Proportional Integral Derivative Controller in Control System

A proportional Integral Derivative controller also called a PID controller, is a widely used feedback control mechanism in industrial automation. It aims to regulate a process variable by adjusting a manipulated variable based on the error between the set point and the actual process variable.

What is a Proportional Integral Derivative Controller?

PID controller or Proportional Integral Derivative Controller is basically a combination of proportional, integral, and derivative action to regulate a process variable by adjusting a manipulated variable.

Important Terms of Proportional Integral Derivative Controller

Here is an explanation of the three terms :

1. Proportional Action

• Role: The P term responds to the current error by generating a control action proportional to the error magnitude.
• Effect: It provides immediate corrective action to reduce the error. A higher Proportional value results in a stronger and quicker response but may lead to overshoot and oscillations.

2. Integral Action

• Role : The I term or Integral term actuates the past error over time and generates a control action to eliminate the accumulated steady-state error.
• Effect : It ensures that even small errors are eventually corrected. It eliminates offset but can lead to sluggish responses and overshooting if too aggressive.

3. Derivative Action

• Role : The D term predicts the future error trend by the rate of change of the error.
• Effect : It adds a dumping effect, reducing oscillation and overshooting caused by rapid changes in error.
  

Mathematical Expression of PID Controller

The PID controller output(Co) is the sum of the proportional, Integral and derivative terms.

If, Co(t) = controller output at time T,

e(t) = error at time t(SP-PV),

K_{p ,} K_i, K_d = tunning constrains for proportional, integral and derivative terms.

So according to proportional action,

Co(t) ∝ e(t)

Co(t) = K_p . e(t) -----------(i)

According to Integral action,

Co(t) ∝ ∫ e(t) dt

Co(t) = K_i . ∫ e(t) dt -----------(ii)

According to Derivation action,

Co(t) ∝ de(t)/dt

Co(t) = K_d . de(t)/dt ------------(iii)

Combining all these three equation we get,

                                                                                      Co(t) = Kp . e(t)   +   Ki . ∫ e(t) dt   +   Kd . de(t)/dt 

Block Diagram of PID controller

Working of PID controller

To understand the working principals of PID controller, it is required to understand what is happening inside the closed loop system. First a setpoint or a target value is given to the PID controller. Then, it takes input of the actuation device through the sensor and then it compares with the setpoint and send feedback to the actuation device according to that. And this loop continues until to get the most desired value. Shortly this is the working principle of a PID controller.

Breakdown of How PID Controller Works?

• Comparison: A PID controller starts its operation by comparing the setpoint value and the process variable value, which it get from the sensor devices. Then it generates a error signal which is the difference between setpoint and the process variable value or the actual value.
• Proportional control: Proportional(P) term responses to the error signal and it aims to bring the system closer to the setpoint.
• Integral control: The integral(I) part is responsible for addressing any long term error from evaluating the past errors. And it eliminates steady-state error of a system.
• Derivative control: The derivative(D) term observe the rate of change of error, and try to calculate the next error and according to that it makes the system stable.
• Control signal: After processing error singal through P, I and D term the controller creates a signal and send it to the actuator to do it's job according to that.

And this loop continues until to get the most desired or the closest value of the process variable of setpoint.

Tuning of PID controller

Tuning of PID controller means adjusting the three variable Proportional(P), Integral(I) and Derivative(D). Through this process we try to achieve the desired controlled position.

Methods that followed to tuning a PID -

• Manual Tuning: It is the simplest way to tune a PID controller. It is done by trial and error method. This process may not lead to the most optimal result also it cannot handle complex system.
• Zeigler - Nicholas Method: This process provides a systematic tunning for PID controller. Tuning rule is as such - first P tune followed by PI tuning followed by PID tuning. This is one of the easiest method which provides good starting point for tuning. Although it has a slow response time as compared to other process.
• Cohen - coon Method: It is an improvement of zeigler - nicholas method. It catches ultimate gain and eliminates period from systems response and utilize that for better tuning.
• Kappa-tau Tuning: This method is also an improvement of Zeigler Nicholas tuning. It eliminates the shortcomings of Ziegler-Nichols, such as high proportional gains and the rules providing poor results for systems with long normalized dead time. It ensures less oscillatory response and results no disturbances with no overshot.
• Lambda Tuning: This process allows user to choose response time of closed loop, which is named as Greek later Lambda( λ). And then it calculate the corresponding time to archive the optimal response time.
• Frequency Response Tuning: This method involves analyzing the frequency of the response of the system.

Although there are several tuning methods for PID controller but mostly used are described.

Applications of Proportional Integral Derivative Controller

1. PID controller are used in various industries such as, manufacturing, process control, robotics, automotive and many more.
2. They regulate temperature, pressure, flow control, motor speed and other variable to maintain desired variable.

Advantages and Disadvantages of PID controller

Here are some advantages of Proportional Integral Derivative Controller :

Advantages

1. PID controller are simple, efficient and widely understood.
2. They offer fast response, stability and adaptability.

Here are some disadvantages of Proportional Integral Derivative Controller :

Disadvantage

1. This controller may struggle in complex linear system.
2. Optimal tuning can be challenging due to the(P, I and D) parameters.

Conclusion

PID controller is known as an instrument that can regulate different process such as flow, temperature, speed, pressure etc. It maintain zero error in the output although their are some issue with tuning and other complexities. inspite of all these PID controller is a workhorse in modern automatic control system.