7. PID Controllers. KEH Process Dynamics and Control 7 1. Process Control Laboratory

Transcription

1 7. PID lers 7.0 Overview 7.1 PID controller variants 7.2 Choice of controller type 7.3 Specifications and performance criteria 7.4 ler tuning based on frequency response 7.5 ler tuning based on step response 7.6 Model-based controller tuning 7.7 ler design by direct synthesis 7.8 Internal model control 7.9 Model simplification KEH Dynamics and 7 1

2 7. PID lers 7.0 Overview PID controller ( pee-i-dee ) is a generic name for a controller containing a linear combination of proportional (P) integral (I) derivative (D) terms acting on a control error (or sometimes the process output). All parts need not be present. Frequently I and/or D action is missing, giving a controller like P, PI, or PD controller It has been estimated that of all controllers in the world 95 % are PID controllers KEH Dynamics and 7 2

3 7. PID lers 7.1 PID controller variants Ideal PID controller An ideal PID controller is described by the control law uu tt = KK c ee tt + 1 tt dee(tt) TT i 0 ee ττ dττ + TTd + uu dtt 0 (7.1) uu(tt) is the controller output ee tt = rr tt yy(tt) is the control error, which is the difference between the setpoint rr(tt) and the measured process output yy(tt) KK c is the proportional gain TT i is the integral time TT d is the derivative time uu 0 is the normal value of the controller output The transfer function of the PID controller is GG PID = UU(ss) EE(ss) = KK c TT i ss + TT dss = KK c TT i ss 1 + TT iss + TT i TT d ss 2 (7.2) UU(ss) is the Laplace transform of uu tt uu 0 EE(ss) is the Laplace transform of the control error KEH Dynamics and 7 3

4 7.1 PID controller variants Ideal PID controller Depending on the values of TT i and TT d, the transfer function of the PID controller can have real or complex-valued zeros Complex zeros might be useful for control of underdamped systems with complex poles. A PI controller is obtained from a PID controller by letting TT d = 0. Its transfer function is GG PI = KK c TT i ss = KK c TT i ss 1 + TT iss (7.3) A PD controller is obtained from a PID controller by letting TT i =. Its transfer function is GG PD = KK c 1 + TT d ss (7.4) The ideal PID controller is sometimes referred to as the parallel form of a PID controller the (ISA) standard form KEH Dynamics and 7 4

5 7. PID lers 7.1 PID controller variants The series form of a PID controller In the pre-digital era it was convenient to implement an analog PID controller as a PI controller and a PD controller in series. This form of a PID controller is called the series form. Occasionally, the terms interactive form or classical form are used. The controller has the transfer function GG PIPD = KK c TT i ss 1 + TT d ss = KK c TT i ss 1 + TT i ss 1 + TT d ss (7.5) where is used to distinguish the parameters from the parameters of the parallel form. The series form of a PID controller can only have real valued zeros. This means that the series form is less general than the parallel form. It is relatively easy to find the controller parameters of the series form by frequency analytic methods by so-called lead-lag design. Exercise 7.1 Which is the control law in the time domain for a series form PID controller? KEH Dynamics and 7 5

6 7. PID lers 7.1 PID controller variants A PID controller with derivative filter A drawback with the ideal PID controller (7.1) is that the derivative part cannot be realized exactly in a real controller. For example, if the control error ee(tt) changes as a step, the derivate in (7.1) becomes infinitely large. This problem can be remedied by filtering the signal to be differentiated. This also has the practical advantage that (high-frequency) noise is filtered before differentiation. The transfer function of a parallel form PID controller with a derivative filter is GG PIDf = KK c TT i ss + TT dss TT f ss+1 The transfer function of a series form PID controller with a derivative filter is usually stated in the form GG PIPDf = KK c TT d ss+1 TT i ss TT f ss+1 (7.6) (7.7) TT f and TT f are filter constants, usually % of corresponding derivative time. KEH Dynamics and 7 6

7 7.1 PID controller variants A PID controller with derivative filter Relationships between parallel and series form If the parameters of the series form are known, the corresponding parameters of the parallel form can be calculated according to TT i = TT i + TT d TT f, TT d = TT d TT i TT i TT f, TT f = TT f, KK c = KK TT i c (7.8) TT i For calculation of the parameters of the series form from the parameters of the parallel form, we define the parameter δδ = 1 4TT i(tt d +TT f ) (TT i +TT f ) 2 (7.9) If δδ 0, the zeros of the parallel PID are real. Then, there exists a series-form PID controller which is equivalent to the parallel form according to TT i = (TT i+tt f ) δδ, TT d = TT i + TT f TT i, TT f = TT f, KK c = KK c TT i The condition for δδ 0 in terms of the controller parameters is TT i (7.10) TT d (TT i TT f ) 2 4TT i (7.11) i.e., the derivative time has to be small enough. KEH Dynamics and 7 7

8 7. PID lers 7.1 PID controller variants Differentiation of the measured output Even if we have a derivative filter, a step change in the setpoint rr(tt) tends to affect the derivative part much more strongly than a disturbance in the output yy(tt). A remedy to this is to differentiate the (filtered) output instead of the control error ee(tt). The ideal control law (7.1) then becomes uu tt = KK c In the Laplace domain we get ee tt + 1 TT i 0 tt ee ττ dττ TTd dyy f (tt) dtt TT f dyy f (tt) dtt + uu 0 (7.12a) + yy f tt = yy(tt) (7.12b) UU ss = KK c TT i ss RR ss KK c TT i ss + TT dss TT f ss+1 which is a combination of a PI controller and a PID controller YY(ss) (7.13) UU ss = GG PI RR ss GG PIDf YY(ss) (7.14) This kind of 2-degrees-of-freedom (2DOF) controller can be tuned separately for setpoint tracking and disturbance rejection. KEH Dynamics and 7 8

9 7.1 PID controller variants Differentiation of the measured output Exercise 7.2 Which is the control law, both in the time domain and the Laplace domain, for the series form of a PID controller with differentiation of the filtered output measurement? Setpoint weighting A simple way of obtaining a 2DOF PID controller is to use setpoint weighting. With the definitions ee p = bbbb yy, ee = rr yy, ee d = cccc yy f (7.15) where bb and cc are setpoint weights, the control law becomes uu tt = KK c ee p tt + 1 TT i 0 tt ee ττ dττ + TTd dee d (tt) dtt + uu 0 (7.16a) TT f dyy f (tt) dtt + yy f tt = yy(tt) (7.16b) KEH Dynamics and 7 9

10 7.1 PID controller variants Setpoint weighting In the Laplace domain the control law with setpoint weighting is where and GG PIDf is as in (7.6). UU ss = GG vpid RR ss GG PIDf YY(ss) (7.17) GG vpid = KK c bb + 1 TT i ss + cctt dss (7.18) With suitable choices of bb and cc, all previously treated PID controllers on parallel form can be obtained. bb and cc do not affect the controller s ability to reject disturbances in the output, only the ability to track setpoint changes. GG vpid can be tuned for setpoint tracking and GG PIDf for disturbance rejection (i.e., KK c, TT i and TT d need not have the same values in GG vpid and GG PIDf ). Exercise 7.3 Include setpoint weighting in the series form of a PID controller. KEH Dynamics and 7 10

11 7. PID lers 7.1 PID controller variants Non-interactive form of a PID controller In the control laws treated so far, the proportional part alone cannot be disconnected by letting KK c = 0 because that would disconnect all parts; it would put the controller on manual with uu tt = uu 0. Tuning the proportional part by adjusting KK c will affect all controller parts (however, this is often a desired feature); hence, it is an interactive controller form. The non-interactive form tt dee uu tt = KK c ee pp tt + KK i 0 ee ττ dττ + d (tt) KKd + uu dtt 0 (7.19) is a more flexible control law. In the Laplace domain it can be written where UU ss = GG vp+i+d RR ss GG P+I+Df YY(ss) (7.20) GG vp+i+d = KK c bb + KK i ss 1 + cckk d ss GG P+I+Df = KK c + KK i ss 1 + KK d ss(tt f ss + 1) 1 (7.21a) (7.21b) Note: It is essential to know which form is used when tuning a controller! KEH Dynamics and 7 11

12 7. PID lers 7.2 Choice of controller type The choice between controller types such as P, PI, PD, PID is considered. In principle, the simplest controller that can do the job should be chosen On-off controller An on-off controller is the simplest type of controller, where the control signal has only two levels. If the variables are defined such that a positive control error ee(tt) should be corrected by an increase of the control signal uu(tt), the control law is uu max if ee tt > ee hi uu tt = uu 0 or unchanged if ee lo ee tt ee hi (7.22) uu min if ee tt < ee lo where uu max, uu 0, uu min are the high, normal, low value of the control signal. The interval [ee lo, ee hi ] is a dead zone. In the simplest case, ee lo = ee hi = 0. The on-off controller is inexpensive, but it causes oscillations in the pro-cess. It is often used for temperature control in simple appliances such as ovens, irons, refrigerators and freezers, where oscillations are tolerated. KEH Dynamics and 7 12

13 7. PID lers 7.2 Choice of controller type P controller A P controller implements the simple control law uu tt = KK c ee tt + uu 0 (7.23) where KK c is the adjustable controller gain and uu 0 is the normal value of the control signal, which is also be adjustable. In principle, uu 0 is selected to make the control error ee tt = 0 at the nominal operating point. If the output is changed by a disturbance or a setpoint change, the P controller is unable to bring the control error to zero, i.e., there will be a remaining control error. The higher the controller gain, the smaller the control error. Thus, P control is used when a (small) control error is allowed and a high controller gain can be used without risk of instability. A typical application for P control is level control in a liquid tank. Another situation when P control is often sufficient is as an inner loop (a secondary loop) in so-called cascade control. KEH Dynamics and 7 13

14 7. PID lers 7.2 Choice of controller type PI controller A PI controller is by far the most common type of controller. The ideal PI controller implements the control law uu tt = KK c ee tt + 1 TT i 0 tt ee ττ dττ + uu0 (7.24) where the gain KK c and the integral time TT i are adjustable parameters; uu 0 is less important due to the integral. The main advantage of the PI controller is that there will be no remaining control error after a setpoint change or a process disturbance. A disadvantage is that there is a tendency for oscillations. PI control is used when no steady-state error is desired and there is no reason to use derivative action. Measurement noise is often a reason for not using derivative action. PI control is suitable for noisy processes, integrating processes and processes resembling first-order systems. The most typical application is flow control. PI control might also be preferable for processes with large time delays. KEH Dynamics and 7 14

15 7. PID lers 7.2 Choice of controller type PD controller The ideal form of a PD controller implements the control law uu tt = KK c ee tt + TT d dee(tt) dtt + uu 0 (7.25) where the gain KK c and the derivative time TT d are adjustable parameters; uu 0 is chosen as for a P controller. A PD controller is preferred when integral action is not needed, but the dynamics of the process are so slow that the predictive nature of derivative action is useful. Many thermal processes, where energy is stored with small heat losses (e.g., ovens), usually have slow dynamics, almost as integrating systems. A PD controller might then be suitable for temperature control. Another typical application for PD control is in servo mechanisms such as electrical motors, which usually behave as second-order integrating systems. KEH Dynamics and 7 15

16 7. PID lers 7.2 Choice of controller type PID controller As has been shown in Section 7.1, there are many variants of PID controllers. The ideal form and the classical series form have 3 adjustable parameters in addition to uu 0 : the proportional gain, the integral time, and the derivative time. If a derivative filter is included, there are 4 adjustable parameters, but the filter time constant is usually selected as a given fraction (e.g., 10 %) of the derivative time. In addition, the setpoint can be weighted in the proportional part and the derivative part. If there is no reason to exclude integral action or derivative action, a PID controller is the natural choice. Typically PID control is used for underdamped processes, processes with slow dynamics and not very large time delays, and systems of second and higher order. Typical applications are control of temperature and chemical composition when the process is not close to an integrating system. KEH Dynamics and 7 16

17 7. PID lers 7.3 Specifications and performance criteria General performance criteria The task of a controller is to control a system to behave in a desired way despite unknown disturbances and an inaccurately known system. The controlled system should satisfy performance criteria such as: The controlled system must be stable; this is absolutely necessary. The effect of disturbances on the controlled output is minimized; this is especially important for regulatory control. The controlled output should follow setpoint changes fast and smoothly; this is especially important for setpoint tracking. The control error is minimized or kept within certain limits, The control signal variations should be moderate or at least not be excessively large; more variations wear out control equipment faster. The control system should be robust (insensitive) to moderate changes in system properties, which introduce model uncertainty. The importance of these criteria varies from case to case. Since many criteria are conflicting, compromises have to be made in the control design. KEH Dynamics and 7 17

18 7. PID lers 7.3 Specifications and performance criteria Fundamental limitations One reason to the fact that there are usually good solutions to the conflicting control criteria is that feedback control is used. However, feedback also introduces limitations because a control error is required for the controller to take action. The fact that the available resources for control are always limited, also limit the achievable performance. In addition to the general limitations above, there are also limitations that depend on the process to be controlled, e.g., the dynamics of the process nonlinearities model and process uncertainty disturbances control signal limitations KEH Dynamics and 7 18

19 7.3 Specifications and performance criteria Fundamental limitations The process dynamics is often the performance-limiting factor. Such factors are time delays as well as RHP (right-half plane) poles and zeros high-order dynamics In practice, all processes are nonlinear. Such a process cannot be described accurately at different operating points by a linear model with constant parameters; thus there is model/process uncertainty. Disturbances such as load disturbances and measurement noise limit how well a variable can be controlled. Efficient control of load disturbances often require derivative action, but measurement noise is bad for the derivative. Large load disturbances can cause the control variable to reach its (physical) maximum or minimum value. This is especially troublesome if the controller contains an integrator. Proportional band and integrator windup are two concepts that deal with this limitation. KEH Dynamics and 7 19

20 7. PID lers 7.3 Specifications and performance criteria Proportional band and integrator windup Proportional band A controller s proportional band (PB) denotes the maximum control error the controller can handle with the available control signal. The PB is defined for a P controller, but it can be extended to a full PID controller. If the control signal is limited by uu min uu(tt) uu max, a P controller can according to (7.23) handle a control error that satisfies uu min uu 0 KK c ee min ee(tt) ee max uu max uu 0 KK c (7.26) The PB is equal to ee max ee min = yy hi yy lo, where yy hi is the highest output (ee min = rr yy hi ) and yy lo is the lowest output (ee max = rr yy lo ) the controller can handle. Usually, the PB is defined in percent of the total measurable output interval yy min, yy max. Then, the PB is PP b = yy hi yy lo yy max yy min 100% = uu max uu min yy max yy min 100% KK c (7.27) KEH Dynamics and 7 20

21 7.3.3 Proportional band and integrator windup Proportional band If the proportional band is known, the controller gain is given by KK c = yy hi yy lo yy max yy min 100% = uu max uu min yy max yy min 100% PP b (7.28) In (old) automation systems, the signals are often expressed as a fraction or percentage of the total signal interval (0-1 or 0-100%). The PB is then PP b = 100%/KK c (7.29) Note that the controller gain here has to be expressed in terms of normalized signals, which means that the controller gain is dimensionless. The practical usefulness of the PB is that it tells something about the size of control errors that can be handled without reaching an input signal constraint. If uu 0 is in the middle of the interval uu min, uu max, a P controller with PP b = 50 % can handle an instantaneous control error equal to ±25 % (i.e., 50 % in total) of the total output signal range. Note that the PB is an adjustable controller parameter if it is to small, it can be increased (corresponding to a decrease of KK c ). KEH Dynamics and 7 21

22 7.3 Specifications and performance criteria PB and integrator windup Integrator windup Usually controllers are tuned for stability and performance, not for signal limits. Therefore, it is not uncommon that a control signal reaches a constraint. If the controller contains integral action, this can be very damaging to the control performance unless the situation is handled properly. Consider the figure, where the PI control law (7.24) is used. A strong disturbance causes the process output to fall well below the set-point. The controller is not able to eliminate the control error (A) because the control signal has reached a constraint. During this time, the positive control error will increase the integral in the controller. If the disturbance later disappears, the controller will still keep the control signal at the constraint due to the large value of the integral, even If the control error goes below zero. This will cause the output (B), which is entirely due to the controller. Illustration of integral windup. KEH Dynamics and 7 22

23 7.3.3 Proportional band and integrator windup Integrator windup The described phenomenon is called integrator windup, integral windup, or reset windup. There are sophisticated as well as simple methods for handling the problem. The term anti-windup is used for such arrangements. A simple solution is to stop integrating when a control signal reaches a constraint. This requires that it is known when the control signal reaches a constraint (e.g., through measurement) there is some built-in logic to interrupt the integration In the case of digital control, which nowadays is customary, automatic antiwindup can be built into the control law. KEH Dynamics and 7 23

24 7. PID lers 7.3 Specifications and performance criteria Design specifications Above, some general performance criteria and fundamental limitations to achievable control performance have been considered. Here, some ways of making more specific design specifications will be introduced. If a process model is available, the specifications make it possible to calculate controller parameters. Step-response specifications It is of often desired that the closed-loop response to a step change in the setpoint resembles an underdamped second-order system. Therefore, parameters familiar from the step response of such a system can be used to specify the desired behaviour. Such parameters are the maximum relative overshoot MM the rise time tt r the settling time tt δδ the relative damping ζζ the ratio between successive relative overshoots (or undershoots) MM R KEH Dynamics and 7 24

25 7.3.4 Design specifications Step-response specifications According to the relationships in Section 5.3.3: The two parameters MM and tt r are sufficient to determine the transfer function of an underdamped second-order system with a given gain. The settling time tt δδ can be used instead of MM or tt r, but the relationships are then only approximate. The relative damping ζζ or the overshoot ratio MM R can be specified instead of MM. Some classical tuning recommendations are based on the specification MM R = 1/4. This may be acceptable for regulatory control, but not for setpoint tracking. MM R = 1/4 corresponds to MM = 0.5 (i.e., a 50 % overshoot) and ζζ = For setpoint tracking, MM 0.1 (ζζ 0.6) is usually more appropriate. If an overdamped closed-loop response is desired, this cannot be achieved with a specification ζζ > 1, because the other parameters require an underdamped system. Instead, the closed-loop transfer function can be directly specified and controller parameters calculated by direct synthesis (Section 7.7), for example. KEH Dynamics and 7 25

26 7.3 Specifications and performance criteria Design specifications Error integrals In principle, a small overshoot, rise time and settling time are desired. In practice, the overshoot and settling time will increase with decreasing rise time, and vice versa. Therefore, compromises have to be made. One way of solving this problem in an optimal way is to specify some error integral to be minimized. Examples of such error integrals are JJ IAE = 0 tt s ee(tt) dtt, JJ ITAE = 0 tt s tt ee(tt) dtt, JJ ISE = 0 tt s ee(tt) 2 dtt JJ ITSE = 0 tt s ttee(tt) 2 dtt (7.30) where the acronyms are IAE = integrated absolute error ISE = integrated square error ITAE = integrated time-weighted absolute error ITSE = integrated time-weighted square error The weighting with time forces the control error towards zero as time increases. In principle, the integration time should be infinite, but because the minimization has to be done numerically, a finite tt s has to be used. KEH Dynamics and 7 26

27 7.3.4 Design specifications Error integrals It is of interest to consider how the error integrals relate to step-response specifications when the closed-loop system is of second order, i.e., GG ss = ωω n 2 ss 2 +2ζζωω n ss+ωω n 2 (7.31) In the figure, IAE and ISE are normalized with ωω n, ITAE and ITSE with ωω n 2. As can be seen, every normalized error integral has a minimum for a given relative damping ζζ. This damping as well as the corresponding relative overshoot MM are shown below. Table 7.1 Optimal relative damping for 2 nd order system. Error integral ζ M (%) ISE ITSE IAE ITAE Error integrals as function of ζζ. KEH Dynamics and 7 27

28 7. PID lers 7.4 Tuning based on frequency response Experimental tuning An ideal PID controller of interactive form can be tuned experimentally by making closed-loop control experiments with the real process. The standard feedback structure is used. G 1. A P controller (GG c = KK c ) is used for the first experiment. A low value is chosen for the gain KK c. Note that KK c must have the same sign as KK p. 2. A change in the setpoint RR is introduced. (Some other disturbance could also be used.) The controller gain KK c is increased until the output YY starts to oscillate with a constant amplitude (see next slide). 3. The value of the controller gain yielding constant oscillations is denoted KK c,max. The period of the oscillations is denoted PP c. 4. The controller gain is changed to KK c = 0.5KK c,max. If the intention was to tune a P controller, this is the final tuning. KEH Dynamics and 7 28

29 7.4 Tuning based on frequency response Experimental tuning 5. To tune a controller with integral action (PI or PID), an experiment is done with a PI controller using KK c = 0.5KK c,max. A large value is initially used for the integral time TT i. 6. A change in the setpoint RR (or some other disturbance) is introduced. The integral time TT i is reduced until YY starts to oscillate with a constant amplitude. This occurs at TT i = TT i,min. 7. The integral time for a PI or PID controller is chosen as TT i = 3TT i,min. 7. To tune the derivative part of a PID (or PD) controller, an experiment is done with such a controller using KK c = 0.5KK c,max, TT i = 3TT i,min (if a PID controller). The derivative time is initially set at TT d = A change in the setpoint RR (or some other disturbance) is introduced. The derivative time TT d is increased until the output YY starts to oscillate with a constant amplitude. This occurs when TT d = TT d,max. 10. The derivative time for a PD or PID controller is set at TT d = 1 TT 3 d,max. KEH Dynamics and 7 29

30 7.4 Tuning based on frequency response Experimental tuning If the control performance obtained by the above tunings turns out to be unsatisfactory, the controller parameters can be adjusted by trial and error. The next figure shows how changes of the controller gain KK c and the integral time TT i typically affect the control performance. The optimal performance is in this case obtained by KK c = 3 and TT i = 11. TT i = 5 TT i = 11 TT i = 20 KK c = 5 KK c = 3 KK c = 1 KEH Dynamics and 7 30

31 7. PID lers 7.4 Tuning based on frequency response Ziegler-Nichols s recommendations In 1942, Ziegler and Nichols suggested tunings for P, PI and PID controllers based on KK c,max and PP c only. To obtain this information, it is sufficient to do steps 1 3 in the experimental procedure. The tunings are primarily intended for regulatory control (i.e., disturbance rejection). For setpoint tracking, setpoint weighting is suggested, e.g. bb = 0.5. The controller tuning should Table 7.2. Ziegler-Nichols s controller preferably not be used out- tuning recommendations based on side the range 0.1 < κκ < 0.5, frequency response (0.1 < κκ < 0.5). where 1 κκ = KKKK c,max. ler Kc / K c,max Ti / P c Td / P c KK is the process gain. The critical frequency ωω c is often used instead of PP c : ωω c = 2ππ/PP c. P 0.5 PI PID KEH Dynamics and 7 31

32 7. PID lers 7.4 Tuning based on frequency response Åström s and Hägglund s correlations In 2006, Åström and Hägglund showed that, in general, KK c,max and PP c alone do not provide sufficient information for good controller tuning. In addition to KK c,max and PP c, Åström and Hägglund also use the parameter κκ = KKKK c,max 1 in their controller tuning correlations. The tuning correlations are primarily intended for regulatory control; for setpoint tracking, setpoint weighting is suggested. The correlations should Table 7.3. Åström-Hägglund s controller not be used below the tuning correlations based on frequency range κκ > 0.1. response (κκ > 0.1). Large time delays are allowed, but clearly underdamped systems are less suitable. ler Kc / K c,max Ti / P c Td / P c PI ( κ) PID κ κ 0.15(1 κ) κ KEH Dynamics and 7 32

33 7. PID lers 7.5 Tuning based on step response A drawback with generating the frequency response is that it is quite cumbersome and time-consuming to generate oscillations with constant amplitude by adjusting a controller parameter. An alternative is to use a step response for the process. The figure illustrates how the needed parameters are obtained from a unit-step response, i.e., a step with size uu step = 1 expressed in the units used for the control variable. The method is based on the (modified) tangent method, but here it is not necessary to wait for the new steady state; only the parameters aa and LL need to be determined. y i L t i Characteristic parameters from a monotonous unit-step response. KEH Dynamics and 7 33

34 T ak i d / c T L 7. PID lers 7.5 Tuning based on step response Instead of taking the aa parameter from the point, where the tangent through the inflexion point (i.e., the point where the slope is highest) of the step response crosses the vertical axis, it can be calculated when the coordinates (tt i, yy i ) of the inflexion point are known. The calculation is valid for any size of uu step. The formula for aa is Another useful parameter is aa = LLyy i uu step (tt i LL) (7.32) θθ = LL/TT eq, TT eq = tt 63 LL (7.33) where TT eq is the equivalent time constant of the system and tt 63 is the time it takes to reach 63 % of the total output change. The step response of a purely integrating system is a ramp that changes linearly with time, i.e., it has a constant slope. Any point on the ramp can then be used as a pair of coordinates (tt i, yy i ) for calculation of aa according to (7.32). KEH Dynamics and 4 34

35 7. PID lers 7.5 Tuning based on step response Ziegler-Nichols s recommendations In 1942, Ziegler and Nichols suggested tunings for P, PI and PID controllers based also on the information that can be obtained from a step test. Their recommendations for an ideal controller are given in Table 7.4. The method requires LL > 0 and preferably 0.1 θθ 1. Table 7.4. Ziegler-Nichols s controller tuning recommendations based on step response. ler ak c T i / L T d / L P 1.0 PI PID Note that Ziegler-Nichols s recommendations based on frequency response and step response do not necessarily give the same controller tuning for the same process. KEH Dynamics and 7 35

36 7. PID lers 7.5 Tuning based on step response The CHR method In 1952, Chien, Hrones and Reswick suggested improvements to Ziegler s and Nichols s recommendations based on a step response. The CHR-method gives different tunings for regulatory control and setpoint tracking tunings for aggressive control (with ~20 % overshoot) and cautious control (no overshoot) The method requires LL > 0 and preferably 0.1 θθ 1. The CHR tunings (even the aggressive one) are less aggressive than the ZN tuning. Note that the different tunings for regulatory control and setpoint tracking can directly be used in a 2DOF controller. KEH Dynamics and 7 36

37 7.5 Tuning based on step response The CHR method Table 7.5. ler tuning for regulatory control by the CHR method. ler No overshoot 20 % overshoot ak c T i / L T d / L ak c T i / L T d / L P PI PID Table 7.6. ler tuning for setpoint tracking by the CHR method. ler No overshoot 20 % overshoot ak c T i / T T d / L ak c T ii / T eq T d / L i eq P 0,3 0,7 PI 0,35 1,2 0,6 1,0 PID 0,6 1,0 0,5 0,95 1,4 0,47 KEH Dynamics and 7 37

38 7. PID lers 7.5 Tuning based on step response Åström s and Hägglund s correlations In 2006, Åström and Hägglund presented improved controller tunings based on a step response. In addition to aa and LL, they use θθ in their correlations, which can be used for all θθ 0. However, for θθ < 0.4, the tunings tend to be conservative. For an integrating process, θθ = 0 is used. The tunings are primarily intended for regulatory control. For setpoint tracking, setpoint weighting can be used as follows: PI control: bb = 1 if θθ > 0.4, bb < 1 if θθ 0.4 (optimal bb is unclear) PID control: bb = 1 if θθ > 1, bb = 0 if θθ 1 Table 7.7. Åström s and Hägglund s controller tuning correlations. ler ak c T i / L T d / L θ 13 PI θ (1 + 2 θ ) 1+ 12θ + 7θ PID θ 8+ 4θ 1+ 10θ θ KEH Dynamics and 7 38

39 7. PID lers 7.6 Model-based controller tuning The controller tuning methods in Sections 7.4 and 7.5 employ parameters that can be determined from an experiment with an existing process. If a process model is known, the same parameters can be determined through a simulation experiment possibly by direct calculation from the process model For example, a first-order system with a time delay has the transfer function GG ss = KK TTTT+1 e LLLL (7.34) from which the parameters aa and θθ can be calculated according to aa = KKKK TT, θθ = LL TT The same tuning methods as in Sections 7.4 and 7.5 can then be used. However, the methods in Sections 7.4 and 7.5 are general purpose methods that are not optimized for any specific model type. For a given model, better controller tunings probably exist. (7.35) KEH Dynamics and 7 39

40 7. PID lers 7.6 Model-based controller tuning First-order system with a time delay The transfer function is defined in (7.34) and the parameter θθ in (7.35). Minimization of error integrals ler tunings that minimize IAE and ITAE when 0.1 θθ 1. Table 7.8. IAE and ITAE minimizing controller tunings for regulatory control. Error integral P controller PI controller PID controller KK c KK c T i / T KK c T i / T T d / T IAE θ θ 1.645θ ITAE θ θ θ θ θ θ 1.188θ θ θ Table 7.9. IAE and ITAE minimizing controller tunings for setpoint tracking. Error integral PI controller PID controller KK c T i / T KK c T i / T T d / T IAE θ ( θ ) 1 ITAE θ ( θ ) θ ( θ ) θ ( θ ) θ θ KEH Dynamics and 7 40

41 7.6 Model-based controller tuning First-order system with time delay Other optimality criteria The controller tunings for minimizing the error integrals IAE and ITAE in Tables 7.8 and 7.9 do not give any robustness guarantees. Thus, the control performance can be bad if the model contains errors. Cvejn (2009) has derived controller tunings that have a certain robustness even for systems with large time delays, i.e., for large θθ values. Table Cvejn s tunings for regulatory control and setpoint tracking. PI controller PID controller KK c T i / T KK c T i / T T d / T Regulatory 1 2θ 5.92θ θ θ 3.91θ θ 4θ θ θ +θ Tracking 1 2θ 1 3+ θ 4θ θ 1+ 3 θ 3 +θ The PI controller tunings tend to give better robustness than the PID controller tunings, which tend to give better performance. KEH Dynamics and 7 41

42 7. PID lers 7.6 Model-based controller tuning Second-order no-zero system with a time delay We shall consider second-order systems with a time delay but no zeros. Such a system has the transfer function GG ss = KKωω n 2 ss 2 +2ζζωω n ss+ωω n 2 e LLLL (7.36) In Cvejn s method for tracking control, the controller GG c (ss) is tuned to give the loop transfer GG l (ss) = GG(ss)GG c (ss) such that or GG l ss = 1 2LLLL e LLLL (7.37) GG l ss = LLLL e LLLL (7.38) Tuning by (7.37) gives better stability, (7.38) gives better performance. Exercise 7.3 Use Cvejn s method for tracking control to tune a PID controller for the system (7.36). KEH Dynamics and 7 42

43 7.6 Model-based controller tuning Second-order system with delay Overdamped system without zeros For an overdamped (or critically damped) second-order system, ζζ 1. In this case, (7.36) is more conveniently written as GG ss = KK (TT 1 ss+1)(tt 2 ss+1) e LLLL, TT 1 TT 2 (7.39) Cvejn s method can be used also in this case, but Åström and Hägglund (2006) suggest the following tuning when the system is overdamped: KKKK c = θθ θθ θθ 1 1 θθ 2 1 KKKK c LL/TT i = θθ θθ θθ 1 1 θθ 2 1 (7.40) where KKKK c TT d /LL = θθ θθ θθ 1 1 θθ 2 1 θθ 1 +θθ 2 θθ 1 +θθ 2 +θθ 1 θθ 2 θθ 1 = LL/TT 1, θθ 2 = LL/TT 2 (7.41) KEH Dynamics and 7 43

44 7.6.2 Second-order system with delay Overdamped system Second-order system including integration A second-order no-zero system including an integrator has the transfer function GG ss = KK ss (TT 2 ss+1) e LLLL (7.42) For this kind of system, Åström and Hägglund (2006) suggest the tuning: KKKK c LL = θθ 2 1 KKKK c LL 2 /TT i = θθ 2 1 (7.43) KKKK c TT d = θθ 2 1 If the system is a double integrator with the transfer function the suggested tuning is GG ss = KK ss 2 e LLLL (7.44) KKKK c LL 2 = 0.02 KKKK c LL 3 /TT i = (7.45) KKKK c TT d LL = 0.28 KEH Dynamics and 7 44

45 7.6 Model-based controller tuning Second-order system with delay Second-order system with a zero An overdamped 2 nd order system with a zero has the transfer function GG ss = KK(TT 3ss+1) (TT 1 ss+1)(tt 2 ss+1) e LLLL (7.46) Such a system can often be approximated by a first-order system or a secondorder system without a zero (see Section 7.9). Integrating second-order system with a zero An IPZ system (1 integrator, 1 pole, 1 zero) has a transfer function GG ss = KK(TT 3ss+1) ss (TT 2 ss+1) e LLLL, TT 3 > TT 2 > 0 (7.47) An IPZ system is difficult to approximate by a simpler one, esp. if TT 3 TT 2. In Table 7.11, θθ 2 = LL/TT 2. For PID control, a derivative filter TT f = 0.1TT d is used. For setpoint tracking, bb < 1 is used. Table Slätteke s regulatory tuning for an IPZ process. ler T3KK c T i / L Td / T 2 1 PI (3θ + 1) PID (3θ + 1) θ θ θ2 2 2 θ θ ( θ ) 2 2 θ2 2 θ2 θ θ ( ) KEH Dynamics and 7 45

46 7. PID lers 7.7 ler design by direct synthesis In the previous sections, equations for controller tuning have been given for firstand second-order no-zero systems. The equations are usually the result of optimization of some criterion that is considered to imply good control. However, what is good control varies from case to case depending on the compromise between stability and performance. A drawback of the tuning equations is that the user cannot influence the tuning according to his/her opinion of good control. In this section, a method is introduced whereby the user can influence the controller tuning in a systematic way according to his/her opinion of good control more model types than in previous sections can be handled, e.g., systems with a zero KEH Dynamics and 7 46

47 7. PID lers 7.7 ler tuning by direct synthesis Closed-loop transfer functions Consider the closed-loop V() s system in the figure with the G d () s + following transfer functions: () GG ss process being controlled G Y() s c () s + G() s Rs+ GG c ss controller GG d ss disturbance system Block diagram of closed-loop system Standard block-diagram algebra gives where YY = GGGG c 1+GGGG c RR + GG d 1+GGGG c VV (7.48) GG r = GGGG c, GG 1+GGGG v = GG d (7.49,50) c 1+GGGG c are the closed-loop transfer functions from the setpoint RR and the disturbance VV to the output YY. The user can specify the desired GG r for setpoint tracking or GG v for regulatory control. For setpoint tracking, the required controller is given by GG c = 1 GG GG r (1 GG r ) (7.51) KEH Dynamics and 7 47

48 7. PID lers 7.7 ler tuning by direct synthesis Low-order minimum-phase systems First-order system A strictly proper first-order system without a time delay has the transfer function GG = KK TTTT+1 Assume that we want the controlled system to behave as a first-order system with the time constant TT r. Then, GG r = 1 GG, which gives TT r ss+1 Substitution of (7.52) and (7.53) into (7.51) gives GG c = TTTT+1 KK 1 = TT TT r ss KKTT r which is a PI controller with the parameters r = 1 1 GG r TT r ss TTTT (7.52) (7.53) (7.54) KK c = TT KKTT r, TT i = TT (7.55) Here, TT r is a design parameter, by which the performance of the control system can be affected. KEH Dynamics and 7 48

49 7.7 ler tuning by direct synthesis Low-order minimum-phase systems Second-order system with no zero A second-order system with no zero and no time delay has the transfer function GG ss = KKωω n 2 ss 2 +2ζζωω n ss+ωω n 2 (7.56) Even if the uncontrolled system is of second order, we can specify the controlled system to be of first order. Substitution of (7.53) and (7.56) into (7.51) then gives GG c = ss2 +2ζζωω n ss+ωω n 2 KKωω n 2 1 TT r ss = 2ζζ KKωω n TT r 1 + ωω n 2ζζss + ss 2ζζωω n (7.57) which is an ideal PID controller with the parameters KK c = 2ζζ, TT KKωω n TT i = 2ζζ, TT r ωω d = 1 (7.58) n 2ζζωω n Also here, TT r is a design parameter which only affects the controller gain. KEH Dynamics and 7 49

50 7.7 ler tuning by direct synthesis Low-order minimum-phase systems Overdamped second-order system with a LHP zero An overdamped second-order system with a zero in the left half of the complex plane (LHP) has the transfer function GG ss = KK(TT 3ss+1) (TT 1 ss+1)(tt 2 ss+1), TT ii 0 (7.59) We can specify the controlled system to be of first order. Substitution of (7.53) and (7.59) into (7.51) gives GG c = (TT 1ss+1)(TT 2 ss+1) KK(TT 3 ss+1) 1 = 1 TT r ss KKTT r ss TT 1 TT 2 ss 2 + TT 1 +TT 2 ss+1 TT 3 ss+1 or where = TT KKTT r ss 1 + TT 2 TT 3 ss + TT 1TT 2 TT 1 +TT 2 TT 3 TT 3 ss 2 TT 3 ss+1 GG c = KK c TT i ss + TT dss TT f ss+1 (7.60) KK c = TT 1+TT 2 TT 3 KKTT r, TT i = TT 1 + TT 2 TT 3, TT d = TT 1TT 2 TT 1 +TT 2 TT 3 TT 3, TT f = TT 3 (7.61) This is a PID controller with a derivative filter. KEH Dynamics and 7 50

51 7. PID lers 7.7 ler tuning by direct synthesis High-order minimum-phase systems A high-order minimum-phase system with real poles and zeros, but with no time delay, has the transfer function GG = KK nn+mm jj=nn+1 nn ii=1 (TTjj ss+1) (TT ii ss+1), TT ii > 0, TT jj > 0, nn > 2 (7.62) If nn = 3 and mm = 0 or 1, a closed-loop system of second order can be obtained by a full PID controller. If nn > 3, it is not possible to obtain a closed-loop system of lower order than 3 by a PID controller and an exact design by specifying GG r is thus not practical. In the case of nn > 3, two possibilities are to specify a closed-loop system of first or second order and then to first calculate a GG c according to (7.51), then to approximate GG c by a PID controller; first approximate GG by a model of at most third order, then to calculate the PID controller according to (7.51). In Section 7.9, the latter approach will be described. KEH Dynamics and 7 51

52 7. PID lers 7.7 ler tuning by direct synthesis Second-order system with RHP zero A second-order system with real poles and a right half-plane (RHP) zero has the transfer function GG ss = KK( TT 3ss+1) (TT 1 ss+1)(tt 2 ss+1), TT ii 0 (7.63) Now division by GG in (7.51) will result in an unstable controller with a RHP pole if GG r is chosen as in the previous sections. One possible solution is to approximate the unstable controller by a stable controller. This tends to result in too aggressive control because the controller is then designed as if there were no RHP zero in GG. Another solution is to include the same RHP zero in GG r as in GG ; it will then be cancelled out in (7.51) and the controller will automatically be stable. This means that the choice of GG r is restricted, but otherwise the control performance tends to be as expected. In this section, the latter approach is used. KEH Dynamics and 7 52

53 7.7 ler tuning by direct synthesis Second-order system with RHP zero Closed-loop system of first order The closed-loop transfer function is chosen as GG r = TT 3ss+1 TT r ss+1, which gives GG r = TT 3ss+1 1 GG r (TT r +TT 3 )ss (7.64) Substitution of (7.63) and (7.64) into (7.51) gives GG c = (TT 1ss+1)(TT 2 ss+1) KK 1 (TT r +TT 3 )ss = TT 1+TT 2 KK(TT r +TT 3 ) which is a PID controller with the parameters TT 1TT 2 ss (7.65) TT 1 +TT 2 ss TT 1 +TT 2 KK c = TT 1+TT 2 KK(TT r +TT 3 ), TT i = TT 1 + TT 2, TT d = TT 1TT 2 TT 1 +TT 2 (7.66) KEH Dynamics and 7 53

54 7.7 ler tuning by direct synthesis Second-order system with RHP zero Closed-loop system of second order A first-order system with a zero is proper, but not strictly proper. If a zero is present, a strictly proper system has to be at least second order. Hence, a more natural choice for GG r is GG r = ( TT 3ss+1)ωω r 2 ss 2 +2ζζ r ωω r ss+ωω r 2, which gives GG r 1 GG r = ( TT 2 3ss+1)ωω r ss(ss+2ζζ r ωω r +TT 3 ωω 2 r ) (7.67) To simplify the derivation of controller parameters, we define TT f = 1/(2ζζ r ωω r + TT 3 ωω r 2 ) (7.68) Substitution of (7.63) and (7.67) into (7.51), gives, with (7.68), GG c = (TT 1ss+1)(TT 2 ss+1)tt f ωω r 2 KK TT f ss+1 ss = TT fωω r 2 KKKK TT 1 TT 2 ss 2 + TT 1 +TT 2 ss+1 TT f ss+1 (7.69) Analogously with the derivation of (7.61), this gives the PID controller parameters KK c = TT fωω r 2 KK (TT 1 + TT 2 TT f ), TT i = TT 1 + TT 2 TT f, TT d = TT 1TT 2 TT 1 +TT 2 TT f TT f (7.70) where TT f, given by (7.68), is the derivative filter time constant in a PID controller (7.60). KEH Dynamics and 7 54

55 7.7.4 Second-order system with RHP zero Closed-loop system of 2 nd order Choice of closed-loop system parameters In (7.67), there are two design parameters, the relative damping ζζ r, and the undamped natural frequency ωω r. The meanings of these parameters are discussed in Section 5.3, especially Subsection The choice of design parameters can be simplified in the following two ways. Let GG r have two equally large real poles at 1/TT r. This corresponds to ζζ r = 1 and ωω r = 1/TT r, which for (7.68) gives TT f = TT r 2 2TT r +TT 3 (7.71) Let GG r have real poles at 1/TT r and 1/TT 3. This corresponds to ζζ r = 0.5(TT r + TT 3 )ωω r and ωω r = 1/ TT r TT 3, which for (7.68) gives TT f = TT rtt 3 TT r +2TT 3 (7.72) KEH Dynamics and 7 55

56 7. PID lers 7.7 ler tuning by direct synthesis First-order system with a time delay To illustrate how systems with a time delay can be handled by direct synthesis, a first-order system with a time delay will be studied. Such a system has the transfer function GG ss = KK TTss+1 e LLLL (7.73) Calculation of a controller by (7.51) will then result in a controller containing a time delay there is no practical way to avoid this by the choice of GG r. There are methods to implement a controller resulting from (7.51) (see Section 7.8), but not by a regular PID controller. If a PID controller is desired, the time delay has to be approximated in some way. KEH Dynamics and 7 56

57 7.7 ler tuning by direct synthesis First-order system with a delay Time-delay approximation in the model A standard way of approximating a time delay is to use a Padé approximation. A first-order Padé approximation gives the model GG ss = e LLLL 1 0.5LLLL 1+0.5LLLL KK( 0.5LLLL+1) (TTss+1)(0.5LLLL+1) (7.74) (7.75) A natural choice for GG r is then GG r = 0.5LLLL+1 (TT r ss+1)(0.5llll+1), which gives GG r 1 GG r = 0.5LLLL+1 ss(0.5tt r LLss+TT r +LL) (7.76) Substitution of (7.75) and (7.76) into (7.51) gives a PID controller with the parameters KK c = TT+0.5LL TT f KK(TT r +LL), TT i = TT + 0.5LL TT f, TT d = 0.5LLLL, TT TT+0.5LL TT f = 0.5LLTT r f TT r +LL (7.77) Here, TT f is the time constant of a derivative filter in the PID controller (7.60). KEH Dynamics and 7 57

58 7.7 ler tuning by direct synthesis First-order system with a delay Time-delay approximation in the controller If e LLLL is retained in the model, it also has to be part of GG r, because it is impossible for the closed-loop system to have a shorter time-delay than the uncontrolled system. If GG r is chosen to be first order with a time delay GG r = 1 TT r ss+1 e LLLL, which gives GG r 1 GG r = e LLLL TT r ss+1 e LLLL (7.78) Substitution of (7.73) and (7.78) into (7.51) gives GG c = TTTT+1 KK(TT r ss+1 e LLLL ) (7.79) Unfortunately, this controller cannot be implemented by a PID controller in a regular feedback loop. In order to do that, the time delay in (7.79) has to be approximated by a rational expression. If the approximation (7.74) is used, the controller parameters will be as in (7.77). The simpler approximation e LLLL 1 LLLL gives a PI controller with KK c = TT, TT KK(TT r +LL) i = TT (7.80) KEH Dynamics and 7 58

59 7. PID lers 7.8 Internal model control Internal model control (IMC) is closely related to direct synthesis (DS). As in DS, a model of the system to be controlled is explicitly built into the controller, but in a different way. An advantage with IMC is that it is easier to implement more complex control laws than regular PID controllers. For example, the controller transfer function (7.79) can easily be implemented exactly with IMC. Even if the controller design is based on IMC, it is often desirable to implement the controller as a regular PID controller. In such cases, the IMC approach offers better possibilities to deal with robustness issues than DS. KEH Dynamics and 7 59

60 7. PID lers 7.8 Internal model control The IMC structure Consider the closed-loop system in the figure with the following transfer functions: GG ss true process GG ss process model GG IMC ss a controller GG d ss disturbance system Standard block diagram algebra gives UU = GG IMC (EE + GGUU) from which Es () The IMC structure. UU = GG EE c = II GG IMC GG 1 1 GG GG IMC = GG IMC II GGGG IMC = IMC (7.81) 1 GGGG IMC Assume that GG IMC = GG 1 GG f (7.82) Gs ˆ () where GG f is a filter. Substitution of (7.82) into (7.81) gives Gs () GG f (1 GG f ) GG c = GG 1 GG f II GG 1 f = 1 GG If the filter is chosen as GG f = GG r (and GG = GG), this is the same as (7.51)! (7.83) KEH Dynamics and 7 60

61 7. PID lers 7.8 Internal model control Handling of time delays without approximation Consider a system modelled as a first-order system with a time delay, i.e., GG = KKe LLLL /(TTTT + 1). Choose the IMC filter as GG f = e LLLL /(TT r ss + 1). Substitution into (7.82) now gives GG IMC = 1 KK TTTT+1 TT r ss+1 = 1 KK 1 + TT TT r ss (7.84) TT r ss+1 which is a PD controller with a derivative filter having the parameters KK cc = 1/KK, TT d = TT TT r, TT f = TT r. Substitution of (7.84) and the model GG into (7.81) gives GG c = TTTT+1 KK(TT r ss+1 e LLLL ) which is identical with (7.79). The difference is that (7.85) can be implemented exactly by the IMC structure without time-delay approximation. Note that there is no integration in GG IMC, but the feedback of GG in the IMC structure introduces integration if GG IMC is calculated using the same GG in (7.82); integration is achieved even if GG GG. Exercise. Calculate the closed-loop transfer function GG r when GG GG. Show that there will be no steady-state error, i.e., that GG r 0 = 1. (7.85) KEH Dynamics and 7 61

62 7. PID lers 7.8 Internal model control The predictive character of the IMC structure The previous block diagram of the IMC structure is drawn to emphasize how GG IMC combined with the feedback of GG is equivalent to GG c. The block diagram can also be drawn to emphasize the predictive character of the IMC structure, as shown below. (Note that the two diagrams are completely equivalent.) The control signal is an input to the real system GG and the model GG. GG predicts the output YY, which is compared with the true output YY. Only the prediction error EE = YY YY is fed back, not the entire YY. The latter property is a clear advantage in controller design. If GG = GG (i.e., EE = 0) GG r = GGGG IMC (7.86) which means that the closedloop transfer function depends linearly on GG IMC making design of GG IMC easier than design of GG c. Gs () ˆ () Gs Predictive nature of IMC structure. KEH Dynamics and 7 62