On the steady-state error of a nonlinear regulator

This paper proves a new property of the nonlinear regulators of [4] and [7]. In both papers the steady-state control is immersed into a linear internal model. In general, the model produces the sinusoidal signals generated by an exosystem, as well as a number of their harmonics, which are induced by the system´s nonlinearities. When the internal model does not account for all harmonics, there will be an error between the steady-state control needed to achieve zero steady-state regulation error and the steady-state control produced by the internal model. If the norm of this error is bounded by a constant δ, it is shown in [4] and [7] that the steady-state regulation error will be of the order O(δ). In this paper we prove a shaper result where the steady-state regulation error is shown to be of the order O(μδ), where μ is a design parameter of the continuously-implemented sliding mode controller of [4] and [7]. Therefore, the regulation error can be reduced by decreasing μ. This result allows us to tradeoff the dimension of the internal model versus the value of μ as means of reducing the regulation error.