Energy control of nonhomogeneous Toda lattices

This paper presents a new approach to handle the problem of energy control of an n-degrees-of-freedom nonhomogeneous Toda lattice with fixed–fixed and fixed–free boundary conditions. The energy control problem is examined from an analytical dynamics perspective, and the theory of constrained motion is used to recast the energy control requirements on the Toda lattice as constraints on the mechanical system. No linearizations and/or approximations of the nonlinear dynamical system are made, and no a priori structure is imposed on the nature of the controller. Given the subset of masses at which control is to be applied, the fundamental equation of mechanics is employed to determine explicit closed-form expressions for the nonlinear control forces. The control provides global asymptotic convergence to any desired nonzero energy state provided that the first mass, or the last mass, or alternatively any two consecutive masses of the lattice are included in the subset of masses that are controlled. To illustrate the ease, simplicity, and efficacy with which the control methodology can be applied, numerical simulations involving a 101-mass Toda lattice are presented with control applied at various mass locations.