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Robust Control in Gap Metric

Encyclopedia of Systems and Control

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Abstract

Robust control needs to start with a model of system uncertainty. What is a good uncertainty model? First it needs to capture the possible system perturbations and uncertainties. Second it needs to be mathematically tractable. The gap metric was introduced by Zames and El-Sakkary for this purpose. Its study climaxed in an award-winning paper by Georgiou and Smith. A modified gap, called the ν-gap, was later discovered by Vinnicombe and was shown to have advantages. With these metrics in hand, robust stabilization issues can be nicely addressed.

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Bibliography

  • Anderson BDO, Brinsmead TS, De Bruyne F (2002) The Vinnicombe metric for nonlinear operators. IEEE Trans Autom Control 47:1450–1465

    Article  Google Scholar 

  • Ball JA, Sasane AJ (2012) Extension of the ν-metric. Complex Anal Oper Theory 6:65–89

    Article  MATH  MathSciNet  Google Scholar 

  • Bian W, French M (2005) Graph topologies, gap metrics, and robust stability for nonlinear systems. SIAM J Control Optim 44:418–443

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Feintuch A (1998) Robust control theory in Hilbert space. Springer, New York

    Book  MATH  Google Scholar 

  • Foias C, Georgiou TT, Smith MC (1993) Robust stability of feedback systems: a geometric approach using the gap metric. SIAM J Control Optim 31:1518–1537

    Article  MATH  MathSciNet  Google Scholar 

  • Georgiou TT (1988) On the computation of the gap metric. Syst Control Lett 11:253–257

    Article  MATH  MathSciNet  Google Scholar 

  • Georgiou TT, Smith MC (1990) Optimal robustness in the gap metric. IEEE Trans Autom Control 35:673–687

    Article  MATH  MathSciNet  Google Scholar 

  • Georgiou TT, Smith MC (1992) Robust stabilization in the gap metric: controller design for distributed plants. IEEE Trans Autom Control 37:1133–1143

    Article  MATH  MathSciNet  Google Scholar 

  • Georgiou TT, Smith MC (1997) Robustness analysis of nonlinear feedback systems: an input-output approach. IEEE Trans Autom Control 42:1200–1221

    Article  MATH  MathSciNet  Google Scholar 

  • Glover K, McFarlane DC (1989) Robust stabilization of normalized coprime factor plant descriptions with \(\mathcal{H}_{\infty }\) bounded uncertainties. IEEE Trans Autom Control 34:821–830

    Article  MATH  MathSciNet  Google Scholar 

  • James MR, Smith MC, Vinnicombe G (2005) Gap metrics, representations and nonlinear robust stability. SIAM J Control Optim 43:1535–1582

    Article  MATH  MathSciNet  Google Scholar 

  • Kato T (1976) Perturbation theory for linear operators, 2nd edn. Springer, Berlin

    Book  MATH  Google Scholar 

  • McFarlane DC, Glover K (1992) A loop shaping design procedure using \(\mathcal{H}_{\infty }\)-synthesis. IEEE Trans Autom Control 37:759–769

    Article  MATH  MathSciNet  Google Scholar 

  • Qiu L, Davison EJ (1992a) Feedback stability under simultaneous gap metric uncertainties in plant and controller. Syst Control Lett 18:9–22

    Article  MATH  MathSciNet  Google Scholar 

  • Qiu L, Davison EJ (1992b) Pointwise gap metrics on transfer matrices. IEEE Trans Autom Control 37:741–758

    Article  MATH  MathSciNet  Google Scholar 

  • Qiu L, Zhang Y, Li CK (2008) Unitarily invariant metrics on the Grassmann space. SIAM J Matrix Anal 27:501–531

    Google Scholar 

  • Qiu L, Zhou K (2013) Preclassical tools for postmodern control. IEEE Control Syst Mag 33(4):26–38

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Vidyasagar M (1985) Control system synthesis: a factorization approach. MIT, Cambridge

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Vinnicombe G (2001) Uncertainty and feedback: \(\mathcal{H}_{\infty }\) loop-shaping and the ν-gap metric. Imperial Collage Press, London

    Google Scholar 

  • Zames G, El-Sakkary AK (1980) Unstable systems and feedback: the gap metric. In: Proceedings of the 16th Allerton conference, Illinois, pp 380–385

    Google Scholar 

  • Zhang Y, Qiu L (2010) From subadditive inequalities of singular values to triangular inequalities of canonical angles. SIAM J Matrix Anal Appl 31:1606–1620

    Article  MATH  MathSciNet  Google Scholar 

  • Zhou K, Doyle JC (1998) Essentials of robust control. Prentice Hall, Upper Saddle River

    Google Scholar 

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Authors and Affiliations

  1. Hong Kong University of Science and Technology, Hong Kong SAR, China

    Li Qiu

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  1. Li Qiu
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Correspondence to Li Qiu .

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  1. Electrical and Computer Engineering, Boston University, Boston, Massachusetts, USA

    John Baillieul

  2. Automation and Control Solutions, Honeywell, Golden Valley, Minnesota, USA

    Tariq Samad

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© 2014 Springer-Verlag London

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Qiu, L. (2014). Robust Control in Gap Metric. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_165-1

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  • DOI: https://doi.org/10.1007/978-1-4471-5102-9_165-1

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  • Online ISBN: 978-1-4471-5102-9

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Chapter history

  1. Latest

    Robust Control in Gap Metric
    Published:
    26 September 2019

    DOI: https://doi.org/10.1007/978-1-4471-5102-9_165-2

  2. Original

    Robust Control in Gap Metric
    Published:
    06 March 2014

    DOI: https://doi.org/10.1007/978-1-4471-5102-9_165-1