Abstract

In this talk we will first introduce what is the idea of an adaptive Lyapunov based control Scheme. And then we will present an adaptive control scheme for a class of one degree of freedom uncertain nonlinear systems. This second part comes from a joint work together with Fayçal Ikhouane and José Rodellar.

Models of physical nonlinear systems are prone to different kind of uncertainties, therefore a control strategy must be efficient even if the "true system" is not known for the control designer . In our scheme a functional description is assumed with uncertain coefficients and an uncertain residual function. These uncertainties are bounded and lump the discrepancies between the adopted description and the real behaviour. The adaptive controller is able to handle these uncertainties and make the closed loop globally uniformly ultimately bounded when the system is subject to an unknown excitation from which a bound is known.

One goal is that the transient and asymptotic performance depend explicitly on the design parameters. This feature of the control scheme establishes the main difference with other control methods used to control nonlinear systems.

Two applications of the control scheme will be presented:

1. A tracking problem for duffing oscillators and
2. a regulation problem for mechanical systems with nonlinear and hysteretic stiffness under external seismic loads.