def _test_normalization(Simulator, sys, rng, normalizer, l1_lower,
lower, radius=5.0, dt=0.0001, T=1.0):
l1_norms = np.empty(len(sys))
for i, sub in enumerate(decompose_states(sys)):
response = impulse(sub, dt=dt, length=int(T / dt))
assert np.allclose(response[-10:], 0)
l1_norms[i] = radius * np.sum(abs(response * dt))
with Network() as model:
stim = nengo.Node(output=lambda t: rng.choice([-radius, radius])
if t < T/2 else radius)
tau = 0.02
subnet = LinearNetwork(sys, n_neurons=1, synapse=tau, dt=dt,
input_synapse=tau,
radii=radius, normalizer=normalizer,
neuron_type=nengo.neurons.Direct())
nengo.Connection(stim, subnet.input, synapse=None)
p = nengo.Probe(subnet.x.output, synapse=None)
assert ((l1_lower*subnet.info['radii'] <= l1_norms) |
((l1_norms <= 1e-6) & (subnet.info['radii'] <= 1e-6))).all()
assert (l1_norms <= subnet.info['radii'] + 1e-4).all()
with Simulator(model, dt=dt) as sim:
sim.run(T)
# lower bound includes both approximation error and the gap between
# random {-1, 1} flip-flop inputs and the true worst-case input
worst_x = np.max(abs(sim.data[p]), axis=0)
assert (lower <= worst_x+1e-6).all()
assert (worst_x <= 1+1e-6).all()
def test_radii(Simulator, seed, plt):
sys = canonical(PureDelay(0.1, order=4))
dt = 0.001
T = 0.3
plt.figure()
# Precompute the exact bounds for an impulse stimulus
radii = []
for sub in decompose_states(sys):
response = impulse(sub, dt=dt, length=int(T / dt))
amplitude = np.max(abs(response))
assert amplitude >= 1e-6 # otherwise numerical issues
radii.append(amplitude)
plt.plot(response / amplitude, linestyle='--')
with Network(seed=seed) as model:
# Impulse stimulus
stim = nengo.Node(output=lambda t: 1 / dt if t <= dt else 0)
# Set explicit radii for controllable realization
subnet = LinearNetwork(sys, n_neurons=1, synapse=0.2,
input_synapse=0.2, dt=dt,
radii=radii, normalizer=Controllable(),
neuron_type=nengo.neurons.Direct())
nengo.Connection(stim, subnet.input, synapse=None)
p = nengo.Probe(subnet.x.output, synapse=None)
assert subnet.info == {}
sim = nengo.Simulator(model, dt=dt)
sim.run(T)
plt.plot(sim.data[p], lw=5, alpha=0.5)
assert np.allclose(np.max(abs(sim.data[p]), axis=0), 1, atol=1e-3)
