FIRST ORDER PLUS DELAY CASE

Assume that the process can be described by the following first order plus delay transfer function 

We refer to as the normaJized desired closed-loop time constant and denote it ais fcj. Therefore, the desired closed-loop time constant and its normaJized form are related by = dfd. For a step setpoint change, we would expect the process output to take approximately (5fci -I- l)d = (5tcj -I- d) time units to reaich the new setpoint vaJue. 

Tb derive the PID tuning rules, we write the Y function defined in Equation (6.36) in terms of the scaled variable s (or w = dw) 

The frequency response of Y jw) is evaluated at wi = dwi = d and W2 = 2wi = 2dit i, where Tg is estimated from Equation (7.5) as (5fa + l)d. Given that the part of the denominator of y(ji5) involving w is dependent only on fa, then, for a fixed fa, we define for the first frequency wi 

Equation (7.6) crm be written in terms of its frequency response at wi and 2u i 

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Prom Equations (7.43)-(7.45), the normalized PID controller parameters can be calculated for a given set of performance parameters. In this case, tuning rules are more straightforward to derive than for the first order plus delay case because the normalized Y jw) is not a function of the process parameters. 

One should be careful to take data when other plant fluctuations are minimized. For many processes, the response of the system Ay (change in the measured value of the controlled variable using a sensor) follows the curve shown in Fig. 18.57 (see case 6 in Fig. 18.50). Also shown is the graphical fit by a first-order plus time delay model, which is described by three parameters ... 

Suppose we have a step response of the process from which we can obtain an estimate of the process settling time Tg and time delay d. In this case, we can crudely approximate the response by that of a first order plus de-... 

By inspection determine which of the following process models can be approximated reasonably accurately by a first-order-plus-time-delay model. For each acceptable case, give your best estimate of 0 and t. 

OT the first-order-plus-time-delay model of Example 12.4, [le PI controller for case (b) provided the best distur-ace response. However, its set-point response had a sig-ficant overshoot. Can set-point weighting significantly educe the overshoot without adversely affecting the set-... 

For a better fit of the system response, the method of Oldenbourg and Sartorius, as described in Douglas (1972), using a combination of two first-order lags plus a time delay, can be used. The method is illustrated in Fig. 2.24. and applies for the case... 

This is a more realistic response than case 1 because it allows for a time delay as well as first-order behavior, and can be applied to many chemical processes. The delay time may not be clearly identified as transport lag but may be imbedded in higher order dynamics. Hence, this is an approximate but very useful model, especially for staged systems such as distillation or extraction columns. An extension of this model is second order plus time-delay, the response of which is equivalent to case 2 or case 4 with an initial time delay. The advantage over first-order models is that an additional model parameter can give greater accuracy in fitting process data. 

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