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(1.17)
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where 7 = V/F. GrGR = 1.0
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If an integrating controller is used to close the loop,
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(2) (2%) = l-O
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(1.18)
Replace the metering pump in Fig. 1.14 with a valve. Then an increase in liquid level would inherently increase the outflow. This action works toward the restoration of equilibrium and is called self-regulation. It is as if a proportional controller were at work within the process. This is a natural form of negative feedback. Although the relationship is in fact not linear, assume for the moment that flow out of the tank is proportional to the head of liquid above the valve: f,, = kh The level will remain steady when f0 = fi, which indicates that every condition of inflow will bring about a new steady-state level:
h=& k
In proceeding from one steady state to another, however, the level will vary with time. With a step increase in fi, the level will start to change at the same rate as in the non-self-regulating case, because outflow has not yet begun to increase. The rate of rise of level will then diminish with time, as j0 approaches fi. As a result, the fina level will only be reached in infinite time. dh - = $ (j-i - fo> dt Substituting for fO, dh - = ; (fi - kh) dt The next step is to solve for h, the controlled variable:
h+idk=f,
Flcdt lc
(1.19)
This is known as a first-order differential equation. The controlled variable h is related to the manipulstcd variable ii, both in the steady st:rt)e
Dynamic Elements in the Control Loop
and with respect to time. form
This particular differential equation is of the
which describes a first-order lag whose time constant is 71 and whose steady-state gain is K. In the level process, 71 = T-/Fk and K = l/k. The solution of the equation for a step input is c = K7,2(1 - e-l l) (1.21)
which is plotted for t)he level process in Fig. l.lG. After an elapsed time equal to TV, 63.2 percent, of t,he distance to the nest steady state \vill have been traversed. After another 71 has elapsed, G3.2 percent of what was left will have been traversed, and so forth. At the beginning of the step response, the self-regulating process resembles the non-self-regulating or integrating process. But after sufficient time, it resembles a process lvithout dynamics. The first-order lag is thus made up of two components, one responsive to a fast-changing input, the ot her responsive to a steady input. This is apparent from examining the differential equation
The relation between level and inflow is the sum of k-o out-of-phase components. The derivative term lends the steady-state term by no , just as integrating produced a 00 phase lag. The gain of the derivntive term to a signal of period 7 is exactly the inverse of the gain of an integrator: G
= 27rV/ F
FIG 1.16. The slope of the response curue equals the departure from steady state divided by T,.
h~fi/k
63.2% i
*4---T1-
n Ud erstanding Feedback Control
FIG 1.17. The vector sum is the gain of inflow with respect to level.
The summation of the two components of h with respect to fi is d grammed in Fig. 1.17. The only difficulty with this vector diagran that the resultant is the ratio of inflow to level. The inverse of resultant represents level vs. inflow, which is the response we are look for:
The steady-state gain l/k may be broken out separately: (1.5 where 71 = V/Fk, as before. A plot of G1 vs. 7O in Fig. 1.18 shows a curve which is complem~enta to that of a proportional-plus-reset controller. Because we are principally concerned with the dynamic behavior the loop, the asymptote containing 7O is of prime importance.
Notice that this dynamic-gain asymptote does not contain Ic. In fa it is identical to the gain of the non-self-regulating process. Althoug the steady-state gain can be changed simply by turning the valve at tl bottom of the tank, this does not affect the dynamic gain.
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