Analytical stability analysis of the fractional-order particle swarm optimization algorithm

Mathematical modeling plays an important role in biology for describing the dynamics of infectious diseases. A useful strategy for controlling infections and disorder conditions is to adopt computational algorithms for determining interactions among their processes. The use of fractional order (FO) calculus has been proposed as one relevant tool for improving heuristic models. The particles memory is captured by the FO derivative and that strategy opens the door for grasping the memory of the long-term particle past behavior. This papers studies the analytical convergence of FO particle swarm optimization algorithm (FOPSO) based on a weak stagnation assumption. This approach allows establishing systematic guidelines for the FOPSO parameters tuning. The FOPSO is formulated on the basis of a control block diagram and the particle dynamics are represented as a nonlinear feedback. To describe the historical evolution of the particles, a state-space representation of different types of the FOPSO is formulated as a delayed discrete-time system. The existence and uniqueness of the equilibrium point of the FOPSO are discussed, and the stability analysis is derived to determine its convergence boundaries. Several simulations confirm the stability region of the FOPSO equilibrium point. The algorithm is also applied to a practical application, namely the minimization of the blood glucose injection in Type I diabetes mellitus patients.