Introduction to Industrial Automation

9.1   PID  Control

A feedback controller is designed to produce an “output”, which acts correctively in one process, in 

order to lead a measured process variable to the desired value, known as the set point. In Figure 9.1, 

a typical feedback control loop is shown, where the blocks represent the dynamics of the whole sys-

tem (controller and controlled process) and the arrows represent the flow of information either in 

the form of electrical signals or in the form of digital data. All the feedback controllers determine 

their output by taking into account the error between the desired and the measured actual value 

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  Introduction to Industrial Automation

of the controlled variable. For example, a home thermostat is a simple ON-OFF controller that 

activates the heating system when the difference (error) between the actual and the desired room 

temperature value exceeds a threshold. A PID (proportional, integral, derivative) controller imple-

ments the same function as a thermostat, but determines the output with a more complex control 

algorithm. In particular, it takes the current value of the error in the series, the integral of the error 

in the latest time period, and the current value of the derivative of the error into account, in order 

to determine not only the size of the correction that should apply, but also the time duration of the 

corrective action. These three quantities are multiplied by three different gains (P, I, and D) with 

their sum as the final controller’s output (CO(t)) according to the following equation:

 

CO t

P e t I

e d

D d

dt

e

t

( )

( )

( )

( )

= ⋅

+

















+









∫

τ τ

0

t

 (9.1)

In Equation 9.1, P is the gain of the analog term, I is the gain of the integration term, D is the 

gain of the differentiation term, and e(t) is the error between the desired value (set point) SP(t) and 

the actual value of the variable PV(t) at time t,

 

e t

SP t PV t

( )

( )

( )

=

−

 (9.2)

If the current error is large and unchanged for some time or is changing rapidly, the PID 

controller will try to make one large correction of the system behavior by producing a respectively 

large output. Conversely, if the process variable is very close to the desired value for some time, the 

PID controller will remain idle. Of course, the issue of the proper operation of a PID controller is 

not as simple as the last paragraph probably implies. The difficult task is that of tuning the PID 

controller, which consists of selecting the values of gains P, I, and D so that the sum of the three 

terms in Equation 9.1 give an output that will drive the process variable with stability to changes 

that will eliminate the error.

How well the PID controller will be tuned depends on how the controlled process responds 

to the corrective actions of the controller. Processes that respond instantaneously and predictably 

don’t require a feedback signal. For example, a car’s lighting system rapidly reaches the desired 

output value (light) when the driver presses the corresponding switch without needing correc-

tions of the real value from a controller. On the other hand, the fixed-economic speed control-

ler (cruise control) of a car cannot accelerate very quickly to the desired speed that the driver 

chooses. Due to friction and inertia of the car, there is always a delay between the time that the 

controller activates the throttle and the desired speed of the car. Generally, a PID controller 

should be adjusted by taking these delays or the physical parameters of the controlled system 

into account.

Setpoint

Error

Controller

Controlled

process

Process

variable

–

+