Lateral Flight Technical Error Estimation Model for Performance Based Navigation

Chinese Journal of Aeronautics 24 (2011) 329-336

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Lateral Flight Technical Error Estimation Model for Performance Based Navigation ZHAO Hongshenga, XU Xiaohaob, ZHANG Juna,*, ZHU Yanboa, YANG Chuansenc, HONG Shengd a

School of Electronics and Information Engineering, Beihang University, Beijing 100191, China b

c

College of Air Traffic Management, Civil Aviation University of China, Tianjin 300300, China

College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China d

School of Reliability and System Engineering, Beihang University, Beijing 100191, China Received 21 July 2010; revised 8 November 2010; accepted 7 December 2010

Abstract Flight technical error (FTE) combined with navigation system error (NSE) is the main part of total system error (TSE) in performance based navigation (PBN). The implementation of PBN requires pre-flight prediction and en-route short-term dynamical prediction of the TSE. Once the sum of predicted lateral FTE and NSE is greater than the specified PBN value, the PBN cannot operate. Thus, accurate modeling and thorough analysis of lateral FTE are indispensible. Multiple-input multiple-output (MIMO) lateral track control system of a transport aircraft is designed using linear quadratic Gaussian and loop transfer recovery (LQG/LTR) method, and the lateral FTE of a turbulence disturbed approach operation is analyzed. The error estimation mapping function of latera FTE and its bound estimation algorithm are proposed based on singular value theory. According to the forming mechanism of lateral FTE, the algorithm considers environmental turbulence fluctuation disturbance, aircraft dynamics and control system parameters. Real-data-based Monte-Carlo simulation validates the theoretical analysis of FTE. It also shows that FTE is mainly caused by turbulence fluctuation disturbance when automatic flight control system (AFCS) is engaged and would increase with escalating environmental turbulence intensity. Keywords: flight technical error; performance based navigation; LQG/LTR; air traffic management; Kalman filters

1. Introduction1 The high accuracy and global coverage of global navigation satellite system (GNSS) have enabled it as the basis of the next generation air traffic management *Corresponding author. Tel.: +86-10-82338282. E-mail address: [email protected] Foundation items: National High-tech Research and Development Program of China (2006AA12A103); National Basic Research Program of China (2010CB731803); Basic Scientific Research Fund of Central Institutions of Higher Education (ZXH2009D006, YWF-10-02-02) 1000-9361/$ - see front matter © 2011 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(11)60039-3

system, of which performance based navigation (PBN) is a fundamental component. Flight technical error (FTE), navigation system error (NSE) and path definition error (PDE) compose the total system error (TSE) of PBN[1]. Lateral FTE is the linear distance between estimated position and defined path. However, PDE is sufficiently small that it could be safely ignored even in accuracy-demanding approach phase of a flight. Thus, the reference position to lateral FTE is desired track/path actually. The safety and efficiency of life-critical flight operation are heavily dependent on the performance of navigation systems, such as accuracy, continuity, integrity, and availability. But in a

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generalized sense of PBN, the satisfying performance of automatic flight control system (AFCS) and pilot is also required. FTE embodies the limitation of AFCS or human-machine closed loop manual control system to track desired flight path and maintain intended altitude as well as target velocity with infinite accuracy. A mixed probability model of vertical FTE during approach is presented in Ref.[2]. The auto-correlation, cross-correlation of vertical and longitudinal FTE are discussed in Ref.[3]. In Ref.[4], vertical FTE of approach phase is measured with position sensors such as surface detection equipment, surface measurement radar etc., and statistical fitting is performed. The FTE in simulation and flight tests for small aircraft transportation system-high volume operation (SATS-HVO) of non-radar, non-tower airports is measured and fitted using probability models in Ref.[5]. The previous literature focuses on field measurement of FTE during approach or the statistical fitting to measured data, which does not reveal the forming mechanism of FTE. The method is also costly because it involves a number of flight tests. However, the successful implementation of PBN requires pre-flight prediction and en-route short-term prediction of FTE, and thus its accurate modeling and thorough analysis are indispensible. 2. System K Design Using Linear Quadratic Gaussian and Loop Transfer Recovery (LQG/LTR) Method 2.1. LQG/LTR method The prominent LQG/LTR theory, originally proposed by Doyle and Stein in Ref.[6], is a multiple-input multiple-output (MIMO) design method based on optimal control theory. It provides guaranteed robust stability and keeps performance robustness comparable to that of state feedback control by the two-step loop-shaping design procedure. The plant model in state-space form is shown in Eqs.(1)-(2), where x is state variable vector, y system output, u system input, A, B, C and * are appropriately dimensioned real constant matrices, w and v zero-mean Gaussian stochastic processes, of which W and V are the corresponding covariance matrices respectively. x Ax + Bu + īw (1) y = Cx  v

E ( ww ) W t 0

(3)

E (vv T ) V ! 0

(4)

T

lim E{³ [( Mx )T QMx  uT Ru]dt}

T of

Q

0

QT t 0 , R u

Kc x

where J is the linear quadratic performance index that minimizes the weighted energy of state variable vector and control vector, Q and R are weighting matrices, M is * T, the optimal state-feedback matrix Kc is given by R 1 B T Pc

Kc

(8)

where Pc satisfies the following algebraic Riccati equation: AT Pc  Pc A  Pc BR 1 B T Pc  M T QM

(9)

0

and Pc = PcT t0. The Kalman filter gain matrix Kf is given by Kf

Pf C TV 1

(10)

where Pf satisfies another algebraic Riccati equation that is dual to Eq.(9): Pf AT  APf  Pf C TV 1CPf  * W T

(11)

0

and Pf = PfT t0. The matrices Kc and Kf exist, and the closed-loop system is internally stable, provided that the systems with state-space realizations ( A, B, Q1/ 2 M ) and (A, * W1/2, C) are stabilizable and detectable. Namely, any uncontrollable or unobservable modes are asymptotically stable. Fig.1 shows the schematic diagram of the structure of LQG compensator interconnected with plant model. For the case of designing return ratio at the output of the plant, the first step of LQG/LTR is to design a Kalman filter by manipulating the covariance matrices W and V until a satisfactory return ratio C(sIA)1Kf is obtained. The second step is to synthesize an optimal state-feedback regulator aimed at recovering the return ratio over a sufficiently large range of frequencies. The regulator is subject to linear quadratic performance index. LTR is achieved by forcing the return ratio at marked Point 1 in Fig.1 to approach that at Point 2 by tuning weighting matrix. The return ratios at Point 1 and at Point 2 are shown in Eqs.(12)-(13) respectively. G ( s) K ( s)

 K c ( sI  A  BK c  K f C ) 1

K f C ( sI  A) B G (s) K ( s)

1

˜ (12)

1

 K c ( sI  A) B

(13)

(2)

T

J

No.3

RT ! 0

(5) (6) (7)

Fig.1 LQG compensator interconnected with plant model.

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2.2. Requirements of lateral track control system The requirements of lateral track control system align with that of lateral-axis autopilot system. The distinction between them is that the latter needs to repeat the design procedure of the former for a dozen of different flight conditions with various flap settings and true air speed[7]. The performance objectives of the compensated system are as follows: good damping and zero steady-state error in the face of step responses or disturbances, and a bandwidth of about 10 rad/s for each loop. Besides, the maximum bank angle during any maneuver shall not exceed 30q, and the lateral acceleration must not exceed 0.05g. 2.3. Lateral track control system design The control configuration is firstly presented. Then, the targeted control system is acquired by sequential design of competent Kalman filter and optimal state feedback regulator. (1) Control configuration design The dynamics of the transport aircraft is linearized during a nominal approach phase[8] to acquire nominal plant data. The air speed is 229.67 ft/s (1 ft/s= 0.304 8 m/s), which is the typical velocity for large jets during final approach phase. With Eqs.(1)-(2) as the state space model of lateral dynamics, the state variable vector, control vector and output vector are chosen as follows: x [E

I \

p r d Ga u [G a G r ]

y [\

Gr ]

(14) (15) (16)

d]

where Eis sideslip angle, I bank angle, \ heading angle, p roll rate, r yaw rate, d lateral displacement, Ga aileron deflection and Gr rudder deflection. Before we proceed to design Kalman filter, the nominal plant is augmented with integrators in both control channels at first to acquire well-damped response. Consequently, two integral variables Ha and Hr are inserted into the state vector: xau

[E

I \

p r

d Ga

Gr

Ha

H r ] (17)

where subscript “au” stands for “augmented”. To avoid difficulty in the recovery step, we choose a sufficiently small pole of the augmented model in the left hand half plane rather than at origin. The augmentation leads to an increase of 60 dB at 0.001 rad/s for the principal gains of the return ratio Cau(sIAau)1Kf. (2) Design of Kalman filter The principal gain shaping technique, which is based on the singular value decomposition of Cau(sIAau)1 * W1/2 at the frequency to be adjusted, is applied to tune open-loop principal gains to obtain 100 dB gain at 0.001 rad/s for V [Cau(sIAau)1Kf],

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and a band width of 10 rad/s. The latter is equivalent to rendering the cross-over frequency of the compensated system being 10 / 2 rad/s. For further information on this technique, refer to Refs.[9]-[11]. Sf ( s ) [ I  C au ( sI  A au ) 1 K f ]1 Tf ( s )

I  Sf ( s )

(18) (19)

The control over the principals of the sensitivity function is exercised to acquire similar behavior of both V (Sf (s)) and V (Sf (s)) (the minimum and maximum singular values of Sf (s)) as they approach 0 dB, with the purpose of reduction in the range of measurement noise being amplified and a more homogenous performance in all signal directions. This is achieved at the expense of a larger bandwidth of Tf (s), but with very little increase of ||Tf||f[12], hence there is hardly any deterioration of stability margins. Fig.2 presents the principal gains of both the sensitivity function and complementary sensitivity function corresponding to the final Kalman filter design.

Fig.2 Principal gains of sensitivity function and complementary sensitivity function.

(3) LQR design and loop transfer recovery The optimal state-feedback matrix Kc is obtained by solving the Riccati equation Eq.(9) with M = Cau, Q = I and R = UI, where U is the parameter to make the return ratio at Point 1 impend over that at Point 2 in Fig.1 as it approaches 0. Eventually, the principal gains of G(s)K(s) are superimposed onto those of Cau(sIAau)1Kf for U = 109, i.e., adequate recovery is exhibited. The time domain response of the closed-loop lateral track control system is shown in Fig.3, in which the input commands are unit step lateral displacement and zero heading angle command. The response of lateral displacement is reasonably damped and exhibits acceptable maximum overshoot (24%) and rise time (0.168 s). It settles to be within 10% of final value after 2.875 s and reaches final value within 2 s. Simultaneously, the heading angle output follows the zero reference pretty well although it takes a longer time for heading angle to reach its final value, for it is necessary to adjust delicate variations of lateral displacement by the corresponding change of heading angle.

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­ Lw h ° h ®L ° v (0.177  0.000 823h)1.2 ¯

Fig.3 Time domain response of lateral track control system to unit step lateral displacement and zero heading angle commands.

3. Turbulence Fluctuation Model

The Dryden turbulence model based on Taylor’s frozen field hypothesis[13] is exploited to account for the environmental turbulence disturbance. With the premise that the turbulence fluctuation is stationary, statistically Gussian distributed with zero mean and homogeneous, the power spectra density (see Fig.4) of lateral-axis Dryden model is as follows: Lv 1  3( Lv : ) 2 ˜ S [1  ( L v: ) 2 ]2

Z

:V

(20) (21)

where subscript “v” stands for the direction along y axis of aircraft, )v the power spectral density of lateral-axis Dryden turbulence, ft3/s2 (1 ft3/s2= 0.028 3 m3/s2); Vv standard deviation of lateral-axis Dryden turbulence, ft/s; Lv scale length for power spectra, ft; V the air speed of aircraft, and : spatial frequency. Eq.(22) holds for the preceding spectral density:

V v2

f

³ 0 )v (: )d:

(23)

­V w 0.1W20 ° (24) 1 ®Vv 0.4 °V ¯ w (0.177  0.000 823h) where W20 is the wind speed at 20 ft and h the flying altitude, subscript “w” stands for the vertical direction. For the altitude range above 2 000 ft, the scale lengths are specified as a constant, while the turbulence intensity is a function of altitude as well as probability of exceedance.

3.2. Forming filter

3.1. Definition of turbulence model

)v (: ) V v2.

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(22)

Fig.4 Power spectral density of lateral-axis Dryden turbulence fluctuation.

In Ref.[13], the scale lengths and the standard deviations are specified for two altitude ranges. When the altitude concerned is below 1 000 ft, the values abide by Eq.(23) and Eq.(24).

The Dryden turbulence[14] is generated by adding white Gaussian noise with specified standard deviation to a forming filter for the corresponding direction. The forming filters are shown in Eq.(25). Fv ( s )

1 1  Lv s

(25)

Eq.(26) prescribes the standard deviation for the corresponding axis:

V wn

V v (2 Lv / Dx)1/ 2

(26)

where sample step in Descartes frame Dx = V·Dt, in which Dt is sample time step. Dt decreases as air speed of aircraft builds up. 4. Lateral FTE Model 4.1. FTE analysis: propagation of covariance The statistical characteristic of FTE could be described as a normal distribution given that FTE is actually a stochastic process[2]. This perspective of FTE could be justified by the fact that it is influenced by a number of random variables with various weighting. If we inject the turbulence disturbance signal at input, it follows that its statistical characteristics will be transmitted to the output through closed-loop system. Thus it will be preferred and beneficial if we analyze the propagation of covariance in a more generalized sense, namely with the approach of vector gain. This perception exposes the formation mechanism of FTE, i.e., the injected turbulence fluctuation accounts for the environmental influence and the linearized plant state space model accounts for the aerodynamical characteristics of aircraft. 4.2. Singular value-based mapping function The power spectral density functions )uu(Z) and )yy(Z) of input vector u(t) and output vector y(t) are

defined as the Fourier transform of their auto-covari-

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auto-covariance as shown in Eqs.(27)-(28). With the vector gain approach, the propagation of covariance is assessed indeed in terms of the sum of the variances of the component signals for mathematical convenience. This would not incur confusion in the case herein, in that the contribution of auto-covariance of the signals other than white Gaussian noise in input vector is minute enough and is ignored.

)uu (Z )

{E[u(t )uT (t  W )]}

(27)

)yy (Z )

{E[ y (t ) y T (t  W )]}

(28)

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where the highest (lowest) gain direction of image hyperellipsoid is used to account for the gains of all the directions in Eq.(33) (Eq.(34)), “u.b.” and “l.b.” stand for “upper bound” and “lower bound” respectively. Note that the bound estimation equation pair involves only the largest and smallest singular values of Gcl(s) as shown in Fig.5 and power spectral density of input vector, therefore could be preferably utilized in practical operation.

where {·} is Fourier function. Gcl(s) is the closedloop transfer function matrix of the integrated system, and thus the following formula holds:

)yy (Z ) Gcl ( jZ ))uu (Z )GclT ( jZ )

(29)

The auto-covariance of the output vector could be obtained with E ( yT y)

1 f tr()yy (Z ))dZ 2S ³ f

(30) Fig.5 Principal gains of disturbed closed-loop compensated lateral track control system.

Besides, according to Ref.[15], we obtain E ( y T y)

1 f tr()yy (Z ))dZ 2S ³ f

1 f tr(Gcl ( jZ ))uu (Z )GclT ( jZ ))dZ 2S ³ f 1 f ¦V i2 ()uu1/ 2 (Z )Gcl ( jZ ))dZ 2S ³ f i

(31)

given that Gcl(s) is stable. In Eq.(31), Vi(·) is singular value. Eq.(31) gives the exact magnitude of auto-covariance of the output vector, i.e. the lateral FTE, and this mapping function is the basis of the bound estimation model of lateral FTE which will be presented in Section 4.3. However, as we can see, it requires every singular value of all the channels. This demands more resource and is also inconvenient in practical use. 4.3. Bound estimation model of lateral FTE PBN implementation can bring more accuracy, but the safety of life-critical aviation is always the primary goal to fulfill. For safety, we need to be conservative enough. Thus, we consider the worst case to define the lateral FTE bound. If Gcl(s) is stable, then the following equation holds: tr()yy (Z )) V 2 (Gcl ( jZ )) d d V 2 (Gcl ( jZ )) (32) tr()uu (Z )) (for proof of Eq.(32), refer to Appendix 2 of Ref.[15]) hence, 1 f 2 E ( y T y) d V (Gcl ( jZ ))tr()uu (Z ))dZ u.b. 2S ³ f (33) 1 f 2 T E ( y y) t V (Gcl ( jZ ))tr()uu (Z ))dZ l.b. 2S ³ f (34)

In most cases, the input signals are statistically independent, thus the spectral density matrix in Eq.(27) degenerates to a diagonal matrix, and we could consequently write Eq.(33) more straightforwardly as Eq.(35), where the subscript “ui” stands for the ith input signal of the input vector. u .b. u .b. |

1 f 2 V (Gcl ( jZ ))¦)uiui (Z )dZ 2S ³ f i

(35)

1 {sup[V (Gcl ( s))]}2 ¦ ³ )uiui (Z )dZ (36) 2S ZBd i ZB d

Because the spectral density functions of turbulence disturbances concentrate the energy within a bandlimited region, the bound estimation model is streamlined as Eq.(36). Bd is the frequency range, within which the power spectral density of dominant disturbance signal concentrates. Generally, Bd should be 2 to 3 times broader than band width in order to be conservative enough. The band width of )u u (Z ) is defined i i

as the frequency range that )u u (Z ) drops down from its i i

peak value

sup ()ui ui (Z )) to 0.707 sup ()ui ui (Z )).

Z( f , f )

Z ( f , f )

The magnitude of auto-covariance of lateral FTE in output vector dominates (10 times larger than that of other signals), consequently E(yT y) in Eq.(36) is indeed the variance of lateral FTE. For the same reason, the variance of white Gaussian noise is considered as the covariance of the input vector. 1 f 2 V (Gcl ( jZ ))¦)uiui (Z )dZ 2S ³ f i

(37)

1 {sup[V (Gcl ( s ))]}2 ¦ ³ )uiui (Z )dZ 2S ZBd i ZB

(38)

l.b. l.b. |

d

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Dually, the lower bound estimation equation is developed in Eqs.(37)-(38). 4.4. Calculation of lateral FTE bound The proposed algorithm, which is based on the aforementioned work, of calculating lateral FTE bound is outlined briefly as (1) Plot the principal gains against frequency of closed loop system. (2) Plot the power spectral density of lateral-axis Dryden turbulence fluctuation. (3) Determine the band width of power spectral density, then define Bd as 2 to 3 times of the bandwidth. (4) Determine the maximum values of the principal gains on the frequency range of Bd. (5) If the wind speed at 20 ft could be acquired, determine the turbulence intensity category. Then calculate the upper bound and lower bound of the lateral FTE variance with Eq.(36) and Eq.(37) respectively. (6) If the wind speed at 20 ft could not be acquired, calculate the upper bound and lower bound of the lateral FTE variance with Eq.(36) and Eq.(38) respectively for different turbulence intensities (light, moderate and severe). (7) Calculate the mean value of the three variance values of different turbulence intensities according to different probabilities[13]. The upper bound values of standard deviation for different turbulence intensities are acquired by Eq.(36) and are shown in Table 1. Bd equals [í5, 5]. Also note that the functions plotted in Fig.4 are even[16]. Table 1

Upper bound of standard deviation of lateral FTE Vmax and probability values for light, moderate and severe turbulence

Parameter

Light

Moderate

Severe

Probability

9.180 0u101

8.200 0u10

1.361 5u10

V max /ft

8.132 1

16.264 0

24.396 0

The expectation of upper bound of standard deviation of lateral FTE during final approach is obtained by E (V max )

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is integrated into the compensated lateral flight control system. Thus the integrated system is driven by zero mean white Gaussian noise. In the integrated system, the state variables of the state-space realization of the forming filter are added to that of the state equation of the compensated system. Consistently, the corresponding output sub-matrix of the state variables inserted has zero elements in order to exclude the effect of turbulence fluctuation other than that added to side-slip angle. The parameters of lateral track control system and environmental turbulence disturbance used in simulation are as follows: flight altitude is 900 ft, W20 is 15 knots for typical light turbulence, and air speed V is 229.67 ft/s for typical approach velocity of transport jets. For the practical estimation of lateral FTE, the cases of moderate as well as severe turbulence are necessary. Correspondingly, W20 is 30 knots and 45 knots respectively. Monte-Carlo simulation is implemented to validate the theoretical algorithm. The simulation operates 50 times for each kind of turbulence intensity (150 runs in total), and each run lasts for 10 s. The sample population has 100 points sampled at 0.1 s time step from each run. Then the statistical analysis including parameter evaluation and 95% confidence level analysis is performed for each category corresponding to light, moderate and severe turbulence. Thereupon, mean values of strongly consistent estimation of standard deviation of the simulated lateral FTE due to each kind of turbulence with different intensities are listed in Table 2 to validate those calculated with the theoretical model. Note that the upper bound is comparatively larger than the simulated value, in that it is calculated based on the worst case. Furthermore, the magnitude of each run is within the range bounded by 3V(i.e. abides by the 3-sgima principle) and centered with mean value. This could be observed from Fig.6. Table 2 Mean values of strongly consistent estimation of standard deviation of lateral FTE due to light, moderate and severe turbulence Parameter

Light

Moderate

Severe

Vmax/ft

6.500 4

12.025 9

14.617 0

Pl ˜V l (d )  Pm ˜V m (d )  Ps ˜V s (d ) 8.799 3 (39)

where Pl, Pm and Ps are the probability values corresponding to light, moderate and severe turbulence, Vl(d), Vm(d) and Vs(d) are the upper bounds of standard deviation of FTE for light, moderate and severe turbulence. 5. Simulation and Discussion 5.1. Simulation configuration and results The forming filter of lateral-axis Dryden turbulence

Fig.6

Time history of lateral displacement and heading angle of light, moderate and severe lateral-axis Dryden turbulence disturbed lateral track control system.

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Fig.6 shows the simulation results of different turbulence intensities in order to visualize the escalating effect of the standard deviation of lateral FTE due to stronger turbulence intensity. It could also be noted in Fig.6 that the magnitude of standard deviation of lateral displacement (lateral FTE) is 10 times larger than that of heading angle. In other words, it is dominant within the output signal vector, which justifies the vector gain approach in analysis of covariance propagation in Section 4. 5.2. Discussion One of the most important requirements in life-critical aviation industry is safety, and with its predominant priority the worst case is always studied and has a fundamental effect on the establishment of regulations. Consistently, the analysis and anticipation of lateral FTE are conducted by this principle. The results visualized in Fig.6 are three realizations with different kinds of turbulence intensities of the FTE. However, the estimated FTE covariance value with the bound estimation algorithm Eqs.(36)-(38) would be generally larger than certain simulated results, in that the algorithm considers the maximum possible lateral FTE covariance within the universal set of all realizations. As such, we believe that the proposed model and algorithm are viable and sound. It is almost impossible to acquire the measured data of FTE for the identical aircraft type we used in this paper, partially due to the confidentiality of AFCS parameters and other data. Another reason is that the aircraft type is not mentioned in the original reference. However, there is an error budget involving FTE available from International Civil Aviation Orgnization (ICAO) Annex 10 to the Convention on International Civil Aviation[17]. The operation scenario is specified in Ref.[17], which is 3q nominal glide slope, microwave landing system, final approach, 200 ft above touch-down zone, then the budgeted lateral error with the exclusion of FTE is 50 ft. Besides, according to Ref.[1], it is suggested that required navigation performance (RNP) 0.1 should be implemented with the aforementioned operation conditions. Consequently, the budgeted lateral FTE is 554 ft, and this could be taken as the statistical maximum. This verifies again that FTE is the main part of TSE and thus it is substantial in PBN operation. Furthermore, the FTE of Boeing 737 could be found in Ref.[18]. To further examine the perspective of the analysis used in this paper, the case that none of the magnitude of covariance of a certain signal in output vector dominates would be discussed briefly. In such a case, it is necessary to distinguish the errors of each output signal of output vector. Thus, we shall acquire the necessary transfer functions of the transfer matrix of integrated system, and the treatment would be modified compared to that in Section 4 in the following way: the upper bound of maximum singular value of

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system transfer matrix would become upper bound of frequency domain gain of single-input single-output (SISO) transfer function, because the transfer matrix of system would degenerate into SISO transfer functions for certain concerned input and output variable pair. Finally, it is quite worthwhile to point out that FTE does not only affect navigation accuracy and its decision threshold in PBN, but also has an influence on the following subjects: airspace efficiency, avionics system certification error budget[19], regulation for certification of airworthiness[20], carrier-based aircraft landing precision, and evaluation of the cockpit vision awareness system[21], etc. Thus, the research on FTE could potentially be referred to whenever the above topics may involve, and that is also what the authors expect. 6. Conclusions (1) The main contributor of lateral FTE of AFCS engaged aircraft is environmental turbulence, and lateral FTE would increase with escalating environmental turbulence intensity. (2) The error estimation mapping function of lateral FTE is proposed based on singular value theory. (3) The bound estimation algorithm of lateral FTE is proposed based on system principal gains and power spectral density of input signal vector. (4) The real-data simulation result justifies the algorithm and the vector approach in analysis of covariance propagation. (5) The bound estimation algorithm proposed in this paper could be utilized to estimate or predict the flight technical error estimation due to atmospheric turbulence in PBN. References [1] [2] [3] [4] [5]

[6] [7] [8]

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Biographies: ZHAO Hongsheng Born in 1982, he received B.S. and M.S. degrees from Civil Aviation University of China in 2003 and 2008 respectively, and he is a Ph.D. candidate in Beihang University now. His main research interest is comunication navigation surveillance/air traffic management (CNS/ATM). E-mail: [email protected] XU Xiaohao

Born in1949, he received M.S. and Ph.D.

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degrees in flight mechanics from Harbin Institute of Technology, Harbin, China in 1986 and 1989 respectively. He is now the Director of Air Traffic Management Research Base of Civil Aviation of China and Senior Expert of Air Traffic Committee of China. His main research interest is CNS/ATM. He has published more than 100 papers for journals as well as meetings. He was also former deputy president of Civil Aviation University of China. E-mail: [email protected] ZHANG Jun Born in 1965, he received M.S. and Ph.D. degrees in electronics and information engineering from Beihang University (BUAA), Beijing, China in 1990 and 2003 respectively. He is now deputy president of BUAA and senior expert of Air Traffic Committee of China. His main research interest is CNS/ATM. He has published more than 60 papers for journals as well as meetings. He has won the Top National Invention Award of China in 2009. E-mail: [email protected] ZHU Yanbo Born in 1970, he received B.S. in engineering of electromagnetic field & microwave technology, M.S. in engineering of communication & electronic systems and Ph.D. degrees in information & communication engineering from Beihang University, Beijing, China in 1992, 2003 and 2008 respectively. He is now the chief engineer of CNS/ATM Key Laboratory of Civil Aviation of China, Deputy General Manager & Chief Engineer of Aviation Data Communication Corporation of China, and also an expert of Air Traffic Committee of China. E-mail: [email protected] YANG Chuansen Born in 1968, he received M.S. degree in electronic & mechanical engineering from Hohai University of China, Nanjing, China in 2003. He is now a Ph.D. candidate in Nanjing University of Aeronautics and Astronautics. His main research interest is global positioning system (GPS) navigation. E-mail: [email protected] HONG Sheng Born in 1981, he received B.S. degree in electronic and information engineering from Harbin Engineering University, Harbin, China in 2003, and Ph.D. degree in communication and information system from Beihang University, Beijing, China in 2009, respectively. He is a graduate student advisor in the same university at the Department of System Engineering of Engineering Technology. His research interests include wireless communication technology, control theory and the design and test of very large scale integrated circuit (VLSI). He is an regular member of the Institute of Electronics, Information and Communication Engineers (IEICE). E-mail: [email protected]