Continuous-time balanced truncation for time-periodic flows using frequential Gramians

Reduced-order models for flows that exhibit time-periodic behavior (e.g., flows in turbomachinery and wake flows) are critical for several tasks, including active control. The existing continuous-time balanced truncation method can be computationally expensive due to slowly-decaying transients and to the need to perform forward and adjoint impulse responses to compute the periodic reachability and observability Gramians. Moreover, this method can only be applied to systems that are stable in the sense of Floquet. To address these issues, we use the frequency-domain representation of the Gramians (henceforth referred to as frequential Gramians). First, these frequential Gramians are well-defined for both stable and unstable dynamics. Second, they can be estimated numerically by solving algebraic systems of equations that lend themselves to space-time parallelism and that deliver the desired post-transient response without following physical transients. We demonstrate our approach on a periodically-forced axisymmetric jet at Reynolds number Re=1500. In this configuration the flow exhibits heavy vortex pairing about an underlying unstable periodic orbit. We use balanced truncation for controller and observer design to restabilize the flow.