Page - (000030) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

20 3 BasicControlArchitecture Fig.3.1 Bodeplotof an intrinsicoscillator,P(s), modulatedbyacontroller, C(s), in anopen loop L(s)=C(s)P(s).Thegold curves followEq.3.3, in which theactual frequency of the internaloscillator is ω˜0 =1.2 rather than the valueω0 =1 that set the designof thecontroller.The underlyingbluecurves show theoutcomewhen the internaloscillator frequency matches thedesign frequency, ω˜0 =ω0 =1 0.1 1 10 -75 -50 -25 0 25 50 0.1 1 10 -200 -100 0 Frequency C(s)= ( ω0 s+ω0 )3(s2+ω20 ω0 ) , (3.1) theopen-loopsystembecomes L(s)=C(s)P(s)= ( ω0 s+ω0 )3 , (3.2) becausethesecondterminC(s)cancelsP(s).ThesystemL(s) is thelow-passfilter inEq.2.11 raised the thirdpower.Withω0 =1, this systemhasaBodeplot similar to the blue curve inFig.2.2e, f, but because of the exponent in L(s), the gain falls morequicklyathigh frequenciesand thephase lag isgreater. As with the low-pass filter illustrated in Fig.2.2, this open-loop system, L(s), tracks environmental signals at frequencyωe ω0 and suppressesnoisy signals at frequencyωn ω0.However,evenifwecouldcreatethiscontrollerovertherequired rangeof frequencies, itmight turnout that this system is fragile tovariations in the parameters. We could study robustness by using the differential equations to calculate the dynamics for many combinations of parameters. However, such calculations are tedious,andtheanalysiscanbedifficult toevaluateformorethanacoupleofparam- eters.UsingBodeplotsprovidesamucheasierwaytoanalyzesystemresponseunder variousconditions. Suppose, for example, that in theabsenceof inputs, the internaloscillator,P(s), actuallyfluctuates at the frequency ω˜0 =ω0.Then, theopen-loopsystembecomes