Abstract
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.
Highlights
It is challenging to ascertain the domain of attraction (DA) for a point in equilibrium in a non-linear dynamical system
It is reasoned that to maximise the size of the Largest Estimation Domain of Attraction (LEDA), an appropriate strategy is to broaden the set Φ(G), which itself contains the LEDA
This paper studies the quadratic Lyapunov function calculation, which expands the DA volume estimate for systems with polynomials
Summary
It is challenging to ascertain the domain of attraction (DA) for a point in equilibrium in a non-linear dynamical system. Using identified relaxations that have been established around the polynomial sum of square, it is revealed that a lower bound of the largest attainable attraction region can be computed through a generalised eigenvalue problem This later is considered to be a quasi-convex LMI optimisation where the solution can be computed efficiently. This paper offers an efficient computational CPSO approach that is validated to estimate the region of stable equilibrium points for the class of nonlinear polynomial systems. The main objective of this work consists of designing a swift optimising Lyapunov-based technique that allows the identification of the Lyapunov function coefficients It exploits the resultant function in a heuristic optimising algorithm to maximise the region of attractions for the class of nonlinear polynomial systems. Standard notation has been used in this paper, where R represents the set of real numbers, AT denotes the transpose of the real matrix A; A > 0 (A ≥ 0) positive definite (semi-definite) matrix; In is the identity matrix n × n; det (A) represents the determinant of A; A ⊗ B refers to the Kronecker product of matrices A and B; s.t. is used to denote subject to
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