Page - (000057) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

6.3 ResonancePeakExample 49 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 ρ Fig. 6.2 RelativeH2 values for the transfer functions inEq. 6.1,withGud =Gηn in red,Gηd in gold, andGun in green.TheH2 value for each transfer function is dividedby the totalH2 values over all four functions. The transfer functionswere derived from the process in Eq. 6.8 and the associatedoptimal controller.Theweightingparameters in the cost functionofEq. 6.4 areμ=1 and ρ varying along the x-axis of the plot. Swapping values ofμ and ρ gives identical results, becauseof the symmetries inEqs.6.1and6.4 The general solution in terms ofμ and ρ provides a simple way in which to obtain the optimal controller for particular values ofμ andρ. For example,when μ=ρ=1, theoptimal controller is C(s)≈ 0.609(s−0.81) s2+1.73s+2.49. Similar controller expressions arise for other values ofμ andρ. Those controllers maybeused in theclosed loopofFig.6.1 to formacomplete system. Figure6.2showstherelativeH2valuesofthefourinput–outputtransferfunctions inEq.6.1.TheH2 valuesexpress sensitivityoverall frequencies. To interpret this figure, look at Eq. 6.4. As the product of theweightings,μρ, increases, theoutputofGηd (goldcurve)playsan increasingly important role in the total cost relative to theoutputofGun (greencurve). As the relativecostweightingofGηd increases, itsH2 valuedeclines.Similarly, as the relativecostweightingofGun decreases, itsH2 value increases.Onceagain, we see the sensitivity tradeoffs in response to the relative importance of different perturbations. ThetoprowofFig.6.3comparestheBodeplotsfortheprocess,P,andtheinput– output transfer functions inEq.6.1.Asρ increases in thecolumnsfromleft to right, the rise in the green curve forGun is the strongest change.Wecanunderstand that changeby examining the cost function inEq. 6.4.BecauseGud =Gηn, a rise inρ reduces theweightingofGun relative toall other terms. The strongest increase in relativeweighting asρ rises occurs forGηd, shown in gold. Themild decline in the gold curvewith increasing ρ is consistent with the increased relativecostweightingof that signal.