Observable System

Observable System
An observable system is one in which every state of the system can be completely identified by measurements of the outputs at a finite time interval. In other words, the internal state of the system can be determined or observed based on the output measurements. This property is crucial for the analysis, control, and estimation of a system.

Controllable System
On the other hand, controllability refers to the ability to manipulate the system's internal state using appropriate inputs. A system is said to be controllable if a control input can be applied to steer the system from any initial state to any desired state within a finite time interval.

Relationship between Observability and Controllability
Observability and controllability are two important concepts in system theory. They are related but distinct properties of a system. While observability deals with the ability to determine the system's internal state based on output measurements, controllability focuses on the ability to manipulate the system's internal state using inputs.

Observable but not Controllable
It is possible for a system to be observable but not controllable. This means that the system's internal state can be fully determined based on output measurements, but it cannot be manipulated to reach any desired state. In such cases, it may not be possible to achieve certain control objectives or design controllers that can steer the system to desired states.

Observable and Controllable
In many practical applications, it is desirable to have both observability and controllability. A system that is both observable and controllable allows for effective control design and analysis. It enables the estimation of the system's internal state, which is crucial for feedback control strategies. Additionally, it provides the ability to manipulate the system's state to achieve desired control objectives.

Conclusion
In the given question, the correct answer is option 'B' - Observable. This means that the system being referred to is observable, indicating that its internal state can be completely determined based on output measurements at a finite time interval.