A Note on a Change of Coordinates for Nonlinear SISO Systems with Singular Points

In the paper titled "Asymptotic Output Tracking Through Singular Points for a class of Uncertain SISO Nonlinear Systems" [2], a transformation T:Rⁿ×R⁺¿R^n+ß which takes (x,t)¿(Z,¿,¿) defined by Z1 = h(x) Z2 = Lfh(x) Z¿ = Lf^¿-1 h(x) ¿¿+1 = a0 (x)+b0 (x)u ¿¿+2 = a1 (x,u)+b0 (x)u\´ +b1 (x,u)u ¿¿+3 = a2 (x,u,u\´) +b2 (x,u,u\´)u+2b1 (x,u)u\´ +b0(x)u" ¿¿+ß = aß- {x,u,¿,u^(ß-2)+bß-{x,u,¿,u^(ß-2))u + ¿ +b0(x)u^(ß-1) (¿1 (x){ⁿ1=¿+1 was proposed in order to get rid of the singularity and being able to compute an admissible control u [1]. There was also discussed that T was formed by at least n linearly independent functions as long as x¿xs i.e., no singular points, (this was assumed when x=xs). The main purpose of this paper is to present a proof where it is shown that T given by (1) provides us with a set of at least n linearly independent functions in case of having singular points.