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TCLab PID Control Tuning

Objective: Tune a discrete PID controller and test the performance with a series of setpoint changes over 10 min in the sequence from 23oC initially, 50oC at 10 sec, and 40oC at 300 sec. Modify the tuning parameters to achieve a low Integral Absolute Error between the measured temperature and the setpoint.

TCLab PID Control Simulator

A simulator is a useful tool to help evaluate changes in tuning before testing on a physical system. Use the PID simulator to find acceptable control performance that minimizes the integral absolute error between the setpoint and measured temperature.

import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
from scipy.integrate import odeint
import ipywidgets as wg
from IPython.display import display

n = 601 # time points to plot
tf = 600.0 # final time

# TCLab FOPDT
Kp = 0.9
taup = 175.0
thetap = 15.0

def process(y,t,u):
    dydt = (1.0/taup) * (-(y-23.0) + Kp * u)
    return dydt

def pidPlot(Kc,tauI,tauD):
    t = np.linspace(0,tf,n) # create time vector
    P = np.zeros(n)          # initialize proportional term
    I = np.zeros(n)         # initialize integral term
    D = np.zeros(n)         # initialize derivative term
    e = np.zeros(n)         # initialize error
    OP = np.zeros(n)        # initialize controller output
    PV = np.ones(n)*23.0    # initialize process variable
    SP = np.ones(n)*23.0    # initialize setpoint
    SP[10:300] = 50.0       # step up
    SP[300:601] = 40.0      # step down    
    y0 = 23.0               # initial condition
    iae = 0.0
    # loop through all time steps
    for i in range(1,n):
        # simulate process for one time step
        ts = [t[i-1],t[i]]         # time interval
        y = odeint(process,y0,ts,args=(OP[max(0,i-int(thetap))],))
        y0 = y[1]                  # record new initial condition
        iae += np.abs(SP[i]-y0[0])
        # calculate new OP with PID
        PV[i] = y[1]               # record PV
        e[i] = SP[i] - PV[i]       # calculate error = SP - PV
        dt = t[i] - t[i-1]         # calculate time step
        P[i] = Kc * e[i]           # calculate proportional term
        I[i] = I[i-1] + (Kc/tauI) * e[i] * dt  # calculate integral term
        D[i] = -Kc * tauD * (PV[i]-PV[i-1])/dt # calculate derivative term
        OP[i] = P[i] + I[i] + D[i] # calculate new controller output
        if OP[i]>=100:
            OP[i] = 100.0
            I[i] = I[i-1] # reset integral
        if OP[i]<=0:
            OP[i] = 0.0
            I[i] = I[i-1] # reset integral            

    # plot PID response
    plt.figure(1,figsize=(15,7))
    plt.subplot(2,2,1)
    plt.plot(t,SP,'k-',linewidth=2,label='Setpoint (SP)')
    plt.plot(t,PV,'r:',linewidth=2,label='Temperature (PV)')
    plt.ylabel(r'T $(^oC)$')
    plt.text(100,30,'Integral Abs Error: ' + str(np.round(iae,2)))
    plt.text(400,30,r'$K_c$: ' + str(np.round(Kc,0)))  
    plt.text(400,27,r'$\tau_I$: ' + str(np.round(tauI,0)) + ' sec')  
    plt.text(400,24,r'$\tau_D$: ' + str(np.round(tauD,0)) + ' sec')  
    plt.legend(loc='best')
    plt.subplot(2,2,2)
    plt.plot(t,P,'g.-',linewidth=2,label=r'Proportional = $K_c \; e(t)$')
    plt.plot(t,I,'b-',linewidth=2,label=r'Integral = ' + \
             r'$\frac{K_c}{\tau_I} \int_{i=0}^{n_t} e(t) \; dt $')
    plt.plot(t,D,'r--',linewidth=2,label=r'Derivative = ' + \
             r'$-K_c \tau_D \frac{d(PV)}{dt}$')  
    plt.legend(loc='best')
    plt.subplot(2,2,3)
    plt.plot(t,e,'m--',linewidth=2,label='Error (e=SP-PV)')
    plt.ylabel(r'$\Delta T$ $(^oC)$')
    plt.legend(loc='best')
    plt.subplot(2,2,4)
    plt.plot(t,OP,'b--',linewidth=2,label='Heater (OP)')
    plt.legend(loc='best')
    plt.xlabel('time')

Kc_slide = wg.FloatSlider(value=5.0,min=0.0,max=50.0,step=1.0)
tauI_slide = wg.FloatSlider(value=120.0,min=20.0,max=180.0,step=5.0)
tauD_slide = wg.FloatSlider(value=0.0,min=0.0,max=20.0,step=1.0)
wg.interact(pidPlot, Kc=Kc_slide, tauI=tauI_slide, tauD=tauD_slide)
print('PID Simulator: Adjust Kc, tauI, and tauD to achieve lowest Integral Abs Error')

TCLab PID Control Test Script

The TCLab PID control validation script implements the best PID tuning values and calculates the integral absolute error with a sequence of setpoint changes from 23oC initially, 50oC at 10 sec, and 40oC at 300 sec. Update the PID tuning parameters before running the script.

# PID Parameters
Kc   = 5.0
tauI = 120.0 # sec
tauD = 2.0   # sec

The PID controller is embedded in the script and includes an animated plot that can be generated by running from the command line or from IDLE (Python.org install).

import tclab
import numpy as np
import time
import matplotlib.pyplot as plt
from scipy.integrate import odeint

# PID Parameters
Kc   = 5.0
tauI = 120.0 # sec
tauD = 2.0   # sec

#-----------------------------------------
# PID Controller
#-----------------------------------------
# inputs ---------------------------------
# sp = setpoint
# pv = current temperature
# pv_last = prior temperature
# ierr = integral error
# dt = time increment between measurements
# outputs --------------------------------
# op = output of the PID controller
# P = proportional contribution
# I = integral contribution
# D = derivative contribution
def pid(sp,pv,pv_last,ierr,dt):
    # Parameters in terms of PID coefficients
    KP = Kc
    KI = Kc/tauI
    KD = Kc*tauD
    # ubias for controller (initial heater)
    op0 = 0
    # upper and lower bounds on heater level
    ophi = 100
    oplo = 0
    # calculate the error
    error = sp-pv
    # calculate the integral error
    ierr = ierr + KI * error * dt
    # calculate the measurement derivative
    dpv = (pv - pv_last) / dt
    # calculate the PID output
    P = KP * error
    I = ierr
    D = -KD * dpv
    op = op0 + P + I + D
    # implement anti-reset windup
    if op < oplo or op > ophi:
        I = I - KI * error * dt
        # clip output
        op = max(oplo,min(ophi,op))
    # return the controller output and PID terms
    return [op,P,I,D]

# save txt file with data and set point
# t = time
# u1,u2 = heaters
# y1,y2 = tempeatures
# sp1,sp2 = setpoints
def save_txt(t, u1, u2, y1, y2, sp1, sp2):
    data = np.vstack((t, u1, u2, y1, y2, sp1, sp2))  # vertical stack
    data = data.T  # transpose data
    top = ('Time,Q1,Q2,T1,T2,TSP1,TSP2')
    np.savetxt('validate.txt', data, delimiter=',',\
               header=top, comments='')

# Connect to Arduino
a = tclab.TCLab()

# Turn LED on
print('LED On')
a.LED(100)

# Run time in minutes
run_time = 10.0

# Number of cycles
loops = int(60.0*run_time)
tm = np.zeros(loops)

# Temperature
# set point (degC)
Tsp1 = np.ones(loops) * a.T1

# Heater set point steps
Tsp1[3:] = 50.0
Tsp1[300:] = 40.0

T1 = np.ones(loops) * a.T1 # measured T (degC)
error_sp = np.zeros(loops)

Tsp2 = np.ones(loops) * a.T2 # set point (degC)
T2 = np.ones(loops) * a.T2 # measured T (degC)

# impulse tests (0 - 100%)
Q1 = np.ones(loops) * 0.0
Q2 = np.ones(loops) * 0.0

print('Running Main Loop. Ctrl-C to end.')
print('  Time     SP     PV     Q1   =  P   +  I  +   D    IAE')
print(('{:6.1f} {:6.2f} {:6.2f} ' + \
       '{:6.2f} {:6.2f} {:6.2f} {:6.2f} {:6.2f}').format( \
           tm[0],Tsp1[0],T1[0], \
           Q1[0],0.0,0.0,0.0,0.0))

# Main Loop
start_time = time.time()
prev_time = start_time
dt_error = 0.0
# Integral error
ierr = 0.0
# Integral absolute error
iae = 0.0

plt.figure(figsize=(10,7))
plt.ion()
plt.show()

try:
    for i in range(1,loops):
        # Sleep time
        sleep_max = 1.0
        sleep = sleep_max - (time.time() - prev_time) - dt_error
        if sleep>=1e-4:
            time.sleep(sleep-1e-4)
        else:
            print('exceeded max cycle time by ' + str(abs(sleep)) + ' sec')
            time.sleep(1e-4)

        # Record time and change in time
        t = time.time()
        dt = t - prev_time
        if (sleep>=1e-4):
            dt_error = dt-1.0+0.009
        else:
            dt_error = 0.0
        prev_time = t
        tm[i] = t - start_time

        # Read temperatures in Kelvin
        T1[i] = a.T1
        T2[i] = a.T2

        # Integral absolute error
        iae += np.abs(Tsp1[i]-T1[i])

        # Calculate PID output
        [Q1[i],P,ierr,D] = pid(Tsp1[i],T1[i],T1[i-1],ierr,dt)

        # Write output (0-100)
        a.Q1(Q1[i])
        a.Q2(0.0)

        # Print line of data
        print(('{:6.1f} {:6.2f} {:6.2f} ' + \
              '{:6.2f} {:6.2f} {:6.2f} {:6.2f} {:6.2f}').format( \
                  tm[i],Tsp1[i],T1[i], \
                  Q1[i],P,ierr,D,iae))

        # Update plot
        plt.clf()
        # Plot
        ax=plt.subplot(2,1,1)
        ax.grid()
        plt.plot(tm[0:i],Tsp1[0:i],'k--',label=r'$T_1$ set point')
        plt.plot(tm[0:i],T1[0:i],'r.',label=r'$T_1$ measured')
        plt.ylabel(r'Temperature ($^oC$)')
        plt.legend(loc=4)
        ax=plt.subplot(2,1,2)
        ax.grid()
        plt.plot(tm[0:i],Q1[0:i],'b-',label=r'$Q_1$')
        plt.ylabel('Heater (%)')
        plt.legend(loc=1)
        plt.xlabel('Time (sec)')
        plt.draw()
        plt.pause(0.05)

    # Turn off heaters
    a.Q1(0)
    a.Q2(0)
    a.close()
    # Save text file
    save_txt(tm[0:i],Q1[0:i],Q2[0:i],T1[0:i],T2[0:i],Tsp1[0:i],Tsp2[0:i])
    # Save figure
    plt.savefig('PID_Control.png')

# Allow user to end loop with Ctrl-C          
except KeyboardInterrupt:
    # Disconnect from Arduino
    a.Q1(0)
    a.Q2(0)
    print('Shutting down')
    a.close()
    save_txt(tm[0:i],Q1[0:i],Q2[0:i],T1[0:i],T2[0:i],Tsp1[0:i],Tsp2[0:i])
    plt.savefig('PID_Control.png')

# Make sure serial connection closes with an error
except:          
    # Disconnect from Arduino
    a.Q1(0)
    a.Q2(0)
    print('Error: Shutting down')
    a.close()
    save_txt(tm[0:i],Q1[0:i],Q2[0:i],T1[0:i],T2[0:i],Tsp1[0:i],Tsp2[0:i])
    plt.savefig('PID_Control.png')
    raise

Solution