Dual inclusion principle for overlapping interconnected systems

Restriction and aggregation are two dual inclusion conditions, which are revealed in the research field of inclusion principle for state space descriptions of overlapping interconnected systems. An input-state-output dual inclusion principle is proposed for linear time invariant systems and their dual systems, with the focus on restriction and aggregation in duality for direct contraction of control laws and state observers. In these two special inclusion cases, dual relationships of restriction and aggregation inclusion conditions are formulated between an original system and its dual system. Overlapping decentralized controller design problems are considered within the framework of the dual inclusion principle. It is shown that, if dual inclusion is satisfied, in the same type of expansions (restriction or aggregation) for both original system and its dual system, direct contraction can be applied to designed control laws or state observers from the expansion to the original space, while dual information of designed state observer or control laws can be contracted directly from the dual expansion to the dual original space.