According to Anstrom's "Computer controlled system" page 37 paragraph above example 2.5, in order there is a corresponding continuous system, the eigenvalue of the state space matrix phi must not have eigenvalues on the negative real axis, why?

Eigenvalues on the negative real axis in the sampled time domain correspond to continuous-time eigenvalues with infinite real parts. That's kind of hard to achieve in practice.

I should also mention that there is an uncertainty in the mapping of eigenvalues back from the z domain to the s domain. The magnitude of the z-domain eigenvalue maps unambiguously to the s domain, but the polar angle of the eigenvalue in the z domain maps to an infinite number of eigenvalues spaced at 2 * pi * sampling frequency apart in the imaginary direction in the s domain.

So you can take a system description in the z domain and come up with _a_ continuous-time system, maybe, but you can't come up with _the_ continuous-time system without some further constraints on the eigenvalues.

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