compared with the XFoil results for different flight cases, expressed as various ..... The SVM-EGD approach was implemented in MATLAB. The results for the CL, ...
New methodology for the prediction of the aerodynamic coefficients of an ATR-42 scaled wing model Abdallah BEN MOSBAH, Ruxandra BOTEZ, Thien My DAO École de technologie supérieure (ÉTS), LARCASE, www.larcase.etsmtl.ca, Montreal, Quebec, H3C1K, Canada
l
Abstract A new approach for the prediction of lift, drag and moment coefficients is presented. This approach is based on the Support Vector Machines methodology, and on a new optimization algorithm, theExtended Great Deluge. The training and validation of this new combination method is realized using the aerodynamic coefficients of an ATR-42 wing model with Xfoil software. The results obtained with our approach are compared with the XFoil results for different flight cases, expressed as various combinations of angles of attack and Mach numbers. The main purpose of this methodology is to rapidly predict aircraft aerodynamic coefficients.
Introduction Supervised learning is a technique in which rules are automatically generated from a database. This learning database is characterized by a pair of inputs-outputs (xn,yn), where yn= f(xn). The objective of the supervised learning method is to determine a representation of the function f, called ''prediction function h''. This new function h provides an output y’=h(x’) for a new input x’. There are two types of problems that we seek to solve by means of supervised learning. The first is a ''regression problem'', in which the output associated with an input is a real number. The second type is a ''classification problem'', in which the output has a finite cardinal and a label should be assigned to a given input [1]. A regression problem is presented in this study, in which the desired outputs are real numbers. There are many supervised learning methods, such as neural networks (NN) and support vector machines (SVM), Page 1 of 8
which is used in this work. SVM, NN and fuzzy logic can be applied to design prediction or identification models. NN and fuzzy logic have been used extensively on control systems in aerospace. Neural networks have been used in multiple domains, including pattern classification, optimal control, and manufacturing [2–6]. In aerospace engineering, NNs can be applied to a large range of complex problems, as in [7]. Neural network have been used to resolve many problems in the aeronautic industry, such as: the detection and identification of structural damage [8], the modeling of aerodynamic characteristics from flight data [9, 10], the detection of unanticipated effects such as icing [11,12,13], and autopilot controllers and advanced control laws for applications such as carefree maneuvering [14, 15], as well as CL and CD aerodynamic coefficient prediction [16]. An experimental study on the use of smart sensing and neural networks to strain loads for different airflow cases are presented by Lunia et al. [17], in which the authors used a fiber-optic sensor to train and verify the neural network performance. They used a multilayer network to map the inputs and the outputs of a nonlinear system, thereby creating a three-layered neural network, with three neurons in the first layer, five in the second, and one neuron in the third. An hyperbolic tangent sigmoid function is used in the first and the second layer; and a linear function in the third layer. The network was developed through supervised learning, in which parameters are adjusted to achieve the target outputs for given inputs. Scott [18] developed an adaptive neural network-based control system that integrated three control systems that were developed and tested. One system used flutter suppression control laws, a second system employed a predictive NN control scheme, and the third system used an NN in an inverse model control scheme. Suresh et al. [19] used recurrent neural networks for the prediction of lift coefficients at high angle of attack. In their
approach, the lift coefficient was obtained from wind tunnel tests. Fei et al. [20] evaluated the air speed, the angle of attack and the angle of sideslip fundamental parameters in the control of flying bodies. They have proposed a new experimental methodology by which the flight parameters are inferred from multiple hot-film flow speed sensors mounted on the surface of the wing of a Micro Air Vehicle (MAV). In order to obtain a good mathematical relationship between the sensor readings and the flight parameters, they propose using micro hot-film flow speed sensor arrays and a back-propagation neural network to determine three flight parameters: air speed, angle of attack and angle of sideslip [20]. Peyada [21] proposes using Feed Forward Neural Networks to estimate aircraft parameters from flight data. That method uses Feed Forward Neural Networks to establish a neural model that can be used th to predict the time histories of motion variables at the (k + 1) instant, given that the measured initial conditions correspond to th the k instant [21]. A neural network based on a flush air data sensing system and demonstrated on a mini air vehicle was presented by Samy et al. [22]. Xuan et al. [23] presented a fuzzy neural network controller, which had the advantages of both fuzzy control and neural network, for the control of uncertain parameters for nonlinear time-varying systems. Fuzzy logic can be used to model highly non-linear, multidimensional systems, including those with parameter variations, or where the sensors’ signals are not accurate enough for other models [24]. De Jesus-Mota et Botez [25] proposed a new technique for helicopter model identification from flight data tests based on neural networks. The dynamic behavior of the helicopter was identified with a recurrence method and an optimization procedure based on neural network theory and tuning of the initial conditions [25]. A aeroservoelastic model was presente by o ly et al. , ], in which neural network and fuzzy logic algorithms identify the multi-input and the multi-output system of the F/A-18 aircraft. A flight parameter control system based on neural networks has been proposed by Mosbah et al. [28], in which the proposed NNs are optimized using a metaheuristic algorithm, called the extended great deluge (EDG). Their approach predicts pressure distributions and aerodynamic coefficients from the known parameters (angle of attack, Mach number, etc.). Kouba et al. [29,30] proposed an identification model, based on fuzzy logic methods, to identify the nonlinear aircraft models for many flight-test cases; their proposed model used an F/A-18 aircraft. A new method for the realization of two neuro-fuzzy controllers for a morphing wing design application is presented by Grigorie et al. [31]. The propose controllers’ main function is to correlate each set of pressure differences, calculated between the optimized and the reference airfoil, with each of the airfoil deformations produce by the actuators’ system [31]. Several other uses of fuzzy logic are presented by the same authors in identification and control areas in [3237].
Support Vector Machines (SVM) The SVM method is used to solve regression and classification problems. The idea is to estimate a prediction function h from the original function f given in the learning phase. y= f(x) Page 2 of 8
y’=h(x’) SVM method
This estimation is made from a training set of N samples. A tube of wi th ε is efine aroun the esire outputs, so that all the predicted values should be inside this tube. However, Smola et al. [38] selected their prediction function to be flat and smooth so that it could be approximated with a linear function.
(1) where ω is weight of the inputs space an b is a billet Є R In order to satisfy these two conditions, the input space is converted to a larger size space, in which the transfer function for this conversion is enote by Φ, as seen in equation (1). As described by Vapnik [39] , the optimization of the prediction function h(x) in the larger space is characterized by the resolution of the following system [1]:
min (, )
N 1 2 C ( i i* ) 2 i 1
(2)
subject to:
((.( xi )) b) yii i*
i , i* 0
i 1...N
where: C is a regularization parameter of the compromise between the complexity of the model, for example the planarity of the function f, and the degree of tolerance for the deviation of the samples against ε; an
i , i*
are variables of samples which are outside of the
ε-tube (Figure 1).
Figure 1. Illustration of the principle of slack variables ζi and ζi* [49]
An optimization problem has a dual form in which the objective function and the constraints are ''strictly convex''. If the conditions are satisfied, the ''dual problem'' becomes equivalent to the ''original problem''. For this reason, we use a Lagrangian function, where the Lagrangian is defined as the sum of the objective function and a linear combination of constraints whose coefficients (αi ≥ 0) are calle Lagrange multipliers [1]. As explained by Smola et al. [38], by introducing Lagrange multipliers, our optimization problem defined in equation (2)
becomes a dual form, and the shape of the ''regression function'' is defined as follows: ∑
(3)
where: αi are the Lagrange multipliers; M is the subset of samples corresponding to the nonzero Lagrange multiplier; and K is the kernel function representing a scalar product in the re-description space. The kernel function is used to construct the decision surface (hyper-plan) in the input space. Some examples of kernel functions are presented in [1]: -Linear kernel function: -Polynomial kernel function: -Gaussian kernel function:
‖
‖
)
Optimization of the SVM parameters To obtain a good regression it is necessary to use the appropriate optimized SVM parameters, such as C, ε and K. In fact the parameters C and ε are optimized. It is also important to use the correct type of kernel function K and to optimize its parameters, such as its degree d, for example, in the case of its polynomial form. To optimize these three parameters, many techniques have been proposed. For example, Keerthi and al.[40] used a technique to minimize the margin. Cherkasky and al.[41] proposed an analytical method to optimize the parameters C and ε when a Gaussian kernel function wasselected. The meta-heuristic methods were also used to optimize the SVM parameters, while Ping and al. [42] used a simulated annealing algorithm to fin the values of C an ε parameters. Since the quality of results necessarily depends on the quality of the SVM parameters, we propose an original hybridization of the SVM method with the ''Extended Great Deluge'' (EGD) algorithm to optimize these parameters. The EGD algorithm is described.
Extended Great Deluge algorithm The local search procedure called "Extended Great Deluge" (EGD) was introduced by Dueck [48] in 1993. It considers a local search algorithm in which bad solutions might be obtained, whose values are smaller than a certain limit B. This limit B decreases monotonically (in the case of minimization problems) during the search. The initial value of B is equal to the "objective function", and for each iteration its value ecreases by a fixe Δ in minimization problems, an increases by the same value of Δ for maximization problems. The step of Δ represents an ''input parameter'' in this approach. During the search, B is the limit between a feasible and a not-feasible area of research, it serves to orientate the solution of the problem towards the ''feasible area''. In other words, the neighborhood of the solution s* is cut by the limit B and the research is only to be conducted on one side, below or above the limit B, depending on the objective function Page 3 of 8
(minimization or maximization of the objective function). The increase or ecrease of with Δ can be consi ere as a control process giving the desired solution. In the beginning stages of research, the actual solution has the ability to move in both directions, and can also be found inside the feasible portion limited by B. Otherwise, there is a great chance of accepting poorsolutions, because the limit B is located at a long distance from the chosen solution s* and a small part of its neighborhood may be cut off. During the search, the limit B moves closer to the value of the current solution, the search space becomes smaller and the possibility of improving the solution becomes lower, leading to the end of the process research [7]. The first application realized with this approach was the optimization of the exam timetable problem treated by Burke et al. [43]. The results proved the effectiveness of this algorithm [43]. Several of these results have been improved using approaches such as ''taboos search''. Figure 2 shows the steps of the EGD algorithm [44]. The second application of the EGD algorithm was the optimization of the group scheduling problem by Mosbah [45, 46, 47, 48] (first author of this paper), in which the EGD algorithm gave better results than genetic algorithm (GA) and simulated annealing (SA). The advantage of the EGD is that only one parameter needs to be a juste , Δ , the rate of increase or decrease of B. Burke et al. [43] have shown that the convergence time of the algorithm epen e on the Δ value. In ee , increase of the Δ value will result in a ecrease of the convergence time, but the quality of solutions can also degrade, which explains the importance of choosing the best Δ to obtain a goo compromise between quality of results and calculation time. The EDG algorithm is a metaheuristic type in which the optimum solution is not guaranteed, because the search process is based on a randomly selected initial solution, which is the initial boundary B and the value of ΔB [48]. The first step of the algorithm, as shown in Figure 2, consists of ran omly initializing Δ an the initial value of the SVM parameters (S). Next, the efficiency α (the Error) of S is calculated, and its value is assigned to B. The neighborhood * N(S) of S is then defined; a neighboring solution S belonging * to N(S) is randomly selected; and the new solution N(S ) is compared with the old solution, N(S) and B. If two conditions * (α(S )≤ α(S) and B) are not satisfied, then a new neighboring solution S is selected; if one of these conditions is true, then * the solution S is accepted and B=B-ΔB is recalculated. Finally, the stopping criterion is tested, and if the number of iterations has been reached, the algorithm ends. A new neighboring solution S is selected and the algorithm is tested again [47].
New proposed SVM-EGD algorithm In this work, a hybrid SVM- EGD method is proposed to control the lift, drag and moment coefficients for different combinations of angles of attack and Mach number values. Figure 3 shows the steps used to obtain the optimal SVM parameters so that the error between the calculated coefficients using XFoil and the calculated coefficients using SVM-EGD remains as small as possible.
In our new algorithm, we use a qualitative performance measure describing the learning abilities of a given trained SVM method, that is, the training error is expressed as the mean sum of the squared residuals (errors) in the training data [40]:
∑
(4)
where t is the time when the y is calculated, y(xk) is the th estimated output of the SVM method for the k input xk and N is the number of data points used in the training set. The performance and results obtained with our proposed algorithm are further evaluated to determine the lift, drag and moment coefficients for varying angles of attack and Mach numbers using Xfoil values.
start
Initialize :∆ , S
Calculate the efficiency of S (s) B(S)
Define the neighborhood N (S)
Select randomly a neighboring solution Figure 2. The Extended Great Deluge [44] ok
no (S*) ≤(S) or (S*) ≤ B
Accept S*and put S=S*
B= B-∆ Stopping criterion
ok END
no
Page 4 of 8
Price-Paidoussis wind tunnel
Start
The validation of this new approach will be achieved by experimental tests using Price-Paidoussis the wind tunnel of LARCASE,shown in Figure 4. This wind tunnel has two test 2 chambers; a first chamber with a section equal to 0.3 x 0.6 m that provides a speed up to 60 m/s, and a second chamber test 2 with a section of 0.6 x 0.9 m that provides a speed up to 40 m/s [50].
Initialize SVM parameters, Δ , number of iteration and error_int
learning vectors :inputs x and outputs y B
error_int
Define the neighborhood N
Select randomly a neighboring solution S*
Learning
Output vector y' Figure 4.
Price-Paidoussis wind tunnel
Error (y,y')
no
Implementation of the SVM-EGD algorithm and preliminary results Error(y,y')≤error(y,y'-1) Or
The objective is to determine the aerodynamic lift (Cl), drag (Cd) and moment (Cm) coefficients for different values of Mach numbers and angles of attack on the ATR-42 wing.
Error(y') < B
ok
For the ''learning'' of the SVM, a database obtained using Xfoil software was used. A total of 101 values of lift (Cl), drag
Accept solution S=S*
Iteration number?
no
ok
SVM-parameters optimized and selected
Predicted outputs y_pred with SVM_EGD
New input data
Figure 3.
Page 5 of 8
Proposed algorithm
B=B- Δ
(Cd) and moment (Cm) coefficients for combinations of angles o o o of attack α between -5 to 5 (0.1 per step) and a Mach -1 number = 0.11 (40 m s ) were used. These test cases were selected so that they could be validated using the PraicePaidoussis wind tunnel. The validation data set was composed of 11 random vectors and the test data set had had 11 random vectors. The optimal values of the SVM parameters (kernel, C, degree d and Ɛ) were obtained using the EGD algorithm. The best results were obtained using the Gaussian kernel with the parameters C= 8036518, d=1 and Ɛ=9.29 10-6. The SVM-EGD approach was implemented in MATLAB. The results for the CL, CD and CM are values are presented in Tables 1 to 3 and Figures 5 to 7.
Table 1. Comparison for lift coefficients CL (original vs prediction) for different airflow cases
0,11757
Original Lift coefficient -0,2676
Predicted Lift coefficient -0,2708196
0,11768
-0,1562
-0,15527248
0,594
-2,7
0,11777
-0,0654
-0,06557326
-0,265 -7,931
α
Mach number
-4,7 -3,6
Error (%) -1,203
-2
0,11784
0,0078
0,00841861
-1,3
0,11791
0,0814
0,07942546
2,426
-0,1
0,11803
0,198
0,19817848
-0,090
0,6
0,1181
0,2648
0,26317644
0,613
1,6
0,1182
0,364
0,36395321
0,013
2,6
0,1183
0,4684
0,46793551
0,099
3,9
0,11843
0,6176
0,61415461
0,558
4,7
0,11851
0,7219
0,72523071
-0,461
Figure 6.
Drag coefficient CD versus angle of attack α
Table 3. Comparison of CM coefficients (original vs prediction) for different airflow cases
Figure 5. Lift coefficient CL versus angle of attack α Table 2. Comparison of CD coefficients (original vs prediction) for different airflow cases α
Mach number
Original Drag coefficient
-4,7
0,11757
0,01101
Predicted Drag coefficient 0,01103101
-3,6
0,11768
0,01082
0,01083617
-0,149
-2,7
0,11777
0,01082
0,01083611
-0,149
-2
0,11784
0,01065
0,010641
0,084
-1,3
0,11791
0,01041
0,01045531
-0,435
-0,1
0,11803
0,00925
0,00922795
0,238
0,6
0,1181
0,00877
0,00875986
0,116
1,6
0,1182
0,00863
0,00865606
-0,302
2,6
0,1183
0,00905
0,00904081
0,102
3,9
0,11843
0,00973
0,00970529
0,254
4,7
0,11851
0,01018
0,01020146
-0,211
0,11757
Original Moment coefficient -0,0342
Predicted Moment coefficient -0,03427513
0,11768
-0,031
-0,03098787
0,039
-2,7
0,11777
-0,0286
-0,02856293
0,130
-2
0,11784
-0,0271
-0,02708922
0,040
-1,3
0,11791
-0,0258
-0,02585505
-0,213
-0,1
0,11803
-0,0221
-0,02206207
0,172
0,6
0,1181
-0,0193
-0,0192996
0,002
1,6
0,1182
-0,0155
-0,0155388
-0,250
2,6
0,1183
-0,0128
-0,01278234
0,138
3,9
0,11843
-0,0128
-0,01249511
2,382
4,7
0,11851
-0,0156
-0,01555607
0,282
α
Mach number
-4,7 -3,6
-0,220
Error (%) -0,191
Figure 7.
Page 6 of 8
Error %
Moment coefficient CM versus angle of attack α
Conclusions Anew algorithm using the SVM-EGD approach was used to optimize the values of SVM parameters. The obtained results were acceptable and can be further improved using experimental wind tunnel test validation. The average percentage error did not exceed a maximum of 1.3 % for the lift coefficient, and most of the error values were lower than 0.2% for the drag and 0.35% for the moment coefficients. Based on these results, the SVM-EGD approach was found to be robust and accurate. Future work will include verifying the performance of the proposed approach with experimental wind tunnel tests.
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Acknowledgments The first author would like to thank Dr Ruxandra Botez for the opportunity to work on this project, for her constant encouragement and support. Thanks are also due to Dr Thien My Dao for his generosity and for his well needed advices. Many thank you to LARCASE team for their cooperation.
