Research of the Aerodynamic Parameter Estimation for the Small Unmanned Aerial Vehicle * Li Chao, Dou Xiuxin, and Wu Liaoni  Abstract—Aircraft parameter estimation is an important problem in aircraft dynamics and control. As standard methods for aircraft parameter estimation, output-error method and equation-error method in the frequency domain are described and examined. The analysis was based on flight data. It was found that the output-error and equation-error method in frequency domain are in statistical agreement. By comparison, the equation-error method has excellent prediction capability similar to the output-error method. z ( ) i  Cx ( ) i  Du ( ) i  v ( ) i i  1,2,..., N E [ (0)] 0 x  E [ (0) x x T (0)]  P 0 (2) (3) are assumed to be independent white ( )iv ( )t and where Gaussian noise sequences with ( ) w t i ( )] 0 t w w E E  [ [ T ( t )] j  Q ( t t  i ) j (4) (5) I. INTRODUCTION [ ( )] 0 v E i  [ ( ) v E i v T ( )] j  R  ij the maximum-likelihood As a very common method for parametric estimation, the an effective unbiased maximum-likelihood method is estimation method. It takes the maximum probability of observed values as the criterion. It acquires the estimates by maximizing function whose arguments are observed value unknown parameters. As the maximum-likelihood method, both the output-error and equation-error methods are used for aircraft parameter estimation. By making simplified assumptions in the general problem formulation, the output-error and equation-error methods in the frequency domain were developed. The difference between them is in the assumptions about noise and the way the physical system response is being matched by the model. The next section introduces the output-error and equation-error methods in the frequency domain. Then flight test procedure is described. After this, the result of flights tests are presented and analyzed. the theory about II. MAXIMUM LIKELIHOOD METHODS The maximum likelihood method[1] is an effective unbiased estimation method proposed by Fisher. It acquires the estimates of the model parameters by minimizing the maximum likelihood function. Dynamic model equations for a linear continuous-time dynamic system with process noise and discrete-time measurements are considered, x  ( ) t  Ax ( ) t  Bu ( ) t B w  w ( ) t (1) *Resrach supported by the fundamental research funds for the central universities (2010121046). is 896291837@qq.com). Xiuxin, Li Chao is with the Xiamen University (e-mail: nextline@foxmail.com). (e-mail: Dou University Xiamen with the Wu Liaoni is with the Department of Aeronautics, Xiamen University, Ximen, Fujian 361005 China (corresponding author to provide phone: 86-13616023191; e-mail: airmud2001@gmail.com). For the complexity and the difficulty of parameters estimation based on the aforementioned model equations, a steady-state Kalman filter is used. The filter equations are d ˆ [ ( | x t dt i  1)]  ˆ Ax ( | t i 1)   Bu ( ) t ˆ x ( | ) i i  ˆ x ( | i i 1)   K ( ) i where ( i 1) t i      t t ˆ Cx   ( ) i z ( ) i  ( | i i 1)   Du ( ) i E [ ( )] 0 i  E T [ ( ) i   ( )] j  ij (6) (7) (8) (9) (10) (11) The innovations ( )i are independent Gaussian vectors. In the frequency domain, equation is transformed to x  j  ( ) ( ) k k ( ) k The cost function to be minimized is Bu  ( ) k Ax     K   ( ) k (12) J ( )   N where N 1   H   0  k ( ) k 1  S    ( ) k  N ln S  (13) N H ( , k  1  )     0 k  1 N  ( , k )  [ ( ) z k   y  ( , k ˆ )]  ˆ S     k  0 z  ( ) k    ( , k ˆ )  y  H   J  is minimized by iteratively computing ( ) ˆ ˆ       0 (14) (15) 

M   E  E [ ; )  ] 2 [  ln ( L V N T    ( ) J   T    ] 2 where 0 is a nominal starting value , and ˆ     M 1  ˆ    0 [ ( ) J     ] ˆ   0  (16) (17) The expressions for the elements of the first-order and second-order gradients of the cost function are ( )  k     ( ) y k     1  H   0 N ( ) J     2 Re[ N 2 Re[ N 1  H   0 ˆ 1  S  ˆ 1  S      ( ) k ( ) k  ] N   k k 2 ( ) J   T     2 Re[ N  2 Re[ N N 1   0 k  1 N   k  0 H ( )  k     H ( ) y  k    ˆ 1  S  ˆ 1  S  ( )  k     ( ) y  k    ] (19) as III. AIRCRAFT PARAMETER ESTIMATION USING OUTPUT-ERROR For aircraft parameter estimation, the output-error method[1-3] is one of the most frequently used techniques. Assuming no process noise is made, the maximum likelihood parameter to output-error method. estimation method simplified can be Assuming there is no process noise, the maximum likelihood estimator in the frequency domain can be substantially simplified to output-error method. Then, the dynamic system is deterministic, ( ) Ax  k ( ) B u k  (20) ( ) k x    j  k ( ) y  k z  ( ) k  G E  Cx  ( ) k  Du  ( ) k ( , k  ) ( ) u k    ( ) k [ ( )] 0  k   S  N ( )] k H [ ( )   k   G ( , k )    ) ( , G k  1 ( I C j   ) A  1 B D    2 k T k / k  0,1,2,..., N  1 E where and vvS is the spectral density of For these model equations, the innovations are reduced to ( )iv . output-error or residuals,   )    ( , k z  z  ( ) k ( ) k   ( , y  k ( , G k )  ) ( ) u  k  (27) The output-error cost function is the negative log-likelihood function, excluding the constant term, J ( )   N N H  1  )      0 ( , k ( , k 1  vv S )   k  N ln S vv (28) The estimates for the parameters in vvS are obtained from ˆ vvS  N  H ) ( , k 1  )     0 1  ( , k   N k [ ( ) z k   y  ( , k ˆ )]    k  0 z  ( ) k    ( , k ˆ )  y  H   (29) (18) ] The modified Newton-Raphson algorithm is used to estimate the unknown parameters in matrices in matrices A, B, C, and D. The expressions for the gradient of the negative log-likelihood function and the information matrix can be derived from the output-error cost function, ] J ( )  N   N H 1  )      0 ˆ 1  S  ( , k ( , k )  k  (30) (31) (32) ( ) J       2 Re[ N N 1   k  0 H S ( ) k ˆ 1 )]  S    vv ( , k M ( )   2 ( ) J   T     2 Re[ N N 1   k  0 H S ( ) k ˆ 1  S S vv ( )] k (21) (22) (23) (24) (25) (26) ( )kS is the n  o n p output sensitivity matrix, given where by S ( ) k  G  ) ( ) ( , u  k k    (33) It’s advantages to use equation-error approach since the nonlinear terms can be generated in the domain before transformation into the frequency domain so that the equation-error approach can use arbitrarily nonlinear terms in the model. However, since the finite Fourier transform is linear operator, the output-error method in the frequency domain is limited to linear dynamic systems. IV. AIRCRAFT PARAMETER ESTIMATION USING EQUATION-ERROR Equation-error method[1-3] is a reduced form of maximum likelihood method. When a system contains process noise and the measurement noise is such small as to be neglected, the state equation is transform into the observation equation.[4] The dynamic model equation in the frequency domain is (34)   ) B w  w (  k  Ax  ( ) k Bu  j  k ( ) x k  ( ) k The model equation then becomes ( ) Bu  k  0,1,2,..., ( ) k k   ( ) k Ax   z  ( ) v  k 1 N  (35) 

where v  ( ) k  w B w  ( )k  (36) ( )kv are the equation errors, which are Gaussian with [ ( )] 0 v E k  (37) [ ( ) v  E k v  H ( )] k  vvS N (38) The cost function is J ( )   N N 1  [ ( ) z k   k  0  Ax  ( ) k  Bu  ( )] k H S -1 vv [ ( ) z k   Ax  ( ) k  Bu  ( )] k where the innovations    ( ) k are equation errors. z  ( ) k  Ax  ( ) k  Bu  ( ) k (39) (40) Figure 2. Pitching moment parameters mC V. AIRCRAFT AND TEST PROCEDURE The test aircraft used for this research of this research is a SUAV (Small Unmanned Aerial Vehicle), see Fig. 1. The aircraft has two pairs of control surfaces: V-tails and ailerons. As common input for aircraft system identification, the 3-2-1-1 inputs with 1 pulse width from 0.1s to 1.0s were applied to V-tails, which were added directly to the appropriate control surface actuator when the feedback control system still operating. The aircraft was equipped with a micro-INS, which provided 3-axis translational accelerometer measurements, angular rate measurements, attitude angles, pitot static pressure and dynamic pressure, and GPS velocity and position. The common sampling rate is 40 Hz, Figure 1. Small unmanned aerial vehicle corresponding to a sampling interval of 0.025s, and the flight data were corrected for systematic instrumentation errors by using a data compatibility analysis. VI. RESULT Using the same model structure, equation-error parameter estimates are plotted along with output-error parameter estimates base on the same data. Estimates for stability and control pitching moment coefficient parameters are plotted in Figs. 2-4. The triangles represent flight estimates identified by equation-error method associated with derivatives Figure 3. Pitching moment parameters de mC Figure 4. Pitching moment parameters q mC in frequent domain and the circles represent estimates calculated by output-error method in frequent domain. Error 

ACKNOWLEDGMENT Supported by the fundamental research funds for the central universities (2010121046) REFERENCES [1] Klein, V. and Morelli, E.A., Aircraft System Identification – Theory and Practice, VA: AIAA, 2006, Chaps. 6, 7, and 9. [2] Cai Jinshi, Aircraft System Identification, Beijing: National Defense Industry Press, 2005. [3] Li yanjun, Zhang ke, Aircraft System Identification Theory and Application, Beijing: National Defense Industry Press, 2011. [4] Morelli, E. A., “Practical Aspects of the Equation-Error Method for Aircraft Parameter Estimation,” AIAA Atmospheric Flight Mechanics Conference, Keystone, CO, Aug. 2006. [5] Eugene A. morelli, “Flight-Test Experiment Design for Characterizing Stability and Control of Hypersonic Vehicles,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 3, May-June 2009, pp. 949-959. bars on all estimates were statistical values, with 95% confidence, assuming a Gaussian distribution.[5] The flight data obtained with pulse width 0.4s and 0.6s are neglected to eliminate their influence on data analysis because of their too obvious differences comparing to the other responding measured data. The flight data obtained with pulse width 0.1s, 0.2s, 0.7s and 1.0s can't satisfy data compatibility check, which means the measurements are inconsistent. The chief cause was likely to be inaccuracy of angle of attack which is computed using by velocity measured by GPS and attitude angles measured by micro-INS. The GPS velocity, particularly in the vertical direction, is inaccurate. For this reason, the responding data will be ignored. f = [0.15, 0.155 ... 1.495, 1.5]Hz The frequencies used for all Fourier transformations were . The lower bound should be chosen as 2/T, where T is the time length of the maneuver. The upper bound should be chosen to include the dynamics of interest. It works well when frequency resolution is chosen as 0.005. In the open loop control cases, the 1 pulse width has been found to work well when equals 0.7/2 nf is the expected natural frequency in hz for dominant mode[1]. However, the analysis of the flight data with 1 pulse width of the 3-2-1-1 input from 0.3s to 0.9s indicates that parameter to be estimated can be identified without evident differences in the closed loop control cases. , where nf The pitching moment parameter associated with the angle of attack isn’t identified accurately, see Fig. 2. It’s because there isn’t sensor that measures the angle of attack and the SUAV is vulnerable to the effects of the wind. The value is obtained indirectly based on attitude angle and the direction of the GPS velocity. The Figs. 2-4 show that the equation-error and output-error estimates are in good agreement and there are not obvious the best point in the term of the Caméra-Rao bound in both methods. Meanwhile, the equation-error method lowered the estimated parameter standard errors compared to the equation-error method. Because of and equation-error parameter estimates, the equation-error method has excellent prediction capability similar to the output-error method. agreement between output-error indicate results that the the VII. CONCLUSION Comparisons were done on an equal basis, and it was found that the output-error and equation-error method in frequency domain are in statistical agreement, considering the estimated error bounds. Using analytical approaches and numerical techniques, the validity and accuracy for aircraft parameter estimation using the two methods are demonstrated. Successful and equation-error method in frequency domain can increase confidence in result of the aerodynamic parameter estimation. comparison between the output-error