Adaptive control of laterally coupled quadruple Lorenz systems

Abstract

The chaotic fluid flows in the laterally interconnected four-loop system are qualitatively modeled by a system of four Lorenz type of equations with momentum coupling/thermal coupling terms in the system.  The flows become chaotic driven by large Rayleigh numbers and the momentum and heat exchanges at the coupling points.  The control configuration consists of single-state feedback proportional controllers to stabilize the flows well into the chaotic regime.  Explicit stability bounds on the feedback gains are derived to guarantee the system’s global asymptotic stability at the equilibrium point, in which the bounds depend on the Rayleigh numbers and the momentum coupling intensity parameter only, but independent of the thermal coupling parameter.  Optimal bounds exist due to the coupling effect.  The adaptive controller design is applied to achieve global stability when the system parameters are unknown.  The performance of the controllers is demonstrated with numerical simulations.