How PID Controller Works

Time is most precious thing human can ever have since we do not have the ability to reutilize it. Being human we always want work to be done automatically, accurately and timely, to do so we need an optimal control strategy which should have command over all tenses of time i.e. Present + Past + Future. One of such control strategy is PID Controller.

How PID Controller Works?

This mode of the controller is a complex combination of proportional-integral-derivative control modes.  PID control mode possesses zero steady state error, oscillations, and high stability. By the addition of the derivative term to PI control mode helps to reduce overshoot, reduce settling time as well as becomes capable of handling sluggish and fast dynamics higher order processes. In this mode, integral terms try to stabilize the lightly damped system, usually, only PD control mode can not do it easily.

Mathematically this is represented as,


  • P = PD controller’s output
  • KP = Proportional Gain
  • KI = Integral Gain (=1/Integral Time (Ti))
  • Ki = KP / Ti
  • ­KD= Derivative Gain (=Derivative Time (Td))
  • Kd = KP x Td
  • ep (t)= Desired Value of controlled variable – Measured Value
  • P­I(0) = Integral term initial value

Since, PID control mode can be utilized in many different ways as shown in above equation, which actually helps to define tunable parameters of PID controller.


PID controller has many industrial as well as domestic applications.
The example we are going to consider here is “maintaining the position of booster rocket at the time of taking off”. To replicate this problem in simplified terms, let’s consider launch pad as a cart and rocket as an inverted pendulum. Now, this a classic example of runaway process, i.e. pinch to the pendulum in normal condition will result in instability of the overall system. Such a system either only PI or only PD controller can not stabilize since one can not handle sudden disturbance and another can not handle initial instability. PID controller maintains it’s position by eliminating steady-state error and predicting error trend.

PID Controller

How to tune a PID Controller: Loop Tuning

Please visit this article to get more understanding on PID Controller Tuning.

Advantages: Why PID controller is used?

  1. No steady state error
  2. Improves overall stability of system
  3. Ability to handle nonlinear higher order unstable system such as CSTR runaway processes
  4. Accurate and quick desired value tracking


  1. Due to its linear nature, it shows poor performance for hysterically nonlinear processes. (Such as HVAC system)
  2. To avoid high-frequency noise accentuation from derivative term we need to add the low pass to the measured value of the controlled variable.
  3. Single PID controller can control only one variable, hence not suitable for coupled system such as quadcopter.
  4. With fixed tuning of parameters, it can not handle processes which has variable dead time.


Please watch below video for more understanding on PID Controller in a simple way.

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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,

PD Controller


  • 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.

PD Controller


  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


Please watch below video for more understanding on PD Controller:

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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,


  • 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.
  3. Removes steady state error.


  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.


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.

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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,


  • 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


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


  1. Effect of transportation lag occurred due to the remote allocation of the sensor can be minimized.
  2. Accumulation of error which will go to affect on actuator saturation can be minimized.
  3. Peak errors are minimized.


  1. Cannot be used when an error is constant. (Derivative of constant value is zero).
  2. A small change in error will affect largely on controller’s output. The high derivative gain will result in heavy overshoots and overall system’s stability.

Integral Controller In Control System

Hello friends, in this article we are going to learn about the Integral controller in Control System with its working.

Integral Controller Working

As we know that, the proportional controller tells us how far to move, to achieve zero error. The integral controller tells us how fast to move to achieve zero error. The proportional controller cannot guarantee zero error in case of a transient change in load dynamics of the system. Integral control accumulates positive and negative errors and tries to eliminate steady-state error.

Integral action is provided by summing the error over time, multiplying that sum by a gain, and adding the result to the present controller output. You can see that if the error makes random excursions above and below zero, the net sum will be zero, so the integral action will not contribute. But if the error becomes positive or negative for an extended period of time, the integral action will begin to accumulate and make changes to the controller output.

Below mathematical expression represents Integral control output,

Integral Controller


  • P(t) = Controller’s output
  • KI= Integral gain usually motioned in terms of integral time (1/Ti)
  • ep (t)= Desired Value of controlled variable – Measured Value
  • dτ = Tiny slices of time
  • P(0) = Initial controller’s output

If we differentiate above equation then,

Above equation shows that when an error occurs, the controller begins to increase or decrease its output at a certain rate that depends on the size of the error and the integral time constant. If the error is zero, the controller output will not change. If there is a positive error, the controller output begins to ramp up at a rate determined.

Integral Controller Applications

There are no specific applications where integral control used individually since integral controller alone will cause transient overshoot and which may result in actuator saturation (means actuator cannot be operated beyond this limit).


  1. Offset error can be eliminated.
  2. Overall system stability increases.


  1. Reset Windup: Due to a sudden change in desired value, there is a huge accumulation of error which will result in high controller output, but as we know that actuator can not react beyond a certain limit, therefore there is no significant change on system’s output. Such a scenario where the action of the controller not able to wind up error due to frequent resetting of actuator known as “Reset wind up” condition.
  1. Phase lag added, which will affect overall settling time of the system.

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Proportional controller In Control System

The proportional controller gives control signal in one to one correspondence with an error over some range. Each value of error about desired value will have corresponding controller output like controller output values are interpolated with respect to error values.

Mathematically this can be represented as


  • P = Controller’s output
  • ep (t)= Desired Value of controlled variable – Measured Value
  • Kp = Proportional gain
  • P0 = Controller output with no error

It indicates that, if the error is zero, the output is constant and equal to P0 and if there is an error, for every 1 % error, a correction of Kp % is added to or subtracted from P0, depending on the sign of error.

Application of proportional controller

Thermostat used in room temperature control expands or compress it’s bimetallic spring as per temperature variation in the room and gives corresponding control signal to heater coil.

Another example is heat exchanger system,

  • If there is sudden large flow of cold fluid enters into heat exchanger, since TT (temperature transmitter) detects this sudden change then,
  • TC (Temperature controller) compares measured value of temperature with desired temperature value,
  • a corresponding error is also large and if we multiply it with proper gain (proportional gain) and subtract from P0,
  • Then control signal gradually opens steam flow through the control valve and maintains the temperature inside the heat exchanger.

proportional controller

Advantages and disadvantages of proportional controller


  1. Processes with slower dynamics are controlled effectively.
  2. Helps to stabilize higher order processes which have transient instability.


  1. The residual error is always required to maintain the desired value of the controlled variable.

E.g. If the controller’s output will increase with falling outlet temperature of the heat exchanger until there is enough steam flow admitted to the heat exchanger to prevent the temperature from falling any further. But in order to maintain this greater flow rate of steam (for greater heating effect), an error must develop between the measured temperature and desired temperature. In other words, the process variable (temperature) must deviate from desired value in order for the controller to call for more steam, in order that the process variable does not fall any further than this. This necessary error between the measured value and the desired value is called ‘offset’ or residual error.

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Types of Controllers in Process Control System

Hello, friends in this article we are going to learn different types of controllers in control system. Classification of controllers is nothing but the different control modes in process control. We will learn different types of controllers in process control along with examples.

What is Controller?

In this universe, there are several types physical changes exists, among those changes, which are under our bounded observation is known as process and device by which we can obtain a desired response from that process is commonly known as the controller.

Types of the controllers

Types of Controllers in Process Control System
Types of Controllers in Process Control System

1. Discontinuous Controller

A. Two position mode controller


Historically, two position controller was abundantly used among the controllers, since it has only two possible positions i.e. 0% or 100%. A simple example of this type of controller is a relay.

Whenever the measured value of the controlled variable is less than desired value i.e. potential difference between two terminal of the coil of relay then, normally open contact gets closed or normally closed contact gets opened.

Similarly, if the measured value is equal to desired value then, there is no change in state.


Room heater, if the temperature of the room goes below the desired temperature then, the heater turns ON and if the temperature is above the desired temperature then heater turns OFF.


  1. Simple
  2. Easily adaptable to large-scale systems which have slower process rates


  1. Neutral zone exists, due to which though the difference between measured and desired value exists, there is no change in controller output.

B. Floating Position Mode Controller

a. Single Speed floating Position Mode Controller


This controller’s output changes at a fixed rate when the difference between desired value and the measured value exceeds neutral zone. Mathematically, it can be represented as,


  • dp/dt = rate change of controller output with respect to time
  • KF= rate constant
  • ∆ep = half of the neutral zone

By integrating equation (1) we get,

Where p(0) is initial controller’s output.

This indicates that current controller’s output keeps a history of previous control outputs. In many cases, such a kind of information is not available.

b. Multiple Speed floating mode controller

Unlike single speed controller, it’s ‘KF’ values increases or decreases as per deviation exceeds certain limits. It means that for large error (|desired value – measured value|) will have large ‘KF’ value and vice versa.


In self-regulating processes such as liquid flow rate control in the pipe (as shown in below diagram) a single speed floating controller is used. The load is determined by the inlet and outlet pressures Pin and Pout, and the flow is determined in part by the pressure P, within the DP cell and control valve. This is an example of a system with self-regulation. We assume that small control valve opening has been found to maintain the desired flow rate inside the pipe. If larger than the neutral zone, the valve begins to open or close at a constant rate until an opening is found that supports the proper flow rate at the new load conditions. Clearly, the rate is very important, because especially fast process lags cause the valve to continue opening or closing beyond that optimum self-regulated position.


  1. Effect of neutral zone can be minimized.
  2. Overshoots and undershoots are reduced compare to two position mode
  3. Gives better performances for self-regulating lower order small dead time systems.


  1. When applied to large-scale systems such as room temperature control, this type of controller shows inevitable cycling i.e. measured value will fluctuate around desired value for a long time.

In upcoming articles, we will learn about Continuous controllers, till then if you like this article, share it with your friends, like our Facebook page and subscribe to our newsletter for future updates. Have a nice day!