A parameter estimation method for second-order nonlinear response model considering ship test data

Abstract

A second-order nonlinear response model parameter estimation method considering real ship test data, comprising: establishing a second-order nonlinear response type mathematical model of a ship according to a turning angle velocity of the ship, a yaw angle of the ship and a rudder angle of the ship to obtain parameters to be estimated; obtaining simplified parameters to be estimated; determining an identification model input and an identification model state variable based on Kalman filtering theory; obtaining a state space equation of the second-order nonlinear response type mathematical model of the ship; obtaining a parameter identification model of the ship; and performing parameter estimation on the parameters to be estimated based on a root mean square unscented Kalman filtering method. The present application obtains a state space equation based on Kalman filtering theory, and performs parameter estimation on the parameters to be estimated based on a root mean square unscented Kalman filtering method through the established parameter identification model of the ship. The present application can effectively avoid filtering divergence, and the identification result has good convergence, generalization and prediction result.

Description

Technical Field

[0001] This invention relates to the field of ship monitoring technology, and in particular to a method for estimating parameters of a second-order nonlinear response model that takes into account actual ship test data. Background Technology

[0002] The development of ship course-keeping control and autopilot control systems both require an accurate responsive mathematical model. When autonomously navigating and operating in complex marine environments, unmanned surface vessels (USVs) should be able to accurately maintain the desired trajectory while also possessing good maneuverability and handling performance. Research in these two areas necessitates the establishment of a highly accurate mathematical model of ship motion; in other words, an accurate mathematical model of ship motion is crucial for ship motion control.

[0003] In the research of ship response mathematical models, the accuracy of existing first-order response mathematical models is lower than that of second-order response mathematical models. However, the maneuverability indices of second-order response mathematical models are difficult to calculate, requiring a large number of complex ship maneuvering tests to determine these indices. Furthermore, the filtering convergence is poor, and the maneuverability indices calculated based on empirical or regression formulas will differ somewhat from the parameters obtained from actual ship test data. Summary of the Invention

[0004] This invention provides a method for estimating parameters of a second-order nonlinear response model that takes into account actual ship test data, in order to overcome the above-mentioned technical problems.

[0005] To achieve the above objectives, the technical solution of the present invention is as follows:

[0006] A method for estimating parameters of a second-order nonlinear response model considering real-ship test data includes the following steps:

[0007] S1: Establish a second-order nonlinear response mathematical model of the ship based on the ship's turning angular velocity, ship's bow roll angle, and ship's rudder angle to obtain the parameters to be estimated, including the ship's first following index, second following index, third following index, nonlinear coefficient, yaw index, and rudder angle.

[0008] S2: Obtain simplified parameters to be estimated based on the parameters to be estimated;

[0009] S3: Based on the simplified parameters to be estimated, and using Kalman filtering theory, determine the input and state variables of the identification model; to obtain the state-space equation of the second-order nonlinear response mathematical model of the ship.

[0010] S4: Based on the state-space equations of the ship's second-order nonlinear response mathematical model, and using the Euler difference discretization method, obtain the ship's parameter identification model.

[0011] S5: Based on the ship's bow turning angular velocity, bow roll angle, rudder angle, and the ship's parameter identification model, the parameters to be estimated are estimated using the root mean square unscented Kalman filter method.

[0012] Beneficial Effects: This invention provides a parameter estimation method for a second-order nonlinear response model considering actual ship test data. Based on the ship's turning angular velocity, bow angle, and rudder angle, a second-order nonlinear response mathematical model is established. Based on Kalman filtering theory, the state-space equation is obtained. Then, using the root mean square unscented Kalman filter method, the parameters to be estimated are estimated through the established ship parameter identification model. This method effectively avoids filter divergence, and the identification results exhibit good convergence, generalization, and prediction results. Attached Figure Description

[0013] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0014] Figure 1 This is a flowchart of the parameter estimation method of the present invention;

[0015] Figure 2 This is a schematic diagram of two nonlinear variations of Kalman filtering in an embodiment of the present invention;

[0016] Figure 3 This is a schematic diagram illustrating the process of establishing a second-order nonlinear response mathematical model in an embodiment of the present invention;

[0017] Figure 4 This is a schematic diagram of the parameter estimation method in an embodiment of the present invention;

[0018] Figure 5 This is a simplified schematic diagram of the convergence curve of the parameters to be estimated for the self-propelled model in an embodiment of the present invention;

[0019] Figure 6 This is a schematic diagram of the bow roll angle prediction results of a 10° / 10° Z-shaped test of a self-propelled aircraft model in an embodiment of the present invention;

[0020] Figure 7 This is a schematic diagram of the generalization results of the bow roll angle of the self-propelled model aircraft in the 20° / 20° Z-shaped test in an embodiment of the present invention;

[0021] Figure 8This is a schematic diagram of the generalization results of the bow roll angle of the self-propelled model aircraft in the 10° / 5° Z-shaped test in an embodiment of the present invention;

[0022] Figure 9 This is a simplified convergence curve diagram of the parameters to be estimated for the "Yu Kun" wheel in an embodiment of the present invention.

[0023] Figure 10 This is a schematic diagram of the bow roll angle prediction results of the 20° / 20° Z-shaped test data of the "Yukun" vessel in an embodiment of the present invention;

[0024] Figure 11 This is a schematic diagram of the bow roll angle prediction results of the 10° / 10° Z-shaped test data of the "Yukun" vessel in an embodiment of the present invention;

[0025] Figure 12 This is a schematic diagram of the bow roll angle prediction results of the "Yukun" vessel in the embodiment of the present invention, based on the -10° / -10° Z-shaped test data. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0027] This embodiment provides a method for estimating parameters of a second-order nonlinear response model that considers actual ship test data, such as... Figure 1 As shown, it includes the following steps:

[0028] S1: Establish a second-order nonlinear response mathematical model of the ship to obtain the parameters to be estimated, including the ship's first following index, second following index, third following index, nonlinear coefficient, yaw index, and rudder angle.

[0029]

[0030] In the formula, r is the ship's bow turning angular velocity; ψ is the ship's bow roll angle; α is the ship's nonlinear coefficient; δ is the ship's rudder angle; δ r T1 is the rudder angle of the ship; HZ is the turning index of the ship; T1, T2, and T3 are the first, second, and third following indices of the ship, respectively. It is a first-order derivative operation; It is a second-order derivative operation;

[0031] The second-order nonlinear response mathematical model in this embodiment is applicable to both self-propelled model aircraft and the "Yu Kun" vessel. The second-order linear maneuverability index of the self-propelled model experimental data is known. Therefore, based on the known maneuverability index, the self-propelled model simulation experimental data is obtained using the fourth-order Runge-Kutta integral. In the fourth-order Runge-Kutta integral, the 10° / 10° Z-shaped test data of the self-propelled model is used as the identification sample, with a simulation duration of 250s and a step size of 0.1s (the shorter the step size, the more accurate the numerical simulation results), resulting in a total of 2500 data points. These 2500 data points are sampled every 10 data points to obtain 250 data sample sets. The 20° / 20° (standard Z-shaped test) and 10° / 5° (non-standard Z-shaped test) Z-shaped test data are used as generalization verification data, with a simulation duration of 200s, ultimately yielding 200 data sample sets. Furthermore, there is no noise in the simulation experimental data. The "Yu Kun" vessel, a dedicated ocean-going vessel integrating scientific research, teaching, and practical training, possesses excellent navigation performance and safety. During the ship trials, the sea area was wide, visibility was good, and sea state was approximately level 2, meeting the standards for maneuverability testing. Data collection was affected not only by the marine environment but also by the measuring instruments, resulting in outliers. Therefore, a local weighted regression method was first used to filter the ship trial data, followed by interpolation to meet the requirements of identification and modeling. The sampling interval for the ship trial data was 1 second, with 20° / 20° Z-shaped test data used as identification samples and ±10° / ±10° Z-shaped test data used as generalization verification data. This approach enabled parameter estimation using a smaller dataset through a second-order nonlinear response mathematical model, significantly shortening the ship trial cycle.

[0032] In essence, the dynamic characteristics of self-propelled models and the "Yu Kun" ship are nonlinear. Therefore, a second-order nonlinear response mathematical model is used for research, which can accurately describe the dynamic response process.

[0033] S2: Obtain simplified parameters to be estimated based on the parameters to be estimated;

[0034] Specifically, to facilitate parameter identification research, the parameters in the second-order nonlinear response model of the ship are summarized as follows:

[0035]

[0036] In the formula: θ represents the simplified parameter vector to be estimated; θ1, θ2, θ3, θ4, θ5, and θ6 are the simplified parameters to be estimated;

[0037] Therefore, HZ, T1, T2, T3, α, and δ r The calculation is as follows:

[0038]

[0039] Where G represents the following index;

[0040] Integrating equations (1) and (2), we get:

[0041]

[0042] Equation (2) shows that there are 6 parameters that need to be identified in the second-order nonlinear response models of the self-propelled model and the "Yu Kun" vessel, meaning the parameters are unknown. Therefore, based on the theory of Kalman filtering, the 6 parameters are estimated together with the state variables of the two ships. The system state variables are further extended to obtain Equation (5). The system input variables are shown in Equation (6).

[0043] S3: Based on the simplified parameters to be estimated, and using Kalman filtering theory, obtain the system input and system state variables; and obtain the state-space equation of the second-order nonlinear response mathematical model of the ship.

[0044] Preferably, the state-space equations of the ship's second-order nonlinear response mathematical model are obtained as follows:

[0045] Obtain the input and state variables of the identification model:

[0046]

[0047]

[0048] In the formula, X represents the state variable vector of the identification model; X1, X2, X3, X4, X5, X6, X7, X8, X9 are all state variables of the identification model; u represents the input of the identification model; u1 and u2 are both inputs of the identification model.

[0049] Based on equations (5) and (6), the state-space equations of the second-order nonlinear response mathematical model of the ship are obtained as follows:

[0050]

[0051] In the formula: express 0 vector; 6×1 Represents a zero vector;

[0052] S4: Based on the state-space equation of the second-order nonlinear response mathematical model of the ship, Equation (7) is discretized using the Euler difference discretization method to obtain a parameter identification model suitable for both the self-propelled model and the actual ship "Yu Kun".

[0053] Preferably, the parameter identification model is obtained as follows:

[0054]

[0055] In the formula, h is the sampling interval; k is the iteration number, k = 1, 2, ..., K, where K is the maximum set number of iterations; X1(k+1) represents X1 in the (k+1)th iteration; X2(k) represents X2 in the kth iteration; θ(k) represents θ in the kth iteration; u1(k) represents u1 in the kth iteration.

[0056] S5: Based on the ship's turning angular velocity, ship's roll angle, ship's rudder angle and the parameter identification model, the parameters to be estimated are estimated using the root mean square unscented Kalman filter method.

[0057] Preferably, the parameter estimation method for the parameters to be estimated is as follows:

[0058] In this embodiment, the square root of the covariance is used instead of the covariance for recursive calculation, which can effectively ensure the non-negativity and numerical stability of the covariance matrix to avoid the problem of filter divergence. Therefore, the parameter identification model of the self-propelled model and the actual ship "Yu Kun" established by Equation (8) is used as the framework, and according to the conversion formula between the identification parameters and the maneuverability index given by Equation (9), the maneuverability index identification of the self-propelled model and the actual ship "Yu Kun" is realized by using root mean square unscented Kalman filtering (SRUKF). In addition, the parameter settings in SRUKF are applicable to the estimation of the maneuverability index of the self-propelled model and the actual ship "Yu Kun", that is, the parameters to be estimated.

[0059] S51: Based on the ship's turning angular velocity, bow angle, and rudder angle (processed self-propulsion simulation experimental data and "Yu Kun" ship test data), the state variables and the square root of the state error covariance matrix of the identification model are initialized according to Equation (9), and the initial state variables X0 and the initial square root of the state error covariance matrix S0 of the identification model are obtained.

[0060]

[0061] In the formula, It is the expectation of the initial state variable X0 of the identification model (i.e., the expectation of x at the 0th iteration).

[0062] S52: Obtain the predicted values ​​of the state variables of the ship identification model: that is, the predicted values ​​of the state variables of the self-propelled model and the actual ship "Yukun".

[0063] Obtain the i-th Sigma point in the k-th iteration.

[0064] First, a sampling strategy suitable for ship motion data is selected to calculate the Sigma point. The calculation method is as shown in equation (10). Based on the data type and number of data points of the ship, the weights are reasonably selected, as shown in equation (11).

[0065]

[0066] In the formula: represents the predicted value of the state variables of the identification model in the k-th iteration; nx is the dimension of the system state variables of the second-order nonlinear response mathematical model of the ship; λ is a scaling factor suitable for ship data selection; S k This represents the state covariance matrix for the k-th iteration;

[0067]

[0068] In the formula, p represents the main scaling factor describing the extent to which the Sigma point set expands near the prior mean, and its value ranges from

[10] . -4 [1] To make it more suitable for ship maneuverability index identification, in this embodiment, it is set to 10. -4 γ is the second scaling factor applicable to ship data selection. To reasonably select the motion data of the two ships, γ is set to 0. The weights represent the mean of the i-th Sigma point; β represents the weight of the covariance of the i-th Sigma point; β is the third scale factor applicable to the selection of data from two ships, which is the combination of prior knowledge and state distribution. In this embodiment, β is set to 2, which can achieve the optimal estimation of the ship maneuverability index.

[0069] The time prediction updates for the state variables of the self-propelled model and the actual ship "Yu Kun" are as follows:

[0070]

[0071] In the formula, Let f(·) represent the predicted value of X obtained in the (k+1)th iteration during the k-th iteration at the i-th Sigma point; f(·) is the second-order nonlinear response model equation of the ship; u k This represents the input vector during the k-th iteration. This represents the predicted value of X for the (k+1)th iteration obtained during the k-th iteration; Represents numbers from 1 to 2n x The sum of the (k+1)th X prediction values ​​obtained during the kth iteration of the Sigma points; qr(·) represents the upper triangular matrix of the state covariance matrix obtained in the (k+1)th iteration during the k-th iteration; qr(·) is the orthogonal triangular decomposition of the ship's state variables and process noise; Q k S is the covariance matrix of the system process noise;k+1|k This represents the state covariance matrix obtained during the k+1 iteration; choleupdate(·) is the Cholesky decomposition of the ship's covariance, state variables, and weights. This represents the predicted value of X for the (k+1)th iteration obtained during the k-th iteration of the Sigma point when i = 0; This represents the weight of the covariance at the Sigma point when i = 0.

[0072] The measurements of the state variables of the self-propelled model and the actual ship "Yu Kun" are updated as follows:

[0073]

[0074] In the formula, Z represents the ship's motion state. The observed values, is the predicted value of Z for the (k+1)th iteration obtained by the i-th Sigma point in the k-th iteration; h(·) is the observation equation of the ship; S represents the upper triangular matrix of the observed covariance matrix; Z R represents the observation covariance matrix; k It is the covariance matrix of the system's observation noise; P XZ For posterior covariance; This represents the (k+1)th predicted value obtained during the k-th iteration of the Sigma point when i = 0; This represents the predicted value of Z obtained in the (k+1)th iteration during the k-th iteration; Represents numbers from 1 to 2n x The sum of the (k+1)th Z-prediction values ​​obtained during the k-th iteration of the Sigma points;

[0075] The filtered results for the state variables of the self-propelled model and the actual "Yu Kun" vessel are updated as follows:

[0076]

[0077] In the formula, K k Z is the Kalman gain matrix of the ship's motion state and the iterative update of the identification parameters at the k-th iteration; k+1 S represents the observation at the (k+1)th iteration; U represents the process matrix; S k+1|k+1 This represents the state covariance matrix at the (k+1)th iteration;

[0078] Get the result when the number of iterations reaches K.

[0079] This allows us to obtain X = [X1X2X3X4X5X6X7X8X9] after K iterations. T This allows us to obtain θ1, θ2, θ3, θ4, θ5, and θ6 after K iterations.

[0080] Therefore, the method for obtaining the parameters to be estimated after K iterations is as follows:

[0081]

[0082] Specifically, based on the self-propelled model applicable to large rudder angles or strongly nonlinear motion processes and the nonlinear response model of the "Yu Kun" vessel, SRUKF iteratively updates its parameters to obtain accurate maneuverability indices for the response model. The self-propelled simulation experimental data is obtained through fourth-order Runge-Kutta integration, free from measurement noise or environmental interference. The "Yu Kun" vessel's actual ship test data is collected using measuring instruments such as fiber optic compasses and Doppler logs. Compared to simulation data, the actual ship test data is affected by environmental interference and measuring instrument noise. Based on the simulation experimental data and the second-order nonlinear response mathematical model, parameter identification studies are conducted using the proposed algorithm to verify its effectiveness and convergence. The simulation duration for the identification samples (10° / 10° Z-shaped test) is 250s, and the simulation duration for the generalization verification samples (20° / 20°, 10° / 5° Z-shaped tests) is 200s, with a sampling interval of 1s for both.

[0083] This embodiment is applicable to both noise-free and noisy ship motion data, providing a model identification scheme for both types of data. For noise-free ship motion data, i.e., self-propelled model simulation experiment data, this invention substitutes the 10° / 10° Z-shaped test data of the self-propelled model into SRUKF for iterative updates to determine the maneuverability index of the responsive model, enabling the prediction of the self-propelled model's heading angle. Furthermore, it utilizes motion data different from the identification samples, including 20° / 20° Z-shaped test (standard Z-shaped test) and 10° / 5° Z-shaped test (non-standard Z-shaped test) data, to verify the algorithm and the generalization of the identification parameters. For noisy ship motion data, specifically the full-scale test data of the "Yu Kun" vessel, this embodiment substitutes the 20° / 20° Z-shaped test data of the "Yu Kun" vessel into SRUKF to determine the maneuverability index of the response model and to achieve the prediction of the heading angle. Since non-standard Z-shaped tests were not conducted during the full-scale test, ±10° / 10° Z-shaped test data are used to verify the algorithm and the generalization of the identification parameters. To better suit the research of this invention,

[0084] To compare and verify the reliability of the proposed algorithm, the results obtained by Extended Kalman Filter (EKF) are used as a comparison. In the initialization stage of the EFK and SRUKF algorithms, based on the characteristics of the simulation experimental data and the actual ship test data, this invention determines the initial values ​​of the state variables X0, the state covariance matrix, and the observation matrix of the two types of data in the algorithm iteration, as shown in equations (15) and (16), respectively.

[0085]

[0086]

[0087] In the formula: H is the observation matrix, I 9×9 It is a 9x9 identity matrix, q is the state noise matrix, Q is the covariance matrix of the system process noise, and Q is the same as Q. k Similarly, J is the observation noise matrix, R is the covariance matrix of the system observation noise, and R... k Similarly; X1(0), X2(0), X3(0) are the initial values ​​of X1, X2, X3 respectively, and S is the state covariance matrix;

[0088] Get the result when the number of iterations reaches K.

[0089] This allows us to obtain X = [X1X2X3X4X5X6X7X8X9] after K iterations. T This allows us to obtain θ1, θ2, θ3, θ4, θ5, and θ6 after K iterations.

[0090] Therefore, the method for obtaining the parameters to be estimated after K iterations is as follows:

[0091]

[0092] Figure 2 This diagram illustrates two Kalman filters applied to state estimation of linear systems. To make filtering algorithms applicable to nonlinear systems, the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) were proposed. EKF uses a Taylor expansion to linearize the nonlinear function. The filtering results show that EKF exhibits some bias during the filtering process. UKF approximates the probability density distribution of the nonlinear function, using a series of deterministic samples to approximate the posterior probability density of the state. The diagram also shows that UKF does not use Taylor expansion to linearize the nonlinear system; instead, it employs an unscented transformation to handle the nonlinear propagation of the mean and covariance.

[0093] Figure 3A flowchart is presented to establish a second-order nonlinear response mathematical model using the Square Root Unscented Kalman Filter (SRUKF) as the algorithm framework and self-propelled simulation experimental data and actual ship test data of the "Yu Kun" vessel as data samples. The maneuverability indices of the second-order nonlinear response mathematical models for two different ship types were determined. Comparing the obtained maneuverability indices with theoretical values ​​verifies the feasibility of the algorithm; comparing the predicted bow angle with the motion data of the two types of ships verifies the generalization ability of the algorithm.

[0094] Figure 4 This diagram illustrates a Bayesian filter approximation method for processing nonlinear systems, capable of approximating any nonlinear system with at least second-order accuracy. However, the UKF algorithm suffers from filter divergence due to the inability to achieve Cholesky decomposition of the state error covariance matrix; specifically, the negative definiteness of the state error covariance leads to imaginary numbers in Sigma sampling. Therefore, to address this challenge, the square root of the covariance is used instead of the covariance itself for recursive calculations. This effectively guarantees the nonnegativity and numerical stability of the covariance matrix, preventing filter divergence. Consequently, a more stable parameter identification scheme based on the Square Root Unscented Kalman Filter (SRUKF) has been proposed and applied to the estimation of maneuverability indices in both self-propelled models and actual ships.

[0095] In this embodiment, based on 10° / 10° Z-shaped test data, the maneuverability index of the second-order nonlinear response mathematical model of the self-propelled aircraft was determined. The convergence curves of the six identification parameters of EKF and SRUKF are shown below. Figure 5 As shown in Table 1, the identified maneuverability indices are also shown. The predicted bow roll angles for the 10° / 10° Z-shaped test are as follows: Figure 6 As shown, the generalization verification results are as follows: Figure 7 and Figure 8 As shown. Based on the 20° / 20° Z-shaped test data, the maneuverability index of the second-order nonlinear response mathematical model of the "Yu Kun" vessel was determined. The convergence curves of the six identification parameters of EKF and SRUKF are shown in the figure. Figure 9 As shown in Table 2, the identified maneuverability indices are also shown. The predicted bow roll angles for the 20° / 20° Z-shaped test are as follows: Figure 10 As shown, the generalization verification results are as follows: Figure 11 and Figure 12 As shown.

[0096] Using SRUKF as the algorithmic framework, the second-order nonlinear maneuverability indices of the self-propelled model and the actual "Yu Kun" vessel were obtained. From the identification results of the self-propelled model's experimental data, [the following data was obtained]. Figures 4-7It can be seen that SRUKF exhibits good performance in both parameter convergence and bow roll angle prediction results. Table 1 shows that the maneuverability index values ​​obtained by SRUKF are close to the theoretical values, with small errors. The feasibility, convergence, and generalization of SRUKF have been verified.

[0097] The self-propelled model experimental data is free from environmental interference and measurement noise. After verifying the effectiveness, convergence, and generalization of SRUKF using the self-propelled model experimental data, this paper uses SRUKF to conduct a study on the identification of the second-order nonlinear maneuverability index of the "Yu Kun" vessel. Actual ship test data is affected by environmental factors, human factors, and instrument measurement factors, which can lead to certain differences between actual ship test data and simulation test data. Figures 9-12 As shown in Table 2, despite the presence of some interference in the experimental data, SRUKF can still obtain a relatively accurate prediction result and identify parameters, while the parameters can also converge to a stable value. The feasibility, convergence, and generalization of SRUKF have been further verified.

[0098] Table 1. Identification Results of Self-propelled Model Airplane Maneuverability Index

[0099]

[0100] Table 2. Identification Results of Maneuverability Indices for the "Yu Kun" Vessel

[0101]

[0102] Based on the above experimental results, the present invention has the following beneficial effects:

[0103] Based on square root unscented Kalman filtering and a second-order nonlinear response mathematical model, the ship maneuvering index estimation method of this invention is applicable to both noisy and noiseless ship data and exhibits strong stability. The proposed square root unscented Kalman filtering algorithm mitigates the divergence problem inherent in the unscented Kalman filtering process to some extent and is applied for the first time to the estimation of the maneuvering index using a ship response mathematical model. It features high prediction accuracy and fast parameter convergence.

[0104] The second-order nonlinear response model can more accurately describe ship motion. Furthermore, the square root unscented Kalman filter, which exhibits strong stability, is used to estimate the maneuverability index for two ship types. The results obtained by the proposed algorithm are compared with those obtained by the extended Kalman filter. The final comparison results verify the effectiveness, convergence, and generalization of the proposed algorithm. The second-order nonlinear response model has wide applications in ship motion control, especially in course-keeping control research. It can effectively and accurately estimate the mathematical model parameters of ship motion under noise interference conditions, laying a model foundation for research on autopilot devices, navigation simulator control modules, and unmanned surface vessel course-keeping control.

[0105] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for estimating parameters of a second-order nonlinear response model considering actual ship test data, characterized in that, Includes the following steps: S1: Establish a second-order nonlinear response mathematical model of the ship based on the ship's turning angular velocity, ship's bow roll angle, and ship's rudder angle to obtain the parameters to be estimated, including the ship's first following index, second following index, third following index, nonlinear coefficient, yaw index, and rudder angle. S2: Obtain simplified parameters to be estimated based on the parameters to be estimated; S3: Based on the simplified parameters to be estimated, and using Kalman filtering theory, determine the input and state variables of the identification model; to obtain the state-space equation of the second-order nonlinear response mathematical model of the ship. S4: Based on the state-space equations of the ship's second-order nonlinear response mathematical model, and using the Euler difference discretization method, obtain the ship's parameter identification model. S5: Based on the ship's turning angular velocity, bow angle, rudder angle, and the ship's parameter identification model, the parameters to be estimated are estimated using the root mean square unscented Kalman filter method. The parameter estimation method for the parameters to be estimated is as follows: The initial state variables of the identification model are obtained by initializing the state variables and the square root of the state error covariance matrix of the identification model based on the ship's turning angular velocity, bow angle, and rudder angle. and the square root of the initial state error covariance matrix ; In the formula, The initial state variables of the identification model Expectations; Obtain the i-th Sigma point in the k-th iteration. : In the formula: This represents the predicted value of the state variables of the identification model in the k-th iteration; It is the dimension of the system state variables in the second-order nonlinear response mathematical model of the ship; It is a scaling factor applicable to the selection of ship data; This represents the state covariance matrix for the k-th iteration; In the formula, This represents the main scale factor describing the extent to which the Sigma point set expands around its prior mean; This is a second scaling factor suitable for selecting ship data; The weights represent the mean of the i-th Sigma point; The weights represent the covariance of the i-th Sigma point; A third scaling factor suitable for selecting data from both ships; In the formula, This represents the predicted value of X for the (k+1)th iteration obtained during the k-th iteration of the i-th Sigma point; It is a second-order nonlinear response model equation for ships; This represents the input vector during the k-th iteration. This represents the predicted value of X for the (k+1)th iteration obtained during the k-th iteration; Indicates from 1 to 2 The sum of the (k+1)th X prediction values ​​obtained during the kth iteration of the Sigma points; It represents the upper triangular matrix of the state covariance matrix obtained in the (k+1)th iteration during the k-th iteration; The orthogonal trigonometric decomposition of the ship's state variables and process noise is used. Let be the covariance matrix of the system process noise; This represents the state covariance matrix obtained during the k+1th iteration. Cholesky decomposition of ship covariance, state variables, and weights This represents the predicted value of X for the (k+1)th iteration obtained during the k-th iteration of the Sigma point when i=0. This represents the weights of the covariance at the Sigma point when i=0; In the formula, It is the ship's motion state ( ) observations, It is the (k+1)th iteration obtained by the i-th Sigma point during the k-th iteration. The forecast value; It is the ship's observation equation; This represents the upper triangular matrix of the observation covariance matrix; Represents the observation covariance matrix; It is the covariance matrix of the system's observation noise; For posterior covariance; This represents the (k+1)th predicted value obtained during the k-th iteration of the Sigma point when i=0. This represents the (k+1)th iteration obtained during the k-th iteration. The forecast value; Indicates from 1 to 2 The (k+1)th iteration obtained during the kth iteration of the Sigma point The sum of forecast values; In the formula, It is the Kalman gain matrix that updates the ship's motion state and identification parameters during the k-th iteration; This represents the observation value at the (k+1)th time. Represents the process matrix; This represents the state covariance matrix at the (k+1)th iteration; Get the result when the number of iterations reaches K. , It can obtain the result after K iterations Thus, the result after K iterations is obtained. ,in, This represents the state variable vector of the identification model. All are state variables of the identification model; All of these are simplified parameters to be estimated; Therefore, the method for obtaining the parameters to be estimated after K iterations is as follows: In the formula: It is the ship's turning index; , , These are the first, second, and third following indices of a ship's following behavior; It is the rudder angle of the ship; It is the nonlinear coefficient of the ship.

2. The method for estimating parameters of a second-order nonlinear response model considering actual ship test data according to claim 1, characterized in that, The second-order nonlinear response mathematical model of the ship is established as follows: (1) In the formula, It is the ship's bow turning angular velocity; It is the bow angle of the ship; It is the nonlinear coefficient of the ship; It is the rudder angle of a ship; It is the rudder angle of the ship; It is the ship's turning index; , , These are the first, second, and third following indices of a ship's following behavior; It is a first-order derivative operation; It is a second-order derivative operation.

3. The method for estimating parameters of a second-order nonlinear response model considering actual ship test data according to claim 1, characterized in that, The simplified parameters to be estimated are obtained as follows: The parameters of the second-order nonlinear response mathematical model of the ship are rearranged to obtain: (2) In the formula: This represents the simplified vector of parameters to be estimated. All of these are simplified parameters to be estimated; then, , , , as well as The calculation is as follows: (3) (4)。 4. The method for estimating parameters of a second-order nonlinear response model considering actual ship test data according to claim 1, characterized in that, The state-space equations of the second-order nonlinear response mathematical model of the ship are obtained as follows: Determine the input and state variables of the identification model: (5) (6) In the formula, Represents the state variable vector of the identification model; All are state variables of the identification model; This represents the input vector of the identification model; All of these are inputs to the identification model; The state-space equations of the second-order nonlinear response mathematical model of the ship are as follows: (7) In the formula: express ; express vector.

5. The method for estimating parameters of a second-order nonlinear response model considering actual ship test data according to claim 1, characterized in that, The parameter identification model of the ship is obtained as follows: (8) In the formula, Sampling interval; For the number of iterations, , The maximum number of iterations is set. Indicates the first During the next iteration ; Indicates the first During the next iteration ; Indicates the first During the next iteration ; Indicates the first During the next iteration .

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