Question

Bounded-input bounded-output stability implies asymptotic stability for
A. Completely controllable system
B. Completely observable system
C. Uncontrollable system
D. Unobservable system
Which of the above statements is/are correct?
Choose the correct answer from the options given below:

• A. A and D only
• B. A and B only
• C. B and C only
• D. C and D only

Hint:

Think of controllability and observability as "windows" into the system's internal states. If a system is fully controllable and observable, the input-output behavior (transfer function) reveals everything about the internal behavior (eigenvalues). Only then does BIBO stability guarantee internal (asymptotic) stability.

Answer 1

The Correct Option is B

Step 1: Define the types of stability.

Asymptotic Stability: An internal property of a system. If un-driven, any initial state decays to zero over time. For an LTI system, this means all eigenvalues of the state matrix A are in the stable (left-half) plane.
BIBO Stability: An input-output property. For any bounded input, the output is also bounded. For an LTI system, this means all poles of the transfer function are in the stable plane.

Step 2: Relate internal (asymptotic) and external (BIBO) stability. The transfer function's poles correspond only to the modes of the system that are both controllable and observable. Asymptotic stability is about all modes (all eigenvalues).

If a system is asymptotically stable, it is always BIBO stable because all its internal modes decay to zero.
However, a system can be BIBO stable but not asymptotically stable. This happens if there is an unstable mode (an eigenvalue in the right-half plane) that is either uncontrollable or unobservable (or both). This unstable mode gets cancelled out when calculating the transfer function, so its pole does not appear, and the system appears stable from an input-output perspective.

Step 3: Determine the condition for equivalence. For BIBO stability to imply asymptotic stability, we must guarantee that there are no such hidden unstable modes. This is achieved if the system is completely controllable and completely observable. In this case, the set of transfer function poles is identical to the set of system eigenvalues, and the two forms of stability become equivalent.