Abstract
The input-output structures of linear, time-invariant, discrete-time systems are investigated. The newly defined perfect reachability and the perfect (null) controllability are discussed together with the perfect observability and the perfect reconstructibility which have been studied by other authors. First the definition is given of those subspaces in the state space which characterize those structural properties, and then it is shown that the subspaces are invariant to the state feedback and the output injection. Making use of this invariance property, the general systems with the direct transmission are reduced to those without the direct transmission.
With the aid of the newly defined subspaces, some formulae related to the perfect observability are rederived, the proof of perfect observability being much simpler than that of other authors. The duality between the perfect observability and the perfect reachability and so forth is also clarified. The perfect observability is equivalent to the conservation of the observability for any state feedback, while the perfect reachability is equivalent to the conservation of the reachability for any output injection. There is no such equivalence relation for the perfect reconstructibility and the perfect controllability. The perfect observability does not necessarily imply the perfect reconstructibility and vice versa. These points are made clear by presenting counterexamples.
The newly defined structural properties are examined with relation to the input observability and the output reproducibility, and to the solution of the Ricatti difference equations.
