Abstract
In the present chapter we focus on a special subclass of nonlinear control systems, which can be called mechanical nonlinear control systems. Roughly speaking these are control systems whose dynamics can be described by the Euler-Lagrangian or Hamiltonian equations of motion. It is well-known that a large class of physical systems admits, at least partially, a representation by these equations, which lie at the heart of the theoretical framework of physics.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R.A. Abraham and J.E. Marsden. Foundations of Mechanics. Benjamin/Cummings, Reading, Mass., 2nd edition, 1978.
V.I. Arnold. Mathematical Methods of Classical Mechanics. Springer, Berlin, 1978. (Translation of the 1974 Russian edition).
R.W. Brockett. Control theory and analytical mechanics. In C.Martin and R. Hermann, editors, Geometric Control Theory, volume VII of Lie groups: History, Frontiers and Applications, pages 1–46. Math. Sci. Press., Brookline, 1977.
P.E. Crouch and M. Irving. On finite Volterra series which admit Hamiltonian realizations. Math. Systems Theory, 17:293–318, 1984.
P.E. Crouch and M. Irving. Dynamical realizations of homogeneous Hamiltonian systems. SIAM J. Contr. Optimiz., 24:374–395, 1986.
P.E. Crouch and A.J. van der Schaft. Variational and Hamiltonian Control Systems, volume 101 of Lect. Notes Contr. Inf. Sci. Springer, Berlin, 1987.
P. Deift, F. Lund, and E. Trubowitz. Nonlinear wave equations and constrained harmonic motion. Comm. Math. Phys., 74:141–188, 1980.
J.W. Grizzle and S.I. Marcus. A Jacobi-Liouville theorem for Hamiltonian control systems. In Proc. 23rd IEEE Conf. Decision Control, Las Vegas, pages 1598–1602, 1984.
J.W. Grizzle and S.I. Marcus. The structure of nonlinear control systems possessing symmetries. IEEE Trans. Autom. Contr., 30:248–258, 1985.
H. Goldstein. Classical Mechanics. Addison-Wesley, Reading, Mass., 1950.
H.J.C. Huijberts and A.J. van der Schaft. Input-output decoupling with stability for Hamiltonian systems. Math. Control, Signals, Systems, 3:125–138, 1990.
B. Jakubczyk. Existence of Hamiltonian realizations of nonlinear causal operators. Bull. Pol. Ac.: Math., 34:737–747, 1986.
B. Jakubczyk. Poisson structures and relations on vectorfields and their Hamiltonians. Bull. Pol. Ac.: Math., 34:713–721, 1986.
P.S. Krishnaprasad and J.E. Marsden. Hamiltonian structures and stability for rigid bodies with flexible attachments. Arch. Rat. Mech. Anal., 98:71–93, 1987.
D.E. Koditschek. Natural motion for robot arms. In Proc. 23rd IEEE Conf. Decision Control, Las Vegas, pages 733–735, 1984.
P.S. Krishnaprasad. Lie-Poisson structures, dual-spin spacecraft and asymptotic stability. Nonl. Anal. Th. Meth. Appl., 9:1011–1035, 1985.
P. Libermann and C.M.Marle. Symplectic geometry and analytical mechanics. Reidel, Dordrecht, 1987.
R. Marino. Stabilization and feedback equivalence to linear coupled oscillators. Int. J. Control, 39:487–496, 1984.
N.H. McClamroch and A.M. Bloch. Control of constrained Hamiltonian systems and applications to control of constrained robots. In F.M.A. Salam and M.L. Levi, editors, Dynamical Systems Approaches to Nonlinear Problems in Systems and Circuits, pages 394–403. SIAM, 1988.
J.E. Marsden and T. Ratiu. Reduction of Poisson manifolds. Lett. Math. Phys., pages 161–169, 1986.
J.E. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys., pages 121–130, 1974.
P.J. Olver. Applications of Lie groups to differential equations. Springer, New York, 1986.
G. Sanchez de Alvarez. Geometric Methods of Classical Mechanics applied to Control Theory. PhD thesis, Dept. Mathematics, Univ. of California, Berkeley, 1986.
N. Sreenath, Y.G. Oh, P.S. Krishnaprasad, and J.E.Marsden. The dynamics of coupled planar rigid bodies, Part I: Reduction, equilibria & stability. Dynamics and Stability of Systems, 3:25–49, 1988.
M. Takegaki and S. Arimoto. A new feedback method for dynamic control of manipulators. J. Dyn. Sys., Meas., Control, 103:119–125, 1981.
J. Tsinias and N. Kalouptsidis. On stabilizability of nonlinear systems. In Proc. 21st IEEE Conf. Decision Control, pages 712–716, 1982.
A.J. van der Schaft. Symmetries and conservation laws for Hamiltonian systems with inputs and outputs: A generalization of Noether’s theorem. Systems Control Lett., 1:108–115, 1981.
A.J. van der Schaft. Controllability and observability for affine nonlinear Hamiltonian systems. IEEE Trans. Autom. Contr., AC-27:490–492, 1982.
A.J. van der Schaft. Hamiltonian dynamics with external forces and observations. Math. Systems Th., 15:145–168, 1982.
A.J. van der Schaft. Symmetries, conservation laws and time-reversibility for Hamiltonian systems with external forces. J. Math. Phys., 24:2095–2101, 1983.
A.J. van der Schaft. System theoretic descriptions of physical systems. CWI Tract 3, CWI, Amsterdam, 1984.
A.J. van der Schaft. Linearization of Hamiltonian and gradient systems. IMA J. Math. Control Information, 1:185–198, 1984.
A.J. van der Schaft. Controlled invariance for Hamiltonian systems. Math. Systems Th., 18:257–291, 1985.
A.J. van der Schaft. On feedback control of Hamiltonian systems. In C.I. Byrnes and A. Lindquist, editors, Theory and Applications of Nonlinear Control Systems, pages 273–290. North-Holland, Amsterdam, 1986.
A.J. van der Schaft. Stabilization of Hamiltonian systems. Nonl. An. Th. Meth. Appl., 10:1021–1035, 1986.
A.J. van der Schaft. Equations of motion for Hamiltonian systems with constraints. J. Phys. A. Math. Gen., 20:3271–3277, 1987.
A.Weinstein. The local structure of Poisson manifolds. J. Differential Geom., 18:523–557, 1983.
E.T. Whittaker. A treatise on the analytical dynamics of particles and rigid bodies. Cambridge University Press, Cambridge, 4th edition, 1959.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 1990 Springer Science+Business Media New York 1990, Corrected printing 2016
About this chapter
Cite this chapter
Nijmeijer, H., van der Schaft, A. (1990). Mechanical Nonlinear Control Systems. In: Nonlinear Dynamical Control Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2101-0_12
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2101-0_12
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-2102-7
Online ISBN: 978-1-4757-2101-0
eBook Packages: Springer Book Archive