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<title>H-infinity loop-shaping</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">H-infinity loop-shaping</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>H-infinity loop-shaping</b> is a design methodology in modern <a href="Control_theory" title="Control theory">control theory</a>. It combines the traditional intuition of classical control methods, such as <a href="Bode's_sensitivity_integral" title="Bode's sensitivity integral">Bode's sensitivity integral</a>, with <a href="H_infinity" class="mw-redirect" title="H infinity">H-infinity optimization techniques</a> to achieve controllers whose stability and performance properties hold despite 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 <a href="Transfer_function" title="Transfer function">transfer function</a> in the <a href="Frequency_domain" title="Frequency domain">frequency domain</a>; 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.

</p><p>H-infinity loop-shaping can be carried out using commercially available software.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>H-infinity loop-shaping has been successfully deployed in industry. In 1995, R. Hyde, K. Glover and G. T. Shanks published a paper<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> describing the successful application of the technique to a VTOL aircraft. In 2008, D. J. Auger, S. Crawshaw and S. L. Hall published another paper<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> describing a successful application to a steerable marine radar tracker, noting that the technique had the following benefits:

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<ul><li>Easy to apply – commercial software handles the hard math.</li>

<li>Easy to implement – standard transfer functions and state-space methods can be used.</li>

<li>Plug and play – no need for re-tuning on an installation-by-installation basis.</li></ul>

<p>A closely related design methodology, developed at about the same time, was based on the theory of the gap metric.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It was applied in 1993 for designing controllers to dampen vibrations in large flexible structures at <a href="Wright-Patterson_Air_Force_Base" title="Wright-Patterson Air Force Base">Wright-Patterson Air Force Base</a> and <a href="Jet_Propulsion_Laboratory" title="Jet Propulsion Laboratory">Jet Propulsion Laboratory</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<ul><li><a href="Control_theory" title="Control theory">Control theory</a></li>

<li><a href="H_infinity" class="mw-redirect" title="H infinity">H-infinity control</a></li>

<li><a href="Gap_metric" title="Gap metric">Gap metric</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The MathWorks, Inc. <i><a rel="nofollow" class="external text" href="http://www.mathworks.com/products/robust/description5.html">Synthesizing Robust Multivariable Controllers</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071012150750/http://www.mathworks.com/products/robust/description5.html">Archived</a> 2007-10-12 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>. Retrieved September 16, 2007.</span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Computing and Control Engineering Journal, 6(1):11–16</span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Proceedings of the UKACC International Conference on Control 2008</span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><i>Optimal robustness in the gap metric,</i> IEEE Transactions on Automatic Control, 35:673-686, June 1990.</span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Buddie S., Georgiou T.T., Ozguner U., and Smith M.C., <i>Flexible Structure Experiments at JPL and WPAFB,</i> International Journal of Control, 58(1): 1-19, 1993.</span>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li>Auger, D. J., Crawshaw, S., and Hall, S. L. (2008). Robust H-infinity Control of a Steerable Marine Radar Tracker. In <i>Proceedings of the UKACC International Conference on Control 2008</i>. Manchester: UKACC.</li>

<li>Chiang, R., Safonov, M. G., Balas, G., and Packard, A. (2007). <i>Robust Control Toolbox</i>, 3rd ed. Natick, MA: The Mathworks, Inc.</li>

<li>Glad, T. and Ljung, L. (2000). <i>Control Theory: Multivariable and Nonlinear Methods</i>. London: Taylor & Francis.</li>

<li>Georgiou T.T. and Smith M.C., <i>Linear systems and robustness: a graph point of view,</i> in Lecture Notes in Control and Information Sciences, Springer-Verlag, 1992, 183, pp. 114–121.</li>

<li>Georgiou T.T. and Smith M.C., <i>Topological Approaches to Robustness,</i> Lecture Notes in Control and Information Sciences, 185, pp. 222–241, Springer-Verlag, 1993.</li>

<li>Hyde, R.A., Glover, K. and Shanks, G. T. (1995). VSTOL first flight of an H-infinity control law. <i>Computing and Control Engineering Journal</i>, 6(1):11–16.</li>

<li>McFarlane, D. C. and Glover, K. (1989). <i>Robust Controller Design Using Normalized Coprime Factor Plant Descriptions (Lecture Notes in Control and Information Sciences)</i>, 1st ed. New York: Springer.</li>

<li>Vinnicombe, G. (2000). <i>Uncertainty and feedback: H-Infinity Loop-Shaping and the V-Gap Metric</i>, 1st ed. London: Imperial College Press.</li>

<li>Zhou, K., Doyle, J. C. and Glover, K. (1995). <i>Robust and Optimal Control</i>. New York: Prentice-Hall.</li>

<li>Zhou, K. and Doyle, J. C. (1998). <i>Essentials of Robust Control</i>. New York: Prentice-Hall.</li></ul>

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