Solve long matrices equation

Hello,

please I would like help to solve this equation with matlab ( you can see it in the screenshot). I want to find the matrix P11. All other matrices in the equation are known.

Answer 1

[cui,xingxing]

From your screenshot formulas, you should first prioritize the formulas Manipulation and Simplification, and then try to solve the analytical solution using some of the functions of the Symbolic Math Toolbox™, and if the analytical solution doesn't exist theoretically, you should try to get a numerical solution, which is valuable in a lot of engineering. Secondly, based on your equation, which is in matrix expression form, you should do a specific analysis based on your specific application context to simplify your process.

Finally, see the following link for generalities about the functionality of solving with the Symbolic Math Toolbox™:

https://ww2.mathworks.cn/help/symbolic/equation-solving.html

[TESSO WOAFO]

Thank you very much for your answer. I solved it using the MATLAB function fsolve for nonlinear equations.

Answer 2

[John D'Errico]

This is a classic algebraic Ricatti equation.

https://en.wikipedia.org/wiki/Algebraic_Riccati_equation

How do you solve it in MATLAB? It looks like the care function does that, which is now obsolete, and superceeded by icare. (I hope they never have to come up with a name to supercede icare. What would they use? Maybe ycare? Ok, I know, who cares?)

help icare

 ICARE  Implicit solver for continuous-time Riccati equations.
 
    [X,K,L] = ICARE(A,B,Q,R,S,E,G) computes the stabilizing solution X of
    the continuous-time algebraic Riccati equation
                                          -1
       A'XE + E'XA + E'XGXE - (E'XB + S) R  (B'XE + S') + Q = 0 .
 
    The matrices Q,R,G must be Hermitian and R,E must be invertible. When 
    omitted or set to [], B,R,S,E,G default to the values B=0, R=I, S=0, 
    E=I, and G=0. Scalar-valued Q,R,G are interpreted as multiples of the 
    identity matrix. ICARE also returns the state-feedback gain K and the 
    closed-loop eigenvalues L given by
            -1
       K = R  (B'XE + S'),    L = EIG(A+G*X*E-B*K,E) .
 
    ICARE returns [] for X,K when there is no finite stabilizing solution.
 
    [X,K,L,INFO] = ICARE(...) also returns a structure INFO with the
    following data:
       * Vectors SX and SR of state and R scalings
       * Basis [U;V;W] of the stable invariant subspace of the associated
         scaled matrix pencil
       * Diagnostic "Report" with value
            0    Accurate solution found (success)
            1    Solution accuracy may be poor
            2    Solution is not finite
            3    Hamiltonian spectrum [L;-L] has eigenvalues on imag. axis.
            4    Pencil is singular ([B;S;R] rank deficient)
    The solution X and gain K are related to DX=diag(SX), DR=diag(SR), and
    U,V,W by
                 -1    -1                -1
       X = DX V U  DX E ,    K = - DR W U  DX .
 
    [...] = ICARE(...,'noscaling') turns off the built-in scaling and
    sets all entries of SX and SR to one. This speeds up computation but 
    can be detrimental to accuracy when A,B,Q,R,S,E,G are poorly scaled.
 
    [...] = ICARE(...,'anti') computes the anti-stabilizing solution X
    that puts all eigenvalues of (A+G*X*E-B*K,E) in the right half-plane.
 
    See the reference page for additional algorithmic details.
 
    See also IDARE.

    Documentation for icare
       doc icare

As you can see, it handles exactly your problem. Just plug in your matrices. You would have something like this:

Q = 0

S = B

G = 0

X = P11

E = I

A = 2*A

I think I got them right, but it would be worth a quick check, as the form in icare does not have B factored out and paired with R^-1.

[TESSO WOAFO]

Thank you very much for your answer. I solved it using the MATLAB function fsolve for nonlinear equations.