2.35 Tuning PID Controllers

Transcription

1 2.35 Tuning PID Controllers P. W. MURRILL (1970) P. D. SCHNELLE, JR. (1985) B. G. LIPTÁK (1995) J. GERRY, M. RUEL, F. G. SHINSKEY (2005) In orer for the reaer to fully unerstan the content an concepts of this section, it is avisable to first become familiar with some basic topics. These inclue gains, time lags an reaction curves (Section 2.22), the PID control moes (Section 2.3), feeback an feeforwar control (Section 2.9), an relative gain calculations (Section 2.25). Controllers are esigne to eliminate the nee for continuous operator attention when controlling a process. In the automatic moe, the goal is to keep the controlle variable (or process variable) on set point. The controller tuning parameters etermine how well the controller achieves this goal when in automatic moe. DISTURBANCES The purpose of a controller is to keep the controlle variable as close as possible to its set point at all times. How well it achieves this objective epens on the responsiveness of the process, its control moes an their tuning, an the size of the isturbances an their frequency istribution. Sources Disturbances arise from three ifferent sources: set point, loa, an noise. Noise is efine as a ranom isturbance whose frequency istribution excees the banwith of the control loop. As such, the controller has no impact on it, other than possibly amplifying it an passing it on to the final actuator, which can cause excessive wear an ultimate failure. Set point an loa changes affect the behavior of the control loop quite ifferently, owing to the ynamics in their path. A controller tune to follow set point changes tens to respon sluggishly to loa variations, an conversely a controller tune to correct isturbances tens to overshoot when its set point is change. Set Point The set point is the esire value of the controlle variable an is subject to ajustment by the operator. In a continuous process plant, most of the control loops operate as regulators, having a set point that remains unchange for ays an even months at a time. Examples of variables hel at constant set points are rum-level an steam temperature of a boiler, most pressure an level variables, ph of process an effluent streams, most prouct-quality variables, an most temperature loops. Setpoint response is of no importance to these loops, but they must conten with loa upsets minute by minute. In fact, the only loops in a continuous plant that must follow set-point changes are flow loops. Batch plants have frequent transitions between steay states, some of which require rapi response to set-point changes with minimal overshoot. However, some of these changes are large enough to saturate the controller, particularly at startup. This can cause integral winup, which requires special means of prevention to overcome. The Loa Only pure-batch processes where no flow into or out of the process takes place operate at constant loa, an that loa is zero. All other processes can expect to encounter variations in loa, which are principally changing flow rates entering an leaving vessels. A liqui-level controller, for example, manipulates the flow of one liqui stream, while other streams represent the loa. Feewater flow to a boiler is manipulate to control rum level an must balance the combine flows of steam an blowown leaving to keep level at set point. The loa changes frequently an often unpreictably, but the set point may never change. In a typical temperature control loop, the loa is the flow of heat require to keep temperature constant. Liqui entering a heat exchanger will require a certain flow of steam to reach a controlle exit temperature. Variations in liqui flow an inlet temperature will change the eman for steam flow manipulate to keep exit temperature at set point. Dynamics The term process ynamics can refer to capacitance, inertia, resistance, time constant, ea time or their combinations. There is no ynamics involve with changing the set point, unless intentionally place there for the purpose of filtering the set point. 414

2 2.35 Tuning PID Controllers 415 Set point SP filler Controller Loa Loa ynamics Process Loop ynamics Noise Controlle variable FIG. 2.35a Loa variables always pass through the ominant ynamic elements. However, there is always ynamics in the loa path. Loa variables are principally the flow rates of streams similar to those manipulate by the controller. Therefore the ynamics in their path to the controlle variable are similar an in most cases ientical to the ynamics in the loop itself. Figure 2.35a presents all the essential elements of a control loop, showing its isturbance sources an ynamics. Most frequently, the ynamics are common to both the loa isturbance an the controller output, meaning that the loa an manipulate streams enter the process at the same point. An example woul be the control of composition of a liqui at the exit of a blener, where both the manipulate an loa streams making up that blen are introuce at a common entry point. Less often encountere is the process where the ynamics in these two paths iffer. An example of this is a shellan-tube heat exchanger, wherein the temperature of a liqui leaving the tube bunle is controlle by manipulating the flow of steam to the shell. The shell may have more heat capacity than the tubes, causing the temperature to respon more slowly to a change in steam flow than to a change in liqui flow. Nonetheless, these two ominant lags will typically not iffer greatly. Steps are also quite common in inustry, representing conitions cause by suen startup an shutown of equipment; starting an stopping of multiple burners, pumps an compressors; an capacity changes of reciprocating compressors. If a control loop can respon aequately to a step isturbance, then a ramp or exponential isturbance will have less of an impact on it. The step is also the easiest test to apply, requiring only a size estimate, an can be aministere manually. Pulses require uration estimates, an oublet pulses require balancing. Step changes in set point are the usual isturbance applie to test or tune a loop, even for loops that operate at constant set point. Figure 2.35b illustrates a step response with 1/4 ecay ratio. The usual result of tuning a controller for set point response is to reuce its performance to variations in loa. Therefore, the effectiveness of a controller an its tuning as a loa regulator nee to be etermine by simulating a step loa change. Simulating a Loa Change Some controllers have an ajustable output bias. An acceptable simulation of loa change, when the controller is in automatic an at steay state, is a step change in the value of this bias. The value of the controller output prior to the step is an inication of the current plant loa because the loop was in a steay state. The step in bias in that case moves the controller output to another value, which isturbs the controlle variable an causes the controller to integrate back to its previous steaystate output. Alternatively, controllers that can be transferre bumplessly between manual an automatic moes (most o all Output Step Responses Step testing is recommene for all control loops where the frequency content of the isturbance variables is not specifie. There are cases of perioic isturbances, an they can pose special problems for control loops that themselves are capable of resonating at a particular perio. They are foun principally in cascae loops an in process interactions where controllers manipulate valves in series or in parallel. These are consiere in other sections of this work. Another perio isturbance is the cyclic operation of such cleaning evices as soot-blowers. For the general case, the step isturbance is the most ifficult test for the controller in that it contains all frequencies, incluing zero. In fact, the frequency content of the step is ientical to that of integrate white noise therefore, it is an excellent test for loops subject to ranom isturbances b P a a/b = 1/4 Time FIG. 2.35b Step response curve of a control loop tune for 1/4 ecay ratio.

3 416 Control Theory shoul) allow simulating a loa change by using that feature. This is one by waiting until the loop is at steay state an on set point (zero eviation). At that point switching to the manual moe an stepping the output by the esire amount in the esire irection, an immeiately (before a eviation evelops) transferring back to the automatic moe. This proceure can be followe for all but the fastest loops, such as flow loops. For them, a step in set point is acceptable, both because flow loops must follow set-point changes, an because for them, set-point tuning gives acceptable loa response. Comparing Set-Point an Loa Responses The steay-state process gain of a flow loop is typically between 1 an 2, as inicate by the controller output being between 50 an 100% when the flow measurement is at full scale. The proportional gain of a typical flow controller is in the range of 0.3 to 1.0, with the higher number associate with the process that has the lower steay-state gain. Therefore, the proportional loop gain for a typical flow loop is in range of 0.6 to 1.0. As a result, a step change in set point will move the controller output approximately the correct amount to prouce the same change in flow, by proportional action alone, that gives excellent set-point response. This is not the case for other loops. Level has the opposite behavior. To maintain a constant level, the controller must match the vessel s inflow an outflow precisely. Changing the set point will cause the controller to change the manipulate flow, but only temporarily when the level reaches the new set point, the manipulate flow must return to its original steay-state value. Therefore, no steay-state change in output is require for a level controller to respon to a set-point change. The Integrate Error (IE) sustaine by a controller following a isturbance varies irectly with the change in output between its initial an final steay states. In response to a set-point change, the level controller has the same initial an final steay-state output values an hence sustains zero integrate error. As a result, the error that is integrate while the level is approaching the new set point will be matche exactly by an equal area of overshoot. In other wors, set-point overshoot is unavoiable in a level loop unless set-point filtering is provie. Most other processes, such as temperature, pressure, an composition, have steay-state gains higher than those of a flow process. But more importantly, they are also ominate by lags, which allows the use of a higher controller proportional gain for tight loa regulation. When this high proportional gain is multiplie by the process steay-state gain, the resulting loop gain can be as high as 5 to 10 or more. A set-point step then moves the controller output far more than require to rive the controlle variable to the new set point, proucing a large overshoot. To minimize set-point overshoot, the controller must be etune, with lower gain Controlle variable 0 Set No filter SP filter FIG. 2.35c Set-point tuning slows loa recovery for lag-ominant processes. an longer integral time than is optimum for loa regulation, or a filter must be applie to the set point. Figure 2.35c compares responses to steps in set point an loa for a process with istribute lag such as a ashe exchanger, istillation column, or stirre tank. The time scale is normalize to Στ, which is the time require for the istribute lag to reach 63.2% of the full response to a step input in the open loop. It is also the resience time of liqui in a stirre tank. If the PID settings are ajuste to minimize the Integrate Absolute Error (IAE) to the set point change, the ashe response curve is prouce (SP tuning). Note that following a step change in loa, the return to set point is sluggish. This is commonly observe with lag-ominant processes. The PID settings that prouce the minimum-iae loa response, shown in black (no filter), result in a large set-point overshoot, however. Set-Point Filtering Loa 1 2 Time, t/st Loa tuning SP tuning 3 4 If optimum loa rejection is esire, without the large setpoint overshoot that it prouces on lag-ominant processes, the proportional response to set-point changes nees to be reuce. Some PID controllers have the option to eliminate proportional action on the set point altogether. This tens to prouce a set-point unershoot, which can significantly ashe the controller response, an it shoul never be use in the seconary controller of a cascae system. (Incientally, erivative action shoul never be applie to the set point, as this always prouces overshoot.) Some controllers can reuce the controller s proportional gain when it acts on set point changes, either through the use of a lea-lag filter, or by the use of a specially structure algorithm. This ajustment allows separate optimization of set-point response, after the PID settings have been tune to optimize the controller s loa response. The filter use for the loop whose response is shown in Figure 2.35c has applie only half the controller s proportional gain to the set point.

4 2.35 Tuning PID Controllers 417 OPEN-LOOP TUNING Applying a step to the process is simple an can be use to tune the loop an to obtain a simple moel for the process. Two methos are wiely use. The first is the process reaction curve, which is not use to calculate the process moel but is use to obtain the tuning parameters for rejecting the upsets cause by loa changes. The secon metho uses the process moel by obtaining a simple process moel; the tuning parameters are calculate from this moel base on either a loa rejection or a set point change criterion. Process Reaction Curve Controlle process variable (% of full scale) L r L r R r t K Time (minutes) FIG Reaction curve of a self-regulating process, cause by a step change of one unit in the controller output. L r = t is ea time, R r is reaction rate, an K is process gain. When a process is at steay state an it is upset by a step change, it usually starts to react after a perio of time calle the ea time (Figure 2.35). After the ea time, most processes will reach a maximum spee (reaction rate), then the spee will rop (self-regulating process) or the spee will remain constant (integrating process). When tuning a loop to remove isturbances cause by loa changes, the controller must react at its maximum rate of reaction, an the strength of the reaction will correspon to the maximum spee. Hence, to tune the loop, it is not necessary to know the process moel. It is sufficient to know the ea time an the maximum spee to calculate the tuning parameters. Figure 2.35e illustrates the response of a temperature loop after a step change in the controller output. This example will be use throughout this section to illustrate the ifferences between the recommene settings arrive at by using the various tuning techniques. From this curve, it is not possible to etermine the process moel since the curve is too short to tell whether the reaction rate remains constant (integrating process) or goes own (self-regulating process). From such a test, the moel cannot be foun but the tuning parameters for loa rejection can be estimate. As can be seen from Figure 2.35e, a 10% change in CO was applie at 10 s an the temperature starte to increase 130 Process Variable s Slope = ( ) eg C (70 40) s Controller Output Time (sec) FIG. 2.35e Process reaction curve in response of a change in controller output (CO). The process variable (PV) range is 0 to 300 egc an the CO range is 0 to 100%.

5 418 Control Theory TABLE 2.35f Equations for Calculating the Ziegler Nichols Tuning Parameters for an Interacting Controller Type of Controller P (gain) I (minutes/repeat) D (minutes) 23 s secons later at 33 s. Hence the ea time (t ) is 23 s an the reaction rate (spee) is The slope is: P CO R r t PI 09. * CO 3.33 t R t r PID CO t 0.5 t R t r R = PV r t 9 eg C 100 PV 132eg C 123eg C 9eg C 300 eg C = = = t 70 s 40 s 30 s 30 s 3% = = 01.%/ s = 60.%/min 30 s 2.35(1) After the ea time (t ) an the reaction rate (R r ) have been etermine, the controller settings are calculate by using the equations in Table 2.35f. If a PI controller is to be use for the process that was teste in Figure 2.35e, the values are: P = 0.9*10%/(0.1%/s X 23 s) = 3.9 I = 3.33 * 23 s = 76.6 s = 1.28 minutes. Ziegler an Nichols recommen using the ratio of the controller output ivie by the prouct of the slope an the ea time to calculate the proportional gain. The ieal process has a small ea time an a small slope, so that the controller can aggressively manipulate the controller output to bring the process back to set point. The integral an erivative are calculate using the ea time. The proportional is calculate using the slope an the ea time. If the slope is high, then the controller gain must be small because the process is sensitive; it reacts quickly. If the ea time is long, the controller gain must be small because the process response is elaye an therefore the controller cannot be aggressive. If one can reuce the slope an the ea time of any process, it will be easier to control. One of the avantages of open-loop tuning over the close-loop tuning technique is its spee because one oes not nee to wait for several perios of oscillation uring several trial-an-error attempts. The other avantage is that one oes not introuce oscillations into the process with unpreictable amplitues. In open-loop tuning, the user selects the upset that is introuce, an it can be small. Yet another avantage is that this test can be performe prior to the installation of the control system. The isavantages are also multiple. The open-loop test is not as accurate as the close-loop one because it isregars the ynamics of the controller. Another isavantage is that the S-shape reaction curve an its inflection point are ifficult to ientify when the measurement is noisy an/or if a small step change was use. Because of the above consierations, a goo approach is to use the open-loop metho of tuning in orer to obtain the first set of initial tuning constants for a loop uring startup. Then, refine these settings once the system is operating by retuning the loop using the close-loop metho. Process Moel There are many ways to use an interpret the ea time (t ) an reaction rate (R r ) values obtaine from the open-loop tuning metho. Most open-loop methos are base on approximating the process reaction curve by a simpler system. Several techniques are available to obtain a moel. The most common approximation by far is a pure time elay (ea time) plus a first-orer lag. One reason for the popularity of this approximation is that a real-time elay of any uration can only be represente by a pure time elay because there is yet no other simple an aequate approximation. Theoretically, it is possible to use higher-than-firstorer lags plus ea time, but accurate approximations are ifficult to obtain. Thus the real process lag is usually approximate by a pure time elay plus a first-orer lag. This approximation is easy to obtain, an it is sufficiently accurate for most purposes. The process s ea time is the time perio following an upset uring which the controlle variable is not yet responing. The time constant is a perio between the time when a response is first etecte an the time when the response has reache 63.2 % of its final (new steay-state) value. The time constant is also the time it woul take for the controlle variable to reach its final value if the initial spee were maintaine. Bump Tests Figure 2.35g shows the Ziegler Nichols proceure to approximate that process reaction curve with a firstorer lag plus a time elay. The first step is to raw a straightline tangent to the process reaction curve at its point of maximum rate of ascent (point of infection). Although this is easy to visualize, it is quite ifficult to o in practice. This is one of the main ifficulties in this proceure, an a consierable number of errors can be introuce at this point. The slope of this line is terme the reaction rate R r. The time between the instant when the bump was applie an the time at which this line intersects the initial value of the controlle variable prior to the test is the ea time, or transport time elay t. Figure 2.35g illustrates the etermination of these values for a one-unit step change ( CO) in the controller output

6 2.35 Tuning PID Controllers 419 Process variable τ = s Controller output Time (sec) FIG. 2.35g Maximum slope curve, Fit 1. (manipulate variable) to a process. If a ifferent-size step change in controller output was applie, the value of τ woul not change significantly. However, the value of R r is essentially irectly proportional to the magnitue of the change in controller output. Therefore, if a two-unit change in output was use instea of a one-unit change, the value of R r woul be approximately twice as large. For this reason the value of R r use in Equation 2.35(2) or any other must be the value that woul be obtaine for a one-unit change in controller output. In aition to the ea time an reaction rate, the value of the process gain K must also be etermine as follows: final steay-state change in controlle variable (%) K = change in controller ouput (control unit) 2.35(2) There is a secon metho to etermine the pure time elay plus the first-orer lag approximation. In orer to istinguish between these two methos, they will be calle Fit 1 (escribe in Figure 2.35g) an Fit 2 (escribe in Figure 2.35h). The only ifference between these two is in how the firstorer time constant is obtaine. In case of Fit 2, the time constant of the process is etermine as the ifference between the time when the ea time ens an the time when the controlle variable has covere 63.2% of the istance between the pre-test steay state an the new one. The ea time etermination by both fits is the same an was alreay escribe. Another metho to etermine the ea time is to measure the time when the PV moves by 2% of the total change. The first-orer lag time constants are given by: Fit1: τ = F KR / 1 r = 1.5/0.1%/s = 150 s (same slope as previous section) 2.35(3) Fit 2: τ F = t t = 155 s 33 s = 122 s 2.35(4) % 0 In Equation 2.35(4), t 63.2% is the time necessary to reach 63.2 % of the final value, an t 0 is the time elapse between the CO change an the beginning of the PV change. Note that the parameters for Fit 1 are base on a single point on the response curve, which is the point of maximum rate of ascent. However, the parameters obtaine with Fit 2 are base on two separate points. Stuies 3 inicate that the open-loop response base on Fit 2 always provies an approximation that is as goo or better than the Fit 1 approximation. A typical curve resulting from the above proceure is shown in Figure 2.35i. The response shown in Figure 2.35i resulte from a tenunit change in controller output. For ifferent step changes, K an R r must be ajuste accoringly. From a curve such as in Figure 2.35i, a number of parameters can be etermine. The controller settings are calculate from the equations in Table 2.23j: Table 2.35k compares the results obtaine in terms of process gain (K), time constant (τ (s)), ea time (t (s)), an the resulting controller gain (K c ) an integral time setting (T i (min)) of a PI controller. In the following iscussion, the process moel etermine by the secon fit will be use to compare against a variety of tuning criteria.

7 420 Control Theory 170 Process variable egc * 45 egc = 148 egc Process gain = ( ) egc (50 40)% = 4.5 egc = 100*(4.5/300)% % % = s 155 s Controller output Time (sec) 800 FIG. 2.35h Bump test, Fit 2 curve. Comparing the Tuning Methos One of the earliest methos for using the process reaction curve was propose by Ziegler an Nichols. When using their process reaction curve metho, which was escribe in connection with Figure 2.35e, only R r an t or t 0 must be etermine. Using these parameters, the empirical equations to be use to preict controller settings to obtain a ecay ratio of 1/4 are given in Table 2.35j in terms of K, t, an τ. In eveloping their equations, Ziegler an Nichols consiere processes that were not self-regulating. To illustrate, Process variable Controller output Time (sec) 800 FIG. 2.35i A typical reaction curve using the ea time an time constant obtaine by a bump test.

8 2.35 Tuning PID Controllers 421 TABLE 2.35j Ziegler Nichols Recommenations to Obtain the Tuning Parameters for an Interacting Controller Base on the Reaings Calculate from a Moel Type of Controller P (gain) I (minutes/repeat) D (minutes) P τ K t PI τ 09. * 3.33 t K t PID τ t 0.5 t K t consier the level control of a tank with a constant rate of liqui outflow. Assume that the tank is initially operating at constant level. If a step change is mae in the inlet liqui flow, the level in the tank will rise until it overflows. This process is not self-regulating. On the other han, if the outlet valve opening an outlet backpressure are constant, the rate of liqui removal increases as the liqui level increases. Hence, in this case, TABLE 2.35k The Tuning Setting Recommenations for a PI Controller Resulting from the Three Methos of Testing Describe Testing Metho Use K τ (s) t (s) K c T i (min) Reaction curve Process moel Fit1, ZN Process moel Fit 2, ZN the level in the tank will rise to some new position but will not increase inefinitely, an therefore system will be selfregulating. To account for self-regulation, Cohen an Coon 4 introuce an inex of self-regulation µ efine as: µ = RL r r / K 2.35(5) Note that this term can also be etermine from the process reaction curve. For processes originally consiere by Ziegler an Nichols, µ equals zero an therefore there is no selfregulation. To account for variations in µ, Cohen an Coon suggeste the equations given in Table 2.35l in terms of t an τ. In case of a proportional control, the requirement that the ecay ratio be 1/4 is sufficient to ensure a unique solution, but for the case of proportional-plus-reset control, this restraint is not sufficient. Another constraint in aition to the 1/4 ecay ratio can be place on the response to etermine the unique values of K c an T i. This secon constraint can be to require that the control area of the response be at its minimum, meaning that the area between the response curve an the set point be the smallest. This area is calle the error integral or the integral of the error with respect to time. With the proportional-plus-reset-plus-rate controller (PID), the same problem of not having a unique solution exists even when the 1/4 ecay ratio an the minimum error integral constraints are applie. Therefore, a thir constraint must be chosen to obtain a unique solution. Base on the work of Cohen an Coon, it has been suggeste that this new constraint coul have a value of 0.5 for the imensionless group K c Kt /τ. The tuning relations that will result from applying these three constraints are given in Table 2.35l. This metho has been referre to as the 3C metho. 5 7 Greg Shinskey suggeste a variation to the above, where the proportional gain an the integral time are increase. TABLE 2.35l Comparison of Equations Recommene by Ziegler Nichols, Shinskey, Cohen Coon, an 3C for the Determination of the Tuning Settings for PID Controllers Ziegler Nichols Shinskey Cohen Coon 3C P KK c = ( t /τ ) 1.0 ( t / τ ) ( t / τ ) ( / τ ) t P KK c = 0.9( t /τ ) ( t / τ ) ( t /) τ ( / τ ) t I Ti = 3.33( t / τ ) 4.0( t / τ τ ) 3.33( t/ τ )[ 1+ ( t/ τ )/ 11] ( t / τ ) ( t / τ ). P KK c = 1.2( t /τ ) ( t / τ ) ( t / τ ) ( t / τ ) I Ti = 2.0( t / τ ) 1.6( t / τ ) 2.50( t/ τ )[ 1+ ( t/ τ )/ 5] τ ( t / τ ) 0.740( t / τ ) D T = 0.5( t τ / τ ) 0.6( t / τ ) 0.37( t / τ ) ( t / τ ) 0.365( t / τ )

9 422 Control Theory Integral Criteria Tuning 8 Table 2.35m provies the controller settings that minimize the respective integral criteria to the ratio t /τ. The settings iffer if tuning is base on loa (isturbance) changes as oppose to set point changes. Settings base on loa changes will generally be much tighter than those base on set point changes. When loops tune to loa changes are subjecte to a set point change, a more oscillatory response is observe. Which Disturbance to Tune for With tuning parameters calculate for loa rejection, the integral time (T i ) an erivative time (T ) will epen mostly on the ea time (t ) of the process. In contrast, if the tuning parameters are calculate for a set-point change, the integral time will be longer an the erivative time will be shorter, an they will epen mostly on the time constant of the process. The relationship between the controller settings base on integral criteria an the ratio t 0 /τ is expresse by the tuning relationship given in Equation 2.35(6). 2.35(6) where Y = KK c for proportional moe, τ/t i for reset moe, T /τ for rate moe; A, B = constant for given controller an moe; t 0, τ = pure elay time an first-orer lag time constant. Hence, using these equations, Lamba Tuning T Y A t B 0 = τ K c = A K t 0 τ 1 A 0 T i t = τ τ 2.35(7) 2.35(8) 2.35(9) Lamba tuning originate from Dahlin in 1968; it is base on the same IMC theory as MPC, 4,5 is moel-base, an uses a moel inverse an pole-zero cancellation to achieve the esire close-loop performance. Lamba tuning is a metho to tune loops base on pole placement. This metho ensures a efine response after a set-point change but is generally too sluggish to properly reject isturbances. Promoters for this metho often claim that all loops shoul be tune on the basis of Lamba tuning. Doing so, the controllers are almost in iling moe an B B = A t 0 τ τ B TABLE 2.35m Tuning Settings for Loa an Set Point Disturbances Loa Change IAE P Set Point Change A B A B P I P I D ITAE P P I P I D ISE P P I P I D ZN P P I P I D CCC P P I P I D Shinskey P P I P I D to 1 ecay P Critical amping (no overshoot, maximum spee) P P I T i =1.16τ P I T i = τ D

10 2.35 Tuning PID Controllers 423 when the process loa changes or other isturbance occurs, the time to eliminate this isturbance is quite long for most processes, because the integral time selecte equals the process time constant. The promoters also suggest the use of a close-loop time constant, which is three times the process time constant. Doing so, the response time in the automatic moe will be three times longer than in manual. Therefore, the response time will be slower in automatic. This is aequate if no isturbance occurs but if no isturbance occurs, the control loop is not neee. Lamba tuning refers to all tuning methos where the control loop spee of response is a selectable tuning parameter. The close loop time constant is referre to as Lamba (λ). Therefore, following a set-point change, the PV will reach set point as a first-orer system (same type of response as in the manual moe when the CO is change). Lamba tuning has been wiely use in the pulp an paper inustry, but control specialists are starting to realize that it is often too sluggish to hanle isturbances. For a firstorer plus ea time moel K c T i = τ 2.35(10) 1 Ti = K λ + t 2.35(11) where λ = close loop time constant; it is recommene to use λ = 3τ. TABLE 2.35n The Tuning Setting Recommenations for a PI Controller Resulting from the Criteria Liste Criteria Tuning K c T i (min) Loa change criteria Ziegler Nichols CCC Shinskey IAE SP change criteria IAE Lamba The performance of lamba tuning is unacceptable for correcting upsets cause by loa changes if the process time constant is larger than the ea time. This is the case with pressure, level, an temperature control applications. With flow loops, the results are similar to other methos since the time constant is in the same orer of magnitue as the ea time. In Table 2.35n, Fit 2 (Figure 2.35h) will be use as the reference to compare the process moels foun using the ifferent tuning criteria. For loa an set-point responses of the ifferent tuning techniques, see Figures 2.35o, p, an q. Ajusting Robustness To remove oscillations in a control loop, hence to increase the robustness, it is necessary to give Process variable IAE SP IAE ZN CCC IAE = SSE= Lamba ExperTune SP ExperTune Loa Shinskey Controller output Time (sec) FIG. 2.35o Loa responses of the ifferent tuning techniques (example).

11 424 Control Theory Process variable IAE = SSE = CCC IAE ZN Shinskey ExperTune Loa IAE (SP) ExperTune Loa Lamba Controller output Time (sec) FIG. 2.35p Set-point responses of the ifferent tuning techniques (example). up some performance. By reucing the proportional gain in a control loop, the robustness will be increase an the oscillations will be reuce or remove. As a rule of thumb, iviing the proportional gain by a factor of two will eliminate the oscillations; iviing again the proportional gain by a factor of two will remove most of the overshoot. (For more on robustness, see Section 2.26.) Digital Control Loops Digital control loops iffer from continuous control loops by having the continuous controller replace by a sampler, a iscrete control algorithm calculate by the computer, an a hol evice (usually a zero-orer hol). In such cases Moore et al. have shown that the open-loop tuning methos presente Process variable CCC IAE (L) ZN IAE (SP) Shinskey Lamba 120 Controller output Time (sec) FIG. 2.35q Set-point responses of the ifferent tuning techniques (example) with a controller where the P is applie only on PV changes.

12 2.35 Tuning PID Controllers 425 previously may be use, consiering that the ea time use is the sum of the true process ea time an one-half of the sampling time, as expresse by Equation 2.35(12): Output t = t + T/ (12) Curve A where T is the sampling time. t 0 is use in the tuning relationships instea of t 0. (Section 2.38 eals with the subject of controller tuning by computer.) Curve B Curve C CLOSED-LOOP RESPONSE METHODS As has been iscusse previously, in the open-loop metho of tuning, the controller oes not even have to be installe in orer for the controller settings to be etermine. When the closeloop metho is use, the controller is in automatic. Describe below are the two most common close-loop methos of tuning, the ultimate metho an the ampe oscillation metho. Ultimate Metho One of the first methos propose for tuning controllers was the ultimate metho, reporte by Ziegler an Nichols 1 in This metho is calle the ultimate metho because its use requires the etermination of the ultimate gain (sensitivity) an the ultimate perio. The ultimate gain K u is the maximum allowable value of gain (for a controller with only a proportional moe) for which the system is stable. The ultimate perio is the response s perio with the gain set at its ultimate value (Figure 2.35r). In orer for a close loop to isplay a quarter of the amplitue amping (Figure 2.35b), its loop gain must be at 0.5. This means that the prouct of the gains of all the components in the loop compose by the process gain (G p ), the sensor gain (K s ), the transmitter gain (K t ), the controller gain (K c = 100/PB), an the control valve gain (K v ) must be at 0.5. When the loop is in sustaine, unampene oscillation, the gain prouct of the loop is 1.0 an the amplitue of cycling is constant (Curve B in Figure 2.35r). The perio when the close loops oscillate epens mostly on the amount of ea time in the loop. The perio of oscillation in flow loops is 1 to 3 secons; for level loops, it is 3 to 30 secons (sometimes minutes); for pressure loops, 5 to 100 secons, for temperature loops; 0.5 to 20 minutes; an for analytical loops, from 2 minutes to several hours. When controlle by analog controllers (no ea time ae by sampling), plain proportional loops oscillate at perios ranging from two to five ea times, PI loops oscillate at perios of three to five ea times, an PID loops at aroun three ea time perios. The settings etermine by this metho will be base on loa isturbance rejection an will not be suitable for set-point changes. The tuning parameters are in fact calculate on the P u Curve A: unstable, runaway oscillation Curve B: continuous cycling, marginal stability Curve C: stable, ampe oscillation Time FIG. 2.35r Ultimate gain is the gain that causes continuous cycling (Curve B) an ultimate perio (Pu) is the perio of that cycling. basis of the istance where the loop will operate from instability. Settings for loa rejection are not too far from instability, but tuning parameters for set-point change are ifferent, an the process moel is neee for their etermination. The optimum integral an erivative settings of controllers vary with the number of moes in the controller (P = 1, PI = 2 moe, PID = 3 moe) an with the amount of ea times in the loop. For noninteracting PI controllers with no noticeable ea time, one woul set the integral (T i ) for about 75% of the perio of oscillation. As the ea time-to-time constant ratio rises, the integral setting becomes a smaller percentage of the oscillation perio aroun 60% when the ea time equals 20% of the time constant, about 50% when their ratio is at 50%, about 33% when they are equal, an about 25% when the ea time excees the time constant. For noninteracting PID loops with no ea time, one woul set the integral minutes/repeat (I) to a value equal to 50 % of the perio of oscillation an the erivative time (D) to about 18% of the perio. As ea time rises to 20% of the time constant, (I ) rops to 45% an (D) to 17% of the perio. At 50% ea time, (I) = 40% an (D) = 16%. When the ea time equals the time constant, (I ) = 33% an (D) = 13%; finally, if ea time is twice the time constant, (I ) = 25% an (D) = 12%.

13 426 Control Theory Tuning Example The same example as was use earlier will be use to illustrate the ultimate close-loop tuning metho. The aim of this tuning process is to etermine the controller gain or proportional ban that woul cause sustaine, unampene oscillation (K u ) an to measure the corresponing perio of oscillation, calle the ultimate perio (P u ). The steps in this tuning sequence are as follows: 1. Set all controller ynamics to zero. In other wors, set the integral to infinite (or maximum) minutes per repeat or zero (or minimum) repeats per minute an set erivative to zero (or minimum) minutes. 2. Set the gain or proportional ban to some arbitrary value near the expecte setting (if known) or at K c = 1 (PB = 100%) if no better information is available. 3. Let the process stabilize. Once the PV is stable, introuce an upset. The simplest way to o that is to move the set point up or own by a safe amount (for example, move it by 2% for half a minute) an then return it to its original value. The result will be an upset in the PV resembling the characteristics of curve A, B, or C in Figure 2.35r. If the response is unampene (curve A), the gain (or proportional) setting is too high (proportional narrow); inversely, if the response is ampe (curve C), the gain (or proportional) setting is too low (proportional wie). Therefore, if the response resembles curve A, the controller gain is increase; if it resembles curve C, the gain is reuce, an the test is repeate until curve B is obtaine. After one or more trials, the state of sustaine, unampene oscillation will be obtaine (curve B), an at that point the test is finishe. (Make sure that the oscillation is a sinusoial an not a limit cycle.) Next, rea the proportional gain that cause the sustaine oscillation. This is calle the ultimate gain (K u ), an the corresponing perio is the ultimate perio of oscillation (P u ). Once the values of K u an P u are known, one might use the recommenations of Ziegler Nichols (Table 2.35s) or the recommenations that were escribe earlier, which also consier the ea time-to-time constant ratio. No one tuning is perfect, an experience process control engineers o come up with their own fuge factors base on experience. TABLE 2.35s Tuning Parameters Base on the Measurement of K u an P u Recommene by Ziegler Nichols for a Noninteracting Controller Type of Controller P (gain) I (minutes/repeat) D (minutes) P 0.5 K u PI 0.45 K u P u /1.2 PID 0.6 K u P u /2 P u /8 Controller output Process variable P u FIG. 2.35t Ultimate cycling response of the same process that was teste in Figure 2.35e. In orer to use the ultimate gain an the ultimate perio to obtain the controller settings for proportional controllers, Ziegler an Nichols correlate the ecay ratio vs. gain expresse as a fraction of the ultimate gain for several systems. From the results they conclue that if the controller gain is set to equal one-half of the ultimate gain, it will often give a ecay ratio of 1/4, i.e., K c u 2.35(13) By analogous reasoning an testing, the equations in Table 2.35s were foun to also give reasonably goo settings for noninteracting two- an three-moe controllers. Again it shoul be note that these equations are empirical an exceptions aboun. For the same example as before, Figure 2.35t illustrates the ultimate cycling response. The ultimate gain an perio obtaine from Figure 2.35t are: K u = 7.75 an P u = 87 s. Hence the recommene tuning settings for the process that was use in the example are: K p = 3.49 an T i = 1.21 minutes. There are a few exceptions to the tuning proceure escribe here because in some cases, ecreasing the gain makes the process more unstable. In these cases, the ultimate metho will not give goo settings. Usually in cases of this type, the system is stable at high an low values of gain but unstable at intermeiate values. Thus, the ultimate gain for systems of this type has a ifferent meaning. To use the ultimate metho for these cases, the lower value of the ultimate gain is sought. Avantages an Disavantages The main avantage of the close-loop tuning metho is that it consiers the ynamics of all system components an therefore gives accurate results at the loa where the test is performe. Another avantage is that the reaings of K u an P u are easy to rea an the perio of oscillation can be accurately rea even if the measurement is noisy Time (sec) = 05. K ( PB = 2PBu)

14 2.35 Tuning PID Controllers 427 TABLE 2.35u Harriott Tuning Parameters for a Noninteracting Controller Calculate from K c1/4 Obtaine to Reach a Quarterof-Amplitue Decay The isavantages of the close-loop tuning metho are that when tuning unknown processes, the amplitues of unampene oscillations can become excessive (unsafe) an the test can take a long time to perform. One can see that when tuning a slow process (perio of oscillation of over an hour), it can take a long time before a state of sustaine, unampene oscillation is achieve through this trial-an-error technique. For these reasons, other tuning techniques have also been evelope an some of them are escribe below. Dampe Oscillation Metho Harriott has propose a slight moification of the previous proceure. For some processes, it is not feasible to allow sustaine oscillations an therefore, the ultimate metho cannot be use. In this moification of the ultimate metho, the gain (proportional control only) is ajuste, using steps analogous to those use in the ultimate metho, until a response curve with 1/4 of the ecay ratio is obtaine. However, with this tuning metho, it is necessary to note only the perio P of the response. Again it shoul be note that the equations in Table 2.35u are empirical an exceptions aboun. Example for Dampe Cycling From Figure 2.35v the gain an perio are foun to be K c1/4 = 7.75 an P = 87 s. Hence the recommene tuning settings for the PI controller are K p = 3.49 an T i = 1.21 minutes. After these moes are set, the sensitivity is again ajuste until a response curve with 1/4 of a ecay ratio is obtaine. Process variable Type of Controller P (gain) I (minutes/repeat) D (minutes) P K c1/4 PID ajuste P/1.5 P/6 Controller output FIG. 2.35v Dampe cycling for the example This metho usually requires about the same amount of work as the ultimate metho since it is often necessary to experimentally ajust the value of the gain to obtain a ecay ratio of 1/4. It is also possible to use this metho to use a ifferent ecay ratio criterion. Avantages an Disavantages In general, there are two major isavantages to the ultimate an ampe oscillation methos. First, both are essentially trial-an-error methos, since several values of gain must be teste before the ultimate gain or the gain to give a 1/4 ecay ratio are to be etermine. To make one test, especially at values near the esire gain, it is often necessary to wait for the completion of several oscillations before it can be etermine whether the trial value of gain is the esire one. Secon, while one loop is being teste in this manner, its output may affect several other loops, thus possibly upsetting an entire unit. While all tuning methos require that some changes be mae in the control loop, other techniques require only one an not several tests, unlike the close-loop methos. Also, if the tuning parameters are too aggressive, the expecte response can be obtaine by increasing the proportional ban (or ecreasing the proportional gain). The integral an erivative settings probably nee to be moifie. The proportional gain has to be reuce to 3.5 to have a quarterof-amplitue ecay. COMPARISON OF CLOSED AND OPEN LOOP Table 2.35w provies a comparison of open-loop an closeloop results for the process example use in Figure 2.35e. FREQUENCY RESPONSE METHODS Frequency response methos for tuning controllers involve first etermining the frequency response of the process, which is a process characteristic. From this, tuning can be evelope. Frequency response methos (FRM) may have several avantages over other methos: These are: 1. FRM require only one process bump to ientify the process. The bump can be a change in automatic or manual, TABLE 2.35w Comparison of Close an Open Loop Test Results Type of Tuning Test K c T i (min) Open loop Reaction curve Moel Fit Close loop Close loop, ultimate cycling Close loop, ampe cycling

15 428 Control Theory an be either a pulse, step, or other type of bump. A setpoint change provies excellent ata from FRM. 2. FRM o not require any prior knowlege of the process ea time or time constant. With the other time response methos, one often nees a ea time estimate an a time constant estimate. 3. FRM o not require any prior knowlege of the process structure. Time response methos often require the user to have such moel structure knowlege, i.e., whether it is first or secon orer or whether it is an integrator. For FRM-base tuning none of this is require; only the process ata are neee. Obtaining the Frequency Response The process frequency response is a graph of amplitue ratio an phase vs. oscillation or sine wave frequency. If one injects a sine wave into a linear process at the controller output, the PV will also isplay a sine wave. The output (PV) sine wave will probably be of smaller height relative to the input an will be shifte in time. The ratio of the heights is the amplitue ratio at the frequency of the input sine wave. The shift in time is the phase shift or phase lag. A time shift resulting in the trough of the output when aligne with the crest of the input is generally thought to be 180 egrees out of phase or 180 egrees of phase lag. By applying a variety of sine wave inputs to a process, one can obtain a table of amplitue ratios an phase lags epenent on the sine wave frequency. If one plots these, the result will be the frequency response of the process. Using Fourier analysis computer software programs one can calculate the process frequency response from a bump, pulse, or any other signal that applies sufficient excitation to the controller output. In the evaluation both the CO an PV trens are use. The ata provie for these programs shoul start from a settle state, experience a quick change, an en settle. Any one of the responses in Figures 2.35e, g, h, i, o, p, q, an v woul provie aequate ata for frequency response base testing. Figure 2.35x shows the typical frequency response arrive at by the use of computer software. PID Tuning Base on Frequency Response In most processes, both the amplitue ratio an the phase angle will ecrease with increasing frequencies. Assuming Amplitue ratio (B) Phase angle (eg) Frequency (raians/sec) FIG. 2.35x Typical moel (soli line) an actual (ash line) process frequency response.

16 2.35 Tuning PID Controllers 429 that the combine phase an amplitue ratio ecreases with frequency when the process an the controller frequency responses are combine, the following general stability rule applies: A control system will be unstable if the open-loop frequency response has an amplitue ratio that is larger than one when the phase lag is 180 egrees. To provie proper tuning, a margin of safety in the gain an phase is esire. Tuning constants are therefore ajuste to result in the highest gain at all frequencies an yet achieve a certain margin of safety or stability. This is best accomplishe using computer software. Deviation l opt 3 l opt l opt l opt Time, t/st FINE TUNING The performance of a controller can be teste by simulating a loa change in the close loop as escribe at the beginning of this section. This can be especially useful if the initial tuning is not satisfactory or if the process characteristics have change. It is simply a trial-an-error metho of recognizing an approaching an optimum or otherwise esirable loa response. Optimum Loa Response As escribe earlier, the optimum loa response is generally consiere to be that which has a minimum IAE, in that it combines minimum peak eviation with low integrate error an short settling time. The secon curve from the bottom in each of Figures 2.35y an z represents a minimum-iae loa response for a istribute lag uner PI control. This secon curve from the bottom has a symmetrical first peak, low overshoot, an effective amping. The time scale of these curves is normalize to the 63.2% open-loop step response of the istribute lag, ientifie as Στ. On either sie of the optimum curve in both figures are other response curves, which were prouce by changing one of the controller settings. Deviation 0 P opt 1.5 P opt 3 P opt 1.5 P opt Time, t/st FIG. 2.35y The proportional setting affects the height of the first peak an its symmetry. FIG. 2.35z The integral setting primarily affects the location of the secon peak. In Figure 2.35y, the proportional ban of the controller is the parameter being ajuste. Note that increasing the proportional ban increases the height of the first peak an also its amping; but along with the increase amping comes a loss of symmetry. In Figure 2.35z, it is the integral time of the controller that is being ajuste. Note that it has no effect at all on the first peak but etermines the location of the secon, i.e., the overshoot or unershoot of the process variable s eviation. The integrate error IE of a stanar PID controller varies irectly with the prouct of its proportional ban an integral time. While increasing either of these settings improves amping, it also increases IE in irect proportion, an therefore it costs performance. Increasing both settings above their optimum compouns this effect. Effect of Loa Dynamics All of the tuning rules escribe in this section to optimize loa regulation will apply only to loops in which the ynamics in the loa path are ientical to those in the path of the controller output. Yet Figure 2.35a shows the possibility that the ynamics are ifferent in the two paths, for example in the case of heat exchangers. Any ea time in the loa path, or variation thereof, will have no effect on the loa response curve, simply elaying it by more or less time. The ominant lag in the loa path is what etermines the shape of the leaing ege of the response curve. As a emonstration, a istribute lag in a PI control loop was simulate to assess the effects of variations in loa ynamics. Three ifferent loa-response curves appear in Figure. 2.35aa representing ratios of Στ q /Στ m varying from 0.5 to 2, where subscripts q an m ientify the loa an manipulate-variable paths, respectively. In all three cases, the PI controller has been tune to minimize IAE following a simulate loa change, simulate by stepping the controller output, which prouces the center curve represente by Στ q /Στ m = 1. However, when a true loa step