Apps

Piecewise Linear

Apps.PiecewiseLinear History

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Attach:pwl_gekko.png
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%width=500px%Attach:pwl_gekko.png
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from scipy import optimize
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----

!!!! PWL in GEKKO Python

Attach:pwl_gekko.png

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

m = GEKKO()
m.options.SOLVER = 1

xz = m.FV(value = 4.5)
yz = m.Var()

xp_val = np.array([1, 2, 3, 3.5,  4, 5])
yp_val = np.array([1, 0, 2, 2.5, 2.8, 3])
xp = [m.Param(value=xp_val[i]) for i in range(6)]
yp = [m.Param(value=yp_val[i]) for i in range(6)]

x = [m.Var(lb=xp[i],ub=xp[i+1]) for i in range(5)]
x[0].lower = -1e20
x[-1].upper = 1e20

# Variables
slk_u = [m.Var(value=1,lb=0) for i in range(4)]
slk_l = [m.Var(value=1,lb=0) for i in range(4)]

# Intermediates
slope = []
for i in range(5):
    slope.append(m.Intermediate((yp[i+1]-yp[i]) / (xp[i+1]-xp[i])))

y = []
for i in range(5):
    y.append(m.Intermediate((x[i]-xp[i])*slope[i]))

for i in range(4):
    m.Obj(slk_u[i] + slk_l[i])

m.Equation(xz == x[0]  + slk_u[0])
for i in range(3):
    m.Equation(xz == x[i+1] + slk_u[i+1] - slk_l[i])
m.Equation(xz == x[4]                - slk_l[3])

m.Equation(yz == yp[0] + y[0] + y[1] + y[2] + y[3] + y[4])

m.solve()

plt.plot(xp,yp,'rx-',label='PWL function')
plt.plot(xz,yz,'bo',label='Data')
plt.show()
(:sourceend:)
July 06, 2018, at 09:49 PM by 45.56.3.173 -
Added lines 17-18:
!!!! APMonitor PWL Object
Changed lines 20-37 from:
Model pwl
to:
  File p.txt
    1,  1
    2,  0
    3,  2
    3.5, 2.5
    4,  2.8
    5,  3
  End File

  Objects
    p = pwl
  End Objects

  Connections
    x = p.x
    y = p.y
  End Connections
 
Deleted lines 40-58:

    ! data points
    xp[1] = 1
    yp[1] = 1

    xp[2] = 2
    yp[2] = 0

    xp[3] = 3
    yp[3] = 2

    xp[4] = 3.5
    yp[4] = 2.5

    xp[5] = 4
    yp[5] = 2.8

    xp[6] = 5
    yp[6] = 3
Deleted lines 43-47:
   ! piece-wise linear segments
    x[1]            <=xp[2]
    x[2:4] >xp[2:4], <=xp[3:5]
    x[5]  >xp[5]

Deleted lines 45-48:

    ! slack variables
    slk_u[1:4]
    slk_l[2:5]
Changed lines 47-92 from:
to:
</pre></font>
(:htmlend:)

----

!!!! Equivalent Problem with Slack Variables

(:html:)<font size=2><pre>
  Parameters
    ! independent variable
    x = 6.0

    ! data points
    xp[1] = 1
    yp[1] = 1

    xp[2] = 2
    yp[2] = 0

    xp[3] = 3
    yp[3] = 2

    xp[4] = 3.5
    yp[4] = 2.5

    xp[5] = 4
    yp[5] = 2.8

    xp[6] = 5
    yp[6] = 3
  End Parameters

  Variables
    ! piece-wise linear segments
    x[1]            <=xp[2]
    x[2:4] >xp[2:4], <=xp[3:5]
    x[5]  >xp[5]

    ! dependent variable
    y

    ! slack variables
    slk_u[1:4]
    slk_l[2:5]
  End Variables

Deleted line 107:
End Model
July 31, 2017, at 07:56 PM by 152.7.224.8 -
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''In Situ'' Adaptive Tabulation (ISAT) is an example of a multi-dimensional piecewise linear approximation.  The piecewise linear segments are built dynamically as new data becomes available.  This way, only regions that are accessed in practice contribute to the function approximation.
to:
[[http://apm.byu.edu/prism/index.php/Projects/InSituAdaptiveTabulation|''In Situ'' Adaptive Tabulation (ISAT)]] is an example of a multi-dimensional piecewise linear approximation.  The piecewise linear segments are built dynamically as new data becomes available.  This way, only regions that are accessed in practice contribute to the function approximation.
February 13, 2013, at 05:09 PM by 69.169.188.188 -
Added lines 63-65:
   minimize slk_u[1:4]
    minimize slk_l[2:5]

Changed line 83 from:
File *.m.csv
to:
File m.csv
August 28, 2010, at 04:16 AM by 206.180.155.75 -
Added lines 68-113:
  End Equations
End Model
</pre></font>
(:htmlend:)


----

(:html:)<font size=2><pre>
! Piece-wise function with the LOOKUP object

! create the csv file
File *.m.csv
  input,  y[1],  y[2]
      1,    2,    4
      3,    4,    6
      5,    -5,  -7
    -1,    1,  0.5
End File

! define lookup object m
Objects
  m = lookup
End Objects

! connect m properties with model parameters
Connections
  x = m.input
  y[1] = m.y[1]
  y[2] = m.y[2]
End Connections

! simple model
Model n
  Parameters
    x = 1
    y[1]
    y[2]
  End Parameters

  Variables
    y
  End Variables

  Equations
    y = y[1]+y[2]
March 06, 2010, at 09:45 AM by 206.180.155.75 -
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(:html:)<font size=2><pre>
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(:htmlend:)
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(:htmlend:)
April 23, 2009, at 04:46 PM by 158.35.225.231 -
Changed lines 15-71 from:
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----

(:html:)<font size=1><pre>
Model pwl
  Parameters
    ! independent variable
    x = 6.0

    ! data points
    xp[1] = 1
    yp[1] = 1

    xp[2] = 2
    yp[2] = 0

    xp[3] = 3
    yp[3] = 2

    xp[4] = 3.5
    yp[4] = 2.5

    xp[5] = 4
    yp[5] = 2.8

    xp[6] = 5
    yp[6] = 3
  End Parameters

  Variables
    ! piece-wise linear segments
    x[1]            <=xp[2]
    x[2:4] >xp[2:4], <=xp[3:5]
    x[5]  >xp[5]

    ! dependent variable
    y

    ! slack variables
    slk_u[1:4]
    slk_l[2:5]
  End Variables

  Intermediates
    slope[1:5] = (yp[2:6]-yp[1:5]) / (xp[2:6]-xp[1:5])
    y[1:5] = (x[1:5]-xp[1:5])*slope[1:5]
  End Intermediates

  Equations
    x = x[1]  + slk_u[1]
    x = x[2:4] + slk_u[2:4] - slk_l[2:4]
    x = x[5]                - slk_l[5]

    y = yp[1] + y[1] + y[2] + y[3] + y[4] + y[5]
  End Equations
End Model
</pre></font>
(:htmlend:)
February 23, 2009, at 02:45 PM by 158.35.225.228 -
Added lines 1-15:
!! Piece-wise Linear Approximation

* %list list-page% [[Attach:pwl.apm | Piece-wise Linear Approximation Example]]

Attach:pwl.gif

A piece-wise linear function is an approximation of a nonlinear relationship.  For more nonlinear relationships, additional linear segments are added to refine the approximation.

As an example, the piecewise linear form is often used to approximate valve characterization (valve position (% open) to flow).  This is a single input-single output function approximation.

''In Situ'' Adaptive Tabulation (ISAT) is an example of a multi-dimensional piecewise linear approximation.  The piecewise linear segments are built dynamically as new data becomes available.  This way, only regions that are accessed in practice contribute to the function approximation.

Attach:mpwl.png