Dual-driver standing wave tube: acoustic impedance matching with robust repetitive control

In some acoustic applications, it may be desirable to make a shorter standing wave tube operate like a longer tube at the same driving frequency. The basic idea is to reduce the length of a long tube, and replace the removed section with a secondary driver. The problem is then to match the acoustic impedance at the boundary where the secondary driver is installed to that of the original system. Two control formulations were investigated for this problem: a two-input-two-output (TITO) and a single-input-single-output (SISO) formulation. The TITO formulation directly tracks the two acoustic variables associated with the desired acoustic impedance, while the SISO formulation minimizes the impedance matching error. The desired impedance containing a very lightly damped mode is embedded in the augmented plant for feedback control design. In addition to the balance realization method, the Schur method was used for model reduction for the high-order plant. Since the standing wave tubes are driven by tonal signals, repetitive control was incorporated into the control frameworks to achieve the desired performance. Good performance of impedance matching was obtained for both formulations. The SISO formulation yielded slightly wider bandwidth of good impedance matching than the TITO. The TITO formulation offered additional control to individual signals related to the acoustic impedance of interest.