Page - (000054) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

46 6 Regulation P d C u y n Fig.6.1 Classic regulationproblemillustratedbyclosed-loop feedbackwithaconstant reference inputsignal,r =0.Thedisturbanceinput,d,perturbsthesystemprocess.Suchperturbationscanbe consideredasstochasticityintheprocess,orasuncertaintywithregardtothetrueprocessdynamics relative to the assumeddynamics.Thenoise input,n, perturbs the sensor that produces theoutput measurement, y, basedon theactualprocessoutput,η. SeeFig.3.2 for context whichweassumethat theother input,n, iszero.Usingourusual rulefor thetransfer functionsofaclosed loop, the four functionsare Gud = −PC 1+PC Gηd = P 1+PC Gun = −C 1+PC Gηn = −PC 1+PC . (6.1) We can express these transfer functions in terms of the sensitivities in Eq. 3.8 by defining theopen loopasL = PC, the sensitivity as S=1/(1+L), and the com- plementary sensitivityasT = L/(1+L), yielding Gud =−T Gηd = PS Gun =−CS Gηn =−T. (6.2) Because S+T =1 at any input, s, these transfer functions highlight the intrinsic design tradeoffs. Wecannowconsider the total cost as thesumof the responsewith respect to the inputd, holdingn atzero,plus the responsewith respect to the inputn, holdingd at zero J =||Gud(s)||22+ρ2||Gηd(s)||22 +||Gun(s)||22+ρ2||Gηn(s)||22. (6.3) For this example, we use impulse function inputs, δ(t), which provide a strong instantaneousshocktothesystem,asdefinedinthecaptionofFig.4.2.Wecandesign thesystemtoberelativelymoreorlesssensitivetodisturbanceinputsrelativetonoise inputsbyweightingthedisturbanceinputbyμ,sothatd(t)=μδ(t)andn(t)= δ(t). Largerμcausesdesignbyoptimization toyieldbetterdisturbance regulationat the expenseofworsenoise regulation.