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Approach I:

Use Kalman Canonical Form Decomposition:

Approach II:

Only check if the modes corresponding to unstable eigenvalues are observable or not. (Since stable modes can always tend to zero asymptotically, we only need to check if the observability of the unstable modes).

  • Find out eigenvalues of transition matrix A \in \mathbb{R}^{n\times n}
  • If all eigenvalues are Hurwitz (continuous-time system) or Schur (discrete-time system), then (C,A) is detectable
  • If there are unstable eigenvalues, say, \lambda, then check

\text{rank}\begin{bmatrix} A-\lambda I\\C\end{bmatrix}

is equal to n or not.