This is a common mistake I see being made. You have a problem that has symmetries in it. x(1), x(2), x(3) all play similar parts in the equations. But when you start all three variables out at the same values, you place the solver in a spot that is essentially a saddle point. It sees a locally perfectly flat surface at that location, and while there are places it could go to improve the result, it thinks the objective is flat.
VDC=5.53; % DC-LINK VOLTAGE
V1=2.596; % THE FIRST HARMONIC OF THE OUTPUT VOLTAGE
V3=0; % THE THIRD HARMONIC OF THE OUTPUT VOLTAGE
V5=0.488; % THE FIFTH HARMONIC OF THE OUTPUT VOLTAGE
% Define the nonlinear equations for the angles
f = @(x) [
(4/(pi))*VDC*(sin(x(1))-sin(x(2))+sin(x(3)))-V1;
(4/(3*pi))*VDC*(sin(3*x(1))-sin(3*x(2))+sin(3*x(3)))-V3;
(4/(5*pi))*VDC*(sin(5*x(1))-sin(5*x(2))+sin(5*x(3)))-V5];
% Solve for angles using fsolve
angles = fsolve(f,[pi/6, pi/6, pi/6])
angles = fsolve(f,[pi/6, 2*pi/6, 3*pi/6])
f(angles)
So we see the solver is now happy at the final point (that result is essentially zero to within tolerance trash), even though it could not decide where to go when you started it out at your initial choice. One solution can be to use a perturbational randomizer as a start point.
fsolve(f,[pi/6, pi/6, pi/6] + randn(1,3)*0.1)
And you see there, we get again a solution, but curiously, the solution is the same one we had before, but in a different sequence, because I started it from a randomly different location. This just reinforces what I said before. The problem is perfectly symmetrical.
