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

This paper presents an analysis of the slow-peaking phenomenon, a pitfall of low-gain designs that imposes basic limitations to large regions of attraction in nonlinear control systems. The phenomenon is best understood on a chain of integrators perturbed by a vector field up(x, u) that satisfies p(x, 0)=0. Peaking may cause a loss of global controllability unless severe growth restrictions are imposed on p(x, u). 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 that achieves global asymptotic stability of x=0. This stabilization result is extended to more general cascade nonlinear systems in which the perturbation p(x, υ)υ, υ=(ξ, u)^T, contains the state ξ and the control u of a stabilizable subsystem ξ˙=a(ξ, u). As an illustration, a control law is derived that achieves global stabilization of the frictionless ball-and-beam model