Authors

A. J. van der Schaft

Abstract

This paper discusses a system theoretic approach to mechanics, regarding Hamiltonian systems as conservative “mechanical m-ports”. Recent results in the Hamiltonian realization problem are surveyed, and generalizations are being indicated. The potential use for control purposes of the Hamiltonian structure of nonlinear control systems is exemplified.

Keywords

hamiltonian structure, hamiltonian system, nonlinear control system, symplectic form, symplectic manifold

Citation

BibTeX

@inbook{van_der_Schaft_1989,
  title={{System theory and mechanics}},
  ISBN={9783540467090},
  ISSN={1610-7411},
  DOI={10.1007/bfb0008472},
  booktitle={{Three Decades of Mathematical System Theory}},
  publisher={Springer Berlin Heidelberg},
  author={van der Schaft, A. J.},
  year={1989},
  pages={426--452}
}

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References

  • Arnold, V. I. Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics (Springer New York, 1978). doi:10.1007/978-1-4757-1693-1 – 10.1007/978-1-4757-1693-1
  • Aubrun, J.-N., Lorell, K., Mast, T. & Nelson, J. Dynamic analysis of the activley controlled segmented mirror of the W. M. Keck ten-meter telescope. IEEE Control Syst. Mag. 7, 3–10 (1987) – 10.1109/mcs.1987.1105386
  • R.A. Abraham, Foundations of Mechanics (1978)
  • H. Abesser, Wiss. Z.TH Ilmenau (1987)
  • B.D.O. Anderson, Network Analysis and Synthesis (1973)
  • R.W. Brockett, Geometric Control Theory (1977)
  • M. Balas, J.Guid.Contr. (1979)
  • V. Belevitch, Classical Network Theory (1968)
  • Gonçalves, J. B. Realization Theory for Hamiltonian Systems. SIAM J. Control Optim. 25, 63–73 (1987) – 10.1137/0325005
  • Byrnes, C. & Isidori, A. A frequency domain philosophy for nonlinear systems, with applications to stabilization and to adaptive control. The 23rd IEEE Conference on Decision and Control (1984) doi:10.1109/cdc.1984.272345 – 10.1109/cdc.1984.272345
  • Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. I. Quart. Appl. Math. 22, 1–33 (1964) – 10.1090/qam/169746
  • R.W. Brockett, Ordinary Differential Equations (1972)
  • A.G. Butkovskii, I & II, Autom.Rem.Contr. (1979)
  • Crouch, P. E. Geometric structures in systems theory. IEE Proc. D Control Theory Appl. UK 128, 242 (1981) – 10.1049/ip-d.1981.0051
  • Crouch, P. E. & Irving, M. On finite volterra series which admit hamiltonian realizations. Math. Systems Theory 17, 293–318 (1984) – 10.1007/bf01744446
  • Crouch, P. E. & Irving, M. Dynamical Realizations of Homogeneous Hamiltonian Systems. SIAM J. Control Optim. 24, 374–395 (1986) – 10.1137/0324022
  • Crouch, P. E. & Lamnabhi-Lagarrigue, F. State space realizations of nonlinear systems defined by input-output differential equations. Lecture Notes in Control and Information Sciences 138–149 (1988) doi:10.1007/bfb0042209 – 10.1007/bfb0042209
  • Variational and Hamiltonian Control Systems. Lecture Notes in Control and Information Sciences (Springer Berlin Heidelberg, 1987). doi:10.1007/bfb0042858 – 10.1007/bfb0042858
  • H. Goldstein, Classical Mechanics (1950)
  • GEVARTER, W. B. Basic relations for control of flexible vehicles. AIAA Journal 8, 666–672 (1970) – 10.2514/3.5739
  • Glad, S. T. Robustness of nonlinear state feedback—A survey. Automatica 23, 425–435 (1987) – 10.1016/0005-1098(87)90072-0
  • B. Jakubczyk, Bull.Pol.Ac.: Math (1986)
  • B. Jakubczyk, Bull.Pol.Ac.: Math (1986)
  • B. Jakubczyk, Theory and Applications of Nonlinear Control Systems (1986)
  • Koditschek, D. Natural motion for robot arms. The 23rd IEEE Conference on Decision and Control (1984) doi:10.1109/cdc.1984.272106 – 10.1109/cdc.1984.272106
  • Krishnaprasad, P. S. & Marsden, J. E. Hamiltonian structures and stability for rigid bodies with flexible attachments. Arch. Rational Mech. Anal. 98, 71–93 (1987) – 10.1007/bf00279963
  • R. Marino, Theory and Applications of Nonlinear Control Systems (1986)
  • Nijmeijer, H. & van der Schaft, A. J. Input-Output Decoupling of Hamiltonian Systems: The Nonlinear Case. Lecture Notes in Control and Information Sciences 300–313 (1986) doi:10.1007/bfb0007567 – 10.1007/bfb0007567
  • Schaft, A. J. Hamiltonian dynamics with external forces and observations. Math. Systems Theory 15, 145–168 (1981) – 10.1007/bf01786977
  • van der Schaft, A. J. Symmetries, conservation laws, and time reversibility for Hamiltonian systems with external forces. Journal of Mathematical Physics 24, 2095–2101 (1983) – 10.1063/1.525962
  • A.J. Schaft van der, System theoretic descriptions of physical systems, CWI Tract 3 (1984)
  • Schaft, A. J. Controlled invariance for hamiltonian systems. Math. Systems Theory 18, 257–291 (1985) – 10.1007/bf01699473
  • van der Schaft, A. J. Stabilization of Hamiltonian systems. Nonlinear Analysis: Theory, Methods & Applications 10, 1021–1035 (1986) – 10.1016/0362-546x(86)90086-6
  • van der Schaft, A. J. Optimal Control and Hamiltonian Input-Output Systems. Algebraic and Geometric Methods in Nonlinear Control Theory 389–407 (1986) doi:10.1007/978-94-009-4706-1_20 – 10.1007/978-94-009-4706-1_20
  • A.J. Schaft van der, Theory and Applications of Nonlinear Control Systems (1986)
  • A.J. Schaft van der, Hamiltonian and quantum mechanical control systems (1987)
  • van der Schaft, A. J. Hamiltonian control systems: decomposition and clamped dynamics. Contemporary Mathematics 441–458 (1989) doi:10.1090/conm/097/1021050 – 10.1090/conm/097/1021050
  • R.M. Santilli, Foundations of Theoretical Mechanics I (1978)
  • G. Sanchez de Alvarez, Geometric Methods of Classical Mechanics applied to Control Theory (1986)
  • Slotine, J.-J. E. Putting physics in control-the example of robotics. IEEE Control Syst. Mag. 8, 12–18 (1988) – 10.1109/37.9164
  • Sussmann, H. J. Existence and uniqueness of minimal realizations of nonlinear systems. Math. Systems Theory 10, 263–284 (1976) – 10.1007/bf01683278
  • H.J. Sussmann, Differential Geometric Control Theory (1983)
  • Takegaki, M. & Arimoto, S. A New Feedback Method for Dynamic Control of Manipulators. Journal of Dynamic Systems, Measurement, and Control 103, 119–125 (1981) – 10.1115/1.3139651
  • Tarn, T. J., Huang, G. & Clark, J. W. Modelling of quantum mechanical control systems. Mathematical Modelling 1, 109–121 (1980) – 10.1016/0270-0255(80)90011-1
  • W.M. Tulczyjew, Symp. Math. (1974)
  • Willems, J. C. Dissipative dynamical systems part I: General theory. Arch. Rational Mech. Anal. 45, 321–351 (1972) – 10.1007/bf00276493
  • Willems, J. C. Realization of systems with internal passivity and symmetry constraints. Journal of the Franklin Institute 301, 605–621 (1976) – 10.1016/0016-0032(76)90081-8
  • J.C. Willems, Ricerche di Automatica (1979)
  • Willems, J. C. From time series to linear system—Part I. Finite dimensional linear time invariant systems. Automatica 22, 561–580 (1986) – 10.1016/0005-1098(86)90066-x
  • E.T. Whittaker, A treatise on the analytical dynamics of particles and rigid bodies (1959)