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Proportional-Integral Control for Set-Point Change
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Again, the controller transfer function is K,(l + l/~~s), and we obtain from Fig. 13. le the transfer function T K,A(1 + lhIs)~l/(m + I)] -= (13.12) 1 + K,A( 1 + l/~~s)[l/(~s + l)] G This equation may be reduced to the standard quadratic form to give T qs,+ 1 -= (13.13) T S + 2Jqs + 1 Tli where ~1 and 4 are the same functions of the parameters as in Eq. (13.9). Introducing a unit-step change (Ti = l/s) into Q. (13.13) gives 71s + 1 T = 1 (13.14) s +2 + 2gqs + 1
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RESPONSE OF SIMF LE CONTROL SYSTEMS
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K,=l 5 =0.2 A=1 7-1 I
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Unit-step response for set point change (PI control).
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obtain the response of T in the time domain, Eq. (13.14) is expanded into two terms: 1 1 71 T = (13.15) TfS + xqs + 1 + STlS 2 +yqs + 1 - 2
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The first term on the right is equivalent to the response of a second-order system to an impulse function of magnitude 71. The second term is the unit-step response of a second-order system. It is convenient to use Figs. 8.2 and 8.5 to obtain the response for Eq. (13.15). For f < 1, an analytic expression for T is
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T = 71 Jhe-lt l
sin J1-52:
(13.16) sin J-i + tan- 47 t ) i The last expression was obtained by combining Eqs. (8.17) and (8.31). A typical response for T is shown in Fig. 13.5. The offset as defined by Eq. (13.3) is zero; thus Offset = T;(m) - T (m) + 1 - J&e-i q
El-l=0
Again notice that the integral action in the controller has eliminated the offset.
Proportional Control of System with Measurement Lag
In the previous examples the lag in the measuring element was assumed to be negligible, for which case the feedback transfer function was taken as 1. We now consider the same control system, the stirred-tank heater of Fig. 13.1, with a firstorder measuring element having a transfer function l/(r,s + 1). The block diagram for the modified system is now shown in Fig. 13.6. By the usual procedure, the transfer function for set-point changes may be written
T Al(~ms + 1) -= r&s2 + 24272s + 1 TA
(13.17)
LINEAR
CLCJSED-LOOP
SYSTEMS
FIGURE 13-6 Control system with measurement lag.
where A, =
&A l+ K,A
rrrn J
1 +K,A 1
7 + Tm = -
2JxJGzJ
We shall not obtain an expression for the transient response for this case, for it will be of the same form as Eq. (13.16). Adding the first-order measuring lag to the control system of Fig. 13.1 produces a second-order system even for proportional control. This means there will be an oscillatory response for an appropriate choice of the parameters T, T,, K,, and A. In order to understand the effect of gain K, and measuring lag rm on the behavior of the system, response curves am shown in Fig. 13.7 for various combinations of K, and T, for a fixed value of T = 1. In general, the response becomes mom oscillatory, or less stable, as K, or r,,, increases.
4 (a)
FIGURE 13-7 Effect of controller gain and measuring lag on system response for unit-step change in set point.
l-RANSIE!NT
RESPONSE OF SIMPLE CONTROL SYSTEMS
For a fixed value of T,,, = 1, Fig. 13.7~ shows that the offset is reduced as K, increases; however, this improvement in steady-state performance is obtained at the expense of a poorer transient response. As K, increases, the overshoot becomes excessive and the response becomes more oscillatory. In general, we shall find that a control system having proportional control will require a value of K, that is based on a compromise between low offset and satisfactory transient response. For a fixed value of controller gain (K, = 8)) Fig. 13.7b shows that an increase in measurement lag produces a poorer transient response in that the overshoot becomes greater and the response more oscillatory as T,,, increases. This behavior illustrates a general rule that the measuring element in a control system should respond quickly if satisfactory response is to be achieved.
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