Chapter7.2: Application of MATLAB in frequency method and stability analysis of frequency method

This series of blogs mainly describes the application of Matlab software in automatic control. If you have no theoretical basis for automatic control, please learn the series of blog posts on automatic control first. This series of blogs will not explain the theoretical knowledge of automatic control in detail.
Links related to the theoretical basis of automatic control: https://blog.csdn.net/qq_39032096/category_10287468.html?spm=1001.2014.3001.5482
Blog reference books: "MATLAB/Simulink and Control System Simulation".

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2. Application of MATLAB in frequency method and stability analysis of frequency method

2.1 Application of MATLAB in frequency method

2.1.1 Calculating and drawing functions related to frequency response curve

1. $Nyquist$ curve drawing function$\mathrm{nyquist}()$。
   • $\mathrm{nyquist}(\mathrm{a},\mathrm{b},\mathrm{c},d)$ : Draw a set of$Nyquist$ curves, each of which corresponds to a continuous state-space system$[\mathrm{a},\mathrm{b},\mathrm{c},d]$ input/output combination pair, in which the frequency range is automatically selected by the function, and more sampling points will be automatically adopted at positions where the response changes rapidly;
   • $\mathrm{nyquist}(\mathrm{a},b,c,\mathrm{d},iu)$ : draw from system iu{\rm iu}Polar plot of$iu\; inputs\; to\; all\; outputs;$
   • $\mathrm{nyquist}(\mathrm{num},den)$ : Draw a polar plot of the system represented by a continuous-time polynomial transfer function;
   • $nyquist(a,b,c,d,\mathrm{iu},\mathrm{w})$或$nyquist(num,the,w)$ : Draw the polar coordinate diagram of the system using the specified angular frequency vector;
2. $Bode$ graph drawing function$\mathrm{bode}()$。
   • $\mathrm{bode}(\mathrm{a},\mathrm{b},\mathrm{c},d)$ : Draw a set of$Bode$ diagram is for continuous state space system$[\mathrm{a},b,c,d]$ for each input$Bode$ diagram, where the frequency range is automatically selected by the function, and more sampling points are automatically used in the position where the response changes rapidly;
   • $bode(a,b,c,\mathrm{d},iu)$ : draw from system iu$iu$ input to all output$Bode$ diagram;
   • $\mathrm{bode}(\mathrm{num},den)$ : plot the system Bode {\rm Bode}in terms of continuous-time polynomial transfer function$Bode$ diagram;
   • $\mathrm{bode}(\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{iu},w)$或$bode(num,the,w)$ : Use the specified angular frequency vector to draw$Bode$ diagram;
3. $Nichols$ curve drawing function$\mathrm{nichols}()$。
   • $\mathrm{nichols}(\mathrm{a},\mathrm{b},\mathrm{c},d)$ : Draw a set of$Nichols$ diagram, for continuous state space system$[\mathrm{a},b,c,d]$ for each input$Nichols$ plots, where the frequency range is automatically selected by the function, and more sampling points are automatically taken at positions where the response changes rapidly;
   • $nichols(a,b,c,\mathrm{d},iu)$ : draw from system iu$iu$ input to N ichols {\rm Nichols}with output$Nichols$ diagram;
   • $\mathrm{nichols}(\mathrm{num},den)$ : Plot the system Nichols {\rm Nichols}in terms of continuous-time polynomial transfer function$Nichols$ diagram;
   • $\mathrm{nichols}(\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{iu},w)$或$nichols(num,the,w)$ : Use the specified angular frequency vector to plot$Nichols$ diagram;
4. plotting etc. $M$ circles and equal$N$ circle function$\mathrm{ngrid}()$。

2.1.2 Actual Combat 1

Experimental requirements: The known first-order link transfer function is: $G(s)=\frac{5}{3s+1}$, draw $Nyquist$ diagram.

untie:

% 实例Chapter7.2.1.2
clc;clear;

% 建立传递函数模型
num=5;den=[3,1];
G=tf(num,den);

% 绘制图形
nyquist(G);grid;
axis([-1,8,-2.5,2.5]);
set(findobj(get(gca,'Children'),'LineWidth',0.5),'LineWidth',1.5);
title('控制系统Nyquist图','FontSize',15);

2.1.3 Actual Combat 2

Experimental requirements: The known second-order link transfer function is: $G\left(s\right)=\frac{{oh}_n^2}{s^2+2z{o}_{n}s+{oh}_n^2}$, in which: ${oh}_n=0.7$ , respectively draw$g=$B ode {\rm Bode}at$0.1$$,$$0.4$$,$$1.0$$,$$1.6$$,$$2.0$$Bode$ diagram.

untie:

% 实例Chapter7.2.1.3
clc;clear;

% 参数定义
w=[0,logspace(-2,2,200)];
wn=0.7;zeta=[0.1,0.4,1.0,1.6,2.0];

% 建立传递函数模型
for j=1:5
    sys=tf([wn*wn],[1,2*zeta(j)*wn,wn*wn]);
    bode(sys);hold on;
end

set(findobj(get(gca,'Children'),'LineWidth',0.5),'LineWidth',1.5);
legend('ζ=0.1','ζ=0.4','ζ=1.0','ζ=1.6','ζ=2.0');
title('控制系统不同ζ下的Bode图','FontSize',15);

2.1.4 Actual Combat 3

Experimental requirements: The transfer function of a known high-order system is: $G(s)=\frac{0.0001s^3+0.028s^2+1.06356s+9.6}{0.0006s^3+0.0286s^2+0.06356s+6}$, draw $Nichols$ diagram.

untie:

% 实例Chapter7.2.1.4
clc;clear;

% 建立模型
num=[0.0001,0.0281,1.06356,9.6];
den=[0.0006,0.0286,0.06356,6];
G=tf(num,den);

% 绘制等M圆和等N圆，Nichols图
ngrid('new');
nichols(G);

axis([-360,0,-20,40]);
set(findobj(get(gca,'Children'),'LineWidth',0.5),'LineWidth',1.5);
title('等M圆、等N圆及Nichols图','FontSize',15);

2.2 Stability analysis of the frequency method

2.2.1 Logarithmic frequency stability criterion

Complex plane $C_{GH}$The curve is generally composed of two parts: the open-loop amplitude-phase curve and the imaginary arc with an infinite radius when the open-loop system has an integral link and an equal-amplitude oscillation link;

Γ GH \Gamma_{GH} in semi-logarithmic coordinates$C_{GH}$The curve determines the number of crossings $N$ or$N_{+}$and $N_{-}$； N N The determination of$N$$A\left(\omega \right)>1$ o'clock$C_{GH}$the number of crossings of the negative real axis;

1. crossing point determination

   设$oh={oh}_c$For example,
   $$\{\begin{array}{ll}& A({oh}_c)=|G(j{\_}_c)H(\mathrm{j}{\omega }_c)|=1\\ & \\ & L\left(o_c\right)=20\lg A\left({oh}_c\right)=0\end{array}$$
   where: ${oh}_c$called the cutoff frequency;

   For the negative real axis and open-loop logarithmic phase-frequency characteristics of the complex plane, when the frequency is taken as the crossing frequency ${oh}_x$时，
   $$f\left(o_x\right)=(2k+1)p,k=0，\pm 1，\pm 2，\dots$$ Set Γ GH \Gamma_{GH}
   in semi-logarithmic coordinates$C_{GH}$The logarithmic amplitude-frequency characteristic curve and logarithmic phase-frequency characteristic curve are respectively: $C_L$和$C_{}$, since $C_L$is equal to $L\left(\omega \right)$ curve, then$C_{GH}$At $A\left(\omega \right)>When\; 1$ , the point crossing the negative real axis is equal to$C_{GH}$In semi-logarithmic coordinates, the logarithmic amplitude-frequency characteristic $L\left(\omega \right)>$The logarithmic phase-frequency characteristic curve at$0$$C_{}$与$(2k+1)p(k=0,\pm 1,\pm 2,\dots )$ , the intersection of parallel lines;

2. $C_{}$Sure

   • When the open-loop system has no poles on the imaginary axis, $C_{}$is equal to $\phi \left(\omega \right)$ curve;
   • The open-loop system has an integral element $\frac{1}{s^n}(n>0)$ ,$C_{GH}$Curve, need from $oh=0_{+}$The corresponding point of the open-loop amplitude-phase characteristic curve $G(\mathrm{j}{0}_{+})H(\mathrm{j}{0}_{+})$ , complement it counterclockwise with$n\times 90°$ imaginary circular arc with infinite radius; it needs to be obtained from the logarithmic phase-frequency characteristic curve$\omega$ is small and$L\left(\omega \right)>$Complementν × 90 °$upward\; at\; the\; point\; of\; 0\; \backslash nu\backslash times90°$$n\times 90°$ imaginary straight line,$The\; \phi \left(\omega \right)$ curve and the supplementary imaginary straight line constitute$C_{}$；
   • The open-loop system has a constant amplitude oscillation link $\frac{1}{(s^2+{oh}_n^2{)}^{n_1}}(n_1>0)$ ,$C_{GH}$Curve, need to start from $oh={oh}_{n^{-}}$The corresponding point $G(\mathrm{j}{\omega }_{n^{-}})H(\mathrm{j}{\omega }_{n^{-}})$ , make clockwise$n_1\times 180°$ imaginary arc with infinite radius to$oh={oh}_{n^{+}}$The corresponding point $G\left(j{\omega }_{n^{+}}\right)H\left(j{\omega }_{n^{+}}\right)$ ; Correspondingly, from the logarithmic phase-frequency characteristic curve$f\left(o_{n^{-}}\right)$ , make up$n_1\times 180°$ imaginary straight line to$f\left(o_{n^{+}}\right)$处，$The\; \phi \left(\omega \right)$ curve and the supplementary imaginary straight line constitute$C_{}$；

3. Calculation of crossing times

   1. Crossing once: $C_{GH}$Cross ( − 1 , j 0 ) from top to bottom $(-1,The\; negative\; real\; axis\; on\; the\; left\; side\; of\; point\; j0)$is equivalent to$L\left(\omega \right)>0$时，$C_{}$Cross $(2k+1)\pi$ line once;
   2. Negative traverse once: $C_{GH}$Cross ( − 1 , j 0 ) from bottom to top $(-1,The\; negative\; real\; axis\; on\; the\; left\; side\; of\; point\; j0)$is equivalent to$L\left(\omega \right)>0$时，$C_{}$Cross $(2k+1)\pi$ line once;
   3. Positive crossing half time: $C_{GH}$End from top to bottom or start from top to bottom at $(-1,The\; negative\; real\; axis\; on\; the\; left\; side\; of\; point\; j0)$, is equivalent to$L\left(\omega \right)>0$时，$C_{}$End at or start at $(2k+1)thread$；
   4. Negative crossing half times: $C_{GH}$End at or start at $(-1,The\; negative\; real\; axis\; on\; the\; left\; side\; of\; point\; j0)$, is equivalent to$L\left(\omega \right)>0$时，$C_{}$End from top to bottom or start from top to bottom at $(2k+1)thread$；
   5. The crossings generated by the supplementary imaginary straight lines are all negative crossings;

4. log frequency stability criterion

   Let $P$ is the number of poles in the positive real part of the open-loop system. The necessary and sufficient condition for the stability of the feedback control system is that$f(o_c)\ne (2k+1)p(k=0,1,2,\dots )$ and$L\left(\omega \right)>0$时，$C_{}$The curve crosses $(2k+1)\; The\; degree\; of\pi$ line:$N=N_{+}-N_{-}$Satisfy: $Z=P-2N\_=0$；

5. Nyquist Stability Criterion

   Nestle's criterion: The necessary and sufficient condition for the stability of the feedback control system is the semi-closed curve $C_{GH}$Does not pass through $(-1,j0)$, and encloses the critical point counterclockwise$(-1,j0)$the number of circles$R$ is equal to the number of poles PPin the positive real part of the open-loop transfer function$P$；

2.2.2 Stability margin

1. phase margin.

   Let ${oh}_c$is the cutoff frequency of the system, then:
   $$A({oh}_c)=|G(j{\_}_c)H(\mathrm{j}{\omega }_c)|=1$$
   Define the phase angle margin:
   $$c=180°+\angle [G(\mathrm{j}{\omega }_c)H(\mathrm{j}{\omega }_c)]$$
   phase angle margin$\gamma meaning$ : For a closed-loop stable system, if the open-loop phase-frequency characteristic of the system lags$\gamma$ degree, the system will be in a critical stable state;

2. Amplitude margin.

   Let ${oh}_x$is the crossing frequency of the system, then the system is at ${oh}_x$工相角：
   $$f(o_x)=\angle \left[G\right(j{\omega }_x\left)H\right(j{\omega }_x)]=(2k+1)p;k=0,\pm 1,...$$
   Define the magnitude margin as:
   $$h=\frac{1}{|G(j{\_}_x)H(\mathrm{j}{\omega }_x)|}$$
   Amplitude margin $The\; meaning\; of\; h$ : For a closed-loop stable system, if the open-loop amplitude-frequency characteristics of the system increase$h$ times, the system will be in a critical stable state;

3. Notes on phase angle margin and amplitude margin.

   • The phase angle margin and amplitude margin of the control system are the polar coordinate diagram pair of the system $(-1+j0)$is a measure of the proximity of points, and these two margins can be used as design criteria;
   • It is not enough to explain the relative stability of the system only by amplitude margin or phase angle margin; in order to determine the relative stability of the system, these two quantities must be given at the same time;
   • For the minimum phase system, the system is stable only when both the phase angle margin and the amplitude margin are positive, and a negative margin indicates that the system is unstable;
   • Appropriate phase angle margin and amplitude margin can prevent the influence of component aging in the system, and the frequency value is specified; in order to obtain satisfactory performance, the phase angle margin should be 30 ° ~ 60 ° 30 ° ~ 60 $30°～60°$ , the amplitude margin should be greater than$6dB$；
   • Requires phase angle margin at $30°$ and$60°$ , that is, in the Bode diagram, the slope of the logarithmic magnitude curve at the cutoff frequency should be greater than$-40dB/dec$ ; in most practical cases, in order to ensure the stability of the system, the slope at the cutoff frequency is required to be$-20dB/dec$ ; if the slope at the cutoff frequency is$-40dB/dec$ , the system may be stable or unstable; if the slope at the cutoff frequency is$-60dB/dec$ , or steeper, the system is likely to be unstable;

2.3 Application of MATLAB in stability analysis

2.3.1 The function margin() to calculate the system stability margin

• $\mathrm{margin}(\mathrm{mag},\mathrm{phase},w)$ : ReasonB ode {\rm Bode}The amplitude mag {\rm mag}obtained by the$Bode\; command$$\mathrm{mag}$、相角$phase$ and angular frequency$w$ vector plotting Bode {\rm Bode}with margin and corresponding frequency display$Bode$ diagram;
• $\mathrm{margin}(\mathrm{num},den)$ : Compute the magnitude margin and phase angle margin represented by the transfer function of the continuous system, and plot the corresponding$Bode$ diagram;
• $\mathrm{margin}(\mathrm{a},\mathrm{b},\mathrm{c},d)$ : Calculate the magnitude margin and phase angle margin represented by the continuous state space system, and plot the corresponding$Bode$ diagram;
• $\mathrm{gm},\mathrm{pm},wcg,wcp=\mathrm{margin}(\mathrm{mag},\mathrm{phase},w)$ : by the magnitude$\mathrm{mag}$、相角$phase$ and angular frequency$w$ vector, calculate the system amplitude margin, phase angle margin and the corresponding phase angle crossover frequency$wcg$ , cutoff frequency$wcp$ , does not directly draw$Bode$ diagram;
• $\mathrm{s}=allmargin\left(sys\right)$ : Calculate the amplitude margin, phase angle margin and corresponding frequency;

2.3.2 Actual Combat 1

Experimental requirements: It is known that the open-loop transfer function of a high-order system is: $G(s)=\frac{5(0.0167s+1)}{s(0.03s+1)(0.0025s+1)(0.001s+1)}$, calculate the phase angle margin and amplitude margin of the system, and draw $Bode$ diagram.

untie:

% 实例Chapter7.2.3.2
clc;clear;

% 建立模型
num=5*[0.0167,1];
den=conv(conv([1,0],[0.03,1]),conv([0.0025,1],[0.001,1]));
G=tf(num,den);

% 频率范围
w=logspace(0,4,50);

% 绘制Bode图
bode(G,w);grid;
set(findobj(get(gca,'Children'),'LineWidth',0.5),'LineWidth',1.5);
title('控制系统Bode图','FontSize',15);

% 计算幅值裕度和相角裕度
[Gm,Pm,Wcg,Wcp]=margin(G)

% 幅值裕度和相角裕度结果
Gm =
  455.2548

Pm =
   85.2751

Wcg =
  602.4232

Wcp =
    4.9620

• Amplitude margin: $K_g=455.2548$ , phase angle margin:$c=85.2751°$ , crossover frequency:${oh}_g=602.4232rad/s$ , cutoff frequency:${oh}_c=4.9620rad/s$。

2.3.3 Actual Combat 2

Experimental requirements: It is known that the open-loop transfer function of a high-order system is: $G(s)=\frac{K(0.0167s+1)}{s(0.03s+1)(0.0025s+1)(0.001s+1)}$, calculate when the open-loop gain $K=5,500,800,3000$ , the change of system stability margin

untie:

% 实例Chapter7.2.3.3
clc;clear;

K=[5,500,800,3000];
den=conv(conv([1,0],[0.03,1]),conv([0.0025,1],[0.001,1]));

for j=1:4
    num=K(j)*[0.0167,1];
    G=tf(num,den);
    y(j)=allmargin(G);               % 计算幅值裕度、相角裕度、频率
end

y(1),y(2),y(3),y(4)

% 计算结果：

% K=5相角裕度、幅值裕度、对应频率
ans = 
  包含以下字段的 struct:
     GainMargin: 455.2548
    GMFrequency: 602.4232
    PhaseMargin: 85.2751
    PMFrequency: 4.9620
    DelayMargin: 0.2999
    DMFrequency: 4.9620
         Stable: 1

% K=500相角裕度、幅值裕度、对应频率
ans = 
  包含以下字段的 struct:
     GainMargin: 4.5525
    GMFrequency: 602.4232
    PhaseMargin: 39.7483
    PMFrequency: 237.7216
    DelayMargin: 0.0029
    DMFrequency: 237.7216
         Stable: 1

% K=800相角裕度、幅值裕度、对应频率
ans = 
  包含以下字段的 struct:
     GainMargin: 2.8453
    GMFrequency: 602.4232
    PhaseMargin: 27.7092
    PMFrequency: 329.9063
    DelayMargin: 0.0015
    DMFrequency: 329.9063
         Stable: 1

% K=3000相角裕度、幅值裕度、对应频率
ans = 
  包含以下字段的 struct:
     GainMargin: 0.7588
    GMFrequency: 602.4232
    PhaseMargin: -6.7355
    PMFrequency: 690.5172
    DelayMargin: 0.0089
    DMFrequency: 690.5172
         Stable: 0

2.3.4 Actual Combat 3

Experimental requirements: The open-loop transfer function of the known system is: $G(s)=\frac{100(s+5)}{(s-2)(s+8)(s+20)}$, draw the polar plot of the system, using $The\; Nyquist$ stability criterion judges the stability of the closed-loop system.

untie:

% 实例Chapter7.2.3.4
clc;clear;

% 建立模型
k=100;z=[-5];p=[2,-8,-20];
sys=zpk(z,p,k);

nyquist(sys);grid;
axis equal;
set(findobj(get(gca,'Children'),'LineWidth',0.5),'LineWidth',1.5);
title('控制系统Nyquist图','FontSize',15);

• According to the open-loop transfer function, the open-loop system has an $Pole\; of\; the\; right\; half\; plane\; of\; s$ , ie$p=1$ , so the open-loop system is unstable;
• N yquist {\rm Nyquist} of an open-loop system$The\; Nyquist$ graph surrounds$(-1,j0)$point$1$ time, according to$The\; Nyquist$ stability criterion shows that the closed-loop system is stable;

2.3.5 Actual Combat 4

Experimental requirements: The open-loop transfer function of the known system is: $G(s)=\frac{100K}{s(s+5)(s+10)}$, respectively draw $K=1,7.8,\; and20$ o'clock, the polar coordinate diagram of the system, using$The\; Nyquist$ stability criterion judges the stability of the closed-loop system.

untie:

% 实例Chapter7.2.3.5
clc;clear;

% 建立控制系统模型
z=[];p=[0,-5,-10];k=100.*[1,7.8,20];

G=zpk(z,p,k(1));[re1,im1]=nyquist(G);
G=zpk(z,p,k(2));[re2,im2]=nyquist(G);
G=zpk(z,p,k(3));[re3,im3]=nyquist(G);

% 绘制图形
plot(re1(:),im1(:),'r-',re2(:),im2(:),'b-',re3(:),im3(:),'g-');
legend('K=1','K=7.8','K=20');
axis([-5,1,-5,1]);grid;
set(findobj(get(gca,'Children'),'LineWidth',0.5),'LineWidth',1.5);
title('不同K值控制系统Nyquist图','FontSize',15);

• According to the open-loop transfer function of the control system, the open-loop system has no $Pole\; of\; the\; right\; half\; plane\; of\; s$ , ie$p=0$ , so the open-loop system is stable;
• 由$It\; can\; be\; seen\; from\; the\; Nyquist$ diagram that when$K=When\; 1$ , the N yquist {\rm Nyquist}of the open-loop system$Nyquist$ graph does not surround$(-1,j0)$, according to$Nyquist$ stability criterion, the system is closed-loop stable; when$K=7.8$ and$K=At\; 20$ o'clock, the N yquist {\rm Nyquist}of the open-loop system$Nyquist$ graph surrounds$(-1,j0)$, according to$Nyquist$ stability criterion, the closed-loop system is unstable;

2.4 Comprehensive example and MATLAB/SIMULINK application

Experimental requirements: The structure diagram of the known thyristor-DC motor open-loop system is shown in the figure below, using $SIMULINK$ dynamic structure diagram for frequency domain analysis and frequency domain performance indicators.

untie:

【$STEP1$】：$Create\; a\; dynamic\; model\; of\; the\; control\; system\; in\; SIMULINK$ , the file name is:$experiment7\_2\_4.mdl$。

【$STEP2$ ]: Find the linear state space model of the system, and find the performance index in the frequency domain.

>> [A,B,C,D]=linmod('experiment7_2_4');
>> sys=ss(A,B,C,D);
>> margin(sys);

• Control system amplitude margin: $\mathrm{GM}=26.4dB$ , frequency:$152rad/s$；
• Control system phase margin: $\mathrm{PM}=54deg$ , frequency:$25.5rad/s$；