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A linear second-order single-input continuous-time system is described by the following set of differential equations :
X1̇ (t) = – 2 X1 (t) + 4 X2 (t)
X2̇ (t) = 2 X1 (t) – X2 (t) + u (t)
where X1 (t) and X2 (t) are the state variables and u (t) is the control variable.
The system is
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- controllable and stable
- controllable but unstable
- uncontrollable and unstable
- uncontrollable but stable
Correct Option: B
In matrix form
 | Ẋ1(t) |  | = | | -2 | | | X1(t) | | + | | 0 | | u(t) | |||
| Ẋ2(t) | 2 | -1 | X2(t) | 1 |
Test for controllability
| [B : AB] = | | | 0 | | | -2 | | | 0 | | | |||||
| 1 | 2 | -1 | 1 |
| = | | | Rank is 2, hence controllable | ||||
For stability, sI – A = 0
| ∴ | | | = 0 | ||||
⇒ s2 + 3s - 4 = 0
Routh criterion is,
| s2 | 1 | -4 |
| s | 3 | |
| s | -4 |
Since sign changes, hence system is not stable.
