TCLab C - Parameter Estimation

The TCLab is a hands-on application of machine learning and advanced temperature control with two heaters and two temperature sensors. The labs reinforce principles of model development, estimation, and advanced control methods. This is the third exercise and it involves estimating parameters in a multi-variate energy balance model. The predictions are aligned to the measured values through an optimizer that adjusts the parameters to minimize a sum of squared error objective. This lab builds upon the TCLab B by using the model of two heaters and two temperature sensors.

Lab Problem Statement

Lab C - Parameter Estimation

Data and Solutions

Batch Parameter Estimation with MATLAB and Python

Dynamic data, 2 heaters

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
from gekko import GEKKO

# Import or generate data
filename = 'tclab_dyn_data2.csv'
try:
data = pd.read_csv(filename)
except:
url = 'https://apmonitor.com/do/uploads/Main/tclab_dyn_data2.txt'
data = pd.read_csv(url)

# Create GEKKO Model
m = GEKKO()
m.time = data['Time'].values

# Parameters to Estimate
U = m.FV(value=10,lb=1,ub=20)
alpha1 = m.FV(value=0.01,lb=0.003,ub=0.03) # W / % heater
alpha2 = m.FV(value=0.005,lb=0.002,ub=0.02) # W / % heater

# STATUS=1 allows solver to adjust parameter
U.STATUS = 1
alpha1.STATUS = 1
alpha2.STATUS = 1

# Measured inputs
Q1 = m.MV(value=data['H1'].values)
Q2 = m.MV(value=data['H2'].values)

# State variables
TC1 = m.CV(value=data['T1'].values)
TC1.FSTATUS = 1 # minimize fstatus * (meas-pred)^2
TC2 = m.CV(value=data['T2'].values)
TC2.FSTATUS = 1 # minimize fstatus * (meas-pred)^2

Ta = m.Param(value=19.0+273.15) # K
mass = m.Param(value=4.0/1000.0) # kg
Cp = m.Param(value=0.5*1000.0) # J/kg-K
A = m.Param(value=10.0/100.0**2) # Area not between heaters in m^2
As = m.Param(value=2.0/100.0**2) # Area between heaters in m^2
eps = m.Param(value=0.9) # Emissivity
sigma = m.Const(5.67e-8) # Stefan-Boltzmann

# Heater temperatures in Kelvin
T1 = m.Intermediate(TC1+273.15)
T2 = m.Intermediate(TC2+273.15)

# Heat transfer between two heaters
Q_C12 = m.Intermediate(U*As*(T2-T1)) # Convective
Q_R12 = m.Intermediate(eps*sigma*As*(T2**4-T1**4)) # Radiative

# Semi-fundamental correlations (energy balances)
m.Equation(TC1.dt() == (1.0/(mass*Cp))*(U*A*(Ta-T1) \
+ eps * sigma * A * (Ta**4 - T1**4) \
+ Q_C12 + Q_R12 \
+ alpha1*Q1))

m.Equation(TC2.dt() == (1.0/(mass*Cp))*(U*A*(Ta-T2) \
+ eps * sigma * A * (Ta**4 - T2**4) \
- Q_C12 - Q_R12 \
+ alpha2*Q2))

# Options
m.options.IMODE = 5 # MHE
m.options.EV_TYPE = 2 # Objective type
m.options.NODES = 2 # Collocation nodes
m.options.SOLVER = 3 # IPOPT

# Solve
m.solve(disp=True)

# Parameter values
print('U : ' + str(U.value[0]))
print('alpha1: ' + str(alpha1.value[0]))
print('alpha2: ' + str(alpha2.value[0]))

# Create plot
plt.figure()
ax=plt.subplot(2,1,1)
ax.grid()
plt.plot(data['Time'],data['T1'],'ro',label=r'$T_1$ measured')
plt.plot(m.time,TC1.value,color='purple',linestyle='--',\
lw=3,label=r'$T_1$ predicted')
plt.plot(data['Time'],data['T2'],'bx',label=r'$T_2$ measured')
plt.plot(m.time,TC2.value,color='orange',linestyle='--',\
lw=3,label=r'$T_2$ predicted')
plt.ylabel('Temperature (degC)')
plt.legend(loc=2)
ax=plt.subplot(2,1,2)
ax.grid()
plt.plot(data['Time'],data['H1'],'r-',\
lw=3,label=r'$Q_1$')
plt.plot(data['Time'],data['H2'],'b:',\
lw=3,label=r'$Q_2$')
plt.ylabel('Heaters')
plt.xlabel('Time (sec)')
plt.legend(loc='best')
plt.show()

[$[Get Code]]

Note: Switch to make_mp4 = True to make an MP4 movie animation. This requires imageio and ffmpeg (install available through Python). It creates a folder named figures in your run directory. You can delete this folder after the run is complete.

import numpy as np
import time
import matplotlib.pyplot as plt
import random
# get gekko package with:
# pip install gekko
from gekko import GEKKO
# get tclab package with:
# pip install tclab
from tclab import TCLab

# Connect to Arduino
a = TCLab()

# Make an MP4 animation?
make_mp4 = False
if make_mp4:
import imageio # required to make animation
import os
try:
os.mkdir('./figures')
except:
pass

# Final time
tf = 10 # min
# number of data points (1 pt every 3 seconds)
n = tf * 20 + 1

# Configure heater levels
# Percent Heater (0-100%)
Q1s = np.zeros(n)
Q2s = np.zeros(n)
# Heater random steps every 50 sec
# Alternate steps by Q1 and Q2
Q1s[3:] = 100.0
Q1s[50:] = 0.0
Q1s[100:] = 80.0

Q2s[25:] = 60.0
Q2s[75:] = 100.0
Q2s[125:] = 25.0

# rapid, random changes every 5 cycles between 50 and 100
for i in range(130,180):
if i%10==0:
Q1s[i:i+10] = random.random() * 100.0
if (i+5)%10==0:
Q2s[i:i+10] = random.random() * 100.0

# Record initial temperatures (degC)
T1m = a.T1 * np.ones(n)
T2m = a.T2 * np.ones(n)
# Store MHE values for plots
Tmhe1 = T1m[0] * np.ones(n)
Tmhe2 = T2m[0] * np.ones(n)
Umhe = 10.0 * np.ones(n)
amhe1 = 0.01 * np.ones(n)
amhe2 = 0.0075 * np.ones(n)

#########################################################
# Initialize Model as Estimator
#########################################################
# Use remote=False for local solve (Windows, Linux, ARM)
# remote=True for remote solve (All platforms)
m = GEKKO(name='tclab-mhe',remote=False)

# 60 second time horizon, 20 steps
m.time = np.linspace(0,60,21)

# Parameters to Estimate
U = m.FV(value=10,name='u')
U.STATUS = 0 # don't estimate initially
U.FSTATUS = 0 # no measurements
U.DMAX = 1
U.LOWER = 5
U.UPPER = 15

alpha1 = m.FV(value=0.01,name='a1') # W / % heater
alpha1.STATUS = 0 # don't estimate initially
alpha1.FSTATUS = 0 # no measurements
alpha1.DMAX = 0.001
alpha1.LOWER = 0.003
alpha1.UPPER = 0.03

alpha2 = m.FV(value=0.0075,name='a2') # W / % heater
alpha2.STATUS = 0 # don't estimate initially
alpha2.FSTATUS = 0 # no measurements
alpha2.DMAX = 0.001
alpha2.LOWER = 0.002
alpha2.UPPER = 0.02

# Measured inputs
Q1 = m.MV(value=0,name='q1')
Q1.STATUS = 0 # don't estimate
Q1.FSTATUS = 1 # receive measurement

Q2 = m.MV(value=0,name='q2')
Q2.STATUS = 0 # don't estimate
Q2.FSTATUS = 1 # receive measurement

# Measurements for model alignment
TC1 = m.CV(value=T1m[0],name='tc1')
TC1.STATUS = 1 # minimize error between simulation and measurement
TC1.FSTATUS = 1 # receive measurement
TC1.MEAS_GAP = 0.1 # measurement deadband gap
TC1.LOWER = 0
TC1.UPPER = 200

TC2 = m.CV(value=T2m[0],name='tc2')
TC2.STATUS = 1 # minimize error between simulation and measurement
TC2.FSTATUS = 1 # receive measurement
TC2.MEAS_GAP = 0.1 # measurement deadband gap
TC2.LOWER = 0
TC2.UPPER = 200

Ta = m.Param(value=23.0+273.15) # K
mass = m.Param(value=4.0/1000.0) # kg
Cp = m.Param(value=0.5*1000.0) # J/kg-K
A = m.Param(value=10.0/100.0**2) # Area not between heaters in m^2
As = m.Param(value=2.0/100.0**2) # Area between heaters in m^2
eps = m.Param(value=0.9) # Emissivity
sigma = m.Const(5.67e-8) # Stefan-Boltzmann

# Heater temperatures
T1 = m.Intermediate(TC1+273.15)
T2 = m.Intermediate(TC2+273.15)

# Heat transfer between two heaters
Q_C12 = m.Intermediate(U*As*(T2-T1)) # Convective
Q_R12 = m.Intermediate(eps*sigma*As*(T2**4-T1**4)) # Radiative

# Semi-fundamental correlations (energy balances)
m.Equation(TC1.dt() == (1.0/(mass*Cp))*(U*A*(Ta-T1) \
+ eps * sigma * A * (Ta**4 - T1**4) \
+ Q_C12 + Q_R12 \
+ alpha1*Q1))

m.Equation(TC2.dt() == (1.0/(mass*Cp))*(U*A*(Ta-T2) \
+ eps * sigma * A * (Ta**4 - T2**4) \
- Q_C12 - Q_R12 \
+ alpha2*Q2))

# Global Options
m.options.IMODE = 5 # MHE
m.options.EV_TYPE = 2 # Objective type
m.options.NODES = 3 # Collocation nodes
m.options.SOLVER = 3 # IPOPT
m.options.COLDSTART = 1 # COLDSTART on first cycle
##################################################################
# Create plot
plt.figure(figsize=(12,7))
plt.ion()
plt.show()

# Main Loop
start_time = time.time()
prev_time = start_time
tm = np.zeros(n)

try:
for i in range(1,n):
# Sleep time
sleep_max = 3.0
sleep = sleep_max - (time.time() - prev_time)
if sleep>=0.01:
time.sleep(sleep-0.01)
else:
time.sleep(0.01)

# Record time and change in time
t = time.time()
dt = t - prev_time
prev_time = t
tm[i] = t - start_time

# Read temperatures in Celsius
T1m[i] = a.T1
T2m[i] = a.T2

# Insert measurements
TC1.MEAS = T1m[i]
TC2.MEAS = T2m[i]
Q1.MEAS = Q1s[i-1]
Q2.MEAS = Q2s[i-1]

# Start estimating U after 10 cycles (20 sec)
if i==10:
U.STATUS = 1
alpha1.STATUS = 1
alpha2.STATUS = 1

# Predict Parameters and Temperatures with MHE
m.solve()

if m.options.APPSTATUS == 1:
# Retrieve new values
Tmhe1[i] = TC1.MODEL
Tmhe2[i] = TC2.MODEL
Umhe[i] = U.NEWVAL
amhe1[i] = alpha1.NEWVAL
amhe2[i] = alpha2.NEWVAL
else:
# Solution failed, copy prior solution
Tmhe1[i] = Tmhe1[i-1]
Tmhe2[i] = Tmhe1[i-1]
Umhe[i] = Umhe[i-1]
amhe1[i] = amhe1[i-1]
amhe2[i] = amhe2[i-1]

# Write new heater values (0-100)
a.Q1(Q1s[i])
a.Q2(Q2s[i])

# Plot
plt.clf()
ax=plt.subplot(3,1,1)
ax.grid()
plt.plot(tm[0:i],T1m[0:i],'ro',label=r'$T_1$ measured')
plt.plot(tm[0:i],Tmhe1[0:i],'k-',label=r'$T_1$ MHE')
plt.plot(tm[0:i],T2m[0:i],'bx',label=r'$T_2$ measured')
plt.plot(tm[0:i],Tmhe2[0:i],'k--',label=r'$T_2$ MHE')
plt.ylabel('Temperature (degC)')
plt.legend(loc=2)
ax=plt.subplot(3,1,2)
ax.grid()
plt.plot(tm[0:i],Umhe[0:i],'k-',label='Heat Transfer Coeff')
plt.plot(tm[0:i],amhe1[0:i]*1000,'r--',label=r'$\alpha_1$x1000')
plt.plot(tm[0:i],amhe2[0:i]*1000,'b--',label=r'$\alpha_2$x1000')
plt.ylabel('Parameters')
plt.legend(loc='best')
ax=plt.subplot(3,1,3)
ax.grid()
plt.plot(tm[0:i],Q1s[0:i],'r-',label=r'$Q_1$')
plt.plot(tm[0:i],Q2s[0:i],'b:',label=r'$Q_2$')
plt.ylabel('Heaters')
plt.xlabel('Time (sec)')
plt.legend(loc='best')
plt.draw()
plt.pause(0.05)
if make_mp4:
filename='./figures/plot_'+str(i+10000)+'.png'
plt.savefig(filename)

# Turn off heaters
a.Q1(0)
a.Q2(0)
# Save figure
plt.savefig('tclab_mhe.png')

# generate mp4 from png figures in batches of 350
if make_mp4:
images = []
iset = 0
for i in range(1,n):
filename='./figures/plot_'+str(i+10000)+'.png'
images.append(imageio.imread(filename))
if ((i+1)%350)==0:
imageio.mimsave('results_'+str(iset)+'.mp4', images)
iset += 1
images = []
if images!=[]:
imageio.mimsave('results_'+str(iset)+'.mp4', images)

# Allow user to end loop with Ctrl-C
except KeyboardInterrupt:
# Disconnect from Arduino
a.Q1(0)
a.Q2(0)
print('Shutting down')
a.close()
plt.savefig('tclab_mhe.png')

# Make sure serial connection still closes when there's an error
except:
# Disconnect from Arduino
a.Q1(0)
a.Q2(0)
print('Error: Shutting down')
a.close()
plt.savefig('tclab_mhe.png')
raise

[$[Get Code]]

Dynamic Optimization