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5.4 RobustnessandStability:H∞ 41 5.4 RobustnessandStability:H∞ A transfer function for a system,G(s), defines the system’s amplification of input signals. For a sinusoidal input at frequency ω, the amplification, or gain, is the absolutevalueof the transfer functionat that frequency, |G(jω)|. Often, the smaller a system’samplificationof inputs, themore robust the system isagainstperturbations.Thus,onecommonoptimizationmethodfordesigningcon- trollers seeks tominimize a system’s greatest amplification of inputs.Minimizing the greatest amplification guarantees a certain level of protection against theworst caseperturbation. In somesituations, onecanalsoguarantee that a systemis stable if itsmaximumsignal amplification isheldbelowakey threshold. Asystem’smaximumamplificationofsinusoidalinputsoverallinputfrequencies, ω, is called itsH∞ norm.For a systemG(s), theH∞ norm iswritten as‖G(s)‖∞. Thenormdescribes themaximumof |G(jω)|over allω. Themaximumis also the peakgainonaBodemagnitudeplot,which is equivalent to the resonancepeak. Systemstabilityandprotectionagainstperturbationsset twofundamentalcriteria for systemdesign.Thus,H∞methods arewidelyused in the engineeringdesignof controllers andsystemarchitectures (ZhouandDoyle1998). OpenAccess This chapter is licensed under the terms of theCreativeCommonsAttribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation,distributionandreproduction inanymediumorformat,as longasyougiveappropriate credit to theoriginal author(s) and thesource,providea link to theCreativeCommons licenseand indicate if changesweremade. The images or other third partymaterial in this chapter are included in the chapter’sCreative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons license and your intended use is not permitted by statutory regulationorexceeds thepermitteduse,youwillneed toobtainpermissiondirectly from thecopyrightholder.