Abstract: Near Real Time estimation of stability (gain and phase) margins using flight test data from development flights of a new aircraft facilitates cost-effective flight envelope expansion. Methods based on FFT, Z-transforms etc. have been proposed to meet this requirement. In this paper a new method combining FFT techniques and parameter estimation in the frequency domain has been used to accurately determine the stability margin of the longitudinal axis of a FBW fighter aircraft. Typically during initial flight tests of a new aircraft only flight data from piloted time domain inputs such as doublets or 3-2-1-1 are available for analysis even though sine sweeps are the preferred inputs for determining stability margins. Further the aircraft response signals due to these inputs have lower bandwidth compared to the typical gain and phase crossover frequencies of the aircraft + control law loop transfer function. Hence estimating the stability margins from these low bandwidth signals is a challenging problem.
In this paper, by noting that the digital
controller is exactly known, the open loop aircraft frequency response is first derived using FFT techniques. However due to poor signal to noise ratios the frequency response is noisy at higher frequencies. Consequently estimation of especially the phase crossover frequency and hence the gain margin tends to become inaccurate. To overcome this problem a smoothening scheme (based on parameter estimation) for the FFT based frequency response data is now used to derive a smooth transfer function in the required frequency range. This is possible since the structure of the short period aircraft transfer function is well known. Thus a parameter estimation procedure based on this model structure is used to derive a best-fit frequency response (gain & phase) from the already estimated aircraft frequency response using FFT signal analysis. This smoothed aircraft frequency response is then concatenated with the known frequency response of the digital controller to derive the loop transfer function of the aircraft + controller from which the desired gain and phase margins are estimated. The method has been initially validated using flight simulator responses derived from actual piloted 3-2-1-1 inputs used during the initial flight trials of India’s Light Combat Aircraft (LCA). The method has subsequently been applied to the flight test data of the LCA to successfully predict the available stability margins at various flight conditions.
1. Introduction Methods have been proposed [7, NASA Report 4598, 1, T. Smith, 2, Mark B. Tishler] for determining the stability margins in near real time. The motivation for developing such capabilities is to allow flight test personnel to give clearance for the pilot (in almost real time) to proceed to the next test point if the stability margins at the present test point are satisfactory. This would permit rapid expansion of the flight envelope and hence a cost-effective flight test plan can be formulated. Methods using purely FFT analysis are generally faster and work well if the excitation signals are sine sweeps covering the gain/phase crossover frequency range of the feedback stabilized aircraft. Methods using parameter estimation procedures postulating state variable models [3, Hamel and Jategaonkar] work well with time domain inputs such as doublets or 3-2-1-1 signals but are computationally intensive due to the iterative nature of the algorithms and analysis. Thus they are ideally suited for off-line accurate determination of stability and control derivatives. During initial flight trials of a new aircraft, computer generated signal injection systems capable of exciting the aircraft with signals such as i) sine sweep signals or ii) well executed high bandwidth 3-2-1-1 signals will generally not have been cleared for flight. Thus flight data analysis has to be invariably be based on manual piloted time domain (low bandwidth) inputs such as 3-2-1-1 and doublets.
Typical fighter aircraft like LCA have gain and phase crossover frequencies around 1 Hz and 3 Hz respectively. The well practiced Piloted 3-2-1-1 inputs typically with an overall time duration of seven seconds have a signal bandwidth of around 1 Hz. Therefore, estimating the stability margins from these low bandwidth 3-2-1-1 signals needs special analysis techniques.
In this paper, three methods with increasing
order of complexity are developed using Fast Fourier Transform (FFT) techniques to estimate in near-real time, stability margins from piloted 3-2-1-1 inputs. “Method I” finds the open loop frequency response from the closed loop system response data. This is the most common method used in the literature. The disadvantage of this method is that the signal-to-noise ratio of the derived information is very low since the high gain controller heavily attenuates the output signals. Moreover, there is insufficient input excitation to the plant beyond 1 Hz, which leads to a
AIAA's Aircraft Technology, Integration, and Operations (ATIO) 2002 Technical 1-3 October 2002, Los Angeles, California
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noisy output frequency response. This method however, will be useful if there is sufficient input excitation over the desired bandwidth of upto 4 Hz. Therefore, it becomes necessary to device alternate methods to estimate the stability margins if excitation inputs are of low bandwidth.
The controller and actuator transfer functions (accounting for all lags) are accurately determined experimentally in ground-based rigs. Thus it follows logically that if the aircraft frequency response can be estimated from flight data using FFT, the controller + actuator transfer function can then be concatenated with the aircraft frequency response to obtain the overall open loop transfer function for estimating the relative stability. This method is referred as “Method II”. In this method the signal-to-noise ratio improves marginally, but the problem of insufficient excitation beyond 1Hz still remains.
The structure of the short period aircraft
transfer functions is well known [6, McRuer]. The short period pitch rate and normal acceleration to elevator transfer functions are of second order. Using parameter estimation techniques in the frequency domain the coefficients of both these transfer functions can be estimated by generating a best fit from the FFT derived frequency response of the aircraft obtained using “Method II”. This amounts to smoothing the noisy frequency response characteristics (at higher frequencies) derived using FFT analysis. The known individual controller transfer functions are then concatenated with the plant transfer functions (Nz & q) estimated using frequency domain parameter estimation to generate the overall loop transfer function. This method is referred to as “Method III”. With “Method III” it will be shown that the longitudinal stability margins can be estimated quite accurately. Since this study using the LCA flight recorded data shows that stability margins can be estimated reliably in near real-time the technique will be very useful for future flight tests when the flight envelope expansion is undertaken.
This paper is divided into five sections. In
section 2 the details of the three methods for estimating the margins from flight test data are highlighted. In Section 3, the canned pitch stick inputs obtained from flight tests are used to generate equivalent responses from the ground based LCA flight simulator. The stability margins are estimated from the simulated data and these margins are compared with the “theoretically correct” margins obtained from linear perturbation model generated at the corresponding flight condition. Thus the results given in this section validate the “Method III”. Section 4, details the flight data analysis and the computation of stability margins from real flight test data. Section 5 provides the summary and concluding remarks.
2. Three Methods for Stability Margin Estimation using FFT
In LCA TD1 first block of flights, the pilot
had given pitch stick 3-2-1-1 inputs at various Mach nos. and altitudes for generating data for conventional aircraft parameter estimation analysis to estimate the longitudinal stability and control derivatives. Table 1, lists the flight segments corresponding to the 3-2-1-1 pitch stick inputs. The 3-2-1-1 input lasts for 7 seconds but for analysis additional time segments before and after the input are considered. This ensures that the aircraft starts from a trim condition and then settles down after the input ends.
Table 1. LCA TD1 first block of flights – Pitch stick 3-2-1-1 flight data segments
Flight No. Seg. No. Under Carriage
Total Duration
(sec) 1 23.9 6 2 Up 17.9 1 19.9 2 21.8 3
Up 20.9 7
4 Down 21.9 1 Down 12.9 2 21.9 3 8.9 4
UP 21.9
8
5 Down 18.9 10 1 Down 8.9
These flight conditions are plotted as points
in the flight envelope (Fig. 1), where ‘∗’ indicates the flight conditions where undercarriage was retracted (UP) whereas ‘∇’ indicate the flight conditions where undercarriage was extended (DN).
7(4)
8(1)
8(5)
10(1)
6(1)
6(2)
7(1) 7(2)
7(3)
8(2) 8(3)
8(4)
Mach
Altit
ude
U/C UP
U/C DN
* U/C Up
∇ U/C DN
Fig. 1 LCA First Block of flight - 3-2-1-1 Pitch Stick Input (Number above the symbol indicates Flight
No.) (Number in brackets indicates flight segment)
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A simplified block schematic of the aircraft closed loop system is shown in Fig. 2. The gain and phase margins for the longitudinal axis are calculated analytically by opening the loop at the actuator consolidation point as shown in Fig. 2.
Even though the gain and phase margins are
defined as two distinct stability margin entities, for ensuring a robust loop transfer function, it is the aircraft industry requirement [1, T. Smith] to define a template on the Nichols plot as shown in Figure 3. The design requirement is that the loop transfer function should not cross this template. This amounts to satisfying simultaneously both the gain and phase margin requirements. However for an unstable aircraft, it is not possible to open the control loops in flight. Hence the margins have to be calculated indirectly from the closed-loop responses.
Denoting the loop transfer function from
“A” to “B” in Fig. 2 by L and the closed-loop transfer function from test points P1 to P2 by G, we get
LG
−=
11 , And hence
GGL 1−
= (1)
Thus the open-loop transfer function L can be determined if one estimates the closed-loop transfer function G from flight data. From this, the margins can then be determined.
Initially to evaluate the applicability of the three stability margin computation methods, the fixed based real time flight simulator of the LCA aircraft called the Engineer-in-loop simulator (ELS) was used as the test platform. To generate the aircraft response trajectories in the simulator, the actual pilot inputs given in the flight tests were injected to the flight simulator after the simulator was trimmed to the reference flight condition. These simulated trajectories were subsequently used for the stability margin calculations.
Method I. Computation of Open Loop Frequency Response from Closed Loop Frequency Response
The ratios of the FFTs of P2 and P1 yield the closed loop frequency response G(jw). Using (1) the open loop frequency response is obtained.
The disadvantage of this method is that the
bandwidth for even an ideal 3-2-1-1 signal of 7 seconds duration is approximately 1 Hz (Appendix A). In addition, a low-pass filter added in the command path of the control laws to meet the handling quality requirements further reduces the bandwidth of the pilot stick input. For a fighter aircraft, the gain margin is generally larger than 6 dB and therefore the signal to noise ratio is very low at P2 (i.e., there is lot of signal attenuation at P2). The gain and phase cross over frequencies are of the order of 1 Hz and 3 Hz respectively. Figure 4 shows a typical Nichols plot of the loop transfer function (L(jw)) estimated by the above method for ELS data and flight test data. From the figure it is seen that even for the simulated (ELS) data, it is difficult to estimate the margins correctly. For flight data the estimated Nichols plot is very “noisy”. Therefore, estimation of gain and phase margins with this method is impossible.
Method II. Calculation of the Plant Frequency Response and then concatenate the Controller Frequency Response
The voted (median of multiple sensor signals – which is less noisy than the raw signals) Pitch rate and Normal Acceleration signals are available from the recorded flight data. The digital controller is generally characterized quite accurately in ground rigs like the ironbird [4]. To utilize this apriori information, only the frequency response of the plant + actuator is determined from flight test data and the known controller frequency response is concatenated in series to find the overall loop transfer function L.
A
Controller
Pitch Stick Path Filters
Pstick
P2
P1
B Plant
q
Nz
de
Actuator
The points B and A are indicated to show the points
where the loop is opened to calculate the margin
Fig 2. Simplified Block Schematic of Longitudinal Axis.
Figure 3. Typical Nichols plot and the Stability Margin Template.
Open loop transfer function
-250 -200 -150 -100 -50 0 -15
-10
-5
0
5
1
1
2
Mag
nitu
de in
dB
Phase
Margin Template
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Thus, the ratios of FFTs of q and P2 yield the plant+actuator transfer function from P2 to q and the ratios of FFTs of Nz and P2 yield the plant + actuator transfer function from P2 to Nz. To this the known frequency response of the controller is concatenated to obtain the loop transfer function.
Figure 5 shows the loop transfer function for
both ELS and flight data. The open loop frequency response obtained from this method is slightly smoother compared to that obtained from “Method I” above. However, it is still impossible to calculate the margins with this method meaningfully.
Thus it can be concluded that by just
performing a FFT signal processing of the aircraft response signals meaningful stability margin estimates cannot be obtained primarily due to the noisy frequency response characteristics in the 1-4Hz frequency range. Hence both Methods I and II are of limited use. From this study it becomes evident that some form of smoothing / filtering of the aircraft transfer function response obtained from the FFT process should be employed. This is the motivation to develop Method III.
Method III. Estimate the short period plant transfer function parameters from the Frequency Response obtained in Method II
The structure of pitch rate and normal acceleration transfer functions for symmetric elevon excitation is known a priori, and only the coefficients in the transfer functions are unknown. The short-period approximations for these two transfer functions are as shown below.
222
2
1
)(nnn
q
q ssT
sK
eqsP
ωωξδθ
++
+
== (2)
2232
_ 2
11
)(nnn
hhnz
cgcgNz ss
Ts
TsK
eNz
sPωωξδ ++
+
+
== (3)
sCGsensor xqNzNz &+= , Where, xs is the forward displacement (in metres) of the accelerometer sensor from the CG in the x direction. For LCA this displacement is 3.485 m.
22232
2
1180
4850.311
)(
nnn
qhh
nz
sensorNz
ss
TssK
Ts
TsK
eNz
sP
ωωξ
πδ
θ
++
+
+
+
+
==
(4)
The elevon actuator transfer function is obtained from rig tests. The frequency response of this actuator transfer function along with the flight computer computation delay of 8 ms modeled as a transportation lag is cascaded with the frequency response of Pq(s) and PNz(s) to obtain the frequency response from P2 to q and Nz.
Denote a=2ξnωn, b=ωn
2, zθ2 =1/Tθ2, zTh2 =1/Th2 and zTh3 =1/Th3. Now, (2) and (4) have seven unknown variables. Denote these seven variables in the vector form as
X=[Kq, zθ2, a, b, KNz, zTh2, zTh3] (5)
To facilitate fast convergence in real-time,
bounds on the parameters are established a-priori. Upper and lower bounds on the parameters in X are obtained from wind tunnel data based aircraft model used in the simulator as follows.
a. The scatter of the parameters at various flight
conditions due to variation in aircraft mass and c.g. can be determined from the linear models generated at these flight conditions. A 2-D look-up table (as a function of Mach and altitude) corresponding to the mean values of the parameters was generated. The scatter was then expressed as a single percentage variation over and above this mean value. However, for parameter b absolute bounds on the scatter were found to be bounded by (b0/3)+15 and (b0/3)-1, where b0 is the mean value obtained from 2-D table. The results of the above analysis is as shown below.
Variation Kq 6 KNz 8 zθ2 4 zTh2 4 a 8 zTh3 5 b (b0/3)+15
(b0/3)-1
b. Since, these parameters are estimated from wind-tunnel data, there is certain amount of uncertainty involved. In order to account for this, an additional ±10% over and above this variation is considered for computing the parameter bounds.
Hence the upper and lower bounds for the
first six parameters were obtained as 1.2x0 and 0.8x0 respectively. For the last parameter in X, the upper bound was (b0/3) + 17 and the lower bound was (b0/3)–3. x0, b0 is the mean value of the particular parameter obtained from the look-up table.
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Denote the frequency response obtained from the flight test data over a frequency range [w1, w2] from P2 to q and P2 to Nz as Fq[w1,w2] and FNz[w1,w2] respectively. Then the problem reduces to finding the optimum parameter vector X * which minimizes a cost function - defined as the root sum square of the error between the frequency response obtained from flight data and the frequency response obtained from the aircraft model as postulated in equations 2 and 4 and denoted by Pq[w1,w2] and PNz[w1,w2] respectively.
Thus the following cost function is
optimized to derive the best-fit q and Nz transfer functions.
(6) The analytical transfer function resulting from the above optimization (equation 6) is now used as the aircraft transfer function for computing the loop transfer function of the aircraft + controller and for subsequent stability margin computations. Appendix A discusses the spectral characteristics of the pilot applied 3-2-1-1 signals and their impact on the FFT signal analysis. From this analysis the flight segment F8_seg4 is found to be the “best” flight segment for validating ‘Method III’ and is used in the analysis discussed in the next section. 3. Validation of “Method III” using Offline Canned Inputs (ELS Simulation)
For verifying the accuracy of the margins computed by “ Method III”, linear perturbation models at the reference flight conditions were used as the ‘Truth Model’ for the aircraft. These models were generated from the six-degree of freedom non-linear simulation in the ELS simulator. The stability margins were then analytically computed by concatenating the digital controller with the linear models while accounting for all the lags in the closed loop system. These margins were used as reference data to compare with those generated using “ Method III ”. As indicated earlier, flight condition F8_seg4 was used for this verification/validation process.
The procedure for computing the stability
margins using ‘Method III’ consists of i) trimming the ELS simulator to the reference flight condition, ii) generating the aircraft model (ELS) responses by injecting the pilot 3-2-1-1 input to the simulator and recording P1, P2, q, Nz responses, iii) performing FFT signal analysis on these signals and computing the aircraft transfer function using ‘Method II’ and iv) finally fitting a best analytical transfer function for the aircraft using the optimization criterion in (6).
Figure 6 shows the time response for P2
(which is the equivalent de command) and pitch rate
(q) signal after removing the bias. For both these signals Fourier Transform is plotted in terms of magnitude and phase upto 4.5 Hz. The dotted horizontal straight line at –35db indicates the frequencies at which the magnitude spectrum for P2 lies above this line and only these segments are considered for optimization. The ratio of pitch rate magnitude spectrum to that of P2 is also plotted. The phase difference between these two signals is plotted in the bottom right graph. Similarly, in Figure 7, P2 and Nz data are plotted. Since Nz is the acceleration signal and noisier than the pitch rate signal, the requirement of excitation level to get reliable output data from Nz was found to be higher when compared to pitch rate. The dotted horizontal straight line at –20db in Fig. 7 indicates the frequencies at which the magnitude spectrum for P2 lies above this line and only these segments are considered for optimization. The cutoff levels of –35dB for q (Fig. 6) and –20 dB (Fig. 7) for Nz were arrived at based on detailed studies over the 12 flight segments.
Figure 8, shows the q and Nz frequency
response obtained from linear models, Method II and Method III. The smoothing of the noisy data in Method III is quite apparent. In the normal acceleration transfer function, Method III frequency response starts to deviate from the linear model beyond 2Hz. This can be attributed to the lower signal to noise ratio in the accelerometer signal. In Fig. 9, the loop transfer functions for method III and the nominal linear model are plotted. The Nichols plots of the loop transfer function generated using the upper and lower bounds (used in optimization) of the parameter vector X are also plotted. From Fig. 9, it can be seen that gain and phase margins of Method III and nominal linear model match very well and thus validating the proposed method. The values of gain and phase margins are compared in Table 2. Table. 2. Gain and Phase Margins from ELS Replay (Flight Segment F8_seg4)
Gain Margin (dB)
Phase Margin (deg)
Estimated (Method III) -9.3749 65.1023
Linear Model -9.3520 64.8873 Error in Estimation -0.0229 0.215
The margins estimated from ELS data for all
twelve data segments along with linear model margins are tabulated in Table 3. It can be seen that maximum absolute error in estimation for ELS replay data is 1.14dB (For F6_seg2) in magnitude and 2.4058 deg in phase (F7_seg1). For both these data segments, in frequency domain, the magnitude and bandwidth is lower when compared to the other segments
2]2,1[]2,1[
2]2,1[]2,1[
],[min wwNzwwNzwwqwwq
XXxPFPF
upperlower
−+−∈
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Table. 3. Gain and Phase Margins from ELS Replay (all flight segments)
4. Estimation of Stability Margins from Flight Tests
The Light Combat Aircraft (LCA) India’s LCA is a single engine tail-less delta
wing supersonic fighter aircraft, which is designed to be aerodynamically unstable in the longitudinal axis. In order to stabilize the airframe and achieve the desired performance over the entire flight envelope it incorporates a quad redundant full authority digital Fly-By-Wire (FBW) Flight Control System (FCS). The control laws resident in the FCS, in addition to guaranteeing stability, optimize the aircraft performance and piloted handling qualities over the entire flight envelope. An overview of the design, development and testing of the flight control laws for the LCA is given in [4, Shyam Chetty et. al.,].
During the LCA initial block of flights, 3-2-
1-1 and doublet inputs were applied for conventional parameter estimation analysis. Due to limited flight test time, sine sweep inputs, which would have been more suitable for stability margin estimation, were not applied. Typical flight test results and comparisons of the flight responses with flight simulator results are given in [4, Shyam Chetty et. al.,
]. While good match in time responses has given the control law designers reasonable confidence in the wind tunnel data based flight models used for design, it is imperative to establish the available stability margins by computing the loop transfer functions from flight data [2, Tishler] Flight Data Analysis
In the previous section, it is seen that the
margins obtained from the linear models and those obtained using “Method III” for 3-2-1-1 canned pilot inputs replayed in ELS match very well. This validates “Method III” and the method is now used to estimate stability margins from flight test data for all flight segments.
Figure 10 shows the comparison of
estimated margins with the margins obtained from a linear model at the same flight condition and mass / C.G configuration for the flight segment F8_seg4. The Nichols plots of the loop transfer function generated using the upper and lower bounds (used in optimization) of the parameter vector X are also plotted. It can be seen that the estimated Nichols plot falls within these plots showing that the estimated values have not saturated at the extreme boundary values during optimization. The results are listed in Table 4 below.
Table 4. Gain and Phase Margins from Flight Data (Flight Segment F8_seg4)
The margins estimated from flight data for
all twelve data segments along with margins obtained from linear models corresponding to the “closest” flight fuel conditions are tabulated in Table 5.
For estimates generated from flight test data
the worst deviation from the linear models is 1.2840 dB in gain and 3.9489 deg in phase. The difference between flight estimated and model predicted stability margins could be chiefly attributed to (among other errors):
a) Accuracy of aerodynamic parameters based on
wind tunnel data. b) The fuel state used for generation of linear
models is not exact; c) it is approximated to the “closest” Flight Fuel
state for which data is available.
Gain Margin (dB)
Phase Margin (deg)
Method III -9.2971 66.1734 Linear Model -10.061 67.1342
Difference between Method III and
linear model 0.7639 -0.9608
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d) Accuracy of mass/ inertia / C.G data e) The 3-2-1-1 inputs as given by pilot being far
from the Ideal 3-2-1-1 input (Poor bandwidth) Despite all these real world effects the
estimated gain and phase margins are indeed quite close to the ones obtained from linear models. This establishes a high degree of confidence in the non–linear simulation model (used for control law design) being a good representation of the actual aircraft in the flight envelope tested. Finally the new margin estimation method proposed (Method III) can be used with confidence for near real time estimation of gain and phase margins and will be a useful tool for rapid envelope expansion in subsequent flight test programs.
4. Conclusions: This paper addresses the estimation of stability margins of India’s Light Combat Aircraft (LCA) from the initial flight trials of the aircraft. The LCA is a longitudinally unstable Fly-by-wire aircraft. Hence estimation of the stability margins, especially in the longitudinal axis, during the initial flight testing of this new aircraft was important to establish
the accuracy of the flight model used for control law design. A new method for estimating the gain and phase margins, in near real time, is proposed. The method uses only the aircraft response signals to pilot applied 3-2-1-1 inputs, which are not of high spectral quality. The two step method consists of first constructing the aircraft frequency response using FFT signal analysis followed by deriving an analytical transfer function of the aircraft which is a best fit to the FFT derived frequency response. This procedure essentially smoothens the FFT derived frequency response, especially near phase cross over frequencies, enabling reliable estimation of both gain and phase margins. For validating the method, the LCA flight simulator is initially used to generate aircraft responses to canned 3-2-1-1 pitch stick inputs, which were actually applied during flight-testing. The stability margins using the new method developed, is estimated based on this response data. The margins derived using the linear perturbation models of the aircraft at the reference flight conditions are used as the ‘truth model’ for determining the accuracy of the margin estimates.
Based on the success of this validation process, the stability margins for LCA are estimated at various flight conditions using flight data. These estimates when compared with the ‘truth model’ reveal that the margins are quite close to the truth model. This indicates that the flight model of the LCA simulator derived from wind tunnel aerodynamic data and all other pertinent aircraft data is a very close representation to the actual aircraft. Thus the stability margins predicted prior to flight have been achieved in the actual flights of the LCA in the flight envelope tested. Based on the success of this study, it is proposed to use this method in future flight-testing of the LCA to estimate the gain and phase margins in near real time.
5. References [1]. T. Smith, “Ground and flight testing of digital flight control systems”, Chapter 6, Flight Control Systems, Edited by Roger W. Pratt, Progress in Astronautics and Aeronautics, Volume 184, Paul Zarchan, Editor-in-Chief, 2000. [2]. Mark B. Tischler, “System identification methods for aircraft flight control development and validation” Chapter 2, Advances in Aircraft Flight Control, Taylor and Francis, 1996. [3] Peter G. Hamel and Ravindra Jategaonkar, “Evolution of Flight Vehicle System Identification”,
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Journal of Aircraft, Vol. 33, No.1, January-February 1996, pp. 9-28. [4] Shyam Chetty, Girish Deodhare and B. B. Misra, “LCA Control Law Design Development and Validation”, Submitted for presentation to AIAA conference on Guidance, Navigation and Control, 2002. [5] Alan V. Oppenheim and Ronald W. Schafer, Digital Signal Processing, Prentice Hall, 1989. [6] McRuer, D. Ashkenas., and Graham, D., Aircraft Dynamics and Automatic Control, Princeton, N. J., 1973. [7] Robert Clarke, John J. Burken, John T. Bosworth, and Jeffery E. Bauer, X-29 Flight Control System: Lessons Learned, NASA Technical Memorandum 4598, June 1994
Appendix A :- Fourier Transform of 3-2-1-1 signal using FFT
Remark 1. If the period T is halved the frequency bandwidth doubles and the magnitude of the spectrum halves. Thus, even though the bandwidth increases, excitation level decreases as T reduces. Remark 2. With 3-2-1-1 input for T=1, at 1 Hz there is no excitation to the plant (which is approximately the gain cross over frequency). Fourier Transforms of Pilot pitch stick 3-2-1-1 input signals.
In order to ascertain the spectral quality of the inputs given by the pilots, the Pitch Stick 3-2-1-1 inputs given in flight and their Fourier Transforms were plotted along with Ideal 3-2-1-1 inputs for all the twelve segments listed in Table 1. It was noticed that there is a lot of deviation between Pilot input and Ideal 3-2-1-1 inputs. Moreover, Pitch Stick 3-2-1-1 input goes through Pitch stick path filter, hence gets modified further. The flight data segment F8_seg4 was found close to an ideal 3-2-1-1 pitch stick input.
Hence, this flight segment was used for initial analysis and to verify the results obtained from various methods. Subsequently the data from other flight conditions were analyzed.
