Page - (000027) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

2.5 BodePlotsofGainandPhase 17 Low-passfiltersareveryimportantbecauselow-frequencyinputsareoftenexter- nal signals that the systembenefits by tracking,whereas high-frequency inputs are oftennoisydisturbances that the systembenefitsby ignoring. Inengineering, adesignercanattacha low-passfilterwithaparticular transition parameter a to obtain the benefits of filtering an input signal. In biology, natural selectionmust often favor appending biochemical processes or physical responses that act as low-pass filters. In this example, the low-pass filter is simply a basic exponential decayprocess. Figure2.2dshowsakeytradeoffbetweenthefastandslowresponses.Inthatpanel, thesysteminput is increased inastepfromzero tooneat timezero.Thefast system respondsquicklybyincreasingitsstate toamatchingvalueofone,whereas theslow system takesmuch longer to increase to amatching value. Thus, the fast system may benefit from its quick response to environmental changes, but itmay lose by itsgreatersensitivity tohigh-frequencynoise.That tradeoffbetweenresponsiveness andnoise rejection formsacommon theme in theoverall performanceof systems. TomaketheBodeplot,wemustcalculatethegainandphaseofatransferfunction’s response to a sinusoidal input of frequencyω.Most control theory textbooks show thedetails (e.g.,Ogata2009).Here, Ibrieflydescribe thecalculations,whichwillbe helpful later. Transfer functions express linear dynamical systems in terms of the complex Laplacevariables=σ + jω. Iuse j for the imaginarynumber tomatchthecontrol theory literature. The gain of a transfer function describes howmuch the functionmultiplies its input toproduceitsoutput.ThegainofatransferfunctionG(s)varieswiththeinput value,s.Forcomplex-valuednumbers,weusemagnitudes toanalyzegain, inwhich themagnitudeofacomplexvalue is |s|=√σ2+ω2. It turns out that the gain of a transfer function in response to a sinusoidal input at frequencyω is simply |G(jω)|, themagnitudeof the transfer functionat s= jω. The phase angle is the arctangent of the ratio of the imaginary to the real parts of G(jω). For theexponential decaydynamics that formthe low-passfilterofEq.2.11, the gainmagnitude,M, andphaseangle,φ, are M =|G(jω)|= a√ ω2+a2 φ=∠G(jω)=−tan−1 ω a . Anystable transfer function’s long-termsteady-state response to a sinewave input at frequencyω is a sinewaveoutput at the same frequency,multiplied by the gain magnitude,M, andshiftedby thephaseangle,φ, as