Page - (000047) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

38 5 PerformanceandRobustnessMeasures 5.1 PerformanceandCost: J Toanalyzeperformance,wemustmeasure the costs andbenefits associatedwith a particularsystem.Weoftenmeasurethosecostsandbenefitsbythedistancebetween a system’s trajectory and some idealized trajectorywith zero cost and perfect per- formance. Squareddeviationsprovideadistancemeasurebetween theactual trajectoryand the idealized trajectory. Consider, for example, the control signal, u(t), which the controllerproduces to feed into the systemprocess, as inFig.2.1c. The value of |u(t)|2 =u2 measures the magnitude of the signal as a squared distance from zero.We can think of u2 as the instantaneous power of the control signal.Typically, thepower requirements for control areacost tobeminimized. The square of the error output signal, |e(t)|2 = e2,measures the distance of the systemfromtheidealperformanceofe=0.Minimizingthesquarederrormaximizes performance.Thus,wemaythinkofperformanceatanyparticularinstant, t, interms of thecost function J(t)=u2+ρ2e2, forwhichminimumcostcorrespondstomaximumperformance.Here,ρ isaweight- ing factor thatdetermines the relativevalueofminimizing thecontrol signalpower, u2, versusminimizing the trackingerror,e2. Typically,wemeasure the cost function over a time interval. Summing up J(t) continuously from t =0 toT yields J = ∫ T 0 (u2+ρ2e2)dt. (5.1) Most squared distance or quadratic performance analyses arise fromextensions of this basic equation. Given thismeasure, optimal design trades offminimizing the energy cost to drive the systemversusmaximizing the benefit of tracking a target goal. 5.2 PerformanceMetrics:EnergyandH2 ThecostmeasureinEq.5.1analyzessignalswithrespecttotime.It isnaturaltothink of inputsandoutputsaschangingover time.With temporaldynamics,wecaneasily incorporatemultivariate signals andnonlinearities. In spiteof thoseadvantages,we oftenobtaingreater insightbyswitching toafrequencyanalysisofsignals, as in the previouschapters. In this section, I present alternativemeasures of cost and performance in terms of transfer functions and complex signals. Those alternativemeasures emphasize frequenciesoffluctuationsratherthanchangesthroughtime.Frequencyandcomplex