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Closed-loop transient response of sampleddata system for a unit step in set point.
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SAMPLED-DATA CONTROL SYSTEMS
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The stability boundaries represented by these three equations have been plotted by Mosler et al. (1966). The stability constraints for the case of n = 2, which require four inequalities, also may be found in Mosler (1966). This demonstrates that as the value of transport lag (UT) increases relative to a fixed sampling period T, the order of the characteristic equation increases and the stability criteria become more and more complex.
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TRANSIENT RESPONSE OF CLOSED-LOOP SAMPLED-DATA SYSTEMS
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We shall consider the transient response for the system shown in Fig. 26.1 for a step change in set point. For this particular disturbance, the block diagram may be drawn as shown in Fig. 26.4. For the special case of a step change in set point, the sampling switch and the hold in Fig. 26.1 can be placed in the forward loop of Fig. 26.4. For a step change in R occurring just before a sampling instant, it does not matter whether the sampling occurs before the comparator or after the comparator. The reason for redrawing the block diagram is that the expression for C(z) for Fig. 26.4 is simpler than the expression for Fig. 26.. 1. Using the method described in Chap. 23, one can obtain for C(z) (26.22) where G(z) = G,G,Gh(z) The expression for C(z ,m) is C(z*m) = G(zmYW 1 + G(z) (26.23)
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Equations (26.22) and (26.23) can also be found from Table 23.1. If one were to obtain expressions for C(z) and C(z ,m) for the system in Fig. 26.1, the result would be as follows: (26.24) and c(z,m) = RG,G,(zm) G(z,mWcG,(z) 1 + G(z) (26.25)
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FIGURE 26-4 Rearranged block diagram.
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S-PLED-DATA CONTROL OF A FIRST-ORDER PROCESS WITH TRANSPORT LAG
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If one were to use Eq. (26.24) or Eq. (26.25) for a step change in R, the result for c(nT) and c,&T) would be, of course, the same as that obtained using Eqs. (26.22) and (26.23); these latter two are simpler than Eqs. (26.24) and (26.25). The diagram in Fig. 26.4 may replace the diagram in Fig. 26.1 for the more general set-point function r(t), which is piecewise constant and where changes in r occur just before sampling instants. This more general function is a stair-step function. The transient response for the system shown in Fig. 26.4 will be considered in detail for two cases: Case I: no transport delay, i.e., a = 0 Case II: 0 < UT I T, i.e., it = 0 The results can be used to establish design criteria.
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Case I: No lkansport Lag
For this case, one can-show that Eq. (26.22) becomes C(z) R(z) where b = e- For a unit-step change in set point, R = z/(z - 1) and the response is expressed by K(l - b)z C(z) = (z - l)[z + K(l - b) - bl Inverting this expression gives
c(nT) = A(1 - [b - K(l - b)] }
K(l - b) z+K(l-b)-b
II = 0,1,2,...
(26.26)
This result was derived in Chap. 23 [Eq. (23.36)] and is presented again for convenience in developing design criteria. The transient response for a specific set of parameters, shown in Fig. 26.5, consists of arcs of exponential functions that intersect at sampling instants. It is possible to make the loop gain of the system (K) so small that the response is overdamped. In fact, the value of K, below which the response is overdamped, can be found by examining Eq. (26.26). When the expression in brackets, b - K( 1 - b), is greater than 0, the system no longer oscillates; therefore, we may write b - K(l- b) > 0 or for overdamped response l - b An overdamped response is also shown in Fig. 26.5.
b K-C-
(26.27)
SAMPLED-DATA
CONTROL
SYSTEMS
An overdamped response for a closed-loop system is usually considered too sluggish; consequently, the design criteria to be developed will be based on the underdamped response, such as the one shown in Fig. 26.5. Since the peaks (maximum and minimum) of the underdamped response occur at sampling instants, Eq. (26.26) may be used to compute overshoot and decay ratio. For a stable response, the ultimate value of c(t) is
This result, which is the same as that for a continuous proportional control system, should not be surprising if one recalls that the system s steady state is determined by steady-state relationships that are the same for both sampled-data control and continuous control. The first peak in the direction of set-point change occurs at t = T and the second peak in the same direction occurs at t = 3T. From this information one may compute from Eq. (26.26) the fractional overshoot and the decay ratio; the results are as follows: Fractional overshoot = Decay ratio = 40 - ~(~1 = K(l - b) - b cCw) (26.29) (26.30)
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