# Understanding PID Controllers: Insights and Applications

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## Chapter 1: Introduction to PID Controllers

PID controllers are essential tools used in various industries to maintain desired outputs by adjusting inputs based on error signals. A survey by Desborough and Miller in 2002 revealed that an impressive 97% of industries incorporate PID controllers in their processes, with data drawn from over 11,000 manufacturing entities, including the pulp industry. This statistic underscores the proven performance and reliability of PID controllers in managing complex systems in practical applications.

In essence, a PID controller is composed of three key actions: Proportional (P), Integral (I), and Derivative (D). While these actions can be utilized in conjunction, they may also be implemented individually based on specific objectives. For instance, a standalone P controller is an option, while a combination of P and I actions forms a PI controller, and a blend of P and D actions results in a PD controller.

### Section 1.1: Proportional Control

The Proportional (P) controller responds to the current error. Its name derives from the fact that its corrective action is directly proportional to the input error signal. For example, if the error signal is zero, the proportional control action will also be zero. The transfer function of a proportional controller can be expressed as follows:

### Section 1.2: Integral Control

The Integral (I) controller focuses on past errors, accumulating error over time. Even if the current error signal is zero, the integral controller continues to act based on historical error values. This characteristic allows it to eliminate steady-state errors or offsets. When tuned appropriately, the integral controller can enhance a system's ability to handle persistent disturbances. However, it may also induce oscillations in the system's response and is sensitive to saturation effects.

### Section 1.3: Derivative Control

The Derivative (D) controller reacts to the rate of error change. Its transfer function can be outlined as follows:

By assessing the error rate, the derivative controller can predict and anticipate future errors through linear extrapolation. For instance, if the rate of error decreases by 5 and the derivative gain is 2, the resulting derivative control action would be -10. This negative action indicates that as the error rate diminishes, the system's response slows down as it approaches the setpoint, which helps mitigate overshooting. Research indicates that the derivative controller has a more pronounced effect on second-order systems compared to first-order systems, although it is sensitive to noise due to its reliance on error rate.

### Section 1.4: PID Controller Structures

PID controllers can be structured in two primary forms: parallel and series. The parallel PID controller (often referred to as the ideal or standard PID controller) is depicted in the earlier figure, with its transfer function defined as:

Conversely, the series PID controller is illustrated in another figure, with its transfer function expressed as follows:

## Chapter 2: Video Resources on PID Controllers

To further enhance your understanding of PID controllers, I recommend watching the following videos:

The first video is titled **"PID Controller Explained - YouTube"**, which provides a comprehensive overview of PID controllers, their functions, and applications.

The second video, **"What is a PID Controller? - YouTube,"** delves into the fundamentals of PID control systems and their significance in various fields.

## References

[1] K. J. Åström and R. M. Murray, *Feedback systems: an introduction for scientists and engineers*. Princeton University Press, 2021.

[2] Y. Li, K. H. Ang, and G. C. Chong, “PID control system analysis and design,” *IEEE Control Systems Magazine*, vol. 26, no. 1, pp. 32–41, 2006.

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