Page - (000066) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

7.2 Uncertainty:DistanceBetweenSystems 59 Read this as theguaranteed stabilitymargin for the alternativeprocess is at least as goodasthestabilitymarginfornominalprocessminusthedistancebetweenthenom- inal and alternative processes.A small distance between processes, δν, guarantees that thealternativeprocess isnearlyas robustly stableas theoriginalprocess. Thedefinitions in this sectiondependon thedistancemeasure, expressedas δ(c1,c2)= |c1−c2|√ 1+|c1|2 √ 1+|c2|2 . Here, c1 and c2 are complexnumbers.Transfer functions return complexnumbers. Thus,wecanuse this function toevaluateδ[P1(jω),P2(jω)]. 7.3 RobustStabilityandRobustPerformance ThestabilitymarginbP,C measures theamountbywhichPmaybealteredandstill allow the system to remain stable.Note thatbP,C inEq. 7.6 expresses aminimum value of δ over all frequencies. Thus,wemay also think of bP,C as themaximum valueof1/δoverall frequencies. Themaximum value of magnitude over all frequencies matches the definition of theH∞ norm, suggesting that maximizing the stability margin corresponds to minimizing some expression for anH∞ norm. Indeed, there is such anH∞ norm expression for bP,C. However, the particular form is beyond our scope. The point hereisthatrobuststabilityviamaximizationofbP,C fallswithintheH∞normtheory, as in the smallgain theorem. Stability is just one aspect of design. Typically, a stable systemmust alsomeet otherobjectives,suchasrejectionofdisturbanceandnoiseperturbations.Thissection shows that increasing the stabilitymarginhas theassociatedbenefitof improvinga system’s rejectionofdisturbanceandnoise.Often,adesign that targets reductionof theH∞normgains thebenefitsofanincreasedstabilitymarginandbetter regulation through rejectionofdisturbanceandnoise. The previous section on regulation showed that a feedback loop reduces its response to perturbations by lowering its various sensitivities, as in Eqs. 6.2 and 6.5.Afeedback loop’s sensitivity isS=1/(1+PC)and itscomplementarysensi- tivity isT = PC/(1+PC). Increasing the stabilitymargin, bP,C, reduces a system’s overall sensitivity.We cansee the relationbetweenstabilityandsensitivityby rewriting theexpression for bP,C as bP,C = [ max ω √ |S|2+|CS|2+|PS|2+|T|2 ]−1 This expression shows that increasingbP,C reduces the totalmagnitudeof the four keysensitivitymeasures fornegative feedback loops.