Page - (000041) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

32 4 PIDDesignExample 0 1 2 3 4 5 -0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020 0 1 2 3 4 5 -6 -4 -2 0 2 4 6 0.0 0.5 1.0 1.5 2.0 -0.04 -0.02 0.00 0.02 0.04 (a) (c) (b) (d) Sensornoise Processdisturbance Fig.4.2 Errorresponse,r−η,of thePIDfeedbacklooptosensornoise,n,orprocessdisturbance, d, fromEq.3.9.Bluecurve for theprocess,P, inEq.4.1andgoldcurve for thealteredprocess, P˜, inEq.4.2.aErrorresponse tosensornoise input,n, foraunit stepinputandb foranimpulse input. cError response to process disturbance input,d, for a unit step input andd for an impulse input. An impulse is u(t)dt =1 at t =0 and u(t)=0 at all other times. The system responses in gold curves reflect the slowerdynamicsof thealteredprocess. If thealteredprocesshad faster intrinsic dynamics, then thealteredprocesswould likelybemoresensitive tonoiseanddisturbance Fig. 4.3 System response output, η= y, to sine wave reference signal inputs, r. Each column shows a different frequency,ω. The rows are (Pr) for reference inputs into the original process, P or P˜, without amodifying controller or feedback loop, and (Rf) for reference inputs into the closed-loop feedback systemwith the PID controller in Eq.4.3. The green curve shows the sine wave input.Thebluecurve showssystemswith thebaseprocess,P, fromEq.4.1.Thegoldcurve showssystemswith thealteredprocess, P˜, fromEq.4.2. In the lower leftpanel, all curvesoverlap. In the lower panel atω=1, the green andblue curves overlap. In the twoupper right panels, the blueandgoldcurvesoverlapnearzero