Nowadays, solving constrained engineering problems related to optimization approaches is an attractive research topic. The chaotic krill herd approach is considered as one of most advanced optimization techniques. An advanced hybrid technique is exploited in this paper to solve the challenging problem of estimating the largest domain of attraction for nonlinear systems. Indeed, an intelligent methodology for the estimation of the largest stable equilibrium domain of attraction established on quadratic Lyapunov functions is developed. The designed technique aims at computing and characterizing a largest level set of a Lyapunov function that is included in a particular region, satisfying some hard and delicate algebraic constraints. The formulated optimization problem searches to solve a tangency constraint between the LF derivative sign and constraints on the level sets. Such formulation avoids possible dummy solutions for the nonlinear optimization solver. The analytical development of the solution exploits the Chebyshev chaotic map function that ensures high search space capabilities. The accuracy and efficiency of the chaotic krill herd technique has been evaluated by benchmark models of nonlinear systems. The optimization solution shows that the chaotic krill herd approach is effective in determining the largest estimate of the attraction domain. Moreover, since global optimality is needed for proper estimation, a bound type meta-heuristic optimization solver is implemented. In contrast to existing strategies, the synthesized technique can be exploited for both rational and polynomial Lyapunov functions. Moreover, it permits the exploitation of a chaotic operative optimization algorithm which guarantees converging to an expanded domain of attraction in an essentially restricted running time. The synthesized methodology is discussed, with several examples to illustrate the advantageous aspects of the designed approach.

A novel technique for estimating the asymptotic stability region of nonlinear autonomous polynomial systems is established. The key idea consists of examining the optimal Lyapunov function (LF) level set that is fully included in a region satisfying the negative definiteness of its time derivative. The minor bound of the biggest achievable region, denoted as Largest Estimation Domain of Attraction (LEDA), can be calculated through a Generalised Eigenvalue Problem (GEVP) as a quasi-convex Linear Inequality Matrix (LMI) optimising approach. An iterative procedure is developed to attain the optimal volume or attraction region. Furthermore, a Chaotic Particular Swarm Optimisation (CPSO) efficient technique is suggested to compute the LF coefficients. The implementation of the established scheme was performed using the Matlab software environment. The synthesised methodology is evaluated throughout several benchmark examples and assessed with other results of peer technique in the literature.

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