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, (2008).   Google Scholar

[4]

R. Douglas, "Banach Algebra Techniques in Operator Theory,", 2nd edition, 179 (1998).   Google Scholar

[5]

J. Garnett, "Bounded Analytic Functions,", revised 1st edition, 236 (2007).   Google Scholar

[6]

Y. Inouye, Parametrization of compensators for linear systems with transfer functions of bounded type,, Technical Report 88-01, (1988), 88.   Google Scholar

[7]

K. Mikkola, Weakly coprime factorization and state-feedback stabilization of discrete-time systems,, Mathematics of Control, 20 (2008), 321.  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 (1986).   Google Scholar

[9]

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

[10]

W. Rudin, "Functional Analysis,", 2nd edition, (1991).   Google Scholar

[11]

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

[12]

M. Smith, On stabilization and the existence of coprime factorizations,, in, 3 (1990), 215.   Google Scholar

[13]

G. Vinnicombe, Frequency domain uncertainty and the graph topology,, IEEE Transactions on Automatic Control, 38 (1993), 1371.  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, (2008).   Google Scholar

[4]

R. Douglas, "Banach Algebra Techniques in Operator Theory,", 2nd edition, 179 (1998).   Google Scholar

[5]

J. Garnett, "Bounded Analytic Functions,", revised 1st edition, 236 (2007).   Google Scholar

[6]

Y. Inouye, Parametrization of compensators for linear systems with transfer functions of bounded type,, Technical Report 88-01, (1988), 88.   Google Scholar

[7]

K. Mikkola, Weakly coprime factorization and state-feedback stabilization of discrete-time systems,, Mathematics of Control, 20 (2008), 321.  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 (1986).   Google Scholar

[9]

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

[10]

W. Rudin, "Functional Analysis,", 2nd edition, (1991).   Google Scholar

[11]

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

[12]

M. Smith, On stabilization and the existence of coprime factorizations,, in, 3 (1990), 215.   Google Scholar

[13]

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

[1]

T. Tachim Medjo, Louis Tcheugoue Tebou. Robust control problems in fluid flows. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 437-463. doi: 10.3934/dcds.2005.12.437

[2]

Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219

[3]

T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201

[4]

T. Tachim Medjo. Robust control problems for primitive equations of the ocean. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 769-788. doi: 10.3934/dcdsb.2011.15.769

[5]

Magdi S. Mahmoud, Omar Al-Buraiki. Robust control design of autonomous bicycle kinematics. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 181-191. doi: 10.3934/naco.2014.4.181

[6]

Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717

[7]

Alexander J. Zaslavski. Stability of a turnpike phenomenon for a class of optimal control systems in metric spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 245-260. doi: 10.3934/naco.2011.1.245

[8]

M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365

[9]

Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183

[10]

T. Tachim Medjo. Robust control of a Cahn-Hilliard-Navier-Stokes model. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2075-2101. doi: 10.3934/cpaa.2016028

[11]

Vadim Azhmyakov, Alex Poznyak, Omar Gonzalez. On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach. Journal of Industrial & Management Optimization, 2013, 9 (3) : 579-593. doi: 10.3934/jimo.2013.9.579

[12]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-15. doi: 10.3934/dcdss.2020066

[13]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017

[14]

Gregory Zitelli, Seddik M. Djouadi, Judy D. Day. Combining robust state estimation with nonlinear model predictive control to regulate the acute inflammatory response to pathogen. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1127-1139. doi: 10.3934/mbe.2015.12.1127

[15]

Pasquale Palumbo, Pierdomenico Pepe, Simona Panunzi, Andrea De Gaetano. Robust closed-loop control of plasma glycemia: A discrete-delay model approach. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 455-468. doi: 10.3934/dcdsb.2009.12.455

[16]

Yanning Li, Edward Canepa, Christian Claudel. Efficient robust control of first order scalar conservation laws using semi-analytical solutions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 525-542. doi: 10.3934/dcdss.2014.7.525

[17]

Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523

[18]

Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420

[19]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[20]

Viktor Levandovskyy, Gerhard Pfister, Valery G. Romanovski. Evaluating cyclicity of cubic systems with algorithms of computational algebra. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2023-2035. doi: 10.3934/cpaa.2012.11.2023

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]