Robust Control Design for a linearized model of a LIGO subsystem

Laser Interferometer Gravitational wave Observatory (LIGO) is a gravitational wave antenna used to detect the infinitesimal contraction and expansion of space by a passing gravitational wave. LIGO comprises several subsystems which isolate the optics from seismic noise. Of these, we focus on the quadruple pendulum suspension system and attempt to design a stabilizing controller for a simpler version of the system, the two wire simple pendulum. A nonlinear model of the two wire simple pendulum based on the Euler-Lagrange theory is derived. This is linearised about an equilibrium and a stabilising controller using a robust control paradigm called $H_{\infty} $ synthesis, is explored as a candidate design paradigm for the two wire simple pendulum. In particular, in this broad framework of $H_{\infty} $ synthesis we adopt a technique called coprime factorisation. Different types of uncertainties like parametric and nonlinear uncertainties have also been considered and an upper bound on the nonlinear uncertainties in the system is determined