March  2012, 2(1): 29-44. doi: 10.3934/mcrf.2012.2.29

Extension of the $\nu$-metric for stabilizable plants over $H^\infty$

1. 

Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom

Received  August 2011 Revised  October 2011 Published  January 2012

An abstract $\nu$-metric was introduced in [1], with a view towards extending the classical $\nu$-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. Here we give an important concrete special instance of the abstract $\nu$-metric, namely the case when the ring of stable transfer functions is the Hardy algebra $H^\infty$, by verifying that all the assumptions demanded in the abstract set-up are satisfied. This settles the open question implicit in [2].
Citation: Amol Sasane. Extension of the $\nu$-metric for stabilizable plants over $H^\infty$. Mathematical Control & Related Fields, 2012, 2 (1) : 29-44. doi: 10.3934/mcrf.2012.2.29
References:
[1]

J. Ball and A. Sasane, Extension of the $\nu$-metric,, preprint, ().   Google Scholar

[2]

J. Ball and A. Sasane, Extension of the $\nu$-metric: The $H^\infty$ case,, preprint, ().   Google Scholar

[3]

T. tom Dieck, "Algebraic Topology," EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008.  Google Scholar

[4]

R. Douglas, "Banach Algebra Techniques in Operator Theory," 2nd edition, Graduate Texts in Mathematics, 179, Springer-Verlag, New York, 1998.  Google Scholar

[5]

J. Garnett, "Bounded Analytic Functions," revised 1st edition, Graduate Texts in Mathematics, 236, Springer, New York, 2007.  Google Scholar

[6]

Y. Inouye, Parametrization of compensators for linear systems with transfer functions of bounded type, Technical Report 88-01, Faculty of Eng. Sci., Osaka University, Osaka, Japan, March 1988. Google Scholar

[7]

K. Mikkola, Weakly coprime factorization and state-feedback stabilization of discrete-time systems, Mathematics of Control, Signals, and Systems, 20 (2008), 321-350. doi: 10.1007/s00498-008-0034-z.  Google Scholar

[8]

N. Nikolski, "Treatise on the Shift Operator. Spectral Function Theory," Grundlehren der Mathematischen Wissenschaften, 273, Springer-Verlag, Berlin, 1986.  Google Scholar

[9]

N. Nikolski, "Operators, Functions, and Systems: An Easy Reading. Vol. 1. Hardy, Hankel, and Toeplitz," Mathematical Surveys and Monographs, 92, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[10]

W. Rudin, "Functional Analysis," 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.  Google Scholar

[11]

D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana University Mathematics Journal, 26 (1977), 817-838. doi: 10.1512/iumj.1977.26.26066.  Google Scholar

[12]

M. Smith, On stabilization and the existence of coprime factorizations, in "Realization and Modelling in System Theory" (Amsterdam, 1989), Progr. Systems Control Theory, 3, Birkhäuser Boston, Boston, MA, (1990), 215-222.  Google Scholar

[13]

G. Vinnicombe, Frequency domain uncertainty and the graph topology, IEEE Transactions on Automatic Control, 38 (1993), 1371-1383. doi: 10.1109/9.237648.  Google Scholar

show all references

References:
[1]

J. Ball and A. Sasane, Extension of the $\nu$-metric,, preprint, ().   Google Scholar

[2]

J. Ball and A. Sasane, Extension of the $\nu$-metric: The $H^\infty$ case,, preprint, ().   Google Scholar

[3]

T. tom Dieck, "Algebraic Topology," EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008.  Google Scholar

[4]

R. Douglas, "Banach Algebra Techniques in Operator Theory," 2nd edition, Graduate Texts in Mathematics, 179, Springer-Verlag, New York, 1998.  Google Scholar

[5]

J. Garnett, "Bounded Analytic Functions," revised 1st edition, Graduate Texts in Mathematics, 236, Springer, New York, 2007.  Google Scholar

[6]

Y. Inouye, Parametrization of compensators for linear systems with transfer functions of bounded type, Technical Report 88-01, Faculty of Eng. Sci., Osaka University, Osaka, Japan, March 1988. Google Scholar

[7]

K. Mikkola, Weakly coprime factorization and state-feedback stabilization of discrete-time systems, Mathematics of Control, Signals, and Systems, 20 (2008), 321-350. doi: 10.1007/s00498-008-0034-z.  Google Scholar

[8]

N. Nikolski, "Treatise on the Shift Operator. Spectral Function Theory," Grundlehren der Mathematischen Wissenschaften, 273, Springer-Verlag, Berlin, 1986.  Google Scholar

[9]

N. Nikolski, "Operators, Functions, and Systems: An Easy Reading. Vol. 1. Hardy, Hankel, and Toeplitz," Mathematical Surveys and Monographs, 92, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[10]

W. Rudin, "Functional Analysis," 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.  Google Scholar

[11]

D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana University Mathematics Journal, 26 (1977), 817-838. doi: 10.1512/iumj.1977.26.26066.  Google Scholar

[12]

M. Smith, On stabilization and the existence of coprime factorizations, in "Realization and Modelling in System Theory" (Amsterdam, 1989), Progr. Systems Control Theory, 3, Birkhäuser Boston, Boston, MA, (1990), 215-222.  Google Scholar

[13]

G. Vinnicombe, Frequency domain uncertainty and the graph topology, IEEE Transactions on Automatic Control, 38 (1993), 1371-1383. doi: 10.1109/9.237648.  Google Scholar

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