Page - (000048) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

5.2 PerformanceMetrics:EnergyandH2 39 analysis allow us to take advantage of transfer functions, Bode plots, and other powerful analytical tools that arisewhenweassume lineardynamics. The assumptionof linearity doesnotmean thatwe think the actual dynamicsof physical and biological processes are linear. Instead, startingwith the linear case providesapowerfulway inwhich togain insight aboutdynamics. In theprevious section,weconsideredhow tomeasure themagnitudeoffluctu- ating control anderror signals.Amagnitude that summarizes somekeymeasure is often called a norm. In the prior section, we chose the sumof squared deviations fromzero,which is related to the2–normofa signal ‖u(t)‖2 = (∫ ∞ 0 |u(t)|2dt )1/2 . (5.2) Theenergyof thesignal is thesquareof the2–norm,‖u(t)‖22.Whenthe timeperiod inthecostfunctionofEq.5.1goestoinfinity,T →∞,wecanwritethecostfunction as J =‖u(t)‖22+ρ2‖e(t)‖22. (5.3) Thesignalu(t) isafunctionoftime.TheassociatedtransferfunctionU(s)describes exactly the same signal, but as a function of the complexnumber, s, rather thanof time, t. It isoftenmucheasier toworkwith the transfer function foranalysis,noting that we can goback and forth between time and transfer function descriptions. For the analysisof squareddistancemetrics, the2–normof the transfer functionexpression is ‖U(s)‖2 = ( 1 2π ∫ ∞ −∞ |U(jω)|2dω )1/2 . (5.4) Thistransferfunction2–normisoftenreferredtoastheH2norm.Theterm |U(jω)|2 is the squareof theBodegainormagnitude, as inFig.2.2e.Thatgaindescribes the amplificationofasinusoidalinputatfrequencyω.TheH2normexpressestheaverage amplificationof input signalsover all input frequencies. If thegoal is tominimize the control input signal,u, or the error deviation from zero,e, then thegreater the amplificationof a signal, thegreater the cost.Thus,we canuse theH2 normtodefineanalternativecost functionas J =‖U(s)‖22+ρ2‖E(s)‖22, (5.5) which leads tomethods that are often calledH2 analysis. This cost describes the amplificationofinputsignalswithrespecttocontrolanderroroutputswhenaveraged overall input frequencies.Minimizing thiscost reduces theaverageamplificationof input signals.