Page - (000019) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

Chapter2 ControlTheoryDynamics Themathematics of classical control theorydependson linear ordinarydifferential equations,whichcommonlyarise inallscientificdisciplines.Control theoryempha- sizes a powerfulLaplace transformexpressionof linear differential equations. The Laplaceexpressionmaybe less familiar inparticulardisciplines, suchas theoretical biology. 2.1 TransferFunctionsandStateSpace Here, I showhowandwhycontrol applicationsuse theLaplace form. I recommend an introductory textoncontrol theory foradditionalbackgroundandmanyexample applications (e.g.,ÅströmandMurray2008;Ogata2009;Dorf andBishop2016). Suppose we have a process, P, that transforms a command input, u, into an output, y. Figure2.1a shows the input–output flow.Typically,wewrite the process asadifferential equation, for example x¨+a1x˙+a2x = u˙+bu, (2.1) inwhich x(t) is an internal state variable of theprocess that dependson time,u(t) is theforcingcommandinputsignal,andoverdotsdenotederivativeswithrespect to time.Here, forsimplicity,welet theoutputbeequivalent totheinternalstate, y≡ x. The dynamics of the input signal, u, may be described by another differential equation, drivenby reference input, r (Fig.2.1b).Mathematically, there is noprob- lemcascadingsequencesofdifferentialequationsinthismanner.However, therapid growthofvarioussymbolsandinteractionsmakesuchcascadesofdifferentialequa- tionsdifficult toanalyzeand impossible tounderstand intuitively. ©TheAuthor(s)2018 S.A.Frank,ControlTheoryTutorial, SpringerBriefs inAppliedSciences andTechnology,https://doi.org/10.1007/978-3-319-91707-8_2 9