Research on closed loop control theory

A general control algorithm has been developed within the framework of Lagrangian mechanics. In the present algorithm the cost function plays the role of the potential energy in the Lagrangian formalism and the model equations are introduced as constraints weighted by a set of Lagrange multipliers. Since the Lagrangian and Hamiltonian descriptions of classical mechanics are equivalent, the new methodology is also, in principle, equivalent to Pontryagin's Maximum Principle (PMP) formulation of optimal control theory. Previous work by March-Leuba and Perez at the University of Tennessee, Knoxville, had shown that for demand-following applications, the usual two-point boundary problem, which arises in the PMP methodology could be converted into an initial value problem. However, in this PMP-based algorithm the sign of the time derivatives in the adjoint equations had to be inverted to avoid numerical instabilities and the formulation could not function in the case of sampled input signals. Nevertheless, for the case in which the plant model and control equations were solved together, the algorithm proved to be flexible and capable of tracking uncertain terms and parameters, as well as working with noise-contaminated signals.

The present work was initiated with the purpose of developing, from first principles, a control algorithm which had to be robust against numerical integration instabilities and capable of working with sampled input signals. To this end the Lagrangian formalism was shown to be more appropriate than its IV equivalent Hamiltonian formalism.

Firstly the formalism is illustrated by using a single nonlinear equation to achieve a clear understanding of the workings of the control algorithm. Secondly the formalism is applied to a realistic simulation of a simplified model of a LMR (Liquid Metal) reactor. In this case two scenarios are considered. The first one deals with signals delayed by the dynamics of the various sensors, being the equations of the plant model and the control solved together. In the second, sampled input signals are introduced with a PID controller for the control of the plant, and the Lagrangian formalism is used to follow the uncertain terms (i.e. the unknown part of the plant dynamics).