Determining the stability of a multifrequency large-scale system using lifting operators

The study of the stability of a multifrequency system usually involves the use of nonlinear operators. It is now shown how the stability of a large-scale multifrequency system can be determined using a linear method. This is possible due to the inclusion of ´lifting´ and ´inverse lifting´ operators in the closed-loop system block diagram. The large-scale system under consideration is the primary mirror of the Gran Telescopio de Canarias telescope. This mirror, with a diameter of 10 m, is segmented into 36 hexagonal pieces. One strategy for the active control of the mirror consists of the simultaneous application of two control actions at different frequencies. Such a strategy is called ´local-global´ control. Its application results in the closed-loop plant being treated as a multifrequency system. The use of the lifting operators reduces the problem to a single sampling frequency, which clearly simplifies the problem of determining the system stability.