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

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

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

Amol Sasane ^1,

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

Received August 2011 Revised October 2011 Published January 2012

Abstract
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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].

Keywords: $\nu$-metric, Hardy algebra., robust control.

Mathematics Subject Classification: Primary: 93B36, 93D15; Secondary: 46J1.

Citation: Amol Sasane. Extension of the $\nu$-metric for stabilizable plants over $H^\infty$. Mathematical Control and 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, ().
[2]	J. Ball and A. Sasane, Extension of the $\nu$-metric: The $H^\infty$ case,, preprint, ().
[3]	T. tom Dieck, "Algebraic Topology," EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008.
[4]	R. Douglas, "Banach Algebra Techniques in Operator Theory," 2^nd edition, Graduate Texts in Mathematics, 179, Springer-Verlag, New York, 1998.
[5]	J. Garnett, "Bounded Analytic Functions," revised 1^st edition, Graduate Texts in Mathematics, 236, Springer, New York, 2007.
[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.
[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.
[8]	N. Nikolski, "Treatise on the Shift Operator. Spectral Function Theory," Grundlehren der Mathematischen Wissenschaften, 273, Springer-Verlag, Berlin, 1986.
[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.
[10]	W. Rudin, "Functional Analysis," 2^nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
[11]	D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana University Mathematics Journal, 26 (1977), 817-838. doi: 10.1512/iumj.1977.26.26066.
[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.
[13]	G. Vinnicombe, Frequency domain uncertainty and the graph topology, IEEE Transactions on Automatic Control, 38 (1993), 1371-1383. doi: 10.1109/9.237648.

show all references

References:

[1]	J. Ball and A. Sasane, Extension of the $\nu$-metric,, preprint, ().
[2]	J. Ball and A. Sasane, Extension of the $\nu$-metric: The $H^\infty$ case,, preprint, ().
[3]	T. tom Dieck, "Algebraic Topology," EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008.
[4]	R. Douglas, "Banach Algebra Techniques in Operator Theory," 2^nd edition, Graduate Texts in Mathematics, 179, Springer-Verlag, New York, 1998.
[5]	J. Garnett, "Bounded Analytic Functions," revised 1^st edition, Graduate Texts in Mathematics, 236, Springer, New York, 2007.
[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.
[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.
[8]	N. Nikolski, "Treatise on the Shift Operator. Spectral Function Theory," Grundlehren der Mathematischen Wissenschaften, 273, Springer-Verlag, Berlin, 1986.
[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.
[10]	W. Rudin, "Functional Analysis," 2^nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
[11]	D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana University Mathematics Journal, 26 (1977), 817-838. doi: 10.1512/iumj.1977.26.26066.
[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.
[13]	G. Vinnicombe, Frequency domain uncertainty and the graph topology, IEEE Transactions on Automatic Control, 38 (1993), 1371-1383. doi: 10.1109/9.237648.

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