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NASA Technical Reports Server (NTRS) 20100004852: Real-Time Dynamic Modeling - Data Information Requirements and Flight Test Results PDF

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Real-Time Dynamic Modeling – Data Information Requirements and Flight Test Results Eugene A. Morelli 1 NASA Langley Research Center, Hampton, Virginia, 23681 Mark S. Smith 2 NASA Dryden Flight Research Center, Edwards, California, 93523 Practical aspects of identifying dynamic models for aircraft in real time were studied. Topics include formulation of an equation-error method in the frequency domain to estimate non-dimensional stability and control derivatives in real time, data information content for accurate modeling results, and data information management techniques such as data forgetting, incorporating prior information, and optimized excitation. Real-time dynamic modeling was applied to simulation data and flight test data from a modified F-15B fighter aircraft, and to operational flight data from a subscale jet transport aircraft. Estimated parameter standard errors, prediction cases, and comparisons with results from a batch output-error method in the time domain were used to demonstrate the accuracy of the identified real-time models. Nomenclature a ,a ,a = body-axis translational accelerometer measurements, g or ft/sec2 x y z b = wing span, ft c = mean aerodynamic chord, ft C ,C ,C = body-axis non-dimensional aerodynamic force coefficients X Y Z C ,C ,C = body-axis non-dimensional aerodynamic moment coefficients l m n E{ } = expectation operator I ,I ,I ,I = mass moments of inertia, slug-ft2 x y z xz j = imaginary number = 1 J = cost function m = aircraft mass, slug M = body-axis pitching moment from engine thrust, ft-lbf T p,q,r = body-axis roll, pitch, and yaw rates, rad/sec or deg/sec q = dynamic pressure, lbf/ft2 s = standard error S = wing reference area, ft2 T = maneuver length, sec T ,T = body-axis engine thrust, lbf x z V = airspeed, ft/sec = angle of attack, rad or deg = sideslip angle, rad or deg , , , = elevator, aileron, rudder, and trailing-edge flap deflections, rad or deg e a r f , , , = canard, differential canard, stabilator, and differential stabilator deflections, rad or deg c dc s ds 1 Research Engineer, Dynamic Systems and Control Branch, MS 308, Associate Fellow 2 Research Engineer, Aerodynamics Branch, P.O. Box 273 1 American Institute of Aeronautics and Astronautics , , = Euler roll, pitch, and yaw angles, rad or deg = parameter vector = covariance matrix = frequency, rad/sec superscripts T = transpose † = complex conjugate transpose ˆ = estimate = time derivative = Fourier transform –1 = matrix inverse subscripts o = reference value I. Introduction DYNAMIC modeling in real time has many important practical uses, such as improving the efficiency of stability and control flight testing, flight envelope expansion, adaptive or reconfigurable control, vehicle health monitoring, and fault detection. Several methods1-5 have been investigated for identifying local linear dynamic models from flight data in real time. One of these methods4,6 is based on a recursive Fourier transform and equation-error modeling in the frequency domain. This method, sometimes called the Fourier Transform Regression (FTR) method, produces very accurate results with valid error measures and has many practical advantages. The FTR method has also been independently evaluated7,8 as the best method available for real-time dynamic modeling. For these reasons, the FTR method was chosen for further study and application. The FTR method has been successfully applied4,6-11 to identify accurate linear dynamic models in real time at individual flight conditions. While this capability is important and useful, further progress requires that this capability be extended to continuous application as the aircraft flies through a wide range of changing flight conditions throughout the flight envelope. Ultimately, local real-time modeling results could be integrated into a global aerodynamic model that could be updated in real time as the aircraft changes flight conditions, changes configuration, ages, or becomes damaged in some way. This vision of real-time global dynamic modeling has many important implications for efficient flight testing, accurate flight simulation, adaptive or reconfigurable control, and aircraft safety. One important aspect of applying real-time dynamic modeling for varying flight conditions and aircraft configurations is determining the data information content requirements for accurate dynamic modeling results. Changing aircraft flight conditions or aircraft configurations means that parameters in the approximating dynamic model change. Dynamic motion of the aircraft, either from ordinary flight operations or from applied control surface excitation, is necessary so that the measured data will exhibit the aircraft dynamics to be modeled. Naturally, if the real-time dynamic modeling is to be done continuously or on a regular basis, it is important that only the minimum necessary aircraft excitation be applied, and the resulting aircraft motion should be as small and unobtrusive as possible. This paper investigates data information requirements for accurate real-time dynamic modeling. Flight experiments on a modified F-15B fighter aircraft are used to illuminate issues related to data information content necessary for accurate real-time modeling. Real-time modeling is also applied to operational flight data from a subscale jet transport model, to evaluate the feasibility of real-time modeling without specific excitation. This is an important step in extending local real-time modeling to the case of changing conditions, interpreted broadly to include flight condition changes, configuration changes, damage, and failure scenarios. Issues such as data information content necessary for fast and accurate local modeling, model validation, necessary excitation, data forgetting, and methods for incorporating prior information are studied. The next section describes the methods used. A model formulation is developed that retains full nonlinear dynamics, with linearized aerodynamic models. The FTR method is described, along with explanations of methods for data forgetting and incorporating prior information into the real-time parameter estimation algorithm. Next, the flight test aircraft are described, including flight instrumentation and characteristics of the flight data. The results section includes simulation and flight test investigations examining data information requirements for accurately 2 American Institute of Aeronautics and Astronautics identifying local dynamic models in real time. Finally, the concluding remarks summarize progress made so far and outline some possible next steps. II. Methods A. Aerodynamic Modeling Non-dimensional aerodynamic force and moment coefficients for an aircraft can be computed from flight measurements as follows6: 1 C ma T (1a) X x x qS 1 C ma T (1b) Z z z qS 1 C I q I I pr I p2 r2 M (1c) m y x z xz T qSc ma y C (2a) Y qS 1 C I p I pq r I I qr (2b) l x xz z y qSb 1 C I r I p qr I I pq (2c) n z xz y x qSb These expressions retain the full nonlinear dynamics in the aircraft equations of motion. For local real-time modeling over a short time period, the force and moment coefficients computed from Eqs. (1) and (2) can be modeled using linear expansions in the aircraft states and controls: qc C C C C C (3a) X X X X X q 2V o qc C C C C C (3b) Z Z Z Z Z q 2V o qc C C C C C (3c) m m m m m q 2V o pb rb C C C C C C (4a) Y Y Y Y Y Y p 2V r 2V o pb rb C C C C C C (4b) l l l l l l p 2V r 2V o pb rb C C C C C C (4c) n n n n n n p 2V r 2V o The notation indicates perturbation from a reference condition. In each expansion, a single term is shown to represent all relevant and similar control terms, to simplify the expressions. For example, in Eq. (3c), the term 3 American Institute of Aeronautics and Astronautics C represents all the control terms for C , e.g., C C C . In Eq. (3c), C m m m m e m f m e f o represents the non-dimensional pitching moment at a reference condition, and similarly for the other expansions. The linear aerodynamic models in Eqs. (3) and (4) contain parameters called stability and control derivatives, such as C and C , which characterize the stability and control of the aircraft. For short periods of time, the m m stability and control derivatives are considered to be constant model parameters to be estimated from flight data. Repeating the parameter estimation at short time intervals produces piecewise constant estimates for the stability and control derivatives, which in general vary with flight condition and changes to the aircraft, such as configuration, age, damage, or failures. The next subsection describes how the unknown stability and control derivatives in the linear aerodynamic models of Eqs. (3) and (4) can be estimated from flight data using equation-error parameter estimation in the frequency domain. B. Stability and Control Derivative Estimation in the Frequency Domain This section describes the FTR method for estimating unknown parameters in a dynamic model in real time. Some of the material presented here can also be found in Refs. 4 and 6. The first step required for modeling in the frequency domain is to transform the measured flight data from the time domain into the frequency domain. The finite Fourier transform is the analytical tool used for this task. For an arbitrary scalar signal x t on the time interval 0,T , the finite Fourier transform is defined by T x t x x t e j tdt (5) 0 which can be approximated by N 1 x t x i e j i t (6) i 0 where x i x i t , T N t , and t is a constant sampling interval. The summation in Eq. (6) is defined as the discrete Fourier transform, N 1 X x i e j i t (7) i 0 so that the finite Fourier transform approximation in Eq. (6) can be written as x X t (8) Some fairly straightforward corrections12 can be made to remove the inaccuracy resulting from the fact that Eq. (8) is a simple Euler approximation to the finite Fourier transform of Eq. (5). However, if the sampling rate is much higher than the frequencies of interest, as is typically the case for dynamic modeling from flight data, then the corrections are small and can be safely ignored. The Fourier transform is applied to the non-dimensional force and moment coefficients computed from Eqs. (1) and (2) using measured time-domain data. This results in the non-dimensional force and moment coefficients in the frequency domain. Often, measurements of the angular accelerations p,q,and r are not available. In the frequency domain, these derivatives can be calculated by multiplying the Fourier transforms of p,q,and r by j . For example, the Fourier transform of the rolling moment coefficient can be computed as: 4 American Institute of Aeronautics and Astronautics C C j Ixp Ixzr Ixzpq Iz Iy qr (9) l l qSb qSb and similarly for C and C . This approach implements the derivative of the body-axis angular momentum in the m n frequency domain, including the time variation in the inertia quantities. Note that the Fourier transform of the nonlinear terms is handled by computing the nonlinear terms in the time domain, then applying the Fourier transform to the resulting time history. Treatment of the dynamic pressure q in Eq. (9) is consistent with an assumption that the dynamic pressure varies slowly, which is a good practical assumption. To obtain the perturbation states and controls in Eqs. (3) and (4), time histories of the measured states and controls are high-pass filtered to remove the steady part of each signal. Then, each perturbation signal is transformed into the frequency domain using the discrete Fourier transform. The break frequency for the high-pass filter is set just below the lowest frequency selected for the modeling. High-pass filtering is implemented with a fourth-order Butterworth digital filter. Similarly, the quantities transformed in Eq. (9) (shown within the square brackets) are high-pass filtered prior to Fourier transformation. This approach effectively drops out the bias terms in the models of Eqs. (3) and (4). The high-pass filtering also prevents leakage from the relatively large spectral component at zero frequency, associated with the steady component of each signal, from polluting transformed data at low frequencies. For each aerodynamic model in Eqs. (3) and (4), the parameter estimation problem can be formulated as a standard least squares regression problem with complex data6, z X e (10) where, for example, using the pitching moment equation (3c), C 1 m C 2 m z (11) C M m 1 q 1 1 1 n e f 2 q 2 2 2 n e f X (12) M q M M M n e f C m C m q (13) C m e C m f The notation q represents qc 2V and e represents the complex equation error vector in the frequency n domain. The symbols k , k 1,2, ,M denote the Fourier transform of the angle of attack perturbation state for each frequency , and similarly for other quantities. Each transformed variable depends on frequency. The k 5 American Institute of Aeronautics and Astronautics frequencies can be chosen arbitrarily, and are therefore chosen to cover the frequency band where the aircraft k dynamics lie, as will be discussed later. The least squares cost function is 1 † J z X z X (14) 2 This cost function contains M squared error terms in summation, corresponding to M frequencies of interest. Similar cost expressions can be written for individual lines from Eqs. (3) and (4). The parameter vector estimate that minimizes the least squares cost function is computed from6 1 ˆ Re X†X Re X†z (15) The estimated parameter covariance matrix is6 Cov ˆ E ˆ ˆ T 2 Re X†X 1 (16) where the equation-error variance can be estimated from the residuals, ˆ 1 z X ˆ † z X ˆ (17) M n p and n is the number of unknown parameters, i.e., the number of elements in parameter vector . Parameter p standard errors are computed as the square root of the diagonal elements of the Cov ˆ matrix from Eq. (16), using ˆ from Eq. (17). Reference 6 explains why the estimated parameter standard errors are computed in this way, and also why this calculation in the frequency domain produces parameter error measures that are consistent with the scatter in parameter estimates from repeated maneuvers. Realistic simulation testing has shown that the accuracy of model parameters estimated with this method is comparable to using a time-domain output-error method employing iterative nonlinear optimization in post-flight batch mode13. The model formulation given here is widely applicable, because the assumption of constant linear aerodynamic models over short time periods is very accurate for non-dimensional stability and control derivatives, where the effects of changing dynamic pressure and mass properties are removed. To implement this least squares parameter estimation in the frequency domain, the parameter estimation calculations in Eqs. (15)-(17) are applied to frequency-domain data at selected times, normally at regular time intervals. The frequency-domain data must therefore be available at any time, so the Fourier transforms are computed using a recursive Fourier transform, described next. C. Recursive Fourier Transform For a given frequency , the discrete Fourier transform in Eq. (7) at time i t, denoted by X , is related to i the discrete Fourier transform at time i 1 t by X X x i e j i t (18) i i 1 where e j i t e j t e j i 1 t (19) 6 American Institute of Aeronautics and Astronautics The quantity e j t is constant for a given frequency and constant sampling interval t. It follows that the discrete Fourier transform can be computed for a given frequency at each time step using one addition in Eq. (18) and two multiplications – one in Eq. (19) using the stored constant e j t for frequency , and one in Eq. (18). There is no need to store the time-domain data in memory when computing the discrete Fourier transform in this way, because the data for each sample time is processed immediately. Time-domain data from the past can be used in all subsequent analysis by simply continuing the recursive calculation of the Fourier transform. In this sense, the recursive Fourier transform acts as memory for the information in the data. More data from more maneuvers improves the quality of the data in the frequency domain without increasing memory requirements to store it. Furthermore, the Fourier transform is available at any time i t. The approximation to the finite Fourier transform is completed using Eq. (8). The recursive computation of the Fourier transform does not use a Fast Fourier Transform FFT algorithm14, and therefore would be comparatively slow, if the entire frequency band up to the Nyquist frequency 1 2 t were of interest. However, rigid-body dynamics of aircraft lie in a rather narrow frequency band of approximately 0.01-2.0 Hz. Since the frequency band is limited, it is efficient to compute the discrete Fourier transform using Eqs. (18) and (19) (which represents a recursive formulation of Eq. (7)) for only the selected frequencies , k 1,2, ,M . With this approach, it is possible to select closely-spaced fixed frequencies for the Fourier k transform and the subsequent modeling and still do the calculations efficiently. Using a limited frequency band for the Fourier transformation confines the data analysis to the frequency band where the dynamics lie, and automatically filters wide band measurement noise or structural responses outside the frequency band of interest. These automatic filtering features are important for real-time applications, where instrumentation error corrections and noise filtering would require additional computational resources. In past work on fighter aircraft short-period modeling, frequency spacing of 0.04 Hz on an interval of approximately [0.1-2] Hz was found to be adequate9-11. Finer frequency spacing requires slightly more computation, but was found to have little effect on the results. When the frequency spacing is very coarse, there is a danger of omitting important frequency components, and this can lead to inaccurate parameter estimates. In general, a good rule of thumb is to use frequencies evenly spaced at 0.04 Hz over the bandwidth for the dynamic system. For good results, the bandwidth should be limited to the frequency range where the signal components in the frequency domain are at least twice the amplitude of the wide band noise level. However, the algorithm is robust to these design choices, so the selections to be made are not difficult. The recursive Fourier transform update need not be done for every sampled time point. Systematically skipping time points effectively lowers the sampling rate of the data prior to Fourier transformation. This saves computation, and does not have a significant adverse impact on the parameter estimation results until the Fourier transform update rate gets below approximately 5 times the highest frequency of interest for the dynamic system. The parameter estimation and covariance calculations in Eqs. (15)-(17) can be done at any time, but are usually done at 1 or 2 Hz, to save computations. Linearized aerodynamic characteristics rarely change faster than this, except in cases of strong nonlinearity, damage, failure, or rapid maneuvering. For these cases, the update rate can be increased, at the cost of more computations. Reference 6 explains that computing standard errors from the covariance matrix in Eq. (16) does not require correction for colored residuals. The standard errors computed from Eq. (16) are therefore a good representation of the error in the estimated parameters. Having high quality error measures is important for problems such as failure detection and control law reconfiguration. D. Data Forgetting The recursive Fourier transform in Eqs. (18) and (19) represents a data information memory for as long as the running sum is incremented. It follows that when the aircraft dynamics change, the older data should be discounted in some way, as has been done for time-domain approaches using a forgetting factor6. If this is not done, then the speed of response for the real-time parameter estimator is progressively degraded, as new information has to overwhelm an increasingly longer memory. Consequently, there is a trade-off between the desired rapid response of the parameter estimator to changes in the aircraft dynamics, versus retaining enough information from past data for sufficiently accurate model parameter estimates. If past values of the Fourier transform X computed from Eq. (18) are saved in computer memory, then it is i possible to implement selective amnesia by simply subtracting past values of the running sum corresponding to the 7 American Institute of Aeronautics and Astronautics Fourier transform, or differences between past values of the running sum. For example, forgetting all data information content older than 10 sec (i.e., removing that data information content from the complex regression problem) could be implemented by subtracting the value of the running sums for the Fourier transforms at 10 sec ago from the current running sums. Similarly, to forget data information content collected between 5 and 7 sec ago, the difference between the running sums at 5 and 7 sec ago would be subtracted from the current running sum. The price to pay for this capability is the computer memory required to store past values of the running sums associated with the Fourier transforms for each signal at each frequency. The memory requirements could be reduced by perhaps only saving the running sums at intervals of 0.5 sec, for example. The simplicity of Eq. (18) also makes it easy to see how exponential data forgetting could be implemented. In exponential data forgetting, each past value of the time-domain signal is multiplied by a forgetting factor 1 at each time step. In this way, old data is gradually devalued and eventually discarded. Usually, is chosen in the range 0.90 1.00. To implement this, Eq. (18) is modified slightly to X X x i e j i t (20) i i 1 and everything else remains the same as before. This simple approach is possible because the Fourier transform is linear with respect to the measured data x i . The challenge with data forgetting is not in the implementation, but rather in deciding how much data information content to forget, and when. At the present time, there are no concrete guidelines, so the choices are made based on analysis of results from simulation and flight data. E. Incorporating Prior Information Model parameter estimates can sometimes be improved by including information based on models identified from prior data. In the case of real-time dynamic modeling, including prior information of this kind can reduce variations in the real-time parameter estimates, and improve convergence speed. One way to incorporate prior information is by using a mixed estimator formulation of the least squares cost function6. Assuming that the fit error variance 2 for the prior modeling is approximately equal to the fit error p variance 2 for a model based on the current data alone, and denoting the vector of parameter estimates from a prior analysis by , with associated covariance matrix , the least squares cost function that incorporates this p p prior information is formulated as6 J 1 z X † z X 1 T 1 (21) p p p 2 2 The vector of parameter estimates that minimize this modified least squares cost function is 1 ˆ Re X†X 1 Re X†z 1 (22) p p p with covariance matrix 1 Cov ˆ 2 Re X†X 1 (23) p where 2 is estimated from Eq. (17), as before. Reference 6 provides further details on the mixed estimator formulation of the least squares parameter estimation problem, and the associated solution. F. Optimized Excitation References 6, 11, and 15 describe a method for designing optimized inputs for use as control surface perturbations to excite aircraft dynamics. The form of each input is a sum of sinusoids with unique frequencies and 8 American Institute of Aeronautics and Astronautics phase shifts. Multiple inputs are designed to be mutually orthogonal in both the time domain and the frequency domain, and are optimized for maximum data information content in multiple axes over a short time period, while minimizing excursions from the nominal flight condition. The mutual orthogonality of the inputs allows simultaneous application of multiple inputs, which helps to minimize excitation time. Optimized inputs of this type were applied to the modified F-15B to collect data for real-time dynamic modeling. The optimized inputs were applied as control surface excitations by summation with the actuator commands from the control system just before the actuator command rate and position limiting. Flight test examples are shown below in the Results section. III. Test Aircraft and Flight Data A. Fighter Aircraft Description The fighter aircraft used for this research is a pre-production Boeing F-15B that has been highly modified to support various test programs. The most visible modification is the inclusion of a set of canards near the pilot station, see Fig. 1. The canards are a set of modified horizontal stabilators from a Boeing F/A-18 aircraft. The purpose of the canard addition was to increase maneuverability and load capability. An additional effect of the canards was to cause the aircraft to be statically unstable longitudinally at most subsonic speeds. The propulsion system consists of two Pratt & Whitney F100-PW-229 engines, each equipped with an axisymmetric thrust Figure 1. Modified F-15B Fighter Jet Aircraft vectoring pitch/yaw balance beam nozzle. The thrust vectoring feature, however, was not used during the flight testing described here. Further information on the modified F-15B aircraft and associated flight test operations can be found in Ref. 16. 1. Control Surfaces The aircraft has five pairs of control surfaces: stabilators, canards, ailerons, trailing edge flaps, and rudders. Flaps and aileron droop are manually set by the pilot and only used for takeoff and landing configurations. Conventional pitch control is provided by symmetric deflection of the all-moving horizontal stabilators and canards. Roll control uses aileron and differential stabilator. Directional control is provided by rudder and differential canard deflection. Definitions of control surface deflections are given below. Trailing edge down is positive deflection for the wing and stabilator surfaces, and trailing edge left is positive for the rudder. 1 1 (24a) s s s c c c 2 right left 2 right left 1 1 1 1 (24b) a a a r r r dc c c ds s s 2 right left 2 right left 2 right left 2 right left For the nominal flight control system, pilot stick and rudder pedal inputs result in high correlation between the symmetric canard and angle of attack, rudder and differential canard, and differential stabilator and aileron. Flight data analysis showed that the symmetric canard and angle of attack exhibited nominal pair-wise correlation of 0.93, rudder and differential canard were correlated at 0.99, and the differential stabilator and aileron were perfectly correlated at 1.00. Given these high levels of correlated inputs, it would be impossible with pilot input maneuvers to distinguish, for example, between rolling moment generated by the aileron and rolling moment generated by the differential stabilator. Consequently, the complete stability and control derivative set cannot be obtained with pilot input maneuvers. 2. Instrumentation and Data Acquisition The modified F-15B aircraft was equipped with a research-quality instrumentation system. A nose boom was installed and calibrated for free stream pitot-static pressure and flow angle measurements. An inertial instrumentation package provided 3-axis linear accelerometer and angular rate measurements. Heading, pitch, and bank angle were obtained from the aircraft inertial navigation system. Control surface positions were measured using variable differential transformer sensors. Fuel measurements were obtained from the three fuselage and two 9 American Institute of Aeronautics and Astronautics wing tanks. Fuel measurements were used to compute total aircraft weight, center of gravity, and mass moments of inertia. Mass and geometry characteristics of the aircraft are given in Table 1. Sensor positions and the center of gravity (c.g.) location were used to correct the linear accelerometer and flow angle measurements to the c.g. Angular accelerations were not measured. Flight data was collected at 40 Hz. B. S-2 Subscale Jet Transport Aircraft Description The S-2 aircraft is a subscale model of a jet transport aircraft A photograph of the aircraft in flight is shown in Fig. 2. The subscale aircraft has a single jet engine mounted in the aft fuselage and retractable tricycle landing gear. Mass and geometry characteristics of the aircraft are given in Table 1. Further information on the S-2 subscale jet aircraft and associated flight test operations can be found in Ref. 17. 1. Control Surfaces Control surfaces on the aircraft are conventional elevator, aileron, rudder, and inboard trailing-edge flaps. Definitions of control surface deflections are given below. Trailing edge down is positive deflection for the wing and elevator surfaces, Figure 2. S-2 Subscale Jet Transport Aircraft and trailing edge left is positive for the rudder: 1 1 (25) e e e a a a 2 right left 2 right left The aircraft can be flown by a safety pilot using direct visual contact and conventional radio control. A research pilot executes flight test maneuvers from inside a mobile control room, using a synthetic vision display drawn from telemetry data and a local terrain database. Inputs from the research pilot and ground-based flight control are used to compute control surface commands which are transmitted by telemetry to the aircraft. 2. Instrumentation and Data Acquisition The S-2 aircraft was equipped with a micro-INS, which provided 3-axis linear accelerometer measurements, angular rate measurements, estimated attitude angles, and GPS velocity and position. Air data probes on each wingtip (visible in Fig. 2) measured angle of attack, sideslip angle, static pressure, and dynamic pressure. Measurements from static pressure sensors and ambient temperature sensors were used to compute air density and altitude. Engine speed in rpm was measured and used as input to an engine model to compute thrust. The engine model was identified from ground test data, with adjustments for ram drag identified from flight data18. Potentiometers on the rotation axes of all control surfaces measured control surface deflections. Mass properties were computed based on measured fuel flow, pre-flight weight and balance, and careful inertia measurements of the aircraft on the ground. The pilot stick and rudder pedal commands and throttle position were also measured and recorded. Flight data was collected at 50 Hz. IV. Results A. Data Information Requirements for Real-Time Dynamic Modeling In this section, results will be presented from investigations concerning data information content necessary for accurate real-time modeling. Figure 3 shows a relationship between signal-to-noise ratio of aircraft measured responses to the mean error in real-time parameter estimates computed using the FTR method described earlier. Data for this analysis was generated with a linear simulation of the modified F-15B lateral dynamics at a flight condition of Mach 0.75, trim angle of attack 2 deg, and 20,000 ft altitude. Optimized multi-sine inputs were applied simultaneously to the lateral control surfaces, as shown in Fig. 4. Repeated simulated data runs with different output signal-to-noise ratios were generated by uniformly reducing the amplitudes of the inputs shown in Fig. 4, generating new simulated outputs, then adding a single realization of noise sequences extracted from flight data to the simulated outputs. This caused the output signal-to-noise ratios to vary in a uniformly decreasing fashion. For each data run, the FTR method was applied, with no prior information, to produce real-time parameter estimates. Final parameter estimates and 10 American Institute of Aeronautics and Astronautics

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