Page - (000068) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

7.4 ExamplesofDistanceandStability 61 P1 = k s+1 P2 = k (s+1)(Ts+1)2, (7.7) when evaluated at k=100 and T =0.025, as shown in Fig.7.2a, b. The distance between these systems is δν(P1,P2)=0.89.That largedistancecorresponds to the verydifferent responsecharacteristicsof the twosystemswheninaclosedfeedback loop. (Åström andMurray (2008) report a value of δν =0.98. The reason for the discrepancyisnotclear.SeethesupplementalMathematicacodeformycalculations, derivations, andgraphicshereand throughout thebook.) In the secondcase, the following twosystemshaveverydifferent responsechar- acteristicsbythemselves inanopenloop,yethaveverysimilarresponses inaclosed feedback loop P1 = k s+1 P2 = k s−1, (7.8) when evaluated at k=100, as shown in Fig.7.2c, d. The distance between these systems is δν(P1,P2)=0.02. That small distance corresponds to the very similar responsecharacteristicsof the twosystemswhen inaclosed feedback loop. 7.5 ControllerDesign forRobustStabilization ThemeasurebP,C describes thestabilitymargin fora feedback loopwithprocessP andcontrollerC.A largermarginmeans that the systemremains robustly stable to variant processes, P′, with greater distance from the nominal process, P. In other words, a larger margin corresponds to robust stability against a broader range of uncertainty. Foragivenprocess,wecanoftencalculatethecontrollerthatprovidesthegreatest stabilitymargin.Thatoptimal controllerminimizesanH∞ norm, so in this casewe mayconsider controllerdesign tobeanH∞optimizationmethod. Often,wealsowish tokeep theH2 normsmall.Minimizing that norm improves a system’s regulationby reducing response toperturbations. Jointly optimizing the stabilitymarginand rejectionofdisturbances leads tomixedH∞ andH2 design. MixedH∞ andH2 optimization is an active area of research (Chen and Zhou 2001; Chang 2017). Here, I briefly summarize an example presented in Qiu and Zhou (2013).That articleprovides analgorithmformixedoptimization that canbe applied toother systems. QiuandZhou(2013)startwiththeprocess,P =1/s2.Theyconsider threecases. First, what controller provides the minimum H∞ norm and associatedmaximum stabilitymargin, b, while ignoring theH2 norm?Second,what controller provides theminimumH2 norm,while ignoring the stabilitymargin andH∞ norm?Third, what controlleroptimizesacombinationof theH∞ andH2 norms?