Step 1: Define stability types.
Asymptotic Stability: A system's intrinsic property where, without external input, any initial state converges to zero over time. For Linear Time-Invariant (LTI) systems, this means all eigenvalues of the state matrix A reside in the left-half plane.
BIBO Stability: An input-output characteristic. Bounded input always results in bounded output. In LTI systems, this requires all transfer function poles to be in the stable plane.
Step 2: Establish the relationship between asymptotic and BIBO stability.The transfer function's poles only represent the controllable and observable system modes. Asymptotic stability considers all modes (all eigenvalues).
Asymptotically stable systems are always BIBO stable because all internal modes decay to zero.
Conversely, a BIBO stable system isn't necessarily asymptotically stable. This occurs with an unstable mode (eigenvalue in the right-half plane) that's either uncontrollable, unobservable, or both. This mode cancels out during transfer function calculation, hiding its pole, making the system appear stable from an input-output perspective.
Step 3: State the equivalence condition.For BIBO stability to guarantee asymptotic stability, there must be no hidden unstable modes. This is ensured when the system is completely controllable and completely observable. Consequently, the transfer function poles and system eigenvalues are identical, making the two stability forms equivalent.