Wave-based planar motion and vibration control of under-actuated mass-spring arrays

This paper considers the problem of position control and active vibration damping of a planar, array (or grid) of masses and springs, by a single actuator, attached to one corner of the array, which is required to translate and rotate the entire system from rest to rest, through target linear and angular displacements, simultaneously. An obvious challenge is that the system has many degrees of freedom, with many undamped vibration modes, and is clearly highly under-actuated. The control technique is a development of "wave-based control," whereby rapid and effective control of the entire system is achieved, robustly, using measurements made only at the actuator, of the actuator's own motion and of the forces between the actuator and the attached flexible system. No detailed system model or system identification is needed. The actuator need not be ideal. The array does not have to be uniform, in its geometry or in the mass and spring values. The control strategy is simple to implement. The 2D array is of interest in itself as a benchmark control challenge, but it can also be considered a model of various lumped structures, or a discretisation of distributed systems.