UNIVERSAL STABILIZATION OF A CLASS OF NONLINEAR-SYSTEMS WITH HOMOGENEOUS VECTOR-FIELDS

Under relative-degree-one and minimun-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 = Lambda(lambda)y, lambda = y(2), where Lambda is a function of Nussbaum type (the terminology ''universal stabilization'' being used in the sense of rendering {0} a global attractor for each member of the underlying class whilst assuring boundedness of the function lambda(.)). A natural generalization of this result to a class H-k of nonlinear control systems (a, b, c), with positively homogeneous(of degree k greater than or equal to 1) drift vector field a, is described. Specifically, under the relative-degree-one (cb not equal 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=Lambda(lambda)\y\(k-1)y, lambda=\y\(k+1) assures H-k-universal stabilization. More generally, the strategy u = Lambda(lambda)exp(\y\)y, lambda = exp(\y\)y(2) assures H-universal stabilization, where H = boolean OR(k greater than or equal to 1)H(k).