Optimal and Robust Control and Estimationof Linear Paths to TransitionBy

Optimal and robust control theories are used to determine eeective, estimator-based feedback control rules for laminar plane channel ows that eeectively stabilize linearly unstable ow perturbations at Re = 10; 000 and linearly stable ow perturbations, characterized by mechanisms for very large disturbance ampliication, at Re = 5; 000. Wall transpiration (unsteady blowing/suction) with zero net mass ux is used as the control, and the ow 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 ow perturbations created by external disturbance forcing. This is important because, when mechanisms for large disturbance ampliication are present, small amplitude external disturbance forcing may excite ow perturbations with suuciently large amplitude to induce nonlinear ow 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 nite \worst case" inputs which are maximally detrimental to the control objective (H 1 or \robust" control). Root loci and transient energy growth analyses are shown to be inadequate measures to characterize overall system performance. Instead, appropriately-deened 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 f; ; `g may be individually adjusted to achieve desired closed-loop characteristics of the resulting systems. These scalar parameters may be intuitively explained, and are deened such that the resulting control equations retain the natural dual structure between the control parameter, `, and the estimation parameter,. The performance of the present systems with respect to these parameters is thoroughly investigated, and comparisons are made to simple proportional schemes where appropriate.