How to use sostools to find S matrix in ARE

This is the parameter

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;

This is the model

% 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);

This is the Q and R matrix for LQR

% Q and R matrices for ARE
Q = [10  0  0  0;
      0 10  0  0;
      0  0 10  0;
      0  0  0 10;
      ];
R = sym(zeros(2,2)); R(1,:) = [1 0]; R(2,:) = [0 1]; display(R);

This is the S matrix that I want to find the value

% 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);

This is the Algebra Ricatti Equation (ARE) in LQR Control

% 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);

I want to use sostools and sedumi to find S semidifinite matrix

 syms s1 s2 s3 s4 s6 s7 s8 s11 s12 s16
[Sol_s] = sosgetsol(left_ARE,right_ARE,sedumi)
disp(Sol_s)

But why there is error like this

Error using sedumi (line 232)
Should have at least (A,b) or (A,c) arguments

Error in Algebraricattieqvpa (line 75)
[Sol_s] = sosgetsol(left_ARE == right_ARE,sedumi)

my left_ARE is A*x

my right_ARE is B

but why can't get the answer?

[Walter Roberson]

It looks to me that you should be creating an sos objective, and creating an options structure that has field solver set to 'SeDuMi', and that you should be calling sossolve() .

https://arxiv.org/pdf/1310.4716.pdf

[Ivan Dwi Putra]

I've read the same instruction manual like that, but still don't really understand it, is there any other reading about sostools? thank you

[Walter Roberson]

http://avalon.caltech.edu/~murray/wiki/images/9/92/SOSTOOLSandControlApplications.pdf

[Ivan Dwi Putra]

ok thank you, I will try first

[Ivan Dwi Putra]

from my understanding i should make a new program that different from this code, can I just change the code in Sol_s to use sostools to find s?

i still don't really understand, can you please help me with the sostools code? Thank you

[Ivan Dwi Putra]

I try to run the example sos10.m but there is an error like this

>> sosdemo10
syms x1 x2;
vartable = [x1 x2];

eps = 1e-6;
% =============================================
% This is the problem data
p = x1^2+x2^2;
gamma = 1;
g0 = [2 0]*[x1;x2];
theta = 1;

% =============================================
% Initialize the sum of squares program
prog = sosprogram(vartable);

% =============================================
% The multiplier
Zmon = monomials(vartable,0:4);
[prog,s] = sospolymatrixvar(prog,Zmon,[1 1]);
Error using mupadengine/feval_internal
Invalid number of arguments.

Error in mupadengine/feval

Error in getdegrees (line 47)
coefmon = feval(symengine,'poly2list',p,vartable);

Error in sosvar (line 55)
Z = getdegrees(ZSym,sos.vartable,sos.symvartable);  %JA edit

Error in sospolyvar (line 56)
	[sos,V] = sosvar(sos,'poly',ZSym);

Error in sospolymatrixvar (line 88)
            [sos,var] = sospolyvar(sos,ZSym);

Error in sosdemo10 (line 23)
[prog,s] = sospolymatrixvar(prog,Zmon,[1 1]);

[Ivan Dwi Putra]

First i initialize square of program and then declare matrix variable as written in the instructions

clear; echo on;

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;

x = [a1_head a_2_head -1 a_2_head].';
prog = sosprogram(x);
[prog,A] = sospolymatrixvar(prog,monomials(x,0),[4 4],'symmetric');

But why there is an error like this

>> sostry
clear; echo on;

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;

x = [a1_head a_2_head -1 a_2_head].';
prog = sosprogram(x);
[prog,A] = sospolymatrixvar(prog,monomials(x,0),[4 4],'symmetric');
Error using mupadengine/feval_internal
Invalid input. Expected ')'.

Error in mupadengine/feval

Error in getdegrees (line 47)
coefmon = feval(symengine,'poly2list',p,vartable);

Error in sosvar (line 55)
Z = getdegrees(ZSym,sos.vartable,sos.symvartable);  %JA edit

Error in sospolyvar (line 56)
	[sos,V] = sosvar(sos,'poly',ZSym);

Error in sospolymatrixvar (line 62)
                    [sos,var] = sospolyvar(sos,ZSym);

Error in sostry (line 32)
[prog,A] = sospolymatrixvar(prog,monomials(x,0),[4 4],'symmetric');
 

I can't move to next instruction because the error

[Walter Roberson]

Sorry, I am having trouble installing sedumi .

[Ivan Dwi Putra]

ok no problem, if use sdpa is same like sedumi?

[Ivan Dwi Putra]

Is this a problem because my version of Matlab is r2020a?

[Walter Roberson]

I do not know at the moment.

The difficulty I am having installing sedumi is probably related to my using MacOS.

[Ivan Dwi Putra]

ok thank you