Page - (000069) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

62 7 Stabilization For thefirst case, thecontroller C(s)= ( 1+√2 ) s+1 s+1+√2 has themaximumstabilitymargin b∗P,C = ( 4+2√2 )−1/2 =0.38. Thecostassociatedwith theH2 normfromEq.6.5 isJ =∞,because thesensitivity functionCShasnonzerogainat infinite frequency. For the secondcase, thecontroller C(s)= 2 √ 2s+1 s2+2√2s+4 hastheminimumH2cost,J∗ =6 √ 2=8.49,withassociatedstabilitymarginbP,C = 0.24. This controller and associated cost match the earlier example of H2 norm minimization inEq.6.7withμ=1. Forthethirdcase,weconstraintheminimumstabilitymargintobeatleastbP,C > 1/ √ 10=0.316,andthenfindthecontrollerthatminimizestheH2normcostsubject to theminimumstabilitymarginconstraint, yielding thecontroller C(s)= 2.5456s+1 0.28s2+1.5274s+2.88, whichhas thecost J =13.9andstabilitymarginbP,C =0.327. In theseexamples,a largerstabilitymargincorresponds toagreaterH2 cost.That relation illustrates the tradeoff between robust stability andperformancemeasured by the rejectionofdisturbanceandnoiseperturbations. 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.