First-order plus deadtime (FOPDT) model system id

This library implements a gradient-approach to fit a first-order plus deadtime (FOPDT) model to input and output data. Specifically, it fits the following model

$$ \tau \frac{d}{dt} y(t) = - y(t) + K u(t - \theta). $$

Where $y$ is the output, $u$ the input, $t$ is time. The unknown parameters that are estimated are

• $\tau$ = process time constant
• $K$ = process gain
• $\theta$ = process dead time

Dependencies

• Numpy
• Scipy (for simulation and filtering in the example)
• Matplotlib (for plotting)

Contents

• FOPDT_fitter.py contains the library for model fitting.
• example.ipynb shows and example using the library.

Usage

To library can be used the following way:

# import the library
import FOPDT_fitter

# initial guess of parameters
tua_0 = 1
K_0 = 0.5
theta_0 = 0

# call the fit_model function with initial guess, 
# and input, output and time vector
tua, K, theta = FOPDT_fitter.fit_model(tua_0, K_0, theta_0, u, y, t)

The graph below shows the result of the example. The blue line is the input signal $u$, and the orange line the measured output $y$. The initial guess is shown in green, and the fitted model in red.

The algorithm

The library makes use of the Gauss–Newton algorithm to solve a non-linear least squares problem in which the residuals are penalized. A backtracking line search is implemented as a globalization strategy.

Because it is a gradient based algorithm, the library is sensitive to the initial guess. It is therefore recommended using a good initial guess. Also, as shown in the example, the algorithm is sensitive to noise in the data. Therefore, it is recommended to filter the data before fitting the model.

Future work

• Expanding it to multiple inputs.