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A second-order linear time-invariant system is described by the following state equations (d/dt) x1(t) + 2 x1(t) = 3 u(t) (d/dt) x2(t) + x2(t) = u(t) where x1(t) and x2(t) are the two state variables and u(t) denotes the input. If the output c(t) = x1(t), then the system is

1. controllable but not observable
2. observable but not controllable
3. both controllable and observable
4. neither controllable nor observable

Correct answer: controllable but not observable

Solution

With x1' = -2x1 + 3u, x2' = -x2 + u, the controllability matrix has full rank 2 (both states driven by u), but with c = x1 the observability matrix has rank 1 (x2 is hidden). So the system is controllable but not observable, option 0, not stored option 1.

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