In [1]:
from controlboros import StateSpaceBuilder
import matplotlib.pyplot as plt
import numpy as np
from scipy import signal
%matplotlib inline
%config InlineBackend.figure_format = 'retina'
In this notebook, we shall simulate the step response of a simple system using scipy.signal and controlboros.
The system is defined by the following transfer function:
$$H(s) = \frac{1}{s^2 + 0.5 s + 1}$$We want to simulate from 0 to 10 seconds:
In [2]:
t_begin, t_end = 0.0, 10.0
We use 0.1 ms sample time for the reference simulation performed with scipy.signal:
In [3]:
dt_ref = 1.0e-4
As for controlboros, it should use 200 ms sample time to highlight the difference:
In [4]:
dt_cb = 2.0e-1
Ok. Now we create a reference system using scipy.signal and compute its step response:
In [5]:
t_ref, y_ref = signal.step(
([1.0], [1.0, 0.5, 1.0]),
T=np.arange(t_begin, t_end, dt_ref),
)
Now we create a controlboros.StateSpace model using the builder pattern.
Notice that discretise() is just a wrapper around scipy.signal.cont2discrete(). It uses the zero-order hold method per default. You can play around and see the difference e.g. when using the Tustin's approximation (method="bilinear").
In [6]:
sys_cb = StateSpaceBuilder().from_tf([1.0], [1.0, 0.5, 1.0])\
.discretise(dt_cb)\
.build()
And we initialise arrays for the step response:
In [7]:
excitation = np.array([1.0]) # Constant step excitation
t_cb = np.arange(t_begin, t_end, dt_cb)
y_cb = np.zeros((len(t_cb),))
We're ready to run the main loop:
In [8]:
# Reset the inital state of the systems,
# helpful if you run this cell multiple times!
sys_cb.set_state_to_zero()
for i in range(len(t_cb)):
y_cb[i] = sys_cb.push_stateful(excitation)
Plot and compare results:
In [9]:
plt.figure(figsize=(7, 5))
plt.plot(t_ref, y_ref)
plt.step(t_cb, y_cb, where="post")
plt.xlabel("Time (s)")
plt.ylabel("Output")
plt.legend(["scipy.signal", "controlboros"])
plt.grid()
plt.show()