Page - (000049) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

40 5 PerformanceandRobustnessMeasures If theenergy2–norminEq.5.2 isfinite, thentheenergy2–normandtheH2 norm areequivalent,‖u(t)‖2 =‖U(s)‖2,andwecanuseEqs.5.3and5.5interchangeably. Often, it ismoreconvenient toworkwith the transfer functionformof theH2 norm. We can use any combination of signals in the cost functions. Andwe can use different weightings for the relative importance of various signals. Thus, the cost functionsprovideamethod toanalyzeavarietyof tradeoffs. 5.3 TechnicalAspectsofEnergyandH2Norms I have given three different cost functions. The first in Eq 5.1 analyzes temporal changes in signals, suchasu(t), overafinite time interval.That cost function is the mostgeneral, in thesensethatwecanapplyit toanyfinitesignals.Wedonotrequire assumptionsabout linearityorotherspecialattributesof theprocessesthatcreate the signals. Thesecondfunction inEq.5.3measurescostoveran infinite time intervaland is otherwise identical to thefirstmeasure.Whyconsider theunrealisticcaseof infinite time? Often,analysisfocusesonaperturbationthatmovesastablesystemawayfromits equilibriumstate.As thesystemreturns toequilibrium, theerrorandcontrol signals go tozero.Thus, the signalshavepositivemagnitudeonlyoverafinite timeperiod, and the signal energy remainsfinite.Asnotedabove, if theenergy2–normisfinite, then theenergy2–normand theH2 normareequivalent, and the thirdcost function inEq.5.5 is equivalent to the secondcost function inEq.5.3. If thesignalenergyof thesecondcost functioninEq.5.3is infinite, thenthatcost functionisnotuseful. Inanunstablesystem,theerroroftengrowswithtime, leading to infinite energy of the error signal. For example, the transfer function 1/(s−1) has temporal dynamics given by y(t)= y(0)et, growing exponentiallywith time. Thesystemcontinuouslyamplifiesaninputsignal,creatinginstabilityandanoutput signalwith infinite energy. Whentheenergyis infinite, theH2 normmayremainfinite.For the transfer func- tion1/(s−1), theH2 norm is1/ √ 2.Theaverage amplificationof signals remains finite. In general, for a transfer function,G(s), theH2 norm remains finite as long asG(jω)doesnotgo to infinity for anyvalueofω, andG(jω)→0asω→±∞. Thus, theH2 normcost inEq.5.5canbeused inawider rangeofapplications. TheH2 norm is related tomany commonaspects of signal processing and time seriesanalysis, suchasFourier analysis, spectraldensity, andautocorrelation.