Grouped weighting on-line least square identification method for longitudinal aerodynamic parameters of aircraft

Abstract

The application provides a grouping weighted recursive least square identification algorithm for aerodynamic parameters of a longitudinal channel of an aircraft, and comprises the following steps: 1, establishing a discrete model for identification of the aerodynamic parameters of the longitudinal channel of the aircraft; 2, determining initial values and unknown hyperparameters related to the identification algorithm; 3, grouping data according to the determined hyperparameters, then calculating indexes of excitation of each group of data and updating least square estimations of the aerodynamic parameters corresponding to each group of data; 4, weighting the least square estimations by using the indexes of excitation of each group of data to obtain final estimations of the aerodynamic parameters. The application is aimed at the online rapid identification of the aerodynamic parameters, uses effective information in the data, designs an online least square algorithm based on data excitation with small sample data, realizes online effective estimation of the aerodynamic parameters, fully utilizes the accurate distribution of the least square estimation, and proposes the grouping weighted average idea, so that the robustness of the parameter estimation is improved compared with the Kalman filter algorithm.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of aircraft dynamics parameter identification, and relates to identification of dynamics parameters by using measurement output and control input. The method proposes the idea of grouping weighting based on the excitation of data and the accurate distribution of least square estimation, and gives high-precision estimation of aerodynamic parameters on line. BACKGROUND

[0002] Due to the limitations of aircraft wind tunnel test and the imperfection of theoretical calculation, it is an important content of aircraft design and research to obtain the aerodynamic parameters of aircraft model from flight test data by using system identification technology. Due to the error of model, especially the change of some parameters with flight conditions, aircraft type, fuel consumption, failure or battle damage, which are usually unpredictable or measured. The purpose of on-line identification of aircraft is to obtain model parameters in real time, to provide model basis for on-line update of various advanced flight control systems, real-time flight stability evaluation, envelope expansion (or boundary control), and fault detection and system reconstruction.

[0003] At present, there are various parameter estimation methods applied to the identification of aerodynamic parameters, including least square method, maximum likelihood method, Bayesian learning method, Kalman filter method, etc. The on-line identification of aircraft parameters has the following difficulties: first, the nonlinearity and uncertainty of aircraft aerodynamic model; second, the non-active excitation and closed-loop of aircraft control system; third, the multi-interruption and uncertain excitation of aircraft control system. These difficulties lead to some problems of existing methods at different levels, for example, Bayesian method and maximum likelihood method mainly deal with off-line data, which is difficult to update model parameters in real time for closed-loop control of aircraft; Kalman filter algorithm is greatly affected by initial value and is not robust; therefore, under the condition of non-active excitation, how to realize the on-line rapid identification of aerodynamic parameters under small sample data is a problem to be solved. SUMMARY

[0004] The technical problem solved by the present application is: for the identification problem of aircraft pitch channel aerodynamic parameters, a small sample data on-line least square algorithm based on data excitation is designed, and high-precision estimation of aerodynamic parameters is realized under the measurement output of angular velocity with noise.

[0005] The present application is a kind of grouping weighting on-line least square identification method of aircraft longitudinal aerodynamic parameters, comprising the following 7 steps:

[0006] Step (one): establish the aircraft aerodynamic parameter model:

[0007]

[0008] In the formula, t is time, x(t)∈R is the angular velocity of pitch channel, is the total disturbance including unknown aerodynamic model and external disturbance, which is usually a slowly time-varying unknown, θ ∈ R is the longitudinal aerodynamic parameter of the aircraft to be identified, u(t) ∈ R is the rudder input of the pitch channel, which can be regarded as the control input, ε(t) ∈ R is the random noise in the dynamic model.

[0009] Step (two): Discretize the model (1) to obtain the discrete form of the system:

[0010] x((k+1) t s ) = x(kt s ) + t s (f(x(kt s ), kt s ) + θu(kt s ) + ε(kt s )) (2)

[0011] where t s is the sampling step, x((k+1) t s ) is the angular velocity of the pitch channel at the (k+1) t s moment, x(kt s ) is the angular velocity of the pitch channel at the kt s moment, f(x(kt s ), kt s ) is the total disturbance at the kt s moment, u(kt s ) is the rudder input of the pitch channel at the kt s moment, and ε(kt s ) is the random noise at the kt s moment;

[0012] Step (three): Transform the system (2) into an equivalent linear regression model:

[0013]

[0014] where the output of the regression model at the kth moment is the regression vector at the kth moment is the input of the model, the regression coefficient at the kth moment is the parameter to be estimated, and the random noise at the kth moment is ε(k). Since f(x(kt s ), kt s ) is a slowly varying unknown, it is directly regarded as an unknown constant f here, so the parameter to be estimated in the regression model can be regarded as a constant vector Therefore, the regression model (3) can be simplified as:

[0015]

[0016] Among them, the regression coefficient is the parameter to be estimated.

[0017] Through the first three steps, the identification problem of the aircraft longitudinal aerodynamic parameter θ in model (1) can be equivalently transformed into the estimation problem of the regression coefficient β in the linear regression model (4). For the estimation problem of the regression coefficient β, we need to obtain some input data of the regression model (4) And the output data y(k), but in the problem of aircraft aerodynamic parameter identification, generally only the measured data of the aircraft pitch channel angular velocity x(t) and the rudder input u(t) are available. Therefore, before estimating β, it is necessary to use y(k) in the regression model (4) to Based on the definition of , the measured data x(t) and u(t) of the angular velocity and steering input are converted into the input and output data of the regression model (4), which are then used to estimate the regression coefficient β. In the following four steps, the input and output data pairs used to estimate the regression coefficient β are collectively referred to as data.

[0018] Step 4: Use offline data to determine the undetermined hyperparameters in the identification method; the undetermined hyperparameters in the identification method include q, m and V * , where q is the number of data sets used for parameter identification, m is the amount of data contained in each set of data, and V * is the motivational threshold, which is used to determine whether the current set of data can be used for identification. These three hyperparameters remain unchanged throughout the online identification process. The specific selection method is:

[0019] 1. Number of groups q: Based on experience, it is generally selected as 10-30;

[0020] 2. The number of data in each data set m: The least squares estimate has an accurate asymptotic normal distribution and can be used to obtain the confidence interval of the least squares estimate. Since this identification method focuses on the identification of the aerodynamic parameter θ, we determine it based on the confidence interval of the second component of the least squares estimate of the regression coefficient β in the regression model (4) (corresponding to the least squares estimate of the aerodynamic parameter θ) and the maximum estimation error that can be tolerated. The specific steps are as follows:

[0021] (1) Obtain confidence intervals for the least squares estimates of the aerodynamic parameters θ from their asymptotically normal distribution;

[0022] (2) Based on the confidence interval obtained in the previous step and the maximum tolerable estimation error, the hyperparameter m is obtained at a 95% confidence level;

[0023] 3. Incentive threshold V* : the variance σ of the random noise ε(k) of the aircraft aerodynamic parameter model (1) 2 .

[0024] Step (five): initialization of the identification method; select the initial value of the excitation index vector as a q-dimensional vector U(0), and the initial value of the least square estimation vector of the aerodynamic parameter θ as a q-dimensional vector where the excitation index vector is used to store the excitation index values of the q sets of data for identification, and the least square estimation vector of the aerodynamic parameter θ is used to store the least square estimation values of θ obtained from the q sets of data.

[0025] Step (six): obtain U(k) at the kth moment, use the new input data at the kth moment and the input of the regression model (4) at the (k-m+1)th to (k-1)th moments to calculate the excitation index value V at the kth moment, specifically as follows:

[0026]

[0027] where are the inputs of the regression model (4) at the ith and (i+1)th moments, respectively, and ||·|| is the Euclidean norm. s s

[0028] compare the value of V with a threshold value V * , if less than V * , then U(k) = U(k-1), where U(k) ∈ R q is the excitation index vector at the kth moment, is the corresponding least square estimation vector of the aerodynamic parameter θ at the kth moment, U(k-1) ∈ R q is the excitation index vector at the (k-1)th moment, is the corresponding least square estimation vector of the aerodynamic parameter θ at the (k-1)th moment; otherwise, update as follows:

[0029] (1) update U(k): remove the minimum value in the vector U(k-1) and add the excitation index value V at the kth moment to the vector to obtain U(k). At the same time, record the element index i corresponding to the minimum value in U(k-1);

[0030] (2) update First, remove the ith element in the vector ; second, according to the regression model (4), use the input-output data from the (k-m+1)th to kth moments to calculate ​​The least square estimation of the regression coefficient β in the model at the kth moment is calculated, wherein y(i) is the output of the regression model (4) at the ith moment, and the calculation can be realized by the following three steps:

[0031] a) the m sets of input-output data pairs are brought into the regression model (4) to obtain the following m equations,

[0032]

[0033] b) the m equations are written into the following vector form of the regression model:

[0034] Y=Φβ+Λ (5)

[0035] wherein the output of the regression model (5) is Y=[y(k-m+1),…,y(k)] T ∈R m , the regression matrix is and the random noise is Λ=[ε(k-m+1),…,ε(k)] T ∈R m .

[0036] c) the least square estimation of β is calculated according to the regression model (5), and is denoted as Specifically,

[0037]

[0038] Finally, the second component of (corresponding to the least square estimation of the aerodynamic parameter θ at the kth moment) is added to to obtain

[0039] Step (seven): the vector U(k) is used to make a weighted average of the vector to obtain the estimation value of the aerodynamic parameter θ at the kth moment Specifically,

[0040] wherein

[0041] is a q-dimensional all-1 vector.

[0042] The present application has the following advantages compared with the prior art:

[0043] 1. The present application is aimed at the online fast identification of the aerodynamic parameter, fully utilizes the effective information in the data, designs a small sample data online least square algorithm based on data excitation, and realizes the online effective estimation of the aerodynamic parameter.

[0044] ​​​2. The application fully utilizes the accurate distribution of least square estimation, proposes the idea of grouping weighted average, and improves the robustness of parameter estimation compared with Kalman filtering algorithm. BRIEF DESCRIPTION OF DRAWINGS

[0045] Figure 1 is a flowchart of the method of the application.

[0046] Figure 2 is a parameter estimation effect diagram of the method of the application under the traditional least square method.

[0047] Figure 3 is a parameter estimation effect diagram of the method of the application under the condition that the hyperparameters are q=10, m=32, V * =10 -2 .

[0048] Figure 4 is a parameter estimation effect diagram of the method of the application under the condition that the hyperparameters are q=20, m=32, V * =10 -2 .

[0049] Figure 5 is a parameter estimation effect diagram of the method of the application under the condition that the hyperparameters are q=30, m=32, V * =10 -2 . DETAILED DESCRIPTION

[0050] In order to illustrate the applicability of the application to the aircraft power system under non-active excitation, we use the pitch channel angular velocity x(t) and rudder input u(t) generated by the real aircraft according to the corresponding control command to perform the following simulation experiment under the condition that the true value of the aerodynamic parameter θ=1.

[0051] In the experiment, 50s of measurement data of the angular velocity of the pitch channel of the aircraft and the rudder input are obtained, the sampling step is 0.005s, and the angular velocity measurement data is added with random noise with a variance σ 2 =10 -2 . The following steps are followed to identify the aerodynamic parameter θ.

[0052] Step (1): Establish the aircraft aerodynamic parameter model:

[0053]

[0054] Step (2): Discretize the model (1) to obtain the discrete form of the system:

[0055] x((k+1)t s )=x(kt s )+0.005(f(x(kt s), kt s )+ θu(kt s )+ ε(kt s )) (2)

[0056] Step (three): transform the system (2) into an equivalent linear regression model:

[0057]

[0058] where, the output of the regression model at the kth time instant is the regression vector at the kth time instant is the input of the model, the regression coefficient at the kth time instant is the parameter to be estimated, the random noise at the kth time instant is ε(k). Since f(x(kt s ), kt s is a slowly varying unknown quantity, it is directly regarded as an unknown constant f here, so the parameter to be estimated in the regression model can be regarded as a constant vector Therefore, the regression model (3) can be simplified as:

[0059]

[0060] where, the regression coefficient is the parameter to be estimated.

[0061] Through the first three steps, the identification problem of the longitudinal aerodynamic parameters θ of the aircraft in the model (1) can be equivalent to the estimation problem of the regression coefficient β in the linear regression model (4). For the estimation problem of the regression coefficient β, we need to obtain some input data and output data y(k) of the regression model (4), so before estimating β, we need to convert all the measured data of the angular velocity x(t) and the rudder input u(t) obtained in 50s into the input and output data of the regression model (4) according to the definition of y(k), and then use them in the estimation of the regression coefficient β. In the following four steps, the input and output data pairs used in the estimation of the regression coefficient β are collectively referred to as data.

[0062] Step (four): determine the undetermined hyperparameters q, m and V in the identification method using offline data * , where the offline data is selected as the input and output data in the regression model (4) converted from the measured data of the angular velocity x(t) and the rudder input u(t) in the pitch channel obtained in the first second; the specific selection method is:

[0063] 1. The number of groups q: here, 10, 20 and 30 are taken respectively to show that the selection of this hyperparameter has little effect on the estimated value;

[0064] 2. The data amount m in each group of data: determined according to the confidence interval of the second component of the least square estimation of the regression coefficient β in the regression model (4) (corresponding to the least square estimation of the aerodynamic parameter θ) and the maximum estimation error that can be tolerated. According to the principle that the relative error of the parameter estimation does not exceed 15%, that is

[0065]

[0066] wherein is the estimation value of θ, it is known that when the real parameter θ = 1, the maximum estimation error that can be tolerated is 0.15.

[0067] According to the following steps (1), (2), the hyperparameter m is determined to be 32 from the selected offline data:

[0068] (1) Obtain the confidence interval of the least square estimation of the aerodynamic parameter θ from the asymptotic normal distribution of the estimation;

[0069] (2) According to the confidence interval obtained in the previous step and the maximum estimation error that can be tolerated, obtain the hyperparameter m according to the confidence level of 95%.

[0070] 4. The incentive threshold value V * : take the variance σ 2 = 10 -2 of the random noise ε(k) of the aircraft aerodynamic parameter model (1) as

[0071] Step (five): initialization of the identification method; select the initial value of the incentive index value vector as the q-dimensional vector U(0) = (0, …, 0, and the initial value of the least square estimation vector of the aerodynamic parameter θ as the q-dimensional vector wherein the incentive index vector is used to store the incentive index values of the q groups of data for identification, and the least square estimation vector of the aerodynamic parameter θ is used to store the least square estimation values of θ obtained from the q groups of data.

[0072] Step (six): obtain U(k), at the kth moment using the new input data

[0073] at the kth moment and the input data from the k-m+1th to the k-1th moment to calculate the incentive index value V at the kth moment, specifically:

[0074]

[0075] wherein are the ith s , (i+1)th sThe input of the moment regression model (4), ||·|| is the Euclidean norm.

[0076] Compare the V value with the threshold V * Compare, if it is less than V * , then U(k)=U(k-1), where U(k)∈R q is the incentive indicator vector at the kth moment, is the least squares estimation vector corresponding to the aerodynamic parameter θ at the kth moment, U(k-1)∈R q is the incentive indicator vector at the k-1th moment, is the least squares estimation vector corresponding to the aerodynamic parameter θ at the k-1th moment; otherwise, the update is performed as follows:

[0077] (1) Update U(k): Remove the minimum value in the vector U(k-1) and add the incentive index value V at the kth moment to the vector to obtain U(k). At the same time, record the element subscript i corresponding to the minimum value in U(k-1);

[0078] (2) Update First, the culling vector The i-th element in; secondly, according to the regression model (4), using the input and output data from the k-m+1th to the kth moment Calculate the least squares estimate of the regression coefficient β in the model at the kth moment, where y(i) is the output of the regression model (4) at the i-th moment. This can be achieved by the following three steps:

[0079] a) Combine these m groups of input and output data Substituted into the regression model (4), we get the following m equations:

[0080]

[0081] b) Write the above m equations into the following vector form regression model:

[0082] Y=Φβ+Λ (5)

[0083] The output of regression model (5) is Y = [y(k-m+1),…,y(k)] T ∈R m , the regression matrix is The random noise is Λ=[ε(k-m+1),…,ε(k)] T ∈R m .

[0084] c) Calculate the least squares estimate of β according to the regression model (5), denoted as Specifically

[0085]

[0086] Finally, add the second component (the least square estimate of the aerodynamic parameter θ at the kth moment corresponding to the aerodynamic parameter θ) to the , to obtain

[0087] Step (seven): use the vector U(k) to make a weighted average of the vector , to obtain the estimate of the aerodynamic parameter θ at the kth moment Specifically,

[0088]

[0089] where is a q-dimensional all-1 vector.

[0090] Therefore, through the above seven steps, using the obtained angular velocity x(t) of the aircraft in the pitch channel and the measured data of the rudder input u(t), the input and output data y(k) of the regression model (5) at the kth moment are obtained, and then the excitation index vector U(k) at the kth moment, the least square estimate vector of the aerodynamic parameter θ and the estimate

[0091] Figure 2 The real value is shown by the dashed line, and the estimate obtained by using the identification method is shown by the solid line. As can be seen from Figure 2 , the estimate obtained by using the traditional least square method greatly deviates from the real value at almost all moments, and the estimation accuracy is poor.

[0092] Figures 3-5 are the real-time estimation effects of the aerodynamic parameter θ using the grouping weighted online least square identification method under the settings that the hyperparameters m, V * remain unchanged and q is respectively set to 10, 20, and 30. The real value is shown by the dashed line, and the estimate obtained by using the identification method is shown by the solid line. As can be seen from Figures 3-5 , for different hyperparameters q, the grouping weighted online least square identification method can obtain a relatively accurate estimate of the aerodynamic parameter θ in real time, which is sufficient to show that the identification method is not very sensitive to the selection of the hyperparameter q. In the three graphs, it can be seen that the estimate obtained by using the identification method fluctuates to different degrees over time, because the data in different time periods contain different information (that is, different excitation), which can show that the identification method makes full use of the effective information in the data.​

Claims

1. A method for on-line least squares identification of longitudinal aerodynamic parameters of an aircraft by grouped weighting, characterized in that, The following seven steps are included: Step (1): Establish the aerodynamic parameter model of the aircraft: where t is time, x(t) e R is the angular velocity of the pitch channel, is the derivative of the angular velocity of the pitch channel with respect to time t, f(x(t), t) e R is the total disturbance including unknown aerodynamic model and external disturbance, which is usually a slowly time-varying unknown quantity, θ e R is the longitudinal aerodynamic parameter of the aircraft to be identified, u(t) e R is the rudder input of the pitch channel, which is regarded as the control input, and ε(t) e R is the random noise in the dynamic model. Step (2): Discretize the model (1) to obtain the discrete form of the system: x((k + 1)t s ) = x(kt s ) + t s (f(x(kt s ), kt s ) + θu(kt s ) + ε(kt s )) (2) where t is the sampling step, x((k+1) t) is the angular velocity of the pitch channel at the (k+1)th time, x(kt) is the angular velocity of the pitch channel at the kth time, f(x(kt), kt) is the total disturbance at the kth time, u(kt) is the rudder input of the pitch channel at the kth time, and ε(kt) is the random noise at the kth time. s s s s s s s s s s s s ​​​​​​​​​​​​ Step (3): Transform the system (2) into an equivalent linear regression model: where the output of the regression model at the kth time instant the regression vector at the kth time instant is the input to the model, the regression coefficient at the kth time instant is is the parameter to be estimated, the random noise at the kth time instant is ε(k); since f(x(kt s ), kt s ) is unknown, it is considered as an unknown constant f, thus the parameter to be estimated in the regression model is considered as a constant vector Therefore, the regression model (3) is simplified as: where the regression coefficients are parameters to be estimated; Through the first three steps, the identification problem of the aircraft longitudinal aerodynamic parameter θ in model (1) is equivalently transformed into the estimation problem of the regression coefficient β in the linear regression model (4); before estimating β, it is necessary to calculate the y(k) and y(k) in the regression model (4). The definition of , the measured data x(t), u(t) of angular velocity and steering deflection input are converted into input and output data in the regression model (4), and then used in the estimation of the regression coefficient β; in the following four steps, the input and output data pairs used for the estimation of the regression coefficient β are collectively referred to as data; Step (four): determining the undetermined hyper-parameters in the identification method using offline data; the undetermined hyper-parameters in the identification method include q, m and V * wherein q is the number of groups of data used for parameter identification, m is the amount of data contained in each group of data, and V * is an incentive threshold value for determining whether the current group of data can be used for identification, and the three hyper-parameters remain unchanged throughout the entire online identification process; Step (five): initialization of the identification method; selecting the initial value of the excitation index vector as a q-dimensional vector U(0), and the initial value of the least square estimation vector of the aerodynamic parameter θ as a q-dimensional vector wherein the excitation index vector is used to store the excitation index values of the q sets of data for identification, and the least square estimation vector of the aerodynamic parameter θ is used to store the least square estimation values of θ obtained from the q sets of data; Step (six): obtain the kth moment of Using the new input data at the kth moment And the input of the regression model (4) at the k-m+1th to k-1th moments, calculate the kth moment of the incentive index value V, specifically: wherein, are the ith s , (i+1)th s inputs of the regression model (4) at time instant, respectively, is the Euclidean norm. Step (seven): using the vector U(k) to weight the vector to obtain the estimate of the aerodynamic parameter θ at the kth moment Specifically, wherein is a q-dimensional all-one vector.

2. The method of claim 1, wherein: In step (4), the specific selection method is: 5.1 Group number q: selected as 10-30 according to experience; 5.2 Data amount m in each group of data: the least square estimation has an accurate asymptotic normal distribution, which can be used to obtain the confidence interval of the least square estimation; since the identification method focuses on the identification problem of the aerodynamic parameter θ, the second component of the least square estimation of the regression coefficient β in the regression model (4) corresponds to the confidence interval of the least square estimation of the aerodynamic parameter θ, and the maximum estimation error that can be tolerated in practice is used to determine; 5.3 Excitatory threshold V * : take the variance σ2 of the random noise ε(k) of the aircraft aerodynamic parameter model (1) 2 .

3. The method of claim 2, wherein: In step 5.2, the specific steps are as follows: obtain the confidence interval of the least square estimation of the aerodynamic parameter θ from the asymptotic normal distribution of the estimation; according to the obtained confidence interval and the maximum estimation error that can be tolerated, the hyperparameter m is obtained according to the confidence level of 95%.

4. The method of claim 2, wherein: In step (six), the value of V is compared with a threshold value V * If less than V * then U(k) = U(k-1), where U(k) e R q is the excitation index vector at the kth time instant, is the least square estimate vector of the aerodynamic parameters θ at the kth time instant, U(k-1) e R q is the excitation index vector at the (k-1)th time instant, is the least square estimate vector of the aerodynamic parameters θ at the (k-1)th time instant.

5. The method of claim 4, wherein: In step (6), conversely, the following updates are performed: (1) Update U(k): remove the minimum value in the vector U(k-1), and add the excitation index value V at the kth moment to the vector to obtain U(k); at the same time, record the element index i corresponding to the minimum value in U(k-1); (2) Update First, the culling vector The i-th element in; secondly, according to the regression model (4), using the input and output data from the k-m+1th to the kth moment Calculate the least squares estimate of the regression coefficient β in the model at time k, where y(i) is the output of the regression model (4) at time i.

6. The method of claim 5, wherein: In step (6), the following three steps are also included: a) the m sets of input-output data pairs are brought into the regression model (4) to obtain the following m equations, b) Write the above m equations into the following vector form of the regression model: Y = Φβ + Λ (5) wherein the output of the regression model (5) is Y = [y(k-m+1),..., y(k)] T ∈ R m and the regression matrix is the random noise is Λ = [ε(k-m+1),..., ε(k)] T ∈ R m ; c) Calculate the least squares estimate of β according to the regression model (5), denoted as Specifically, Finally, to the estimate of the second component corresponding to the aerodynamic parameter θ at the k-th time instant, the least squares estimate of the first component is added, obtaining

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