Vehicle dynamics Kupman modeling and predictive stability expansion control method

Abstract

The invention discloses a vehicle dynamics Kupman modeling and predictive stability expansion control method, and belongs to the technical field of vehicle dynamics modeling and safety technology control. The invention aims to provide a vehicle dynamics Kupman modeling and predictive stability expansion control method designed by combining a model predictive control method based on a closed-loop data promoted Kupman operator modeling method. According to the method, firstly, a linear two-degree-of-freedom reference model of the vehicle is designed, then a nonlinear closed-loop control system is established, Kupman lifting is carried out, a high-dimensional observation function is constructed, and finally, a linear MPC controller is designed. According to the method, linearization of vehicle dynamics is achieved through the Kupman model, the calculation complexity is remarkably reduced, the real-time performance of the controller is greatly improved, and a control system can rapidly respond under the dynamic working condition.

Description

Technical Field

[0001] The present invention belongs to the technical field of vehicle dynamics modeling and safety technology control. Background Art

[0002] Under complex working conditions such as ice and snow roads with low adhesion coefficients, vehicle lateral dynamics exhibit significant nonlinear characteristics and high dynamic complexity. Traditional control strategies face many limitations when dealing with such complex nonlinear systems and are difficult to meet the requirements of real-time performance and global accuracy simultaneously.

[0003] Specifically, the current technology has the following deficiencies: 1. To more accurately describe the tire characteristics under low-adhesion roads, many studies use complex nonlinear tire models to characterize vehicle dynamics. However, such methods significantly increase the solution difficulty of the controller, resulting in an increased computational burden and reduced real-time performance of the control algorithm. In a rapidly changing dynamic environment, this method is difficult to meet the high requirements of vehicle stability control for real-time performance and response speed.

[0004] 2. Most current studies simplify nonlinear dynamics problems through local linearization techniques and can only work effectively under specific working conditions. This limitation makes such methods difficult to capture the global nonlinear dynamic characteristics under low-friction conditions such as ice and snow roads, thus restricting the control performance of the controller in complex working conditions.

[0005] 3. Existing modeling methods based on the Koopman operator mostly rely on open-loop data and fail to fully exploit the potential of the control input data and state data generated during the operation of the closed-loop controller. This deficiency restricts the performance of the Koopman model in global prediction ability, thus restricting the actual application effect of the control system in a complex dynamic environment. Summary of the Invention

[0006] The purpose of the present invention is a vehicle dynamics Koopman modeling and predictive stability augmentation control method designed based on a Koopman operator modeling method enhanced by closed-loop data and combined with a model predictive control method.

[0007] The steps of the present invention are as follows: S1. Design a two-degree-of-freedom reference model: According to the steering angle input by the driver and combined with vehicle dynamics parameters, establish an ideal state equation: The steady-state gain G of the sideslip angle at the center of mass β and the steady-state gain G of the yaw rate γ are defined as follows: where m is the vehicle mass, L fis the distance from the front axle to the vehicle's center of mass, L r is the distance from the rear axle to the vehicle's center of mass, and the distance between the front axle and the rear axle is L = L f + L r , C r is the rear wheel cornering stiffness; The stability factor K of the vehicle is: where C f is the front wheel cornering stiffness; The differential coefficients are defined respectively as: where I z is the moment of inertia of the vehicle about the vertical axis; The oscillation frequency ω of the system n and the damping coefficient ξ are defined as follows: S2. Establish a non - linear closed - loop control system: Considering only the lateral and yaw motions of the vehicle, a two - degree - of - freedom vehicle dynamics model is obtained: where F y represents the lateral force of the tire, and the subscripts f, r represent the front and rear wheels respectively, and fl, fr, rl, rr represent the left front, right front, left rear, and right rear wheels; The expression of the tire lateral force determined by the slip angle α is as follows: where k s is the lateral force fitting coefficient, F z is the vertical load of the tire, C 1 , C 2 , C 3 are the Burckhardt tire model fitting coefficients; The tire slip angle α ij , where ij = fl, fr, rl, rr: where d is the vehicle's left - right wheelbase; The vertical load F of the tire zij has the following expression: where g is the acceleration due to gravity, h cg is the height of the vehicle's center of mass, a y is the lateral acceleration; According to formulas (6) - (9), the vehicle dynamics model is described in the form of a general non - linear expression: Among them, the state variable \(x = [x 1 x 2 T = [V y γ] T , and the control variable \(u=\Delta M z \) is the additional yaw moment of the tire, and the front wheel steering angle \(\delta f \) is used as the disturbance input; Define the state variables of the system as normalized state variables: Among them, \(V ym = |V x |\arctan(0.02\mu g)|\) and are the upper limit values of the lateral velocity and the yaw rate respectively, and \(\mu\) is the road surface friction coefficient; The external input variable is defined as \(d = \delta f \), and the control variable is defined as Among them, \(\Delta M zm \) represents the upper limit value of the yaw moment; Use the first-order Euler formula to discretize the state-space equation of the system: x(k + 1)=x(k)+T s f(x(k),\delta f (k),u(k)) (12) Among them, \(T s \) is the sampling time of the system, and \(x k \) is used as the starting point of the prediction model; The prediction model is expressed as: x(k + i|k)=x(k + i - 1|k)+T s f(x(k + i - 1|k),\delta f (k + i - 1),u(k + i - 1)) (13) Among them, \(k + 1|k\) represents the predicted system state at the \(k + 1\) moment at the current \(k\) moment; Set the prediction horizon of the system to \(N p \), and the control horizon of the system to \(N c \), then the output state sequence \(X(k)\) of the system in the future period of time and the sequence \(U(k)\) of the control input of the system are expressed as: The reference value obtained from step S1 Define the reference value sequence within the prediction horizon as: Obtain the optimization problem: ​ where \(Q = \text{diag}(\Gamma vy ,\Gamma γ )\) are the weights of the vehicle lateral velocity state and the vehicle yaw rate state respectively, and \(W=\text{diag}(\Gamma mz )\) is the weight of the additional yaw moment of the vehicle control quantity; S3. Obtain the high-dimensional linearized Koopman model using closed-loop data Expand the perturbation into a new state variable where \(\chi(k)\) is the expanded state variable, including the system state \(x(k)\) and the perturbation input \(\delta k (k)\); Define the observation value of the new state variable \(\chi(k)\) to be lifted to a high-dimensional space through the non-linear mapping function \(\varPsi:\mathbb{R} n \to\mathbb{R} N . Here, \(n\) is the dimension of the original expanded state space, \(N\) is the dimension of the lifted high-dimensional state space, and \(N\gg n\). The Koopman operator is: \varPsi(\chi(k + 1))=K d \varPsi(\chi(k))\ (18) where \(K d is the discrete Koopman operator; Collect \(p\) groups of discrete state measurement values in the form of snapshot pairs \([\chi(j),\chi(j + 1)],j = 1,2,\cdots,p\), and integrate all snapshot pairs into two matrices with a mapping relationship, denoted as \(X 1 and \(X 2 X 1 =[\chi(1)\ \chi(2)\cdots\chi(p)] n×p X 2 =[\chi(2)\ \chi(3)\cdots\chi(p + 1)] n×p (19) where the subscript represents the dimension of each matrix; Select a set of basis functions \(\varPsi(\chi)\) to lift \(X 1 and \(X 2 : Obtain the lifted state data pairs \(X 1lift and \(X 2lift : X 1lift =[\varPsi(\chi(1))\ \varPsi(\chi(2))\cdots\varPsi(\chi(p))] N×p X 2lift =[\varPsi(\chi(2))\ \varPsi(\chi(3))\cdots\varPsi(\chi(p + 1))]N×p (21) For X 1lift and X 2lift Perform further expansion to construct a state matrix X with enhanced control 1 and X 2 : where U = [u(1) u(2)…u(p)], and m is the dimension of the control input; Determine the approximate Koopman operator Koopman state space model: z d (k + 1) = A d z d (k) + B d u(k) x(k) = C d z d (k) (24) where z d (k) = Ψ(χ(k)), which is the lifted high-dimensional state; The linear coefficient matrices A d and B d are obtained from : where I represents the identity matrix and O represents the all-zero matrix; The matrix C d is defined as: C d = [I (n-1)×(n-1) O (n-1)×(N-n+1) (n-1)×N (26) Design the candidate nonlinear feature function library Θ(X 1 ): where θ j (X 1 )(j = 1, 2,..., S, S >> N) represents the nonlinear candidate function terms of the nonlinear system; Extract the dynamically dominant feature functions: X 2 = ΞΘ(X 1 ) (28) where the sparse matrix Ξ = [ξ 1 ξ 2 …ξ q T ​​It is obtained by using the LASSO algorithm: Among them, ξ k is a sparse row vector, representing the k-th row of Ξ, and X 2k represents the k-th row of X 2 , and the parameter λ is the regularization factor; S4. Design a linear MPC controller After the system is lifted to a high-dimensional linear space, the optimization problem is expressed as a standard quadratic programming problem: Among them, U k =[u(k|k) u(k + 1|k)…u(k + N c -1|k)] T is the optimization variable at time k, H is the quadratic weight matrix, and f(k + 1|k) is the linear weight vector; The relevant matrix derivation is as follows: Expand the prediction model to the prediction time domain N p length to obtain the following matrix form: Z = Αz d (k) + ΒU (31) Among them, Z is the predicted high-dimensional state sequence, Α is the state transition matrix, and Β is the input influence matrix; The expressions of Z, Α, and Β are as follows: Map the lifted state sequence Z back to the low-dimensional space to obtain the output expression matrix as: Among them, X is the predicted initial state sequence; The expressions of the quadratic weight matrix H and the linear weight vector f(k + 1|k): Combine the state variable constraints and the control input constraints, and the complete constraints are expressed as: Take the first element of U * (k): u = [1 0…0]U * (k) as the control input and apply it to the control system; Corresponding constraint condition matrix: The control input constraint matrix is The state variable constraint is only for the first state variable V y , so the state variable constraint matrix is

[0008] The positive effects of the present invention are as follows: 1. Aiming at the problems of increased computational burden and insufficient real-time performance caused by traditional complex non-linear tire models, the present invention linearizes vehicle dynamics through the Koopman model, significantly reducing the computational complexity and greatly improving the real-time performance of the controller, enabling the control system to respond quickly under dynamic conditions; 2. By constructing a high-dimensional observation function, the Koopman model of the present invention can accurately capture the global non-linear dynamic characteristics under low-adhesion road surfaces (such as ice and snow road surfaces). Compared with traditional local linearization techniques, the present invention effectively expands the stable driving area of the vehicle under low-adhesion road surface conditions and significantly enhances the applicability of the controller in complex working conditions; 3. Different from existing open-loop modeling methods based on the Koopman operator, the present invention utilizes high-quality control inputs and state data generated during closed-loop operation, significantly improving the global prediction ability and accuracy of the Koopman model, further enhancing the performance of the control system in complex dynamic environments, and thus enhancing the practical application effect of the controller based on the Koopman framework. Description of the Drawings

[0009] Figure 1 is a flowchart of the Koopman MPC control method based on closed-loop data improvement according to the present invention; Figure 2 is a schematic diagram of the Koopman theory based on closed-loop data improvement according to the present invention; Figure 3 is a schematic diagram of the vehicle dynamics model described in the present invention; Figure 4 is a verification diagram of the Koopman model based on closed-loop data improvement according to the present invention, with the working condition set as a double lane change condition at a friction coefficient of 0.35 and a speed of 60 km / h. The blue solid line is the state curve output by the Koopman model, and the red dashed line is the real vehicle data curve; Figure 5 is the verification and comparison result of the Koopman MPC control based on closed-loop data improvement according to the present invention under the double lane change condition with a friction coefficient of 0.35 and a speed of 60 km / h. Figure 6 is a comparison of the solution times of three controllers. Detailed Embodiment

[0010] The present invention specifically relates to a linearization modeling method based on the Koopman Operator combined with model predictive control (MPC) to achieve real-time stability control. The present invention focuses on the problem of lateral dynamics stability control of vehicles under low-friction road surfaces (such as ice and snow road surfaces), aiming to solve the real-time performance and global linearization problems of non-linear dynamic systems, and providing theoretical support and practical tools for vehicle stability control under complex working conditions.

[0011] In view of the problems of non - linearity, complexity and insufficient real - time performance faced by vehicle stability control under low - adhesion conditions, the present invention proposes a Koopman operator modeling method based on closed - loop data enhancement, and designs a linear and efficient real - time stability control strategy in combination with the model predictive control method. Based on the non - linear two - degree - of - freedom vehicle dynamics model, by using the control input and state data generated during closed - loop operation, a high - dimensional observation function is constructed to achieve the global linearization of the non - linear dynamic system, providing a high - precision prediction model for MPC. The simulation results show that the Koopman MPC controller based on closed - loop data enhancement proposed by the present invention can effectively suppress the lateral speed of the vehicle and achieve accurate tracking of the yaw angular velocity. Compared with traditional methods, the present invention expands the stable driving area of the vehicle under low - friction road surface conditions, significantly improves the solution efficiency and real - time performance of the controller while accurately describing the non - linear characteristics of the system.

[0012] A vehicle stability control method based on a closed - loop Koopman model includes the following steps: Step 1: Design a vehicle linear two - degree - of - freedom reference model to obtain the expected values of the vehicle yaw angular velocity and vehicle lateral velocity considering the road surface adhesion coefficient, which are used to describe the ideal motion state of the vehicle and provide a dynamic target for the subsequent MPC controller design.

[0013] Step 2: Establish a non - linear closed - loop control system. Use a non - linear tire model to describe the adhesion characteristics between the tire and the road surface, construct a vehicle dynamics model, and design a non - linear MPC controller. During the closed - loop operation of the controller, the state data and control input data of the vehicle are collected in real - time to capture the non - linear dynamic characteristics of the system.

[0014] Step 3: Perform Koopman enhancement, construct a high - dimensional observation function, and use the state data and control input data generated by the closed - loop controller to achieve the global linearization of the non - linear system through Koopman operator enhancement. The generated Koopman model can accurately describe the non - linear behavior of vehicle dynamics in a linear form globally.

[0015] Step 4: Based on the Koopman model obtained in Step 3 and the reference model in Step 1, design a linear MPC controller to achieve real - time stability control of the vehicle under low - friction road surface conditions. This controller significantly reduces the solution complexity through the global linearization model, ensuring the real - time performance and accuracy of the control strategy.

[0016] Functionally, the present invention includes the following parts: reference model, non - linear closed - loop data acquisition, Koopman enhancement, linear MPC controller. The functions of each part are described in detail below: The reference model is based on a two-degree-of-freedom linear vehicle dynamics model, calculates the desired yaw rate and lateral velocity, determines the ideal motion state of the vehicle, and provides dynamic targets for the nonlinear MPC controller and the linear MPC controller.

[0017] The nonlinear closed-loop data acquisition part simulates the real vehicle dynamics behavior with a nonlinear dynamics model, collects vehicle state data and control input data during the closed-loop operation of the nonlinear controller, better captures the nonlinear characteristics in the system, provides high-quality data for Koopman lifting, and ensures the accuracy of subsequent modeling.

[0018] The Koopman lifting part performs Koopman lifting on the closed-loop collected data through a high-dimensional observation function, calculates the approximate Koopman operator using the extended dynamic mode decomposition (EDMD), expands the region where the vehicle can stably travel under low adhesion road surface conditions, realizes the global linearization description of the system. This module improves the global prediction ability of the Koopman model using the closed-loop data, enabling it to more accurately reflect the nonlinear dynamic characteristics of vehicle dynamics and providing support for the linear MPC controller.

[0019] The linear MPC controller based on the Koopman model realizes real-time vehicle stability control. While accurately tracking the vehicle target state, it significantly reduces the computational complexity, improves the real-time response performance, and ensures the safety and stability of the vehicle under low-friction road surface conditions.

[0020] To explain the technical content, structural features, implementation purposes, etc. of the present invention in detail, the present invention will be comprehensively explained below with reference to the accompanying drawings: The flow of the Koopman model predictive control method based on closed-loop data improvement of the present invention is as Figure 1 shown. The design of the controller combines a reference model, a Koopman model, and an MPC controller to achieve precise dynamic control of the vehicle under low adhesion road surface conditions. The driver input module in the figure provides an external operation signal through the steering wheel angle (which can be converted into the front wheel angle δ f ) and the vehicle longitudinal speed V x . The reference model calculates the desired yaw rate γ x and the desired vehicle lateral velocity V f based on the vehicle longitudinal speed V ref and the front wheel angle δ yref to determine the ideal dynamic behavior of the vehicle. The controller combines the actually measured vehicle state variables (including V x , V y , γ, and δ f ) with the expected values provided by the reference model, uses the globally linearized Koopman model to predict the system, and generates the control input additional yaw moment ΔM zThe additional yaw moment output by the controller is converted into the actual control commands ΔT for the four wheels through the torque distribution module ij , realizing the precise regulation of the vehicle's power distribution. The core of the controller design lies in improving and optimizing the Koopman model through closed-loop data, expanding the stable driving area of the vehicle under low adhesion road surface conditions, and improving the accuracy of global prediction.

[0021] Figure 2 Fig. shows the schematic diagram of the Koopman theory based on closed-loop data improvement of the present invention. First, the vehicle state and control input data are collected in real time during operation through a closed-loop controller. These data contain the dynamic characteristics of the nonlinear dynamic system during closed-loop operation. The front wheel steering angle input by the driver is used as a perturbation quantity to participate in the modeling. Since there is a complex nonlinear relationship between it and the evolution of the nonlinear system state, it is added as an extended state quantity to the high-dimensional space improvement. The arrow in the central part of the figure represents the process of the state variable being lifted from the low-dimensional initial state space to the high-dimensional state space. By constructing a high-dimensional observation function (such as polynomial, trigonometric function, etc.), the initial state variable is mapped to the high-dimensional feature space. This process retains the nonlinear characteristics of the original system and provides a basis for subsequent global linearization. The high-dimensional state space is linearized by the approximate Koopman operator calculated by EDMD. The operator is generated by training with the collected closed-loop data and can describe the dynamic characteristics of the nonlinear system in a global linear form. Finally, the high-dimensional linear model generated by the approximate Koopman operator is used to describe the global dynamic behavior of the system. This model combines the results of closed-loop data improvement and can be used to design an efficient linear MPC controller.

[0022] Through the closed-loop data improvement method, the present invention significantly improves the global accuracy and prediction ability of the Koopman model. By making full use of the dynamic data generated by the closed-loop controller, under complex dynamic conditions such as low-friction road surfaces, the stable driving area of the vehicle is expanded, a more accurate global linearization description is provided, and the stability and real-time performance of the vehicle under complex conditions are effectively guaranteed.

[0023] The present invention provides a co-simulation model based on the above operating principles and processes. Its construction and operation processes are as follows: 1. Software selection The simulation models of the controller of the control system and the controlled object are built through the software MATLAB / Simulink and CarSim respectively. The software versions are MATLAB R2022a and CarSim 2020.0 respectively, and the simulation step size is 0.02 s. Among them, CarSim is a commercial software specifically for vehicle dynamics simulation. Its high fidelity enables it to accurately simulate the dynamic characteristics of vehicles under various working conditions. In the present invention, the main role of CarSim is to provide a simulation environment for vehicle dynamics, as a virtual substitute for a real four-wheel drive electric vehicle, and at the same time to simulate the dynamic behavior under low adhesion conditions, providing a reliable experimental platform for control strategy verification. MATLAB / Simulink is used for the design and implementation of the simulation model of the controller, that is, the operation of the controller in this control system is completed through Simulink programming.

[0024] 2. Co-simulation settings To achieve the co-simulation of MATLAB / Simulink and CarSim, first set the working path of CarSim to the specified Simulink Model. Then, in CarSim, select the preset vehicle model and road information and add them to the Simulink model. By running Simulink, the co-simulation and real-time communication between the two can be achieved. If it is necessary to modify the structure or parameters of the CarSim model, the updated model needs to be sent to Simulink again to ensure the simulation consistency.

[0025] 3. Construction of the four-wheel in-wheel motor drive electric vehicle model in the co-simulation software In the co-simulation environment, the electric vehicle integrated model in CarSim includes main subsystems such as the body, driveline, steering system, braking system, tires, suspension, aerodynamics, and working condition configuration. The vehicle model selected in this study is a four-wheel drive electric vehicle, and its power system consists of four in-wheel motors. The additional motor torque inputs are selected as IMP_MYUSM_L1, IMP_MYUSM_L2, IMP_MYUSM_R1, and IMP_MYUSM_R2. The vehicle working condition information and the steering angle input by the driver are configured in CarSim. The parameters of the electric vehicle are shown in Table 1.

[0026] Table 1 Electric vehicle parameter table Symbol Physical description Value / Unit m Gross vehicle mass 1430 / kg <![CDATA[R e > Wheel radius 0.325 / m <![CDATA[L f > Distance from vehicle center of mass to front axle 1.05 / m <![CDATA[L r > Distance from vehicle center of mass to rear axle 1.61 / m d Track width between left and right wheels 1.55 / m <![CDATA[C f > Cornering stiffness of front tire <![CDATA[90700 / N·m -1 > <![CDATA[C r > Cornering stiffness of rear tire <![CDATA[109000 / N·m -1 > <![CDATA[I z > Moment of inertia of vehicle about z-axis <![CDATA[2059.2 / kg·m -2 > <![CDATA[h cg > Vehicle center of mass height 0.54m

[0027] 4. The Koopman modeling and predictive stability augmentation control principle based on closed-loop data improvement of the present invention The controlled object of the present invention is a four-wheel in-wheel motor drive electric vehicle, and the control objective is to improve the handling stability of the vehicle. The main design process of the control method is described as follows: First, design a two-degree-of-freedom reference model. Based on the two-degree-of-freedom vehicle dynamics model, derive the expected values of the vehicle's yaw rate and lateral velocity to describe the ideal dynamic behavior of the vehicle. Secondly, based on the Burckhardt tire model, establish a non-linear vehicle dynamics model, combine with the simulation software CarSim for parameter identification, determine the key dynamic parameters of the vehicle, and design a non-linear controller to form a closed-loop circuit. During the closed-loop operation, collect the state variables and control input data of the vehicle in real time to capture the non-linear characteristics of the vehicle dynamics. Then, construct a high-dimensional observation function, solve the approximate Koopman operator through the EDMD method, realize the global linearization of the non-linear system, and generate a Koopman model that can accurately describe the global dynamic characteristics of the vehicle. Finally, based on the Koopman model and the reference model, with the goal of ensuring vehicle lateral stability, design a linear MPC controller in combination with lateral safety constraints, optimize and solve to obtain the final control variable of the additional motor torque, greatly improving the real-time performance of vehicle stability control.

[0028] The following introduces the specific steps of the control method of the present invention: A vehicle lateral prediction and stability augmentation control method under low adhesion conditions, comprising the following steps: Step 1: Design a two-degree-of-freedom reference model. Take the ideal linear two-degree-of-freedom steady-state steering model as the reference model, use its transient response as the expected dynamic behavior, and provide the lateral velocity and yaw rate as the ideal reference targets of the vehicle. According to the steering angle input by the driver and combined with the vehicle dynamic parameters, establish the following ideal state equations:

[0029] The steady-state gain G of the sideslip angle at the center of mass β and the steady-state gain G of the yaw rate γ are defined as follows: where m is the vehicle mass, L f is the distance from the front axle to the center of mass of the vehicle, L r is the distance from the rear axle to the center of mass of the vehicle, and the distance L from the front axle to the rear axle is L = L f + L r , C r is the rear wheel cornering stiffness.

[0030] The stability factor K of the vehicle is: where C f is the front wheel cornering stiffness.

[0031] The differential coefficients are respectively defined as: where I zis the moment of inertia of the vehicle about the vertical axis.

[0031] The oscillation frequency ω of the system n and the damping coefficient ξ are defined as follows:

[0032] Step 2: Establish a non-linear closed-loop control system. Build a non-linear two-degree-of-freedom vehicle dynamics model to simulate the real controlled object, and adopt the non-linear Burckhardt tire model to describe the adhesion characteristics between the tire and the road surface. The schematic diagram of the vehicle dynamics model is as Figure 3 shown. Only considering the lateral and yaw motions of the vehicle, the two-degree-of-freedom vehicle dynamics model is obtained: where, F y represents the lateral force of the tire, and the subscripts f and r represent the front and rear wheels respectively, and fl, fr, rl, and rr represent the left front, right front, left rear, and right rear wheels respectively.

[0033] The tire lateral force is given by the Burckhardt tire model. Since only the lateral and yaw motions of the vehicle are concerned, it is assumed that the longitudinal speed is constant and the longitudinal slip ratio is taken as zero. In this case, the combined slip ratio is only determined by the side slip angle α, and the expression of the tire lateral force is as follows: where, k s is the lateral force fitting coefficient, F z is the vertical load of the tire, C 1 , C 2 , C 3 are the fitting coefficients of the Burckhardt tire model. By adjusting the values of C 1 , C 2 , C 3 the road surfaces with different adhesion coefficients are modeled.

[0034] The tire side slip angle α ij (ij = fl, fr, rl, rr) is calculated as follows: where, d is the wheelbase of the vehicle.

[0035] The vertical load F zij of the tire is expressed as follows: where, g is the acceleration due to gravity, h cg is the height of the vehicle's center of mass, and a y is the lateral acceleration.

[0036] According to formulas (6)-(9), the vehicle dynamics model is described in the form of a general non-linear expression: where the state variable x = [x 1 x 2 T = [V y γ] T , which consists of the vehicle's lateral velocity and yaw rate. The control variable u = ΔM z is the additional yaw moment of the tire, and the front wheel steering angle δ f is used as a disturbance input to characterize the change in the vehicle's operating conditions.

[0037] To build a closed-loop control system, a non-linear MPC controller is designed. Model predictive control is an optimization-based control method that can explicitly handle constraints and has multi-input multi-output optimization characteristics. It has been widely applied in the field of vehicle electronic stability control. In this invention, the CasADi tool is used for optimization and solution to ensure the efficiency and real-time performance of the control system.

[0038] Furthermore, the state variables of the system are defined as normalized state variables: where V ym = |V x arctan(0.02μg)| and are the upper limit values of the lateral velocity and yaw rate respectively, used to achieve the normalization of the state variables, where μ is the road surface friction coefficient. The external input variable is defined as d = δ f , and the control variable is defined as where ΔM zm represents the upper limit value of the yaw moment.

[0039] To facilitate the design of the controller, it is necessary to discretize the state space equation of the system. Since model predictive control has a large computational burden, to reduce the computational amount of the controller, this invention uses the first-order Euler formula to discretize the state space equation of the system. The discretized expression form is: x(k + 1) = x(k) + T s f(x(k), δ f (k), u(k)) (12) where T s is the sampling time of the system, and x k serves as the starting point of the prediction model.

[0040] The expression form of the prediction model can be further expressed as: x(k + i|k) = x(k + i - 1|k) + T​s f(x(k+i-1|k), δ f (k+i-1), u(k+i-1)) (13) Among them, k+1|k in the brackets represents predicting the system state at the (k + 1)th moment at the current kth moment, and so on.

[0041] According to the basic principle of model predictive control, the prediction model needs to predict the system state in the future for a period of time at the current moment. Set the prediction horizon of the system to N p , and the control horizon of the system to N c , then the output state sequence X(k) of the system in the future for a period of time and the sequence U(k) of the control input of the system can be expressed as:

[0042] To ensure the handling stability of the vehicle under low adhesion conditions, the main control objective of the MPC controller is to ensure that the yaw rate and lateral velocity can accurately track their reference values, so as to maintain the ideal dynamic behavior of the vehicle. At the same time, to improve the economy and ride comfort of the control system, the controller also needs to constrain the control actions, avoid excessive control inputs, reduce energy consumption, and thus achieve efficient and stable vehicle stability control under complex conditions. The reference value obtained from Step 1 Define the reference value sequence within the prediction horizon as:

[0043] The following optimization problem is obtained: Among them, Q = diag(Γ vy , Γ γ ), which are the weights of the vehicle lateral velocity state and the vehicle yaw rate state respectively, and W = diag(Γ mz ) is the weight of the additional yaw moment of the vehicle control quantity.

[0044] The parameters and weight coefficients used in the simulation experiment are shown in Table 2 Table 2 Simulation Experiment Parameter Table Symbol Definition Value / Unit <![CDATA[ΔM zmax > Upper limit value of additional yaw moment of vehicle 1200 / Nm <![CDATA[T s > Simulation time 0.02 / s <![CDATA[Γ vy > Lateral velocity suppression weight in MPC 1 <![CDATA[Γ γ > Yaw rate tracking weight in MPC 0.8 <![CDATA[Γ mz > Additional yaw moment tracking weight in MPC 0.2

[0045] During the closed-loop operation process, collect the state quantities and control inputs of the vehicle to capture the nonlinear characteristics in vehicle dynamics.

[0046] Step 3: Use the closed-loop data to obtain a high-dimensional linearized Koopman model. For a nonlinear system (10) with input and disturbance, since the evolution of the state is nonlinearly related to the disturbance input and linearly related to the control input, it is only necessary to expand the disturbance into a new state quantity, defined as follows: Among them, χ(k) is the extended state variable, which includes the system state x(k) and the perturbation input δ k (k). The observed value of the newly defined state variable χ(k) is lifted to a high-dimensional space through the nonlinear mapping function Ψ: R n →R N where n is the dimension of the original extended state space, N is the dimension of the lifted high-dimensional state space, and N >> n.

[0047] At this time, the Koopman operator, as an infinite-dimensional linear operator, is used to advance the evolution of the observed value, and its expression is: Ψ(χ(k + 1)) = K d Ψ(χ(k)) (18) where K d is the discrete Koopman operator.

[0048] Since the Koopman operator is essentially infinite-dimensional, it is very difficult to directly solve and apply. Therefore, the present invention approximates it through the extended dynamic mode decomposition (EDMD) method. The EDMD method calculates a finite-dimensional approximate Koopman operator by constructing a finite-dimensional basis function library to approximate the infinite-dimensional operator K d .

[0049] First, p groups of discrete state measurement values are collected in the form of snapshot pairs [χ(j), χ(j + 1)], j = 1, 2,..., p, and all snapshot pairs are integrated into two matrices with a mapping relationship, denoted as X 1 and X 2 , as the state sequences at the current moment and the next moment, used to capture the evolution characteristics of the system state at different time points and provide a data basis for subsequent dynamic modeling and operator estimation. The specific form is: X 1 = [χ(1) χ(2)... χ(p)] n×p X 2 = [χ(2) χ(3)... χ(p + 1)] n×p (19) where the subscript represents the dimension of each matrix.

[0050] A set of basis functions Ψ(χ) is selected to lift X 1 and X 2 . This set of basis functions consists of a group of nonlinear real-valued functions ψ(χ) and is used to map the extended state variables to a high-dimensional space. The definition is as follows:

[0051] The enhanced state data pair X 1lift and X 2lift : X 1lift = [Ψ(χ(1)) Ψ(χ(2))...Ψ(χ(p))] N×p X 2lift = [Ψ(χ(2)) Ψ(χ(3))...Ψ(χ(p + 1))] N×p (21)

[0052] Since the system under study is a controlled system, in order to introduce the control input into the state evolution model, it is necessary to further expand X 1lift and X 2lift to construct a control-enhanced state matrix X 1 and X 2 : where U = [u(1) u(2)...u(p)], and m is the dimension of the control input.

[0053] Finally, the approximate Koopman operator is determined by solving the following minimization problem

[0054] The solution of this problem is completed by the least squares method to ensure obtaining an optimal finite-dimensional linear approximation operator.

[0055] Based on the obtained approximate Koopman operator Based on the high-dimensional Koopman global linear model enhanced by closed-loop data, it is used to describe the dynamic behavior of the system, and finally a Koopman state space model in the following form is formed, hereinafter referred to as the Koopman model: z d (k + 1) = A d z d (k) + B d u(k) x(k) = C d z d (k) (24) where z d (k) = Ψ(χ(k)), which is the enhanced high-dimensional state and represents the state of the system in the high-dimensional feature space.

[0056] The linear coefficient matrices A d and B d can be obtained from : Among them, I represents the identity matrix, and O represents the all-zero matrix.

[0057] To ensure that the high-dimensional state can be correctly mapped back to the original state, the matrix C d is defined as: C d = [I (n-1)×(n-1) O (n-1)×(N-n+1) (n-1)×N (26)

[0058] To more accurately describe and control the behavior of the nonlinear system, the present invention combines the sparse regression method to identify and screen the nonlinear dynamics characteristic function library. By extracting the main dynamic characteristics, redundant information is effectively reduced, and the dimension of the Koopman model is appropriately reduced, thereby significantly improving the efficiency of subsequent control calculations while ensuring the model accuracy.

[0059] Design a candidate nonlinear characteristic function library Θ(X 1 ), select the radial basis function, polynomial function, and trigonometric function as the basis, and combine these functions with the original state variables to form a candidate function set. Similar to the Taylor series and Fourier series, the elements in this library can fit any nonlinear characteristics. At the same time, this library allows for flexible adaptation to various system dynamic characteristics and is an important basis for constructing a global linearization model. Its specific expression is as follows: Among them, θ j (X 1 )(j = 1, 2,..., S, S >> N) represents the nonlinear candidate function term of the nonlinear system (10). Although the function library may contain many candidate features, usually only a few play a dominant role in the dynamic characteristics of the system.

[0060] Therefore, the dynamic dominant characteristic function can be extracted by solving the following sparse regression optimization problem: X 2 = ΞΘ(X 1 ) (28) Among them, the sparse matrix Ξ = [ξ 1 ξ 2 …ξ q T is obtained by using the LASSO algorithm: Among them, ξ k is the sparse row vector, representing the k-th row of Ξ, X 2k represents the k-th row of X 2 , and the parameter λ is the regularization factor, which is used to control the strength of the sparsity constraint.

[0061] The verification of the Koopman model based on closed-loop data improvement obtained by the present invention is as Figure 4 shown, and the improved state space dimension is 10 dimensions.

[0062] Step 4: Design a linear MPC controller. The original non-linear system is mapped to a high-dimensional linear space through the Koopman operator, and a linear MPC controller is designed using the obtained global linear model, so as to expand the stable driving area of the vehicle on low-adhesion roads and achieve efficient and real-time vehicle dynamic control.

[0063] The state variables of the prediction model consist of the lateral velocity V y and the yaw rate γ, the control quantity is the additional yaw moment ΔM z , the disturbance quantity is the front wheel steering angle δ f . To ensure the effective description of the disturbance by the model, the disturbance quantity is taken as part of the extended state. The prediction model is as shown in Equation (24), where the matrices A d , B d , C d are obtained in Step 3. The optimization objective is to achieve accurate tracking of the state variables to the reference values by minimizing the cost function, and at the same time avoid excessive control actions to reduce energy consumption, which is the same as in Step 2.

[0064] After lifting the system to a high-dimensional linear space, the optimization problem can be expressed as a standard quadratic programming problem: where, U k =[u(k|k) u(k + 1|k)…u(k + N c -1|k)] T is the optimization variable at time k, H is the quadratic weight matrix, and f(k + 1|k) is the linear weight vector. The relevant matrix derivations are as follows...

[0065] Expand the prediction model to the prediction time domain N p length, and the following matrix form is obtained: Z = Αz d (k) + ΒU (31) where, Z is the predicted high-dimensional state sequence, Α is the state transition matrix, and Β is the input influence matrix.

[0066] The specific expressions are as follows:

[0067] Map the lifted state sequence Z back to the low-dimensional space, and the output expression matrix is: where, X is the predicted initial state sequence.

[0068] In the objective function, the quadratic optimization structure is used for expansion to obtain the expressions of the quadratic weight matrix H and the linear weight vector f(k + 1|k):

[0069] To satisfy the constraint conditions of the normalized system, a corresponding constraint condition matrix is constructed. Among them, the control input constraint matrix is The state variable constraint only applies to the first state variable V y , so the state variable constraint matrix is Combining the state variable constraint and the control input constraint, the complete constraint expression is:

[0070] After the optimization solution is completed, take the first element of U * (k): u = [1 0...0]U * (k) as the control input and apply it to the control system to achieve precise dynamic control of the vehicle. The parameters and weight coefficients used in the simulation experiment are the same as those in Table 2.

[0071] Figure 5 shows the simulation results of the comparative experiment of the present invention under the double lane change condition with a friction coefficient of 0.35 and a speed of 60 km / h. Among them, Figure 5(a) is the comparison diagram of the lateral velocity suppression effect, and Figure 5(b) is the comparison diagram of the yaw rate tracking effect. The yellow solid line represents the vehicle response without control, the green dashed line represents the control effect of the open-loop Koopman MPC, the orange solid line represents the control effect of the closed-loop Koopman MPC, the blue solid line represents the control effect of the NMPC, and the red dashed line represents the reference value. It can be seen from the figure that the closed-loop Koopman MPC and NMPC control strategies perform excellently in the control of the yaw rate and lateral velocity, and both can effectively achieve the optimization goal, while the open-loop Koopman MPC is slightly worse.

[0072] Figure 6 The comparison of the solution times of the three controllers is shown. The green solid line represents the open-loop Koopman MPC, the orange solid line represents the closed-loop Koopman MPC, and the blue solid line represents the NMPC. The specific values in the figure are listed in Table 3. According to the comparison results, it can be seen that the solution times of the open-loop Koopman MPC and the closed-loop Koopman MPC are significantly shorter than that of the NMPC, indicating the advantage of the Koopman MPC method in terms of computational efficiency and being able to better meet the real-time control requirements.

[0073] Table 3 Comparison of controller solution times Controller Peak (ms) Mean (ms) Open-loop Koopman MPC 12.56 2.914 Closed-loop Koopman MPC 7.248 1.968 NMPC 529.2 30.35

[0074] Based on the above experimental results, the conclusion can be drawn that the closed-loop Koopman MPC strategy proposed by the present invention can significantly improve the handling stability of the vehicle. By constructing a global linear representation of the nonlinear system, the present invention effectively expands the stable driving area of the vehicle under low-friction road surface conditions. At the same time, on the premise of accurately describing the nonlinear characteristics of the system, the present invention shows significant advantages in solving efficiency and is applicable to actual real-time control applications.

Claims

1. A vehicle dynamics Koopman modeling and predictive stabilization control method, characterized in that: The steps are: S1. Design a two-degree-of-freedom reference model: According to the steering angle input by the driver and combined with the vehicle dynamics parameters, the ideal state equation is established: Steady-state gain of center of mass sideslip angle G β and the yaw rate steady-state gain G γ The definition is as follows: Where m is the vehicle mass, L f is the distance from the front axle to the center of mass of the vehicle, L r is the distance from the rear axle to the center of mass of the vehicle, and the distance from the front axle to the rear axle is L = L f +L r , C r is the rear wheel cornering stiffness; The stability factor K of the vehicle is: Among them, C f is the front wheel cornering stiffness; The differential coefficients are defined as: Among them, I z is the moment of inertia of the vehicle about the vertical axis; The oscillation frequency of the system ω n And the damping coefficient ξ is defined as follows: S2. Establish a nonlinear closed-loop control system: Considering only the lateral and yaw motions of the vehicle, the two-degree-of-freedom vehicle dynamics model is obtained: Among them, F y represents the lateral force of the tire, the subscripts f and r represent the front and rear wheels respectively, and fl, fr, rl, rr represent the left front, right front, left rear, and right rear wheels respectively; The expression of tire lateral force determined by the sideslip angle α is as follows: Among them, k s is the lateral force fitting coefficient, F z is the vertical load of the tire, C1, C2, C3 are the fitting coefficients of the Burckhardt tire model; Tire slip angle α ij , where ij = fl, fr, rl, rr: Where d is the left and right wheelbase of the vehicle; Vertical load F of tire zij The expression is as follows: Where g is the acceleration due to gravity, h is the cg is the height of the vehicle's center of mass, a y is the lateral acceleration; according to formulas (6)-(9), the vehicle dynamics model is described as a general nonlinear expression: Among them, the state quantity x=[x1 x2] T =[V y γ] T , control amount u=ΔM z is the additional yaw moment of the tire, the front wheel turning angle δ f As disturbance input; The state variables of the system are defined as normalized state variables: Among them, V ym =|V x arctan(0.02μg)|and are the upper limits of lateral velocity and yaw rate, respectively, μ is the road friction coefficient; The external input is defined as d = δ f , the control quantity is defined as Where ΔM zm Indicates the upper limit of the yaw moment; The state space equation of the system is discretized using the first-order Euler formula: x(k+1)=x(k)+T s f(x(k),δ f (k),u(k)) (12) Among them, T s is the sampling time of the system, x k As a starting point for predictive models; The prediction model is expressed as: x(k+i|k)=x(k+i-1|k)+T s f(x(k+i-1|k),δ f (k+i-1),u(k+i-1)) (13) Among them, k+1|k means predicting the system state at time k+1 at the current time k; Set the prediction time domain of the system to N p , the control time domain of the system is N c , then the system output state sequence X(k) and the system control input sequence U(k) in the future are expressed as: The reference value obtained in step S1 The reference value sequence in the prediction time domain is defined as: The optimization problem is: Where, Q = diag(Γ vy ,Γ γ ), are the weights of the vehicle lateral velocity state and the vehicle yaw rate state, respectively, W = diag (Γ mz ) is the weight of the vehicle control variable plus the yaw moment; S3. Using closed-loop data to obtain a high-dimensional linearized Koopman model Expand the perturbation into a new state quantity Among them, χ(k) is the extended state quantity, including the system state x(k) and the disturbance input δ k (k); Define the observation value of the new state quantity χ(k) through the nonlinear mapping function Ψ:R n →R N Lifted to a high-dimensional space, where n is the dimension of the original expanded state space, N is the dimension of the lifted high-dimensional state space, and N>>n, the Koopman operator is: Ψ(χ(k+1))=K d Ψ(χ(k)) (18) Among them, K d is the discrete Koopman operator; Collect p groups of discrete state measurements in the form of snapshot pairs [χ(j), χ(j+1)], j = 1, 2, ..., p, and integrate all snapshot pairs into two matrices with mapping relationships, denoted as X1 and X2 respectively. X1=[χ(1)χ(2)…χ(p)] n×p X2=[x(2) x(3) … x(p+1)] n×p (19) Here, the subscripts represent the dimensions of each matrix; Select a set of basis functions Ψ(χ) to improve X1 and X2: The improved status data pair X 1lift and X 2lift : X 1lift =[Ψ(χ(1)) Ψ(χ(2)) … Ψ(χ(p))] N×p X 2lift =[Ψ(χ(2)) Ψ(χ(3)) … Ψ(χ(p+1))] N×p (21) X 1lift and X 2lift Further expansion is carried out to construct the state matrices X1 and X2 for control enhancement: Where U = [u(1) u(2) … u(p)], m is the dimension of the control input; Determine the approximate Koopman operator Koopman state-space model: z d (k+1)=A d z d (k)+B d u(k) x(k)=C d z d (k) (24) Among them, z d (k) = Ψ(χ(k)), which is the high-dimensional state after promotion; Linear coefficient matrix A d and B d Depend on get: Among them, I represents the identity matrix, O represents the all-zero matrix; Matrix C d Defined as: C d =[I (n-1)×(n-1) O (n-1)×(N-n+1) ] (n-1)×N (26) Design candidate nonlinear feature function library Θ(X1): Among them, θ j (X1)(j=1,2,...,S,S>>N) represents the nonlinear candidate function term of the nonlinear system; extract the dynamic dominant characteristic function: X2=ΞΘ(X1) (28) Among them, the sparse matrix Ξ=[ξ1 ξ2 … ξ q ] T It is obtained using the LASSO algorithm: Among them, ξ k is a sparse row vector, representing the kth row of Ξ, X 2k represents the kth row of X2, and the parameter λ is the regularization factor; S4. Design of linear MPC controller After the system is promoted to a high-dimensional linear space, the optimization problem is expressed as a standard quadratic programming problem: Among them, U k =[u(k|k) u(k+1|k) … u(k+N c -1|k)] T is the optimization variable at time k, H is the quadratic weight matrix, and f(k+1|k) is the linear weight vector; The correlation matrix is ​​derived as follows: Expand the prediction model to the prediction time domain N p length, and obtain the following matrix form: Z=Αz d (k)+ΒU (31) Among them, Z is the predicted high-dimensional state sequence, Α is the state transfer matrix, and Β is the input influence matrix; The expressions of Z, Α, and Β are as follows: Mapping the boosted state sequence Z back to the low-dimensional space, the output expression matrix is: in, X is the predicted initial state sequence; Expressions for the quadratic weight matrix H and the linear weight vector f(k+1|k): Combining the state variable constraints and control input constraints, the complete constraint expression is: Take U * The first element of (k): u = [1 0 … 0] U * (k) used as control input in control systems; Corresponding constraint matrix: The control input constraint matrix is The state variable constraint is only for the first state variable V y , so the state variable constraint matrix is

Images