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