Minimum time controller in a class of chemical reactors based on Lagrangian approach

The main goal of this work is the construction of a class of controller, which employs directly a Lagrangian formulation to resolve the classical brachistochrone problem, this allows to obtain an optimal controller which reaches in a minimum time the stabilization of an isothermal continuous stirred tank reactor, whose chemical kinetic model is based on the power law. The proposed methodology is compared with an input/output linearizing which achieve asymptotic and exponential closed-loop convergence, sliding-mode controller with a finite time convergence and an exact gradient optimal control to compare the time convergence performance. Numerical experiments show the satisfactory performance of the proposed controller, despite sustained disturbances in the concentration input feed.

Brachistochrone on a velodrome

The Brachistochrone problem, which describes the curve that carries a particle under gravity in a vertical plane from one height to another in the shortest time, is one of the most famous studies in classical physics. There is a similar problem in track cycling, where a cyclist aims to find the trajectory on the curved sloping surface of a velodrome that results in the minimum lap time. In this paper, we extend the classical Brachistochrone problem to find the optimum cycling trajectory in a velodrome, treating the cyclist as an active particle. Starting with two canonical cases of cycling on a sloping plane and a cone, where analytical solutions are found, we then solve the problem numerically on the reconstructed surface of the velodrome in Montigny le Bretonneux, France. Finally, we discuss the parameters of the problem and the effects of fatigue.