Your guess is correct. The target system and the PI controller is integrated, you can't separate it into two odeint. I modified your code, that the system has two status variables: one for the velocity of the system, one for the integration of the control error:
import matplotlib.pylab as plt
import numpy as np
import scipy.integrate as integrate
##Parameters
kp = .5 #proportional gain
ki = .1 #integral gain
vr = 25 #desired velocity in m/s
Tm = 190 #Max Torque in Nm
wm = 420 #engine speed
B = 0.4 #Beta
an = 12 #at gear 4
p = 1.3 #air density
Cd = 0.32 #Drag coefficient
Cr = .01 #Coefficient of rolling friction
A = 2.4 #frontal area
##Variables
m = 18000.0 #weight
v0 = 20. #starting velocity
t = np.linspace(61, 500, 5000) #time
theta = np.radians(4) #Theta
def torque(v):
return Tm * (1 - B*(an*v/wm - 1)**2)
def vderivs(status, t):
v, int_err = status
err = vr - v
control = kp * err + ki * int_err
v1 = an * control * torque(v)
v2 = m*Cr*np.sign(v)
v3 = 0.5*p*Cd*A*v**2
v4 = m*np.sin(theta)
vtot = v1-v2-v3-v4*(t>=200)
return vtot/m, err
def velocity(desired, theta, time):
return integrate.odeint(vderivs, [desired, 0], time)[:, 0]
t0l = [i for i in range(61)]
vf=[v0 for i in range(61)]+[v for v in velocity(v0,theta,t)]
tf=t0l+[time for time in t]
plt.plot(tf, vf, 'k-', label=('V(0) = '+str(v0)))
v0=35.
vf=[v0 for i in range(61)]+[v for v in velocity(v0,theta,t)]
plt.plot(tf, vf, 'b-', label=('V(0) = '+str(v0)))
v0=vr
vf=[v0 for i in range(61)]+[v for v in velocity(v0,theta,t)]
plt.plot(tf, vf, 'g-', label=('V(0) = Vr'))
plt.axhline(y=vr, xmin=0, xmax=1000, color='r', label='Desired Velocity')
plt.legend(loc = "upper right")
plt.axis([0,500,18,36])
plt.show()
output: