**1. Introduction**

Feedback control is a control mechanism that uses information from measurements. In a feedback control system, the output is sensed. There are two main types of feedback control systems: 1) positive feedback 2) negative feedback. The positive feedback is used to increase the size of the input but in a negative feedback, the feedback is used to decrease the size of the input. The negative systems are usually stable. A PID is widely used in feedback control of industrial processes on the market in 1939 and has remained the most widely used controller in process control until today. Thus, the PID controller can be understood as a controller that takes the present, the past, and the future of the error into consideration. After digital implementation was introduced, a certain change of the structure of the control system was proposed and has been adopted in many applications. But that change does not influence the essential part of the analysis and design of PID controllers. A proportional– integral–derivative controller (PID controller) is a method of the control loop feedback. This method is composing of three controllers [1]:


## **1.1 Role of a Proportional Controller (PC)**

The role of a proportional depends on the present error, I on the accumulation of past error and D on prediction of future error. The weighted sum of these three actions is used to adjust Proportional control is a simple and widely used method of control for many kinds of systems. In a proportional controller, steady state error tends to depend inversely upon the proportional gain (ie: if the gain is made larger the error goes down). The proportional response can be adjusted by multiplying the error by a constant *Kp*, called the proportional gain. The proportional term is given by:

$$P = K\_P.error(t) \tag{1}$$

A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is very high, the system can become unstable. In contrast, a small gain results in a small output response to a large input error. If the proportional gain is very low, the control action may be too small when responding to system disturbances. Consequently, a proportional controller (Kp) will have the effect of reducing the rise time and will reduce, but never eliminate, the steady-state error.

In practice the proportional band (PB) is expressed as a percentage so:

$$PB\% = \frac{100}{K\_P} \tag{2}$$

PID Control Theory 215

derivative actions, with the resulting signals weighted and summed to form the control

where u(t) is the input signal to the multivariable processes, the error signal e(t) is defined as

A standard PID controller structure is also known as the ''three-term" controller. This principle mode of action of the PID controller can be explained by the parallel connection of

<sup>2</sup> 1 .. ( ) (1 ) .

*TT S Gs K*

*P*

*I D*

<sup>=</sup> <sup>1</sup> (1 ) *P D*

where *KP* is the proportional gain, *TI* is the integral time constant, *TD* is the derivative time constant, *KI* =*KP* /*TI* is the integral gain and *KD* =*KPTD* is the derivative gain. The ''three-

The proportional term: providing an overall control action proportional to the error

The integral term: reducing steady state errors through low frequency compensation by

The derivative term: improving transient response through high frequency

*i K Ts T s*

(5)

*I*

term" functionalities are highlighted below. The terms *KP , TI* and *TD* definitions are:

*T S*

signal u(t) applied to the plant model.

Fig. 1. A PID control system

Fig. 2. A structure of a PID control system

the P, I and D elements shown in Figure 3.

signal through the all pass gain factor.

compensation by a differentiator.

Block diagram of the PID controller

an integrator.

e(t) =r(t) − y(t), and r(t) is the reference input signal.

Thus a PB of 10% ⇔ Kp=10

#### **1.2 Role of an Integral Controller (IC)**

An Integral controller (IC) is proportional to both the magnitude of the error and the duration of the error. The integral in in a PID controller is the sum of the instantaneous error over time and gives the accumulated offset that should have been corrected previously. Consequently, an integral control (Ki) will have the effect of eliminating the steady-state error, but it may make the transient response worse.

The integral term is given by:

$$I = K\_I \int\_0^t error(t)dt\tag{3}$$

#### **1.3 Role of a Derivative Controller (DC)**

The derivative of the process error is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain Kd. The derivative term slows the rate of change of the controller output.A derivative control (Kd) will have the effect of increasing the stability of the system, reducing the overshoot, and improving the transient response. The derivative term is given by:

$$D = K\_D.\frac{derror(t)}{dt} \tag{4}$$

Effects of each of controllers Kp, Kd, and Ki on a closed-loop system are summarized in the table shown below in tableau 1.
