Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators

Abstract

We present an estimator-based control design procedure for flow control, using reduced-order models of the governing equations, linearized about a possibly unstable steady state. The reduced-order models are obtained using an approximate balanced truncation method that retains the most controllable and observable modes of the system. The original method is valid only for stable linear systems, and in this paper, we present an extension to unstable linear systems. The dynamics on the unstable subspace are represented by projecting the original equations onto the global unstable eigenmodes, assumed to be small in number. A snapshot-based algorithm is developed, using approximate balanced truncation, for obtaining a reduced-order model of the dynamics on the stable subspace.The proposed algorithm is used to study feedback control of two-dimensional flow over a flat plate at a low Reynolds number and at large angles of attack, where the natural flow is vortex shedding, though there also exists an unstable steady state. For control design, we derive reduced-order models valid in the neighborhood of this unstable steady state. The actuation is modeled as a localized body force near the leading edge of the flat plate, and the sensors are two velocity measurements in the near-wake of the plate. A reduced-order Kalman filter is developed based on these models and is shown to accurately reconstruct the flow field from the sensor measurements, and the resulting estimator-based control is shown to stabilize the unstable steady state. For small perturbations of the steady state, the model accurately predicts the response of the full simulation. Furthermore, the resulting controller is even able to suppress the stable periodic vortex shedding, where the nonlinear effects are strong, thus implying a large domain of attraction of the stabilized steady state.

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

“…Nonetheless, we omit the subscript h wherever possible. The snapshots are typically equispaced in time along the entire period T and obtained after discarding the initial transient of the flow until a stable regime is reached and the flow can be modelled as a stochastic process that is stationary in time 1 .…”

“…So far we have not mentioned the treatment of boundary conditions that need to be imposed on (1). In the case of homogeneous boundary conditions, the snapshots as well as their linear combinations will naturally satisfy the same boundary conditions so that nothing special needs to be done.…”

“…[132] for further details). This strategy was used to propose reduced-order controllers around linearly unstable steady states for flow around a cylinder in [122] and for flow past a flat plate in [1].…”

“…In addition, suitable boundary conditions need to be prescribed in order to solve problem (1), see e.g. [51,80,118].…”

“…In this case we deal with a parametric model reduction problem. We denote µ ∈ P ⊂ R P a vector of P parameters of interest for a given fluid dynamics problem, as in the case of the Reynolds number appearing in (1). Other typical examples deal with different physical parametrizations (e.g.…”