GATE 2018 EE Q.33: solution and answer

GATE 2018 · EEQ.33 · 2 marks · MCQ

Consider a system governed by the following equations

$\frac{d x _{1} ( t )}{d t} = x_{2} (t) - x_{1} (t)$
$\frac{d x _{2} ( t )}{d t} = x_{1} (t) - x_{2} (t)$

The initial conditions are such that $x_{1} (0) < x_{2} (0) < \infty$ . Let $x_{1 f} = lim_{t \to \infty} x_{1} (t)$ and $x_{2 f} = lim_{t \to \infty} x_{2} (t)$ . Which one of the following is true?

(A)
$x_{1 f} < x_{2 f} < \infty$
(B)
$x_{2 f} < x_{1 f} < \infty$
(C)
$x_{1 f} = x_{2 f} < \infty$
(D)
$x_{1 f} = x_{2 f} = \infty$

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Answer and solution

ANSWER

Both states settle at the same finite value, the average of the two initial values: $x_{1 f} = x_{2 f} = \frac{x _{1} ( 0 ) + x _{2} ( 0 )}{2} < \infty$ . The answer is (C).

Official answer: CSource: Official answer key, GATE 2018 EE (IIT Guwahati)

Write the equations in state-space form: $\overset{x}{˙} = A x$ with $x = [x_{1} x_{2}]$ and $A = [- 1 1 1 - 1]$ .
Eigenvalues: $det (λ I - A) = (λ + 1)^{2} - 1 = λ (λ + 2) = 0$ , so $λ = 0, - 2$ . Neither eigenvalue has a positive real part, and $λ = 0$ is a simple (non-repeated) root, so it contributes only a constant term. No state can grow without bound, which rules out (D).
Conserved quantity: adding the two equations gives $\frac{d}{d t} (x_{1} + x_{2}) = (x_{2} - x_{1}) + (x_{1} - x_{2}) = 0$ , so $x_{1} (t) + x_{2} (t) = x_{1} (0) + x_{2} (0)$ for all $t$ . This is the $λ = 0$ mode.
Difference mode: let $e = x_{2} - x_{1}$ . Then $\overset{e}{˙} = \overset{x}{˙}_{2} - \overset{x}{˙}_{1} = (x_{1} - x_{2}) - (x_{2} - x_{1}) = - 2 e$ , so $e (t) = e (0) e^{- 2 t}$ . This is the $λ = - 2$ mode.
Since $e (0) = x_{2} (0) - x_{1} (0) > 0$ , the gap stays positive for every finite $t$ but decays to zero, so $x_{2 f} - x_{1 f} = 0$ . That rules out (A) and (B).
Solving the sum and the difference together gives the final values: $x_{1 f} = x_{2 f} = \frac{x _{1} ( 0 ) + x _{2} ( 0 )}{2}$ This is finite because both initial values are finite.
So $x_{1 f} = x_{2 f} < \infty$ , which is option (C).

KEY IDEA

Break the system into its modes. The sum $x_{1} + x_{2}$ is conserved (the $λ = 0$ mode) and the difference $x_{2} - x_{1}$ decays as $e^{- 2 t}$ (the $λ = - 2$ mode). The states therefore meet at the average of their initial values. The system is marginally stable, not asymptotically stable: it stays bounded but does not return to zero.

WATCH OUT FOR

Reading $λ = 0$ as instability and choosing (D). A simple eigenvalue at the origin gives a constant mode, not a growing one; only a repeated eigenvalue at zero (with a Jordan block) would give a term that grows like $t$ . The other mistake is assuming the ordering $x_{1} < x_{2}$ survives to the limit and choosing (A). The gap stays positive for every finite $t$ but goes to zero as $t \to \infty$ .

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GATE 2018 EE Q.33: solution and answer

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