Page - (000040) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

4.1 OutputResponse toStep Input 31 The gold curve, based onEq.4.2, rises evenmore slowly, because that alternative process, P˜, hasaneven longer timehorizon foraveraging inputsof1/a=100. Panel (b) shows the responseof the full feedback loopofFig.3.2awith thePID controller inEq.4.3andnofeedforwardfilter,F =1.Note that thesystemresponds muchmore rapidly,with amuch shorter time spanover the x-axis than in (a). The rapidresponsefollowsfromtheveryhighgainof thePIDcontroller,whichstrongly amplifies low-frequency inputs. The PID controller was designed tomatch the base process P in Eq.4.1, with response in blue. When the actual base process deviates as in P˜ of Eq.4.2, the response is still reasonablygood, although the systemhasagreaterovershootupon first response and takes longer to settle down andmatch the reference input. The reasonablygoodresponseinthegoldcurveshowstherobustnessofthePIDfeedback loop tovariations in theunderlyingprocess. Panel (c) shows the response of the systemwith a feedforward filter, F, from Eq.4.4.Notethat thesysteminbluewiththebaseprocess,P, improvessignificantly, with lowerovershootandlessoscillationwhensettling tomatch thereference input. By contrast, the system in gold with the alternative base process, P˜, changes its responseverylittlewith theadditional feedforwardfilter.Thisdifferencereflects the fact that feedforwardworkswell onlywhen one has very good knowledge of the underlying process, whereas feedbackworks broadly and robustlywith respect to manykindsofperturbations. 4.2 ErrorResponse toNoiseandDisturbance Figure4.2 illustrates the system error in response to sensor noise, n, and process disturbance,d. Panel (a) shows the error in response to aunit step change inn, the inputnoise to thesensor.Thatstepinput to thesensorcreatesabiasedmeasurement, y, of the system output, η. The biased measured value of y is fed back into the control loop. A biased sensor produces an error response that is equivalent to the output response for a reference signal.Thus,Fig.4.2amatchesFig.4.1b. Panel (b) shows theerror response toan impulse input at the sensor.An impulse causes a brief jolt to the system. The systembriefly responds by a large deviation from its setpoint, but then returns quickly to stable zero error, atwhich the output matches the reference input.An impulse to the reference signal produces anequiv- alentdeviation in the systemoutputbutwithopposite sign. The error response to process disturbance in panels (c) and (d) demonstrates that the systemstrongly rejects disturbancesoruncertainties to the intrinsic system process.