## Main.DynamicOptimizationBenchmarks History

May 15, 2021, at 05:02 AM by 136.36.4.38 -
Changed line 1 from:
(:title Optimal Control Problems:)
to:
(:title Optimal Control Benchmark Problems:)
February 10, 2021, at 01:14 AM by 10.35.117.248 -
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February 10, 2021, at 01:14 AM by 10.35.117.248 -
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February 10, 2021, at 01:04 AM by 10.35.117.248 -
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'''Objective:''' Solve the [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^2^' and [[MoreDynamicOptimizationBenchmarks]]. For each problem, create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''
to:
'''Objective:''' Solve the [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^2^' and [[Main/MoreDynamicOptimizationBenchmarks|more dynamic optimization benchmark problems]]. Complete the 9 exercises as shown in the Jupyter Notebook link below. For each problem, create a program to optimize and display the results. ''Estimated Time (each): 10-30 minutes''
February 10, 2021, at 01:03 AM by 10.35.117.248 -
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'''Objective:''' Set up and solve '''three''' of the five [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^2^'. Create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''
to:
'''Objective:''' Solve the [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^2^' and [[MoreDynamicOptimizationBenchmarks]]. For each problem, create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''

February 09, 2021, at 11:52 PM by 10.35.117.248 -
Changed line 139 from:
{$\frac{dx_4}{dt}=x_1^2+x_2^2+0.0005 \left(x_2 + 16 \, t -8 -0.1x_3\,u^2\right)^2$}
to:
{$\frac{dx_4}{dt}=x_1^2+x_2^2+0.005 \left(x_2 + 16 \, t -8 -0.1x_3\,u^2\right)^2$}
(:title Optimal Control Problems:)
(:keywords nonlinear control, optimal control, dynamic optimization, engineering optimization, MATLAB, Python, GEKKO, differential, algebraic, modeling language, university course:)
(:description Optimal control problems solved with Dynamic Optimization in MATLAB, Excel, and Python.:)

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Changed lines 383-431 from:
# Show source
to:
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

nt = 101
m.time = np.linspace(0,12,nt)

# Parameters
u = m.MV(value=1,ub=1,lb=0)
u.STATUS = 1
u.DCOST = 0

# Variables
x1 = m.Var(value=1)
x2 = m.Var(value=0)

p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)

# Equations
m.Equation(x1.dt()==u*(10*x2-x1))
m.Equation(x2.dt()==-u*(10*x2-x1)-(1-u)*x2)

# Objective Function
m.Obj(-final*(1-x1-x2))

m.options.IMODE = 6
m.solve()

print('Objective: ' + str(1-x1[-1]-x2[-1]))

plt.figure(1)

plt.subplot(2,1,1)
plt.plot(m.time,x1.value,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$')
plt.ylabel('Value')
plt.legend(loc='best')

plt.subplot(2,1,2)
plt.plot(m.time,u.value,'r-',LineWidth=2,label=r'$u$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')

plt.show()
Changed lines 304-356 from:
# Show source
to:
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

nt = 101
m.time = np.linspace(0,1,nt)

# Parameters
T = m.MV(value=362,ub=398,lb=298)
T.STATUS = 1
T.DCOST = 0

# Variables
x1 = m.Var(value=1)
x2 = m.Var(value=0)

p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)

# Intermediates
k1 = m.Intermediate(4000*m.exp(-2500/T))
k2 = m.Intermediate(6.2e5*m.exp(-5000/T))

# Equations
m.Equation(x1.dt()==-k1*x1**2)
m.Equation(x2.dt()==k1*x1**2 - k2*x2)

# Objective Function
m.Obj(-x2*final)

m.options.IMODE = 6
m.solve()

print('Objective: ' + str(x2[-1]))

plt.figure(1)

plt.subplot(2,1,1)
plt.plot(m.time,x1.value,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$')
plt.ylabel('Value')
plt.legend(loc='best')

plt.subplot(2,1,2)
plt.plot(m.time,T.value,'r--',LineWidth=2,label=r'$T$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')

plt.show()
Changed lines 235-275 from:
# Show source
to:
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

nt = 101
m.time = np.linspace(0,1,nt)

# Parameters
u = m.MV(value=1,ub=5,lb=0)
u.STATUS = 1

# Variables
x1 = m.Var(value=1)
x2 = m.Var(value=0)

p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)

# Equations
m.Equation(x1.dt()==-(u+0.5*u**2)*x1)
m.Equation(x2.dt()==u*x1)

# Objective Function
m.Obj(-x2*final)

m.options.IMODE = 6
m.solve()

print('Objective: ' + str(x2[-1]))

plt.figure(1)
plt.plot(m.time,x1.value,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,u.value,'r--',LineWidth=2,label=r'$u$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()
Changed line 244 from:
{$\min_{T(t)} x_2 \left( t_f \right)$}
to:
{$\max_{T(t)} x_2 \left( t_f \right)$}
Changed line 217 from:
{$\min_{u(t)} x_2 \left( t_f \right)$}
to:
{$\max_{u(t)} x_2 \left( t_f \right)$}
Changed lines 68-70 from:
plt.plot(m.time,x1,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,u,'r--',LineWidth=2,label=r'$u$')
to:
plt.plot(m.time,x1.value,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,u.value,'r--',LineWidth=2,label=r'$u$')
Changed lines 111-113 from:
plt.plot(m.time,x1,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,u,'r--',LineWidth=2,label=r'$u$')
to:
plt.plot(m.time,x1.value,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,u.value,'r--',LineWidth=2,label=r'$u$')
Changed lines 201-204 from:
plt.plot(m.time,x1,'r--',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2,'g:',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,x3,'k-',LineWidth=2,label=r'$x_3$')
plt.plot(m.time,x4,'b-',LineWidth=2,label=r'$x_4$')
to:
plt.plot(m.time,x1.value,'r--',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'g:',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,x3.value,'k-',LineWidth=2,label=r'$x_3$')
plt.plot(m.time,x4.value,'b-',LineWidth=2,label=r'$x_4$')
Deleted lines 29-31:
(:toggle hide solution1 button show="Show Solutions for Problem 1a and 1b":)
(:div id=solution1:)

Deleted line 123:
(:divend:)
Changed lines 28-32 from:
!!!! Solution to Benchmarks 1a and 1b
to:
!!!! Solutions to Benchmarks 1a and 1b

(:toggle hide solution1 button show="Show Solutions for Problem 1a and 1b":)
(:div id=solution1:)

(:divend:)

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Changed line 268 from:
{$\min_{u(t)} 1 - x_1 \left( t_f \right) - x_2 \left( t_f \right)$}
to:
{$\max_{u(t)} \left(1 - x_1 \left( t_f \right) - x_2 \left( t_f \right) \right)$}
Changed lines 267-275 from:
->Attach:dynopt_5.png
to:

{$\min_{u(t)} 1 - x_1 \left( t_f \right) - x_2 \left( t_f \right)$}
{$\mathrm{subject \; to}$}
{$\frac{dx_1}{dt}=u \left(10 \, x_2 - x_1 \right)$}
{$\frac{dx_2}{dt}=-u \left(10 \, x_2 - x_1 \right)-\left(1-u\right) x_2$}
{$x(0) = [1 \; 0]^T$}
{$0 \le u \le 1$}
{$t_f=12$}

Changed lines 238-248 from:
->Attach:dynopt_4.png
to:

{$\min_{T(t)} x_2 \left( t_f \right)$}
{$\mathrm{subject \; to}$}
{$\frac{dx_1}{dt}=-k_1 \, x_1^2$}
{$\frac{dx_2}{dt}=k_1 \, x_1^2 - k_2 \, x_2$}
{$k_1 = 4000 \, \exp{\left(-\frac{2500}{T}\right)}$}
{$k_2 = 6 .2e5 \, \exp{\left(-\frac{5000}{T}\right)}$}
{$x(0) = [1 \; 0]^T$}
{$298 \le T \le 398$}
{$t_f=1$}

Changed lines 211-219 from:
->Attach:dynopt_3.png
to:

{$\min_{u(t)} x_2 \left( t_f \right)$}
{$\mathrm{subject \; to}$}
{$\frac{dx_1}{dt}=-\left(u+0 .5u^2\right) x_1$}
{$\frac{dx_2}{dt}=u \, x_1$}
{$x(0) = [1 \; 0]^T$}
{$0 \le u \le 5$}
{$t_f=1$}

February 07, 2018, at 05:43 AM by 10.37.117.169 -
Changed lines 145-203 from:
# Show source
to:
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

nt = 101
m.time = np.linspace(0,1,nt)

# Parameters
u = m.MV(value=9,lb=-4,ub=10)
u.STATUS = 1
u.DCOST = 0

# Variables
t = m.Var(value=0)
x1 = m.Var(value=0)
x2 = m.Var(value=-1)
x3 = m.Var(value=-np.sqrt(5))
x4 = m.Var(value=0)

p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)

# Equations
m.Equation(t.dt()==1)
m.Equation(x1.dt()==x2)
m.Equation(x2.dt()==-x3*u+16*t-8)
m.Equation(x3.dt()==u)
m.Equation(x4.dt()==x1**2+x2**2 \
+0.005*(x2+16*t-8-0.1*x3*(u**2))**2)

# Objective Function
m.Obj(x4*final)

m.options.IMODE = 6
m.options.NODES = 4
m.options.MV_TYPE = 1
m.options.SOLVER = 3
m.solve()

print(m.path)

print('Objective = min x4(tf): ' + str(x4[-1]))

plt.figure(1)
plt.subplot(2,1,1)
plt.plot(m.time,u,'r-',LineWidth=2,label=r'$u$')
plt.legend(loc='best')
plt.subplot(2,1,2)
plt.plot(m.time,x1,'r--',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2,'g:',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,x3,'k-',LineWidth=2,label=r'$x_3$')
plt.plot(m.time,x4,'b-',LineWidth=2,label=r'$x_4$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()
February 07, 2018, at 04:53 AM by 10.37.117.169 -
Changed line 44 from:
nt = 501
to:
nt = 101
Changed lines 124-133 from:
->Attach:dynopt_2.png
to:

{$\min_{u(t)} x_4 \left( t_f \right)$}
{$\mathrm{subject \; to}$}
{$\frac{dx_1}{dt}=x_ 2$}
{$\frac{dx_2}{dt}=-x_3 \, u + 16 \, t - 8$}
{$\frac{dx_3}{dt}=u$}
{$\frac{dx_4}{dt}=x_1^2+x_2^2+0 .0005 \left(x_2 + 16 \, t -8 -0.1x_3\,u^2\right)^2$}
{$x(0) = [0 \; -1 \; -\sqrt{5} \; 0]^T$}
{$-4 \le u \le 10$}
{$t_f=1$}
February 07, 2018, at 04:48 AM by 10.37.117.169 -
Changed line 14 from:
{$x(0) = [1 0]^T$}
to:
{$x(0) = [1 \; 0]^T$}
Changed lines 17-35 from:
(:toggle hide gekko1a button show="Show GEKKO (Python) Code":)
to:
!!!! Example 1b
* Nonlinear, unconstrained, minimize final state with terminal constraint

{$\min_{u(t)} x_2 \left( t_f \right)$}
{$\mathrm{subject \; to}$}
{$\frac{dx_1}{dt}=u$}
{$\frac{dx_2}{dt}=x_1^2 + u^2$}
{$x(0) = [1 \; 0]^T$}
{$x_1 \left( t_f \right)=1$}
{$t_f=1$}

!!!! Solution to Benchmarks 1a and 1b

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":)
Changed lines 77-87 from:
!!!! Example 1b
* Nonlinear, unconstrained, minimize final state with terminal constraint
->Attach:dynopt_1b.png

!!!! Solution to Benchmarks 1a and 1b

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to:
(:toggle hide gekko1b button show="Show GEKKO (Python) Code for 1b":)
(:div id=gekko1b:)
(:source lang=python:)
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

nt = 101
m.time = np.linspace(0,1,nt)

# Variables
x1 = m.Var(value=1)
x2 = m.Var(value=0)
u = m.Var(value=-0.48)

p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)

# Equations
m.Equation(x1.dt()==u)
m.Equation(x2.dt()==x1**2 + u**2)
m.Equation(final*(x1-1)==0)

# Objective Function
m.Obj(x2*final)

m.options.IMODE = 6
m.solve()

plt.figure(1)
plt.plot(m.time,x1,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,u,'r--',LineWidth=2,label=r'$u$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()
(:sourceend:)
(:divend:)

(:toggle hide gekko2 button show="Show GEKKO (Python) Code":)
(:div id=gekko2:)
(:source lang=python:)
# Show source
(:sourceend:)
(:divend:)

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(:div id=gekko3:)
(:source lang=python:)
# Show source
(:sourceend:)
(:divend:)

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(:div id=gekko4:)
(:source lang=python:)
# Show source
(:sourceend:)
(:divend:)

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(:div id=gekko5:)
(:source lang=python:)
# Show source
(:sourceend:)
(:divend:)
February 07, 2018, at 04:44 AM by 10.37.117.169 -
Changed lines 9-15 from:
->Attach:dynopt_1a.png
to:

{$\min_{u(t)} x_2 \left( t_f \right)$}
{$\mathrm{subject \; to}$}
{$\frac{dx_1}{dt}=u$}
{$\frac{dx_2}{dt}=x_1^2 + u^2$}
{$x(0) = [1 0]^T$}
{$t_f=1$}
February 06, 2018, at 03:24 PM by 10.37.117.169 -
Changed line 3 from:
'''Objective:''' Set up and solve five [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^2^'. Create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''
to:
'''Objective:''' Set up and solve '''three''' of the five [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^2^'. Create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''

(:toggle hide gekko1a button show="Show GEKKO (Python) Code":)
(:div id=gekko1a:)
(:source lang=python:)
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

nt = 501
m.time = np.linspace(0,1,nt)

# Variables
x1 = m.Var(value=1)
x2 = m.Var(value=0)
u = m.Var(value=-0.75)

p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)

# Equations
m.Equation(x1.dt()==u)
m.Equation(x2.dt()==x1**2 + u**2)

# Objective Function
m.Obj(x2*final)

m.options.IMODE = 6
m.solve()

plt.figure(1)
plt.plot(m.time,x1,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,u,'r--',LineWidth=2,label=r'$u$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()
(:sourceend:)
(:divend:)

Changed line 3 from:
'''Objective:''' Set up and solve several [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^2^'. Create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''
to:
'''Objective:''' Set up and solve five [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^2^'. Create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''

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# Hedengren, J. D. and Asgharzadeh Shishavan, R., Powell, K.M., and Edgar, T.F., Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Computers and Chemical Engineering, Volume 70, pg. 133–148, 2014. [[https://dx.doi.org/10.1016/j.compchemeng.2014.04.013|Article]]

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----
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!!!! Solution
to:
!!!! Solution to Benchmarks 1a and 1b
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to:
----
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!!!! Solution
to:
!!!! Solution to Benchmark 2
----
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!!!! Solution
to:
!!!! Solution to Benchmark 3
----
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!!!! Solution
to:
!!!! Solution to Benchmark 4
----
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!!!! Solution
to:
!!!! Solution to Benchmark 5

----
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* Example 1a - Nonlinear, unconstrained, minimize final state
to:
!!!! Example 1a
* Nonlinear, unconstrained, minimize final state
Changed lines 9-10 from:
* Example 1b - Nonlinear, unconstrained, minimize final state with terminal constraint
to:
!!!! Example 1b
* Nonlinear, unconstrained, minimize final state with terminal constraint
Changed lines 12-20 from:
* Example 2 - Nonlinear, constrained, minimize final state
->Attach:dynopt_2.png
* Example 3 - Tubular reactor with parallel reaction
->Attach:dynopt_3.png
* Example 4 - Batch reactor with consecutive reactions A->B->C
->Attach:dynopt_4.png
Example 5 - Catalytic reactor with A<->B->C
->Attach:dynopt_5.png

to:
Changed lines 14-16 from:

to:

!!!! Example 2
* Nonlinear, constrained, minimize final state
->Attach:dynopt_2.png
!!!! Solution

!!!! Example 3
* Tubular reactor with parallel reaction
->Attach:dynopt_3.png
!!!! Solution

!!!! Example 4
* Batch reactor with consecutive reactions A->B->C
->Attach:dynopt_4.png
!!!! Solution

!!!! Example 5
* Catalytic reactor with A<->B->C
->Attach:dynopt_5.png
!!!! Solution

!!!! Exercise

'''Objective:''' Set up and solve several [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^2^'. Create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''

* Example 1a - Nonlinear, unconstrained, minimize final state
->Attach:dynopt_1a.png
* Example 1b - Nonlinear, unconstrained, minimize final state with terminal constraint
->Attach:dynopt_1b.png
* Example 2 - Nonlinear, constrained, minimize final state
->Attach:dynopt_2.png
* Example 3 - Tubular reactor with parallel reaction
->Attach:dynopt_3.png
* Example 4 - Batch reactor with consecutive reactions A->B->C
->Attach:dynopt_4.png
Example 5 - Catalytic reactor with A<->B->C
->Attach:dynopt_5.png

!!!! Solution