【Advanced Chapter】Detailed Explanation of MATLAB Transfer Functions

# **Advanced Chapter** Detailed Explanation of MATLAB Transfer Functions

Transfer functions are mathematical functions representing linear time-invariant (LTI) systems, describing the relationship between the system's input and output. Transfer functions are typically represented as follows:

H(s) = Y(s) / X(s)

where:

- H(s) is the transfer function
- Y(s) is the system's output Laplace transform
- X(s) is the system's input Laplace transform
- s is the complex frequency variable

The transfer function can be expressed as the ratio of polynomials, where the numerator polynomial represents the system's zeros, and the denominator polynomial represents the system's poles.

## 2. Theoretical Analysis of Transfer Functions

### 2.1 Definition and Representation of Transfer Functions

A transfer function is a mathematical function that describes the relationship between the input and output of a linear time-invariant (LTI) system. It is defined as the ratio of the system's output Laplace transform to the input Laplace transform:

H(s) = Y(s) / X(s)

where:

- `H(s)` is the transfer function
- `Y(s)` is the Laplace transform of the system's output
- `X(s)` is the Laplace transform of the system's input

The transfer function can be represented as a ratio of polynomials:

H(s) = N(s) / D(s)

where:

- `N(s)` is the numerator polynomial, representing the system's zeros
- `D(s)` is the denominator polynomial, representing the system's poles

### 2.2 Properties and Theorems of Transfer Functions

Transfer functions have the following properties:

- **Linearity:** Transfer functions describe the linear relationship between input and output.
- **Time-invariance:** Transfer functions do not change with time.
- **Causality:** The output of the transfer function depends only on the current and past inputs.

Here are some important theorems concerning transfer functions:

- **Final Value Theorem:** When `t` approaches infinity, the system's output equals the residue at `s = 0` of the transfer function's poles.
- **Initial Value Theorem:** When `t` approaches 0, the system's output equals the residue at `s = 0` of the transfer function's numerator.
- **Analytic Theorem:** Transfer functions are analytic across the entire complex plane, except at poles.

### 2.3 Time-domain and Frequency-domain Analysis of Transfer Functions

Transfer functions can be used to analyze a system's time-domain and frequency-domain characteristics.

**Time-domain Analysis**

A transfer function can be transformed into time-domain response through the inverse Laplace transform. The time-domain response describes the system's response to the input signal.

**Frequency-domain Analysis**

Transfer functions can be analyzed in the frequency domain by calculating their magnitude and phase responses. The frequency-domain response describes the system's response to input signals at various frequencies.

The following code block demonstrates how to use MATLAB to calculate the time-domain and frequency-domain responses of a transfer function:

% Define transfer function
num = [1 2 3];
den = [1 4 5 6];
H = tf(num, den);

% Time-domain analysis
t = 0:0.01:10;
x = sin(2*pi*1*t);
y = lsim(H, x, t);

% Frequency-domain analysis
[mag, phase] = bode(H,