Page - (000074) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

8.2 StabilityMargins 67 8.2 StabilityMargins Minimizing a quadratic cost function or anH2 normmay lead to a poor stability margin.Forexample,closetrackingofareferencesignalmayrequirealargecontrol signalfromthecontroller.Suchhighgainfeedbackcreatesrapidlyrespondingsystem dynamics,whichcanbesensitive touncertainties. InFig.8.3,thestabilitymarginsforthethreerowsassociatedwithρ = (1,10,100) arebP,C = (0.285,0.023,0.038).Arobuststabilitymargintypicallyrequiresavalue greater thanapproximately1/3orperhaps1/4. In this case, the systemassociatedwithρ=1has a reasonable stabilitymargin, whereas thesystemsassociatedwithhigherρ haveverypoor stabilitymargins.The poor stability margins suggest that those systems could easily be destabilized by perturbationsof theunderlyingprocessorcontrollerdynamics. Wecouldminimize the cost function subject to a constraint on the lower bound of the stabilitymargin.However, numericalminimization for that problem can be challenging.See the supplementalMathematicacode foranexample. 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.