Page - (000060) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

52 6 Regulation 0.1 1 10 -20 -10 0 10 0.1 1 10 -20 -10 0 10 (a) (b) Fig. 6.5 Role of frequencyweighted inputs in the design of optimalH2 controllers for system regulation, illustratedbyBodemagnitudeplots.aPlot for theunweightedcase,matching theplot inFig.6.3c.bPlotfor thefrequencyweightedexampleinthetext,whichemphasizes theregulation of low-frequency loaddisturbances,d, andhigh-frequencysensornoise,n Byminimizing theweightedH2 cost inEq.6.9,weobtain theoptimalcontroller C(s)= 2.02(s+1.52) s2+1.17s+6.3. I calculated thevalues for thiscontrollerbyusing thenumericalminimizationfunc- tioninMathematicatominimizetheH2cost,subjecttotheconstraintthatall transfer functions inEq.6.1are stable.See the supplementalMathematicacode. Figure6.5 compares the optimized system response for the unweighted and weighted cases. Panela shows theBodemagnitude responseof theoptimized sys- tem for the unweighted case, equivalent to the plot in Fig.6.3c. Panelb shows the responseof theoptimizedsystemfor theweightedcase in this section. The weighted case emphasizes low-frequency load disturbances and high- frequency sensor noise,with lowweight onmidrange frequencies. Comparing the unweightedcase in (a)with theweightedcase in (b),wesee twokeydifferences. First,theweightedcaseallowsalargeriseinmagnitudesandassociatedsensitivity to perturbations formidrange frequencies. That rise occurs because the particular weighting functions in this examplediscountmidrangeperturbations. Second, the gold curve shows that the weighted case significantly reduces the low-frequency sensitivity of system outputs, η, to load disturbances, d. The gold curvedescribes the responseof the transfer function,Gηd.Note that, becauseof the log scaling formagnitude, almost all of the costs arise in theupperpart of theplot. The lowrelativemagnitude for the lowerpart contributes little to theoverall cost.