# PID Control. 6.1 Introduction

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1 6 PID Control 6. Introduction The PID controller is the most common form of feedback. It was an essential element of early governors and it became the standard tool when process control emerged in the 94s. In process control today, more than 95% of the control loops are of PID type, most loops are actually PI control. PID controllers are today found in all areas where control is used. The controllers come in many different forms. There are stand-alone systems in boxes for one or a few loops, which are manufactured by the hundred thousands yearly. PID control is an important ingredient of a distributed control system. The controllers are also embedded in many special-purpose control systems. PID control is often combined with logic, sequential functions, selectors, and simple function blocks to build the complicated automation systems used for energy production, transportation, and manufacturing. Many sophisticated control strategies, such as model predictive control, are also organized hierarchically. PID control is used at the lowest level; the multivariable controller gives the setpoints to the controllers at the lower level. The PID controller can thus be said to be the bread and butter t t of control engineering. It is an important component in every control engineer s tool box. PID controllers have survived many changes in technology, from mechanics and pneumatics to microprocessors via electronic tubes, transistors, integrated circuits. The microprocessor has had a dramatic influence on the PID controller. Practically all PID controllers made today are based on microprocessors. This has given opportunities to provide additional features like automatic tuning, gain scheduling, and continuous adaptation. 26

2 6.2 The Algorithm 6.2 The Algorithm We will start by summarizing the key features of the PID controller. The textbook version of the PID algorithm is described by: u(t) =K ( e(t)+ T i t ) de(t) e(τ)dτ +T d dt (6.) where y is the measured process variable, r the reference variable, u is the control signal and e is the control error (e = y sp y). The reference variable is often called the set point. The control signal is thus a sum of 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 (which is proportional to the derivative of the error). The controller parameters are proportional gain K, integral time T i, and derivative time T d. The integral, proportional and derivative part can be interpreted as control actions based on the past, the present and the future as is illustrated in Figure 2.2. The derivative part can also be interpreted as prediction by linear extrapolation as is illustrated in Figure 2.2. The action of the different terms can be illustrated by the following figures which show the response to step changes in the reference value in a typical case. Effects of Proportional, Integral and Derivative Action Proportional control is illustrated in Figure 6.. The controller is given by (6.) with T i = and T d =. The figure shows that there is always a steady state error in proportional control. The error will decrease with increasing gain, but the tendency towards oscillation will also increase. Figure 6.2 illustrates the effects of adding integral. It follows from (6.) that the strength of integral action increases with decreasing integral time T i. The figure shows that the steady state error disappears when integral action is used. Compare with the discussion of the magic of integral action in Section Section 2.2. The tendency for oscillation also increases with decreasing T i. The properties of derivative action are illustrated in Figure 6.3. Figure 6.3 illustrates the effects of adding derivative action. The parameters K and T i are chosen so that the closed-loop system is oscillatory. Damping increases with increasing derivative time, but decreases again when derivative time becomes too large. Recall that derivative action can be interpreted as providing prediction by linear extrapolation over the time T d. Using this interpretation it is easy to understand that derivative action does not help if the prediction time T d is too large. In Figure 6.3 the period of oscillation is about 6 s for the system without derivative 27

3 Chapter 6. PID Control K = 5 K = 2 K = K = 5 K = 2 K = Figure 6. Simulation of a closed-loop system with proportional control. The process transfer function is P(s) =/(s + ) 3. T i = T i = 2 T i = 5 T i = T i = T i =2 T i =5 T i = Figure 6.2 Simulation of a closed-loop system with proportional and integral control. The process transfer function is P(s) =/(s + ) 3, and the controller gain is K =. action. Derivative actions ceases to be effective when T d is larger than a s(one sixth of the period). Also notice that the period of oscillation increases when derivative time is increased. A Perspective There is much more to PID than is revealed by (6.). A faithful implementation of the equation will actually not result in a good controller. To obtain a good PID controller it is also necessary to consider 28

4 6.3 Filtering and Set Point Weighting T d =. T d =.7 T d = T d =.7 T d = 4.5 T d = Figure 6.3 Simulation of a closed-loop system with proportional, integral and derivative control. The process transfer function is P(s) =/(s + ) 3, the controller gain is K = 3, and the integral time is T i = 2. Noise filtering and high frequency roll-off Set point weighting and 2 DOF Windup Tuning Computer implementation In the case of the PID controller these issues emerged organically as the technology developed but they are actually important in the implementation of all controllers. Many of these questions are closely related to fundamental properties of feedback, some of them have been discussed earlier in the book. 6.3 Filtering and Set Point Weighting Filtering Differentiation is always sensitive to noise. This is clearly seen from the transfer function G(s) =sof a differentiator which goes to infinity for large s. The following example is also illuminating. 29

5 Chapter 6. PID Control EXAMPLE 6. DIFFERENTIATION AMPLIFIES HIGH FREQUENCY NOISE Consider the signal y(t) =sin t + n(t) =sin t + a n sin ω n t where the noise is sinusoidal noise with frequency ω. The derivative of the signal is dy(t) dt = cos t + n(t) =cos t + a n ω cos ω n t The signal to noise ratio for the original signal is /a n but the signal to noise ratio of the differentiated signal is ω /a n. This ratio can be arbitrarily high if ω is large. In a practical controller with derivative action it is therefor necessary to limit the high frequency gain of the derivative term. This can be done by implementing the derivative term as D = skt d + st d /N Y (6.2) instead of D = st d Y. The approximation given by (6.2) can be interpreted as the ideal derivative st d filtered by a first-order system with the time constant T d /N. The approximation acts as a derivative for low-frequency signal components. The gain, however, is limited to KN. This means that high-frequency measurement noise is amplified at most by a factor KN. Typical values of N are 8 to 2. Further limitation of the high-frequency gain The transfer function from measurement y to controller output u of a PID controller with the approximate derivative is C(s) = K This controller has constant gain ( + ) st d + st i + st d /N lim C(s) = K(+N) s at high frequencies. It follows from the discussion on robustness against process variations in Section 5.5 that it is highly desirable to roll-off the 22

6 6.3 Filtering and Set Point Weighting controller gain at high frequencies. This can be achieved by additional low pass filtering of the control signal by F(s) = (+st f ) n where T f is the filter time constant and n is the order of the filter. The choice of T f is a compromise between filtering capacity and performance. The value of T f can be coupled to the controller time constants in the same way as for the derivative filter above. If the derivative time is used, T f = T d /N is a suitable choice. If the controller is only PI, T f = T i /N may be suitable. The controller can also be implemented as ( C(s) = K + ) + st d (6.3) st i ( + st d /N) 2 This structure has the advantage that we can develop the design methods for an ideal PID controller and use an iterative design procedure. The controller is first designed for the process P(s). The design gives the controller parameter T d. An ideal controller for the process P(s)/(+sT d /N) 2 is then designed giving a new value of T d etc. Such a procedure will also give a clear picture of the trade-off between performance and filtering. Set Point Weighting When using the control law given by (6.) it follows that a step change in the reference signal will result in an impulse in the control signal. This is often highly undesirable therefor derivative action is frequently not applied to the reference signal. This problem can be avoided by filtering the reference value before feeding it to the controller. Another possibility is to let proportional action act only on part of the reference signal. This is called set point weighting. A PID controller given by (6.) then becomes ( u(t) =K br(t) y(t)+ T i t e(τ)dτ +T d ( c dr(t) dt dy(t) ) ) (6.4) dt where b and c are additional parameter. The integral term must be based on error feedback to ensure the desired steady state. The controller given by (6.4) has a structure with two degrees of freedom because the signal path from y to u is different from that from r to u. The transfer function from r to u is ) U(s) R(s) = C r(s) =K ( b+ st i + cst d (6.5) 22

7 Chapter 6. PID Control Output y.5.5 Input u Time t Figure 6.4 Response to a step in the reference for systems with different set point weights b = dashed, b =.5 full and b =. dash dotted. The process has the transfer function P(s) =/(s+) 3 and the controller parameters are k = 3, k i =.5 and k d =.5. and the transfer function from y to u is ( U(s) Y(s) = C y(s) =K + ) + st d st i (6.6) Set point weighting is thus a special case of controllers having two degrees of freedom. The system obtained with the controller (6.4) respond to load disturbances and measurement noise in the same way as the controller (6.). The response to reference values can be modified by the parameters b and c. This is illustrated in Figure 6.4, which shows the response of a PID controller to setpoint changes, load disturbances, and measurement errors for different values of b. The figure shows clearly the effect of changing b. The overshoot for setpoint changes is smallest for b =, which is the case where the reference is only introduced in the integral term, and increases with increasing b. The parameter c is normally zero to avoid large transients in the control signal due to sudden changes in the setpoint. 6.4 Different Parameterizations The PID algorithm given by Equation (6.) can be represented by the 222

8 transfer function G(s) =K 6.4 Different Parameterizations ( + ) + st d st i (6.7) A slightly different version is most common in commercial controllers. This controller is described by ( G (s) =K + ) st i ( + st d) ( =K + T d T i + ) st i + st d (6.8) The controller given by Equation (6.7) is called non-interacting, and the one given by Equation (6.8) interacting. The interacting controller Equation (6.8) can always be represented as a non-interacting controller whose coefficients are given by K = K T i + T d T i T i = T i + T d T d = T i T d T i + T d (6.9) An interacting controller of the form Equation (6.8) that corresponds to a non-interacting controller can be found only if T i 4T d The parameters are then given by K = K ( + ) 4T d /T i 2 T i = T i ( + ) 4T d /T i 2 T d = T i ( ) 4T d /T i 2 (6.) The non-interacting controller given by Equation (6.7) is more general, and we will use that in the future. It is, however, sometimes claimed that the interacting controller is easier to tune manually. It is important to keep in mind that different controllers may have different structures when working with PID controllers. If a controller is replaced by another type of controller, the controller parameters may have to be changed. The interacting and the non-interacting forms differ only 223

9 Chapter 6. PID Control when both the I and the D parts of the controller are used. If we only use the controller as a P, PI, or PD controller, the two forms are equivalent. Yet another representation of the PID algorithm is given by G (s) =k+ k i s +sk d (6.) The parameters are related to the parameters of standard form through k = K k i = K T i k d = KT d The representation Equation (6.) is equivalent to the standard form, but the parameter values are quite different. This may cause great difficulties for anyone who is not aware of the differences, particularly if parameter /k i is called integral time and k d derivative time. It is even more confusing if k i is called integration time. The form given by Equation (6.) is often useful in analytical calculations because the parameters appear linearly. The representation also has the advantage that it is possible to obtain pure proportional, integral, or derivative action by finite values of the parameters. The PIPD Controller The controller with set point weighting can also be represented by the block diagram in Figure 6.5. To see this we introduce the transfer functions of the blocks G PI (s)=k + k i s G PD (s)=+k ds Notice that the proportional gain of the PD controller must be one in order to have zero steady state error. The input-output relation of the complete controller is U(s) =k R(s)+ k i s (R(s) Y(s)) (k + k dh i)y(s) k k dsy(s) Which shows that the controller is thus identical to the controller given 224

10 6.4 Different Parameterizations d r PI u Process y PD r e y PI P PD Figure 6.5 Block diagram of a PI-PD controller. This controller is equivalent to a conventional PID controller with set point weighting. by (6.4). The parameters are related in the following way k = k + k d k i k i = k i k d = k k d T i = k + k d k i k i = k k i T d = k k d k i k b = k + k d h i c = Following the same pattern the controller with b = andc= is sometimes called an I-PD controller, and the controller with b = andc= is called a PI-D controller. Notice, however, that the representation given by (6.4) is much better suitable for tuning, because the parameters k, T i and T d canfirstbedetermined to deal with load disturbances, measurement noise and process 225

11 Chapter 6. PID Control uncertainty. When this is done the response to set points can be adjusted by choosing the parameters b and c. The controller parameters appear in a much more complicated way in the PIPD controller. 6.5 Windup Although many aspects of a control system can be understood based on linear theory, some nonlinear effects must be accounted for in practically all controllers. Windup is such a phenomena, which is caused by the interaction of integral action and saturations. All actuators have limitations: a motor has limited speed, a valve cannot be more than fully opened or fully closed, etc. For a control system with a wide range of operating conditions, it may happen that the control variable reaches the actuator limits. When this happens the feedback loop is broken and the system runs as an open loop because the actuator will remain at its limit independently of the process output. If a controller with integrating action is used, the error will continue to be integrated. This means that the integral term may become very large or, colloquially, it winds up. It is then required that the error has opposite sign for a long period before things return to normal. The consequence is that any controller with integral action may give large transients when the actuator saturates. We will illustrate this by an example. EXAMPLE 6.2 ILLUSTRATION OF INTEGRATOR WINDUP The wind-up phenomenon is illustrated in Figure 6.6, which shows control of an integrating process with a PI controller. The initial setpoint change is so large that the actuator saturates at the high limit. The integral term increases initially because the error is positive; it reaches its largest value at time t = when the error goes through zero. The output remains saturated at this point because of the large value of the integral term. It does not leave the saturation limit until the error has been negative for a sufficiently long time to let the integral part come down to a small level. Notice that the control signal bounces between its limits several times. The net effect is a large overshoot and a damped oscillation where the control signal flips from one extreme to the other as in relay oscillation. The output finally comes so close to the setpoint that the actuator does not saturate. The system then behaves linearly and settles. The example show integrator windup which is generated by a change in the reference value. Windup may also be caused by large disturbances or equipment malfunctions. It can also occur in many other situations. The phenomenon of windup was well known to manufacturers of analog controllers who invented several tricks to avoid it. They were described 226

12 6.5 Windup 2 y y sp u I Figure 6.6 Illustration of integrator windup. The diagrams show process output y, setpoint y sp, control signal u, and integral part I. under labels like preloading, batch unit, etc. Although the problem was well understood, there were often restrictions caused by the analog technology. The ideas were often kept as trade secrets and not much spoken about. The problem of windup was rediscovered when controllers were implemented digitally and several methods to avoid windup were presented in the literature. In the following section we describe some of the methods used to avoid windup. Setpoint Limitation One attempt to avoid integrator windup is to introduce limiters on the setpoint variations so that the controller output never reaches the actuator limits. This frequently leads to conservative bounds and poor performance. Furthermore, it does not avoid windup caused by disturbances. Incremental Algorithms In the early phases of feedback control, integral action was integrated with the actuator by having a motor drive the valve directly. In this case windup is handled automatically because integration stops when the valve stops. When controllers were implemented by analog techniques, 227

13 Chapter 6. PID Control y e = r y KT d s K Actuator model Actuator K T i s + e s T t Figure 6.7 Controller with anti-windup where the actuator output is estimated from a mathematical model. and later with computers, many manufacturers used a configuration that was an analog of the old mechanical design. This led to the so-called velocity algorithms. A velocity algorithm first computes the rate of change of the control signal which is then fed to an integrator. In some cases this integrator is a motor directly connected to the actuator. In other cases the integrator is implemented internally in the controller. With this approach it is easy to avoid windup by inhibiting integration whenever the output saturates. This method is equivalent to back-calculation, which is described below. If the actuator output is not measured, a model that computes the saturated output can be used. It is also easy to limit the rate of change of the control signal. Back-Calculation and Tracking Back-calculation works as follows: When the output saturates, the integral term in the controller is recomputed so that its new value gives an output at the saturation limit. It is advantageous not to reset the integrator instantaneously but dynamically with a time constant T t. Figure 6.7 shows a block diagram of a PID controller with anti-windup based on back-calculation. The system has an extra feedback path that is generated by measuring the actual actuator output and forming an error signal (e s ) as the difference between the output of the controller (v) and the actuator output (u). Signal e s is fed to the input of the integrator through gain /T t. The signal is zero when there is no saturation. Thus, it will not have any effect on the normal operation when the actuator does not saturate. When the actuator saturates, the signal e s is different from zero. The normal feedback path around the process is broken because the 228

14 6.5 Windup.5 y y sp u I Figure 6.8 Controller with anti-windup applied to the system of Figure 6.6. The diagrams show process output y, setpoint y sp, control signal u, and integral part I. process input remains constant. There is, however, a feedback path around the integrator. Because of this, the integrator output is driven towards a value such that the integrator input becomes zero. The integrator input is e s + K e T t T i where e is the control error. Hence, e s = KT t T i in steady state. Since e s = u v, it follows that e v = u lim + KT t T i where u lim is the saturating value of the control variable. This means that the signal v settles on a value slightly out side the saturation limit and the control signal can react as soon as the error changes time. This prevents e 229

15 Chapter 6. PID Control y sp T t = 3 T t = 2 T t =.,T t = 2 3. T t =. T t = T t = 3 T t = Figure 6.9 The step response of the system in Figure 6.6 for different values of the tracking time constant T t. The upper curve shows process output y and setpoint y sp, and the lower curve shows control signal u. the integrator from winding up. The rate at which the controller output is reset is governed by the feedback gain, /T t, where T t can be interpreted as the time constant, which determines how quickly the integral is reset. We call this the tracking time constant. It frequently happens that the actuator output cannot be measured. The anti-windup scheme just described can be used by incorporating a mathematical model of the saturating actuator, as is illustrated in Figure 6.7. Figure 6.8 shows what happens when a controller with anti-windup is applied to the system simulated in Figure 6.6. Notice that the output of the integrator is quickly reset to a value such that the controller output is at the saturation limit, and the integral has a negative value during the initial phase when the actuator is saturated. This behavior is drastically different from that in Figure 6.6, where the integral has a positive value during the initial transient. Also notice the drastic improvement in performance compared to the ordinary PI controller used in Figure 6.6. The effect of changing the values of the tracking time constant is illustrated in Figure 6.9. From this figure, it may thus seem advantageous to always choose a very small value of the time constant because the integrator is then reset quickly. However, some care must be exercised when introducing anti-windup in systems with derivative action. If the time constant is chosen too small, spurious errors can cause saturation of the output, which accidentally resets the integrator. The tracking time constant T t should be larger than T d and smaller than T i. A rule of thumb 23

16 6.5 Windup y sp b K P y skt d + st d / N D e K T i s I v T t w + y sp y w SP MV TR PID v Figure 6. Block diagram and simplified representation of PID module with tracking signal. that has been suggested is to choose T t = T i T d. Controllers with a Tracking Mode A controller with back-calculation can be interpreted as having two modes: the normal control mode, when it operates like an ordinary controller, and a tracking mode, when the controller is tracking so that it matches given inputs and outputs. Since a controller with tracking can operate in two modes, we may expect that it is necessary to have a logical signal for mode switching. However, this is not necessary, because tracking is automatically inhibited when the tracking signal w is equal to the controller output. This can be used with great advantage when building up complex systems with selectors and cascade control. Figure 6. shows a PID module with a tracking signal. The module has three inputs: the setpoint, the measured output, and a tracking signal. The new input TR is called a tracking signal because the controller output will follow this signal. Notice that tracking is inhibited when w = v. Using the module the system shown in Figure 6.7 can be presented as shown in Figure

17 Chapter 6. PID Control SP MV TR PID A Actuator SP MV TR PID v B Actuator model u Actuator Figure 6. Representation of the controllers with anti-windup in Figure 6.7 using the basic control module with tracking shown in Figure Tuning All general methods for control design can be applied to PID control. A number of special methods that are tailor-made for PID control have also been developed, these methods are often called tuning methods. Irrespective of the method used it is essential to always consider the key elements of control, load disturbances, sensor noise, process uncertainty and reference signals. The most well known tuning methods are those developed by Ziegler and Nichols. They have had a major influence on the practice of PID control for more than half a century. The methods are based on characterization of process dynamics by a few parameters and simple equations for the controller parameters. It is surprising that the methods are so widely referenced because they give moderately good tuning only in restricted situations. Plausible explanations may be the simplicity of the methods and the fact that they can be used for simple student exercises in basic control courses. The Step Response Method One tuning method presented by Ziegler and Nichols is based on a process information in the form of the open-loop step response obtained from a bump test. This method can be viewed as a traditional method based on modeling and control where a very simple process model is used. The step response is characterized by only two parameters a and L, as shown in 232

18 6.6 Tuning Figure 6.2 Characterization of a step response in the Ziegler-Nichols step response method. Table 6. method. PID controller parameters obtained for the Ziegler-Nichols step response Controller K T i T d T p P /a 4L PI.9/a 3L 5.7L PID.2/a 2L L/2 3.4L Figure 6.2. The point where the slope of the step response has its maximum is first determined, and the tangent at this point is drawn. The intersections between the tangent and the coordinate axes give the parameters a and L. The controller parameters are then obtained from Table 6.. An estimate of the period T p of the closed-loop system is also given in the table. The Frequency Response Method A second method developed by Ziegler and Nichols is based on a simple characterization of the the frequency response of the process process dynamics. The design is based on knowledge of only one point on the Nyquist curve of the process transfer function P(s), namely the point where the Nyquist curve intersects the negative real axis. This point can be characterized by two parameters the frequency ω 8 and the gain at that frequency k 8 = P(iω 8 ). For historical reasons the point has been called the ultimate point and characterized by the parameters K u = /K 8 and T u = 2π /ω 8, which are called the ultimate gain and the ultimate period. These parameters can be determined in the following way. Connect a controller to the process, set the parameters so that control action is proportional, i.e., T i = and T d =. Increase the gain slowly until the 233

19 Chapter 6. PID Control Figure 6.3 Characterization of a step response in the Ziegler-Nichols step response method. Table 6.2 Controller parameters for the Ziegler-Nichols frequency response method. Controller K T i T d T p P.5K u T u PI.4K u.8t u.4t u PID.6K u.5t u.25t u.85t u process starts to oscillate. The gain when this occurs is K u and the period of the oscillation is T u. The parameters of the controller are then given by Table 6.2. An estimate of the period T p of the dominant dynamics of the closed-loop system is also given in the table. The frequency response method can be viewed as an empirical tuning procedure where the controller parameters are obtained by direct experiments on the process combined with some simple rules. For a proportional controller the rule is simply to increase the gain until the process oscillates and then reduce it by 5%. Assessment of the Ziegler Nichols Methods The Ziegler-Nichols tuning rules were developed to give closed loop systems with good attenuation of load disturbances. The methods were based on extensive simulations. The design criterion was quarter amplitude decay ratio, which means that the amplitude of an oscillation should be reduced by a factor of four over a whole period. This corresponds to closed loop poles with a relative damping of about ζ =.2, which is too small. 234

20 6.6 Tuning Controllers designed by the Ziegler-Nichols rules thus inherently give closed loop systems with poor robustness. It also turns out that it is not sufficient to characterize process dynamics by two parameters only. The methods developed by Ziegler and Nichols have been been very popular in spite of these drawbacks. Practically all manufacturers of controller have used the rules with some modifications in recommendations for controller tuning. One reason for the popularity of the rules is that they are simple and easy to explain. The tuning rules give ball park figures. Final tuning is then done by trial and error. Another (bad) reason is that the rules lend themselves very well to simple exercises for control education. With the insight into controller design that has developed over the years it is possible to develop improved tuning rules that are almost as simple as the Zigler-Nichols rules. These rules are developed by starting with a solid design method that gives robust controllers with effective disturbance attenuation. We illustrate with some rules where the process is characterized by three parameters. An Improved Step Response Method This method characterizes the unit step response by three parameters K, L and T for stable processes and Kv = K/T and L for integrating processes. This parameterization matches the transfer functions P (s) = k p +st e sl P 2 (s)= k v s e sl The transfer function P (s), which is called a first order system with time delay or a KLT model. Parameter L is determined from the intercept of the tangent with largest slope with the time axis as was described in Figure 6.2. Parameter T is also determined as shown in the figure as the difference between the time when the step response reaches 63% of its steady state value. Parameter k p is the static gain of the system. The parameter k v is the largest slope of the unit step response. Parameter L is called the apparent time delay and parameter T the apparent time constant or the apparent lag. The adverb apparent is added to indicate that parameters are based on approximations. The parameter τ = L L + T is called the relative time delay. This parameter is a good indicator of process dynamics. 235

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