Control of linear instabilities by dynamically consistent order reduction on optimally time-dependent modes

Abstract

Identification and control of transient instabilities in high-dimensional dynamical systems remain a challenge because transient (non-normal) growth cannot be accurately captured by reduced-order modal analysis. Eigenvalue-based methods classify systems as stable or unstable on the sole basis of the asymptotic behavior of perturbations and therefore fail to predict any short-term characteristics of disturbances, including transient growth. In this paper, we leverage the power of the optimally time-dependent (OTD) modes, a set of time-evolving, orthonormal modes that capture directions in phase space associated with transient and persistent instabilities, to formulate a control law capable of suppressing transient and asymptotic growth around any fixed point of the governing equations. The control law is derived from a reduced-order system resulting from projecting the evolving linearized dynamics onto the OTD modes and enforces that the instantaneous growth of perturbations in the OTD-reduced tangent space be nil. We apply the proposed reduced-order control algorithm to several infinite-dimensional systems, including fluid flows dominated by normal and non-normal instabilities, and demonstrate unequivocal superiority of OTD control over classical modal control.