Hanging Chain Optimization
Main.HangingChain History
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plt.xlabel('Time'); plt.ylabel('u')
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plt.xlabel('Distance'); plt.ylabel('u')
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%width=550px%Attach:hanging_chain.png
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The objective of the hanging chain problem is to find the position of a chain of uniform density and length suspendend between two points {`a`} and {`b`}. The problem is solved by minimizing the potential energy. This problem is from the [[http://www.mcs.anl.gov/~more/cops/|COPS benchmark set]]. The equations are
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The objective of the hanging chain problem is to find the position of a chain of uniform density and length suspended between two points {`a`} and {`b`}. The problem is solved by minimizing the potential energy. This problem is from the [[http://www.mcs.anl.gov/~more/cops/|COPS benchmark set]]. The equations are
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The [[https://gekko.readthedocs.io/en/latest/|Gekko Optimization Suite]] is a machine learning and optimization package in Python for mixed-integer and differential algebraic equations. The GEKKO package is available in Python with '''pip install gekko'''. There are [[link|additional example problems]] for equation solving, optimization, regression, dynamic simulation, model predictive control, and machine learning.
to:
The [[https://gekko.readthedocs.io/en/latest/|Gekko Optimization Suite]] is a machine learning and optimization package in Python for mixed-integer and differential algebraic equations.
📄 [[https://apmonitor.com/wiki/index.php/Main/APMonitorReferences|Gekko Publication and References]]
The GEKKO package is available in Python with '''pip install gekko'''. There are [[https://apmonitor.com/wiki/index.php/Main/GekkoPythonOptimization|additional example problems]] for equation solving, optimization, regression, dynamic simulation, model predictive control, and machine learning.
📄 [[https://apmonitor.com/wiki/index.php/Main/APMonitorReferences|Gekko Publication and References]]
The GEKKO package is available in Python with '''pip install gekko'''. There are [[https://apmonitor.com/wiki/index.php/Main/GekkoPythonOptimization|additional example problems]] for equation solving, optimization, regression, dynamic simulation, model predictive control, and machine learning.
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(:title Hanging Chain Optimization:)
(:keywords optimization, optimal control, benchmark, nonlinear control, dynamic optimization, engineering optimization, Python, GEKKO, differential, algebraic, modeling language:)
(:description Hanging chain optimization solved with Python Gekko.:)
%width=550px%Attach:hanging_chain.png
The objective of the hanging chain problem is to find the position of a chain of uniform density and length suspendend between two points {`a`} and {`b`}. The problem is solved by minimizing the potential energy. This problem is from the [[http://www.mcs.anl.gov/~more/cops/|COPS benchmark set]]. The equations are
{$ \begin{array}{llcl} \min_{x, u} & x_2(t_f) \\ \mbox{s.t.} & \dot{x}_1 & = & u \\ & \dot{x}_2 & = & x_1 (1+u^2)^{1/2} \\ & \dot{x}_3 & = & (1+u^2)^{1/2} \\ & x(t_0) &=& (a,0,0)^T \\ & x_1(t_f) &=& b \\ & x_3(t_f) &=& Lp \\ & x(t) &\in& [0,10] \\ & u(t) &\in& [-10,20] \end{array} $}
The parameters of this model are defined as {`a=1`}, {`b=3`}, and {`Lp=4`} with {`t`} from 0 to 1.
%width=550px%Attach:hanging_chain_solution.png
(:toggle hide solution button show="Hanging Chain Source":)
(:div id=solution:)
(:source lang=python:)
from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt
m = GEKKO(remote=False)
n = 101; Lp = 4; b = 3; a = 1
m.time = np.linspace(0,1,n)
x1,x2,x3 = m.Array(m.Var,3,lb=0,ub=10)
x1.value = a
u = m.MV(value=0, lb=-10, ub=20) # integer=True
z = np.zeros(n); z[-1] = 1
final = m.Param(value=z)
m.Equation(x1.dt() == u)
m.Equation(x2.dt() == x1 * (1+u**2)**0.5)
m.Equation(x3.dt() == (1+u**2)**0.5)
m.Minimize(1000*(x1-b)**2*final)
m.Minimize(1000*(x3-Lp)**2*final)
m.options.SOLVER = 3
m.options.NODES = 3
m.options.IMODE = 6
m.Minimize(x2*final)
# initialize
u.STATUS = 0
m.solve()
# optimize
m.options.TIME_SHIFT = 0
u.STATUS = 1; u.DCOST = 1e-3
m.solve()
plt.figure(figsize=(6,4))
plt.subplot(2,1,1)
plt.plot(m.time,x1.value,'r--',label=r'$x_1$')
plt.plot(m.time,x2.value,'g-',label=r'$x_2$')
plt.plot(m.time,x3.value,'k:',label=r'$x_3$')
plt.ylabel('x'); plt.xlim([0,1])
plt.legend(); plt.grid()
plt.subplot(2,1,2)
plt.plot(2,1,2)
plt.plot(m.time,u.value,'k-',label=r'$u$')
plt.grid(); plt.xlim([0,1]); plt.legend()
plt.xlabel('Time'); plt.ylabel('u')
plt.tight_layout()
plt.savefig('chain.png',dpi=300); plt.show()
(:sourceend:)
(:divend:)
----
%width=200px%Attach:gekko.png
The [[https://gekko.readthedocs.io/en/latest/|Gekko Optimization Suite]] is a machine learning and optimization package in Python for mixed-integer and differential algebraic equations. The GEKKO package is available in Python with '''pip install gekko'''. There are [[link|additional example problems]] for equation solving, optimization, regression, dynamic simulation, model predictive control, and machine learning.
(:keywords optimization, optimal control, benchmark, nonlinear control, dynamic optimization, engineering optimization, Python, GEKKO, differential, algebraic, modeling language:)
(:description Hanging chain optimization solved with Python Gekko.:)
%width=550px%Attach:hanging_chain.png
The objective of the hanging chain problem is to find the position of a chain of uniform density and length suspendend between two points {`a`} and {`b`}. The problem is solved by minimizing the potential energy. This problem is from the [[http://www.mcs.anl.gov/~more/cops/|COPS benchmark set]]. The equations are
{$ \begin{array}{llcl} \min_{x, u} & x_2(t_f) \\ \mbox{s.t.} & \dot{x}_1 & = & u \\ & \dot{x}_2 & = & x_1 (1+u^2)^{1/2} \\ & \dot{x}_3 & = & (1+u^2)^{1/2} \\ & x(t_0) &=& (a,0,0)^T \\ & x_1(t_f) &=& b \\ & x_3(t_f) &=& Lp \\ & x(t) &\in& [0,10] \\ & u(t) &\in& [-10,20] \end{array} $}
The parameters of this model are defined as {`a=1`}, {`b=3`}, and {`Lp=4`} with {`t`} from 0 to 1.
%width=550px%Attach:hanging_chain_solution.png
(:toggle hide solution button show="Hanging Chain Source":)
(:div id=solution:)
(:source lang=python:)
from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt
m = GEKKO(remote=False)
n = 101; Lp = 4; b = 3; a = 1
m.time = np.linspace(0,1,n)
x1,x2,x3 = m.Array(m.Var,3,lb=0,ub=10)
x1.value = a
u = m.MV(value=0, lb=-10, ub=20) # integer=True
z = np.zeros(n); z[-1] = 1
final = m.Param(value=z)
m.Equation(x1.dt() == u)
m.Equation(x2.dt() == x1 * (1+u**2)**0.5)
m.Equation(x3.dt() == (1+u**2)**0.5)
m.Minimize(1000*(x1-b)**2*final)
m.Minimize(1000*(x3-Lp)**2*final)
m.options.SOLVER = 3
m.options.NODES = 3
m.options.IMODE = 6
m.Minimize(x2*final)
# initialize
u.STATUS = 0
m.solve()
# optimize
m.options.TIME_SHIFT = 0
u.STATUS = 1; u.DCOST = 1e-3
m.solve()
plt.figure(figsize=(6,4))
plt.subplot(2,1,1)
plt.plot(m.time,x1.value,'r--',label=r'$x_1$')
plt.plot(m.time,x2.value,'g-',label=r'$x_2$')
plt.plot(m.time,x3.value,'k:',label=r'$x_3$')
plt.ylabel('x'); plt.xlim([0,1])
plt.legend(); plt.grid()
plt.subplot(2,1,2)
plt.plot(2,1,2)
plt.plot(m.time,u.value,'k-',label=r'$u$')
plt.grid(); plt.xlim([0,1]); plt.legend()
plt.xlabel('Time'); plt.ylabel('u')
plt.tight_layout()
plt.savefig('chain.png',dpi=300); plt.show()
(:sourceend:)
(:divend:)
----
%width=200px%Attach:gekko.png
The [[https://gekko.readthedocs.io/en/latest/|Gekko Optimization Suite]] is a machine learning and optimization package in Python for mixed-integer and differential algebraic equations. The GEKKO package is available in Python with '''pip install gekko'''. There are [[link|additional example problems]] for equation solving, optimization, regression, dynamic simulation, model predictive control, and machine learning.