## Piecewise Linear

## Apps.PiecewiseLinear History

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from scipy import optimize

#### PWL in GEKKO Python

(: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:)

#### APMonitor PWL Object

Model pwl

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

! 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

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

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

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#### Equivalent Problem with Slack Variables

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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

End Model

*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.

*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.

minimize slk_u[1:4] minimize slk_l[2:5]

File *.m.csv

File m.csv

End Equations

End Model </pre></font> (:htmlend:)

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# 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]

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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:)

## Piece-wise Linear Approximation

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.