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GATE 2014 EC (EC1) – Question 56

Control Systems · State Variable Model and Solution of State Equation · 2 marks · Multiple choice

A state-space model is $\dot x=\begin{pmatrix}-1&1&0\\0&-1&0\\0&0&-2\end{pmatrix}x+\begin{pmatrix}0\\4\\0\end{pmatrix}u$, $y=\begin{pmatrix}1&1&1\end{pmatrix}x$. The system is ______.

  1. controllable and observable
  2. uncontrollable and observable
  3. uncontrollable and unobservable
  4. controllable and unobservable

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Show answer and explanation

Correct answer: (B) uncontrollable and observable

Explanation

The controllability columns are B=(0,4,0), AB=(4,−4,0) and A²B=(−8,4,0), of rank2: the third state cannot be driven. The observability rows are C=(1,1,1), CA=(−1,0,−2), CA²=(1,−1,4), whose determinant is1 and rank3. Hence the system is **uncontrollable and observable,B**.