Estimating Regions of Attraction for Transitional Flows Using Quadratic Constraints

Regional Stability Analysis of Transitional Fluid Flows

“…Globally, when considering x ∈ R n , losslessness is one of the only properties satisfied by ϕ(x); however, when considering a regional analysis with x ∈ E ⊂ R n , the nonlinearity ϕ(x) can be locally bounded. The benefits of introducing local bounds for ϕ(•) was identified in [7], with the bounds obtained from the Cauchy-Schwartz Lemma allowing the conservatism of their stability certificates to be reduced compared to the earlier results of [8] and [9]. However, in general, bounds produced by the Cauchy-Schwartz Lemma are not tight, which suggests that there may be room to reduce this conservatism still further.…”

“…, n, with G ξ = {x ∈ R n : x(t) ⊤ Gx(t) ≤ ξ 2 } defining an ellipsoid over R n and with e i ∈ R n being the i th standard basis vector. Contrasting with the formulation of [7], in this paper, the matrices ∆ i = λ K+i ξ 2 G −1 are defined as matrix decision variables in the optimisation problem of the stability conditions instead of being fixed at each iteration.…”

“…The stability analysis of even these reduced-order models can still be challenging, owing to their nonlinear dynamics and non-normal state-transition matrices [5], [6]. However, progress has been made, notably in [7]- [9] where it was observed that the nonlinear model dynamics exhibit some structure that can be exploited. These studies observed that the model dynamics could be understood in terms of the feedback interconnection of a linear system with an energy persevering or lossless nonlinear gain, as shown in Figure 1, allowing the powerful and scalable techniques of passive systems theory [10, Chapter VI] to be applied.…”

“…Contribution: Motivated by the passive systems theory results [7]- [9], this paper extends the approach by introducing an algorithm that allows the axes lengths of the ellipsoids bounding the state trajectories to be defined as matrix variables to be optimised over. To evaluate the performance of the method, the obtained stability conditions were applied to both the 4-state WKH model for shear flow and the 9state model of Couette flow and demonstrated a reduction in conservatism compared to [7]- [9] (see Section VII).…”

A structured input-output approach to characterizing optimal perturbations in wall-bounded shear flows

“…In order to provide a more complete characterization of the flow, a number of researchers have sought to include nonlinear effects in the input-output approach; e.g., through harmonic balance methods [20] that build upon techniques for analyzing systems with spatiotemporal periodic coefficients [21], [22], [23], [24], [25]. Nonlinearity has also been included in stability analysis using quadratic constraints within a linear matrix inequality formulations [26], [27], [28], [29], [30]. Liu & Gayme [31] proposed an alternative approach that employs an input-output model of the nonlinearity placed within a feedback interconnection with the linearized dynamics (in the spirit of a Luré decomposition [32], [33] of the problem [34], [2]).…”