109/tac.2010.2052711 – 10.1109/tac.2010.2052711
Friswell MI, Inman DJ (1999) The relationship between positive position feedback and output feedback controllers. Smart Mater Struct 8(3):285–291. https://doi.org/10.1088/0964-1726/8/3/301 – 10.1088/0964-1726/8/3/301
GOH CJ, CAUGHEY TK (1985) On the stability problem caused by finite actuator dynamics in the collocated control of large space structures. International Journal of Control 41(3):787–802. https://doi.org/10.1080/0020718508961163 – 10.1080/0020718508961163
Jeltsema D, Ortega R, M.A. Scherpen J (2004) An energy-balancing perspective of interconnection and damping assignment control of nonlinear systems. Automatica 40(9):1643–1646. https://doi.org/10.1016/j.automatica.2004.04.007 – 10.1016/j.automatica.2004.04.007
Kerber F, van der Schaft A (2011) Compositional properties of passivity. In: IEEE Conference on Decision and Control and European Control Conference. IEEE, pp 4628–4633 – 10.1109/cdc.2011.6160591
Lanzon A, Petersen IR (2008) Stability Robustness of a Feedback Interconnection of Systems With Negative Imaginary Frequency Response. IEEE Trans Automat Contr 53(4):1042–1046. https://doi.org/10.1109/tac.2008.919567 – 10.1109/tac.2008.919567
maschke, Port-controlled Hamiltonian systems: modelling origins and system theoretic properties. Proceedings 2nd IFAC Symposium on Nonlinear Control Systems (NOLCOS 2004) (1992)
Nijmeijer H, van der Schaft A (1990) Nonlinear Dynamical Control Systems. Springer New York, New York, NY – 10.1007/978-1-4757-2101-0
Ortega R, van der Schaft A, Maschke B, Escobar G (2002) Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38(4):585–596. https://doi.org/10.1016/s0005-1098(01)00278-3 – 10.1016/s0005-1098(01)00278-3
Padthe AK, JinHyoung Oh, Bernstein DS (2006) Counterclockwise Dynamics of a Rate-Independent Semilinear Duhem Model. In: Proceedings of the 44th IEEE Conference on Decision and Control. IEEE, pp 8000–8005 – 10.1109/cdc.2005.1583456
(2010) Feedback Control of Negative-Imaginary Systems. IEEE Control Syst 30(5):54–72. https://doi.org/10.1109/mcs.2010.937676 – 10.1109/mcs.2010.937676
Wang J, Lanzon A, Petersen IR (2015) Robust Output Feedback Consensus for Networked Negative-Imaginary Systems. IEEE Trans Automat Contr 60(9):2547–2552. https://doi.org/10.1109/tac.2015.2395472 – 10.1109/tac.2015.2395472
Angeli D (2007) Multistability in Systems With Counter-Clockwise Input–Output Dynamics. IEEE Trans Automat Contr 52(4):596–609. https://doi.org/10.1109/tac.2007.894507 – 10.1109/tac.2007.894507
Venkatraman A, van der Schaft A (2010) Energy Shaping of Port-Hamiltonian Systems by Using Alternate Passive Input-Output Pairs. European Journal of Control 16(6):665–677. https://doi.org/10.3166/ejc.16.665-677 – 10.3166/ejc.16.665-677
Angeli D (2006) Systems With Counterclockwise Input–Output Dynamics. IEEE Trans Automat Contr 51(7):1130–1143. https://doi.org/10.1109/tac.2006.878747 – 10.1109/tac.2006.878747
Borja P, Cisneros R, Ortega R (2015) Shaping the energy of port-Hamiltonian systems without solving PDE’s. In: 2015 54th IEEE Conference on Decision and Control (CDC). IEEE, pp 5713–5718 – 10.1109/cdc.2015.7403116
Wang J, Lanzon A, Petersen IR (2015) Robust output feedback consensus for multiple heterogeneous negative-imaginary systems. In: 2015 54th IEEE Conference on Decision and Control (CDC). IEEE, pp 2371–2376 – 10.1109/cdc.2015.7402562
Arcak M (2007) Passivity as a Design Tool for Group Coordination. IEEE Trans Automat Contr 52(8):1380–1390. https://doi.org/10.1109/tac.2007.902733 – 10.1109/tac.2007.902733
Crouch PE, van der Schaft AJ (eds) (1987) Variational and Hamiltonian Control Systems. Springer Berlin Heidelberg, Berlin, Heidelberg – 10.1007/bfb0042858
brockett, Control theory and analytical mechanics. Geometric Control Theory (1977)
angeli, On systems with counter-clockwise input/output dynamics. Proceedings 43th IEEE Conference on Decision and Control (2004)
Dalsmo M, van der Schaft A (1998) On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM J Control Optim 37(1):54–91. https://doi.org/10.1137/s0363012996312039 – 10.1137/s0363012996312039
abraham, Foundations of Mechanics (1978)
Schaft AJ (1981) Hamiltonian dynamics with external forces and observations. Math Systems Theory 15(1):145–168. https://doi.org/10.1007/bf01786977 – 10.1007/bf01786977
van der schaft, $L_ 2 $-Gain and Passivity Techniques in Nonlinear Control (2000)
van der Schaft AJ (1982) Observability and Controllability for Smooth Nonlinear Systems. SIAM J Control Optim 20(3):338–354. https://doi.org/10.1137/0320026 – 10.1137/0320026
van der Schaft AJ, Maschke BM (2013) Port-Hamiltonian Systems on Graphs. SIAM J Control Optim 51(2):906–937. https://doi.org/10.1137/110840091 – 10.1137/110840091
van der schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv Elektronik und Übertragungstechnik (1995)
van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2-3):173–378. https://doi.org/10.1561/2600000002 – 10.1561/2600000002
van der Schaft AJ (2011) Positive feedback interconnection of Hamiltonian systems. In: IEEE Conference on Decision and Control and European Control Conference. IEEE, pp 6510–6515 – 10.1109/cdc.2011.6160395