Robust internal models with a star-shaped attractor are linear

In linear regulation theory, it is well-known that embedding in the control loop a suitable internal model of the exogenous disturbances and references permits to achieve perfect regulation of the desired variables robustly with respect to parametric uncertainties in the plant's equations. However, it was recently proved that this principle does not extend, in general, to nonlinear systems or non-parametric perturbations. Indeed, there exist systems for which no smooth finite-dimensional regulator can exist that regulates the desired variables to zero in spite of unstructured uncertainties affecting the plant's dynamics. This article complements such a negative result by proving that, in the canonical context of minimum-phase normal forms, a nonlinear regulator of the Luenberger type that guarantees robust asymptotic regulation with respect to unstructured uncertainties and possesses a star-shaped attractor necessarily behaves as a linear system on such an attractor. This result further strengthens the conjecture that robust regulation is essentially a linear property.