Cross-directional control on paper machines using Gram polynomials

A control algorithm for cross-directional basis weight control based on orthogonal polynomials is presented. Instead of controlling the cross-directional values at each point, the parameters of the orthogonal polynomials are controlled, taking into account the maximum bending moment of the slice lip and the relative stiffness of the slice with respect to the actuators. The Gram polynomials have inherent ability to limit bending of the slice lip in a way that conventional control does not. The slice lip will have smooth deflection, governed by the maximum order of the polynomial, which can be lowered if the bending limits are exceeded. By modelling the measurements and the actuator positions by polynomials, the mapping between the actuators and the sensors, which is the key element of the current control systems, is effectively eliminated. Computational requirements are greatly reduced by avoiding inverses of large matrices, or the inverses are found for a diagonal matrix. The performance of the Gram polynomial control method is shown to be superior to conventional control methods, especially for badly conditioned matrices.