Design of H∞ controller for a lightly damped system using a bilinear pole shifting transform

An H^∞ control law design is presented for a benchmark problem consisting of an undamped pair of spring-coupled masses with a sensor and actuator which are not co-located. This simple mechanical system captures many of the salient features of more complex aircraft and space structure vibration control problems. The H^∞ problem formulation enables the issue of stability robustness in the face of large mass and spring constant variation to be directly addressed. Constraints on closed-loop dominant pole locations and settling time are accommoodated via a simple s-plane bilinear transform s = s+¿/¿s+1. The parameter "σ" can give direct control of closed-loop disturbance settling time and controller open loop pole-zero locations. A 4-state H^∞ minimum phase controller was found to meet the given specifications of each design. Detailed analysis on the trade-offs of sensor noise versus control energy are presented in a full-length version of the paper.