Orthogonal Collocation on Finite Elements

Main.OrthogonalCollocation History

Show minor edits - Show changes to output

Changed line 11 from:
(:toggle hide ex1 button show="Automation Collocation with Python Gekko":)
to:
(:toggle hide ex1 button show="Automated Collocation with Python Gekko":)
May 14, 2021, at 04:00 AM by 136.36.4.38 -
Changed line 86 from:
%width=30px%Attach:github.png [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Both Notebooks on GitHub]]
to:
%width=30px%Attach:github.png [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Collocation Notebooks on GitHub]]
May 14, 2021, at 03:59 AM by 136.36.4.38 -
Changed lines 9-14 from:
%width=50px%Attach:colab.png [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity.ipynb|Collocation Activity on Google Colab]]

%width=50px%Attach:colab.png
[[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Solution on Google Colab]]

%width=30px%Attach:github.png [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Both Notebooks on GitHub
]]
to:
%width=50px%Attach:colab.png [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity.ipynb|Collocation Activity on Google Colab]] and [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Solution]]
Added lines 85-86:

%width=30px%Attach:github.png [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Both Notebooks on GitHub]]
May 14, 2021, at 03:58 AM by 136.36.4.38 -
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%width=30px%Attach:colab.png [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity.ipynb|Collocation Activity on Google Colab]]

%width=30px%Attach:colab.png [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Solution on Google Colab]]
to:
%width=50px%Attach:colab.png [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity.ipynb|Collocation Activity on Google Colab]]

%width=50px%Attach:colab.png [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Solution on Google Colab]]
May 14, 2021, at 03:55 AM by 136.36.4.38 -
Changed lines 9-10 from:
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity.ipynb|Collocation Activity on Google Colab]]
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Solution on Google Colab]] with [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Both Notebooks on GitHub]]
to:
%width=30px%Attach:colab.png [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity.ipynb|Collocation Activity on Google Colab]]

%width=30px%Attach:colab.png [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Solution on Google Colab]]

%width=30px%Attach:github.png
[[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Both Notebooks on GitHub]]
May 14, 2021, at 03:52 AM by 136.36.4.38 -
Added lines 11-85:

(:toggle hide ex1 button show="Automation Collocation with Python Gekko":)
(:div id=ex1:)
(:source lang=python:)
from gekko import GEKKO
m = GEKKO(remote=False) # create GEKKO model
y = m.Var(5.0,name='y') # create GEKKO variable
m.Equation(y.dt()==-y)  # create GEKKO equation
m.time = [0,1,2,3]      # time points
m.options.IMODE = 4    # simulation mode
m.options.NODES = 4    # set NODES=4
m.options.CSV_WRITE=2  # write results_all.json
                        #  with internal nodes
m.solve(disp=False)    # solve
(:sourceend:)
(:divend:)

(:toggle hide ex2 button show="Manual Collocation with Python Gekko":)
(:div id=ex2:)
(:source lang=python:)
import numpy as np
N = np.array([[0.436,-0.281, 0.121], \
              [0.614, 0.064, 0.0461], \
              [0.603, 0.230, 0.167]])

time = np.array([0.0, \
                0.5-np.sqrt(5)/10.0, \
                0.5+np.sqrt(5)/10.0, \
                1.0])

from gekko import GEKKO
m = GEKKO()
y0 = 5
y  = m.Array(m.Var,3,value=0)
dy = m.Array(m.Var,3,value=0)

Ndy = np.dot(N,dy)
m.Equations([Ndy[i]==(y[i]-y0) for i in range(3)])
m.Equations([dy[i]+y[i]==0 for i in range(3)])
m.solve(disp=False)
print(y)
(:sourceend:)
(:divend:)

(:toggle hide ex3 button show="Manual Collocation with Python Scipy":)
(:div id=ex3:)
(:source lang=python:)
import numpy as np
N = np.array([[0.436,-0.281, 0.121], \
              [0.614, 0.064, 0.0461], \
              [0.603, 0.230, 0.167]])

time = np.array([0.0, \
                0.5-np.sqrt(5)/10.0, \
                0.5+np.sqrt(5)/10.0, \
                1.0])

from scipy.optimize import fsolve

y0 = 5
zGuess = np.zeros(6) # y1-y3 and dydt1-dydt3

def myFunction(z):
    y  = z[0:3]
    dy = z[3:6]

    F = np.empty(6)
    F[0:3] = np.dot(N,dy) - (y-y0)
    F[3:7] = dy + y
    return F

z = fsolve(myFunction,zGuess)
print(z)
(:sourceend:)
(:divend:)
March 21, 2021, at 05:07 AM by 10.35.117.248 -
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%width=550px%Attach:collocation.png
January 22, 2021, at 09:36 PM by 10.35.117.248 -
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(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/Avwn53krkgY" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
(:htmlend:)
January 21, 2021, at 10:45 PM by 10.35.117.248 -
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* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Solution on Google Colab]]
* [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Notebooks on GitHub]]
to:
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Solution on Google Colab]] with [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Both Notebooks on GitHub]]
January 21, 2021, at 10:43 PM by 10.35.117.248 -
Changed lines 8-9 from:
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Collocation Activity Solution on Google Colab]]
* [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Collocation Activity Notebooks on GitHub]]
to:
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Solution on Google Colab]]
* [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Notebooks on GitHub]]
January 21, 2021, at 10:43 PM by 10.35.117.248 -
Changed line 8 from:
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Collocation Activity on Google Colab]]
to:
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Collocation Activity Solution on Google Colab]]
January 21, 2021, at 10:43 PM by 10.35.117.248 -
Changed lines 7-9 from:
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity.ipynb|Collocation Activity on Google Colab]] with [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Collocation Activity on Google Colab]] or [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|source]].
to:
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity.ipynb|Collocation Activity on Google Colab]]
* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Collocation Activity on Google Colab]]
* [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|Collocation Activity Notebooks on GitHub]]
January 21, 2021, at 10:42 PM by 10.35.117.248 -
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Discretization of a continuous time representation allow large-scale nonlinear programming (NLP) solvers to find solutions at specified intervals in a time horizon. There are many names and related techniques for obtaining mathematical relationships between derivatives and non-derivative values. Some of the terms that are relevant to this discussion include orthogonal collocation on finite elements, direct transcription, Gauss pseudospectral method, Gaussian quadrature, Lobatto quadrature, Radau collocation, Legendre polynomials, Chebyshev polynomials, Jacobi polynomials, Laguerre polynomials, any many more. There are many papers that discuss the details of the derivation and theory behind these methods'^1-5^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations. See the documentation on [[https://apmonitor.com/wiki/index.php/Main/OptionApmNodes|Nodes]] for additional details on displaying the internal nodes.
to:
Discretization of a continuous time representation allow large-scale nonlinear programming (NLP) solvers to find solutions at specified intervals in a time horizon.

* [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity.ipynb|Collocation Activity on Google Colab]] with [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/CollocationActivity/Collocation_Activity_Solution.ipynb|Collocation Activity on Google Colab]] or [[https://github.com/APMonitor/dynopt/tree/master/CollocationActivity|source]].

There are many names and related techniques for obtaining mathematical relationships between derivatives and non-derivative values. Some of the terms that are relevant to this discussion include orthogonal collocation on finite elements, direct transcription, Gauss pseudospectral method, Gaussian quadrature, Lobatto quadrature, Radau collocation, Legendre polynomials, Chebyshev polynomials, Jacobi polynomials, Laguerre polynomials, any many more. There are many papers that discuss the details of the derivation and theory behind these methods'^1-5^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations. See the documentation on [[https://apmonitor.com/wiki/index.php/Main/OptionApmNodes|Nodes]] for additional details on displaying the internal nodes.
January 21, 2021, at 10:32 PM by 10.35.117.248 -
Added lines 369-372:

----

Attach:download.png [[https://colab.research.google.com/github/APMonitor/dynopt/blob/master/OrthogonalCollocation.ipynb|Collocation Solutions on Google Colab]]
August 09, 2020, at 03:45 AM by 136.36.211.159 -
Changed line 5 from:
Discretization of a continuous time representation allow large-scale nonlinear programming (NLP) solvers to find solutions at specified intervals in a time horizon. There are many names and related techniques for obtaining mathematical relationships between derivatives and non-derivative values. Some of the terms that are relevant to this discussion include orthogonal collocation on finite elements, direct transcription, Gauss pseudospectral method, Gaussian quadrature, Lobatto quadrature, Radau collocation, Legendre polynomials, Chebyshev polynomials, Jacobi polynomials, Laguerre polynomials, any many more. There are many papers that discuss the details of the derivation and theory behind these methods'^1-5^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations.
to:
Discretization of a continuous time representation allow large-scale nonlinear programming (NLP) solvers to find solutions at specified intervals in a time horizon. There are many names and related techniques for obtaining mathematical relationships between derivatives and non-derivative values. Some of the terms that are relevant to this discussion include orthogonal collocation on finite elements, direct transcription, Gauss pseudospectral method, Gaussian quadrature, Lobatto quadrature, Radau collocation, Legendre polynomials, Chebyshev polynomials, Jacobi polynomials, Laguerre polynomials, any many more. There are many papers that discuss the details of the derivation and theory behind these methods'^1-5^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations. See the documentation on [[https://apmonitor.com/wiki/index.php/Main/OptionApmNodes|Nodes]] for additional details on displaying the internal nodes.
December 09, 2019, at 11:06 AM by 216.129.163.16 -
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   plt.plot(tc(n),yc,'o',markersize=20)
to:
   plt.plot(tc(n),yc,'o',markersize=10)
Changed lines 151-163 from:
try:
    from APMonitor import *
except:
    # Automatically install APMonitor
    import pip
    pip.main(['install','APMonitor'])
    try:
        from APMonitor import *
    except:
        print('pip installation unsuccessful')
        print('Visit apmonitor.com to download apm.py')
        from
apm import *
to:
from APMonitor.apm import *
Deleted line 187:
  
March 13, 2019, at 01:47 PM by 10.37.126.112 -
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(:toggle hide python button show="APM Python Source":)
to:
(:toggle hide python button show="Show APM Python Source":)
March 27, 2018, at 05:00 PM by 10.5.113.167 -
Deleted lines 330-339:
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

Changed lines 334-343 from:
   #m.clear()#apm(s,a,'clear all') # clear prior application
to:
   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
Added line 345:
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   plt.plot(tc(n),yc,'o',markersize=20)
to:
   plt.plot(tc(n),yc,'o',markersize=10,label='Nodes = '+str(i))
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to:
plt.legend(loc='best')
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   m.clear()#apm(s,a,'clear all') # clear prior application
to:
   #m.clear()#apm(s,a,'clear all') # clear prior application
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(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/0UL-Y_mEIuI" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>
(:htmlend:)
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(:toggle hide solution1 button show="Show Solution 1":)
(:div id=solution1:)
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(:divend:)
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(:toggle hide solution2 button show="Show Solution 2":)
(:div id=solution2:)
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(:divend:)
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(:toggle hide solution1 button show="Show Solution 1":)
(:div id=solution1:)
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(:toggle hide solution2 button show="Show Solution 2":)
(:div id=solution2:)
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(:divend:)
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(:toggle hide python button show="Python Source":)
to:
(:toggle hide python button show="APM Python Source":)
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(:toggle hide gekko button show="Show GEKKO (Python) Code":)
(:div id=gekko:)
(:source lang=python:)
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=20)
    # store just the last x[n] value
    sol_py[i-2] = z[n-2]

# ----------------------------------------------------
# Approach #2 - Write model in APMonitor and let
#  modeling language create the collocation equations
# ----------------------------------------------------
# load GEKKO
from gekko import GEKKO

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

sol_apm = np.empty(5) # store 5 results
i = 0
for nodes in range(2,7):
    m.clear()#apm(s,a,'clear all') # clear prior application
    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()
(:sourceend:)
(:divend:)
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="315" src="https://www.youtube.com/embed/DBjmW4Lwpjc?rel=0" frameborder="0" allowfullscreen></iframe>
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":)
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(:toggle hide python button show="Python Source":)
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Attach:download.png [[Attach:collocation_exercise.zip|Collocation in MATLAB and Python (APM Solution)]]

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/DBjmW4Lwpjc?rel=0" frameborder="0" allowfullscreen></iframe>
(:htmlend:)

(:toggle hide python button show="Buy or Build TCLab":)
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Attach:download.png [[Attach:collocation_exercise.zip|Collocation in MATLAB and Python (APM Solution)]]

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<iframe width="560" height="315" src="https://www.youtube.com/embed/DBjmW4Lwpjc?rel=0" frameborder="0" allowfullscreen></iframe>
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'''Objective:''' Solve a differential equation with orthogonal collocation on finite elements. Create a MATLAB or Python script to simulate and display the results. ''Estimated Time: 1 hour''
to:
'''Objective:''' Solve a differential equation with orthogonal collocation on finite elements. Create a MATLAB or Python script to simulate and display the results. ''Estimated Time: 2-3 hours''
Added lines 28-200:

(:source lang=python:)
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=20)
    # store just the last x[n] value
    sol_py[i-2] = z[n-2]

# ----------------------------------------------------
# Approach #2 - Write model in APMonitor and let
#  modeling language create the collocation equations
# ----------------------------------------------------
# load APM package
try:
    from APMonitor import *
except:
    # Automatically install APMonitor
    import pip
    pip.main(['install','APMonitor'])
    try:
        from APMonitor import *
    except:
        print('pip installation unsuccessful')
        print('Visit apmonitor.com to download apm.py')
        from apm import *

s = 'https://byu.apmonitor.com'
a = 'node_test'

fid = open('collocation.apm','w')
# write model
fid.write('Parameters        \n')
fid.write(' u = 4            \n')
fid.write('Variables          \n')
fid.write(' x = 0            \n')
fid.write('Equations          \n')
fid.write(' 5 * $x = -x^2 + u \n')
# close file
fid.close()

fid = open('collocation.csv','w')
# write data file
fid.write('time \n')
fid.write('0 \n')
fid.write(str(tf)+'\n')
# close file
fid.close()

sol_apm = np.empty(5) # store 5 results
i = 0
for nodes in range(2,7):
    apm(s,a,'clear all') # clear prior application
    apm_load(s,a,'collocation.apm') # load model
    csv_load(s,a,'collocation.csv') # load data
    apm_option(s,a,'nlc.imode',4)  # imode = 4, simulation
    apm_option(s,a,'nlc.nodes',nodes) # nodes (2-6)
    apm(s,a,'solve') # solve problem
    y = apm_sol(s,a) # retrieve solution
    sol_apm[i] = y['x'][-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()
(:sourceend:)
January 25, 2017, at 07:24 AM by 173.117.254.58 -
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Attach:download.png [[Attach:orthogonal_collocation_python.zip|Orthogonal Collocation in Python]]
January 25, 2017, at 07:23 AM by 173.117.254.58 -
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Compare orthogonal collocation on finite elements with 3 nodes with a numerical integrator (e.g. ODE15s in MATLAB or ODEINT in Python). Calculate the error at each of the solution points.
to:
Compare orthogonal collocation on finite elements with 3 nodes with a numerical integrator (e.g. ODE15s in MATLAB or ODEINT in Python). Calculate the error at each of the solution points for the equation (same as for Exercise 1):

 5 dx/dt = -x'^2^' + u
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Attach:download.png [[Attach:collocation_compare.zip|Collocation Solution (APM, ODE15s (MATLAB), ODEINT (Python) Compared)]]
to:
Attach:download.png [[Attach:collocation_compare.zip|Collocation Solution (APM, ODE15s (MATLAB), ODEINT (Python))]]
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----
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----
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----
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!!!! Exercise
to:
!!!! Exercise 1
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!!!! Solution

Attach:download.png [[Attach:collocation_exercise.zip|Collocation Exercise in MATLAB and Python]]
to:
!!!! Solution 1

Attach:download.png [[Attach:collocation_exercise.zip|Collocation in MATLAB and Python (APM Solution)]]
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!!!! Exercise 2

Compare orthogonal collocation on finite elements with 3 nodes with a numerical integrator (e.g. ODE15s in MATLAB or ODEINT in Python). Calculate the error at each of the solution points.

!!!! Solution 2

Attach:download.png [[Attach:collocation_compare.zip|Collocation Solution (APM, ODE15s (MATLAB), ODEINT (Python) Compared)]]
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# Kameswaran, S., Biegler, L.T., Convergence rates for direct transcription of optimal control problems using collocation at Radau points, Computational Optimization and Applications, 41 (1), pp. 81-126, 2008, DOI: 10.1007/s10589-007-9098-9.

May 13, 2015, at 04:18 PM by 10.10.147.4 -
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Discretization of a continuous time representation allow large-scale nonlinear programming (NLP) solvers to find solutions at specified intervals in a time horizon. There are many names and related techniques for obtaining mathematical relationships between derivatives and non-derivative values. Some of the terms that are relevant to this discussion include:

* Orthogonal
collocation on finite elements
* Direct transcription
* Gauss pseudospectral method
* Gaussian quadrature
* Lobatto quadrature
* Radau collocation
* Legendre polynomials
* Chebyshev polynomials
* Jacobi polynomials
* Laguerre polynomials
* Any many more...

There are many papers that discuss the details of the derivation and theory behind these methods'^1-5^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations.
to:
Discretization of a continuous time representation allow large-scale nonlinear programming (NLP) solvers to find solutions at specified intervals in a time horizon. There are many names and related techniques for obtaining mathematical relationships between derivatives and non-derivative values. Some of the terms that are relevant to this discussion include orthogonal collocation on finite elements, direct transcription, Gauss pseudospectral method, Gaussian quadrature, Lobatto quadrature, Radau collocation, Legendre polynomials, Chebyshev polynomials, Jacobi polynomials, Laguerre polynomials, any many more. There are many papers that discuss the details of the derivation and theory behind these methods'^1-5^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations.
May 04, 2015, at 04:09 PM by 45.56.3.184 -
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Attach:download.png [[Attach:collocation_exercise.zip|Collocation Exercise in MATLAB]]
to:
Attach:download.png [[Attach:collocation_exercise.zip|Collocation Exercise in MATLAB and Python]]
May 04, 2015, at 03:04 PM by 45.56.3.184 -
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(:htmlend:)

!!!! Exercise

'''Objective:''' Solve a differential equation with orthogonal collocation on finite elements. Create a MATLAB or Python script to simulate and display the results. ''Estimated Time: 1 hour''

Solve the following differential equation from time 0 to 1 with orthogonal collocation on finite elements with 4 nodes for discretization in time.

 5 dx/dt = -x'^2^' + u

Specify the initial condition for ''x'' as 0 and the value of the input, ''u'', as 4. Compare the solution result with 2-6 time points (nodes). Report the solution at the final time for each and comment on how the solution changes with an increase in the number of nodes.

!!!! Solution

Attach:download.png [[Attach:collocation_exercise.zip|Collocation Exercise in MATLAB]]

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/0Et07u336Bo?rel=0" frameborder="0" allowfullscreen></iframe>
April 21, 2015, at 03:33 PM by 10.10.33.68 -
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There are many papers that discuss the details of the derivation and theory behind these methods'^1^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations.
to:
There are many papers that discuss the details of the derivation and theory behind these methods'^1-5^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations.
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# Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Hedengren, J. D. and Asgharzadeh Shishavan, R., Powell, K.M., and Edgar, T.F., Computers and Chemical Engineering, Volume 70, pg. 133–148, 2014. Available at: [[https://dx.doi.org/10.1016/j.compchemeng.2014.04.013|ScienceDirect]] or [[https://www.researchgate.net/publication/262191158_Nonlinear_Modeling_Estimation_and_Predictive_Control_in_APMonitor|ResearchGate]]
to:
# Hedengren, J. D. and Asgharzadeh Shishavan, R., Powell, K.M., and Edgar, T.F., Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Computers & Chemical Engineering, Volume 70, pg. 133–148, 2014. Available at: [[https://dx.doi.org/10.1016/j.compchemeng.2014.04.013|ScienceDirect]] or [[https://www.researchgate.net/publication/262191158_Nonlinear_Modeling_Estimation_and_Predictive_Control_in_APMonitor|ResearchGate]]

# Cizniar, M., Salhi, D., Fikar, M., and Latifi, M.A., A Matlab Package for Orthogonal Collocations on Finite Elements in Dynamic Optimization, 15th Int. Conference Process Control, June 7-10, 2015. [[https://www.kirp.chtf.stuba.sk/~fikar/research/dynopt/fik05oc.pdf | Article]]

# Renfro, J., Morshedi, A., Asbjornsen, O., Simultaneous Optimization and Solution of Systems Described by Differential/Algebraic Equations, Computers & Chemical Engineering, Volume 11, Number 5, pp. 503-517, 1987.

# Finlayson, B., [[https://faculty.washington.edu/finlayso/ebook/pde/OC/OCFE.htm|Orthogonal Collocation on Finite Elements for PDE Discretization and Solution]]

# Biegler, L.T., Solution of Dynamic Optimization Problems by Successive Quadratic Programming and Orthogonal Collocation, Computers & Chemical Engineering, Volume 8, Number 3/4, pp. 243-248, 1984.

April 11, 2015, at 01:18 PM by 45.56.12.124 -
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# Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Hedengren, J. D. and Asgharzadeh Shishavan, R., Powell, K.M., and Edgar, T.F., Computers and Chemical Engineering, Volume 70, pg. 133–148, 2014. Available at: https://dx.doi.org/10.1016/j.compchemeng.2014.04.013
to:
# Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Hedengren, J. D. and Asgharzadeh Shishavan, R., Powell, K.M., and Edgar, T.F., Computers and Chemical Engineering, Volume 70, pg. 133–148, 2014. Available at: [[https://dx.doi.org/10.1016/j.compchemeng.2014.04.013|ScienceDirect]] or [[https://www.researchgate.net/publication/262191158_Nonlinear_Modeling_Estimation_and_Predictive_Control_in_APMonitor|ResearchGate]]
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April 11, 2015, at 01:06 AM by 10.5.113.179 -
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There are many papers that discuss the details of the derivation and theory behind these methods'^1^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements for the numerical solution of differential algebraic equations.
to:
There are many papers that discuss the details of the derivation and theory behind these methods'^1^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations.
April 11, 2015, at 01:04 AM by 10.5.113.179 -
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There are many excellent papers that discuss the details of the derivation and theory behind these methods. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements for the numerical solution of differential algebraic equations.
to:
There are many papers that discuss the details of the derivation and theory behind these methods'^1^'. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements for the numerical solution of differential algebraic equations.
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!!!! References

# Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Hedengren, J. D. and Asgharzadeh Shishavan, R., Powell, K.M., and Edgar, T.F., Computers and Chemical Engineering, Volume 70, pg. 133–148, 2014. Available at: https://dx.doi.org/10.1016/j.compchemeng.2014.04.013
April 11, 2015, at 01:01 AM by 10.5.113.179 -
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(:title Orthogonal Collocation on Finite Elements:)
(:keywords direct transcription, orthogonal collocation, modeling language, differential, algebraic, tutorial:)
(:description Collocation methods are used to convert a differential algebraic model into a nonlinear programming problem that can be solved with NLP solvers:)

Discretization of a continuous time representation allow large-scale nonlinear programming (NLP) solvers to find solutions at specified intervals in a time horizon. There are many names and related techniques for obtaining mathematical relationships between derivatives and non-derivative values. Some of the terms that are relevant to this discussion include:

* Orthogonal collocation on finite elements
* Direct transcription
* Gauss pseudospectral method
* Gaussian quadrature
* Lobatto quadrature
* Radau collocation
* Legendre polynomials
* Chebyshev polynomials
* Jacobi polynomials
* Laguerre polynomials
* Any many more...

There are many excellent papers that discuss the details of the derivation and theory behind these methods. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements for the numerical solution of differential algebraic equations.

Attach:download.png [[Attach:orthogonal_collocation.zip|Orthogonal Collocation in MATLAB]]

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/2FeOaGUQwKA?rel=0" frameborder="0" allowfullscreen></iframe>
(:htmlend:)