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Reduced Order Nonlinear System Identification Methodology

Peter Joseph Attar, Earl H. Dowell, John R. White, Jeffrey P. Thomas

2006
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AIAA Journal
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A new method is presented which enables the identification of a reduced order nonlinear ordinary differential equation (ODE) which can be used to model the behavior of nonlinear fluid dynamics. The method uses a harmonic balance technique and proper orthogonal decomposition to compute reduced order training data which is then used to compute the unknown coefficients of the nonlinear ODE. The method is used to compute the Euler compressible flow solutions for the supercritical two-dimensional
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... two-dimensional NLR-7301 airfoil undergoing both small and large pitch oscillations at three different reduced frequencies and at a Mach number of 0.764. Steady and dynamic lift coefficient data computed using a three equation reduced order system identification model compared well with data computed using the full CFD harmonic balance solution. The system identification model accurately predicted a nonlinear trend in the lift coefficient results (steady and dynamic) for pitch oscillation magnitudes greater than 1 deg. Overall the reduction in the number of nonlinear differential equations was 5 orders of magnitude which corresponded to a 3 order of magnitude reduction in the total computational time. Nomenclature linear function of the u i C = matrix of unknown coefficients E t = total energy F = vector which contains x, y, and z fluxes F i = nonlinear function of theâ i _ f = x component of unsteady control volume motion G ij = coefficients of the function B i _ g = y component of unsteady control volume motion _ h = z component of unsteady control volume motion N C = number of unknown coefficients per equation N P = number of training data sets N h = number of harmonics kept in Fourier expansion n = surface normal vector P = number of modes kept in modal expansion p = pressure S = snapshot matrix T = fundamental period of motion TD = matrix of CFD training data t = dimensional time U = vector of conservative fluid variables U 0 = steady flow solution u = x component of fluid velocity Vt = control volume at time t v = y component of fluid velocity w = z component of fluid velocity _ x = unsteady motion of control volume t = time dependent pitching motion 0 = pitch magnitude = fluid density = nondimensional time = matrix of POD modes i = ith POD fluid mode !, ! = frequency and reduced frequency

doi:10.2514/1.16221
fatcat:bru3dpjnijbfbfo6ci4kfef47q