Abstract
We extend the ν-metric introduced by Vinnicombe in robust control theory for rational plants to the case of infinite-dimensional systems/classes of nonrational transfer functions.
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Bernstein, D.: Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press, Princeton (2005)
Brudnyi A., Sasane A.J.: Sufficient conditions for the projective freeness of Banach algebras. J. Funct. Anal. 257(12), 4003–4014 (2009)
Böttcher A.: On the corona theorem for almost periodic functions. Integr. Equ. Oper. Theory 33(3), 253–272 (1999)
Böttcher, A., Karlovich, Y.I.: Carleson curves, Muckenhoupt weights and Toeplitz operators. In: Progress in Mathematics, vol. 154. Birkhäuser, Basel (1997)
Böttcher, A., Karlovich, Y.I., Spitkovsky, I.: Convolution operators and factorization of almost periodic matrix functions. In: Operator Theory Advances and Applications, vol. 131. Birkhäuser, Basel (2002)
Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators. Springer, Berlin (1990) (Second edition (prepared jointly with A. Karlovich), Springer, Berlin, 2006)
Callier F.M., Desoer C.A.: A graphical test for checking the stability of a linear time-invariant feedback system. IEEE Trans. Automat. Contr. AC-17(6), 773–780 (1972)
Callier F.M., Desoer C.A.: An algebra of transfer functions for distributed linear time-invariant systems. Special issue on the mathematical foundations of system theory. IEEE Trans. Circuits Syst. 25(9), 651–662 (1978)
Davis J.H.: Encirclement conditions for stability and instability of feedback systems with delays. Int. J. Contr. 15(4), 793–799 (1972)
DeCarlo R.A., Murray J., Saeks R.: Multivariable Nyquist theory. Int. J. Contr. 25(5), 657–675 (1977)
Gamelin T.W.: Uniform Algebras. Prentice-Hall, Englewood Cliffs (1969)
Gelfand, I., Raikov, D., Shilov, G.: Commutative Normed Rings. Translated from the Russian, with a supplementary chapter. Chelsa, New York (1964)
Gohberg I.C., Fel’dman I.A.: Integro-difference Wiener–Hopf equations. (Russian) Acta Sci. Math. (Szeged) 30, 199–224 (1969)
Hille, E., Phillips, R.S.: Functional analysis and semi-groups. In: Third Printing of the Revised Edition of 1957. American Mathematical Society Colloquium Publications, vol. XXXI. American Mathematical Society, Providence (1974)
Jessen B., Tornehave H.: Mean motions and zeros of almost periodic functions. Acta Math. 77, 137–279 (1945)
Murphy G.J.: Topological and analytical indices in C*-algebras. J. Funct. Anal. 234(2), 261–276 (2006)
Quadrat A.: A lattice approach to analysis and synthesis problems. Math. Contr. Signals Syst. 18(2), 147–186 (2006)
Rudin W.: Function Theory in Polydiscs. W.A. Benjamin, New York (1969)
Rudin W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)
Saeks R.: On the encirclement condition and its generalization. IEEE Trans. Circuits Syst. CAS-22(10), 780–785 (1975)
Sasane, A.J.: An abstract Nyquist criterion containing old and new results (2010, submitted)
Treil S.: A counterexample on continuous coprime factors. IEEE Trans. Automat. Contr. 39(6), 1262–1263 (1994)
Ullrich, D.C.: Complex made simple. In: Graduate Studies in Mathematics, vol. 97. American Mathematical Society, Providence (2008)
Vidyasagar M.: Control System Synthesis: a Factorization Approach. MIT Press, Cambridge (1985)
Vinnicombe G.: Frequency domain uncertainty and the graph topology. IEEE Trans. Automat. Contr. 38(9), 1371–1383 (1993)
Young, N.: Some function-theoretic issues in feedback stabilization. In: Holomorphic Spaces (Berkeley, CA, 1995). Publications of the Mathematical Science Research Institute, vol. 33, pp. 337–349. Cambridge University Press, Cambridge (1998)
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Communicated by Daniel Aron Alpay.
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Ball, J.A., Sasane, A.J. Extension of the ν-metric. Complex Anal. Oper. Theory 6, 65–89 (2012). https://doi.org/10.1007/s11785-010-0097-y
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DOI: https://doi.org/10.1007/s11785-010-0097-y