【Advanced Chapter】 Detailed Explanation of MATLAB Zero-Pole Model Related Functions

## [Advanced] Detailed Explanation of MATLAB Pole-Zero Model Related Functions

### 2.1 Mathematical Principles of the Pole-Zero Model

#### 2.1.1 Definition and Properties of Zeros

A zero is the value at which the numerator polynomial of a transfer function equals zero. It represents the initial conditions of oscillations or decay in a system. The number of zeros determines the number of oscillations in the system's response. A negative real part of a zero indicates that the system's response is decaying, while a positive real part indicates that the response is oscillatory.

#### 2.1.2 Definition and Properties of Poles

A pole is the value at which the denominator polynomial of a transfer function equals zero. It represents points of energy storage or release in a system. The number of poles determines the stability of the system's response. A negative real part of a pole means the system is stable, while a positive real part indicates instability.

# 2. Theoretical Foundations of Pole-Zero Model Functions

### 2.1 Mathematical Principles of the Pole-Zero Model

#### 2.1.1 Definition and Properties of Zeros

A zero is a point where the numerator polynomial of the transfer function equals zero, indicating undamped oscillation modes in the system response. Mathematically, zeros are defined as:

z = -a / b

where `a` and `b` are the coefficients of the numerator polynomial of the transfer function.

Properties of zeros include:

- **Amplitude Response:** Zeros cause infinite gain at the zero frequency in amplitude response.
- **Phase Response:** Zeros cause a 180° phase shift in the phase response at the zero frequency.
- **Stability:** When zeros are in the left half-plane (LHP), the system is stable; when in the right half-plane (RHP), the system is unstable.

#### 2.1.2 Definition and Properties of Poles

A pole is a point where the denominator polynomial of the transfer function equals zero, indicating damped oscillation modes in the system response. Mathematically, poles are defined as:

p = -c / d

where `c` and `d` are the coefficients of the denominator polynomial of the transfer function.

Properties of poles include:

- **Amplitude Response:** Poles cause infinite attenuation at the pole frequency in amplitude response.
- **Phase Response:** Poles cause a 180° phase shift in the phase response at the pole frequency.
- **Stability:** When poles are in the LHP, the system is stable; when in the RHP, the system is unstable.

### 2.2 Model Order and System Response

#### 2.2.1 Order and System Stability

The model order refers to the sum of the highest powers of the numerator and denominator polynomials of the transfer function. Order is closely related to system stability:

- **Even Order:** System stability is determined by the number of poles.
- **Odd Order:** System stability is determined by the number of zeros.

#### 2.2.2 Order and System Response Time

Order also affects the system response time:

- **Higher Order:** Longer system response time.
- **Lower Order:** Shorter system response time.

This is because higher-order systems have more oscillation modes, requiring more time to stabilize.

# 3. Practical Application of MATLAB Pole-Zero Model Functions

### 3.1 Extraction and Analysis of Poles and Zeros

#### 3.1.1 Pole-Zero Extraction Functions

MATLAB provides the `zero` and `pole` functions to extract the poles and zeros of a system. These functions accept tran