Aircraft robust trajectory optimization method based on SRSM and NIPC

Abstract

The application provides an aircraft robust trajectory optimization method based on SRSM and NIPC, belongs to the field of aircrafts, and can quickly solve a robust trajectory optimization problem containing multiple parameter uncertainties, and comprises the following steps: constructing an aircraft robust trajectory optimization problem; constructing an uncertainty quantization propagation model based on NIPC and SRSM, converting the above-mentioned aircraft robust trajectory optimization problem into an equivalent high-dimensional deterministic optimal control problem; performing convexization and discretization processing on the problem, converting the problem into a parameter convex optimization problem, and then performing optimization and solving on the problem based on a sequential convex optimization algorithm to obtain optimal control quantities and complete aircraft trajectory optimization. The application can obtain an optimized trajectory with reliability and robustness, and has considerable precision and significant computational efficiency advantages.

Description

Technical Field

[0001] This invention belongs to the field of aircraft technology, specifically relating to a robust trajectory optimization method for aircraft based on SRSM and NIPC. Background Technology

[0002] As a key technology in aircraft design, trajectory optimization has gradually become a hot research topic in aerospace fields such as hypersonic flight. However, existing trajectory optimization research usually only focuses on improving trajectory performance under nominal conditions, without considering the robustness and reliability of the trajectory under real-world conditions. In actual flight missions, there are many uncertainties in initial states, dynamics, and environmental parameters, which inevitably lead to significant process and terminal deviations when flying along the nominal trajectory. This increases the risk of constraint violations and reduces flight reliability and terminal mission accuracy. Therefore, in order to improve the reliability and anti-interference capability of optimized trajectories, some scholars have gradually focused their research on robust trajectory optimization that considers uncertainties. That is, they consider the impact of uncertainties in advance during the trajectory design stage, thereby reducing the time consumption of trajectory planning iterations and the burden on the guidance and control system caused by trajectory deviation correction.

[0003] A typical strategy for solving robust trajectory optimization problems is to first transform the stochastic optimal control problem containing stochastic differential equations into a deterministic optimal control problem containing extended deterministic differential equations, and then use efficient trajectory optimization algorithms to numerically solve this high-dimensional problem. This involves two key techniques: uncertainty quantization propagation and deterministic trajectory optimization. Typical uncertainty quantization propagation theories and methods include Monte Carlo (MC), linear covariance analysis, unscented transformation, and polynomial expansion (PC). Monte Carlo methods are widely used for uncertainty quantification and propagation due to their simplicity and ease of implementation, but extensive numerical simulations can lead to slow convergence of statistical results. Linear covariance analysis can quantify and propagate uncertainties in linear Gaussian process models and has high computational efficiency, but it is not suitable for highly nonlinear trajectory optimization problems. In contrast, PCE represents random variables as a weighted sum of random orthogonal polynomials, which can achieve a relatively accurate approximation of random variables with arbitrary distribution types, while achieving accuracy comparable to the Monte Carlo method at a lower computational cost. In recent years, it has been widely used in the quantification of uncertainty in Mars entry trajectories and the analysis of fluid dynamic uncertainties. There are two ways to combine PCE methods with system models: embedded and non-embedded. Embedded chaotic polynomial (IPC) substitutes the chaotic polynomial expansion of random variables into the dynamic differential equations and uses the Galerkin projection method to transform the stochastic differential equations into a set of high-dimensional deterministic differential equations with chaotic polynomial coefficients as state variables. Finally, uncertainty propagation is completed by solving this set of differential equations, which has high computational efficiency. Non-embedded chaotic polynomial (NIPC) treats the original system model as a "black box" and then approximates the chaotic polynomial coefficients according to the input and output of the model. Therefore, it can handle the uncertainty quantification and propagation problem of general nonlinear systems and has good versatility and engineering application value.

[0004] For another key technology, deterministic trajectory optimization, common methods mainly include indirect and direct methods. Indirect methods are difficult to widely apply in complex-constrained trajectory optimization problems because they are sensitive to guessing initial values, require derivation of necessary optimality conditions, and cannot handle complex constraints. In recent years, direct methods, represented by pseudospectral methods and convex optimization, have been widely used in trajectory optimization tasks in aerospace fields such as hypersonic navigation, demonstrating good numerical computation advantages and universality. Among them, convex optimization methods, due to their polynomial complexity, fast convergence speed, and the fact that their local optimal solutions are theoretically also global optimal solutions, have been gradually used to solve trajectory optimization problems under complex constraints and high-dimensional state extension optimization problems considering uncertainties.

[0005] Given the advantages of PCE and direct methods, many studies have organically combined the two, designing various robust trajectory optimization algorithms and solution strategies based on PCE and direct methods. Fisher et al. first proposed a framework for numerically solving stochastic optimal control problems. This framework uses IPC to transform stochastic optimal control problems into high-dimensional deterministic optimal control problems and employs the collocation method to solve them. Li et al. combined NIPC with the Gaussian pseudospectral method to solve the climb robust trajectory optimization problem for hypersonic aircraft with uncertain aerodynamic coefficients. Jiang et al. combined NIPC with the hp pseudospectral method, using the OpenCossan toolbox for uncertainty quantization propagation and employing parallel computing technology to improve the solution efficiency of high-dimensional deterministic optimization problems, solving the robust trajectory optimization design problem for Mars reentry with uncertain initial states and model parameters. Wang et al. proposed a robust trajectory optimization method combining IPC and convex optimization, and compared it with methods based on IPC and the Gaussian pseudospectral method, demonstrating that the proposed method significantly improves computational efficiency while maintaining comparable computational accuracy. Yang Ben et al. established a robust optimization model for reentry trajectory that combines NIPC and convex optimization based on the least squares method, which improved the reliability of the trajectory under uncertain aerodynamic parameters.

[0006] While stochastic problem-solving techniques based on PCE and direct methods have been gradually applied to robust trajectory optimization, they still suffer from low computational efficiency, poor versatility, and difficulty in handling multidimensional uncertainties. This limits their application in complex constraints and strongly nonlinear robust optimization problems (such as robust trajectory optimization in aerospace). Firstly, while traditional direct methods, such as the Gaussian pseudospectral method, have solved some robust trajectory optimization problems, they still face limitations such as low computational efficiency and difficulty in quickly solving multidimensional uncertainties. Furthermore, the combination of convex optimization techniques and PCE remains imperfect. Secondly, while IPC-based optimization methods have high computational efficiency, they require derivation, modeling, and code rewriting for different problems. Due to model simplification and approximation in the derivation process, their versatility is poor, and they are only applicable to polynomial nonlinear problems. Although NIPC-based optimization methods can avoid the above problems, the PCE coefficient solution is mostly based on quadrature and sampling methods, such as Gaussian quadrature (GQ) and sparse grid quadrature. As the dimension and nonlinearity of the system's random variables increase, the state expansion dimension of these methods will grow exponentially, resulting in a significant increase in the number of discrete state nodes. This makes it very difficult to solve large-scale NLP problems, with high computational costs and low efficiency. Especially when the dimension of uncertain factors n>3, the large amount of memory consumption and computational costs will seriously hinder the optimization process and even lead to optimization failure.

[0007] It is worth noting that, in addition to the aforementioned quadrature-based methods, NIPC coefficients can also be calculated using the Stochastic Response Surface Method (SRSM). This method, based on the surrogate model concept, obtains a chaotic polynomial model by fitting the relationship between the system output and input. Combined with typical sampling methods, a small number of samples are sufficient to calculate the polynomial coefficients, significantly reducing the transformation dimension and scale of the model, thus achieving high accuracy and computational efficiency. Currently, SRSM is widely used in fluid dynamics uncertainty analysis, but it has not yet been widely applied in robust trajectory optimization design. Summary of the Invention

[0008] To address the aforementioned technical problems, this invention proposes a robust trajectory optimization method for aircraft based on SRSM and NIPC.

[0009] To achieve the above objectives, the present invention provides the following technical solution:

[0010] A robust trajectory optimization method for aircraft based on SRSM and NIPC includes the following steps:

[0011] Constructing a robust trajectory optimization problem for an aircraft under conditions of parameter uncertainty P R ;

[0012] An uncertainty quantification propagation model is constructed based on the non-embedded chaotic multinomial (NIPC) and stochastic response surface methodology (SRSM) to address the robust trajectory optimization problem of aircraft. R Transformed into a high-dimensional deterministic optimal control problem P ED ;

[0013] Using successive linearization to solve the high-dimensional deterministic optimal control problem P ED Convexification to a convex optimization problem P ED-Convex ;

[0014] The convex optimization problem P ED-Convex Discretization is performed to obtain the parametric convex optimization problem P. ED-disConvex Furthermore, an optimization algorithm based on sequential convex optimization is constructed to iteratively solve the convex optimization problem P. ED-disConvex Approaching the robust trajectory optimization problem of a spacecraft P R The solution;

[0015] The overall parameters, constraint parameters, uncertain parameters, and weight coefficients of the aircraft are used as input parameters. The optimal control quantity is obtained by calling the optimization solution algorithm based on sequential convex optimization, thus completing the aircraft trajectory optimization.

[0016] Preferably, the robust trajectory optimization problem P of the aircraft R The model is:

[0017]

[0018] In equation (1), x(t) is the state vector, u(t) is the control vector, and w is the system random parameter vector; min with u max The upper and lower boundaries of the control variable are defined by: J, f, g, and σ. The objective function is defined by: J, f by: f, g by: g ... J k g Let x be the weighting coefficient to be designed. 0μ With x 0σ The mean and standard deviation of the initial state are ε, respectively. f This is the upper bound of the standard deviation of the final value. It is a random differential vector of the state vector.

[0019] Preferably, the uncertainty quantification propagation model constructed based on the non-embedded chaotic multinomial (NIPC) and stochastic response surface methodology (SRSM) is used to address the robust trajectory optimization problem of the aircraft. R Transformed into a high-dimensional deterministic optimal control problem P ED Specifically:

[0020] By introducing a Latin hypercube sampling strategy and extending the stochastic differential vector of the state vector containing uncertain parameters into a high-dimensional ordinary differential equation with respect to the coefficients of the PC polynomial based on the stochastic response surface method (SRSM), the problem P is calculated using the PC polynomial coefficients. R The mean and standard deviation of the state, constraints, and objective function are used to further analyze the robust trajectory optimization problem P of the aircraft. R Transformed into a high-dimensional deterministic optimal control problem P ED .

[0021] Preferably, the robust trajectory optimization problem P of the aircraft... R Transformed into a high-dimensional deterministic optimal control problem P ED Specifically, it includes the following steps:

[0022] (1) PC expansion of the stochastic differential vector of the state vector

[0023] Assuming the random state vector x is n-dimensional and the system parameter vector w is s-dimensional, then each state variable x i (t) and parameter w k Expanded into a PC polynomial:

[0024]

[0025]

[0026] Where Δ={Δ1,…,Δ d} is a d-dimensional random vector, Φ j (Δ) is an orthogonal polynomial function, obtained through the univariate orthogonal basis function φ corresponding to the Δ distribution type. j (Δ l ), l=1,...,d, obtained by taking the tensor product; x ij (t) and w kj For the PC coefficients, the expansion term P+1 = [(p+d)!] / (p!d!), which is derived from the orthogonal basis functions φ. j The order p of (Δ) is determined by the dimension d of the random variable;

[0027] PC coefficient w kj and x ij (t0) According to w k and x i The distribution of (t0) is determined, and the coefficients x are derived using the random response surface method. ij The expression for (t);

[0028] Substituting equations (2) and (3) into the differential equation of equation (1), we get:

[0029]

[0030] Where i = 1, 2, ..., n, k = 1, 2, ..., s;

[0031] Using the Latin hypercube method to perform n s Random sampling was performed, and the random sample points were... Substituting into equation (4), we obtain the system of differential equations:

[0032]

[0033] In equation (5), X and W represent the PC coefficient vectors of the random state vector x and parameter w, respectively, and F(X,u,t,W) is a nonlinear function of the extended state X, and we have:

[0034]

[0035] W = [w 10 ,…,w 1P ,w 20 ,…,w 2P ,…,w s0 ,…,w sP ] T (7)

[0036]

[0037] Let Ω(Δ) m)=[Φ0(Δ m ),Φ1(Δ m ),…,Φ P (Δ m (m=1,2,...,n) s ),definition:

[0038]

[0039] but

[0040] If n s If ≥P+1, then the system of differential equations in equation (5) can be solved by the least squares regression method, i.e.:

[0041]

[0042] In equation (10), the constant matrix

[0043] sampling points n s =2(P+1), and the PC coefficient vector X(t) at each time step is obtained by integrating equation (10);

[0044] (2) Process constraints and PC expansion of the objective function

[0045] Since both the process constraint function g(x(t),u(t),t,w) and the objective function J(x(t),u(t),t,w) are functions of the state vector x(t), the chaotic polynomial expansion described above is used, as follows:

[0046]

[0047]

[0048] In equations (11) and (12), g j (x(t),u(t),t) and J j (x(t),u(t),t) are the PC polynomial coefficients of the corresponding function;

[0049] random sample points Substituting into equations (11) and (12), we get:

[0050]

[0051]

[0052] In equations (13) and (14), and Let g(x(t),u(t),t,w) and J(x(t),u(t),t,w) represent the PC coefficient vectors of functions g(x(t),u(t),t,w) and J(x(t),u(t),t,w) respectively, and let J(x(t),u(t),t,w) represent the functions of g(x(t),u(t),t,w) respectively.

[0053]

[0054]

[0055]

[0056]

[0057] Similarly, there are:

[0058] ψ=[Ω(Δ1) T ,Ω(Δ2) T ,…,Ω(Δ ns ) T ] T (19)

[0059] Similarly, the coefficient vector can be solved using the least squares regression method. and

[0060]

[0061]

[0062] In equations (20) and (21), the matrix The functions G(X,u,t,W) and J(X,u,t,W) are determined by reference equations (17) and (18);

[0063] (3) Statistical calculation

[0064] The mean and variance of the random state vector x, process constraint function g(x(t),u(t),t,w) and objective function J(x(t),u(t),t,w) in equation (1) are expressed as functions of the coefficients of the PC polynomial.

[0065]

[0066]

[0067]

[0068] Where <,> represent inner product operations; g j J j (j=1,2,...,P) is expressed as a nonlinear function of X and u through equations (20) and (21), and then the statistics of state, process constraints and objective function are calculated through equations (22), (23) and (24).

[0069] Preferably, the method of successive linearization is used to solve the high-dimensional deterministic optimal control problem P. ED Convexification to a convex optimization problem P ED-Convex Specifically, it combines the uncertainty quantification propagation model with convex optimization methods, and uses successive linearization methods to solve the high-dimensional deterministic optimal control problem P. ED The high-dimensional deterministic differential equations, non-convex constraints, and objective functions in the problem are transformed into convex ones, thus transforming the high-dimensional deterministic optimal control problem P... ED Transform into a convex optimization problem P ED-Convex .

[0070] Preferably, the high-dimensional deterministic optimal control problem P ED Convexification to a convex optimization problem P ED-Convex Specifically, it includes the following steps:

[0071] (1) Convexation

[0072] Equation (10) is made convex using a successive linearization method;

[0073] Equation (10) is applied to the reference trajectory {X} k ,u k Performing a first-order Taylor expansion near}, we get:

[0074]

[0075] definition:

[0076]

[0077] in, and These are the partial derivatives of the function F(X,u,t,W) with respect to the state X and the control quantity u, respectively, on the reference trajectory.

[0078] To ensure the accuracy of linearization, a trust region constraint is added:

[0079]

[0080] In equation (27), δ X δ u Indicates the radius of the trust region;

[0081] Considering that the process constraints and objective function in equation (1) are both nonlinear functions of the extended state X and the control quantity u, the successive linearization method is still used to make them convex.

[0082] In the reference trajectory {X k ,u k Performing a first-order Taylor expansion near}, we get:

[0083]

[0084]

[0085] definition:

[0086]

[0087]

[0088] At this point, problem P ED Transformed into the following equivalent high-dimensional deterministic trajectory convex optimization problem P ED-Convex :

[0089]

[0090] Preferably, the convex optimization problem P ED-Convex Discretization is performed to obtain the parametric convex optimization problem P. ED -disConvex Specifically:

[0091] Discretization of extended state variables and control variables based on the trapezoidal rule is applied to the convex optimization problem P. ED-Convex Transform into a parametric convex optimization problem P ED-disConvex Furthermore, an optimization strategy based on sequential convex optimization is constructed to iteratively solve problem P. ED-disConvex Approximating the robust optimization problem P R The solution.

[0092] Preferably, the discretization of the extended state variables and control variables based on the trapezoidal rule transforms the convex optimization problem P... ED-Convex Transform into a parametric convex optimization problem P ED-disConvex Furthermore, an optimization strategy based on sequential convex optimization is constructed to iteratively solve problem P. ED-disConvex Approximating the robust optimization problem P R The solution specifically includes the following steps:

[0093] (1) Discretization processing

[0094] Based on the trapezoidal rule, the domain of the independent variable is [t0, t...]. f The data is discretized into N intervals at equal intervals, and the independent variable is discretized into {t0, t1, ..., t...}. N The state variables are discretized as {X0, X1, ..., X}. N The control quantity is discretely defined as {u0, u1, ..., u}. N};

[0095] Discretizing the linear differential equation of equation (25), we get:

[0096]

[0097] in, T=(t f -t0) / N;

[0098] For question P ED-Convex Other constraints and the objective function are discretized to obtain the final problem P. ED-disConvex :

[0099]

[0100] (2) Sequence convex optimization solution strategy

[0101] Aircraft robust trajectory optimization problem P R The solution is obtained by iteratively solving problem P. ED-disConvex To approximate the trajectory, and continuously update the reference trajectory using the currently obtained trajectory, that is:

[0102]

[0103] Until the constraint is satisfied:

[0104]

[0105] In equation (36), ε X ε u This is the convergence error limit.

[0106] Preferably, the process of taking the overall parameters, constraint parameters, uncertain parameters, and weighting coefficients of the aircraft as input parameters, and then using an optimization algorithm based on sequential convex optimization to obtain the optimal control quantity and complete the aircraft trajectory optimization specifically includes the following steps:

[0107] Set the overall parameters of the aircraft and construct the motion model of the aircraft's center of mass;

[0108] Set the distribution type and standard deviation of the uncertain parameter w, and then determine the basis function expression of the PC polynomial, the dimension of the random variable d, the order of the polynomial p, and the Latin hypercube sampling points;

[0109] The initial flight trajectory, terminal state constraints, control quantity constraints, process constraints, objective function, and weight coefficients are set to construct a specific trajectory optimization problem.

[0110] The trajectory optimization problem is solved by calling an optimization algorithm to obtain the optimal control quantity and complete the aircraft trajectory optimization.

[0111] Preferably, the overall parameters of the aircraft include mass, reference area, and aerodynamic coefficient.

[0112] The robust trajectory optimization method for aircraft based on SRSM and NIPC provided by this invention has the following beneficial effects:

[0113] By constructing an uncertainty quantization propagation model based on NIPC and SRSM, an effective transformation of uncertain stochastic problems was achieved. Subsequently, this model was combined with convex optimization methods to design a solution strategy based on sequential convex optimization algorithms, enabling rapid solution of this high-dimensional deterministic problem. This strategy yields optimization trajectories with both reliability and robustness, exhibiting considerable accuracy and significant computational efficiency advantages. Based on the constructed uncertainty quantization propagation model based on stochastic response surface method and non-embedded chaotic polynomials, the effective transformation of multi-parameter uncertain stochastic problems such as initial state and process dynamics can be achieved, with advantages of low state expansion dimension and high computational efficiency. Attached Figure Description

[0114] To more clearly illustrate the embodiments and design schemes of the present invention, the accompanying drawings required for this embodiment will be briefly described below. The drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0115] Figure 1 This is a block diagram illustrating the principle of the robust trajectory optimization method for aircraft based on SRSM and NIPC according to Embodiment 1 of the present invention.

[0116] Figure 2 The results represent the optimized gliding trajectory under nominal conditions.

[0117] Figure 3 To optimize the target shooting results with control quantities under nominal conditions;

[0118] Figure 4 The results of comparing the state mean values ​​for quantifying the propagation of uncertainty;

[0119] Figure 5 Comparison of state standard deviation results for quantifying the propagation of uncertainty;

[0120] Figure 6 Comparison of robust trajectory optimization results between GQ-PC-CO and SRSM-PC-CO;

[0121] Figure 7 Comparison of parameters for DO, RBO, and RO;

[0122] Figure 8 The probability density function curves for DO, RBO, and RO shooting results are shown. Detailed Implementation

[0123] To enable those skilled in the art to better understand and implement the technical solutions of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. The following embodiments are only used to more clearly illustrate the technical solutions of the present invention and should not be construed as limiting the scope of protection of the present invention.

[0124] Example 1

[0125] This invention addresses the problems existing in robust trajectory optimization algorithms in the aerospace field by proposing a general and efficient robust trajectory optimization design method. By constructing an uncertainty quantization propagation model based on NIPC and SRSM, an effective transformation of the uncertain stochastic problem is achieved. Subsequently, this model is combined with convex optimization methods to design a solution strategy based on a sequential convex optimization algorithm, enabling rapid solution to this high-dimensional deterministic problem. Finally, simulations using hypersonic glide reentry as an example are conducted to verify the effectiveness of the proposed method.

[0126] Based on the above principles, this embodiment provides a robust trajectory optimization method for aircraft based on SRSM and NIPC, specifically including:

[0127] Step 1: Construct the robust trajectory optimization problem P of the aircraft under parameter uncertainty. R This problem is a stochastic optimization problem.

[0128] The general trajectory optimization problem can be reduced to the following deterministic optimal control problem P D :

[0129]

[0130] In the formula, x and u are the state and control vectors, respectively, and w is the system parameter vector; The system dynamic equations are: g(x(t),u(t),t,w)≤0 represents the process constraints, which also include state boundary constraints; u min with u max The upper and lower boundaries of the control quantity; x(t0) = x0, x(t f )=x f For endpoint constraints.

[0131] When the system parameter vector w and the initial state value x(t0) are uncertain, the objective function J, the dynamic function f, and the constraint function all become stochastic functions. Therefore, the original deterministic optimization problem P D This transforms into a stochastic optimization problem. To reduce the sensitivity of the optimized trajectory to uncertainty—that is, to minimize the statistically significant dispersion of the expected trajectory when uncertainty exists—it is typically necessary to introduce statistical moments of random quantities into the objective function and constraint functions. Therefore, the general robust trajectory optimization problem for aircraft, P... R It can be described as:

[0132]

[0133] In equation (1), x(t) is the state vector, u(t) is the control vector, and w is the system random parameter vector; min with u max The upper and lower boundaries of the control variable are defined by: J, f, g, and σ. The objective function is defined by: J, f by: f, g by: g ... J k g Let x be the weighting coefficient to be designed. 0μ With x 0σ The mean and standard deviation of the initial state are ε, respectively. f This is the upper bound of the standard deviation of the final value. Let be the stochastic differential vector of the state vector. When using the above model for trajectory optimization, it is necessary to introduce uncertainty quantization propagation techniques to calculate relevant statistics and handle stochastic dynamic equations.

[0134] Note: (1) ε f It needs to be determined based on prior statistical information. If the value is inappropriate, it may lead to an unsolvable problem. In order to improve the convergence of the optimization algorithm, x can be... σ (t f (1) The penalty function is introduced into the objective function; (2) The above robust trajectory optimization model considers both the robustness and reliability of the trajectory. The mean and standard deviation terms are introduced into the objective function and endpoint constraints to improve the robustness of the trajectory. The process constraint model considering statistical moments is introduced to ensure that the process constraints are met with a certain probability, thereby improving the reliability of the trajectory; (3) By designing the weight coefficient k J k g A trade-off between robustness and reliability of the trajectory can be achieved.

[0135] like Figure 1 As shown, the robust trajectory optimization algorithm considering the uncertainties of the initial state x(t0) and the system parameter vector w will be given in detail below. First, the original robust trajectory optimization problem P of the spacecraft is solved by using the chaotic polynomial expansion technique based on the response surface methodology. R The problem is transformed into a high-dimensional deterministic optimization problem with state extension; then, the problem is convexified and discretized, and solved using a sequential convex optimization algorithm.

[0136] Step 2: Based on the non-embedded chaotic polynomial (NIPC) and stochastic response surface methodology (SRSM), the robust trajectory optimization problem of the aircraft P is solved. R Transformed into a high-dimensional deterministic optimal control problem P ED We will construct an uncertainty quantification propagation model.

[0137] Specifically, by introducing a Latin hypercube sampling strategy and using the Stochastic Response Surface Method (SRSM), the state vector x(t) containing uncertain parameters is extended into a high-dimensional ordinary differential equation with respect to PC polynomial coefficients. The problem P is then calculated using the PC polynomial coefficients. R The mean and standard deviation of the state, constraints, and objective function are used to further analyze the robust trajectory optimization problem P of the aircraft. R Transformed into an equivalent high-dimensional deterministic optimal control problem P ED .

[0138] The following section combines NIPC with SRSM and introduces a Latin hypercube sampling strategy to establish an uncertainty quantization propagation model, thereby transforming the robust trajectory optimization problem of the aircraft into an equivalent deterministic optimization problem in a high-dimensional state space.

[0139] (1) PC expansion of state x

[0140] Assuming the random state vector x is n-dimensional and the system parameter vector w is s-dimensional, then each state variable x i (t) and parameter w k It can be expanded into a PC polynomial:

[0141]

[0142]

[0143] Where Δ={Δ1,…,Δ d} is a d-dimensional random vector, Φ j (Δ) is an orthogonal polynomial function, which can be obtained through the univariate orthogonal basis function φ corresponding to the Δ distribution type. j (Δ l ), l=1,...,d, obtained by taking the tensor product; x ij (t) and w kj For the PC coefficients, the expansion term P+1 = [(p+d)!] / (p!d!), which is derived from the orthogonal basis functions φ. j The order p of (Δ) and the dimension d of the random variable are determined; the orthogonal basis functions {φ j (Δ l )} and random variable Δ l The relationship is referenced.

[0144] PC coefficient w kj and x ij (t0) can be determined according to w k and x i The distribution of (t0) is determined, while the random state vector x i PC coefficients x of (t) ij (t) is unknown. The following uses the random response surface methodology to derive the solution coefficients x.ij The expression for (t).

[0145] Substituting equations (2) and (3) into the differential equation of equation (1), we get:

[0146]

[0147] Where i = 1, 2, ..., n, k = 1, 2, ..., s.

[0148] Using the Latin hypercube method to perform n s Random sampling was performed, and the random sample points were... Substituting into equation (4) above, we obtain a system of differential equations:

[0149]

[0150] In equation (5), X and W represent the PC coefficient vectors of the random state vector x and parameter w, respectively, and we have:

[0151]

[0152] W = [w 10 ,…,w 1P ,w 20 ,…,w 2P ,…,w s0 ,…,w sP ] T (7)

[0153]

[0154] Let Ω(Δ) m )=[Φ0(Δ m ),Φ1(Δ m ),…,Φ P (Δ m (m=1,2,...,n) s ),definition:

[0155]

[0156] but

[0157] If n s If ≥P+1, then the system of differential equations in equation (5) can be solved by the least squares regression method, that is:

[0158]

[0159] In equation (10), the matrix To ensure the approximation accuracy of the model, this invention uniformly uses sampling points n. s=2(P+1), which will not be elaborated further. The PC coefficient vector X(t) at each time step can be obtained by integrating the above equation.

[0160] (2) Process constraints and PC expansion of the objective function

[0161] Since both the nonlinear constraint function g(x(t),u(t),t,w) and the objective function J(x(t),u(t),t,w) are functions of the state vector x(t), a similar method can be used for chaotic polynomial expansion, as follows:

[0162]

[0163]

[0164] In equations (11) and (12), g j (x(t),u(t),t) and J j (x(t),u(t),t) are the PC polynomial coefficients of the corresponding function.

[0165] random sample points Substituting into equations (11) and (12), we get:

[0166]

[0167]

[0168] In equations (13) and (14), and Let g(x(t),u(t),t,w) and J(x(t),u(t),t,w) represent the PC coefficient vectors of functions g(x(t),u(t),t,w) and J(x(t),u(t),t,w) respectively, and let J(x(t),u(t),t,w) represent the functions of g(x(t),u(t),t,w) respectively.

[0169]

[0170]

[0171]

[0172]

[0173] Similarly, there are

[0174]

[0175] Similarly, the coefficient vector can be solved using the least squares regression method. and

[0176]

[0177]

[0178] In equations (20) and (21), the matrix The functions G(X,u,t,W) and J(X,u,t,W) are determined by reference equations (17) and (18).

[0179] (3) Statistical calculation

[0180] The mean and variance of the random state vector x, process constraint function g(x(t),u(t),t,w) and objective function J(x(t),u(t),t,w) in equation (1) are expressed as functions of the coefficients of the PC polynomial.

[0181]

[0182]

[0183]

[0184] Where <,> represent inner product operations; g j J j (j=1,2,...,P) can be expressed as nonlinear functions of X and u through equations (20) and (21), and then the statistics of state, process constraints and objective function can be calculated through the above formulas (22), (23) and (24).

[0185] At this point, the robust trajectory optimization problem P of the aircraft has been solved. R Transformed into a high-dimensional deterministic optimization problem P with respect to the extended state X and the control variable u. ED Therefore, convex optimization algorithms can be used to process and solve the problem. Due to space limitations, problem P... ED I will not go into details here.

[0186] As mentioned earlier, when using non-embedded chaotic polynomial NIPC for parameter uncertainty quantization propagation, two typical model transformation methods are often used: the stochastic response surface method (SRSM) and the Gaussian quadrature method (GQ). The computational efficiency of these two methods is compared through quantitative analysis below.

[0187] Generally speaking, for uncertain problems with d-dimensional random variables, if the order of the PC polynomial expansion is p, the state expansion factor of the GQ method is (p+1). d The state expansion factor of the SRSM method is 2(p+d)! / (p!d!). Table 1 shows the state expansion of the two methods under different values ​​of p and d.

[0188] Table 1 Comparison of state extension performance of GQ and SRSM under different conditions

[0189]

[0190] Table 1 shows that when p and d are small, the state expansion factors of the two methods are similar. As p increases, the state expansion factors of both methods increase, but the growth rate of the SRSM method is slower than that of the GQ method. With the increasing dimension d of the random variable, the expansion factor of the GQ method grows exponentially, while that of the SRSM method grows more slowly. Therefore, the SRSM method used in this invention has higher computational efficiency than the GQ method, especially in handling high-dimensional uncertainties, while the GQ method is generally only applicable to cases where d ≤ 3. Furthermore, the simulation section below provides a further comparative analysis of the two methods in terms of accuracy and computational efficiency.

[0191] Step 3: Use convex optimization methods to quantify the uncertainty propagation model of the high-dimensional deterministic optimal control problem P. ED Convexification to a convex optimization problem P ED-Convex .

[0192] Specifically, this involves combining an uncertainty quantization propagation model with convex optimization methods, and using a successive linearization method to solve the high-dimensional deterministic optimal control problem P. ED The high-dimensional deterministic differential equations, non-convex constraints, and objective functions in the problem are transformed into convex ones, thus transforming the high-dimensional deterministic optimal control problem P... ED Transform into a convex optimization problem P ED-Convex .

[0193] Step 4: Transform the convex optimization problem P ED-Convex Discretization is performed to obtain the parametric convex optimization problem P. ED-disConvex And based on the parametric convex optimization problem P ED-disConvex Construct a sequential convex optimization algorithm to solve problem P iteratively. ED-disConvex Approximating the robust optimization problem P R The solution.

[0194] Specifically, the extended state variables and control variables are discretized based on the trapezoidal rule, transforming the convex optimization problem P... ED -Convex Transform into a parametric convex optimization problem P ED-disConvex Furthermore, an optimization strategy based on sequential convex optimization is constructed to iteratively solve problem P. ED-disConvex Approximating the robust optimization problem P R The solution.

[0195] Since most complex trajectory optimization problems are non-convex, the key to solving the optimal trajectory using convex optimization techniques is to transform non-convex optimization problems into convex ones. This has attracted many scholars to conduct in-depth research on this topic, and the relevant results have been widely applied in the field of aerospace for optimal control and trajectory optimization.

[0196] Here, techniques such as successive linearization are introduced to address the high-dimensional deterministic optimization problem P from the previous section. ED The problem is convexified and discretized into a parametric convex optimization problem. A corresponding sequential convex optimization iterative algorithm is then established to solve this problem. The sequential convex optimization method iteratively updates the reference trajectory through the outer loop and solves the convex optimization problem through the inner loop, gradually approximating the optimal solution of the original problem. The convexification, discretization process, and the principles and implementation steps of the iterative algorithm are given below.

[0197] (1) Convexation

[0198] Analysis shows that H is a constant matrix and F(X,u,t,W) is a nonlinear function of the extended state X. Therefore, this equation is a nonlinear differential equation. The successive linearization method is then used to convexify equation (10).

[0199] Equation (10) is applied to the reference trajectory {X} k ,u k Performing a first-order Taylor expansion near}, we get:

[0200]

[0201] definition

[0202]

[0203] in, and These are the partial derivatives of the function F(X,u,t,W) with respect to the state X and the control quantity u, respectively, on the reference trajectory.

[0204] To ensure the accuracy of linearization, a trust region constraint is added:

[0205]

[0206] In the formula, δ X δ u Indicates the radius of the trust region.

[0207] Similarly, considering that the process constraints and objective function in equation (1) are both nonlinear functions of the extended state X and the control quantity u, the successive linearization method is still used to make them convex.

[0208] In the reference trajectory {X k ,u k Performing a first-order Taylor expansion near}, we get:

[0209]

[0210]

[0211] definition:

[0212]

[0213]

[0214] The definition of the relevant partial derivatives is similar to that in the previous text, and will not be repeated here.

[0215] At this point, problem P ED Transformed into the following equivalent high-dimensional deterministic trajectory convex optimization problem P ED-Convex :

[0216]

[0217] It should be noted that question P ED-Convex The state to be optimized is X, which is expanded into problem P. R 1 times P.

[0218] (2) Discretization

[0219] In order to apply numerical methods to problem P ED-Convex To solve this problem, the state and control variables need to be discretized. Based on the trapezoidal rule, the domain of the independent variable [t0, t...] is... f The data is discretized into N intervals at equal intervals, and the independent variable is discretized into {t0, t1, ..., t...}. N The state variables are discretized as {X0, X1, ..., X}. N The control quantity is discretely defined as {u0, u1, ..., u}. N}

[0220] Discretizing the linear dynamic equation of equation (25), we get:

[0221]

[0222] in, T=(t f -t0) / N.

[0223] Similarly, for problem P ED-Convex Discretize the other constraints and objective function to obtain the final problem P. ED-disConvex :

[0224]

[0225] At this point, P ED-Convex This transforms into a parametric convex optimization problem, which can be solved using the classical interior point method.

[0226] (3) Sequence convex optimization solution strategy

[0227] In summary, the robust trajectory optimization problem P of an aircraft R That is, the solution to the nonlinear stochastic optimal control problem can be obtained by iteratively solving problem P. ED-disConvex To improve the convergence and accuracy of the solution, we approximate the current trajectory and continuously update the reference trajectory. That is, we set:

[0228]

[0229] Until the constraint is satisfied:

[0230]

[0231] In equation (36), ε X ε u This is the convergence error limit.

[0232] Step 5: Using the overall aircraft parameters, constraint parameters, uncertain parameters, and weighting coefficients as input, optimize the solution using a sequential convex optimization algorithm, and output the control quantity for trajectory optimization. This specifically includes the following steps:

[0233] (1) Set the overall parameters of the aircraft and construct the motion model of the aircraft's center of mass;

[0234] (2) Set the distribution type and standard deviation of the uncertainty of the initial state and environmental parameters, and then determine the basis function expression of the PC polynomial, the dimension of the random variable d, the order of the polynomial p, and the Latin hypercube sampling points.

[0235] (3) Set the initial and terminal state constraints of the flight trajectory, control constraints, process constraints, objective function and weight coefficients.

[0236] (5) Substitute the reference trajectory into the uncertainty quantization propagation model to obtain the initial value of the extended state. and initial value of control quantity To construct a parametric convex optimization problem P ED-disConvex .

[0237] (5) Based on the sequential convex optimization solution strategy, the problem P is solved iteratively. ED-disConvex Approximating the original robust optimization problem P R The solution outputs the robust trajectory optimization control quantity u. l .

[0238] The following example uses the optimization of robust trajectory for hypersonic gliding to verify the optimization algorithm provided in this embodiment.

[0239] Hypersonic gliding flight is characterized by long range, high speed, and complex flight environment, and faces various complex constraints, parameter uncertainties, and disturbances. This makes trajectory optimization a typical fast time-varying, nonlinear, strongly coupled, and non-convex stochastic optimal control problem. Studies have shown that reentry gliding trajectory optimization under nominal conditions has significant challenges and application value, while robust trajectory optimization considering multi-parameter uncertainties in real-world conditions presents even greater difficulties and challenges. Therefore, this paper takes typical hypersonic gliding flight as an example to conduct rapid optimization simulations of robust trajectories under multiple uncertainties to verify the effectiveness of the algorithm designed in this invention. Simultaneously, simulation results based on a Gaussian quadrature strategy and a convex optimization method using chaotic polynomials (denoted as GQ-PC-CO) are presented, and the comparison demonstrates the computational efficiency advantages of the algorithm in this invention.

[0240] First, a description of the robust trajectory optimization problem in hypersonic gliding.

[0241] Assuming the Earth is a rotating sphere, the hypersonic gliding center of mass motion model under nominal conditions adopts an existing form. For the shortest flight time gliding problem with a fixed target point, if the angle of attack profile is pre-selected according to mission requirements, and only the roll angle is used as the control variable to be optimized, the deterministic optimal control problem corresponding to this trajectory optimization can be described as follows:

[0242]

[0243] In the formula: the independent variable is the dimensionless energy e = 1 / rv 2 / 2, dimensionless state quantity x=[r,θ,φ,γ,ψ] T r is the dimensionless geocentric distance, v is the dimensionless velocity, θ is the geocentric longitude, φ is the geocentric latitude, γ is the local velocity inclination angle, ψ is the track yaw angle, υ is the roll angle, and r min To account for process constraints, the distance from the geocenter to the lower boundary can be specifically determined based on the maximum heat flux density. Maximum dynamic pressure q max Maximum overload n max calculate.

[0244] The uncertainties faced by reentry gliding flight mainly include the initial reentry state and uncertainties in dynamic parameters such as atmospheric density and aerodynamic coefficients, which can be specifically modeled as follows:

[0245]

[0246] And there are:

[0247]

[0248] c = [ρ, C L C D] T (40)

[0249]

[0250] Where c is the dynamic parameter vector, ρ is the atmospheric density, and C L C D Let σ be the lift and drag coefficients, σ ​​be the standard deviation, and δ be the ratio of the standard deviation to the nominal value. The underscore "-" indicates the nominal value. The random variables Δ are independent of each other and follow a standard normal distribution.

[0251] To minimize the dispersion of terminal landing points, the standard deviation of the terminal latitude and longitude is introduced into the objective function of the robust trajectory optimization problem, i.e.:

[0252] J = μ(t) f )+k J [σ(t f )+σ(θ f )+σ(φ f (42)

[0253] For the aforementioned robust optimization problem of gliding trajectories considering uncertainties, the initial state uncertainty is 5, and the process parameter uncertainty is 3. If these uncertainties are considered simultaneously, the dimension of the random variable d = 8. Setting the chaotic polynomial order p = 2, if the GQ-PC-CO method is used for robust trajectory optimization, the state variables to be optimized need to be extended to the nominal case of 3. 8 =6561 times, the original stochastic ordinary differential equation is thus expanded into a 6561×5 dimensional coupled deterministic differential equation system. If the discretization process in the convex optimization process is considered, the number of parameters to be optimized will be even larger. The above high-dimensional expansion will lead to computational difficulties such as decreased computational efficiency or even failure to converge. However, when using the convex optimization (SRSM-PC-CO) method based on SRSM and NIPC proposed in this invention to solve the problem, the state to be optimized only needs to be expanded by (8+2)! / (8!2!) = 45 times. In comparison, the number of variables to be optimized is significantly reduced, thus having good optimization computational efficiency and convergence.

[0254] Simulation Analysis

[0255] (1) Simulation condition settings

[0256] The CAV-H aircraft was used for simulation. The nominal boundary conditions are set as shown in Table 2, and the parameter uncertainties are set as shown in Table 3. The main constraints are set as follows: q max =100kPa, n max =3,υ max =80°.

[0257] Taking into account both range capability and heat protection requirements, the angle of attack profile is preset as follows:

[0258]

[0259] Table 2 Boundary Condition Values

[0260]

[0261] Table 3 Parameter Uncertainty Settings

[0262]

[0263] In the simulation, a fixed number of discrete points, N=300, are used. The trust region radius and convergence error limit of the sequential convex optimization algorithm are set as follows:

[0264]

[0265] All simulations were performed on a desktop computer equipped with an Intel Core i7-8700 3.20GHz Intel processor, using the MATLAB 2016b platform. Uncertainty quantization and propagation were performed using self-developed code. A trajectory optimization algorithm was developed using the CVX toolkit, and the SDPT3 solver was used to solve the convex optimization subproblem P. ED-disConvex .

[0266] Second, simulation analysis of uncertainty quantification propagation.

[0267] First, a deterministic gliding trajectory optimization simulation was performed based on nominal conditions, and the results are as follows: Figure 2 As shown, (a) HV profile, (b) process constraints, and (c) roll angle command. Figure 2 As can be seen from (a) and (b), the obtained velocity-height curves are continuous and smooth and meet the process constraints. However, the heat flux density and dynamic pressure curves are close to the constraint boundaries for a period of time, indicating a potential risk of exceeding the constraint boundaries. Figure 2 As can be seen from (a) and (c), the optimization results are consistent with the integral results, and the number of control command reversals is small, indicating that the optimization results have good feasibility.

[0268] Considering only the uncertainties of three dynamic parameters—atmospheric density, lift, and drag coefficient—uncertainty propagation simulations using the GQ-PC and SRSM-PC methods were conducted based on the optimization results. The simulations were compared with Monte Carlo firing results to analyze the impact of uncertainty on the optimized trajectory and verify the effectiveness of the proposed uncertainty quantification algorithm. The PC polynomial order was set to p=2, and the number of Monte Carlo firings was 3000. Some simulation results are shown below. Figures 3-5 As shown.

[0269] Figure 3 (a) Terminal latitude and longitude distribution, (b) heat flow constraint, by Figure 3 It is known that trajectory optimization results under nominal conditions are easily affected by parameter uncertainties, leading to problems such as process constraints exceeding limits and decreased terminal accuracy. In other words, trajectory reliability and robustness are reduced to varying degrees. Therefore, further robust trajectory optimization under uncertain conditions is needed to improve the trajectory's anti-interference capability and reliability, thereby reducing the burden on the guidance and control system and the risk of mission failure.

[0270] Figures 4-5 Comparison curves of partial state means and standard deviations for different methods are given. Figure 4 (a) Mean values ​​of altitude and velocity deflection, (b) Mean values ​​of longitude and latitude; Figure 5 (a) Standard deviations of altitude and velocity skew angles, (b) Standard deviations of longitude and latitude. It can be seen that the statistical results obtained by the three methods are quite consistent, indicating that both GQ-PC and SRSM-PC methods can approximate the target firing results and can replace the target firing method for uncertainty quantization propagation. In a single quantization propagation process with comparable computational accuracy, the SRSM-PC method only requires 20 integrations of the dynamic equations, while the GQ-PC method requires 27 integrations, making its computational load 1.35 times that of SRSM-PC. If the uncertainty dimension is further increased and the iterative computation process of the optimization algorithm is considered, the computational efficiency advantage of the SRSM-PC method will become even more pronounced.

[0271] Third, a simulation comparison of robust trajectory optimization between the two algorithms.

[0272] To verify the robust trajectory optimization capability of the designed algorithm, the proposed SRSM-PC-CO algorithm is used for robust trajectory optimization simulation. Simultaneously, the algorithm is compared with existing GQ-PC-CO optimization methods and deterministic optimization methods (DO) to verify its effectiveness. The weight coefficient k is set... J =k g =3, other simulation conditions are the same as above, and the results are as follows: Figure 6 As shown, Figure 6 (a) Height profile, (b) Process constraints, (c) Ground track, (d) Bank angle command. Subsequently, using... Figure 6 The tilt angle command in (d) was used in Monte Carlo target simulation, and the statistical results of some parameters are shown in Table 4. Among them, Δθ μf , Δφ μf θ represents the mean deviation of the terminal longitude and latitude, respectively. σf φ σf These represent the standard deviations of the terminal longitude and latitude, respectively; t μf t σf Let P represent the mean and standard deviation of the flight time, respectively.e This represents the probability that the trajectory exceeds the process constraint boundary, i.e., the accident occurrence rate.

[0273] Table 4 Comparison of target shooting data from the optimization results of the two methods

[0274]

[0275] Depend on Figure 6 As shown in (a) to (b), the optimization results of the two sets of robust trajectory are basically consistent, and the height, heat flux density, and dynamic pressure curves are further away from the process constraint boundary than those of DO, indicating a significant improvement in the reliability of the optimized trajectory. Table 5 shows that the target statistics of the two sets of robust optimized control variables are quite consistent. Compared to DO, the terminal latitude and longitude distribution is more concentrated, and the mean deviation and standard deviation are both reduced, significantly lowering the probability of exceeding process constraints. These results demonstrate that the optimization results of the method of this invention are largely consistent with those of GQ-PC-CO, and both the robustness and reliability of the trajectory are significantly improved, verifying the effectiveness of the method of this invention.

[0276] In addition, by Figure 6 (d) It can be seen that, compared to DO, robust optimization keeps the bank angle command basically around 0° during the initial descent phase, and then continues to fly with a relatively small amplitude to increase the normal climb overload in the longitudinal plane, thereby increasing the altitude of the initial trajectory and ultimately reducing the heat flux density to comply with process constraints. However, in order to complete the heading adjustment, a relatively larger bank angle amplitude is required in the later stages of flight to achieve rapid turns, such as... Figure 6 As shown in (c).

[0277] Table 5 presents a comparison of the numerical processing performance of the two algorithms. It is evident that, under the same conditions, the algorithm of this invention significantly reduces the number of variables to be optimized, and reduces the computation time by approximately 62% when the number of iterations is comparable. This further demonstrates the significant advantage of the SRSM-PC-CO robust trajectory optimization algorithm in terms of computational efficiency.

[0278] Table 5 Comparison of numerical computation performance of the two optimization algorithms

[0279]

[0280] Fourth, reliability and robustness analysis of trajectory optimization based on weight parameter adjustment.

[0281] To further explore the trade-off between reliability and robustness of the proposed trajectory optimization algorithm and better meet the needs of different tasks, the following section presents robust optimization simulation and analysis based on weight parameter adjustment.

[0282] As we know from the robust optimization modeling discussed earlier, adjusting the weight parameter k g k JIt can meet different reliability and robustness requirements. If k g =3,k J If k = 0, the optimization model evolves into reliability-based trajectory optimization (RBO); if k = 0, then k = 0. g =k J =3, then the problem evolves into Robust Optimization (RO). Considering the uncertainties in dynamic parameters and initial state, the SRSM-PC-CO algorithm is used for simulation, and the results are compared with the deterministic optimization (DO) results to explore the changes in trajectory characteristics caused by parameter adjustments. Table 6 shows the weight coefficient values ​​under different conditions; other simulation conditions are consistent with those described above. Simulation results are as follows: Figure 7 As shown, Figure 7 (a) Height profile, (b) Process constraints, (c) Tilt angle command, (d) Ground track.

[0283] Table 6 Weighting Coefficient Settings

[0284]

[0285] Depend on Figure 7 As shown in (a) and (b), the altitude, heat flux density, and dynamic pressure curves corresponding to RO and RBO are further away from the process constraint boundary than those of DO, indicating a significant improvement in the reliability of the optimized trajectory. Compared to RBO, the altitude corresponding to RO is higher at the first peak and the final stage, suggesting that this trajectory form helps reduce the impact of random uncertainties and improves the robustness of terminal position and flight time indicators. Figure 7 (c) As can be seen from the previous analysis, in order to reduce the heat flux peak at the first trough of the altitude curve and thus improve trajectory reliability, both RBO and RO have relatively small and roughly the same roll angle amplitude during the initial descent phase, indicating that RBO and RO have comparable reliability for the heat flux peak. After the trough, RO continues to maintain a smaller roll angle amplitude to maintain a higher flight altitude; however, in the later stage of flight, RO uses a larger roll angle amplitude to achieve rapid heading adjustment. This phenomenon is observed in... Figure 7 This is further confirmed in the ground trajectory curve results of (d). In fact, the improvement of trajectory reliability and robustness places higher demands on the control capabilities of the aircraft at different flight stages, and there is no unified rule to follow, requiring specialized analysis for different problems.

[0286] use Figure 7 (c) The tilt angle optimization results were used for Monte Carlo target simulation to obtain statistical results for some parameters, such as... Figure 8And as shown in Table 7, (a) terminal longitude, (b) terminal latitude, (c) peak heat flux density, (d) peak dynamic pressure, (e) peak overload, and (f) flight time; among which, Figure 8 Approximate probability density functions for some parameters in the target shooting results are given. Figure 8 As shown in (a), (b), and Table 7, the terminal latitude and longitude statistical accuracy of RBO and RO is basically the same, and significantly improved compared to DO. However, considering the robustness of terminal performance and optimization indicators, the terminal latitude and longitude and flight time dispersion corresponding to RO are the smallest among the three. Figure 8 As shown in (c) to (e) and Table 7, the peak distributions of heat flux, dynamic pressure, and overload for both RBO and RO are far from the constraint boundary. The probability of constraint exceedance decreases from 67.4% for DO to 0.90% and 0.88%, respectively, indicating improved reliability for both. These results demonstrate that, compared to RBO, RO exhibits the lowest sensitivity to uncertainty while maintaining trajectory reliability, thus demonstrating better robustness. Figure 8 (f) and Table 7 show that, compared to the DO results, the mean flight time for RBO and RO is larger. This is because the introduction of uncertainty and the application of reliability constraints reduce the feasible region for optimization, forcing an increase in trajectory altitude and thus increasing flight time. Conversely, the mean flight time for RO is even larger than that for RBO. This is because the objective function of RO considers both the mean and standard deviation of flight time, while the objective function of RBO only considers the mean. Therefore, improving trajectory robustness and reliability comes at the cost of sacrificing some optimality. In practical applications, the weighting coefficients can be adjusted according to design requirements to meet the comprehensive requirements of trajectory reliability, robustness, and optimality.

[0287] Table 7 Comparison of DO, RBO and RO target data

[0288]

[0289] Table 8 compares the computation time and the number of variables to be optimized under three simulation conditions. It can be seen that even when considering the initial state and parameter uncertainty, the optimization time of RBO and RO is within an acceptable range. If parallel computing is used, the optimization time can be further shortened.

[0290] Table 8 Comparison of numerical computation performance of DO, RBO and RO

[0291]

[0292] In summary, for trajectory optimization problems with multiple parameter uncertainties, this invention studies a general and efficient robust trajectory optimization method. Simulation analysis and comparative verification were conducted using hypersonic reentry gliding trajectory optimization as an example, yielding the following results:

[0293] (1) Compared with deterministic trajectory optimization and traditional robust trajectory optimization algorithms, the designed robust trajectory optimization method can obtain optimized trajectories that are both reliable and robust, and has considerable accuracy and significant computational efficiency advantages.

[0294] (2) The improvement of trajectory robustness and reliability comes at the cost of sacrificing a certain degree of optimality. In practical applications, the weight coefficients can be adjusted according to design requirements to comprehensively meet the requirements of trajectory reliability, robustness and optimality.

[0295] (3) Based on the constructed uncertainty quantization propagation model based on the random response surface method and non-embedded chaotic polynomial, it can effectively transform multi-parameter uncertain random problems such as initial state and process dynamics, and has the advantages of low state expansion dimension and high computational efficiency.

[0296] (4) Compared with optimization methods based on embedded chaotic polynomials, the method of the present invention treats the original system model as a "black box" and does not require derivation and modeling for different problems. Therefore, it has high versatility and engineering application value. In addition, the method of the present invention is not only applicable to general robust trajectory optimization problems, but can also be extended to general nonlinear robust dynamic optimization problems.

[0297] The above-described embodiments are merely preferred embodiments of the present invention, and the scope of protection of the present invention is not limited thereto. Any simple changes or equivalent substitutions of the technical solutions that can be obviously obtained by those skilled in the art within the scope of the technology disclosed in the present invention shall fall within the scope of protection of the present invention.

Claims

1. A robust trajectory optimization method for aircraft based on SRSM and NIPC, characterized in that, Includes the following steps: Constructing a robust trajectory optimization problem for an aircraft under conditions of parameter uncertainty P R ; An uncertainty quantification propagation model is constructed based on the non-embedded chaotic multinomial (NIPC) and stochastic response surface methodology (SRSM) to address the robust trajectory optimization problem of aircraft. R Transformed into a high-dimensional deterministic optimal control problem P ED ; Using successive linearization to solve the high-dimensional deterministic optimal control problem P ED Convexification to a convex optimization problem P ED-Convex ; The convex optimization problem P ED-Convex Discretization is performed to obtain the parametric convex optimization problem P. ED-disConvex Furthermore, an optimization algorithm based on sequential convex optimization is constructed to iteratively solve the convex optimization problem P. ED-disConvex Approaching the robust trajectory optimization problem of a spacecraft P R The solution; The overall parameters, constraint parameters, uncertain parameters, and weight coefficients of the aircraft are used as input parameters. The optimal control quantity is obtained by calling the optimization solution algorithm based on sequential convex optimization, thus completing the aircraft trajectory optimization.

2. The robust trajectory optimization method for aircraft based on SRSM and NIPC according to claim 1, characterized in that, The robust trajectory optimization problem of the aircraft P R The model is: In equation (1), x(t) is the state vector, u(t) is the control vector, and w is the system random parameter vector; min with u max The upper and lower boundaries of the control variable are defined by: J, f, g, and σ. The objective function is defined by: J, f by: f, g by: g ... J k g Let x be the weighting coefficient to be designed. 0μ With x 0σ The mean and standard deviation of the initial state are respectively, ε f This is the upper bound of the standard deviation of the final value. It is a random differential vector of the state vector.

3. The robust trajectory optimization method for aircraft based on SRSM and NIPC according to claim 2, characterized in that, The uncertainty quantification propagation model, constructed based on the non-embedded chaotic multinomial (NIPC) and stochastic response surface methodology (SRSM), addresses the robust trajectory optimization problem of the aircraft. R Transformed into a high-dimensional deterministic optimal control problem P ED Specifically: By introducing a Latin hypercube sampling strategy and extending the stochastic differential vector of the state vector containing uncertain parameters into a high-dimensional ordinary differential equation with respect to the coefficients of the PC polynomial based on the stochastic response surface method (SRSM), the problem P is calculated using the PC polynomial coefficients. R The mean and standard deviation of the state, constraints, and objective function are used to further analyze the robust trajectory optimization problem P of the aircraft. R Transformed into a high-dimensional deterministic optimal control problem P ED .

4. The robust trajectory optimization method for aircraft based on SRSM and NIPC according to claim 3, characterized in that, The robust trajectory optimization problem of the aircraft P R Transformed into a high-dimensional deterministic optimal control problem P ED Specifically, it includes the following steps: (1) PC expansion of the stochastic differential vector of the state vector Assuming the random state vector x is n-dimensional and the system parameter vector w is s-dimensional, then each state variable x i (t) and parameter w k Expanded into a PC polynomial: Where Δ={Δ1,…,Δ d } is a d-dimensional random vector, Φ j (Δ) is an orthogonal polynomial function, obtained through the univariate orthogonal basis function φ corresponding to the Δ distribution type. j (Δ l ), l=1,...,d, obtained by taking the tensor product; x ij (t) and w kj For the PC coefficients, the expansion term P+1 = [(p+d)!] / (p!d!), which is derived from the orthogonal basis functions φ. j The order p of (Δ) is determined by the dimension d of the random variable; PC coefficient w kj and x ij (t0) According to w k and x i The distribution of (t0) is determined, and the coefficients x are derived using the random response surface method. ij The expression for (t); Substituting equations (2) and (3) into the differential equation of equation (1), we get: Where i = 1, 2, ..., n, k = 1, 2, ..., s; Using the Latin hypercube method to perform n s Random sampling was performed, and the random sample points were... Substituting into equation (4), we obtain the system of differential equations: In equation (5), X and W represent the PC coefficient vectors of the random state vector x and parameter w, respectively, and F(X,u,t,W) is a nonlinear function of the extended state X, and we have: In=[in 10 ,…,In 1P ,In 20 ,…,In 2P ,…,In s0 ,…,In sP ] T (7) set(D m )=[Φ0(Δ m ),Φ1(D m ),…,Φ P (D m )],(m=1,2,...,n s ), definition: but If n s If ≥P+1, then the system of differential equations in equation (5) can be solved by the least squares regression method, i.e.: In equation (10), the constant matrix sampling points n s =2(P+1), and the PC coefficient vector X(t) at each time step is obtained by integrating equation (10); (2) Process constraints and PC expansion of the objective function Since both the process constraint function g(x(t),u(t),t,w) and the objective function J(x(t),u(t),t,w) are functions of the state vector x(t), the chaotic polynomial expansion described above is used, as follows: In equations (11) and (12), g j (x(t),u(t),t) and J j (x(t),u(t),t) are the PC polynomial coefficients of the corresponding function; random sample points Substituting into equations (11) and (12), we get: In equations (13) and (14), and Let g(x(t),u(t),t,w) and J(x(t),u(t),t,w) represent the PC coefficient vectors of functions g(x(t),u(t),t,w) and J(x(t),u(t),t,w) respectively, and let J(x(t),u(t),t,w) represent the functions of g(x(t),u(t),t,w) respectively. Similarly, there are: Similarly, the coefficient vector can be solved using the least squares regression method. and In equations (20) and (21), the matrix (3) Statistical calculation The mean and variance of the random state vector x, process constraint function g(x(t),u(t),t,w) and objective function J(x(t),u(t),t,w) in equation (1) are expressed as functions of the coefficients of the PC polynomial. Where <,> represent inner product operations; g j J j (j=1,2,...,P) is expressed as a nonlinear function of X and u through equations (20) and (21), and then the statistics of state, process constraints and objective function are calculated through equations (22), (23) and (24).

5. The robust trajectory optimization method for aircraft based on SRSM and NIPC according to claim 4, characterized in that, The method of successive linearization is used to solve the high-dimensional deterministic optimal control problem P. ED Convexification to a convex optimization problem P ED-Convex Specifically, it combines the uncertainty quantification propagation model with convex optimization methods, and uses successive linearization methods to solve the high-dimensional deterministic optimal control problem P. ED The high-dimensional deterministic differential equations, non-convex constraints, and objective functions in the problem are transformed into convex ones, thus transforming the high-dimensional deterministic optimal control problem P... ED Transform into a convex optimization problem P ED-Convex .

6. The robust trajectory optimization method for aircraft based on SRSM and NIPC according to claim 5, characterized in that, The high-dimensional deterministic optimal control problem P ED Convexification to a convex optimization problem P ED-Convex Specifically, it includes the following steps: (1) Convexation Equation (10) is made convex using a successive linearization method; Equation (10) is applied to the reference trajectory {X} k ,u k Performing a first-order Taylor expansion near}, we get: definition: in, and These are the partial derivatives of the function F(X,u,t,W) with respect to the state X and the control quantity u, respectively, on the reference trajectory. To ensure the accuracy of linearization, a trust region constraint is added: In equation (27), δ X δ u Indicates the radius of the trust region; Considering that the process constraints and objective function in equation (1) are both nonlinear functions of the extended state X and the control quantity u, the successive linearization method is still used to make them convex. In the reference trajectory {X k ,u k Performing a first-order Taylor expansion near}, we get: definition: At this point, problem P ED Transformed into the following equivalent high-dimensional deterministic trajectory convex optimization problem P ED-Convex :

7. The robust trajectory optimization method for aircraft based on SRSM and NIPC according to claim 6, characterized in that, The convex optimization problem P ED-Convex Discretization is performed to obtain the parametric convex optimization problem P. ED-disConvex Specifically: Discretization of extended state variables and control variables based on the trapezoidal rule is applied to the convex optimization problem P. ED-Convex Transform into a parametric convex optimization problem P ED-disConvex Furthermore, an optimization strategy based on sequential convex optimization is constructed to iteratively solve problem P. ED-disConvex Approximating the robust optimization problem P R The solution.

8. The robust trajectory optimization method for aircraft based on SRSM and NIPC according to claim 7, characterized in that, The trapezoidal rule is used to discretize the extended state variables and control variables, transforming the convex optimization problem P... ED-Convex Transform into a parametric convex optimization problem P ED-disConvex Furthermore, an optimization strategy based on sequential convex optimization is constructed to iteratively solve problem P. ED -disConvex Approximating the robust optimization problem P R The solution specifically includes the following steps: (1) Discretization processing Based on the trapezoidal rule, the domain of the independent variable [t0, t] is defined. f The data is discretized into N intervals at equal intervals, and the independent variable is discretized into {t0, t1, ..., t...}. N The state variables are discretized as {X0, X1, ..., X}. N The control quantity is discretely defined as {u0, u1, ..., u}. N }; Discretizing the linear differential equation of equation (25), we get: in, T=(t f -t0) / N; For question P ED-Convex Other constraints and the objective function are discretized to obtain the final problem P. ED-disConvex : (2) Sequence convex optimization solution strategy Aircraft robust trajectory optimization problem P R The solution is obtained by iteratively solving problem P. ED-disConvex To approximate the trajectory, and continuously update the reference trajectory using the currently obtained trajectory, that is: Until the constraint is satisfied: In equation (36), ε X ε u This is the convergence error limit.

9. The robust trajectory optimization method for aircraft based on SRSM and NIPC according to claim 8, characterized in that, The process involves taking the overall parameters, constraint parameters, uncertain parameters, and weighting coefficients of the aircraft as input parameters, and then using an optimization algorithm based on sequential convex optimization to obtain the optimal control quantity, thereby completing the aircraft trajectory optimization. Specifically, this includes the following steps: Set the overall parameters of the aircraft and construct the motion model of the aircraft's center of mass; Set the distribution type and standard deviation of the uncertain parameter w, and then determine the basis function expression of the PC polynomial, the dimension of the random variable d, the order of the polynomial p, and the Latin hypercube sampling points; The initial flight trajectory, terminal state constraints, control quantity constraints, process constraints, objective function, and weight coefficients are set to construct a specific trajectory optimization problem. The trajectory optimization problem is solved by calling an optimization algorithm to obtain the optimal control quantity and complete the aircraft trajectory optimization.

10. The robust trajectory optimization method for aircraft based on SRSM and NIPC according to claim 9, characterized in that, The overall parameters of the aircraft include mass, reference area, and aerodynamic coefficient.

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