Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems

Authors

Romeo Ortega, Arjan van der Schaft, Fernando Castanos, Alessandro Astolfi

Abstract

The dynamics of many physical processes can be suitably described by Port-Hamiltonian (PH) models, where the importance of the energy function, the interconnection pattern and the dissipation of the system is underscored. To regulate the behavior of PH systems it is natural to adopt a Passivity-Based Control (PBC) perspective, where the control objectives are achieved shaping the energy function and adding dissipation. In this paper we consider the PBC techniques of Control by Interconnection (CbI) and Standard PBC. In CbI the controller is another PH system connected to the plant (through a power-preserving interconnection) to add up their energy functions, while in Standard PBC energy shaping is achieved via static state feedback. In spite of the conceptual appeal of formulating the control problem as the interaction of dynamical systems, the current version of CbI imposes a severe restriction on the plant dissipation structure that stymies its practical application. On the other hand, Standard PBC, which is usually derived from a uninspiring and non-intuitive ldquopassive output generationrdquo viewpoint, is one of the most successful controller design techniques. The main objectives of this paper are: (1) To extend the CbI method to make it more widely applicable-in particular, to overcome the aforementioned dissipation obstacle. (2) To show that various popular variants of Standard PBC can be derived proceeding from a unified perspective. (3) To establish the connections between CbI and Standard PBC proving that the latter is obtained restricting the former to a suitable subset-providing a nice geometric interpretation to Standard PBC-and comparing the size of the set of PH plants for which they are applicable.

Citation

• Journal: IEEE Transactions on Automatic Control
• Year: 2008
• Volume: 53
• Issue: 11
• Pages: 2527–2542
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tac.2008.2006930

BibTeX

@article{Ortega_2008,
  title={{Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems}},
  volume={53},
  ISSN={0018-9286},
  DOI={10.1109/tac.2008.2006930},
  number={11},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Ortega, Romeo and van der Schaft, Arjan and Castanos, Fernando and Astolfi, Alessandro},
  year={2008},
  pages={2527--2542}
}

Download the bib file

References

• Geometric Network Modeling and Control of Complex Physical Systems (2008)
• Sepulchre, R., Janković, M. & Kokotović, P. V. Constructive Nonlinear Control. Communications and Control Engineering (Springer London, 1997). doi:10.1007/978-1-4471-0967-9 – 10.1007/978-1-4471-0967-9
• Perez, M., Ortega, R. & Espinoza, J. Passivity-Based PI Control of Switched Power Converters. IEEE Transactions on Control Systems Technology vol. 12 881–890 (2004) – 10.1109/tcst.2004.833628
• ortega, ?Asymptotic Stabilization of Port-Hamiltonian Systems via Control by Interconnection ? (2007)
• Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
• Wang, Y., Cheng, D. & Ge, S. S. Approximate dissipative Hamiltonian realization and construction of local Lyapunov functions. Systems & Control Letters vol. 56 141–149 (2007) – 10.1016/j.sysconle.2006.08.005
• van der schaft, -Gain and Passivity Techniques in Nonlinear Control (1999)
• Power-factor compensation of electrical circuits. IEEE Control Systems vol. 27 46–59 (2007) – 10.1109/mcs.2007.338263
• Hill, D. J. & Moylan, P. J. Dissipative Dynamical Systems: Basic Input-Output and State Properties. Journal of the Franklin Institute vol. 309 327–357 (1980) – 10.1016/0016-0032(80)90026-5
• Jayawardhana, B., Ortega, R., Garcia-Canseco, E. & Castanos, F. Passivity of Nonlinear Incremental Systems: Application to PI Stabilization of Nonlinear RLC Circuits. Proceedings of the 45th IEEE Conference on Decision and Control 3808–3812 (2006) doi:10.1109/cdc.2006.377132 – 10.1109/cdc.2006.377132
• Jeltsema, D. & Scherpen, J. M. A. A dual relation between port-Hamiltonian systems and the Brayton–Moser equations for nonlinear switched RLC circuits. Automatica vol. 39 969–979 (2003) – 10.1016/s0005-1098(03)00070-0
• Jeltsema, D. & Scherpen, J. M. A. A power-based description of standard mechanical systems. Systems & Control Letters vol. 56 349–356 (2007) – 10.1016/j.sysconle.2006.10.015
• Jeltsema, D., Ortega, R. & M.A. Scherpen, J. An energy-balancing perspective of interconnection and damping assignment control of nonlinear systems. Automatica vol. 40 1643–1646 (2004) – 10.1016/j.automatica.2004.04.007
• jeltsema, energy-balancing and ida pbc of nonlinear systems. Proc IFAC Workshop Lagrangian Hamiltonian Methods Nonlin Syst (2003)
• johnsen, interconnection and damping assignment passivity-based control of a four-tank system. Proc IFAC Lagrangian Hamiltonian Methods Nonlin Control (2006)
• McLachlan, R. I., Quispel, G. R. W. & Robidoux, N. Unified Approach to Hamiltonian Systems, Poisson Systems, Gradient Systems, and Systems with Lyapunov Functions or First Integrals. Physical Review Letters vol. 81 2399–2403 (1998) – 10.1103/physrevlett.81.2399
• Marsden, J. E. & Ratiu, T. S. Introduction to Mechanics and Symmetry. Texts in Applied Mathematics (Springer New York, 1994). doi:10.1007/978-1-4612-2682-6 – 10.1007/978-1-4612-2682-6
• Ortega, R., Jeltsema, D. & Scherpen, J. M. A. Power shaping: A new paradigm for stabilization of nonlinear RLC circuits. IEEE Transactions on Automatic Control vol. 48 1762–1767 (2003) – 10.1109/tac.2003.817918
• Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. I. Quarterly of Applied Mathematics vol. 22 1–33 (1964) – 10.1090/qam/169746
• Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
• Bloch, A. M., Leonard, N. E. & Marsden, J. E. Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem. IEEE Transactions on Automatic Control vol. 45 2253–2270 (2000) – 10.1109/9.895562
• Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998) – 10.1137/s0363012996312039
• Ortega, R. & García-Canseco, E. Interconnection and Damping Assignment Passivity-Based Control: A Survey. European Journal of Control vol. 10 432–450 (2004) – 10.3166/ejc.10.432-450
• Cheng, D., Astolfi, A. & Ortega, R. On feedback equivalence to port controlled Hamiltonian systems. Systems & Control Letters vol. 54 911–917 (2005) – 10.1016/j.sysconle.2005.02.005
• Fujimoto, K. & Sugie, T. Canonical transformation and stabilization of generalized Hamiltonian systems. Systems & Control Letters vol. 42 217–227 (2001) – 10.1016/s0167-6911(00)00091-8
• desoer, Feedback Systems Input-Output Properties (1975)
• Blankenstein, G. Power balancing for a new class of non-linear systems and stabilization of RLC circuits. International Journal of Control vol. 78 159–171 (2005) – 10.1080/00207170500036191
• garcia-canseco, power shaping control of nonlinear systems: a benchmark example. Proc IFAC Lagrangian Hamiltonian Methods Nonlin Control (2006)
• Blankenstein, G., Ortega, R. & Van Der Schaft, A. J. The matching conditions of controlled Lagrangians and IDA-passivity based control. International Journal of Control vol. 75 645–665 (2002) – 10.1080/00207170210135939
• 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
• Polderman, J. W. & Willems, J. C. Introduction to Mathematical Systems Theory. Texts in Applied Mathematics (Springer New York, 1998). doi:10.1007/978-1-4757-2953-5 – 10.1007/978-1-4757-2953-5
• Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
• Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
• Ortega, R., Loría, A., Nicklasson, P. J. & Sira-Ramírez, H. Passivity-Based Control of Euler-Lagrange Systems. Communications and Control Engineering (Springer London, 1998). doi:10.1007/978-1-4471-3603-3 – 10.1007/978-1-4471-3603-3
• Ortega, R. & Spong, M. W. Adaptive motion control of rigid robots: A tutorial. Automatica vol. 25 877–888 (1989) – 10.1016/0005-1098(89)90054-x
• Ortega, R., Spong, M. W., Gomez-Estern, F. & Blankenstein, G. Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Transactions on Automatic Control vol. 47 1218–1233 (2002) – 10.1109/tac.2002.800770