**2. PID controller (PIDC)**

A typical structure of a PID control system is shown in Fig.1. Fig.2 shows a structure of a PID control system. The error signal e(t) is used to generate the proportional, integral, and


Table 1. A PID controller in a closed-loop system

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

> <sup>100</sup> % *P*

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

0

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

> *D*. *derror t D K*

Effects of each of controllers Kp, Kd, and Ki on a closed-loop system are summarized in the

A typical structure of a PID control system is shown in Fig.1. Fig.2 shows a structure of a PID control system. The error signal e(t) is used to generate the proportional, integral, and

*t*

( )

( )

*<sup>K</sup>* (2)

*<sup>I</sup> I K error t dt* (3)

*dt* (4)

*PB*

and will reduce, but never eliminate, the steady-state error.

Thus a PB of 10% ⇔ Kp=10

The integral term is given by:

table shown below in tableau 1.

**2. PID controller (PIDC)** 

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

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

error, but it may make the transient response worse.

transient response. The derivative term is given by:

Table 1. A PID controller in a closed-loop system

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

derivative actions, with the resulting signals weighted and summed to form the control signal u(t) applied to the plant model.

Fig. 1. A PID control system

Fig. 2. A structure of a PID control system

where u(t) is the input signal to the multivariable processes, the error signal e(t) is defined as e(t) =r(t) − y(t), and r(t) is the reference input signal.

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

Block diagram of the PID controller

$$\mathbf{G(s)} = \mathbf{K\_P}(\mathbf{1} + \frac{\mathbf{1} + T\_I \mathbf{1}\_D \mathbf{S}}{T\_I \mathbf{S}}) \ = \mathbf{K\_P}(\mathbf{1} + \frac{\mathbf{1}}{T\_{\hat{\mathbf{s}}} \mathbf{s}} + T\_D \mathbf{s}) \tag{5}$$

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


PID Control Theory 217

( ) ( ) ( ) *U s G s*

<sup>2</sup> ( )

*<sup>K</sup> <sup>K</sup> Ks s*

*s*

2 2 0 0

> 2 0

<sup>0</sup> <sup>2</sup> *<sup>p</sup> d K <sup>K</sup>*

() () *Kd GsHs*

*s s* 

*p i*

*d d*

 

*E s* (7)

(9)

(10)

(11)

(12)

*<sup>s</sup>* (13)

(8)

<sup>2</sup> *KS KS K D PI S*

Fig. 4. a) Step response of PID ideal formb) Step response of PID real form

( ) *<sup>I</sup>*

*P D <sup>K</sup> Gs K K S S* =

( )

*H s*

When this form is used it is easy to determine the closed loop transfer function.

*d*

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

*i d K <sup>K</sup>* 

*K K G s*

**2.1 The transfer function of the PID controller**  The transfer function of the PID controller is

**2.2 PID pole zero cancellation** 

If

Then

The PID equation can be written in this form:

Fig. 3. Parallel Form of the PID Compensator

These three variables *KP* , *TI* and *TD* are usually tuned within given ranges. Therefore, they are often called the *tuning parameters* of the controller. By proper choice of these tuning parameters a controller can be adapted for a specific plant to obtain a good behaviour of the controlled system.

The time response of the controller output is

$$\int\_{t}^{t} e(t)dt$$

$$dL(t) = K\_P(e(t) + \frac{0}{T\_i} + T\_d \frac{de(t)}{dt}) \tag{6}$$

Using this relationship for a step input of *e t*( ) , i.e. *et t* () () , the step response r(t) of the PID controller can be easily determined. The result is shown in below. One has to observe that the length of the arrow *K TP D* of the D action is only a measure of the weight of the impulse.

OR

These three variables *KP* , *TI* and *TD* are usually tuned within given ranges. Therefore, they are often called the *tuning parameters* of the controller. By proper choice of these tuning parameters a controller can be adapted for a specific plant to obtain a good behaviour of the

> 0 ( ) ( ) () (() )

controller can be easily determined. The result is shown in below. One has to observe that the length of the arrow *K TP D* of the D action is only a measure of the weight of the

*de t Ut K et T*

 

*P d i*

*e t dt*

*T dt*

(6)

, the step response r(t) of the PID

*t*

Fig. 3. Parallel Form of the PID Compensator

The time response of the controller output is

Using this relationship for a step input of *e t*( ) , i.e. *et t* () ()

controlled system.

impulse.

Fig. 4. a) Step response of PID ideal formb) Step response of PID real form

#### **2.1 The transfer function of the PID controller**

The transfer function of the PID controller is

$$G(\mathbf{s}) = \frac{\mathcal{U}(\mathbf{s})}{E(\mathbf{s})} \tag{7}$$

$$\mathbf{G(s)} = \mathbf{K\_P} + \frac{\mathbf{K\_I}}{\mathbf{S}} + \mathbf{K\_D}\mathbf{S} \quad \mathbf{=} \frac{\mathbf{K\_D}\mathbf{S}^2 + \mathbf{K\_P}\mathbf{S} + \mathbf{K\_I}}{\mathbf{S}} \tag{8}$$

#### **2.2 PID pole zero cancellation**

The PID equation can be written in this form:

$$G(s) = \frac{K\_d(s^2 + \frac{K\_p}{K\_d}s + \frac{K\_i}{K\_d})}{s} \tag{9}$$

When this form is used it is easy to determine the closed loop transfer function.

$$H(\mathbf{s}) = \frac{1}{s^2 + 2\xi a\_0 s + o^2\_{-0}} \tag{10}$$

If

$$\frac{K\_i}{K\_d} = \alpha^2\_{\;\;\;d} \tag{11}$$

$$\frac{K\_p}{K\_d} = 2\xi a\_0 \tag{12}$$

Then

$$\mathbf{G}(\mathbf{s})H(\mathbf{s}) = \frac{K\_d}{s} \tag{13}$$

PID Control Theory 219

1. Circuit diagram below (figure.5) shows an analog PID controller. In this figure, we present an analog PID controller with three simple op amp amplifier, integrator and

We can realise a PID controller by two methods:

Fig. 5. Electronic circuit implementation of an analog PID controller

First, an analog PID controller Second, a digital PID controller

differentiator circuits.

This can be very useful to remove unstable poles.

There are several prescriptive rules used in PID tuning. The most effective methods generally involve the development of some form of process model, and then choosing P, I, and D based on the dynamic model parameters.
