## Practice Final Exam

## Main.FinalExam History

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(: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[4])])

print(['u2 = ' + str(ymat[5]) + \

' (Matrix) vs ' + str(u.value[-1]) + ' (GEKKO)'])

print(['x1 = ' + str(ymat[2])])

print(['x2 = ' + str(ymat[3]) + \

' (Matrix) vs ' + str(x.value[-1]) + ' (GEKKO)'])

print(' ')

print('Derivatives with Orthogonal Collocation')

print(['d(x11)/dt = ' + str(ymat[0])])

print(['d(x21)/dt = ' + str(ymat[1])])

(:sourceend:)

(:divend:)

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Attach:download.png [[Attach:final_solution3_steps_python.zip|Dual Tank: Step Test for Model Identification (Python/Excel)]]

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

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Attach:download.png [[Attach:final_solution3_linear_mpc.zip|Dual Tank: Linear MPC Solution (MATLAB/Simulink)]]

Attach:download.png [[Attach:final_solution3_python_mpc.zip|Dual Tank: Linear MPC Solution (Python)

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Attach:download.png See [[Main/LevelControl|Level Control]]

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(:title ~~2015~~ Final Exam~~ for Dynamic Optimization~~:)

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(:title Practice Final Exam:)

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Attach:inverted_pendulum_wgts.gif

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Attach:inverted_pendulum2.gif

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!!!! Solution

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!!!! Solution 1

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

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!!!! Solution 2

Attach:download.png See [[Main/InvertedPendulum|Inverted Pendulum Solution]]

Attach:download.png See [[Main/InvertedPendulum|Inverted Pendulum Solution]]

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!!!! Solution 3

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

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

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Attach:download.png [[Attach:final_solution3_steps_python.zip|Dual Tank: Step Test for Model Identification (Python)]]

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Attach:download.png [[Attach:final_solution3_steps_python.zip|Dual Tank: Step Test for Model Identification (Python/Excel)]]

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Attach:download.png [[Attach:final_solution3_steps_simulink.zip|Dual Tank: Step Test for Model Identification (~~MATLAB/~~Simulink/Excel)]]

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Attach:download.png [[Attach:final_solution3_steps_simulink.zip|Dual Tank: Step Test for Model Identification (Simulink/Excel)]]

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Attach:download.png [[Attach:final_solution3_~~step~~_~~test~~.zip|Dual Tank: Step Test for Model Identification]]

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Attach:download.png [[Attach:final_solution3_steps_simulink.zip|Dual Tank: Step Test for Model Identification (MATLAB/Simulink/Excel)]]

Attach:download.png [[Attach:final_solution3_steps_python.zip|Dual Tank: Step Test for Model Identification (Python)]]

Attach:download.png [[Attach:final_solution3_steps_python.zip|Dual Tank: Step Test for Model Identification (Python)]]

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Attach:download.png [[Attach:final_solution3_linear_mpc.zip|Dual Tank: Linear MPC Solution]]

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Attach:download.png [[Attach:final_solution3_linear_mpc.zip|Dual Tank: Linear MPC Solution (MATLAB/Simulink)]]

Attach:download.png [[Attach:final_solution3_python_mpc.zip|Dual Tank: Linear MPC Solution (Python)]]

Attach:download.png [[Attach:final_solution3_python_mpc.zip|Dual Tank: Linear MPC Solution (Python)]]

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Attach:download.png [[Attach:final_solution3_step_test.zip|Dual Tank: Step Test for Model Identification]]

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Attach:download.png [[Attach:final_solution3_~~files~~.zip|Dual Tank~~ MPC~~: ~~Solution Files~~]]

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Attach:download.png [[Attach:final_solution3_linear_mpc.zip|Dual Tank: Linear MPC Solution]]

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(:title Final Exam for Dynamic Optimization:)

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(:title 2015 Final Exam for Dynamic Optimization:)

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(:description Final exam for the graduate level course on dynamic estimation and optimization.:)

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(:description Final exam for the graduate level course on dynamic estimation and optimization in 2015.:)

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Attach:download.png [[Attach:final_~~solution~~_~~key~~.~~pdf~~|~~Final Exam Solution Key (PDF)~~]]

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Attach:download.png [[Attach:final_solution1_files.zip|KKT Conditions for Dynamic Optimization: Solution Files]]

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Attach:download.png [[Attach:final_~~solution1~~_files.zip|~~KKT Conditions for Dynamic Optimization~~ Solution Files]]

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Attach:download.png [[Attach:final_solution2_files.zip|Inverted Pendulum: Solution Files]]

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Attach:download.png [[Attach:final_~~solution2~~_files.zip|~~Inverted Pendulum~~ Solution Files]]

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

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Attach:download.png [[Attach:final_solution3_files.zip|Dual Tank MPC Solution Files]]

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

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Attach:download.png [[Attach:final_solution3_files.zip|Dual Tank MPC Solution Files]]

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Attach:download.png [[Attach:final_solution3_files.zip|Dual Tank MPC: Solution Files]]

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!!!! Solution

Attach:download.png [[Attach:final_solution_key.pdf|Final Exam Solution Key (PDF)]]

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Attach:download.png [[Attach:final_solution1_files.zip|KKT Conditions for Dynamic Optimization Solution Files]]

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

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Attach:download.png [[Attach:final_solution2_files.zip|Inverted Pendulum Solution Files]]

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

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Attach:download.png [[Attach:final_solution3_files.zip|Dual Tank MPC Solution Files]]

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(: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~~.:)

(:keywords Python, MATLAB, Simulink, nonlinear control, model predictive control, exam,

(:description

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(: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.:)

(:keywords Python, MATLAB, Simulink, nonlinear control, model predictive control, exam, final:)

(:description Final exam for the graduate level course on dynamic estimation and optimization.:)

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Attach:download.png [[Attach:final_~~exam1~~.pdf|Final Exam]]

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Attach:download.png [[Attach:final_exam.pdf|Final Exam]]

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Attach:download.png [[Attach:final_exam1.pdf|Final Exam]]

Attach:download.png [[Attach:final_exam_files1.zip|Final Exam Files]]

Attach:download.png [[Attach:final_exam_files1.zip|Final Exam Files]]

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(: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.:)

!!!! Final Exam

Attach:download.png [[Attach:final_exam.pdf|Final Exam]]

Attach:download.png [[Attach:final_exam_files.zip|Final Exam Files]]

(: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.:)

!!!! Final Exam

Attach:download.png [[Attach:final_exam.pdf|Final Exam]]

Attach:download.png [[Attach:final_exam_files.zip|Final Exam Files]]