Page - (000102) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

13.4 DelaysDestabilizeSimpleExponentialDecay 99 Fig. 13.3 Greater process gain, k, can be destabilized by smaller feedback lag, δ. Combinations ofgainand lagbelowthecurveare stable.Combinationsabove thecurveareunstable.Stability is determinedby themaximumrealpartof theeigenvalues forEq.13.4withconstant reference input Inearlierchapters,Ishowedthathighgainfeedbacksystemsmoverapidlytoward theirsetpointbutmaysuffersensitivitytodestabilizingperturbationsoruncertainties. Feedback lagmaybe thoughtof asakindofperturbationoruncertainty. Figure13.3 shows how the system gain, k, enhances the destabilizing effect of feedback lag,δ.Combinationsofgainand lagbelowthecurvearestable.Combina- tions above the line are unstable. Systemswith greater gain canbedestabilized by smaller feedback lag. Process delays differ fromfeedbackdelays only in the extra lag associatedwith the reference input. For the process delay systemgiven by the transfer function in Eq.13.2, thedynamicsare x˙(t)= k[r(t−δ)−x(t−δ)], which describe an error integrator lagged by t−δ. For constant reference input, r(t)= rˆ, theprocessdelaydynamicsarethesameasforthefeedbackdelaydynamics inEq.13.4. 13.5 SmithPredictor Compensatingfora timedelayrequiresprediction.Suppose, forexample, that there is a process delay between input and output, as in Fig.13.1b. The Smith predictor provides oneway to compensate for the delay. To understand the Smith predictor, wefirst reviewtheprocessdelayproblemandhowwemight solve it. InFig.13.1b, the time-delay transfer function in theprocess,e−δs,mapsan input signalattimet toanoutputthatistheinputsignalatt−δ.Thus,theopenloopCPe−δs transformsthecurrentinput,r(t), totheoutput,y(t−δ).Themeasurederrorbetween inputandoutput,r(t)−y(t−δ), givesan incorrect signal for thefeedbackrequired topush the trackingerror, r(t)−y(t), towardzero. One way to obtain an accurate measure of the tracking error is to predict the output, y(t), caused by the current input, r(t). The true systemprocess,Pe−δs, has