Abstract
Various notions of passivity are introduced for a lossless circuit, or equivalently, for a rational matrix function θ which is J-unitary on the unit circle. These notions, as well as how they are related to each other, are analyzed from several points of view: energy bookkeeping in the circuit, analytic conditions on θ and on the associated scattering matrix U, geometry of shift invariant subspaces, positive definiteness conditions on associated reproducing kernel functions, connections with classical interpolation problems, and state space representations. This gives a circuit theoretic interpretation for several modern approaches to interpolation such as the geometric one of Ball-Helton.
Supported in part by the National Science Foundation, the Air Force Office of Scientific Research and the Office of Naval Research.
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
D. Alpay and H. Dym, Hilbert spaces of analytic functions inverse scattering, and operator models I, Integral Equations and Operator theory, 7 (1984), 589–641.
D. Alpay and H. Dym, On applications of reproducing kernel spaces to the Schur algorithm J-unitary factorization, in: Methods in Operator Theory and Signal Processing (ed. I. Gohberg), Operator Theory: Advances and Applications, 18 Birkhäuser (Basel), 1986.
D. Alpay and I. Gohberg, Unitary rational matrix functions and orthogonal matrix polynomials, Integral Equations and Operator Theory, to appear in Operator Theory: Advances and Applications volume, Birkhäuser (Basel).
J. A. Ball, N. Cohen and A. C. M. Ran, Inverse spectral problems for regular improper rational matrix functions, to appear in Operator Theory: Advances and Applications volume, Birkhäuser (Basel).
J. A. Ball, I. Gohberg and L. Rodman, Interpolation problems for matrix valued functions, Part I: rational functions, monograph in preparation.
J. A. Ball and J. W. Helton. A Beurling-Lax Theorem for the Lie group U(m, n) which contains most classical interpolation theory, J. Operator Theory, 9 (1983), 107–142.
J. A. Ball and J. W. Helton. Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parametrization of the set of all solutions, Integral Equations and Operator Theory, 9 (1986), 155–203.
J. A. Ball and J. W. Helton, Lie groups over the field of rational functions, signed spectral factorization, signed interpolation, and amplifier design, J. Operator Theory, 8 (1982), 19–64.
J. A. Ball and J. W. Helton. Beurling-Lax representations using classical Lie groups with many applications III: groups preserving forms, Amer. J. Math., 108 (1986), 95–174.
J. A. Ball and A. C. M. Ran. Local inverse spectral problems for rational matrix functions, Integral Equations and Operator Theory, 10 (1987), 349–415.
J. A. Ball and A. C. M. Ran. Global inverse spectral problems for rational matrix functions, Lin. Alg. and Appl., 86 (1987), 237–282.
I. I. Fedchin, Description of solutions of the tangential Nevanlinna-Pick problem, Akad. Nauk. Armjan. SSR. Dokl., 60:1 (1975), 37–42 (in Russian).
I. I. Fedchin, Tangential Nevanlinna-Pick problem with multiple points, Akad. Nauk. Armjan. SSR. Dokl., 60:1 (1975), 37–42 (in Russian).
C. Foias and A. Frazho, On the Schur representation in the commutant lifting theorem, I, Operator Theory, Advances and Applications, Birkhäuser 18 (1986).
K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L∞-error bounds, Inter. J. Control, 39 (1984), 1115–1193.
I. Gohberg and M. A. Kaashoek, An inverse spectral problem for rational matrix functions and minimal divisibility, Integral Equations and Operator Theory, 10 (1987), 437–465.
I. Gohberg, M. A. Kaashoek, L. Lerer and L. Rodman, Minimal divisors of rational matrix functions with prescribed zero and pole struture, in Topics in Operator Theory Systems and Networks, (ed. H. Dym and I. Gohberg), OT 12 Birkhäuser (Basel) (1983), 241–275.
T. Kailath, Linear Systems, Prentice Hall, Engelwood Cliffs, New Jersey, 1980.
H. Kimura, Directional interpolation in the state space, Lecture presented at SIAM Workshop on Linear Systems and Signal Processing, Stanford University, September 1987.
D. J. N. Limebeer and B.D. O. Anderson, An interpolation theory approach to H ∞ controller degree bounds, Lin. Alg. and Appl., to appear.
A. A. Nudelman, On a new problem of moment problem type, Soviet Math. Doklady, 18 (1977), 507–510 [Doklady Akademii Nauk SSSR (1977)].
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Ball, J.A., Helton, J.W. (1988). Shift Invariant Subspaces, Passivity, Reproducing Kernels and H ∞-Optimization. In: Gohberg, I., Helton, J.W., Rodman, L. (eds) Contributions to Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 35. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9284-1_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9284-1_12
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9978-9
Online ISBN: 978-3-0348-9284-1
eBook Packages: Springer Book Archive