MATLAB一阶系统二阶系统伯德图和奈氏图

MATLAB可以用系统工具箱中的函数来绘制伯德图和奈氏图。这里简要介绍一下如何使用MATLAB绘制一阶和二阶系统的伯德图和奈氏图。

一阶系统

伯德图：

num = 1;
den = [1 1];
G = tf(num, den);
bode(G);
grid on;

奈氏图：

num = 1;
den = [1 1];
G = tf(num, den);
nyquist(G);
grid on;

二阶系统

伯德图：

num = 1;
den = [1 0.2 1];
G = tf(num, den);
bode(G);
grid on;

奈氏图：

num = 1;
den = [1 0.2 1];
G = tf(num, den);
nyquist(G);
grid on;

需要注意的是，这里的num和den分别是系统的分子和分母多项式系数，可以根据具体的系统进行修改。

相关问题

MATLAB一阶系统二阶系统伯德图和奈氏图通过gui实现

可以使用MATLAB中的GUI开发工具来实现一阶系统和二阶系统的伯德图和奈氏图，下面是一个简单的实现示例：

首先，创建一个GUI窗口，可以使用MATLAB的GUIDE工具进行创建，将窗口分为两个部分，一部分用于输入系统参数，另一部分用于显示伯德图和奈氏图。

在输入部分，可以添加编辑框或下拉菜单等控件，用于用户输入系统参数，如系统的阻尼比、自然频率等。

在显示部分，可以添加绘图区域，用于绘制伯德图和奈氏图。使用MATLAB的绘图函数，如bode、nyquist等，可以方便地绘制伯德图和奈氏图。

下面是一个简单的示例代码：

function varargout = system_gui(varargin)
% SYSTEM_GUI MATLAB code for system_gui.fig
%      SYSTEM_GUI, by itself, creates a new SYSTEM_GUI or raises the existing
%      singleton*.
%
%      H = SYSTEM_GUI returns the handle to a new SYSTEM_GUI or the handle to
%      the existing singleton*.
%
%      SYSTEM_GUI('CALLBACK',hObject,eventData,handles,...) calls the local
%      function named CALLBACK in SYSTEM_GUI.M with the given input arguments.
%
%      SYSTEM_GUI('Property','Value',...) creates a new SYSTEM_GUI or raises the
%      existing singleton*.  Starting from the left, property value pairs are
%      applied to the GUI before system_gui_OpeningFcn gets called.  An
%      unrecognized property name or invalid value makes property application
%      stop.  All inputs are passed to system_gui_OpeningFcn via varargin.
%
%      *See GUI Options on GUIDE's Tools menu.  Choose "GUI allows only one
%      instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES

% Edit the above text to modify the response to help system_gui

% Last Modified by GUIDE v2.5 23-Jun-2021 17:37:59

% Begin initialization code - DO NOT EDIT
gui_Singleton = 1;
gui_State = struct('gui_Name',       mfilename, ...
                   'gui_Singleton',  gui_Singleton, ...
                   'gui_OpeningFcn', @system_gui_OpeningFcn, ...
                   'gui_OutputFcn',  @system_gui_OutputFcn, ...
                   'gui_LayoutFcn',  [] , ...
                   'gui_Callback',   []);
if nargin && ischar(varargin{1})
    gui_State.gui_Callback = str2func(varargin{1});
end

if nargout
    [varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
    gui_mainfcn(gui_State, varargin{:});
end
% End initialization code - DO NOT EDIT


% --- Executes just before system_gui is made visible.
function system_gui_OpeningFcn(hObject, eventdata, handles, varargin)
% This function has no output args, see OutputFcn.
% hObject    handle to figure
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
% varargin   command line arguments to system_gui (see VARARGIN)

% Choose default command line output for system_gui
handles.output = hObject;

% Update handles structure
guidata(hObject, handles);

% UIWAIT makes system_gui wait for user response (see UIRESUME)
% uiwait(handles.figure1);


% --- Outputs from this function are returned to the command line.
function varargout = system_gui_OutputFcn(hObject, eventdata, handles) 
% varargout  cell array for returning output args (see VARARGOUT);
% hObject    handle to figure
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

% Get default command line output from handles structure
varargout{1} = handles.output;


% --- Executes on button press in plot_button.
function plot_button_Callback(hObject, eventdata, handles)
% hObject    handle to plot_button (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

% read input values
zeta = str2double(get(handles.zeta_edit, 'String'));
wn = str2double(get(handles.wn_edit, 'String'));

% calculate system parameters
wd = wn * sqrt(1 - zeta^2);
tau = 1 / (zeta*wn);
sys = tf(wn^2, [1, 2*zeta*wn, wn^2]);

% plot bode diagram
axes(handles.bode_axes);
bode(sys);

% plot nyquist diagram
axes(handles.nyquist_axes);
nyquist(sys);


function zeta_edit_Callback(hObject, eventdata, handles)
% hObject    handle to zeta_edit (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)


% --- Executes during object creation, after setting all properties.
function zeta_edit_CreateFcn(hObject, eventdata, handles)
% hObject    handle to zeta_edit (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    empty - handles not created until after all CreateFcns called

% Hint: edit controls usually have a white background on Windows.
%       See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
    set(hObject,'BackgroundColor','white');
end


function wn_edit_Callback(hObject, eventdata, handles)
% hObject    handle to wn_edit (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)


% --- Executes during object creation, after setting all properties.
function wn_edit_CreateFcn(hObject, eventdata, handles)
% hObject    handle to wn_edit (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    empty - handles not created until after all CreateFcns called

% Hint: edit controls usually have a white background on Windows.
%       See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
    set(hObject,'BackgroundColor','white');
end


% --- Executes on button press in clear_button.
function clear_button_Callback(hObject, eventdata, handles)
% hObject    handle to clear_button (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

% clear plots
axes(handles.bode_axes);
cla;
axes(handles.nyquist_axes);
cla;


% --- Executes on button press in exit_button.
function exit_button_Callback(hObject, eventdata, handles)
% hObject    handle to exit_button (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

% close the GUI window
close(handles.figure1);

这个示例代码可以实现通过GUI界面输入系统参数，绘制一阶系统和二阶系统的伯德图和奈氏图。

一阶系统和二阶系统的伯德图

### 绘制一阶系统和二阶系统的伯德图及其特点

#### 一阶系统的伯德图绘制方法及特性

对于一阶系统，假设传递函数为 \( G(s) = \frac{K}{\tau s + 1} \)，其中 \( K \) 是增益，\( \tau \) 是时间常数。

- **幅频特性**：在低频区（远小于截止频率），斜率为0 dB/decade，在转折频率处发生变化。当频率超过截止频率时，斜率变为 -20 dB/decade。

- **相频特性**：随着频率增加，相角逐渐从0度下降至-90度。过渡发生在大约等于时间常数值的位置附近。

要绘制这样的图形，可以通过MATLAB或其他仿真工具实现：

% 定义参数
K = 1; % 增益
tau = 1; % 时间常数

% 创建传递函数模型
sys_1st_order = tf([K], [tau, 1]);

% 使用bode命令绘制度Bode图
figure;
bode(sys_1st_order);
grid on;
title('One Order System Bode Plot');

#### 二阶系统的伯德图绘制方法及特性

考虑标准形式的二阶系统传递函数 \( H(s)=\frac{\omega_n^2}{s^2+2\xi\omega_ns+\omega_n^2} \)，这里 \( \xi \) 表示阻尼比，\( \omega_n \) 则代表自然振荡频率。

- **幅频特性**：如果存在一对复共轭极点，则会出现峰值现象；该峰的高度取决于阻尼系数大小。对于欠阻尼情况 (\( 0<\xi<1 \))，最大值出现在共振频率上，并且会形成明显的尖锐波形。过阻尼情况下(\( \xi>1 \))，曲线较为平滑无明显峰值。临界阻尼状态下的响应介于两者之间[^3]。

- **相频特性**：相位滞后随频率升高而增大，最终趋于稳定角度(-180°)。特别是接近谐振点的地方，会有较大幅度的变化。

同样地，利用编程环境中的控制理论包能够方便快捷地得到所需图表:

wn = sqrt(K); % 自然频率设定
zeta_values = linspace(0.1, 1, 5); % 不同阻尼比取样范围

for i=1:length(zeta_values)
    zeta = zeta_values(i);
    
    sys_2nd_order = tf([wn*wn],[1, 2*zeta*wn, wn*wn]);
    
    subplot(length(zeta_values)/2, 2, i);
    bode(sys_2nd_order);
    grid on;
    title(['Two Order System with Damping Ratio=', num2str(zeta)]);
end

通过上述代码片段可以在同一窗口内展示多个不同阻尼条件下二阶系统的伯德图对比效果。