  This paper proposes a non-adaptive control solution framework to the
practical output regulation problem (PORP) for a class of nonlinear systems
with uncertain parameters, unknown control directions and uncertain exosystem
dynamics. The concurrence of the unknown control directions and uncertainties
in both the system dynamics and the exosystem pose a significant challenge to
the problem. In light of a nonlinear internal model approach, we first convert
the robust PORP into a robust non-adaptive stabilization problem for the
augmented system with integral Input-to-State Stable (iISS) inverse dynamics.
By employing an extremum-seeking control (ESC) approach, the construction of
our solution method avoids the use of Nussbaum-type gain techniques to address
the robust PORP subject to unknown control directions with time-varying
coefficients. The stability of the non-adaptive output regulation design is
proven via a Lie bracket averaging technique where uniform ultimate boundedness
of the closed-loop signals is guaranteed. As a result, both the estimation and
tracking errors converge to zero exponentially, provided that the frequency of
the dither signal goes to infinity. Finally, a simulation example with unknown
coefficients is provided to exemplify the validity of the proposed control
solution frameworks.
