Frequency domain stability and relaxed convergence conditions for filtered error adaptive feedforward

The convergence of filtered error and filtered reference adaptive feedforward is limited by three effects: model mismatch, unintended input-disturbance interaction and too fast parameter adaptation. In this article, the first two effects are considered for MIMO systems under the slow parameter adaptation assumption. The convergence with model mismatch is conventionally guaranteed using a strictly positive-real condition. This condition can be easily verified in the frequency domain, but due the high-frequency parasitic dynamics of real systems, it is hardly ever satisfied. Nevertheless, filtered error and filtered reference adaptive feedforward have successfully been implemented in numerous applications without satisfying the strictly positive-real condition. It is shown in this article that the strictly positive-real condition can be relaxed to a power-weighted integral condition, that is less conservative and provides a practical check for the convergence of filtered error adaptive feedforward for real systems in the frequency domain. The effects of input-disturbance interaction are analysed and conditions for the stability are given in the frequency domain. Both conditions give clear indicators for frequency domain filter tuning, and are verified on an experimental active vibration isolation system.