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

uThis paper presents an analysis of the slow-peaking phenomenon, a pitfall of low-gain designs that imposes basic limitations to large regions of attr action 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. Because small controls (or low-gain designs) are sufficient to stabilize the unperturbed chain of integrators, it may seem that smaller co ntrols, which attenuate the perturbation up(x, u) in a larger compact set, can be employed to achieve larger regions of attraction, This intuition is false, however, and peaking may cause a loss of global controllability unle ss severe growth restrictions are imposed on p(x, u), These growth restrict ions 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 d erived that achieves global asymptotic stability of x = 0. This stabilizati on result is extended to more general cascade nonlinear systems in which th e perturbation p(x, v)v, v = (xi, u)(T), contains the state xi and the cont rol u of a stabilizable subsystem xi = a(xi, u), As an illustration, a cont rol law is derived that achieves global stabilization of the frictionless b all-and-beam model.