Connectability and structural controllability of composite systems

The notion of connectability for multivariable composite systems consisting of a number of subsystems interconnected in an arbitrary way is introduced in this paper. It is shown that connectability plays a fundamental role in composite systems; in particular, it is shown that under certain mild conditions, almost all composite interconnected systems are controllable and observable from any nontrivial input and output if and only if the resultant composite system is connectable. A class of composite systems called general input-output hierarchical systems, which has the property of always being connectable is then defined. Since such systems are almost always controllable, this observation perhaps gives some insight in explaining why so many real world systems have as their basis a hierarchical structure. An application of the previous results is then made to show that a system (C, A, B) is structurally controllable and observable if and only if it is connectable and a certain non-pathological rank condition holds.