Page - (000067) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

60 7 Stabilization 0 1 2 3 4 5 20 40 60 80 100 0.0 0.5 1.0 1.5 2.0 100 200 300 400 0.0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 0.00 0.02 0.04 0.06 0.08 0.10 0.2 0.4 0.6 0.8 1.0 (a) (b) (c) (d) Fig. 7.2 Comparisonbetween the responses of two systems to a unit step input, r =1.Theblue curvesshowP1 andthegoldcurvesshowP2.a,bSystemsinEq.7.7,withk=100andT =0.025. Thetopplotshowstheopen-loopresponseforeachsystem.Thebottomplotshowstheclosed-loop feedback responsewith unit feedback, P/(1+P), inwhich the error signal into the system, P, is 1− y for systemoutput, y. c,dOpen (top) andclosed (bottom) loop responses for the systems in Eq. 7.8, with k=100. Redrawn from Fig. 12.3 of Åström andMurray (2008), ©Princeton UniversityPress 7.4 ExamplesofDistanceandStability Themeasure, δν(P1,P2), describes the distance between processeswith respect to their response characteristics in a negative feedback loop. The idea is that P1 and P2mayhavedifferent responsecharacteristicswhenbythemselves inanopen loop, yet havevery similar responses in a feedback loop.OrP1 andP2mayhave similar responsecharacteristicswhenby themselves, yet haveverydifferent responses ina feedback loop. Thus,wecannotsimplyusetheresponsecharacteristicsamongasetofalternative systemstounderstandhowvariationsinaprocessinfluencestabilityorperformance. Instead,wemustuseameasure,suchasδν, thatexpresseshowvariationsinaprocess affect feedback loopcharacteristics. ThissectionpresentstwoexamplesfromSect.12.1ofÅströmandMurray(2008). Inthefirstcase, thefollowingtwosystemshaveverysimilarresponsecharacteristics bythemselvesinanopenloop,yethaveverydifferentresponsesinaclosedfeedback loop