Page - (000029) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

Chapter3 BasicControlArchitecture 3.1 Open-LoopControl Supposeasystembenefitsbytrackingrelativelyslowoscillatoryenvironmentalfluc- tuationsat frequencyωe and ignoringmuch fasternoisyenvironmentalfluctuations atfrequencyωn.Assumethat thesystemhasanintrinsicdailyoscillatoratfrequency ω0 =1,with timemeasured in days.Howcan a systembuild a control circuit that uses its intrinsic daily oscillator to track slower environmental signals and ignore fasternoisy signals? Wecanbeginbyconsideringcircuit designs that follow thecascade inFig.2.1b. That cascade is a single direct path from input to output,matching the cascade in Eq.2.3.Thatpath is anopen loopbecause there isnoclosed-loop feedback. Using thecomponents inFig.2.1b, the internaloscillator isgivenby P(s)= ω0 s2+ω20 , and theexternal reference signal isgivenby R(s)= ωe s2+ω2e + ωn s2+ω2n , the sumofone low- andonehigh-frequency sinewave.FromFig. 2.1b, thedesign goal seeks to create a preprocess controlling filter,C(s), that combines with the intrinsic internal oscillator, P(s), to transform the reference input, R(s), into an output,Y(s)≈ωe/(s2+ω2e), thatfluctuates atωe and ignoresωn. In this case,weknowexactly the intrinsicdynamics,P(s).Thus,wecanuse the open-looppath inFig.2.1b tofindacontroller,C(s), such that the transfer function C(s)P(s)givesapproximately the input–output relation thatweseekbetweenR(s) andY(s). Forexample, byusing thecontroller ©TheAuthor(s)2018 S.A.Frank,ControlTheoryTutorial, SpringerBriefs inAppliedSciences andTechnology,https://doi.org/10.1007/978-3-319-91707-8_3 19