Controllability and observabliity stand in a dual relationship to one another. This is clear in the necessary and sufficient conditions: that [B AB ⋯An−1B] must be rank n (for controllability), and
[C CA⋮CAn−1] must be of rank n (for observability).
This duality is also nicely exhibited in the starred question on Example Sheet 3, Question 11. You should now be able to do this question.
Notice that a system can be stabilizable without being controllable. E.g. the scalar system with xt+1=(1/2)xtis trivially stabilizable, but it is not controllable.
[C CA⋮CAn−1] must be of rank n (for observability).
This duality is also nicely exhibited in the starred question on Example Sheet 3, Question 11. You should now be able to do this question.
Notice that a system can be stabilizable without being controllable. E.g. the scalar system with xt+1=(1/2)xtis trivially stabilizable, but it is not controllable.
