ISRO EC 2023 Q.61: solution and answer

ISRO 2023 · ECQ.61 · 1 mark · MCQ

When converting the following system into the standard state space form $\overset{x}{¨} + 3 \overset{x}{˙} - 9 x = 3 sin (w t)$ the resultant $A$ matrix is

(A)
$[0 - 9 1 - 3]$
(B)
$[09 - 1 - 3]$
(C)
$[09 1 - 3]$
(D)
$[0 - 9 - 1 3]$

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Answer and solution

ANSWER

$A = [09 1 - 3]$ , option (c).

Official answer: CSource: ISRO's revised and final answer key of 16 February 2024

Choose the phase variables $x_{1} = x$ and $x_{2} = \overset{x}{˙}$ , so that $\overset{x}{˙}_{1} = x_{2}$ .
Solve the equation for the highest derivative: $\overset{x}{¨} = 9 x - 3 \overset{x}{˙} + 3 sin (w t)$ .
Substitute the states: $\overset{x}{˙}_{2} = 9 x_{1} - 3 x_{2} + 3 sin (w t)$ .
Write $\dot{x} = A x + B u$ with $u = sin (w t)$ : $[\overset{x}{˙}_{1} \overset{x}{˙}_{2}] = [09 1 - 3] [x_{1} x_{2}] + [03] u$
So $A = [09 1 - 3]$ , which is option (c).
Option (a) has $- 9$ in place of $+ 9$ . Options (b) and (d) have $- 1$ in the top-right entry, and (d) also has $- 9$ and $+ 3$ in the bottom row. None of these matches the derived rows $[01]$ and $[9 - 3]$ .

KEY IDEA

Take $x_{1} = x$ and $x_{2} = \overset{x}{˙}$ , then move every term except the highest derivative to the right-hand side. The coefficients of $x_{1}$ and $x_{2}$ in that equation form the last row of $A$ .

WATCH OUT FOR

Copying the left-hand-side coefficients ( $- 9$ , $+ 3$ ) straight into the bottom row. They must change sign when $\overset{x}{¨}$ is isolated, giving $+ 9$ and $- 3$ .

From the ISRO EC 2023 paper. Every question of this paper · The full paper (PDF, ISRO) · All 2 ISRO State space questions, with notes