First-Order Plus Dead-Time Model: A Worked Example

A first-order plus dead-time model describes a response using a process gain K, a time constant τ and a dead time θ. In deviation variables, its transfer function is G(s) = K exp(−θs)/(τs + 1). It is a simplified representation, not proof that the physical process contains only one mechanism or one lag.

The example below is synthetic, linear and time invariant, with one input, one output and no added noise or competing disturbance. Its numbers are chosen to make the interpretation checkable.

At time zero, the input changes from 10 to 12 input units. The output starts at 20 output units and eventually approaches 30. The response remains at its initial value until 4 seconds and then follows a first-order rise with an 8-second time constant.

For t ≥ 4 seconds, the constructed output is y(t) = 20 + 10[1 − exp(−(t − 4)/8)]. Before that delay, y(t) = 20. The formula generates this table; the table is not an independently measured validation dataset.

The input change is 12 − 10 = 2. The final output change is 30 − 20 = 10. Process gain is therefore K = 10/2 = 5 output units per input unit. The stated dead time is θ = 4 seconds.

One time constant after the delay, the output has completed about 63.2% of its final change. The corresponding level is 20 + 0.6321 × 10 ≈ 26.321, reached at t = 12 seconds. Therefore τ = 12 − 4 = 8 seconds. Subtracting the delay matters: 12 seconds is not the time constant in this example.

These values describe process response. They are not PID gain, integral time or recommended controller settings.

A real record may contain drift, noise, nonlinear response, simultaneous input changes or insufficient settling. Those conditions can make a visual first-order approximation misleading. An input change produced inside a feedback loop also needs its operating context; it is not automatically equivalent to this idealized input.

• Record initial and final conditions and the input/output units.
• Distinguish the input-change time from the first observable output response.
• Check whether another input or disturbance changed during the record.
• Use separate data and operating conditions to examine whether the model remains useful.
• Inspect structured residuals and mismatched dynamics instead of relying only on visual fit.

This is a model-reading exercise, not an instruction to step an operating plant.

The system-identification guide separates estimation and validation. Review STA200 transfer-function identification training for the published syllabus. If your next subject is predictive control, compare the MPC horizon timing example without treating a single fitted curve as a ready-to-use controller model.

Technical definitions are supported by the primary references below. The labelled numerical records, timelines and checklists are original teaching examples for this article. They do not establish site-specific design approval or a new course entitlement.

• APMonitor: FOPDT model interpretation
• APMonitor: model assumptions and fitting limits

Browse the process-control learning library for the wider subject guides and course routes.