Vehicle dynamics modeling method based on adaptive depth Koopman operator

Abstract

The invention belongs to the technical field of autonomous vehicles, and particularly relates to a vehicle dynamics modeling method based on an adaptive depth Koopman operator. The method comprises the following steps: firstly, re-representing vehicle dynamics by applying a Koopman operator theory, lifting a nonlinear vehicle dynamics system to an infinite-dimensional space, and completing linear evolution in the space through a Koopman operator; secondly, an optimal dimension raising function of a vehicle dynamics system is trained offline through an extended dynamic mode decomposition method based on deep learning, and a finite dimension approximation matrix of a Koopman operator is obtained; thirdly, designing a recursive updating strategy based on a real-time data set to solve a self-adaptive Koopman operator approximate matrix, and realizing online updating of the vehicle dynamics model to adapt to dynamic data flow; according to the vehicle dynamics modeling method based on the adaptive depth Koopman operator provided by the invention, the model prediction precision of the vehicle dynamics model during online prediction is effectively improved, and a set of solution is provided for precise control of an automatic driving vehicle in a complex scene in the future.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of autonomous vehicles, and in particular to a vehicle dynamics modeling method based on an adaptive deep Koopman operator. BACKGROUND

[0002] Autonomous vehicles apply advanced sensors and artificial intelligence technology, and have made important contributions in improving road safety, optimizing traffic flow, and reducing environmental pollution. As a core element of the autonomous driving system, the prediction ability of the vehicle's dynamic response characteristics under various driving conditions will directly affect the accuracy and robustness of the core algorithms such as speed planning and trajectory tracking. Therefore, establishing an accurate vehicle dynamics model is an important technical foundation for ensuring the reliable, safe, and efficient operation of the autonomous driving system.

[0003] The dynamics modeling of autonomous vehicles faces nonlinear problems such as longitudinal-lateral dynamics coupling and system parameter uncertainty disturbance. Existing researches through T-S fuzzy theory, hybrid affine model, differential flatness technology, etc. can help vehicles improve the accuracy of dynamics modeling to a certain extent. However, the vehicle dynamics model processed by these approximate linearization methods inevitably has some unmodeled dynamic deviations, and the physical parameters such as the built-in cornering stiffness coefficient in the model are also difficult to accurately measure or estimate.

[0004] Compared with the physical modeling method with high parameter dependence, data-driven methods such as multilayer perceptron and long short-term memory network have attracted widespread attention in vehicle dynamics modeling due to their strong nonlinear fitting ability. Although these data-driven models can well represent the nonlinear dynamics characteristics of the vehicle, the implicit nonlinear structure will lead to nonlinear or even non-convex optimization problems in the controller solving process, which is complex and computationally intensive, and difficult to meet the real-time requirements of vehicle control. On the other hand, data-driven models usually lack interpretability, making it difficult to establish an explicit relationship between model parameters and vehicle controller design. This limitation makes it difficult to optimize or adjust the control strategy by analyzing the internal mechanism of the model when the vehicle control performance deviates, which poses new challenges to the safety requirements of vehicle motion control.

[0005] The Koopman operator, which is independent of specific physical parameters and has interpretability, provides a new way for the nonlinear dynamics modeling of vehicles. The main idea of the Koopman operator theory is to map the state space of a nonlinear system into an infinite-dimensional linear space, which linearizes the system while revealing its global dynamic characteristics. Considering that the infinite-dimensional Koopman operator is difficult to implement in engineering applications, some studies have begun to use dynamic mode decomposition (DMD) and extended dynamic mode decomposition (EDMD) to obtain a finite-dimensional approximation of the Koopman operator. However, the high-dimensional observation functions constructed in the EDMD method are mostly manually selected based on prior knowledge or expert experience of the system, lacking a systematic selection method and theoretical guidance, which to some extent limits the application of the Koopman operator in complex vehicle dynamics systems. Therefore, some studies have begun to use learning-based methods to effectively learn the observation function. These studies can effectively reduce the complexity of constructing a vehicle nonlinear dynamics model using a deep learning-based Koopman operator, helping advanced control theory to improve vehicle motion control performance with lower computational resources. However, they usually assume that the data are independent and identically distributed when applying deep learning to train the optimal observation function. However, in actual driving scenarios, the dynamic changes in the environment may cause the data distribution of autonomous vehicles to drift, leading to model mismatch and deterioration of vehicle motion control performance. SUMMARY

[0006] To solve the above problems, the present application provides a vehicle dynamics modeling method based on an adaptive deep Koopman operator, which effectively improves the model prediction accuracy of the vehicle dynamics model during online prediction, and provides a solution for future intelligent vehicle chassis dynamics optimal integrated control.

[0007] The technical scheme of the present application is described below in combination with the drawings:

[0008] The present application provides a vehicle dynamics modeling method based on an adaptive deep Koopman operator, comprising the following steps:

[0009] Step 1: Apply the Koopman operator theory to the controlled system to re-describe the vehicle dynamics, lift the vehicle nonlinear dynamics system with lateral-longitudinal coupling to an infinite-dimensional space, and complete linear evolution in the space through the Koopman operator;

[0010] Step 2: Based on the offline data set of the vehicle system, apply the deep learning-based extended dynamic mode decomposition (EDMD) method to obtain a finite-dimensional approximation of the Koopman operator. First, learn the optimal dimensionality function vector of the system through a self-encoder neural network, and then solve the nominal analytical solution of the finite-dimensional approximation matrix of the Koopman operator through the least squares method;

[0011] Step three, based on the real-time data set of vehicle system, the online update of the model is realized through the recursive update strategy of adaptive Koopman operator.

[0012] Further, the specific method of step one is as follows:

[0013] 11) Select the system state quantity Control input u = [T x δ f ] T ;

[0014] In the formula, v x is the longitudinal speed of the vehicle; β is the vehicle centroid side slip angle; ω is the vehicle yaw rate; Y and are the lateral displacement and heading angle of the vehicle in the earth coordinate system respectively; T x is the longitudinal total moment of the vehicle; δ f is the front wheel steering angle of the vehicle;

[0015] A discrete distributed drive electric vehicle lateral-longitudinal coupling nonlinear dynamics system is constructed, which is represented as:

[0016] x k+1 = f(x k ,u k ) (1)

[0017] In the formula, is the state quantity of the system at time k; is the state quantity of the system at time k+1; is the control quantity of the system at time k; is the nonlinear mapping function of the system; is an N x dimensional real number vector space, which is the state space of the system; is an N u dimensional real number vector space, which is the control input space of the system; is the constraint set of the control quantity; N x = 5 is the dimension of the state space of the system; N u = 2 is the dimension of the control input space of the system;

[0018] 12) Recharacterize the nonlinear vehicle dynamics system by applying Koopman operator theory;

[0019] First, define an extended state space as the product of the original state space and the space composed of all control sequences , that is wherein, is the constraint set of the control quantity, The extended system representation of the discrete distributed drive electric vehicle lateral-longitudinal coupled nonlinear dynamics system in state space is given as

[0020]

[0021] where χ k = [x k u k ] T is the extended state of the system at time k; x k is the state of the system at time k; u k is the control sequence of the system at time k; χ k+1 = [x k+1 u k+1 ] T is the extended state of the system at time k+1; x k+1 is the state of the system at time k+1; u k+1 is the control sequence of the system at time k+1; is the nonlinear mapping function of the extended system; is the nonlinear mapping function of the original system; is the state space of the original system; is the control input space of the original system; is the left shift operator for updating the control sequence, i.e., satisfying u k (0) is the first element of the control sequence u at time k, i.e.,

[0022] For the extended system, the Koopman operator is defined as an infinite-dimensional linear operator acting on the observation function φ in the Hilbert space

[0023]

[0024] where is the Koopman operator; χ k is the extended state of the system at time k; χ k+1 is the extended state of the system at time k+1; F is the nonlinear mapping function of the extended system; φ is a real-valued function in the space

[0025] 13) Under the action of the observation function φ, the distributed drive electric vehicle lateral-longitudinal coupled nonlinear dynamics system is lifted to the infinite-dimensional space, and the linear evolution is completed in the infinite-dimensional space by the Koopman operator

[0026] ​​​Further, the specific method of step two is as follows:

[0027] 21) Use the extended dynamic mode decomposition method based on deep learning to solve the finite-dimensional approximation matrix of the Koopman operator;

[0028] First, assume that the observation function φ has the form:

[0029]

[0030] In the formula, x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; χ k is the extended state quantity of the system at time k; is the dimension-increasing function vector of the state quantity of the system at time k; ψ i (i=1,…,N z ) is the dimension-increasing function of the system; N z is the dimension of the state space after dimension-increasing;

[0031] Simplify formula (3) to the following form:

[0032]

[0033] In the formula, x k is the state quantity of the system at time k; ψ(x k ) is the dimension-increasing function vector of the state quantity of the system at time k; x k+1 is the state quantity of the system at time k+1; ψ(x k+1 ) is the dimension-increasing function vector of the state quantity of the system at time k+1; u k is the control quantity of the system at time k; u k+1 is the control quantity of the system at time k+1; K is the approximation matrix of the Koopman operator ;

[0034] Based on the state quantity and control quantity collected in advance in the actual vehicle system, define the data set where x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; x k+1 is the state quantity of the system at time k+1; N is the sample capacity of the data set; the data in the data sets X and X + satisfy where f is the nonlinear mapping function of the system;

[0035] Ignore the last N u row components of formula (5), and define [AB] as the first N z row of the approximation matrix K of the Koopman operator, where A is Nz ×N z is a real matrix; B is a real matrix; N z ×N u is a real matrix; N u is the dimension of the control input space; N z is the dimension of the lifted state space; then the optimization problem to solve [AB] is defined based on the dataset {X, X + , U}:

[0036]

[0037] where ψ is the lifted function vector of the system;

[0038] 22) The method of training deep neural network offline is used to learn the optimal lifted function vector;

[0039] The deep neural network is designed as the structure of autoencoder; the encoder and the decoder are each composed of 4 fully connected layers, in which the hidden layer adopts linear rectifier unit as the activation function, and the output layer does not set the activation function to keep the linear mapping; the approximation matrix [A B] of Koopman operator is the trainable parameter of linear layer, which is used to obtain the evolved lifted state function vector In addition, the lifted function vector ψ is designed to contain the state quantity of the original system:

[0040]

[0041] where x is the state quantity of the original system at time k; N x is the dimension of the state space of the original system; is the basis function vector to be learned; is the basis function to be learned; N z is the dimension of the lifted state space; N z -N x is the dimension of the basis function vector to be learned;

[0042] 23) The state prediction error loss function L pre is designed:

[0043]

[0044] where ψ is the lifted function vector; is the evolved lifted state function vector; A and B are the approximation matrix of Koopman operator; x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; x k+1 is the state quantity of the system at time k+1; is the reconstruction of the decoder the obtained state quantity of the system at time k+1;

[0045] 24) design a multi-step linear error loss function L mlin and a multi-step state prediction error loss function L mpre to reduce the cumulative error:

[0046]

[0047] where p is the prediction step; ψ is the lifted function vector; is the evolved lifted state function vector, x k+i (i=1,…,p) is the state quantity of the system at time k+i; is the decoder reconstruction the obtained state quantity of the system at time k+i;

[0048] the evolved lifted state function vector is obtained by the following recursion:

[0049]

[0050] where p is the prediction step; A and B are the approximation matrices of the Koopman operator; ψ is the lifted function vector; x k+i (i=0,…,p) is the state quantity of the system at time k+i; u k+i (i=0,…,p-1) is the control quantity of the system at time k+i;

[0051] 25) design a reconstruction error loss function L rec :

[0052]

[0053] where Decoder is the decoder; ψ is the lifted function vector; x k is the state quantity of the system at time k;

[0054] 26) the total loss function Loss composed of each loss is represented as:

[0055]

[0056] where a i (i=1,…,6) is the weight coefficient of each partial loss; L lin is the linear error loss function; L pre is the state prediction error loss function; L mlin is the multi-step linear error loss function; L mpre is the multi-step state prediction error loss function; L rec is the reconstruction error loss function; and are L2 regularization terms for the encoder and decoder networks, respectively, to avoid overfitting of the networks;

[0057] 27) The encoder part obtained by training the autoencoder neural network is used to obtain the optimal function vector ψ, i.e., Encoder(x) = ψ(x), where Encoder is the encoder; combined with the data set {X, X + , U}, the least squares method is applied to solve the optimization problem (6) to obtain the nominal analytical solution of [AB] in the form of normal equations:

[0058]

[0059] where [AB] is the approximate matrix of the Koopman operator; is the Moore-Penrose pseudo-inverse; ψ is the function vector of the dimensionality increase; X and U are data sets composed of state quantities and control quantities collected in advance in actual vehicle systems; N is the sample capacity of the data set; x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; x k+1 is the state quantity of the system at time k+1.

[0060] Further, the specific method of step three is as follows:

[0061] 31) Define the real-time data set of the vehicle dynamics system at time k > 0 Ω k = [ψ(x1) … ψ(x k )], where ψ is the function vector of the dimensionality increase of the system; x i (i = 0, …, k) is the state quantity of the system at time i; u i (i = 0, …, j-1) is the control quantity of the system at time i; Ξ k and Ω k are dynamic data streams updated in real time according to the actual driving state of the vehicle;

[0062] 32) Design a recursive update strategy for the approximate matrix of the Koopman operator:

[0063] According to formula (15), the least squares solution of the Koopman operator approximate matrix [A k+1 B k+1 ] at time j+1 can be expressed based on the Koopman operator approximate matrix [A k B k ] at time j as:

[0064]

[0065] where, is the Moore-Penrose pseudo-inverse; ψ is the lifted function vector; Ξ k and Ω k is the real-time data set of the vehicle at time k; Ξ k+1 and Ω k+1 is the real-time data set of the vehicle at time k+1; x k is the state of the system at time k; u k is the control of the system at time k; x k+1 is the state of the system at time k+1;

[0066] 33) According to the Sherman-Morrison formula, the following equation is obtained:

[0067]

[0068] where, is the Moore-Penrose pseudo-inverse; Ξ k is the real-time data set of the vehicle at time k; Ξ k+1 is the real-time data set of the vehicle at time k+1; x k is the state of the system at time k; u k is the control of the system at time k; ψ is the lifted function vector, is a scalar;

[0069] 34) Define the initial value at time k = 0 where x0and u0are the initial state and initial control of the system, respectively; ψ is the lifted function vector; is the Moore-Penrose pseudo-inverse; combining (18) and (19), the least squares solution of the recursive form of the adaptive Koopman operator approximation matrix is obtained:

[0070]

[0071] where, [A k B k ] is the Koopman operator approximation matrix of the system at time k; [A k+1 B k+1 ] is the Koopman operator approximation matrix of the system at time k+1; ψ(x k+1 ) is the lifted function vector of the system at time k+1 predicted by [A k B k ]; x kis the state quantity of the system at time k; u k is the control quantity of the system at time k; ψ(x k is the dimension-lifting function vector of the system at time k; P k+1 is obtained through formula (19);

[0072] 35) the nominal analytical solution of the Koopman operator approximation matrix obtained offline through the deep learning-based extended dynamic mode decomposition method is taken as the recursive initial value of the Koopman operator approximation matrix, denoted as [A0 B0]; finally, the online updated linear approximation dynamics model of the distributed drive electric vehicle based on the adaptive Koopman operator and the distributed drive electric vehicle transverse-longitudinal coupling nonlinear dynamics system is written as:

[0073]

[0074] In the formula, A k is the state matrix of the system at time k; B k is the control matrix of the system at time k; x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; ψ(x k is the dimension-lifting function vector of the system at time k; ψ(x k+1 is the dimension-lifting function vector of the system at time k+1 predicted; is the state reconstruction matrix of the system; is a unit matrix with a size of N x × N x ; is a zero matrix with a size of N x × N z ; N x is the dimension of the state space of the original system; N z is the dimension of the state space after dimension lifting; is the state quantity of the original system at time k+1 predicted through the linear approximation dynamics model.

[0075] The beneficial effects of the present application are:

[0076] 1) the present application applies the Koopman operator theory to represent vehicle dynamics, can promote the nonlinear vehicle dynamics system to an infinite-dimensional space, and realize linear evolution in the space through the Koopman operator;

[0077] 2) the present application obtains the nominal analytical solution of the Koopman operator finite-dimensional approximation matrix through the deep learning-based extended dynamic mode decomposition method, and can effectively learn the optimal dimension-lifting function of the vehicle dynamics system;

[0078] 3) The application can realize online updating of the model by solving the adaptive Koopman operator finite-dimensional approximation matrix based on the recursive updating strategy of the real-time data set, so that the constructed vehicle dynamics model can better adapt to the dynamic data flow. BRIEF DESCRIPTION OF DRAWINGS

[0079] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings needed in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present application, and therefore should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can also be obtained without creative labor on the basis of these drawings.

[0080] Figure 1 Architectural diagram of vehicle dynamics modeling method based on adaptive deep Koopman operator;

[0081] Figure 2 Schematic diagram of dynamics characteristics of distributed drive electric vehicle;

[0082] Figure 3 Koopman operator learning framework based on autoencoder structure;

[0083] Figure 4a Schematic diagram of longitudinal total moment input of step working condition of longitudinal vehicle speed change;

[0084] Figure 4b Schematic diagram of front wheel angle input of step working condition of longitudinal vehicle speed change;

[0085] Figure 5a Schematic diagram of longitudinal speed state prediction result;

[0086] Figure 5b Schematic diagram of center of mass side slip angle state prediction result;

[0087] Figure 5c Schematic diagram of yaw rate state prediction result;

[0088] Figure 5d Schematic diagram of lateral displacement state prediction result;

[0089] Figure 5e Schematic diagram of heading angle state prediction result. DETAILED DESCRIPTION

[0090] The present application will be further described in detail below in combination with the drawings and embodiments. It can be understood that the specific embodiments described herein are only used to explain the present application, and not to limit the present application. In addition, it should be noted that, for the convenience of description, only the parts related to the present application are shown in the drawings, not all the structures.

[0091] Embodiment one

[0092] Referring to Figure 1 , the embodiment provides a vehicle dynamics modeling method based on an adaptive deep Koopman operator, comprising the following steps:

[0093] Step one, re-describe the vehicle dynamics by applying the Koopman operator theory extended to a controlled system, lift the vehicle nonlinear dynamics system coupled in the lateral and longitudinal directions to an infinite-dimensional space, and complete linear evolution in the space through the Koopman operator, as follows:

[0094] 11) The present application takes a front-wheel steering and four-wheel independent driving distributed drive electric vehicle as the research object, and the dynamic characteristics thereof are as shown in Figure 2 , the longitudinal motion, lateral motion and yaw motion of the vehicle are comprehensively considered, and the system state quantity is selected x δ f ] T ;

[0095] In the formula, v x is the longitudinal speed of the vehicle; β is the vehicle mass center side slip angle; ω is the vehicle yaw angular velocity; Y and are the lateral displacement and heading angle of the vehicle in the ground coordinate system, respectively; T x is the total longitudinal moment of the vehicle; δ f is the front wheel steering angle;

[0096] A discrete distributed drive electric vehicle lateral-longitudinal coupled nonlinear dynamics system is constructed and represented as:

[0097] x k+1 =f(x k ,u k ) (1)

[0098] In the formula, is the state quantity of the system at time k; is the state quantity of the system at time k+1; is the control quantity of the system at time k; is the nonlinear mapping function of the system; is an N x -dimensional real number vector space, which is the state space of the system; is an N u -dimensional real number vector space, which is the control input space of the system; is the constraint set of the control quantity; N x =5 is the dimension of the state space of the system; N u =2 is the dimension of the control input space of the system;

[0099] 12) In order to reduce the complexity of the vehicle model, the present application applies Koopman operator theory to re-characterize the nonlinear vehicle dynamics system;

[0100] Firstly, define an extended state space as the original state space and all control sequences constitute the space , that is where, is the constraint set of control variables, is all control variables of the system, then the extended system representation of the discrete distributed drive electric vehicle lateral-longitudinal coupled nonlinear dynamics system in the state space is:

[0101]

[0102] In the formula, χ k = [x k u k ] T is the extended state variable of the system at time k; x k is the state variable of the system at time k; u k is all control sequences of the system at time k; χ k+1 = [x k+1 u k+1 ] T is the extended state variable of the system at time k+1, x k+1 is the state variable of the system at time k+1; u k+1 is all control sequences of the system at time k+1; is the nonlinear mapping function of the extended system; is the nonlinear mapping function of the original system; is the state space of the original system; is the control input space of the original system; is the left shift operator for updating control sequences, that is, it satisfies u k (0) is the first element of the control sequence u at time k, that is

[0103] For the extended system, define the Koopman operator in the infinite-dimensional Hilbert space as an infinite-dimensional linear operator acting on the observation function φ:

[0104]

[0105] In the formula, is the Koopman operator; χ k is the extended state variable of the system at time k; χk+1 is the extended state quantity at time k+1 of the system; F is the nonlinear mapping function of the extended system; and φ is a real-valued function in space;

[0106] 13) Under the action of the observation function φ, the lateral-longitudinal coupled nonlinear dynamics system of the distributed drive electric vehicle is lifted to an infinite-dimensional space, and the Koopman operator is completed.

[0107] Step two, based on the offline data set of the vehicle system, the extended dynamic mode decomposition method based on deep learning is applied to the finite-dimensional approximation of the Koopman operator, first, the optimal lifting function vector of the system is learned through the self-encoder neural network, and then the nominal analytical solution of the finite-dimensional approximation matrix of the Koopman operator is solved through the least square method, which is as follows:

[0108] 21) In order to apply the infinite-dimensional Koopman operator to the actual vehicle control algorithm, the extended dynamic mode decomposition method based on deep learning is used to solve the finite-dimensional approximation matrix of the Koopman operator;

[0109] First, it is assumed that the observation function φ has the form:

[0110]

[0111] In the formula, x k is the state quantity at time k of the system; u k is the control quantity at time k of the system; χ k is the extended state quantity at time k of the system; is the lifting function vector of the state quantity at time k of the system; ψ i (i=1,…,N z ) is the system lifting function; N z is the dimension of the state space after lifting;

[0112] Formula (3) is simplified as follows:

[0113]

[0114] In the formula, x k is the state quantity at time k of the system; ψ(x k ) is the lifting function vector of the state quantity at time k of the system; x k+1 is the state quantity at time k+1 of the system; ψ(x k+1 ) is the lifting function vector of the state quantity at time k+1 of the system; u k is the control quantity at time k of the system; u k+1is the control quantity of the system at time k+1; K is the Koopman operator The approximate matrix of

[0115] Define the data set based on the state and control quantities collected in the actual vehicle system in advance where x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; x k+1 is the state quantity of the system at time k+1; N is the sample size of the data set; data sets X and X + The data in Where f is the nonlinear mapping function of the system;

[0116] Since the state prediction of the vehicle system does not need to consider the evolution of the control sequence, the last N in formula (5) is ignored. u row components, and define [AB] as the first N of the Koopman operator approximation matrix K z Rows where A is N z ×N z A real matrix; B is N z ×N u The real matrix of N u To control the dimension of the input space; N z is the dimension of the state space after dimensionality increase; based on the data set {X,X + ,U} is defined as the optimization problem to solve [AB]:

[0117]

[0118] Where, ψ is the system's dimensionality-raising function vector;

[0119] 22) The above optimization problem shows that the selection of the dimension-raising function vector ψ is directly related to the approximation effect of the Koopman operator. The traditional extended dynamic mode decomposition method generally constructs the dimension-raising function vector by manually selecting basis functions. In order to improve the accuracy of the established vehicle dynamics model, the present invention uses an offline training method of a deep neural network to learn the optimal dimension-raising function vector; the deep neural network is designed as an autoencoder structure; Figure 3 As shown in Figure 1, the encoder and decoder each consist of four fully connected layers, where the hidden layer uses the rectified linear unit (ReLU) as the activation function, and the output layer does not set an activation function to maintain the linear mapping; the approximate matrix [AB] of the Koopman operator is the trainable parameter of the linear layer, which is used to obtain the evolved dimensional state function vector In addition, in order to facilitate the reconstruction of the state quantity of the original vehicle dynamics system from the high-dimensional space, the dimension-raising function vector ψ is designed to contain the state quantity of the original system:

[0120]

[0121] where, is the state of the original system at time k; N x is the dimension of the state space of the original system; is the basis function vector to be learned; is the basis function to be learned; N z is the dimension of the state space after dimensionality increase; N z -N x is the dimension of the basis function vector to be learned;

[0122] 23) A linear error loss function L lin is designed considering the linear property of Koopman operator in high-dimensional space, in order to ensure the prediction accuracy of the vehicle dynamics model. A state prediction error loss function L pre is designed:

[0123]

[0124] where ψ is the dimensionality-increased function vector; is the evolved dimensionality-increased state function vector; A and B are the approximation matrices of Koopman operator; x k is the state of the system at time k; u k is the control quantity of the system at time k; x k+1 is the state of the system at time k+1; is the state of the system at time k+1 reconstructed by the decoder; obtained by the decoder;

[0125] 24) A multi-step linear error loss function L mlin and a multi-step state prediction error loss function L mpre are designed to reduce the cumulative error, considering the prediction effect of the vehicle dynamics model at a longer time step:

[0126]

[0127] where p is the prediction step; ψ is the dimensionality-increased function vector; is the evolved dimensionality-increased state function vector, x k+i is the state of the system at time k+i; (i=1,…,p) is the state of the system at time k+i reconstructed by the decoder; obtained by the decoder;

[0128] The evolved dimensionality-increased state function vector can be obtained by the following recursion:

[0129]

[0130] where p is the prediction step size; A and B are the approximated matrices of Koopman operator; ψ is the lifted function vector; x k+i are the state variables of the system at time k+i; u k+i are the control variables of the system at time k+i;

[0131] 25) To ensure the accuracy of state reconstruction, the reconstruction error loss function L rec is designed as:

[0132]

[0133] where Decoder is the decoder; ψ is the lifted function vector; x k is the state variable of the system at time k;

[0134] 26) The total loss function Loss composed of each loss is expressed as:

[0135]

[0136] where a i are the weight coefficients of each part of loss; L lin is the linear error loss function; L pre is the state prediction error loss function; L mlin is the multi-step linear error loss function; L mpre is the multi-step state prediction error loss function; L rec is the reconstruction error loss function; and are the L2 regularization terms of the encoder and decoder networks, respectively, to avoid network overfitting;

[0137] 27) The encoder part obtained by training the autoencoder neural network is used to obtain the optimal lifted function vector ψ, i.e. Encoder(x) = ψ(x), where Encoder is the encoder; combined with the data set {X, X + , U}, the least squares method is applied to solve the optimization problem (6) to obtain the nominal analytical solution of [AB] in the form of normal equation:

[0138]

[0139]

[0140] where [AB] is the approximated matrix of Koopman operator; is the Moore-Penrose pseudo-inverse; ψ is the lifted function vector; are the data sets composed of state variables and control variables collected in advance in actual vehicle systems; N is the sample capacity of the data sets; x k is the state variable of the system at time k; u k is the control variable of the system at time k; x k+1 is the state variable of the system at time k+1.

[0141] Step three, based on the real-time data set of the vehicle system, the model is updated online through the recursive updating strategy of the adaptive Koopman operator, specifically as follows:

[0142] In order to solve the model mismatch problem caused by data distribution drift during online prediction, the application designs an adaptive Koopman operator based on dynamic data flow to obtain an online updated vehicle dynamics model.

[0143] 31) define the real-time data set of the vehicle dynamics system at time k>0 Ω k = [ψ(x1) … ψ(x k )], wherein ψ is a dimension increasing function vector of the system; x i (i=0,…,k) is the state variable of the system at time i; u i (i=0,…,k-1) is the control variable of the system at time i; Ξ k and Ω k are dynamic data streams updated in real time according to the actual driving state of the vehicle;

[0144] 32) in order to avoid the calculation efficiency decline caused by the recalculation of Moore-Penrose pseudo-inverse at each k time in the prediction process, the recursive updating strategy of the approximate matrix of the adaptive Koopman operator is designed:

[0145] According to formula (15), the least square solution of the Koopman operator approximate matrix [A k+1 B k+1 ] at time k+1 can be expressed based on the Koopman operator approximate matrix [A k B k ] at time k as:

[0146]

[0147] In the formula, is the Moore-Penrose pseudo-inverse; ψ is the dimension increasing function vector; Ξ k and Ω k are the real-time data sets of the vehicle at time k; Ξ k+1 and Ω k+1 are the real-time data sets of the vehicle at time k+1; xk is the state of the system at time k; u k is the control of the system at time k; x k+1 is the state of the system at time k+1;

[0148] 33) According to the Sherman-Morrison formula, the following equation is obtained:

[0149]

[0150] where, is the Moore-Penrose pseudo-inverse; Ξ k is the real-time data set of the vehicle at time k; Ξ k+1 is the real-time data set of the vehicle at time k+1; x k is the state of the system at time k; u k is the control of the system at time k; ψ is the lifted function vector, is a scalar;

[0151] 34) Define the initial value at time k=0 where x0and u0are the initial state and initial control of the system, respectively; ψ is the lifted function vector; is the Moore-Penrose pseudo-inverse; combining (18) and (19), the least squares solution of the recursive form of the adaptive Koopman operator approximation matrix is obtained:

[0152]

[0153] where, [A k B k ] is the Koopman operator approximation matrix of the system at time k; [A k+1 B k+1 ] is the Koopman operator approximation matrix of the system at time k+1; ψ(x k+1 ) is the lifted function vector of the system at time k+1 predicted by [A k B k ]; x k is the state of the system at time k; u k is the control of the system at time k; ψ(x k ) is the lifted function vector of the system at time k; P k+1 is calculated by equation (19);

[0154] 35) The nominal analytical solution of the Koopman operator approximation matrix obtained offline by the deep learning-based extended dynamic mode decomposition method is taken as the recursive initial value of the Koopman operator approximation matrix, denoted as [A0 B0]; finally, the online updated linear approximation dynamics model of the distributed drive electric vehicle based on the adaptive Koopman operator of the transverse-longitudinal coupled nonlinear dynamics system of the distributed drive electric vehicle is written as:

[0155]

[0156] wherein, A k is the state matrix of the system at k moment; B k is the control matrix of the system at k moment; x k is the state quantity of the system at k moment; u k is the control quantity of the system at k moment; ψ(x k ) is the dimension-increasing function vector of the system at k moment; ψ(x k+1 ) is the dimension-increasing function vector of the system at k+1 moment predicted; is the state reconstruction matrix of the system; is an identity matrix with the size of N x ×N x ; is a zero matrix with the size of N x ×N z ; N x is the dimension of the state space of the original system; N z is the dimension of the state space after dimension-increasing; is the state quantity of the original system at k+1 moment predicted by the linear approximation dynamics model.

[0157] Example Two

[0158] This embodiment is based on the joint simulation platform built by MATLAB / Simulink and vehicle dynamics software CarSim, which is used to collect the data set of the state quantity and control quantity of the vehicle system, and test the vehicle dynamics modeling method designed by the application. The autoencoder neural network in the deep learning-based extended dynamic mode decomposition method is built based on the PyTorch framework, and trained using the NVIDIA GeForce RTX 3060 GPU. The data set involved in the training is normalized to [-1, 1] to eliminate the scale difference between the data. The step working condition of longitudinal vehicle speed change is selected to verify the prediction performance of the vehicle dynamics model based on the adaptive deep Koopman operator, and the experiment is carried out on the road surface with adhesion coefficient of 0.75. The control input is as shown in Figure 4a and Figure 4b , and the state prediction result is as shown in Figures 5a-5eThe results are shown in FIG. 6. It can be seen that the vehicle dynamics model established by applying the adaptive deep Koopman operator can more accurately predict the trend of change of each state over a long period of time.

[0159] Although embodiments of the present application have been shown and described, it is to be understood that various modifications, substitutions, replacements and changes can be made to these embodiments without departing from the principles and spirit of the present application, and the scope of the present application is defined by the appended claims and their equivalents.

Claims

1. A vehicle dynamics modeling method based on an adaptive deep Koopman operator, characterized in that: The following steps are involved: Step 1: Apply the Koopman operator theory generalized to the controlled system to redescribe the vehicle dynamics, elevate the lateral-longitudinal coupled vehicle nonlinear dynamic system to an infinite-dimensional space, and complete the linear evolution in the infinite-dimensional space using the Koopman operator; Step 2: Based on the offline dataset of the vehicle system, the extended dynamic mode decomposition method based on deep learning is applied to perform a finite-dimensional approximation of the Koopman operator. First, the optimal dimensionality-increasing function vector of the system is learned through an autoencoder neural network. Then, the nominal analytical solution of the Koopman operator's finite-dimensional approximation matrix is ​​solved using the least squares method. Step 3: Based on the real-time data set of the vehicle system, the online update of the model is achieved through the recursive update strategy of the adaptive Koopman operator.

2. The vehicle dynamics modeling method based on the adaptive deep Koopman operator according to claim 1, characterized in that: The specific method of step one is as follows: 11) Select the system state quantity Control input u=[T x δ f ] T ; Where, v x is the longitudinal velocity of the vehicle; β is the sideslip angle of the vehicle's center of mass; ω is the yaw rate of the vehicle; Y and are the lateral displacement and heading angle of the vehicle in the geodetic coordinate system; T x is the total longitudinal moment of the vehicle; δ f is the vehicle's front wheel turning angle; Construct a discrete distributed drive electric vehicle lateral-longitudinal coupled nonlinear dynamic system, which can be expressed as: x k+1 =f(x k ,u k ) (1) Where, is the state quantity of the system at time k; is the state quantity of the system at time k+1; is the control quantity of the system at time k; f: is the nonlinear mapping function of the system; is an N x The real vector space of dimension is the state space of the system; is an N u dimensional real vector space, which is the control input space of the system; is the constraint set of the control quantity; N x =5 is the dimension of the system state space; N u =2 is the dimension of the system control input space; 12) Apply Koopman operator theory to re-characterize nonlinear vehicle dynamics systems; First, define an extended state space as the original state space and all control sequences The space composed The product of in, is the set of constraints for the control quantity, If is all the control variables of the system, the extended system of the discrete distributed drive electric vehicle transverse-longitudinal coupled nonlinear dynamic system in the state space is expressed as: Where, χ k =[x k u k ] T is the extended state quantity of the system at time k; x k is the state quantity of the system at time k; u k is all control sequences of the system at time k; k+1 =[x k+1 u k+1 ] T is the extended state quantity of the system at time k+1, x k+1 is the state quantity of the system at time k+1; u k+1 is all control sequences of the system at time k+1; F: is the nonlinear mapping function of the extended system; f: is the nonlinear mapping function of the original system; is the state space of the original system; is the control input space of the original system; L is the left shift operator used to update the control sequence, that is, u k (0) is the first element of the control sequence u at time k, that is, For extended systems, in infinite-dimensional Hilbert space The Koopman operator is defined as an infinite-dimensional linear operator acting on the observation function φ: Where, is the Koopman operator; k is the extended state quantity of the system at time k; k+1 is the extended state quantity of the system at time k+1; F is the nonlinear mapping function of the extended system; φ is A real-valued function in space; 13) Under the action of the observation function φ, the horizontal-vertical coupled nonlinear dynamic system of the distributed drive electric vehicle is elevated to an infinite-dimensional space and is solved by the Koopman operator in the infinite-dimensional space. The linear evolution is completed.

3. The vehicle dynamics modeling method based on the adaptive deep Koopman operator according to claim 1, characterized in that: The specific method of step 2 is as follows: 21) Using the extended dynamic mode decomposition method based on deep learning to solve the finite-dimensional approximation matrix of the Koopman operator; First, assume that the observation function φ has the form: Where x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; k is the extended state quantity of the system at time k; is the dimensional function vector of the system state quantity at time k; ψ i (i=1,…,N z ) is the system dimension-raising function; N z is the dimension of the state space after dimensionality increase; Simplify formula (3) into the following form: Where x k is the state quantity of the system at time k; ψ(x k ) is the dimensional function vector of the system state quantity at time k; x k+1 is the state quantity of the system at time k+1; ψ(x k+1 ) is the dimension-raising function vector of the system state quantity at time k+1; u k is the control quantity of the system at time k; u k+1 is the control quantity of the system at time k+1; K is the Koopman operator The approximate matrix of Define the data set based on the state and control quantities collected in the actual vehicle system in advance where x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; x k+1 is the state quantity of the system at time k+1; N is the sample size of the data set; data sets X and X + The data in Where f is the nonlinear mapping function of the system; Ignore the last N in formula (5) u row components, and define [AB] as the first N of the Koopman operator approximation matrix K z Rows where A is N z ×N z A real matrix; B is N z ×N u The real matrix of N u To control the dimension of the input space; N z is the dimension of the state space after dimensionality increase; based on the data set {X,X + ,U} is defined as the optimization problem to solve [AB]: Where, ψ is the system's dimensionality-raising function vector; 22) Using offline training of deep neural networks to learn the optimal dimensionality-increasing function vector; The deep neural network is designed as an autoencoder structure; the encoder and decoder each consist of 4 fully connected layers, where the hidden layer uses a linear rectified unit as the activation function, and the output layer does not set an activation function to maintain linear mapping; the approximate matrix [AB] of the Koopman operator is the trainable parameter of the linear layer, which is used to obtain the evolved dimensional state function vector In addition, the dimension-raising function vector ψ is designed to contain the state quantity of the original system: Where, is the state quantity of the original system at time k; N x is the dimension of the state space of the original system; is the basis function vector that needs to be learned; is the basis function to be learned; N z is the dimension of the state space after dimensionality increase; N z -N x is the dimension of the basis function vector to be learned; 23) Design state prediction error loss function L pre : Where, ψ is the dimension-raising function vector; is the dimensional state function vector after evolution; A and B are the approximate matrices of the Koopman operator; x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; x k+1 is the state quantity of the system at time k+1; Refactoring for the decoder The state quantity of the system at time k+1 is obtained; 24) Design a multi-step linear error loss function L mlin and the multi-step state prediction error loss function L mpre To reduce the cumulative error: Where p is the prediction step size; ψ is the dimension-raising function vector; is the dimensional state function vector after evolution, x k+i (i=1,…,p) is the state quantity of the system at time k+i; Refactoring for the decoder The state quantity of the system at time k+i is obtained; Evolved dimensional state function vector Obtained by the following recursion: Where p is the prediction step size; A and B are the approximate matrices of the Koopman operator; ψ is the dimension-raising function vector; x k+i (i=0,…,p) is the state of the system at time k+i; u k+i (i=0,…,p-1) is the control quantity of the system at time k+i; 25) Design reconstruction error loss function L rec : Where Decoder is the decoder; ψ is the dimension-raising function vector; x k is the state quantity of the system at time k; 26) The total loss function Loss composed of each loss is expressed as: Where a i (i=1,…,6) is the weight coefficient of each part loss; L lin is the linear error loss function; L pre is the state prediction error loss function; L mlin is the multi-step linear error loss function; L mpre is the multi-step state prediction error loss function; L rec is the reconstruction error loss function; and are the L2 regularization terms of the encoder and decoder networks, respectively, to avoid overfitting of the network; 27) The encoder part obtained by training the autoencoder neural network is used to obtain the optimal dimension-raising function vector ψ, that is, Encoder(x)=ψ(x), where Encoder is the encoder; combined with the data set {X,X + ,U}, apply the least squares method to solve the optimization problem (6) and obtain the nominal analytical solution of [AB] in the form of normal equation: Where [AB] is the approximate matrix of the Koopman operator; is the Moore-Penrose pseudo-inverse; ψ is the dimension-raising function vector; are all data sets consisting of state variables and control variables collected in advance in the actual vehicle system; N is the sample size of the data set; x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; x k+1 is the state quantity of the system at time k+1.

4. The vehicle dynamics modeling method based on the adaptive deep Koopman operator according to claim 3 is characterized in that: The specific method of step three is as follows: 31) Define the real-time data set of the vehicle dynamics system at time k>0 Ω k =[ψ(x1)…ψ(x k )], where ψ is the system's dimensionality-raising function vector; x i (i=0,…,k) is the state quantity of the system at time i; u i (i=0,…,k-1) is the control quantity of the system at time i; k and Ω k It is a dynamic data stream that is updated in real time according to the actual driving status of the vehicle; 32) Design a recursive update strategy for the adaptive Koopman operator approximation matrix: According to formula (15), the Koopman operator approximation matrix at time k+1 [A k+1 B k+1 The least squares solution of ] can be based on the Koopman operator approximation matrix [A k B k ] is represented as: Where, is the Moore-Penrose pseudo-inverse; ψ is the dimension-raising function vector; Ξ k and Ω k is the real-time data set of vehicles at time k; k+1 and Ω k+1 is the real-time data set of vehicles at time k+1; x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; x k+1 is the state quantity of the system at time k+1; 33) According to the Sherman-Morrison formula, we get the following equation: Where, is the Moore-Penrose pseudoinverse; Ξ k is the real-time data set of vehicles at time k; k+1 is the real-time data set of vehicles at time k+1; x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; ψ is the dimension-raising function vector, is a scalar; 34) Define the initial value at k = 0 Where x0 and u0 are the initial state and initial control variables of the system respectively; ψ is the dimension-raising function vector; is the Moore-Penrose pseudo-inverse; combining equations (18) and (19), we obtain the least squares solution of the recursive adaptive Koopman operator approximation matrix: In the formula, [A k B k ] is the Koopman operator approximation matrix of the system at time k; [A k+1 B k+1 ] is the Koopman operator approximation matrix of the system at time k+1; ψ(x k+1 ) is passed [A k B k ]The predicted k+1 time system dimension-raising function vector; x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; ψ(x k ) is the dimension-raising function vector of the system at time k; P k+1 Calculated by formula (19); 35) The nominal analytical solution of the Koopman operator approximation matrix obtained offline by the extended dynamic mode decomposition method based on deep learning is used as the recursive initial value of the Koopman operator approximation matrix, denoted as [A0 B0]; Finally, based on the adaptive Koopman operator, the distributed drive electric vehicle linear approximate dynamic model of the distributed drive electric vehicle horizontal-longitudinal coupled nonlinear dynamic system is updated online as follows: Where A k is the state matrix of the system at time k; B k is the control matrix of the system at time k; x k is the state quantity of the system at time k; u k is the control quantity of the system at time k; ψ(x k ) is the dimension-raising function vector of the system at time k; ψ(x k+1 ) is the predicted k+1 time system dimension-raising function vector; Reconstruct the matrix for the system's state; For size N x ×N x The identity matrix of For size N x ×N z The zero matrix of N x is the dimension of the original system state space; N z is the dimension of the state space after dimensionality increase; is the state quantity of the original system at time k+1 predicted by the linear approximate dynamic model.

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