NARX modelling of unsteady separation control

J. Dandois, E. Garnier and P.-Y. Pamart NARX modelling of unsteady separation control

Experiments in Fluids , Vol. 54, No. 2, 2013

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R E S E A R C H A R T I C L E

NARX modelling of unsteady separation control

J. Dandois^•E. Garnier ^•P.-Y. Pamart

Received: 2 April 2012 / Revised: 19 November 2012 / Accepted: 11 December 2012 ÓSpringer-Verlag Berlin Heidelberg 2013

Abstract This paper presents the application of Nonlin-ear Auto-Regressive with eXogenous input (NARX) modelling to model the flow behaviour in response of a periodic forcing. In the first part, the NARX black-box model is presented. The model coefficients are obtained by least-square fitting. The resolution of the associated linear system being ill-conditioned, a Tikhonov regularization is employed. The first application presented is the identifi-cation of a NARX model of separation control by a syn-thetic jet slot over a rounded ramp. It is shown that the pressure at a particular location is representative of the flow state (attached or separated). This quantity will be the output of the model, the input being the forcing signal of the synthetic jet. Then, an identification signal is designed to explore as many flow states as possible with a short signal duration. The next step is dedicated to the selection of the NARX model structure. Input/output correlations, partial output autocorrelations and the Akaike Information Criterion minimization are used. A fit coefficient of 84 % is obtained. Finally, the accuracy of the NARX model, both on the steady and on the unsteady components of the output, is checked by comparison of the results with signals

not used in the identification phase. The third part deals with the application of the same methodology to an experiment of separation control by pulsed fluidic vortex generators in a highly curved duct. A fit coefficient of 60 % is obtained in this case.

1 Introduction

Separated flows occur in a variety of engineering applica-tions and have generally a negative impact on performance.

For external aerodynamics, flow separation leads to a lift decrease and a concomitant drag increase. Moreover, flow unsteadiness associated with separation increases structural loads and noise levels. In internal aerodynamics, separation raises pressure losses in ducts and more generally affects the propulsive system efficiency. Many flow control con-cepts have been successfully proposed in the literature to control separation (see for example ref. Miller and Joslin (2009) for a review). Most of them are based on actuators which generate either longitudinal (mechanical vortex generator, Air Jet Vortex generator) or transverse vorticity (blowing or suction trough slots). A review of existing actuators for active flow control can be found in ref.

Cattafesta and Sheplak (2011). Even if open-loop control has been successful in suppressing separation, the practical implementation of such techniques has often been pre-vented by the resulting penalties on the global system (added mass, added drag, excessive removal to the pro-pulsion system). More advanced closed-loop active flow control techniques are seen as a promising way to reduce the control cost thanks to a real-time adaptation of flow perturbations. The identification of a ‘‘flow model’’ is a prerequisite for most of closed-loop control techniques. In the literature, a vast community has used physically based This article is part of the collection Topics in Flow Control. Guest

Editors J. P. Bonnet and L. Cattafesta.

J. Dandois (&)E. Garnier

Applied Aerodynamics Department, ONERA - The French Aerospace Lab, 8 rue des Vertugadins, 92190 Meudon, France e-mail: julien.dandois@onera.fr

flow model both for experiments and simulations. In many cases, snapshots of a combination of the non-controlled and open-loop controlled flow are projected on an orthogonal basis thanks to POD-like techniques [Sirovich (1987), Schelgel et al. (2012)]. A Galerkin projection then permits to reduce the dynamic system to a small number of non-linear differential ordinary equations (ODE). Moreover, the ODE are often linearized around an equilibrium state to make possible the use of classical tools of control theory.

In experiments, the flow state is estimated by building a correlation matrix between the temporal coefficients of the POD decomposition and the unsteady wall information [Taylor and Glauser (2004), Pinier et al. (2007)]. Appli-cations of such techniques can be found, for example, in ref. Samimy et al. (2007) and Caraballo et al. (2008). In simulations, global modes which derive directly from the linearization of the Navier–Stokes equations can be com-puted. Methods that combine POD representing the stable subspace and global modes accounting for the unstable subspace have been developed [see for example Barbagallo et al. (2011)]. Recently, empirical nonlinear Galerkin methods have been developed. They have in particular been successfully applied to model the behaviour of mas-sively separated flows: a high-lift configuration (Luchten-burg et al.2009) and a cylinder wake Tadmor et al.2011;

Aleksic et al.2010. A comprehensive description of these techniques can be found in ref. Noack et al. (2011).

The aforementioned approaches are difficult to imple-ment and aim at describing the dynamics of a large spatial extent of the considered flow. In complement to these techniques, black-box models have been proposed in the literature. In most of the cases, they are used to describe the control effect on the flow with a limited number of inputs (actuators signals) and outputs (sensors signals), usually less than a tens of each. These models can be nonlinear and do not require any knowledge of the flow. Neural networks, adaptive filters and more generally any types of linear or nonlinear auto-regressive filters with exogenous input belong to this category. Application of a NARMAX (Nonlinear Auto-Regressive Moving Average with eXogenous input, see Chen and Billings (1989) for more details) model can be found in ref. Kim (2005; Kim et al. 2005) where this method has been applied to the identification of the flow over a backward-facing step. The SISO (single input–

single output) relationship between a pressure sensor and the excitation of a synthetic jet has been successfully identified. Comparisons between linear and nonlinear adaptive filters have been performed on a nonlinear drag law in ref. Pillarisetti and Cattafesta (2001). An AR-MARKOV algorithm has been used by Tian et al. (2006) for the on-line identification of the separated flow over a NACA0025 airfoil. On the same case, equivalent perfor-mances at a reduced computational cost have been obtained

by Song et al. (2007) with a Generalized Predictive Control (GPC) algorithm. A numerical illustration of black-box model identification performed on CFD-based signals can be found in ref. Huang and Kim (2008) where the superi-ority of a subspace system identification method from ref.

(Van Overschee and De Moor1996) over a more classical ARX model is demonstrated. Moreover, the subspace identification allows the problem to be expressed in a classical state space form. Such a method has also been used experimentally on a backward-facing step by Henning and King (2007) for a MIMO controller. It is eventually used to build a fourth-order transfer function for each of the step excitation dedicated to the system identification.

Identification of low-order models with time delay based on step excitations was previously employed by Becker et al. (2005) on the same configuration and by Henning et al. (2007) on a bi-dimensional base flow. Moreover, it is worth mentioning that a black-box model can be used in combination with POD-like techniques, and the model is then used to simulate the temporal evaluation of a reduced set of POD coefficients. Example of this technique can be found in Cohen et al. (2008), the black-box model being an artificial neural network. A more recent paper of the same team uses wavelet network (Fagley et al.2011).

In the present study, the objective is to define a reduce-order model of the flow response to an actuation being a good compromise between simplicity, cost and accuracy.

Our scope is restricted to a SISO control, and a black-box-type model is selected. Nevertheless, it was considered important to be able to represent a nonlinear dynamics. For these reasons, a NARX model has been selected. The present paper is divided into two main parts. In the first part, a polynomial NARX model is identified in the case of a rounded ramp controlled by a synthetic jet. The input is the synthetic jet outflow velocity and the output is a properly chosen pressure sensor. Unsteady Reynolds-Averaged Navier–Stokes (URANS) computations are performed to obtain the flow unsteady response to the synthetic jet actuation. In a first step, the location of the pressure sensor is justified, and the relationship between the pressure information and the separation bubble geom-etry is evidenced. In a second step, the structure of the NARX model is chosen. A rigorous methodology is then developed to justify the choice of each of these model parameters, and both the dynamic and the steady-state responses of the model are assessed. The proposed meth-odology is inspired by the one developed by Box and Jenkins (1976) for linear ARMAX models. The choice of the polynomial power which introduces the nonlinearity is specific to the present study. In the second part, the same methodology is applied to the case of an experiment of separation control by pulsed fluidic vortex generators in a highly curved duct.

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2 Black-box modelling

The first step in the design of a feedback controller is the definition of a mathematical model of the system. The designed controller may achieve the specified performance only if the resulting model captures the system dynamics.

For example, the aim of the control could be to maximize pressure on a model since it is well-known that in the inviscid case without separation, the pressure is higher. The model consists in modelling the pressure signal at a well-chosen point as a function of the forcing signal.

Due to the governing equation nature, a nonlinear model is studied. Nevertheless, its accuracy will be compared to a linear model for each application to justify the gain brought by the nonlinear modelling. The idea is not to capture the flow physics but only to represent the pressure output as a function of the forcing signal. That is the reason why a black-box model is used. The NARMAX model (Chen and Billings1989) is a class of model which links inputsuand outputs ywith nonlinear relationships. Its general formu-lation is written in Eq. (1) where G denotes a nonlinear function,kis the discrete time,n_uandn_yare the number of past inputs and past outputs, respectively,eandnstand for possible noise and residual error.

yðkÞ ¼Gðyðk1Þ;...;yðknyÞ;uðk1Þ; ...;uðk

nuÞ;eðk1Þ;...;eðknyÞÞ þnðxkÞ ð1Þ The nonlinear function G can be a polynomial, a neural network, a wavelet network or any other function subject to nonlinear constraints.

In the present study, a polynomial NARX (a NARMAX with the noise terms excluded) is computed. The advantage of the polynomial NARX model is that the model is linear with respect to the coefficients of the modelh. The specific model structure used is described in Eq. (2).

yðkÞ ¼h0þX^p where the hij are the coefficients of the model, p is the polynomial power, and lag_uis the delay between the output and the input to take into account, for example, the con-vection time of the vortices between the actuator location and the sensor one. The cross-terms of order over 2 on the

input u or the output y are not taken into account in the model. Moreover, it must be noticed that the constant term h₀is particularly important since it has been observed that it enables the mean output to be different from zero.

Therefore, it allows the model to reproduce the static map hPi ¼fðF^þÞwherehPiis the mean wall pressure.

The total number of regressors is:

n_h¼1þpðnuþnyÞ þnyðny1Þ=2þnuðnu1Þ=2 ð3Þ and corresponds to the number of coefficients h to be determined.

The identification phase consists in computing the NARX regressors h. For this purpose, an identification signal u needs to be defined to explore the system response to an actuation. In the present study, a periodic forcing is considered. In order to model the flow response, the identification signal has to randomly explore the whole frequency range of interest. Then, the resulting model is not expected to be valid out of this frequency range.

LetY_Mbe the URANS identification simulation pressure outputs vector, H be the regressors vector and M be the NARX relation matrix betweenY_MandHdefined such that MH=Y_Mwith the property that:

Y_M^T ¼ ðyðkÞ;yðk1Þ; ...;yðknkÞÞ ð4Þ and

H^T¼ ðh0;h1; ...;hn_hÞ ð5Þ wheren_kis the number of samples.

Generally, the number of measurements is larger than the number of regressors. The standard approach to solve an overdetermined system of linear equations given as MH=Y_M is known as linear least squares and seeks to minimize the residual kMHY_Mk²where k k: is the Euclidean norm.

The matrixMis generally ill-conditioned. The Tikhonov regularization (Tikhonov and Arsenin 1977) is the most commonly used regularization method for ill-posed prob-lems. In order to give preference to a particular solution with desirable properties, the regularization term is inclu-ded in this minimization:kMHYMk²þkCHk² for some suitably chosen Tikhonov matrix C. In many cases, this matrix is chosen as the identity matrix C=I, giving preference to solutions with the smallest norm. In other cases, high-pass operators (e.g., a difference operator or a weighted Fourier operator) may be used to enforce smoothness if the underlying vector is believed to be mostly continuous. This regularization improves the con-ditioning of the problem, thus enabling the existence of a numerical solution. An explicit solution, denoted byH, is^ given by:

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