Metaheuristic Solution for Stability Analysis of Nonlinear Systems Using an Intelligent Algorithm with Potential Applications

• Published in: Fractal and fractional Vol. 7; no. 1; p. 78
• Main Authors: Hamidi, Faiçal, Jerbi, Houssem, Alharbi, Hadeel, Leiva, Víctor, Popescu, Dumitru, Rajhi, Wajdi
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.01.2023
• Subjects: Algorithms; Attraction; Control engineering; Control theory; Design; Equilibrium; fractional differential equations; fractional systems; Global optimization; heuristic algorithms; Heuristic methods; Jaya algorithm; Liapunov functions; Linear matrix inequalities; Lyapunov theory; Nonlinear systems; Optimization techniques; Partial differential equations; Polynomials; Stability analysis
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract7010078

In this article, we provide a metaheuristic-based solution for stability analysis of nonlinear systems. We identify the optimal level set in the state space of these systems by combining two optimization phases. This set is in a definite negative region of the time derivative for a polynomial Lyapunov function (LF). Then, we consider a global optimization problem stated in two phases. The first phase is an external optimization to search for a definite positive LF, whose derivative is definite negative under linear matrix inequalities. The candidate LF coefficients are adjusted using a Jaya metaheuristic optimization algorithm. The second phase is an internal optimization to ensure an accurate estimate of the attraction region for each candidate LF that is optimized externally. The key idea of the algorithm is based mainly on a Jaya optimization, which provides an efficient way to characterize accurately the volume and shape of the maximal attraction domains. We conduct numerical experiments to validate the proposed approach. Two potential real-world applications are proposed.