Characterization of strong observability and construction of an observer

For given matrices A, B, C there is considered the time-invariant linear system x = Ax + Bu, y = Cx with state x, input u, and output y. It is called strongly observable if x = Ax + Bu, Cx(t) ≡ 0 with a piecewise continuous control u(t) always implies x(t) ≡ 0. This means that, for any piecewise continuous input u(t), the output y(t) can vanish identically only if the state x(t) vanishes already, so that the state x(t) can be expressed (“observed”) by the output y(t) alone [without knowing u(t)]. The derivation of such a formula (observer), which expresses x(t) in terms of y(t) alone, for time-invariant systems (i.e. constant matrices A, B, C) is one part of the contents of this note. The other part consists of characterizations of strong observability by rank conditions concerning the matrices A, B, and C (similarly to the well-known rank condition for controllability or observability).