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Bounded-input bounded-output stability implies asymptotic stability for 1. Completely controllable system 2. Completely observable system 3. Uncontrollable system 4. Unobservable system Which of the above statements are correct?

Bounded-input bounded-output stability implies asymptotic stability for 1. Completely controllable system 2. Completely observable system 3. Uncontrollable system 4. Unobservable system Which of the above statements are correct? Correct Answer 1 and 2 only

BIBO stability: A linear system is said to be BIBO stable if the output is bounded for an arbitrary bounded input.

Asymptotic stability: It is the same as BIBO stability, except pole-zero cancellation should not be there.

If a system is asymptotic stable, then the system is BIBO stable but not vice versa.

Asymptotic stability → BIBO stability

BIBO stability + no pole-zero cancellation → Asymptotic stability

To have no pole-zero cancellation, the state space representation is minimal, i.e. both controllable and observable.

BIBO stability + controllable and observable system → Asymptotic stability

Therefore, bounded-input bounded-output stability implies asymptotic stability for a completely controllable and observable system.

This also implies that a marginally stable system with minimal realization is not BIBO stable.

Important Points:

Controllability:

  • The concept of controllability of a system which is related to the transfer of any initial state of the system to any other desired state, in a finite length of time by application of proper inputs.
  • A system is said to completely state controllable if it is possible to transfer the system state from any initial state x(t0) to any other desired state X(tf) in a specified time interval (tf) by a control vector u(t).
  • A system is said to completely output controllable if it is possible to construct an unconstrained input vector u(t) which will transfer any given initial output y(t0) to any final output Y(tf) in a finite time interval t0 ≤ t ≤ tf.

 

Observability:

  • The observability is related to the problem of determining the system state by measuring the output for a finite length of time.
  • A system is said to be completely observable if every state X(t0) can be completely identified by measurements of the outputs Y(t) over a finite time interval.
  • If the system is not completely observable means that few of its state variables are not practically measurable and are shielded from the observation.

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