Page - (000062) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

Chapter7 Stabilization The previous chapter assumed that the intrinsic process, P, has a given unvarying form.Theactualprocessmaydiffer fromthegivenformormayfluctuateover time. If a system is designedwith respect to a particular formof P, then variation in P awayfromtheassumedformmaycause the systemtobecomeunstable. We can take into account the potential variation in P by altering the optimal design problem. The new design problem includes enhanced stability guarantees against certainkindsofvariation inP. Variation in an intrinsic process is an inevitable aspect of design problems. In engineering, theprocessmaydiffer fromtheassumedformbecauseof limitedinfor- mation,variability inmanufacturing,orfluctuatingaspectsof theenvironment. In biology, a particular set of chemical reactionswithin an individualmayvary stochasticallyovershort timeperiods.That reactionsetmayalsovarybetweenindi- viduals because of genetic and environmental fluctuations. In all cases, actual pro- cesses typically follownonlinear, time-varyingdynamics that often differ from the assumedform. Wemayalsohavevariationinthecontrollerorothersystemprocesses.Ingeneral, howmuchvariability canbe toleratedbefore a stable systembecomesunstable? In otherwords,howrobust is agivensystem’s stability toperturbations? Wecannotanswerthosequestionforalltypesofsystemsandalltypesofperturba- tions.However, theH∞normintroducedearlierprovidesinsightformanyproblems. Recall that theH∞normisthepeakgaininaBodeplot,whichisatransferfunction’s maximumgainoverallfrequenciesofsinusoidalinputs.Thesmallgaintheorempro- videsanexampleapplicationof theH∞norm. ©TheAuthor(s)2018 S.A.Frank,ControlTheoryTutorial, SpringerBriefs inAppliedSciences andTechnology,https://doi.org/10.1007/978-3-319-91707-8_7 55