Rest-to-Rest Motion of an Experimental Flexible Structure subject to Friction: Linear Programming Approach

Abstract

A linear programming approach designed to eliminate the residual vibration of the two-mass harmonic system subject to friction and undergoing a point-to-point maneuver is implemented. Techniques for non-robust and robust open loop controller design are explored. It is shown that consistent results can be obtained from experiments and the robustness against frequency uncertainty results in reduction in residual vibration as well as steady-state error. I. I NTRODUCTION Linear programming (LP) is a powerful numerical op- timization technique that is able to handle hundreds of constraints efficiently. It requires all constraints and co sts to be linear in the unknown variable. This restriction on the use of LP may seem to eliminate practical use on real engineering problem. It is possible, however, to linearize some complex nonlinear systems to fit the required form of LP (1). It is also possible to manipulate the original proble m definition to fit the LP format as is done in (2), where a nonlinear constraint due to fuel usage is re-written as two linear constraints. In system dynamics, LP can be used on linear discrete systems for time optimal and fuel limited time optimal control (3), (4), (2), where constraints linear in the discr ete input sequence u(k), are imposed to result in an input profile. The significance of friction to the control community is in its effects on positioning systems and velocity tracking operation. Positioning applications include telescopes, an- tennas, machine tools, disk drives and robot arm position- ing. Velocity control is also relevant in machine tool, disk drive and robot arm industrial applications which require the accurate tracking of a pre-determined trajectory. The effect of friction becomes accentuated in the low velocitie s region near the reference position. The majority of work done on control of frictional systems is on rigid body systems. Yang and Tomizuka (5) exploited a simple relationship between a pulse input and the displacement of the rigid body. This utilizes the fact th at the rigid body subject to a pulse input never changes the sign of the velocity and thus the Coulomb friction acts like a bias input. This scheme, known as Pulse Width Control (PWC), is presented in an adaptive control setting where an estimate of the friction is determined in real-time. Wijdeven and Singh (6), modified the PWC approach to increase accuracy in actual discrete implementation of the input. Their technique modulates the pulse height to compensate for a rounded up pulse width and is called Pulse Amplitude Pulse Width Control (PAPWC). Additional schemes developed for rigid body systems include internal-model following error control (7), PID and state feedback linearization control (8) and variable structure control in order to try to handle qualitatively different friction regimes (9), (10). Nonlinear PID contro l has also been developed to overcome the stick-slip behavior