Aerodynamic parameter estimation using adaptive unscented Kalman filter M. Majeed and Indra Narayan Kar Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India Abstract Purpose – The purpose of this paper is to estimate aerodynamic parameters accurately from flight data in the presence of unknown noise characteristics. Design/methodology/approach – The introduced adaptive filter scheme is composed of two parallel UKFs. At every time-step, the master UKF estimates the states and parameters using the noise covariance obtained by the slave UKF, while the slave UKF estimates the noise covariance using the innovations generated by the master UKF. This real time innovation-based adaptive unscented Kalman filter (UKF) is used to estimate aerodynamic parameters of aircraft in uncertain environment where noise characteristics are drastically changing. Findings – The investigations are initially made on simulated flight data with moderate to high level of process noise and it is shown that all the aerodynamic parameter estimates are accurate. Results are analyzed based on Monte Carlo simulation with 4000 realizations. The efficacy of adaptive UKF in comparison with the other standard Kalman filters on the estimation of accurate flight stability and control derivatives from flight test data in the presence of noise, are also evaluated. It is found that adaptive UKF successfully attains better aerodynamic parameter estimation under the same condition of process noise intensity changes. Research limitations/implications – The presence of process noise complicates parameter estimation severely. Since the non-measurable process noise makes the system stochastic, consequently, it requires a suitable state estimator to propagate the states for online estimation of aircraft aerodynamic parameters from flight data. Originality/value – This is the first paper highlighting the process noise intensity change on real time estimation of flight stability and control parameters using adaptive unscented Kalman filter. Keywords Aircraft, Aerodynamics, Noise, Stability (control theory), Adaptive filter, Parameter estimation, Unscented Kalman filter, Aerodynamic parameters Paper type Research paper
u Q a1 ; b1 ; k sT n h
Nomenclature CDð · Þ ; CLð · Þ ; Cmð · Þ Fe Iy m P q Q R S u V w x z c q
¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼
non-dimensional derivatives thrust (N) inertia about y-axis (kg m2) mass of the aircraft (kg) state error covariance pitch rate (radian/s) state noise covariance measurement noise covariance surface area of aircraft (m2) input vector true airspeed (m/s2) state noise state vector observation vector wing chord (m) dynamic pressure (pascal)
pitch angle (radian) unknown parameters scaling parameters of UKF engine inclination (radian) additive measurement noise diagonal elements of noise covariance
Subscripts 0 m k
¼ initial value ¼ measured quantity ¼ index
Superscripts a c m
¼ augmented state vector ¼ computation of covariance ¼ computation of mean
Introduction
Greek symbols a b
¼ ¼ ¼ ¼ ¼ ¼
Recent advances in computational power have allowed the use of online aerodynamic parameter estimation techniques in varied applications such as reconfigurable or adaptive flight control, and system health monitoring. Online modelling is a key technology for autonomous controller to maintain flight stability and high performance in uncertain environment and in the presence of process and measurement noise. Robust and adaptive control methods suffer from issues related to the real time convergence, and the complications involved in their real-time implementation (Wittenmark and Astrom, 1984). These problems necessitate the development
¼ angle of attack (radian) ¼ aerodynamic parameters
The current issue and full text archive of this journal is available at www.emeraldinsight.com/1748-8842.htm
Aircraft Engineering and Aerospace Technology: An International Journal 85/4 (2013) 267– 279 q Emerald Group Publishing Limited [ISSN 1748-8842] [DOI 10.1108/AEAT-Mar-2011-0038]
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Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
of a new online estimation algorithm for the purpose of controller design that addresses the situation more directly. To this end, autonomous control methods on the basis of model-reference have become the focus of research, and basic technology and online modelling method has attracted more and more research attention. Neural networks (NN) and NN-based self-learning were proposed as the most effective approaches for the online modeling of an unmanned vehicle as it does not require initial values (Pesonen et al., 2004; Pedro and Kantue, 2011). However, the problems involved in NN, such as training data selection, online convergence, robustness, reliability, and real time implementation, limit its application in real systems. In recent years, sequential estimation has become an important approach for online modelling and model-reference control with encouraging achievements (Haykin and deFreitas, 2004). Online or recursive system identification techniques handle flight data as it is measured through onboard sensors and estimate the required aerodynamic derivatives in real-time. Measured flight data can contain considerable amount of noise, furthermore there might be biases and unobserved states in the system model which must be estimated; hence filtering techniques are generally employed. The most popular nonlinear filtering technique is the extended Kalman filter (EKF) (Grewal and Andrews, 1993). Although widely used, EKFs have some deficiencies, including the requirement of differentiability of the state dynamics as well as susceptibility to bias, hard to tune and implement when dealing with significant nonlinearities and exhibits divergence in estimates. On the contrary, unscented Kalman filter (UKF) uses the nonlinear model directly instead of linearizing it (Julier and Uhlmann, 2004). The UKF was developed with the underlying assumption that approximating a Gaussian distribution is easier than approximating a nonlinear transformation (Julier and Uhlmann, 2004). The UKF uses deterministic sampling to approximate the state distribution as a Gaussian random variable (GRV). The UKF has the same level of computational complexity as that of EKF, both of which are within the order O (L3) (Julier and Uhlmann, 2004). Since the nonlinear models are used without linearization, the UKF does not need to calculate Jacobians or Hessians, and can achieve second-order accuracy, whereas the accuracy of the EKF is of the first order (Xiong et al., 2006). However, the performance of UKF closely depends on the prior knowledge of noise covariance. Recent study shows the use of UKF has possible advantages for aerodynamic parameter estimation from flight data (Chowdhary and Jategaonkar, 2009). The UKF can only achieve a good performance under the assumption that such prior knowledge of noise meets the situation well (Julier and Uhlmann, 1995). But in practice, the assumptions are usually not totally satisfied, and the performance of the UKF might be seriously downgraded from the theoretical performance or could even diverge. To avoid these problems, an adaptive filter may be applied, which automatically tunes the filter parameter to adapt insufficiently known a priori filter statistics. There have been many investigations in the area of adaptive filter and parameter estimation. Maybeck (1979) used a maximum-likelihood estimator for designing an adaptive filter that could estimate the system-error covariance matrix. Katebi et al. (1985) introduced state space LQG adaptive autopilot for ship control. Lee and Alfriend (2004) modified the Maybeck’s methods by introducing a window-scale factor. A multiple
Kalman filter based adaptive stochastic filtering method was applied for the online aircraft flight path reconstruction with estimation of noise statistics (Celso-Braga et al., 2007). The estimator was introduced in UKF to estimate its noise statistics through maximizing the posteriori density function (Zhao and Wang, 2009), and modified Sage estimator was also utilized to estimate the system noise variance adaptively (Shi et al., 2009). Song and Han (2008) were designed a MIT rule based adaptive UKF algorithm to update the covariance of the process uncertainties online by minimizing the cost function. But it introduces a relative large computational burden. Recently, Kalman filter based adaptive UKF algorithm is introduced for the actuator failure estimation of Rotorcraft UAV (Juntong et al., 2012). Moreover, Chanying and Guo (2010) have established a new critical theorem for global stabilization of adaptive nonlinear system. The new automated adaptive algorithms are integrated into the UKF and can be applied to the nonlinear system. Nonlinear dynamics of aerospace vehicles and the presence of considerable noise and biases in measurements demand that a nonlinear filtering algorithm be used (Jategaonkar and Plaetschke, 1989). Traditionally, the EKF has been used for parameter estimation purposes. In this paper, an online innovation-based adaptive UKF is used to estimate aerodynamic parameters from flight data. The used adaptive filter composed of two parallel UKFs, called as master UKF and slave UKF. The master UKF estimates both the states and aerodynamic parameters of aircraft while the slave one estimates the diagonal elements of the noise covariance matrix for the master UKF. This approach of noise covariance identification based adaptive UKF is previously applied in mobile robot systems (Song et al., 2007). By estimating the noise covariance, this adaptive method is able to compensate the estimation errors resulting from the insufficient knowledge of the noise statistics. Performance evaluation of adaptive UKF was done with use of both simulated and flight test data. The simulated data were generated with moderate to high level of process noise. A 4,000 samples Monte Carlo simulation was done and the results were very encouraging to apply on flight test data. The main contribution of this paper is the application of adaptive filter method to estimate aerodynamic parameters recursively from flight data, and successfully reduces the dependency of the estimation performance on the accurate knowledge of the system noise statistics. The adaptive UKF was compared with filtering algorithms of the EKF (Grewal and Andrews, 1993), standard UKF (the simplified version of the UKF), augmented UKF (the complete version of the UKF) to estimate aerodynamic derivatives from flight test data. It was shown that adaptive UKF is superior to classical UKF in terms of fast convergence and estimation accuracy under the variation of noise characteristics. The paper is organized as follows. A brief review of the recursive parameter estimation (RPE) is presented in second section. The applied adaptive UKF structure is illustrated in third section. The flight simulated and flight test data have applied to the adaptive filter and other RPE methods in fourth section. The estimation results are analyzed in fifth section and concluding remarks are presented in the last section.
Recursive parameter estimation The basic ideas of practical engineering applications of RPE originate from control applications in chemical and thermal 268
Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
power-generating industrial process, where online adaptation of model parameters is desired to increase the overall plant efficiency. In such applications, we encounter not only changing operating conditions but also several different types of system disturbances. In the case of aircraft, the primary motivation for recursive estimation could be to obtain immediate knowledge about an aircraft model, which is essential for designing adaptive control. These real time applications necessarily call for recursive/online estimation methods. The system dynamics are represented in generic continuous-discrete state space form in equation (1) (Jategaonkar, 2006): 8 x_ ðtÞ ¼ f ½xðtÞ; uðtÞ; b þ FwðtÞ; xðt 0 Þ ¼ x0 > > < yðtÞ ¼ h½xðtÞ; uðtÞ; b ð1Þ > > : zðkÞ ¼ yðkÞ þ GnðkÞ; k ¼ 1; . . . ; N
method is normally employed If no other suitable method is found. Since the inaccurate assumptions of F will lead to poor performance or even divergence of the filter (Julier and Uhlmann, 1995), it is necessary to identify the process noise covariance by using adaptive algorithm, which attempts to adapt the unknown noise characteristics (Julier and Uhlmann, 2004).
The adaptive UKF structure
where x is state vector with initial value x0 at time t 0 , u is the input vector, y is the observation vector, f and h are the general nonlinear functions, z is the measurement vector sampled at N discrete time steps with a fixed sampling time Dt and k is discrete time index. The measurement noise vector n is assumed to be sequence of independent zero mean white Gaussian noise. The matrices F and G represent the distribution matrices for additive state and measurement noise, respectively. The unknown parameter vector Q consists of system parameters b, the measurement biases Dz, and the trim estimates formulated as input biases Du and is represented in equation (2):
The master UKF
QT ¼ ½bT ; DzT ; DuT
The applied adaptive scheme is composed of two parallel UKFs as shown in Figure 1. At every time-step, the master UKF estimates the states and parameters using the noise covariance obtained by the slave UKF, while the slave UKF estimates the noise covariance using the innovations generated by the master UKF. The master UKF can also work independently without the slave one. Thus, dual-UKF structure is reduced to a standard UKF with fix noise covariance. The setting of master UKF does not need any updates while activating/deactivating the slave one, which indicates that the slave UKF can be shut down to reduce the computational burden when the system statistics do not change a lot.
In the adaptive scheme, the calculation of the master UKF is the same as that of a classical UKF. To illustrate the UKF procedure, we consider a discrete time representation widely used in the applications of the UKF. The discrete time nonlinear state space model is given by: ( xkþ1 ¼ f d ðxk ; Qk ; uk Þ þ wk ð6Þ z k ¼ yk þ v k
ð2Þ
Estimation of these parameters through the filtering approach is an indirect procedure, consisting of transforming the parameter estimation problem into a state estimation problem. This is done by augmenting the system state vector by intentionally defining the unknown parameters as additional state variables. So the constant system parameters Q considered as output of an auxiliary dynamic system presented in equation (3): _ ¼0 Q
yk ¼ hd ðxk; Qk ; uk Þ
ð7Þ
where xk is the (nk £ 1) state vector, Qk the (nq £ 1) vector of unknown parameters, uk the (nu £ 1) vector of exogenous inputs, yk the (ny £ 1) model output vector at time k,f d and hd the corresponding state and output functions. wk and nk are, respectively, the disturbance and sensor noise vector, which are assumed to be zero mean Gaussian white noise with their covariance matrices Q and R. The measurement vector is denoted by zk. The augmented state vector of the size (na £ 1) is given by: h i T T T ð8Þ xak ¼ xk Qk
ð3Þ
The augmented state vector is then defined in equation (4) as: " # x xa ¼ ð4Þ Q where the augmented variables are denoted by subscript “a”. The extended system is represented as: 8 x_ a ¼ f a ½xa ðtÞ; uðtÞ þ F a W a ðtÞ > > > > #" # " # " > > F 0 wðtÞ f ½xðtÞ; uðtÞ; b > > > > þ < ¼ 0 0 0 0 ð5Þ > > > > yðtÞ ¼ h ½x ðtÞ; uðtÞ > a a > > > > > : zðkÞ ¼ yðkÞ þ GvðkÞ
where the superscript “a” denotes the augmented state vector, and na ð¼ nx þ nq Þ is the total number of states: Figure 1 The adaptive UKF structure Master UKF Time update
Measurement update zk Measurement
The value of F and G, i.e. the process and measurement noise distribution matrices must be specified a priori. The measurement noise matrix G can be calibrated using laboratory measurements of sensors to ensure good noise filtering. The process noise matrix F is however more difficult to determine and a trial and error
Time update
Slave UKF
269
Measurement update
State vk Innovation
Noise Covariance
Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
The slave UKF
Master UKF: initialization 8 T > x^ a0 ¼ E xa0 ¼ E x^ T0 QT > 0 > > > T > > > ¼ x^ T0 QT > 0 > > > > n o < a P^ 0 ¼ E ðxa0 2 x^ a0 Þðxa0 2 x^ a0 ÞT > > > " # > > P x0 0 > > > > ¼ > > 0 P Q0 > > :
There are six parameters in UKF, which are the initial state x^ 0 , initial covariance P^ 0 , process-noise covariance Q, measurement-noise covariance R, and unscented transform (UT) parameters a1 and b1 . The influence of the initial state and covariance will become asymptotically negligible as more and more data are processed. The selection for a1 and b1 has little impact on improving the estimate accuracy of the UKF if the order of the system under consideration is not high. As a priori knowledge, initial covariance P 0 , the covariance matrices Q and R are most important to the performance and stability of the UKF. The P 0 can be chosen based on the physical knowledge of variation in state variables, but Q and R are associated with uncertainties in the environment. In principle, an adaptive filter can estimate both R and Q, and their theoretical justification were well described by Myers and Tapley (1976). However, adaptive filtering algorithms that try to update both the observational noise and the system noise are not robust, since it is not easy to distinguish between errors in R and Q (Blanchet et al., 1977). The measurementnoise statistics are relatively well known compared to the system-model error. Therefore, selecting noise covariance matrix Q is most important to maintain the performance and stability of the UKF. In this section, the slave UKF is described to estimate the noise covariance matrix Q. The criterion for estimation of Q is to minimize the difference between the filter-computed and the actual innovation covariance. From equation (12) of the master UKF, the computed innovation covariance can be obtained as:
ð9Þ
Master UKF: sigma points calculation and time update 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a > a a a > > x~ak ¼ x^ k ; x^ k 2 ðna þ lÞP^ k ; x^ k þ ðna þ lÞP^ k > > > > > R > > < x~akþ1 ¼ x^ ak þ t tkþ1 f ½xa ðtÞ; uðkÞdt k ð10Þ P2na m a > x~ kþ1 ¼ i¼0 W i x~i;kþ1 > > > > > h ih iT > P a > a > ~ kþ1 ¼ 2na W c x~a > 2 x x 2 x þQx P ~ ~ ~ kþ1 kþ1 i;kþ1 i;kþ1 i : i¼0 where: 8 m W 0 ¼ nalþl > > > > > > < W c0 ¼ nalþl þ ð1 2 a21 þ b1 Þ
ð11Þ
c 1 > i ¼ 1; . . . ; 2na Wm > i ¼ W i ¼ 2ðna þlÞ > > > > 2 : l ¼ a1 ðna þ kÞ 2 na
" S~ k ¼ P y~ y~ k ¼ diag Master UKF: measurement update Y kþ1 ¼ hd x~akþ1 ; ukþ1 y~ kþ1 ¼
2na X
2na X
# W ci ½Y i;k
T
2 y~ k ½Y i;k 2 y~ k þ R
xa
ð13Þ
i¼0
The time-averaged approximation of actual innovation covariance is defined as:
Wm i Y i;kþ1
i¼0
P y~ y~ kþ1 ¼
2na X
a
Sk ¼
W ci ½Y i;kþ1 2 y~ kþ1 ½Y i;kþ1 2 y~ kþ1 T þ Rx
i¼0
P x~ y~ kþ1 ¼
2na X
h i W ci x~ ai;kþ1 2 x~ kþ1 ½Y i;kþ1 2 y~ kþ1 T
k21 1 X mi mTi N i¼k2N
ð14Þ
ð12Þ where N is the size of the estimation window, mi is the innovation and can be written as:
i¼0
K Kþ1 ¼ P x~ y~ kþ1 P 21 y~ y~ kþ1
mi ¼ zi 2 y~ i
mkþ1 ¼ zkþ1 2 y~ kþ1 x^ kþ1 ¼ x~ kþ1 þ K kþ1 mkþ1 P^ kþ1 ¼ P~ kþ1 2 K kþ1 P y~ y~ kþ1 K Tkþ1
ð15Þ
where zi and y~ i are, respectively, the real measurement and its estimated value. Thus, for the estimated Q is the condition the minimization of S k 2 S~ k , i.e. ðS k 2 S~ k Þ < 0:
The variables in equations (9) – (12) are defined as followings. W i is a set of scalar weights, a1 determines the spread of the sigma points around the estimated xa and is usually set as 0.0001 # a1 # 1 whereas k is secondary scaling parameter usually set to zero for state or (3 2 na ) for parameter estimation. The constant b1 is used to incorporate part of the prior knowledge of the statistics of x, while b1 ¼ 2 is optimal for Gaussian distributions. The process and measurement a a noise covariance are represented by Qx and Rx , respectively, whose diagonal elements are required to be estimated by the slave UKF.
S k ¼ E mk mTk ¼ P y~ y~ k
ð16Þ
From equation (12), K k P y~ y~ k K Tk ¼ P~ k 2 P^ k can be written as: 21 T 21 K k ðP~ k 2 P^ k ÞK k K Tk K k P y~ y~ k ¼ K Tk K k
ð17Þ
Referring to the master UKF equations, we can get P~ k ; P^ k ; K k as follows: 270
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Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
P y~ y~ k ¼
K Tk K k
21
2na X
K Tk
h ih iT W ci x~ai;k 2 x~ k x~ai;k 2 x~ k þQ 2 P^ k
Slave UKF: initialization 8 < h^0 ¼ E½h0
!
i¼0 2na h ih iT X 21 K k K Tk K k ¼ K k W ci x~ai;k 2 x~ k x~ai;k 2 x~ k 2P^ k
: P^ h0 ¼ E½ðh0 2 h^0 Þðh0 2 h^0 ÞT
!
ð25Þ
i¼0
Slave UKF: sigma points calculation and time update qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qk ¼ h^k ; h^k 2 ðn þ lÞP^ hk ; h^k þ ðn þ lÞP^ hk
K Tk þ K k QK Tk ð18Þ
21
K Tk so that ðK k Þ21 ¼ K k . From where K k ¼ K Tk K k equations (16) and (18), innovation sequence can be written as:
q~K þ1 ¼ qk
¼ K k
2n X
~ Wm hi qi;K þ1
i¼0
E{mk mTk } ¼ P y~ y~ k 2na X
h~kþ1 ¼
P~ hðkþ1Þ ¼
!
h
ih iT W ci x~ai;k 2 x~ k x~ai;k 2 x~ k 2P^ k K Tk
2n X
W chi ½q~i;K þ1 2 h~kþ1 ½q~i;K þ1 2 h~kþ1 T þ Qh
i¼0
ð19Þ
ð26Þ
i¼0
k QK Tk þK
where: 8 m W h0 ¼ nþl l > > > > > > < W ch0 ¼ nþl l þ 1 2 a21 þ b1
Thus, the noise covariance matrix Q is estimated from equation (19) as: ^ ¼ K k E mk mT K T Q k k ð20Þ ! 2na h ih iT X 2 W c x~ a 2 x~ k x~a 2 x~ k 2P^ k i
i;k
c 1 Wm > hi ¼ W hi ¼ 2ðnþlÞ > > > > > : l ¼ n a21 2 1
i;k
i ¼ 1; . . . ; 2n
i¼0
Usually, the process-noise covariance Q is a diagonal matrix. Therefore, the estimation of Q can be simplified as the estimation of its diagonal elements. Thus, we use the hik to denote the diagonal elements of matrix Qxk , i.e.: 3 2 n1 hk · · · 0 " x # 6 . Qk 0 .. .. 7 7 6 . x xa Qk ¼ 6 . ð21Þ . . 7 and Qk ¼ 5 4 0 0 nx 0 · · · hk
Slave UKF: measurement update 8 ~ > d ½qkþ1 > > 6kþ1 ¼ gP > > 2n > > S~ kþ1 ¼ i¼0 W m > hi 6i;kþ1 > > > P > T c h > ~ ~ P S~ S~ kþ1 ¼ 2n > i¼0 W hi ½6i;kþ1 2 Skþ1 ½6i;kþ1 2 Skþ1 þ R > > > < P T c ~ ~ P h~S~ kþ1 ¼ 2n i¼0 W hi ½qi;kþ1 2 h~kþ1 ½6i;kþ1 2 Skþ1 > > 21 >K > hðkþ1Þ ¼ P h~S~ kþ1 P S~ S~ kþ1 > > > > > > ~ > > h^kþ1 ¼ h~kþ1 þ K hðkþ1Þ ðS kþ1 2 Skþ1 Þ > > > > T > : P^ hðkþ1Þ ¼ P~ hðkþ1Þ 2 K hðkþ1Þ P ~ ~ K
If the dynamics of h is known, state equation of the slave UKF is:
hkþ1 ¼ f ðhk Þ þ whk
ð22Þ
SSkþ1
hðkþ1Þ
If the dynamics of h is unknown, it can be modelled as a noncorrelated random drift vector:
hkþ1 ¼ hk þ whk
ð27Þ where n is state dimension. Qh and Rh are the process and measurement noise covariance, respectively.
ð23Þ
where whk is the Gaussian white noise with zero mean. The innovation covariance generated by the master UKF is taken as the observation signal for the slave UKF, and then according to equation (12), the observation model can be described as: S~ k ¼ gðhk Þ " ¼ diag
2na X
# W ci ½Y i;k 2 y~ kþ1 ½Y i;k 2 y~ k T þ Rx
Aerodynamic parameter estimation from simulated flight data To evaluate the efficacy of the adaptive UKF algorithm, aircraft responses pertaining to longitudinal motion are generated through the simulation of aircraft dynamics models. An independent process and measurement noise vectors are generated using pseudo-random noise generators (Jategaonkar and Plaetschke, 1989). The state noise matrix is assumed to be diagonal and an appreciable level of process noise is incorporated in the system dynamics. Note that state variables are only affected by the state noise, and the control input is noise free. A total of 30 s of data with a sampling time 0.025 s are generated.
ð24Þ
a
i¼0
The measurement of S~ k received by the slave UKF is: S k ¼ diag mk mTk ; mk ¼ zk 2 y~ k Therefore, the recursive algorithm of the slave UKF can be formulated as: 271
Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
Aircraft dynamic model for parameter estimation
Figure 3 compares the performance of the adaptive UKF algorithm over the simple UKF for the purpose of aerodynamic parameter estimation. There is initial mismatch in marginal level of their estimates due to the use of very low initial values of Q in adaptive filter. But time progresses, the accurate agreement is observed between adaptive UKF and classical UKF estimates of the parameters. It shows that adaptive UKF does not match correctly with all classical UKF estimates, namely C La and C mde as shown in Figure 3, and there is an error between them, which is within the limits of considering the noise variation. It is worth to note that the classical UKF uses accurate value of Q, but adaptive UKF estimates accurately compared to the classical UKF even without knowledge of Q. The performance of the adaptive filter was further analyzed based on a Monte Carlo simulation with 4,000 realizations. In the state estimation model, diagonal elements of noise matrix are selected randomly for the 4,000 realizations. Figure 4 shows the consistency of aerodynamic parameter estimation from simulated data using adaptive UKF for the varied process noise properties. Mean value of all estimated parameters for 4,000 iterations are listed in Table I. The good agreement between all estimated aerodynamic parameters and their true value is very encouraging to apply on flight test data.
The longitudinal motion of aircraft responses are represented using the following postulated aircraft models (Klein and Moreli, 2006): 8 q S Fe > a_ ¼ 2 mV C L þ q þ Vg cosða 2 uÞ 2 mV sinða þ sT Þ > > < _ u¼q ð28Þ > > q Sc Fe > _ q ¼ C þ ðl sin s þ l cos s Þ : m tx T tz T Iy Iy where the lift, and pitching moment coefficients (C L ; C m ) are modelled as: 8 qc þ C Lde de < C L ¼ C L0 þ C La a þ CLq 2V ð29Þ qc : C m ¼ C m0 þ C ma a þ Cmq 2V þ C mde de Here the state variables are angle of attack a, pitch angle u, and pitch rate q. The elevator deflection de and thrust F e are considered to be control input to aircraft system dynamics equation (28). The mass of aircraft is m and its geometrical parameters consist of wing area S, wing chord c, moment of inertia I y , and the inclination angle of the engines sT . The dynamic pressure q is given by q ¼ 0:5rV 2, where r is the density of air and V is the true airspeed. The estimation algorithm has used equations (28) and (29) as state equations, and corresponding observation equations are given in equation (30):
Aerodynamic parameter estimation from flight data Open accessible flight test data of the research aircraft HFB320 (Figure 5) were used to estimate aircraft lift, drag, and pitching moment coefficients (Jategaonkar, 2006) and demonstrated the efficacy of the adaptive UKF algorithm for aerodynamic parameter estimation from flight data. For the use of flight test data, flight tests were carried out to excite longitudinal motion of research aircraft through a multi-step elevator input resulting in short period motion and a pulse input leading to phugoid motion (Jategaonkar, 2006). These recorded flight test data has been used without any smoothing or filtering; the angle of attack (a) was calibrated using flight path reconstruction techniques (Majeed and Kar, 2010). The thrust was computed prior to parameter estimation and used in the estimation process as an input variable.
am ¼ a um ¼ u qm ¼ q q Sc Fe C m þ ðltx sinsT þ ltz cossT Þ Iy Iy q S Fe C L cosa 2 sinsT ¼2 m mV
ð30Þ
q_ m ¼ 2 azm
The unknown parameter vector Q need to be estimated is consisting of the aerodynamic derivatives, which is given by: h iT Q ¼ CL0 C La C Lq C m0 C ma C mq C mde
Aircraft dynamic model for parameter estimation The longitudinal model of the research aircraft is represented in equation (31) and contains one additional state variable of true airspeed V to the model equation (28) used for generating the flight simulated data in the previous section. This state variable V leads to additional nonlinearities due to its inversion in the state equation of a , and increases the number of parameters by incorporating drag derivatives as given in the following postulated aircraft state models (Jategaonkar, 2006; Chowdhary and Jategaonkar, 2009): 8 V_ ¼ 2 qmS CD þ g sinða 2 uÞ þ Fme cosða þ sT Þ > > > > > q S Fe > < a_ ¼ 2 mV C L þ q þ Vg cosða 2 uÞ 2 mV sinða þ sT Þ ð31Þ _ u ¼ q > > > > > > q_ ¼ q Sc C m þ F e ðltx sinsT þ ltz cossT Þ : Iy Iy
Results and analysis for simulated data In the absence of flight test data, generated flight simulated data with moderate to high level of process noise to estimate the aerodynamic parameters using adaptive UKF algorithm. The simulated data contain adequate level of process noise similar to aircraft flying in turbulence (Iliff, 1978). Figure 2 shows the flight simulated and model estimated responses. This time history plots for the model output from Adaptive UKF show good agreement except at the first data point for the estimated signals “az”. This is because, the initial state error covariance matrix (Po) specified may be too large for the system states or parameters. The sigma points are computed based on initially specified state and state error covariance matrix. For the subsequent data points the sigma points are generated using the updated covariance, which are computed by the adaptive UKF algorithm, and are more realistic. This problem is specific to UKF only.
where the lift, drag, and pitching moment coefficients (C L ; C D ; C m ) are modelled as: 272
Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
α (°)
20 0 –20
θ (°)
50 0 –50
q (°/s)
40 20 0 –20 –40
qp (°/s2)
50 0 –50
az (m/s2)
Figure 2 Histories of output variables for the generic transport aircraft
40 20 0 –20 –40
dele (°)
10 5 0 0
5
10
15 Time in sec
20
25
30
estimated
Simulated, and
Figure 3 Performance of adaptive and standard UKF algorithms 5.65 CLLa
CL0
0.22
Adap UKF
0.2
5.6
True Value 0
10
40
UKF
0
10
20
30
40
10
20
30
40
12 CLLa
CLq
0.6 11 10
0.4 0
10
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40
0 –1.155 Cmmδe
Cm0
0.229 0.2285
–1.16 –1.185
0.228 0
10
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40
0
–22.8
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–1.382 Cmmδe
Cmq
0.5
–23 0
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20 Time in sec
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–1.384 0
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20 Time in sec
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Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
Figure 4 Performance of adaptive UKF algorithm in Monte Carlo simulation CL0
0.215 0.21
CLα CLq
11 10.5 10 0.48 0.47 0.46
Cmα
–1.161 –1.1612 –1.1614
Cmq
0.2283 0.2282
–22.98 –22.99 –23
Cmδe
Cm0
5.62 5.61 5.6
CLδe
0.205
–1.3825 –1.383 –1.3835 0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
Number of iterations
Table I Lift force and pitching moment derivatives
Figure 5 The DLR HFB-320 research aircraft
Parameters
True value
M 5 0.25, H 5 5,000 ft Mean estimated values
CL0 CLa CLq CLde Cm0 Cma Cmq Cmde
0.2102 5.6275 10.5249 0.4671 0.22826 21.1613 222.9920 21.3831
0.2095 (7.14 £ 102 04) * 5.6165 (0.0017) 10.5584 (0.1097) 0.4712 (0.0076) 0.2283 (1.9 £ 102 04) 2 1.1613 (4.603 £ 102 04) 2 22.9920 (0.0289) 21.3830 (0.002)
Note: *Mean values of standard deviation of estimates
8 C D ¼ CD0 þ C DV VVo þ CDa a > > > < C L ¼ CL0 þ C LV VVo þ CLa a > > qc V >C ¼ C þ C : m m0 mV V o þ C ma a þ C mq 2V o þ C mde de
Vm ¼ V am ¼ a um ¼ u qm ¼ q
ð32Þ
Fe q Sc C m þ ðltx sinsT þ ltz cossT Þ Iy Iy q S Fe ¼ C X þ cossT m m Fe q S ¼ C Z þ sinsT m m
q_ m ¼ 2
where V 0 is initial value of V . The dynamic pressure q ð¼ 0:5rV 2 Þ contains state variable V which introduces nonlinearity in estimation process as it multiplies with all derivatives needs to be estimated. The following measurement model is used to estimate the aerodynamic derivatives including drag coefficients:
axm azm
274
ð33Þ
Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
where the longitudinal and vertical force coefficients C X and C Z are given by: ( C X ¼ C L sina 2 CD cosa ð34Þ C Z ¼ 2CL cosa 2 C D sina
initial state error covariance matrix (P 0 ) specified may be too large for the system states or parameters. Similar type of initial mismatch was shown in Figure 2 for the case of simulated data. Apart from this, the agreement between flight measured and model estimated responses is good. The similar parameter estimation procedure was carried out using EKF, UKF, and UKFaug algorithms and compared their performance for the purpose of aerodynamic parameter estimation (Chowdhary and Jategaonkar, 2009). Here, we repeated the applications of these EKF, UKF, and UKFaug algorithms for the purpose of comparison with Adaptive UKF algorithm and showing the agreement in estimated parameters. The details of the estimation methods (EKF, and UKFaug) are not described here for brevity, and they are illustrated in (Chowdhary and Jategaonkar, 2009). Figure 7 shows the performance of these RPE methods for the purpose of aerodynamic parameter estimation from flight data. It shows that convergence of the estimates for the adaptive UKF is similar to the usual UKF and other methods, so that the use of adaptive UKF is appreciable without knowing Q. Table II compares the numerical values in four decimal places of the estimates of the parameters arrived at with the different methods. It is clearly seen that the parameter estimates of all these methods are in close vicinity of one another, and all numerical values are in good agreement. The performance of these methods
Thus, the unknown parameter vector Q need to be estimated is consisting of the aerodynamic derivatives and is given by: h iT Q ¼ C D0 CDV C Da C L0 C LV C La C m0 C mV C ma C mq C mde
Results and analysis for flight test data The adaptive UKF algorithm was applied to estimate the aerodynamic parameters from the flight data. We have introduced initial state x0 ¼ (106.02, 0.11, 0.15, 20.003) for master UKF and x20 ¼ (1.2856, 5.86 £ 102 005, 2.01 £ 102 006, 3.02 £ 102 005) for slave UKF. Similarly, state propagation error covariance matrix P 0 ¼ diag(10, 10, 10, 10) for master UKF and P 20 ¼ diag (1, 1, 1, 1) for slave UKF. Figure 6 shows the flight measured and model estimated responses for the HFB-320 research aircraft. The model estimated signals “ax” and “az” show large deviations from the first few data points. It may be due to the
v (m/s)
Figure 6 Histories of output variables for HFB-320 aircraft 114
104 α (°)
7.5 5 θ (°)
10
q (°/s)
2 2.5
qdot (°/s )
–2.5 8 0
δe (°)
az (m/s2)
ax (m/s2)
–8 1.2 0.7 –7.5 –11.5 0 0
10
20
30 Time in sec
Measure and
40
Estimated
275
50
60
Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
Figure 7 Performances of RPE methods
Table II Comparison of parameter estimates Parameters Computational time
CD0 CDV CDa CL0 CLV CLa Cm0 CmV Cma Cmq Cmde
RPE methods FEM –
Adapt. UKF 7.55
UKF 5.83
UK Faug 11
EKF 2.56
0.1226 (2.45) 2 0.0645 (3.95) 0.3201 (2.26) 2 0.0929 (21.10) 0.1487 (11.10) 4.3278 (1.08) 0.1119 (3.27) 0.0040 (82.10) 2 0.9679 (1.12) 234.7098 (2.27) 2 1.5291 (1.27)
0.1226 (2.64) 2 0.0642 (4.27) 0.3197 (2.40) 2 0.0871 (22.91) 0.1468 (11.42) 4.2894 (1.14) 0.1115 (4.29) 0.0045 (92.19) 2 0.9703 (1.54) 235.3628 (2.82) 2 1.5395 (1.65)
0.1226 (2.64) 2 0.0642 (4.27) 0.3197 (2.40) 2 0.0871 (22.91) 0.1468 (11.42) 4.2894 (1.14) 0.1115 (4.29) 0.0045 (92.19) 2 0.9703 (1.54) 235.3628 (2.82) 2 1.5395 (1.65)
0.1239 (2.55) 2 0.0653 (4.09) 0.3160 (2.37) 2 0.0990 (20.03) 0.1569 (10.61) 4.3032 (1.13) 0.1152 (3.40) 0.0022 (153.88) 2 0.9832 (1.24) 235.0981 (2.27) 2 1.5517 (1.31)
0.1235 (2.50) 20.0652 (4.01) 0.3191 (2.33) 20.0853 (23.49) 0.1440 (11.69) 4.3027 (1.14) 0.1115 (4.28) 0.0046 (90.48) 20.9712 (1.54) 2 34.9372 (2.85) 21.5328 (1.65)
Note: The values in parenthesis denote standard deviation values in percent
8 QTo ¼ diag{1:2856 5:8612 £ 1025 > > > > > QT ¼ diag{1:2856 £ 1023 5:8612 £ 1028 > > > > :0:2035 £ 1028 3:0199 £ 1028 }; t $ 20sec
ð35Þ
Filter error method (FEM) represents the most general stochastic approach to aircraft parameter estimation (Jategaonkar and Plaetschke, 1989, 1990; Jategaonkar, 1993) and estimates state noise distribution matrix F additional to the system parameters to calculate process noise covariance Q ¼ FF T Dt 21 , Dt is sampling time. In UKF, the prior knowledge of the process noise covariance is selected as Q ¼ QT 0 , which is the true process noise covariance computed
The dynamic changes of process noise The estimation accuracy of the adaptive UKF with respect to the situation of changing process-noise covariance is tested. The change of the true process-noise intensity is assumed as: 276
Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
by offline FEM (Jategaonkar and Plaetschke, 1990) and it is get changed after 20 s, i.e. Q ¼ QT . But in the case of adaptive UKF, process-noise intensity fixed to the very small quantity for the credibility to select arbitrarily small value of Q:
“Q fixed” due to the violation of the optimality conditions. On the contrary, the estimation errors in the adaptive UKF are quickly overcome at the time of 20 s and are almost the same as their previous value for the cost of additional computation in estimation of process noise (Figure 8(b)). The estimated Q value in average for the last ten iterations of adaptive UKF is given by:
8 adp QTo ¼ diag{1:2856 £ 10214 5:8612 £ 10219 > > > > > < 0:2035 £ 10219 3:0199 £ 10219 }; t , 20 sec > ¼ diag{1:2856 £ 10217 5:8612 £ 10221 Qadp > T > > > : 0:2035 £ 10221 3:0199 £ 10221 }; t $ 20 sec
^ adp ¼ diag{1:2856 5:8612 £ 1025 2:0035 Q avg
ð36Þ
£ 1026 3:0199 £ 1025 }
ð37Þ
The aerodynamic pitching moment derivatives estimates by standard UKF, and the adaptive UKF under the same condition of the process-noise intensity change, are shown in Figures 9 and 10, respectively. We can see that, by the classical UKF, the aerodynamic parameter estimations happen to divergence after the time of 20 s due to the priori covariance setting of UKF fails to meet the true values. By the application of adaptive UKF, on the other hand, the estimated parameters
Thus, the Q of the adaptive filter is randomly selected to very small value as Q ¼ Qadp T 0 without a priori knowledge of noise statistics and it is get changed to Q ¼ Qadp T after the time of 20 s. The stateestimation errors of the classical UKF, and the adaptive UKF, under the same condition of the process-noise intensity change, are shown in Figure 8. It shows that, under incorrect noise information for the case of “Q change” after 20 s, the classical UKF cannot produce optimal estimates as showing the case of Figure 8 State-estimation errors with the time varying process noise
(a) Standard UKF
(b) Adaptive UKF
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Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
Figure 9 UKF-divergence of parameter estimates due to changing process noise 0 UKF –Q change UKF +σ UKF –σ
Cmα
–0.95 –1
FEM
–1
–2
–1.05
–3
25
30
35
40
45
50
55
60
–20
Cmq
–25 –30 –35 0
10
20
30
40
50
60
Time in sec
Figure 10 Adaptive UKF-convergence of parameter estimates while changing process noise 2
Cmα
1
–0.9
0 –1 –1
–2
AdpUKE AdpUKF+σ AdpUKF–σ FEM
–3 –1.1 –20
25
30
35
40
45
50
60
AdpUKE AdpUKF+σ
–25 Cmq
55
AdpUKF–σ
–30
FEM
–35 –40
0
10
20
30
40
50
60
Time in sec
converge to same as before (Figure 10) even though the process noise covariance was changed. In real application, there is a possibility of the error in the assumption of the prior knowledge of noise (say order of 1023 ). This leads to the fixing of very low/high value of Q with reference to its true value, and degrades the performance of the UKF through parameter divergence as shown in Figure 9. It is a major issue in online flight stability and control parameter estimation, which has addressed by the adaptive UKF. It is ensure that by online estimating the process noise covariance, the adaptive UKF successfully reject the influence caused by the incorrect priori covariance of process noise and achieve better estimates of aerodynamic parameters under varying noise statistics.
noisy flight data. Initially, it was successfully applied to the flight simulated data with moderate to high level of process noise and the results were analyzed based on a Monte Carlo simulation with 4,000 realizations. Subsequently, the performance of the adaptive UKF was demonstrated to identify the aircraft system dynamics from flight test data, which was superior to the classical UKF in terms of fast convergence and estimation accuracy while changing the noise statistics. Therefore, the potential use of adaptive UKF is favoured to the design of adaptive flight controller for the recursive estimation of unobserved states in the aircraft system model and unknown aerodynamic parameters. It is worth to note that the two UKFs are independent in the adaptive UKF structure. Thus, the slave UKF can be replaced by another simple filter such as Kalman filter to save the computational burden further. This may be considered as a future work, if the efficacy of proposed adaptive filter is significant for the cost of additional computation in estimation of process noise.
Conclusions This paper described the application of adaptive UKF algorithm to estimate aircraft aerodynamic parameters from 278
Aerodynamic parameter estimation using adaptive UKF
Aircraft Engineering and Aerospace Technology: An International Journal
M. Majeed and Indra Narayan Kar
Volume 85 · Number 4 · 2013 · 267 –279
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About the authors M. Majeed completed his B.Tech (Instrumentation & Control) from Calicut University in 1998 and M.Tech (Control Engineering) from IIT Delhi, India in 2002. He is a research scholar at IIT Delhi and working as senior scientist in CSIR-National Aerospace Laboratories, Bangalore. His research interests are aircraft parameter estimation, nonlinear system identification, contraction based stability analysis, and data fusion. M. Majeed is the corresponding author and can be contacted at:
[email protected] Indra Narayan Kar completed his M.Tech and PhD (Electrical) from IIT Kanpur in 1991 and 1997, respectively. He is working as Professor at the Department of Electrical Engineering, Indian Institute of Technology Delhi. His research interests are robust and intelligent control, nonlinear control, and system identification.
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