Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Abstract

Under relative-degree-one and minimum-phase assumptions, it is well known that the class L of finite-dimensional, linear, single-input ( u), single-output ( y) systems ( A,b,c) is universally stabilized by the feedback strategy u = Λ( λ) y, λ = y 2, where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class H k of nonlinear control systems ( a,b,c), with positively homogeneous (of degree k ⩾ 1) drift vector field a, is described. Specifically, under the relative-degree-one ( cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ( λ)/ vby/ vb k−1 y, dot λ = /vby/vb k+1 assures H k-universal stabilization. More generally, the strategy u = Λ( λ)exp(/ vby/ vb) y, dot λ = exp(/vby/vb)y 2 assures H -universal stabilization, where H = ∪ k ⩾ 1 H k.