Page - (000100) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

13.3 ProcessDelay 97 13.3 ProcessDelay Figure13.1b illustrates a feedback systemwith a process delay. The full process, Pe−δs, requires δ time units to transform its input to its output. Thus, the process output lagsbehind theassociatedcontrol input to theprocessbyδ timeunits. The open loop in Fig.13.1b isLe−δs =CPe−δs.We can derive the closed-loop systemresponseby themethodused toderiveEqs.3.4and13.1,yielding G(s)= L(s)e −δs 1+L(s)e−δs . (13.2) Thesimpletransferfunctiondescriptionforsignaldelaysallowsonetotracethecon- sequencesofdelays throughasystemwithmanycomponents that areeachapprox- imately linear. 13.4 DelaysDestabilizeSimpleExponentialDecay Thissectionillustrateshowdelayscandestabilizeasystem.Ianalyzeasimpleopen- loopintegrator,L(s)= k/s.Thattransferfunctioncorrespondstodynamicsgivenby x˙(t)= kr(t), for reference input r,whichhas solutionx(t)= k∫ t0 r(τ)dτ for initial condition x0 =0.Thus, theoutput ofL is the integral of its inputmultipliedby the gain, k. I assume throughout this section that the output equals the system state, y(t)= x(t). AstandardnegativefeedbacksystemhastransferfunctionG =L/(1+L),which forL= k/s is G(s)= k k +s, whichhasdynamics x˙(t)=−kx(t)+kr(t)= k[r(t)−x(t)]. Theerror signal is r(t)−x(t).Thesolution is the integralof theerror signal. For constant input, rˆ = r(t), the solution is a constant exponential decay toward theequilibriumsetpointat ratek.Without lossofgenerality,wecantakethesetpoint as rˆ =0andwrite the solutionas x(t)= x0e−kt. We can apply the same approach for the sensor delay system in Eq.13.1. For L= k/s, the systemtransfer function is