Solve function is unable to find an explicit solution

Hello,

the code is as follows.

clc;clear all
A = [0.7 0.2;
    0.05 0.64];
C = [0.5 -0.8;
    0 0.7];
Q = [0.5 0;
    0 0.7];
R = [1 0;
    0 0.8];
T = sym('T',[2 2]);
L = sym('L',[2 2]);
[Pba,L2,G] = dare(A',C',Q,R);
P = C*Pba*C'+R;
f = trace(P^(-1)*(T*P*T'+L-P));
g = L(1,2)-L(2,1);
assume(L(1,2)==L(2,1) & L(1,1)>=0 & det(L)>=0);
x = solve(f,[T,L],"ReturnConditions",true)

The output is

Warning: Unable to find explicit solution. For options, see help. 
> In sym/solve (line 317)
In main_240331 (line 20) 

x = 

struct with fields:

T1_1: [0×1 sym]
T2_1: [0×1 sym]
T1_2: [0×1 sym]
T2_2: [0×1 sym]
L1_1: [0×1 sym]
L2_1: [0×1 sym]
L1_2: [0×1 sym]
L2_2: [0×1 sym]
parameters: [1×0 sym]
conditions: [0×1 sym]

I already know that the solutions exist, such as T=[1 0;0,1], L=[0,0;0,0]. Why is the solve function here unable to find the explicit solution?

Or, are there any other methods to get the solutions?

Thanks!

Answer 1

[Torsten]

If you only want one possible solution, use

x = solve(f,[T,L])

If you remove your assumptions, you get the general form of the solution.

[Joey Qi]

Hi, I need to get all the possible solutions, and the assumptions here are also needed. In this case, the matlab is unable to find explicit solutions.

[Torsten]

In this case, the matlab is unable to find explicit solutions.

It is unable to find all possible solutions. If you remove "ReturnConditions",1 , it gives you a special solution.

But after generating the general solution without the assumptions, you can simply include your assumptions by demanding L1_1 >= 0, L1_2 = L2_1 and L1_1*L2_2 - L1_2*L2_1 >= 0 where L1_1, L2_2, L1_2 and L2_1 are the representations obtained by the code.

clc;clear all
A = [0.7 0.2;
    0.05 0.64];
C = [0.5 -0.8;
    0 0.7];
Q = [0.5 0;
    0 0.7];
R = [1 0;
    0 0.8];
T = sym('T',[2 2]);
L = sym('L',[2 2]);
[Pba,L2,G] = dare(A',C',Q,R);
P = C*Pba*C'+R;
f = trace(P^(-1)*(T*P*T'+L-P));
x = solve(f,[T,L],'ReturnConditions',1)

x = struct with fields:
          T1_1: z
          T2_1: z1
          T1_2: z2
          T2_2: z3
          L1_1: conj(z2)*((3973518972162105*z)/9007199254740992 - (175580345583681*z2)/140737488355328) - (8353311061136183*z5)/23623227723860540 - (1591643986987335*z6)/118116...
          L2_1: z4
          L1_2: z5
          L2_2: z6
    parameters: [z    z1    z2    z3    z4    z5    z6]
    conditions: symtrue

Thus you have 7-parametric solution where the parameters are constrained by the conditions

imag(conj(z2)*(...)) = 0
real(conj(z2)*(...)) >= 0 
z4 = z5
imag(conj(z2)*(...)*z6 - z4*z5) = 0
real(conj(z2)*(...)*z6 - z4*z5) >= 0    

[Joey Qi]

Thank you, Torsten! You inspired me to solve the problem. I have already got some solutions using your method.