Preservation of controllability-observability in expanded systems

The result contributed by this note is that controllability-observability o f an original continuous-time LTI dynamic system can always be simultaneons ly preserved in expanded systems within the Inclusion Principle when using block structured complementary matrices. This new structure offers more deg rees of freedom for the selection of specific complementary matrices than w ell known used cases, such as aggregations and restrictions, which enable s uch preservation only in certain special cases. A complete unrestricted tra nsmission of these qualitative properties from the original controllable-ob servable system to its expansion is a basic requirement on the expansion/co ntraction process, mainly when controllers/observers are designed in expand ed systems to be consequently contracted for implementation in initially gi ven systems. An original system composed of two overlapped subsystems is ad opted as a general prototype case. A numerical example is supplied.