幅频特性曲线分析及使用WPF绘制

1、一阶惯性环节的幅频特性曲线分析及绘制

这里的a和b可以根据系统的不同修改,然后在0-50Hz内以1/10000的分辨率取点，可得对数幅频特性曲线（1/0.5s+1）：

MATLAB脚本实现传递函数（1/0.5s+1）的伯德图绘制：

% 0.001 - 10^1.5  10000个点
w = logspace(-3,2,10000);
num = [0 1];
f = [0.5 1];
sys = tf(num,f);
P = bodeoptions;
% 横坐标为Hz
P.FreqUnits = 'Hz';
bode(sys,w,P)

2、二阶系统的幅频特性曲线分析及绘制

MATLAB脚本实现传递函数（2/(0.01s+1)(0.2s+1)的伯德图绘制：

% 0.001 - 10^1.5  10000个点
w = logspace(-3,2,10000);
num = [0 2];
f1 = [0.01 1];
f2 = [0.2 1];
den = conv(f1,f2);
sys = tf(num,den);
P = bodeoptions;
% 横坐标为Hz
P.FreqUnits = 'Hz';
bode(sys,w,P)

3、一般的系统

4、上位机代码实现

源码及Oxyplot源码下载地址： WPF实现bode图demo源码

4.1 一阶惯性系统

<Page x:Class="WPF_Demo_V2.View.LogChartPage"
      xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
      xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
      xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006" 
      xmlns:d="http://schemas.microsoft.com/expression/blend/2008" 
      xmlns:local="clr-namespace:WPF_Demo_V2.View"
      xmlns:oxyplot="http://oxyplot.org/wpf"
      mc:Ignorable="d" 
      d:DesignHeight="450" d:DesignWidth="800"
      Title="LogChartPage">

    <Grid>
        <Grid.RowDefinitions>
            <RowDefinition/>
            <RowDefinition/>
        </Grid.RowDefinitions>
        <Grid.ColumnDefinitions>
            <ColumnDefinition Width="5*"/>
            <ColumnDefinition/>
        </Grid.ColumnDefinitions>
        <oxyplot:PlotView Model="{Binding MyPlotModelUp}"/>
        <oxyplot:PlotView Grid.Row="1"  Model="{Binding MyPlotModelDown}"/>
        <StackPanel Grid.Row="0" Grid.Column="1" Orientation="Vertical">
            <Grid>
                <Grid.ColumnDefinitions>
                    <ColumnDefinition Width="0.8*"/>
                    <ColumnDefinition/>
                </Grid.ColumnDefinitions>
                <Grid.RowDefinitions>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                </Grid.RowDefinitions>
                <TextBlock Text="a:" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Column="1" Text="{Binding A}" Margin="2"></TextBox>
                <TextBlock Text="b:" Grid.Row="1" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="1" Grid.Column="1" Text="{Binding B}" Margin="2"></TextBox>
                <TextBlock Text="X max:" Grid.Row="2" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="2" Grid.Column="1" Text="{Binding X_max}" Margin="2"></TextBox>
                <TextBlock Text="factor:" Grid.Row="3" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="3" Grid.Column="1" Text="{Binding Factor}" Margin="2" ToolTip="X_max/factor 为最小分辨率"></TextBox>
                <TextBlock Text="Y1 min:" Grid.Row="4" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="4" Grid.Column="1" Text="{Binding Y1_min}" Margin="2"></TextBox>
                <TextBlock Text="Y1 max:" Grid.Row="5" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="5" Grid.Column="1" Text="{Binding Y1_max}" Margin="2"></TextBox>
                <TextBlock Text="Y2 min:" Grid.Row="6" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="6" Grid.Column="1" Text="{Binding Y2_min}" Margin="2"></TextBox>
                <TextBlock Text="Y2 max:" Grid.Row="7" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="7" Grid.Column="1" Text="{Binding Y2_max}" Margin="2"></TextBox>
                <Button Grid.Row="8" Grid.ColumnSpan="2" Content="确定" Margin="20,2" Command="{Binding sureCommand}"/>
            </Grid>
            <Image Source="../Resource/Image/function.png"/>
        </StackPanel>
        
    </Grid>
</Page>

using CommunityToolkit.Mvvm.ComponentModel;
using CommunityToolkit.Mvvm.Input;
using OxyPlot;
using OxyPlot.Axes;
using OxyPlot.Series;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace WPF_Demo_V2.ViewModel
{
    partial class LogChartViewModel: ObservableObject
    {
        #region 全局变量
        //UInt32 LEN = 50;
        //UInt32 FACT = 10000;
        double[] Gain;
        double[] Phase;
        FunctionSeries upFunctionSeries=new FunctionSeries();
        FunctionSeries downFunctionSeries=new FunctionSeries();
        #endregion

        #region 构造函数
        public LogChartViewModel()
        {
            upFunctionSeries.Title = "Gain";
            downFunctionSeries.Title = "Phase";
        }
        #endregion

        #region 属性
        [ObservableProperty]
        public PlotModel _MyPlotModelUp;

        [ObservableProperty]
        public PlotModel _MyPlotModelDown;

        [ObservableProperty]
        public double _A = 0.5;

        [ObservableProperty]
        public double _B = 1;

        [ObservableProperty]
        public int _X_max = 50;

        [ObservableProperty]
        public int _Factor = 10000;

        [ObservableProperty]
        public double _Y1_max = 0;

        [ObservableProperty]
        public double _Y1_min = -50;

        [ObservableProperty]
        public double _Y2_max = 0;

        [ObservableProperty]
        public double _Y2_min = -90;
        #endregion

        #region 方法
        public PlotModel PlotModelInit(string xtitle,string ytitle,double ymin,double ymax)
        {
            var plotModel = new PlotModel();
            //plotModel.Title = "Log Paper";
            var logarithmicAxis1 = new LogarithmicAxis();
            logarithmicAxis1.MajorGridlineColor = OxyColor.FromArgb(40, 0, 0, 139);
            logarithmicAxis1.MajorGridlineStyle = LineStyle.Solid;
            logarithmicAxis1.Maximum = X_max;
            logarithmicAxis1.Minimum = 1/ (double)Factor;
            logarithmicAxis1.MinorGridlineColor = OxyColor.FromArgb(20, 0, 0, 139);
            logarithmicAxis1.MinorGridlineStyle = LineStyle.Solid;
            logarithmicAxis1.Position = AxisPosition.Bottom;
            if (!string.IsNullOrEmpty(xtitle)) logarithmicAxis1.Title = xtitle;
            plotModel.Axes.Add(logarithmicAxis1);
            var linearAxis1 = new LinearAxis();
            linearAxis1.MajorGridlineStyle = LineStyle.Solid;
            linearAxis1.MinorGridlineStyle = LineStyle.Dot;
            linearAxis1.Maximum = ymax;
            linearAxis1.Minimum = ymin;
            linearAxis1.Title = ytitle;
            plotModel.Axes.Add(linearAxis1);
            return plotModel;
        }

        [RelayCommand]
        private void sure()
        {
            if(upFunctionSeries.Points.Count>0) upFunctionSeries.Points.Clear();
            if (downFunctionSeries.Points.Count > 0)downFunctionSeries.Points.Clear();
            if (MyPlotModelUp != null) 
            {
                if (MyPlotModelUp.Series.Count > 0) MyPlotModelUp.Series.Clear();
            }
            if(MyPlotModelDown != null)
            {
                if (MyPlotModelDown.Series.Count > 0) MyPlotModelDown.Series.Clear();
            }
            
            MyPlotModelUp = PlotModelInit("", "Magnitude[dB]", Y1_min, Y1_max);
            MyPlotModelDown = PlotModelInit("Frequency[Hz]", "Phase[deg]", Y2_min, Y2_max);

            Gain = new double[X_max * Factor];
            Phase = new double[X_max * Factor];


            for (int i = 0; i < X_max * Factor; i++)
            {
                double temp_a = 2 * Math.PI * A * (i / (double)Factor);
                double temp_2 = Math.Pow(temp_a, 2) + Math.Pow(B, 2);
                double temp_sqrt = 1 / Math.Sqrt(temp_2);
                double temp = Math.Abs(temp_sqrt);
                Gain[i] = 20 * Math.Log10(temp);
                Phase[i] = -Math.Atan(temp_a / (double)B) / Math.PI * 180;

                upFunctionSeries.Points.Add(new DataPoint(i / (double)Factor, Gain[i]));
                downFunctionSeries.Points.Add(new DataPoint(i / (double)Factor, Phase[i]));
            }
            MyPlotModelUp.Series.Add(upFunctionSeries);
            MyPlotModelDown.Series.Add(downFunctionSeries);
            MyPlotModelUp.ResetAllAxes();
            MyPlotModelUp.InvalidatePlot(true);
            MyPlotModelDown.ResetAllAxes();
            MyPlotModelDown.InvalidatePlot(true);
        }
        #endregion
    }
}

4.2 二阶系统

<Page x:Class="WPF_Demo_V2.View.Log3ChartPage"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006" 
            xmlns:d="http://schemas.microsoft.com/expression/blend/2008" 
                xmlns:local="clr-namespace:WPF_Demo_V2.View"
                    xmlns:oxyplot="http://oxyplot.org/wpf"
                        mc:Ignorable="d" 
                            d:DesignHeight="450" d:DesignWidth="800"
                                Title="Log3ChartPage">

    <Grid>
        <Grid.RowDefinitions>
            <RowDefinition/>
            <RowDefinition/>
        </Grid.RowDefinitions>
        <Grid.ColumnDefinitions>
            <ColumnDefinition Width="5*"/>
            <ColumnDefinition/>
        </Grid.ColumnDefinitions>
        <oxyplot:PlotView Model="{Binding MyPlotModelUp}"/>
        <oxyplot:PlotView Grid.Row="1"  Model="{Binding MyPlotModelDown}"/>
        <StackPanel Grid.Row="0" Grid.Column="1" Orientation="Vertical">
            <Grid>
                <Grid.ColumnDefinitions>
                    <ColumnDefinition Width="0.8*"/>
                    <ColumnDefinition/>
                </Grid.ColumnDefinitions>
                <Grid.RowDefinitions>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                    <RowDefinition/>
                </Grid.RowDefinitions>
                <TextBlock Text="a:" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Column="1" Text="{Binding A}" Margin="2"></TextBox>
                <TextBlock Text="b:" Grid.Row="1" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="1" Grid.Column="1" Text="{Binding B}" Margin="2"></TextBox>
                <TextBlock Text="c:" Grid.Row="2" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="2" Grid.Column="1" Text="{Binding C}" Margin="2"></TextBox>
                
            <TextBlock Text="X max:" Grid.Row="3" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="3" Grid.Column="1" Text="{Binding X_max}" Margin="2"></TextBox>
                <TextBlock Text="factor:" Grid.Row="4" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="4" Grid.Column="1" Text="{Binding Factor}" Margin="2" ToolTip="X_max/factor 为最小分辨率"></TextBox>
                <TextBlock Text="Y1 min:" Grid.Row="5" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="5" Grid.Column="1" Text="{Binding Y1_min}" Margin="2"></TextBox>
                <TextBlock Text="Y1 max:" Grid.Row="6" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="6" Grid.Column="1" Text="{Binding Y1_max}" Margin="2"></TextBox>
                <TextBlock Text="Y2 min:" Grid.Row="7" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="7" Grid.Column="1" Text="{Binding Y2_min}" Margin="2"></TextBox>
                <TextBlock Text="Y2 max:" Grid.Row="8" Foreground="White" HorizontalAlignment="Right"/>
                <TextBox Grid.Row="8" Grid.Column="1" Text="{Binding Y2_max}" Margin="2"></TextBox>
                <Button Grid.Row="9" Grid.ColumnSpan="2" Content="确定" Margin="20,2" Command="{Binding sureCommand}"/>
            </Grid>
            <Image Source="../Resource/Image/function.png"/>
        </StackPanel>

    </Grid>
</Page>

using CommunityToolkit.Mvvm.ComponentModel;
using CommunityToolkit.Mvvm.Input;
using OxyPlot.Axes;
using OxyPlot.Series;
using OxyPlot;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace WPF_Demo_V2.ViewModel
{
    partial class Log3ChartViewModel: ObservableObject
    {
        #region 全局变量
            //UInt32 LEN = 50;
            //UInt32 FACT = 10000;
            double[] Gain;
        double[] Phase;
        FunctionSeries upFunctionSeries = new FunctionSeries();
        FunctionSeries downFunctionSeries = new FunctionSeries();
        #endregion

            #region 构造函数
            public Log3ChartViewModel()
        {
            upFunctionSeries.Title = "Gain";
            downFunctionSeries.Title = "Phase";
        }
        #endregion

            #region 属性
            [ObservableProperty]
            public PlotModel _MyPlotModelUp;

        [ObservableProperty]
        public PlotModel _MyPlotModelDown;

        [ObservableProperty]
        public double _A = 0.5;

        [ObservableProperty]
        public double _B = 1;

        [ObservableProperty]
        public double _C = 1;

        [ObservableProperty]
        public int _X_max = 50;

        [ObservableProperty]
        public int _Factor = 10000;

        [ObservableProperty]
        public double _Y1_max = 0;

        [ObservableProperty]
        public double _Y1_min = -100;

        [ObservableProperty]
        public double _Y2_max = 100;

        [ObservableProperty]
        public double _Y2_min = -100;
        #endregion

            #region 方法
            public PlotModel PlotModelInit(string xtitle, string ytitle, double ymin, double ymax)
        {
            var plotModel = new PlotModel();
            //plotModel.Title = "Log Paper";
            var logarithmicAxis1 = new LogarithmicAxis();
            logarithmicAxis1.MajorGridlineColor = OxyColor.FromArgb(40, 0, 0, 139);
            logarithmicAxis1.MajorGridlineStyle = LineStyle.Solid;
            logarithmicAxis1.Maximum = X_max;
            logarithmicAxis1.Minimum = 1 / (double)Factor;
            logarithmicAxis1.MinorGridlineColor = OxyColor.FromArgb(20, 0, 0, 139);
            logarithmicAxis1.MinorGridlineStyle = LineStyle.Solid;
            logarithmicAxis1.Position = AxisPosition.Bottom;
            if (!string.IsNullOrEmpty(xtitle)) logarithmicAxis1.Title = xtitle;
            plotModel.Axes.Add(logarithmicAxis1);
            var linearAxis1 = new LinearAxis();
            linearAxis1.MajorGridlineStyle = LineStyle.Solid;
            linearAxis1.MinorGridlineStyle = LineStyle.Dot;
            linearAxis1.Maximum = ymax;
            linearAxis1.Minimum = ymin;
            linearAxis1.Title = ytitle;
            plotModel.Axes.Add(linearAxis1);
            //var logarithmicAxis2 = new LogarithmicAxis();
            //logarithmicAxis2.MajorGridlineColor = OxyColor.FromArgb(40, 0, 0, 139);
            //logarithmicAxis2.MajorGridlineStyle = LineStyle.Solid;
            //logarithmicAxis2.Maximum = 180;
            //logarithmicAxis2.Minimum = -180;
            //logarithmicAxis2.MinorGridlineColor = OxyColor.FromArgb(20, 0, 0, 139);
            //logarithmicAxis2.MinorGridlineStyle = LineStyle.Solid;
            //logarithmicAxis2.Title = "Y";
            //plotModel.Axes.Add(logarithmicAxis2);
            return plotModel;
        }

        [RelayCommand]
        private void sure()
        {
            if (upFunctionSeries.Points.Count > 0) upFunctionSeries.Points.Clear();
            if (downFunctionSeries.Points.Count > 0) downFunctionSeries.Points.Clear();
            if (MyPlotModelUp != null)
            {
                if (MyPlotModelUp.Series.Count > 0) MyPlotModelUp.Series.Clear();
            }
            if (MyPlotModelDown != null)
            {
                if (MyPlotModelDown.Series.Count > 0) MyPlotModelDown.Series.Clear();
            }

            MyPlotModelUp = PlotModelInit("", "Magnitude[dB]", Y1_min, Y1_max);
            MyPlotModelDown = PlotModelInit("Frequency[Hz]", "Phase[deg]", Y2_min, Y2_max);

            Gain = new double[X_max * Factor];
            Phase = new double[X_max * Factor];

            double previousPhase = 0;

            for (int i = 0; i < X_max * Factor; i++)
            {
                double w = 2 * Math.PI * (i / (double)Factor);
                double temp_a = 1 - A*B * Math.Pow(w, 2);
                double temp_b = (A+B)*w;
                double temp_2 = Math.Pow(temp_a, 2) + Math.Pow(temp_b, 2);
                double temp_sqrt = C / Math.Sqrt(temp_2);
                double temp = Math.Abs(temp_sqrt);
                Gain[i] = 20 * Math.Log10(temp);

                double currentPhaseA = Math.Atan(A * w) / Math.PI * 180;
                double currentPhaseB = Math.Atan(B * w) / Math.PI * 180;

                Phase[i] = 0-(currentPhaseA + currentPhaseB);

                upFunctionSeries.Points.Add(new DataPoint(i / (double)Factor, Gain[i]));
                downFunctionSeries.Points.Add(new DataPoint(i / (double)Factor, Phase[i]));
            }
            MyPlotModelUp.Series.Add(upFunctionSeries);
            MyPlotModelDown.Series.Add(downFunctionSeries);
            MyPlotModelUp.ResetAllAxes();
            MyPlotModelUp.InvalidatePlot(true);
            MyPlotModelDown.ResetAllAxes();
            MyPlotModelDown.InvalidatePlot(true);
        }
        #endregion
        }
}

5、稳定裕度

5.1 幅值裕度

5.2 相角裕度

参考

【1】第五章 线性系统的频域分析法：
https://buckyi.github.io/Note-Automation/%E7%BB%8F%E5%85%B8%E6%8E%A7%E5%88%B6%E7%90%86%E8%AE%BA/%E7%AC%AC05%E7%AB%A0%20%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F%E7%9A%84%E9%A2%91%E5%9F%9F%E5%88%86%E6%9E%90%E6%B3%95.html#33
【2】【自动控制原理】第5章 幅相频率特性曲线（奈氏图）及其绘制：
https://blog.csdn.net/persona5joker/article/details/140055111
【3】相角裕度与幅值裕度：
https://blog.csdn.net/qq_38972634/article/details/119787996
【4】【自动控制原理】第五章 奈奎斯特稳定判据，相角裕度和幅值裕度，对数频率特性及其绘制：
https://blog.csdn.net/persona5joker/article/details/140059710?utm_medium=distribute.pc_relevant.none-task-blog-2_defaultbaidujs_baidulandingword~default-4-140059710-blog-119787996.235^v43pc_blog_bottom_relevance_base8&spm=1001.2101.3001.4242.3&utm_relevant_index=7
【5】自控理论 第6章 II 相对稳定性、伯德图和闭环频率响应
https://www.cnblogs.com/harold-lu/p/15740368.html