Eigenvalues and BIBO stability

Question

Could someone please explain to me the relationship between eigenvalues of a system matrix A and BIBO stability?

I've studied control engineering, and for example in modern control, we say that a system is BIBO satble if the real parts of eigenvalues (of matrix A) are all negative.

But today I had a lecture on digital signal processing, and the professor said that the system would have BIBO stability if all the eigenvalues are smaller than 1. Why is that? How does the condition change for discrete-time systems?

Answer 1

In control engineering, the system matrix A is given in Laplace transformation and indeed, the real parts of the eigenvalues of matrix A must be all negative for BIBO-stability.

In digital signal processing, the system matrix A is given in z-Transformation and there, indeed, the absolute values of the eigenvalues of matrix A must be less than 1 for BIBO-stability.

You compared two different system descriptions, hence the confusion.

Answer 2

For continuous systems, the real part of the eigenvalues must be less than one for strict stability. For discrete systems, the eigenvalues must lie inside the unit circle, that is, their modulus must be less than one.