Mechanical vibration is a common phenomenon observed in the operation of many machines and arises from the inertia effect of machine parts in motion. While many control system design methods for distributed parameter systems have already been proposed in the literature, generally they are either based on truncated models and, as a result, suffer from computational and “spillover” difficulties or require distributed parameter actuators which are rarely available in reality. Therefore, there is a definite need for the development of a class of controllers which can be realized by spatially discrete sensors and actuators and whose design specifically includes stabilization and control of all the higher frequency vibration modes. To address this need, we propose the design of linear compensators whose design is based on root locus arguments for infinite dimensional systems. Since the design is not based on finite dimensional models of the plant to be controlled, we expect it to perform well for those distributed parameter systems for which sufficiently accurate data on pole and zero locations can be obtained. In this paper we apply this approach to control mechanical vibrations in those physical systems which can be accurately modeled as a flexible circular disk. Computer simulation results indicate that all the predominant lower frequency vibrations can be efficiently eliminated by just a few pairs of colocated sensor and actuator.
Finite-dimensional control of distributed parameter systems by Galerkin approximation of infinite dimensional controllers
