Optimal control of microalgae growth using linear quadratic regulator method with particle swarm optimization

Abstract

Microalgae represent a significant and promising resource for a multitude of industrial applications. The lipids extracted from microalgae are used to produce biofuels, as dietary supplements for humans and animals, and as sources of vitamins or proteins in the pharmaceutical industry. Furthermore, microalgae serve as agents for carbon dioxide capture and wastewater treatment (Kumar et al., 2010). This study examines the optimal regulation of microalgal proliferation by controlling variables such as carbon dioxide and nutrients using the Linear Quadratic Regulator (LQR) methodology. Within the framework of the LQR approach, the weight matrices Q and R are commonly established through trial and error, a time-consuming process that does not guarantee a globally optimal solution. This manuscript advocates implementing Particle Swarm Optimization (PSO) to precisely determine the matrix values Q and R, thereby supplanting traditional manual tuning. PSO is a metaheuristic algorithm predicated on the collective behaviors observed in avian flocking and piscine schooling. Compared with the Firefly Algorithm previously applied to this problem, PSO offers a simpler velocity-position update mechanism, fewer algorithmic control parameters to calibrate, and a lower computational burden per iteration, while still maintaining a strong balance between exploration and exploitation of the search space. The principal novelty of this study lies in the application of PSO rather than the Firefly Algorithm to automatically tune the LQR weight matrices for the Thornton microalgae growth model, accompanied by an explicit, iteration-by-iteration account of the convergence behavior of the swarm, an aspect that has not been reported in prior LQR-based microalgae control studies. The use of PSO is expected to enhance microalgal growth, resulting in substantially increased biomass production. Simulation findings indicate that the LQR approach in isolation yields a 21.7% increase in algal concentration from the baseline concentration, whereas the LQR method, when augmented by PSO optimization, results in an approximate 64.5% enhancement in algal concentration from the initial level, thereby underscoring the advantage of PSO-assisted weight optimization over both the unoptimized LQR controller and the previously reported LQR–Firefly Algorithm combination.