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CALTECH CDS 101 - PID Control

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Chapter 8PID ControlBased on a survey of over eleven thousand controllers in the refining, chemi-cals and pulp and paper industries, 97% of regulatory controllers utilize PIDfeedback.Desborough Honeywell, 2000.This chapter describes the PID controller which unquestionably the mostcommon way of solving practical control problem. Practical implementationissues are also discussed particularly mechanisms for avoiding integratorwindup. Methods for automatic tuning of a PID controller are also dis-cussed.8.1 IntroductionThe PID controller is by far the most common control algorithm. Mostpractical feedback loops are based on PID control or some minor variations ofit. Many controllers do not even use derivative action. The PID controllersappear in many different forms, as a stand-alone controllers, they can also bepart of a DDC (Direct Digital Control) package or a hierarchical distributedprocess control system or they are built into embedded systems. Thousandsof instrument and control engineers worldwide are using such controllers intheir daily work. The PID algorithm can be approached from many differentdirections. It can be viewed as a device that can be operated with a fewempirical rules, but it can also be approached analytically.This chapter gives an introduction to PID control. The basic algorithmand various representations are presented in detail. A description of theproperties of the controller in a closed loop based on intuitive arguments isgiven. The phenomenon of reset windup, which occurs when a controller201202 CHAPTER 8. PID CONTROLwith integral action is connected to a process with a saturating actuator,is discussed, including several methods to avoid it. Filters to reduce noiseinfluence and means to improve reference responses are also provided.Implementation aspects of the PID controller are presented in Chap-ter ??.8.2 The PID ControllerThe textbook version of the PID controller isu(t) = ke(t) + kiZt0e(τ)dτ + kddedt, (8.1)where u is the control signal and e is the control error (e = r − y). Thereference value is also called the setpoint. The control signal is thus a sumof three terms: the P-term (which is proportional to the error), the I-term(which is proportional to the integral of the error), and the D-term (whichis proportional to the derivative of the error). The controller parameters areproportional gain k, integral gain kiand derivative gain kd. The controllercan also be parameterized asu(t) = kµe(t) +1TitZ0e(τ)dτ + Tdde(t)dt¶, (8.2)where Tiis called integral time and Tdderivative time. The proportionalpart acts on the present value of the error, the integral represent and averageof past errors and the derivative can be interpreted as a prediction of futureerrors based on linear extrapolation, see Figure 8.1.Proportional ActionFigure 8.2 shows the response of the output to a unit step in the commandsignal for a system with pure proportional control. The output never reachesthe steady state error. Let the process and the controller have transferfunctions P (s) and C(s). The transfer function from reference to output isGyr(s) =P (s)C(s)1 + P (s)C(s)(8.3)The steady state gain with proportional control C(s) = k isGyr(0) =P (0)k1 + P (0)k8.2. THE PID CONTROLLER 203Figure 8.1: A PID controller takes control action based on past, present andprediction of future control errors.The steady state error for a unit step is thus 1/(1+kP (0). For the system inFigure 8.2 with gains k = 1, 2 and 5 the steady state error is 0.5, 0.33 0.17.The error decreases with increasing gain, but the system also becomes moreoscillatory. Notice in the figure that the initial value of the control signalequals controller gain. To avoid having a steady state error the proportionalcontroller can be change tou(t) = Ke(t) + ub. (8.4)where ubis a bias or reset term which is adjusted to give the desired steadystate value.Integral ActionIntegral action guarantees that the process output agrees with the referencein steady state. This can be shown as follows. Assume that the system is insteady state with a constant control signal (u0) and a constant error e06= 0.It follows from Equation (8.1) thatu0= ke0+ kie0t.The left hand side is constant but the right hand side is a function of t. Wethus have a contradiction and e0must be zero. Notice that in this argumentthe only assumption made is that there exist a steady state. Nothing specificis said about the process, it can for example be nonlinear.204 CHAPTER 8. PID CONTROL0 5 10 15 2000.511.50 5 10 15 20−2024PSfrag replacementstyuPPIPID0 5 10 15 2000.511.50 5 10 15 20012PSfrag replacementstyuPPIPID0 5 10 15 2000.511.50 5 10 15 20024PSfrag replacementstyuPPIPIDFigure 8.2: Responses to step changes in the command signal for propor-tional (left), PI (middle) and PID controllers (right). The process has thetransfer function P (s) = 1/(s + 1)3, the controller parameters are k = 1(dashed), 2 and 5 (dash-dotted) for the P controller, k = 1, ki= 0 (dashed),0.2, 0.5 and 1 (dash-dotted) for the PI controller and k = 2.5, ki= 1.5 andkd= 0 (dashed), 1, 2, 3 and 4 (dash-dotted) for the PID controller.ΣKIe u 11+ sTiFigure 8.3: Implementation of integral action as automatic bias adjustment.Another argument is that the transfer function of a controller with inte-gral action has infinite gain at zero frequency (C(0) = ∞). It then followsfrom (8.3) that Gyr(0) = 0. This argument requires however that the systemis linear.Integral action can also be viewed as a method for generating the biasterm ubin the proportional controller (8.4) automatically. This is illustratedin Figure 8.3, where the bias ubis generated by low pass filtering the output.This implementation, called automatic reset, was one of the early inventionsof integral control. The transfer function of the system in Figure 8.3 isobtained by loop tracing. Assuming exponential signals and tracing them8.2. THE PID CONTROLLER 205around the loop givesu = ke +11 + sTu.Solving for u givesu = k1 + sTsTe =³k +ksT¢,which is the transfer function of a PI controller.The properties of integral action are illustrated in Figure 8.2. The pro-portional gain is constant, k = 1, and the integral gain is changed. The caseki= 0 corresponds to pure proportional control, with a steady state error is50%. The steady state error is removed when integral gain kiis increased.The response creeps slowly towards the reference for small values of ki. Theapproach is faster for larger integral gains but the system also becomes


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