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GATE 2014 EE (EE2) – Question 56

Control Systems · State space model, Solution of state equations of LTI systems · 2 marks · Multiple choice

$\dot X=PX+Qu$, y = RX, with $P=\begin{pmatrix}-1&1\\0&-3\end{pmatrix}$, $Q=\begin{pmatrix}0\\1\end{pmatrix}$ and R = [0,1]. Find controllability and observability.

  1. Controllable and observable
  2. Not controllable but observable
  3. Controllable but not observable
  4. Neither controllable nor observable

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Correct answer: (C) Controllable but not observable

Explanation

The controllability matrix[Q,PQ]=[[0,1],[1,−3]]has determinant−1and rank2. The observability matrix[R;RP]=[[0,1],[0,−3]]has rank1. Therefore the system is **controllable but not observable,C**. The first state can affect the internal dynamics without appearing in the chosen output.