First-Order Plus Dead Time (FOPDT)

Sometimes also referred to as the First-Order Plus Time Delay model.

It’s often used to model the dynamic behavior of systems.

\tau_p \frac{dy(t)}{dt} = -y(t) + K_p u(t - \theta_p)

where:

$y(t)$ : The output of the system.
$u(t)$ : The input to the system.
$K_p$ : The process gain. Ratio of the change in the output to the change in the input ( $\frac{\Delta y}{\Delta u}).$
$\tau_p$ : The time constant. How quickly the system’s output responds to a change in the input. The time it takes to go from one steady state to another.
$\theta_p$ : The dead time (or time delay). The time lag between a change in the input and the start of the output’s response.

The values of $K_p$ , $\tau_p$ and $\theta_p$ are usually found with data from a step response, by minimizing the error.

For tuning a FOPDT based controller, the values of $K_p$ , $\tau_p$ and $\theta_p$ can be used for a correlation.

There are two variants:

Set point tracking (servo control): $K_c = \frac{0.20}{K_p}\left(\frac{\tau_p}{\theta_p}\right)^{1.22}$
Disturbance rejection (regulatory control): $K_c = \frac{0.50}{K_p}\left(\frac{\tau_p}{\theta_p}\right)^{1.08}$

$K_c = \frac{1}{K_p}\frac{\tau_p}{\left( \theta_p + \tau_c \right)}$

with: $\tau_I = \tau_p$

Set point tracking (servo control):
- $K_c = \frac{0.586}{K_p}\left(\frac{\theta_p}{\tau_p}\right)^{-0.916}$
- $\tau_I = \frac{\tau_p}{1.03-0.165\left(\theta_p/\tau_p\right)}$
Disturbance rejection (regulatory control):
- $K_c = \frac{0.859}{K_p}\left(\frac{\theta_p}{\tau_p}\right)^{-0.977}$
- $\tau_I = \frac{\tau_p}{0.674}\left(\frac{\theta_p}{\tau_p}\right)^{0.680}$

$u(t) = K_P \cdot e(t) + K_I\cdot \int_0^t e(t)\;dt + K_D \frac{d\,e(t)}{dt}$

$K_c = \frac{1}{K_p}\frac{\tau_p}{\left( \theta_p + \tau_c \right)}$

$\tau_I = \tau_p + 0.5\theta_p$

$\tau_D = \frac{\tau_p\theta_p}{2\tau_p + \theta_p}$