State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions

A.E. Frazho; S. ter Horst; M.A. Kaashoek

doi:10.1007/978-3-319-10335-8_8

State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions

A.E. Frazho
, S. ter Horst
, M.A. Kaashoek

Mathematics

Research output: Chapter in Book / Report / Conference proceeding › Chapter › Academic › peer-review

Abstract

For the strictly positive case (the suboptimal case), given stable rational matrix functions G and K, the set of all H∞ solutions X to the Leech problem associated with G and K, that is, G(z)X(z)=K(z) and sup|z|≤1∥X(z)∥≤1, is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions G and K. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations.

Original language	English
Title of host publication	Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes
Subtitle of host publication	A Collection of Papers Dedicated to Lev Sakhnovich
Editors	Daniel Alpay, Bernd Kirstein
Publisher	Birkhäuser
Pages	149-179
Number of pages	31
ISBN (Electronic)	9783319103358
ISBN (Print)	9783319103341, 9783319376721
DOIs	https://doi.org/10.1007/978-3-319-10335-8_8
Publication status	Published - 2015

Publication series

Name	Operator Theory: Advances and Applications (OT)
Publisher	Springer
Volume	244

Name	Linear Operators and Linear Systems (LOLS)
Publisher	Springer

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10.1007/978-3-319-10335-8_8

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Frazho, A. E., ter Horst, S., & Kaashoek, M. A. (2015). State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions. In D. Alpay, & B. Kirstein (Eds.), Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes: A Collection of Papers Dedicated to Lev Sakhnovich (pp. 149-179). (Operator Theory: Advances and Applications (OT); Vol. 244), (Linear Operators and Linear Systems (LOLS)). Birkhäuser. https://doi.org/10.1007/978-3-319-10335-8_8

Frazho, A.E. ; ter Horst, S. ; Kaashoek, M.A. / State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions. Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes: A Collection of Papers Dedicated to Lev Sakhnovich. editor / Daniel Alpay ; Bernd Kirstein. Birkhäuser, 2015. pp. 149-179 (Operator Theory: Advances and Applications (OT)). (Linear Operators and Linear Systems (LOLS)).

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title = "State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions",

abstract = "For the strictly positive case (the suboptimal case), given stable rational matrix functions G and K, the set of all H∞ solutions X to the Leech problem associated with G and K, that is, G(z)X(z)=K(z) and sup|z|≤1∥X(z)∥≤1, is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions G and K. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations.",

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Frazho, AE, ter Horst, S & Kaashoek, MA 2015, State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions. in D Alpay & B Kirstein (eds), Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes: A Collection of Papers Dedicated to Lev Sakhnovich. Operator Theory: Advances and Applications (OT), vol. 244, Linear Operators and Linear Systems (LOLS), Birkhäuser, pp. 149-179. https://doi.org/10.1007/978-3-319-10335-8_8

State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions. / Frazho, A.E.; ter Horst, S.; Kaashoek, M.A.
Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes: A Collection of Papers Dedicated to Lev Sakhnovich. ed. / Daniel Alpay; Bernd Kirstein. Birkhäuser, 2015. p. 149-179 (Operator Theory: Advances and Applications (OT); Vol. 244), (Linear Operators and Linear Systems (LOLS)).

Research output: Chapter in Book / Report / Conference proceeding › Chapter › Academic › peer-review

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AU - ter Horst, S.

AU - Kaashoek, M.A.

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N2 - For the strictly positive case (the suboptimal case), given stable rational matrix functions G and K, the set of all H∞ solutions X to the Leech problem associated with G and K, that is, G(z)X(z)=K(z) and sup|z|≤1∥X(z)∥≤1, is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions G and K. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations.

AB - For the strictly positive case (the suboptimal case), given stable rational matrix functions G and K, the set of all H∞ solutions X to the Leech problem associated with G and K, that is, G(z)X(z)=K(z) and sup|z|≤1∥X(z)∥≤1, is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions G and K. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations.

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M3 - Chapter

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SN - 9783319376721

T3 - Operator Theory: Advances and Applications (OT)

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EP - 179

BT - Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes

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Frazho AE, ter Horst S, Kaashoek MA. State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions. In Alpay D, Kirstein B, editors, Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes: A Collection of Papers Dedicated to Lev Sakhnovich. Birkhäuser. 2015. p. 149-179. (Operator Theory: Advances and Applications (OT)). (Linear Operators and Linear Systems (LOLS)). Epub 2015 Apr 30. doi: 10.1007/978-3-319-10335-8_8

State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions

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