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Open-loop feedforward test to determine parameters for Cf.
transient for C shown as Curve I in Fig, 18.21. Comparing the shape of the transient with those of Fig. 18.19, we see that lead must predominate in Gf(s). The peak value occurs at tp = 2. Applying the rules in Table 18.1 gives TI = lSt, = 3 T2 = 0.7t, = 1.4 The feedforward controller transfer function is therefore Gf(s) = -(3s + 1)/(1.4s + 1) (18.12)
It is of interest to show the response of C for feedforward only when the feedforward transfer function of Eq. (18.12) is used; the result for a unit-step change in Ci is shown as Curve II in Fig. 18.21. When the Gf(s) of Eq. (lg.12) is used and the controller parameters for G,(s) are Kc = 2.84 and rl = 5.0, the feedforward-feedback response to a unitstep change in Ci is shown as Curve II in Fig. 18.22. For comparison, the response for feedback control only is also shown in Fig. 18.22.
RATIO CONTROL
An important control problem in chemical industry is the combining of two or
more streams to provide a mixture having a desired ratio of components. Examples of this mixing operation include the blending of reactants entering a chemical reactor or for the injection of a fuel-air mixture into a furnace. In Fig. 18.23~ is shown a control system for blending two liquid streams A and B to produce a mixed stream having the ratio K, in units of mass B/mass A. Stream A, which is uncontrolled, is used to adjust the flow of stream B so that the desired ratio is maintained. The measured signal for stream A is multiplied by
JTlGURE 18-21
Open-loop response for step change in C; for Example 18.3. CurveI: Gf = -1
Curve II: Gf = -(3s + 1)/(1.4s + 1)
266 PROCESS APPLICAT lONS
FIGURE 18-22 Comparison of conventional feedback control with feedforward-feedback control for Example 18.3. Curve I: PI control with
K, = 2.84,q = 5.0
Curve II: Feedforward-feedback control with K, = 2.84, r1 = 5.0, and Gf = -(3s + 1)/(1.4s + 1)
the desired ratio K, to provide a signal that is the set point for the flow-control loop for stream B. The parameter K, can be adjusted to the desired value. Control hardware is available to perform the multiplication of two control signals. A block diagram of the ratio control system is shown in Fig. 18.23b. In a flow-control loop, the dynamic elements consist of the controller, the flowI I
flow-hmasnHng
Ruid
Controller
flow-measuring element Lb Fluid B pB = Supply pressure
) 48
measured variable
FIGURE 18-23 (a) Ratio control system; (b) block diagram for ratio control (set point = G,,K,QA).
ADVANCEDCONTROLSTRATEGIES
measuring element, and the control valve. For incompressible fluids, there is no lag between the change in valve position and the corresponding flow rate. For this reason, the transfer function between the valve and the measurement of flow rate is simply unity. The block diagram also shows a transfer function G, that relates. the flow rate of B to the supply pressure of B. A transfer function G,, is also shown that represents the dynamic lag of the flow measuring element for stream A. From the block diagram, the flow of B may be written:
QB = GmtKrGcGv QA + 1 + GG; G PB 1 + Gc'SGm, c Y m2
The control action for a flow-control system is usually PI. The integral action is needed to eliminate offset and thereby establish a precise ratio of the mixed streams of A and B. Derivative action is usually avoided in flow control because the signal from a flow-measuring element is inherently noisy. The presence of derivative action would amplify the noise and give poor control.
DEAD-TIME COMPENSATION (SMITH PREDICTOR)
Processes that contain a large transport lag [exp ( -T~s)] can be difficult to control because a disturbance in set point or load does not reach the output of the process until TO units of time have elapsed. The control strategy to be described here, which is also known as dead-time compensation, attempts to reduce the deleterious effect of transport lag. Dead-time compensation, which is also referted to as a Smith predictor, was first described by 0. J. M. Smith (1957). Consider the single-loop control system of Fig. 18.24 in which the process transfer function Gp(s) is to be modeled by G,(s) = G(s)e-DS (18.13)
The right side of Q. (18.13) is the product of a transport lag [exp( - rgs)] and a transfer function G(s), which has minimum phase characteristics, such as l/(rs + 1). For convenience in developing the dead-time compensation method, only a change in set point R will be considered. If a step change is made in R,
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