How to find steady state solution of recatti equation

hi everyone,

I want to find solution of algebric equations as below:

-A_Star'*x-x*A-Q+x*G*x ... where x is the solution of this equation

where A_Star, A Q and G are defined as below:

N=50;
a1=1; a2=1;
tau=1;
A=zeros(N+1,N+1);
A=diag(-N/tau*ones(N+1,1)) + diag(N/tau*ones(N,1),-1);
A(1,1)=a1;
A(1,N+1)=a2;
B=zeros(N+1,1);
B(1)=1;
Q=zeros(N+1,N+1);Q(1,1)=1;
R=1;
W=zeros(N+1,N+1);
W=diag(tau/N*ones(N+1,1));
W(1,1)=1;
G=B*R*B'; 
A_Star=inv(W)*A'*W; 

Thank you in advance for your helps,

Answer 1

[Torsten]

If it's a Riccati equation, use "icare" or "idare".

[N/A]

Thank you, but my reccatii equation is not as same as MATLAB calculate. Please see the the equation again.

-A_Star'*x-x*A-Q+x*G*x

I want to solve this equation directly.

[Torsten]

If your equation does not fit in the MATLAB scheme for the Riccati equation, try "fsolve":

fun = @(X)-Astar'*X-X*A-Q+X*G*X;
X0 = ones(N+1);
X = fsolve(fun,X0)

[N/A]

Thank you for your resporse. Actually I want to solve this

A_Star'*x-x*A-Q+x*G*x=0,

Do we still need initial condition for solving the problem?

Thank you in advance for your help .

[Torsten]

Every nonlinear solver needs a point from where to start ...

These are not initial conditions in the usual sense, but only a starting guess for X.

But it is quite important for that you get convergence - so if you have a good approximation, you should use it here.

But I just saw that X will be 51x51. I think you can forget about "fsolve".

Try to get a solver for the Riccati equation that includes your form of the equation.

[N/A]

Thank you for your response. I appreciate it.

[Torsten]

The following code seems to work:

N=50;
a1=1; a2=1;
tau=1;
A=zeros(N+1,N+1);
A=diag(-N/tau*ones(N+1,1)) + diag(N/tau*ones(N,1),-1);
A(1,1)=a1;
A(1,N+1)=a2;
B=zeros(N+1,1);
B(1)=1;
Q=zeros(N+1,N+1);Q(1,1)=1;
R=1;
W=zeros(N+1,N+1);
W=diag(tau/N*ones(N+1,1));
W(1,1)=1;
G=B*R*B';
A_Star=inv(W)*A'*W;

X0 = ones((N+1)^2,1);
X = fsolve(@(X)fun(X,N,A_Star,A,Q,G),X0)
X = reshape(X,N+1,N+1);
norm(-A_Star'*X-X*A-Q+X*G*X)

function res = fun(X,N,A_Star,A,Q,G)
  X = reshape(X,N+1,N+1);
  res = -A_Star'*X-X*A-Q+X*G*X;
  res = res(:);
end

[N/A]

thank you very much, it works!

Answer 2

[Sam Chak]

@shahin sharafi

You can find the solution for x with the Implicit algebraic Riccati equation solver:

[X, K, L] = icare(A_Star, [], Q, [], [], [], G)

For older versions of MATLAB (before R2019a), then use this:

[X, L, G] = care(A_Star, B, Q)

[N/A]

Thank you, but my reccatii equation is not as same as MATLAB calculate. Please see the the equation again.

-A_Star'*x-x*A-Q+x*G*x

I want to solve this equation directly.

[N/A]

Thank you for your response.