MIMO Model Identification

Main.ModelIdentification History

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February 10, 2023, at 06:10 PM by 10.35.117.248 -
Changed line 50 from:
m = GEKKO(remote=False) # remote=True for MacOS
to:
m = GEKKO(remote=False)
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March 02, 2019, at 07:28 PM by 45.56.3.173 -
Changed lines 50-51 from:
m = GEKKO(remote=False)
to:
m = GEKKO(remote=False) # remote=True for MacOS
Changed line 55 from:
yp,a,b,c = m.sysid(t,u,y,na,nb)
to:
yp,p,K = m.sysid(t,u,y,na,nb,diaglevel=1)
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March 02, 2019, at 07:19 PM by 45.56.3.173 -
Added lines 74-168:

(:toggle hide gekko_step button show="Python TCLab MIMO ID with Step Test Validation":)
(:div id=gekko_step:)
(:source lang=python:)
from gekko import GEKKO
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# load data and parse into columns
url = 'http://apmonitor.com/do/uploads/Main/tclab_dyn_data2.txt'
data = pd.read_csv(url)
t = data['Time']
u = data[['H1','H2']]
y = data[['T1','T2']]

# generate time-series model
m = GEKKO(remote=False)

# system identification
na = 3 # output coefficients
nb = 4 # input coefficients
yp,p,K = m.sysid(t,u,y,na,nb)

plt.figure()
plt.subplot(2,1,1)
plt.plot(t,u)
plt.legend([r'$H_1$',r'$H_2$'])
plt.ylabel('MVs')
plt.subplot(2,1,2)
plt.plot(t,y)
plt.plot(t,yp)
plt.legend([r'$T_{1meas}$',r'$T_{2meas}$',\
            r'$T_{1pred}$',r'$T_{2pred}$'])
plt.ylabel('CVs')
plt.xlabel('Time')
plt.savefig('sysid.png')

# step test model
yc,uc = m.arx(p)

# steady state initialization
m.options.IMODE = 1
m.solve(disp=False)

# dynamic simulation (step tests)
m.time = np.linspace(0,240,241)
m.options.TIME_SHIFT=0
m.options.IMODE = 4
m.solve(disp=False)

plt.figure()

# step for first MV (Heater 1)
uc[0].value = np.zeros(len(m.time))
uc[0].value[5:] = 100
uc[1].value = np.zeros(len(m.time))
m.solve(disp=False)

plt.subplot(2,2,1)
plt.title('Step Test 1')
plt.plot(m.time,uc[0].value,'b-',label=r'$H_1$')
plt.plot(m.time,uc[1].value,'r-',label=r'$H_2$')
plt.ylabel('Heater (%)')
plt.legend()
plt.subplot(2,2,3)
plt.plot(m.time,yc[0].value,'b--',label=r'$T_1$')
plt.plot(m.time,yc[1].value,'r--',label=r'$T_2$')
plt.ylabel('Temperature (K)')
plt.xlabel('Time (sec)')
plt.legend()

# step for second MV (Heater 2)
uc[0].value = np.zeros(len(m.time))
uc[1].value = np.zeros(len(m.time))
uc[1].value[5:] = 100
m.solve(disp=False)
plt.subplot(2,2,2)
plt.title('Step Test 2')
plt.plot(m.time,uc[0].value,'b-',label=r'$H_1$')
plt.plot(m.time,uc[1].value,'r-',label=r'$H_2$')
plt.ylabel('Heater (%)')
plt.legend()
plt.subplot(2,2,4)
plt.plot(m.time,yc[0].value,'b--',label=r'$T_1$')
plt.plot(m.time,yc[1].value,'r--',label=r'$T_2$')
plt.ylabel('Temperature (K)')
plt.xlabel('Time (sec)')
plt.legend()

plt.show()
(:sourceend:)
(:divend:)

%width=550px%Attach:gekko_sysid_tclab_step.png
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February 25, 2019, at 07:40 PM by 10.37.235.51 -
Changed lines 25-26 from:
!!!! Solution
to:
!!!! Solution (Excel, MATLAB, Python)
Added lines 32-73:

!!!! Solution (GEKKO Python)

(:toggle hide gekko_mimo button show="Python TCLab MIMO ID":)
(:div id=gekko_mimo:)
(:source lang=python:)
from gekko import GEKKO
import pandas as pd
import matplotlib.pyplot as plt

# load data and parse into columns
url = 'http://apmonitor.com/do/uploads/Main/tclab_dyn_data2.txt'
data = pd.read_csv(url)
t = data['Time']
u = data[['H1','H2']]
y = data[['T1','T2']]

# generate time-series model
m = GEKKO(remote=False)

# system identification
na = 2 # output coefficients
nb = 2 # input coefficients
yp,a,b,c = m.sysid(t,u,y,na,nb)

plt.figure()
plt.subplot(2,1,1)
plt.plot(t,u)
plt.legend([r'$u_0$',r'$u_1$'])
plt.ylabel('MVs')
plt.subplot(2,1,2)
plt.plot(t,y)
plt.plot(t,yp)
plt.legend([r'$y_0$',r'$y_1$',r'$z_0$',r'$z_1$'])
plt.ylabel('CVs')
plt.xlabel('Time')
plt.savefig('sysid.png')
plt.show()
(:sourceend:)
(:divend:)

%width=550px%Attach:gekko_sysid_tclab.png
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February 15, 2017, at 12:36 PM by 10.10.144.27 -
Changed lines 17-20 from:
Attach:download.png  [[Attach:dynamic_MIMO_identification.zip|Download Dynamic Model Identification Files (Excel and MATLAB)]]

Attach:download.png [[Attach:dynamic_MIMO_id_Python.zip|Download Dynamic Model Identification Files (Python)]]
to:
Attach:download.png  [[Attach:dynamic_MIMO_identification.zip|Download Dynamic Model Identification Files (Excel, MATLAB, Python)]]
Added lines 28-29:

Attach:download.png [[Attach:dynamic_MIMO_id_Python.zip|Solution to Dynamic Model Identification Files (Python)]]
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February 15, 2017, at 12:33 PM by 10.10.144.27 -
Changed lines 19-20 from:
Attach:download.png  
[[Attach:dynamic_MIMO_id_Python.zip|Download Dynamic Model Identification Files (Python)]]
to:
Attach:download.png [[Attach:dynamic_MIMO_id_Python.zip|Download Dynamic Model Identification Files (Python)]]
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February 15, 2017, at 12:33 PM by 10.10.144.27 -
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Attach:download.png 
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February 15, 2017, at 12:33 PM by 10.10.144.27 -
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[[Attach:dynamic_MIMO_id_Python.zip|Download Dynamic Model Identification Files (Python)]]
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April 13, 2016, at 02:04 PM by 45.56.3.173 -
Added lines 30-34:

!!!! References

* Nikbakhsh, S., Hedengren, J.D., Darby, M., Udy, J., Constrained Model Identification Using Open-Equation Nonlinear Optimization, AIChE Spring Meeting, Houston, TX, April 2016. [[Attach:AIChE_2016_Model_ID.pdf|Presentation]] [[https://aiche.confex.com/aiche/s16/webprogram/Paper444955.html|Abstract]]
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January 22, 2016, at 12:38 AM by 10.5.113.123 -
Changed line 27 from:
Attach:download.png  [[Attach:dynamic_MIMO_identification.zip|Solution to Dynamic Model Identification (Excel and MATLAB)]]
to:
Attach:download.png  [[Attach:dynamic_MIMO_solution.zip|Solution to Dynamic Model Identification (Excel and MATLAB)]]
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January 22, 2016, at 12:38 AM by 10.5.113.123 -
Changed lines 27-29 from:
In class exercise
to:
Attach:download.png  [[Attach:dynamic_MIMO_identification.zip|Solution to Dynamic Model Identification (Excel and MATLAB)]]

Attach:dynamic_MIMO_solution.png
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January 20, 2016, at 04:35 PM by 45.56.3.184 -
Changed line 15 from:
The following data set was generated in class from two voltage inputs (u) that changed the temperature (x) of two thermisters.
to:
The following data set was generated in class from two voltage inputs (u) that changed the temperature (x) of two thermistors  from the [[Main/ArduinoLab|Arduino Lab]].
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January 20, 2016, at 04:28 PM by 45.56.3.184 -
Changed lines 17-18 from:
Attach:download.png -> [[Attach:dynamic_MIMO_identification.zip|Download Dynamic Model Identification Files (Excel and MATLAB)]]
to:
Attach:download.png  [[Attach:dynamic_MIMO_identification.zip|Download Dynamic Model Identification Files (Excel and MATLAB)]]
Added line 20:
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January 20, 2016, at 04:27 PM by 45.56.3.184 -
Added lines 17-18:
Attach:download.png -> [[Attach:dynamic_MIMO_identification.zip|Download Dynamic Model Identification Files (Excel and MATLAB)]]
Changed lines 20-21 from:
Attach:download.png [[Attach:dynamic_MIMO_identification.zip|Download Dynamic Model Identification Files (Excel and MATLAB)]]
-> Attach:dynamic_MIMO_id.zip
to:
Attach:dynamic_MIMO_data.png
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January 20, 2016, at 04:26 PM by 45.56.3.184 -
Changed lines 18-20 from:
-> Attach:dynamic_MIMO_data.png

Determine a linear dynamic model that best describes the input to output relationship between voltage and temperature. Compute the steady state gain for each input to output relationship.
to:
Attach:download.png [[Attach:dynamic_MIMO_identification.zip|Download Dynamic Model Identification Files (Excel and MATLAB)]]
-> Attach:dynamic_MIMO_id.zip

Determine a linear dynamic model that best describes the input to output relationship between voltage and temperature. Compute the steady state gain for each input to output relationship. Data and example script files are posted above in Excel and MATLAB.
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January 20, 2016, at 04:08 PM by 45.56.3.184 -
Added lines 1-24:
(:title MIMO Model Identification:)
(:keywords dynamic data, model identification, validation, differential, algebraic, tutorial, Simulink:)
(:description Multiple input, multiple output model identification for dynamic and empirical identification with an example exercise in Excel, MATLAB, and Python.:)

Multiple Input, Multiple Output (MIMO) systems can be empirically described by several linear dynamic system models. MIMO systems are more complicated than [[Main/DataSimulation|Single Input, Single Output (SISO)]] systems because of several factors including multivariate interaction, potential co-linearity of inputs, and large data processing requirements. Some common MIMO model forms include:

* Discrete transfer functions (z or q - Time series)
* Continuous transfer functions (s - Laplace variables)
* State space (A,B,C,D - time domain, linear differential equations)

Model identification in these forms typically involves fitting unknown coefficients in the model followed by an analysis to determine how many parameters are statistically significant.

!!!! Exercise

The following data set was generated in class from two voltage inputs (u) that changed the temperature (x) of two thermisters.

Attach:download.png [[Attach:dynamic_MIMO_data.zip|Download Dynamic Data]]
-> Attach:dynamic_MIMO_data.png

Determine a linear dynamic model that best describes the input to output relationship between voltage and temperature. Compute the steady state gain for each input to output relationship.

!!!! Solution

In class exercise
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