Linear time‐varying control of the vibrations of flexible structures

Recent results on pole placement for linear time‐varying (LTV) systems are exploited here for the control of flexible structures. The infinite‐dimensional system is approached as usual by a restricted number of modes of interest, according to the frequency range in which the system is exploited. The difference with the previous approaches (e.g. gain scheduling) is that only one finite dimension linear model is used and its parameters (frequency and damping of the modes) are varying according to the operating conditions. The control model is thus LTV and a regulator is analytically synthesised to ensure closed‐loop stability and performances. This regulator is also LTV and thus automatically (without any special switching action) tracks the operating conditions of the system. The case of a flexible beam is studied in simulation as well as in the laboratory experimentation. Hence, a suitable application of the proposed LTV synthesis is done to find a compromise between the complexity of interpolation and its efficiency. Furthermore, the bad dissipation factor of flexible systems leads them to be good candidates to proof the efficiency of the proposed gain scheduling strategy. This work validates on a real system the LTV pole placement approach and opens the way to a new gain scheduling strategy for the control of LTV, non‐linear or infinite‐dimensional systems.