Conditions on shifted passivity of port-Hamiltonian systems

Authors

Nima Monshizadeh, Pooya Monshizadeh, Romeo Ortega, Arjan van der Schaft

Abstract

In this paper, we examine the shifted passivity property of port-Hamiltonian systems. Shifted passivity accounts for the fact that in many applications the desired steady-state values of the input and output variables are nonzero, and thus one is interested in passivity with respect to the shifted signals. We consider port-Hamiltonian systems with strictly convex Hamiltonian, and derive conditions under which shifted passivity is guaranteed. In case the Hamiltonian is quadratic and state dependency appears in an affine manner in the dissipation and interconnection matrices, our conditions reduce to negative semidefiniteness of an appropriately constructed constant matrix. Moreover, we elaborate on how these conditions can be extended to the case when the shifted passivity property can be enforced via output feedback, thus paving the path for controller design. Stability of forced equilibria of the system is analyzed invoking the proposed passivity conditions. The utility and relevance of the results are illustrated with their application to a 6th order synchronous generator model as well as a controlled rigid body system.

Keywords

Passivity; Shifted passivity; Incremental passivity; Port-Hamiltonian systems; Stability theory

Citation

• Journal: Systems & Control Letters
• Year: 2019
• Volume: 123
• Issue:
• Pages: 55–61
• Publisher: Elsevier BV
• DOI: 10.1016/j.sysconle.2018.10.010

BibTeX

@article{Monshizadeh_2019,
  title={{Conditions on shifted passivity of port-Hamiltonian systems}},
  volume={123},
  ISSN={0167-6911},
  DOI={10.1016/j.sysconle.2018.10.010},
  journal={Systems & Control Letters},
  publisher={Elsevier BV},
  author={Monshizadeh, Nima and Monshizadeh, Pooya and Ortega, Romeo and van der Schaft, Arjan},
  year={2019},
  pages={55--61}
}

Download the bib file

References

• van der Schaft, A. L2-Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer International Publishing, 2017). doi:10.1007/978-3-319-49992-5 – 10.1007/978-3-319-49992-5
• Ortega, (2013)
• Bai, (2011)
• Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
• Jayawardhana, B., Ortega, R., García-Canseco, E. & Castaños, F. Passivity of nonlinear incremental systems: Application to PI stabilization of nonlinear RLC circuits. Systems & Control Letters vol. 56 618–622 (2007) – 10.1016/j.sysconle.2007.03.011
• Maschke, B., Ortega, R. & Van Der Schaft, A. J. Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Transactions on Automatic Control vol. 45 1498–1502 (2000) – 10.1109/9.871758
• Bregman, L. M. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics vol. 7 200–217 (1967) – 10.1016/0041-5553(67)90040-7
• Alonso, A. A. & Ydstie, B. E. Stabilization of distributed systems using irreversible thermodynamics. Automatica vol. 37 1739–1755 (2001) – 10.1016/s0005-1098(01)00140-6
• Keenan, J. H. Availability and irreversibility in thermodynamics. British Journal of Applied Physics vol. 2 183–192 (1951) – 10.1088/0508-3443/2/7/302
• Wen, A unifying passivity framework for network flow control. (2003)
• Persis, Bregman storage functions for microgrid control. IEEE Trans. Automat. Control (2017)
• Trip, S., Bürger, M. & De Persis, C. An internal model approach to (optimal) frequency regulation in power grids with time-varying voltages. Automatica vol. 64 240–253 (2016) – 10.1016/j.automatica.2015.11.021
• Monshizadeh, N. & De Persis, C. Agreeing in networks: Unmatched disturbances, algebraic constraints and optimality. Automatica vol. 75 63–74 (2017) – 10.1016/j.automatica.2016.09.008
• Wei, J. & van der Schaft, A. J. Load balancing of dynamical distribution networks with flow constraints and unknown in/outflows. Systems & Control Letters vol. 62 1001–1008 (2013) – 10.1016/j.sysconle.2013.08.001
• Hines, G. H., Arcak, M. & Packard, A. K. Equilibrium-independent passivity: A new definition and numerical certification. Automatica vol. 47 1949–1956 (2011) – 10.1016/j.automatica.2011.05.011
• Bürger, M., Zelazo, D. & Allgöwer, F. Duality and network theory in passivity-based cooperative control. Automatica vol. 50 2051–2061 (2014) – 10.1016/j.automatica.2014.06.002
• Simpson-Porco, (2017)
• van der Schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. AEU. Arch. Elektron. Übertrag. (1995)
• van der Schaft, (2014)
• van der Schaft, A. J. & Maschke, B. M. Port-Hamiltonian Systems on Graphs. SIAM Journal on Control and Optimization vol. 51 906–937 (2013) – 10.1137/110840091
• Ferguson, Disturbance rejection via control by interconnection of port-Hamiltonian systems. (2015)
• Ortega, R., Monshizadeh, N., Monshizadeh, P., Bazylev, D. & Pyrkin, A. Permanent magnet synchronous motors are globally asymptotically stabilizable with PI current control. Automatica vol. 98 296–301 (2018) – 10.1016/j.automatica.2018.09.031
• Desoer, (2009)
• Arnol’d, (2013)
• Rockafellar, (2015)
• Ryu, Primer on monotone operator methods. Appl. Comput. Math. (2016)
• Nesterov, (2013)
• Ortega, R., Stanković, A. & Stefanov, P. A Passivation Approach to Power Systems Stabilization. IFAC Proceedings Volumes vol. 31 309–313 (1998) – 10.1016/s1474-6670(17)40353-3
• Fiaz, S., Zonetti, D., Ortega, R., Scherpen, J. M. A. & van der Schaft, A. J. A port-Hamiltonian approach to power network modeling and analysis. European Journal of Control vol. 19 477–485 (2013) – 10.1016/j.ejcon.2013.09.002
• Caliskan, S. Y. & Tabuada, P. Compositional Transient Stability Analysis of Multimachine Power Networks. IEEE Transactions on Control of Network Systems vol. 1 4–14 (2014) – 10.1109/tcns.2014.2304868
• van der Schaft, A. & Stegink, T. Perspectives in modeling for control of power networks. Annual Reviews in Control vol. 41 119–132 (2016) – 10.1016/j.arcontrol.2016.04.017
• Forni, F. & Sepulchre, R. A Differential Lyapunov Framework for Contraction Analysis. IEEE Transactions on Automatic Control vol. 59 614–628 (2014) – 10.1109/tac.2013.2285771
• Forni, On differential passivity of physical systems. (2013)
• Sontag, Input to state stability: Basic concepts and results. (2008)