Time-varying state feedback and output injection: duality and distinctions

The classical results on the duality of controllability and observability of linear systems are very well known and deeply analyzed in both cases of time invariant and time-varying linear systems. However the significant difference appears in the important properties of the responses in the problems of stabilization by state feedback (controller) and by output injection (observer) even in cases when only the gains become time-varying. Namely, it is possible to design a time-varying observer with fast decaying error almost without peaking (or equivalently to stabilize the system by output injection providing a fast decay without large deviations). But it is impossible to get the analogous result by using state feedback due to indispensable connection of fast decaying and large peaking. This difference is the main topic of the article and as a motivation for our interest we emphasize that the results of stabilization obtainable by output injection and by state feedback seem to be dual