On the steady-state error of a nonlinear regulator


Correspondence to: Ranran Li, School of Information Science and Engineering, Northeastern University, Shenyang, 110004, China.

E-mail: liranran.neu@gmail.com


This paper proves a new property of the nonlinear regulators that proposed in two previous papers by the second author with co-workers. 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 or when the model's characteristic frequencies are not exactly those of the exosystem, 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 has been shown 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 a continuously implemented sliding mode controller. Therefore, the regulation error can be reduced by decreasing μ. This result allows us to trade off the accuracy of the internal model versus the value of μ as means of reducing the regulation error.Copyright © 2012 John Wiley & Sons, Ltd.