Quantitative measures of controllability for descriptor systems

Controllability and observability are a structural property that carries information useful for control system design. There are many measures to determine the location of actuator and sensor like degree of controllability(DOC), degree of observability(DOO) and degree of disturbance rejectability(DODR). However, Those measures are mostly defined in standard state space systems, not in descriptor systems. Descriptor systems, or singular systems, consist of slow and fast systems, which mean there are singularities in the systems. In this paper, DOC for descriptor systems which is defined as minimum input energy to change the states is proposed and the result is same with the minimum energy to change the states of slow systems and that means the fast systems does not affect the input energy. A simple numerical example is used to verify the result.