Page - (000070) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

Chapter8 Tracking Thepreviouschaptersonregulationandstabilization ignored the reference input,r. In thosecases,wefocusedonasystem’sability torejectperturbationsandtoremain stablewithrespect touncertainties.However,asystem’sperformanceoftendepends stronglyon its ability to trackexternal environmentalor reference signals. To study tracking of a reference input, let us return to the basic feedback loop structure in Fig. 2.1c, shown again in Fig.8.1. Good tracking performancemeans minimizing theerror,e= r− y, thedifferencebetween the reference input and the systemoutput. Typically,wecanreduce trackingerrorbyincreasing thecontrol signal,u,which increases the speedatwhich the systemchanges itsoutput tobecloser to the input. However, in a real system, a larger control signal requiresmore energy. Thus, we must consider the tradeoff betweenminimizing the error and reducing the cost of control. Ipreviouslyintroducedacostfunctionthatcombinesthecontrolanderrorsignals inEq.5.1as J = ∫ T 0 (u2+ρ2e2)dt, (8.1) inwhich u(t) and e(t) are functions of time, andρ is aweighting for the relative importanceof theerror signal relative to thecontrol signal. InotedinEq.5.2 that thesquareof theH2 normisequal to theenergyofasignal, for example, ‖e(t)‖22 = ∫ ∞ 0 |e(t)|2dt. In this chapter,wewill consider reference signals that changeover time.A system will typically not track a changing reference perfectly. Thus, the errorwill not go to zero over time, and the energywill be infinite. For infinite energy,we typically cannotuse theH2 norm. Instead,wemayconsider theaverageof thesquaredsignal per unit time,which is the power. Orwemay analyze the error over a finite time period, as inEq.8.1. ©TheAuthor(s)2018 S.A.Frank,ControlTheoryTutorial, SpringerBriefs inAppliedSciences andTechnology,https://doi.org/10.1007/978-3-319-91707-8_8 63