GATE 2025 EE Q.62: solution and answer

GATE 2025 · EEQ.62 · 2 marks · NAT

Consider the state-space model

$\dot{x} (t) = A x (t) + B r (t),$
$y (t) = C x (t)$

where $x (t)$ , $r (t)$ , $y (t)$ are the state, input and output, respectively. The matrices $A, B, C$ are given below

A = [0 - 2 1 - 3], B = [01], C = [10]

The sum of the magnitudes of the poles is ________ (round off to nearest integer).

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Answer and solution

ANSWER

The poles are $s = - 1$ and $s = - 2$ , so the sum of their magnitudes is $3$ .

Official answer: 3Source: Official answer key, GATE 2025 EE (IIT Roorkee)

The poles are the eigenvalues of $A$ , which are the roots of the characteristic equation $det (s I - A) = 0$ .
$s I - A = [s 2 - 1 s + 3] \Rightarrow det (s I - A) = s (s + 3) + 2 = s^{2} + 3 s + 2$
Factorise: $s^{2} + 3 s + 2 = (s + 1) (s + 2) = 0$ , so $s = - 1$ and $s = - 2$ .
Check for pole-zero cancellation. With $adj (s I - A) = [s + 3 - 2 1 s]$ , we get $G (s) = C (s I - A)^{- 1} B = \frac{1}{s ^{2} + 3 s + 2}$ . The numerator is 1, so nothing cancels and both eigenvalues are poles of the transfer function too. (The matrix is in controllable canonical form, so you can also read the coefficients 2 and 3 straight from the last row.)
Sum of magnitudes: $∣ - 1∣ + ∣ - 2∣ = 1 + 2 = 3$ .

KEY IDEA

The poles of a state-space system are the eigenvalues of $A$ , the roots of $det (s I - A) = 0$ . For a companion (controllable canonical) matrix, the last row $[- a_{0} - a_{1}]$ gives the characteristic polynomial $s^{2} + a_{1} s + a_{0}$ directly. Check $C (s I - A)^{- 1} B$ for cancellation before treating every eigenvalue as a transfer-function pole.

WATCH OUT FOR

Adding the poles themselves, $(- 1) + (- 2) = - 3$ , which is the trace of $A$ , instead of adding their magnitudes. Another common slip is a sign error in the last row: reading the polynomial as $s^{2} - 3 s - 2$ instead of $s^{2} + 3 s + 2$ .

From the GATE 2025 EE paper. Every question of this paper · Download the full paper · All 16 State space questions, with notes

GATE 2025 EE Q.62: solution and answer

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