  The innocuousness property of a controller is that property that makes the
closed-loop system stable regardless the values of the controller parameters.
In other words, the closed-loop system exhibits some structural stability
property with respect to the parameters of the controller. The innocuousness
property was first emphasized in [Briat, Gupta, and Khammash, Cell Systems,
2016] where it was shown that for stochastic unimolecular networks, the
Antithetic Integral Controller (AIC) is innocuous under very mild conditions on
the controlled network; namely the ergodicity and the output-controllability of
the open-loop network, and the admissibility of the set-point value. We show
here that the class of p-type AIC controllers arising in the use of Antithetic
Integral Rein Controllers (AIRC) also exhibit such a property. It is shown in
the unimolecular reaction network case that the closed-loop network is
structurally stable with respect to the controller parameters provided that the
open-loop dynamics is stable and that the set-point is admissible. Those
results are then extended to the case of so-called output unstable linear
systems and to stable nonlinear networks. Analogous results are obtained for
exponential and logistic integral controllers. Some examples are given for
illustration.
