I have a linear system on standard state space form
$\dot{x}(t) = Ax(t) + Bu(t)$
I would like to find the solution to this, which usually is
$x(t) = e^{At}x(0) + \int\limits_0^t e^{A(t-\tau)}Bu(\tau)d\tau$
Since, in my case, $u(t) = u$ is constant (but undetirmened) from $0$ to $t$, it can be moved outside the integral, and the integral can be computed symbolically
$x(t) = e^{At}x(0) + A^{-1}(e^{At} - I)Bu$
Now, unfortunately my $A$ matrix is singular, so the formula above can not be computed. I've tried doing a coordinate transformation, to no avail.
I also tried to expand the matrix exponential to remove the inverse
$A^{-1}(e^{At} - I)Bu = A^{-1}(I + At + \frac{A^2t^2}{2} + \frac{A^3t^3}{6} + \frac{A^4t^4}{24} + ... - I)Bu = (It + \frac{At^2}{2} + \frac{A^2t^3}{6} + \frac{A^3t^4}{24} + ...)Bu$
But since $A$ is singular, the exponentiation of it breaks down numerically.
Any ideas?