Optimal and robust control and estimation of linear paths to transition

Optimal and robust control theories are used to determine effective, estimator-based feedback control rules for laminar plane channel flows that effectively stabilize linearly unstable flow perturbations at Re = 10000 and linearly stable flow perturbations, characterized by mechanisms for very large disturbance amplification, at Re = 5000. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control, and the flow measurement is derived from the wall skin friction. The control objective, beyond simply stabilizing any unstable eigenvalues (which is relatively easy to accomplish), is to minimize the energy of the flow perturbations created by external disturbance forcing. This is important because, when mechanisms for large disturbance amplification are present, small-amplitude external disturbance forcing may excite flow perturbations with sufficiently large amplitude to induce nonlinear flow instability. The control algorithms used in the present work account for system disturbances and measurement noise in a rigorous fashion by application of modern linear control techniques to the discretized linear stability problem. The disturbances are accounted for both as uncorrelated white Gaussian processes (H-2 or 'optimal' control) and as finite 'worst case' inputs which are maximally detrimental to the control objective (H-infinity or 'robust' control). Root loci and transient energy growth analyses are shown to be inadequate measures to characterize overall system performance. Instead, appropriately defined transfer function norms are used to characterize all systems considered in a consistent and relevant manner. In order to make a parametric study tractable in this high-dimensional system, a convenient new scaling to the estimation problem is introduced such that three scalar parameters {gamma, alpha, l} may be individually adjusted to achieve desired closed-loop characteristics of the resulting systems. These scalar parameters may be intuitively explained, and are defined such that the resulting control equations retain the natural dual structure between the control parameter, l, and the estimation parameter, alpha. The performance of the present systems with respect to these parameters is thoroughly investigated, and comparisons are made to simple proportional schemes where appropriate.