4.2 First-Order + Dead Time

Recall that a first-order + dead-time process, represented by the transfer function relationship

Equation 4.1

has the following output response to a step input change,

Equation 4.2

where the measured output is in deviation variable form. The three process parameters can be estimated by performing a single step test on the process input.

The gain is found as simply the long-term change in process output divided by the change in process input. Also the time delay is the amount of time, after the input change, before a significant output response is observed. There are several easy ways to estimate the time constant for this type of model.

Time for 63.2% Approach to New Steady State

In Equation (4.2), if we set t = t_p + q, we find y(t = t_p + q) = k_pDu(1 - exp(–1)) = 0.632k_pDu. Now, since k_pDu is simply the long-term change in the process output, then t_63.2% = t_p + q is the amount of time it takes for the output to make 63.2% of its ultimate change. This method of estimating the time constant is illustrated in Figure 4-1.

Example 4.1: Numerical Application of 63.2% Method

Consider the response to a step input change at t = 1 minute, shown in Figure 4-2. The measured output is shown as the noisy solid curve, and a "best fit" first-order + time-delay model is shown as the dashed curve. Here we find the process gain from

The time delay is observed to be 1 minute, and the time constant is the time it takes (after the time delay) for the output to change by 0.632(-2) = -1.3°C. That is, the time when y = 23.7°C. In this case,

and the process transfer function is

Maximum Slope Method

We can also find (see Exercise 1) that the maximum slope of the output response to a step input change at t = 0 occurs at t = q and is

So the time constant can be estimated from

A major disadvantage is that the slope estimate may not be accurate if there is significant measurement noise. This approach is particularly useful for higher-order responses that are approximated as a first-order + time-delay response, as shown in the following example.

Example 4.2: Maximum Slope Method

The response of a high order process is shown in Figure 4-3. Here, the dashed line is drawn through the maximum slope of the output response. The slope is

Since the process gain is k_p = Dy/Du = 6°C/2 gpm = 3°C/gpm, and the time-delay is 8-1 = 7 min, the first-order + time-delay transfer function is estimated as

Two-Point Method for Estimating Time Constant

A technique similar to the first one, but using two points on the output response, is shown in Figure 4-4. Here the time required for the process output to make 28.3% and 63.2% of the long-term change is denoted by t_28.3% and t_63.2%, respectively. The time constant and time delay can be estimated from (see Exercise 2)

Limitation to FODT Models

The primary limitation to using step responses to identify first-order + dead-time transfer functions is the amount of time required to assure that the process is approaching a new steady state. That is, the major limit is the time required to determine the gain of the process. For large time constant processes it is often desirable to use a simpler model that does not require a long step test time. The integrator + dead-time model shown next will often be good enough for control-system design.