Stabilization of nonlinear systems by similarity transformations. (1998) - Zuber, Irina E.
Abstract : For a system x ˙ = A ( x ) + b ( x ) u, u ( x ) = s ∗ ( x ) x, x ∈ ℝ n, 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 { ∂ k A ( x ) ∂ x k, ∂ k b ( x ) ∂ x k }, k = 0, n − 1 ¯ . The stabilization of the transformed system is subject to the choice of the constant vector g .