System Modeling · Lesson

English translation

Execution setup

The Python option uses python-control. If the module is missing, run %pip install control==0.10.2 in a Colab or Jupyter code cell.

Topics covered

Use the model to determine whether changing mass or resistance affects the transient response, the steady-state value, or both.

Overall workflow

Physical equations and units → state-space model → measured response → coefficient estimation and residuals

Choosing inputs and states from physical equations

In the cruise model $m\dot v+bv=F$ , the input is force, and the state and output are velocity. $A=-\frac bm$ , $B=\frac1m$ . For constant force, the final velocity is $\frac Fb$ and the time constant is $\frac mb$ .

Increasing mass leaves the final velocity unchanged but makes the velocity change more slowly.

Checking the model against measurements

In the unit-step response $y(t)=K(1-e^{-t/\tau})$ , $K$ is the steady-state value, and $\tau$ is the time to reach approximately 63.2% of that value. Fit this equation to the measured response to estimate both parameters.

The first 3 s of this example capture only the rising response. Compare how much the estimated final value and time constant change when the 15 s record is also used.

Worked example

$m=1000$ kg, $b=50$ N·s/m, force 500 N. The state and output are velocity [m/s].

import numpy as np
import matplotlib.pyplot as plt
import control as ctl
s = ctl.tf('s')
m=1000.; b=50.; force=500.
G=ctl.ss([[-b/m]],[[1/m]],[[1]],[[0]])
t=np.linspace(0,150,1501)
t,v=ctl.step_response(force*G,t)
plt.plot(t,v); plt.xlabel('Time [s]'); plt.ylabel('Speed [m/s]'); plt.grid()

Official documentation and data

SciPy · curve_fit

Reference example

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Reference materials