to = t0.632DO - t (2-5.3)

The units of t and to are the same time units as those used by the control system.

Now that t and to have been evaluated, we proceed to evaluate K. Following the

de¬nition of gain,

(156 - 150)∞C 6∞C ∞C

K= = = 1.2

(55 - 50) %CO 5 %CO %CO

31

OBTAINING PROCESS CHARACTERISTICS FROM PROCESS DATA

This gain says that at the present operating condition, a change of 1%CO results in

a change of 1.2°C in outlet process temperature. This gain correctly describes the

sensitivity of the outlet process temperature to a change in controller output.

However, this gain is only a partial process gain and not the total process gain. The

total process gain is the one that says how much the process output, c(t) in %TO,

changes per change in process input, m(t) in %CO; the reader may refer to Fig. 2-

1.1b to understand this point. That is, the process output is given by the transmitter

output and it is not the temperature. Therefore, we are interested in how much the

transmitter output changes per change in controller output, or

Dc change in transmiter™s output, %TO

K= = (2-5.4)

Dm change in controller™s output, %CO

The change in transmitter output is calculated as follows:

Ê 6 ∞C ˆ

Dc = 100% = 4 %TO

Ë 150∞C ¯

or, in general,

Ê DPV ˆ Ê change in process variable in engineering unitsˆ

Dc = 100%TO = 100%TO

Ë span ¯ Ë ¯

span of transmitter

(2-5.5)

Therefore, the total process gain for this example is

4%TO %TO

K= = 0.8

5%CO %CO

If the process variable had been recorded in percent of transmitter output, there

would be no need for any extra calculation.

We can now write the transfer function for this process as

C ( s) 0.8 e - to s

=

M ( s) t s+1

This transfer function describes the relation between the transmitter output and the

controller output. If a transfer function describing the relation between the trans-

mitter output and any other process input (other than the controller output) is

desired, the same procedure is then followed to evaluate K, t, and to. In this case

the units of the K will be different than before; that is, they will not be %TO/%CO.

The units will depend on the units of the particular input.

To illustrate the approximation using the two-point method, consider a third-

order process described by the following transfer function:

32 PROCESS CHARACTERISTICS

Figure 2-5.2 Response of process and FOPDT model.

O(s) 1.2

=

I (s) (3s + 1)(3s + 1)(4 s + 1)

Suppose that the input I(t) is changed by 10 input units at time 10 time units. The

output O(t) is recorded, and the mathematical calculations described above are

followed. The FOPDT model calculated is

O( s) 1.2 e -4.25 s

=

I ( s) 6.75 s + 1

Figure 2-5.2 shows the responses of the process and of the model to the same step

change in input. The responses are quite close. Note that the model shows a longer

dead time than the “apparent dead time” in the process. This will always be the case,

and it is done in an effort to minimize the area between the two curves.

As mentioned earlier, the procedure just presented provides the “best” approx-

imation of a higher-order (or multicapacitance) process by a ¬rst-order-plus-dead-

time (or single-capacitance) process. It provides an important tool to process control

personnel.

2-6 QUESTIONS WHEN PERFORMING PROCESS TESTING

The following questions must be answered when performing process testing.

1. In what direction should the controller output be moved?

Safety is the most important consideration. You always want to go in a safe

direction.

33

REFERENCE

2. By how much (%) should the controller output be moved?

Move by the smallest amount that gives you a good (readable) answer. There are

two reasons for this: (1) if you move the process far from its present steady state,

this makes the operating personnel nervous; and (2) you want to obtain the char-

acteristics close to the operating point, because much away from it, nonlinearities

may start having an effect.

3. How many tests should be performed?

We want to have repeatability in the results; therefore, we could say that we

should have as many tests as possible to ensure repeatability. However, many tests

are not realistic either. To start, perhaps two tests, in different directions, are enough.

Once the numerical values of the characteristics are obtained, they can be com-

pared, and if not similar, more tests may be justi¬ed. If they are similar, an average

can then be calculated.

4. What about noise?

Noise is a fact in many processes. Once a recording is obtained, an average

process curve can be drawn freehand to obtain an average curve. This in itself is a

way to ¬lter the noise.

5. What if an upset enters the process while it is being tested?

The purpose is to learn how the manipulated variable affects the controlled vari-

able. If a disturbance enters the process while it is being tested, the results will be

due to both inputs (manipulated variable and disturbance). It is very dif¬cult to

deconvolute the effects. This disturbance may be the reason why two tests may

provide much different results. The test should be done under the most possible

steady-state conditions.

2-7 SUMMARY

In this chapter we have only considered the process. We have looked at the char-

acteristics, or personality, of processes. Speci¬cally, we de¬ned and discussed the fol-

lowing terms: process, self-regulating and non-self-regulating processes, integrating

and open-loop unstable processes, single- and multicapacitance processes, process

gain, process time constant, process dead time, process nonlinearities, and transfer

functions. Finally, we presented and discussed a method to evaluate the process

characteristics empirically.

What we learned in this chapter is most useful in tuning controllers (Chapters 3

and 4), in evaluating control system stability (Chapter 6), and in designing advanced

control strategies (Chapters 7 to 9).