Adaptive output regulation of uncertain nonlinear systems with unknown control directions

Dear editor, The output regulation problem for nonlinear systems with unknown exosystems has been an important research topic over the last few decades [1]. The early results commonly assume that the sign of the high-frequency gain is known. It is shown [2] that a nonlinear system can be transformed into a special observer form under geometric coordinatefree conditions. In this case, with a known control direction, the problem can be solved by a robust control method such as high-gain feedback. However, this method is not applicable when the high-frequency gain sign is unknown. It has been proved that the standard way to deal with the unknown sign of the high-frequency gain is the Nussbaum gain technique [3]. The Nussbaum gain is a kind of oscillating function which allows the controller to try both positive and negative directions. It is still a challenge to solve the nonlinear output regulation problem with both unknown nonlinear exosystems and unknown high-frequency gain sign. This motivates us to develop a new approach, which combines the circle criterion with Nussbaum gain. Recently, this problem has attracted extensive attention. In [4], the Nussbaum gain technique is integrated with the backstepping method for solving adaptive consensus output regulation of a class of network-connected nonlinear systems, and some progresses are reported on cooperative output regulation of a class of nonlinear multiagent systems [5] without any knowledge of the high frequency gain [6]. We consider adaptive output regulation of output feedback systems with both unknown nonlinear exosystems and unknown high-frequency gain sign. A nonlinear adaptive internal model is proposed on a basis of circle criterion to reproduce the feedforward input term. As for the nonlinear exosystems, some specific assumptions are determined such that the circle criterion can be used in the stability analysis for the proposed internal model. Because of the lack knowledge of parameters in the system and the exosystem, a Nussbaum gain is then used to tackle the unknown sign of high-frequency gain. The proposed internal model, Nussbaum gain and nonlinear adaptive backstepping technique together provide a solution to the output regulation with nonlinear exosystems. The control input is finally obtained by a recursive procedure. The proposed control scheme guarantees the global asymptotic convergence of the tracking error.