Identification of a Hammerstein Model for Wing Flutter Analysis Using CFD Data and Correlation Method

Transcription

1 2 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 2 WeB53 Identification of a Hammerstein Model for Wing Flutter Analysis Using CFD Data and Correlation Method Kai-Yew Lum and Kwok Leung Lai Abstract A correlation approach for the identification of a Hammerstein model of unsteady aerodynamics using CFD data is presented The main idea is to use correlation functions of the input-output processes as data for the identification problem It is shown in a numerical example of the AGARD 4456 wing that the proposed method can more accurately approximate the linear aerodynamic response, which is needed for a first-order prediction of wing flutter boundaries I INTRODUCTION Aircraft wing flutter is an aeroelastic phenomenon caused by feedback interaction between air flow and wing structure At critical airspeeds this interaction causes undamped or limit-cycle oscillations of the wing Prediction of flutter boundaries through flight tests is a critical and costly phase of aircraft development, and poses danger to the pilot and machine Typically, the aircraft is flown at a known stable test point, where data is collected and analyzed to predict the likely speed at which flutter may occur Then the test is repeated by increasing the speed successively, until flutter or near flutter is encountered As an alternative to flight tests, computational aeroelasticity (CAE), coupling computational uid dynamics (CFD) and computational structural dynamics (CSD) tools, has been developed and proven to be viable for flutter simulation Using high-order flow models such as those based on the Navier-Stokes equations and the Euler equations, CAE is capable of reproducing the nonlinear interactions between fluid flows and structural dynamic However, full-order CAE approach remains computationally expensive, primarily due to the computing time needed to accurately simulate the unsteady aerodynamics For this reason, reduced-order model (ROM) techniques have drawn much attention; see [] and [2] for comprehensive surveys In ROM construction, the accurate modeling capability of fullorder CAE is exploited to obtain the aerodynamic response to prescribed structural excitations The ROM constructed using the simulated data gives a dynamical representation of the modeled system, its accuracy depending on the excitation signal and model employed Proper orthogonal decomposition (POD) is one of the approaches employed, which consists in projecting the discretized unsteady flow equation onto a modal space of chosen dimension Coupling the POD-based ROM with the CSD model greatly reduces the integration time, and have been K-Y Lum and KL Lai 2 are with Temasek Laboratories, National University of Singapore kaiyew lum@nusedusg, 2 tsllaikl@nusedusg This work is supported by DSTA fundings POD63447 and POD846 used in linearized flutter analysis as well as numerical Hopf bifurcation analysis [3], [4] The other, more system-theoretic approach to flutter ROM is system identification Prescribed time-domain perturbations of the structural modes are used as input to unsteady simulation, and the resulting generalized aerodynamics forces as output Different types of models have been employed in this approach, including frequency-domain model [5], nonlinear model represented by Volterra series [6], [7], [8], and direct identification of linear state-space models [9], [], [] The latter involves estimation of a large number of Markov parameters or state-matrix coefficients Comparatively, linear autoregressive moving-average (ARMA) models require far fewer parameters and have been shown to offer good approximations of unsteady aerodynamics [2], [3], [4] Recently, Hammerstein and Wiener models have also been employed to approximate nonlinear aeroelastic and flutter systems [5], [6] See [7] for a comparison of POD and system identification Besides replacing CFD simulation, ROM of aerodynamic response also offers computationally efficient, first-order flutter prediction through eigenvalue analysis of the aerostructure closed-loop system [6], [8] In such analysis, however, it is important that the identified linear model does not merely match the input-output data; because the underlying unsteady aerodynamics are nonlinear, straight-forward identification of a linear model using input-output data is input dependent, and may not yield a good approximation of the local dynamics [9] For this reason, the Hammerstein and Wiener models appear to be more suited for an explicit separation of linear and nonlinear properties Many classical works have been devoted to the identification of the Hammerstein and Wiener models, notably the iterative Narendra-Gallman algorithm [2] A recent noniterative method based on frequency-domain estimation of the poles, and a linear block expressed as a linear expansion on an orthonormal basis of transfer functions derived from the estimated poles, has also been proposed [2] In the correlation method, using Gaussian noise as input, the linear block can be identified separately from the static nonlinearly in a two-step procedure [22], [23] The present work is an extension of [8], where a statespace model was constructed from an identified matrix ARMA model Now, we consider a Hammerstein model consisting of a cubic static nonlinearity followed by a matrix ARMA linear block To obtain input-output data using Gaussian noise as input perturbation is clearly infeasible in CFD However, the correlation method is really based //$26 2 AACC 34

2 on a more fundamental result due to Nuttal [24] With this, it is simple to show that, given a separable process as input, the cross-correlation function of the input and output of the Hammerstein model is proportional to the input autocorrelation function passed through the linear block In other words, the linear dynamics of the Hammerstein model can be identified, up to a constant factor, by using the correlation functions as input-output data instead of the raw data The unknown constant factor can next be corrected by identifying the static nonlinearity as a cubic function Using this two-step procedure and filtered impulses as input, we are able to more accurately identify an ARMA model of linear aerodynamic response Then, similarly as in [8], the ARMA model is converted via the Hankel method [25], [26] to a continuous time state-space model which, when coupled with the state-space model of the linear structural dynamics, yields a linear parameter-varying (LPV) flutter system Prediction of the flutter boundaries of the AGARD 4456 wing by eigenvalue analysis of the resulting LPV system yields results that are very close to experimentally obtained flutter parameters, and are an improvement over our previous work II MODELING OF WING FLUTTER DYNAMICS The flutter dynamical system is essentially a closed-loop interaction between the dynamics of air flowing over the wing, and those of the elastic structure constituting the wing The air movement exerts aerodynamic forces on the structure, which reacts by deforming This deformation results in a time-varying boundary of the flow domain and, thus, a feedback mechanism between the structural dynamics and aerodynamics (Fig ) Aside from experimental studies or flight tests, flutter analysis can be performed using computational models of the flutter dynamic system Numerous works have dealt with flutter modeling, thus we shall give a brief description here For a detailed exposé see, for example, Leyland et al [27] The structural dynamics of the wing can be written in modal equations of motion, of the following normalized form: η i + 2ζ i ω i η i + ω 2 i η i = f i, i =,,N () where η i is the i th modal or generalized displacement, N is the number of structural modes, ζ i and ω i are the corresponding damping factor and natural frequency, and f i u = η modal displacements Aerodynamic response (M ) η Structural dynamics Fig y = F Flutter dynamic system generalized aerodynamic forces f αv 2 (a) Mode, f = 96Hz (b) Mode 3, f 3 = 4835Hz (c) Mode 5, f 5 = 8Hz Fig 2 (d) Mode 2, f 2 = 387Hz (e) Mode 4, f 4 = 954Hz Modal shapes and frequencies of the AGARD 4456 wing is the i th component of the generalized aerodynamic forces In this study, we consider the AGARD 4456 wing and its first five dominant modes as shown in Fig 2 [8], [28] Using a vector notation, the physical forces F exerted by air movement is a nonlinear function of the fluid state vector U and the displacement vector together with its derivatives: F = F ( ) U, η, η, η, (2) whereas the fluid state vector U = (ρ,ρ v,ρe), of density ρ, fluid momentum ρ v and total energy ρe, is determined by the solution of the governing flow equation For this study, the Euler equation is employed for modeling threedimensional inviscid unsteady flow, which can be expressed in integral form as t Ω(t) UdΩ + F n dσ =, (3) Ω(t) where F is the flux vector, Ω(t) is the flow domain with boundary Ω(t), and n denotes the outward normal vector on the boundary Notice in particular that the flow domain and its boundary are time-varying due to structural deformation, and are functions of the vector ξ of wing-surface displacements in spatially discretized representation The relation between ξ and η is determined by a mapping Φ : ξ = Φ η, called the fluid-structure coupling, which transforms modal displacements into wing-surface displacements Hence, one can in fact write: Ω(t) = Ω( η(t)), Ω(t) = Ω( η(t)) (4) In a previous work [8] we have described the numerical methods employed for solving the system Eq(2)-Eq(4) In brief, they consist of a Cartesian-grid Euler solver using a time-accurate second-order implicit scheme and multigrid procedure to accelerate convergence, and an algebraic transformation for the fluid-structure coupling operator based on the constant-volume tetrahedron method [29] 35

3 Eq(2)-Eq(4) describe the aerodynamic response of the wing as an N-input, N-output system, with the input being the modal displacements η, and output being the physical forces F in modal coordinates Its solution is obtained at a given free-stream Mach number M By non-dimensionalizing time with respect to Mach number, the generalized aerodynamic force vector f acting on the structure is related to F by f = αv 2 F, V = V b s ω α µ ; (5) here, α is a constant depending on the dimensions and mass of the wing, V is the speed index, where V is the freestream velocity, b s is the semi-chord length, ω α is the natural frequency of wing s first uncoupled torsion mode, and µ is the wing-air mass ratio The smallest value V f of V such that the closed-loop is unstable is called the flutter speed index, and the frequency ω f of oscillation is the flutter frequency The traditional definition of flutter boundaries are thus the graphs V f (M ) and ω f (M ) [28] Thus, at a given M, a full-order flutter simulation consists in increasing V, effectively the feedback gain, until oscillation appears in the solution [4], [27], [3] However, this is computationally onerous Since the structural dynamics are linear, a faster, system-theoretic method consists in obtaining a reduced-order linear model of the aerodynamic input-output system, and analyzing the resulting closed-loop eigenvalues as V varies Such a model can be obtained by system identification using suitably simulated input and output data, u = η and y = F The key, however, lies in identifying a model that represents the true linear behavior of the aerodynamic response III IDENTIFICATION OF HAMMERSTEIN MODEL USING u Fig 3 CORRELATION METHOD N[ ] x h(τ) Hammerstein model for aerodynamic response We consider the Hammerstein model [23], [5] composed of a static input nonlinearity N[ ] following by a linear block represented by its impulse response function h(τ) (Fig 3), as an approximation of the nonlinear aerodynamic response The nonlinearity N[ ] is assumed to take the form x(t) = N[u(t)] = γ u(t) + γ 2 u 2 (t) + + γ p u p (t) (6) We are interested in identifying the linear portion of the Hammerstein model, ie γ h(τ), which can be represented in a matrix ARMA form: y(k) = A y(k ) A na y(k n a ) (7) + B o u(k) + B u(k ) + + B nb u(k n b ), with y(k) R N and u(k) R N The above can also be written in the compact form y(k) = A(q)y(k)+B(q)u(k), where q denotes the delay operator y A Correlation Method The result of Nuttall [24] states that the cross-correlation function φ xu of the input and output of a wide class of static nonlinearity N[ ] is related to the autocorrelation function φ uu of the input in the following manner φ xu (τ) = C N φ uu, (8) provided that the input is separable In Eq(8), C N is a constant depending only on N[ ] and the input statistics [3] Separability in the sense of Nuttall means that the conditional expectation of u satisfies E{u(t τ) u(t)} = c(τ)u(t), with c(τ) = φ xx (τ)/φ xx () Nuttall [24] showed that the Gaussian, sine-wave, and random phase-modulated sinewave processes are separable, among others One can also establish a general relationship between the input and output correlation functions of the Hammerstein model Indeed, since h(τ) is linear, one has φ yw (τ) = h(τ) φ xw (τ) for any signal w(t), where denotes the convolution operation In particular, letting w = u and using Eq(8), one obtains the following lemma Lemma : Provided the input is a separable process, the cross-correlation function of the input and output of the Hammerstein model is proportional to the input autocorrelation function passed through h(τ): φ yu (τ) = C N h(τ) φ uu (τ) (9) The above property means that to identify the linear block, one should use the correlation data (φ uu (k),φ yu (k)) as input-output data instead of (u(k),y(k)) This then sets up the following two-step algorithm for the identification of the ARMA model (7) B Two-Step Identification Algorithm Step : C N h(τ) is identified as ĥ(τ) in its ARMA representation {Âi, ˆB j} i=,,na;j=,,n b, using least-squares estimation and the correlation functions (φ uu (k),φ yu (k)) as input-output data Step 2: Estimate the γ j s according to the equation [I + Â(q)]y(k) = ˆB (q)[γ u(k) + + γ p u p (k)] + e(k) Stacking up the data points in rows yields a matrix equation Z = Φθ + E where θ = (γ,,γ p ) T A least-squares estimate ˆθ can be obtained Finally, correct for the unknown factor C N by replacing the moving-average part with ˆB j = ˆγ ˆB j, j =,,n b IV CONSTRUCTION OF STATE-SPACE FLUTTER MODEL After obtaining a matrix ARMA model of the aerodynamic response for suitably chosen orders (n a,n b ), it is more convenient to convert it to a continuous state-space realization for flutter analysis, as the structural properties and flutter parameters are defined in continuous time 36

4 A State-Space Realization of Aerodynamic Response In the case of computational data with negligible numerical noise, minimal state-space realizations for the ARMA model Eq(7) are of the order n = Nn a Eq(7) has associated Markov parameters H i given by: i H = B, H i = B i A j H i j, i =,, j= where, for ease of presentation, A j and B i are zeros for indices larger than n a and n b, respectively The associated Hankel matrices for any m > n are defined by: T = H H m H m H 2m, T = H 2 H m+ H m+ H 2m Then, T and T have ranks n; denote the singular-value decomposition of T (after removing the zero blocks) by T = K ΣL, with Σ = diag{λ,λ 2,,λ n }, λ > > λ n >, K R m n, L R n m One has the following: Theorem 2 (Hankel method [25], [26]): An irreducible discrete-time state-space realization {A d,b d,c d,d d } of Eq(7) is given by: A d = V + TU + R n n, B d = U( :, : N) (first N columns) R 2N N, C d = V ( : N, : ) (first N rows) R N 2N, D d = H R N N, with V = K Σ 2, U = Σ 2 L, and + is the pseudo-inverse Now, a continuous-time equivalent denoted by {A c,b c,c c,d c } can be obtained by Tustin s method, provided there are no discrete-time poles at - [32] This latter condition can be ensured by proper selection of the sampling frequency and the model orders (n a,n b ), ie avoiding over-dimensioning B Linear Flutter Closed-Loop Equation Referring to Fig, Eq() and Eq(5), the structural dynamics can also be represented in continuous-time statespace form by defining the following variables: x s = (η, η,,η N, η N ) R 2N, u = (η,,η N ) R N and y = (f,,f N ) R N Then, ẋ s (t) = A s x s (t) + V 2 B s y(t), u(t) = C s x s (t), where A s R 2N 2N, B s R 2N N, C s R N 2N are blockdiagonal, with the respective blocks (i =,,N) given by: [ ] [ ] A i s =, B i 2ζ i ω s =, C i i α s = [ ], ω 2 i Combining the structural-dynamics model {A s,b s,c s,d s } and the reduced-order aerodynamic model {A e,b e,c e,d e }, a closed-loop linear model of the flutter system can be established: ] [ ][ ] [ẋc A = e (M ) B e (M )C s xc ẋ s V 2 B s C e (M ) A s + V 2, B s D e (M )C s x s or, in a more compact form, ẋ = A(M,V )x () We write Eq() to depend on the Mach number M besides the speed index V, because the data used for model identification are simulated at fixed Mach numbers For each given Mach number, the flutter boundary is found by searching the scalar V until a pair of eigenvalues crosses the imaginary axis at ±jω f ; the cross-over value of V is then the flutter speed index V f and the flutter frequency is ω f Remark : The system Eq() is similar to those in recent works on LPV aeroelastic models based on the structured singular-value framework [5], [6], [33] which considered parametric uncertainty and exogenous disturbance such as varying dynamic pressure Here, using computed data and without model uncertainty, the flutter closed loop is linear time-invariant (for a fixed Mach number), where V 2 is the only varying scalar parameter This analysis is thus simpler V NUMERICAL RESULTS: AGARD 4456 WING The method described in the previous sections is applied to the AGARD 4456 wing, which have been extensively studied in experiments and flight tests [28], [34] As shown in Fig 2, N = 5 dominant modes are considered with the modal frequencies ω i = (632, 2398, 338, 575, 742) rad/sec or, (96, 387, 4835, 954, 8) Hz Mode 2 is the first uncoupled torsion mode, ie ω α = 2398 rad/sec In order to compare with experimental data, four Mach numbers are investigated: M = (5, 678, 85, 96) The structural damping is assumed to be zero, which corresponds to the weakened 25-foot panel-span model in [28] For this study, a combination of filtered-impulse signals are employed as input excitation: u i (t) = u exp k(ωit ψc)2 sinω i t, i =,,5; () where the ω i s are the modal frequencies Here, ψ c affects the symmetry of the signal (symmetrical at ψ c = π) and the peak frequency of the power spectrum To cover the structural frequency range of the 4456 wing, we choose ψ c = 35 and k = Moreover, the five input signals are staggered by half a period of the preceding ones The input signals, and output at Mach 5 are shown in Fig 4 Remark 2: The filtered impulse has been found to be suitable for reduced-order modeling of flutter based on the idea that the conventional V-g method for flutter analysis only requires accurate frequency-domain aerodynamics near certain distinct frequencies where flutter is likely to happen [4] This insight is exploited using signals that are smooth and provide excitation for frequencies in the neighborhood of the structural frequencies Admittedly, the filtered impulse is not a separable process In fact, Nuttall [24] showed that amplitude-modulated signals of the form x(t)sin(ωt τ) are separable if x(t) is zero-mean and separable, while exponential modulation such as Eq() is not separable 37

5 However, as shown later, flutter prediction is not severely affected by this shortcoming At each Mach number, a Hammerstein model with cubic nonlinearity is identified using the proposed correlation method for (n a,n b ) = (,2) However, increasing n a to 2 yields unstable poles and a Hankel matrix that is rankdeficient, indicating an over-dimensioned model To compare with standard identification using input-output data, two models with (n a,n b ) = (,2) and (n a,n b ) = (2,) have been obtained using the Marquardt-Levenberg algorithm; in particular, the second, higher-order model was reported in our previous paper [8] This suggests that the correlation method is more discriminating of the model s order Fig 5 plots the errors of estimated outputs at Mach 5, and Table I compares the root-mean-squares (RMS) errors between the model and original output data It can been seen that the linear part of the Hammerstein model yields worse errors in terms of fitting the output signals However, the opposite is true when the three models are used to predict the flutter boundaries based on eigenvalue analyses of Eq() Indeed, Fig 6 shows the predicted flutter speed indexes and frequency ratios, as well as the experimental data given in [28] It can be clearly seen that the linear part of the Hammerstein model gives much more accurate estimates of the flutter frequency ratios at Mach 5, 678 and 96 than the other linear models, as well as overall better prediction of the flutter speed indexes At transonic Mach 85, flow behavior is more difficult to simulate as well as experiment; nevertheless, the values obtained with the proposed model fall within the ball-park of the experimental data The above results show that, while the flutter system can be approximated by linear models, the proposed Hammerstein-model identification by the correlation method is better at capturing the local linear behavior of the aerodynamics VI CONCLUSION A method using correlation functions as input-output data for the identification problem has been proposed for the identification of a Hammerstein model of unsteady aerodynamics Its merit over standard identification lies in a more accurate approximation of the local linear aerodynamic response, and has clearly been demonstrated on flutter-boundary prediction of the AGARD 4456 wing The proposed method is simple and only requires computation of the correlation Model (n a, n b ) rank(t ) Average RMS error 5X ǫ i rms 5 i= M5 M678 M85 M96 Hammerstein (,2) Hammerstein (2,) Standard 2 (,2) Standard 2 (2,) Linear part of Hammerstein model (p=3), correlation method 2 Linear model, standard identification method TABLE I IDENTIFICATION RESULTS u(t) y(t) Time (sec) Fig 4 CFD simulation of aerodynamic response (in modal decomposition) with filtered-impulse inputs (Mach 5) 5 ε (t) ε2 (t) ε3 (t) ε4 (t) ε5 (t) t (sec) Fig 5 Output errors of model by correlation method (Mach 5) functions as an additional task, and does not need iteration or frequency-domain analysis Admittedly, the theory requires that the input be a separable process, a condition which the filtered impulses employed here do not satisfy As future improvement, one can investigate separable processes that are acceptable to CFD and that address the frequency-range considerations mentioned earlier Particularly, viability of the method under the more difficult conditions of transonic and supersonic Mach numbers is yet to be demonstrated REFERENCES [] E Dowell and K Hall, Modeling of fluid-structure interaction, Annual Review of Fluid Mechanics, vol 33, pp , 2 [2] W A Silva, P S Beran, C E S Cesnik, R E Guendel, A Kurdila, R J Prazenica, L Librescu, P Marzocca, and D E Raveh, Reducedorder modeling: Cooperative research and development at the NASA Langley Research Center, in Proc 42nd AIAA/ASME/ASCE/AHS/ASC Structure and Structural Dynamics and Materials Conference and Exhibit, Seatle, WA, USA, 2 38

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