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The state representation of a second order system is
x1 = – x1 u, x2 = x1 – 2x2 + u
Consider the following statements regarding the above system:
1. The system is completely state controllable.
2. If x1 is the output, then the system is completely output controllable.
3. If x2 is the output, then the system is completely output controllable. Of these statements—
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- 1, 2 and 3 are correct
- 1 and 2 are correct
- 2 and 3 are correct
- 1 and 3 are correct
- 1, 2 and 3 are correct
Correct Option: C
The given state equation can be written as
 |  | = | | | | | + | | | u | |||||
| x2 | 1 | -2 | x2 | 1 |
The controllability matrix is
QC = [B: AB] ≠ 0
B = [1 1]
| AB = | | | | | = | | | ||||
| 1 | -2 | 1 | 1 |
| AB = | | | | | = | | | ||||
| 1 | -2 | 1 | -1 |
| Qc = | | | ||
| 1 | -1 |
Since determinant QC = 0, the system is not state controllable. Hence statement (C) is correct.
