## First Order Plus Dead Time (FOPDT)

A first-order linear system with time delay is a common empirical description of many stable dynamic processes. The equation

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

has variables *y(t)* and *u(t)* and three unknown parameters.

$$K_p \quad \mathrm{= Process \; gain}$$

$$\tau_p \quad \mathrm{= Process \; time \; constant}$$

$$\theta_p \quad \mathrm{= Process \; dead \; time}$$

#### Process Gain, `K_p`

The process gain is the change in the output *y* induced by a unit change in the input *u*. The process gain is calculated by evaluating the change in *y(t)* divided by the change in u(t) at steady state initial and final conditions.

$$K_p = \frac{\Delta y}{\Delta u} = \frac{y_{ss_2}-y_{ss_1}}{u_{ss_2}-u_{ss_1}}$$

The process gain affects the magnitude of the response, regardless of the speed of response.

#### Process Time Constant, `\tau_p`

Given a change in `u(t)` = `\Delta u`, the solution to the linear first-order differential (without time delay) becomes:

$$y(t) = \left( e^{-t / \tau_p} \right) y(0) + \left( 1 - e^{-t / \tau_p} \right) K_p \Delta u $$

If the initial condition *y(0)=0* and at *t=*`\tau_p`, the solution is simplified to the following.

$$y\left( \tau_p \right) = \left( 1 - e^{-\tau_p / \tau_p} \right) K_p \Delta u = \left( 1 - e^{-1} \right) K_p \Delta u = 0.632 K_p \Delta u$$

The process time constant is therefore the amount of time needed for the output to reach *(1-exp(-1))* or *63.2%* of the way to steady state conditions. The process time constant affects the speed of response.

#### Process Time Delay, `\theta_p`

The time delay is expressed as a time shift in the input variable *u(t)*.

$$u\left(t-\theta_p\right)$$

Suppose that that input signal is a step function that normally changes from 0 to 1 at time=0 but this shift is delayed by *5 sec*. The input function *u(t)* and output function *y(t)* are time-shifted by *5 sec*. The solution to the first-order differential equation with time delay is obtained by replacing all variables `t` with `t-\theta_p` and applying the conditional result based on the time in relation to the time delay, `\theta_p`.

$$y(t \lt \theta_p) = y(0)$$

$$y(t \ge \theta_p) = \left( e^{-\left(t - \theta_p \right) / \tau_p}\right) y(0) + \left( 1 - e^{-\left(t - \theta_p \right) / \tau_p} \right) K_p \Delta u $$

#### Fit Lower Order Model to Data

Two popular methods to fit data or more complex models to an FOPDT systems are with a graphical method and step data or with optimization techniques.