Linearization of Non Linear with the editor

How can i linearizate my non linear mimo system? This is my code

clc;
clear;
close all;

syms x1 x2 x3 u u1 u2 u3 x_punkt x1_punkt x2_punkt x3_punkt;
%% Modellparameter
a1 = 0.00751;
a2 = 0.00418;
a3 = 0.05;
a4 = 0.03755;
a5 = 0.02091;
a6 = 0.00315;

b1 = 0.00192;
b2 = 0.05;
b3 = 0.00959;
b4 = 0.1866;
b5 = 0.14;

k1 = 0.01061;
k2 = 2.5;
k3 = 6.84;
k4 = 2.5531;
%% nicht lineares System
x1_punkt = a1*x3 + a2*x2 - b1*u2 -b2*u2 - k1;
x2_punkt = -a3*x2*u2 + k2;
x3_punkt = -a4*x3 - a5*x2 +b3*u1 + ((a6*x3+b4)/(b5*u3+k3))*u3 + k4;
y1 = x1;
y2 = x2;
y3 = x3;
%% Ruhelage
xR = [1; 15; 70];
uR = [214.13; 3.33; 65.40];
%% Linearisierung
x = [x1 x2 x3];
y = [y1 y2 y3];
n = 3;
m = 3;
p = 3;
x_punkt = [x1_punkt x2_punkt x3_punkt]';
linear_A = jacobian(x_punkt, x);
linear_B = jacobian(x_punkt, u);
linear_C = jacobian(y, x);

A= subs(linear_A,x',xR');
A= subs(A,u',uR');
A = double(A)

B= subs(linear_B,x',xR');
B= subs(B,u',uR');
B = double(B)

C= subs(linear_C,x',xR');
C= subs(C,u',uAP');
C = double(C)

D = zeros(p,m);
sys = ss(A,B,C,D);

Answer 1

[Paul]

Hi Hamza,

The code will run after fixing a few things, mostily mismatched dimensions. All I did was get it to run; did not check the underlying math and implementation.

Might not matter in this problem, but best to declare variables as real if they are, which I assume they are from the context of this problem.

syms x1 x2 x3 u u1 u2 u3 real

%% Modellparameter
a1 = 0.00751;
a2 = 0.00418;
a3 = 0.05;
a4 = 0.03755;
a5 = 0.02091;
a6 = 0.00315;
b1 = 0.00192;
b2 = 0.05;
b3 = 0.00959;
b4 = 0.1866;
b5 = 0.14;
k1 = 0.01061;
k2 = 2.5;
k3 = 6.84;
k4 = 2.5531;
%% nicht lineares System
x1_punkt = a1*x3 + a2*x2 - b1*u2 -b2*u2 - k1;
x2_punkt = -a3*x2*u2 + k2;
x3_punkt = -a4*x3 - a5*x2 +b3*u1 + ((a6*x3+b4)/(b5*u3+k3))*u3 + k4;
y1 = x1;
y2 = x2;
y3 = x3;
%% Ruhelage
xR = [1; 15; 70];
uR = [214.13; 3.33; 65.40];
%% Linearisierung
x = [x1 x2 x3];
y = [y1 y2 y3];

Define the input vector

u = [u1 u2 u3];
n = 3;
m = 3;
p = 3;
x_punkt = [x1_punkt x2_punkt x3_punkt]';
linear_A = jacobian(x_punkt, x);
linear_B = jacobian(x_punkt, u);
linear_C = jacobian(y, x);

Here, x is a row vector and xR is a column vector. So don't tanspose xR in the subs commands. Same thing with uR.

A= subs(linear_A,x',xR);  
A= subs(A,u',uR);         
A = double(A)

A = 3×3

0 0.0042 0.0075 0 -0.1665 0 0 -0.0209 -0.0247

B= subs(linear_B,x',xR);  
B= subs(B,u',uR);         
B = double(B)

B = 3×3

0 -0.0519 0 0 -0.7500 0 0.0096 0 0.0109

C= subs(linear_C,x',xR);

uAP is not defined, so I used uR instead, consistent with the with subs above.

% C= subs(C,u',uAP');
C= subs(C,u',uR); 

C = double(C)

C = 3×3

1 0 0 0 1 0 0 0 1

D = zeros(p,m);
sys = ss(A,B,C,D)

sys =
 
  A = 
             x1        x2        x3
   x1         0   0.00418   0.00751
   x2         0   -0.1665         0
   x3         0  -0.02091  -0.02467
 
  B = 
             u1        u2        u3
   x1         0  -0.05192         0
   x2         0     -0.75         0
   x3   0.00959         0   0.01088
 
  C = 
       x1  x2  x3
   y1   1   0   0
   y2   0   1   0
   y3   0   0   1
 
  D = 
       u1  u2  u3
   y1   0   0   0
   y2   0   0   0
   y3   0   0   0
 
Continuous-time state-space model.

[Hamza Zaoui]

thank you very much. it worked.