Passivity of Lotka–Volterra and quasi-polynomial systems

Abstract

This study approaches the stability analysis and controller design of Lotka–Volterra and quasi-polynomial systems from the perspective of passivity theory. The passivity based approach requires to extend the autonomous system model with a suitable input structure. The condition of passivity for Lotka–Volterra systems is less strict than the classic asymptotic stability criterion. It is shown that each Lotka–Volterra system is feedback equivalent to a passive system and a passifying state feedback controller is proposed. The passivity based approach enables the design of novel state feedback controllers to Lotka–Volterra systems. The asymptotic stability can be achieved by applying an additional diagonal state feedback having arbitrarily small gains. This result was further explored to achieve rate disturbance attenuation in controlled Lotka–Volterra systems. By exploiting the dynamical similarities between the Lotka–Volterra and quasi-polynomial systems, it was shown that the passivity related results, developed for Lotka–Volterra systems, are also valid for a large class of quasi-polynomial systems. The methods and tools developed have been illustrated through simulation case studies.

Universal embedding of autonomous dynamical systems into a Lotka-Volterra-like format

“…This implies that the above GLV → LV general route can be straightforwardly applied to the specific case of interest. By making the correct identification of the indexes, equations (A4)-(A5) directly lead to the quadratic format equation (18) with the matrix given in equation (19).…”

“…For instance, in the (deterministic) chemical kinetics context, in which the mass-action rate equations already have a GLV format, we mention the works of Gouzé [5], of Fairén and Hernández-Bermejo [6], and of the present author and co-workers for both closed [7] and open [8] chemical reaction networks. Quasi-polynomial (GLV) and LV formats have been widely studied in terms of stability of the stationary points [9][10][11][12][13][14][15][16], boundedness of the solutions [5,12], stabilizing feedback control in process systems [17][18][19], integrability of ODEs with polynomial nonlinearities [20], connection with stochastic urn processes [21], and connection with abstract Lie algebra [22]. This brief overview is by no means complete and should only give the idea of the broad interest in such topic across decades of research.…”

On the Stability of Four Dimensional Lotka-Volterra Prey-Predator System

“…More details on the modified Lotka-Volterra models and prey-predator models can be found in [10][11][12][13][14]. In [15], a modified for Lotka-Volterra model was proposed and studied, and it represents a food chain consisting of three species.…”