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CALVIN ENGR 315 - PID Tuning Methods-Automation Study with MathCad-a

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PID Tuning MethodsAn Automatic PID Tuning Study with MathCadNeil KuyvenhovenCalvin CollegeENGR. 315December 19, 2002AbstractThere are several methods for tuning a PIDcontroller. This paper takes a qualitative lookat three common methods, with comparisonsof accuracy and effectiveness. These threemethods include a guided Trial and Errormethod, the Ziegler-Nichols method, and theCohen-Coon method. For an exceptionallyresponsive system the Trial and Error methodis often used after the Ziegler-Nichols orCohen-Coon so as to enhance the roughresults of these two methods. Using thesemethods in cooperation will result in a finelytuned control system. A study is completedusing MathCad to implement automating PIDtuning. Due to the nature of MathCad, theprocess is not fully automated due to somelimitations of MathCad, rather the processsolves the solution for manually inputtingPID coefficients, much like a MatLab process.IntroductionModels will never emulate their actualphysical counterpart perfectly because themathematical formulae applied is completelypredictable; whereas, the physical systembeing modeled will change over time and dueto unaccounted for disturbances. Controlsystems, specifically PID control systemsattempt to reduce the error due to unknowndisturbances by designing for typicaldisturbances that ideally will include anyunsuspected disturbances in the physicalsystem being modeled.PID systems are very unique to eachapplication. As one set of settings may beideal for one system these same settings canthrow another system horribly off. For thisreason, multiple methods for tuning the PIDcoefficients have been made. As somemethods are better than others for givenapplications, each method has its advantagesand disadvantages. This paper will outline andcompare the three methods known as Trialand Error, Ziegler-Nichols, and Cohen-Coon.PID BasicsBefore explaining the methods for tuning PIDcontrol systems, the effects of changing thedifferent components must be examined. Forthe analysis, the system in figure one will beused, here the plant is shown to be G(S). PIDcontrollers consist of three components; theproportional, integral, and derivative controls.Figure 1 – System with a PID controllerEach of these components has very distincteffects on the system. Table one outlines theeffects of the PID components on a particularsystem. Though there is no set standard forthe way a PID controller is set up, there ishowever three main types. These include theideal, parallel and series controllers. Thesetypes can be seen in figure 1.5 along withthere corresponding frequency domainequations as equations one, two and three. The difference between these types of PIDcontrollers is only seen when attempting totune them. The outcome can be made similarwith different values, but as is realized fromthe equations, one set of parameters for thecontroller will result in drastically differenteffects on the the different types ofcontrollers. This difference is most significanton the derivative control in the series andparallel controllers. With the seriescontroller, the derivative control is operatingon the partially fixed error as it has alreadygone through the proportional and integralcontrol. The is juxtaposed by the parallelcontroller where the parameters are orderindependent. The effect of the derivativecontrol is not amplified by proportionalcontrol which results in a less fine tunedsystem. The series system begins byamplifying the error which promotes a fasterresponse from the integral and derivativecontrol. The ideal system creates a sensitiveresponse in the proportional control as smallchanges have big effects on the integral andderivative portions separately. This isdifferent from the series model due to the lackof direct interaction between the integral andseries control.This paper uses the parallel PID control modelfor analysis.To increase the rise time of the system, theproportional, integral, and derivativecomponents are tweaked so as to improve therise time, steady state error, and overshootrespectively. Despite the fact that changingone component effects all the characteristicsas seen in table one, the system can still bebrought to stability with the threecomponents together; although sometimesfewer than all three are needed.Eq. 1Table 1 – Effects of PIDEq. 2Figure 2 – Ideal, series, and parallel PID configurationsEq. 3Rise Time Steady State OvershootProportional Decrease Decrease IncreaseIntegral Decrease Eliminate IncreaseDerivative None None DecreaseMethod 1: Trial and ErrorThe Trail and Error method requires a closedloop system, it steps through the system fromproportional to integral to derivative. Thismethod is a divide and conquer approach, firstit puts the system into a rough solution fromwhich small tweaks are performed to perfectthe response. To begin, each coefficient of thePID controller is set to zero. The proportionalcomponent is now considered by increasingits value until a steady oscillation is obtainedas in figure two. Scaling the currentproportional value down by a factor of twowill give the resulting proportional value.Applying this proportional value will dismissthe steady oscillations. Next the integralcoefficient is increased until steadyoscillations are again obtained. The presentvalue of the integral coefficient is scaled up bya factor of three and applied to the integral asthe final value. This once again setsoscillations off, which brings up thederivative control, this value is increased untilfor a final time the oscillations are at aconstant period and amplitude. Thecoefficient of the derivative is then scaleddown by a factor of three and applied as thefinal value for the derivative control. Theresulting output may still have some noiseassociated with it, this must now be tuned byhand with small educated tweak of thedifferent coefficients.Method 2: Ziegler-NicholsThe Ziegler Nichols method takes twoapproaches depending on the system at hand.First, with the closed method or begins inmuch the same way as the Trial and Errormethod as a steady oscillation is desired withonly a proportional influence present. Theproportional value at which the oscillationsbecome constant is coined the term 'ultimategain'. The period of oscillations at theultimate gain is termed 'ultimate period'. Theultimate gain can be found in an simpler waywith the root locus of the open loop transferfunction. The ultimate gain and ultimateperiod as noted in figure 2, are applied to


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CALVIN ENGR 315 - PID Tuning Methods-Automation Study with MathCad-a

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