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0.5 - - - - - B c 0 !L 0 4 20 input to valve, ma
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Plant start-up illustrating reset windup (tanks are initially empty): (a) process, (b) linear valve with saturation limits, (c) block diagram of process.
M I C R O P R O C E S S O R - B A S E D CONTROLLERS
AND DISTRIBUTED CONTROL
Fig. 35.3b. Upon start-up with the PI controller in automatic mode, the tanks are empty, and the error (R - C) is large and positive. The action of the controller on this error will result in a large output M due to proportional action and a rising contribution to M due to the integral action. The output of the controller will be at its saturation value, which is typically about 10% above the top of the 4 to 20 ma scale (i.e., 22 ma). The large saturated value of M will in turn cause the valve to reach its saturation value, which has been taken as 0.5. During the initial phase of the operation, the tanks are being filled at the maximum rate of flow provided by the upper limit of the control valve. During this filling stage of operation the controller is not exercising any control since the valve is at its limit. As the level rises toward the set point, the large error that existed at start-up gradually diminishes toward zero. If only proportional action were present in the controller, the output of the controller would return quickly to a mid-scale value; however, because of the integral action, the controller output remains high, at its saturation value, long after the process variable first reaches the set point. To reduce the output M, the integral action must be applied to-negative error so that the integration can Iower the output to mid-scale. This negative error occurs as a result of the tank level remaining above the set point for some time after the tank level reaches the set point. Other causes of reset windup and some methods to prevent it are discussed by Shinskey (1979). The control system shown in Fig. 35.3~ was simulated for a start-up transient with the tanks initially empty; the transient is shown as Curve I in Fig. 35.4. The large overshoot in tank level after the level reaches the set point is clearly illustrated. Now that the problem of reset windup has been described, we focus our attention on how to reduce or eliminate it. The development that follows on the use of external feedback to eliminate reset windup is based on the work of Shunta and Klein (1979). A feature of microprocessor-based controllers is the availability of external feedback in the configuration of a PI or PID controller. The block diagram of a PI controller with external feedback is shown in Fig. 35.5. The output of this controller is given by
M(t) = K& (r) + -
71 I0
e(t)dt + $ j, [F(t) - M(t)]dt
(35.1)
I No
external feedback
dback FIGURE 35-4
Start-up transients for system in Fig. 35.3 with and
t without external feedback.
COMFVTERS
IN PROCESS CONTROL
Controller Set point -+I[-+ M Control variable -
FIGURE 35-5 Controller with external feedback for use in anti-reset
where M(t) = controller output e(t) = error = setpoint - control variable F(t) = external feedback signal If the Laplace transform of both sides of Eq. (35.1) is taken, the result is &e(s) M(s) = K,e(s) + -
+ $w - M(s)1
If the feedback signal is the controller output, F(s) = M(s), Eq. (35.2) becomes the usual transfer function for a PI controller: M(s) = K, 1 + $ e(s) i i The feedback signal F(t) can be any signal available to the microprocessor-based controller. When F(t) is not equal to M(t), Eq. (35.2) can be solved for M(s) to give F(s) M(s) = K,e(s) + 71s + 1 A controller following this equation provides a signal consisting of proportional action plus first-order tracking of F(t). If F(t) in Fq. (35.1) is taken as the output of the valve (or the output signal of the current-to-pressure transducer that goes to the valve) in our example in Fig. 35.3c, we have the basis for eliminating reset windup. During the filling stage of the tank, the feedback signal F(t) will be constant at the saturation value of the valve output. When the tank level reaches the set point, the error will be zero and the only contribution from the controller output will be the tracked signal represented by the second term on the right side of Eq. (35.4). This value will be less than would be the case if external feedback were not employed. The overall result is that the controller output is less with the external feedback at the time the level first equals the set point and the overshoot is reduced. The transient using external feedback is also shown in Fig. 35.4 as Curve II. Notice that the overshoot is less when external feedback is used. To emphasize the benefit of external feedback for eliminating reset windup, no limits were placed on the output of the controller in the simulation of Fig. 35.3. In practice, there are physical limits on the controller output, and when this is the case, the reduction of overshoot with the use of external feedback may not be so pronounced as shown in Fig. 35.4.
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