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Robust Control in Gap Metric

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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. When the plant and controller communicate through a non-ideal bidirectional channel, cascaded two-port networks can be utilized to model such a channel whose uncertainties are measured by the \(\mathcal {H}_\infty \) norm. With these descriptions of uncertainties 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  MathSciNet  Google Scholar 

  • Åström KJ, Murray RM (2008) Feedback fundamentals – control & dynamical systems. Princeton University Press, Princeton

    Google Scholar 

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

    Article  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  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  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  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  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  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  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

    Google Scholar 

  • Halsey KM, Glover K (2005) Analysis and synthesis of nested feedback systems. IEEE Trans Autom Control 50:984–996

    Article  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  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  • Kimura H (1996) Chain-scattering approach to \(\mathcal {H}_\infty \) control. Springer Science & Business Media, New York

    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  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  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  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  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  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  Google Scholar 

  • Zhao D, Chen C, Khong SZ, Qiu L (2018) Robust control against uncertainty quartet: a polynomial approach. In: Başar T (ed) Uncertainty in complex networked systems. Springer International Publishing, Cham, pp 149–178

    Chapter  Google Scholar 

  • Zhao D, Qiu L, Gu G (2020, to appear) Stabilization of two-port networked systems with simultaneous uncertainties in plant, controller, and communication channels. IEEE Trans Autom Control. https://doi.org/10.1109/TAC.2019.2918121

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

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong SAR, China

    Li Qiu & Di Zhao

Authors
  1. Li Qiu
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  2. Di Zhao
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Correspondence to Li Qiu .

Editor information

Editors and Affiliations

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

    John Baillieul

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

    Tariq Samad

Section Editor information

  1. Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, USA

    Kemin Zhou

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Qiu, L., Zhao, D. (2019). 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_165-2

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

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-5102-9

  • Online ISBN: 978-1-4471-5102-9

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

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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

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