APMonitor
commented
4 years ago Gekko uses orthogonal collocation on finite elements. Check out https://apmonitor.com/do/index.php/Main/OrthogonalCollocation Here is one of the examples from that material that shows how the collocation methods are implemented.
from __future__ import division
import numpy as np
from scipy.optimize import fsolve
from scipy.integrate import odeint
import matplotlib.pyplot as plt
# final time
tf = 1.0
# solve with ODEINT (for comparison)
def model(x,t):
u = 4.0
return (-x**2 + u)/5.0
t = np.linspace(0,tf,20)
y0 = 0
y = odeint(model,y0,t)
plt.figure(1)
plt.plot(t,y,'r-',label='ODEINT')
# ----------------------------------------------------
# Approach #1 - Write the model equations in Python
# ----------------------------------------------------
# define collocation matrices
def colloc(n):
if (n==2):
NC = np.array([[1.0]])
if (n==3):
NC = np.array([[0.75,-0.25], \
[1.00, 0.00]])
if (n==4):
NC = np.array([[0.436,-0.281, 0.121], \
[0.614, 0.064, 0.0461], \
[0.603, 0.230, 0.167]])
if (n==5):
NC = np.array([[0.278, -0.202, 0.169, -0.071], \
[0.398, 0.069, 0.064, -0.031], \
[0.387, 0.234, 0.278, -0.071], \
[0.389, 0.222, 0.389, 0.000]])
if (n==6):
NC = np.array([[0.191, -0.147, 0.139, -0.113, 0.047],
[0.276, 0.059, 0.051, -0.050, 0.022],
[0.267, 0.193, 0.252, -0.114, 0.045],
[0.269, 0.178, 0.384, 0.032, 0.019],
[0.269, 0.181, 0.374, 0.110, 0.067]])
return NC
# define collocation points from Lobatto quadrature
def tc(n):
if (n==2):
time = np.array([0.0,1.0])
if (n==3):
time = np.array([0.0,0.5,1.0])
if (n==4):
time = np.array([0.0, \
0.5-np.sqrt(5)/10.0, \
0.5+np.sqrt(5)/10.0, \
1.0])
if (n==5):
time = np.array([0.0,0.5-np.sqrt(21)/14.0, \
0.5,0.5+np.sqrt(21)/14.0, 1])
if (n==6):
time = np.array([0.0, \
0.5-np.sqrt((7.0+2.0*np.sqrt(7.0))/21.0)/2.0, \
0.5-np.sqrt((7.0-2.0*np.sqrt(7.0))/21.0)/2.0, \
0.5+np.sqrt((7.0-2.0*np.sqrt(7.0))/21.0)/2.0, \
0.5+np.sqrt((7.0+2.0*np.sqrt(7.0))/21.0)/2.0, \
1.0])
return time*tf
# solve with SciPy fsolve
def myFunction(z,*param):
n = param[0]
m = param[1]
# rename z as x and xdot variables
x = np.empty(n-1)
xdot = np.empty(n-1)
x[0:n-1] = z[0:n-1]
xdot[0:n-1] = z[n-1:m]
# initial condition (x0)
x0 = 0.0
# input parameter (u)
u = 4.0
# final time
tn = tf
# function evaluation residuals
F = np.empty(m)
# nonlinear differential equations at each node
# 5 dx/dt = -x^2 + u
F[0:n-1] = 5.0 * xdot[0:n-1] + x[0:n-1]**2 - u
# collocation equations
# tn * NC * xdot = x - x0
NC = colloc(n)
F[n-1:m] = tn * np.dot(NC,xdot) - x + x0 * np.ones(n-1)
return F
sol_py = np.empty(5) # store 5 results
for i in range(2,7):
n = i
m = (i-1)*2
zGuess = np.ones(m)
z = fsolve(myFunction,zGuess,args=(n,m))
# add to plot
yc = np.insert(z[0:n-1],0,0)
plt.plot(tc(n),yc,'o',markersize=10,label='Nodes = '+str(i))
# store just the last x[n] value
sol_py[i-2] = z[n-2]
plt.legend(loc='best')
# ----------------------------------------------------
# Approach #2 - Write model in APMonitor and let
# modeling language create the collocation equations
# ----------------------------------------------------
# load GEKKO
from gekko import GEKKO
sol_apm = np.empty(5) # store 5 results
i = 0
for nodes in range(2,7):
m = GEKKO()
u = m.Param(value=4)
x = m.Var(value=0)
m.Equation(5*x.dt() == -x**2 + u)
m.time = [0,tf]
m.options.imode = 4
m.options.time_shift = 0
m.options.nodes = nodes
m.solve() # solve problem
sol_apm[i] = x.value[-1] # store solution (last point)
i += 1
# print the solutions
print(sol_py)
print(sol_apm)
# show plot
plt.ylabel('x(t)')
plt.xlabel('time')
plt.show()You could do something similar but with Runga-Kutta method, solved as IMODE=3 where you do the differential to algebraic equation conversion.
If I understand well, Gekko uses a euler discretization technics for ode. Is it possible to choose different disctretization technics like Runge-Kutta discretization (RK4 for exemple)? If it is not implemented, how can I implement it?