Is there a way to extract my 3 transfer function in matlab given these equations.

I have these 3 equations

X1(s)*[m1*s^2 + c1*s+(k01+k12)] = F(s) + k12*X2(s)

X2(s)*[m2*s^2 + c2*s +(k12+k23)] = k12*X1(s)+k23*X3(s)

X3(s) [m3*s^2 + c3*s +(k23+k34)] = k23*X2(s)

and I am wondering if there is a possibility to be able to extract my 3 transfer functions, X1(s)/F(s), X2(s)/F(s), and X3(s)/F(s).

Ive really havent explore this part of Matlab and am interested to know if it is possible. I ask because currently I am doing it by hand and setting the left side S polynomial to be P1, P2, and P3 respectively.

but it is alot of algabra and I fear I will make a simple arithmetic mistake and ruin the whole thing.

Can this be done.

[Sam Chak]

I think the equations can be expressed in this form:

where you can algebraically solve

If exists, then

, ,

[Jorje German]

Interesting concept. I just showed my wife and she said, this is why you are still a student. I will try this method and play around in Matlab. Thank you. You have given me a direction to go to.

Answer 1

[Sam Chak]

Hi Jorje German,

The transfer functions for your case in given in the first column of the matrix. You can use the functions in Symbolic Math toolbox to simplify the rational expressions. Hope my answer helps.

[Jorje German]

Would you be able to code that. I was able to input it in my calculator and selecting the rref(A) I was able to extract X1(s)/F(s) = (p2*p3-(k23)^2)/(p1*(p2*p3-(k23)^2)-p3*(k12)^2 this is the correct transfer function, which can be found below From input 1 to output..1...BUT how can I extract this so I can specifically use that transfer function and use it when sys=p41

M =
 
  From input 1 to output...
   1:  s^2 - s + 100
 
   2:  -100
 
   3:  0
 
  From input 2 to output...
   1:  100
 
   2:  -s^2 + s - 110
 
   3:  10
 
  From input 3 to output...
   1:  0
 
   2:  10
 
   3:  -s^2 + s - 10
 
Continuous-time transfer function.


ans =
 
  From input 1 to output...
                                                    
                s^4 - 2 s^3 + 121 s^2 - 120 s + 1000
                                                    
   1:  -------------------------------------------------------
                                                              
       s^6 - 3 s^5 + 223 s^4 - 441 s^3 + 3220 s^2 - 3000 s    
                                                              
                                                   - 3.659e-12
                                                              
 
                                              
                       -100 s^2 + 100 s - 1000
                                              
   2:  -------------------------------------------------------
                                                              
       s^6 - 3 s^5 + 223 s^4 - 441 s^3 + 3220 s^2 - 3000 s    
                                                              
                                                   - 3.659e-12
                                                              
 
                                     
                                -1000
                                     
   3:  -------------------------------------------------------
                                                              
       s^6 - 3 s^5 + 223 s^4 - 441 s^3 + 3220 s^2 - 3000 s    
                                                              
                                                   - 3.659e-12
                                                              
 
  From input 2 to output...
                                             
                       100 s^2 - 100 s + 1000
                                             
   1:  -------------------------------------------------------
                                                              
       s^6 - 3 s^5 + 223 s^4 - 441 s^3 + 3220 s^2 - 3000 s    
                                                              
                                                   + 5.192e-11
                                                              
 
                                                     
                -s^4 + 2 s^3 - 111 s^2 + 110 s - 1000
                                                     
   2:  -------------------------------------------------------
                                                              
       s^6 - 3 s^5 + 223 s^4 - 441 s^3 + 3220 s^2 - 3000 s    
                                                              
                                                   + 5.192e-11
                                                              
 
                                             
                        -10 s^2 + 10 s - 1000
                                             
   3:  -------------------------------------------------------
                                                              
       s^6 - 3 s^5 + 223 s^4 - 441 s^3 + 3220 s^2 - 3000 s    
                                                              
                                                   + 5.192e-11
                                                              
 
  From input 3 to output...
                                    
                                1000
                                    
   1:  -------------------------------------------------------
                                                              
       s^6 - 3 s^5 + 223 s^4 - 441 s^3 + 3220 s^2 - 3000 s    
                                                              
                                                   + 5.545e-13
                                                              
 
                                             
                        -10 s^2 + 10 s - 1000
                                             
   2:  -------------------------------------------------------
                                                              
       s^6 - 3 s^5 + 223 s^4 - 441 s^3 + 3220 s^2 - 3000 s    
                                                              
                                                   + 5.545e-13
                                                              
 
                                                     
                -s^4 + 2 s^3 - 211 s^2 + 210 s - 1000
                                                     
   3:  -------------------------------------------------------
                                                              
       s^6 - 3 s^5 + 223 s^4 - 441 s^3 + 3220 s^2 - 3000 s    
                                                              
                                                   + 5.545e-13
                                                              
 
Continuous-time transfer function.

[Jorje German]

Here is my Original code. As you can see I only know how to use it in this form p41 = (p2*p3-(k23)^2)/(p1*(p2*p3-(k23)^2)-p3*(k12)^2); so I can use sys=p41 and plot certain plots. How can I extract p41 from the previous comment which is From input 1 to output..1 so I can plot it like the one below.

clear all

m1=1;
m2=1;
m3=1;
c1=1;
c2=1;
c3=1;
k01=0;
k12=100;
k23=10;
k34=0;
t = 0:0.01:10;
U1 = 10+2*t;
U2 = sin(t);

s = tf('s');

p1=(m1*s^2 + c1*s +(k01+k12)); 
p2=(m2*s^2 + c2*s +(k12+k23)); 
p3=(m3*s^2 + c3*s +(k23+k34));

p41 = (p2*p3-(k23)^2)/(p1*(p2*p3-(k23)^2)-p3*(k12)^2);
sys=p41;

p41

OPT = stepDataOptions('StepAmplitude', 100); % OPT means option to put in a step

[Y1, t1]=impulse(sys, 10);
[Y2, t2]=step(sys,10,OPT);
[Y3, t3]=lsim(sys,U1,t);
[Y4, t4]=lsim(sys,U2,t);

subplot(2,2,1)
plot(t1,Y1,'r')
hold on 
title ('Impulse Response')
xlabel ('Time (S)')
ylabel("Amplitude)")
grid on

subplot(2,2,2)
plot(t2,Y2,'r')
hold on 
title ('Step Response')
xlabel ('Time (S)')
ylabel("Amplitude)")
grid on

subplot(2,2,3)
plot(t3,Y3,'r')
hold on 
title ('Step Ramp Response')
xlabel ('Time (S)')
ylabel("Amplitude)")
grid on

subplot(2,2,4)
plot(t4,Y4,'r')
hold on 
title ('Sin Response')
xlabel ('Time (S)')
ylabel("Amplitude)")
grid on

[Sam Chak]

Hi Jorje German,

In your original problem, if the parameters are given or known, then you can surely determine the transfer functions by following mathematical procedures as described above. Your new problem seems to be a MATLAB programming problem, where you want to extract the algebraic results from the Symbolic Math Toolbox for immediate system analysis in the Control System Toolbox in a one-shot run of the M-script file.

[Jorje German]

What is Symbolic math toolbox. Are those build in functions.

[Jorje German]

Also. This methods works for Amy system correct? It's ready going through my calculator and doing rref

[Sam Chak]

The method works as long as the matrix inverse exists. For more info about Symbolic Math Toolbox, please visit:

https://www.mathworks.com/products/symbolic.html