Page - (000032) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

22 3 BasicControlArchitecture Theerror,E, inresponsetotheenvironmentalreferenceinput,R,canbeobtained byasimilar approach,yielding E(s)= 1 1+L(s)R(s). (3.5) If the open loop, L(s), has a large gain, that gainwill divide the error by a large number and cause the system to track closely to the reference signal.A large gain forL =CP canbeachievedbymultiplyingthecontroller,C,byalargeconstant,k. The largegaincauses thesystemtorespondrapidly todeviationsfromthereference signal. Feedback,withitspowerfulerrorcorrection,typicallyprovidesgoodperformance evenwhen the actual systemprocess,P, or controller,C, differs from theassumed dynamics.Feedbackalsotendstocorrectforvarioustypesofdisturbancesandnoise, andcanalso stabilizeanunstableopen-loopsystem. Feedbackhastwopotentialdrawbacks.First, implementingfeedbackmayrequire significantcosts for thesensors todetect theoutputandfor theprocesses that effec- tively subtract theoutput value from the reference signal. In electronics, the imple- mentationmaybe relatively simple. Inbiology, feedbackmayrequirevariousaddi- tional molecules and biochemical reactions to implement sensors and the flow of information through the system. Simple open-loop feedforward systems may be moreefficient for someproblems. Second, feedback can create instabilities. For example, when L(s)→−1, the denominator of the closed-loop system inEq.3.4 approaches zero, and the system blowsup.For a sinusoidal input, if there is a frequency,ω, atwhich themagnitude, |L(jω)|, is one and the phase is shifted byone-half of a cycle,φ=±π =±180◦, thenL(jω)=−1. Theproblemofphasearises fromthe time lag (or lead)between input and feed- back.When the sinusoidal input is at a peak value of one, the output is shifted to a sinusoidal trough value ofminus one. The difference between input and output combines inanadditive,expansionarywayrather thanprovidinganerrorsignal that can shrink towardanaccurate trackingprocess. Ingeneral, timedelays in feedback cancreate instabilities. Instabilitiesdonot requireanexacthalf cyclephaseshift.Suppose, forexample, that theopen loop is L(s)= k (s+1)3 . This system is stable, because its eigenvalues are the rootsof thepolynomial in the denominator, in this case s=−1, corresponding to a strongly stable system. The closed loophas the transfer function G(s)= L(s) 1+L(s) = k k+(s+1)3,