On time scaling for nonlinear systems: Application to linearization

In this note, we propose a method to analyze systems in a time scale which is varied depending on the state such as
dt/d\tau = s(x)
(where
t
and τ are the actual time scale and that of new one, respectively, and
s(x)
is the function which we call time scaling function). Analysis of the system in the new time scale τ enables us to investigate the intrinsic structure of the system. A linearization problem in the new time scale is formulated as wide-sense feedback equivalence and is solved. It is also shown that the time scaling function which makes the system linear is derived as the solution of differential equations.