Nonlinear stability analysis of transitional flows using quadratic constraints

Optimizing the Kreiss Constant

“…or equivalently, to minimization of the numerical abscissa ω(A + BKC), defined as ω(M ) := 1/2λ max (M + M T ). This is in line with the results in [24] for transitional flow studies. On the other end, when n ϕ = 0, the plant is linear and the controller can be of full order.…”

“…Chaos dynamics: design with the QC approach. Here we assess the stability properties of the closed loop (25) using the Lyapunov Quadratic Constraints (QC) approach of [36,24,30].…”

“…This is clearly a sufficient possibly conservative condition because of the chosen quadratic in V (x cl ), and also because the specific dependence of w on the states x is ignored. Using a S-procedure argument [24,12] to aggregate the lossless constraint (27), this is rewritten as…”

“…Theorem 5. There exist a linear time-invariant controller (24) such that the sufficient global stability conditions (28) hold, if and only if there exist solutions X = X T and…”

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]).…”