Page - (000071) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

64 8 Tracking Fig.8.1 Systemwithabasic feedback loop in response to a reference input,r C(s) y P(s) ur e Toanalyzeparticularproblems,webeginbyexpressing the transfer function for theerror fromEq.3.5as E(s)= R(s)−Y(s)= 1 1+C(s)P(s)R(s). Wemaywrite the transfer function for thecontrol signal as U(s)=C(s)E(s)= C(s) 1+C(s)P(s)R(s). These equations express the key tradeoff between the error signal and the control signal.Acontroller,C, that outputs a large control signal reduces the error,E, and increases the control signal,U. The following example illustrates this tradeoff and thepotential consequences for instability. 8.1 VaryingInputFrequencies Toanalyze thecostoveraparticular timeperiod, as inEq.8.1,wemust express the transfer functions as differential equations that describe change over time.We can usethebasicrelationbetweentransferfunctionsinEq.2.5anddifferentialequations inEq.2.6. In this example, Iuse theprocess inEq.4.1 that I analyzed inearlier chapters P(s)= 1 (s+0.1)(s+10). Iuse thecontroller C(s)= q0s 2+q1s+q2 p0s2+ p1s+ p2 . (8.2) OurgoalistofindacontrollerofthisformthatminimizesthecostfunctioninEq.8.1. I use a reference signal that is the sum of three sine waves with frequencies ωi = ( ψ−1,1,ψ ) . I weight each frequency by κi = (1,1,0.2), such that the high frequencymaybeconsidereda rapid, relatively low-amplitudedisturbance.Thus,