Aerodynamic Parameter Estimation for Launch Vehicles

Abstract

Aerodynamic force plays an important role in the flight of space launch vehicles. Therefore, obtaining accurate aerodynamic characteristics is the basis and prerequisite for establishing an aerodynamic model and designing a vehicle with excellent characteristics.

7.1 Introduction

7.1.1 What is Aerodynamic Parameter Estimation

Aerodynamic force plays an important role in the flight of space launch vehicles. Therefore, obtaining accurate aerodynamic characteristics is the basis and prerequisite for establishing an aerodynamic model and designing a vehicle with excellent characteristics.

Aerodynamic parameter identification is to establish the aerodynamic mathematical model reflecting the flight state of the vehicle and identify the coefficients based on the input and the measured output. Normally, the approaches for obtaining aerodynamic characteristics are theoretical calculation, wind tunnel test and the flight experiment. The method of theoretical calculation is based on the development of computer technology and aerodynamics, which is important for the initial design stage. The wind tunnel test is the basic method of aerodynamics research in recent years, which can establish the experimental database of aerodynamic model and provide data support for the establishment and development of aerodynamic model in the real application. The flight test is the closest way to the real flight mode, which can get the first-hand flight data and provide technical support for the aerodynamic model. However, these methods have certain defects. The theoretical calculation is hampered by limited theoretical knowledge and computing power of machine. The approximation error between the mathematical model and the real model cannot be eliminated. The accuracy of the established aerodynamic model is poor. The wind tunnel test uses the scaled-down model of real vehicles. The experiment results may be influenced by the tunnel size, structure and so on. Hence, the generated flight data may differ from the real flight data. The flight experiment is the closest to the real flight state. However, the cost of the launch experiment is quite high, and the aerodynamic parameter can only be indirectly estimated based on the motion measurement of the vehicle.

The process of aerodynamic parameter identification is a systematic process, including four aspects: [1]

(1) Experimental design. Through the experiment to obtain a sufficient amount of information and a sufficient number of experimental data.

(2) Aerodynamic model determination. A suitable mathematical model is constructed according to certain experience and guidelines.

(3) Aerodynamic parameter identification process. Determine a set of model parameter values based on the experimental data and the established model. Then, the numerical results calculated by the model can best fit the test data.

(4) Model check. Through the operation to check whether the established model conforms to the flight dynamics model of the vehicle.

There are many factors that affect the accuracy of aerodynamic parameter identification, such as sensor accuracy, wind speed influence and control system design. Also, there are many difficulties in the estimation algorithm design, in addition to the parameter estimation criterion. The most widely used methods for aerodynamic parameter identification are the least squares method, the Kalman filter, and the maximum likelihood estimation method.

7.1.2 Approaches for Aerodynamic Parameters Estimation

In system identification, least squares method is one of the most basic estimation methods that can be used for static systems as well as dynamic systems. In practical application, measurement data are often given in time order. In order to reduce the computation cost and the memory requirement, recursive least squares is usually used. The basic recursive least squares method has the following advantages:

(1) Unknown parameters can be easily found and the sum of the squares of the errors between the predicted and the actual measurements can be minimized.

(2) It gives the best parameter fit in a statistical sense when the measurement noise follows the Gaussian distribution and each measurement is independent.

However, the basic recursive least squares method has some drawbacks and limitations in dealing with the complicated system:

(1) It requires the calculation of inverse of matrix, which may not exist.

(2) If the number of samples is relatively large, the cost for computing the inverse matrix is large.

(3) It cannot be directly used for nonlinear system.

There are many works for identifying aerodynamic parameters by using least squares method. Zanette et al. [2] proposed a new calculation tool named RealSysId, which used the recursive orthogonal least square method for aerodynamic parameter identification. The recursive orthogonalization process obtained the matrices Q and R through the Givens rotation method. In this process, it was not necessary to store all the data matrices, which reduced the calculation amount of the algorithm and improved the real-time performance of system identification. Yang et al. [3] proposed a least-squares algorithm to identify the aerodynamic parameters of the projectile. The pathological problems existing in the traditional least squares method were effectively solved by dividing the whole launching process into three dynamic processes and establishing the aerodynamic model separately. Tang et al. [4] proposed a numerically robust least squares algorithm based on vector orthogonal polynomials. This method used matrix scores to describe the model. Expanding the numerator and denominator polynomial matrices on the basis of vector orthogonality, a very suitable numerical substitution was found. This method overcame the numerical problem of the least squares estimator. Guibert et al. [5] proposed an aerodynamic parameter identification method using constrained least squares. It was used in identifying the lift coefficients of the aircraft based on a new segmented polynomial model. The method improved computational accuracy compared to similar segmented models.

Kalman filtering is a recursive filtering method derived from the principle of linear unbiased minimum variance estimation. Chowdhary et al. [6] conducted an analysis and research on the unscented Kalman filter method used for aerodynamic parameter identification. This method could handle nonlinear systems and reduce approximation error of linearization. At the same time, it could speed up system convergence and improve the reliability of system estimation. When the system had considerable complexity, this method was better than the extended Kalman filter method in performance. Although it could be a good alternative to the extended Kalman filtering method, it would significantly increase the amount of calculations in the system. The method should be further explored and improved. Li et al. [7] used the aerodynamic parameter identification method of unscented Kalman filtering to verify that nonlinear models could be approximated by linear models. The state of the system was expanded by adding the estimated parameters. The application of this method proved that if the accuracy loss was ignored, the linear model could be used for the control design of the aircraft, and the control performance was well. This method could effectively improve the convergence speed and robustness of the system. Compared with the method proposed by Chowdhary et al., the method proposed by Li et al. used an augmented state unscented Kalman filter method, it was an improvement on the former method. This method proved that not only the nonlinear model could be used for system identification, but also the simplified linear model could be, and they had similar identification results. It could simplify the system modeling process, reduce costs, and shorten the design cycle. An adaptive unscented Kalman filtering method for aerodynamic parameters identification was proposed by Majeed et al. [8]. It utilized two parallel unscented Kalman filters (UKF). The master UKF estimated the states and parameters using the noise covariance obtained by the slave UKF, while the slave UKF estimated the noise covariance using the innovations generated by the master UKF. The system could identify the system with unknown noise and determine the vehicle parameters in the uncertain environment where the noise characteristics change rapidly. However, the computational cost for this kind of two-level approach may be large and it may not be suitable for online application. Ding et al. [9] proposed a Bayesian adaptive unscented Kalman filter for aerodynamic parameter identification. This method combined Bayesian inference method and unscented Kalman filter method to jointly estimate the covariance coefficient of unknown noise. In the process of using the unscented Kalman filter, the Gauss-Newton method was used to maximize the posterior likelihood function of the unknown parameter estimation. This method had good filtering characteristics, high estimation accuracy and calculation efficiency. Compared with the adaptive unscented Kalman filter method proposed by Majeed et al., the Bayesian adaptive unscented Kalman filter method proposed by Ding et al. could reduce the calculation cost of the system and improve the calculation efficiency of the system. This method had better real-time performance and a wider range of applications. Zhang et al. [10] proposed a real-time estimation algorithm for unsteady aerodynamics. The Kalman filter was used in the algorithm based on the constant acceleration model to identify the unsteady terms in the aerodynamic parameter identification model. The non-Gaussian noise was dealt with by introducing the maximum correlation entropy criterion. The fast time-varying characteristics of the system could be effectively tracked and the accuracy of aerodynamic parameter estimation could be improved in this real-time estimation method. Although the Kalman filter method has been widely used in aerodynamic parameter identification, there are several disadvantages. The first is that the state model and noise statistical properties of the system must be known precisely, which is difficult in practice. When inaccurate or incorrect system models or noise statistical properties are used, the filtering can make the state estimation error large. Second, the computation cost of Jacobi matrices is heavy for high nonlinear system. Finally, the convergence of Kalman filter for nonlinear system should be improved.

In addition to the least squares and Kalman filter, there are many other kind of approaches for aerodynamic parameters estimation. In order to study the effect of icing on the tail of the aircraft on the flight status, Xu et al. [11] used the maximum likelihood estimation method. According to the flight data of the aircraft, the aerodynamic parameters of the normal model and the two interference models are identified. The identification results proved that the maximum likelihood estimation method was effective and also the lift and drag characteristics would be greatly affected by the icing on the tail of the aircraft. It provided a reference for the model improvement of the aircraft. Kumar et al. [12] estimated the aerodynamic parameters of the vehicle by analyzing flight data using traditional methods and neural network-based methods. The state of the system model should be estimated before system identification by the traditional estimation methods, while the neural network-based method had the advantage that a priori mathematical model was not necessary. Both methods had the similar estimation accuracy. Wang et al. [13] proposed an RBF neural network that automatically added hidden nodes to identify the aerodynamic parameters of the reentry vehicle. The initial weights were optimized during neural network training, the identification accuracy was improved and the training time was shortened. At the same time, satisfactory identification results could be obtained by using this method to identify the parameters in the severely nonlinear state equation. Morelli [14] proposed a method for real-time estimation of aerodynamic parameters using flight data. More accurate modeling results could be provided without measuring the airflow angle of the aircraft. The airflow angle was reconstructed in the time domain model, after removing the constant deviation and drift from the time domain data, and then Fourier transform was applied. Real-time parameter estimation was performed in the frequency domain. In this identification method, the flight cost of the aircraft was effectively reduced, and the stability and safety of the aircraft were improved. An aerodynamic parameter identification method using traditional linear theory of nonlinear programming was proposed by Burchett [15]. After the linear theory solution was reformulated, the aerodynamic parameters and initial conditions were easier to be distinguished. The aerodynamic parameters and angular rate derivatives were obtained in a closed form. The derivatives were used to improve the gradient-based parameter estimation. Simulation results showed that the estimation accuracy of the system parameters could be effectively improved. Bagherzadeh et al. [16] proposed a global nonlinear aerodynamic parameter identification method. The amplitude and frequency models were decomposed and analyzed through an empirical mode decomposition algorithm. The non-linear behavior of the system could be effectively predicted and analyzed. The stability of the system also could be improved. In order to eliminate the error caused by the linearization of the model, an output error method proposed by Tu et al. [17] was used to identify the aerodynamic parameters of the nonlinear model. The nonlinear model with measurement noise was simulated and analyzed, and the influence of model error on the accuracy of the identification result was reduced. Dou et al. [18] proposed a method with a combination of maximum expectation (EM) and extended Kalman filtering (EKF) methods for aerodynamic parameter identification of the return segment of a reusable vehicle. The EKF was used to identify the system, which could effectively filter out the noise and estimate the unknown aerodynamic parameters, and the EM was used to estimate the a priori statistical distribution of measurement noise and process noise in the EKF process, which could reduce the influence of measurement and process noise on the system.

In this work, the aerodynamic parameter estimation for launch vehicles is considered based on simultaneous approach and maximum likelihood principle. Also, the prediction accuracy of the model with estimated values is focused on, not accuracy of the estimation for the aerodynamic parameters themselves.

7.2 Statistic Criterion Based Aerodynamic Parameter Estimation

The rocket is assumed to be a symmetric cylindrical shape without considering the side slip angle. In the velocity coordinate, the motion of the rocket can be described as follows, [19]

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{dx}{dt}=\frac{R_{e}}{R_{e}+h}v\cos \theta \\ \frac{dh}{dt}=v\sin \theta \\ \frac{dv}{dt}=\frac{P_{e}\cos \alpha }{m}-\frac{\rho v^{2}S_{m}C_{D}}{2m}-g\sin \theta \\ \frac{d\theta }{dt}=\frac{P_{e}\sin \alpha }{mv}+\frac{\rho v^{2}S_{m}C_{L}}{2mv}-\frac{g\cos \theta }{v}+\frac{v\cos \theta }{R_{e}+h} \\ \end{array}\right. } \end{aligned}$$

(7.1)

where v is the speed, $P_{e}$ is the axial thrust, $S_{m}$ is the rocket reference area, $\theta$ is the flight path angle, $R_{e}$ is the radius of the earth, h is the flight altitude, m is the mass of the rocket, x is the distance of the rocket. The attack angle $\alpha$ is a control variable. The considered time span is $\mathrm{{[0, }}\,\,{t}_{f}\mathrm{{]}}$. The atmosphere density $\rho$ is calculated based on Yang’s model [20]. The drag coefficient $C_{D}$ and lift coefficient $C_{L}$ are considered as time dependent parameters. Simultaneous approach [21] is applied for aerodynamic parameter estimation and the motion model of rocket should be discretized. Formulation (7.1) can be converted into algebraic equations by approximating the state and control variables using a family of polynomials on finite elements $\mathrm{{[}}{\mathrm{{t}}_{i - 1}}\mathrm{{, }}\,\,{\mathrm{{t}}_i}\mathrm{{]}}$, which satisfied

$$\begin{aligned} {t_0}< {t_1}< \mathrm{{ }} \cdots < {t_N} = {t_f} \end{aligned}$$

(7.2)

where N is the number of finite elements. Lagrange interpolation polynomials are used as [21]

$$\begin{aligned} {\left\{ \begin{array}{ll} x\approx x^{\left( K\right) }= \sum \limits _{j=0}^{K} l_{j}\left( \tau \right) x_{ij},h\approx h^{\left( K\right) }=\sum \limits _{j=0}^{K} l_{j}\left( \tau \right) h_{ij} \\ v\approx v^{\left( K\right) }= \sum \limits _{j=0}^{K} l_{j}\left( \tau \right) v_{ij},\theta \approx \theta ^{\left( K\right) }= \sum \limits _{j=0}^{K} l_{j}\left( \tau \right) \theta _{ij} \\ \alpha \approx \alpha ^{\left( K\right) }= \sum \limits _{j=1}^{K} \bar{l}_{j}\left( \tau \right) \alpha _{ij} \\ l_{j}(\tau )=\sum \limits _{k=0, \ne j}^{K}\frac{(\tau -\tau _{k})}{(\tau _{j}-\tau _{k})},\bar{l}_{j}(\tau )=\sum \limits _{k=1, \ne j}^{K}\frac{(\tau -\tau _{k})}{(\tau _{j}-\tau _{k})} \end{array}\right. } \end{aligned}$$

(7.3)

where K is the number of collocation points. The drag coefficient $C_{D}$ and lift coefficient $C_{L}$ are approximated by piecewise linear function

$$\begin{aligned} {\left\{ \begin{array}{ll} C_{D}\approx C_{D}^{\left( m-1\right) }+\frac{\left( C_{D}^{\left( m\right) }-C_{D}^{\left( m-1\right) } \right) }{\left( t_{i_{m}^{D}}-t_{i_{m-1}^{D}}\right) }\left( t-t_{i_{m-1}^{D}}\right) ,m=1,\ldots ,N_{D} \\ C_{L}\approx C_{L}^{\left( m-1\right) }+\frac{\left( C_{L}^{\left( m\right) }-C_{L}^{\left( m-1\right) } \right) }{\left( t_{i_{m}^{L}}-t_{i_{m-1}^{L}}\right) }\left( t-t_{i_{m-1}^{L}}\right) ,m=1,\ldots ,N_{L} \\ \end{array}\right. } \end{aligned}$$

(7.4)

$N_D$ and $N_L$ are the number of segments for $C_D$ and $C_L$. $i_m^D$ and $i_m^L$ are elements from 1...N. It is assumed that the measured outputs are distance x, height h and velocity v. Based on Eqs. (7.1)–(7.4), the formulation for aerodynamic parameter estimation can be established. For simplifying the following discussion, the resulting nonlinear programming for aerodynamic parameter estimation is represented as follows,

$$\begin{aligned} \begin{aligned} \begin{array}{l} \min \,J = \frac{1}{2}{(\mathbf{{y}} - {\mathbf{{y}}^{(meas)}})^T}{} \mathbf{{C}}_y^{ - 1}(\mathbf{{y}} - {\mathbf{{y}}^{(meas)}})\\ s.t.\left\{ \begin{array}{l} \mathbf{{c}}({\boldsymbol{\gamma }},\mathbf{{p}}) = 0\\ \mathbf{{y}} = \mathbf{{h}}({\boldsymbol{\gamma }}) \end{array} \right. \end{array} \end{aligned} \end{aligned}$$

(7.5)

where $\boldsymbol{\gamma \in R}$ $^{ng}$ represents the vector of distance x, height h, velocity v, flight path angle $\theta$ at all the discretization points. $\mathbf {c:R}$ $^{ng}\mathbf {\rightarrow R}$ $^{ng}$ represents the discretization of (7.1) based on Eqs. (7.2)–(7.4) and $\mathbf {h:R}$ $^{ng}\mathbf {\rightarrow R}$ $^{ny}$ represents the relationship between system output and state variables. $\mathbf {y \in R}$ $^{ny}$represents the vector of distance x, height h, velocity v at all the sampling points. $\textbf{y}$ $^{(meas)}$ $\mathbf {\in R}$ $^{ny}$ represents the measurements vector corresponding to $\textbf{y}$. $\textbf{C}$ $_{y}$ is a diagonal matrix, and each diagonal element is the variance of the measurement noise corresponding to $\textbf{y}$. The Jacobian of $\textbf{c}$ with respect to $\boldsymbol{\gamma }$ is a square matrix. For all physically meaningful parameters $\textbf{p}$ $= \mathrm{{ }}{[{C_D}^{\left( 1 \right) },{C_D}^{\left( 2 \right) },\ldots ,{C_D}^{(ND)},{C_L}^{\left( 1 \right) },{C_L}^{\left( 2 \right) },\ldots ,{C_L}^{(NL)}]^T}$, the Jacobian is usually assumed to be non-singular, and it satisfies the linear independence constraint qualification (LICQ). The reduced Hessian of the Lagrange function for (7.5), denoted by $\textbf{H}$ $r$ which is calculated as [22]

$$\begin{aligned} \mathbf{{Z}} = \left[ {\begin{array}{*{20}{c}} \mathbf{{\textrm{I}}}\\ { - {{\left[ {\begin{array}{*{20}{c}} {{\nabla _\mathbf{{\gamma }}}{\mathbf{{c}}^T}}&{}\mathbf{{0}}\\ {{\nabla _\mathbf{{\gamma }}}{\mathbf{{h}}^T}}&{}{ - \mathbf{{I}}} \end{array}} \right] }^{ - 1}}\left[ {\begin{array}{*{20}{c}} {{\nabla _\mathbf{{\gamma }}}{\mathbf{{c}}^T}}\\ {{\nabla _\mathbf{{\gamma }}}{\mathbf{{h}}^T}} \end{array}} \right] } \end{array}} \right] ,{\mathbf{{H}}_\mathbf{{r}}} = \left( {{\mathbf{{Z}}^T}\left[ {\begin{array}{*{20}{c}} \mathbf{{0}}&{}\mathbf{{0}}&{}\mathbf{{0}}\\ \mathbf{{0}}&{}\mathbf{{0}}&{}\mathbf{{0}}\\ \mathbf{{0}}&{}\mathbf{{0}}&{}{\mathbf{{C}}_\mathbf{{y}}^{ - 1}} \end{array}} \right] \mathbf{{Z}}} \right) \end{aligned}$$

(7.6)

In this work, reduced Hessian based transformed parameter strategy [22] is used for eliminating the correlation among the aerodynamic parameters. The reduced Hessian $\textbf{H}$ $r$ is decomposed as

$$\begin{aligned} {\mathbf{{H}}_\mathbf{{r}}} = \mathbf{{V}\left[ {\begin{array}{*{20}{c}} {{\lambda _1}}&{}{}&{}{}&{}{}\\ {}&{}{{\lambda _2}}&{}{}&{}{}\\ {}&{}{}&{} \ddots &{}{}\\ {}&{}{}&{}{}&{}{{\lambda _{{N_D} + {N_L}}}} \end{array}} \right] }{} \mathbf{{V}}^{T} \end{aligned}$$

(7.7)

where $\textbf{V}$ is the orthogonal matrix and the ${\lambda _1},{\lambda _2},\mathrm{{}}\ldots ,\;{\lambda _{ND+NL}}$ with descending order are eigenvalues. The aerodynamic parameters $\textbf{p}$ can be transformed as

$$\begin{aligned} \mathbf{{q = V}}^{T}{} \mathbf{{p}} \end{aligned}$$

(7.8)

where $\textbf{q}$ is the vector of transformed parameters. Different from the mean squared error criterion [23], the number of estimated transformed parameters is determined based on the modified E-optimal design criterion [24] and $\textbf{q}$ is divided into ${[{\mathbf{{q}}_1}^T,{\mathbf{{q}}_2}^T]^T}$, where only ${\mathbf{{q}}_1}$ are estimated. Usually, the unestimated parameters ${\mathbf{{q}}_2}$ are fixed at the nominal values ${\mathbf{{q}}_2}^{(trial)}$ and the following problem is solved,

$$\begin{aligned} \begin{array}{l} \min \,J = \frac{1}{2}{(\mathbf{{y}} - {\mathbf{{y}}^{(meas)}})^T}{} \mathbf{{C}}_y^{ - 1}(\mathbf{{y}} - {\mathbf{{y}}^{(meas)}})\\ s.t.\left\{ \begin{array}{l} \mathbf{{c}}({\boldsymbol{\gamma }},\mathbf{{p}}) = 0\\ \mathbf{{y}} = \mathbf{{h}}({\boldsymbol{\gamma }})\\ \mathbf{{q}} = {[\mathbf{{q}}_1^T,\mathbf{{q}}_2^T]^T} = {\mathbf{{V}}^T}{} \mathbf{{p}}\\ {\mathbf{{q}}_\mathrm{{2}}} = \mathbf{{q}}_2^{(trial)} \end{array} \right. \end{array} \end{aligned}$$

(7.9)

However, unreasonable nominal values for the fixed parameters significantly affect the prediction of the model. Statistic criterion based approach is used for determining the nominal values for unestimated parameters and the statistic criterion is designed as [22]

$$\begin{aligned} \begin{array}{l} \eta = |skewnes{s_1} - 0| + |kurtosi{s_1} - 3|\\ + |skewnes{s_1} - skewnes{s_2}| + |kurtosi{s_1} - kurtosi{s_2}|\\ skewnes{s_1} = \frac{{\frac{1}{{ny}}\sum \limits _{i = 1}^{ny} {{{\left( {{\varepsilon _i} - 0} \right) }^3}} }}{{{\sigma ^3}}},skewnes{s_2} = \frac{{{z_3}}}{{z_2^{1.5}}}\\ kurtosi{s_1} = \frac{{\frac{1}{{ny}}\sum \limits _{i = 1}^{ny} {{{\left( {{\varepsilon _i} - 0} \right) }^4}} }}{{{\sigma ^4}}},kurtosi{s_2} = \frac{{{z_4}}}{{{z_2}^2}}\\ {\varepsilon _i} = {\mathbf{{y}}_i}({\boldsymbol{\alpha }}({\boldsymbol{\alpha }}_2^{(trial)})) - \mathbf{{y}}_i^{(m)},\bar{\varepsilon }= \frac{1}{{ny}}\sum \limits _{i = 1}^{ny} {{\varepsilon _i}} \\ {z_2} = \frac{1}{{ny}}\sum \limits _{i = 1}^{ny} {{{\left( {{\varepsilon _i} - \bar{\varepsilon }} \right) }^2}} ,{z_3} = \frac{1}{{ny}}\sum \limits _{i = 1}^{ny} {{{\left( {{\varepsilon _i} - \bar{\varepsilon }} \right) }^3}} ,{z_4} = \frac{1}{{ny}}\sum \limits _{i = 1}^{ny} {{{\left( {{\varepsilon _i} - \bar{\varepsilon }} \right) }^4}} \end{array} \end{aligned}$$

(7.10)

where the subscript i in (7.10) represents the i-th component of the vector $\mathbf{{y}}$, ny is the dimension of $\mathbf{{y} }$ and the measurement noise follows Gauss probability distribution $N(0,{\sigma ^2})$. The statistic criterion based approach [22] combined with the modified E-optimal design criterion for estimating the aerodynamic parameters are described as

Step 1: Let $\mathbf{{p}}^{(trial)} = \mathbf{{p}}^{(0)}$ and initialize n_iter.

Step  2:  Calculate the reduced Hessian matrix at ${\mathbf{{p}}^{(trial)}}$ and the orthogonal matrix $\textbf{V}$.

Step 3: Calculate the number of estimated parameters np based on the modified E-optimal design criterion and the transformed parameter vector $\textbf{q}$ is partitioned into $\mathbf{{q}}_{1}$ and $\mathbf{{q}}_{2}$ according to np.

Step 4: Let $iter=0$ and $\eta ^{(opt)}=Infinity$.

Step 5: Obtain $\mathbf{{p}}^{(trial)}$ by sampling based on the given range and the distribution of $\mathbf{{p}}$ and then $\mathbf{{q}}^{(trial)} = [\mathbf{{q}}_{1}^{(trial)T}, \mathbf{{q}}_{2}^{(trial)T}]^{T}$ is equal to $\mathbf{{V}}^{T}{} \mathbf{{p}}^{(trial)}$.

Step 6: Fix $\mathbf{{q}}_{2}$ at ${\mathbf{{q}}_{2}}^{(trial)}$, and estimate the selected parameters $\mathbf{{q}}_{1}$ by solving the NLP problem (7.9).

Step 7: Calculate the criterion $\eta$ based on Eq. (7.10). If $\eta ^{(opt)} > \eta$, go to Step 8, otherwise go to Step 9.

Step 8: Let $\eta ^{(opt)} = \eta$, $\mathbf{{q}}_{2}^{(opt)}=\mathbf{{q}}_{2}^{(trial) }$, $iter=0$.

Step 9: Let $iter=iter+1$.

Step 10: If $iter==$ n_iter, print the solution $\mathbf{{q}}$ corresponding to $\eta ^{(opt)}$ and stop, otherwise return to Step 5.

7.3 Numerical Results

Only the ascending stage of rocket launching is considered and the total flying time is 66 s. The measurement sampling time is 10 ms and the standard deviations of the measurement error for x, h and v are 1m, 1m and 1m/s respectively. For simplicity, the true values for drag coefficient $C_{D}$ and lift coefficient $C_{L}$ are assumed to be 0.5 and 0.1 respectively. Although the aerodynamic parameters are set to be constant here, they can be handled as time dependent. $N_{D}$ is set to 6, i.e. the drag coefficient is approximated by 6-segment linear function. $N_{L}$ is set to 1 and $C_{L}^{(0)}=C_{L}^{(1)}$ is required. Hence, there are 8 parameters should be estimated. The initial guesses for drag coefficient and lift coefficient are ${C_D}^{\left( 0 \right) } = {0.1_,}\;{C_D}^{\left( 1 \right) } = {0.6_,}\;{C_D}^{\left( 2 \right) } = {0.1_,}\;{C_D}^{\left( 3 \right) } = {0.6_,}\;{C_D}^{\left( 4 \right) } = {0.1_,}\;{C_D}^{\left( 5 \right) } = {0.6_,}\;{C_D}^{\left( 6 \right) } = {0.1_,}\;{C_L}^{\left( 0 \right) } = 0.12$. The threshold for the modified E-optimal design criterion is set to 3000. The procedure for determining $C_{D}$ and $C_{L}$ is terminated if the criterion $\eta$ has not been updated n_iter ($=$2000) times from the last update. The lower and upper bound for $C_{D}$ and $C_{L}$ is set to 1.0$\times 10 ^{-4}$ and 1.0 respectively.

Based on the modified E-optimal design criterion, $np=5$ is obtained. Namely, there are 5 estimated parameters and 3 parameters are fixed. The results corresponding to the initial guess $\mathbf{{p}}^{(0)}$ (with fixed $\mathbf{{q}}_{2}^{(0)}$ ) are shown in the Table 7.1 with index 0. Since there are random sampling involved in the algorithm, 10 experiments are performed. The estimated values for $C_{D}$ and $C_{L}$ corresponding to the 10 experiments are shown in the Table 7.1 with index from 1 to 10.

Table 7.1 The estimation results corresponding to the initial guess $\mathbf{{p}}^{(0)}$ and ten runs of experiments

The average values for ${C_D}^{(0)} \sim {C_D}^{(6)}$ and $C_{L}^{(0)}$ in ten runs of experiments are 0.0184218, 0.625591, 0.463646, 0.493976, 0.508575, 0.474668, 0.549004 and 0.393578 respectively. Compared with the true values, the relative errors of the average estimated values for $C_{D}^{(0)}$, $C_{D}^{(1)}$ and $C_{L}^{(0)}$ are over $20\%$, while others are less than $10\%$.

There are total 6600 samplings for each state variable. The average errors of the state variables x, h and v corresponding to each group of estimated values are shown in the Table 7.2.

Table 7.2 The average errors of the state variables x, h and v corresponding to each group of estimated values

Table 7.3 The prediction accuracy of the model based on the aerodynamic parameters from the initial guess $\mathbf{{p}}^{(0)}$ (with fixed $\mathbf{{q}}_{2}^{(0)}$) and the 9th group in Table 7.1

From Table 7.2, we can see that any group (from index 1 to 10) of estimated values obtained by the algorithm is much better than those of the results corresponding to the initial guess $\mathbf{{p}}^{(0)}$ (with fixed $\mathbf{{q}}_{2}^{(0)}$). The mean values for “average error for x”, “average error for h” and “average error for v” in ten runs of experiments are 0.014094, 0.161 and 0.06484, which are better than the results corresponding to the initial guess $\mathbf{{p}}^{(0)}$ (0.0309, 0.497, 0.152).

The 9th group of estimated values correspond to the least statistic criterion $\eta ^{(opt)}$. Although it is not the best group of estimated values for $C_{D}$ and $C_{L}$, the prediction accuracy of the model based on the aerodynamic parameters from the initial guess $\mathbf{{p}}^{(0)}$ (with fixed $\mathbf{{q}}_{2}^{(0)}$) and the 9th group in Table 7.1 are demonstrated with perturbations on the initial state of x, h and v (the true initial states are 0, 0 and 0 respectively). The randomly generated initial states (following Gauss distribution ${N\mathrm {(0,1)}}$) and the corresponding results with the initial guess $\mathbf{{p}}^{(0)}$ (with fixed $\mathbf{{q}}_{2}^{(0)}$) and the 9th group in Table 7.1 are shown in the Table 7.3.

From Table 7.3, we can see that the performance of the aerodynamic parameters from the 9th group in Table 7.1 is still superior to the initial guess $\mathbf{{p}}^{(0)}$ (with fixed $\mathbf{{q}}_{2}^{(0)}$) even if the initial states are perturbed.

7.4 Conclusions

It is difficult to estimate all the aerodynamic parameters of launch vehicles based on the measurements of the distance, height and velocity. The traditional approach is to fix the unestimated variables on the nominal values before estimation, however the selection of the nominal values have significant impact on the prediction accuracy of the model. In this work, the modified E-optimal design criterion and the statistic criterion based approach is used for estimating the aerodynamic parameters. The numerical results show that the prediction accuracy can be remarkably improved even when the initial states are perturbed.