Smooth global stabilization for a class of nonlinear systems using homogeneity with monotone degrees

Abstract

This paper considers the problem of designing C 1 (continuously differentiable) state feedback stabilizers for a class of 3-dimensional nonlinear systems whose linearizations around the origin may contain uncontrollable modes. Based on a new definition of homogeneity with monotone degrees, we not only propose conditions of constructing C 1 and C ∞ (smooth) controllers, but also provide explicit design schemes for such systems. Several examples are investigated to show the advantages of the generalized homogeneity compared to the traditional unified homogeneity.

Stabilization Via Generalized Homogeneous Approximations

“…Bxn and α defined in (11), together with propositions 4.1 and 4.2, we readily obtain a stabilization result on a chain of power integrators with a nonlinear additive perturbation. Throughout this section we consider A P R nˆn , B P R n , pA, Bq in Brunowski form and q P R n ą,odd .…”

“…This proves the second part of the proposition. Remark 4.1: If aj`1pr8,j `d8,j q aj qj r8,j ą aj`1pr0,j `d0,j q aj qj r0,j for some j P r1, ns, then the design of α j in (11)) can be simplified as follows: α j :" α…”

“…As a second, step, we establish that the derivative of V along the trajectories of the closed-loop system (8) with u " αpxq, where α is defined in (11), is negative definite for a suitable choice of γ 1 , . .…”

“…Notice that Ṽ (as V ) is positive definite and radially unbounded for each L ą 0. Moreover, let α be defined in (11)…”

“…x " Ax q `Bu `cgpxq with u " αpxq and α defined in (11), is globally asymptotically stable for all c P r0, c ˚s and γ j ě γ j , j P r1, ns.…”