Linear Quadratic Gaussian control for vibration isolation systems in a gravitational wave detector Integration internship

Abstract

Gravitational waves are a type of wave that is a deformation of space-time travelling at the speed of light. This has been theorized by Albert Einstein, and it is only 100 years later that the first direct detection has been made. The detection has been made by two detectors of LIGO, using ultra long Michelson’s interferometer. Since then, the interest of building gravitational waves detectors has grown. It is in this context that the project of building a third generation gravitational waves detector, called the Einstein Telescope, has been launched by the European Union.

To be able to make detection, it is important to isolate the mirror from all sorts of noises. One of which that is noise coming from the ground. Pushing the isolation capabilities of a system is one of the goal of the E-TEST project run by the Precision Mechatronics Laboratory. In the objective of increasing the bandwidth of isolation while keeping the overall height of the system small, the E-TEST prototype is combining passive and active isolation. Multiple control strategies can be used to isolate the mirror from ground, and there is an interest in using an optimal control strategy. One of the strategies is the Linear Quadratic Gaussian method.

This kind of controller is composed of a Linear Quadratic Regulator and of a Kalman filter. This work is explaining the design methodology used to create the controller. This is followed by the design of the controller, loop gain and closed loop designed. Then the method used to assess the robustness and the margin of the feedback system is presented.

The results showed that the design of the Linear Quadratic Regulator is straight forward, since the states that need to be isolated are the ones penalized. The design of the Kalman filter showed that the states are not well estimated for frequencies lower than the resonance frequency of the inertial sensor, due to the dynamics of the latter. The analysis of the controller design has shown that the dynamics of thesensor has been inverted. This creates a problem for the robustness and margins, since small uncertainties lead to instability. Both robustness and margins were highly dependent on the inertial sensor. The loop gain analysis has shown that the static gain was very high, but changing the values of the ground process noise covariance decrease this effect. The analysis of the closed loop showed that the feedback gave similar results than for the Linear Quadratic Regulator, but it was however limited by the error on the estimations of the states. Then, a comparison was done using two inertial sensors instead of an inertial sensor and gyroscope. In this configuration, the performances were degraded due to the fact that none of the two sensors were able to reconstruct the states at low frequencies.

Finally, an experimental run on the E-TEST prototype was done. A damping loop was implemented in the vertical direction using three inertial sensors. The experimental test showed the feasibility of implementing such control strategy on a real plant. The test was however limited by high frequency peaks that were not modeled in the system.

Overall, the thesis showed that the Linear Quadratic Gaussian is a control strategy that is a good candidate for isolation. But the method relies on good knowledge of the plant, and it leads to small margins and poor robustness.