Problem with singular Q matrix in discrete-time Riccati equations

[olof3]

Something like the following has to be solved for discrete-time LQR problems with a penalty on the output signal.

A = randn(2,2)
B = randn(2)
C = [-100 1]
Q = C'*C
dare(A, B, Q, 1)

However this code throws the error Q must be positive semi-definite.

The problem is that Q has a negative eigenvalue that is on the order of -eps(). I believe that this condition is too restrictive and a meaningful solution is found by both MatLab and MatrixEquations.jl for most A and B.

The numerics of RIccati equations might be a bit too much to get into, but this could be another reason to consider using MatrixEquations.jl (note that its ared function use the opposite order of the Q and R arguments).

A quick fix would be to just use something like Q = C'*C + eps()*I