From resolvent to Gramians: forcing and response modes for control

During the last decade, forcing and response modes produced by resolvent analysis have promised to have an impact on flow control applications due to their potential to guide sensor and actuator placement and design. However, resolvent modes are frequency-dependent, which, although responsible for their success in identifying scale interactions in turbulence, complicates their use for control purposes. In this work, we seek orthogonal bases of forcing and response modes that are the most responsive and receptive, respectively, across all frequencies. We show that these frequency-independent bases of representative resolvent modes are given by the eigenvectors of the observability and controllability Gramians of the system considering full state inputs and outputs. We present several numerical examples where we leverage these eigenbases by building orthogonal or interpolatory projectors. Forcing modes are used to identify dynamically relevant disturbances, to place point sensors to measure disturbances, and to design spatially-distributed and spatially-localized actuators for feedforward control. Response modes are used for point sensor placement aiming at flow reconstruction.