# Find 16 variables in matrix with 16 equation in matrix

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Ivan Dwi Putra on 10 Jun 2020
Commented: Ivan Dwi Putra on 10 Jun 2020
This is my code
clear; close; clc;
%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);
b = 1;
% Model data
A = sym(zeros(4,4));
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

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Ivan Dwi Putra on 10 Jun 2020
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 on 10 Jun 2020
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 on 10 Jun 2020
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.

R2020a

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