Page - (000101) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

98 13 TimeDelays 0 1 2 3 4 5 6 -1 0 1 2 3 1 10 100 -20 -10 0 10 20 1 10 100 -100 0 100 200 (a) (b) (c) Fig. 13.2 Feedback delay destabilizes a simple integrator process. a Temporal dynamics from Eq.13.4,with gain k =5 andunit step input r(t)=1.The feedbackdelays are δ=0,0.25,0.33 shownin theblue,gold, andgreencurves, respectively.bBodegainplotof theassociated transfer functioninEq.13.3.Greaterfeedbacklagincreasestheresonantpeak.cBodephaseplot.Notehow thedestabilizing feedback lag (greencurve) createsa largephase lag in the frequency response G(s)= k ke−δs +s, (13.3) inwhich the term e−δs expresses the delay by δ. The differential equation for this systemis x˙(t)= k[r(t)−x(t−δ)], (13.4) which, for reference input rˆ =0, is x˙(t)=−kx(t−δ). This systemexpresses a delaydifferential process.Although this delaydifferential system is very simple in structure, there is nogeneral solution.A sufficiently large delay, δ, destabilizes the systembecause the rate of change toward the equilibrium setpoint remains toohighwhenthat ratedependsonapastvalueof thesystemstate. Inparticular, thedynamics inEq.13.4describeasimple laggedfeedbacksystem. At each time, t, the error between the target value and the systemstate from δ time units ago is rˆ−x(t−δ).That laggederror,multipliedby the feedbackgain,k, sets the rateatwhich the systemmoves toward the setpoint. Because the systemstateused for the feedbackcalculationcomes froma lagged timeperiod, the feedbackmaynot accurately reflect the true systemerror at time t. Thatmiscalculationcandestabilize the system. Figure13.2a shows how feedback lag can destabilize simple exponential decay towardanequilibriumsetpoint.Withnotimelag,thebluecurvemovessmoothlyand exponentially toward the setpoint.Thegold curve illustrates howa relatively small feedbacklagcausesthissystemtomovetowardthesetpointwithdampedoscillations. Thegreencurveshowshowalarger feedback lagdestabilizes thesystem.TheBode plots in Fig.13.2b, c illustrate how feedback delay alters the frequency and phase responseof the systemindestabilizingways.