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ControllersI 93
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error-so this criterion is only applicable to those systems without a steady state. Technically, the IAE and the ISE are the only all-encompassing indices of performance. The principal distinction between them is the weight placed on large errors. Two response curves with the same IAE would have different values of ISE if t,here were a difference in error magnitude. For this reason, the ISE criterion is seen to be a combination of error magnitude and IAE. For the case where the response curve lies wholly on one side of zero error, the integrated error equals the IAE. But this is not the limit of usefulness of the integrated error, for it represents the average error that has existed over a particular time span. The average error or integrated error is a valid basis for comparing response curves with equal damping, like those comparisons shown in Fig. 1.13. By specifying the damping, the objection raised in number 1 above is overruled. Integrated error will therefore be used as a performance index throughout the balance of the book, and in every case >i-amplitude damping will be meant, unless otherwise indicated.
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The choice of integrated error as a performance index has a very practical aspect, in that it can be readily calculated from controller settings. In a proportional-plus-reset controller,
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Prior to a load change, at time t 1, the output will be stationary at a level VL~, and the error will be zero. After the transient from a load change has subsided, i.e., at time tz, the output will come to rest at a new level ~122, at which the error will again be zero. Then, subtract,ing the two outputs,
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Let the integrated error resulting from the load change AH be designated B: I!: = I;f e dt
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1 Selecting the Feedback Controller
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Then the load respouse of a giver1 control loop cau be assessed on the basis of integrated error per unit load change: E PR - =AT % 100 Again this is integrated error and not ME; the damping of the loop must be assured before this ndex can be used. The load-response criteriou E/A/H depends ou the proportioual baud and reset t,irne which, in t,urn, depend on the characteristics of the plant. This is another way of illustrutiug the difficulty of coutrol which was described in Chap. 1. If the proportional baud can be made to approach zero because of he ease with which the process can be contcolled, or the reset t,irue because of its speed of response, E/AM will approach zero. The integrated error will be found useful iu evaluating not ouly the difficulty of a process, but also the effectiveness of the means used in its control.
Error Magnitude
The magnitude of an error is a fuuctiou of how fast the load change takes place. If the load change is very gradual, several orders of nmgnitude longer in duration thau the reset time, it may produce no nieasurable error magnitude; the ntcgmted error, however, does not depend on rate of change of load. An instantaneous load chauge will be countered by proportional control action, aud derivative, if used. If the proportional-plus-reset control equation is written in the differeutial forni, dm dt=
(4.3)
A plot of e versus t,he rate of change of load dw/dt can be constructed from it (see Fig. 4.1). The maximum value of e is limited by proportional action to P A~,/100. Derivative nctiou can reduce the effect of a rapid load change principally by allowing a reduction in the proportioual baud setting.
FIG 4.1. The magnitude of the error is a function of the rate of change of load as well as its magnitude.
Linear Controllers 95 I
TWO- AND THREE-MODE CONTROLLERS
Although the primary functions of proportional, derivative, and reset have already been introduced, many of their features remain to be defined. The discussion will be restricted to the commonly available controllers, i.e.,. proportional-plus-derivative, proportional-plus-reset, and proportional-plus-reset-plus-derivative. (Reset controllers are rarely used in process work and are not available as standard items from most manufacturers. Derivative by itself is not recognized as a controlling mode.)
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