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FIG 1.23. If the proportional band is widened to 200 T /T,, the intermediate variable will not overshoot.
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n Ud erstanding Feedback Control
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zone is still twice the difference between tank level and measurement, just as it was at 50 percent load, so the results will be the same. Therefore the proportional band should always be 2004~~ for critical damping, regardless of the load. Only the bias need be changed. Critical damping makes for sluggish response, however. In most cases, some overshoot is not detrimental. It is important that we determine what is necessary to achieve >i-amplitude damping. Knowing that the period at which the two-capacity loop naturally oscillates is zero, we can be sure t hat any oscillation at a period of 2.572 will be damped. The period of 2.5~~ is chosen as it seems to be the natural period of the first cycle (Fig. 1.21). Since we know that oscillations cannot be sustained, let the loop gain at 70 = 2.5~~ be 1.0: G(+!!=lO 1 P .
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Substituting for the dynamic gains of ~~ and 72,
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Substituting 2.5~~ for TV, 2.5~2 2.5~2 P = 100 ~ ~ 2TTl 2lrr2 r=1s; (1.25)
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This is the proportional band which will produce >i-amplitude damping. If the method of arriving at these conditions seems somewhat arbitrary, compare the results against those previously established :
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> & a m p l i t u d e Overshoot.. Critical.
P, % of n/n . 0
The proport ional band of l&2/71 fits right in with t he rest of the table. Gross changes in P are required to affect the damping of the two-capacity process. It is doubtful whether any difference would be discernible between the response of a loop at 30 percent and that at 16 percent. Unfortunately, this is not always so. The two-capacity process has more tolerance for proportional band setting than any more difficult process. Earlier in the chapter it was noted that the damping of the dead-time loop is changed from zero to >i-amplitude by doubling the proportional
T2/T1
Dynamic Elements in the Control Loop
band. With the two-capacity process, however, the multiplication is infinite. Another important factor must be brought out. By definition of the primary and secondary capacities, 72 is never greater than ~1, regardless of their relative positions in the loop. This means that the most difficult two-capacity process will be one where 72/~1 = 1.0. For >/4-amplitude damping, P would be 16 percent. By comparison, the dead-time process is 209{,3 or 12.5 times more difficult to control than the most difficult two-capacity process. Notice also that as 72 approaches zero, the process approaches single capacity and P for any damping approaches zero. It is wise therefore, in the design of the process, to make T~/T~ as low as possible. Since the natural period of the loop varies as r2 only, this should be done by reducing 72 where possible, instead of increasing 71.
Proportional-plus-derivative Control
Adding derivative to a proportional controller relates output to the rate of change of error:
rn=$ (e+D$)+b
(1.26)
where D is the derivative time. The parenthetic part of this expression is the inverse of a first-order lag-it is called a first-order lead. In the two-capacity-level process,
de c+rzdt=h
where c is the result of changes in h. In the proportional-plus-derivative controller, m is the result of changes in e-the derivative term is on the input side of the equation. Since c = r - e, the lag may be written in terms of e:
If the set point is constant, dr/dt = 0.
e+r2a=r-
Rearranging,
If the derivative time of the controller is set equal to 72, the above expression can be substituted into the proportional-plus-derivative controller equa,tion, with the result
nz = F (T - h) + b
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We now have proportional control of the intermediate variable. Adding derivative has caused cancelation of the secondary lag, making the process appear to be single-capacity. In theory, the proportional band may then be reduced to zero and still produce critical damping. In practice, it is not possible. The gain of a derivative term, 2aD/r,, approaches infinity as the period of the input approaches zero. Noise is a mixture of random periodic signals. A small amount of noise at a high frequency (low period) would be amplified tremendously by a perfect derivative unit. In addition, controllers are made of mechanical or electrical parts that have certain inherent properties of phase lag. Consequently, a high limit is always placed on GD, preventing high-frequency instability within the controller. This high limit is usually about 10. A real derivative unit is actually a combination of a lead whose time constant is D and a lag whose time constant is D/10. In the two-capacity process, then, setting D = 72 will not completely cancel 72, but will replace it with a lag equal to ~~/10. The effect is considerable, however, in that the characteristics of the same process under proportional control are improved tenfold. For pi-amplitude damping with proportional-plus-derivative control, P = 1.672
D = 72
70 = 0.2572
(1.27)
Being able to reduce P by 10 also reduces offset by 10. And as a bonus, the loop cycles 10 times as fast as before. Derivative always has this effect, although nowhere else is it so pronounced as in a two-capacity process. There is one best value of derivative for a given control loop. TO O high a setting can be as harmful as none at all. The object is to cancel the secondary lag in the process. If D > TV, the controller will lead the intermediate variable, causing premature throttling of the valve. Figure 1.24 shows the effect of three different derivative settings on the same process.
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