THE PEAKING PHENOMENON AND THE GLOBAL STABILIZATION OF NONLINEAR-SYSTEMS

The problem of global stabilization is considered for a class of cascade systems. The first part of the cascade is a linear controllable system and the second part is a nonlinear system receiving the inputs from the states of the first part. With zero input, the equilibrium of the nonlinear part is globally asymptotically stable (GAS). The design problem is to stabilize the linear part using linear feedback from its own states only. The design must stabilize the whole cascade, that is, the inputs into the nonlinear part must not destroy the GAS property. It may seem that a way to achieve this is to drive the states of the linear part to zero with a very fast exponential rate and thus make the behavior of the nonlinear part close to its zero input behavior. This intuition is false, however, because it disregards the effects of peaking. In linear system, a peaking phenomenon occurs when high-gain feedback is used to produce eigenvalues with very negative real parts. Then some states peak to very large values, before they rapidly decay to zero. Such peaking states act as destabilizing inputs to the nonlinear part and may even cause some of its states to escape to infinity in finite time. The destabilizing effects of peaking can be reduced if the nonlinearities have sufficiently slow growth, as established in Sections IV, V, and VI of the paper. The remaining sections provide a detailed analysis of the peaking phenomenon and examine the tradeoffs between linear peaking and nonlinear growth conditions.