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Slow peaking and low-gain designs for global stabilization of nonlinear systems

This paper addresses the global stabilization of a chain of integrators perturbed by a vector field up(x,u) which satisfies p(x, 0)=0. As small controls (or low-gain designs) are sufficient to stabilize the unperturbed chain of integrators, it may seem that smaller controls which attenuate the perturbation up(x, u) in a larger compact set can be employed to achieve larger regions of attraction. However, this intuition is false due to the slow peaking phenomenon which imposes fundamental limitations to nonlinear feedback designs. To overcome the effect of peaking, we must impose on p(x, u) certain growth restrictions. These growth restrictions are expressed as a higher-order condition with respect to a particular weighted dilation related to the peaking exponents of the nominal system. When this higher-order condition is satisfied, an explicit control law is derived which achieves global asymptotic stability of x=0.