Controllability of a System

Controllability refers to the ability of a control system to transfer the system state from any initial state to any desired state in a finite interval of time. In other words, it is possible to steer the system from any starting point to any endpoint using a control input. A system is said to be controllable if it satisfies the following conditions:

1. The system is linear and time-invariant.
2. The system is described by a set of first-order ordinary differential equations.
3. The control input can be applied directly to the system.

Observable of a System

Observable refers to the ability of a control system to determine the internal state of a system based on its outputs. In other words, it is possible to reconstruct the system state using the available measurement data. A system is said to be observable if it satisfies the following conditions:

1. The system is linear and time-invariant.
2. The system is described by a set of first-order ordinary differential equations.
3. The output measurements are complete and noise-free.

Difference between Controllability and Observability

Controllability and observability are two important concepts in control theory. While controllability deals with the ability to control the system, observability deals with the ability to observe the system. The main difference between controllability and observability is that controllability deals with the internal state of the system, while observability deals with the external behavior of the system.

Conclusion

In conclusion, a system is said to be controllable if it is possible to transfer the system state from any initial state to any desired state in a finite interval of time. This property is important in control theory as it allows us to design control systems that can steer the system to a desired state. Observability, on the other hand, refers to the ability to determine the internal state of the system based on its outputs, which is important in control systems where the internal state cannot be directly measured.