Test: PID Controllers

PID controller:

PID stands for Proportional, Integral, and Derivative.

It is used to increase the stability and accuracy of the system.

The PID controller's circuit diagram is shown below:

Tuning methods:

Some of the tuning methods are mentioned below

• Trial and Error method.
• Process reaction curve technique
• Ziegler-Nicholas method.

Process reaction curve:

• It is an open-loop tuning technique.
• It produces a response when a step input is applied to the system.
• Initially, we have to apply some control output to the system manually and have to record the response curve.

The PID controllers are tuned by using the open-loop gain corresponding to marginal stability.

​Important points:

• PID controller acts as a Band rejection filter.
• It reduces rise time and increases the Bandwidth.
• It decreases peak overshoot and increases the damping ratio.
• It works like the Lag-Lead compensator.

Proportion + Integral + Derivative (PID):

 

The PID controller produces on output, which is the combination of outputs of proportional, integral, and derivative controllers.

This is defined in terms of differential equations as:

Applying Laplace transform, we get:

Transfer function will be:

Integral control:

It is the control mode where the controller Output is proportional to the integral of the error with respect to time.

Integral controller output = k × integral of error with time, i.e.

Proportional + Derivate:

 

The additive combination of proportional & Derivative control is known as P-D control.

Overall transfer function for a PD controller is given by:

It is equivalent to a High-pass filter.

Concept:
Proportional + Derivate(PD):

The additive combination of proportional & Derivative control is known as P-D control.

The overall transfer function for a PD controller is given by:

PD controller is nothing but a differentiator (or) a High Pass Filter.

The frequency of noise is very high. 

So this high pass filter will allow noise into the system which results in noise amplification. 

PD Controllers reduce the response time and thus improve transient response

Effects of Proportional Derivative (PD) controllers:

• Decreases the type of the system by one
• Reduces the rise time and settling time
• It has high sensitivity.
• Rise time and settling time decreases and Bandwidth increases
• The speed of response is increased i.e. the transient response is improved
• Improves gain margin, phase margin, and resonant peak
• Increases the input noise
• Improves the stability

​Proportional Integral Controller:

This controller resembles the combination of the proportional and integral controller.

The structure of this controller is shown below:

 

This is used to decrease the steady-state error without affecting the stability of the system.

The transfer function is defined as:

Analysis:

The transfer function of the system is calculated as:

Disadvantages: Slow reaction to the disturbances.

Advantages:

•  It provides the zero control error
• It is insensitive to the interference of the measurement channel.
• PI Controllers increase the type of the system and thus reduce steady-state error and improve steady state
  response

(a) Addition of pole reduces stability

Consider system =

Adding Pole [say at origin]


 

(b) Addition of zero increase ξ

Consider system with Transfer function

Now add one zero to left half say at -2

(c) Integral controller adds one pole at origin

As type of system increase steady state error reduce

(d) Derivative controllers are not used Alone because with sudden changes in the system the derivative controller will compensate the output fast therefore in long term effects the isolated controller will produce huge steady state errors.

Damping factor is increased for reducing the oscillations and increasing the stability and speed of response which are the essential requirements of the control system and damping factor is increased by the rate compensation.

Lag-Lead compensation is a second order control system which has lead and lag compensation both and thus has combined effect of both lead and lag compensation this is obtained by the differential equation.

High pass filter is similar to the phase lead compensation and leads to increase in bandwidth and also increase in speed of response.

Phase compensation can be lead, lag or lead lag compensation and integral compensation is also known as lag compensation and leads to attenuation which has least effect on the speed but the accuracy is increased.

Damping constant reduces the gain, as it is inversely proportional to the gain and as increasing the damping gain reduces and hence the speed of response and bandwidth are both increased.

Derivative controller is the controller that is also like high pass filter and is also phase lead controller and it is used to increase the speed of response of the system by increasing the damping coefficient.

Stability of the system can be determined by various factors and for a good control system the stability of the system must be more and this can be increased by adding zero to the system and improves the transient response.

Phase lag controller is the integral controller that creates the phase lag and does not affect the value of the damping factor and that tries to reduce the steady state error.

Controller is the block in the control system that control the input and provides the output and this is the first block of the system having the input as the error signal.