Optimal Boundary Control of an Axially Moving Material System

The objective of this paper is to develop an optimal boundary control strategy for the axially moving material system through a mass-damper-spring (MDS) controller at its right-hand-side (RHS) boundary. The partial differential equation (PDE) describing the axially moving material system is combined with an ordinary differential equation (ODE), which describes the MDS. The combination provides the opportunity to suppress the flexible vibration by a control force acting on the MDS. The optimal boundary control laws are designed using the output feedback method and maximum principle theory. The output feedback method only includes the states of displacement and velocity at the RHS boundary, and does not require any model discretization thereby preventing the spillover associated with discrete parameter models. By utilizing the maximum principle theory, the optimal boundary controller is expressed in terms of an adjoint variable, and the determination of the corresponding displacement and velocity is reduced to solving a set of differential equations involving the state variable, as well as the adjoint variable, subject to boundary, initial and terminal conditions. Finally, a finite difference scheme is used to validate the theoretical results.