Ship uncertainty optimization method based on improved unscented transformation and application

Abstract

The invention belongs to the technical field of ships, and discloses a ship uncertainty optimization method based on improved unscented transformation and application, the unscented transformation method is introduced into statistical moment estimation of any uncertainty system, and uncertainty optimization is further carried out by taking the statistical moment as a target; compared with a traditional statistical moment estimation method, the unscented transformation method can consider both precision and efficiency, and efficient calculation of the statistical moment of the random system is achieved; the unscented transformation method provided by the invention can be suitable for any random variable type and uncertainty that only a small amount of sample data is available and accurate probability distribution cannot be established; when the unscented transformation method is applied to uncertainty optimization, the uncertainty does not need to be repeatedly sampled in the uncertainty analysis link, the statistical moment is calculated only through a small number of fixed node response values, the uncertainty optimization structure is simplified, and the uncertainty optimization efficiency is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of ships, and particularly relates to a ship uncertainty optimization method and application based on improved unscented transformation. Background Technique

[0002] During the navigation of a ship, there are a large number of uncertain factors in the ship system itself and the operating environment, which may cause the ship performance not to reach the expected optimal design value. Traditional ship deterministic optimization (DO) does not consider the influence of uncertain factors and mostly optimizes based on a deterministic mathematical model at a fixed design point, unable to ensure excellent ship performance when uncertain factors fluctuate. In order to meet the increasingly high ship design requirements and enable the ship design scheme to adapt to the changes of uncertain factors, uncertainty-based design optimization has gradually become a hot research topic.

[0003] Compared with deterministic optimization, the difference and focus of ship uncertainty optimization lie in the uncertainty analysis link. The analysis index changes from the objective function value at the design point to the statistical characteristics of the ship design scheme when uncertain factors fluctuate. Usually, the low-order moment information (such as expectation, variance) of the output performance uncertainty distribution is used to describe the average level and fluctuation magnitude of ship performance changes. In some cases, it is also necessary to calculate the complete probability density function (PDF) of the performance function to describe its change law. Diez proposed a ship optimization method considering uncertainty with the mathematical expectation of the total ship cost function as the optimization goal. Hou Yuanhang robustly optimized the scale parameters of an autonomous underwater vehicle (AUV) with the expectation and variance of the resistance of the AUV under speed uncertainty as the goals. Wei Xiao performed uncertainty optimization on the KCS (KRISO Container Ship) ship type with the mean and variance of the energy efficiency design index (EEDI) as the goals, and obtained the PDF of the constraint function and further the failure probability by combining the maximum entropy method. For most practical engineering problems, it is very difficult to solve the probability density function, while the low-order statistical moments are easier to calculate and can characterize random information. Currently, a variety of methods for solving the probability density function using statistical moments have been developed. Therefore, the uncertainty analysis problem can be reduced to the problem of calculating the statistical moments of the performance function under uncertain factors.

[0004] When the distribution of uncertainty factors is known, the statistical moments of the ship performance function can be obtained through a large number of samplings and statistical analyses. The Monte Carlo (MC) method estimates the required statistical characteristics, such as statistical moments and probability density functions, by performing repeated samplings and simulations, and is usually used as a benchmark for evaluating the advantages and disadvantages of new uncertainty analysis techniques. However, the high computational cost caused by a large number of samplings makes the MC method inefficient in practical engineering and difficult to apply in ship design optimization that requires repeated simulation calculations.

[0005] The statistical moments of the ship performance function are essentially multi-dimensional integrals of the probability density function. In practical engineering problems, this integral is usually difficult to calculate, and numerical integration methods are often used to approximately calculate the integral value. Based on the idea of numerical integration, scholars have conducted a large number of studies to develop numerical methods for calculating complex integrals. Gaussian quadrature methods and other numerical quadrature methods approximate multi-dimensional integrals by the weighted sum of the integrand at a set of discrete integration points within the integration region. The Laplace approximation method approximates the integrand by minimizing the second-order Taylor series expansion to calculate the integral result. The Unscented Transform (UT) method combines the characteristics of numerical integration methods and the MC method. By assuming quadrature nodes and weight coefficients according to a certain point selection strategy, it takes into account both the efficiency of numerical integration methods and the distribution characteristics of random variables. Julier derived the statistical moment constraint equations for the normal distribution by using the symmetry of the normal distribution weight function and gave the analytical solutions of the 4th-order unscented transform nodes and weight coefficients. Adurthi proposed the Conjugate Unscented Transform (CUT) method, which adds a conjugate axis during the point selection process and develops a high-order unscented transform method. Aiming at the problem that the unscented transform method is only applicable to the normal distribution, Wang Tao used the Rosenblatt transform to transform any random variable into a normal random variable and then used the quadrature nodes of the normal distribution for statistical moment estimation, developing a more widely applicable CUT method.

[0006] Through the above analysis, the problems and defects existing in the prior art are as follows:

[0007] Traditional ship deterministic optimization (Deterministic Optimization, DO) does not consider the influence of uncertainty factors and mostly optimizes based on a deterministic mathematical model at fixed design points, unable to ensure excellent ship performance when uncertainty factors fluctuate. Most of the existing uncertainty analysis methods have low computational efficiency, making it difficult for ship uncertainty optimization methods to be applied in practical engineering.

[0008] The present invention improves the unscented transformation method. By constructing statistical moment constraint equations for various uncertainty distributions and establishing accurate and efficient statistical moment solution formulas, efficient uncertainty analysis and optimization are achieved. Summary of the Invention

[0009] In view of the problems existing in the prior art, the present invention provides a ship uncertainty optimization method and application based on improved unscented transformation.

[0010] The present invention is implemented as follows. A ship uncertainty optimization method based on improved unscented transformation includes:

[0011] Step 1: Use the unscented transformation method to calculate the statistical moments of the ship performance function under the influence of uncertainty factors.

[0012] Step 2: Generalize the method to uncertainties with arbitrary distributions by reconstructing the statistical moment constraint equations of uncertainties.

[0013] Step 3: Apply any unscented transformation method to numerical examples and the uncertainty optimization of bulk carriers.

[0014] Furthermore, the unscented transformation method:

[0015] (1) The basic principle of the unscented transformation method;

[0016] (2) The traditional unscented transformation method;

[0017] (3) Any unscented transformation method.

[0018] Furthermore, the basic principle of the unscented transformation method:

[0019] Considering the response function f(x) under the influence of random uncertainty x, its expected value is defined as

[0020]

[0021] where p(x) is the probability density function of the random uncertainty x; the unscented transformation method expresses Equation (1) as a numerical integration formula through a series of quadrature nodes and weight coefficients

[0022]

[0023] where x i and w i are the i-th quadrature node and its weight coefficient respectively; performing Taylor expansions on Equation (1) and Equation (2) at x = 0, we have

[0024]

[0025] By comparing Equation (3) and Equation (4), it can be seen that for the numerical integration formula (2) of the unscented transformation method to meet the accuracy requirements, its variables must satisfy the following system of equations

[0026]

[0027] The system of equations in Equation (5) is called the statistical moment constraint equation; when r = 0, the equation indicates that the sum of the weight coefficients of all quadrature nodes is 1; when r ≠ 0, the equation represents the numerical calculation formula for the r-th origin moment of the random variable x; similarly, the quadrature formula for the higher-order moment E(f r (x)) of the response function can be derived by the unscented transformation.

[0028] Furthermore, the traditional unscented transformation method is as follows:

[0029] 1) Coordinate origin: The two-dimensional standard normal random variables are all located at the mean value, that is, (0, 0), and its weight coefficient is w0;

[0030] 2) Main axis: The orthogonal coordinate axes centered at the origin are the main axes. There is at most one non-zero element in the node coordinates on the main axes. The node coordinates are (s1, 0), (-s1, 0), (0, s1), (0, -s1), and the weight coefficients are all w1;

[0031] 3) Axis of symmetry: The angle bisectors of the coordinate axes are the axes of symmetry. The absolute values of the non-zero elements in the node coordinates on the axes of symmetry are equal. The node coordinates are (s2, s2), (s2, -s2), (-s2, s2), (-s2, s2), and the weight coefficients are all w2;

[0032] Obviously, the nodes on the main axis and the axis of symmetry are all distributed on circles with different radii centered at the origin. The symmetry assumption ensures that the statistical moment constraint equations of odd-order normal distributions are automatically satisfied, greatly simplifying the solution process of the statistical moment constraint equations; for two-dimensional standard normal random variables, the statistical moment constraint equation in Equation (5) can be expanded into the following equation

[0033]

[0034] After solving Equation (6) to obtain the nodes and weight coefficients, the quadrature formula in Equation (2) can be constructed to calculate the statistical moments of the system response function.

[0035] Furthermore, the arbitrary unscented transformation method is as follows:

[0036] 1) Quadrature node and weight coefficient assumption

[0037] For a random variable X subject to an arbitrary distribution, to make the selection of quadrature nodes more general, regardless of whether it has symmetry or not, all node coordinates x i (i = 1, 2,..., N) and all weight coefficients wi (i = 1, 2, …, N) are all set as unknowns, and each node should satisfy the following constraint equations:

[0038]

[0039] In Equation (7), when k = 0, M0 = 1, and the equation represents that the sum of the weight coefficients of each node is 1; when k = 1, 2, …, r, M k is the k-th origin moment of the random variable X;

[0040] Taking a two-dimensional random system as an example, let x 1i , x 2j be the quadrature nodes of the random variables X1 and X2 respectively, and the corresponding weight coefficients are w 1i , w 2j , for any one-dimensional case, its nodes respectively satisfy:

[0041]

[0042] Orthogonally combine the two-variable nodes to form a node network, and each grid point is used as a two-dimensional node;

[0043] In the figure, the coordinates of the node X ij and its weight coefficient W ij can be calculated by the formula

[0044]

[0045] Taking the variable X1 as an example, conduct a statistical moment constraint test on the two-dimensional nodes. From the formula, it can be obtained that the two-dimensional nodes still satisfy the corresponding distribution in the direction of the variable X1;

[0046]

[0047] 2) Calculation of statistical moments and solution of constraint equations

[0048] The statistical moments of a general random distribution can be directly obtained by integrating its probability density function. The statistical moments of the extreme value distribution are represented by the statistical moments of t ~ EV(0, 1), and its calculation formula is shown in Equation (12);

[0049]

[0050] In Equation (12), ζ(s) is the Riemann Zeta function, and its function value can be obtained by looking up the table. When s = 1, the Zeta function diverges. To ensure the uniformity of the general formula (12), ζ(1) = γ is defined here; γ = 0.57721566490153286060651209… is the Euler–Mascheroni constant. The summation is for all non-negative integers m1, m2, …, m that satisfy Equation (13) nperformed, that is

[0051]

[0052] Substitute the calculation result into Equation (5) to solve the statistical moment constraint equation, and the one-dimensional node coordinates and weight coefficients corresponding to any random distribution can be obtained;

[0053] For the random variable X, if its statistical moments of all orders are known, then there exists a set of constants C0, C1, …, C N-1 , such that its origin moments of orders N - 1, N, …, 2N - 1 satisfy

[0054] C0M N-1 + C1M N + … + C N-1 M 2N-2 + M 2N-1 =0 (14)

[0055] Express its statistical moments of all orders in terms of nodes and weights, and we have

[0056]

[0057] To make the value of Equation (15) equal to 0, the unscented transformation quadrature nodes x1, x2, …, x of the random variable X N must simultaneously satisfy C0 + C1x + … + C N-1 x N-1 + x N =0, that is, the zeros of the polynomial (16) are the node coordinates;

[0058]

[0059] Therefore, by constructing the polynomial (16), the quadrature node coordinates can be obtained; the polynomial coefficients can be calculated through the linear equation system (17);

[0060]

[0061] After obtaining the node coordinates through Equations (16) and (17), substitute the node coordinates into the statistical moment constraint equation (5), and the weight coefficients can be obtained by solving the linear equation system;

[0062] 3) Standardization transformation of the random variable.

[0063] Furthermore, the standardization transformation of the random variable:

[0064] Taking the normal distribution as an example, assume that x follows a normal distribution with mean μ and standard deviation σ, x ∼ N(μ, σ 2 ), and t follows a normal distribution with mean 0 and standard deviation 1, t ∼ N(0, 1); according to the definition of statistical moments, the r - order origin moments of the two are respectively:

[0065]

[0066] Perform a variable substitution on t There is

[0067]

[0068] Let the standard normal variable t satisfy the statistical moment constraint equation, and its quadrature nodes and weight coefficients be w i , t i , there is

[0069]

[0070] Substitute the formula into the formula, then the statistical moment of the normal variable under any parameter can be expressed as

[0071]

[0072] Therefore, the quadrature nodes and weight coefficients of the normal variable x are w i , σt i + μ; Similarly, for other types of random distributions, the proof process is as follows;

[0073] Uniform distribution:

[0074]

[0075] Exponential distribution:

[0076]

[0077] Log-normal distribution:

[0078]

[0079] Extreme value distribution:

[0080]

[0081] Another object of the present invention is to provide a ship uncertainty optimization system based on an improved unscented transform, including:

[0082] A calculation module for calculating the statistical moment of the ship performance function under the influence of uncertainty factors by using the unscented transform method;

[0083] A promotion module for generalizing this method to uncertainties of any distribution by reconstructing the statistical moment constraint equation of uncertainty;

[0084] An application module for applying any unscented transform method to numerical examples and the uncertainty optimization of bulk carriers.

[0085] Another object of the present invention is to provide a computer device, which includes a memory and a processor. The memory stores a computer program. When the computer program is executed by the processor, the processor is caused to execute the steps of the ship uncertainty optimization method based on the improved unscented transform.

[0086] Another object of the present invention is to provide a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, the processor is caused to execute the steps of the ship uncertainty optimization method based on the improved unscented transform.

[0087] Another object of the present invention is to provide an information data processing terminal for implementing the ship uncertainty optimization system based on the improved unscented transform.

[0088] Combined with the above technical solutions and the solved technical problems, the advantages and positive effects of the technical solution to be protected by the present invention are as follows:

[0089] First, the present invention improves the unscented transform method and reconstructs the statistical moment constraint equation for uncertain quantities with any random distribution. In addition, for the uncertainty where the sample data is insufficient to establish an accurate probability distribution, in order to avoid the uncertainty analysis error caused by inaccurate distribution parameters, the present invention further applies the unscented transform method to the statistical moment estimation of small sample data, forming an unscented transform method for arbitrary uncertainty (Arbitrary Unscented Transform, AUT). The present invention verifies the effectiveness of this method through numerical examples and finally applies it to ship uncertainty optimization to obtain a robust solution less affected by uncertainty factors.

[0090] Ship uncertainty optimization often describes the impact of uncertainty on ship performance with low-order moment (expectation, variance) information of the performance function and optimizes with this as the goal. The present invention introduces the unscented transform method into the statistical moment estimation of arbitrary uncertainty systems and further conducts uncertainty optimization with statistical moments as the goal, and the main conclusions are as follows:

[0091] 1) Compared with traditional statistical moment estimation methods, the unscented transform method can balance accuracy and efficiency and achieve efficient calculation of the statistical moments of random systems;

[0092] 2) The unscented transform method proposed by the present invention is applicable to any type of random variable and uncertainty with only a small amount of sample data insufficient to establish an accurate probability distribution;

[0093] 3) When applying the unscented transformation method to uncertainty optimization, there is no need to repeatedly sample uncertainties in the uncertainty analysis step. Instead, statistical moments are calculated only through the response values of a small number of fixed nodes, simplifying the uncertainty optimization structure and improving the efficiency of uncertainty optimization.

[0094] Second, by adopting an improved unscented transformation method in uncertainty optimization, the present invention can significantly reduce the computational amount, save high-performance computing resources and R & D investment. For the design and analysis of large and complex systems, this technology can reduce the number of iterations and computational overhead while ensuring accuracy, thereby shortening the project cycle and improving the R & D efficiency. In the industrial application field, enterprises can use the solution of the present invention for rapid and multiple-iteration product optimization and risk assessment, thus significantly enhancing economic benefits and market competitiveness.

[0095] Existing unscented transformations mainly focus on uncertainty quantification for normal distributions and are difficult to handle complex distribution forms. The present invention extends the applicable range of the unscented transformation method from normal distributions to arbitrary random distributions and further covers the case of epistemic uncertainty that only contains a limited number of data points and for which an accurate probability distribution cannot be established. This extensibility breaks the limitations of traditional methods in distribution assumptions, fills the technical gaps that have not been solved in the domestic and international industries, and provides a feasible and efficient solution for more types of uncertainty analysis.

[0096] In actual engineering scenarios, random uncertainty and epistemic uncertainty often coexist, and the latter is more difficult to quantify due to data scarcity or subjective judgment. The present invention innovatively incorporates the two into the same analysis framework and effectively models limited samples or epistemic uncertainty through an improved unscented transformation method, which can not only ensure the comprehensiveness of the analysis but also take into account computational feasibility, thus providing a new technical means for the optimal design, fault tolerance control, and risk assessment of complex systems.

[0097] In the presence of coexisting mixed uncertainties, traditional methods often face the dilemmas of a sharp increase in computational amount, an overly long analysis period, or even being unable to complete. The improved unscented transformation method proposed by the present invention can still maintain a high degree of accuracy while significantly compressing the computational amount, which helps to quickly complete uncertainty optimization in engineering practice. By reducing the dependence on high-performance computing resources, the present invention enables more small and medium-sized enterprises or research institutions to carry out the design and analysis of complex systems, thereby promoting the popularization and in-depth development of uncertainty optimization technology.

[0098] Thirdly, in the design of ship engineering and ocean structures, most traditional uncertainty analysis methods are based on the assumption of normal distribution, and their accuracy is limited when dealing with multi-source signal data or finite samples. At the same time, for the actual working conditions with a mixture of random and epistemic uncertainties, existing technologies often require a large number of samplings or high-complexity operations, resulting in an excessive amount of calculation and making it difficult to meet the requirements of engineering design and optimization. Therefore, the present invention effectively overcomes the limitations of the normal distribution assumption in industrial applications and solves the technical problem of being difficult to balance the amount of calculation and accuracy.

[0099] The present invention expands the unscented transformation method from a single normal distribution to an arbitrary distribution by reconstructing the distribution constraint equation of uncertainty factors, and is compatible with multi-source signal data and epistemic uncertainties with only limited data points. Compared with the traditional Monte Carlo method that is only applicable to Gaussian distribution or requires a large number of samplings, the present invention can maintain a high analysis accuracy with fewer sampling points, making the uncertainty analysis of non-normal, non-linear or multi-modal distributions in the engineering field more feasible and efficient.

[0100] Through the improved unscented transformation method, the present invention can quickly approximate the statistical characteristics of the performance function in a multi-dimensional uncertainty scenario, and complete the optimization process without a large computing cluster or a long iteration time. Compared with the traditional high-dimensional sampling or numerical integration methods, the present invention significantly reduces the amount of calculation at the same accuracy level, shortens the engineering R & D cycle, reduces the investment in hardware resources, saves costs for industrial applications and improves efficiency.

[0101] When dealing with the uncertainty optimization problems of ships and ocean structures, the present invention takes into account the utilization of multi-source signal data and the modeling requirements of epistemic uncertainties, and still maintains reliable calculation accuracy and efficiency in an environment with a mixture of random and epistemic uncertainties. By reducing the engineering trial-and-error cost and improving the design reliability, the present invention provides a new technical path for the optimization design and risk assessment of ship and ocean engineering, reflecting significant technological progress in the field of uncertainty analysis. Brief Description of the Drawings

[0102] Figure 1 is a flowchart of a ship uncertainty optimization method based on an improved unscented transformation provided by an embodiment of the present invention.

[0103] Figure 2 is a structural block diagram of a ship uncertainty optimization system based on an improved unscented transformation provided by an embodiment of the present invention.

[0104] Figure 3 is a two-dimensional node schematic diagram of a traditional unscented transformation method provided by an embodiment of the present invention.

[0105] Figure 4 is a two-dimensional node schematic diagram of an improved unscented transformation method provided by an embodiment of the present invention.

[0106] Figure 5 It is the flowchart of the uncertainty optimization of the bulk carrier provided by the embodiment of the present invention.

[0107] Figure 6 It is the graph of the objective function varying with the uncertainty factors provided by the embodiment of the present invention. Specific embodiments

[0108] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below in conjunction with embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0109] As Figure 1 shown, a ship uncertainty optimization method based on the improved unscented transformation provided by the embodiment of the present invention includes the following steps:

[0110] S101, using the unscented transformation method to calculate the statistical moments of the ship performance function under the influence of uncertainty factors;

[0111] S102, by reconstructing the statistical moment constraint equation of uncertainty, extending this method to uncertainties with arbitrary distributions;

[0112] S103, applying an arbitrary unscented transformation method to numerical examples and the uncertainty optimization of bulk carriers.

[0113] The unscented transformation method provided by the embodiment of the present invention:

[0114] (1) The basic principle of the unscented transformation method;

[0115] (2) The traditional unscented transformation method;

[0116] (3) The arbitrary unscented transformation method.

[0117] The embodiment of the present invention aims at the performance analysis and optimization problems of ships under the influence of uncertainty factors, and proposes a ship uncertainty optimization method based on the improved unscented transformation. Compared with the traditional normal distribution assumption or sampling method, this method can be applied to uncertainties with arbitrary distributions, and can significantly reduce the calculation amount and improve the accuracy. This method has been verified in numerical examples and actual ship optimization scenarios, providing an efficient and accurate uncertainty analysis and optimization tool for complex ocean engineering and ship design.

[0118] The unscented transformation method is a numerical method that uses deterministic sampling points to approximate the high-order moments of a random distribution. The core idea is to select a small number of representative sampling points (also known as sigma points) and solve or simulate them in the performance function to approximate the statistical characteristics of the objective function under the influence of uncertainty. This method avoids a large number of Monte Carlo samplings, thus effectively reducing the computational amount while ensuring accuracy, and has strong scalability when analyzing multi-dimensional uncertainty.

[0119] The traditional unscented transformation method is generally based on the assumption of a normal distribution. After linearly or non-linearly transforming the random variables, the statistical quantities such as the mean and variance of the performance function can be estimated using sigma points. However, in practical engineering applications, uncertain factors often exhibit non-normal or multi-modal distributions, and some uncertainties even lack sufficient data to construct an accurate probability distribution. The accuracy and applicability of the traditional unscented transformation method are greatly limited in this case.

[0120] In the embodiments of the present invention, by reconstructing the statistical moment constraint equation of uncertainty, the unscented transformation method is extended from a single normal distribution to any distribution, including the cognitive uncertainty scenario with only a finite number of data points. The specific approach is to comprehensively consider the high-order moment information such as skewness and kurtosis of different distributions when selecting sigma points, so as to more accurately reflect the probability distribution characteristics of random variables in each region. This can not only effectively sample non-normal distributions, but also construct an approximate distribution model based on limited data or subjective judgment in the case of lack of a complete probability distribution.

[0121] In the numerical example, the present invention uses the arbitrary unscented transformation method to estimate the performance function under multi-dimensional non-normal distributions. The results show that compared with traditional sampling or normal assumption methods, it can reduce a large amount of computational amount at a similar accuracy level. In the uncertainty optimization of bulk carriers, by comprehensively considering the main performance indicators of the ship (such as stability, wave resistance, and structural strength) and making arbitrary distribution assumptions for relevant uncertain factors (such as load distribution and sea conditions), the optimization method proposed by the present invention can more accurately obtain the optimal design parameters, reduce the design risk and improve the economy.

[0122] The improved unscented transformation method of the present invention can not only effectively adapt to the uncertainties of various distribution forms, but also significantly reduce the computational cost brought by large-scale simulation or sampling, and has high generality and engineering value. Its successful application in the ship industry shows that this method can be extended to fields such as aerospace, ocean engineering, and mechanical manufacturing that require uncertainty optimization, providing efficient and accurate decision-making support for complex system design.

[0123] The basic principle of the unscented transformation method provided by the embodiments of the present invention:

[0124] Consider the response function \(f(x)\) under the influence of random uncertainty \(x\), and its expected value is defined as

[0125]

[0126] where \(p(x)\) is the probability density function of the random uncertainty \(x\); the unscented transformation method expresses Equation (1) as a numerical integration formula through a series of quadrature nodes and weight coefficients

[0127]

[0128] where \(x\) i and \(w\) i are the \(i\)-th quadrature node and its weight coefficient respectively; perform Taylor expansions on Equations (1) and (2) at \(x = 0\), and we have

[0129]

[0130] By comparing Equation (29) and Equation (30), it can be seen that for the numerical integration formula (28) of the unscented transformation method to meet the accuracy requirements, its variables must satisfy the following system of equations

[0131]

[0132] The system of equations in Equation (31) is called the statistical moment constraint equation; when \(r = 0\), the equation indicates that the sum of the weight coefficients of all quadrature nodes is 1; when \(r\neq0\), the equation represents the numerical calculation formula for the \(r\)-th origin moment of the random variable \(x\); similarly, the unscented integration formula for the higher-order moment \(E(f\) r (x)) can be derived.

[0133] The traditional unscented transformation method provided by the embodiments of the present invention:

[0134] 1) Coordinate origin: The two-dimensional standard normal random variables are all located at the mean value, that is, \((0, 0)\), and its weight coefficient is \(w_0\);

[0135] 2) Main axis: The orthogonal coordinate axes centered at the origin are the main axes. There is at most one non-zero element in the node coordinates on the main axis, and the node coordinates are \((s_1, 0), (-s_1, 0), (0, s_1), (0, -s_1)\), and the weight coefficients are all \(w_1\);

[0136] 3) Axis of symmetry: The angle bisectors of the coordinate axes are used as the axes of symmetry. The absolute values of the non-zero elements of the node coordinates on the axis of symmetry are equal, and the node coordinates are \((s_2, s_2), (s_2, -s_2), (-s_2, s_2), (-s_2, s_2)\), and the weight coefficients are all \(w_2\);

[0137] Obviously, the nodes on the main axis and the symmetry axis are all distributed on circles with different radii centered at the origin. The symmetry assumption ensures that the constraint equations of the odd-order statistical moments of the normal distribution are automatically satisfied, greatly simplifying the solution process of the statistical moment constraint equations. For two-dimensional standard normal random variables, the statistical moment constraint equation in Equation (31) can be expanded into the following equations

[0138]

[0139] After solving Equation (32) to obtain the nodes and weight coefficients, the quadrature formula in Equation (28) can be constructed to calculate the statistical moments of the system response function.

[0140] The any unscented transformation method provided by the embodiments of the present invention

[0141] 1) Quadrature node and weight coefficient assumption

[0142] For a random variable X subject to an arbitrary distribution, to make the selection of quadrature nodes more general, regardless of whether it has symmetry, all node coordinates x i (i = 1, 2,..., N) and all weight coefficients w i (i = 1, 2,..., N) are set as unknowns, and each node should satisfy the following constraint equations

[0143]

[0144] In Equation (33), when k = 0, M0 = 1, and the equation represents that the sum of the weight coefficients of each node is 1; when k = 1, 2,..., r, M k is the k-th origin moment of the random variable X

[0145] Taking a two-dimensional random system as an example, let x 1i , x 2j be the quadrature nodes of the random variables X1 and X2 respectively, and the corresponding weight coefficients be w 1i , w 2j , for any one-dimensional case, its nodes respectively satisfy

[0146]

[0147] The nodes of the two variables are orthogonally combined to form a node network, and each grid point is used as a two-dimensional node

[0148] The coordinates of the node X ij and its weight coefficient W ij can be calculated by the formula

[0149]

[0150] Taking the variable X1 as an example, the statistical moment constraint test is performed on the two-dimensional nodes. From the formula, it can be obtained that the two-dimensional nodes still satisfy the corresponding distribution in the direction of the variable X1;

[0151]

[0152] 2) Calculation of statistical moments and solution of constraint equations

[0153] The statistical moments of a general random distribution can be directly obtained by integrating its probability density function. The statistical moments of the extreme value distribution are represented by the statistical moments of t~EV(0,1), and its calculation formula is shown in Equation (38);

[0154]

[0155] In Equation (38), ζ(s) is the Riemann Zeta function, and its function value can be obtained by looking up the table. When s = 1, the Zeta function diverges. To ensure the unity of the general formula (38), ζ(1) = γ is defined here; γ = 0.57721566490153286060651209… is the Euler–Mascheroni constant. The summation is carried out for all non-negative integers m1, m2, …, m that satisfy Equation (39), that is n performed, that is

[0156]

[0157] Substitute the calculation results into Equation (31) to solve the statistical moment constraint equation, and the one-dimensional node coordinates and weight coefficients corresponding to any random distribution can be obtained;

[0158] For the random variable X, if its statistical moments of all orders are known, then there exists a set of constants C0, C1, …, C N-1 , such that its origin moments of orders N - 1, N, …, 2N - 1 satisfy

[0159] C0M N-1 +C1M N +…+C N-1 M 2N-2 +M 2N-1 =0 (40)

[0160] Express its statistical moments of all orders in terms of nodes and weights, and we have

[0161]

[0162] To make the value of Equation (41) equal to 0, the unscented transformation quadrature nodes x1, x2, …, x of the random variable X N must simultaneously satisfy C0 + C1x + … + C N-1 x N-1 +x N =0, that is, the zeros of the polynomial (42) are the node coordinates;

[0163]

[0164] Therefore, by constructing the polynomial (42), the coordinates of the quadrature nodes can be obtained; the polynomial coefficients can be calculated through the linear equations (43).

[0165]

[0166] After obtaining the node coordinates through equations (42) and (43), substituting the node coordinates into the statistical moment constraint equation (31), the weight coefficients can be obtained by solving the linear equations.

[0167] 3) Standardization transformation of random variables.

[0168] The standardization transformation of random variables provided by the embodiments of the present invention:

[0169] Taking the normal distribution as an example, let x follow a normal distribution with a mean of μ and a standard deviation of σ, x ∼ N(μ, σ 2 ), and t follow a normal distribution with a mean of 0 and a standard deviation of 1, t ∼ N(0, 1); according to the definition of statistical moments, the r-th origin moments of the two are respectively:

[0170]

[0171] Perform a variable substitution on t There is

[0172]

[0173] Let the standard normal variable t satisfy the statistical moment constraint equation, and its quadrature nodes and weight coefficients be w i , t i , there is

[0174]

[0175] Substitute the equation into the equation, and the statistical moments of the normal variable under any parameters can be expressed as

[0176]

[0177] Therefore, the quadrature nodes and weight coefficients of the normal variable x are w i , σt i + μ; similarly, for other random distribution types, the proof process is as follows;

[0178] Uniform distribution:

[0179]

[0180] Exponential distribution:

[0181]

[0182] Logarithmic normal distribution:

[0183]

[0184] Extreme value distribution:

[0185]

[0186] As Figure 2 shown, a ship uncertainty optimization system based on improved unscented transform provided by an embodiment of the present invention includes:

[0187] A calculation module, configured to calculate the statistical moments of a ship performance function under the influence of uncertainty factors by using the unscented transform method;

[0188] A generalization module, configured to generalize this method to uncertainties with arbitrary distributions by reconstructing the statistical moment constraint equation of the uncertainty;

[0189] An application module, configured to apply an arbitrary unscented transform method to numerical examples and the uncertainty optimization of bulk carriers.

[0190] Another object of the present invention is to provide a computer device, where the computer device includes a memory and a processor, and the memory stores a computer program. When the computer program is executed by the processor, the processor executes the steps of the ship uncertainty optimization method based on improved unscented transform.

[0191] Another object of the present invention is to provide a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, the processor executes the steps of the ship uncertainty optimization method based on improved unscented transform.

[0192] Another object of the present invention is to provide an information data processing terminal, where the information data processing terminal is used to implement the ship uncertainty optimization system based on improved unscented transform.

[0193] Specific implementation of the present invention:

[0194] 1. The present invention improves the unscented transformation method, reconstructs the statistical moment constraint equation for any randomly distributed uncertain quantity. In addition, for the uncertainty where the sample data is insufficient to establish an accurate probability distribution, in order to avoid the uncertainty analysis error caused by inaccurate distribution parameters, the present invention further applies the unscented transformation method to the statistical moment estimation of small sample data, forming the Arbitrary Unscented Transform (AUT) for any uncertainty. The present invention verifies the effectiveness of this method through numerical examples, and finally applies it to the uncertainty optimization of ships to obtain a robust solution with less influence from uncertain factors.

[0195] 2 Unscented Transformation Method for Statistical Moment Estimation of Uncertainty Systems

[0196] 2.1 Basic Principle of Unscented Transformation Method

[0197] Consider the response function f(x) affected by random uncertainty x, and its expected value is defined as

[0198]

[0199] where p(x) is the probability density function of the random uncertainty x. The unscented transformation method expresses Equation (1) as a numerical integration formula through a series of quadrature nodes and weight coefficients

[0200]

[0201] where x i and w i are the i-th quadrature node and its weight coefficient respectively. Perform Taylor expansions on Equations (1) and (2) at x = 0, and we have

[0202]

[0203] By comparing Equations (55) and (56), it can be seen that for the numerical integration formula (54) of the unscented transformation method to meet the accuracy requirements, its variables must satisfy the following system of equations

[0204]

[0205] The system of equations in Equation (57) is called the statistical moment constraint equation. When r = 0, the equation means that the sum of the weight coefficients of all quadrature nodes is 1; when r ≠ 0, the equation represents the numerical calculation formula for the r-th origin moment of the random variable x. Similarly, the unscented quadrature formula for the higher-order moment E(f r (x)) of the response function can be derived.

[0206] As can be seen from the formula, as a numerical integration method, the unscented transformation method depends on the selection of quadrature nodes for its calculation accuracy. Note that in Eqs. (55) and (56), to ensure the numerical calculation accuracy of the expected value E(f(x)), the expansion terms of the Taylor series are infinite. However, considering the computational complexity, in practical applications, usually according to a certain accuracy requirement, only the first r-order statistical moment constraint equations are retained, which is called the r-order unscented transformation method. Obviously, more nodes and higher orders can improve the accuracy of the unscented transformation method, but at the same time, it will increase the difficulty of solving the equations and the computational complexity.

[0207] 2.2 Traditional Unscented Transformation Method

[0208] The traditional unscented transformation method establishes statistical moment constraint equations for the normal distribution and omits the odd-order equations in Eq. (57) by using the symmetry of the normal distribution, reducing the difficulty of solving the statistical moment constraint equations. However, the obtained quadrature nodes are not applicable to the statistical moment calculation of any random distribution.

[0209] The unscented transformation method essentially belongs to a numerical integration method. The quadrature nodes and their weight coefficients are determined through two steps: (1) Node hypothesis: Regarding the quadrature node coordinates and weight coefficients as unknowns, based on the symmetry of the normal distribution, the nodes are placed on the coordinate axes and their angle bisectors, and the weight coefficients of the symmetric nodes are equal, so that the odd-order statistical moment constraint equations are automatically satisfied; (2) Quantitative solution: Substitute the assumed node coordinates and weight coefficients into the statistical moment constraint equations to make them meet the order accuracy requirements, solve the equations to obtain the node coordinates and weight coefficients, and construct the quadrature formula. Obviously, the higher the order and the higher the dimension, the more equations are required to determine the nodes and weight coefficients, and the more difficult it is to solve. Taking the unscented transformation method of two-dimensional standard normal random variables as an example, the determination criteria for the quadrature nodes and weight coefficients are as follows:

[0210] 1) Coordinate origin: The two-dimensional standard normal random variables are both located at the mean value, that is, (0, 0), and its weight coefficient is w0;

[0211] 2) Main axis: The orthogonal coordinate axes centered at the origin are the main axes. There is at most one non-zero element in the node coordinates on the main axes. The node coordinates are (s1, 0), (-s1, 0), (0, s1), (0, -s1), and the weight coefficients are all w1;

[0212] 3) Axis of symmetry: Taking the angle bisector of the coordinate axes as the axis of symmetry, the absolute values of the non-zero elements in the node coordinates on the axis of symmetry are equal. The node coordinates are (s2, s2), (s2, -s2), (-s2, s2), (-s2, s2), and the weight coefficients are all w2.

[0213] Figure 3 Schematic diagram of two-dimensional nodes of the unscented transformation method

[0214] Obviously, the nodes on the main axis and the symmetry axis are all distributed on circles with different radii centered at the origin, and the symmetry assumption ensures the normal distribution.

[0215] The constraint equations of the odd-order statistical moments are automatically satisfied, greatly simplifying the solution process of the statistical moment constraint equations. For two-dimensional standard normal random variables, the statistical moment constraint equation in Equation (57) can be expanded into the following equations

[0216]

[0217] After solving Equation (58) to obtain the nodes and weight coefficients, the quadrature formula in Equation (54) can be constructed to calculate the statistical moments of the system response function.

[0218] 2.3 Arbitrary Unscented Transformation Method

[0219] Although certain progress has been made in the calculation accuracy and efficiency of the existing unscented transformation methods, their application scope is limited. If they are to be extended to general random uncertainty systems, the following problems need to be urgently solved:

[0220] 1) The traditional unscented transformation method utilizes the symmetry of the random distribution and is not applicable to the uncertainty analysis of arbitrary random variables;

[0221] 2) The existing unscented transformation methods can obtain analytical solutions in low-order problems, but are complex to solve and difficult to obtain analytical solutions in high-order problems.

[0222] Therefore, in order to solve the above problems, the present invention has developed an unscented transformation method applicable to arbitrary random systems.

[0223] 2.3.1 Quadrature Node and Weight Coefficient Assumptions

[0224] For a random variable X subject to an arbitrary distribution, to make the selection of quadrature nodes more general, regardless of whether it has symmetry, all node coordinates x i (i = 1, 2,..., N) and all weight coefficients w i (i = 1, 2,..., N) are set as unknowns, and each node should satisfy the following constraint equations:

[0225]

[0226] In Equation (59), when k = 0, M0 = 1, and the equation represents that the sum of the weight coefficients of each node is 1; when k = 1, 2,..., r, M k is the k-th origin moment of the random variable X.

[0227] For a multi-dimensional random system, an increase in the number of nodes will lead to a large number of unknowns in the statistical moment constraint equations, making it difficult to directly solve. Therefore, based on the known one-dimensional quadrature nodes and weight coefficients, the present invention proposes a method for determining multi-dimensional nodes to avoid complex equation-solving problems.

[0228] Taking a two-dimensional random system as an example, let x 1i , x 2j be the quadrature nodes of random variables X1 and X2 respectively, and the corresponding weight coefficients be w 1i , w 2j . For any one-dimensional case, its nodes respectively satisfy:

[0229]

[0230] Orthogonally combine the nodes of the two variables to form a node network, and each grid point serves as a two-dimensional node.

[0231] In the figure, the coordinates of node X ij and its weight coefficient W ij can be calculated by the formula

[0232]

[0233] Taking variable X1 as an example to conduct a statistical moment constraint test on the two-dimensional nodes, it can be obtained from the formula that the two-dimensional nodes still satisfy the corresponding distribution in the direction of variable X1.

[0234]

[0235] By analogy, the nodes still satisfy the statistical moment constraint equations in each dimension; further, for a random system with a higher-dimensional case, the orthogonal method proposed in this section can be used to directly obtain the quadrature nodes and weight coefficients of the multi-dimensional random system from the one-dimensional nodes.

[0236] 2.3.2 Calculation of Statistical Moments and Solution of Constraint Equations

[0237] According to the distribution type of the random variable, the statistical moments of the corresponding random distribution can be calculated respectively, and the constraint equations can be solved to obtain the unscented transformation quadrature nodes under this random distribution. For a random variable with a known distribution type and parameters, the statistical moments can be obtained by integrating the probability density function; for the uncertainty with too few sample data to establish an accurate probability distribution, the statistical moments of the sample data can be estimated and then substituted into the statistical moment constraint equations for solution. Table 1 shows the statistical moment calculation formulas for 5 common random distributions:

[0238] Table 1 Statistical Moment Calculation Formulas for Common Random Distributions

[0239]

[0240] In Table 1, the calculation formulas of statistical moments for the first four random distributions can be easily obtained from their distribution parameters. 1 , while the statistical moments of the extreme value distribution are expressed by the statistical moments of t ~ EV(0,1), and its calculation formula is shown in Equation (64).

[0241]

[0242] In Equation (64), ζ(s) is the Riemann Zeta function, and its function values can be obtained from the table. When s = 1, the Zeta function diverges. To ensure the unity of the general formula (64), ζ(1) = γ is defined here; γ = 0.57721566490153286060651209… is the Euler–Mascheroni constant. The summation is carried out for all non-negative integers m1, m2, …, m that satisfy Equation (65), that is n as follows

[0243]

[0244] After calculating the statistical moments of the random variable using Table 1, substitute the calculation results into Equation (57) to solve the statistical moment constraint equation, and the one-dimensional node coordinates and weight coefficients corresponding to any random distribution can be obtained.

[0245] Since the statistical moment constraint equation (5) is a system of high-order nonlinear equations, it is often difficult to obtain an accurate analytical solution in the actual solution process. Therefore, the present invention adopts the polynomial method to transform the statistical moment constraint equation into a linear equation to improve the solution accuracy.

[0246] For the random variable X, if its statistical moments of each order are known, then there exists a set of constants C0, C1, …, C N-1 , such that its origin moments of order N - 1, N, …, 2N - 1 satisfy

[0247] C0M N-1 + C1M N + … + C N-1 M 2N-2 + M 2N-1 = 0 (66)

[0248] Expressing its statistical moments of each order in terms of nodes and weights, we have

[0249]

[0250] To make the value of Equation (67) equal to 0, the unscented transformation quadrature nodes x1, x2, …, x of the random variable X N must simultaneously satisfy C0 + C1x + … + C N-1 x N-1 + x N = 0, that is, the zeros of the polynomial (16) are the node coordinates.

[0251]

[0252] Therefore, by constructing the polynomial (68), the coordinates of the quadrature nodes can be obtained. The polynomial coefficients can be calculated from the system of linear equations (69).

[0253]

[0254] After obtaining the node coordinates from Eqs. (68) and (69), substituting the node coordinates into the statistical moment constraint equation (57), the weight coefficients can be obtained by solving the system of linear equations. The method for solving the statistical moment constraint equation adopted in this section transforms the non-linear system of equations into a simple system of linear equations and the problem of solving polynomial zeros, ensuring the calculation accuracy of the node coordinates and weight coefficients. The accuracy of the statistical moment order calculated for N nodes is r = 2N - 1.

[0255] Applying the statistical moment calculation and constraint equation solving method in this section to the 5 common random distributions in Table 1, the 9th-order 5-node coordinates and weight coefficients under the corresponding distribution parameters are shown in Table 2.

[0256] Table 2 9th-order 5-node coordinates and weight coefficients of common distributions

[0257]

[0258] Using the quadrature nodes and weight coefficients in Table 2, construct the unscented transform quadrature formula shown in the equation to achieve efficient calculation of the statistical moments of the response function.

[0259] 2.3.3 Standardization transformation of random variables

[0260] The unscented transform method proposed in the present invention is applicable to uncertainty variables with arbitrary distributions. If the statistical moment constraint equations are repeatedly solved for each variable, the computational amount of the analysis process will increase; and when the distribution parameters are large, the order of magnitude difference between the statistical moments of the variables may be too large, resulting in difficulty in obtaining a convergent solution for the statistical moment constraint equations. Therefore, in order to simplify the solution of the statistical moment constraint equations, the statistical moment constraint equations are solved according to the distribution type in this section. The random variables with the same distribution type are standardized, and the transformed standard distribution is used as the reference distribution for the corresponding distribution type. Based on the unscented transform quadrature nodes under the reference distribution, the transformation relationship under different parameters of the same distribution is established to quickly obtain the quadrature nodes of any random distribution.

[0261] Taking the normal distribution as an example, assume that x follows a normal distribution with mean μ and standard deviation σ, x ~ N(μ, σ 2 ), and t follows a normal distribution with mean 0 and standard deviation 1, t ~ N(0, 1). According to the definition of statistical moments, the r-th order origin moments of the two are respectively:[[]]

[0262]

[0263] Perform a variable substitution on t There is

[0264]

[0265] Let the standard normal variable t satisfy the statistical moment constraint equation, and its quadrature nodes and weight coefficients be w i , t i , there is

[0266]

[0267] Substitute the formula into the formula, then the statistical moments of the normal variable under any parameters can be expressed as

[0268]

[0269] Therefore, the quadrature nodes and weight coefficients of the normal variable x are w i , σt i + μ. Similarly, for other types of random distributions, the quadrature nodes and weight coefficients under any distribution parameters can be quickly obtained according to the transformation relationship shown in Table 3. The proof process is as follows.

[0270] Uniform distribution:

[0271]

[0272] Exponential distribution:

[0273]

[0274] Log-normal distribution:

[0275]

[0276] Extreme value distribution:

[0277]

[0278] Table 3 Transformation relationship between different parameters of random distributions and reference distributions

[0279]

[0280] In Table 3, the transformations of normal, uniform, exponential, and extreme value distributions are all linear transformations, which do not affect the node calculation accuracy. However, the transformation type of the log-normal distribution is a non-linear transformation, and the order may exceed the order of the constraint equation during the transformation process, resulting in calculation errors. Therefore, for any uncertainty variable, the calculation method of node coordinates and weight coefficients can be selected according to its distribution type.

[0281] 2.3.4 Implementation process

[0282] In summary, the specific implementation process of the arbitrary unscented transformation method proposed in this section is as follows:

[0283] 1) Determine the random distribution type and distribution parameters of the uncertain quantity;

[0284] 2) Calculate the statistical moments of all orders of the uncertain quantity according to the probability density function or sample data of the random distribution;

[0285] 3) Assume the quadrature node coordinates and weight coefficients, solve the statistical moment constraint equation by the polynomial method, and calculate the one-dimensional quadrature nodes of the uncertain quantity;

[0286] 4) Obtain the multi-dimensional node coordinates and weight coefficients according to the one-dimensional nodes;

[0287] 5) Calculate the statistical moments of the random system.

[0288] 3 Numerical Examples

[0289] 3.1 Numerical Examples

[0290] In this section, 6 function examples are selected to verify the accuracy, efficiency, and applicability of the proposed unscented transformation method. The objective functions include linear functions and non-linear functions. The uncertainty variables involved include random uncertainties with known random distributions and unknown distribution variables with only point data. The applicability of this method to different uncertainty systems is studied by analyzing the results.

[0291] The expressions of the objective functions and the distributions of the uncertainty variables included in each function are as follows.

[0292] 1) Linear function: f1 = x1 + 2x2 + 2x3 + x4 - 5x5 - x6, and the distribution of the random variables is shown in Table 4.

[0293] Table 4 Statistical Characteristics of Random Variables of f1

[0294]

[0295] 2) Polynomial function: The distribution of the random variables is shown in Table 5;

[0296] Table 5 Statistical Characteristics of Random Variables of f2

[0297]

[0298] 3) Polynomial function: In this example, x i (i = 1, 2, 3, 4) respectively follow two different types of random distributions, and the specific distribution parameters are shown in Table 6;

[0299] Table 6 Statistical Characteristics of Random Variables of f3

[0300]

[0301] 4) Structural displacement function: The distribution of random variables is shown in Table 7;

[0302] Table 7 Statistical characteristics of f4 random variables

[0303]

[0304] 5) Ishigami function: The distribution of random variables is shown in Table 8;

[0305] Table 8 Statistical characteristics of f5 random variables

[0306]

[0307]

[0308] 6) Polynomial function with unknown distribution: In this example, the distribution of the uncertain quantity is unknown, and the existing sample point data is shown in Table 9;

[0309] Table 9 Data table of f6 sample points

[0310]

[0311] The present invention adopts the unscented transformation method proposed in Section 2, selects 9th order 5 nodes to perform uncertainty analysis on the above 6 functions, and the calculated expectation, standard deviation and number of calculations of the objective function are shown in Table 10. Among them, the expectations and standard deviations of functions f1, f2, f3, f5 are easy to obtain analytical solutions. The calculated results of the expectations and standard deviations of f4 by the MC method are used as accurate values for comparison, and the calculated results of the expectations and standard deviations of f6 by sample data are used as accurate values for comparison.

[0312] Table 10 Results of uncertainty analysis of numerical examples

[0313]

[0314] As can be seen from Table 10, in terms of accuracy, for linear functions, polynomial functions, and non-linear structural displacement functions, the unscented transformation method proposed in the present invention can accurately estimate their expectations and standard deviations. However, for the Ishigami function, the calculation error is relatively large, and the calculation error of its standard deviation is 15.92%, which is caused by insufficient node orders used. By performing Taylor expansions on the trigonometric function terms in the Ishigami function respectively, it can be seen that the more expansion terms and the higher the order, the higher the fitting accuracy. Table 10 shows the calculation results of the 9th-order 5-node unscented transformation method, and the calculation accuracy of higher-order moments beyond the constraint order cannot be guaranteed. To study the influence of node accuracy on the results of uncertainty analysis, the mean and standard deviation of the Ishigami function are calculated using the 13th-order 7-node in Table 11 and compared with the calculation results of the 9th-order 5-node. From the result comparison in Table 12, it can be seen that the error of the standard deviation is reduced from 15.92% to 3.89%. Increasing the number of nodes can effectively improve the accuracy of uncertainty analysis, but the calculation amount increases from 30 times to 56 times. In practical problems, the number of nodes should be reasonably selected according to the order of the objective function.

[0315] Table 11 9th-order 5-node and 13th-order 7-node of Ishigami function

[0316]

[0317]

[0318] Table 12 Comparison of uncertainty analysis results of Ishigami function

[0319]

[0320] For uncertainties of different distribution types, from the comparison of two sets of working conditions of f3, it can be seen that whether it is a common normal distribution or any other random distribution, AUT can accurately calculate the expectation and standard deviation of the objective function. For the function f6, the type of its uncertainty distribution is unknown, and there are only 50 point data. After AUT captures the statistical moment characteristics of the sample points, the quadrature nodes that meet the sample distribution characteristics are obtained, reducing the calculation amount of statistical analysis while accurately calculating the higher-order moments of the objective function.

[0321] In terms of the calculation amount, for functions f1, f2, f3, f5, when the distribution of random variables is known, it is easy to obtain the analytical solutions of their statistical moments. Generally, however, the form of the objective function of an uncertain system is complex (such as f4), or even a black-box problem with an unknown expression form, making it difficult to obtain analytical solutions. The commonly used MC method requires a large number of random samplings and statistical calculations to obtain its statistical moment information, while the number of calculations of AUT is greatly reduced.

[0322] 3.2 Uncertainty optimization example of bulk carrier

[0323] 3.2.1 Optimization Problem Description

[0324] In this section, the AUT method is applied to an example of the optimization design of ship uncertainty. Considering the engineering optimization model of bulk carriers proposed by Hannapel, according to the literature, the steel weight of the bulk carrier is

[0325] W S = 0.034L 1.7 B 0.7 D 0.4 C B 0.5 (79)

[0326] The calculation formula for the steel weight in this model is obtained by statistical regression of actual ship data. Considering that the sample size of actual ship data is limited and the regression method is unknown, resulting in uncontrollable errors, the exponential value obtained by regression cannot accurately reflect the actual engineering situation. Therefore, the exponents of the calculation formula are considered uncertain quantities. Define the length exponent and the beam exponent as uncertain quantities, i.e.:

[0327] W S = 0.034L X B 0.7 D 0.4 C B 0.5 (80)

[0328] where X follows a normal distribution with a mean of 1.7 and a standard deviation of 0.1.

[0329] During the navigation of the ship at sea, due to wind and wave disturbances and human factors, the actual draft and the actual speed are not fixed values, but uncertain quantities that fluctuate around the design values. By monitoring the change of speed over a period of time and considering the uncertainty of ship speed, its expression is as follows:

[0330] T = T k + Y (81)

[0331] V = V k + Z (82)

[0332] where, T represents the actual draft, T k represents the design draft, Y represents the draft fluctuation, and follows a uniform distribution in the interval [-0.2, 0.2]; V represents the actual speed, V k represents the design speed, Z represents the speed fluctuation, and follows a normal distribution with a mean of 0 and a standard deviation of 0.1V k .

[0333] In this example, deterministic optimization (DO) and uncertainty optimization were carried out respectively. In deterministic optimization, the influence of uncertainty was not considered, and the uncertain quantities were fixed at the mean values respectively, and optimization was carried out with the goal of minimizing the freight cost in the design state. In uncertainty optimization, the ship speed fluctuation, the length exponent and the breadth exponent in the steel calculation formula were considered as uncertain quantities, and multi-objective optimization was carried out with the goal of minimizing the expectation and standard deviation of the freight cost under the influence of uncertainty. The optimization model is shown in Table 13.

[0334] Table 13 Engineering Optimization Model of Bulk Carrier

[0335]

[0336]

[0337] 3.2.2 Uncertainty Optimization Process

[0338] In this section, the AUT method was used for uncertainty optimization, and the MC method was used to verify the optimization scheme. The optimization process is shown in the figure.

[0339] Figure 5 The specific steps of the uncertainty optimization process shown are as follows:

[0340] 1) Preparation: Using any unscented transformation method in Section 2.3, according to the distribution type of uncertain quantities, obtain the unscented transformation quadrature nodes and weight coefficients, and construct the statistical moment quadrature formula;

[0341] 2) Optimization definition: Determine the design variables and their value ranges;

[0342] 3) Modify the design variables: The optimizer generates a set of design variable values and passes them into the uncertainty analysis module;

[0343] 4) Uncertainty analysis: When the design variables are fixed, calculate the response values of the objective function at each node of the uncertain quantities. In the unscented transformation method, the unscented transformation quadrature formula is used to calculate the expectation and standard deviation; in the MC method, statistical analysis is carried out on the response values of each sample point to calculate the expectation and standard deviation of the objective function

[0344] 5) Convergence determination: Taking the maximum number of iterations as the convergence criterion, judge whether the optimization is completed. If the maximum number of iterations is reached, output the optimal design scheme; otherwise, repeat steps 3) - 5).

[0345] 3.2.3 Analysis of Optimization Results

[0346] In this example, deterministic optimization uses the multi-island genetic algorithm with a subgroup size of 20, 20 islands, and 50 generations. For the outer optimization loop of uncertainty, the NSGA-II optimization algorithm is used with a population size of 100 and 100 generations. In the inner uncertainty analysis, the MC method has 105 sampling points, and the unscented transformation method uses 9th-order nodes. The objective is optimized for uncertainty when the uncertain variables X, Y, and Z vary individually and simultaneously. The optimization schemes are denoted as UT-X, UT-Y, UT-Z, and UT-XYZ respectively. The optimization results are shown in Table 14, and the variation of the optimization schemes with uncertainty factors is shown in Figure 6 .

[0347] Table 14 Engineering Optimization Results of Bulk Carrier

[0348]

[0349] As can be seen from Table 14, the objective function values of the DO and the four UDO schemes are all smaller than the initial scheme under the design conditions, achieving certain optimization effects. When optimizing considering a single uncertainty factor, under the influence of the uncertainty factors X, Y, and Z, the variation trends of the objective functions of the optimization schemes UT-X, UT-Y, and UT-Z are generally smoother, and the mean value μ(TC) and standard deviation σ(TC) are both smaller than those of the DO scheme, indicating significant uncertainty optimization effects. When considering the simultaneous variation of the three uncertainty factors X, Y, and Z, since the influence of the three uncertainty factors needs to be considered simultaneously, the optimization conditions are more stringent, and the optimization effect on a single uncertain variable is not as significant as that of the single uncertain variable optimization scheme. Only the variation curve of the UT-XYZ scheme becomes smoother under the influence of the uncertain variable X, effectively reducing the fluctuation of the objective function when the uncertain variable X is large. However, under the simultaneous action of the three uncertainty factors, since the uncertainty factor X has the greatest influence on TC, X contributes the most to the mean value and standard deviation of TC, and the mean value μ(TC) and standard deviation σ(TC) of the objective function are still smaller than those of DO. Generally speaking, although the objective function of the DO scheme is better under the design conditions, under the influence of uncertainty factors, the mean value μ(TC) and standard deviation σ(TC) of the objective functions of the four UDO schemes are smaller, indicating that UDO reduces the influence degree of uncertainty factors, making the optimization scheme have a better average level and smaller fluctuation under the influence of uncertainty factors. In this example, its practical significance is that when there are errors in the hull weight estimation and fluctuations in draft and speed during actual navigation, the freight cost TC will not change violently within its fluctuation range and will always be at a relatively low level, ensuring the robustness of the optimization objective and thus better adapting to changes in the external environment.

[0350] From the analysis of the computational complexity of the UDO method, in this example, the 9th-order unscented transformation method used has 5 nodes in one dimension and 125 nodes in three dimensions, both of which are much smaller than the 105 sample points required by MC. Since in uncertainty optimization, uncertainty analysis needs to be performed on each design scheme during each iteration, a large number of samplings and statistical calculations will lead to unbearable computational costs. The method proposed in the present invention effectively captures the statistical characteristics of uncertainty factors through a set of carefully selected nodes, while reducing the computational complexity of uncertainty analysis, and effectively obtains the uncertainty optimization solution during engineering optimization.

[0351] 4 Conclusions

[0352] The uncertainty optimization of ships often describes the impact of uncertainty on ship performance with the low-order moments (expectation, variance) information of the performance function, and optimizes with this as the goal. The present invention introduces the unscented transformation method into the statistical moment estimation of any uncertain system, and further performs uncertainty optimization with the statistical moment as the goal, and the main conclusions are as follows:

[0353] 1) Compared with the traditional statistical moment estimation method, the unscented transformation method can balance accuracy and efficiency and achieve efficient calculation of the statistical moments of the random system;

[0354] 2) The unscented transformation method proposed in the present invention can be applied to any type of random variable, as well as the uncertainty with only a small amount of sample data that is not enough to establish an accurate probability distribution;

[0355] 3) When applying the unscented transformation method to uncertainty optimization, there is no need to repeatedly sample the uncertainty in the uncertainty analysis link. Only the response values of a small number of fixed nodes are used to calculate the statistical moments, which simplifies the uncertainty optimization structure and improves the uncertainty optimization efficiency.

[0356] I. The specific application fields or related products of the present invention.

[0357] Example of Uncertainty Optimization of Bulk Carrier

[0358] 1 Optimization Problem Description

[0359] In this section, the AUT method is applied to the example of ship uncertainty optimization design. Considering the engineering optimization model of the bulk carrier proposed by Hannapel, according to the literature, the steel weight of the bulk carrier is

[0360] W S = 0.034L 1.7 B 0.7 D 0.4 C B 0.5 (83)

[0361] The calculation formula for the weight of the steel material in this model is obtained by statistical regression of actual ship data. Considering that the sample size of actual ship data is limited and the regression method is unknown, the error cannot be controlled, and the exponential value obtained by regression cannot accurately reflect the actual engineering situation. Therefore, the exponents in the calculation formula are considered uncertain quantities. Define the length exponent and the breadth exponent as uncertain quantities, that is:

[0362] W S = 0.034L X B 0.7 D 0.4 C B 0.5 (84)

[0363] where X follows a normal distribution with a mean of 1.7 and a standard deviation of 0.1.

[0364] During the navigation of the ship at sea, due to wind and wave disturbances and human factors, the actual draft and the actual speed are not fixed values, but uncertain quantities that fluctuate around the design values. By monitoring the change of the speed over a period of time and considering the uncertainty of the ship speed, its expression is as follows:

[0365] T = T k + Y (85)

[0366] V = V k + Z (86)

[0367] where T represents the actual draft, T k represents the design draft, Y represents the draft fluctuation, which follows a uniform distribution in the interval [-0.2, 0.2]; V represents the actual speed, V k represents the design speed, and Z represents the speed fluctuation, which follows a normal distribution with a mean of 0 and a standard deviation of 0.1V k of.

[0368] In this example, deterministic optimization (Deterministic optimization, DO) and uncertainty optimization are carried out respectively. In deterministic optimization, the influence of uncertainty is not considered, and the uncertain quantities are fixed at the mean values respectively, and the optimization is carried out with the minimum freight cost under the design state as the goal. In uncertainty optimization, considering the ship speed fluctuation, the length exponent and the breadth exponent in the steel material calculation formula as uncertain quantities, multi-objective optimization is carried out with the minimum expectation and standard deviation of the freight cost under the influence of uncertainty as the goal. The optimization model is shown in Table 13.

[0369] Table 13 Engineering Optimization Model of Bulk Carrier

[0370]

[0371] 2 Analysis of Optimization Results

[0372] In this example, the deterministic optimization adopts the multi-island genetic algorithm, with a subgroup size of 20, 20 islands, and 50 generations. The outer optimization loop for uncertainty adopts the NSGA-II optimization algorithm, with a population size of 100 and 100 generations. In the inner uncertainty analysis, the number of samples for the MC method is 105, and the unscented transformation method uses 9th-order nodes. The objective is optimized for uncertainty when the uncertain variables X, Y, and Z vary individually and simultaneously. The optimization schemes are denoted as UT-X, UT-Y, UT-Z, and UT-XYZ respectively. The optimization results are shown in Table 14, and the variation of the optimization scheme with uncertainty factors is shown in Figure 6 .

[0373] Table 14 Engineering Optimization Results of Bulk Carrier

[0374]

[0375]

[0376] As can be seen from Table 14, the objective function values of the DO and four UDO schemes are all smaller than those of the initial scheme under the design conditions, achieving certain optimization effects. When optimizing considering a single uncertainty factor, under the influence of the uncertainty factors X, Y, and Z, the change trends of the objective functions of the optimization schemes UT-X, UT-Y, and UT-Z are generally smoother, and the mean μ(TC) and standard deviation σ(TC) are both smaller than those of the DO scheme, indicating significant uncertainty optimization effects. When considering the simultaneous variation of the three uncertainty factors X, Y, and Z, since the influence of the three uncertainty factors needs to be considered simultaneously, the optimization conditions are more stringent, and the optimization effect on a single uncertain variable is not as significant as that of the single uncertain variable optimization scheme. Only the change curve of the UT-XYZ scheme becomes smoother under the influence of the uncertain variable X, effectively reducing the fluctuation of the objective function when the uncertain variable X is large. However, under the simultaneous action of the three uncertainty factors, since the uncertainty factor X has the greatest influence on TC, X contributes the most to the mean and standard deviation of TC, and the mean μ(TC) and standard deviation σ(TC) of the objective function are still smaller than those of DO. Generally speaking, although the objective function of the DO scheme is better under the design conditions, under the influence of uncertainty factors, the mean μ(TC) and standard deviation σ(TC) of the objective functions of the four UDO schemes are smaller, indicating that UDO reduces the influence degree of uncertainty factors, making the optimization scheme better in terms of the average level and smaller fluctuation under the influence of uncertainty factors. In this example, its practical significance is that when there are errors in the hull weight estimation and fluctuations in draft and speed during the actual voyage, the freight cost TC will not change violently within its fluctuation range and will always be at a relatively low level, ensuring the robustness of the optimization objective and thus better adapting to changes in the external environment.

[0377] From the analysis of the computational complexity of the UDO method, in this example, the 9th-order unscented transformation method used has 5 nodes in one dimension and 125 nodes in three dimensions, both of which are much smaller than the 105 sample points required by MC. Since in uncertainty optimization, it is necessary to perform uncertainty analysis on each design scheme during each iteration, a large number of samplings and statistical calculations will make the computational cost unbearable. The method proposed in the present invention effectively captures the statistical characteristics of uncertainty factors through a set of carefully selected nodes, while reducing the computational complexity of uncertainty analysis, and effectively obtains the uncertainty optimization solution during engineering optimization.

[0378] II. Evidence related to the technical effects obtained in the embodiments of the present invention.

[0379] Table 14 Engineering optimization results of bulk carriers

[0380]

[0381] Figure 6 is a graph of the objective function varying with uncertainty factors.

[0382] As can be seen from Table 14, at the design state, the objective function values of DO and the 4 UDO schemes are all smaller than those of the initial scheme, achieving the optimization effect. When optimizing considering a single uncertainty factor, under the influence of uncertainty factors X, Y, and Z, the change trends of the objective functions of the optimization schemes AUT-X, AUT-Y, and AUT-Z are generally flatter, and the mean μ(TC) and standard deviation σ(TC) are both smaller than those of the DO scheme, and the uncertainty optimization effect is significant. When considering the simultaneous variation of the three uncertainty factors X, Y, and Z, the optimization conditions are more stringent, and the optimization effect on a single uncertain quantity is not as significant as that of the single uncertain quantity optimization scheme. The change curve of the AUT-XYZ scheme becomes flatter only under the influence of the uncertain quantity X, effectively reducing the fluctuation of the objective function when the uncertain quantity X is large. Since the uncertainty factor X contributes the most to the mean and standard deviation of TC, the mean μ(TC) and standard deviation σ(TC) of the objective function under the combined action of the three uncertainty factors are still smaller than those of DO. Generally speaking, although the objective function of the DO scheme is better at the design state, under the influence of uncertainty factors, the mean μ(TC) and standard deviation σ(TC) of the objective functions of the 4 UDO schemes are smaller, indicating that UDO reduces the influence degree of uncertainty factors, making the optimization scheme better in terms of average level and smaller in terms of fluctuation change under the influence of uncertainty factors. In this example, its practical significance is that when there are errors in the hull weight estimation and fluctuations in draft and speed during actual navigation, the change in the freight cost TC of the UDO design scheme is smaller, reducing the influence of uncertainty factors on the design scheme and ensuring the robustness of the optimization objective, thus being able to better adapt to changes in the external environment.

[0383] It should be noted that the embodiments of the present invention can be implemented by hardware, software, or a combination of software and hardware. The hardware part can be implemented using dedicated logic; the software part can be stored in a memory and executed by an appropriate instruction execution system, such as a microprocessor or dedicated design hardware. Those of ordinary skill in the art can understand that the above devices and methods can be implemented using computer-executable instructions and / or included in processor control code, such as provided on a carrier medium such as a disk, CD, or DVD-ROM, a programmable memory such as a read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. The devices and modules of the present invention can be implemented by hardware circuits such as very large scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, etc., or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., can also be implemented by software executed by various types of processors, or can be implemented by a combination of the above hardware circuits and software such as firmware.

[0384] As described above, the above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present invention, any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention should be covered within the protection scope of the present invention.

Claims

1. An optimization method for ship uncertainty based on improved unscented transform, characterized in that, It includes the following steps: 1) Obtain signal data from multi-source sensors or measurement systems to characterize the uncertain factors affecting the ship performance function; 2) Use the unscented transformation method to calculate the statistical moments of the ship performance function under various uncertain factors; 3) By reconstructing the distribution constraint equation of the uncertain factors, extend the unscented transformation method from the normal distribution to an arbitrary distribution; 4) Apply the unscented transformation method of the arbitrary distribution to numerical examples and the uncertainty optimization of bulk carriers.

2. The ship uncertainty optimization method based on the improved unscented transform according to claim 1, characterized in that, The distribution constraint equation described in step 3) is constructed by combining the high-order moment information of the uncertain factors with the finite data points extracted from the signal data.

3. The ship uncertainty optimization method based on the improved unscented transform according to claim 1, characterized in that The unscented transformation method: (1) Basic principle of the unscented transformation method; (2) Traditional unscented transformation method; (3) Arbitrary unscented transformation method.

4. The ship uncertainty optimization method based on the improved unscented transform according to claim 3, wherein The basic principle of the unscented transformation method: Consider the response function f(x) affected by the random uncertainty x, and its expected value is defined as where p(x) is the probability density function of the random uncertainty x; the unscented transformation method expresses Equation (1) as a numerical integration formula through a series of quadrature nodes and weight coefficients where x i and w i are the i-th quadrature node and its weight coefficient respectively; performing Taylor expansions on equations (1) and (2) at x = 0, we have By comparing Equation (89) and Equation (90), it can be seen that for the numerical integration formula (88) of the unscented transformation method to meet the accuracy requirements, its variables must satisfy the following system of equations The system of equations in Equation (91) is called the statistical moment constraint equation; when r = 0, the equation indicates that the sum of the weight coefficients of all quadrature nodes is 1; when r ≠ 0, the equation represents the numerical calculation formula for the r-th origin moment of the random variable x; similarly, the quadrature formula for the high-order moment E(f r (x)) of the response function can be derived by unscented transformation.

5. The ship uncertainty optimization method based on the improved unscented transform according to claim 3, characterized in that The traditional unscented transformation method: 1) Coordinate origin: The two-dimensional standard normal random variables are all located at the mean value, that is, (0, 0), and its weight coefficient is w0; 2) Main axis: The orthogonal coordinate axes centered at the origin are the main axes. There is at most one non-zero element in the node coordinates on the main axis. The node coordinates are (s1, 0), (-s1, 0), (0, s1), (0, -s1), and the weight coefficients are all w1; 3) Axis of symmetry: The angular bisector of the coordinate axes is the axis of symmetry. The absolute values of the non-zero elements of the node coordinates on the axis of symmetry are equal. The node coordinates are (s2, s2), (s2, -s2), (-s2, s2), (-s2, s2), and the weight coefficients are all w2; Obviously, the nodes on the main axis and the axis of symmetry are all distributed on circles with different radii centered at the origin. The symmetry assumption ensures that the odd-order statistical moment constraint equations of the normal distribution are automatically satisfied, greatly simplifying the solution process of the statistical moment constraint equations; for two-dimensional standard normal random variables, Equation (91) of the statistical moment constraint equation can be expanded as the following equation After solving Equation (92) to obtain the nodes and weight coefficients, the quadrature formula of Equation (88) can be constructed to calculate the statistical moments of the system response function.

6. The ship uncertainty optimization method based on the improved unscented transform according to claim 3, characterized in that The arbitrary unscented transformation method: 1) Quadrature node and weight coefficient assumption For a random variable X that follows an arbitrary distribution, to make the selection of quadrature nodes more general, regardless of whether it has symmetry or not, all node coordinates x i (i = 1, 2, …, N) and all weight coefficients w i (i = 1, 2, …, N) are set as unknowns, and each node should satisfy the following constraint equations: In Equation (16), when k = 0, M0 = 1, and the equation represents that the sum of the weight coefficients of each node is 1; when k = 1, 2, …, r, M k is the k-th order origin moment of the random variable X; Taking a two-dimensional random system as an example, let x 1i , x 2j be the quadrature nodes of the random variables X1 and X2 respectively, and the corresponding weight coefficients be w 1i , w 2j . For any one-dimensional case, its nodes respectively satisfy: Orthogonally combine the two-variable nodes to form the node network shown in Figure 4, and each grid point is used as a two-dimensional node; In the figure, node X ij and its coordinate and weight coefficient W ij can be calculated by the formula Taking the variable X1 as an example, conduct a statistical moment constraint test on the two-dimensional nodes. It can be obtained from the equation that the two-dimensional nodes still satisfy the corresponding distribution in the direction of the variable X1; 2) Statistical moment calculation and solution of the constraint equation The extreme value distribution statistical moment is represented by the statistical moment of t~EV(0, 1), and its calculation formula is shown in Equation (17); In formula (17), γ = 0.57721566490153286060651209… is the Euler–Mascheroni constant; ζ(s) is the Riemann Zeta function, and its function values can be obtained by looking up tables. When s = 1, the Zeta function diverges. To ensure the uniformity of the general formula (17), ζ(1) is defined as γ here; the summation is carried out over all non-negative integers m1, m2, …, m that satisfy formula (18), that is n performed Substitute the calculation results into Equation (5) to solve the statistical moment constraint equation, and the one-dimensional node coordinates and weight coefficients corresponding to any random distribution can be obtained. For the random variable X, if its statistical moments of all orders are known, then there exists a set of constants C0, C1, …, C N-1 , such that its origin moments of orders N-1, N, …, 2N-1 satisfy C0M N-1 +C1M N +…+C N-1 M 2N-2 +M 2N-1 =0 (100) Express its various order statistical moments using nodes and weights, and there is To make the value of formula (15) equal to 0, the quadrature nodes x1, x2, …, x of the unscented transformation of the random variable X N must simultaneously satisfy C0 + C1x + … + C N-1 x N-1 + x N = 0, that is, the zeros of the polynomial (16) are the node coordinates; Therefore, by constructing polynomial (16), the coordinates of the quadrature nodes can be obtained; the polynomial coefficients can be calculated through the linear equation system (17); After obtaining the node coordinates through equations (16) and (17), substitute the node coordinates into the statistical moment constraint equation (5), and the weight coefficients can be obtained by solving the linear equation system; 3) Standardization transformation of random variables.

7. The ship uncertainty optimization method based on the improved unscented transform according to claim 6, characterized in that, The standardization transformation of the said random variable: Taking the normal distribution as an example, let \(x\) follow a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), i.e., \(x\sim N(\mu,\sigma)\) 2 ), and \(t\) follow a normal distribution with mean \(0\) and standard deviation \(1\), i.e., \(t\sim N(0,1)\); according to the definition of statistical moments, the \(r\)-th raw moments of the two are respectively: Perform a variable substitution on t There is Let the standard normal variable t satisfy the statistical moment constraint equation, and its quadrature nodes and weight coefficients be w i ,t i , there is Substitute the equation into the equation, then the statistical moments of the normal variable under any parameter can be expressed as Therefore, the quadrature nodes and weight coefficients of the normal variable x are w i , σt i + μ; Similarly, for other types of random distributions, the proof process is as follows; Uniform distribution: Exponential distribution: Log-normal distribution: Extreme value distribution:

8. An improved unscented transform-based ship uncertainty optimization system for implementing the improved unscented transform-based ship uncertainty optimization method according to any one of claims 1-6, characterized in that, The ship uncertainty optimization system based on the improved unscented transform includes: A calculation module, which is used to calculate the statistical moments of the ship performance function under the influence of uncertainty factors by using the unscented transform method; A generalization module, which is used to generalize this method to the uncertainty of any distribution by reconstructing the statistical moment constraint equation of uncertainty; An application module, which is used to apply any unscented transform method to numerical examples and the uncertainty optimization of bulk carriers.

9. A computer-readable storage medium stores a computer program. When the computer program is executed by a processor, the processor is made to execute the steps of the ship uncertainty optimization method based on the improved unscented transform according to any one of claims 1-7.

10. An information data processing terminal, characterized in that, The information data processing terminal is used to implement the ship uncertainty optimization system according to claim 8.

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