# Flow Control: Reduced Order Modelling, Nonlinear Analysis and Control Design

## Final Report Summary - FLOCON (Flow Control: Reduced Order Modelling, Nonlinear Analysis and Control Design)

This project has two main goals:

1) To build reduced-order models for flow processes on which control design can be performed easily.

Fluid flows are generally modelled by complicated partial differential equations such as the Navier-Stokes (NS) partial differential equations (PDEs). These equations can model the flow behaviour very accurately and are therefore very suitable for simulation purposes. However, from the analysis and control design point of view these PDEs cannot be utilised since they are highly complicated and nonlinear. Hence, it is necessary to come up with some means to simplify these equations to create approximations on which researchers can work easily. There are certain techniques in literature towards this goal such as Proper Orthogonal Decomposition (POD)/Galerkin Projection (GP) which do produce reduced order models; however these models do not have the control effect as an explicit term which is necessary for control design purposes. Also the standard POD/GP models are usually nonlinear, which complicates any analysis and control design to be carried out. In addition, the GP procedure requires extensive manipulations on the NS PDEs, which is a potential source of numerical errors.

The first goal of the project is to solve this issue by coming up with reduced order methods that can be used for control design. It is usually easier to perform control design on linear models so the initial attempt will be to produce linear models, followed later by nonlinear (yet still simple) models. It is also desirable to study and alternative approach to the GP procedure which will be less prone to numerical errors.

2) Developing analysis and control design methods for flow control problems, with eventual emphasis on nonlinear methods.

The second part of the project will deal with developing methods to analyse the certain characteristics of the flow using the models developed, such as stability, performance and so on. The first task towards this direction is to determine how well the reduced order models represent the original flow. This is necessary to make sure that the conclusions obtained from the models will be valid on the large scale systems. For this purpose one needs to build reconstructions from the models obtained and compare these with the original flow. This needs to be done both qualitatively and quantitatively; the former can be performed by visually comparing the original flow snapshots with the reconstructions, whereas the latter can be performed by proposing a suitable metric such as mean squared error (MSE). Once the confidence for the models are established, one can utilise standard tools from control theory on these models to reveal important characteristics of the flow such as stability, time constant, period of oscillations, existence and properties of limit cycles and so on. Finally one can define control problems on the flow process and carry out the controller design on the reduced order models. For instance, it may be desirable to suppress unwanted oscillations, drive the flow field close to a desired profile, maximize/minimize lift/drag forces on certain bodies within the flow field and so on. Again, standard tools from linear control theory can be utilised on the linear models, but nonlinear methods are problem specific and require revealing and exploiting certain structures present within the dynamics. The final task in the second objective is to deal with these nonlinear aspects and come up with solutions that can be used towards this direction.

The first activity within the project was data collection, from which the process models would be built. The flow data is in the form of snapshots, i.e. the instantaneous values for the velocity field collected at certain time intervals. There are two main channels through which this data was collected: through physical experiments using a particle image velocimetry (PIV) setup and through computational fluid dynamics (CFD) simulations. The latter data collection (CFD) was performed through the software FLUENT and also through an open license MATLAB NS simulator tool. The main test case was the flow over a pitching NACA airfoil. Both slow and fast pitching cases were studied and the data from both were collected in the form of flow snapshots. The POD/GP procedure was applied to these data to obtain reduced order models and reconstructions using these snapshots were compared with the original data, where it was observed that the model outputs can satisfactorily represent the flow. In addition, an alternate modelling approach, namely POD followed by system identification (SI) was developed and tried as a novel means of input separation (i.e. obtaining the actuation effect as a standalone term in the mathematical model). It was seen that the models obtained through this procedure can also represent the original flow dynamics with acceptable accuracy.

In the second half of the project the main task was to develop analysis and control design methods for flow control problems, including nonlinear approaches. For this purpose we have chosen to rely on CFD simulations as our PIV setup lacks actuators and thus the ability to induce a control effect into the system. As a test case we considered nonlinear modelling and feedback control of vorticity behind an immersed circular cylinder system. First a number of control input points over the cylinder and some measurement points for vorticity past the cylinder were assigned. A type of nonlinear dynamic model (namely a Hammerstein-Wiener (HW) model) of the flow field was estimated via system identification techniques using measurement data obtained from a chirp input function. Once the dynamical model of the system was estimated, a controller for the linear block of the HW model was designed using internal model control method, and this controller was then mapped to the HW model by reversing the input/output nonlinearity functions. The procedure described was implemented and tested numerically in MATLAB and CFD computations performed on the closed-loop system showed that the controller was capable of achieving significant reduction in the vorticity levels past the cylinder. A similar approach was also applied successfully to a different test case, namely the nonlinear dynamic modelling and feedback control of the drag to lift coefficient ratio for the airfoils NACA 23012, ag13 and b737a.

As a whole, the project results constitute a full systematic methodology of attacking flow control problems starting with data collection and ending with a closed-loop feedback controlled system achieving a desired flow control task.

It has been understood for quite a while that flow control is very important from a technological viewpoint and offers many potential benefits to the community. These potential benefits are spread among diverse areas and disciplines, and include the reduction fuel costs for vehicles, improvement of the effectiveness of industrial processes, providing increased safety and comfort in land and air transportation, reduction of structural damage in pipelines and space vehicles, and optimization of flow of therapeutic drugs in the bloodstream.

The results of the project will enable us to approach one step further towards the above-mentioned goals over the long haul. As to the short-term, the results have the potential of starting new projects, possibly with some local industrial companies that have stated interest in the topic of flow control. The results are also disseminated to the scientific community through conference and journal publications, which have the potential of generating new collaborations and further results in the field.