zn closed loop method - TechTeach

1 Article:Ziegler-Nichols closed -Loop MethodFinn HaugenTechTeach17. July 20101 IntroductionZiegler and Nichols published in 1942 a paper [1] where they described twomethods for tuning the parameters of P-, PI- and PID controllers. Thesetwo methods are theZiegler-Nichols closed loop method1, and theZiegler-Nichols open loop method2. The present article describes theclosed-loop method , while the open-loop method is described in anotherarticle (available at ).Ziegler and Nichols [1] used the following definition of acceptable stabilityas a basis for their contoller tuning rules: The ratio of the amplitudes ofsubsequent peaks in the same direction (due to a step change of thedisturbance or a step change of the setpoint in the control loop) isapproximately 1/4, see Figure 1:A2A1=14(1)However, there is no guaranty that the actual amplitude ratio of a givencontrol system becomes 1/4 after tuning with one of the Ziegler andNichols methods , but it should not be very different from 1 that the Ziegler-Nichols closed loop method can be applied only toprocesses having a time delay or having dynamics of order higher than are a few examples of process transfer function models for which themethod cannotbe used.

2 H(s) =Ks(integrator)(2)H(s) =KTs+1(first order system)(3)H(s) =K(s 0)2+2 s 0+1(second order system)(4)1 Also denoted theUltimate gain theProcess reaction curve method1yyrA1A2ttvFigure 1:IfA2/A1 1/4the stability of the system is ok, according to Zieglerand Nichols2 The Ziegler-Nichols PID tuning procedureThe Ziegler-Nichols closed loop method is based on experiments executedon an established control loop (a real system or a simulated system), seeFigure tuning procedure is as follows:1. Bring the process to (or as close to as possible) the specifiedoperating pointof the control system to ensure that the controllerduring the tuning is feeling representative process dynamic3and tominimize the chance that variables during the tuning reach can bring the process to the operating point by manuallyadjusting the control variable, with the controller in manual mode,until the process variable is approximately equal to the Turn the PID controller into aP controllerby setting setTi= 4andTd= 0.

3 Initially set gainKp= 0. Close the control loop bysetting the controller in automatic may be important for nonlinear some commercial controllersTi= 0is a code that is used to deactivate the I-term,corresponding toTi= .2 ProcessSensorvyMeasured yySPuePIDu0 AutoManualControllerTpFigure 2:The Ziegler-Nichols closed loop method is executed on an establishedcontrol IncreaseKpuntil there aresustained oscillationsin the signals in thecontrol system, in the process measurement, after an excitationof the system. (The sustained oscillations corresponds to the systembeing on the stability limit.) ThisKpvalue is denoted theultimate(or critical) gain, excitation can be a step in the setpoint. This step must besmall, for example 5% of the maximum setpoint range, so that theprocess is not driven too far away from the operating point where thedynamic properties of the process may be different. On the otherhand, the step must not be too small, or it may be difficult toobserve the oscillations due to the inevitable is important thatKpuis foundwithout the control signal beingdriven to any saturation limit(maximum or minimum value) duringthe oscillations.

4 If such limits are reached, you will find that therewill be sustained oscillations for any (large) value ofKp, ,and the resultingKp-value (as calculated from the Ziegler-Nichols formulas, cf. Table 1) is useless (the control system will probably beunstable). One way to say this is thatKpumust be the smallestKpvalue that drives the control loop into sustained Measure theultimate (or critical) periodPuof the the controller parameter valuesaccording to Table 1, anduse these parameter values in the the stability of the control loop is poor, try to improve the stabilityby decreasingKp, for example a 20% 0PI 1:Formulas for the controller parameters in the Ziegler-Nichols closedloop 1 Tuning a PI controller with the Ziegler-Nichols closed loop methodI have tried the Ziegler-Nichols closed loop method on a level controlsystem for a wood-chip tank with feed screw and conveyor beltwhich runswith constant speed, see Figure 6 The purpose of the control system isto keep the chip level of the tank equal to the actual, measured level control system works as follows.

5 The controller tries to keep themeasured level equal to the level setpoint by adjusting the rotational speedof the feed screw as a function of the control error (which is the differencebetween the level setpoint and the measured level).Figure 4 shows the signals after a step in the setpoint from 9 mto mwith a ultimate gain ofKpu= The ultimate period is approximatelyPu= 1100s. From Table 1 we get the following PI parameters:Kp= = (5)Ti= 917s(6)Td= 0s(7)Figure 5 shows signals of the control system with the above PID parametervalues. The control system has satisfactory stability. Theamplitude ratioin the damped oscillations is less than 1/4, that is, which means that thestability is a little better than prescribed by Ziegler and Nichols .[End of Example 1]5 This example is based on an existing system in the paper pulp factory S dra Cell Toftein Norway. The tank with conveyor belt is in the beginning of the paper pulp simulator of the system is available at [m]Wood chipWood chip tankud[kg/min]LTLCFeed screwProcess(tank with belt and screw)Sensor(LevelTransmitter- LT )ySPuePID controllerydBlock diagram:Process & Instrumentation (P &I) Diagram:Level controllerSensor (Level transmitter)ControlvariableProcessoutput variableProcess disturbance(environmental variable)ControlerrorProcess measure-mentymProcessmeasure -mentymnMeasurement noiseReferenceorSetpointControlvariableP rocess output variableLevel controller (LC)ControlloopMeasure-mentfilterym,fFil teredmeasure-mentConveyor beltReferenceorSetpointySPMeasurement noisenProcess (tank with belt and screw)Process disturbance(environmental variable)Figure 3.

6 P&I (Process and Instrumentation) diagram and block diagram of alevel control system for a wood-chip tank in a pulp factory3 Some comments to the Ziegler-Nichols closedloop do not know in advance the amplitude of the sustainedoscillations. The amplitude depends on the size of the excitations ofthe control If the operating point varies and if the process dynamic propertiesdepends on the operating point, you should consider using some kindofadaptive control or gain scheduling, where the PID parameter areadjusted as functions of the operating the controller parameters shall have fixed value, they should be5A2A1 PuFigure 4:Example 1: The tuning phase of the Ziegler-Nichols in the worst case as stability is regarded. This ensures properstability if the operation point varies. The worst operating point isthe operation point where the process gain has its greatest valueand/or the time delay has its greatest responses in the control system may become unsatisfactorywiththe Ziegler-Nichols method .

7 1/4 decay ratio may be too much, thatis, the damping in the loop is too small. A simple re-tuning inthiscase is to reduce theKpsomewhat, for example by 20%.References[1] J. G. Ziegler and N. B. Nichols:Optimum Settings for AutomaticControllers,Trans. ASME, Vol. 64, 1942, s. 759-7686 Figure 5:Example 1: Time responses with PI parameters tuned using theZiegler-Nichols closed loop method7