Hybrid uncertainty propagation method for nonlinear dynamic system based on probability box

Abstract

The invention discloses a hybrid uncertainty propagation method for a nonlinear dynamic system based on a probability box, and the method is characterized in that a non-parametric P-box is not only suitable for describing an imprecise probability, but also can build a unified description model for the imprecise probability and randomness, and can be used as a hybrid uncertainty model which is more generalized than an interval-probability model. Then, a propagation analysis method of a'random boundary Chebyshev (RBC) 'is provided. An interval analysis problem in propagation analysis is efficiently solved by using a Chebyshev method, and a random analysis problem of random probability distribution and non-parameterization is effectively solved by using data-driven polynomial chaos expansion (DD-PCE). The DD-PCE only needs an uncertain statistical moment when constructing a polynomial, does not depend on a complete probability distribution function, is suitable for solving the common problem that the probability distribution is arbitrary and incomplete in the non-parameterized P-box, and the RBC method is suitable for various nonlinear dynamic systems, and can achieve very high precision while improving the efficiency.

Description

Technical Field

[0001] The present invention belongs to the technical field of uncertainty propagation analysis, and relates to a method for hybrid uncertainty propagation of a nonlinear dynamic system based on a probability box. Background Art

[0002] The dynamic response evaluation of nonlinear systems is crucial in most engineering problems in the aerospace field, such as flight dynamics analysis, structural dynamics analysis, and rotor dynamics analysis, etc. Due to the inevitable uncertainties in practical applications such as design, testing, and service, it is not enough to evaluate the response only under deterministic and precise conditions. Therefore, in recent years, uncertainty propagation in nonlinear dynamics has become a research focus in the aerospace field. Uncertainties are divided into stochastic uncertainties caused by the inherent physical randomness of systems and excitations, and epistemic uncertainties caused by limited or low-quality information, and usually present as imprecision. In engineering practice, the two often occur simultaneously, forming hybrid uncertainties.

[0003] Regarding hybrid uncertainties, which refer to the coexistence of imprecision and randomness, but there are the following two situations for the specific "hybrid mode": 1) "Imprecise-random" problem: imprecision and randomness appear in different parameters respectively; 2) Imprecise probability problem: imprecision and randomness appear together in the same parameter. Most existing studies only focus on the imprecise-random problem and use the "interval-probability" model to describe hybrid uncertainties, and accordingly, some uncertainty propagation analysis methods based on the interval-probability model have been developed. For the imprecise probability problem, the P-box model is one of the mainstream description models and has been studied in the uncertainty problems of many static systems. However, it has only recently attracted attention in the dynamics field.

[0004] Most of the existing studies on the hybrid uncertainties of nonlinear dynamic systems only focus on the imprecise-random problem, and the research on the imprecise probability problem is still very limited. For the non-parametric P-box variable describing imprecise probability, there is still a lack of effective uncertainty propagation analysis methods so far, which limits the application of the P-box model in nonlinear dynamic system problems. Summary of the Invention

[0005] The present invention provides a method and system for hybrid uncertainty propagation of nonlinear dynamic systems based on probability boxes, aiming to solve two major limitations existing in the prior art of hybrid uncertainty propagation for nonlinear dynamic systems: First, the prior art mainly focuses on the problem of imprecise-stochastic hybrid uncertainty, and the research on imprecise probability problems is very limited; Second, for non-parametric P-box variables that describe imprecise probability problems, there is still a lack of effective global uncertainty propagation methods, which limits the application of P-box models in nonlinear dynamic system problems. Especially in analysis scenarios involving hybrid uncertainty, it is difficult for existing methods to achieve a balance between computational accuracy and efficiency.

[0006] To achieve the above object, the present invention adopts the following technical solutions:

[0007] The method for hybrid uncertainty propagation of nonlinear dynamic systems based on probability boxes includes the following steps:

[0008] Construct a nonlinear dynamic equation, and based on the nonlinear dynamic equation, use different P-box variables to describe three types of uncertain parameters in the nonlinear dynamic system, and obtain the dimension of the P-box variables;

[0009] Construct a DD-PCE model, calculate the number of samples required for the DD-PCE model based on the dimension of the P-box variables, and generate corresponding samples;

[0010] Construct an interval domain of Chebyshev polynomials, calculate the interval domain of Chebyshev polynomials based on the corresponding samples, and calculate the interpolation sample vector of the interval domain;

[0011] Solve the nonlinear dynamic equation based on the interpolation sample vector, obtain the interpolation samples of the response of the nonlinear dynamic system at discrete time series, and calculate the coefficients of the Chebyshev polynomials based on the interpolation samples of the response;

[0012] Perform interval scanning on the Chebyshev polynomials based on the coefficients of the Chebyshev polynomials, obtain the upper and lower boundaries of the response on each interval domain, and calculate the DD-PCE coefficients of the upper and lower boundaries of the system response according to the obtained upper and lower boundaries;

[0013] Calculate the CDF boundaries of the system response based on the DD-PCE coefficients of the upper and lower boundaries of the system response, and complete the analysis of hybrid uncertainty propagation of the nonlinear dynamic system.

[0014] A further improvement of the present invention lies in:

[0015] The construction of the nonlinear dynamic equation uses different P-box variables based on the nonlinear dynamic equation to describe three types of uncertain parameters in the nonlinear dynamic system, including:

[0016] Considering the M-dimensional P-box variable, denoted as

[0017] Given the system parameters, the P-box variable W P.B is represented by the boundaries of the probability distribution function as where, and represent the lower bound and upper bound of the CDF respectively.

[0018] The construction of the DD-PCE model calculates the number of samples required for the DD-PCE model based on the number of P-box variables and generates the corresponding samples, including:

[0019] For each set of given probability boundaries, generate orthogonal polynomial basis functions not higher than the d-th order, and construct the DD-PCE model according to the basis functions. Define the number of required samples N according to the dimension M s ;

[0020] Execute the Latin hypercube sampling method in the interval [0,1] to generate M data sets containing N s sample points, and transform each data set into a sample of W

[0021] The construction of the interval domain of the Chebyshev polynomial calculates the interval domain of the Chebyshev polynomial based on the corresponding samples and calculates the interpolation sample vector of the interval domain, including:

[0022] Calculate the interval domain for constructing the Chebyshev polynomial:

[0023]

[0024] Obtain the N M interpolation point vectors θ in [0,π]∈IR p ; p ;

[0025] Transform the interpolation point vector θ p into the interval domain I to obtain the interpolation sample vector

[0026] The solution of the nonlinear dynamic equation based on the interpolation sample vector to obtain the interpolation samples of the response of the nonlinear dynamic system at discrete time series, including:

[0027] Define the time discretization of the numerical method of the nonlinear dynamics, i.e., {tj | j = 1, 2, …, N t};

[0028] Collect samples of the response at each discrete time point, denoted as {g(w p (k) , t j ) | k = 1, 2, …, N p , j = 1, 2, …, N t}.

[0029] The interval scanning of the Chebyshev polynomial based on the coefficients of the Chebyshev polynomial to obtain the upper and lower boundaries of the response in each interval domain includes:

[0030]

[0031] where N s represents the number of samples; and both represent samples of W; g(w, t j ) represents the non - linear function that transforms the response of the system at a given time t j into a P - box variable.

[0032] The calculation of the CDF boundary of the system response based on the upper and lower boundaries of the system response includes:

[0033] Calculate the coefficients of the DD - PCE by the least - squares method, and then perform Monte Carlo simulation in the DD - PCE model to obtain the CDF boundary.

[0034] The hybrid uncertainty propagation system of the non - linear dynamic system based on the probability box includes:

[0035] The non - linear dynamic equation construction module is used to construct the non - linear dynamic equation, and based on the non - linear dynamic equation, different P - box variables are used to describe three types of uncertainty parameters in the non - linear dynamic system, and the dimension of the P - box variable is obtained;

[0036] The sample generation module is used to construct the DD - PCE model, calculate the number of samples required for the DD - PCE model based on the dimension of the P - box variable, and generate corresponding samples;

[0037] The polynomial interval domain calculation module is used to construct the interval domain of the Chebyshev polynomial, calculate the interval domain of the Chebyshev polynomial based on the corresponding samples, and calculate the interpolation sample vector of the interval domain;

[0038] A module for solving non - linear dynamic equations, which is used to solve non - linear dynamic equations based on interpolation sample vectors, obtain interpolation samples of the response of the non - linear dynamic system at discrete time series, and calculate the coefficients of Chebyshev polynomials based on the interpolation samples of the response;

[0039] An interval domain boundary calculation module, which is used to perform interval scanning on Chebyshev polynomials based on the coefficients of Chebyshev polynomials, obtain the upper and lower boundaries of the response on each interval domain, and calculate the DD - PCE coefficients of the upper and lower boundaries of the system response according to the obtained upper and lower boundaries;

[0040] A CDF boundary calculation module, which is used to calculate the CDF boundary of the system response based on the DD - PCE coefficients of the upper and lower boundaries of the system response, and complete the analysis of the propagation of hybrid uncertainties in the non - linear dynamic system.

[0041] A terminal device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of any of the methods of the present invention.

[0042] A computer - readable storage medium stores a computer program. When the computer program is executed by a processor, it implements the steps of any of the methods of the present invention.

[0043] Compared with the prior art, the present invention has the following beneficial effects:

[0044] The present invention discloses a method for propagating hybrid uncertainties in a non-linear dynamic system based on probability boxes, and uses a non-parametric P-box to establish a unified description model for hybrid uncertainties. The non-parametric P-box is not only applicable to describing imprecise probabilities, but also capable of establishing a unified description model for imprecision, imprecise probabilities, and randomness, and can be used as a more generalized hybrid uncertainty model than the "interval-probability" model. Then, a propagation analysis method of "random boundary Chebyshev (RBC)" is provided. The Chebyshev method is used to efficiently solve the interval analysis problem in the propagation analysis, and the data-driven polynomial chaos expansion (DD-PCE) is used to effectively solve the random analysis problem with arbitrary and non-parametric probability distributions. DD-PCE only requires the statistical moments of uncertainties when constructing polynomials and does not depend on the complete probability distribution function, which is very suitable for solving the problems of arbitrary and incomplete probability distributions commonly found in non-parametric P-boxes. The RBC method is applicable to various non-linear dynamic systems and can achieve very high precision while improving efficiency. This method can be applied to the problem of propagating hybrid uncertainties including factors such as imprecision, imprecise probabilities, and randomness, and helps the application of the P-box model in non-linear dynamic system problems. Description of the Drawings

[0045] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for use in the embodiments will be briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present invention and should not be regarded as limiting the scope. For those of ordinary skill in the art, other related drawings can also be obtained based on these drawings without creative efforts.

[0046] Figure 1 is the flowchart of the present invention;

[0047] Figure 2 is a schematic diagram of the probability box (P-box) model in the present invention (where a is a schematic diagram of the parametric P-box model; b is a schematic diagram of the non-parametric P-box model);

[0048] Figure 3 is a schematic diagram of the unified description model of imprecision-imprecise probability-randomness based on the non-parametric P-box in the present invention (where a is an interval model describing imprecision; b is a P-box model describing imprecise probability with imprecision and randomness mixed together; c is a probability model describing randomness);

[0049] Figure 4 is the CDF boundary of imprecise probability in the application case of the vehicle trajectory of the present invention (where a is the CDF boundary of thrust; b is the CDF boundary of atmospheric density; c is the CDF boundary of sound speed);

[0050] Figure 5 For the velocity V P.B. (t) in the trajectory application case of the vehicle of the present invention (where a is the propagation result of the P-box; b is the comparison of the error bands between the RBC method and the reference solution IMC method; c is the comparison of the CDF boundaries of the RBC method and the reference solution IMC method at 5 time instants);

[0051] Figure 6 For the dynamic pressure × angle of attack QA P.B. (t) in the trajectory application case of the vehicle of the present invention (where a is the propagation result of the P-box; b is the comparison of the error bands between the RBC method and the reference solution IMC method; c is the comparison of the CDF boundaries of the RBC method and the reference solution IMC method at 5 time instants);

[0052] Figure 7 For the altitude H P.B. (t) in the trajectory application case of the vehicle of the present invention (where a is the propagation result of the P-box; b is the comparison of the error bands between the RBC method and the reference solution IMC method; c is the comparison of the CDF boundaries of the RBC method and the reference solution IMC method at 5 time instants). Detailed implementation manners

[0053] To make the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Apparently, the described embodiments are some, but not all, of the embodiments of the present invention. Usually, the components of the embodiments of the present invention described and illustrated herein can be arranged and designed in various different configurations.

[0054] Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed present invention, but merely represents selected embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts fall within the scope of protection of the present invention.

[0055] It should be noted that: like reference numerals and letters denote like items in the following drawings, and thus, once an item is defined in one drawing, it does not need to be further defined and explained in subsequent drawings.

[0056] In the description of the embodiments of the present invention, it should be noted that if terms such as "upper", "lower", "horizontal", "inner", etc. are used to indicate the orientation or positional relationship, it is based on the orientation or positional relationship shown in the drawings, or the orientation or positional relationship in which the product of the invention is usually placed when in use. This is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. Therefore, it should not be construed as a limitation to the present invention. In addition, terms such as "first", "second", etc. are only used for distinguishing descriptions and cannot be understood as indicating or implying relative importance.

[0057] In addition, if the term "horizontal" appears, it does not mean that the component is required to be absolutely horizontal, but it can be slightly inclined. For example, "horizontal" only means that its direction is more horizontal relative to "vertical", and it does not mean that the structure must be completely horizontal, but it can be slightly inclined.

[0058] In the description of the embodiments of the present invention, it should also be noted that unless otherwise clearly specified and limited, if terms such as "set", "installed", "connected", "coupled" are used, they should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be directly connected or indirectly connected through an intermediate medium, and it can be the communication inside two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific situations.

[0059] The following further describes the present invention in detail with reference to the drawings:

[0060] See Figure 1 , the embodiments of the present invention disclose a method for propagating hybrid uncertainties in a nonlinear dynamic system based on probability boxes, including the following steps:

[0061] Step 1: Describe three types of uncertainty parameters in the nonlinear dynamic system with different P-box variables, and obtain the dimension M of the P-box variables.

[0062] Given the uncertainty parameters to be considered in the nonlinear dynamic system, and divide them into three categories: imprecise parameters, random parameters, and imprecise probability parameters according to whether the data is sufficient. Different P-box variables are used to describe the imprecise parameters, random parameters, and imprecise probability parameters respectively.

[0063] The P-box variable W P.B. can be represented by the boundaries of the cumulative density function (CDF) as where and respectively represent the lower and upper bounds of the CDF. P-box variables are usually divided into parametric P-boxes and non-parametric P-boxes, as shown in Figure 2 a and Figure 2 b respectively. Non-parametric P-boxes lack precise information about the distribution type and parameters. Therefore, they can only be expressed as:

[0064]

[0065] The P-box model can not only describe imprecise probabilities in which imprecision and randomness are mixed, but also be transformed into interval models and probability models, thus establishing a unified description of imprecision, imprecise probability, and randomness, as shown in Figure 3 respectively.

[0066] Specifically, for an M-dimensional P-box variable, denoted as When the nonlinear dynamic system contains W P.B. it can be expressed as:

[0067]

[0068] where f ODE (·) represents a nonlinear vector function.

[0069] The solution of Equation (1) can be expressed as a nonlinear vector function of the P-box variable W P.B. and time t, that is:

[0070]

[0071] When there exists W P.B. , the response of the system at a given time t is transformed into the P-box variable X P.B. (t). Therefore, the problem of uncertainty propagation is defined as calculating the characteristics of X P.B. (t). The primary goal of mixed uncertainty propagation is to calculate the CDF boundaries of any system response X at time t according to the given P-box parameters P.B. ∈ X P.B. , denoted as and which can be expressed as:

[0072]

[0073] Step 2: Calculate the number of samples N s required to construct the DD-PCE model according to the dimension M of the P-box variable and the order d of the PCE basis function to be adopted,

[0074] The core of DD-PCE is to construct a set of orthogonal polynomial bases for any distribution based on a finite number of statistical moments. For the d basis functions corresponding to the d-order one-dimensional polynomial, they can be expressed as:

[0075]

[0076] where, represents the sub-coefficient of the basis function. As a basis function, P (k) (w) should satisfy orthogonality;

[0077] Based on Equation (4), orthogonal polynomial basis functions of no higher than the d-order are generated, that is

[0078] Using the basis functions to construct PCE is shown in the following equation:

[0079]

[0080] Furthermore, according to the dimension M, the number of required samples N s is defined;

[0081] If the terms Φ 2M-1 +i 2M ≤d are selected as the basis functions of the d-order PCE, then the number of basis functions is: i1,i2,i3,i4,…,i2M-1,i2M (W), then the number of basis functions is:

[0082]

[0083] In this embodiment, the defined number of required samples N s is twice the value given by formula (6) for use in least squares calculation.

[0084] Furthermore, the LHS method is executed in the interval [0, 1] to generate M data sets containing N s sample points, and each data set is transformed into a sample of W through formula (7)

[0085]

[0086] Step 3: Calculate the interval domain for constructing the Chebyshev polynomials required for trajectory analysis based on the samples of W, and calculate N p sets of interpolation sample vectors in the interval domain

[0087] Calculate the interval domain for constructing the Chebyshev polynomials:

[0088]

[0089] Calculate \(N\) in \([0, \pi] \in \mathbb{R}\) according to formula (9). M in p interpolation point vectors \(\theta\) of p :

[0090]

[0091] Furthermore, transform the interpolation point vector \(\theta\) into the interval domain \(I\) through formula (10), that is p

[0092]

[0093] Step 4: Perform nonlinear dynamic system analysis at each group of interpolation sample vectors to obtain the response \(\{g(w p (k) , t j )|k = 1, 2, \ldots, N p , j = 1, 2, \ldots, N t}\) of the nonlinear dynamic system at discrete time series. Numerically solve the nonlinear dynamic equations corresponding to each discrete point \(w p (k) \), and collect the samples of the system response from \(t_1\) to where \(g(\cdot)\) represents the nonlinear vector function of the P - box variable \(W\) in formula (2) P.B. and time \(t\).

[0094] Step 5: At each discrete time series, calculate the coefficients of the Chebyshev polynomial according to the interpolation samples and the corresponding responses.

[0095] Establish the Chebyshev polynomial approximation of any component \(g(\cdot) \in g(\cdot)\) of the nonlinear vector function \(g(\cdot)\) in formula (2), as shown in formula (11):

[0096]

[0097] where \(a i1,i2,…,iM \) represents the coefficients of the polynomial; \(C i1,i2,…,iM (\theta)\) represents the \(M\) - dimensional Chebyshev polynomial, as shown in formula (12):

[0098]

[0099] The steps to calculate the coefficients according to \(\theta p \) and \(g(w p )\) can be expressed as formula (13):

[0100] ​​

[0101] Substitute formula (10) into formula (13), and since θ p is an interpolation sample vector determined in the interval [0, π] ∈ IR M then formula (13) can be further expressed as formula (14):

[0102]

[0103] Step 6: Use the constructed Chebyshev polynomial for interval scanning to find the upper and lower bounds of the system response at each discrete time series in each interval domain, and based on the upper and lower bounds of the response, calculate the DD-PCE coefficients corresponding to the upper and lower bounds of the response by combining the least squares method.

[0104] Taking the calculation of the upper bound of the CDF of any component X of the response vector X as an example for introduction, the subscripts of this component X and the corresponding nonlinear function g(·) in formula (2) are omitted.

[0105] To obtain the upper bound of the CDF of X, it is necessary to calculate the lower bound of g(·). Based on the Chebyshev method, the analysis of the lower bound of g(·) can be expressed as:[[]]

[0106]

[0107] where W L and W U represent two random boundaries, and the corresponding CDFs are respectively and In this way, the P-box problem is transformed into a Chebyshev interval analysis problem with random input boundaries. Based on the PCE polynomial, the nonlinear function of formula (15) can be expanded as:[[]]

[0108]

[0109] It should be noted that the present invention selects the term Φ i1,i2 (·) that satisfies i1 + i2 ≤ d as the basis function of the d-order PCE. Next, calculate the DD-PCE polynomial coefficient b i1,i2 by the least squares method, where samples of the random variables W L and W U are required.

[0110] The P-box problem has an additional condition, that is, each sample of W L and W U should be the two boundaries of the same interval in the IMC analysis, and its mathematical expression is as follows:[[]]

[0111]

[0112] Among them, w L and w U represent the samples of W L and W U respectively. Geometrically speaking, on the function surface only the curves that satisfy formula (17) need to be regressed Therefore, the regression samples should be obtained on the "curve", rather than being generated in the entire two-dimensional space corresponding to W L and W U respectively.

[0113] Then, calculate the function values of f LC (·) at the generated samples, that is, from i = 1 to N s , and calculate the lower bound of g(w) using the Chebyshev interval analysis shown in formula (18) according to the given input interval .

[0114]

[0115] Step 7: Based on the DD-PCE coefficients, perform an MC model to obtain the CDF boundary of the system response, thus completing the hybrid uncertainty propagation analysis of the nonlinear dynamic system.

[0116] In each iterative step of the interval analysis based on the Chebyshev method, the nonlinear dynamics needs to be solved N p times. In step 6, based on the function value samples calculated from f LC (·), the coefficient b i1,i2 can be regressively calculated, thereby establishing the PCE of f LC (·). Finally, a fast MC analysis can be performed based on this PCE to obtain the CDF upper bound. Thus, the hybrid uncertainty propagation analysis of the nonlinear dynamic system is completed.

[0117] The steps of the conventional MC analysis are as follows:

[0118] Calculate the CDF of X at time t as

[0119]

[0120] where N MC represents the total number of MC analyses, w i represents the i-th sample vector of the random variable W, and I[·] represents the indicator function, that is, if [·] is true, then I[·] = 1, otherwise it is 0.

[0121] Taking the ascent trajectory of a certain vehicle as an example, the method disclosed in this embodiment is applied to the uncertainty propagation analysis in the "core stage + booster" phase, and the performance of the method is fully tested, and the applicability of the method to non-linear dynamic systems is demonstrated. Specifically as follows:

[0122] In the test of the embodiment, the uncertainties of structural mass and thrust eccentricity are defined as random variables with a normal distribution, and the specific distribution parameters are shown in Table 1. There are still many inaccuracies for which it is always difficult to obtain probability distribution information, or which are not random in themselves but simply cannot be accurately described. Therefore, they can only continue to be described as interval variables, including lift, drag coefficient error, additional angle of attack and sideslip angle, and gravitational acceleration error, and the inaccuracies are defined as shown in Table 2.

[0123] Table 1 Definition of randomness in the vehicle trajectory application case

[0124]

[0125] Table 2 Definition of inaccuracy in the vehicle trajectory application case

[0126]

[0127] In addition, due to various conditions, the probability distribution information of some uncertainties cannot be accurately obtained. In the engineering practice of vehicle trajectories, engine thrust tests and atmospheric environment measurements are all complex tests with high difficulty, high cost or long cycle, and the known probability distribution information is often not accurate enough. In this case, the uncertainties of thrust deviation, atmospheric density and sound speed are defined as non-parametric P-box variables, as shown in Table 3. Among them, and The CDF boundaries of Figure 4 are as shown in

[0128] Since they are all non-parametric P-boxes and the CDF boundaries cannot be parametrically defined, they are defined in the form of discrete points, and the specific data are shown in Tables 4 to 6.

[0129]

[0130] Table 4 In the vehicle trajectory application case Definition

[0131]

[0132] Table 5 In the vehicle trajectory application case Definition

[0133]

[0134]

[0135] In Table 6, for the application cases of the vehicle trajectory Definition

[0136]

[0137] According to the booster working time of the vehicle, this problem is solved within the time range of 0 - 173 s. The variable-step RK algorithm is used to solve the ODEs, and the relative error tolerance is less than 1×10 -10 . In the RBC method, the orders of the Chebyshev polynomials and DD-PCE used are defined as 2. And the results of 1×10 5 times of IMC analysis are used as the reference solutions.

[0138] The P-box distributions of the vehicle flight speed, altitude, and dynamic pressure × angle of attack are denoted as V P.B. , H P.B. , QA P.B. . The uncertainty propagation results of V P.B. and QA P.B. are shown respectively as Figure 5 and Figure 6 . The uncertainty propagation result of H P.B. is shown as Figure 7 . Among them, the error band proportionality coefficients are all taken as 1; the calculation accuracy and efficiency are shown in Tables 7, 8, and 9 respectively. The results show that the relative errors of the RBC method in calculating the speed V P.B. and altitude H P.B. are less than 0.01%, and the relative error of calculating the dynamic pressure × angle of attack QA P.B. is less than 0.5%. This indicates that for the hybrid uncertainty propagation analysis of the vehicle trajectory problem, the RBC method has satisfactory accuracy in calculating various vehicle parameters. In terms of calculation efficiency, in the uncertainty problem considering 5 non-exactnesses, 3 non-exact probabilities, and 4 randomnesses, the RBC method only needs to call the original nonlinear system 182 times, and the number is only 7.2×10 -4 % of the IMC method, indicating that this method can achieve the hybrid uncertainty propagation analysis of the vehicle trajectory with extremely high efficiency. Combining efficiency and accuracy, the RBC method can provide a powerful analysis tool for the uncertainty propagation of the vehicle trajectory under multi-source static uncertainties.

[0139] Table 7 Precision and efficiency of different methods for calculating V P.B. (t) in the application cases of the vehicle trajectory

[0140]

[0141] Table 8 Precision and efficiency of calculating H P.B.(t) Precision and efficiency of different methods

[0142]

[0143] Table 9 Calculation of QA in the application case of the vehicle trajectory P.B. (t) Precision and efficiency of different methods

[0144]

[0145] In view of the problem of hybrid uncertainty coexisting with imprecision, imprecise probability and randomness in non - linear dynamic systems, the present invention, based on non - parametric probability boxes (P - boxes), establishes a propagation model for hybrid uncertainty and proposes a propagation analysis method of "random boundary Chebyshev (RBC)". A unified description model for hybrid uncertainty is established using non - parametric P - boxes; the Chebyshev method is used to efficiently solve the interval analysis problem in propagation analysis, and the data - driven polynomial chaos expansion (DD - PCE) is used to effectively solve the random analysis problem with arbitrary and non - parametric probability distributions. The following performances of the method are verified through the vehicle trajectory analysis example:

[0146] (1) The non - parametric P - box is not only applicable to describing imprecise probability, but also can establish a unified description model for imprecision, imprecise probability and randomness, and can be used as a more general hybrid uncertainty model than the "interval - probability" model.

[0147] (2) DD - PCE only requires the statistical moments of uncertainty when constructing polynomials and does not depend on the complete probability distribution function, which is very suitable for solving the problem of arbitrary and incomplete probability distributions commonly found in non - parametric P - boxes.

[0148] (3) For the vehicle trajectory problem, the RBC method achieves very high precision while improving efficiency. Compared with 1×10 5 times of interval Monte Carlo analysis, the relative error is less than 0.5%, and the calculation amount is reduced to 7×10 -4 %.

[0149] This embodiment also discloses a hybrid uncertainty propagation system for non - linear dynamic systems based on probability boxes, including:

[0150] A non - linear dynamic equation construction module for constructing non - linear dynamic equations, using different P - box variables based on the non - linear dynamic equations to describe three types of uncertainty parameters in the non - linear dynamic system, and obtaining the dimension of the P - box variables;

[0151] A sample generation module, configured to construct a DD-PCE model, calculate the number of samples required for the DD-PCE model based on the dimension of the P-box variable, and generate corresponding samples;

[0152] A polynomial interval domain calculation module, configured to construct an interval domain of Chebyshev polynomials, calculate the interval domain of Chebyshev polynomials based on the corresponding samples, and calculate an interpolation sample vector of the interval domain;

[0153] A module for solving the nonlinear dynamics equation, configured to solve the nonlinear dynamics equation based on the interpolation sample vector, obtain interpolation samples of the response of the nonlinear dynamics system at discrete time series, and calculate the coefficients of Chebyshev polynomials based on the interpolation samples of the response;

[0154] An interval domain boundary calculation module, configured to perform interval scanning on the Chebyshev polynomials based on the coefficients of the Chebyshev polynomials, obtain the upper and lower boundaries of the response on each interval domain, and calculate the DD-PCE coefficients of the upper and lower boundaries of the system response according to the obtained upper and lower boundaries;

[0155] A CDF boundary calculation module, configured to calculate the CDF boundary of the system response based on the DD-PCE coefficients of the upper and lower boundaries of the system response, and complete the analysis of the hybrid uncertainty propagation of the nonlinear dynamics system.

[0156] A schematic diagram of a terminal device provided by an embodiment of the present invention. The terminal device in this embodiment includes: a processor, a memory, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, the steps in the above-mentioned method embodiments are implemented. Alternatively, when the processor executes the computer program, the functions of each module / unit in the above-mentioned device embodiments are implemented.

[0157] The computer program may be divided into one or more modules / units, and the one or more modules / units are stored in the memory and executed by the processor to complete the present invention.

[0158] The terminal device may be a computing device such as a desktop computer, a notebook, a palm computer, and a cloud server. The terminal device may include, but is not limited to, a processor and a memory.

[0159] The processor may be a Central Processing Unit (CPU), or may also be other general-purpose processors, Digital Signal Processors (DSPs), Application Specific Integrated Circuits (ASICs), Field-Programmable Gate Arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc.

[0160] The memory can be used to store the computer program and / or module. By running or executing the computer program and / or module stored in the memory, and by invoking the data stored in the memory, the processor realizes various functions of the terminal device.

[0161] If the modules / units integrated in the terminal device are implemented in the form of software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, to implement all or part of the processes in the above-described embodiment methods of the present invention, it can also be completed by a computer program instructing relevant hardware. The computer program can be stored in a computer-readable storage medium. When the computer program is executed by the processor, the steps of the above-described various method embodiments can be realized. Among them, the computer program includes computer program code, and the computer program code can be in the form of source code, object code, executable file, or some intermediate form, etc. The computer-readable medium can include: any entity or device capable of carrying the computer program code, recording medium, USB flash drive, mobile hard disk, magnetic disk, optical disc, computer memory, Read-Only Memory (ROM), Random Access Memory (RAM), electrical carrier signal, telecommunication signal, and software distribution medium, etc. It should be noted that the content included in the computer-readable medium can be appropriately increased or decreased according to the requirements of legislation and patent practice in the jurisdiction. For example, in some jurisdictions, according to legislation and patent practice, the computer-readable medium does not include electrical carrier signals and telecommunication signals.

[0162] The above are only the preferred embodiments of the present invention and are not used to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for propagating hybrid uncertainties in a nonlinear dynamic system based on probability boxes, characterized in that, It includes the following steps: Construct a non - linear dynamics equation. Based on the non - linear dynamics equation, use different P - box variables to describe three types of uncertainty parameters in the non - linear dynamics system, and obtain the dimension of the P - box variables; Construct a DD - PCE model. Based on the dimension of the P - box variables, calculate the number of samples required for the DD - PCE model, and generate corresponding samples; Construct the interval domain of Chebyshev polynomials. Based on the corresponding samples, calculate the interval domain of Chebyshev polynomials, and calculate the interpolation sample vector of the interval domain; Solve the non - linear dynamics equation based on the interpolation sample vector to obtain the interpolation samples of the response of the non - linear dynamics system at discrete time series. Based on the interpolation samples of the response, calculate the coefficients of Chebyshev polynomials; Perform interval scanning on the Chebyshev polynomials based on the coefficients of the Chebyshev polynomials to obtain the upper and lower boundaries of the response on each interval domain. Calculate the DD - PCE coefficients of the upper and lower boundaries of the system response according to the obtained upper and lower boundaries; Calculate the CDF boundaries of the system response based on the DD - PCE coefficients of the upper and lower boundaries of the system response, and complete the analysis of the propagation of hybrid uncertainties in the non - linear dynamics system.

2. The method for propagating hybrid uncertainties of a non-linear dynamic system based on probability boxes according to claim 1, wherein, The construction of the non - linear dynamics equation, based on the non - linear dynamics equation, using different P - box variables to describe three types of uncertainty parameters in the non - linear dynamics system, includes: Consider the M-dimensional P-box variable, denoted as Given system parameters, the P-box variable W P.B is represented by the boundaries of the probability distribution function as where and represent the lower bound and upper bound of the CDF, respectively.

3. The method for propagating hybrid uncertainties of a non-linear dynamic system based on probability boxes according to claim 1, wherein The construction of the DD - PCE model, based on the number of P - box variables, calculating the number of samples required for the DD - PCE model, and generating corresponding samples, includes: For each given set of probability bounds, generate orthogonal polynomial basis functions of order not higher than d, construct a DD-PCE model based on the basis functions, and define the number of required samples N according to the dimension M s ; Execute the Latin hypercube sampling method in the interval [0, 1] to generate M data sets containing N s sample points, and transform each data set into a sample of W 4. The method for propagating hybrid uncertainties of a non-linear dynamic system based on probability boxes according to claim 3, characterized in that The construction of the interval domain of Chebyshev polynomials, based on the corresponding samples, calculating the interval domain of Chebyshev polynomials, and calculating the interpolation sample vector of the interval domain, includes: Calculate the interval domain used to construct Chebyshev polynomials: Obtain \(N\) in \([0,\pi]\in\mathbb{R}\). M p interpolation point vectors \(\theta\) p ;​ Transform the interpolation point vector θ p into the interval domain I to obtain the interpolation sample vector 5. The method for hybrid uncertainty propagation of a non-linear dynamic system based on probability boxes according to claim 1, characterized in that The solution of the non - linear dynamics equation based on the interpolation sample vector to obtain the interpolation samples of the response of the non - linear dynamics system at discrete time series, includes: Define the time discretization of the numerical method for nonlinear dynamics, i.e., {t j | j = 1, 2, …, N t}; Samples of the response are collected at each discrete time point, denoted as {g(w p (k) ,t j )|k = 1, 2, …, N p ,j = 1, 2, …, N t}.

6. The method for propagating hybrid uncertainties of a non-linear dynamic system based on probability boxes according to claim 1, wherein The interval scanning of the Chebyshev polynomials based on the coefficients of the Chebyshev polynomials to obtain the upper and lower boundaries of the response on each interval domain, includes: Among them, N s represents the number of samples; and both represent samples of W; g(w, t j ) represents a non-linear function that transforms the response of the system at a given time t j into a P-box variable.

7. The method for propagating hybrid uncertainties of a non-linear dynamic system based on probability boxes according to claim 1, wherein The calculation of the CDF boundaries of the system response based on the DD - PCE coefficients of the upper and lower boundaries of the system response, includes: Calculate the coefficients of DD - PCE by the least - squares method, and then perform Monte Carlo simulation in the DD - PCE model to obtain the CDF boundaries.

8. A hybrid uncertainty propagation system for nonlinear dynamic systems based on probability boxes, characterized in that, It includes: A non - linear dynamics equation construction module, used to construct a non - linear dynamics equation, based on the non - linear dynamics equation, using different P - box variables to describe three types of uncertainty parameters in the non - linear dynamics system, and obtain the dimension of the P - box variables; A sample generation module, used to construct a DD - PCE model, based on the dimension of the P - box variables, calculate the number of samples required for the DD - PCE model, and generate corresponding samples; A polynomial interval domain calculation module, which is used to construct the interval domain of Chebyshev polynomials, calculate the interval domain of Chebyshev polynomials based on corresponding samples, and calculate the interpolation sample vector of the interval domain; A module for solving non - linear dynamic equations, which is used to solve non - linear dynamic equations based on the interpolation sample vector, obtain the interpolation samples of the response of the non - linear dynamic system at discrete time series, and calculate the coefficients of Chebyshev polynomials based on the interpolation samples of the response; An interval domain boundary calculation module, which is used to perform interval scanning on Chebyshev polynomials based on the coefficients of Chebyshev polynomials, obtain the upper and lower boundaries of the response on each interval domain, and calculate the DD - PCE coefficients of the upper and lower boundaries of the system response according to the obtained upper and lower boundaries; A CDF boundary calculation module, which is used to calculate the CDF boundary of the system response based on the DD - PCE coefficients of the upper and lower boundaries of the system response, and complete the analysis of the mixed uncertainty propagation of the non - linear dynamic system.

9. A terminal device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 - 7.

10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of the method according to any one of claims 1 - 7.

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