X. Yuan

Feedback control of subsonic cavity flows using reduced-order models

Development, experimental implementation, and the results of reduced-order model based feedback control of subsonic shallow cavity flows are presented and discussed. Particle image velocimetry (PIV) data and the proper orthogonal decomposition (POD) technique are used to extract the most energetic flow features or POD eigenmodes. The Galerkin projection of the Navier-Stokes equations onto these modes is used to derive a set of nonlinear ordinary differential equations, which govern the time evolution of the eigenmodes, for the controller design. Stochastic estimation is used to correlate surface pressure data with flow field data and dynamic surface pressure measurements are used to estimate the state of the flow. Five sets of PIV snapshots of a Mach 0.3 cavity flow with a Reynolds number of 10 5 based on the cavity depth are used to derive five different reduced-order models for the controller design. One model uses only the snapshots from the baseline (unforced) flow while the other four models each uses snapshots from the baseline flow combined with snapshots from an open-loop sinusoidal forcing case. Linear-quadratic optimal controllers based on these models are designed to reduce cavity flow resonance and evaluated experimentally. The results obtained with feedback control show a significant attenuation of the resonant tone and a redistribution of the energy into other modes with smaller energy levels in both the flow and surface pressure spectra. This constitutes a significant improvement in comparison with the results obtained using open-loop forcing. These results affirm that reduced-order model based feedback control represents a formidable alternative to open-loop strategies in cavity flow control problems even in its current state of infancy.

Exploring strategies for closed-loop cavity flow control

One of the current three main thrust areas of the Collaborative Center of Control Science (CCCS) at The Ohio State University is feedback control of aerodynamic flows. Synergistic capabilities of the flow control team include all of the required multidisciplinary areas of flow simulations, low-dimensional and reduced-order modeling, controller design, and experimental integration and implementation of the components along with actuators and sensors. The initial application chosen for study is closed-loop control of shallow subsonic cavity flows. We have made significant progress in the development of various components necessary for reduced-order model based control strategy, which will be presented and discussed in this paper. Stochastic estimation was used to show that surface pressure measurements along with the reduced-order model based on flow-field variables can be used for closed-loop control. Linear controllers such as H ∞ , Smith predictor, and PID were implemented experimentally with various degrees of success. The results showed limitations of linear controllers for cavity flow with inherent nonlinear dynamics. Detailed experimental work further explored the physics and showed the highly non-linear nature of the cavity flow and the effects of forcing on the flow structure.

Experimental study of linear closed-loop control of subsonic cavity flow

A study is presented of the modeling and implementation of different concepts for linear feedback control of a single-mode resonance shallow cavity flow. When a physics-based linear model is used for cavity pressure oscillations, an H ∞ controller was designed and tested experimentally. It significantly reduced the main Rossiter mode for which it was designed, while leading to strong oscillations at other Rossiter modes. Other linear control methods such as Smith predictor controller and proportional integral derivative (PID) controller exhibited similar results. The ineffectiveness of using fixed linear models in the design of controllers for the cavity flows is discussed. A modification of the PID design produced a parallel-proportional with time-delay controller that remedied this problem by placing zeros at the frequencies corresponding to other resonance states. Interestingly, it was observed that introducing the same zero to the H ∞ controller can also successfully avoid the strong oscillations at other Rossiter modes otherwise observed in the single-mode-based design. The parallel-proportional with time-delay controller was compared to a very effective open-loop method for reducing cavity resonance and exhibited superior robustness with respect to departure of the Mach number from the design conditions. An interpretation is presented for the physical mechanisms by which the open-loop forcing and the parallel-proportional with time-delay controllers reduce the cavity flow noise. The results support the idea that both controls induce in the system a rapid switching between modes competing for the available energy that can be extracted from the mean flow.

Feedback Control of Cavity Flow Using Experimental Based Reduced Order Model

We present preliminary results on subsonic cavity flow control using reduced-order model based feedback control derived from experimental measurements. The reduced-order model was developed using the Proper Orthogonal Decomposition of PIV results in conjunction with the Galerkin projection of the Navier-Stokes equations onto the resulting spatial eigenfunctions. The stochastic estimation method was used for real-time estimate of the model time coefficients from dynamic surface pressure measurements. Equilibrium analysis led to the linearization of the reduced-order model around the equilibrium point and a model for controller design was obtained by shifting the origin of the coordinates to the equilibrium point. A linear-quadratic optimal controller was then designed and tested in the experiments. The results obtained are very promising and show that control is capable of reducing the cavity flow resonance not only at the Mach 0.3 flow, for which the reducedorder model was specifically derived, but also at other flows with some variation of the Mach number. These preliminary results indicate that the control switches the flow from a single mode resonance to a multi-mode resonance.

Closed-Loop Active Flow Control - A Collaborative Approach

The Collaborative Center of Control Science (CCCS) at The Ohio State University was founded very recently with funding from the Air Force Research Laboratory to conduct multidisciplinary research in the area of feedback control, with applications such as cooperative control of unmanned air vehicles (UAVs), guidance and control of hypersonic vehicles, and closed-loop active flow control. The last topic is the subject of this paper. The goal of this effort is to develop tools and methodologies for the use of closedloop aerodynamic flow control to manipulate the flow over maneuvering air vehicles and ultimately to control the maneuvers of the vehicles themselves. It is well known in the scientific community that this is a challenging task and requires expertise in flow simulation, low dimensional modeling of the flow, controller design, and experimental integration and implementation of these components along with actuators and sensors. The CCCS flow control team possesses synergistic capabilities in all these areas, and all parties have been intimately involved in the project from the beginning, a radical departure from the traditional approach whereby an experiment is designed and constructed, data are collected, a model is developed, and a control law is designed, i.e. the system is assembled for validation in a sequential fashion. The first problem chosen for study, control of the noise created by a shallow cavity placed in a flow, has specific relevance to the needs of the Air Force. For example, significant pressure fluctuations in an aircraft weapon bay can lead to structural damage to the air vehicle, to the stores carried in the cavity, and especially to the electronics carried onboard the stores. The team has been working together for a relatively short period of time. Nevertheless, significant progress has been made in the development of various components of the closed-loop cavity flow control problem. The paper will present and discuss the progress made to date and future plans.

Reduced-order Model-based Feedback Controller Design for Subsonic Cavity Flows

This paper explores feedback controller design for cavity ∞ows based on reduced-order models derived using Proper Orthogonal Decomposition (POD) along with Galerkin projection method. Our preliminary analysis shows that the equilibrium of the POD model is unstable and a static output feedback controller cannot stabilize it. We develop Linear Quadratic (LQ) optimal state feedback controllers and LQ optimal observers for the linearized models. The linear controllers and observers are applied to the nonlinear system using simulations. The controller robustness is numerically tested with respect to difierent POD models generated at difierent forcing frequencies. An estimation for the region of attraction of the linear controllers is also provided.

CONTROLLER DESIGN FOR ACTIVE CLOSED-LOOP CONTROL OF CAVITY FLOWS

In this paper we discuss feedback controller design issues for active control of shallow cavity flows. Linear controllers, such as H ∞ , PID, and Smith predictor based controllers are designed and tested experimentally. The ineffectiveness of using fixed linear models in the design of linear controllers for the cavity flows is demonstrated via experimental results. In order to better address this problem, we are in the process of developing a nonlinear model of the cavity flow dynamics using Proper Orthogonal Decomposition (POD). We briefly discuss control issues related to the class of feedback systems involving this type of nonlinear plants.

On Delay-Based Linear Models and Robust Control of Cavity Flows

Design and implementation of now control problems pose challenging difficulties as the now dynamics are governed by coupled nonlinear equations. Recent research outcomes stipulate that the problem can be studied either from a reduced order modeling point of view or from a transfer function point of view. The latter idcntifies the physics of the problem on the basis of separate components such as scattering, acoustics, shear layer etc. This chapter uses the transfer function representation and demonstrates a good match between the real-time observations and a well-tuned transfer function can be obtained. Utilizing the devised model, an H∞ controller based on Toker-Ozbay formula is presented. The simulati on results illustrate that the effect of the noise can be eliminated significantly by appropriately exciting the now dynamics.

Reduced-Order Model-Based Feedback Control of Subsonic Cavity Flows — An Experimental Approach

The results of an ongoing research activity in the development and implementation of reduced-order model-based feedback control of subsonic cavity flows are presented and discussed. Particle image velocimetry data and the proper orthogonal decomposition technique are used to extract the most energetic flow features or POD eigenmodes. The Galerkin projection of the Navier-Stokes equations onto these modes is used to derive a set of ordinary nonlinear differential equations, which govern the time evolution of the modes, for the controller design. Stochastic estimation is used to correlate surface pressure data with flow field data and dynamic surface pressure measurements are used for real-time state estimation of the flow model. Three sets of PIV snapshots of a Mach 0.3 cavity flow were used to derive three reduced-order models for controller design: (1) snapshots from the baseline (no control) flow, (2) snapshots from an open-loop forced flow, and (3) combined snapshots from the cases 1 and 2. Linear-quadratic optimal controllers based on all three models were designed and tested experimentally. Real-time implementation shows a remarkable attenuation of the resonant tone and a redistribution of the energy into various modes with much lower energy levels.

Experimental Results and Bifurcation Analysis on Scaled Feedback Control for Subsonic Cavity Flows

In this paper, we present the latest results of our ongoing research activities in the development of reduced-order models based feedback control of subsonic cavity flows. The model was developed using the proper orthogonal decomposition of particle image velocimetry images in conjunction with the Galerkin projection of the Navier-Stokes equations onto the resulting spatial eigenfunctions. Stochastic estimation method was used to obtain the state estimation of the Galerkin system from real time surface pressure measurements. A linear-quadratic optimal controller was designed to reduce cavity flow resonance and tested in the experiments. Real-time implementation shows a significant reduction of the sound pressure level within the cavity, with a remarkable attenuation of the resonant tone and a redistribution of the energy into various modes with lower energy levels. A mathematical analysis of the performance of the LQ control, in agreement with the experimental results, is presented and discussed