Nonlinear optimal and flatness-based control for the multivariable Lotka-Volterra dynamics

The multivariable Lokta-Volterra dynamic model has been widely used to describe interacting and competing populations and has found applications in biology, ecology, agricultural production management and in economics. In this article a nonlinear optimal (H-infinity) control method is developed for the Lotka-Volterra system. First, differential flatness properties are proven for the associated state-space model. Next, the state-space description undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a time-varying operating point which is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. For the approximately linearized model of the system a stabilizing H-infinity feedback controller is designed. To compute the controller's gains an algebraic Riccati equation has to be repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. Finally, the performance of the nonlinear optimal control method is compared against a flatness-based control approach implemented in successive loops.