Stabilization of nonlinear systems by similarity transformations

Abstract

For a system x˙=A(x)+b(x)u, u(x)=s∗(x)x, x∈ℝn, 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,n−1¯. The stabilization of the transformed system is subject to the choice of the constant vector g.