# 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
    system an 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)

    U array_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.

    T array_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].

    X0 array_like (1D), optional

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

    kwargs dict

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

  Returns
    T 1D ndarray

      The time values for the output.

    yout ndarray

      The response of the system.

    xout ndarray

      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.

      >>> 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()

scipy.signal.lsim scipy.signal.impulse