GATE 2015 EE (EE2) – Question 62

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

The numerical oscillator equations are $(x_{k+1}-x_k)/h=y_k$ and $(y_{k+1}-y_k)/h=-x_k$. Which stability statement is true?

unstable for all $h>0$
stable for $h>1/\pi$
stable for $0<h<1/(2\pi)$
stable for $1/(2\pi)<h<1/\pi$

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Show answer and explanation

Correct answer: (A) unstable for all $h>0$

Explanation

The forward-Euler update matrix is
$$A_d=\begin{bmatrix}1&h\\-h&1\end{bmatrix}.$$
Its eigenvalues are 1±jh, whose magnitudes are √(1+h²)>1 for every h>0.

Thus the discrete state grows even though the underlying continuous undamped oscillator has bounded trajectories. The numerical system is **unstable for every positive h, A**.

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