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 delta, it is shown in [4] and [7] that the steady-state regulation error will be of the order O(delta). In this paper we prove a shaper result where the steady-state regulation error is shown to be of the order O(mu delta), where mu is a design parameter of the continuously-implemented sliding mode controller of [4] and [7]. Therefore, the regulation error can be reduced by decreasing mu. This result allows us to tradeoff the dimension of the internal model versus the value of mu as means of reducing the regulation error.