Extremely large telescopes are characterized by high degree of freedom control systems used to coordinate multiple segments and mirrors. The dynamics can interact so that single loop requirements do not provide sufficient stability and performance robustness. This paper reviews the relevant multivariable robustness and performance methods, and presents examples from Giant Magellan Telescope (GMT) motion control systems.
Singular value bounds of multivariable frequency responses are well developed computational tools that provide a methodology that can be used for telescope analysis. The singular value bounds are relevant because they give the maximum sensitivity for coupled, multivariable systems. Singular values are recommended for analysis, and can be considered for requirements. With sufficient numbers of sensors, these multivariable bounds are measurable and hence can be validated. There is a practical reason for using multivariable tools, to combine many, perhaps thousands of transfer functions and/or measurements that can be compared against singular value bounds.
The first example is the AZ/EL mount control. Coupling tends to be small, hence single-input analysis tools suffice, nevertheless the mount control system provides a good introduction to multivariable methodology. The maximum singular value of both the sensitivity and complementary sensitivity functions provide a good bound for crossover robustness near the position control bandwidth, typically +6 dB near 1 Hz. The high frequency region of the complementary sensitivity function provides a good bound on robustness with respect to unmodeled structural dynamics, typically–40dB above the maximum frequency of the finite element modes.
Similar multivariable stability robustness bounds can be applied to position control of the M2 assembly, for both the macrocell relative to the top end assembly, and each mirror subassembly relative to the macrocell. The latter includes control of the Fast Steering Mirror, where 21 PZT actuators control the tip and tilt of seven secondary mirrors. The risk is the 21 PZT control loops meet good classical phase and gain margin robustness metrics when measured as individual, single-input-single-output systems, but the multivariable bound exceeds either the +6 dB or – 40 dB bound. This can occur due to interaction in the macrocell, the structure used to support the individual segments. Whether or not this interaction occurs depends on the bandwidth of the control system relative to the structural modes of the macrocell. This tradeoff is important, and the maximum singular value is a good tool to test for this sensitivity.