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Implementation of PID controller in Python

What is PID control?

Proportional-integral-derivative (PID) controllers have been used in manufacturing industry since the 1940s. PID control is similar to proportional control, but with the addition of integral and derivative components. The integral feature eliminates the offset while the derivative element provides a fast response. 
Let’s start with a bit of theory. The following equation is used by PID controller to compute the output value u(t):

    \[u={{K}_{p}}e(t)+{{K}_{i}}\int\limits_{0}^{t}{e(\tau )d\tau }+{{K}_{d}}\frac{de(t)}{dt}\]

Kp, Ki and Kd denote the coefficients for the proportional, integral, and derivative gains respectively; e denotes error and t – time.

In this tutorial you will learn how to use Python programming language to implement PID controller.

PID controller – Example of application

The system

Let’s consider a tank used to continuously heat up the liquid stream flowing through this tank with heating jacket shown in the figure. We can use the PID controller to obtain temperature set. We will use a simple mathematical model of this system (check more detailed description in this lecture).

    \[\frac{dT}{dt}=\frac{1}{1+\varepsilon }\left[ \frac{1}{\tau }({{T}_{f}}-T)+Q({{T}_{q}}-T) \right]\]

    \[\varepsilon =\frac{{{m}_{s}}{{c}_{s}}}{V\rho {{c}_{p}}},\ Q=\frac{{{A}_{q}}{{k}_{q}}}{V\rho {{c}_{p}}},\ \tau =\frac{V}{{{F}_{V}}}\]

Continuous tank with heating jacket - scheme
Continuous tank with heating jacket

Python code of a system with PID control

Let’s start writing the program with importing necessary python libraries.

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

We will need several global variables as the calculations require the history of the object.

Python
time = 0
integral = 0
time_prev = -1e-6
e_prev = 0

The function below calculates the value of the manipulated variable (MV) based on the measured value (in example it is the temperature of the liquid) and setpoint value (temperature we want to obtain). The function also accepts parameters of the controller.

Python
def PID(Kp, Ki, Kd, setpoint, measurement):
    global time, integral, time_prev, e_prev

    # Value of offset - when the error is equal zero
    offset = 320
    
    # PID calculations
    e = setpoint - measurement
        
    P = Kp*e
    integral = integral + Ki*e*(time - time_prev)
    D = Kd*(e - e_prev)/(time - time_prev)

    # calculate manipulated variable - MV 
    MV = offset + P + integral + D
    
    # update stored data for next iteration
    e_prev = e
    time_prev = time
    return MV

Next, we create a function defining the system.

Python
def system(t, temp, Tq):
    epsilon = 1
    tau = 4
    Tf = 300
    Q = 2
    dTdt = 1/(tau*(1+epsilon)) * (Tf-temp) + Q/(1+epsilon)*(Tq-temp)
    return dTdt

Let’s see the results for the system without the automatic control. We obtain them by solving the differential equations shown before.

Python
tspan = np.linspace(0,10,50)
Tq = 320,
sol = odeint(system,300, tspan, args=Tq, tfirst=True)
plt.xlabel('Time')
plt.ylabel('Temperature')
plt.plot(tspan,sol)
Temperature response of uncontrolled system
Temperature change in the system without automatic control

We see that the temperature in the tank is equal 300 K at the initial time and it rises to about 317.5 steadily. Now, we will make a simulation of the tank with automatic control. Let’s say that we want to reach the temperature of 310 K. This is our setpoint. The controller constants are set arbitrarily – they can be adjusted to get a better control. However, it is out of the scope of this tutorial. Temperature of steam was chosen as a manipulated variable (Tq). You can see that its value is calculated in every step of simulation with the use of previously written PID function.

Python
# number of steps
n = 250

time_prev = 0
y0 = 300
deltat = 0.1
y_sol = [y0]
t_sol = [time_prev]

# Tq is chosen as a manipulated variable
Tq = 320,

q_sol = [Tq[0]]
setpoint = 310
integral = 0
for i in range(1, n):
    time = i * deltat
    tspan = np.linspace(time_prev, time, 10)
    Tq = PID(0.6, 0.2, 0.1, setpoint, y_sol[-1]),
    yi = odeint(system,y_sol[-1], tspan, args = Tq, tfirst=True)
    t_sol.append(time)
    y_sol.append(yi[-1][0])
    q_sol.append(Tq[0])
    time_prev = time

plt.plot(t_sol, y_sol)
plt.xlabel('Time')
plt.ylabel('Temperature')
Temperature response of controlled system
Temperature change in the system with automatic control

You can see that the controller makes the temperature tend to the desired value.

Download Jupiter notebook file with PID implementation


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