ISRO EC 2023 Q.62: solution and answer

ISRO 2023 · ECQ.62 · 1 mark · MCQ

The matrix used for calculating the controllability is

(A)
$[B A B A^{2} B \dots]$
(B)
$[B B A B A^{2} \dots]$
(C)
$[B A B 2 A B \dots]$
(D)
$[B - A B - A^{2} B \dots]$

See the question in the paper

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Answer and solution

ANSWER

The controllability matrix is $[B A B A^{2} B \dots]$ , option (a).

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

For $\overset{x}{˙} = A x + B u$ with $n$ states and $x (0) = 0$ , the state at time $t$ is $x (t) = \int_{0}^{t} e^{A (t - τ)} B u (τ) d τ$ .
By the Cayley-Hamilton theorem, $e^{A s} = \sum_{k = 0}^{n - 1} α_{k} (s) A^{k}$ , so every reachable state is a linear combination of the columns of $B, A B, A^{2} B, \dots, A^{n - 1} B$ .
Hence the controllability matrix is $C = [B A B A^{2} B \dots A^{n - 1} B],$ and the system is controllable exactly when $rank C = n$ . This is option (a).
Option (b) is not valid: for an $n \times m$ input matrix $B$ with $m \neq = n$ (for example a single column), $B A$ is not even a defined product.
Option (c) uses $2 A B$ , a scalar multiple of $A B$ , which adds no new direction; the columns must follow successive powers of $A$ acting on $B$ .
Option (d) inserts minus signs that are not part of the standard form. They would not change the rank, but the matrix actually used is the one in (a).

KEY IDEA

Reachable states from the origin are spanned by $B, A B, A^{2} B, \dots, A^{n - 1} B$ ; the system is controllable iff this matrix has rank $n$ .

WATCH OUT FOR

Writing the products in the wrong order ( $B A$ instead of $A B$ ) or confusing the controllability matrix with the observability matrix, which is built from $C, C A, C A^{2}, \dots$ .

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