I have a continuous-time state-space system $\dot{x} = Ax + Bu$ with
$A = \begin{bmatrix}0 & 0 & 0 & 1.0000 & 0 & 0\\ 0 & 0 & 0 & 0 & 1.0000 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1.0000 \\ 0.5000 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2.0000 & 0 & 0 & 0 & 0 \\ & 0 & -2.0000 & 0 & 0 & 0\end{bmatrix}$ and
$B = \begin{bmatrix}0 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1\end{bmatrix}$.
The controllability matrix $\mathcal{C}_{contin}$ has eigenvalues 8.3955, 9.4014, -2.4786, 0.2951 + 1.3889i, 0.2951 - 1.3889i, and -0.8557, so it is of full rank and the system is controllable.
I'm trying to implement deadbeat control, so I use a digital sampler with a sampling time of $T_s = 0.1$ seconds and zero-order hold. The resulting system $x_{k+1} = Gx_k + Hu_k$ has
$G = \begin{bmatrix} 1.0025 & 0 & 0 & 0.1001 & 0 & 0 \\ 0 & 1.0100 & 0 & 0 & 0.1003 & 0 \\ 0 & 0 & 0.9900 & 0 & 0 & 0.0997 \\ 0.0500 & 0 & 0 & 1.0025 & 0 & 0 \\ 0 & 0.2007 & 0 & 0 & 1.0100 & 0 \\ 0 & 0 & -0.1993 & 0 & 0 & 0.9900 \end{bmatrix}$ and
$H = \begin{bmatrix} 0.1051 \\ 0.1053 \\ 0.1047 \\ 0.1026 \\ 0.1104 \\ 0.0897\end{bmatrix}$.
The controllability matrix $\mathcal{C}_{discrete}$ has eigenvalues 0.7082, -0.0576, 0.0013, -0.0010, 0, and 0, so it is evidently NOT of full rank and the system is evidently NOT controllable.
My questions is, is this possible? How can a continuous-time system be controllable but its corresponding discrete-time equivalent not be controllable? Would changing the sampling time or $B$ matrix help make the discrete system controllable? I've never experienced this phenomenon before.
