scipy.signal.lsim#
- scipy.signal.lsim(system, U, T, X0=None, interp=True)[source]#
Simulate output of a continuous-time linear system.
- Parameters
- systeman instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and the interpretation:
1: (instance of
lti
)2: (num, den)
3: (zeros, poles, gain)
4: (A, B, C, D)
- Uarray_like
An input array describing the input at each time T (interpolation is assumed between given times). If there are multiple inputs, then each column of the rank-2 array represents an input. If U = 0 or None, a zero input is used.
- Tarray_like
The time steps at which the input is defined and at which the output is desired. Must be nonnegative, increasing, and equally spaced.
- X0array_like, optional
The initial conditions on the state vector (zero by default).
- interpbool, optional
Whether to use linear (True, the default) or zero-order-hold (False) interpolation for the input array.
- Returns
- T1D ndarray
Time values for the output.
- yout1D ndarray
System response.
- xoutndarray
Time evolution of the state vector.
Notes
If (num, den) is passed in for
system
, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g.s^2 + 3s + 5
would be represented as[1, 3, 5]
).Examples
We’ll use
lsim
to simulate an analog Bessel filter applied to a signal.>>> from scipy.signal import bessel, lsim >>> import matplotlib.pyplot as plt
Create a low-pass Bessel filter with a cutoff of 12 Hz.
>>> b, a = bessel(N=5, Wn=2*np.pi*12, btype='lowpass', analog=True)
Generate data to which the filter is applied.
>>> t = np.linspace(0, 1.25, 500, endpoint=False)
The input signal is the sum of three sinusoidal curves, with frequencies 4 Hz, 40 Hz, and 80 Hz. The filter should mostly eliminate the 40 Hz and 80 Hz components, leaving just the 4 Hz signal.
>>> u = (np.cos(2*np.pi*4*t) + 0.6*np.sin(2*np.pi*40*t) + ... 0.5*np.cos(2*np.pi*80*t))
Simulate the filter with
lsim
.>>> tout, yout, xout = lsim((b, a), U=u, T=t)
Plot the result.
>>> plt.plot(t, u, 'r', alpha=0.5, linewidth=1, label='input') >>> plt.plot(tout, yout, 'k', linewidth=1.5, label='output') >>> plt.legend(loc='best', shadow=True, framealpha=1) >>> plt.grid(alpha=0.3) >>> plt.xlabel('t') >>> plt.show()
In a second example, we simulate a double integrator
y'' = u
, with a constant inputu = 1
. We’ll use the state space representation of the integrator.>>> from scipy.signal import lti >>> A = np.array([[0.0, 1.0], [0.0, 0.0]]) >>> B = np.array([[0.0], [1.0]]) >>> C = np.array([[1.0, 0.0]]) >>> D = 0.0 >>> system = lti(A, B, C, D)
t and u define the time and input signal for the system to be simulated.
>>> t = np.linspace(0, 5, num=50) >>> u = np.ones_like(t)
Compute the simulation, and then plot y. As expected, the plot shows the curve
y = 0.5*t**2
.>>> tout, y, x = lsim(system, u, t) >>> plt.plot(t, y) >>> plt.grid(alpha=0.3) >>> plt.xlabel('t') >>> plt.show()