LMI-based distributed H∞H∞ control of the Thirty Meter Telescope’s primary mirror

This paper investigates linear matrix inequality (LMI)-based distributed H∞H∞ control of the proposed Thirty Meter Telescope (TMT), in which 492 hexagonal segments will be employed to create a 30 m diameter circular optical aperture. The vast number of control parameters introduced by the segmented mirror design makes stabilizing and aligning the segments against disturbances a very challenging task to handle by the classical centralized control techniques. Although a decentralized design can simplify the control problem, the objective of bounding the relative displacements at the segment edges couples the neighboring segments at the control objective level. Also, since the segments are installed on a common support structure, any control action in one segment can excite the structure’s natural modes and affect the neighboring segments due to their coupling at the structural level. In order to account for these two levels of couplings, a distributed control scheme is proposed. The main contribution of this work is the proof-of-concept demonstration of distributed control of the TMT’s 492 mirror segments. First, a distributed model compatible with linear matrix inequalities (LMIs) is extracted via the finite element analysis (FEA) of the proposed TMT. Then, a distributed controller is designed by using the LMI approach. Closed-loop simulations of the 492-segment system with the synthesized controller are carried out, and compared against the performance of a Fourier-based distributed H∞H∞ controller. The simulation results show that both distributed controllers can satisfy the stringent imaging performance requirements of the TMT.