[classical control theory] summary of automatic control experiment

※ Automatic control system script writing

1、 Variable

sym, syms Symbolic variables

sym a;
syms a b;
f = str2sym('a+ b^2');

2、 Basic operation

1. Derivation

   dfa = diff(f function ,a For which variable （ partial ） guide );
   

2. Accumulated faintness

   intb = int(f function ,b Which variable to integrate ,b1,b2 Section );
   

3. Laplace transform

   %t You still have to define it in advance ,s Finding Laplace transform will bring 
   ft = exp(2*t);
   L1 = laplace(ft);
   

4. Inverse Laplace transform

   ftt = ilaplace(L1);
   

3、 Control system model

1. Polynomial form （ Transfer function model ）

   

   The coefficients are stored continuously in an array , An array of numerators and denominators , Pass from high to low

   num = [1 2 3];  % The coefficient corresponding to the molecule 
   den  = [4 5 6 7]; % The coefficient corresponding to the denominator 
   gs = tf(num,den);
   

   

2. Zero pole model

   

   Two arrays are required , ditto , Add one more k gain

   z = [1 2 3]';  % zero , Pay attention to add transpose 
   p = [4 5 6 7]; % pole 
   k = 9;
   gs2 = zpk(z,p,k);
   

   

3. Interconversion

   [num,den] = zp2tf(z,p,k);
   % Zero pole transfer function 
   

4. Other expressions

   molecular / The denominator expression has zpk type , But need tf When the format , Use conv（ It's polynomial multiplication ） Unified conversion to polynomial type , for example

   num = [1 2.9 2];
   den = conv([1 0],[1 10 25]);
   gs = tf(num,den);    % The root locus diagram should be drawn later 
   

   

4、 Connection of models

1. Series connection — series

   [num3,den3] = series(num,den,num2,den2);
   

2. parallel connection — parallel

   [num3,den3] = parallel(num,den,num2,den2);
   

3. feedback — feedback

   [num3,den3] = feedback(numg,deng,numh,denh,sign);
   % First give G(s) Transmission characteristics of , Give again H(s) Transmission characteristics of , Last sign Is the default -1（ Negative feedback ）, It can be changed +1（ The positive feedback ）
   

4. Unit feedback — cloop

   [num3,den3] = cloop(numg,deng,sign);
   % Less than above H(s)
   

5、 Typical input function

1. Unit step function — step

   t = [0:0.1:100];
   num = 1;
   den = [1 0.4 1];
   step(num, den, t);
   %t The main thing is to explain how long to draw t Of ct Images ,num,den Directly transfer the characteristic array of the closed-loop transfer function 
   

2. Unit pulse function — impulse

   t = [0:0.1:100];
   num = 1;
   den = [1 0.4 1];
   impulse(num, den, t);
   %t The main thing is to explain how long to draw t Of ct Images ,num,den Directly transfer the characteristic array of the closed-loop transfer function 
   

3. Custom input — lsim

   t = 0:0.1:10;
   
   %u Contain with t The expression of , such as 
   u = t;
   % perhaps 
   
   lsim(num,den,u,t);
   lsim(z,p,k,u,t);
   % It is still the beginning to give the characteristics of the closed-loop transfer function of the system , The above two forms are acceptable ,u It's input r(t), Contain with t Expression of integer ,t Is time 
   
   % for example , This is used for frequency characteristic test , Input a sine function to the system 
   t = [0:0.001:10];
   rt = 1* sin(15 * t);
   lsim(nums,dens,rt,t);
   
   
   

4. Be careful ： After using all the above functions , If you use its return value , It defaults to direct single window drawing ; If you use a variable to store the return value （ It is the result of the system output , It's a one-dimensional array ）, Then it can be used to draw multiple graphs

6、3 A system characteristic image

1. Root locus drawing

   gs = tf(num,den);
   rlocus(gs);  % What is given is the complete function expression , And here the first expression of the open-loop transmittal （ Without root track gain K*）
   

2. Nyquist

   matlab The painting contains w from 0 To -∞ Track of , Completely symmetrical with the positive part , Never mind

   nyquist(gs);   % Pass reference pass   Open loop transmittal expression 
   

3. bode chart

   bode(gs);    % Same as before , Generally, it draws an open-loop logarithmic amplitude frequency diagram , Then give the expression of open loop 
   

   for example , Left root locus , Right bode chart

   

7、matlab Drawing method

1. Basic drawing settings

   grid;  % Adding grid 
   xlabel(" Title name ");  % Set the name of the abscissa 
   ylabel(" ditto ");  % Vertical axis name 
   title("abb");   % The name of the drawn image 
   

2. Multigraph drawing

   （1） Draw a picture in one window , Be careful ,grid and legend Such special requirements do not apply to all windows , Every figure Add it separately if necessary

   figure;
   %code, Draw a picture 
   
   figure;
   %code, Draw another picture , And so on 
   
   % give an example , First draw a bode chart , Draw another Nyquist diagram 
   figure;
   z = [-0.2];
   p = [0 -0.1 -4];
   k = 16;
   [num,den] = zp2tf([],[0 -4*sqrt(2)],k);
   tf_gs = tf(num,den);
   bode(tf_gs);
   
   figure;
   nyquist(tf_gs);
   grid; %attention
   

   

   （2） Multiple graphs are drawn in one window

   ​ First of all, we need not let matlab Draw the response directly , Just use variables to receive the return value

   t = 0:0.1:10;
   y_step = step(numfi,denfi,t);  % Receive return value 
   
   u = t;
   y_ramp = lsim(numfi,denfi,u,t);   % Receive return value 
   

   ​ Then use a format similar to the previous one ,subplot Refer to this article for the parameters of ：http://t.csdn.cn/pXIZ5

   subplot(x1,y1,z1);
   plot(t1,c1);
   axis normal;
   
   subplot(x2,y2,z2);
   plot(t2,c2);
   axis normal;
   
   % give an example , Continue with the previous content 
   subplot(2,1,1);
   plot(t,y_step);
   xlabel(" Time /s");
   ylabel("c(t)");
   grid;
   axis normal
   
   subplot(2,1,2);
   plot(t,y_ramp);
   xlabel(" Time /s");
   ylabel("c(t)");
   axis normal
   grid;
   

※simulink Plug in emulation

Name of common component library

And here is , Directly check the corresponding... In the component library name Search and you'll get

• gain K gain

• Transfer function

  • Polynomial type transfer fcn：

    

  

  • Zero pole model Zero-Pole

    

• Oscilloscope scope

• Ramp signal ramp

• Step signal step

• Sine signal sine wave

• Sum up sum

• Dead zone module dead zone

• Limiting module saturation

• Analysis of system characteristic diagram of transfer function

  First use of in and out modular , Build the following model

  

  For a transmittal module , Right click , find Linearize block

  

  Select the source of input and output as shown in the following figure , Then click the system characteristic curve you want to obtain , You can draw directly

  

  If you want to draw multiple curves on the same graph （ It is mostly used for system calibration ）, Just after the first drawing ,（ I'm right simulink After simply modifying the parameter value of the model ）, Then click the current plot The button , There are two pictures , Stack in this way

  

  

• PID modular PID controller

  Corresponding modification PID Three parameters are sufficient

  You can also use PID tunner Instantly adjust rapidity and volatility , See the picture below

  

  

※ Scripts and simulink Dream linkage

1. take simulink Simulation data is used .m Script processing

   Directly use the transfer function model , Through one more simout To transmit the data is

   

   

   Several setting modes

   1. structure with time

      This is Time And corresponding Output

      plot(out.so.time, out.so.signals.values);
      %out It should be added so to simout The name of life , This can be customized , Because a .slx There can be many songs simout modular 
      

      

   2. structure

      Only Output

      yt = out.so.signals.values;
      

   3. array

      Only Output , Return... As an array

      plot(out.so);
      

   4. timeseries

      plot(out.so.Time,out.so.Data);
      

2. use .m Draw Lissajou figure / Time domain response curve

   Here we directly use the code of the eighth experiment of automatic control

   

   figure;
   plot(x.signals.values,y.signals.values);
   title(" Lissajous figure ");
   grid;
   
   figure;
   plot(x.time,x.signals.values);
   hold on;
   plot(y.time,y.signals.values);
   legend(" Input "," Output ");
   title(" Time domain curve ");
   grid;
   

3. It can be used x（ Corresponding to the input of the system ）, and y（ Corresponding to the output of the system ） Direct use matlab fitting

   Be careful ： First the x,y The data type of is set to 2-D array, It's a two-dimensional array , Then follow the steps in the following article Data import ：

   http://t.csdn.cn/NrfDU

   Next, analyze the link ：operations Personally, I don't think it's necessary , The analysis method is the transmittal mode , You need to give the number of zeros and poles ,matlab Direct fitting according to the number of zeros and poles

   

   Take the screenshot of your eighth experiment as an example , No zero point was selected at that time ,3 A pole

   

※ The principle of automatic control makes up the leak ~~（ A vegetable dog didn't understand until he finished the exam ）~~

Mainly with The higher-order system uses the dominant pole reduction method and Frequency test of , Combined with the specific experimental content of automatic control, briefly explain

1、 Order reduction of dominant poles of higher-order systems

1. subject （ The first question is ignore ）：

2. Personal understanding ： The given transfer function is the open-loop transfer function of this system , And the corresponding root locus gain is 160, If you want to draw, you will get no 160 The transmittal of is passed as a parameter to rlocus Just draw ;

   And for order reduction , You need to get in K*=160 All the roots of , Actually use matlab Find the root locus diagram drawn in the diagram 4 It's better to go directly to Casio 991 Solve the equation fragrant , Get more accurate results directly , See tutorial

   fx-991CN X equation （ Group ） Use of solving function - Dianzhuo Yuanya Jiliang's article - You know https://zhuanlan.zhihu.com/p/28795401

3. Theoretical knowledge supplement ： First film a wave of explanation from the giant of Harbin Industry ：https://www.bilibili.com/video/BV1cK4y1U7jR?share_source=copy_web

   For the reduction of higher-order systems , It is necessary to ensure the stability of the system before and after order reduction Closed loop gain unchanged , So the best way is to get the closed-loop transmittal of the system , Written as closed-loop zero pole form , Then analyze the position of closed-loop zero pole , There are a pair of very close zeros and poles （ Dipole , In the blue box ）, Can be removed , remainder The real part of the pole does not 2-3 Times more , Therefore, none of them can be omitted

   

   

4. result ： according to K The unchanging principle , Whoever gets rid of it will leave the corresponding constant term , Finally, the step response curve is obtained

   

   

2、 Frequency characteristic test

1. subject ：

2. Add ：

   （1） The solution of phase and amplitude is as follows , So just need to be in Lissajou image （x For input ,y For export ） Found in Image and x=0 The intersection of the axes , Find in the step response curve Peak value of input / output image Put it in a table , Just draw a picture
   $$\begin{gathered} R(t)=A_0\ast sin(wt) \\ C(t)=A_1\ast sin(wt+\phi ) \\ A(w)=A1/A0 \\ \phi (w)=acrsin(C(0)/A1) \end{gathered}$$
   （2） There are also aspects , Lissajou's image is elliptical , The inclination angle of its long axis （ And right x Included angle of shaft ） It's probably the phase difference , So we need to Pay attention to phase override 90°/180°/270° etc. , Modify the corresponding phase solution formula in time

   Like this picture , It's a positive ellipse , The phase is close 90°

   

   This is more than 90° 了

   

   （3） But you can find that the curve above is not smooth enough , So I copied the similar model settings sent by the teacher before , The curve is instantly beautiful ： stay simulink Look for this setting in the interface

   

   （4） Next is excel processing section , Get organized w/A1/A0/C0 And use excel Function of , Using the type of scatter plot with asymptote , You can get the image of frequency characteristic test , And according to the maximum phase 、 The turning point 、 Initial phase 、 Initial amplitude, etc

   

   （5） Although obviously 3 Order system , But the analysis method of second-order system is also ok 、、、 But this is not recommended ,system identification Doesn't it smell good ; Other ways , For example, some information can be obtained from the curve of step response to fit the function

3. It's also a doubt that will be solved after the exam （ Pure is because he sprinkled money ）：bode The picture is only for Transfer function The expression of , It has nothing to do with open-loop and closed-loop , It only uses this function The sinusoidal response result of A、φ, So all kinds of typical transmittal links in the book （ inertia 、 Oscillate 、 First and second order differential 、 Integral, etc. ） These results are applicable to the observation of frequency characteristics

automatic control 1 End of the flower **ヽ(°▽°)ノ*, Already a self-control person