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How can i solve Algebraic Ricatti Equation for SDRE controller?

Asked by burak ergocmen on 3 Sep 2017
Latest activity Edited by burak ergocmen on 8 Dec 2017
I have a model in simulink. I use SDRE controller for that model . I can obtain SDC matrices(A(x) and B(x)) by matlab function in simulink . One of the example in simulink matlab function is :
function P = fcn(u,w)
%#codegen
B = [ 0 2 2 0 ;
1 0 0 3 ;
0 4 5 0 ;
0 0 3 0 ;
0 4 0 0 ] ;
A = [2 0 4 u 0 ;
0 5 0 0 0 ;
6 0 7 w 0 ; 1 0 0 0 0 ; 0 1 0 0 0 ] ;
Q= diag([0.6 0.6 0.6 2.6 0.5]);
R= diag([0.00001 1000 1000 1000]);
n = size(A,1);
P = zeros(n,n);
H = [A -(B/R)*B'; -Q -A'];
[T,L] = eig(H);
lambda = diag(L);
ord_lambda = zeros(1,2*n);
for k = 1 : 2*n;
if real(lambda(k)) < 0
ord_lambda(k) = -1;
elseif real(lambda(k)) > 0
ord_lambda(k) = 1;
else
%disp('!!!');
end
end
[~, ord_index] = sort(ord_lambda, 'descend');
T_ord = T(:,ord_index);
T22 = T_ord(n+1:2*n,n+1:2*n);
T12 = T_ord(1:n,n+1:2*n);
P = real(T22/T12);

  1 Comment

1. The ARE can be solved by some methods .These are iterative and numerical ways. Iterative is the best but numerical is also work well. Some numerical example : Schur decomposition of Hamiltonian matrix , Spectral Factorization Some Iterative example : Matrix sign function 2. For real time application of solving algebraic ricatti equation for SDRE controller there are some ways one of that is using S-function (write codes in C programming and use it in the simulink block). There is a really good thesis about that, also there is C codes in thesis : "ANALYSIS AND REAL-TIME IMPLEMENTATION OF STATE-DEPENDENT RICCATI EQUATION CONTROLLED SYSTEMS-EVRIN BILGE ERDEM " another way is using matlab function in Simulink environment . Some functions does not work but i found one of that which is working but some calculation problems which about re ordering schur decomposition (all eigenvalues have to be in the upper triangle of the schur matrix). I post it in here because "[U_ordered,S_ordered] = ordschur(U,S,'lhp')" code does not work in script.So if anybody have an idea pls post it in here These matlab codes are about schur decomposition of hamiltonian matrix . A and B can easily convertible to SDC matrices which shows the system dynamics
function y = fcn(x1,x2)
A = [x1 x2; x1 x2];
B = [2; 1];
C = eye(2);
D = [0; 0];
Q = [1 1; 3 0];
R = [3];
H = [A -(B*inv(R)*B'); -Q -A'];
[U, S] = schur(H);
[m,n] = size(U);
U11 = U(1:(m/2), 1:(n/2));
U21 = U((m/2+1):m, 1:(n/2));
P = U21*inv(U11) ;
y = P;

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