Frequency-Response Method for Rotorcraft System Identification:
Flight Applications to BO-105 Coupled Fuselage/Rotor Dynamics
Mark B. Tischler
U.S. Army Aviation Development Directorate - Ames
Moffett Field, California
Mavis G. Cauffman
Sterling Federal Systems, Inc.
Palo Alto, California
Abstract
A comprehensive frequency-response method for rotorcraft system identification is presented. The overall concept is to (1) extract a complete set of non-parametric input-to-output frequency responses that fully characterizes the coupled helicopter dynamics, and (2) conduct a nonlinear search for a state-space model that matches the frequency-response data set. Each major element of the procedure is reviewed. A new method for combining the results of multi-input frequency-response analyses obtained from a range of spectral windows into a single optimized response is presented. This method eliminates the need for manual optimization of windows and significantly improves the dynamic range and accuracy of the identified frequency-responses relative to single window methods.
An integrated user-oriented software package for the frequency-response method is described: Comprehensive Identification from FrEquency Responses (CIFER). CIFER is used to identify a 9-DOF hybrid model of the DLR BO-105 dynamics from flight test data at 80 knots. The identified model includes coupled body/rotor-flapping and lead-lag dynamics, and is accurate to 30 rad/sec. The 9-DOF results are compared with 6-DOF (quasi-steady) identification results. An application of the model to flight control design shows that the maximum roll rate gain is limited by the destabilization of the lead-lag dynamics.
Introduction
System identification is a procedure by which a mathematical description of vehicle dynamic behavior is extracted from flight test data. System identification can be thought of as an inverse of simulation. Simulation requires adoption of a priori engineering assumptions for the formulation of model equations. These simulation models allow the prediction of aircraft motion. In contrast, system identification begins with measured aircraft motion and "inverts" the responses to extract a model that accurately reflects the measured aircraft motion, without making a priori assumptions. Applications of system identification results include: (1) comparing wind tunnel and flight test measured characteristics, (2) validating and updating simulation models, (3) demonstrating handling-qualities specification compliance, and (4) optimizing automatic flight-control systems.
Reliable methods for conducting system identification have been available for some time in the fixed-wing community,^{1} but the methods have only recently become mature for rotorcraft applications.^{2} The identification of rotorcraft dynamic characteristics is especially difficult because of the high level of measurement noise, generally unstable vehicle dynamics, high degree of inter-axis coupling, and the high order of the helicopter dynamical system. Conventional time-domain techniques are often not well suited to these difficult aspects of rotorcraft identification as discussed by Tischler,^{3} and Fu and Kaletka.^{4}
The U.S. Army/NASA and Sterling Federal Systems have jointly developed an integrated software facility (CIFER) for system identification based on a comprehensive frequency-response approach that is uniquely suited to the difficult rotorcraft problem. The overall concept is to (1) extract a complete set of non-parametric input-to-output (pilot control-to-vehicle response) frequency responses that fully characterize the coupled behavior of the system without a priori assumptions, and (2) conduct a nonlinear search for a state-space model that matches the input/output frequency-response data set. Key features of this approach which address the issues discussed above are
. Generation of high quality multi-input/multi-output frequency responses
. Elimination of uncorrelated noise sources via the frequency-response calculation
. Fitting each identified frequency response only in the region of highest accuracy
. Weighting fit errors based on frequency-response accuracy
. Linearized analysis for extracting parameter accuracy metrics
. Use of confidence ellipsoid theory for multiple correlation evaluations
. Integrated procedure for identification and model structure determination
. Fitting responses only in a frequency range where the model structure is appropriate
This paper describes this integrated frequency-response approach and the CIFER software facility. A new method for combining spectral analysis results to achieve significant improvement in dynamic range and accuracy of the frequency-response identification is presented. System identification results based on flight test data of the BO-105 at 80 knots are presented in detail. Emphasis is placed on the techniques for the identification of models suitable for application to high-bandwidth flight control system design. A 9-DOF hybrid model is extracted which accounts for rotor-flap/body coupling and
lead-lag dynamics, and is accurate for frequencies of up to 30 rad/sec. Finally, an application of the identification results to flight control system design is presented. More detail on the identification methods, background, and additional results is presented in the original version of this paper.^{3} The present results are slightly improved over the original results of Ref. 3 through some recent refinements in the numerical techniques.
Frequency-Response Identification Method
The identification procedure developed by the U. S. Army/NASA is based heavily on the use of frequency-responses and is depicted in Fig. 1. Pilot-generated frequency-sweep inputs^{5} are used to obtain broadband excitation of the vehicle dynamics of interest, including all of the rigid body and lower-frequency (regressive) rotor dynamic modes. A comprehensive data compatibility analysis is performed using the Kalman filter/smoother program SMACK (Smoothing for Aircraft Kinematics^{6}) in a procedure developed by Fletcher.^{7} Using this procedure, reduced parameter-set models of measurement system errors involving unknown scale factors and biases are determined, and estimates of unknown states and/or noisy measurements are reconstructed. For example, in the case of the BO-105, the body-axis airspeeds (u,v,w) were estimated to remove the effects of rotor wake interference on the LASSIE (Low Air Speed Sensing and Indicating Equipment) airspeed measurements. Results from the compatibility analysis were cataloged for later use in identification processing in CIFER.
SISO and MISO Frequency-Response Calculations
The key step in the identification procedure is the extraction of accurate frequency-responses for each input/output pair. Time histories for multiple frequency sweeps on a particular control are concatenated. Single-input/single-output (SISO) frequency responses for each input/output pair are determined using the Chirp-Z transform (an advanced Fast Fourier Transform) and overlapped/windowed spectral averaging.^{8} When n_{c} multiple control inputs are present in the excitation, as is the case for the BO-105 data and most other open-loop helicopter tests, the contaminating effects of partially correlated inputs must be removed. This is accomplished by inverting the spectral matrices of all inputs x to a single output y at each frequency point f_{k}. The required "conditioned transfer-function matrix" T(f_{k}) is obtained as
See Equation (1)
where G_{xy }is the n_{c} X 1 matrix of SISO cross-spectra between each control input and the single output, and G_{xx} is the n_{c} X n_{c }matrix of auto- and cross-spectra between the inputs.^{9}This matrix solution is determined at each frequency point f_{k} and then again for each output to yield a set of "conditioned frequency responses." These conditioned
multi-input/single-output (MISO) responses are the same as the SISO frequency responses that would have been obtained had no correlated controls been present during the frequency sweep of a single control.
Associated with each conditioned frequency response are partial coherence functions, which are important measures of the accuracy and linearity of the identified conditioned frequency responses at each frequency point:
See Equation (2)
where G_{xiy*(nc-1)} is the cross-spectrum of control input-x_{i} to output-y conditioned for the remaining n_{c}-1 control inputs as calculated by the matrix solution of Ref. 9. Analogous definitions for the conditioned input autospectrum G_{xix*(nc-1)} and conditioned autospectrum G_{yiy*(nc-1)} are also given in Ref. 9.
Composite Windows
The selection of window size in the SISO frequency-response identification step (Chirp-Z transform) involves a fundamental tradeoff between dynamic range and accuracy of the identification. The window length T sets the minimum frequency w_{min} achieved in the identification:
See Equation (3)
The level of random error associated with the identified frequency response is inversely proportional to the number of window segments^{10}:
See Equation (4)
where n_{d} = (concatenated run length in seconds)/T and a 50% window overlap is used.
As can be seen from Eq. (3), long windows are required for frequency-response identification of low-frequency dynamics. The identification of high-frequency dynamics is degraded by low signal-to-noise levels. For these high-frequency dynamics, considerably smaller windows are used because of the noise suppression afforded by the averaging effect in Eq. (4). In previous applications of the frequency-response method (e.g., Ref. 11), the window sizes were manually optimized for each input/output pair. This manual optimization is very laborious and time consuming. Furthermore, the requirement to select a single "overall-best" window limits the overall achievable dynamic range of the identification because of the above tradeoff between dynamic range and random error.
A new method has been developed that combines the conditioned frequency-response identification results obtained using several window sizes. Composite curves of the conditioned input, output, and cross-spectral functions are generated using a weighted nonlinear Least-Squares procedure to achieve a composite conditioned frequency response that has good coherence and low random error over the entire frequency range of interest. The cost function for this procedure is
See Equation (5)
n_{w }= number of windows included in procedure
W_{i }= weighting function (internally calculated) dependent on random error (Eq. (4)) and blended at the ends of each window's frequency range
Subscript i refers to individual window conditioned results
Subscript c refers to "composite" conditioned data
The weighting factor of 5 on the coherence term was selected empirically to ensure that the composite partial coherence tracks the best partial coherence values of the individual windows.
The startup values for the nonlinear solution are obtained by the linear least-squares problem that results from dropping the last term in Eq. (5).
The composite window procedure allows the user to calculate conditioned frequency responses from a number of windows (up to 5 are allowed in the current computer implementation), without the need to compromise in any particular frequency range. The largest and smallest windows are selected based on the minimum and maximum frequency desired for the identification, and the remaining windows are generally distributed evenly between these limits. Experience to date indicates that the results are not very sensitive to the specific selection of windowing spacing.
In addition to improving the dynamic range of the identification significantly, the composite window technique eliminates the laborious job of manually optimizing the windows, which previously consumed the large percentage of the engineer's time. The overall result of using this technique is the rapid identification of a set of broadband frequency responses for all input/output pairs for which there is dynamic excitation. This set of composite conditioned frequency responses and associated coherence functions forms the core of the frequency-response identification method.
Transfer-Function Modeling
Direct transfer-function fitting of individual input/output frequency responses leads to single-axis, transfer-function models. This provides a direct and minimal-dimensional realization of the input-to-output dynamical behavior of a system, useful for many applications, such as handling-qualities analyses^{12} and classical design of flight control systems.^{13} A key application of transfer-function modeling is in the determination of an appropriate model structure for state-space model formulation. Systematic evaluation of the matching quality of candidate transfer-function models over desired frequency ranges provides valuable information on the order of the system, level of coupling, and initial guesses for many state-space parameters (as indicated by the dashed line in Fig. 1). A detailed application of this approach to model structure determination is presented in Ref. 14. The results of this study were used in the present paper to establish the structure of the 9-DOF hybrid model discussed later.
State-Space Model Identification
State-space models are often the desired end product of system identification. Such models are needed for example in: (1) control system design and optimization; (2) simulation model development, troubleshooting, and improvement; and (3) comparison of wind tunnel and flight characteristics. A complete set of transfer functions can be transformed into an estimate of a state-space model, as was demonstrated for the XV-15 aircraft.^{15} However, this transfer-function modeling and inversion procedure is too cumbersome when vehicle dynamics are highly coupled as in the case of the BO-105. In the current frequency-response approach, stability-derivative identification is achieved directly through iterative multi-input/multi-output matching of the identified conditioned frequency responses with those of the following linear model:
See Equation (6)
where the matrix M_{m} has been included to allow the direct identification of stability derivatives that are dependent on state rates (e.g., side-wash lag derivative, N_{v}). The elements of M_{m}, F_{m}, G_{m}, H_{m}, and j_{m} are the unknown stability and control derivatives. Some of these elements may be known from physical considerations and/or direct transfer function modeling.^{15}
Taking the Laplace transform of Eq. (6) results in the following transfer function:
See Equation (7a)
To account for time delays associated with unmodeled higher-order states, a matrix of time delays, t_{m}(s), may be incorporated. Also, the control feedthrough term j_{m} may be eliminated by allowing H_{m }to be a function of s. The result is
See Equation (7b)
The unknown state-space model parameters are determined by minimizing J, a weighted function of the error E between the identified (composite) frequency responses T(s) and the model responses T_{m}(s) over a selected frequency range:
See Equation (8)
The frequency ranges for the identification criterion (w_{1}, w_{2}, ... , w_{n}) are selected individually for each input/output pair according to their individual ranges of good coherence. In this way, only valid data are used in the fitting process. Within these frequency ranges, the points are selected linearly across the logarithmic frequency range. In an improvement since the publication of Ref. 3, the spacing is altered if data points with low coherence can be avoided by slightly shifting the frequency points. The weighting matrix W is based on the values of coherence at each frequency point to emphasize the most accurate data. By way of contrast to Eq. (8), other existing output-error methods have tried to match all data over the same frequency range. Furthermore, the use of the identified frequency response in the cost function of Eq. (8) (rather than simply the output error) eliminates the effects of (uncorrelated) process and measurement noise,^{8 }which is a key advantage for rotorcraft applications.
A sophisticated iterative non-linear pattern search algorithm^{16} is used to adjust both the stability and control derivatives and the time delays in the model until convergence on a minimum criterion of Eq. (8) is achieved. The pattern search method has been found to be highly robust for very large problem sizes associated with the BO-105 identification. Up to 60 unknown parameters have been extracted in matching of 40 frequency responses with 20 points in each response. The user may fix or free each parameter in the model, add a constant to a parameter, or constrain parameters to be constant multiples of
other parameters. The latter capability is useful in handling known physical constraints between parameters.
Model Structure Determination
An integrated procedure for parameter identification and model structure determination (MSD) is an important feature of the frequency-response method and is described in detail in Ref. 15. In the present context, "model structure determination" refers to the elimination of insignificant or unidentifiable parameters. Model order is already fixed based on transfer-function modeling results or on physical principles.
The fundamental quantity for the MSD analysis is the Hessian matrix:
See Equation (9)
which is estimated from numerically linearizing the gradients of the frequency-response errors with respect to the unknown model parameters. Three key metrics of parameter accuracy and correlation are calculated (see Ref. 15 for equations) from the Hessian matrix:
1. Parameter insensitivity- a direct measure of the insensitivity of the cost function to changes in individual parameters, taking into account the correlation with the remaining parameters.
2. Cramer-Rao bound- an estimate of the minimum achievable standard deviation in the parameter estimates and a reflector of high parameter insensitivity and/or parameter correlation.
3. Confidence ellipsoid- a multi-variable measure of parameter correlation (the conventional two-dimensional correlation matrix is not reliable when multiple correlations exist).
The above metrics provide a reliable absolute measure of parameter accuracy and correlation because the effects of colored noise are eliminated in the frequency-response calculation.^{15}Using these three metrics, the parameters that are determined to be insignificant or highly correlated to other parameters may be systematically deleted (or fixed at a priori values), resulting in a final model structure which consists of a smaller number of significant parameters. This approach was successfully used to achieve an improved model structure for the XV-15 and BO-105 identification studies.^{11,15}
Time-Domain Verification
The last step in the procedure of Fig. 1 is model verification. For this step, the identified state-space model is driven with flight data not used in the identification process, in order to check the model's predictive capability. A key concern is that the model, which was identified based on one input form, must be capable of predicting the response characteristics to other input forms. A multiple-input/multiple-output time-domain program integrates the state-space model Eq. (6), and determines the unknown state equation biases and zero shifts in the data. This is done by minimizing the weighted least-squares error between the model and vehicle responses.
Comprehensive Identification from Frequency Responses (CIFER)
The identification procedure described above has been implemented in a comprehensive package of user-oriented programs referred to as CIFER.^{17} Such software packages are critical in the application of system identification tools; CIFER is the first such integrated package for the frequency-response methodology. A functional layout of CIFER is shown in Fig. 2. The programs FRESPID, MISOSA, and COMPOSITE allow up to 10 control inputs, 20 outputs, and 5 spectral windows to be processed in single jobs. State-space model descriptions in DERIVID and VERIFY can consist of up to 20 states, 100 unknown derivatives, and 80 frequency-response matches. Up to 50 frequency points for each response can be included in the identification cost function in DERIVID.
The CIFER user interface was designed and implemented in such a way as to relieve the user of bookkeeping concerns. All the information needed to run a particular program is saved in a database that can be reaccessed by simply specifying the "case name." The program parameters comprising this case are then presented to the user on a series of screens, through which the user steps (forward or backward), changing default and/or saved values as needed. An example of the user interface is shown in Fig. 3 for a typical session using DERIVID. This example is representative of the complexity and flexibility of the state-space model that can be identified. As shown in Fig. 3, the section of the F matrix displayed consists of parameters with fixed and variable parts. Each element can be given a derivative name, and each derivative can be fixed or freed. Furthermore, any fixed derivative can be constrained to be a linear function of any free derivative. Changes in the model definition are achieved by simply moving the cursor around on the user screens and changing the default on previously saved setups. The user can easily update the database with the changes that are made at each screen.
Case information from the various programs as well as all output results are stored in a series of databases that are accessed by CIFER. Utilities have been developed to allow quick inspection, plotting, and tabulated output of the contents of the database. Databasing is a key requirement for organizing the large amounts of data that are generated for a difficult coupled identification problem such as the one considered here.
BO-105 Flight Data
Flight data used in this study were obtained from flight tests of the BO-105 helicopter conducted by the DLR^{18} for use by AGARD Working Group 18 on Rotorcraft System Identification. The flight condition was 80 knots at a density altitude of 3000 ft. Tests were flown in conditions of calm air. Test inputs consisted of pilot-generated frequency sweeps, 3211-multistep inputs, and doublets. The frequency sweeps are used for model identification with CIFER, while the 3211 and doublet data are reserved for verification.
Three concatenated lateral cyclic frequency sweep flight records are shown in Fig. 4. The dominant input, lateral cyclic, reaches amplitudes of about 10% of full control deflection. Pilot inputs to the longitudinal cyclic have approximately half the amplitude of that used in the lateral cyclic, while the collective and pedal inputs are less than 10% of the lateral cyclic. The presence of partially correlated off-axis inputs of these magnitudes necessitates the use of MISOSA to condition the extracted frequency responses.
All flight data were processed for compatibility using the techniques described in Ref. 7. Angular measurement consistency was found to be excellent with only small biases on rate gyro measurements. Significant disturbances in the lateral velocity signals occur with the passage of the rotor wake over the LASSIE measurement system. This motivated an investigation into the use of reconstructed velocities in the identification, rather than simply adjusting the flight data for the extracted scale factors. Uncorrected flight records were used for the remaining measurement signals. The following sections present in detail BO-105 identification results for a 9-DOF (coupled rotor/body) model using CIFER. Results for a quasi-steady 6-DOF models are presented in Refs. 3 and 11.
Frequency-Response Identification
Frequency-response identification (using FRESPID) and multi-input conditioning (using MISOSA) were conducted to obtain a full matrix of input-to-output frequency responses for a range of window sizes. The roll rate response to lateral stick (conditioned for the remaining inputs, ) obtained from the flight data of Fig. 4 is shown in Figs. 5a and 5b. This response was obtained from MISOSA using a 25-sec window. The partial coherence for lateral stick inputs shown in (Fig. 5c) indicates excellent identification for this window has been achieved over a frequency range of about 0.8-12 that includes rigid-body and some rotor dynamics.
Partial coherence plots for the remaining controls also (Figs. 5d-5f) indicate that all controls contribute to the roll-rate response and must be included in the multi-variable spectral analysis. In fact, the longitudinal input is nearly as significant as the lateral input near 1 rad/sec, which indicates considerable control cross-coupling. The multiple coherence of Fig. 5g is nearly unity over a broad range, indicating that the total roll-rate response can be linearly accounted for by considering the four pilot inputs. This indicates that the turbulence and nonlinear effects are not significant in this response.
FRESPID and MISOSA calculations were repeated for window lengths of 50, 35, 25, 10, and 5 sec. The resulting lateral cyclic partial coherences for the 50-, 25-, and 5-sec windows are shown in Fig. 6. The 50-sec window is seen to produce the best coherence at low frequency, where poorer results are obtained for the 25-sec window. In the mid-frequency range, the coherence begins to oscillate for the 50-sec window due to reduced spectral averaging. This indicates degradation of the identification quality. For mid-frequencies, the 25-sec window shows the best performance, although its coherence again begins to oscillate in the high-frequency range due to insufficient spectral averaging. The 5-sec window has poor performance in mid-frequency range due to reduced low-frequency content, but performs well at high frequency where it allows a maximum amount of spectral averaging. The composite coherence result of for 5 windows is shown in Fig. 7. -- ** The notation for conditioned response to remaining inputs is implied throughout the remainder of the paper, but is dropped for notational convenience. -- The composite result indicates excellent identification (g^{2} > 0.8) over a wide frequency range (0.4-30 rad/sec) with considerable improvement in low-frequency accuracy and suppression of spectral oscillation compared to the single window result of Fig. 6.
The final frequency response of shown in Fig. 7 demonstrates a number of important dynamic characteristics. First, the frequency response is fairly flat over a broad frequency range, as expected from the high value of roll damping (L_{p}) for the hingeless BO-105 rotor. The peak in magnitude at around 2.5 rad/sec reflects the influence of the Dutch roll mode. The resonance clearly visible at 15 rad/sec is the very lightly damped regressive lead-lag mode. This mode is superimposed on the coupled body-roll/rotor-regressing flapping response at 13.5 rad/sec.
Frequency-response identification following the above procedure was conducted for all remaining input/output pairs. In each case, the frequency response was identified from the sweep that corresponds to the respective input (i.e., the response was obtained from the longitudinal stick sweep). Of the 36 possible frequency-response combinations, a satisfactory identification was achieved for 33 responses.
9-DOF Hybrid-Model Formulation
A quasi-steady formulation in 6-degrees-of-freedom is commonly used in rotorcraft system identification for application to simulations or handling-qualities studies that do not require high-frequency validity. The 6-DOF formulation accounts for rotor dynamics as simple time delays, absorbing the steady-state effects of rotor flapping into the conventional stability and control derivatives. Such models can adequately describe the low- and mid-frequency dynamics of the BO-105 up to about 13 rad/sec as discussed in detail in Ref. 14.
For flight-control system applications, the model must be accurate in the frequency range of about 0.3 to 3 times the expected crossover frequency. Dynamic modes within this frequency range will have a significant influence on the magnitude and phase characteristics near the crossover frequency and will thus be important in predicting the closed-loop behavior.^{14} A typical high-bandwidth control system for an advanced combat rotorcraft will have a crossover frequency of about 6 rad/sec.^{13} Therefore, the flight control system design model must be accurate up to about 20 rad/sec, which is clearly beyond the range of applicability of the 6-DOF model. A higher-order model is needed that includes the coupled body/roll and lead-lag modes that dominate the responses in the frequency range of 2-3 Hz.
A study was conducted to determine the appropriate order and structure for a higher-order model that are accurate for frequencies of up to about 30 rad/sec.^{14} Two key conclusions were reached in this study.
1. Body-roll and rotor-flapping responses are highly coupled for the BO-105 and form a damped second-order response associated with the regressive rotor mode.
2. The regressive lead-lag dynamics may be treated as a one-way coupled parasitic complex dipole attached to the roll-rate response to lateral stick. Physically, this dipole models the rolling moment caused by the lateral offset of the rotor disk center-of-gravity which occurs in the lead-lag motion. This lead-lag excitation is due to the Coriolis forces that accompany blade flapping when the lateral cyclic is applied.^{19} The modal damping is nearly zero owing to the lack of lead-lag dampers on the BO-105. These characteristics of the parasitic complex lead-lag dipole are analogous to a structural mode on a fixed-wing aircraft.
Since the quasi-steady formulation is suitable in the low-frequency range (below 2 Hz), and the dynamics associated with regressive rotor flapping and lead-lag motion are associated with higher frequency dynamics (above 2 Hz), a "hybrid formulation" was developed. In this approach, highly simplified rotor equations are used that capture the on-axis regressive flapping response for cyclic control and angular body rate inputs in the high-frequency range (w > 10 rad/sec). All quasi-steady (6-DOF) derivatives except for
, are retained to account for the coupling terms and low-frequency dynamics. In the current study, improved results were achieved by absorbing the off-axis rotor flapping response into quasi-steady control derivatives. Retention of the complete coupled rotor equations is probably warranted when rotor state measurements are available for the identification, or when very accurate off-axis response modeling is required at high frequency. The rotor and fuselage dynamics are coupled through effective rotor spring terms L_{b1s} and M_{a1s}. Analogous simple equations for the lead-lag response cannot be formulated, but this is not a severe limitation because of the one-way coupled characteristics of these dynamics. Instead, the lead-lag motion is modeled using a modal approximation.
The desired simplified equations-of-motion for the regressive rotor response from stick and angular rate inputs are obtained from the full equations of Chen^{20} by dropping (1) flapping acceleration, (2) nonlinear terms, (3) forward flight terms, (4) terms involving perturbation velocities, and (5) off-axis flapping response. The resulting first-order equation for rotor flapping due to roll rate and lateral stick inputs is
See Equation (10)
Equation (10) is coupled into the roll acceleration response through a flapping spring derivative (L_{b1s}):
See Equation (11)
where all of the quasi-steady derivatives have been retained, except for the roll damping and roll control derivatives whose effects have been replaced by the flapping equation. The combination of Eqs. (10) and (11) leads to a model that captures the rotor regressing response at high frequency and reduces to the quasi-steady response of the 6-DOF model at low frequency. The hybrid model development follows the model given by Heffley,^{21} although the hybrid model will be more accurate because it retains all of the quasi-steady derivatives.
At higher frequencies, where the quasi-steady terms are not significant, the coupled roll/flapping response has a second-order characteristic. This is the appropriate dynamic behavior based on the model structure discussion above. At frequencies below the regressive rotor flapping mode (about 13 rad/sec), the effective roll damping and roll control derivatives are determined from the steady-state flapping response (b_{1s} = 0 in Eq. (10)):
See Equation (12)
Thus, the hybrid model structure reduces to the quasi-steady formulation and will achieve the same level of accuracy for frequencies below about 10 rad/sec. However, at high frequency much improved model matching is achieved.
An analogous equation to that developed above for the lateral flapping response can also be developed for the longitudinal flapping response (a_{1s}), where it is again assumed that no lateral to longitudinal coupling exists in the flapping dynamics. The longitudinal flapping time constant (t_{a1s}) has the same theoretical value as the lateral flapping time constant (t_{b1s}). The pitch spring constant (M_{a1s}) is approximately one-third of the roll spring constant (L_{b1s}), reflecting the fact that the helicopter's pitch inertia is about three times its roll inertia. As a result, the body pitch response and longitudinal flapping remains as distinct real modes.^{21}
Simple physical models for the lead-lag dynamics, such as those for the flapping dynamics above, are not available in the literature. Therefore, a modal approach was taken based on the results of Ref. 14. As described above, a second-order dipole is appended to the identified model of roll-rate response to lateral stick:
See Equation (13)
Hybrid Model Identification Techniques
The 9-DOF parametric model is identified by matching the 33 frequency responses obtained from the nonparametric identification results of the Frequency-Response Identification section. Twenty frequency points were selected linearly across the logarithmic frequency range of good coherence for each response pair. The maximum fitting frequency for the responses was 30 rad/sec, which includes rotor flapping and lead-lag dynamics.
In the most general formulation, the hybrid model may comprise 36 unknown parameters in the F matrix, 24 parameters in the G matrix, 4 lead-lag parameters, and 4 time delays (actuator effects), for a total of 68 possible unknown parameters. While, in principle, the rotor parameters of Eqs. (10)-(13) should be identifiable from matching the high-frequency angular responses, excessive correlations among the parameters makes it difficult to achieve a model structure with acceptable accuracy metrics. Analysis indicated that flapping response measurements were needed to alleviate this correlation problem. For the present study, rotor state measurements were not available, so the correlation problems were corrected by enforcing some physical constraints:
1. The lateral and longitudinal flapping time constants were constrained to be equal, as predicted by theory:
2. The lateral cyclic time delay was fixed at the value obtained from a transfer-function fit of (Refs. 11 and 14). The value of = 0.022 sec corresponds to the known effective delay of hydraulic actuator dynamics.
3. The lead-lag dipole parameters (damping ratios and natural frequencies) were obtained by fixing the parameters of the final coupled body/flapping model (8-DOF), and then optimizing only the lead-lag parameters in a match of the roll response (one-way coupled dynamics model). Finally, the lead-lag transfer function was converted to phase canonical form and coupled to Eq. (10) to achieve a complete 9-DOF (12th-order) state-space representation.
Using the above constraints (1-3) on the rotor parameters, the accuracy analysis described earlier in the Frequency-Response Identification Method section was conducted on the full 68-parameter hybrid model to expose insensitivity and correlation problems. Parameters in F, G, and t were sequentially dropped and the identification was repeated until insensitivities and Cramer-Rao bounds were near the satisfactory guidelines (10% and 20% respectively of the parameter values) and just before a significant rise in the cost function was detected. This procedure resulted in a reduction of the parameter set from 68 to 50 unknowns. The 9-DOF model parameters are listed in Table 1 along with the associated Cramer-Rao bounds and insensitivities. The full 9-DOF state-space model is shown in Table 2 to illustrate the conversion of the lead-lag transfer function to canonical form and its inclusion in the final model.
The 9-DOF model parameters relate some important characteristics of the BO-105. First, the roll spring (L_{b1s} = 163.6), sets the frequency of the coupled roll/flapping second order mode:
See Equation (14)
which is quite close to the value identified by Kaletka.^{18} The presence of this mode at
w = 12.8 rad/sec sets the upper-frequency limit on the 6-DOF model, as discussed earlier. The inverse of the flapping time constant (1/t_{f} = 20.3 rad/sec) is also quite close to the value identified by Kaletka,^{18} but it has a value about twice the simple theoretical result based on articulated rotor theory and an effective hinge offset of 12%.^{21} The fact that the associated Cramer-Rao value is very low (4.7%) and the parameter value is in close agreement with Kaletka indicates that the flight test estimate is quite accurate. The theoretical result obtained by approximating the BO-105 hingeless rotor using articulated rotor theory is apparently not adequate. Furthermore, simple dynamics theory based on inertia ratios would predict the flapping spring constant for pitch to be about one-third of the value for roll, while the results show a ratio of about 2.4. Flapping data would again be useful here in resolving these discrepancies.
The effective quasi-steady derivatives are calculated from Eq. (12) as
See Equation (15)
which are close to the quasi-steady values obtained for the 6-DOF model formulation of Ref. 3 as expected. The remaining quasi-steady derivatives of the 9-DOF model are also close to those obtained previously for the 6-DOF model. Thus, as described earlier, the hybrid model has essentially the same characteristics in the low-frequency range as the quasi-steady model, while retaining the improved high-frequency description afforded by the relatively simple rotor equations.
As expected for the BO-105 hingeless rotor, the effective roll-damping derivative (L_{p}) has a high value, and the response coupling ratios are significant |L_{q}/ (L_{p})_{e }| = 0.50 and |M_{p}/ (M_{q})_{e}| = 0.23. The off-axis control response of rolling moment due to longitudinal stick is also quite significant , while the associated pitching moment due to lateral control is of negligible importance and was dropped from the identification. The pitch control derivatives to collective and longitudinal stick inputs are close in value , which is a fact well appreciated by BO-105 pilots, who report that pitch control can be achieved equally through either control at the forward speed analyzed (80 knots).
The natural modes of the 9-DOF model are presented in Table 3.
Table 3. 9-DOF BO-105 model eigenvalues |
||||
Mode |
Real |
Imag |
Omega |
Zeta |
Pitch phugoid |
0.1191 |
0.2778 |
0.3022 |
0.3942 |
Dutch roll |
0.5713 |
2.546 |
2.609 |
0.2190 |
Roll/flapping |
9.904 |
7.740 |
12.57 |
0.7880 |
Lead-lag |
0.8680 |
15.57 |
15.59 |
0.05567 |
Spiral |
0.05068 |
0.0 |
||
Pitch-1 |
0.4480 |
0.0 |
||
Pitch-2 |
5.843 |
0.0 |
||
Long. flapping |
15.93 |
0.0 |
The BO-105 has an unstable, low-frequency, phugoid oscillation, and a lightly-damped dutch roll oscillation. The coupled roll/flapping mode is well damped ( = 0.8) and the natural frequency (w_{rf}= 12.6 rad/sec) agrees closely with that predicted by Eq. (14). The natural frequency of the lead-lag response corresponds to the peak in the response at 15.5 rad/sec displayed in Fig. 7. This mode is very lightly damped = 0.056 as expected for the hingeless rotor without lead-lag dampers. The total damping () is slightly higher than that obtained previously from a transfer-function fit of the isolated roll response
(s_{ll} = 0.67 in Ref. 14); however, both values are within the bounds of previously published flight test results.^{22} Once again, better estimates of rotor parameters require the inclusion of blade deflection measurements. The body pitch mode and longitudinal flapping mode are distinct real roots as compared to the coupled roll/flapping oscillation, due to the larger pitch inertia compared to the roll inertia. These results show that modeling the rotor dynamics as an actuator in series with the quasi-steady fuselage modes is a reasonable approximation for the pitch response (as done in Ref. 13), but not for the roll response where the modes form a coupled oscillation.
Figure 8 shows frequency-response comparisons of the hybrid model, the quasi-steady model (from Ref. 3), and flight data for six typical response pairs. In these figures, only 50 frequency points are plotted, which causes the data to have a more jagged appearance than the earlier response shown in Fig. 7. As can be seen from the roll rate response (), for example, the two models have essentially the same response characteristics for frequencies below about 9 rad/sec, confirming that the hybrid model reduces to the quasi-steady model at low frequency. In the higher frequency range of 9-30 rad/sec, the hybrid model matches the roll frequency-response data very well (especially compared to the 6-DOF model as shown in the figure) indicating that the simple rotor modeling for the high-frequency response is adequate.
A magnitude discrepancy is apparent in the high-frequency responses for both the 6- and 9-DOF models. This discrepancy also occurred in the time domain identification results.^{4}The phase fit for the 9-DOF is somewhat improved, however. A more complete rotor model which includes lateral/longitudinal flapping cross-coupling and a lead-lag dipole in the pitch response was evaluated; however these enhancements did not improve the fit. The source of this discrepancy cannot be ascertained until rotor flapping measurements are introduced in the identification. A second problem with the hybrid model is the failure to improve the match of the roll response in the region of the dutch roll mode. Detailed studies were conducted to evaluate the source of this discrepancy without significant success. Because the coherence is very high in the dutch roll frequency range, nonlinearities cannot be considered a cause of the problem. One possible cause is unmodeled dynamic states such as side-wash lag between the main rotor and the tail rotor.^{23} However, this derivative is highly correlated with the control derivatives , and cannot be identified without flapping data.
A time history comparison of the identified 9-DOF hybrid model and flight data for records not used in the identification is shown in Fig. 9 for the four control axes. Overall, the predictive capability of the identified model is quite good in both the on- and off-axis responses, especially considering the dynamically unstable and highly coupled nature of the BO-105 helicopter. The overall time-domain cost functions for the 6-DOF and 9-DOF models are within 5% for all 4 inputs. Clearly, the extracted models generally reflect the fundamental dynamic characteristics of the helicopter. The discrepancies in the on-axis pitch rate response and the dutch roll coupling in the on-axis roll rate response discussed earlier for the frequency-response comparisons show up in the time-domain plots as well. The model predicts a lightly damped yaw response due to longitudinal stick that appears to be suppressed in the verification flight data. This discrepancy may be due to unmodeled interference effects between the main rotor and empennage surfaces.
The need for the higher-order model is best seen in the short-term roll rate and roll acceleration responses shown in Fig. 10. Here we see, in contrast to the 6-DOF model, that the hybrid model does a very good job in matching the peak roll acceleration, peak roll rate, and the lightly damped oscillation associated with the lag dynamics. Thus, the hybrid model will be well suited for high-bandwidth applications.
Hybrid Model Applicability for High-Bandwidth Control
The importance of accurate broadband design models for high-bandwidth flight control can be seen by incorporating the 9-DOF hybrid identification results in the analysis of a simple roll rate feedback system. An additional 50 msec of time delay is added to dynamic models to account for a digital computer, zero-order hold, and additional digital system elements. Figure 11 shows the root locus as a function of roll rate gain to swashplate, and it indicates the feedback level is limited by the destabilization of the lead-lag rotor mode, an effect which would be completely missed by the quasi-steady model. The point of incipient instability of the lead-lag mode occurs for a rate feedback gain of 0.081 deg/deg/sec. Previous analytical studies by Curtiss,^{19} Diftler,^{24} and Miller and White^{25} have shown that this potential for destabilizing the lead-lag rotor mode exists in articulated rotor helicopters for feedback gains of 0.1-0.5 deg/deg/sec. The problem is further exaggerated in the case of the BO-105 because of the strong roll/flap coupling due to the high effective hinge offset, low roll inertia, and the lack of lead-lag dampers. The current results emphasize that design models for advanced flight-control system studies must include these important rotor dynamic modes. Further discussion of this topic is presented in Ref. 14.
Summary
A frequency-response method for rotorcraft system identification has been presented. Key aspects of this method that were highlighted in this paper are
1. Single-input and multi-input frequency-response calculations to account for partially correlated inputs.
2. Composite window technique which combines conditioned spectrum results of different windows to obtain a set of broadband frequency-responses. The laborious job of manually optimizing window size is eliminated.
3. Integrated approach to state-space model identification and model structure determination.
4. Time-domain verification of models using dissimilar flight data.
5. Integrated software package for the frequency-response method: Comprehensive Identification from FrEquency Responses (CIFER).
Application of the frequency-response method to the identification of BO-105 helicopter dynamics (80 knots) has shown that
1. A 9-DOF hybrid model that includes coupled body/rotor-flapping and lead-lag dynamics was formulated and identified that is accurate for frequencies up to 30 rad/sec. Rotor parameters closely match previously published results. At low frequencies (below 13 rad/sec), the hybrid model response reduces to the 6-DOF quasi-steady response.
2. Unresolved deficiencies exhibited by both the 6-DOF and 9-DOF models are (a) the dutch roll influence in the roll response to lateral stick and yaw response to longitudinal stick, and (b) the mid-frequency pitch response to longitudinal stick (3-8 rad/sec). Rotor state measurements are needed to resolve these deficiencies and eliminate the need for a priori model constraints in the 9-DOF identification.
3. Maximum achievable roll-rate gain is limited by the destabilization of the lead-lag dynamics, when 0.05 sec of time delay is included in the control system. Advanced flight-control designs for helicopters must be based on models that include the flapping and lead-lag dynamics.
References
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^{4}Fu, K. H. and Kaletka, J., "Frequency-Domain Identification of BO 105 Derivitive Models with Rotor Degrees of Freedom," 16th European Rotorcraft Forum, Glasgow, UK, Sep 1990.
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^{12}Tischler, M. B., Fletcher, J. W., Morris, P. M., and Tucker, G. E., "Flying Quality Analysis and Flight Evaluation of a Highly Augmented Combat Rotorcraft," Journal of Guidance, Control, and Dynamics, Vol. 14, No. 5, Sep-Oct 1991.
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^{14}Tischler, M. B., "System Identification Requirements for High-Bandwidth Rotorcraft Flight Control System Design," AIAA Journal of Guidance, Control, and Dynamics, Vol. 13, (5), Sep-Oct 1990.
^{15}Tischler, M. B., "Advancements in Frequency-Domain Methods for Rotorcraft System Identification," 2nd International Conference on Rotorcraft Basic Research, University of Maryland, College Park, MD, Feb. 1988. (Also, Vertica Vol. 13, (3), 1989.)
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^{18}Kaletka, J. and von Grunhagen, W., "Identification of Mathematical Derivative Models for the Design of a Model Following Control System," American Helicopter Society 45th Annual Forum, Boston, Mass., May 1989. (Also, Vertica, Vol. 13, (3), 1989.)
^{19}Curtiss, H. C., Jr., "Stability and Control Modeling," paper no. 41, 12th European Rotorcraft Forum, Garmisch-Partenkirchen, Federal Republic of Germany, Sep 1986.
^{20}Chen, R. T. N., "Effects of Primary Rotor Parameters on Flapping Dynamics," NASA TP-1431, Jan 1980.
^{21}Heffley, R. K., Bourne, S. M., Curtiss, H. C., Jr., Hindson, W. S., and Hess,
R. A., "Study of Helicopter Roll Control Effectiveness Criteria," NASA CR-177407, April 1986.
^{22}Warmbrodt, W. and Peterson, R. L., "Hover Test of a Full-Scale Hingeless Rotor," NASA TM-85990, Aug 1984.
^{23}McRuer, D. T., Ashkenas, I., and Graham, D., Aircraft Dynamics and Automatic Control, Princeton University Press, 1974.
^{24}Diftler, M. A., "UH-60A Helicopter Stability Augmentation Study," paper no. 74, 14th European Rotorcraft Forum, Milano, Italy, Sep 1988.
^{25}Miller, D. G. and White, F., "A Treatment of the Impact of Rotor-Fuselage Coupling on Helicopter Handling Qualities," American Helicopter Society 43rd Annual National Forum, St. Louis, MO, May 1987.