We provide a new method for building strict Lyapunov functions for two dimensional chains of exponential integrators, using nested exponential functions. One challenge is that the right sides of the systems saturate, so they are not completely controllable. The strictness of the Lyapunov functions is key to proving input-to-state stability (or ISS) properties with respect to additive uncertainty on the controls. We show how a large class of tracking problems for nonlinear systems with positivity constraints on the states can be solved using our theory. (C) 2016 Elsevier Ltd. All rights reserved.