Model Independent Numerical Procedure for the Diagonalization of a Multiple Input Multiple Output Dynamic System

Seismic noise limits Earth based gravitational wave interferometric detectors at low frequencies. The detection threshold can be lowered down to a few Hz using a seismic attenuation system based on Inverted Pendulum (IP) which sustains interferometer optical components by means of a chain of pendulums. The IP, acting as a mechanical low pass filter, is able to filter out seismic noise in the horizontal plane and at the same time it provides a quasi-inertial stage where the suspension point of the chain of pendulums lies. The main degrees of freedom of an IP are three: two translational modes and one rotational mode. Therefore, to fully determinate its position, three independent sensors are mounted at the periphery of the IP top table. For the same reason, three independent actuators are used to move the IP. The geometrical position of the sensors is different from actuator positions, in addition, both of them are not connected to the normal modes of the IP. Each sensor will be sensitive in all the three IP normal modes and each actuator will generate movements which are a mix of the three modes. To take advantage of controlling a Single Input Single Output (SISO) system instead of a Multiple Input Multiple Output (MIMO) system, a diagonalization of the actuation and detection system is needed. An original and model independent experimental procedure for determining the system dynamics, giving an effective diagonalization has been developed and tested.