  The application of control tools to complex flows frequently requires
approximations, such as reduced-order models and/or simplified forcing
assumptions, where these may be considered low-rank or defined in terms of
simplified statistics (e.g. white noise). In this work, we propose a
resolvent-based control methodology with causality imposed via a Wiener-Hopf
formalism. Linear optimal causal estimation and control laws are obtained
directly from full-rank, globally stable systems with arbitrary disturbance
statistics, circumventing many drawbacks of alternative methods. We use
efficient, matrix-free methods to construct the matrix Wiener-Hopf problem, and
we implement a tailored method to solve the problem numerically. The approach
naturally handles forcing terms with space-time colour; it allows inexpensive
parametric investigation of sensor/actuator placement in scenarios where
disturbances/targets are low rank; it is directly applicable to complex flows
disturbed by high-rank forcing; it has lower cost in comparison to standard
methods; it can be used in scenarios where an adjoint solver is not available;
or it can be based exclusively on experimental data. The method is particularly
well-suited for the control of amplifier flows, for which optimal control
approaches are typically robust. Validation of the approach is performed using
the linearized Ginzburg-Landau equation. Flow over a backward-facing step
perturbed by high-rank forcing is then considered. Sensor and actuator
placement are investigated for this case, and we show that while the flow
response downstream of the step is dominated by the Kelvin-Helmholtz mechanism,
it has a complex, high-rank receptivity to incoming upstream perturbations,
requiring multiple sensors for control.
