On stability of nonzero set-point for non linear impulsive control systems

TL;DR

This paper investigates the stability of non-zero set-points in nonlinear impulsive control systems, establishing conditions for asymptotic stability and proposing a model predictive control approach with applications in medical treatments.

Contribution

It introduces a stability analysis framework for non-zero equilibrium orbits in nonlinear impulsive systems and develops a feasible impulsive Model Predictive Control method.

Findings

Conditions for asymptotic stability of control orbits are established.

The proposed MPC stabilizes systems in medical treatment scenarios.

Applications include drug administration for HIV and Lithium treatments.

Abstract

The interest in non-linear impulsive systems (NIS) has been growing due to its impact in application problems such as disease treatments (diabetes, HIV, influenza, 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 allows the system to keep orbiting (between two consecutive pulses) in some state space regions out of the origin, according to usual objectives of most real applications. Although such regions can be characterized by means of a discrete-time system obtained by sampling the NIS at the impulsive times, no agreements have reached about their asymptotic stability (AS). This paper studies the asymptotic stability of control equilibrium orbits for NSI, 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. Finally, the proposed stable MPC is applied to two control problems of interest: the intravenous bolus administration of Lithium and the administration of antiretrovirals for HIV treatments.