Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 103 (2017) 67 – 74
XIIth International Symposium «Intelligent Systems», INTELS’16, 5-7 October 2016, Moscow, Russia
Real-time aerodynamic parameter identification for the purpose of aircraft intelligent technical state monitoring O.N. Korsuna *, M.H. Omb, K.Z. Lattb, A.V. Stulovskiia b
a State Institute of Aviation Systems, Moscow, 7 Victorenko Street, 125319, Russia Moscow Aviation Institute (National Research University), Moscow, 7 Volokolamsk Road, 125993, Russia
Abstract The report deals with the problem of design the new generation of aircraft technical state monitoring systems using methods of artificial intellect. The key point in the intellectual support is the theory of system identification, applied to the proble m of aerodynamic parameter estimation from the flight data. The report considers the specific aspects of the parameter identification problem as a part of real-time intelligent monitoring system. The aircraft model with the object noise and the identification algorithm with estimation of identification accuracy are formulated. The presented algorithms are tested through processing the data generated using a modern aircraft simulation facility. © 2017 2017The TheAuthors. Authors.Published Published Elsevier © by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium «Intelligent Systems». Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” Keywords: technical state; system identification; aerodynamic parameter estimation.
Identification of aerodynamic coefficients can be an effective method of estimating the technical state of the aircraft. In this paper the aircraft movement model is formulated which includes also the atmospheric turbulence, flight parameter sensors and the aircraft control system. The identification of the aircraft aerodynamic parameters using the continuous-discrete extended Kalman filter is also discussed. For the simulation of atmospheric turbulence, it is recommended to use, for example, the Dryden model, according to which the spectral density of the turbulent wind in the vertical, longitudinal and lateral directions is given by:
*Corresponding author E-mail address:
[email protected]
1877-0509 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” doi:10.1016/j.procs.2017.01.014
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2 L VW W SV §
SW (Z )
2
§L · 1 3¨ W Z ¸ © V ¹ 2·
,
2
§L · ¨1 ¨ W Z ¸ ¸ ¨ © V ¹ ¸ © ¹
2 L VU U SV
SU (Z )
1
§L · 1 ¨ X Z¸ ©V ¹
2 L VV V SV §
SV (Z )
2
(1)
,
2
§L · 1 3¨ V Z ¸ © V ¹ 2·
2
,
§L · ¨1 ¨ V Z ¸ ¸ ¨ © V ¹ ¸ © ¹
where V - aircraft velocity, m/s; Z - angular frequency, 1/s; V W , V U , V V - std. deviations of wind gust velocities, m/s; LW , LU , LV - turbulence scales, m; The turbulence scales at the flight altitude H ! 525m LW
LW
H , LU
LV
43, 5 H
V W2
V U2
V V2
LW
LU
LV
1
3
LU
LV
525m . At altitude H 525m
In addition, we have the relation:
(2)
To estimate the influence of turbulence on the closed-loop contour the models of aircraft movement and aircraft control system must be coupled with the Dryden turbulence model. In this case, the linearized model is acceptable, for example, in the vicinity of a straight and horizontal flight. Let us consider the longitudinal motion of the aircraft equipped with the fly-by-wire system in turbulence. The simplified model of the fly-by-wire system can be approximated by elements providing feedback signals of overload and angular velocity. The appropriate transfer functions are as follows
Wn ( p )
Kn Tn p 1
, WZ ( p )
KZ
Т2 p 1 T1 p 1
.
In this case, the deviation of stabilizer is given by:
M B (t ) M P (t ) Mn (t ) MZ (t )
(3)
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O.N. Korsun et al. / Procedia Computer Science 103 (2017) 67 – 74
Where, M P (t ) - deviations, due to the actions of the pilot. After transforming the transfer functions into the model in the state space, we obtain:
M n' (t ) ec(t )
K 1 M n (t ) n n y (t ), Tn Tn K 1 e (t ) Z Z z (t ), Tn T1
§
M Z (t ) ¨ 1 ©
T2 · T2 ¸ e (t ) K Z Z z (t ) T1 ¹ T1
Equations which form filter for the vertical component of turbulent gusts of wind are the following
u1c (t )
u 2 (t )
u2c (t )
P 2u1 (t ) 2 P u2 (t ) XW (t )
Let us combine the above equations with the model of short-period longitudinal motion given in deviations relative to steady motion. For accounting the dynamic components of measurement errors, let us additionally introduce models of the angular velocity and overload sensors as a second order differential equation. The sensor models are also used as forming filters for measurement noises. As a result, we obtain the following system of equations.
D c(t ) Z z (t ) cDy
qS mV
§ 1 ¨ D (t ) V ©
P qS § ·· u1 (t ) ¸ ¸ c My M B (t ), ¨ u 2 (t ) mV 3 © ¹¹
qSbA § m Zz z qSbA bA 1§ P ·· M qSbA Z z (t ), Z (t ) m M B (t ) u1 (t ) ¸ ¸ m z ¨ a (t ) ¨ u 2 (t ) Jz © V© Jz Jz V 3 ¹¹ u1c (t ) u 2 (t ) ' z
a z
u2c (t )
P 2 u1 (t ) 2 P u 2 (t ) XW (t ),
Z zc изм (t ) Z1 (t ), a0Z Z z изм (t ) a1Z Z1 (t ) a0Z Z z (t ) XZ (t ),
Z1c(t ) ncy изм (t )
n1 (t ),
n1c (t )
a0n n y изм (t ) a1n n1 (t ) a0n n y (t ) X n (t ),
M n' (t )
K 1 M n (t ) n n y изм (t ) Tn Tn K 1 e (t ) Z Z z изм (t ) T1 T1
ec(t )
where V - aircraft velocity, m/s; q - dynamic pressure, Pa; m - mass, kg; S - equivalent wing surface, m2; bA - mean aerodynamic chord, m;
(4)
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O.N. Korsun et al. / Procedia Computer Science 103 (2017) 67 – 74
Jz - moment of inertia, kg m ; 2
P
M / LW - turbulence spatial frequency, 1/s;
Z1 (t ), n1 (t ), e(t ) - supplementary variables; a0Z
ZZ2 , a1Z
2[Z ZZ , a0n
Z n2 , a1n
2[ n Z n - parameters of the angular velocity and overload
sensors,
[Z , [ n - damping coefficients of sensors; Z z изм (t ), n y изм (t ) - sensor output signals;
XW (t ), XZ (t ), Xn (t ) - normal random processes of the white noise type with zero means and intensities. 2 a0Z a1ZV W2 , S n
SW
3PV W2 , SZ
2 a0n a1nV n2
where
V W - std. deviation velocity of the vertical component turbulence m/s;
V Z , V n - std. deviation of random errors of the angular velocity sensor (1/s) and of the overload sensor (overload units); n y (t ) - the deviation of the normal overload relative to steady-state path, defined by the formula
n y (t )
cDy
qS mV
§ 1 ¨ D (t ) V ©
P qS § ·· V § · V u1 (t ) ¸ ¸ u ¨ c Iy I B (t ) ¸ u . ¨ u 2 (t ) 3 ¹ g © ¹ ¹ g © mV
In the differential equations system (4) the stabilizer deviations are calculated through the formula (3), which feed-back component MZ (t ) has the form:
§
M Z (t ) ¨ 1 ©
T2 · T2 ¸ e(t ) K Z Z z изм (t ) T1 ¹ T1
In equation (4), it is also assumed that the angle of attack, caused by the influence of turbulence is given by the expression:
D W (t ) |
' (t ) , where ' (t ) V
u 2 (t )
P
3
u1 (t ) - velocity of vertical gusts of turbulent wind, m/s.
It is assumed that the correction of systematic measurement errors of angle of attack is carried out according to the method [4,5]. Simulation of stochastic systems (4) are carried out as follows. This system is a case of the following systems:
yc(t )
f ( y (t ), a, u (t )) Z (t )
When performing the simulation it’s necessary to solve the following equations at each discretization interval
t > t k , t k 1 @
dy (t t k ) dt
f ( y (t t k ), a , u (t )),
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dF1 (t , tk )
A11 (t , t k ) F1 (t , t k ), where,
dt
df ( y (t t k ), a (t ), u (t )
A11 (t , t k )
dW (t , tk ) dt
A11 (t , tk ) W (t , tk ) W (t , t k ) A11T (t , t k ) SZ (t ) .
SZ (t )
Matrix
(5)
dy (t t k )
of
the
intensities
of
object
noise
for
the
system
(4)
has
the
form
diag (0 , 0 , 0 , 3PV W2 , 0, 2 a0Z a1ZV W2 , 0, 2 a0Z a1nV n2 , 0, 0) After calculating the prediction estimate of the state vector y (t k 1 t k ) and the correlation matrix of the equivalent discrete noise W (t k 1 t k ) the value of state vector is calculated according to the formula S Z (t )
SZ
y (t k 1 t k ) GkK k
y (t k 1 )
(6)
where K k - independent sequence of n-dimensional normal vectors with zero mean and unit correlation matrix K k N (0, E )
G k - triangular matrix corresponding to equation
W (t k 1 t k )
Gk G kT
(7)
For identification algorithm the continuous-discrete extended Kalman filter is chosen because it takes into account the object noise. Model of object can be formulated as;
yc(t )
a
f ( y (t ), a, u (t ) ) Z (t )
(8)
0
It’s obvious that in equation (8) the assumption of constancy of the identification parameters is introduced. Discrete model of the observations is given by the equation:
z (ti )
h( y (ti ), a, u (ti ) ) K (ti )
(9)
Prior to identification algorithm, it is necessary to set the initial values of the generalized state vector > y (t0 ) a0 @ and the corresponding correlation matrix V (t0 ) . To find the estimates xˆ (t0 ) the following computing operations are performed at each discretization interval t > tk , tk 1 @ , k 0, N 1 determination of prediction estimates of the state vector and parameters
x T (t 0 )
dy (t tk ) dt
a (t k 1 t k )
f ( y (t t k ) , aˆ (t k ) , u (t ))
aˆ (t k )
with initial condition y (t k t k )
(10)
yˆ (t k ) ;
the calculation of the transition matrix F1 (tk 1 , tk ) of dimension n u (n p ) through the solution of the equation
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O.N. Korsun et al. / Procedia Computer Science 103 (2017) 67 – 74
dF1 (t , t k ) dt
A11 (t , t k ) F1 (t , t k ) ª¬0 ( nun ) | A12 (t , t k ) @ ,
with initial condition F1 (t k , t k )
(11)
ª¬ E( nu n ) | 0 ( nu p ) º¼ ,
where,
A11 (t , t k ) ( nu n )
A12 (t , t k ) ( nu p )
w f ( y (t t k ), aˆ (t k ), u (t )) w y (t t k )
,
w f ( y (t t k ), aˆ (t k ), u (t )) w aˆ (t k )
– determination of the correlation matrix W (t k 1 , t k ) of
equivalent discrete noise of object by solving the equation
dW (t , tk ) dt
A11 (t , tk ) W (t , t k ) W (t , t k ) A11T (t , t k ) SZ (t )
(12)
with initial condition W (t k 1 , t k ) 0 ( nun ) ; determination of the prediction error correlation matrix
Fk Vk FkT Wk
Pk 1
(13)
where,
Fk
ª F1 (t k 1 , t k ) « 0 ¬
K k 1
0º , Wk E »¼
ªW (t k 1 , t k ) « 0 ( pu n ) ¬
0 ( nu p ) º - filter gain calculation: 0 ( pu p ) »¼
Pk 1 H kT1 ( H k 1 Pk 1 H kT1 Rk 1 ) 1
where H k 1
w h ( x (t k 1 t k ), u (t k 1 )) w x (t k 1 t k )
(14)
- matrix of observations of dimension r u (n p) ; calculation of the
correlation matrix of the error of current estimates of the state and parameters
Vk 1
( K k 1 H k 1 E ) Pk 1 ( K k 1 H k 1 E )T K k 1 Rk 1 K kT1
(15)
determination of the current estimation of the state vector and the identifiable parameters
xˆ (t k 1 )
x (t k 1 t k ) K k 1 ( z (t k 1 ) h ( x (t k 1 t k , u (t k 1 )))
(16)
The processing of the simulation data confirmed the stability of identification algorithm, especially in turbulent conditions. Let us consider the classical problem of parametric identification - identification of aerodynamic coefficients. Assume, that the lateral roll moment is defined by a linear model:
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O.N. Korsun et al. / Procedia Computer Science 103 (2017) 67 – 74
mx
mxE E mZx x
L L Z Z x mx y Z y mGx FLG FL mGx Н G Н , V V
Where E - slip angle, Z x , Z y - angular velocities, G FL - Differential deflection of flaperons G Н - Deflection of the rudder. To obtain estimates of the parameters Kalman filter which has been discussed above, multiple linear regression, maximum likelihood method [1,4] or frequency domain maximum likelihood method [6] can be used. Since the object is generally a non-stationary, the identification should be performed on a sliding interval. We investigate the relation of the accuracy of estimates and the length of the interval. The estimates correctness can be checked by comparing them with the corresponding values obtained by the processing the aircraft wind tunnel aerodynamic characteristics. In this case we assume that the match of the estimates obtained from two different sources, shows their correctness. Let us examine relation between the aerodynamic coefficients estimates and length of the sliding interval according to the flight experiment. 0.005 0.004 0.003 0.002 mx(Dn)(bank) 165.75
156.00
146.25
136.50
126.75
117.00
97.50
107.25
87.75
78.00
68.25
58.50
48.75
39.00
-0.002
29.25
9.75
-0.001
19.50
0 0.00
dmx
0.001
(mxDn)*Dn
-0.003 -0.004 -0.005 t, s Fig. 1. Relation of time paths of the roll moment estimates due to deflection of the rudder, obtained at 9 s sliding interval (purple line), and from the bank of aerodynamic characteristics (blue line).
0.005 0.004 0.003 0.002 mx(Dn)(bank)
165.75
156.00
146.25
136.50
126.75
117.00
107.25
97.50
87.75
78.00
68.25
58.50
48.75
39.00
29.25
-0.002
19.50
-0.001
9.75
0
0.00
dmx
0.001
(mxDn)*Dn
-0.003 -0.004 -0.005 t, s
Fig. 2 Relation of time e paths of the roll moment estimates due to deflection of the rudder , obtained at 20 s sliding interval (purple line), and from the bank of aerodynamic characteristics (blue line).
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O.N. Korsun et al. / Procedia Computer Science 103 (2017) 67 – 74
0.004 0.003 0.002
0
165.75
156.00
146.25
136.50
126.75
117.00
97.50
107.25
87.75
78.00
68.25
58.50
48.75
39.00
9.75
29.25
-0.002
mx(Dn)(bank)
19.50
-0.001
0.00
dmx
0.001
(mxDn)*Dn
-0.003 -0.004 -0.005 t, s
Fig. 3. Relation of time paths of the roll moment estimates due to deflection of the rudder, obtained at the interval length equal to the whole area (168 s), (the purple line) and from the bank of aerodynamic characteristics (blue line).
We can see in Figures 1 - 3, the length of the selected interval has a significant effect in evaluating the signal. When the length of the interval 9 seconds, we can see a good enough match to signal peaks, but significant noises generated by the instability of estimates are noticeable. By increasing the length of the moving interval to 20 seconds significant noises are decreased and the quality of the desired signal is high enough. By increasing the length of the interval to 168 seconds (the entire area) noises are almost completely smoothed out, but there are mismatches in the useful signal. What indicates the presence of nonstationarity. Thus, it is possible to choose the optimal length of the interval, which ensures low noise level and good accounting of unsteadiness on the treated area. In this example, the optimal length of the sliding interval is 20 seconds. Presented example also shows a good match of available bank of aerodynamic characteristics with flight data. This work was supported by the Russian Foundation for Basic Research (RFBR), 14-08-01109-a project. References 1. Klein V, Morelli EA. Aircraft system identification: theory and practice. Reston: American Institute of Aeronautics and Astronautics, 2006. 499 р. 2. Jategaonkar RV. Flight vehicle system identification: A time domain methodology. Reston: American Institute of Aeronautics and Astronautics, 2006. 410 р. 3. Korsun ON. Poplavsky BK. Approaches for flight tests aircraft parameter identification. Proc. of the 29 Congress of International Council of the Aeronautical Sciencies. Saint Petersbourg. 2014. Paper № 2014-0210. 4. Korsun ON, Poplavskii BK. Estimation of systematic errors of onboard measurement of angle of attack and sliding angle based on integration of data of satellite navigation system and identification of wind velocity. Journal of Computer and Systems Sciences International. 2011. Т. 50. № 1. p. 130-143. 5. Korsun ON, Nikolaev SV, Pushkov SG. An algorithm for estimating systematic measurement errors for air velocity, angle of attack, and sliding angle in flight testing. Journal of Computer and Systems Sciences International. 2016-55 (3), pp. 446-457. 6. Кorsun ON. An identification algorithm for dynamic systems with a functional in the frequency domain. Automation and Remote Control. 2003- 64 (5), pp. 772-781.
