Time-Domain Analysis of Control Systems Using MATLAB: Response, Error, and Stability

# MATLAB Control System Time-Domain Analysis: Response, Error, and Stability

## 1. The Role of MATLAB in Control System Analysis

Control systems, as a core component of modern engineering and science, play a crucial role in the performance and stability of systems. MATLAB, as a high-level mathematical computing software, is widely used in the modeling, simulation, analysis, and design of control systems. This chapter will explore the role of MATLAB in control system analysis and delve deeper into its specific applications in time-domain response analysis through subsequent chapters.

MATLAB plays a vital role in control system analysis. First and foremost, MATLAB offers a comprehensive set of toolboxes, especially the Control System Toolbox, which includes a wide range of functions and commands tailored for control system design and analysis, providing engineers and researchers with a powerful platform. For instance, using MATLAB makes it convenient to create system models, solve equations, simulate system responses, and conduct stability analyses.

Engineers need to be familiar with MATLAB's programming environment and the functions within its toolboxes to effectively utilize MATLAB for control system analysis. The scripting and function writing in MATLAB allow users to transform complex mathematical models and algorithms into executable code, which not only improves work efficiency but also makes complex control system analysis more intuitive and understandable. With a deeper understanding of MATLAB's control analysis capabilities, we can further explore the theoretical foundations of control system time-domain responses and how to apply MATLAB in practice.

## 2. Theoretical Foundations of Control System Time-Domain Response

### 2.1 Definition and Characteristics of Linear Time-Invariant Systems

#### 2.1.1 Time-Domain Description of Systems

Linear Time-Invariant Systems (LTI systems) are one of the core concepts in control systems. In the time domain, the output response of an LTI system can be described by the convolution of the system's input signal and the system's impulse response. Specifically, if the impulse response of a system is h(t), then for any input signal x(t), the system output y(t) can be calculated using the following convolution integral:

\[ y(t) = \int_{-\infty}^{\infty} x(\tau) \cdot h(t - \tau) \, d\tau \]

The physical meaning of this integral is that the output of the system is the "weighted" sum of the input signal at all past moments, where the weights are the system's impulse response. Due to the system's time-invariant characteristics, the impulse response depends only on the time interval (t-τ), which simplifies the analysis process.

To represent and manipulate such convolution operations in MATLAB, we can use the built-in function conv() to simulate the convolution process in discrete-time systems, and for continuous-time signals, we typically use symbolic computation or numerical integration methods (such as the trapz() function) to perform the calculations.

#### 2.1.2 Classification of System Responses

The response of an LTI system can be divided into zero-input response and zero-state response. The zero-input response refers to the system response when the initial state is not zero, while the zero-state response refers to the system response caused solely by the input signal when the initial state is zero. The sum of these two constitutes the system's full response.

Additionally, responses can be classified based on the type of input signal, such as impulse response, step response, and sinusoidal response. Each type of response provides important information for understanding the system's behavior under different inputs.

In MATLAB, zero-input and zero-state responses can be solved using functions from the linear algebra toolbox, such as using the linsolve() function to solve linear equations, and for specific types of responses, custom functions can be written to simulate the system behavior under different input signals.

### 2.2 Basic Concepts of Time-Domain Analysis

#### 2.2.1 Impulse Response and Step Response

Impulse response and step response are key indicators for analyzing the time-domain performance of linear time-invariant systems. The impulse response describes the system's reaction to an ideal impulse input, while the step response describes the system's behavior under a step input. These responses are directly related to the system's time-domain characteristics, such as stability and transient behavior.

In MATLAB, we can obtain the time-domain impulse response by transforming the system's transfer function H(s) into the s-domain and applying the inverse Laplace transform. Similarly, the step response can be obtained by multiplying the transfer function by 1/s and then performing the inverse Laplace transform. These transformations can be executed using MATLAB's built-in laplace() and ilaplace() functions.

#### 2.2.2 Fundamentals of Error Analysis

Error analysis is a key step in control system design. It mainly focuses on steady-state error, which is the difference between the system'***mon error coefficients for a unit step input include the position error coefficient (Kp), the velocity error coefficient (Kv), and the acceleration error coefficient (Ka).

In MATLAB, we can use functions from the Control System Toolbox, such as stepinfo(), to calculate these error coefficients. Additionally, we can analyze the steady-state error manually by performing numerical integration over a long time-domain range on the unit step response, finding the steady-state value, and then calculating the difference from the ideal output.

#### 2.2.3 Stability Criteria

System stability is a crucial indicator in control system design. A system is considered stable if its output is finite for any finite input. The stability of a linear time-invariant system can be determined by the location of its poles. According to the Laplace transform, a system is stable if all its poles are located in the left half of the s-plane.

In MATLAB, the pole() function can be used to find the system's poles. In addition to the pole criterion, MATLAB's Control System Toolbox provides stability analysis tools such as the Routh criterion and the Hurwitz criterion, which can help determine whether a system is stable and how to adjust system parameters to ensure stability.

Through the content of the above二级 chapters, we can see that a fundamental understanding of linear time-invariant systems includes system time-domain description, response classification, error analysis, and stability criteria. These contents lay the theoretical foundation for subsequent chapters on MATLAB applications and provide core principles for actual control system analysis and design.

## 3. Applications of MATLAB in Time-Domain Response Analysis

In the field of control systems, time-domain response analysis is an important means of evaluating system performance. Through time-domain analysis, we can observe the system's response to input signals, assess its stability, and dynamic performance. MATLAB, as a powerful engineering computation and simulation platform, offers a wealth of functions and toolboxes, making time-domain response analysis simple and intuitive. This chapter will delve into how to use MATLAB for time-domain response analysis, including solving system responses, time-domain error analysis, and determining system stability.

## 3.1 Solving System Responses Using MATLAB

### 3.1.1 Solving Ordinary Differential Equations

Control systems are often described by ordinary differential equations (ODEs), which reflect the laws governing the variation of the system's internal state variables over time. To solve the