H-infinity loop-shaping

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H-infinity loop-shaping is a design methodology in modern control theory. It combines the traditional intuition of classical control methods, such as Bode's sensitivity integral, with H-infinity optimization techniques to achieve controllers whose stability and performance properties hold good in spite of bounded differences between the nominal plant assumed in design and the true plant encountered in practice. Essentially, the control system designer describes the desired responsiveness and noise-suppression properties by weighting the plant transfer function in the frequency domain; the resulting 'loop-shape' is then 'robustified' through optimization. Robustification usually has little effect at high and low frequencies, but the response around unity-gain crossover is adjusted to maximise the system's stability margins. H-infinity loop-shaping can be applied to multiple-input multiple-output (MIMO) systems.

H-infinity loop-shaping can be carried out using commercially available software.[1]

H-infinity loop-shaping has been successfully deployed in industry. In 1995, R. Hyde, K. Glover and G. T. Shanks published a paper[2] describing the successful application of the technique to a VSTOL aircraft. In 2008, D. J. Auger, S. Crawshaw and S. L. Hall published another paper[3] describing a successful application to a steerable marine radar tracker, noting that the technique had the following benefits:

  • Easy to apply – commercial software handles the hard math.
  • Easy to implement – standard transfer functions and state-space methods can be used.
  • Plug and play – no need for re-tuning on an installation-by-installation basis.

See also[edit]


  1. ^ The MathWorks, Inc. Synthesizing Robust Multivariable Controllers. Retrieved September 16, 2007.
  2. ^ Computing and Control Engineering Journal, 6(1):11–16
  3. ^ Proceedings of the UKACC International Conference on Control 2008

Further reading[edit]

  • Auger, D. J., Crawshaw, S., and Hall, S. L. (2008). Robust H-infinity Control of a Steerable Marine Radar Tracker. In Proceedings of the UKACC International Conference on Control 2008. Manchester: UKACC.
  • Chiang, R., Safonov, M. G., Balas, G., and Packard, A. (2007). Robust Control Toolbox, 3rd ed. Natick, MA: The Mathworks, Inc.
  • Glad, T. and Ljung, L. (2000). Control Theory: Multivariable and Nonlinear Methods. London: Taylor & Francis.
  • Hyde, R.A., Glover, K. and Shanks, G. T. (1995). VSTOL first flight of an H-infinity control law. Computing and Control Engineering Journal, 6(1):11–16.
  • McFarlane, D. C. and Glover, K. (1989). Robust Controller Design Using Normalized Coprime Factor Plant Descriptions (Lecture Notes in Control and Information Sciences), 1st ed. New York: Springer.
  • Vinnicombe, G. (2000). Uncertainty and feedback: H-Infinity Loop-Shaping and the V-Gap Metric, 1st ed. London: Imperial College Press.
  • Zhou, K., Doyle, J. C. and Glover, K. (1995). Robust and Optimal Control. New York: Prentice-Hall.
  • Zhou, K. and Doyle, J. C. (1998). Essentials of Robust Control. New York: Prentice-Hall.