Journal of Applied Mathematics and Stochastic Analysis
Volume 11 (1998), Issue 4, Pages 519-526

Stabilization of nonlinear systems by similarity transformations

Irina E. Zuber

“Ecology”, Shpalernaya St. 36, St. Petersburg, Russia

Received 1 March 1996; Revised 1 November 1997

Copyright © 1998 Irina E. Zuber. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For a system x˙=A(x)+b(x)u, u(x)=s(x)x, xn, where the pair (A(x),b(x)) is given, we obtain the feedback vector s(x) to stabilize the corresponding closed loop system. For an arbitrarily chosen constant vector g, a sufficient condition of the existence and an explicit form of a similarity transformation T(A(x),b(x),g) is established. The latter transforms matrix A(x) into the Frobenius matrix, vector b(x) into g, and an unknown feedback vector s(x) into the first unit vector. The boundaries of A˜(y,g) are determined by the boundaries of {kA(x)xk,kb(x)xk}, k=0,n1¯. The stabilization of the transformed system is subject to the choice of the constant vector g.