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GATE 2014 EE (EE2) – Question 43

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

$\mathbf X(k+1)=\begin{pmatrix}a&a-1\\a+1&a\end{pmatrix}\mathbf X(k)$, with X₁(0) = 1 and X₂(0) = 0. Find poles for a = 1.

  1. 1 ± j0
  2. −1 ± j0
  3. ±1 + j0
  4. 0 ± j1

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Correct answer: (A) 1 ± j0

Explanation

Substitution gives the state matrix [[1,0],[2,1]]. Its characteristic determinant det(zI−A)=(z−1)²,so both eigenvalues are1. Thus the pole locations are **1±j0,A**. The repeated eigenvalue has a Jordan block;initial conditions do not change these characteristic poles.