Active rear wheel steering adaptive control optimization method based on MRAC-LQR

Abstract

The invention relates to the technical field of active rear wheel steering, in particular to an MRAC-LQR-based active rear wheel steering adaptive control optimization method, which comprises the following steps: S1, modeling to obtain a two-degree-of-freedom four-wheel steering vehicle state space model; s2, constructing an ideal reference model based on the steady-state two-degree-of-freedom front wheel steering vehicle model; s3, designing an LQR optimal controller to obtain an LQR optimal control law; s4, performing MRAC adaptive compensation design based on the LQR optimal control law to obtain an MRAC adaptive compensation item; s5, combined control law synthesis is carried out based on the LQR optimal control law and the MRAC adaptive compensation item, and a synthesized rear wheel steering angle control law is obtained; and S6, calculating a corresponding rear wheel steering angle control quantity based on the rear wheel steering angle control law in combination with the real-time state parameters of the target vehicle, and realizing active rear wheel steering control of the target vehicle through the rear wheel steering angle control quantity. According to the invention, the accuracy, robustness and adaptability of active rear wheel steering adaptive control can be improved.

Description

Optimization Method for Active Rear-Wheel Steering Adaptive Control Based on MRAC-LQR Technical Field

[0001] This invention relates to the field of active rear-wheel steering technology, and more specifically to an active rear-wheel steering adaptive control optimization method based on MRAC-LQR. Background Technology

[0002] Vehicle handling stability is a core performance indicator for ensuring high-speed vehicle safety, and it is related to the vehicle's ride quality and driving reliability. Front-wheel steering vehicles, when facing emergency obstacle avoidance and changes in road surface adhesion coefficient at high speeds, struggle to coordinate yaw response and center of gravity sideslip characteristics, easily leading to decreased handling stability, response lag, and insufficient disturbance rejection capabilities. Therefore, the introduction of active rear-wheel steering technology increases the degree of freedom in adjusting the vehicle's lateral dynamics, improving handling stability and driving safety.

[0003] However, most existing active rear-wheel steering control methods are based on fixed parameters or ideal models. When vehicle parameters, tire lateral slip characteristics, and road adhesion conditions change, the control performance is easily affected, making it difficult to simultaneously meet the adaptability requirements under multiple operating conditions. Therefore, conducting research on adaptive active rear-wheel steering control is of great significance. It can achieve online adjustment and dynamic compensation for time-varying parameters and model uncertainties during vehicle operation, thereby further improving the consistency of vehicle lateral response, stability, and handling safety.

[0004] Currently, research on active rear-wheel steering control mainly focuses on model predictive control, sliding mode control, optimal control, and multi-actuator cooperative control. Zhang H et al. proposed an active rear-wheel steering control method considering driver steering characteristics and designed a model predictive controller based on a driver-vehicle joint model, achieving comprehensive optimization of driver workload and vehicle stability. Deng Z et al. proposed a comprehensive stability control method combining active aerodynamics and active rear-wheel steering, improving the handling stability and path-following ability of high-speed vehicles. Wu Y et al. proposed a distributed rear-wheel steering control algorithm and extended it to nonlinear conditions by introducing sliding mode control, verifying its effectiveness under conditions of adhesion changes and external disturbances. Ahangarnejad AH et al. proposed a coordinated control strategy for active rear-wheel steering and multi-chassis actuators, achieving a comprehensive improvement in vehicle stability and comfort. Kanchwala H et al. proposed a comprehensive control method integrating torque distribution and active rear-wheel steering, improving vehicle path-following ability and lateral stability. Sahin H et al. studied the active rear-wheel steering control of a tractor under high-speed obstacle avoidance conditions, and the results showed that this method can effectively suppress the folding phenomenon of articulated vehicles. Yim S et al. proposed a comparative analysis framework for active four-wheel steering and independent four-wheel steering, revealing the performance differences of different steering systems in vehicle stability control. Perozzi G et al. used high-order sliding mode control to design a lane keeping system for four-wheel steering vehicles, verifying its effectiveness in lateral deviation control. Alves JAV et al. proposed a method for constructing the vehicle's lateral stability region, providing a constraint basis for stability control. Zhang C et al. proposed a sliding mode predictive control method based on model predictive optimization, improving the control chattering problem of active rear-wheel steering systems. Ariff MHM et al. proposed a feedforward-feedback four-wheel active steering strategy based on optimal control, balancing low-speed maneuverability and high-speed handling stability. Park K et al. constructed a rear-wheel steering control framework that considers both steady-state and transient responses, reducing the dependence of active rear-wheel steering on tire parameters. Bredthauer et al. studied the impact of active rear-wheel steering on the transient lateral dynamics of vehicles, showing that it can significantly improve the transient lateral response characteristics of vehicles.

[0005] The applicant found that while existing active rear-wheel steering control methods have achieved certain results in improving vehicle yaw response, suppressing center of gravity sideslip angle, and enhancing path tracking performance, most of them are still based on nominal models or fixed parameter designs, which are highly dependent on model accuracy and working condition matching. Under working conditions such as high-speed steering and changes in road surface adhesion, control performance is prone to decline, and therefore their accuracy, robustness, and adaptability still need to be improved. Summary of the Invention

[0006] To address the shortcomings of the existing technologies, the technical problem this invention aims to solve is: how to provide an active rear-wheel steering adaptive control optimization method based on MRAC-LQR, combining model reference adaptive control with LQR control, introducing model reference adaptive compensation on the basis of linear quadratic optimal control, designing an adaptive law using Lyapunov stability theory, and adaptively compensating the rear wheel steering angle based on the yaw rate error and the center of gravity sideslip angle error as controller inputs, thereby improving the accuracy, robustness, and adaptability of the active rear-wheel steering adaptive control.

[0007] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0008] The active rear-wheel steering adaptive control optimization method based on MRAC-LQR includes:

[0009] S1: Based on the structural state parameters of the target vehicle, vehicle dynamics modeling is performed to obtain a two-degree-of-freedom four-wheel steering vehicle state space model;

[0010] S2: Based on the steady-state two-degree-of-freedom front-wheel steering vehicle model, a reference model is constructed to obtain an ideal reference model;

[0011] S3: Based on the state-space model of a two-degree-of-freedom four-wheel steering vehicle and an ideal reference model, an optimal LQR controller is designed, and the optimal LQR control law is obtained.

[0012] S4: Based on the LQR optimal control law, MRAC adaptive compensation design is performed to obtain the MRAC adaptive compensation term;

[0013] S5: Based on the LQR optimal control law and the MRAC adaptive compensation term, a joint control law is synthesized to obtain the synthesized rear wheel steering angle control law;

[0014] S6: Based on the rear wheel steering angle control law and the real-time state parameters of the target vehicle, calculate the corresponding rear wheel steering angle control quantity, and realize the active rear wheel steering control of the target vehicle through the rear wheel steering angle control quantity.

[0015] Preferably, in step S1, the structural state parameters of the target vehicle are obtained by constructing a Carsim vehicle model; wherein the structural state parameters of the target vehicle include the vehicle wheelbase, the distance from the center of mass to the front axle, the distance from the center of mass to the rear axle, the vehicle mass, the steering wheel to front wheel transmission ratio, the front axle lateral stiffness, the rear axle lateral stiffness and / or the moment of inertia about the Z-axis.

[0016] Preferably, in step S1, the processing steps for constructing the state space model of a two-degree-of-freedom four-wheel steering vehicle include:

[0017] S101: Establish the vehicle coordinate system: with the vehicle's center of mass as the origin. The axle is along the direction of vehicle travel. The axis points to the driver's left. The axis is vertically upward;

[0018] S102: Calculate the front and rear wheel slip angles based on the vehicle coordinate system:

[0019] ;

[0020] In the formula: , These represent the slip angles of the front and rear wheels, respectively. Indicates the vehicle's sideslip angle; Indicates the vehicle's yaw rate; , This represents the distance from the center of mass to the front and rear axles; This indicates the vehicle's forward speed, i.e., its longitudinal speed; , These represent the turning angles of the front and rear wheels, respectively, with counterclockwise as the positive direction.

[0021] S103: Establish vehicle protection along Resultant force on the axis And around Resultant torque of the shaft The equilibrium equations are:

[0022] ;

[0023] in:

[0024] ;

[0025] ;

[0026] In the formula: Indicates the total mass of the vehicle; Indicates vehicles go around Moment of inertia of the shaft; express On-axis acceleration; , These represent the lateral forces on the front and rear wheels of the vehicle, respectively. For front axle lateral stiffness; Rear axle lateral stiffness; Indicates the lateral speed of the vehicle;

[0027] S104: Constructing the differential equations for a two-degree-of-freedom, four-wheel-steering vehicle:

[0028] ;

[0029] S105: Constructing a state-space model of a two-degree-of-freedom four-wheel steering vehicle based on its differential equations:

[0030] ;

[0031] The two-degree-of-freedom four-wheel steering state-space model is transformed into the following form:

[0032] ;

[0033] in:

[0034] ;

[0035] ;

[0036] ;

[0037] ;

[0038] In the formula: This is the system state matrix; The state matrix of a two-degree-of-freedom four-wheel steering model; , These are the input matrices for the front and rear wheel rotation angles, respectively.

[0039] Preferably, in step S2, the formula for the steady-state two-degree-of-freedom front-wheel steering vehicle model is expressed as:

[0040] .

[0041] Preferably, in step S2, the formula for the ideal state model is expressed as:

[0042] ;

[0043] in:

[0044] ;

[0045] ;

[0046] In the formula: Indicates the ideal centroid sideslip angle; This represents the ideal yaw rate; The ideal transfer function representing the front wheel steering angle to the center of gravity sideslip angle; This represents the steady-state yaw rate gain of a vehicle with front-wheel steering. Indicates the stability factor; This indicates the vehicle's wheelbase.

[0047] Preferably, in step S3, the processing steps for constructing the LQR optimal control law include:

[0048] S301: Constructing a quadratic performance index function based on a four-wheel steering vehicle state-space model and an ideal reference model. :

[0049] ;

[0050] ;

[0051] ;

[0052] In the formula: , It is a weighted diagonal matrix. This indicates the degree of importance attached to the centroid sideslip angle and yaw rate of the state vector. This indicates the degree of importance attached to the control variable, the rear wheel steering angle; This is the weight for the centroid sideslip angle error; This is the weighting for the yaw rate error; The weights for the control of the rear wheel steering angle; , These represent the start and end times of the performance index integration interval, respectively.

[0053] S302: Constructing the Hamlittion function based on a quadratic performance index function and a state-space model of a four-wheel steering vehicle. :

[0054] ;

[0055] In the formula: Indicates the accompanying variable;

[0056] S303: Based on the Hamlittion function Calculate the control law using the control equations:

[0057] The governing equations are: ;

[0058] The control law is: ;

[0059] The regular equation is: ;

[0060] The adjoint equation is: ;

[0061] in Differentiating, we get:

[0062] In the formula: , All are intermediate calculation variables;

[0063] S304: Considering moments , , , For a constant matrix, when , , Substitute this condition into the control law to solve for the state variable. From the equation, we obtain the Riccati algebraic equation:

[0064] ;

[0065] The results were:

[0066] ;

[0067] S305: Obtained from the above formula , Substituting into the Riccati algebraic equation, we obtain the LQR optimal control law:

[0068] .

[0069] Preferably, the LQR front wheel steering angle feedforward matrix defining the LQR optimal control law is:

[0070] ;

[0071] The LQR state variable feedback matrix of the LQR optimal control law is defined as follows:

[0072] ;

[0073] The optimal control law is expressed as:

[0074] .

[0075] Preferably, in step S4, the processing steps for constructing the MRAC adaptive compensation term include:

[0076] S401: Equivalently representing the error as an uncertain term in the input, we obtain Formula 1:

[0077] ;

[0078] In the formula: Indicates an equivalent unknown input;

[0079] S402: Using a linear parameterized approximation, let , For an unknown constant parameter vector, the MRAC adaptive compensation term is defined as:

[0080] ;

[0081] In the formula: For online parameter estimation, This is the system state matrix;

[0082] Integrating Formula 1 and the MRAC adaptive compensation term, we obtain Formula 2:

[0083] ;

[0084] S403: Define the reference model:

[0085] ;

[0086] in:

[0087] ;

[0088] ;

[0089] In the formula: For reference state parameters; Here, is the Hurwitz matrix, which is the state matrix of the reference model; Input matrix for reference model; The state matrix of a two-degree-of-freedom four-wheel steering model; , These are the input matrices for the front and rear wheel rotation angles, respectively. This is the LQR state variable feedback matrix; This is the LQR front wheel steering angle feedforward matrix;

[0090] S404: Define tracking error:

[0091] ;

[0092] Combining Formula 2, we obtain the error dynamics formula:

[0093] ;

[0094] In the formula: For parameter estimation error;

[0095] S405: Take any symmetric positive definite matrix ,make If the solution is the only solution to the Lyapunov equation, then we have Formula 3:

[0096] ;

[0097] For any There exists a unique symmetric positive definite solution. ;

[0098] Constructing Lyapunov functions :

[0099] ;

[0100] In the formula: This is the adaptive gain matrix;

[0101] Taking the derivative of the Lyapunov function, we obtain Formula 4:

[0102] ;

[0103] S406: Substituting the error dynamics formula into Formula 4, we obtain Formula 5:

[0104] ;

[0105] Combining Formula 3 and Formula 5, we obtain Formula 6:

[0106] ;

[0107] S407: Based on Formula 6, consider Let be a constant, let To cancel out the cross terms, the adaptive law is chosen as follows:

[0108] ;

[0109] S408: Through adaptive law Integrating yields online parameter estimates Generate the final MRAC adaptive compensation term:

[0110] .

[0111] Preferably, in step S5, the formula for the rear wheel steering angle control law is expressed as:

[0112] .

[0113] Preferably, in step S6, the real-time state parameters of the target vehicle include the vehicle's center of gravity sideslip angle and the vehicle's yaw rate.

[0114] Compared with existing technologies, the active rear-wheel steering adaptive control optimization method based on MRAC-LQR in this invention has the following advantages:

[0115] The benefits of obtaining a state-space model for a four-wheel steering vehicle by modeling vehicle dynamics based on the structural state parameters of the target vehicle:

[0116] This invention relates to a four-wheel steering vehicle that uses the vehicle's center of gravity as the origin of the coordinate system. It comprehensively considers key structural parameters such as wheelbase, center of gravity position, vehicle mass, and front and rear axle lateral stiffness, accurately describing the coupling relationship between the vehicle's lateral velocity and yaw motion. Furthermore, by introducing the rear wheel steering angle as an independent control input, it overcomes the limitations of the traditional two-degree-of-freedom front-wheel steering model, more comprehensively reflecting the impact of rear-wheel steering on the vehicle's handling stability. This invention can accurately describe the vehicle's actual dynamic characteristics under normal speed and small steering angle conditions, ensuring the theoretical reliability of the control strategy design and thus improving the accuracy of active rear-wheel steering adaptive control.

[0117] The ideal reference model of this invention comprehensively considers the dual control requirements of suppressing the center of gravity sideslip angle and tracking the ideal yaw rate. By using parameters such as the steady-state yaw rate gain and stability factor of a front-wheel steering vehicle, it defines the ideal response characteristics that the vehicle should achieve during high-speed cornering. Simultaneously, the ideal reference model uses the steady-state center of gravity sideslip angle as one of the control objectives, which helps to reduce the steady-state center of gravity sideslip angle under high-speed cornering conditions, while effectively tracking the ideal yaw rate, thereby improving lateral robustness and reducing the risk of instability. Furthermore, the ideal reference model provides a benchmark for subsequent model-referenced adaptive control, enabling the system to evaluate the deviation between the actual and desired responses in real time, enhancing the adaptability of the control strategy to complex operating conditions.

[0118] This invention designs an optimal LQR controller based on a four-wheel steering vehicle state-space model and an ideal reference model. By constructing a quadratic performance index function, using the center of gravity sideslip angle and yaw rate as state variables, and the rear wheel steering angle as the control input, it achieves comprehensive optimization of multi-state control objectives. Through reasonable tuning of the weighting matrix, the emphasis on the center of gravity sideslip angle error, yaw rate error, and rear wheel steering angle control variables can be flexibly adjusted, achieving a balance between control performance and system stability. Simultaneously, the LQR controller obtains the optimal feedback gain matrix by solving the Riccati algebraic equation, enabling optimal state feedback under given weights, achieving center of gravity sideslip angle suppression and yaw response improvement within a certain range. Furthermore, the control law includes a front wheel steering angle feedforward matrix and a state variable feedback matrix, simultaneously considering feedforward compensation for the front wheel input and feedback adjustment of the vehicle state, improving control response speed and accuracy.

[0119] This invention introduces model reference adaptive compensation based on LQR optimal control. By adjusting the compensation amount online, it corrects model parameter deviations and effectively addresses the influence of uncertainties such as changes in vehicle tire side stiffness, altered road adhesion conditions, and unmodeled dynamics. Specifically, it employs Lyapunov stability theory to design an adaptive law, incorporating yaw rate and centroid sideslip angle errors into the controller's adaptive compensation of the rear wheel steering angle. This ensures the stability of the closed-loop system and keeps the tracking error bounded. Compared to robust designs that rely on precise model parameters, the joint control strategy of this invention is insensitive to key parameters that are difficult to obtain accurately, and better reflects the actual situation of vehicle characteristic fluctuations under different speeds and adhesion conditions. Simulation verification under different steering conditions and road adhesion conditions shows that the control strategy of this invention can adaptively adjust yaw rate and centroid sideslip angle errors, more quickly approaching the reference model after the steering transient, reducing yaw response deviation and decreasing the centroid sideslip angle. This makes the vehicle's state more stably follow the ideal reference model, providing an efficient and reliable control strategy for active rear-wheel steering control of vehicles. Attached Figure Description

[0120] To make the objectives, technical solutions, and advantages of the invention clearer, the invention will now be described in further detail with reference to the accompanying drawings, wherein:

[0121] Figure 1 is a logic block diagram of the active rear-wheel steering adaptive control optimization method based on MRAC-LQR.

[0122] Figure 2 shows a two-degree-of-freedom four-wheel steering vehicle model.

[0123] Figure 3 is a schematic diagram of the MRAC-LQR co-simulation.

[0124] Figure 4 shows the simulation results of the double lane change condition: (a) yaw rate curve; (b) center of gravity sideslip angle curve; (c) rear wheel steering angle curve; (d) center of gravity sideslip angle phase trajectory curve; (e) yaw rate error curve; (f) center of gravity sideslip angle error curve.

[0125] Figure 5 shows the simulation results of the angular step condition: (a) Steering wheel angle curve; (b) Lateral acceleration curve; (c) Yaw rate curve; (d) Center of gravity sideslip angle curve; (e) Rear wheel angle curve; (f) Yaw rate error curve; (g) Center of gravity sideslip angle error curve.

[0126] Figure 6 shows the simulation results of the double lane change condition: (a) yaw rate curve; (b) center of gravity sideslip angle curve; (c) rear wheel steering angle curve; (d) center of gravity sideslip angle phase trajectory curve; (e) yaw rate error curve; (f) center of gravity sideslip angle error curve.

[0127] Figure 7 shows the simulation results of the angular step condition: (a) Steering wheel angle curve; (b) Lateral acceleration curve; (c) Yaw rate curve; (d) Center of gravity sideslip angle curve; (e) Rear wheel angle curve; (f) Yaw rate error curve; (g) Center of gravity sideslip angle error curve. Detailed Implementation

[0128] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but only to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0129] The following detailed explanation illustrates the specific implementation methods:

[0130] Example:

[0131] This embodiment discloses an active rear-wheel steering adaptive control optimization method based on Model Reference Adaptive (MRAC)-Linear Quadratic Optimal Control (LQR).

[0132] As shown in Figure 1, the active rear-wheel steering adaptive control optimization method based on Model Reference Adaptive (MRAC)-Linear Quadratic Optimal Control (LQR) includes:

[0133] S1: Based on the structural state parameters of the target vehicle, vehicle dynamics modeling is performed to obtain a two-degree-of-freedom four-wheel steering vehicle state space model;

[0134] S2: Based on the steady-state two-degree-of-freedom front-wheel steering vehicle model, a reference model is constructed to obtain an ideal reference model;

[0135] S3: Based on the state-space model of a two-degree-of-freedom four-wheel steering vehicle and an ideal reference model, an optimal LQR controller is designed, and the optimal LQR control law is obtained.

[0136] S4: Based on the LQR optimal control law, MRAC adaptive compensation design is performed to obtain the MRAC adaptive compensation term;

[0137] S5: Based on the LQR optimal control law and the MRAC adaptive compensation term, a joint control law is synthesized to obtain the synthesized rear wheel steering angle control law;

[0138] S6: Based on the rear wheel steering angle control law and the real-time state parameters of the target vehicle, calculate the corresponding rear wheel steering angle control quantity, and realize the active rear wheel steering control of the target vehicle through the rear wheel steering angle control quantity.

[0139] In this embodiment, the real-time state parameters of the target vehicle include the vehicle's center of gravity sideslip angle and the vehicle's yaw rate.

[0140] This invention proposes a model reference adaptive optimal control. Based on linear quadratic optimal control, it introduces model reference adaptive compensation and designs an adaptive law using Lyapunov stability theory. The rear wheel steering angle is adaptively compensated based on the yaw rate error and the center of gravity sideslip angle error. Simulations were conducted under different steering conditions and road adhesion conditions. The proposed model reference adaptive optimal control can adaptively adjust the yaw rate and center of gravity sideslip angle errors, achieving the best control effect and providing an effective control strategy for active rear wheel steering control of vehicles.

[0141] To better illustrate the technical solution of the present invention, this embodiment will be described in more detail through the following parts.

[0142] I. Carsim vehicle models

[0143] In the specific implementation process, a Carsim vehicle model of the target vehicle is constructed to obtain its structural state parameters. The main structural state parameters of the target vehicle include wheelbase, distance from the center of gravity to the front axle, distance from the center of gravity to the rear axle, vehicle mass, steering wheel-to-front wheel transmission ratio, front axle lateral stiffness, rear axle lateral stiffness, and moment of inertia about the Z-axis. The Z-axis moment of inertia is set to 1536.7 kg·m. 2 .

[0144] Carsim is a professional simulation tool for vehicle dynamics research. This software employs parametric modeling, allowing users to set vehicle body parameters such as geometry, mass distribution, and moment of inertia. It also supports configuration of parameters for key subsystems such as steering, suspension, braking, and tires. Furthermore, it provides a co-simulation interface with the Matlab / Simulink platform, facilitating the coupling between control algorithms and vehicle dynamics models.

[0145] Obtaining accurate all model parameters for a vehicle is difficult, time-consuming, and labor-intensive. This invention selects a model in Carsim that best matches the actual vehicle of a partner company. Some parameters in the model are modified according to the actual vehicle parameters provided by the company, while the remaining system parameters retain the original Carsim parameter settings to ensure model integrity. The main vehicle parameters are shown in Table 1.

[0146] Table 1 Main Vehicle Parameters

[0147]

[0148] II. State-space model of a two-degree-of-freedom four-wheel steering vehicle

[0149] In the development of vehicle handling stability control strategies, linear two-degree-of-freedom models are widely used. Extensive theoretical analysis and experimental results show that, under normal vehicle speeds and small steering angles, this model can accurately describe the actual dynamic characteristics of the vehicle and reflect the fundamental coupling relationship between lateral velocity and yaw motion, providing an effective modeling basis for analyzing the influence of steering input on the vehicle's lateral response. This invention studies rear-wheel steering control strategies, requiring attention to the impact of rear-wheel steering angle changes on overall vehicle handling stability. Therefore, it is necessary to add rear-wheel steering angle to the traditional two-degree-of-freedom front-wheel steering model, establishing a two-degree-of-freedom four-wheel steering vehicle model. The two-degree-of-freedom four-wheel steering model is shown in Figure 2.

[0150] Specifically, the steps for constructing a state-space model of a four-wheel steering vehicle include:

[0151] S101: Establish the vehicle coordinate system: with the vehicle's center of mass as the origin. The axle is along the direction of vehicle travel. The axis points to the driver's left. The axis is vertically upward;

[0152] To accurately describe the vehicle's motion state and force relationships, and to provide a unified reference for the subsequent establishment of dynamic models, a vehicle coordinate system is established with the vehicle's center of mass as the origin.

[0153] S102: Calculate the front and rear wheel slip angles based on the vehicle coordinate system:

[0154] (1)

[0155] In the formula: , These represent the slip angles of the front and rear wheels, respectively. Indicates the vehicle's sideslip angle; Indicates the vehicle's yaw rate; , This represents the distance from the center of mass to the front and rear axles; This indicates the vehicle's forward speed, i.e., its longitudinal speed; , These represent the turning angles of the front and rear wheels, respectively, with counterclockwise as the positive direction.

[0156] S103: Establish vehicle protection along Resultant force on the axis And around Resultant torque of the shaft The equilibrium equations are:

[0157] (2)

[0158] To solve equation (2), the relationship between tire lateral force and slip angle needs to be given. This invention uses a linear slip stiffness model to describe the tire lateral force:

[0159] (3)

[0160] The vehicle is On-axis center of mass acceleration It can be determined by longitudinal velocity and lateral velocity get:

[0161] (4)

[0162] In the formula: Indicates the total mass of the vehicle; Indicates vehicles go around Moment of inertia of the shaft; express On-axis acceleration; , These represent the lateral forces on the front and rear wheels of the vehicle, respectively. For front axle lateral stiffness; Rear axle lateral stiffness; Indicates the lateral speed of the vehicle;

[0163] S104: Combining the above formulas, construct the differential equations for a two-degree-of-freedom four-wheel steering vehicle:

[0164] (5)

[0165] S105: Constructing a state-space model of a two-degree-of-freedom four-wheel steering vehicle based on its differential equations:

[0166] (6)

[0167] Since the matrix form is not conducive to the subsequent description, the matrix part of the two-degree-of-freedom four-wheel steering state-space model is replaced with symbols, and the two-degree-of-freedom four-wheel steering state-space model is transformed into the following form:

[0168] ;

[0169] in:

[0170] ;

[0171] ;

[0172] ;

[0173] ;

[0174] In the formula: This is the system state matrix; The state matrix of a two-degree-of-freedom four-wheel steering model; , These are the input matrices for the front and rear wheel rotation angles, respectively.

[0175] III. Ideal Reference Model

[0176] To achieve rear-wheel steering control, scholars have proposed proportional feedforward control, proportional feedback control, and proportional integrated control. Among them, proportional integrated control, which introduces proportional yaw rate compensation on the basis of proportional feedforward control, has the best control effect.

[0177] Proportional integrated control can be expressed as:

[0178] (7)

[0179] In the formula, This is the feedforward scaling factor. This is the feedback ratio coefficient.

[0180] Proportional integrated control uses the steady-state centroid sideslip angle as the control target, and sets... Combining equation (7) and equation (5) and rearranging, we can obtain:

[0181] (8)

[0182] From equation (8), we can obtain:

[0183] (9)

[0184] (10)

[0185] Considering that during high-speed vehicle cornering, it is generally desirable to reduce the steady-state center of gravity sideslip angle while effectively tracking the ideal yaw rate to improve lateral stability and reduce the risk of instability, further research is needed on control methods that can integrate multi-state control objectives.

[0186] To simultaneously meet the control requirements of suppressing sideslip angle and tracking ideal yaw rate, an active rear-wheel steering controller is designed using the Linear Quadratic Regulation (LQR) method. This method uses sideslip angle and yaw rate as state variables and rear wheel angle as control input, constructing a quadratic performance index function. Subsequently, a state error model is established, the Riccati equation is solved, and the optimal control law for the rear wheel angle and the feedback gain matrix are obtained. The tuning principle of the weighted matrix is ​​also given.

[0187] Specifically, the steps for constructing an ideal reference model include:

[0188] S201: Establish a steady-state two-degree-of-freedom front-wheel steering vehicle model, and determine the yaw rate in steady state. If it is a constant, then we have:

[0189] ;

[0190] Combine the two equations and eliminate the lateral velocity. The steady-state yaw rate gain can then be obtained. ;

[0191] S202: Establishing an ideal state model:

[0192] ;

[0193] In an ideal state, the sideslip angle of the center of gravity is 0, and the yaw rate is the steady-state yaw rate of a two-degree-of-freedom front-wheel steering vehicle.

[0194] in:

[0195] ;

[0196] ;

[0197] In the formula: Indicates the ideal centroid sideslip angle; This represents the ideal yaw rate; The ideal transfer function representing the front wheel steering angle to the center of gravity sideslip angle; This represents the steady-state yaw rate gain of a vehicle with front-wheel steering. Indicates the stability factor; This indicates the vehicle's wheelbase.

[0198] IV. LQR Optimal Control Law

[0199] In specific implementation, the steps for constructing the LQR optimal control law include:

[0200] S301: In order to find the optimal control law Performance indicators should be To obtain the minimum value, considering that the terminal error has little impact on the entire active rear-wheel steering system, the first term in the performance index is disregarded. A quadratic performance index function is constructed based on the four-wheel steering vehicle state-space model and the ideal reference model. :

[0201] (17)

[0202] , (18)

[0203] In the formula: , It is a weighted diagonal matrix. This indicates the relative importance of the centroid sideslip angle and yaw rate of the state vector; the higher their proportion, the better the control effect. This indicates the degree of importance attached to the control variable, the rear wheel steering angle. Increasing its proportion can reduce system oscillations and improve system stability, but weaken the control effect on the control target. This is the weight for the centroid sideslip angle error; This is the weighting for the yaw rate error; The weights for the control of the rear wheel steering angle; , These represent the start and end times of the performance index integration interval, respectively.

[0204] S302: Constructing the Hamlittion function based on a quadratic performance index function and a state-space model of a four-wheel steering vehicle. Substituting the input variables, the front and rear wheel angles, the Hamiltonation function is constructed as follows:

[0205] (19)

[0206] In the formula: Indicates the accompanying variable;

[0207] S303: Based on the Hamlittion function Calculate the control law using the control equations:

[0208] The governing equations are: (20)

[0209] The control law is: (twenty one)

[0210] The regular equation is: (twenty two)

[0211] The adjoint equation is: (twenty three)

[0212] in Differentiating, we get:

[0213] (twenty four)

[0214] In the formula: , All are intermediate calculation variables;

[0215] S304: Considering moments , , , For a constant matrix, when , , Substitute this condition into the control law to solve for the state variable. From the equation, we obtain the Riccati algebraic equation:

[0216] (25)

[0217] The results were:

[0218] (26)

[0219] S305: Obtained from the above formula , Substituting into the Riccati algebraic equation, we obtain the LQR optimal control law:

[0220] (27)

[0221] Specifically, the LQR front wheel steering angle feedforward matrix for defining the LQR optimal control law is:

[0222] (28)

[0223] The LQR state variable feedback matrix of the LQR optimal control law is defined as follows:

[0224] (29)

[0225] The optimal control law is expressed as:

[0226] .

[0227] V. MRAC Adaptive Compensation Term

[0228] The LQR controller, based on a linear model, obtains optimal state feedback under given weights, achieving centroid sideslip angle suppression and yaw response improvement within a certain range. This invention, building upon the LQR active rear-wheel steering control law, further considers the time-varying parameter issues encountered in high-speed conditions. Changes in vehicle tire sideslip stiffness and road surface adhesion can lead to discrepancies between the theoretical model and the actual system, making it difficult for the fixed-gain state feedback to maintain consistent dynamic performance under different conditions. To enhance the control strategy's adaptability to these uncertainties, this invention introduces Model Reference Adaptive Control (MRAC) compensation into the LQR framework. By adjusting the compensation amount online, it corrects the deviation of the model state parameters, enabling the vehicle to maintain the desired lateral response characteristics during high-speed steering. A schematic diagram of the MRAC-LQR co-simulation is shown in Figure 3.

[0229] The selection of model reference adaptive control is primarily based on its ability to target the desired dynamics of the reference model. By adjusting parameters online, the closed-loop response continuously converges towards the reference target, thereby mitigating the impact of time-varying parameters on control quality. Compared to robust design that relies on precise model parameters, this method is less sensitive to key parameters such as tire lateral stiffness, which are difficult to obtain accurately, and is more suitable for the actual situation where vehicle characteristics fluctuate significantly under different speeds and adhesion conditions.

[0230] The MRAC compensated LQR control structure can be expressed as:

[0231] (30)

[0232] In the formula, This is the MRAC adaptive compensation term.

[0233] Specifically, the processing steps for constructing the MRAC adaptive compensation term include:

[0234] S401: Considering the influence of factors such as changes in vehicle tire lateral stiffness, unmodeled dynamics, and external disturbances on rear wheel steering, the error is equivalent to an input uncertainty term, resulting in Formula 1:

[0235] (31)

[0236] In the formula: Indicates an equivalent unknown input;

[0237] S402: To obtain an implementable adaptive law, a linear parameterized approximation is used, letting... , For an unknown constant parameter vector, the MRAC adaptive compensation term is defined as:

[0238] (32)

[0239] In the formula: For online parameter estimation, This is the system state matrix;

[0240] S403: To ensure the reference model is consistent with the dynamics of the LQR control closed loop, define the reference model:

[0241] (33)

[0242] in:

[0243] , (34)

[0244] In the formula: For reference state parameters; Here, is the Hurwitz matrix, which is the state matrix of the reference model; Input matrix for reference model; The state matrix of a two-degree-of-freedom four-wheel steering model; , These are the input matrices for the front and rear wheel rotation angles, respectively. This is the LQR state variable feedback matrix; This is the LQR front wheel steering angle feedforward matrix;

[0245] S404: Define tracking error:

[0246] (35)

[0247] Integrating Formula 1 and the MRAC adaptive compensation term, we obtain Formula 2:

[0248] (36)

[0249] Combining Formula 2, we obtain the error dynamics formula:

[0250] (37)

[0251] In the formula: For parameter estimation error;

[0252] S405: Take any symmetric positive definite matrix ,make If the solution is the only solution to the Lyapunov equation, then we have Formula 3:

[0253] (38)

[0254] For any There exists a unique symmetric positive definite solution. ;

[0255] Constructing Lyapunov functions :

[0256] (39)

[0257] In the formula: This is the adaptive gain matrix;

[0258] Taking the derivative of the Lyapunov function, we obtain Formula 4:

[0259] (40)

[0260] S406: Substituting the error dynamics formula into Formula 4, we obtain Formula 5:

[0261] (41)

[0262] Combining Formula 3 and Formula 5, we obtain Formula 6:

[0263] (42)

[0264] S407: Based on Formula 6, consider Let be a constant, let To cancel out the cross terms, the adaptive law is chosen as follows:

[0265] (43)

[0266] Substituting equation (43) into equation (42), we get:

[0267] (44)

[0268] Therefore, the derivative of the Lyapunov function under the selected adaptive law is semi-negative definite, which ensures the stability of the closed-loop system and keeps the tracking error bounded.

[0269] S408: Through adaptive law Integrating yields online parameter estimates Generate the final MRAC adaptive compensation term:

[0270] .

[0271] VI. MRAC Compensated LQR Control Law

[0272] In practical implementation, the formula for the rear wheel steering angle control law (MRAC compensated LQR control law) is expressed as follows:

[0273] (45)

[0274] VII. Simulation Analysis of Active Rear Wheel Steering

[0275] To verify the improvement effect of the designed active rear-wheel steering control strategy on vehicle handling and high-speed stability, this embodiment conducts comparative verification on the MATLAB / Simulink and Carsim co-simulation platform. The simulation conditions are set according to the test methods of ISO3888(1)-2018 and GB / T6323-2014. The double lane change condition and the angular step input condition are selected as typical operating inputs to observe the changes in vehicle yaw response and lateral stability. This embodiment conducts simulations under different road surface adhesion conditions and different steering conditions to evaluate the control effect of the control strategy. The high-speed steering condition in this embodiment does not consider the low-adhesion road surface coefficient.

[0276] 1. High-adhesion road surface simulation

[0277] (1) Double line shifting condition

[0278] A double lane change simulation test was conducted under high-adhesion road surface conditions. During the simulation, the vehicle was set to travel at a constant speed of 100 km / h, and the road surface adhesion coefficient was set to 0.85, corresponding to the characteristics of typical high-adhesion road surfaces such as dry asphalt. To verify the actual control effect of the controller, this simulation compared and analyzed four schemes: rear-wheel control without rear wheels, proportional integrated control, LQR rear-wheel control, and MRAC-LQR control. The corresponding simulation results are shown in Figure 4.

[0279] As shown in Figures 4(a)-(c), when the vehicle is traveling at high speed, the yaw rate and sideslip angle fluctuation amplitude of the vehicle without rear wheel control are significantly greater than those of the vehicle with rear wheel control, indicating insufficient stability at high speeds. All three control strategies can effectively suppress yaw rate and sideslip angle, with the MRAC-LQR control exhibiting the smallest amplitude and correspondingly a larger active adjustment range for the rear wheel steering angle. This strategy can more effectively optimize the vehicle's driving posture through active adjustment of the rear wheel steering angle.

[0280] The center-of-gravity sideslip angle phase plane method is an effective way to identify the stable state of a vehicle. A phase trajectory starting from zero and eventually returning to the origin indicates that the vehicle is in a stable state. The smaller the area enclosed by the phase trajectory and the phase plane, the better the vehicle's stability. As shown in Figure 4(d), all three control phase trajectories start from zero and eventually return to the origin, without any instability. Comparing the three control methods, the center-of-gravity sideslip angle phase trajectory using MRAC-LQR control has the smallest area enclosed on the phase plane, indicating the best vehicle stability.

[0281] For the three control algorithms designed in this invention, since LQR control and MRAC-LQR control are significantly superior to proportional integrated control, to further observe the advantages of the proposed MRAC-LQR control, the vehicle state parameters obtained from LQR control and MRAC-LQR control are subtracted from the ideal reference vehicle model state parameters to obtain the vehicle state parameter error. As shown in Figures 4(e) and 4(f), under the same operating conditions, the yaw rate error and centroid sideslip angle error of LQR control and MRAC-LQR control are basically coincident in the initial stage of the operating condition, indicating that the two control strategies have similar tracking levels in the initial stage. As the input changes continuously during the steering process, the MRAC-LQR error amplitude gradually decreases. In contrast, MRAC-LQR can continuously maintain a smaller error amplitude and fluctuation range, thus exhibiting superior performance compared to LQR. MRAC compensation has a better online adjustment effect on error changes.

[0282] (2) Angular step condition

[0283] On a high-adhesion road surface with an adhesion coefficient of 0.85, the vehicle travels at a constant straight speed of 100 km / h. After the vehicle enters a stable driving state, a step steering angle input is applied to the steering wheel. The magnitude of the steering angle is determined by the target value of steady-state lateral acceleration of 3 m / s². The entire step input process is completed within 0.1 s. The simulation results of the step steering condition are shown in Figure 5.

[0284] As shown in Figures 5(a)-(b), referring to GB / T6323-2014 "Test Methods for Handling Stability of Automobiles", a step input lateral acceleration of 3 m / s² is applied to the steering wheel angle of a front-wheel steering vehicle. 2 The steering wheel angle was set to respond. Under the same steering wheel input, the lateral acceleration of the rear-wheel steering vehicle was significantly reduced, which meets the design requirements.

[0285] As shown in Figures 5(c)-(e), the vehicle without rear-wheel steering control exhibits the largest yaw rate and sideslip angle. All three control strategies with rear-wheel steering effectively improve vehicle dynamics response. Among them, the peak yaw rate and sideslip angle deviation of MRAC-LQR control and LQR control are smaller than those of the proportional integrated control and uncontrolled vehicles. The rear-wheel steering curves show that LQR control and MRAC-LQR control produce larger steady-state rear-wheel steering angles, effectively optimizing the vehicle's driving posture under angular step inputs through active adjustment of the rear-wheel steering angle.

[0286] To further observe the advantages of the proposed MRAC-LQR control, the vehicle state parameters obtained from LQR control and MRAC-LQR control were subtracted from the ideal reference vehicle model state parameters to obtain the yaw rate error and sideslip angle error for both control methods. As shown in Figure 4(f), the peak error amplitudes of the two controls are similar in the step transient phase. The MRAC-LQR control returns to near zero faster after the peak, maintaining a steady-state error near zero. The steady-state error of the MRAC-LQR control is 83.4% lower than that of the LQR control. As shown in Figure 4(g), the steady-state error of the MRAC-LQR control is slightly smaller than that of the LQR control, suppressing the sideslip angle by 1.5%. Under the premise that LQR control provides basic stability, the introduction of MRAC compensation can correct the rear wheel control quantity online, allowing the system to approach the reference model more quickly after the steering transient, thereby reducing the deviation of the yaw response and decreasing the sideslip angle.

[0287] 2. Simulation of medium-adhesion road surface

[0288] (1) Double line shifting condition

[0289] On a medium-adhesion road surface with an adhesion coefficient of 0.5, a simulation of a double lane change condition was conducted. The vehicle traveled at a constant speed of 80 km / h. Under these conditions, the lateral force of the vehicle was limited and its stability was more easily affected. To observe the differences in controller performance, simulations were performed on three control strategies: no rear wheel control and rear wheel control. The simulation results are shown in Figure 6.

[0290] As shown in Figures 6(a)-(c), under medium-adhesion road conditions with an adhesion coefficient reduced to 0.5, the vehicle without rear-wheel control exhibits larger yaw rate and sideslip angle amplitudes during double lane change, resulting in higher response amplitudes during steering transitions. Introducing rear-wheel steering helps suppress yaw response, with LQR and MRAC-LQR showing significantly smaller fluctuations than no control and proportional integrated control. The vehicle without rear-wheel control has the smallest rear-wheel steering angle. Proportional integrated control effectively adjusts the rear-wheel steering angle, but the sideslip angle remains relatively large after adjustment. The proportional integrated control with LQR and MRAC-LQR phase advance of the rear-wheel steering angle demonstrates better overall performance.

[0291] As shown in Figure 6(d), all three control phase trajectories start from zero and eventually return to the origin without any instability. Comparing the three control methods, the MRAC-LQR control phase trajectory with the smallest center of gravity sideslip angle encloses the smallest area on the phase plane, resulting in the best vehicle stability.

[0292] As shown in Figures 6(e) and 6(f), the yaw rate error and centroid sideslip angle error of the LQR and MRAC-LQR control strategies are basically coincident in the initial stage, exhibiting similar tracking capabilities when the steering input begins to change. However, as model uncertainty and nonlinear effects increase, the error peak and oscillations under LQR control become more pronounced, while MRAC-LQR control can reduce the overall error peak and subsequent oscillation amplitude in the same stage, with faster error decay and smoother zero return. The centroid sideslip angle error also shows that MRAC-LQR control has a slightly smaller peak-to-valley error than LQR control. In the mid-adhesion double lane change condition, the adaptive adjustment of MRAC compensation effectively weakens the tracking deviation and oscillations caused by changes in adhesion conditions, making the vehicle state more stable in following the ideal reference model, and its control performance is superior to simple LQR control.

[0293] (2) Angular step condition

[0294] On a low-friction road surface with a coefficient of friction of 0.5, the vehicle travels at a constant straight speed of 80 km / h. A typical transient steering condition is constructed by applying a step input to the steering wheel angle. The magnitude of the steering wheel angle is determined by the target value of steady-state lateral acceleration of 3 m / s², and the angle step change is completed within 0.1 s. The simulation results are shown in Figure 7, comparing the vehicle parameters of no rear-wheel control, proportional integrated control, LQR control, and MRAC-LQR control.

[0295] As can be seen from Figures 7(a)-(b), the lateral acceleration of a vehicle steering with its front wheels is 3 m / s². 2 The steering wheel angle is set to the target response to meet the requirements of transient steering conditions. Under the same steering wheel input, the lateral acceleration of the rear-wheel steering vehicle is significantly reduced, which meets the design requirements.

[0296] As shown in Figures 7(c)-(e), the vehicle without rear-wheel steering control exhibits the highest peak yaw rate and the largest deviation in the center of gravity sideslip angle. All three control strategies with rear-wheel steering effectively improve vehicle dynamics response. Among them, the peak yaw rate and the deviation in the center of gravity sideslip angle of MRAC-LQR control and LQR control are both smaller than those of the proportional integrated control and the vehicle without control, with MRAC-LQR control showing the best effect. The rear-wheel steering angle curves show that proportional integrated control experiences the largest fluctuation in rear-wheel steering angle at the moment of steering, followed by a smaller adjustment angle than LQR control and MRAC-LQR control. MRAC-LQR control, through more comprehensive active adjustment of the rear-wheel steering angle, effectively optimizes the vehicle's driving posture under angular step input.

[0297] To further observe the advantages of the proposed MRAC-LQR control, the yaw rate error and sideslip angle error of LQR control and MRAC-LQR control were analyzed. As shown in Figure 6(f), the peak error amplitudes of the two controls are similar in the step transient phase, but MRAC-LQR returns to near zero faster after the peak. Compared to LQR control, the steady-state yaw rate error of MRAC-LQR control is reduced by 0.122 deg / s. As shown in Figure 6(g), the two curves have a high degree of overlap. Compared to LQR control, the steady-state error of MRAC-LQR is reduced by 30.5%. Under the premise that LQR control provides basic stability, the introduction of MRAC compensation can correct the rear wheel control quantity online, allowing the system to approach the reference model more quickly after the steering transient, thereby reducing the deviation of the yaw response and decreasing the sideslip angle.

[0298] In summary, this invention focuses on the research of an active rear-wheel steering adaptive control strategy. Through vehicle modeling, controller design, and co-simulation analysis, the main effects are as follows:

[0299] 1) This invention establishes a two-degree-of-freedom four-wheel steering vehicle dynamics model and clarifies the influence of the rear wheel steering angle on the vehicle's yaw response and center of gravity sideslip angle; at the same time, it combines the CarSim whole vehicle model to build a joint simulation platform, providing a model basis for the design and performance verification of the active rear wheel steering controller.

[0300] 2) This invention proposes an MRAC-LQR adaptive control strategy. Based on the comprehensive adjustment of yaw rate and sideslip angle using LQR, this method introduces model reference adaptive compensation, enabling online correction of control deviations and improving the control system's adaptability to parameter changes and complex operating conditions.

[0301] 3) The overall performance of the MRAC-LQR control in this invention is superior to both proportional integrated control and LQR control. Simulation results show that, on a road surface with a friction coefficient of 0.85, with a step input of the steering wheel angle, compared to LQR control, MRAC-LQR reduces the steady-state error of the yaw rate by 83.4% and the steady-state sideslip angle error by 1.5%; on a road surface with a friction coefficient of 0.5, with a step input of the steering wheel angle, the steady-state error of the sideslip angle is reduced by 30.5%. The designed adaptive control strategy has significant advantages in yaw response tracking and sideslip suppression, and can effectively improve the lateral stability of the vehicle under complex conditions.

[0302] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit the technical solutions. Those skilled in the art should understand that any modifications or equivalent substitutions to the technical solutions of the present invention without departing from the spirit and scope of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. An active rear-wheel steering adaptive control optimization method based on MRAC-LQR, characterized in that, include: S1: Based on the structural state parameters of the target vehicle, vehicle dynamics modeling is performed to obtain a two-degree-of-freedom four-wheel steering vehicle state space model; S2: Construct a reference model based on the steady-state two-degree-of-freedom front-wheel steering vehicle model to obtain an ideal reference model; S3: Design an LQR optimal controller based on the two-degree-of-freedom four-wheel steering vehicle state-space model and the ideal reference model to obtain the LQR optimal control law; S4: Design MRAC adaptive compensation based on LQR optimal control law to obtain MRAC adaptive compensation term; S5: Synthesize joint control law based on LQR optimal control law and MRAC adaptive compensation term to obtain synthesized rear wheel steering angle control law; S6: Based on the rear wheel steering angle control law and the real-time state parameters of the target vehicle, calculate the corresponding rear wheel steering angle control quantity, and realize the active rear wheel steering control of the target vehicle through the rear wheel steering angle control quantity.

2. The active rear-wheel steering adaptive control optimization method based on MRAC-LQR as described in claim 1, characterized in that: In step S1, the structural state parameters of the target vehicle are obtained by constructing a Carsim vehicle model; the structural state parameters of the target vehicle include the vehicle wheelbase, the distance from the center of mass to the front axle, the distance from the center of mass to the rear axle, the vehicle mass, the steering wheel to front wheel transmission ratio, the front axle lateral stiffness, the rear axle lateral stiffness and / or the moment of inertia about the Z-axis.

3. The active rear-wheel steering adaptive control optimization method based on MRAC-LQR as described in claim 1, characterized in that: In step S1, the processing steps for constructing the state space model of a two-degree-of-freedom four-wheel steering vehicle include: S101: Establishing the vehicle coordinate system: with the vehicle's center of mass as the origin, The axle is along the direction of vehicle travel. The axis points to the driver's left. The axis is vertically upward; S102: Calculate the front and rear wheel slip angles based on the vehicle coordinate system: In the formula: 、 These represent the slip angles of the front and rear wheels, respectively. Indicates the vehicle's sideslip angle; Indicates the vehicle's yaw rate; 、 This represents the distance from the center of mass to the front and rear axles; This indicates the vehicle's forward speed, i.e., its longitudinal speed; 、 S103: Establish the turning angles of the front and rear wheels respectively, with counterclockwise rotation defined as the positive direction; Resultant force on the axis And around Resultant torque of the shaft The equilibrium equations are: ;in: ； In the formula: Indicates the total mass of the vehicle; Indicates vehicles go around Moment of inertia of the shaft; express On-axis acceleration; 、 These represent the lateral forces on the front and rear wheels of the vehicle, respectively. For front axle lateral stiffness; Rear axle lateral stiffness; S104: Describe the lateral velocity of the vehicle; Construct the differential equations for a two-degree-of-freedom four-wheel steering vehicle: S105: Constructing a state-space model of a two-degree-of-freedom four-wheel steering vehicle based on its differential equations: The two-degree-of-freedom four-wheel steering state-space model is transformed into the following form: ;in: ； ； ； In the formula: This is the system state matrix; The state matrix of a two-degree-of-freedom four-wheel steering model; 、 These are the input matrices for the front and rear wheel rotation angles, respectively.

4. The active rear-wheel steering adaptive control optimization method based on MRAC-LQR as described in claim 3, characterized in that: In step S2, the formula for the steady-state two-degree-of-freedom front-wheel steering vehicle model is expressed as follows: 。 5. The active rear-wheel steering adaptive control optimization method based on MRAC-LQR as described in claim 4, characterized in that: In step S2, the formula for the ideal state model is expressed as: ;in: ； In the formula: Indicates the ideal centroid sideslip angle; This represents the ideal yaw rate; The ideal transfer function representing the front wheel steering angle to the center of gravity sideslip angle; This represents the steady-state yaw rate gain of a vehicle with front-wheel steering. Indicates the stability factor; This indicates the vehicle's wheelbase.

6. The active rear-wheel steering adaptive control optimization method based on MRAC-LQR as described in claim 5, characterized in that: In step S3, the processing steps for constructing the LQR optimal control law include: S301: Constructing a quadratic performance index function based on the four-wheel steering vehicle state-space model and ideal reference model. ： ； ； In the formula: 、 It is a weighted diagonal matrix. This indicates the degree of importance attached to the centroid sideslip angle and yaw rate of the state vector. This indicates the degree of importance attached to the control variable, the rear wheel steering angle; This is the weight for the centroid sideslip angle error; This is the weighting for the yaw rate error; The weights for the control of the rear wheel steering angle; 、 S302: Construct the Hamlittion function based on the quadratic performance index function and the state-space model of a four-wheel steering vehicle. These represent the start and end times of the performance index integration interval, respectively; ： In the formula: S303: Based on the Hamiltonian function; (This refers to the accompanying variable; S303: based on the Hamiltonian function) Calculate the control law from the control equations: The control equations are: The control law is: The regular equation is: The adjoint equation is: ;in Differentiating, we get: In the formula: 、 All are intermediate calculation variables; S304: Considering moments 、 、 、 For a constant matrix, when ， ， Substitute this condition into the control law to solve for the state variable. From the equation, we obtain the Riccati algebraic equation: The results of the analysis are as follows: S305: Obtained through the above formula 、 Substituting into the Riccati algebraic equation, we obtain the LQR optimal control law: 。 7. The active rear-wheel steering adaptive control optimization method based on MRAC-LQR as described in claim 6, characterized in that: The LQR front wheel steering angle feedforward matrix for the LQR optimal control law is defined as follows: The LQR state variable feedback matrix of the LQR optimal control law is defined as follows: The optimal control law is expressed as: 。 8. The active rear-wheel steering adaptive control optimization method based on MRAC-LQR as described in claim 7, characterized in that: In step S4, the processing steps for constructing the MRAC adaptive compensation term include: S401: Equivalently converting the error into an input uncertainty term, resulting in Formula 1: In the formula: S402: Use a linear parameterized approximation, let ， For an unknown constant parameter vector, the MRAC adaptive compensation term is defined as: In the formula: For online parameter estimation, The system state matrix is ​​given; integrating Formula 1 and the MRAC adaptive compensation term, Formula 2 is obtained: S403: Define the reference model: ;in: ； In the formula: For reference state parameters; Here, is the Hurwitz matrix, which is the state matrix of the reference model; Input matrix for reference model; The state matrix of a two-degree-of-freedom four-wheel steering model; 、 These are the input matrices for the front and rear wheel rotation angles, respectively. This is the LQR state variable feedback matrix; S404: Defines the front wheel steering angle feedforward matrix for LQR; S404: Defines the tracking error. Combining Formula 2, we obtain the error dynamics formula: In the formula: For parameter estimation error; S405: Take any symmetric positive definite matrix ,make If the solution is the only solution to the Lyapunov equation, then we have Formula 3: For any There exists a unique symmetric positive definite solution. Construct Lyapunov functions ： In the formula: The adaptive gain matrix is ​​given; taking the derivative of the Lyapunov function, we obtain Formula 4: S406: Substituting the error dynamics formula into Formula 4, we obtain Formula 5: Combining Formula 3 and Formula 5, we obtain Formula 6: S407: Based on Formula 6, consider... Let be a constant, let To cancel out the cross terms, the adaptive law is chosen as follows: S408: Through adaptive law Integrating yields online parameter estimates Generate the final MRAC adaptive compensation term: 。 9. The active rear-wheel steering adaptive control optimization method based on MRAC-LQR as described in claim 8, characterized in that: In step S5, the formula for the rear wheel steering angle control law is expressed as follows: 。 10. The active rear-wheel steering adaptive control optimization method based on MRAC-LQR as described in claim 1, characterized in that: In step S6, the real-time state parameters of the target vehicle include the vehicle's center of gravity sideslip angle and the vehicle's yaw rate.

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