Exact Modal Decomposition of Nonlinear Hamiltonian Systems

Energy dissipation in damped autonomous Hamiltonian systems is a fundamental asymptotic convergence principle which underlies many physical and technical applications. While for linear time-invariant systems, trajectories converge if and only if the real part of the symmetric eigenvalue matrix is negative-definite, as of yet such sy stematic stability decomposition has no analog for nonlinear damped Hamiltonian systems. This paper shows that, by analyzing the stability of a time-varying, non-linear and damped Hamiltonian system in geodesic coordinates, the differential analysis tools of contraction theory 14‐18, 26 allow an exact modal decomposition to be performed. Accordingly, specifying only the nonlinear potential energy and damping is

[1]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[2]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[3]  D. Jordan,et al.  Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers , 1979 .

[4]  Jean-Jacques E. Slotine,et al.  On partial contraction analysis for coupled nonlinear oscillators , 2004, Biological Cybernetics.

[5]  P. Hartman.  Ordinary Differential Equations , 1965 .

[6]  G. Seifert,et al.  ON STABILITY IN THE LARGE FOR PERIODIC SOLUTION OF DIFFERENTIAL SYSTEMS , 1958 .

[7]  J. Slotine,et al.  On metric observers for nonlinear systems , 1996, Proceeding of the 1996 IEEE International Conference on Control Applications IEEE International Conference on Control Applications held together with IEEE International Symposium on Intelligent Contro.

[8]  Winfried Lohmiller,et al.  Shaping State-Dependent Convergence Rates in Nonlinear Control System Design , 2008 .

[9]  W. Lohmiller,et al.  Contraction analysis of non-linear distributed systems , 2005 .

[10]  Yong Zhao,et al.  Discrete nonlinear observers for inertial navigation , 2005, Syst. Control. Lett..

[11]  Jean-Jacques E. Slotine,et al.  Modularity, evolution, and the binding problem: a view from stability theory , 2001, Neural Networks.

[12]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[13]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[14]  John Weston,et al.  Strapdown Inertial Navigation Technology , 1997 .

[15]  Nikolaus Moritz,et al.  Three-Stage Gradient-Based Optimization Scheme in Design of Feedback Gains within Eurofighter Primary Control Laws , 2006 .

[16]  Soon-Jo Chung,et al.  Cooperative Robot Control and Concurrent Synchronization of Lagrangian Systems , 2007, IEEE Transactions on Robotics.

[17]  Henry D. I. Abarbanel,et al.  Analysis of Observed Chaotic Data , 1995 .

[18]  Jean-Jacques E. Slotine,et al.  Nonlinear process control using contraction theory , 2000 .

[19]  Thomas Kailath,et al.  Linear Systems , 1980 .

[20]  Shun-ichi Amari,et al.  Natural Gradient Works Efficiently in Learning , 1998, Neural Computation.

[21]  R. Tolman.  Relativity, Thermodynamics, and Cosmology , 1934 .

[22]  D. C. Lewis.  Metric Properties of Differential Equations , 1949 .

[23]  Jean-Jacques E. Slotine,et al.  Control system design for mechanical systems using contraction theory , 2000, IEEE Trans. Autom. Control..

[24]  David Angeli,et al.  A Lyapunov approach to incremental stability properties , 2002, IEEE Trans. Autom. Control..

[25]  I. Prigogine,et al.  Thermodynamics and cosmology , 1989 .