Continuous-time balanced truncation for time-periodic fluid 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 and optimization. One well-known procedure to obtain the desired reduced-order model in the proximity of a periodic solution of the governing equations is continuous-time balanced truncation. Within this framework, the periodic reachability and observability Gramians are usually estimated numerically via quadrature using the forward and adjoint post-transient response to impulses. However, this procedure can be computationally expensive, especially in the presence of slowly decaying transients. Moreover, it can only be performed if the periodic orbit is stable in the sense of Floquet. In order to address these issues, we use the frequency-domain representation of the Gramians, which we henceforth refer to as frequential Gramians. First, these frequential Gramians are well-defined for both stable and unstable dynamics. In particular, we show that when the underlying system is unstable, these Gramians satisfy a pair of allied differential Lyapunov equations. Second, they can be estimated numerically by solving algebraic systems of equations that lend themselves to heavy computational parallelism and that deliver the desired post transient response without having to follow physical transients. The computational gains that we can achieve by using the frequency domain are demonstrated on a simple three-dimensional toy model that exhibits time-periodic dynamics. We then demonstrate this method on a periodically forced axisymmetric jet at Reynolds numbers Re = 1250 and Re = 1500. At the lower Reynolds number, the flow strongly amplifies subharmonic perturbations and exhibits vortex pairing about a Floquet-stable T-periodic solution. At the higher Reynolds number, the underlying T-periodic orbit is unstable and the flow naturally settles onto a 2T-periodic limit cycle characterized by pairing vortices. At both Reynolds numbers, we compute a reduced-order model and we use it to design a feedback controller and a state estimator capable of suppressing vortex pairing.