Control of uncertain non-linear systems via adaptive backstepping

Models of physical non-linear systems are prone to different kinds of uncertainties. This paper presents a backstepping-based adaptive control designed for a class of one degree-of-freedom uncertain non-linear systems. The true system does not need to be known for the control design. 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 performances depend explicitly on the design parameters. This feature of the control scheme establishes the main difference with other control methods used to control non-linear systems, in particular chaotic systems. The efficiency of the approach is tested by numerical simulations on Duffing oscillators and systems with non-linear and hysteretic stiffness under external loads.