Designing a PID Controller in MATLAB: Theory, Code, and Debugging

# 1. Overview of PID Controllers

A PID controller, which stands for Proportional-Integral-Derivative controller, is a feedback loop device widely used in industrial control. It calculates and outputs a corrective measure based on the deviation between the actual output of the system and the desired output to achieve dynamic control of the system. In this chapter, we will initially understand the basic concepts of PID controllers and their importance in control systems.

## 1.1 The Role and Significance of PID Controllers

PID controllers convert deviation signals into control signals through three control actions: proportion, integration, and differentiation, enabling the output of the controlled object to reach the desired value quickly and accurately. Whether it's a simple first-order system or a complex multivariable system, PID controllers stand out for their simple structure, ease of understanding and implementation, making them a core technology in the field of automation control.

## 1.2 The Origin and Development of PID Controllers

Since the invention of PID controllers at the beginning of the 20th century, they have been a cornerstone in automatic control systems. After a century of development, PID control theory has matured in industrial practice. The advent of modern digital technology has made automatic adjustment and optimization of PID parameters possible, greatly enhancing the performance and reliability of control systems.

## 1.3 The Application of PID Controllers in Modern Industry

In modern industry, PID controllers are widely used in various process controls, such as temperature, pressure, flow, speed, and more. They are not only found in traditional manufacturing and processing fields but also penetrate into advanced technology fields such as robotics, aerospace, and biomedicine.

*This chapter introduces the core concepts of PID controllers to the reader, laying the foundation for subsequent chapters that delve into the theoretical and practical applications of PID controllers.*

In the next chapter, we will explore in detail the working principles of PID controllers and their theoretical foundations in the MATLAB environment.

# 2. Theoretical Foundations of PID in MATLAB

## 2.1 The Working Principle of PID Controllers

### 2.1.1 Analysis of Proportional (P), Integral (I), and Derivative (D) Actions

In the field of automatic control, PID controllers are widely used feedback controllers. Their core function is to reduce the steady-state and dynamic errors of the system, thereby improving the system's response speed, stability, and robustness. A PID controller consists of three parts: Proportional (P), Integral (I), and Derivative (D), each of which has a specific impact on the system's output.

- **Proportional (P) Action**: Responsible for the direct proportional relationship between output and error, it can immediately respond to changes in error and provide basic control action. The larger the proportional coefficient (Kp), the more sensitive the system is to errors, but too large a proportional coefficient may cause system oscillations.

- **Integral (I) Action**: Mainly used to eliminate the steady-state error of the system. The integral action continuously accumulates errors and converts the accumulated value into a control signal. Therefore, when there is a persistent error, the integral term gradually increases the control action until the error is eliminated. The integral time constant (Ti) determines the speed of accumulation, with a smaller Ti resulting in a stronger integral action.

- **Derivative (D) Action**: Predicts the trend of error changes and pre-acts on the control system, which helps to reduce overshoot and oscillations in the system. The derivative term is sensitive to the rate of change of error. The larger the derivative coefficient (Kd), the faster the response to changes in error, but it may also cause a significant noise amplification effect.

### 2.1.2 Transfer Function and Time-Domain Analysis of PID Controllers

After understanding the individual components of PID, we can combine them to form the transfer function of the PID controller. In the Laplace transform domain, the transfer function of the PID controller can be represented as:

\[ C(s) = Kp + \frac{Ki}{s} + Kd s \]

Where, \(C(s)\) is the controller output, \(Kp\), \(Ki\), and \(Kd\) are the proportional, integral, and derivative gains, respectively, and \(s\) is the complex variable in the Laplace transform.

From a time-domain analysis perspective, the dynamic performance of a PID controller can be evaluated in three aspects:

1. **Transient Response**: The system's response to initial disturbances. This includes rise time, peak time, and settling time. The integral term can increase the system's response time, while the derivative term helps reduce overshoot and increase response speed.
2. **Steady-State Response**: The system's response after long-term operation. The integral term ensures that the system's steady-state error is zero, while the proportional and derivative terms provide immediate feedback for errors.
3. **Stability**: The ability of the system to maintain its output stability in the absence of external disturbances. The system's stability depends on the configuration of the PID controller parameters; inappropriate parameter settings may lead to instability.

## 2.2 PID Parameter Tuning Theory

### 2.2.1 Classical PID Parameter Adjustment Methods

Adjusting PID controller parameters is a critical step in achieving control objectives. There are several classical methods to adjust PID parameters, mainly divided into:

- **Trial and Error Method**: This is an experimental and error-based method that involves observing system responses and manually adjusting PID parameters until satisfactory results are achieved. Although this method is intuitive and easy to implement, it is inefficient and cannot guarantee finding the optimal solution.

- **Ziegler-Nichols (Z-N) Method**: The Z-N method is a systematic adjustment method that determines PID parameters by observing the system's critical oscillation point. This method is simple and easy to use, but the system must be able to safely enter an oscillatory state, and it may not be accurate for complex systems.

- **Cohen-Coon Method**: This is a parameter adjustment method based on the system's step response, providing a more accurate parameter estimation than the Z-N method. It is suitable for systems with delays, but the calculation process is relatively complex.

### 2.2.2 The Impact of PID Parameters on System Performance

Adjusting PID parameters has a clear impact on system performance, with the detailed effects of each parameter as follows:

- **Proportional Gain (Kp)**:直接影响系统的响应速度和稳定性。较大的比例增益可能会导致系统的快速响应，但也容易引起系统的振荡。因此，Kp的选择需要在快速响应和稳定性之间权衡。

- **Integral Gain (Ki)**: 影响系统消除稳态误差的能力。增大的Ki可以减少稳态误差，但过高的Ki值可能导致系统响应缓慢或振荡。因此，Ki的调整必须确保系统稳定性和合适的响应速度。

- **Derivative Gain (Kd)**: 在系统中引入微分作用可以预测误差的变化趋势，进而提前进行调整。提高微分增益有助于减少系统的超调和振荡，同时加速系统达到稳定状态。然而，过高的Kd值会放大测量噪声，使系统对噪声的敏感度增加。

PID parameter tuning is a multi-objective optimization problem that requires a balance between fast response, accuracy, and stability in the control system. To achieve optimal system performance, advanced adaptive or optimization algorithms may be needed to automatically adjust PID parameters.

# 3. Design and Code Implementation of PID Controllers in MATLAB

## Designing PID Controllers with MATLAB Simulink

### 3.1.1 Simulink Interface and Module Introduction

Simulink is an integrated environment in MATLAB for simulating dynamic systems. It provides a visual interface that allows users to build system models by dragging and dropping different modules. To design a PID controller, first, open Simulink and create a new model.

- **Signal Sources**: Modules used to generate input signals, such as Step and Sine Wave.
- **Continuous Modules**: Include modules for continuous dynamic systems, such as transfer functions and integrators.
- **Discrete Modules**: Such as Discrete Integrator, used for establishing discrete-time systems.
- **Sinks**: Modules used to observe system output signals, such as Scope and To Workspace.

### 3.1.2 Constructing a PID Control Model and Simulation Process

Before designing a PID controller, a thorough understanding of the controlled object is necessary. Once the system's behavior is understood, a basic closed-loop control model can be constructed. Here are the steps for building the model and running simulations:

1. Open Simulink and create a new model.
2. Drag a Step function module from the signal source library into the model as the reference input.
3. Drag a transfer function module from the continuous module library to represent the controlled system (Plant). Configure it with known transfer function parameters (such as numerator and denominator coefficients).
4. Drag a PID controller module from the library into the model and connect it to the controlled system module.
5. Connect the output of the Step function module to the reference input of the PID controller.
6. Connect the output of the PID controller back to the input of the controlled system to form a closed-loop system.
7. Connect an Oscilloscope module to the output of the closed-loop system to observe the system's response.
8. Configure the start and end times of the simulation and then run the simulation.
9. After the simulation, view the system response through the Oscilloscope mo