The first-order plus time-delay(FOPTD) model and the second-order plus time-delay (SOPTD) model have been widely used to design and implement process controllers. And the terms related to the low-order plus time-delay processes are very useful for describing the dynamic characteristics of the process or the closed-loop control system. In this section, the dynamic behaviors of the FOPTD and SOPTD processes for the step input are derived and the important terms are introduced.
The FOPTD process is
where k, τ and θ are called the static gain, the time constant and the time delay respectively. The step input response (3.3) can be obtained by solving the differential equation (3.1) for the initial conditions (3.2):
The following equations can be obtained from (3.3):