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

- AUTHOR: Rodolphe Sepulchre
- ABSTRACT: This paper presents an analysis of the slow peaking phenomenon, a pitfall of low-gain designs which 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 $u \, p(x,u)$ which satisfies $p(x,0)=0$. Because
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 and
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 which achieves global asymptotic stability of $x=0$. This stabilization result is extended to more general cascade nonlinear systems where the perturbation $p(x,v)v$ , $v=(\xi,u)^T$, contains the state $\xi$ and the control $u$ of a stabilizable subsystem $\dot \xi=a(\xi,u)$. As an illustration, a control law is derived which achieves global stabilization of the frictionless ball-and-beam model.
- DATE OF ENTRY: April 1997.
- STATUS: IEEE Transactions on
Automatic Control, Vol. 45, No 3, pp. 453-461, 2000.

- Paper: PS
file.
- Short version published in the Proceedings of the 36th IEEE Conference on Decision
and Control, pp. 3491-3496, 1997.
PS
file.