2.9 Feedback and Feedforward Control

Transcription

1 2.9 Feedback and Feedforward Control M. F. HORDESKI (985) B. G. LIPTÁK (995) F. G. SHINSKEY (970, 2005) Feedback control is the action of oving a anipulated variable in response to a deviation or error e between the controlled variable c and its set point r in such a way as to reduce, or if possible, to eliinate the error. Feedback control cannot anticipate and thereby prevent errors, because it can only initiate its corrective action after an error has already developed. Because of dynaic lags and delays in the response of the controlled variable to anipulation, soe tie will elapse before the correction will take effect. During this interval, the deviation will tend to grow, before eventually diinishing. Feedback control cannot therefore achieve perfect results; its effectiveness is liited by the responsiveness of the process to anipulation. By contrast, feedforward correction can be initiated as soon as any change is detected in a load variable as it enters the process; if the feedforward odel is accurate and the load dynaics are favorable, the upset caused by the load change is canceled before it affects the controlled variable. Because feedforward odels and sensors are both iperfect, feedforward loops are usually corrected by feedback triing. FEEDBACK CONTROL Set, r Controller Load, q Valve or flowloop Controller var., c FIG. 2.9a Load changes can enter through different gain and dynaics fro the controller output. The purpose of any process control syste is to aintain the controlled variable at a desired value, the set point, in the face of disturbances. The control syste regulates the process by balancing the variable load(s) with equivalent changes in one or ore anipulated variables. For the controlled variable to reain stationary, the controlled process ust be in a balanced state. Regulation through feedback control is achieved by acting on the change in the controlled variable that was induced by a change in load. Deviations in the controlled variable are converted into changes in the anipulated variable and sent back to the process to restore the balance. Figure 2.9a shows the backward flow of inforation fro the output of the process back to its anipulated input. The load q is a flow of ass or energy entering or leaving a process, which ust be balanced by an equal flow leaving or entering. It ay have ore than one coponent for exaple in a teperature loop, both the flow and teperature of a strea are coponents of its energy deand, but they ay be balanced by a single anipulated variable such as stea flow. The steadystate gain K and dynaic gain vector g in the paths of the anipulated and load variables ay differ, and therefore have been given different subscripts and q. Liitations of Feedback Control Feedback, by its nature, is incapable of correcting a deviation in the controlled variable at the tie of detection. In any process, a finite delay exists between a changing of the anipulated variable and the effect of the change on the controlled variable. Perfect control is not even theoretically attainable because a deviation in the controlled variable ust appear before any corrective action can begin. In addition, the value of the anipulated variable needed to balance the load ust be sought by trial and error, with the feedback controller observing the effect of its output on the controlled variable. Best-Possible Feedback Control An absolute liitation to the perforance of a feedback control loop is iposed by any deadtie in the loop. Figure 2.9b describes the results of a step-load change applied to a process whose dynaics consist of deadtie and a single lag in both the load path g q and the anipulated-variable path g. The tie scale is given in units of deadtie τ d in the path of the anipulated variable. To siplify the illustration, the deadties in both paths are identical (this is not essential any K q g q Process K g 73

2 74 Control Theory c r e b E b Manual These values of e b and E b are the best that can be obtained for any feedback controller on a process whose dynaics consist of deadtie and a single lag. A real controller will yield a soewhat larger peak deviation and an area at least twice as great, depending on its odes and their tuning. If the process contains two or ore lags, the value of E b reains the sae as estiated above but is ore difficult to approach with a real controller. q Tie, t/t d FIG. 2.9b The best-possible load response for a process having deadtie and a single lag. deadtie in the load path, or none, will produce the sae response, siply shifting its location in tie). Also for siplification, the steady-state gains in both paths are ade equal. At tie, a step decrease in load q enters the process. After the deadtie in the load path expires, at tie zero, the controlled variable c responds, beginning to rise along an exponential curve deterined by the gain and lag in the load path. If the controller were left in Manual, it would continue to a new steady state. In this exaple, K q is 2.0, leaving the final value of c in Manual having changed twice as uch as load q; the tie constant τ q of the load lag in this exaple is 2.0τ d. Also shown is the possible return trajectory of c to set point r if the anipulated variable were to step as soon as any deviation is detected, i.e., at tie zero, by an aount = q( K / K )( + / ε τ τ ) q d q 2.9() where ε is the exponential operator This ove turns out to be the best that is possible by a feedback controller, as it causes the deviation to decay to zero during the next deadtie. The peak deviation reached during this excursion is e b d q = qk ( / ε τ τ ) q 2.9(2) At the tie the peak is reached, the controller output ust be stepped to atch the new steady-state load. The leading edge of the curve, i.e., up to tie.0, is deterined copletely by the load step and the gain and dynaics in its path; the trailing edge is deterined by the controller and its tuning. The leading and trailing edges of the best response are copleentary, so that the area that they enclose is d q E = e τ = qk τ ε τ / ( τ ) b b d q d 2.9(3) Integrated Error The actual effectiveness of feedback control depends on the dynaic gain of the controller, which is a function of its control odes and their tuning. Although a high controller gain is desirable, the dynaic gain of the closed loop at its period of oscillation ust be less than unity if the loop is to reain stable. In effect, then, the dynaic gain of the process dictates the allowable dynaic gain of the controller. For each process and controller, optiu settings exist that iniize soe objective function such as Integrated Absolute Error (IAE). However, ost controllers are not tuned optially, for various reasons, such as process nonlinearities. Therefore, controller perforance needs to be stated in ters of the actual settings being used. As an exaple, the integrated error produced by a load change to a process under ideal Proportional- Integral-Derivative (PID) control will be evaluated based only on its ode settings. The controller output at tie t is related to the deviation e between the controlled variable and its set point by 00 P e t edt D de = + + C I + t0 dt 2.9(4) where P, I, and D are the percent Proportional Band, Integral tie and Derivative tie, respectively, and C is the output of the controller at tie, t 0 when it was first placed in Autoatic. Let t be a steady state, so that e = 0 and its derivative is also zero. Then a load change arises, causing the controller to change its output to return the deviation to zero. When a new steady state is reached, the controller will have an output 2 : t 00 P e 2 edt D de 2 = + + C I t dt (5) where e 2 and its derivative are again zero. The difference in output values between the two steady states is t 00 2 = 2 = edt PI t Solving for the integrated error: t2 E = e dt = PI t (6) 2.9(7)

3 2.9 Feedback and Feedforward Control 75 For any sustained load change, there ust be a corresponding change in the controller output, and in fact the current load is often reported as the steady-state controller output. The wider the proportional band and the longer the integral tie, the greater will be the integrated error per unit load change. (While the derivative setting does not appear in the integrated-error function, its use allows a lower integral tie than is optiu for a PI controller.) Therefore, loops with controllers having large PI products are candidates for feedforward control. FEEDFORWARD CONTROL Feedforward provides a ore direct solution to the control proble than finding the correct value of the anipulated variable by trial and error, as occurs in feedback control. In the feedforward syste, the ajor coponents of load are entered into a odel to calculate the value of the anipulated variable required to aintain control at the set point. Figure 2.9c shows how inforation flows forward fro the load to the anipulated variable input of the process. The set point is used to give the syste a coand. (If the controlled variable were used in the calculation instead of the set point, a positive feedback loop would be fored.) A syste, rather than a single control device, is norally used for feedforward loops because it is not always convenient to provide the coputing functions required by the forward loop with a single device or function. Instead, the feedforward syste consists of several devices if ipleented in hardware or several blocks of software if ipleented digitally. The function of these blocks is to ipleent a atheatical odel of the process. Load Balancing A dynaic balance is required to keep the control variable at set point. It is achieved by solving the process aterial- and/or energy-balance equations continuously. When a change in load is sensed, the anipulated variable is autoatically adjusted to the correct value at a rate that keeps the process continually in balance. Although it is theoretically possible to achieve such Set, r Feedfwd syste Load, q Valve or Flowloop K g FIG. 2.9c Feedforward calculates the value of the anipulated variable based on current load and set point. K q g q Process c perfect control, in practice the syste cannot be ade to duplicate the process balance exactly. The aterial- and energy-balance equations are not usually difficult to write for a given process. Variations in nonstationary paraeters such as heat-transfer and ass-transfer coefficients do not ordinarily affect the perforance of a feedforward syste. The load coponents are usually inflow or outflow when level or pressure is to be controlled, feed flow and feed coposition where product coposition is to be controlled, and feed flow and teperature where product teperature is to be controlled. Flow is the priary coponent of load in alost every application because it can change widely and rapidly. Coposition and teperature are less likely to exhibit such wide excursions, and their rate of change is usually liited by upstrea capacity. In ost feedforward systes these secondary load coponents are left out; their effects are accoodated by the feedback controller. The output of a feedforward syste should be an accurately controlled flow rate, if possible. This controlled flow rate cannot usually be obtained by anipulating the control valve directly, since valve characteristics are nonlinear and inconsistent, and the delivered flow is subject to such external influences as upstrea and downstrea pressure variations. Therefore, ost feedforward systes depend on soe easureent and feedback control of flow to obtain an accurate anipulation of the flow rate. Only when the range of the anipulated variable exceeds that which is available in floweters, or the required speed of response exceeds that of a flow loop, should one consider having the valves positioned directly, and in such cases care ust be taken to obtain a reproducible response. Steady-State Model The first step in designing a feedforward control syste is to for a steady-state atheatical odel of the process. The odel equations are solved for the anipulated variable, which is to be the output of the syste. Then the set point is substituted for the controlled variable in the odel. This process will be deonstrated by using the exaple of teperature control in a heat exchanger (Figure 2.9d). 2 A liquid flowing at rate W is to be heated fro teperature T to controlled teperature T 2 by stea flowing at rate W s. The energy balance, excluding any losses, is WC ( T T ) = W λ p 2 2.9(8) Coefficient C p is the heat capacity of the liquid and λ is the latent heat given up by the stea in condensing. Solving for W s yields W WK ( T T ) s = s 2set 2.9(9)

4 76 Control Theory X W FY T FY T 2set T W s FIC Contr. var. Set Offset fro 0% error Accurate static odel Liquid FT TT TT HIC T 2set Heat Exchanger FIG. 2.9d Stea flow is calculated here to satisfy the steady-state heat balance. where K = C p /λ, set point T 2set replaces T 2, and W and T are load coponents to which the control syste ust respond. Note that this energy balance is iperfect, because it disregards inor loads such as the heat losses to the environent and any variations in the enthalpy of the stea. Another factor is that even the two ajor loads cannot be easured perfectly, as there are no error-free sensors of flow and teperature. The net result is a odel that is approxiately an energy balance around this process. An ipleentation of Equation 2.9(9) is given in Figure 2.9d. A control station introduces T 2set into the difference block [ ] where input T is subtracted. The gain of this block is adjusted to provide the constant K. Its output is then ultiplied by the liquid flow signal W to produce the required set point for stea flow. During startup, if the actual value of the controlled variable does not equal the set point, an adjustent is ade to gain K. Then, after aking this adjustent, if the controlled variable does not return to the set point following a change in load, the presence of an error in the syste is indicated. The error ay be in one of the calculations, a sensor, or soe other factor affecting the heat balance that was not included in the syste. Heat losses and variations in stea pressure are two possible sources of error in the odel. Since any error can cause a proportional offset, the designer ust weigh the various sources of error and copensate for the largest or ost changeable coponents where practical. For exaple, if stea pressure variations were a source of error, the stea floweter could be pressure copensated. Because of all these errors, feedforward control as shown in Figure 2.9d is seldo used without feedback tri. Later in this section, a feedback controller will be added to autoatically correct the loop for the errors in the odel. T FT Stea Load Manip. var. Tie FIG. 2.9e The dynaic response of the uncopensated feedforward syste when the controlled variable responds faster to changes in load than to changes in anipulated flow. control systes. Two typical dynaic responses of the controlled variable are shown in Figure 2.9e; the offset in the dashed curve results fro a 0% error in the static calculations. If there is no static error, the teperature returns to set point eventually. However, the dynaic balance is issing. The decreased stea flow does not result in an instantaneous decrease in heat-transfer rate, because the teperature difference across the heat-transfer surface ust first be decreased, and this requires a decrease in shell pressure. The lower shell pressure at lower loads eans that the shell contains less stea as the load decreases. Since the static feedforward syste does not overtly lower the stea inventory in the shell, it is lowered through a lagging heat-transfer rate, resulting in a transient rise in exit teperature on a falling load; conversely, a transient teperature drop follows a rising load. This transient error reveals a dynaic ibalance in the syste on a drop in load, the exchanger teporarily needs less heat than the stea flow controller is allowing. In order to correct the transient teperature error, a further cutback in the flow of energy ust be applied to the exchanger beyond what is required for the steady-state balance. A ore general explanation is provided through the use of Figure 2.9f. This figure shows the load and the anipulated Feedforward dynaics + t s + t q s + t s Dynaic Model With the feedforward odel as ipleented in Figure 2.9d, a step decrease in liquid flow results in a siultaneous and proportional step decrease in stea flow. Transient errors following a change in load are to be expected in feedforward Load q + t q s Process FIG. 2.9f A feedforward dynaic odel for a general process. c

5 2.9 Feedback and Feedforward Control 77 variable as entering the process at different points, where they encounter different dynaic eleents. In the heatexchanger exaple, the liquid enters the tubes while the stea enters the shell. The heat capacities of the two locations are different. As a result, the controlled variable (the liquid teperature) responds ore rapidly to a change in liquid flow than to a change in stea flow. Thus, the lag tie of the load input is less than that of the anipulated variable. The objective of the feedforward syste is to balance the anipulated variable against the load, providing the forward loop with copensating dynaic eleents. Neglecting steadystate considerations, Figure 2.9f shows what the dynaic odel ust contain to balance a process having parallel firstorder lags. The lag tie τ q of the load input ust be duplicated in the forward loop, and the lag tie τ in the anipulated input of the process ust be cancelled. Thus the forward loop should contain a lag divided by a lag. Since the inverse of a lag is a lead, the dynaic copensating function is a leadlag. The lead tie constant should be equal to τ, and the lag tie constant should equal τ q. In the case of the heat exchanger, the fact that τ is longer than τ q causes the teperature rise on a load decrease. This is the direction in which the load change would drive the process in the absence of control. In transfer function for, the response of a lead-lag unit is s Gs ()= + τ + τ s 2.9(0) where τ is the lead tie constant, τ 2 is the lag tie constant, and s is the Laplace operator. The frequency response of a feedforward loop is usually not significant, since forward loops cannot oscillate. The ost severe test for a forward loop is a step change in load. The response of the lead-lag unit output y to a step in its input x is τ τ2 yt () = xt () + ε τ 2.9() Its axiu dynaic gain (at t = 0) is the lead/lag ratio τ /τ 2. The response curve then decays exponentially fro this axiu at the rate of the lag tie constant τ 2, with 63.2% recovery reached at t = τ 2 as shown in Figure 2.9g. The figure shows soe deadtie copensation added to the lead-lag function however, it is not required for the heat exchanger. When properly adjusted, the dynaic copensation afforded by the lead-lag unit can produce the controlled variable response having the for shown in Figure 2.9h. This is a signature curve of a feedforward-controlled process since ost processes do not consist of siple first-order lags, a first-order lead-lag unit cannot produce a perfect dynaic balance. Yet a ajor reduction in peak deviation is achieved over the uncopensated response, and the area can be equally distributed on both sides of the set point. 2 2 t/ τ2 t t2 Load t2 FIG. 2.9g The step response of a lead-lag unit with deadtie copensation. The lead-lag unit is applied to the flow signal as [ f(t)] in Figure 2.9i. Its settings are particular to that load input, and therefore it ust be placed where it can odify that signal and no other. No dynaic copensator is included for T, as it is not expected to change quickly. Tuning the Lead-Lag Unit First, a load change should be introduced without dynaic copensation by setting the lead and lag tie constants at equal values or zero to observe the direction of the error. If the resulting response is in the direction that the load would drive it, the lead tie should exceed the lag tie; if not, the lag tie should be greater. Next, easure the tie required for the controlled variable to reach its axiu or iniu value. This tie should be introduced into the lead-lag unit as the saller tie constant. Thus, if the lead doinates, this would be the lag setting. For exaple, in Figure 2.9e the teperature responds in the direction the load would drive it, and therefore the lead tie setting should exceed the lag tie setting. The tie required for the teperature in Figure 2.9e to rise to its peak should be set equal to the saller tie constant, in this case equal to the lag tie setting. Set the greater tie constant at twice Contr. var. Manip. var. Load Manip. var. Deadtie 63% resp. Set Tie Tie FIG. 2.9h The typical step response of a dynaically copensated feedforward syste.

6 78 Control Theory Liquid f(t) FY FT TT T TT FIG. 2.9i Feedforward feedback control of the heat exchanger, with dynaic copensation. this value and repeat the load change. If the peak deviation is reduced, but the curve is still not equally distributed around the set point, increase the greater tie constant and repeat the load change. The area (integrated error) of the response curve will be equally distributed about the set point if the difference between the lead and lag settings is correct. Once this equalization is obtained, both settings should be increased or decreased with their difference constant until a iniu error aplitude is achieved. When the controller is properly tuned, the stea energy equivalent of the transient (area between teperature and its set point) in Figure 2.9e should atch the extra stea introduced (area above input step change) in Figure 2.9g. If there is a significantly greater deadtie in the load path than in the anipulated-variable path, deadtie copensation ay be added, which will reduce the peak deviation. However, if the deadtie in the anipulated-variable path is the longer of the two, exact copensation is ipossible, and this is the case in the heat exchanger. Still, careful adjustent of the lead-lag copensator can result in zero integrated error. Adding a Feedback Loop X W FY FY T TIC Heat Exchanger FIC Any offset resulting fro steady-state errors can be eliinated by adding feedback. This can be done by replacing the feedforward set point with a controller, as shown in Figure 2.9i. The feedback controller adjusts the set point of the feedforward syste in cascade, while the feedforward syste adjusts the set point of the anipulated flow controller in cascade. The feedback controller should have the sae control odes as it would without feedforward control, but the settings need not be as tight. The feedback controller reacts to a disturbance by creating another disturbance in the opposite direction one-half cycle later. However, the feedforward syste positions the anipulated variable so that the error in the controlled variable disappears. If acted upon by a tightly W s T FT Stea set feedback controller, the correct position calculated by feedforward will be altered, producing another disturbance that prolongs the settling tie of the syste. The feedback controller ust have the integral ode to eliinate any steady-state offset that ight be caused by errors in the sensors, the odel, or the calculations. However, the integrated error it sustains following a load change is arkedly reduced copared to what it would be following a load change without feedforward. Now its output changes only an aount equal to the change in the error in the feedforward calculation. For sall load changes, this change in error would approach zero, leading to an integrated error of zero, with or without dynaic copensation. In the presence of feedback, but without dynaic copensation, the transient in Figure 2.9e would be balanced by a following transient on the other side of set point, until the area is equalized. This usually prolongs the response without reducing the peak deviation. The dynaic copensator therefore needs to be tuned with the feedback controller in anual, to approach zero integrated error before adding feedback. Linear and Bilinear Feedforward Systes The relationship shown in the energy balance of Equation 2.9(8) is bilinear: stea flow is related to both liquid flow and to teperature rise in a linear anner, but as their product rather than their su or difference. This distinction is crucial to the successful functioning of feedforward control because a bilinear syste has variable gains. The feedforward gain the ratio of stea flow to liquid flow varies directly with the required teperature rise. This variable gain is accoodated in the syste by the ultiplier [ ]. If inlet teperature were to vary so slowly that it did not require feedforward copensation or if it were not easured, then feedforward fro flow alone would be required. Yet the required feedforward gain the ratio W s /W would vary directly with T 2set T. If either T 2set or T were to change, the feedforward gain should change with it. For a bilinear process, then, a linear feedforward syste having a constant gain is a isfit it is accurate at only one cobination of secondary load and set point. To be sure, feedback control can tri any errors in the feedforward calculation and return the deviation to zero. But if the error is in the gain ter, the feedback controller should adjust the feedforward gain, through a ultiplier otherwise the gain will be incorrect for all subsequent flow changes, resulting in only partial feedforward correction or even overcorrection. Eliinating inlet teperature as a variable in Figure 2.9i also eliinates the difference block [ ]. The controller output then goes directly to the ultiplier, and in reducing the teperature deviation to zero, will assue a value representing K(T 2set T ). If either T 2set or T were then to vary, the feedback controller would respond to the resulting deviation by changing its output until the deviation returns to zero. In so doing, it has changed the gain of the ultiplier to the new value of K(T 2set T ).

7 2.9 Feedback and Feedforward Control 79 Control of coposition is also a bilinear process. The required ratio of two flows entering a blender, for exaple, is a function of their individual copositions and that of the blend. Siilarly, the ratio of product flow to feed rate or stea flow to feed rate in a distillation colun or evaporator also varies with feed and product copositions. In general, teperature and coposition loops are bilinear, and their flow ratios should always be adjusted through a ultiplier. By contrast, pressure and level loops are linear, and their feedforward gain can be constant. An exaple of the latter is three-eleent dru-level control, where each unit of stea reoved fro the dru ust be replaced by an equal unit of feedwater the feedforward gain is constant at.0. This distinction is crucial because any advanced ultivariable control systes are based on a linear atrix with constant coefficients. The plant is tested to develop the coefficients that relate the variables, and the control atrix is built fro these results. When the plant operating conditions ove away fro the original test conditions, the fixed coefficients in the control atrix ay no longer represent the process relationships accurately, degrading the perforance of the feedforward loops. This ay require frequent retesting and recalibration of the atrix. Soe ultivariable systes contain ultiple atrices, each corresponding to its own set of operating conditions and switched into service when those conditions develop. Self-tuning feedforward control 3 is also available, applied either in a linear (additive) or bilinear (ultiplicative) anner, as configured anually. The paraeters that are tuned adaptively are the steady-state gain and a lag copensator. The steady-state gain is adjusted to iniize integrated error following a change in easured load, and the lag is adjusted to iniize integral-square error. Load changes ust be sharp, clean steps, with settling tie allowed between the, for tuning to proceed effectively. But once tuned, rando disturbances can be accoodated. However, self-tuning is only recoended where precise odeling and flow control are unavailable. Perforance The use of feedback in a feedforward syste does not detract fro the perforance iproveent that was gained by feedforward control. Without feedforward, the feedback controller was required to change its output to follow all changes in load. With feedforward, the feedback controller ust only change its output by an aount equal to what the feedforward syste fails to correct. If a feedforward syste applied to a heat exchanger could control the stea flow to within 2% of that required by the load, the feedback controller would only be required to adjust its output to copensate for 2% of a load change, rather than the full aount. This reduction of in Equation 2.9(7) by 50/ results in the reduction of E by the sae ratio. Reductions by 0/ in errors resulting fro load changes are relatively coon, and iproveents of 00/ have been achieved in soe systes. Teperature PID control Dynaic feedforward Tie, s FIG. 2.9j Coparison of feedforward control of a heat exchanger against optially tuned PID feedback control. Figure 2.9j illustrates the control perforance of a steaheated exchanger that has experienced a step decrease in process flow. The static feedforward odel reduces the peak deviation substantially but extends settling tie and does not iprove on the integrated error. Dynaic copensation is seen to be essential in axiizing perforance in this exaple. It is not perfect because the deadtie in the path of the anipulated variable is soewhat longer than that in the load path, and that lost tie cannot be ade up. Still, the lead-lag copensator has been tuned to eliinate any integrated error. The feedforward syste is ore costly and requires ore engineering effort than a feedback syste does, so prior to design and installation, the control iproveent it brings ust be deterined to be worthwhile. Most feedforward systes have been applied to processes that are very sensitive to disturbances and slow to respond to corrective action and to product streas that are relatively high in value. Distillation coluns of 50 trays or ore have been the principal systes controlled with this technology. Boilers, ultipleeffect evaporators, direct-fired heaters, waste neutralization plants, solids dryers, copressors, and other hard-to-control processes have also benefited fro feedforward control. Variable Paraeters Static feedforward Heat exchangers are characterized by gain and dynaics that vary with flow. The settings of both the PID controller and the feedforward copensator that produced the results shown in Figure 2.9j were optiized for the final flow and would not be optiu for any other flow. Differentiation of the heat-balance equation, Equation 2.9(8), shows the process gain to vary inversely with process flow W: dt 2 = λ dw WC s 2.9(2) If PID control alone is applied, this gain variation can be copensated through the use of an equal-percentage stea p 240

8 80 Control Theory valve, whose gain varies directly with stea flow. However, when the PID controller is cobined with bilinear feedforward, its output passes through the ultiplier shown in Figure 2.9i, where it is ultiplied by process flow. This operation keeps the PID loop gain constant, while anipulating stea flow linearly. Heat-exchanger dynaics deadtie and lags also vary inversely with process flow, as is typical of once-through processes, i.e., where no recirculation takes place. This proble is less easily solved. The PID controller ust have its integral tie set relative to the slowest response (lowest expected flow) and its derivative tie set relative to its fastest response (highest expected flow), or ideally, prograed as a function of easured flow. Otherwise, instability ay result at extree flow rates. Ideally, lead and lag settings of the feedforward dynaic copensator should also be prograed as a function of easured flow. However, the penalty for not doing so is not severe. The lead-lag ratio is not subject to change because the lags on both sides of the process change in the sae proportion with flow. The priary purpose of the dynaic copensator is to iniize the peak deviation following the load change, and this is accoplished by the dynaic gain of the copensator, which is its lead-lag ratio. Any subsequent variation in integrated error is eliinated by the PID controller. Although paraeter variations can also be accoodated by a self-tuning feedforward copensator and feedback controller, these ethods are less accurate in arriving at optiu settings and are always late, having tuned for the last flow condition and not for the next one. References. Shinskey, F. G., Process Control Systes, 4th ed., New York: McGraw- Hill, 996, pp Shinskey, F. G., Feedforward Control Applied, ISA Journal, Noveber Hansen, P. D., Badavas, P. C., and Kinney, T. B., New Adaptive Multivariable Controller, presented at the 3rd Intl. Control Eng. Conference, Chicago, 994. Bibliography Butterworths, W. R., Advanced Process Control, New York: McGraw-Hill, 98. Gore, F., Add Copensation for Feedforward Control, Instruents and Control Systes, March 979. Paler, R., Nonlinear Feedforward Can Reduce Servo Settling Tie, Control Engineering, March 978. Seborg, D., et al., Process Dynaics and Control, New York: Wiley, 989. Shinskey, F.G., Reflections on CPC-III, Cheical Process Control-CPC III, T. J. McAvoy, et al. (Eds.), Asterda: Elserier Sci. Pub., 986. Shinskey, F. G., Soothing Out Copressor Control, Cheical Engineering, February 999.