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The second order dynamic system dx/dt = PX + Qu y = RX has the matrices P, Q and R as follows: P = [ -1 1 0 -3 ], Q = [ 0 1 ], R = [ 0 1 ] The system has the following controllability and observability properties:

1. Controllable and observable
2. Not controllable but observable
3. Controllable but not observable
4. Not controllable and not observable

Correct answer: Controllable but not observable

Solution

Controllability matrix [Q PQ] = [[0,1],[1,-3]] has determinant -1, so the system is controllable. Observability matrix [R; RP] = [[0,1],[0,-3]] has determinant 0, so it is not observable. Hence controllable but not observable.

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