def test_state_norm(plt):
# Choose a filter, timestep, and number of simulation timesteps
sys = Alpha(0.1)
dt = 0.000001
length = 2000000
# Modify the state-space to read out the state vector
A, B, C, D = sys2ss(sys)
old_C = C
C = np.eye(len(A))
D = np.zeros((len(A), B.shape[1]))
response = np.empty((length, len(C)))
for i in range(len(C)):
# Simulate the state vector
response[:, i] = impulse((A, B, C[i, :], D[i, :]), dt, length)
# Check that the power of each state equals the H2-norm of each state
# The analog case is the same after scaling since dt is approx 0.
actual = norm(response, axis=0) * dt
assert np.allclose(actual, state_norm(cont2discrete(sys, dt)))
assert np.allclose(actual, state_norm(sys) * np.sqrt(dt))
plt.figure()
plt.plot(response[:, 0], label="$x_0$")
plt.plot(response[:, 1], label="$x_1$")
plt.plot(np.dot(response, old_C.T), label="$y$")
plt.legend()
def test_grams():
sys = 0.6*Alpha(0.01) + 0.4*Lowpass(0.05)
A, B, C, D = sys2ss(sys)
P = control_gram(sys)
assert np.allclose(np.dot(A, P) + np.dot(P, A.T), -np.dot(B, B.T))
assert np.linalg.matrix_rank(P) == len(P) # controllable
Q = observe_gram(sys)
assert np.allclose(np.dot(A.T, Q) + np.dot(Q, A), -np.dot(C.T, C))
assert np.linalg.matrix_rank(Q) == len(Q) # observable
def test_grams():
sys = 0.6 * Alpha(0.01) + 0.4 * Lowpass(0.05)
A, B, C, D = sys2ss(sys)
P = control_gram(sys)
assert np.allclose(np.dot(A, P) + np.dot(P, A.T), -np.dot(B, B.T))
assert matrix_rank(P) == len(P) # controllable
Q = observe_gram(sys)
assert np.allclose(np.dot(A.T, Q) + np.dot(Q, A), -np.dot(C.T, C))
assert matrix_rank(Q) == len(Q) # observable
def test_similarity_transform():
sys = Alpha(0.1)
TA, TB, TC, TD = similarity_transform(sys, np.eye(2)).ss
A, B, C, D = sys2ss(sys)
assert np.allclose(A, TA)
assert np.allclose(B, TB)
assert np.allclose(C, TC)
assert np.allclose(D, TD)
T = [[1, 1], [-0.5, 0]]
TA, TB, TC, TD = similarity_transform(sys, T).ss
assert not np.allclose(A, TA)
assert not np.allclose(B, TB)
assert not np.allclose(C, TC)
assert np.allclose(D, TD)
assert sys_equal(sys, (TA, TB, TC, TD))
