On time scaling for nonlinear systems: Application to linearization

Abstract

In this note, we propose a method to analyze systems in a time scale which is varied depending on the state such as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dt/d\tau = s(x)</tex> (where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> and τ are the actual time scale and that of new one, respectively, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(x)</tex> 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.