On the stability analysis of nonlinear systems using polynomial Lyapunov functions

In the stability study of nonlinear systems, not to found feasible solution for the LMI problem associated with a quadratic Lyapunov function shows that it doesn’t exist positive definite quadratic Lyapunov function that proves stability of the system, but doesn’t show that the system isn’t stable. So, we must search for other Lyapunov functions. That's why, in the present paper, the construction of polynomial Lyapunov candidate functions is investigated and sufficient conditions for global asymptotic stability of analytical nonlinear systems are proposed to allow computational implementation.The main keys of this work are the description of the nonlinear studied systems by polynomial state equations, the use of an efficient mathematical tool: the Kronecker product; and the non-redundant state formulation. These notations allow algebraic manipulations and make easy the extension of the stability analysis associated to quadratic or homogeneous Laypunov functions towards more general functions.The advantage of the proposed approach is that the derived conditions proving the stability of the studied systems can be presented as linear matrix inequalities (LMIs) feasibility tests and the obtained results can show in some cases how the polynomial Lyapunov functions leads to less conservative results than those obtained via quadratic (QLFs) or monomial Laypunov functions. This contribution to the stability analysis of high order nonlinear continuous systems can be extended to the stability, robust analysis and control of other classes of systems.