Control strategy using bang–bang and minimax principle for FRF with ATMDs

An active control strategy for vibration control of structures using ATMDs is proposed. The control strategy incorporates the minimax criteria for shaping the frequency response function (FRF) of the structure–TMD system and the bang–bang control law with a control force applied to a TMD. In the control scheme, the control law using bang–bang control is obtained by minimising the time derivative of a Lyapunov function of the open-loop structural system [Wu Z, Soong TT. Modified bang–bang control law for structural control implementation. Journal of Engineering Mechanics 1996;122(8):771–7] and is used to control the ATMD. In addition, the corresponding FRF for the coupled structure–ATMD system is reshaped by changing the tuning parameter of the TMD (when there is no control force applied), such that peaks in the FRF of the structure are equal in magnitude. When the active control is ineffective or is turned off, the damper system acts as a passive TMD. The proposed strategy is compared to another conventional control strategy, the Linear Quadratic Controller (LQC) applied to the structure–TMD system and the advantages are highlighted. The system is examined under harmonic and random excitations.