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I wish to model the biweekly HFMD cases in Malaysia.
Then, I want to show that the model using the Squared Error Objective and L1-Norm Objective can better model the biweekly HFMD cases than the model without objectives.
My question is, is it possible to model the biweekly HFMD cases without using the Squared Error Objective and L1-Norm Objective?
With this, I have attached the coding below:
from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt
m1 = GEKKO(remote=False)
m2 = GEKKO(remote=False)
m = m1
# Known parameters
nb = 26 # Number of biweeks in a year
ny = 3 # Number of years
biweeks = np.zeros((nb,ny*nb+1))
biweeks[0][0] = 1
for i in range(nb):
for j in range(ny):
biweeks[i][j*nb+i+1] = 1
# Write csv data file
tm = np.linspace(0,78,79)
# case data
# Malaysia weekly HFMD data from the year 2013 - 2015
cases = np.array([506,506,700,890,1158,1605,1694,1311,1490,1310,1368,\
1009,1097,934,866,670,408,481,637,749,700,648,710,\
740,627,507,516,548,636,750,1066,1339,1565,\
1464,1575,1759,1631,1601,1227,794,774,623,411,\
750,1017,976,1258,1290,1546,1662,1720,1553,1787,1291,1712,2227,2132,\
2550,2140,1645,1743,1296,1153,871,621,570,388,\
347,391,446,442,390,399,421,398,452,470,437,411])
data = np.vstack((tm,cases))
data = data.T
# np.savetxt('measles_biweek_2.csv',data,delimiter=',',header='time,cases')
np.savetxt('hfmd_biweek_2.csv',data,delimiter=',',header='time,cases')
# Load data from csv
# m.time, cases_meas = np.loadtxt('measles_biweek_2.csv', \
m.time, cases_hfmd = np.loadtxt('hfmd_biweek_2.csv', \
delimiter=',',skiprows=1,unpack=True)
# m.Vr = m.Param(value = 0)
# Variables
# m.N = m.FV(value = 3.2e6)
# m.mu = m.FV(value = 7.8e-4)
# m.N = m.FV(value = 3.11861e7)
# m.mu = m.FV(value = 6.42712e-4)
m.N = m.FV(value = 3.16141e7) # Malaysia average total population (2015 - 2017)
m.mu = m.FV(value = 6.237171519e-4) # Malaysia scaled birth rate (births/biweek/total population)
m.rep_frac = m.FV(value = 0.45) # Percentage of underreporting
# Beta values (unknown parameters in the model)
m.beta = [m.FV(value=1, lb=0.1, ub=5) for i in range(nb)]
# Predicted values
m.S = m.SV(value = 0.162492875*m.N.value, lb=0,ub=m.N) # Susceptibles (Kids from 0 - 9 YO: 5137066 people) - Average of 94.88% from total reported cases
m.I = m.SV(value = 7.907863896e-5*m.N.value, lb=0,ub=m.N) #
# m.V = m.Var(value = 2e5)
# measured values
m.cases = m.CV(value = cases_hfmd, lb=0)
# turn on feedback status for CASES
m.cases.FSTATUS = 1
# weight on prior model predictions
m.cases.WMODEL = 0
# meas_gap = deadband that represents level of
# accuracy / measurement noise
db = 100
m.cases.MEAS_GAP = db
for i in range(nb):
m.beta[i].STATUS=1
#m.gamma = m.FV(value=0.07)
m.gamma = m.FV(value=0.07)
m.gamma.STATUS = 1
m.gamma.LOWER = 0.05
m.gamma.UPPER = 0.5
m.biweek=[None]*nb
for i in range(nb):
m.biweek[i] = m.Param(value=biweeks[i])
# Intermediate
m.Rs = m.Intermediate(m.S*m.I/m.N)
# Equations
sum_biweek = sum([m.biweek[i]*m.beta[i]*m.Rs for i in range(nb)])
# m.Equation(m.S.dt()== -sum_biweek + m.mu*m.N - m.Vr)
m.Equation(m.S.dt()== -sum_biweek + m.mu*m.N)
m.Equation(m.I.dt()== sum_biweek - m.gamma*m.I)
m.Equation(m.cases == m.rep_frac*sum_biweek)
# m.Equation(m.V.dt()==-m.Vr)
# options
m.options.SOLVER = 1
m.options.NODES=3
# imode = 5, dynamic estimation
m.options.IMODE = 5
# ev_type = 1 (L1-norm) or 2 (squared error)
m.options.EV_TYPE = 2
# solve model and print solver output
m.solve()
[print('beta['+str(i+1)+'] = '+str(m.beta[i][0])) \
for i in range(nb)]
print('gamma = '+str(m.gamma.value[0]))
# export data
# stack time and avg as column vectors
my_data = np.vstack((m.time,np.asarray(m.beta),m.gamma))
# transpose data
my_data = my_data.T
# save text file with comma delimiter
beta_str = ''
for i in range(nb):
beta_str = beta_str + ',beta[' + str(i+1) + ']'
header_name = 'time,gamma' + beta_str
##np.savetxt('solution_data.csv',my_data,delimiter=',',\
## header = header_name, comments='')
np.savetxt('solution_data_EVTYPE_'+str(m.options.EV_TYPE)+\
'_gamma'+str(m.gamma.STATUS)+'.csv',\
my_data,delimiter=',',header = header_name)
plt.figure(num=1, figsize=(16,8))
plt.suptitle('Estimation')
plt.subplot(2,2,1)
plt.plot(m.time,m.cases, label='Cases (model)')
plt.plot(m.time,cases_hfmd, label='Cases (measured)')
if m.options.EV_TYPE==2:
plt.plot(m.time,cases_hfmd+db/2, 'k-.',\
lw=0.5, label=r'$Cases_{db-hi}$')
plt.plot(m.time,cases_hfmd-db/2, 'k-.',\
lw=0.5, label=r'$Cases_{db-lo}$')
plt.fill_between(m.time,cases_hfmd-db/2,\
cases_hfmd+db/2,color='gold',alpha=.5)
plt.legend(loc='best')
plt.ylabel('Cases')
plt.subplot(2,2,2)
plt.plot(m.time,m.S,'r--')
plt.ylabel('S')
plt.subplot(2,2,3)
[plt.plot(m.time,m.beta[i], label='_nolegend_')\
for i in range(nb)]
plt.plot(m.time,m.gamma,'c--', label=r'$\gamma$')
plt.legend(loc='best')
plt.ylabel(r'$\beta, \gamma$')
plt.xlabel('Time')
plt.subplot(2,2,4)
plt.plot(m.time,m.I,'g--')
plt.xlabel('Time')
plt.ylabel('I')
plt.subplots_adjust(hspace=0.2,wspace=0.4)
name = 'cases_EVTYPE_'+ str(m.options.EV_TYPE) +\
'_gamma' + str(m.gamma.STATUS) + '.png'
plt.savefig(name)
plt.show()
To define a custom objective, use the m.Minimize() or m.Maximize() functions instead of the squared error or l1-norm objectives that are built into the m.CV() objects. To create a custom objective, use m.Var() instead of m.CV() such as:
from gekko import GEKKO
import numpy as np
m = GEKKO()
x = m.Array(m.Var,4,value=1,lb=1,ub=5)
x1,x2,x3,x4 = x
# change initial values
x2.value = 5; x3.value = 5
m.Equation(x1*x2*x3*x4>=25)
m.Equation(x1**2+x2**2+x3**2+x4**2==40)
m.Minimize(x1*x4*(x1+x2+x3)+x3)
m.solve()
print('x: ', x)
print('Objective: ',m.options.OBJFCNVAL)
Here is a similar problem with disease prediction (Measles) that uses m.CV().
import numpy as np
from gekko import GEKKO
import matplotlib.pyplot as plt
# Import Data
# write csv data file
t_s = np.linspace(0,78,79)
# case data
cases_s = np.array([180,180,271,423,465,523,649,624,556,420,\
423,488,441,268,260,163,83,60,41,48,65,82,\
145,122,194,237,318,450,671,1387,1617,2058,\
3099,3340,2965,1873,1641,1122,884,591,427,282,\
174,127,84,97,68,88,79,58,85,75,121,174,209,458,\
742,929,1027,1411,1885,2110,1764,2001,2154,1843,\
1427,970,726,416,218,160,160,188,224,298,436,482,468])
# Initialize gekko model
m = GEKKO()
# Number of collocation nodes
nodes = 4
# Number of phases (years in this case)
n = 3
#Biweek periods per year
bi = 26
# Time horizon (for all 3 phases)
m.time = np.linspace(0,1,bi+1)
# Parameters that will repeat each year
N = m.Param(3.2e6)
mu = m.Param(7.8e-4)
rep_frac = m.Param(0.45)
Vr = m.Param(0)
beta = m.MV(2,lb = 0.1)
beta.STATUS = 1
gamma = m.FV(value=0.07)
gamma.STATUS = 1
gamma.LOWER = 0.05
gamma.UPPER = 0.5
# Data used to control objective function
casesobj1 = m.Param(cases_s[0:(bi+1)])
casesobj2 = m.Param(cases_s[bi:(2*bi+1)])
casesobj3 = m.Param(cases_s[2*bi:(3*bi+1)])
# Variables that vary between years, one version for each year
cases = [m.CV(value = cases_s[(i*bi):(i+1)*(bi+1)-i],lb=0) for i in range(n)]
for i in cases:
i.FSTATUS = 1
i.WMODEL = 0
i.MEAS_GAP = 100
S = [m.Var(0.06*N,lb = 0,ub = N) for i in range(n)]
I = [m.Var(0.001*N, lb = 0,ub = N) for i in range(n)]
V = [m.Var(2e5) for i in range(n)]
# Equations (created for each year)
for i in range(n):
R = m.Intermediate(beta*S[i]*I[i]/N)
m.Equation(S[i].dt() == -R + mu*N - Vr)
m.Equation(I[i].dt() == R - gamma*I[i])
m.Equation(cases[i] == rep_frac*R)
m.Equation(V[i].dt() == -Vr)
# Connect years together at endpoints
for i in range(n-1):
m.Connection(cases[i+1],cases[i],1,bi,1,nodes)#,1,nodes)
m.Connection(cases[i+1],'CALCULATED',pos1=1,node1=1)
m.Connection(S[i+1],S[i],1,bi,1,nodes)
m.Connection(S[i+1],'CALCULATED',pos1=1,node1=1)
m.Connection(I[i+1],I[i],1,bi,1,nodes)
m.Connection(I[i+1],'CALCULATED',pos1=1, node1=1)
# Solver options
m.options.IMODE = 5
m.options.NODES = nodes
m.EV_TYPE = 1
m.options.SOLVER = 1
# Solve
m.Obj(2*(casesobj1-cases[0])**2+(casesobj3-cases[2])**2)
m.solve()
# Calculate the start time of each phase
ts = np.linspace(1,n,n)
# Plot
plt.figure()
plt.subplot(4,1,1)
tm = np.empty(len(m.time))
for i in range(n):
tm = m.time + ts[i]
plt.plot(tm,cases[i].value,label='Cases Year %s'%(i+1))
plt.plot(tm,cases_s[(i*bi):(i+1)*(bi+1)-i],'.')
plt.legend()
plt.ylabel('Cases')
plt.subplot(4,1,2)
for i in range(n):
tm = m.time + ts[i]
plt.plot(tm,beta.value,label='Beta Year %s'%(i+1))
plt.legend()
plt.ylabel('Contact Rate')
plt.subplot(4,1,3)
for i in range(n):
tm = m.time + ts[i]
plt.plot(tm,I[i].value,label='I Year %s'%(i+1))
plt.legend()
plt.ylabel('Infectives')
plt.subplot(4,1,4)
for i in range(n):
tm = m.time + ts[i]
plt.plot(tm,S[i].value,label='S Year %s'%(i+1))
plt.legend()
plt.ylabel('Susceptibles')
plt.xlabel('Time (yr)')
plt.show()
everyone and Professor John
We are using gekko to do MPC on tclab simulation model. We try to emulate the situation that on site the actuator deviates from MV calculated by gekko because of the problems of actuator.
If the deviation is in the fixed pattern, for example a quite big constant deviation happens for a long time and may come back then work well for a long time. We can deal with it by extra logic to detect deviation and add the deviation value to the mv calculated by gekko.
one day, I noticed that there could be meas for MV when fstatus = 1. So I gave it a try. I hope gekko could deal with the deviation by itself. for example, if mv from gekko is 10 and the measurement is 5 and the pattern continues, gekko may spit out a higher MV value than 10, for example 15 and measurement is 10.
In the simulation, when I set MV's fstatus=1, the MV's curve becomes to :
q1a is the q1 with manual deviation. In the above pic, q1a == q1. It looks like gekko takes one more step thinking about the MV's effect.
In the below pic, there are two times range, one with "q1a == q1+20" and the other with "q1a == q1 -20". q1a's value is fed to tclab and mv(q1)'s meas.
I do not understand why the q1 calculated by gekko is going up or going down when meas deviates despite the t1 is going far away from sp.
Edit: Example Code
See the screen shot below from "normal" HMI. The sluggish MV disappeared, so it maybe caused by bug in my code. But the up-going or down-going could still be seen.
See my code below:
from random import random
from random import randrange
import tclab
from tclab import labtime
from tclab import TCLabModel
import numpy as np
import time
import matplotlib.pyplot as plt
from gekko import GEKKO
import json
from tclab import TCLabModel
make_mp4 = True
if make_mp4:
import imageio # required to make animation
import os
try:
os.mkdir('./figures')
except:
pass
class tclab_heaterpipe():
def __init__(self,d1,d2,model):
if(d1 >= 1 and d2 >=1):
self.delay_q1_step = int(d1)
self.delay_q2_step = int(d2)
self.q1_buffer = [0] * self.delay_q1_step
self.q2_buffer = [0] * self.delay_q2_step
self.m = model
else:
self.delay_q1_step =0
self.delay_q2_step =0
return
def Q1_delay(self,q1):
if(self.delay_q1_step == 0):
self.m.Q1(q1)
self.q1_buffer.insert(0,q1)
self.m.Q1(self.q1_buffer.pop())
def Q2_delay(self,q2):
if(self.delay_q2_step == 0):
self.m.Q1(q2)
self.q2_buffer.insert(0,q2)
self.m.Q2(self.q2_buffer.pop())
# Connect to Arduino
connected = False
theta1 = 1
theta2 = 1
T = tclab.setup(connected)
a = T()
tclab_delay = tclab_heaterpipe(theta1,theta2,a)
# Turn LED on
print('LED On')
a.LED(100)
# Simulate a time delay
# Run time in minutes
run_time = 80.0
# Number of cycles
loops = int(60.0*run_time)
# Temperature (K)
t1sp = 45.0
t2sp = 35.0
#########################################################
# Initialize Model
#########################################################
# use remote=True for MacOS
m = GEKKO(name='tclab-mpc',remote=False)
m.time = np.linspace(0,400,41)
step = 10
T1 = np.ones(int(loops/step)+1) * a.T1 # temperature (degC)
T2 = np.ones(int(loops/step)+1) * a.T2 # temperature (degC)
Tsp1 = np.ones(int(loops/step)+1) * t1sp # set point (degC)
Tsp2 = np.ones(int(loops/step)+1) * t2sp # set point (degC)
# heater values
Q1s = np.ones(int(loops/step)+1) * 0.0
Q2s = np.ones(int(loops/step)+1) * 0.0
# Parameters
Q1_ss = m.Param(value=0)
TC1_ss = m.Param(value=a.T1)
Q2_ss = m.Param(value=0)
TC2_ss = m.Param(value=a.T2)
Kp1 = m.Param(value= 0.7)
tau1 = m.Param(value=160.0)
Kp2 = m.Param(value=0.05)
tau2 = m.Param(value=160.0)
Kp3= m.Param(value=0.05)
tau3 = m.Param(value=160.0)
Kp4 = m.Param(value=0.4)
tau4 = m.Param(value=200.0)
sp1 = m.Param(value=a.T1)
sp2 = m.Param(value=a.T2)
# Manipulated variable
Q1 = m.MV(value=0, name='q1')
Q1.STATUS = 1 # use to control temperature
Q1.FSTATUS = 1 # no feedback measurement
Q1.LOWER = 0.0
Q1.UPPER = 100.0
Q1.DMAX = 10.0
Q1.DCOST = 5.0
Q2 = m.MV(value=0, name='q2')
Q2.STATUS = 1 # use to control temperature
Q2.FSTATUS = 1 # no feedback measurement
Q2.LOWER = 0.0
Q2.UPPER = 100.0
Q2.DMAX = 10.0
Q2.DCOST = 5.0
# Controlled variable
TC1 = m.CV(value=a.T1, name='tc1')
TC1.STATUS = 1 # minimize error with setpoint range
TC1.FSTATUS = 1 # receive measurement
TC1.TR_INIT = 2 # reference trajectory
# TC1.COST = 0.1
TC1.WSPHI = 20
TC1.WSPLO = 20
TC1.TAU = 50 # time constant for response
#TC1.TR_OPEN = 3
TC2 = m.CV(value=a.T2, name='tc2')
TC2.STATUS = 1 # minimize error with setpoint range
TC2.FSTATUS = 1 # receive measurement
TC2.TR_INIT = 2 # reference trajectory
# TC2.COST = 0.1
TC2.WSPHI = 20
TC2.WSPLO = 20
TC2.TAU = 30 # time constant for response
#kTC2.TR_OPEN = 3
# 添加延时
Q1d=m.Var()
m.delay(Q1, Q1d, theta1)
Q2d=m.Var()
m.delay(Q2, Q2d, theta2)
# Equation
#m.Equation(tau1 * TC1.dt() + (TC1 - TC1_ss) == Kp1 * (Q1d - Q1_ss))
# m.Equation(tau2 * TC2.dt() + (TC2 - TC2_ss) == Kp2 * (Q1d - Q1_ss))
# m.Equation(tau3 * TC1.dt() + (TC1 - TC1_ss) == Kp3 * (Q2d - Q2_ss))
# m.Equation(tau2 * TC2.dt() + (TC2 - TC2_ss) == Kp4 * (Q2d - Q2_ss))
m.Equation(0.5 * (tau1 * TC1.dt() + (TC1 - TC1_ss) + tau3 * TC1.dt() + (TC1 - TC1_ss)) == Kp1 * (Q1d - Q1_ss) + Kp3 * (Q2d -Q2_ss))
m.Equation(0.5 * (tau2 * TC2.dt() + (TC2 - TC2_ss) + tau4 * TC2.dt() + (TC2 - TC2_ss)) == Kp4 * (Q2d - Q2_ss) + Kp2 * (Q1d - Q1_ss))
# Steady-state initializations
m.options.IMODE = 1
m.options.SOLVER = 1 # 1=APOPT, 3=IPOPT
m.solve()
sp1.VALUE = 45
sp2.VALUE = 35
# Global Options
m.options.IMODE = 6 # MPC
m.options.CV_TYPE = 3 # Objective type
m.options.NODES = 2 # Collocation nodes
m.options.MAX_TIME = 10
m.options.SOLVER = 1 # 1=APOPT, 3=IPOPT
#m.options.CV_WGT_START = 2*theta
#m.options.CV_WGT_SLOPE = theta
# m.options.MV_STEP_HOR = 5
##################################################################
# Create plot
plt.figure()
plt.ion()
plt.show()
# Main Loop
a.Q1(0)
a.Q2(0)
Q2s[0:] = 0
start_time = time.time()
tm = np.linspace(1,loops,int(loops/step)+1)
j=0
try:
time_start = time.time()
labtime_start = labtime.time()
if(not connected):
labtime.set_rate(10)
for i in tclab.clock(loops,adaptive=False):
i = int(i)
if(i == 0):
continue
print("-----------------------")
t_real = time.time() - time_start
t_lab = labtime.time() - labtime_start
print("real time = {0:4.1f} lab time = {1:4.1f} m.time = {1:4.1f}".format(t_real, t_lab,m.time))
#print("real time = {0:4.1f} m.time = {1:4.1f}".format(t_real, m.time))
if(i%step != 0):
continue
j = i/step
j = int(j)
print(j)
T1[j:] = a.T1
T2[j:] = a.T2
tm[j] = i
###############################
### MPC CONTROLLER ###
###############################
TC1.MEAS = T1[j]
TC2.MEAS = T2[j]
print("T1 meas:{0:4.1f} ".format(a.T1))
print("T2 meas:{0:4.1f} ".format(a.T2))
# input setpoint with deadband +/- DT
DT =0.5
TC1.SPHI = Tsp1[j] +DT
TC1.SPLO = Tsp1[j] -DT
TC2.SPHI = Tsp2[j] +DT
TC2.SPLO = Tsp2[j] -DT
try:
# stop model time to solve MPC in cast the solver takes too much time
if(not connected):
labtime.stop()
m.solve(disp=False)
#start model time
if(not connected):
labtime.start()
except Exception as e:
if(not connected):
if(not labtime.running):
labtime.start()
print("sovle's exception:")
print(e)
if(j != 0):
Q1s[j] = Q1s[j-1]
Q2s[j] = Q2s[j-1]
continue
# test for successful solution
if (m.options.APPSTATUS==1):
# retrieve the first Q value
Q1s[j:] = np.ones(len(Q1s)-j) * Q1.NEWVAL
Q2s[j:] = np.ones(len(Q2s)-j) * Q2.NEWVAL
#a.Q1(Q1.NEWVAL)
#a.Q2(Q2.NEWVAL)
print("Q1 applied with delay: {0:4.1f} ".format(Q1.NEWVAL))
print("Q2 applied with delay: {0:4.1f} ".format(Q2.NEWVAL))
with open(m.path+'//results.json') as f:
results = json.load(f)
else:
# not successful, set heater to zero
print("APPSTATUS is not 1,set Q to 0")
#Q1s[j] = 0
#Q2s[j] = 0
if i> 300 and i < 600:
Q1s[j] = Q1s[j] - 20
Q2s[j] = Q2s[j] - 20
if i>= 600:
Q1s[j] = Q1s[j] + 20
Q2s[j] = Q2s[j] + 20
Q1.meas= Q1s[j]
Q2.meas= Q2s[j]
tclab_delay.Q1_delay(Q1s[j])
tclab_delay.Q2_delay(Q2s[j])
print("calc:"+str(Q1s[j]))
print("calc:"+str(Q2s[j]))
#apply disturbance on 50s, 200s,
#if(i == 600):
# Q2s[j] = 100
#if(i == 1400):
# Q2s[j] = 0
#Q2s[j] = 20 - randrange(20)
#Q2s[j:] = np.ones(len(Q2s)-j) * Q2s[j]
#restore Q2 to 0
#if(i == 300):
#Q2s[j:] = 0
#a.Q2(Q2s[j])
#tclab_delay.Q2_delay(Q2s[j])
#take Q2 to FV
#Q2.MEAS = Q2s[j]
if(not connected):
labtime.stop()
# Plot
try:
plt.clf()
ax=plt.subplot(2,1,1)
ax.grid()
plt.plot(tm[0:j],T1[0:j],'ro',markersize=3,label=r'$T_1$')
plt.plot(tm[0:j],Tsp1[0:j],'r-',markersize=3,label=r'$T_1 Setpoint$')
plt.plot(tm[0:j],T2[0:j],'bo',markersize=3,label=r'$T_2$')
plt.plot(tm[0:j],Tsp2[0:j],'b-',markersize=3,label=r'$T_2 Setpoint$')
plt.plot(tm[j]+m.time,results['tc1.bcv'],'r-.',markersize=1,\
label=r'$T_1$ predicted',linewidth=1)
plt.plot(tm[j]+m.time,results['tc2.bcv'],'b-.',markersize=1,\
label=r'$T_2$ predicted',linewidth=1)
plt.plot(tm[j]+m.time,results['tc1.tr_hi'],'k--',\
label=r'$T_1$ trajectory')
plt.plot(tm[j]+m.time,results['tc1.tr_lo'],'k--')
plt.plot(tm[j]+m.time,results['tc2.tr_hi'],'k--',\
label=r'$T_2$ trajectory')
plt.plot(tm[j]+m.time,results['tc2.tr_lo'],'k--')
plt.ylabel('Temperature (degC)')
plt.legend(loc='best')
ax=plt.subplot(2,1,2)
ax.grid()
plt.plot(tm[0:j],Q1s[0:j],'r-',linewidth=3,label=r'$Q_1$')
plt.plot(tm[0:j],Q2s[0:j],'b-',linewidth=3,label=r'$Q_2$')
plt.plot(tm[j]+m.time,Q1.value,'r-.',\
label=r'$Q_1$ plan',linewidth=1)
plt.plot(tm[j]+m.time,Q2.value,'b-.',\
label=r'$Q_2$ plan',linewidth=1)
#plt.plot(tm[0:i],Q2s[0:i],'b:',LineWidth=3,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(j+10000)+'.png'
plt.savefig(filename)
except Exception as e:
print(e)
pass
if(not connected):
labtime.start()
# Turn off heaters
a.Q1(0)
a.Q2(0)
print('Shutting down')
input("Press Enter to continue...")
a.close()
# Allow user to end loop with Ctrl-C
except KeyboardInterrupt:
# Disconnect from Arduino
a.Q1(0)
a.Q2(0)
print('Shutting down')
a.close()
if make_mp4:
images = []
iset = 0
for i in range(1,int(loops/step)+1):
filename='./figures/plot_'+str(i+10000)+'.png'
if os.path.exists(filename):
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)
# 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()
raise
Regards
Tibalt
Is the FSTATUS also ON for the CVs such as t1.FSTATUS=1? If you update the measurement such as:
t1.MEAS = lab.T1
t2.MEAS = lab.T2
then this updates the BIAS for t1 and t2 (BIAS documentation). This should take care of any process / model mismatch that you are introducing by arbitrarily increasing or decreasing the heater by 20%. If t1.FSTATUS is OFF (0) then it is not able to compensate for the mismatch.
Another thing to try is to adjust the reference trajectory. The controller can appear sluggish if TAU is too high. Here is an example application with MPC and a linear model.
One additional way to compensate for the mismatch is to use Moving Horizon Estimation as shown here.
It looks like you have created a nice interface!
Response to Edit
Thanks for adding the code. The problem is that Q1.DMAX=10 and Q2.DMAX=10. When the Q1 and Q2 values are shifted up by 20 each cycle, the most that the controller can shift down is 20-10=10 so the controller appears that it is ramping in the wrong direction. Changing to DMAX=100 fixes the problem. There is still offset from the setpoint because the recommended Q1 and Q2 are shifted each cycle. The true recommended values are never implemented. Another thing to try is to impose an offset on the measured values such as TC1.MEAS = T1[j] + 20. The model bias will remove the offset in this case.
from random import random
from random import randrange
import tclab
from tclab import labtime
from tclab import TCLabModel
import numpy as np
import time
import matplotlib.pyplot as plt
from gekko import GEKKO
import json
from tclab import TCLabModel
make_gif = True
make_mp4 = True
if make_gif or make_mp4:
# pip install imageio-ffmpeg with imageio to make MP4
import imageio # required to make animation
import os
try:
os.mkdir('./figures')
except:
pass
class tclab_heaterpipe():
def __init__(self,d1,d2,model):
if(d1 >= 1 and d2 >=1):
self.delay_q1_step = int(d1)
self.delay_q2_step = int(d2)
self.q1_buffer = [0] * self.delay_q1_step
self.q2_buffer = [0] * self.delay_q2_step
self.m = model
else:
self.delay_q1_step =0
self.delay_q2_step =0
return
def Q1_delay(self,q1):
if(self.delay_q1_step == 0):
self.m.Q1(q1)
self.q1_buffer.insert(0,q1)
self.m.Q1(self.q1_buffer.pop())
def Q2_delay(self,q2):
if(self.delay_q2_step == 0):
self.m.Q1(q2)
self.q2_buffer.insert(0,q2)
self.m.Q2(self.q2_buffer.pop())
# Connect to Arduino
connected = False # switch to connected=True with physical hardware
theta1 = 1
theta2 = 1
T = tclab.setup(connected)
a = T()
tclab_delay = tclab_heaterpipe(theta1,theta2,a)
# Turn LED on
print('LED On')
a.LED(100)
# Simulate a time delay
# Run time in minutes
run_time = 20.0
# Number of cycles
loops = int(60.0*run_time)
# Temperature (K)
t1sp = 45.0
t2sp = 35.0
#########################################################
# Initialize Model
#########################################################
# use remote=True for MacOS
m = GEKKO(name='tclab-mpc',remote=False)
m.time = np.linspace(0,400,41)
step = 10
T1 = np.ones(int(loops/step)+1) * a.T1 # temperature (degC)
T2 = np.ones(int(loops/step)+1) * a.T2 # temperature (degC)
Tsp1 = np.ones(int(loops/step)+1) * t1sp # set point (degC)
Tsp2 = np.ones(int(loops/step)+1) * t2sp # set point (degC)
# heater values
Q1s = np.ones(int(loops/step)+1) * 0.0
Q2s = np.ones(int(loops/step)+1) * 0.0
# Parameters
Q1_ss = m.Param(value=0)
TC1_ss = m.Param(value=a.T1)
Q2_ss = m.Param(value=0)
TC2_ss = m.Param(value=a.T2)
Kp1 = m.Param(value= 0.7)
tau1 = m.Param(value=160.0)
Kp2 = m.Param(value=0.05)
tau2 = m.Param(value=160.0)
Kp3= m.Param(value=0.05)
tau3 = m.Param(value=160.0)
Kp4 = m.Param(value=0.4)
tau4 = m.Param(value=200.0)
sp1 = m.Param(value=a.T1)
sp2 = m.Param(value=a.T2)
# Manipulated variable
Q1 = m.MV(value=0, name='q1')
Q1.STATUS = 1 # use to control temperature
Q1.FSTATUS = 1 # no feedback measurement
Q1.LOWER = 0.0
Q1.UPPER = 100.0
Q1.DMAX = 100.0
Q1.DCOST = 1e-3
Q2 = m.MV(value=0, name='q2')
Q2.STATUS = 1 # use to control temperature
Q2.FSTATUS = 1 # no feedback measurement
Q2.LOWER = 0.0
Q2.UPPER = 100.0
Q2.DMAX = 100.0
Q2.DCOST = 1e-3
# Controlled variable
TC1 = m.CV(value=a.T1, name='tc1')
TC1.STATUS = 1 # minimize error with setpoint range
TC1.FSTATUS = 1 # receive measurement
TC1.TR_INIT = 2 # reference trajectory
# TC1.COST = 0.1
TC1.WSPHI = 20
TC1.WSPLO = 20
TC1.TAU = 50 # time constant for response
#TC1.TR_OPEN = 3
TC2 = m.CV(value=a.T2, name='tc2')
TC2.STATUS = 1 # minimize error with setpoint range
TC2.FSTATUS = 1 # receive measurement
TC2.TR_INIT = 2 # reference trajectory
# TC2.COST = 0.1
TC2.WSPHI = 20
TC2.WSPLO = 20
TC2.TAU = 30 # time constant for response
#kTC2.TR_OPEN = 3
# 添加延时
Q1d=m.Var()
m.delay(Q1, Q1d, theta1)
Q2d=m.Var()
m.delay(Q2, Q2d, theta2)
# Equation
#m.Equation(tau1 * TC1.dt() + (TC1 - TC1_ss) == Kp1 * (Q1d - Q1_ss))
# m.Equation(tau2 * TC2.dt() + (TC2 - TC2_ss) == Kp2 * (Q1d - Q1_ss))
# m.Equation(tau3 * TC1.dt() + (TC1 - TC1_ss) == Kp3 * (Q2d - Q2_ss))
# m.Equation(tau2 * TC2.dt() + (TC2 - TC2_ss) == Kp4 * (Q2d - Q2_ss))
m.Equation(0.5 * (tau1 * TC1.dt() + (TC1 - TC1_ss) + tau3 * TC1.dt() + (TC1 - TC1_ss)) == Kp1 * (Q1d - Q1_ss) + Kp3 * (Q2d -Q2_ss))
m.Equation(0.5 * (tau2 * TC2.dt() + (TC2 - TC2_ss) + tau4 * TC2.dt() + (TC2 - TC2_ss)) == Kp4 * (Q2d - Q2_ss) + Kp2 * (Q1d - Q1_ss))
# Steady-state initializations
m.options.IMODE = 1
m.options.SOLVER = 1 # 1=APOPT, 3=IPOPT
m.solve()
sp1.VALUE = 45
sp2.VALUE = 35
# Global Options
m.options.IMODE = 6 # MPC
m.options.CV_TYPE = 3 # Objective type
m.options.NODES = 2 # Collocation nodes
m.options.MAX_TIME = 10
m.options.SOLVER = 1 # 1=APOPT, 3=IPOPT
#m.options.CV_WGT_START = 2*theta
#m.options.CV_WGT_SLOPE = theta
# m.options.MV_STEP_HOR = 5
##################################################################
# Create plot
plt.figure(figsize=(12,8))
plt.ion()
plt.show()
# Main Loop
a.Q1(0)
a.Q2(0)
Q2s[0:] = 0
start_time = time.time()
tm = np.linspace(1,loops,int(loops/step)+1)
j=0
try:
time_start = time.time()
labtime_start = labtime.time()
if(not connected):
labtime.set_rate(10)
for i in tclab.clock(loops,adaptive=False):
i = int(i)
if(i == 0):
continue
print("-----------------------")
t_real = time.time() - time_start
t_lab = labtime.time() - labtime_start
print("real time = {0:4.1f} lab time = {1:4.1f} m.time = {1:4.1f}".format(t_real, t_lab,m.time))
#print("real time = {0:4.1f} m.time = {1:4.1f}".format(t_real, m.time))
if(i%step != 0):
continue
j = i/step
j = int(j)
print(j)
T1[j:] = a.T1
T2[j:] = a.T2
tm[j] = i
###############################
### MPC CONTROLLER ###
###############################
TC1.MEAS = T1[j]
TC2.MEAS = T2[j]
print("T1 meas:{0:4.1f} ".format(a.T1))
print("T2 meas:{0:4.1f} ".format(a.T2))
# input setpoint with deadband +/- DT
DT =0.5
TC1.SPHI = Tsp1[j] +DT
TC1.SPLO = Tsp1[j] -DT
TC2.SPHI = Tsp2[j] +DT
TC2.SPLO = Tsp2[j] -DT
try:
# stop model time to solve MPC in cast the solver takes too much time
if(not connected):
labtime.stop()
m.solve(disp=False)
#start model time
if(not connected):
labtime.start()
except Exception as e:
if(not connected):
if(not labtime.running):
labtime.start()
print("sovle's exception:")
print(e)
if(j != 0):
Q1s[j] = Q1s[j-1]
Q2s[j] = Q2s[j-1]
continue
# test for successful solution
if (m.options.APPSTATUS==1):
# retrieve the first Q value
Q1s[j:] = np.ones(len(Q1s)-j) * Q1.NEWVAL
Q2s[j:] = np.ones(len(Q2s)-j) * Q2.NEWVAL
#a.Q1(Q1.NEWVAL)
#a.Q2(Q2.NEWVAL)
print("Q1 applied with delay: {0:4.1f} ".format(Q1.NEWVAL))
print("Q2 applied with delay: {0:4.1f} ".format(Q2.NEWVAL))
with open(m.path+'//results.json') as f:
results = json.load(f)
else:
# not successful, set heater to zero
print("APPSTATUS is not 1,set Q to 0")
#Q1s[j] = 0
#Q2s[j] = 0
if i> 300 and i < 600:
Q1s[j] = max(0,Q1s[j] - 20)
Q2s[j] = max(0,Q2s[j] - 20)
if i>= 600:
Q1s[j] = min(100,Q1s[j] + 20)
Q2s[j] = min(100,Q2s[j] + 20)
Q1.meas= Q1s[j]
Q2.meas= Q2s[j]
tclab_delay.Q1_delay(Q1s[j])
tclab_delay.Q2_delay(Q2s[j])
print("calc:"+str(Q1s[j]))
print("calc:"+str(Q2s[j]))
if(not connected):
labtime.stop()
# Plot
try:
plt.clf()
ax=plt.subplot(2,1,1)
ax.grid()
plt.plot(tm[0:j],T1[0:j],'ro',markersize=3,label=r'$T_1$')
plt.plot(tm[0:j],Tsp1[0:j],'r-',markersize=3,label=r'$T_1 Setpoint$')
plt.plot(tm[0:j],T2[0:j],'bo',markersize=3,label=r'$T_2$')
plt.plot(tm[0:j],Tsp2[0:j],'b-',markersize=3,label=r'$T_2 Setpoint$')
plt.plot(tm[j]+m.time,results['tc1.bcv'],'r-.',markersize=1,\
label=r'$T_1$ predicted',linewidth=1)
plt.plot(tm[j]+m.time,results['tc2.bcv'],'b-.',markersize=1,\
label=r'$T_2$ predicted',linewidth=1)
plt.plot(tm[j]+m.time,results['tc1.tr_hi'],'k--',\
label=r'$T_1$ trajectory')
plt.plot(tm[j]+m.time,results['tc1.tr_lo'],'k--')
plt.plot(tm[j]+m.time,results['tc2.tr_hi'],'k--',\
label=r'$T_2$ trajectory')
plt.plot(tm[j]+m.time,results['tc2.tr_lo'],'k--')
plt.ylabel('Temperature (degC)')
plt.legend(loc=1)
ax=plt.subplot(2,1,2)
ax.grid()
plt.plot(tm[0:j],Q1s[0:j],'r-',linewidth=3,label=r'$Q_1$')
plt.plot(tm[0:j],Q2s[0:j],'b-',linewidth=3,label=r'$Q_2$')
plt.plot(tm[j]+m.time,Q1.value,'r-.',\
label=r'$Q_1$ plan',linewidth=1)
plt.plot(tm[j]+m.time,Q2.value,'b-.',\
label=r'$Q_2$ plan',linewidth=1)
#plt.plot(tm[0:i],Q2s[0:i],'b:',LineWidth=3,label=r'$Q_2$')
plt.ylabel('Heaters')
plt.xlabel('Time (sec)')
plt.legend(loc=1)
plt.draw()
plt.pause(0.05)
if make_mp4:
filename='./figures/plot_'+str(j+10000)+'.png'
plt.savefig(filename)
except Exception as e:
print(e)
pass
if(not connected):
labtime.start()
# Turn off heaters
a.Q1(0)
a.Q2(0)
print('Shutting down')
input("Press Enter to continue...")
a.close()
# make gif
if make_gif:
images = []
iset = 0
for i in range(1,int(loops/step)+1):
filename='./figures/plot_'+str(i+10000)+'.png'
if os.path.exists(filename):
images.append(imageio.imread(filename))
if ((i+1)%350)==0:
imageio.mimsave('results_'+str(iset)+'.gif', images)
iset += 1
images = []
if images!=[]:
imageio.mimsave('results_'+str(iset)+'.gif', images)
if make_mp4:
images = []
iset = 0
for i in range(1,int(loops/step)+1):
filename='./figures/plot_'+str(i+10000)+'.png'
if os.path.exists(filename):
images.append(imageio.imread(filename))
if ((i+1)%350)==0:
imageio.mimsave('results_'+str(iset)+'.gif', images)
iset += 1
images = []
if images!=[]:
imageio.mimsave('results_'+str(iset)+'.gif', 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()
if make_gif:
images = []
iset = 0
for i in range(1,int(loops/step)+1):
filename='./figures/plot_'+str(i+10000)+'.png'
if os.path.exists(filename):
images.append(imageio.imread(filename))
if ((i+1)%350)==0:
imageio.mimsave('results_'+str(iset)+'.gif', images)
iset += 1
images = []
if images!=[]:
imageio.mimsave('results_'+str(iset)+'.gif', images)
if make_mp4:
images = []
iset = 0
for i in range(1,int(loops/step)+1):
filename='./figures/plot_'+str(i+10000)+'.png'
if os.path.exists(filename):
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)
# 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()
raise
I am making a numerical problem to show as an example and am trying to find an optimal control using gekko for the following problem:
minimize the integral of a*x(t) from 0 to T, where T is the first time x(t) is 0, i.e., it is a random time. The constraints are such that x(t) follows some dynamic f(x(t),u(t)), x(t) >= 0, and u(t) is between 0 and 1.
I followed the tutorials on GEKKO website and youtube for fixed final time, but I could not find any information on a random final time. The following is the current code I have, but how would I be able to move from a fixed final time to a random final time? Any help would be appreciated! Thanks!
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from gekko import GEKKO
# Initial conditions
xhh0 = 3; xhi0 = 0;
xvh0 = 30; xvi0 = 0;
hin0 = 0; vin0 = 0;
tt0 = 0
# Parameters
a1 = 0.1; a2 = 0.1;
b1 = 0.01; b2 = 0.5;
delta1 = 0.1; delta2 = 0.5;
rho1 = 0.3; rho2 = 0.01
mu = 1
# Gekko
m = GEKKO()
# Control variable
u = m.MV(0.5, lb = 0, ub = 1)
# Final time <------------------------ currently a fixed final time
T = 10
# Initialize
xhh, xhi, xvh, xvi, Ah, Av = m.Array(m.Var, 6)
xhh.value = xhh0; xhi.value = xhi0;
xvh.value = xvh0; xvi.value = xvi0;
Ah.value = hin0; Av.value = vin0;
# System dynamics
m.Equations([xhh.dt() == -a1*xhh - mu*u - b1*xhi*xhh,\
xhi.dt() == a1*xhh + b1*xhi*xhh - delta1*xhi - rho1*xhi,\
xvh.dt() == -a2*xvh - mu*(1-u) - b2*xvi*xvh,\
xvi.dt() == a2*xvh + b2*xvi*xvh - delta2*xvi - rho2*xvi,\
Ah.dt() == a1*xhh,\
Av.dt() == a2*xvh])
# Time space
t = np.linspace(0, T, 101)
m.time = t
# initialize with simulation
m.options.IMODE = 7
m.options.NODES = 3
m.solve(disp = False)
# optimization
m.options.IMODE = 6
xhh.LOWER = 0; xhi.LOWER = 0; xvh.LOWER = 0; xvi.LOWER = 0
u.STATUS = 1
m.options.SOLVER = 3
xhh.value = xhh.value.value
xhi.value = xhi.value.value
xvh.value = xvh.value.value
xvi.value = xvi.value.value
Ah.value = Ah.value.value
Av.value = Av.value.value
# Objective function
m.Minimize(Ah + Av)
m.solve()
The final time is adjustable with T = m.FV() and T.STATUS=1 when each differential is divided by T. This scales the problem to any arbitrary final time when t = np.linspace(0,1).
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from gekko import GEKKO
# Initial conditions
xhh0 = 3; xhi0 = 0;
xvh0 = 30; xvi0 = 0;
hin0 = 0; vin0 = 0;
tt0 = 0
# Parameters
a1 = 0.1; a2 = 0.1;
b1 = 0.01; b2 = 0.5;
delta1 = 0.1; delta2 = 0.5;
rho1 = 0.3; rho2 = 0.01; mu = 1
# Gekko
m = GEKKO()
# Control variable
u = m.MV(0.5, lb = 0, ub = 1)
# Final time
T = m.FV(10,lb=1e-2,ub=100); T.STATUS = 1
# Initialize
xhh, xhi, xvh, xvi, Ah, Av = m.Array(m.Var, 6)
xhh.value = xhh0; xhi.value = xhi0;
xvh.value = xvh0; xvi.value = xvi0;
Ah.value = hin0; Av.value = vin0;
xhh.LOWER = 0; xhi.LOWER = 0; xvh.LOWER = 0; xvi.LOWER = 0
u.STATUS = 1
# System dynamics
m.Equations([xhh.dt()/T == -a1*xhh - mu*u - b1*xhi*xhh,\
xhi.dt()/T == a1*xhh + b1*xhi*xhh - delta1*xhi - rho1*xhi,\
xvh.dt()/T == -a2*xvh - mu*(1-u) - b2*xvi*xvh,\
xvi.dt()/T == a2*xvh + b2*xvi*xvh - delta2*xvi - rho2*xvi,\
Ah.dt()/T == a1*xhh,\
Av.dt()/T == a2*xvh])
# Time space
t = np.linspace(0, 1, 101)
m.time = t
# optimization
m.options.IMODE = 6
m.options.SOLVER = 3
# Objective function
m.Minimize(Ah + Av)
m.solve()
print('Final time: ', T.value[0])
There may be a missing constraint or some other information because the optimal final time always goes to the lower bound. The Jennings problem is a related example with variable final time.
Attached below is my PYTHON code:
I have a CSTR and im trying to control the height of the tank and the temperature while manipulating the inlet flow and the cooling temperature. The problem is that the CV's are not tracking their respective setpoints. I tried doing the problem for only 1 CV and 1 MV, it worked really well.
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from gekko import GEKKO
# Steady State Initial Condition
u1_ss = 280.0
u2_ss=100.0
# Feed Temperature (K)
Tf = 350
# Feed Concentration (mol/m^3)
Caf = 1
# Steady State Initial Conditions for the States
Ca_ss = 1
T_ss = 304
h_ss=94.77413303
V_ss=8577.41330293
x0 = np.empty(4)
x0[0] = Ca_ss
x0[1] = T_ss
x0[2]= h_ss
x0[3]= V_ss
#%% GEKKO nonlinear MPC
m = GEKKO(remote=False)
m.time = [0,0.02,0.04,0.06,0.08,0.1,0.12,0.15,0.2]
c1=10.0
Ac=100.0
# Density of A-B Mixture (kg/m^3)
rho = 1000
# Heat capacity of A-B Mixture (J/kg-K)
Cp = 0.239
# Heat of reaction for A->B (J/mol)
mdelH = 5e4
# E - Activation energy in the Arrhenius Equation (J/mol)
# R - Universal Gas Constant = 8.31451 J/mol-K
EoverR = 8750
# Pre-exponential factor (1/sec)
k0 = 7.2e10
# U - Overall Heat Transfer Coefficient (W/m^2-K)
# A - Area - this value is specific for the U calculation (m^2)
UA = 5e4
# initial conditions
Tc0 = 280
T0 = 304
Ca0 = 1.0
h0=94.77413303
q0=100.0
V0=8577.41330293
tau = m.Const(value=0.5)
Kp = m.Const(value=1)
m.Tc = m.MV(value=Tc0,lb=250,ub=350)
m.T = m.CV(value=T_ss)
m.h= m.CV(value=h0)
m.rA = m.Var(value=0)
m.Ca = m.CV(value=Ca_ss,lb=0,ub=1)
m.V= m.CV(value=V_ss,lb=0,ub=100000)
m.q=m.MV(value=q0,lb=0,ub=100000)
m.Equation(m.rA == k0*m.exp(-EoverR/m.T)*m.Ca)
m.Equation(m.T.dt() == m.q/m.V*(Tf - m.T) \
+ mdelH/(rho*Cp)*m.rA \
+ UA/m.V/rho/Cp*(m.Tc-m.T))
m.Equation(m.Ca.dt() == m.q/m.V*(Caf - m.Ca) - m.rA)
m.Equation(m.h.dt()==(m.q-c1*m.h**0.5)/Ac)
m.Equation(m.V.dt() == m.q- c1*m.h**0.5)
#MV tuning
m.Tc.STATUS = 1
m.Tc.FSTATUS = 0
m.Tc.DMAX = 100
m.Tc.DMAXHI = 20
m.Tc.DMAXLO = -100
m.q.STATUS = 1
m.q.FSTATUS = 0
m.q.DMAX = 10
#CV tuning
m.T.STATUS = 1
m.T.FSTATUS = 1
m.T.TR_INIT = 1
m.T.TAU = 1.0
DT = 0.5 # deadband
m.h.STATUS = 1
m.h.FSTATUS = 1
m.h.TR_INIT = 1
m.h.TAU = 1.0
m.Ca.STATUS = 1
m.Ca.FSTATUS = 0 # no measurement
m.Ca.TR_INIT = 0
m.V.STATUS = 1
m.V.FSTATUS = 0 # no measurement
m.V.TR_INIT = 0
m.options.CV_TYPE = 1
m.options.IMODE = 6
m.options.SOLVER = 3
#%% define CSTR model
def cstr(x,t,u1,u2,Tf,Caf,Ac):
# Inputs (3):
# Temperature of cooling jacket (K)
Tc = u1
q=u2
# Tf = Feed Temperature (K)
# Caf = Feed Concentration (mol/m^3)
# States (2):
# Concentration of A in CSTR (mol/m^3)
Ca = x[0]
# Temperature in CSTR (K)
T = x[1]
# the height of the tank (m)
h=x[2]
V=x[3]
# Parameters:
# Density of A-B Mixture (kg/m^3)
rho = 1000
# Heat capacity of A-B Mixture (J/kg-K)
Cp = 0.239
# Heat of reaction for A->B (J/mol)
mdelH = 5e4
# E - Activation energy in the Arrhenius Equation (J/mol)
# R - Universal Gas Constant = 8.31451 J/mol-K
EoverR = 8750
# Pre-exponential factor (1/sec)
k0 = 7.2e10
# U - Overall Heat Transfer Coefficient (W/m^2-K)
# A - Area - this value is specific for the U calculation (m^2)
UA = 5e4
# reaction rate
rA = k0*np.exp(-EoverR/T)*Ca
# Calculate concentration derivative
dCadt = q/V*(Caf - Ca) - rA
# Calculate temperature derivative
dTdt = q/V*(Tf - T) \
+ mdelH/(rho*Cp)*rA \
+ UA/V/rho/Cp*(Tc-T)
# Calculate height derivative
dhdt=(q-c1*h**0.5)/Ac
if x[2]>=300 and dhdt>0:
dhdt = 0
dVdt= q-c1*h**0.5
# Return xdot:
xdot = np.zeros(4)
xdot[0] = dCadt
xdot[1] = dTdt
xdot[2]= dhdt
xdot[3]= dVdt
return xdot
# Time Interval (min)
t = np.linspace(0,8,401)
# Store results for plotting
Ca = np.ones(len(t)) * Ca_ss
V=np.ones(len(t))*V_ss
T = np.ones(len(t)) * T_ss
Tsp = np.ones(len(t)) * T_ss
hsp=np.ones(len(t))*h_ss
h=np.ones(len(t))*h_ss
u1 = np.ones(len(t)) * u1_ss
u2 = np.ones(len(t)) * u2_ss
# Set point steps
Tsp[0:100] = 330.0
Tsp[100:200] = 350.0
Tsp[230:260] = 370.0
Tsp[260:290] = 390.0
hsp[0:100] = 30.0
hsp[100:200] =60.0
hsp[200:250]=90.0
# Create plot
plt.figure(figsize=(10,7))
plt.ion()
plt.show()
# Simulate CSTR
for i in range(len(t)-1):
# simulate one time period (0.05 sec each loop)
ts = [t[i],t[i+1]]
y = odeint(cstr,x0,ts,args=(u1[i+1],u2[i+1],Tf,Caf,Ac))
# retrieve measurements
Ca[i+1] = y[-1][0]
T[i+1] = y[-1][1]
h[i+1]= y[-1][2]
V[i+1]= y[-1][3]
# insert measurement
m.T.MEAS = T[i+1]
m.h.MEAS=h[i+1]
# solve MPC
m.solve(disp=True)
m.T.SPHI = Tsp[i+1] + DT
m.T.SPLO = Tsp[i+1] - DT
m.h.SPHI = hsp[i+1] + DT
m.h.SPLO = hsp[i+1] - DT
# retrieve new Tc value
u1[i+1] = m.Tc.NEWVAL
u2[i+1] = m.q.NEWVAL
# update initial conditions
x0[0] = Ca[i+1]
x0[1] = T[i+1]
x0[2]= h[i+1]
x0[3]= V[i+1]
#%% Plot the results
plt.clf()
plt.subplot(6,1,1)
plt.plot(t[0:i],u1[0:i],'b--',linewidth=3)
plt.ylabel('Cooling T (K)')
plt.legend(['Jacket Temperature'],loc='best')
plt.subplot(6,1,2)
plt.plot(t[0:i],u2[0:i],'b--',linewidth=3)
plt.ylabel('inlet flow')
plt.subplot(6,1,3)
plt.plot(t[0:i],Ca[0:i],'b.-',linewidth=3,label=r'$C_A$')
plt.plot([0,t[i-1]],[0.2,0.2],'r--',linewidth=2,label='limit')
plt.ylabel(r'$C_A$ (mol/L)')
plt.legend(loc='best')
plt.subplot(6,1,4)
plt.plot(t[0:i],V[0:i],'g--',linewidth=3)
plt.xlabel('time')
plt.ylabel('Volume of Tank')
plt.subplot(6,1,5)
plt.plot(t[0:i],Tsp[0:i],'k-',linewidth=3,label=r'$T_{sp}$')
plt.plot(t[0:i],T[0:i],'b.-',linewidth=3,label=r'$T_{meas}$')
plt.plot([0,t[i-1]],[400,400],'r--',linewidth=2,label='limit')
plt.ylabel('T (K)')
plt.xlabel('Time (min)')
plt.legend(loc='best')
plt.subplot(6,1,6)
plt.plot(t[0:i],hsp[0:i],'g--',linewidth=3,label=r'$h_{sp}$')
plt.plot(t[0:i],h[0:i],'k.-',linewidth=3,label=r'$h_{meas}$')
plt.xlabel('time')
plt.ylabel('tank level')
plt.legend(loc='best')
plt.draw()
plt.pause(0.01)
Attached is the code I wrote: When it runs, the level controlled variable is not tracking its setpoint.
On the other hand, the Temperature controlled variable is tracking its setpoint very well. I am using manipulating the cooling temperature and inlet flow rate. I am trying to control the level of the tank, temperature and concentration.
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from gekko import GEKKO
# Steady State Initial Condition
u1_ss = 300.0
u2_ss=100.0
Ca_ss = 0.87725294
T_ss = 324.47544313
h_ss=75.82018806
# Feed Temperature (K)
Tf = 350
# Feed Concentration (mol/m^3)
Caf = 1
# Steady State Initial Conditions for the States
x0 = np.empty(2)
x0[0] = Ca_ss
x0[1] = T_ss
p0=np.empty(1)
p0[0]=h_ss
#%% GEKKO nonlinear MPC
m = GEKKO(remote=False)
m.time = [0,0.02,0.04,0.06,0.08,0.1,0.12,0.15,0.2]
c1=10
Ac=400.0
# Volume of CSTR (m^3)
V = 100
# Density of A-B Mixture (kg/m^3)
rho = 1000
# Heat capacity of A-B Mixture (J/kg-K)
Cp = 0.239
# Heat of reaction for A->B (J/mol)
mdelH = 5e4
# E - Activation energy in the Arrhenius Equation (J/mol)
# R - Universal Gas Constant = 8.31451 J/mol-K
EoverR = 8750
# Pre-exponential factor (1/sec)
k0 = 7.2e10
# U - Overall Heat Transfer Coefficient (W/m^2-K)
# A - Area - this value is specific for the U calculation (m^2)
UA = 5e4
# initial conditions
Tc0 = 300
T0 = 324.47544313
Ca0 = 0.87725294
h0=75.82018806
q0=100.0
tau = m.Const(value=0.5)
Kp = m.Const(value=1)
m.Tc = m.MV(value=Tc0,lb=250,ub=350)
m.T = m.CV(value=T_ss)
m.rA = m.Var(value=0)
m.Ca = m.CV(value=Ca_ss,lb=0,ub=1)
m.h=m.CV(value=h_ss)
m.q=m.MV(value=q0,lb=0,ub=1000)
m.Equation(m.rA == k0*m.exp(-EoverR/m.T)*m.Ca)
m.Equation(m.T.dt() == m.q/V*(Tf - m.T) \
+ mdelH/(rho*Cp)*m.rA \
+ UA/V/rho/Cp*(m.Tc-m.T))
m.Equation(m.Ca.dt() == (m.q)/V*(Caf - m.Ca) - m.rA)
m.Equation(m.h.dt()==(m.q-c1*pow(m.h,0.5))/Ac)
#MV tuning
m.Tc.STATUS = 1
m.Tc.FSTATUS = 0
m.Tc.DMAX = 100
m.Tc.DMAXHI = 20
m.Tc.DMAXLO = -100
m.q.STATUS = 1
m.q.FSTATUS = 0
m.q.DMAX = 10
#CV tuning
m.T.STATUS = 1
m.T.FSTATUS = 1
m.T.TR_INIT = 1
m.T.TAU = 1.0
DT = 0.5 # deadband
m.h.STATUS = 1
m.h.FSTATUS = 1
m.h.TR_INIT = 1
m.h.TAU = 1.0
m.Ca.STATUS = 1
m.Ca.FSTATUS = 0 # no measurement
m.Ca.TR_INIT = 0
m.options.CV_TYPE = 1
m.options.IMODE = 6
m.options.SOLVER = 3
# define CSTR model
def cstr(x,t,u1,u2,Tf,Caf):
# Inputs (3):
# Temperature of cooling jacket (K)
Tc = u1
q=u2
# Tf = Feed Temperature (K)
# Caf = Feed Concentration (mol/m^3)
# States (2):
# Concentration of A in CSTR (mol/m^3)
Ca = x[0]
# Temperature in CSTR (K)
T = x[1]
# Parameters:
# Volume of CSTR (m^3)
V = 100
# Density of A-B Mixture (kg/m^3)
rho = 1000
# Heat capacity of A-B Mixture (J/kg-K)
Cp = 0.239
# Heat of reaction for A->B (J/mol)
mdelH = 5e4
# E - Activation energy in the Arrhenius Equation (J/mol)
# R - Universal Gas Constant = 8.31451 J/mol-K
EoverR = 8750
# Pre-exponential factor (1/sec)
k0 = 7.2e10
# U - Overall Heat Transfer Coefficient (W/m^2-K)
# A - Area - this value is specific for the U calculation (m^2)
UA = 5e4
# reaction rate
rA = k0*np.exp(-EoverR/T)*Ca
# Calculate concentration derivative
dCadt = q/V*(Caf - Ca) - rA
# Calculate temperature derivative
dTdt = q/V*(Tf - T) \
+ mdelH/(rho*Cp)*rA \
+ UA/V/rho/Cp*(Tc-T)
# Return xdot:
xdot = np.zeros(2)
xdot[0] = dCadt
xdot[1] = dTdt
return xdot
def tank(p,t,u2,Ac):
q=u2
h=p[0]
dhdt=(q-c1*pow(h,0.5))/Ac
if p[0]>=300 and dhdt>0:
dhdt = 0
return dhdt
# Time Interval (min)
t = np.linspace(0,10,410)
# Store results for plotting
Ca = np.ones(len(t)) * Ca_ss
T = np.ones(len(t)) * T_ss
Tsp=np.ones(len(t))*T_ss
hsp=np.ones(len(t))*h_ss
h=np.ones(len(t))*h_ss
u1 = np.ones(len(t)) * u1_ss
u2 = np.ones(len(t)) * u2_ss
# Set point steps
Tsp[0:100] = 330.0
Tsp[100:200] = 350.0
hsp[200:300] = 150.0
hsp[300:] = 190.0
# Create plot
plt.figure(figsize=(10,7))
plt.ion()
plt.show()
# Simulate CSTR
for i in range(len(t)-1):
ts = [t[i],t[i+1]]
y = odeint(cstr,x0,ts,args=(u1[i+1],u2[i+1],Tf,Caf))
y1=odeint(tank,p0,ts,args=(u2[i+1],Ac))
Ca[i+1] = y[-1][0]
T[i+1] = y[-1][1]
h[i+1]=y1[-1][0]
# insert measurement
m.T.MEAS = T[i+1]
m.h.MEAS= h[i+1]
# solve MPC
m.solve(disp=True)
m.T.SPHI = Tsp[i+1] + DT
m.T.SPLO = Tsp[i+1] - DT
m.h.SPHI = hsp[i+1] + DT
m.h.SPLO = hsp[i+1] - DT
# retrieve new Tc value
u1[i+1] = m.Tc.NEWVAL
u2[i+1]= m.q.NEWVAL
# update initial conditions
x0[0] = Ca[i+1]
x0[1] = T[i+1]
p0[0]=h[i+1]
plt.clf()
# Plot the results
plt.subplot(5,1,1)
plt.plot(t[0:i],u1[0:i],'b--',linewidth=3)
plt.ylabel('Cooling T (K)')
plt.legend(['Jacket Temperature'],loc='best')
plt.subplot(5,1,2)
plt.plot(t[0:i],u2[0:i],'g--')
plt.xlabel('time')
plt.ylabel('flow in')
plt.subplot(5,1,3)
plt.plot(t[0:i],Ca[0:i],'r-',linewidth=3)
plt.ylabel('Ca (mol/L)')
plt.legend(['Reactor Concentration'],loc='best')
plt.subplot(5,1,4)
plt.plot(t[0:i],Tsp[0:i],'r-',linewidth=3,label=r'$T_{sp}$')
plt.plot(t[0:i],T[0:i],'k.-',linewidth=3,label=r'$T_{meas}$')
plt.ylabel('T (K)')
plt.xlabel('Time (min)')
plt.legend(loc='best')
plt.subplot(5,1,5)
plt.plot(t[0:i],hsp[0:i],'g--',linewidth=3,label=r'$h_{sp}$')
plt.plot(t[0:i],h[0:i],'k.-',linewidth=3,label=r'$h_{meas}$')
plt.xlabel('time')
plt.ylabel('tank level')
plt.legend(loc='best')
plt.draw()
plt.pause(0.01)
One problem is that the function pow is not supported by Gekko and is evaluating that part to a constant. Here is a modified version of your equation that should work better:
m.Equation(m.h.dt()==(m.q-c1*m.h**0.5)/Ac)
One other issue is that your similar is broken into two parts and should be one model:
def tank(p,t,u2,Ac):
q=u2
h=p[0]
dhdt=(q-c1*pow(h,0.5))/Ac
if p[0]>=300 and dhdt>0:
dhdt = 0
return dhdt
You should add a third state to your simulator
# Return xdot:
xdot = np.zeros(3)
xdot[0] = dCadt
xdot[1] = dTdt
xdot[2] = dhdt
return xdot
When you have a variable height, the volume is changing so you can't assume that it is constant in the other equations. You'll need to modify your energy balance and species balance as shown in the material on balance equations.