MATLAB: Error struct with field

lqrMATLAB

This i smy matlab code
clear; close; clc;
syms a1_head a2_head b hstar 
%Parameter Massa
m1 = 8095; % massa train set 1 dalam kg
m2 = 8500; % massa train set 2 dalam kg
g  = 10;
c_0_1 = 0.01176;
c_1_1 = 0.00077616;
c_2_1 = 4.48 ;
c_0_2 = 0.01176 ;
c_1_2 = 0.00077616;
c_2_2 = 4.48;
v_0 = 300;
hstar = 120;
a2_head
a_1   = -1./m1.*(c_1_1 + 2.*c_2_1.*v_0);  
a_2   = -1./m2.*(c_1_2 + 2.*c_2_2.*v_0); 
a_1_head = 1-(a_1.*hstar);
a_2_head = 1-(a_2.*hstar);
b = 1;
% Model data
A = sym(zeros(4,4));
A(1,2) = a_1_head;
A(3,2) = (a_2_head) - 1;  A(3,4) = a_2_head;
display(A);
B = sym(zeros(4,2));
B(1,1) = -b*hstar;
B(2,1) = b;
B(3,2) = -b*hstar   ;
B(4,1) = -b; B(4,2) = b;
display(B);
% Q and R matrices for ARE
Q = sym(eye(4)); display(Q);
R = 1;
% Matrix S to find
svar = sym('s',[1 16]);
S = [svar(1:4); svar(5:8); svar(9:12); svar(13:16)];
S(2,1) = svar(2);
S(2,2) = svar(1);
S(2,4) = svar(2);
S(3,1) = svar(3);
S(3,2) = svar(7);
S(3,3) = svar(1);
S(4,1) = svar(4);
S(4,2) = svar(2);
S(4,3) = svar(12);
S(4,4) = svar(1);
display(S);
% LHS of ARE: A'*S + S*A' - S*B*Rinv*B'*S
left_ARE = transpose(A)*S + S*A - S*B*inv(R)*transpose(B)*S; 
display(left_ARE);
% RHS of ARE: A'*S + S*A' - S*B*Rinv*B'*S
right_ARE = -Q;
display(right_ARE);
% Find S in ARE

[Sol_S] = solve(left_ARE == right_ARE,'Real',true)
from my code get S symmetric 4×4
S =
 
[ s1, s2,  s3,  s4]
[ s2, s1,  s7,  s2]
[ s3, s7,  s1, s12]
[ s4, s2, s12,  s1]
I use solve function like this to fidn all element S
% Find S in ARE
[Sol_S] = solve(left_ARE == right_ARE,'Real',true)
But cannot get all element S, I get error like this
Sol_S = 
  struct with fields:
     s1: [0×1 sym]
     s2: [0×1 sym]
     s3: [0×1 sym]
     s4: [0×1 sym]
     s7: [0×1 sym]
    s12: [0×1 sym]
Please help me to find all element S
Q and R in this code can change because i want find eigenvalue in S matrix is positive.

Best Answer

You have 16 equations in 6 variables; there will rarely be a solution to such systems.
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