  During the last decade, forcing and response modes produced by resolvent
analysis have demonstrated great potential to guide sensor and actuator
placement and design in flow control applications. 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
\emph{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 bases by building orthogonal or interpolatory projectors onto the
dominant forcing and response subspaces. Gramian-based forcing modes are used
to identify dynamically relevant disturbances, to place point sensors to
measure disturbances, and to design actuators for feedforward control in the
subcritical linearized Ginzburg--Landau equation. Gramian-based response modes
are used to identify coherent structures and for point sensor placement aiming
at state reconstruction in the turbulent flow in a minimal channel at
$\mathrm{Re}_{\tau}=185$. The approach does not require data snapshots and
relies only on knowledge of the steady or mean flow.
