Question

The state equation of a dynamical system is: \(\dot{X} = AX + Bu\), where \([A]\) is a \(3 \times 3\) system matrix and \([B]\) is a \(3 \times 1\) input matrix. Then the controllability matrix \(Q_c\) is

• A. \(Q_c = [B \mid AB \mid A^2B]\)
• B. \(Q_c = [A \mid AB \mid A^2B]\)
• C. \(Q_c = [B \mid AB \mid AB^2]\)
• D. \(Q_c = [B \mid A \mid A^2]\)

Hint:

The controllability matrix always starts with the input matrix \(B\) and progresses by premultiplying by \(A\) at each step until you reach \(A^{n-1}B\). For a 3rd-order system, this sequence is always \(B, AB, A^2B\).

Answer 1

The Correct Option is A

Concept: In state-space control analysis, a system described by the state equations is completely state controllable if it is possible to transfer the system from any initial state to any other desired state within a finite time interval using an unconstrained control vector. Kalman’s test for controllability requires forming the composite controllability matrix \(Q_c\). For an \(n \times n\) matrix \(A\) and an \(n \times m\) matrix \(B\), the structural design of \(Q_c\) is defined as: \[ Q_c = \begin{bmatrix} B & AB & A^2B & \cdots & A^{n-1}B \end{bmatrix} \]

Step 1: Determine system dimensional parameters. The system is defined with:
• Order of the system matrix \(A\) is \(n = 3\) (since it is a \(3 \times 3\) matrix).
• Input matrix \(B\) has dimensions \(3 \times 1\).

Step 2: Construct the tracking sequence matrix up to power limit \(n-1\). Since \(n = 3\), the maximum power index of matrix \(A\) inside our matrix block is: \[ n - 1 = 3 - 1 = 2 \] Substituting this limit into Kalman's expression: \[ Q_c = \begin{bmatrix} B & AB & A^2B \end{bmatrix} \] Writing this configuration using vertical partitioning breaks gives exactly: \[ Q_c = [B \mid AB \mid A^2B] \] This perfectly aligns with Option (A).