Abstract
In this paper we show how the Total Least Squares identification problem may be viewed as a steepest descent problem on a Kähler manifold. The Kähler structure gives a method of explicitly deriving the steepest descent equations from a corresponding Hamiltonian flow associated with the problem. In the line-fitting case the steepest descent flow itself is shown also to be equivalent to the flow of a constrained Hamiltonian system — the Toda system.
This is a preview of subscription content, log in via an institution to check access.
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, “Foundation of Mechanics,” Benjamin Cummings, 1978.
A. Bloch, Estimation, Principal Components and Hamiltonian Systems, Systems and Control Letters 6 (1985), 103–108.
A. Bloch, An Infinite-Dimensional Hamiltonian System on Projective Hilbert Space, Trans. A.M.S. 302 (1987), 787–796.
A. Bloch, Steepest Descent, Linear Programming and Hamiltonian Flows, to appear in Cont Math. A.M.S.
A. Bloch, R. W. Brockett and T. Ratiu, A New Formulation of the Generalized Toda Lattice Equations and their Fixed Point Analysis via the Moment Map, to appear.
R. W. Brockett, A Geometrical Matching Problem, to appear in J. of Linear Algebra and its Applications.
R. W. Brockett, Dynamical Systems that Sort Lists and Solve Linear Programming Problems, in, Proc 27th IEEE Conf. on Decision and Control, IEEE, 1988, 799–803.
C. I. Byrnes and J. C. Willems, Least Squares Estimation, Linear Programming and Momentum, preprint.
P. Deift, T. Nanda and C. Tomei, Differential Equations for the Symmetric Eigenvalue Problem, SIAM J. on Numerical Analysis 20 (1983), 1–22.
P. Deift, F. Lund and E. Trubowitz, Nonlinear Wave Equations and Constrained Harmonic Motion, Communications in Math. Physics 74 (1980), 141–188.
H. Flaschka, The Toda Lattice, Phys. Rev. B 9 (1976), 1924–1925.
J. Moser, Finitely Many Mass Points on the Line under the Influence of an Exponential Potential, Battelles Recontres, Springer Notes in Physics (1976), 417–497.
A. Pressley and G. Segal, Loop Groups, Oxford University Press, 1986.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Birkhäuser Boston
About this chapter
Cite this chapter
Bloch, A.M. (1990). The Kähler Structure of the Total Least Squares Problem, Brockett’s Steepest Descent Equations, and Constrained Flows. In: Kaashoek, M.A., van Schuppen, J.H., Ran, A.C.M. (eds) Realization and Modelling in System Theory. Progress in Systems and Control Theory, vol 3. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3462-3_7
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3462-3_7
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8033-0
Online ISBN: 978-1-4612-3462-3
eBook Packages: Springer Book Archive