The matrix used for calculating the controllability is

Controllability Matrix in Control Systems

Definition: Controllability is a fundamental concept in control system theory, which determines whether a system's state can be driven to a desired state using appropriate control inputs over a finite duration. The controllability matrix is a mathematical tool used to check the controllability of a linear time-invariant (LTI) system.

State-Space Representation:

A linear time-invariant (LTI) system can be represented in state-space form as:

dx/dt = Ax + Bu

y = Cx + Du

• Here, x is the state vector, u is the input vector, y is the output vector, A is the system matrix, B is the input matrix, C is the output matrix, and D is the feedthrough matrix.

Controllability Matrix:

To determine the controllability of a system, the controllability matrix is defined as:

[B AB A²B ... A^n-1B]

• Here, A is the system matrix and B is the input matrix.
• The columns of the controllability matrix represent the influence of the input u through the dynamics of the system.

Explanation of Correct Option:

The correct option is:

Option 1: [B AB A²B...]

This representation is the standard form of the controllability matrix used in control system theory. The matrix is constructed by appending B, AB, A²B, and so on, up to A^n-1B, where n is the order of the system.