Page - (000059) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

6.4 FrequencyWeighting 51 Ifweweight this process transfer function byW(s)=1/(s+1), then the new system becomesWG=10/(s+10). Now, the gain declines with increasing fre- quency, fromamaximumofoneat zero frequency toaminimumofzeroat infinite frequency. Byweighting the original system,G, by theweighting function,W, we cause theH2 normof the combined system,WG, to be relativelymore sensitive to low- frequency disturbances.Whenwedesign a controller byminimizing theH2 norm associatedwithWG,wewill typicallyfinda system that is better at rejecting low- frequency loaddisturbances thanadesignminimizing theH2 normassociatedwith G. For the weighted system, optimization will avoid controllers that reject high- frequencyloaddisturbances,becausetheweightedsystemalreadyrejectsthosehigh- frequency inputs. Roughly speaking, a weighting function instructs the optimization method to reducethegainandsensitivityforcertainfrequenciesandtoignorethegainforother frequencies.Theweighting functionsdonot alter the actual system.Theweighting functionsareonlyusedtoalter thecost functionandoptimizationmethodthatdeter- mine theoptimal controller. Figure6.4 shows the regulation feedback system of Fig.6.1 with additional weightingsforthedisturbanceandnoiseinputs.Theweightingsmodifythefoursys- tem transfer functions andassociated sensitivities inEq. 6.2 tobeWdGud,WdGηd, WnGun, andWnGηn.Thecost function inEq.6.5becomes J =μ2||WdT||22+μ2ρ2||WdPS||22 +||WnCS||22+ρ2||WnT||22. (6.9) Consider an example in which we begin with the process, P, in Eq. 6.8. To emphasize low-frequency loaddisturbances, setWd =1/(s+0.1) tobea low-pass filter.Thatweightingfiltersoutdisturbances that are significantlygreater thanω= 0.1.Toemphasizehigh-frequencysensornoise,setWn = s/(s+10).Thatweighting filters out noise that is significantly less thanω=10. By using these two filters, the optimization method puts very low weight on any disturbances in midrange frequenciesofω= (0.1,10). n P d C u y n Wd d ~ Wn ~ Fig.6.4 Basic regulationfeedback loop inFig.6.1withadditionalweightings fordisturbanceand noise inputs. Theweightings alter the cost function to emphasize particular frequency bands for disturbanceandnoise,yieldingamodifiedoptimal controller