This notebook contains material from CBE32338; content is available on Github.

< 4.1 Relay Control | Contents | 4.10 Lab Assignment: PID Control >

4.3 PID Control

References

Karl J. Astrom and Richard Murray (2016). Feedback Systems: An Introduction for Scientists and Engineers, 2nd Edition. Web version available.

4.3.1 PID Simulation

!pip install slycot
!pip install control

Collecting slycot
Installing collected packages: slycot
Successfully installed slycot-0.2.0
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Installing collected packages: control
Successfully installed control-0.7.0

%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt
import control.matlab as control

# control constants
Kc = 0.85
tauI = 10000

# control transfer function
Gc = Kc*control.tf([tauI,1],[tauI,0])

# model transfer functions
Gp = control.tf([0.31],[16,1])*control.tf([1],[135,1])

t = np.linspace(0,1000)
y,t = control.step(Gp,t)
plt.plot(t,50*y + 22)

[<matplotlib.lines.Line2D at 0x1c1dd7e550>]

# control constants
Kc = 2
tauI = 60

# control transfer function
Gc = Kc*control.tf([tauI,1],[tauI,0])

t = np.linspace(0,1000)

H = Gp*Gc/(1+Gp*Gc)
y,t = control.step(H,t)
plt.plot(t,y)

[<matplotlib.lines.Line2D at 0x1c1df2b080>]

4.3.2 PID Implementation

Reference Conditions

One of the implementation issues for PID control of the Temperature Control Lab is choice of reference conditions. One reason is that the linearization on which PID analysis is based is only valid in some 'neighborhood' of a nominal operating condition. But perhaps a more typical situation in most practical

%matplotlib inline

import sys
sys.path.append('..')
from TCLab import TCLab, clock, pid

# ambient and reference values
Tamb = 20
Tref = 50
uref = (Tref - Tamb)/0.85

# control parameters
b = 1              # setpoint weighting
kp = 0.8          # proportional control gain
ki = kp/60

# sampling period
tf = 1200           # experiment length (sec.)
h = 1               # sample time (sec.)

# setpoint function
def Tset(t):
    if t <= 900:
        return 50
    else:
        return 35


bi = ki*h

r = Tset(0) - Tref
y = Tamb - Tref

P = kp*(b*r - y)
I = 0

uref,P,I,r

(35.294117647058826, 24.0, 0, 0)

# device initialization
with TCLab() as a:
    a.initplot(tf)
    for t in clock(tf,h):
        r = Tset(t) - Tref
        y = a.T1 - Tref
    
        P = kp*(b*r - y)
        v = P + I
    
        u = max(0,min(200,v + uref))
        I += bi*(r-y)
    
        a.Q1 = u
        a.updateplot()
        

TCLab disconnected successfully.

< 4.1 Relay Control | Contents | 4.10 Lab Assignment: PID Control >