## Main.ControllerObjective History

Changed line 141 from:
{$y,x,u=\mathrm{outputs\,states\,and\,inputs}$}
to:
{$y,x,u=\mathrm{outputs,\,states,\,and\,inputs}$}
Changed lines 91-92 from:
{$\min_{x,y,u} \Phi = \left(y-y_t\right)^T W_t \left(y-y_t\right) + y^T c_y + u^T c_u + \Delta u^T c_{\Delta u} \Delta u$}
to:
{$\min_{x,y,u} \Phi = \left(y-y_t\right)^T W_t \left(y-y_t\right) + y^T w_y + u^T w_u + \Delta u^T W_{\Delta u} \Delta u$}
Changed lines 105-106 from:
{$\min_{x,y,u} \Phi = w_{hi}^T e_{hi} + w_{lo}^T e_{lo} + y^T c_y + u^T c_u + w_{\Delta u}^T \left(\Delta u_U+\Delta u_L\right)$}
to:
{$\min_{x,y,u} \Phi = w_{hi}^T e_{hi} + w_{lo}^T e_{lo} + y^T w_y + u^T w_u + w_{\Delta u}^T \left(\Delta u_U+\Delta u_L\right)$}
Changed lines 137-139 from:
{$c_{\Delta u}=\mathrm{manipulated\,variable\,movement\,penalty\,(DCOST)}$}
to:
{$W_{\Delta u}, w_{\Delta u}=\mathrm{manipulated\,variable\,movement\,penalty\,(DCOST)}$}

{$w_{u}, w_{y}=\mathrm{weight\,on\,input\,and\,output\,(COST )}$}
Changed lines 109-111 from:
{$0 = f\left(\frac{dx}{dt},x,y,p\right)$}

{$0 \le g\left(\frac{dx}{dt},x,y,p\right)$}
to:
{$0 = f\left(\frac{dx}{dt},x,y,u\right)$}

{$0 \le g\left(\frac{dx}{dt},x,y,u\right)$}
Changed line 133 from:
{$y_t,y_{t,hi},y_{t,lo}=\mathrm{reference trajectory or range}$}
to:
{$y_t,y_{t,hi},y_{t,lo}=\mathrm{reference\,trajectory\,or\,range}$}
{$\tau_c \frac{dy_{t,hi}}{dt} + y_{t,hi} = sp_{hi}$}

{$\tau_c \frac{dy_{t,lo}}{dt} + y_{t,lo} = sp_{lo}$}

Changed lines 129-147 from:
%width=400px%Attach:obj_ctrl_nomenclature.png
to:
{$\Phi=\mathrm{Objective\,Function}$}

{$y=\mathrm{model\,predictions}$}

{$y _t,y_{t,hi},y_{t,lo}=\mathrm{reference trajectory or range}$}

{$W_t,w_{hi}, w_{lo}=\mathrm{penalty\,outside\,reference\,trajectory\,(WSP,WSPHI,WSPLO)}$}

{$c_{\Delta u}=\mathrm{manipulated\,variable\,movement\,penalty\,(DCOST)}$}

{$y,x,u=\mathrm{outputs\,states\,and\,inputs}$}

{$\Delta u=\mathrm{manipulated\,variable\,change}$}

{$f,g=\mathrm{equality\,and\,inequality\,constraints}$}

{$e_{hi},e_{lo}=\mathrm{upper\,and\,lower\,error\,outside\,dead-band}$}

{$\Delta u_U,\Delta u_L=\mathrm{upper\,and\,lower\,manipulated\,variable\,change}$}
Changed lines 117-121 from:
{$\Delta p_U \ge p - \hat p$}

{$\Delta p_L \ge \hat p - p$}

{$e_{hi}, e_{lo}, \Delta p_U, \Delta p_L \ge 0$}
to:
{$\Delta u_U \ge u_i - u_{i-1}$}

{$\Delta u_L \ge u_{i-1} - u_i$}

{$e_{hi}, e_{lo}, \Delta u_U, \Delta u_L \ge 0$}
Changed lines 105-121 from:
%width=300px%Attach:obj_ctrl_l1_norm.png
to:
{$\min_{x,y,u} \Phi = w_{hi}^T e_{hi} + w_{lo}^T e_{lo} + y^T c_y + u^T c_u + w_{\Delta u}^T \left(\Delta u_U+\Delta u_L\right)$}

{$\mathrm{subject\;\;to}$}

{$0 = f\left(\frac{dx}{dt},x,y,p\right)$}

{$0 \le g\left(\frac{dx}{dt},x,y,p\right)$}

{$e_{hi} \ge y - y_{t,hi}$}

{$e_{lo} \ge y_{t,lo} - y$}

{$\Delta p_U \ge p - \hat p$}

{$\Delta p_L \ge \hat p - p$}

{$e_{hi}, e_{lo}, \Delta p_U, \Delta p_L \ge 0$}
Changed lines 95-99 from:
{$0 = f\left(\frac{d\;x}{d\;t},x,y,u\right)$}

{$0 \le g\left(\frac{d\;x}{d\;t},x,y,u\right)$}

{$\tau_c \frac{d\;y_t}{d\;t} + y_t = sp$}
to:
{$0 = f\left(\frac{dx}{dt},x,y,u\right)$}

{$0 \le g\left(\frac{dx}{dt},x,y,u\right)$}

{$\tau_c \frac{dy_t}{dt} + y_t = sp$}
Changed lines 91-99 from:
%width=300px%Attach:obj_ctrl_sq_error.png
to:
{$\min_{x,y,u} \Phi = \left(y-y_t\right)^T W_t \left(y-y_t\right) + y^T c_y + u^T c_u + \Delta u^T c_{\Delta u} \Delta u$}

{$\mathrm{subject\;\;to}$}

{$0 = f\left(\frac{d\;x}{d\;t},x,y,u\right)$}

{$0 \le g\left(\frac{d\;x}{d\;t},x,y,u\right)$}

{$\tau_c \frac{d\;y_t}{d\;t} + y_t = sp$}

December 28, 2018, at 10:34 PM by 10.37.144.156 -
Some applications require a custom reference trajectory that does not fit a standard form. A custom reference trajectory is specified by creating a new error (e) variable that is the difference between the specified trajectory (sinusoidal, sawtooth, random, etc) and the model output. This error is specified as a controlled variable (CV) with an upper and lower dead-band denoted as SPHI and SPLO. The CV can also be a value of zero with a squared error objective (e.SP=0, m.options.CV_TYPE=2) to drive to a target instead of a dead-band range.
Changed line 180 from:
e.TR_INIT = 1 #setpoint trajectory
to:
e.TR_INIT = 0 #setpoint trajectory
December 28, 2018, at 10:28 PM by 10.37.144.156 -
Changed lines 124-128 from:
!!!! Reactor Tuning Example

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<iframe width="560" height="315" src="https://www.youtube.com/embed/qH55ym1g-NM" frameborder="0" allowfullscreen></iframe>
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to:
!!!! Custom Reference Trajectory

%width=500px%Attach:gekko_traj.png

(:toggle hide gekko_traj button show="Custom Reference Trajectory with GEKKO":)
(:div id=gekko_traj:)
(:source lang=python:)
import numpy as np
from random import random
from gekko import GEKKO
import matplotlib.pyplot as plt

# 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=-100, ub=100)

# Reference trajectory
sine = 10*np.sin(m.time/20*4*np.pi)
traj = m.Param(value=sine)

# Controlled Variable
v = m.SV(value=0,name='v')

# Error
e = m.CV(value=0,name='e')

# Equations
m.Equation(mass*v.dt() == -v*b + K*b*p)
m.Equation(e==v-traj)

m.options.IMODE = 6 # control

# MV tuning
p.STATUS = 1 #allow optimizer to change
p.DCOST = 0.1 #smooth out MV
p.DMAX = 50 #slow down change of MV

# CV tuning
e.STATUS = 1 #add the CV to the objective
db = 2
e.SPHI = db #set point
e.SPLO = -db #set point
e.TR_INIT = 1 #setpoint trajectory
e.TAU = 5 #time constant of setpoint trajectory

# Solve
m.solve()

import json
with open(m.path+'//results.json') as f:

# Plot solution
plt.figure()
plt.subplot(3,1,1)
plt.plot(m.time,p.value,'b-',LineWidth=2,label='MV')
plt.legend(loc='best')
plt.ylabel('MV')
plt.subplot(3,1,2)
plt.plot(m.time,sine+db,'k-',label='SPHI')
plt.plot(m.time,sine-db,'k-',label='SPLO')
plt.plot(m.time,v.value,'r--',LineWidth=2,label='CV')
plt.legend(loc='best')
plt.ylabel('CV')
plt.subplot(3,1,3)
plt.plot(m.time,results['e.tr_hi'],'k-',label='SPHI')
plt.plot(m.time,results['e.tr_lo'],'k-',label='SPLO')
plt.plot(m.time,e.value,'r--',LineWidth=2,label='Error')
plt.legend(loc='best')
plt.ylabel('Error')
plt.xlabel('time')
plt.show()
(:sourceend:)
(:divend:)

December 28, 2018, at 09:29 PM by 10.37.144.156 -
Changed lines 40-41 from:
v = m.CV(value=0)
to:
v = m.CV(value=0,name='v')
Changed lines 75-76 from:
plt.plot(m.time,results['v1.tr_lo'],'k-',label='SPHI')
plt.plot(m.time,results['v1.tr_hi'],'k-',label='SPLO')
to:
plt.plot(m.time,results['v.tr_lo'],'k-',label='SPHI')
plt.plot(m.time,results['v.tr_hi'],'k-',label='SPLO')

(:toggle hide gekko_mpc button show="Model Predictive Control with GEKKO":)
(:div id=gekko_mpc:)

%width=500px%Attach:gekko_mpc.png

(:source lang=python:)
import numpy as np
from random import random
from gekko import GEKKO
import matplotlib.pyplot as plt

# 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)

m.options.IMODE = 6 # control

# MV tuning
p.STATUS = 1 #allow optimizer to change
p.DCOST = 0.1 #smooth out MV
p.DMAX = 20 #slow down change of MV

# CV tuning
v.STATUS = 1 #add the CV to the objective
v.SPHI = 42 #set point
v.SPLO = 38 #set point
v.TR_INIT = 1 #setpoint trajectory
v.TAU = 5 #time constant of setpoint trajectory

# Solve
m.solve()

import json
with open(m.path+'//results.json') as f:

# Plot solution
plt.figure()
plt.subplot(2,1,1)
plt.plot(m.time,p.value,'b-',LineWidth=2,label='MV')
plt.legend(loc='best')
plt.ylabel('gas')
plt.subplot(2,1,2)
plt.plot(m.time,results['v1.tr_lo'],'k-',label='SPHI')
plt.plot(m.time,results['v1.tr_hi'],'k-',label='SPLO')
plt.plot(m.time,v.value,'r--',LineWidth=2,label='CV')
plt.legend(loc='best')
plt.ylabel('velocity')
plt.xlabel('time')
plt.show()
(:sourceend:)
(:divend:)

!!!! Reactor Tuning Example

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/dM8gX54jWW4" frameborder="0" allowfullscreen></iframe>
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Changed lines 5-9 from:
The dynamic control objective function is a mathematical statement that is minimized or maximized to find a best solution among all possible feasible solutions for a controller. The form of this objective function is critical to give desirable solutions for driving a system to a desirable state or along a desired trajectory. Common objective statements relate to economic, safety, operability, environmental, or related objectives. Two common objective functions are shown below as squared error and l'_1_'-norm forms'^1^'.
to:
The dynamic control objective function is a mathematical statement that is minimized or maximized to find a best solution among all possible feasible solutions for a controller. The form of this objective function is critical to give desirable solutions for driving a system to a desirable state or along a desired trajectory. Common objective statements relate to economic, safety, operability, environmental, or related objectives.

Two common objective functions are shown below as squared error and l'_1_'-norm forms'^1^'.
Changed line 21 from:
%width=300px%Attach:obj_ctrl_nomenclature.png
to:
%width=400px%Attach:obj_ctrl_nomenclature.png
Changed lines 11-12 from:
%width=200px%Attach:obj_ctrl_sq_error.png
to:
%width=300px%Attach:obj_ctrl_sq_error.png
Changed lines 17-18 from:
%width=200px%Attach:obj_ctrl_l1_norm.png
to:
%width=300px%Attach:obj_ctrl_l1_norm.png
Changed line 21 from:
Attach:obj_ctrl_nomenclature.png
to:
%width=300px%Attach:obj_ctrl_nomenclature.png
Changed lines 11-12 from:
Attach:obj_ctrl_sq_error.png
to:
%width=200px%Attach:obj_ctrl_sq_error.png
Changed lines 15-17 from:
The l'_1_'-norm objective is the absolute value of the difference between the current measured value of the controlled variable and the desired target value. The absolute value operator is

Attach
:obj_ctrl_l1_norm.png
to:
The l'_1_'-norm objective is the absolute value of the difference between the current measured value of the controlled variable and the desired target value. The absolute value operator (abs(x)) does not have a continuous first and second derivative at x=0 (see [[https://apmonitor.com/me575/index.php/Main/LogicalConditions|absolute value function alternatives]]). [[https://apmonitor.com/wiki/index.php/Main/SlackVariables|Slack variables]] give the same absolute value function but with continuous first and second derivatives.

%width=200px%
Attach:obj_ctrl_l1_norm.png
The squared error objective is the most common form used in the literature. It penalizes deviation of the current measured value of the controlled variable from the desired target value.
The l'_1_'-norm objective is the absolute value of the difference between the current measured value of the controlled variable and the desired target value. The absolute value operator is

In formulating an objective function and model equations follow the following tips for improved convergence.

# Rearrange to equation in residual form to:
## Avoid divide by zero
## Minimize use of functions like sqrt, log, exp, etc.
## Have continuous first and second derivatives
## Fit the equation into a linear or quadratic form
# Bounds
## Include variable bounds to exclude infeasible solutions
## Variable bounds to avoid regions of strong nonlinearity
## Caution: watch for infeasible solutions
# Scaling:
## Scale absolute value of variables to 1e-3 to 1e3
## Scale absolute value of equation residuals to 1e-3 to 1e3
## Better that 1st derivative values are closer to 1.0
# Good initial conditions:
## Starting near a solution can improve convergence
## Try multiple initial conditions to verify global solution (non-convex problems)
## Explicitly calculate intermediate values
# Check iteration summary for improved convergence

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Deleted lines 18-42:
!!!! 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

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Deleted line 22:
# 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]]

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Changed line 33 from:
Example 5 - Catalytic plug flow reactor with A->B->C
to:
Example 5 - Catalytic reactor with A<->B->C
May 18, 2015, at 03:29 PM by 45.56.3.184 -
!!!! 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 plug flow reactor with A->B->C
->Attach:dynopt_5.png

!!!! Solution

Changed line 44 from:
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]]
May 18, 2015, at 03:21 PM by 45.56.3.184 -
Deleted lines 18-35:
!!!! Exercise

'''Objective:''' Design a model predictive controller with a custom objective function that satisfies a specific problem criterion. Simulate and optimize a pendulum system with an adjustable overhead cart. ''Estimated time: 3 hours.''

A pendulum is described by the following dynamic equations:

Attach:pendulum_crane_equations.png

where ''epsilon'' is ''m'_2_'/(m'_1_'+m'_2_')'', ''y'' is the position of the overhead cart, ''v'' is the velocity of the overhead cart, ''theta'' is the angle of the pendulum relative to the cart, and ''q'' is the rate of angle change'^2^'.

Attach:pendulum_crane.png

The objective of the controller is to adjust the force on the cart to move the pendulum mass to a new final position. Ensure that initial and final velocities and angles of the pendulum are zero. The position of the pendulum mass is initially at -1 and it is desired to move it to the new position of 0 within 6.2 seconds. Demonstrate controller performance with changes in the pendulum position and that the final pendulum mass remains at the final position without oscillation.

!!!! Solution

Deleted line 22:
# Bryson, A.E., Dynamic Optimization, Addison-Wesley, 1999.
May 13, 2015, at 06:36 PM by 10.5.113.160 -
Changed lines 27-28 from:
where ''epsilon'' is ''m'_2_'/(m'_1_'+m'_2_')'', ''y'' is the position of the overhead cart, ''v'' is the velocity of the overhead cart, ''theta'' is the angle of the pendulum relative to the cart, and ''q'' is the rate of angle change.
to:
where ''epsilon'' is ''m'_2_'/(m'_1_'+m'_2_')'', ''y'' is the position of the overhead cart, ''v'' is the velocity of the overhead cart, ''theta'' is the angle of the pendulum relative to the cart, and ''q'' is the rate of angle change'^2^'.
Deleted lines 30-31:

# Bryson, A.E., Dynamic Optimization, Addison-Wesley, 1999.
May 13, 2015, at 06:22 PM by 10.5.113.160 -
!!!! Exercise

'''Objective:''' Design a model predictive controller with a custom objective function that satisfies a specific problem criterion. Simulate and optimize a pendulum system with an adjustable overhead cart. ''Estimated time: 3 hours.''

A pendulum is described by the following dynamic equations:

Attach:pendulum_crane_equations.png

where ''epsilon'' is ''m'_2_'/(m'_1_'+m'_2_')'', ''y'' is the position of the overhead cart, ''v'' is the velocity of the overhead cart, ''theta'' is the angle of the pendulum relative to the cart, and ''q'' is the rate of angle change.

Attach:pendulum_crane.png

The objective of the controller is to adjust the force on the cart to move the pendulum mass to a new final position. Ensure that initial and final velocities and angles of the pendulum are zero. The position of the pendulum mass is initially at -1 and it is desired to move it to the new position of 0 within 6.2 seconds. Demonstrate controller performance with changes in the pendulum position and that the final pendulum mass remains at the final position without oscillation.

!!!! Solution

May 11, 2015, at 10:58 PM by 10.5.113.160 -
Changed line 21 from:
# 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. [[Article|https://dx.doi.org/10.1016/j.compchemeng.2014.04.013]]
to:
# 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]]
May 11, 2015, at 10:57 PM by 10.5.113.160 -
Changed lines 5-21 from:
The dynamic control objective function is a mathematical statement that is minimized or maximized to find a best solution among all possible feasible solutions for a controller. The form of this objective function is critical to give desirable solutions for driving a system to a desirable state or along a desired trajectory. Common objective statements relate to economic, safety, operability, environmental, or related objectives.
to:
The dynamic control objective function is a mathematical statement that is minimized or maximized to find a best solution among all possible feasible solutions for a controller. The form of this objective function is critical to give desirable solutions for driving a system to a desirable state or along a desired trajectory. Common objective statements relate to economic, safety, operability, environmental, or related objectives. Two common objective functions are shown below as squared error and l'_1_'-norm forms'^1^'.

!!!! Squared Error Objective

Attach:obj_ctrl_sq_error.png

!!!! l'_1_'-norm Objective

Attach:obj_ctrl_l1_norm.png

!!!! Nomenclature

Attach:obj_ctrl_nomenclature.png

!!!! References

# 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. [[Article|https://dx.doi.org/10.1016/j.compchemeng.2014.04.013]]