# 1812_02c Advanced Control Unleashed - Plant Performance Management for Optimum Benefit

60 Advanced Control Unleashed in the process. The integral time factor is decreased and can be as small as 1/8 of the ultimate period for a pure dead time process. Figures 2-25a through 2-25f show the effect of the three mode settings on a PID controller for a process with a time lag 4 times larger than the loop time delay. Note that a gain setting too large increases overshoot due to the presence of reset action. An integral (reset) time setting that is too small causes a greater overshoot and increases the period of oscillation. Too much derivative action causes an oscillatory response and a shorter period but no real overshoot. Figures 2-26a through 2-26d show the effect of two mode settings on a PID controller for a process with a time lag 4 times smaller than the loop time delay. The doubling of the gain setting is much more disruptive. Figure 2-25a. Effect of Gain on Set point Response of PID for a Large Time Lag-to-Time Delay Ratio A starting point is needed on tuning settings, particularly for new plants or existing plants with upgraded instrumentation and control valves. In Table 2-5, the first number outside the parentheses provides a tuning, scan time, and filter time setting suitable for a download. Within the parenthe ses is a range of typical settings [2.5]. An auto tuner should be run as soon as the plant is up to normal operating conditions. If the calculated settings are outside the range noted in Table 2-5, the auto tuner test should be run again. If the results are still outside of the range, it means either that it is an unusual loop or that there is a problem with the measurement or control Previous PageChapter 2 - Setting the Foundation 61 Figure 2-25b. Effect of Gain on 40% Load Upset to PID for a Large Time Lag-to-Time Delay Ratio Figure 2-25c. Effect of Reset on Set point Response of PID for a Large Time Lag-to-Time Delay Ratio valve. Whenever a loop is commissioned or tuned, it should be closely watched for several days for a variety of operating conditions to make sure the loop is stable and the performance is acceptable. 62 Advanced Control Unleashed Figure 2-25d. Effect of Reset on 40% Load Upset to PID for a Large Time Lag-to-Time Delay Ratio Figure 2-25e. Effect of Rate on Set point Response of PID for a Large Time Lag-to-Time Delay Ratio Some loops will fail during an auto tuner pretest because the loop response is too small or too large within the allowable time frame of the pretest, or because the valve has too much stick-slip. Also, fast runaway Chapter 2 - Setting the Foundation 63 Figure 2-25f. Effect of Rate on 40% Load Upset to PID for a Large Time Lag-to-Time Delay Ratio Figure 2-26a. Effect of Gain on Set point Response of PID for a Small Time Lag-to-Time Delay Ratio and integrating loops cannot be safely taken out of the auto mode. For these loops, a closed-loop tuning method is best because it is the fastest method for a large time constant, it keeps the controller in service with 64 Advanced Control Unleashed Figure 2-26b. Effect of Gain on a 20% Load Upset to PID for a Small Time Lag-to-Time Delay Ratio Figure 2-26c. Effect of Reset on Set point Response of PID for a Small Time Lag-to-Time Delay Ratio maximum gain, (safest for processes that can get into trouble quickly or have a non-self-regulating response), and it includes the effect of poor valve response in the tuning [2.29]. Chapter 2 - Setting the Foundation 65 Figure 2-26d. Effect of Reset on a 20% Load Upset to PID for a Small Time Lag-to-Time Delay Ratio Table 2-5. Typical Tuning Settings [2.5] Application Type Scan Gain Reset Rate Method (seconds) (seconds) (seconds) Liquid Flow/Press 1(0.2-2) 0.3(0.2-0.8) 6(1-12) 0(0-2) λ Tight Liquid Level 5 (1.0-30) 5.0(0.5-25)* 600(120-6000) 0(0-60) CLM Gas Pressure (psig) 0.2(0.02-1) 5.0(0.5-20) 300(60-600) 3(0-30) CLM Reactor pH 2(1.0-5) 1.0(0.001-50) 120(60-600) 30(6-30) SCM Neutralizer pH 2(1.0-5) 0.1(0.001-10) 300(60-600) 70(6-120) SCM Inline pH 1 (0.2-2) 0.2 (0.1-0.3) 30 (15-60) 0 (0-3) λ Reactor Temperature 5(2.0-15) 5.0(1.0-15) 300(300-3000) 70(30-300) CLM Inline Temperature 2(1.0-5) 0.5(0.2-2.0) 60(12-120) 12(12-40) λ Column Temperature 10(2.0-30) 0.5(0.1-10) 300(300-3000) 70(30-600) SCM * An error square algorithm or gain scheduling should be used for gains Fo*Cr + U*A, then we have a negative-feedback time constant and positive-feedback time constant, respectively. The time constant is not constant but is proportional to reaction mass and inversely proportional to the outlet flow and the product of the heat transfer coefficient and area for the jacket. For a batch reactor (no outlet flow) with a negligible heat release (dQr /dTr = 0), we have the general form of the equation to approximate the thermal time lag of a closed volume. It can also be used for a thermowell by substi tution of the proper parameter values. τp= (Mr*Cr) /(U*A) (2-10b) The process gain depends upon the input under consideration. If the manipulated or disturbance variable is feed flow: Kp = (Cf*Tf) / (Fo*Cr + U*A - dQr /dTr) (2-11) If the manipulated or disturbance variable is jacket temperature: Kp = (U*A) / (Fo*Cr+ U*A - dQr /dTr) (2-12) 78 Advanced Control Unleashed In either case, we see that the process gain is not constant and is inversely proportional to the outlet flow and the product of the heat transfer coefficient and area for the jacket. For a jacket coolant system with recirculation where there is equal volume displacement of coolant return flow by coolant makeup flow, so that jacket flow is constant, we have the following equation for the temperature of a mixture of makeup and recirculation flow: Tm=To- (To- Tc)*(Fc/ Fj) (2-13) The process gain (Kp) for the control of the jacket inlet temperature (Ti) by manipulation of coolant makeup flow (Fc) in a secondary loop is the partial derivative dTm / dFc: dTm/dFc = (To-Tc)/Fj (2-14) We see that the process gain is proportional to the temperature difference between the jacket outlet and the makeup coolant and is inversely proportional to the jacket flow. The process time constant for this secondary loop is negligible and the largest time constant becomes the thermal time lag of the thermowell. If we were to choose the outlet jacket temperature as the controlled vari able of the secondary loop, and the jacket is sufficiently mixed to do an energy balance on the entire coolant mass in the jacket, we would end up with: Mj*Cj*dTj/dt = Fj*Cj*Ti - Fj*Cj*Tj + U*A*(Tr-Tj) (2-15) If we combine terms, and divide through by the coefficient of the process feedback, we can identify the process time constant and gain for jacket temperature control. τp = (Mj*Cj) / [(Fj*Cj + U*A) (2-16) Kp = [(Fj*Cj) / (Fj*Cj + U*A)] * dTi / dFc (2-17) If we substitute in Equation 2-14, we end up with the process gain proportional to the temperature difference and inversely proportional to the jacket flow. Kp = [(To - Tc)*Cj ] / (Fj*Cj + U*A) (2-18) Chapter 2 - Setting the Foundation 79 For a total mass balance, the rate of accumulation of mass is equal to the mass flow in minus the mass flow out: p*Ar*dL/dt = Fi-Fo (2-19) Here we see that the integrator gain (Ki) for the level response is inversely proportional to the product of the fluid density (p) and the cross sectional area of the reactor (Ar). To derive the material balance for component A for a first-order reaction, the rate of accumulation of the mass of component A is equal to the mass flow in minus the mass flow out of component A minus the consumption of the mass of component A by the reaction. The concentration term (CA) is the mass fraction of component A. Mr*dCAo/dt = CAf*F - M*k*CAo (2-20) If we combine terms, and divide through by the coefficient of the process feedback, we can identify the process time constant and gain for reactor concentration control. τp = Mr /(Fo + Mr*k) (2-21) Kp = CAf /(Fo + Mr*k) (2-22Similar to what we saw for reactor temperature control, both the process time constant and gain are inversely proportional to the outlet flow and the reaction rate. Process Time Delay For sheet lines, webs, fibers, and conveyors, the time delay is simply the distance between the manipulated variable and the process variable divided by the speed. For example, the transportation delay for average thickness control of a sheet is the distance between the die bolts and the thickness sensor divided by the sheet speed. For plug flow, the entire residence time is a transportation delay. The time delay is the volume divided by the flow. The flow in pipelines, sample lines, static mixers, coils, and heat exchanger tubes can be considered to be essentially plug flow. For perfect mixing, the entire residence time is a process time constant. In well-mixed volumes of proper geometry and with baffles, the portion of the residence time that shows up as time delay can be estimated as half the 80 Advanced Control Unleashed turnover time, as shown in Equations 2-23 and 2-24, and most of the resi dence time becomes a process time constant [2.14]. τd = (0.5*ρ*ρ*V)/(Fo + Fa) (2-23) τp = [(ρ*V) / Fo]-τd (2-24Lastly, time constants in series create time delay. When the flow of material or energy can reverse direction depending upon the sign of the driving force, the time constants are interactive. Conductive heat transfer, gas flow in pipelines, and the tray response in columns all have interactive time constants. As the number of equal interactive and non-interactive time constants in series increases, the time delay increases, from 0.02 to 0.16 and 0.14 to 0.88 times the sum of the time constants respectively. The portion of the sum not converted to time delay is the process time constant for a first order-plus-dead time approximation. Thus, interactive time constants do not create much time delay and the time delay-to-time constant ratio is always rather nice. A large number of non-interactive time constants creates an extremely difficult to control dead-time-dominant system. Ultimate Gain and Period The ultimate gain (Ku is the controller gain that causes equal amplitude oscillations. It is equal to the inverse of the product of the open loop (static) gain (Ko) and the amplitude ratio at a phase shift of -180 degrees (AR-180) [2.20]. (2-25) The amplitude ratio for a sine wave applied to a self-regulating process with an open loop negative feedback time constant (τo) is: [2.20] (2-26) If Equation 2-26 is substituted into Equation 2-25, we have the ultimate gain as function of the open loop negative feedback time constant (τo) and the natural frequency (ωn). (2-27) Chapter 2 - Setting the Foundation 81 Since the natural frequency in radians per minute (ωn) is 2Π divided by the ultimate period (Tu), we can express the ultimate gain (Ku)as a function of the ultimate period. (2-28) For a time constant much larger than the time delay (τ »τd), the ultimate gain is: (2-29) Since for this case the ultimate period is about 4 times the time delay (Tu \?\ 4 * τd), the ultimate gain can be simplified to a ratio of the time constant to time delay. (2-30) Since the controller gain is a factor of the ultimate gain (Kc = 0.25*Ku), the controller gain is proportional to the time constant and inversely proportional to the time delay and the open loop gain. (2-31) If the time delay is much larger than the time constant (τd »τo), it can be shown that Equation 2-27 reduces to the ultimate gain being the inverse of the open loop gain. This relationship can also be realized from the amplitude ratio being 1 for a pure time delay. (2-32) The phase shift (f) from a time delay is: [2.20] \?\ = -360 * fn* τd (2-33) 82 Advanced Control Unleashed If we substitute in the relationship between natural frequency in cycles per minute and the ultimate period (fn = 1/Tu), we have: Tu = (-360/\?\)*τd (2-34) For a time constant much larger than the time delay there is a -90 phase shift from the time constant, which leaves only -90 phase shift (\?\) needed from the time delay to reach the -180 total phase shift; the ultimate period becomes simply 4 times the time delay. For τo » τd: Tu = 4*τd (2-35) If, on the other hand, the time constant is so much smaller than the time delay that essentially all -180 phase shift (\?\) comes from the time delay, the ultimate period approaches 2 times the time delay. For τd»τo: Tu = 2*τd (2-36) The following curve fit shows how the ultimate period changes from a multiple of 2 to 4 of the time delay and as a function of the relative sizes of the time constant and time delay. (2-37) Peak and Integrated Error Since a controller can neither sense nor compensate for an upset until after one dead time, the minimum peak error (Ex) is the exponential response after one dead time. Ex=[l-e(-τd / τo )]*Eo (2-38) If the time delay is less than the time constant, we can simplify the relationship. For τd τo: (2-39) Chapter 2 - Setting the Foundation 83 If the time delay is much less than the time constant, we end up with the ratio of the time delay to the time constant. For τd τo: (2-40) The minimum integrated error (Ei) is approximately the peak error (Ex) multiplied by the time delay and is thus proportional to the time delay squared. For τd τo : (2-41) If we use Equation 2-31, we see the peak error (Ex) is inversely proportional to the controller gain (Kc): For τd τo: (2-42) If we further realize that the integral time (Ti) is a factor of the ultimate period that is a multiple of the time delay, we have a relationship where the integrated error (Ei) is proportional to the integral time and inversely proportional to the controller gain. For τd τo: (2-43) For dead time-dominant loops, the peak error approaches the open loop error (Eo), and the integrated error approaches the product of the open loop error and the integral time. For τd»τo: (2-44) 84 Advanced Control Unleashed Feedforward Control The equations for feedforward control are derived from the steady-state material or energy balance at the point of control. For a heat exchanger where the bypass is throttled for temperature control, the feedforward equation for the hot bypass flow (Fhb) is: [2.19] Fhb= [(Tp- Thi) /(Thi -Tho)] * (Fhb) (2-45) Similarly, a steady state energy balance can be used to develop the following feedforward equation for the cold inlet flow (Fd) for a heat exchanger without a bypass. [2.19] Fd = [ Fho*Ch*(Thi- Tsp)]/ [Cc*(Tci- Tco )] (2-46) Dead Time from Valve Dead Band The time delay (τdv) from a controller output that must transverse the dead band after a change in direction is simply the valve dead band (DB) divided by the rate of change of the controller output (ΔCO / Δt): [2.19] (2-47) The rate of change of controller output depends upon the controller tuning and the error. If we consider the effect of just controller gain (Kc) for a loop dominated by a large time lag so that the amount of reset action used is small [2.19]: ΔCO / Δt = Kc * (ΔCV / Δt) (2-48) If we use Equation 2-31 for the controller gain with a detuning factor (Kx) and real