## Dynamic Control Introduction

## Main.DynamicControl History

Hide minor edits - Show changes to output

Added lines 120-125:

* The first step to solve a model predictive control (MPC) in Python Gekko is to define the MPC problem. This may involve setting the time horizon and decision variables, constraints, objective function, and other parameters.

* Once the problem is defined, the next step is to create the MPC object. This is done by setting ''m.options.IMODE=6''. The parameters of the MPC object can be set in ''m.options''.

* The third step is to solve the MPC problem. This is done by calling the ''m.solve()'' command. This will solve the MPC problem and return the optimal solution.

* The fourth step is to process the solution. This can involve extracting the optimal decisions, evaluating the objective function, and other processing steps.

* Finally, the fifth step is to apply the optimal decisions. This can involve setting the plant state or inputs based on the optimal decisions.

Changed line 111 from:

plt.plot(m.time,p.value,'b-',~~LineWidth~~=2)

to:

plt.plot(m.time,p.value,'b-',lw=2)

Changed line 114 from:

plt.plot(m.time,v.value,'r--',~~LineWidth~~=2)

to:

plt.plot(m.time,v.value,'r--',lw=2)

Added lines 53-119:

(:toggle hide gekko button show="Show GEKKO (Python) Code":)

(:div id=gekko:)

(:source lang=python:)

#%%Import packages

import numpy as np

from random import random

from gekko import GEKKO

import matplotlib.pyplot as plt

#%% Build model

#initialize GEKKO model

m = GEKKO()

#time

m.time = np.linspace(0,20,41)

#constants

mass = 500

#Parameters

b = m.Param(value=50)

K = m.Param(value=0.8)

#Manipulated variable

p = m.MV(value=0, lb=0, ub=100)

#Controlled Variable

v = m.CV(value=0)

#Equations

m.Equation(mass*v.dt() == -v*b + K*b*p)

#%% Tuning

#global

m.options.IMODE = 6 #control

#MV tuning

p.STATUS = 1 #allow optimizer to change

p.DCOST = 0.1 #smooth out gas pedal movement

p.DMAX = 20 #slow down change of gas pedal

#CV tuning

#setpoint

v.STATUS = 1 #add the SP to the objective

m.options.CV_TYPE = 2 #L2 norm

v.SP = 40 #set point

v.TR_INIT = 1 #setpoint trajectory

v.TAU = 5 #time constant of setpoint trajectory

#%% Solve

m.solve()

#%% Plot solution

plt.figure()

plt.subplot(2,1,1)

plt.plot(m.time,p.value,'b-',LineWidth=2)

plt.ylabel('gas')

plt.subplot(2,1,2)

plt.plot(m.time,v.value,'r--',LineWidth=2)

plt.ylabel('velocity')

plt.xlabel('time')

plt.show()

(:sourceend:)

(:divend:)

Changed line 17 from:

'''Objective:''' Implement a model predictive controller that automatically regulates vehicle velocity. Implement the controller in Excel, MATLAB, Python, or Simulink and tune the controller for acceptable performance. Discuss factors that may be ~~import~~ for evaluating controller performance. ''Estimated time: 1 hour''

to:

'''Objective:''' Implement a model predictive controller that automatically regulates vehicle velocity. Implement the controller in Excel, MATLAB, Python, or Simulink and tune the controller for acceptable performance. Discuss factors that may be important for evaluating controller performance. ''Estimated time: 1 hour''

Changed line 17 from:

'''Objective:''' Implement a model predictive controller that automatically regulates vehicle velocity. Implement the controller in ~~MATLAB~~, ~~Python~~, ~~and~~ Simulink and tune the controller for acceptable performance. Discuss factors that may be import for evaluating controller performance. ''Estimated time: ~~2 hours~~''

to:

'''Objective:''' Implement a model predictive controller that automatically regulates vehicle velocity. Implement the controller in Excel, MATLAB, Python, or Simulink and tune the controller for acceptable performance. Discuss factors that may be import for evaluating controller performance. ''Estimated time: 1 hour''

Changed lines 21-22 from:

to:

The dynamic relationship between a vehicle gas pedal position (MV) and velocity (CV) is given by the following set of conditions and a single dynamic equation.

Changed line 36 from:

to:

Discuss the controller performance and how it could be tuned to meet multiple objectives including:

Changed lines 15-16 from:

to:

A method to solve dynamic control problems is by numerically integrating the dynamic model at discrete time intervals, much like measuring a physical system at particular time points. The numerical solution is compared to a desired trajectory and the difference is minimized by adjustable parameters in the model that may change at every time step. The first control action is taken and then the entire process is repeated at the next time instance. The process is repeated because objective targets may change or updated measurements may have adjusted parameter or state estimates.

!!!! Exercise

'''Objective:''' Implement a model predictive controller that automatically regulates vehicle velocity. Implement the controller in MATLAB, Python, and Simulink and tune the controller for acceptable performance. Discuss factors that may be import for evaluating controller performance. ''Estimated time: 2 hours''

# The dynamic relationship between a vehicle gas pedal position (MV) and velocity (CV) is given by the following set of conditions and a single dynamic equation.

Constants

m = 500 ! Mass (kg)

Parameters

b = 50 ! Resistive coefficient (N-s/m)

K = 0.8 ! Gain (m/s-%pedal)

p = 0 >= 0 <= 100 ! Gas pedal position (%)

Variables

v = 0 ! initial condition

Equations

m * $v = -v * b + K * b * p

Implement a model predictive controller that adjusts gas pedal position to regulate velocity. Start at an initial vehicle velocity of 0 m/s and accelerate to a velocity of 40 m/s.

# Discuss the controller performance and how it could be tuned to meet multiple objectives including:

* minimize travel time

* remain within speed limits

* improve vehicle fuel efficiency

* discourage excessive gas pedal adjustments

* do not accelerate excessively

There is no need to implement these advanced objectives in simulation for this second part of the exercise, only discuss the possible competing objectives.

!!!! Solution

!!!! Exercise

'''Objective:''' Implement a model predictive controller that automatically regulates vehicle velocity. Implement the controller in MATLAB, Python, and Simulink and tune the controller for acceptable performance. Discuss factors that may be import for evaluating controller performance. ''Estimated time: 2 hours''

# The dynamic relationship between a vehicle gas pedal position (MV) and velocity (CV) is given by the following set of conditions and a single dynamic equation.

Constants

m = 500 ! Mass (kg)

Parameters

b = 50 ! Resistive coefficient (N-s/m)

K = 0.8 ! Gain (m/s-%pedal)

p = 0 >= 0 <= 100 ! Gas pedal position (%)

Variables

v = 0 ! initial condition

Equations

m * $v = -v * b + K * b * p

Implement a model predictive controller that adjusts gas pedal position to regulate velocity. Start at an initial vehicle velocity of 0 m/s and accelerate to a velocity of 40 m/s.

# Discuss the controller performance and how it could be tuned to meet multiple objectives including:

* minimize travel time

* remain within speed limits

* improve vehicle fuel efficiency

* discourage excessive gas pedal adjustments

* do not accelerate excessively

There is no need to implement these advanced objectives in simulation for this second part of the exercise, only discuss the possible competing objectives.

!!!! Solution

Added lines 50-51:

Excel, MATLAB, Python, and Simulink are used in this example to both solve the differential equations that describe the velocity of a vehicle as well as minimize the control objective function.

Deleted line 54:

Deleted lines 22-45:

'''Objective:''' Set up and solve several [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^1^'. 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 plug flow reactor with A->B->C

->Attach:dynopt_5.png

!!!! Solution

Attach:download.png [[Attach:dynamic_optimization_benchmarks.zip|Dynamic Optimization Benchmarks in MATLAB and Python]]

!!!! References

# M. Čižniar, M. Fikar, M.A. Latifi: A MATLAB Package for Dynamic Optimisation of Processes, 7th International Scientific – Technical Conference – Process Control 2006, June 13 – 16, 2006, Kouty nad Desnou, Czech Republic. [[Attach:DynOpt_Benchmarks.pdf|Article]]

Changed lines 25-26 from:

'''Objective:''' Set up and solve several dynamic optimization benchmark problems'^1^'. Create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''

to:

'''Objective:''' Set up and solve several [[Attach:Dynamic_Optimization_Benchmarks.pdf|dynamic optimization benchmark problems]]'^1^'. Create a program to optimize and display the results. ''Estimated Time (each): 30 minutes''

Changed line 42 from:

Attach:download.png [[Attach:~~dynopt~~_benchmarks.zip|Dynamic Optimization Benchmarks in MATLAB and Python]]

to:

Attach:download.png [[Attach:dynamic_optimization_benchmarks.zip|Dynamic Optimization Benchmarks in MATLAB and Python]]

Changed line 46 from:

# M. Čižniar, M. Fikar, M.A. Latifi: A MATLAB Package for Dynamic Optimisation of Processes, 7th International Scientific – Technical Conference – Process Control 2006, June 13 – 16, 2006, Kouty nad Desnou, Czech Republic.

to:

# M. Čižniar, M. Fikar, M.A. Latifi: A MATLAB Package for Dynamic Optimisation of Processes, 7th International Scientific – Technical Conference – Process Control 2006, June 13 – 16, 2006, Kouty nad Desnou, Czech Republic. [[Attach:DynOpt_Benchmarks.pdf|Article]]

Added lines 23-46:

!!!! Exercise

'''Objective:''' Set up and solve several dynamic optimization benchmark problems'^1^'. 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 plug flow reactor with A->B->C

->Attach:dynopt_5.png

!!!! Solution

Attach:download.png [[Attach:dynopt_benchmarks.zip|Dynamic Optimization Benchmarks in MATLAB and Python]]

!!!! References

# M. Čižniar, M. Fikar, M.A. Latifi: A MATLAB Package for Dynamic Optimisation of Processes, 7th International Scientific – Technical Conference – Process Control 2006, June 13 – 16, 2006, Kouty nad Desnou, Czech Republic.

'''Objective:''' Set up and solve several dynamic optimization benchmark problems'^1^'. 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 plug flow reactor with A->B->C

->Attach:dynopt_5.png

!!!! Solution

Attach:download.png [[Attach:dynopt_benchmarks.zip|Dynamic Optimization Benchmarks in MATLAB and Python]]

!!!! References

# M. Čižniar, M. Fikar, M.A. Latifi: A MATLAB Package for Dynamic Optimisation of Processes, 7th International Scientific – Technical Conference – Process Control 2006, June 13 – 16, 2006, Kouty nad Desnou, Czech Republic.

Changed line 17 from:

to:

Attach:download.png [[Attach:model_predictive_control.zip|MPC in Excel, MATLAB, Python, and Simulink]]

Added lines 6-7:

* [[Attach:Intro_Dynamic_Control.pdf|Introduction to Dynamic Control/Optimization (pdf)]]

Changed line 18 from:

<iframe width="560" height="315" src="https://www.youtube.com/embed/~~umxAfu44kWo~~?rel=0" frameborder="0" allowfullscreen></iframe>

to:

<iframe width="560" height="315" src="https://www.youtube.com/embed/dqm2OqXYLR8?rel=0" frameborder="0" allowfullscreen></iframe>

Added lines 6-9:

(:html:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/DFqOf5wbQtc?rel=0" frameborder="0" allowfullscreen></iframe>

(:htmlend:)

Changed lines 5-16 from:

Dynamic control is a method to use model predictions to plan an optimized future trajectory for time-varying systems. It is often referred to as Model Predictive Control (MPC) or Dynamic Optimization.

to:

Dynamic control is a method to use model predictions to plan an optimized future trajectory for time-varying systems. It is often referred to as Model Predictive Control (MPC) or Dynamic Optimization.

!!!!Model Predictive Control Example

A method to solve dynamic control problems is by numerically integrating the dynamic model at discrete time intervals, much like measuring a physical system at particular time points. The numerical solution is compared to a desired trajectory and the difference is minimized by adjustable parameters in the model that may change at every time step. The first control action is taken and then the entire process is repeated at the next time instance. The process is repeated because objective targets may change or updated measurements may have adjusted parameter or state estimates. Excel, MATLAB, Python, and Simulink are used in the following example to both solve the differential equations that describe the velocity of a vehicle as well as minimize the control objective function.

* [[Attach:model_predictive_control.zip|MPC in Excel, MATLAB, Python, and Simulink]]

(:html:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/umxAfu44kWo?rel=0" frameborder="0" allowfullscreen></iframe>

(:htmlend:)

!!!!Model Predictive Control Example

A method to solve dynamic control problems is by numerically integrating the dynamic model at discrete time intervals, much like measuring a physical system at particular time points. The numerical solution is compared to a desired trajectory and the difference is minimized by adjustable parameters in the model that may change at every time step. The first control action is taken and then the entire process is repeated at the next time instance. The process is repeated because objective targets may change or updated measurements may have adjusted parameter or state estimates. Excel, MATLAB, Python, and Simulink are used in the following example to both solve the differential equations that describe the velocity of a vehicle as well as minimize the control objective function.

* [[Attach:model_predictive_control.zip|MPC in Excel, MATLAB, Python, and Simulink]]

(:html:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/umxAfu44kWo?rel=0" frameborder="0" allowfullscreen></iframe>

(:htmlend:)

Added lines 1-5:

(:title Dynamic Control Introduction:)

(:keywords control, dynamics, dynamic optimization, simulation, modeling language, differential, algebraic, tutorial:)

(:description Dynamic control in MATLAB and Python for use in real-time or off-line applications:)

Dynamic control is a method to use model predictions to plan an optimized future trajectory for time-varying systems. It is often referred to as Model Predictive Control (MPC) or Dynamic Optimization.

(:keywords control, dynamics, dynamic optimization, simulation, modeling language, differential, algebraic, tutorial:)

(:description Dynamic control in MATLAB and Python for use in real-time or off-line applications:)

Dynamic control is a method to use model predictions to plan an optimized future trajectory for time-varying systems. It is often referred to as Model Predictive Control (MPC) or Dynamic Optimization.