import numpy as np
import matplotlib.pyplot as plt
import control

def dynamic_function(t, x, u, param=None):
    """
    Dynamic function for the system.
    """
    A = np.array([[0, 1], [-1, -1]])
    B = np.array([[0], [1]])
    return A @ x + B @ u

def output_function(t, x, u, param=None):
    """
    Output function for the system.
    """
    C = np.array([[1, 0]])
    return C @ x

sys = control.NonlinearIOSystem(
    dynamic_function, output_function,
    inputs=1,
    outputs=1,
    states=2
    )

t = np.linspace(0, 10, 100)
u = np.zeros_like(t)


dirac_index = 6


u[dirac_index] = 1  # Dirac input at t=0

results = control.forced_response(sys, T=t, U=u)

plt.figure()
plt.plot(t, results.y[0], label='Output')
plt.title('Response of Nonlinear System to Dirac Input')
plt.xlabel('Time (s)')
plt.ylabel('Output')
plt.grid()
plt.legend()
plt.show()