On stability of nonzero set-point for nonlinear impulsive control systems

The interest in non-linear impulsive systems (NIS) has been growing due to their impact on application problems such as disease treatments (diabetes, HIV, influenza, COVID-19, among many others), where the control action (drug administration) is given by short-duration pulses followed by time periods of null values. Within this framework, the concept of equilibrium needs to be extended (redefined) to allow the system to keep orbiting (between two consecutive pulses) in some state-space sets out of the origin, according to the usual objectives of most real applications. Although such sets can be characterized by means of a discrete-time system obtained by sampling the NIS at the impulsive times, no agreement has been reached on their asymptotic stability (AS) under optimizing control strategies. This paper studies the asymptotic stability of control equilibrium orbits for NIS, based on the underlying discrete-time system, in order to establish the conditions under which the AS for the latter leads to the AS for the former. Furthermore, based on the latter AS characterization, an impulsive Model Predictive Control (i-MPC) that feasibly stabilizes the non-linear impulsive system is presented. The biomedical problems of intravenous bolus administration of Lithium and antiretrovirals administration for HIV treatments are considered as simulation examples to demonstrate the controller performance