Sunil Ahuja

Reduced-order models for control of fluids using the eigensystem realization algorithm

As sensors and flow control actuators become smaller, cheaper, and more pervasive, the use of feedback control to manipulate the details of fluid flows becomes increasingly attractive. One of the challenges is to develop mathematical models that describe the fluid physics relevant to the task at hand, while neglecting irrelevant details of the flow in order to remain computationally tractable. A number of techniques are presently used to develop such reduced-order models, such as proper orthogonal decomposition (POD), and approximate snapshot-based balanced truncation, also known as balanced POD. Each method has its strengths and weaknesses: for instance, POD models can behave unpredictably and perform poorly, but they can be computed directly from experimental data; approximate balanced truncation often produces vastly superior models to POD, but requires data from adjoint simulations, and thus cannot be applied to experimental data. In this article, we show that using the Eigensystem Realization Algorithm (ERA) (Juang and Pappa, J Guid Control Dyn 8(5):620–627, 1985) one can theoretically obtain exactly the same reduced-order models as by balanced POD. Moreover, the models can be obtained directly from experimental data, without the use of adjoint information. The algorithm can also substantially improve computational efficiency when forming reduced-order models from simulation data. If adjoint information is available, then balanced POD has some advantages over ERA: for instance, it produces modes that are useful for multiple purposes, and the method has been generalized to unstable systems. We also present a modified ERA procedure that produces modes without adjoint information, but for this procedure, the resulting models are not balanced, and do not perform as well in examples. We present a detailed comparison of the methods, and illustrate them on an example of the flow past an inclined flat plate at a low Reynolds number.

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

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 neighbourhood of this unstable steady state. The actuation is modelled as a localized body force near the trailing 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.

Low-Dimensional Models for Control of Leading-Edge Vortices: Equilibria and Linearized Models

When an airfoil is pitched up rapidly, a dynamic stall vortex forms at the leading edge and produces high transient lift before shedding and stall occur. The aim of this work is to develop low-dimensional models of the dynamics of these leading-edge vortices, which may be used to develop feedback laws to stabilize these vortices using closed-loop control, and maintain high lift. We first perform a numerical study of the two-dimensional incompressible flow past an airfoil at varying angles of attack, finding steady states using a timestepper-based Newton/GMRES scheme, and dominant eigenvectors using ARPACK. These steady states may be either stable or unstable; we develop models linearized about the stable steady states using a method called Balanced Proper Orthogonal Decomposition, an approximation of balanced truncation that is tractable for large systems. The balanced POD models dramatically outperform models using the standard POD/Galerkin procedure, and are used to develop observers that reconstruct the flow state from a single surface pressure measurement.

Low-Dimensional Models for Feedback Stabilization of Unstable Steady States

When the angle of attack of an airfoil is gradually increased at low Reynolds numbers, the flow undergoes a Hopf bifurcation wherein the steady state loses stability in a transition to periodic vortex shedding. The goal of this work is to stabilize these unstable steady states using feedback control. For that purpose, we derive reduced order models valid in the neighborhood of these unstable steady states, using an approximate balanced truncation method valid for large dimensional systems. The original method is valid only for stable systems, and here we present a modification to derive models for unstable sytems, and use the same along with standard linear control techniques such as Linear Quadratic Regulator (LQR) to obtain stabilizing control laws. We use full-state feedback to stabilize an unstable steady state of a two-dimensional flow past a past flat plate at an angle of attack of 35 degrees. We also present observers that reconstruct the entire flow-field based on the flat-plate lift and drag measurements.

Closed-Loop Control of Leading Edge Vorticity on a 3D Wing: Simulations and Low-Dimensional Models

We study model-based feedback control of the low-Reynolds-number flow over a flat plate at large angles of attack, in both two and three dimensions. Our long-term goal is to be able to manipulate the leading-edge vortices that form on low-aspect-ratio wings at high angles of attack, and that often contribute to exceptionally large lift coefficients. In two-dimensional simulations, we present a model-based feedback controller that uses an observer to reconstruct the entire flow field from velocity measurements at three locations, and stabilizes the flow at an angle of attack for which the natural flow state is periodic shedding. In three-dimensional simulations, we use open-loop forcing to study actuator placement, and conclude that trailing-edge actuation is more effective than leading-edge actuation in influencing the forces on the plate, as well as the wake structures. Finally, we present initial results towards extending our model-based control design to the 3D setting, and apply a selective frequency damping method to find unstable equilibrium flow fields in 3D simulations.

Template-Based Stabilization of Relative Equilibria in Systems with Continuous Symmetry

We present an approach to the design of feedback control laws that stabilize relative equilibria of general nonlinear systems with continuous symmetry. Using a template-based method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. We present two different approaches of control design. The first approach assumes that the closed loop system is affine in the control and that the actuation is equivariant. We derive feedback laws for the reduced system that minimize a quadratic cost function. The second approach is more general; here the actuation need not be equivariant, but the actuators can be translated in the symmetry direction. The controller resulting from this approach leaves the dynamics associated with the symmetry variable unchanged. Both approaches are simple to implement, as they use standard tools available from linear control theory. We illustrate the approaches on three examples: a rotationally invariant planar ODE, an inverted pendulum on a cart, and the Kuramoto-Sivashinsky equation with periodic boundary conditions.

Reduced-order models for control of stratified flows in buildings

Building heating and cooling systems have potential for energy savings by employing passive devices that exploit thermal stratification and buoyancy. The resulting thermal-fluid flow patterns from such systems tend to be sensitive to disturbances, and advanced flow control techniques are important to maintain occupant comfort. In this work, we employ Eigensystems Realization Algorithm (ERA) to obtain low-order models of airflow in buildings, which capture relevant dynamics and are amenable for control design. We present an alternative interpretation that allows one to obtain models using ERA, without resorting to lifting, an approach that is typically used to introduce boundary control and boundary disturbances explicitly in the reduced-order model. Using this reduced-order model we derive an optimal control law in closed form (which is composed of a feedforward and a feedback term) for rejecting a known disturbance, while minimizing a quadratic cost related to occupant discomfort and energy consumption. We demonstrate this approach using closed loop CFD simulations of airflow in a room with a passively cooled radiant ceiling and a displacement vent.

Computational coarse graining of a randomly forced one-dimensional Burgers equation

We explore a computational approach to coarse graining the evolution of the large-scale features of a randomly forced Burgers equation in one spatial dimension. The long term evolution of the solution energy spectrum appears self-similar in time. We demonstrate coarse projective integration and coarse dynamic renormalization as tools that accelerate the extraction of macroscopic information (integration in time, self-similar shapes, and nontrivial dynamic exponents) from short bursts of appropriately initialized direct simulation. These procedures solve numerically an effective evolution equation for the energy spectrum without ever deriving this equation in closed form.

Template-based stabilization of relative equilibria

We present an approach to the design of feedback control laws that stabilize the relative equilibria of general nonlinear systems with continuous symmetry. Using a template-based method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original system are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. Assuming equivariant actuation, we derive feedback laws for the reduced system that are optimal in the sense that they minimize a quadratic cost function. We illustrate the method by stabilizing unstable traveling waves of a dissipative PDE possessing translational invariance