Page - (000033) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

3.2 FeedbackControl 23 whichhasaneigenvaluewithrealpartgreaterthanzerofork>8,causingthesystem tobeunstable.Anunstablesystemtends toexplode inmagnitude, leadingtosystem failureordeath. 3.3 Proportional, Integral, andDerivativeControl Open loop systems cannot use information about the error difference between the target reference input and theactual output.Controllersmust bedesignedbasedon informationabout the intrinsicprocessand the likely inputs. By contrast, feedback provides information about errors, and controller design focuses primarily onusing the error input.Given the error, the controller outputs a new command reference input to the intrinsic system process. Precise knowledge about the intrinsic systemdynamics ismuch less importantwith feedbackbecause the feedback loopcanself-correct. This section discusses controller design for feedback systems. A controller is a process that modulates system dynamics. For the simplest feedback shown in Fig.2.1c,westartwithan intrinsicprocess,P(s), andendupwith feedbacksystem dynamics G(s)= C(s)P(s) 1+C(s)P(s) = L(s) 1+L(s), inwhichC(s) is the controller.Theproblem ishow tochooseaprocess,C(s), that balances the tradeoffs between variousmeasures of success, such as tracking the reference input and robustness toperturbationsanduncertainties. Figure3.2a includes two kinds of perturbations. The input d describes the load disturbance, representinguncertainties about the internal process,P(s), anddistur- bances to that internal process. Traditionally, one thinks of d as a relatively low- frequencyperturbation thatalters the intrinsicprocess.Theinputndescribespertur- bations that addnoise to the sensor thatmeasures theprocessoutput,η, toyield the finaloutput, y.Thatmeasuredoutput, y, is used for feedback into the system. To analyze alternative controller designs, it is useful to consider how different controllers alter theopen-loopdynamics,L(s)=C(s)P(s).Howdoes aparticular change in thecontroller,C(s),modulate the intrinsicdynamics,P(s)? First,wecansimply increase thegainby lettingC(s)= kp >1,amethodcalled proportional control. The systembecomesG= kpP/(1+kpP). For large kp and positiveP(s), the system transfer function isG(s)→1,whichmeans that the sys- temoutput tracksveryclosely to the system input. Proportional control cangreatly improvetrackingatall frequencies.However,bestperformanceoftenrequires track- ing low-frequency environmental inputs and ignoring noisy high-frequency inputs from the reference signal. In addition, largekp values cancause instabilities, and it maybe thatP(s)<0 for some inputs.