Preservation of controllability-observability in expanded systems

The result contributed by this note is that controllability–observability of an original continuous-time LTI dynamic system can always be simultaneously preserved in expanded systems within the Inclusion Principle when using block structured complementary matrices. This new structure offers more degrees of freedom for the selection of specific complementary matrices than well known used cases, such as aggregations and restrictions, which enable such preservation only in certain special cases. A complete unrestricted transmission of these qualitative properties from the original controllable–observable system to its expansion is a basic requirement on the expansion/contraction process, mainly when controllers/observers are designed in expanded systems to be consequently contracted for implementation in initially given systems. An original system composed of two overlapped subsystems is adopted as a general prototype case. A numerical example is supplied.