JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 26, No. 2, March–April 2003 Uncertainty Estimation from Volterra Kernels for Robust Flutter Analysis Richard J. Prazenica, ¤ Rick Lind, † and Andrew J. Kurdila ‡ University of Florida, Gainesville, Florida 32611-6250 The utterometer is a tool used for predicting the onset of utter during ight testing. This tool uses robust utter analysis to consider a model with an associated uncertainty description. The utterometer is particularly useful because the uncertainty description is determined by ight data. However, the standard method of uncertainty estimation is somewhat suspect because of the effects of nonlinearities in the ight data. A method is introduced to estimate uncertainties by considering only the linear component of the ight data. The linear component is extracted by representing the system in terms of Volterra kernels. The rst-order kernel describes the linear component of the data and, thus, can be used by the utterometer. Flight data from the aerostructures test wing is used to demonstrate this procedure. The analysis using the rst-order kernel is shown to generate a more accurate description of the modeling error than standard analysis of the measured ight data. Nomenclature A = area of domain a = scaling function lter b = wavelet lter D = length of Volterra kernel g = function h = Volterra kernel K = set of multi-indices [ P ] = matrix of integral values T = operator [ T ] = matrix form of wavelet transform operator t = time [U 1 ] = matrix of discrete inputs [U 2 ] = matrix of products of discrete inputs u = input V = scaling function approximationspace W = wavelet detail space y = response y = vector of discrete output values ® = coefcient of scaling function ® = vector of single-scale kernel coefcients ¯ = coefcient of wavelet ¯ = vector of multiscale kernel coefcients ° = mapping ´, » = time Á = orthonormal scaling function ’ = scaling function  = characteristicfunction à = wavelet Ä = domain of second-orderkernel © = summation of vector spaces ª = subtraction of vector spaces Received 11 March 2002; presented as Paper 2002-1650 at the AIAA/ ASME/ASCE/AHS/ASC 43rd Structures, Structural Dynamics, and Materi- als Conference, Denver, CO, 22–25 April 2002;revision received 22 Novem- ber 2002;accepted for publication25 November 2002.Copyright c ° 2003by the authors. Published by the American Institute of Aeronautics and Astro- nautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/03 $10.00 in correspondence with the CCC. ¤ Graduate Student, Department of Mechanical and Aerospace Engineer- ing; currently National Research Council Fellow, NASA Dryden Flight Re- search Center, Edwards, CA, 93523; chad.prazenica@dfrc.nasa.gov. † Assistant Professor, Department of Mechanical and Aerospace Engineer- ing; rick@aero.u.edu. Senior Member AIAA. ‡ Professor, Department of Mechanical and Aerospace Engineering; ajk@aero.u.edu. Associate Fellow AIAA. Subscripts and Superscripts i ; P = counter j = discretizationlevel k = value in f0; 1; 2; 3g r = value in f1; 2; 3g · = multi-index Introduction T HE analysis of ight data is obviously important for any ight test. The measurements are usually corrupted by noise and imperfections; however, these data are often the best indicator of the true dynamics of the aircraft. The dependencyon data exists for all types of ight testing, but it is especially prevalent when ight utter testing for envelope expansion. A tool called the utterometer has been developed for predicting the onset of utter during a ight test. 1 This tool is a model-based utility, but it is directly dependent on ight data. The utterometer computes a utter speed for an analytical model that is robust with respect to an uncertainty description. 2 The tool uses ight data to generate that uncertainty description.Essentially, the uncertainty is a mathematicaloperatorthat describesdifferencesbetweentransfer functions of the model and data. The utterometer predicts a utter speed dependent on characteristics of the uncertainty description and consequentlydependent on characteristicsof the ight data. A particularconcernfor testing with the utterometeris the qual- ity of the uncertainty description. A description that does not con- sider a sufcient level of modeling error may overpredict the utter speed. Conversely, a description that considers too much modeling error may underpredictthe utter speed. Either situation is adverse to conducting a safe and efcient ight test. Anaccurateassessmentof modelingerror,usingtheutterometer approach, can only result from comparing the transfer function of the modelto a transferfunctionthataccuratelyrepresentsthe aircraft dynamics.Such an accuratetransferfunctionis difcult to compute. The ight data used to generate that transfer function often contains componentsthatviolateassumptions,suchaslinearityandstatistical properties,associated with standard spectral analysis. A technique was developed to analyze ight data and assess an accurate measure of modeling error. 3 This technique actually iden- tied model parameters and their associated variances simultane- ously. The approach used wavelets for the signal analysis and a min–max optimization for the estimation. This method was shown to generate reasonable results using ight data; however, the results are somewhat limited in that uncertaintyis only associated with the observation matrix of the model. This paper introducesa new techniquefor estimatinguncertainty descriptions.The techniqueis also a wavelet-basedapproach,but it 331