## Main.FinalExam History

March 21, 2019, at 03:59 PM by 10.35.117.63 -

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

# Check solution with GEKKO
m = GEKKO()
m.time=[0,1]
u = m.Var(value=1)
x = m.Var(value=0)
m.Obj((x-3)**2)
m.Equation(5*x.dt()==-x+2*u)
m.options.IMODE = 6
m.options.NODES = 3
m.solve()

# Matrix solution for Orthogonal Collocation on Finite Elements
#
# min (x-3)^2
# s.t. 5 * dx/dt = -x + 2 * u
#
# Equations 1-2 (subject to equations at each node)
# 5 * xdot1 = -x1 + 2*u1
# 5 * xdot2 = -x2 + 2*u2
# Equations 3-4 (collocation equations)
# t2 * 0.75 * xdot1 - t2 * 0.25 * xdot2 = x1 - x0
# t2 * 1.00 * xdot1 = x2 - x0
# Equations 5-6 (derivative of objective = 0)
# 2 * (x1-3) = 0
# 2 * (x2-3) = 0
#
# Rearrange to put all variables on left-hand side
# Equations 1-2 (subject to equations at each node)
# 5 * xdot1 + x1 - 2*u1 = 0
# 5 * xdot2 + x2 - 2*u2 = 0
# Equations 3-4 (collocation equations)
# t2 * 0.75 * xdot1 - t2 * 0.25 * xdot2 - x1 = -x0
# t2 * 1.00 * xdot1 - x2 = -x0
# Equations 5-6 (derivative of objective = 0)
# 2 * x1 = 6
# 2 * x2 = 6
#
# Set-up and solve A y = b
# y = [xdot1 xdot2 x1 x2 u1 u2]
# Matrix A
A = np.array([[5,0,1,0,-2,0],\
[0,5,0,1,0,-2],\
[0.75,-0.25,-1,0,0,0],\
[1,0,0,-1,0,0],\
[0,0,2,0,0,0],\
[0,0,0,2,0,0]])
# Column vector b
b = np.array([0,0,0,0,6,6])
# Solve A y = b as y = A^-1 * b
ymat = np.linalg.solve(A,b)

print('Variables with Orthogonal Collocation')
print(['u1 = ' + str(ymat)])
print(['u2 = ' + str(ymat) + \
' (Matrix) vs ' + str(u.value[-1]) + ' (GEKKO)'])
print(['x1 = ' + str(ymat)])
print(['x2 = ' + str(ymat) + \
' (Matrix) vs ' + str(x.value[-1]) + ' (GEKKO)'])
print(' ')
print('Derivatives with Orthogonal Collocation')
print(['d(x11)/dt = ' + str(ymat)])
print(['d(x21)/dt = ' + str(ymat)])
(:sourceend:)
(:divend:)
March 23, 2018, at 03:42 PM by 10.5.113.167 -
Changed lines 30-41 from:
!!!! Solution 3

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/U7uyj9BaNKg" frameborder="0" allowfullscreen></iframe>
(:htmlend:)

]]
to:
Deleted lines 33-34:
(:html:)
(:htmlend:)
March 23, 2018, at 01:46 PM by 10.5.113.167 -
Changed line 1 from:
(:title 2015 Final Exam for Dynamic Optimization:)
to:
(:title Practice Final Exam:)
March 22, 2018, at 08:45 PM by 10.5.113.167 -
Changed line 26 from:
Attach:inverted_pendulum2.gif
to:
Attach:inverted_pendulum_wgts.gif
March 22, 2018, at 08:44 PM by 10.5.113.167 -
Changed line 26 from:
Attach:inverted_pendulum.gif
to:
Attach:inverted_pendulum2.gif
March 22, 2018, at 12:31 AM by 10.37.35.33 -

Attach:inverted_pendulum.gif
March 22, 2018, at 12:04 AM by 10.37.35.33 -

----
March 22, 2018, at 12:03 AM by 10.37.35.33 -
Changed lines 10-11 from:
!!!! Solution
to:
!!!! Solution 1
Changed lines 20-21 from:
See [[Main/InvertedPendulum|Inverted Pendulum Solution]]
to:
!!!! Solution 2

See [[Main/InvertedPendulum|Inverted Pendulum Solution]]

!!!! Solution 3
March 22, 2018, at 12:02 AM by 10.37.35.33 -
Changed lines 20-24 from:

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/iW52CoG33pk" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
to:
See [[Main/InvertedPendulum|Inverted Pendulum Solution]]
March 26, 2016, at 01:02 AM by 10.5.113.125 -

Attach:2tank_linear_MPC.png
March 23, 2016, at 03:06 PM by 45.56.3.173 -
Changed line 29 from:
to:
March 23, 2016, at 01:22 PM by 45.56.3.173 -
Changed line 28 from:
to:
March 23, 2016, at 01:21 PM by 45.56.3.173 -
Changed lines 28-29 from:
to:
]]
Changed lines 35-36 from:
to:
]]
March 23, 2016, at 01:18 PM by 45.56.3.173 -

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/U7uyj9BaNKg" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
March 21, 2016, at 03:45 PM by 45.56.3.173 -
Changed line 28 from:
to:
February 09, 2016, at 09:55 PM by 174.148.97.76 -
Changed line 1 from:
(:title Final Exam for Dynamic Optimization:)
to:
(:title 2015 Final Exam for Dynamic Optimization:)
Changed line 3 from:
(:description Final exam for the graduate level course on dynamic estimation and optimization.:)
to:
(:description Final exam for the graduate level course on dynamic estimation and optimization in 2015.:)
February 09, 2016, at 09:53 PM by 174.148.97.76 -
Changed line 5 from:
!!!! Final Exam
to:
!!!! Final Exam 2015
June 15, 2015, at 02:27 PM by 45.56.3.184 -
Changed lines 12-13 from:
to:

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/n4OzENKziR4" frameborder="0" allowfullscreen></iframe>
(:htmlend:)

Changed lines 20-21 from:
to:
Changed line 23 from:
<iframe width="560" height="315" src="https://www.youtube.com/embed/n4OzENKziR4" frameborder="0" allowfullscreen></iframe>
to:
<iframe width="560" height="315" src="https://www.youtube.com/embed/iW52CoG33pk" frameborder="0" allowfullscreen></iframe>
Changed lines 28-36 from:

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/iW52CoG33pk" frameborder="0" allowfullscreen></iframe>
(:htmlend:)

----

to:
June 15, 2015, at 02:17 PM by 45.56.3.184 -

!!!! Solution

----

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/n4OzENKziR4" frameborder="0" allowfullscreen></iframe>
(:htmlend:)

----

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/iW52CoG33pk" frameborder="0" allowfullscreen></iframe>
(:htmlend:)

----

(:html:)
(:htmlend:)
June 05, 2015, at 11:15 PM by 10.5.113.170 -
Changed lines 1-3 from:
(:title Midterm Exam for Dynamic Optimization:)
(:keywords Python, MATLAB, Simulink, nonlinear control, model predictive control, exam, midterm:)
(:description Mid-term exam for dynamic estimation and optimization as a graduate-level course.:)
to:
(:title Final Exam for Dynamic Optimization:)
(:keywords Python, MATLAB, Simulink, nonlinear control, model predictive control, exam, final:)
(:description Final exam for the graduate level course on dynamic estimation and optimization.:)
June 05, 2015, at 08:29 PM by 10.10.144.142 -
Changed lines 7-8 from:
to:
June 05, 2015, at 04:57 PM by 45.56.3.184 -
Changed lines 7-8 from:
to: