CHARACTERIZATION OF STRONG OBSERVABILITY AND CONSTRUCTION OF AN OBSERVER

For given matrices A, B, C there is considered the time-invariant line ar 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 piecewi se continuous control Il(t) always implies x(t) = 0. This means that, for any piecewise continuous input u(t), the output y(t) can vanish id entically 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 expr esses x(t) in terms of y(t) alone, for time-invariant systems (i.e. co nstant matrices A, B, C) is one part of the contents of this note. The other part consists of characterizations of strong observability by r ank conditions concerning the matrices A, B, and C (similarly to the w ell-known rank condition for controllability or observability).