Quasi-LPV positive observer-based control of closed-loop deep brain stimulation systems

Abstract

Parkinson’s disease is a neurodegenerative disorder in which dopamine (a specific type of neurotransmitter) production by neurons is destroyed in the brain. Therefore, there is less or no production of dopamine in the human brain and the patient has motor and non-motor complications. The main reason for Parkinson’s disease is the basal ganglia (BG) system imbalance. Therefore, it is imperative to find effective control methods to stabilize and regulate the BG system. To control the BG system within the normal oscillating performance, the opinion of closed-loop deep brain stimulation (DBS) system is used as the most innovative surgical treatment. In this paper, a quasi-linear parameter-varying (LPV) model is extracted from two known nonlinear models (the RT model and the modified RT model with 19 dynamics as the state variables of the BG system) as a comprehensive model to result in a more accurate control for the DBS system. The theoretical contributions in this paper are novel because this model is considered as a positive system and the related theories and properties are discussed. In view of this, quasi-LPV positive observer-based control method is designed for two situations; when all parameters are available for estimation (exact observer) and when all accurate parameters cannot be easily accessed (inexact observer). By designing such controllers, the provided algorithm does not need the availability of the scheduling parameters of the model, which is the novel challenge of this paper. The positivity and asymptotic stability of the proposed controllers have been studied based on linear matrix inequalities. The feasibility of the proposed closed-loop system and the effectiveness of the controllers are verified by numerical simulation. The designed controllers make the closed-loop DBS system stable and positive in both situations. It is while, without this controller, there is no such oscillating performance for Parkinsonian patients.