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Block diagram of temperature-control system.
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The block diagram is shown in Fig. 13. Id. The block representing the process is taken directly from Fig. 9.3. To reduce the number of symbols I/WC has been replaced by A in Fig. 13.1.~. Throughout this chapter, we shall assume that the valve does not have any dynamic lag, for which case the transfer function of the valve (G t in Fig. 13.1) will be taken as a constant K,. This assumption was shown to be reasonable in Chap. 10. To simplify the discussion further, K, has been taken as 1. (If K ,, were other than 1, we may simply replace G, by G,K, in the ensuing discussion.) In the first part of the chapter, we shall also assume that there is no dynamic lag in the measuring element (rm = 0), so that it may be represented by a transfer function that is simply the constant 1. A bare thermocouple will have a response that is so fast that for all practical purposes it can be assumed to follow the slowly changing bath temperature without lag. When the feedback transfer function is unity, the system is called a unity-feedback system. Introducing these assumptions leads to the simplified block diagram of Fig. 13.le, for which we shall obtain overall transfer functions for changes in set point and load when proportional and proportional-integral control are used.
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For proportional control, G, = K,. Using the methods developed in the previous chapter, the overall transfer function in Fig. 13. le is T K,A/(m + 1) &A -= 1 + K&m + 1) = rs + 1 + K,A G (13.1)
TRANSIENT
RESPONSE OF SlMPLE
CONTROL SYSTEMS
This may be rearranged in the form of a first-order lag to give T Al -= 71s + 1 TA where ~1 = A1 =
(13.2)
1 + K,A 1 1 + K,A = 1 + l/K,A
According to this result, the response of the tank temperature to change in set point is first-order. The time constant for the control system, ~1, is less than that of the stirted tank itself, r. This means that one of the effects of feedback control is to speed up the response. We may use the results of Chap. 5 to find the response to a variety of inputs. The response of the system to a unit-step change in set point Ti is shown in Fig. 13.2. (We have selected a unit change in set point for convenience; responses to steps of other magnitudes am obtained by superposition.) For this case of a unit-step change in set point, T approaches A1 = K&( 1 + K,A), a fraction of unity. The desired change is, of course, 1. Thus, the ultimate value of the temperature T (a) does not match the desired change. This discrepancy is called o&t and is defined as (13.3) Offset = T;(w) - T (m) In terms of the particular control system parameters 1 (13.4) Offset = 1 - 1 :tA = 1 + K,A c This discrepancy between set point and tank temperature at steady state is characteristic of proportional control. In some cases offset cannot be tolerated. However, notice from Eq. (13.4) that the offset decreases as K, increases, and in theory the offset could be made as small as desired by increasing K, to a sufficiently large value. To give a full answer to the problem of eliminating offset by high controller gain requires a discussion of stability and the response of the system when other lags, which have been neglected, are included in the system. Both these subjects are to be covered later. For the present we shall simply say that whether or not proportional control is satisfactory depends on the amount of offset that can be tolerated, the speed of response of the system, and the amount of gain that can be provided by the controller without causing the system to go unstable.
00 t
FIGURE W-2 Unit-step response for set-point change (P control).
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