On Transforming Single Input Linear Time-Varying Systems into High-order Fully Actuated Systems

For a linear time-invariant (LTI) system, controllability is equivalent to that it can be transformed into a high-order fully actuated system (HOFA). In this note, it is demonstrated that this is not the case for linear time-varying (LTV) systems even in the single input case. This paper points out that an LTV system can be transformed into a standard HOFA system if and only if its controllability matrix defined in the Silverman criterion is nonsingular for any time, which is only sufficient but not necessary for the controllability of the system. In addition, if the controllability matrix defined in the Silverman criterion is not nonsingular for any time, a degenerated HOFA system might be obtained. Some simple examples are given to demonstrate the reported facts.