scipy.signal.lsim2

scipy.signal.lsim2(system, U=None, T=None, X0=None, **kwargs)

Simulate output of a continuous-time linear system, by using the ODE solver scipy.integrate.odeint.

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 (1D or 2D), optional

An input array describing the input at each time T. Linear interpolation is used between given times. If there are multiple inputs, then each column of the rank-2 array represents an input. If U is not given, the input is assumed to be zero.

Tarray_like (1D or 2D), optional

The time steps at which the input is defined and at which the output is desired. The default is 101 evenly spaced points on the interval [0,10.0].

X0array_like (1D), optional

The initial condition of the state vector. If X0 is not given, the initial conditions are assumed to be 0.

kwargsdict

Additional keyword arguments are passed on to the function odeint. See the notes below for more details.

Returns:

T1D ndarray

The time values for the output.

youtndarray

The response of the system.

xoutndarray

The time-evolution of the state-vector.

See also

lsim

Notes

This function uses scipy.integrate.odeint to solve the system’s differential equations. Additional keyword arguments given to lsim2 are passed on to odeint. See the documentation for scipy.integrate.odeint for the full list of arguments.

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 lsim2 to simulate an analog Bessel filter applied to a signal.

>>> import numpy as np
>>> from scipy.signal import bessel, lsim2
>>> 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 lsim2.

>>> tout, yout, xout = lsim2((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 input u = 1. We’ll use the state space representation of the integrator.

>>> from scipy.signal import lti
>>> A = np.array([[0, 1], [0, 0]])
>>> B = np.array([[0], [1]])
>>> C = np.array([[1, 0]])
>>> D = 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 = lsim2(system, u, t)
>>> plt.plot(t, y)
>>> plt.grid(alpha=0.3)
>>> plt.xlabel('t')
>>> plt.show()