Control of Discrete-Time Linear Systems with Charge-Balanced Inputs

TL;DR

This paper explores how charge-balanced inputs can control discrete-time linear systems, providing new reachability conditions and validating them through numerical control input design, with applications in safe neurostimulation.

Contribution

It introduces novel reachability and controllability conditions for charge-balanced inputs in discrete-time systems, applicable to neurostimulation protocols.

Findings

Derived reachability conditions for periodic charge-balanced inputs

Established controllability criteria for non-repetitive charge-balanced inputs

Validated theoretical results with numerical minimum-energy control designs

Abstract

Electrical brain stimulation relies on externally applied currents to modulate neural activity, but safety constraints require each stimulation cycle to be charge-balanced, enforcing a zero net injected charge. However, how such charge-balanced stimulation works remains poorly understood. This paper investigates the ability of charge-balanced inputs to steer state trajectories in discrete-time linear systems. Motivated by both open-loop and adaptive neurostimulation protocols, we study two practically relevant input structures: periodic (repetitive) charge-balanced inputs and non-repetitive charge-balanced inputs. For each case, we derive novel reachability and controllability conditions. The theoretical results are further validated through numerical demonstrations of minimum-energy control input design.