Abstract
Abstract This paper concerns the computational stability analysis of locally stable Lotka-Volterra (LV) systems by searching for appropriate Lyapunov functions in a general quadratic form composed of higher order monomial terms. The Lyapunov conditions are ensured through the solution of linear matrix inequalities. The stability region is estimated by determining the level set of the Lyapunov function within a suitable convex domain. The paper includes interesting computational results and discussion on the stability regions of higher (3,4) dimensional LV models as well as on the monomial selection for constructing the Lyapunov functions. Finally, the stability region is estimated of an uncertain 2D LV system with an uncertain interior locally stable equilibrium point.
Highlights
Approximating the domain of attraction (DOA) is often a fundamental task in the analysis and control of nonlinear systems
Due to their advantageous properties and the availability of efficient numerical solvers, the use of linear matrix inequalities (LMI) and semi-definite programming (SDP) techniques has become popular in the field of system and control theory
An optimization-based method for DOA estimation was published [4], where the authors use Finsler’s lemma and affine parameter-dependent LMIs to compute rational Lyapunov functions for a wide class of locally asymptotically stable nonlinear systems. Based on these results an improved method was published [5,6], where the transformation of the model to the form required for optimization is done automatically using the linear fraction transformation (LFT) and further automatic model simplification steps, which results in the dimension reduction of the optimization task
Summary
Approximating the domain of attraction (DOA) is often a fundamental task in the analysis and control of nonlinear systems. An optimization-based method for DOA estimation was published [4], where the authors use Finsler’s lemma and affine parameter-dependent LMIs to compute rational Lyapunov functions for a wide class of locally asymptotically stable nonlinear systems. Based on these results an improved method was published [5,6], where the transformation of the model to the form required for optimization is done automatically using the linear fraction transformation (LFT) and further automatic model simplification steps, which results in the dimension reduction of the optimization task. The main motivation of our work was to evaluate the applicability of the approach [4] on a general polynomial system class consisting of low-degree monomials, and to study the limits of the method as the number of dimensions of the state space increase
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