How to calculate gain in a PID controller

To optimize the performance of a PID, you can learn how to calculate gains (Kp Ki Kd). This guide includes formulas, references, and examples.

1. Introduction

Proportional-Integral-Derivative (PID) controllers are widely used in industrial control systems to maintain a desired setpoint by adjusting the control input. It is important to understand how gains (Kp Ki Kd) are calculated in order to optimize the performance of PID controllers. The article below will walk you through the steps of calculating the gains, and give examples that illustrate these concepts.

2. Understanding PID Components

Three main components make up PID controls: Proportional Gains (Kp), Integral Gains (Ki) and Derivative Gains (Kd). Each component has a distinct role to play in the process of control:

The proportional gain (Kp ) determines how the system will respond to the error. Higher Kp leads to a more powerful response.

The Integral Gain (Ki). This component is designed to address the accumulation of errors. This component helps to eliminate errors in steady state by integrating them over time.

Derived Gain (Kd): This component is used to predict future errors by analyzing the rate at which the error changes. This component helps to dampen oscillations, and improve the stability of the system.

3. Identification of System Parameters

It is important to know the parameters of the system before calculating gains.

Gain of Process (Kp_process).: the gain that is achieved by controlling a process.

Process Time Constant (t). indicates the speed at which the process reacts to changes.

Desired Setting Time (T_settling).: the desired time that the system settles to its final values.

Desired overshoot (OS). The percentage of acceptable overshoot, which shows how far the system is allowed to exceed its setpoint.

4. Calculating Proportional Gain (Kp)

You can calculate the proportional gain by using this formula.

Kp=Kpprocessx(Tsettling2)Kp = Kp_process \times \left(\fracT_settling2\right)

Calculating Integral Gain (Ki)

You can calculate the integral gain (Ki), using this formula.

Ki=Kpprocessx(1t)Ki = Kp_process \times \left(\frac1t\right)

For example, if the process gain KpprocessKp_process is 10 and the time constant tt is 5 seconds, the integral gain KiKi would be:

Ki=10x(15)=2Ki = 10 \times \left(\frac15\right) = 2

Calculating Derivative Gain (Kd)

You can calculate the derivative gain (Kd), using this formula.

Kd=Kix(Tsettling2)Kd = Ki \times \left(\fracT_settling2\right)

For example, if the integral gain KiKi is 2 and the desired settling time TsettlingT_settling is 30 seconds, the derivative gain KdKd would be:

Kd=2x(302)=30Kd = 2 \times \left(\frac302\right) = 30

4. Adjusting and Verifying Gains

It is important to test the effectiveness of gains using real-world tests or simulations. The gains can be fine-tuned based on how the system responds. This will help you achieve your desired performance.

The process of calculating the gain in a PID is systematic and involves understanding PID components as well as identifying the system parameters. Follow these steps to optimize your PID controller's performance and achieve accurate and stable control.

5. Further Reading

PID Controller Explained

PID Tuning via Classical Methods

Calculating Controller Gain: Kp. Ki. and Kd