American Institute of Mathematical Sciences

Advanced
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

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 & 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

[1]

T. Tachim Medjo, Louis Tcheugoue Tebou. Robust control problems in fluid flows. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 437-463. doi: 10.3934/dcds.2005.12.437
[2]

Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021011
[3]

Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219
[4]

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

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
[6]

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
[7]

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
[8]

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
[9]

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
[10]

Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029
[11]

Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183
[12]

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
[13]

Peng Cheng, Feng Pan, Yanyan Yin, Song Wang. Probabilistic robust anti-disturbance control of uncertain systems. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2441-2450. doi: 10.3934/jimo.2020076
[14]

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
[15]

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
[16]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066
[17]

Paul Breiding, Türkü Özlüm Çelik, Timothy Duff, Alexander Heaton, Aida Maraj, Anna-Laura Sattelberger, Lorenzo Venturello, Oǧuzhan Yürük. Nonlinear algebra and applications. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021045
[18]

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
[19]

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
[20]

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

2020 Impact Factor: 1.284

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy