HOMOGENEOUS FEEDBACK CONTROLS FOR HOMOGENEOUS SYSTEMS

摘要

Let X(x) = SIGMA(j=2)n a(j)(x) partial derivative/partial derivativex(j) be a real analytic, n-dimensional vector field with X(0) = 0 which is homogeneous of degree m less-than-or-equal-to 0 with respect to a dilation delta(epsilon)rx = (epsilon(r1) x1,..., epsilon(rn)xn) and Y = partial derivative/partial derivativex1. We show that if the control system x = X(x) + uY admits a continuous, asymptotically stabilizing, feedback control u(x), then a one-dimensional dynamic extension of the system admits a homogeneous, asymptotically stabilizing (dynamic) feedback control u(tau, x) such that the vector field W(tau, x) = alphatau1-mpartial derivative/partial derivativetau + u(tau, x)partial derivative/partial derivativex1 + SIGMA(j=2)n a(j)(x)partial derivative/partial derivativex(j) describing the extended, controlled, system is homogeneous of degree m with respect to the extended dilation delta(epsilon)r(tau, x) = (epsilontau, epsilon(r1) x1,..., epsilon(rn) x(n)). [References: 14]