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, (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]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327
[2]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076
[3]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[4]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123
[5]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274
[6]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158
[7]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[8]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020
[9]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046
[10]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[11]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[12]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347
[13]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014
[14]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011
[15]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
[16]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[17]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032
[18]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences