Page - (000020) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

10 2 ControlTheoryDynamics P(s) u yU(s) Y(s) C(s) y P(s) ur C(s) y P(s) ur e (a) (b) (c) Fig. 2.1 Basic process and control flow. aThe input–output flow inEq.2.2. The input,U(s), is itselfa transfer function.However, forconvenience indiagramming, lowercase lettersare typically usedalongpathwaystodenoteinputsandoutputs.Forexample, ina,ucanbeusedinplaceofU(s). Inb,onlylowercaselettersareusedfor inputsandoutputs.Panelb illustrates theinput–outputflow ofEq.2.3.Thesediagramsrepresentopen-looppathwaysbecausenoclosed-loopfeedbackpathway sends a downstreamoutput back as an input to an earlier step. cAbasic closed-loopprocess and control flowwith negative feedback.The circle between r and e denotes additionof the inputs to produce theoutput. In thisfigure,e= r− y Wecanuseamuchsimplerwaytotraceinput–outputpathwaysthroughasystem. If the dynamics of P follow Eq.2.1, we can transform P from an expression of temporaldynamicsinthevariable t toanexpressioninthecomplexLaplacevariable s as P(s)= Y(s) U(s) = s+b s2+a1s+a2 . (2.2) Thenumeratorsimplyuses thecoefficientsof thedifferentialequation inu fromthe right sideofEq.2.1 tomakeapolynomial in s. Similarly, thedenominator uses the coefficients of the differential equation in x from the left side of Eq.2.1 tomake a polynomial in s. The eigenvalues for the process, P, are the roots of s for the polynomial in thedenominator.Control theoryrefers to theeigenvaluesas thepoles of the system. From this equation and thematching picture in Fig.2.1, wemaywriteY(s)= U(s)P(s). Inwords, theoutput signal,Y(s), is the input signal,U(s),multipliedby the transformation of the signal by the process, P(s). Because P(s)multiplies the signal,wemaythinkofP(s)asthesignalgain, theratioofoutput toinput,Y/U.The signal gain is zero at the roots of the numerator’s polynomial in s. Control theory refers to thosenumerator roots as the zerosof the system.