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A Generalized Chordal Metric in Control Theory Making Strong Stabilizability a Robust Property

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Abstract

An abstract chordal metric is defined on linear control systems described by their transfer functions. Analogous to a previous result due to Partington (Linear Operators and Linear Systems. An Analytical Approach to Control Theory. London Mathematical Society Student Texts, vol. 60, Cambridge University Press, Cambridge, 2004) for \(H^\infty \), it is shown that strong stabilizability is a robust property in this metric.

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References

  1. Ball, J.A., Sasane, A.J.: Extension of the \(\nu \)-metric. Complex Anal. Oper. Theory 6(1), 65–89 (2012)

    Article  MathSciNet  Google Scholar 

  2. Bose, N.K.: Multidimensional Systems Theory and Applications, 2nd edn. Kluwer Academic Publishers, Dordrecht (2003) (with contributions by B. Buchberger and J. P. Guiver)

  3. El-Sakkary, A.K.: The gap metric: robustness of stabilization of feedback systems. IEEE Trans. Autom. Control 30(3), 240–247 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hayman, W.K.: Meromorphic Functions. Oxford Mathematical Monographs Clarendon Press, Oxford (1964)

    MATH  Google Scholar 

  5. Partington, J.R.: Robust control and approximation in the chordal metric. In: Hosoe, S. (ed.) Robust Control. Proceedings of the workshop held in Tokyo. Lecture Notes in Control and Information Sciences, vol. 183. Springer, Berlin (1992)

  6. Partington, J.R.: Linear Operators and Linear Systems. An Analytical Approach to Control Theory. London Mathematical Society Student Texts, vol. 60. Cambridge University Press, Cambridge (2004)

  7. Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1991)

    Google Scholar 

  8. El-Sakkary, A.K.: Estimating robustness on the Riemann sphere. Int. J. Control 49(2), 561–567 (1989)

    MathSciNet  MATH  Google Scholar 

  9. Sasane, A.J.: Extension of the \(\nu \)-metric for stabilizable plants over \(H^\infty \). Math. Control Relat. Fields 2(1), 29–44 (March 2012)

  10. Vidyasagar, M.: The graph metric for unstable plants and robustness estimates for feedback stability. IEEE Trans. Autom. Control 29(5), 403–418 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Vinnicombe, G.: Frequency domain uncertainty and the graph topology. IEEE Trans. Autom. Control 38(9), 1371–1383 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zames, G., El-Sakkary, A.K.: Unstable systems and feedback: the gap metric. In: Proceedings of the eighteenth Allerton conference on communication, control and computing, pp. 380–385. University of Illinois, Department of Electrical Engineering, Urbana-Champaign, IL (1980)

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Acknowledgments

The author thanks Jonathan Partington for kindly providing a copy of [5], and Rudolf Rupp for useful comments on a previous draft of the article.

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Authors and Affiliations

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

    Amol Sasane

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  1. Amol Sasane
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Correspondence to Amol Sasane.

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Communicated By: Joseph Ball.

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Sasane, A. A Generalized Chordal Metric in Control Theory Making Strong Stabilizability a Robust Property. Complex Anal. Oper. Theory 7, 1345–1356 (2013). https://doi.org/10.1007/s11785-012-0239-5

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  • DOI: https://doi.org/10.1007/s11785-012-0239-5

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