## PD Controller (Proportional-Derivative) Controller in Control System

### How does a PD controller work?

Prediction of the behavior of error will always result in better stability. In order to avoid effects of the sudden change in load, the derivative of the error signal taken in this mode to predict the trend of a controlled variable. So let us see in detail, how does PD Controller work.

Almost all physical processes have transportation lag (Dead Time) in their system (usually due to improper allocation of the sensor) since only proportional controller’s output will react after some time to sudden change in load and which may result in a huge transient error.  But, with the addition of a derivative controller, this mode becomes capable of predicting error with consideration of dead time. So that, sudden jerks or spike signals are not given to actuator, hence improves the life span of actuators.

Mathematically this is represented as,

Where,

• P = PD controller’s output
• KP = Proportional Gain
• ­KD= Derivative Gain
• ep (t)= Desired Value of controlled variable – Measured Value
• P­(0) = Controller’s output when error is zero

From the equation, we can say that this mode cannot eliminate the steady state error of proportional controller. However, It can handle fast process load changes as long as the load change steady state error is acceptable.

### Applications of PD Controller

Maintaining a level of liquid inside the tank is a sluggish and integrating process and many cases due to improper allocation of level sensor (in this case which is measured as a function of flow) result into the significant addition of transportation lag. PD controller mode has the capability to predict future of error, hence the effect of additional dead time is reduced. A sudden change in desired value of level will result in high overshoot in the case of PI control mode, but in the case of PD control mode, this integrating effect will be reduced by addition derivative term with the proportional term.

1. Overall stability of system improves
2. Capable of handling processes with time lag
3. Reduces settling time by improving damping and reducing overshoot

1. Not suited for fast responding systems which are usually lightly damped or initially unstable.
2. Amplifies noise at higher frequencies which result in improper handling of actuators.
3. Does not eliminate steady state error

### Video

Please watch below video for more understanding on PD Controller:

## PI Controller (Proportional Integral) Controller in Control System

### How does PI Controller Work?

The control mode of PI Controller has a one-to-one correspondence of the proportional mode as well as the integral mode which eliminates that inherent offset. This controller is mostly used in areas where the speed of the system is not an issue. Since proportional controller can not provide new nominal controller’s output in case of new load condition, but in this new configuration necessity of fixed (offset) error has been replaced by the accumulation of error term i.e integral term. Mathematically, this can be represented as,

Where,

• P = PI controller’s output
• KP = Proportional Gain
• ­KI= Integral Gain
• ep (t)= Desired Value of controlled variable – Measured Value
• I(0) = Integral term initial value

Form PI controller’s equation we can say that when an error is zero, but the controlled variable is oscillating about desired value, then integral action tries to eliminate error and reaches desired value.

When an error is not zero and only accumulated error is not sufficient for resulting in the quick ramp up, in that case, the proportional controller reduces rise time and tries to achieve optimal controller’s output at new load conditions.

### Applications of PI Controller

Flow control of any liquid is a dynamic process, improper prediction of error might result in control value saturation or extended flow of liquid which usually happens when we apply derivative controller to such a system.

In this case, the Proportional controller gives proper ramp up to achieve desired value quickly as well as the occurrence of offset error or steady state error about desired value has been eliminated by the integral term.

1. Desired value can be achieved accurately.
2. Ease to apply for fast response processes as well as processes in which load change is large and frequent.

1. The speed of response of system becomes sluggish due to the addition of integral term.
2. During start-up of a batch process, the integral action causes an overshoot.
3. Since PI controller doesn’t have the ability to predict the future errors of the system, therefore it cannot eliminate steady state oscillations and reduces settling time. Hence, overall stability system is comparatively low.

### Video

Please watch below video for more understanding on PI Controller. In this video, they have explained how we can eliminate steady state error using PI Controller.

## Derivative Controller in Control System

### How does Derivative Controller work?

With an integral controller, we can calculate accumulated error, but with the derivative control, we can calculate the ratio of error change per unit time, hence act as a predictor. Derivative controller action responds to the rate at which the difference between desired value and the measured value is changing that is derivative of the error.  Mathematically represented as below,

Where,

• KD­ = Derivative gain
• The derivative controller is also known as Rate action controller or anticipatory controller.
• ep (t)= Desired Value of controlled variable – Measured Value

### Applications

The derivative controller is not used alone because it provides no output when the error is constant.