get strange result from scipy.signal.lsim when run twice - scipy

Question

i run scipy.signal.lsim 10 times, it seems that the x0 only be used in the first time, why?
t=np.linspace(0.0,100,100*100)
transfun=[]
for i in range(10):
transfun.append(signal.lti([1],[1+i,1]))
y=[]
for i in range(10):
y.append(np.sin(2*np.pi*300*t)+np.random.normal(0,1,10000)+50)
sensor_output=[]
for i in range(10):
tout, yout, xout =signal.lsim(transfun[i],y[i],t,X0=[50.0])
sensor_output.append(yout)
fig=plt.figure()
for i in range(10):
plt.subplot(10,1,i+1)
plt.plot(t,y[i])
plt.plot(t,sensor_output[i])
plt.show()

Answer 1

lsim takes initial state vector as an argument, not initial output.
Transfer functions don't really have state vectors, but under the hood lsim is converting the transfer function to a state-space realization (which does have a state vector), and using that to simulate the system.
One problem is that, for a given transfer function, there's no unique realization. lsim doesn't say how it converts transfer functions to state-space realizations, but given your results I took a guess which happened to work (see below), but it's not robust.
To solve this for general transfer functions (i.e., not just first-order), you'd need to work with a specific state-space realization, and also specify more than just initial output, or the problem is under-constrained (I guess a typical approach would be to require d(y)/dt = 0, and similarly for all higher derivatives).
Below is a quick-and-dirty fix for your problem, and a sketch of how to do this for first-order state-space realizations.
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
nex = 3
t = np.linspace(0, 40, 1001)
taus = [1, 5, 10]
transfun = [signal.lti([1],[tau,1])
for tau in taus]
u = np.tile(50, t.shape)
yinit = 49
sensor_output = [signal.lsim(tf,u,t,X0=[yinit*tau])[1]
for tf, tau in zip(transfun, taus)]
fig=plt.figure()
for i in range(nex):
plt.subplot(nex,1,i+1)
plt.plot(t,u)
plt.plot(t,sensor_output[i])
plt.savefig('img1.png')
# different SS realizations of the same TF need different x0
# to get the same initial y0
g = signal.tf2ss([1], [10, 1])
# create SS system with the same TF
k = 1.234
g2 = (g[0], k*g[1], g[2]/k, g[3])
# desired initial value
y0 = 321
#solve for initial state for the two SS systems
x0 = y0 / g[2]
x0_2 = y0 / g2[2]
output = [signal.lsim(g,u,t,X0=x0)[1],
signal.lsim(g2,u,t,X0=x0_2)[1]]
fig=plt.figure()
for i,out in enumerate(output):
plt.subplot(len(output),1,i+1)
plt.plot(t,u)
plt.plot(t,out)
plt.savefig('img2.png')
plt.show()