Contents:

Approximating time constants 3

Damping 4

Simple performance measures 6

The integral criteria 7

Task performance 12

References 13

1. Approximating time constants

Developing a low-order model for a process invariably involves approxima-tions. In some cases, we approximate a dead time by using multiple small timeconstants. Conversely, we can approximate the contribution of one or moresmall time constants with a dead time. The responses in Figure 3.18 illustrate approximating a dead time using multiple time constants. For example, a dead time of 1.0 min could be approximated by 100 time constants of 0.01 min each. As the number of time constants increases, the approximation becomes better and better. One extreme is using one time constant

(n = 1) to approximate the dead time. This is not a very good approximation. Although increasing the value of n improves the approximation, it also increases the complexity of the model that uses the approximation.We rarely approximate a dead time with a large number of time constants, however, this approximation goes both ways. That is, several small time constants in series can be approximated by a dead time.

Apparent Dead Time.

True dead time is the result of a transportation delay within the process. But when interpreting process behavior from response data, several time constants in series can give the appearance of dead time.One such situation involves trays in a distillation column. The feed is all-liquid, so increasing the feed rate will increase the liquid rate from the bottom tray. But when the feed rate to the column is changed, some time will elapse before the feed rate change is reflected in the bottoms level. When the liquid flow to a tray increases, the amount of liquid retained on the tray (the tray holdup) increases slightly, which results in a lag (called the hydraulic lag) in the liquid flow leaving the tray. The hydraulic time constant for each tray is usually between 5 and 10 seconds. A column with 24 trays below the feed tray will have a total lag between 120 and 240 seconds.

The response in the bottoms level will give the appearance of a dead time in the range of 2 to 4 minutes. This dead time is not true transportation lag, but it is the cumulative effect of the small hydraulic time constants associated with the trays.

Approximating High - Order Time Constants with a Dead Time.

Althoughapproximating a dead time with multiple time constants is unusual, approximating several small time constants by a dead time is common. The two responses in Figure 3.20 illustrate the use of a dead time to approximate several small time constants.The original process has five time constants. In the approximation, the three smallest time constants have been replaced with a dead time whose value is the sum of the three smallest time constants. The approximation is very close to the response of the original process.When developing process models, it is customary to determine the largest (or major) process time constant and possibly the next largest (minor) processtime constant. A dead time is then added to represent the remaining timeconstants plus the true transportation lag, if any, in the process.