Utilizing Under-Determined Solutions for MIMO Vibration Control

A typical constraint in multiple-input/multiple-output (MIMO) vibration testing is the number of available control gauges and associated MIMO-compatible specifications. This can arise due to the difficulty in obtaining field response data with high channel counts which would be used to create MIMO-compatible specifications at multiple control gauge locations. As MIMO vibration control is typically posed as an over-determined (i.e., more outputs than inputs) regression problem, having limited controls (outputs) limits the number of inputs that can be used. To allow a MIMO vibration test to be run with a sufficient number of inputs, it may be necessary to utilize fewer outputs than inputs, which switches the control to an under-determined regression problem. Under-determined regression problems have different behavior than over-determined problems, and it is not immediately clear how those differences translate to quantities of interest in MIMO vibration control. This paper aims to explore under-determined problems in the context of MIMO vibration control using examples to provide guidance to the MIMO vibration practitioner.