Find 16 variables in matrix with 16 equation in matrix

This is my 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;

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 = sym(zeros(2,2)); R(1,:) = [1 1]; R(2,:) = [1 2]; display(R);

% 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(3,1) = svar(3);
% S(3,2) = svar(7);
% S(4,1) = svar(4);
% S(4,2) = svar(8);
% S(4,3) = svar(12);
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: -Q
right_ARE = -Q;
display(right_ARE);

% Find S
[Sol_s] = solve(left_ARE == right_ARE)

% % Find S
% X = linsolve(left_ARE,right_ARE);

I try use solve and linsolve still can't get the variable and the process is very long

[KSSV]

Why you want to use syms? You can make equation :

Ax = b

and use

x = A\b ; 

[Ivan Dwi Putra]

because in ARE LQR the S matrix must be symmetric. in my left_ARE equation it have s1-s16 variables in 16 equation because i have S matrix 4x4. I want to find s1-s16 with left_ARE equation = right_equation

[Ivan Dwi Putra]

eft_ARE =
 
[                                                                               s5*(240*s1 - 2*s2 - 120*s3 + 3*s4) - s13*(360*s1 - 3*s2 - 240*s3 + 5*s4) + s9*(14400*s1 - 120*s2 - 14400*s3 + 240*s4) - s1*(28800*s1 - 240*s2 - 14400*s3 + 360*s4),                                                                                (2874340168023969*s1)/70368744177664 + (2670370432474805*s3)/70368744177664 + s6*(240*s1 - 2*s2 - 120*s3 + 3*s4) - s14*(360*s1 - 3*s2 - 240*s3 + 5*s4) + s10*(14400*s1 - 120*s2 - 14400*s3 + 240*s4) - s2*(28800*s1 - 240*s2 - 14400*s3 + 360*s4),                                                                                s7*(240*s1 - 2*s2 - 120*s3 + 3*s4) - s15*(360*s1 - 3*s2 - 240*s3 + 5*s4) + s11*(14400*s1 - 120*s2 - 14400*s3 + 240*s4) - s3*(28800*s1 - 240*s2 - 14400*s3 + 360*s4),                                                                                (2740739176652469*s3)/70368744177664 + s8*(240*s1 - 2*s2 - 120*s3 + 3*s4) - s16*(360*s1 - 3*s2 - 240*s3 + 5*s4) + s12*(14400*s1 - 120*s2 - 14400*s3 + 240*s4) - s4*(28800*s1 - 240*s2 - 14400*s3 + 360*s4)]
[ (2874340168023969*s1)/70368744177664 + (2670370432474805*s9)/70368744177664 + s5*(240*s5 - 2*s6 - 120*s7 + 3*s8) - s13*(360*s5 - 3*s6 - 240*s7 + 5*s8) + s9*(14400*s5 - 120*s6 - 14400*s7 + 240*s8) - s1*(28800*s5 - 240*s6 - 14400*s7 + 360*s8), (2874340168023969*s2)/70368744177664 + (2874340168023969*s5)/70368744177664 + (2670370432474805*s7)/70368744177664 + (2670370432474805*s10)/70368744177664 + s6*(240*s5 - 2*s6 - 120*s7 + 3*s8) - s14*(360*s5 - 3*s6 - 240*s7 + 5*s8) + s10*(14400*s5 - 120*s6 - 14400*s7 + 240*s8) - s2*(28800*s5 - 240*s6 - 14400*s7 + 360*s8), (2874340168023969*s3)/70368744177664 + (2670370432474805*s11)/70368744177664 + s7*(240*s5 - 2*s6 - 120*s7 + 3*s8) - s15*(360*s5 - 3*s6 - 240*s7 + 5*s8) + s11*(14400*s5 - 120*s6 - 14400*s7 + 240*s8) - s3*(28800*s5 - 240*s6 - 14400*s7 + 360*s8), (2874340168023969*s4)/70368744177664 + (2740739176652469*s7)/70368744177664 + (2670370432474805*s12)/70368744177664 + s8*(240*s5 - 2*s6 - 120*s7 + 3*s8) - s16*(360*s5 - 3*s6 - 240*s7 + 5*s8) + s12*(14400*s5 - 120*s6 - 14400*s7 + 240*s8) - s4*(28800*s5 - 240*s6 - 14400*s7 + 360*s8)]
[                                                                   s5*(240*s9 - 2*s10 - 120*s11 + 3*s12) - s13*(360*s9 - 3*s10 - 240*s11 + 5*s12) + s9*(14400*s9 - 120*s10 - 14400*s11 + 240*s12) - s1*(28800*s9 - 240*s10 - 14400*s11 + 360*s12),                                                                   (2874340168023969*s9)/70368744177664 + (2670370432474805*s11)/70368744177664 + s6*(240*s9 - 2*s10 - 120*s11 + 3*s12) - s14*(360*s9 - 3*s10 - 240*s11 + 5*s12) + s10*(14400*s9 - 120*s10 - 14400*s11 + 240*s12) - s2*(28800*s9 - 240*s10 - 14400*s11 + 360*s12),                                                                    s7*(240*s9 - 2*s10 - 120*s11 + 3*s12) - s15*(360*s9 - 3*s10 - 240*s11 + 5*s12) + s11*(14400*s9 - 120*s10 - 14400*s11 + 240*s12) - s3*(28800*s9 - 240*s10 - 14400*s11 + 360*s12),                                                                   (2740739176652469*s11)/70368744177664 + s8*(240*s9 - 2*s10 - 120*s11 + 3*s12) - s16*(360*s9 - 3*s10 - 240*s11 + 5*s12) + s12*(14400*s9 - 120*s10 - 14400*s11 + 240*s12) - s4*(28800*s9 - 240*s10 - 14400*s11 + 360*s12)]
[                        (2740739176652469*s9)/70368744177664 + s5*(240*s13 - 2*s14 - 120*s15 + 3*s16) - s13*(360*s13 - 3*s14 - 240*s15 + 5*s16) + s9*(14400*s13 - 120*s14 - 14400*s15 + 240*s16) - s1*(28800*s13 - 240*s14 - 14400*s15 + 360*s16),                      (2740739176652469*s10)/70368744177664 + (2874340168023969*s13)/70368744177664 + (2670370432474805*s15)/70368744177664 + s6*(240*s13 - 2*s14 - 120*s15 + 3*s16) - s14*(360*s13 - 3*s14 - 240*s15 + 5*s16) + s10*(14400*s13 - 120*s14 - 14400*s15 + 240*s16) - s2*(28800*s13 - 240*s14 - 14400*s15 + 360*s16),                        (2740739176652469*s11)/70368744177664 + s7*(240*s13 - 2*s14 - 120*s15 + 3*s16) - s15*(360*s13 - 3*s14 - 240*s15 + 5*s16) + s11*(14400*s13 - 120*s14 - 14400*s15 + 240*s16) - s3*(28800*s13 - 240*s14 - 14400*s15 + 360*s16),                       (2740739176652469*s12)/70368744177664 + (2740739176652469*s15)/70368744177664 + s8*(240*s13 - 2*s14 - 120*s15 + 3*s16) - s16*(360*s13 - 3*s14 - 240*s15 + 5*s16) + s12*(14400*s13 - 120*s14 - 14400*s15 + 240*s16) - s4*(28800*s13 - 240*s14 - 14400*s15 + 360*s16)]
 

that is my left_ARE

in ARE

A^T.S + S.A. - S.B.R^-1.B^T.S + Q = 0

if i get the S matrix i can find K matrix and find the eigen value S matrix. I know K matrix can find with the K = lqr(A,B,Q,R), but i need S matrix to compare with K from the function, so i know the K is right with the eigenvalue from S matrix is all positive

[Ivan Dwi Putra]

in my graph that simulate the train without control LQR and in my LQRtry image with control lqr. The lqr control image still wrong, i want to the distance from x1 x3 is same after control same like x2 and x4.