Control of Positive Systems with an Unknown State-Dependent Power Law Input Delay and Input Saturation

This paper is motivated by a class of positive systems with an input that is subject to an unknown state-dependent power law delay as well as saturation. For example, biological networks have non-negative protein concentration states. Mass action kinetics in these systems result in power law behavior, while complex interactions cause signal propagation delays. Incomplete network characterization makes delay state-dependence unknown. Manipulating network activity via modulated protein concentrations to attain desired performance is restricted by upper-bounds on concentration actuator authority. Here, an innovative control law exploits system dynamics to compensate for control domain restrictions. A Lyapunov stability analysis establishes that the reference tracking error of the closed-loop system is uniformly ultimately bounded. Numerical simulations on a human coagulation model show controller efficacy and better performance compared to the relevant literature. This example application steps toward personalized, closed-loop treatments for trauma coagulopathy, which currently has 30% mortality with open-loop clinical approaches.