Optimal backstepping tracking control method of one-class automatic vehicle system

Abstract

The invention provides an optimal backstepping tracking control method of an automatic vehicle system. Although a traditional backstepping method can realize tracking control, when system parameters are uncertain or are subjected to external disturbance, the control performance of the traditional backstepping method is obviously reduced. In addition, a single backstepping method cannot achieve optimal performance, and the high-performance requirement of an automatic vehicle system is difficult to meet. The adaptive dynamic programming (ADP) can solve the optimal control problem by virtue of a strong self-learning capability, and an optimal control strategy under unknown dynamics can be trained and approached through a neural network. Therefore, the invention provides an automatic vehicle system optimal tracking control method combining a backstepping method and an ADP, and a new path is provided for realizing high-performance optimal tracking control of the automatic vehicle system. Finally, MATLAB is used for simulation analysis, and the simulation result verifies the effectiveness of the method.

Description

Technical Field

[0001] This invention belongs to the field of automatic vehicle tracking control. Background Technology

[0002] Transportation is the lifeblood of the national economy, and traffic safety is closely related to the safety of people's lives and property, social stability and long-term peace, and the high-quality development of the national economy. Autonomous vehicle systems play a significant role in improving vehicle safety, alleviating traffic pressure, improving quality of life, and increasing work efficiency. With the development of next-generation artificial intelligence, autonomous driving technology has become a strategic research hotspot both domestically and internationally. Autonomous vehicle systems (such as...) Figure 1 As shown, this type of system (multi-input multi-output underactuated system) exhibits strong nonlinear dynamic characteristics and high parameter uncertainty, making it a complex system. Furthermore, vehicle operation is often accompanied by numerous external disturbances and uncertainties. Therefore, robust, high-precision, and fast-response motion control strategies remain crucial for the successful implementation of autonomous driving technology. With the continuous development of autonomous vehicles, extremely high demands are placed on the accuracy, stability, smoothness, and adaptability of vehicle path tracking control.

[0003] Backstepping is a systematic design and method for systems with uncertain parameters, and it is widely used in the field of automated vehicles. Combining backstepping with other control methods such as preset performance control, finite-time tracking control, and sliding mode control can achieve stable control of automated vehicle systems. However, achieving optimal control based on stable control remains a challenging problem to be solved.

[0004] A single backstepping method typically relies on an accurate mathematical model. Its performance degrades significantly when parameter uncertainties, unmodeled dynamics, or mismatched disturbances are present. Furthermore, backstepping methods focus more on system stability and often do not directly consider the optimality of comprehensive performance indicators such as energy consumption and tracking error during the control process. With the ever-increasing performance requirements of autonomous vehicle systems, a single backstepping method is insufficient to meet these high-performance demands. Adaptive Dynamic Programming (ADP), with its powerful self-learning capabilities, is widely used to solve optimal control problems. ADP can train neural networks online or offline, allowing the system to gradually approach the optimal control strategy under unknown dynamics.

[0005] Therefore, this invention proposes combining backstepping with ADP to achieve optimal tracking control of autonomous vehicle systems. Compared with the traditional ADP method, the proposed method's neural network weight update rate does not depend on Bellman residuals, eliminating the need for continuous excitation conditions to guarantee weight convergence. Furthermore, as a typical high-order system, autonomous vehicle systems require differentiation of the virtual control law during backstepping design, leading to exceptionally complex expressions and increased computational complexity. This invention utilizes a neural network as a function approximator to learn complex virtual control laws, avoiding direct calculation of higher-order derivatives and reducing computational complexity. Summary of the Invention

[0006] This invention addresses the optimal tracking control problem in automated vehicle systems by designing a control method that integrates evaluation network control and backstepping. A high-order system is divided into several low-cost subsystems, and the ADP algorithm is incorporated into each subsystem to obtain the optimal virtual control law, thereby yielding the optimal actual control law. Figure 2 The optimal backstepping tracking control structure diagram of the automated vehicle system is shown. This method mainly consists of the following steps:

[0007] Step 1: Establish the optimal tracking control problem for the automated vehicle system; Step 2: Design the optimal virtual control law and the optimal tracking controller. The specific implementation process of each step is then described below.

[0008] Step 1: Establish the optimal tracking control problem for the autonomous vehicle system. Consider the longitudinal dynamics of the vehicle as described by the following nonlinear third-order model:

[0009]

[0010] in: , , These represent the vehicle's position, speed, and acceleration, respectively. , , These represent the first derivatives of the vehicle's position, velocity, and acceleration, respectively. , , This represents the unknowns caused by parameter variations and modeling errors; Input for the engine; This represents the disturbance caused by gusts of wind; It is the engine time constant, which is usually unknown.

[0011] Note: All forms represent The first derivative.

[0012]

[0013]

[0014] in: This represents the mass of the car; Represents the air drag coefficient; Represents air quality; Represents the cross-sectional area; This is mechanical resistance. Based on engineering experience, the engine input is usually... Designed for

[0015]

[0016] in: To control the input. As can be seen from the above analysis...

[0017]

[0018] Define the position tracking error as Speed ​​tracking error is Acceleration tracking error is Its expression is

[0019]

[0020]

[0021]

[0022] in: For position reference signal; and These are the optimal virtual control laws for the first and second steps, respectively. and They represent the optimal virtual control law respectively. and The estimated value.

[0023] Step 2: In the conventional backstepping algorithm, the virtual control quantity can be directly calculated by introducing the Lyapunov function, thus designing the tracking controller for the autonomous vehicle system. To enhance the system's anti-interference capability, this invention introduces the ADP method at each step of the backstepping method to obtain the optimal virtual control law and the optimal actual controller, enabling the autonomous vehicle system to move to the specified position with minimal cost.

[0024] Define about Optimal cost function

[0025]

[0026] in, For definition in compact set The virtual control law on, and satisfies . This is the optimal virtual control law for the first step. Let this be the utility function for the first step. The Hamiltonian function is defined as follows:

[0027]

[0028] Among them, regarding Optimal cost function The gradient is .about The optimal cost function satisfies the following HJB equation:

[0029]

[0030] Therefore, the optimal virtual control law for the first step can be obtained.

[0031]

[0032] For ease of calculation, this invention sets the unknown function... And set

[0033]

[0034]

[0035] in, It is a constant greater than 0. To introduce a new unknown term. Because... and Since all terms are unknown, a radial basis function neural network is used to approximate the unknowns. A recognition neural network is introduced for approximation. And introduce unknowns in the approximate optimal virtual control law of the evaluation neural network. .

[0036]

[0037]

[0038] in, Ideal recognition neural network weights and For the ideal evaluation of neural network weights, all weights are unknowns; and For activation functions; and This represents the approximation error of the neural network. and These represent the number of neurons in the identification neural network and the number of neurons in the evaluation neural network, respectively, and their values ​​are given in the parameter section.

[0039] Since the ideal weights are unknown, we construct a recognition neural network and a judgment neural network respectively.

[0040]

[0041]

[0042] in, and They are respectively and The estimated value.

[0043] Therefore, the approximate optimal virtual control law Can be rewritten as

[0044]

[0045] Define the Lyapunov function for the first step as follows:

[0046]

[0047] in, This represents the weight error of the recognition neural network. This represents the weight error of the evaluation neural network. This represents a positive definite matrix.

[0048] The rules for identifying neural network weights and evaluating their update are as follows:

[0049]

[0050]

[0051] in, It is a constant greater than zero. It is also a constant greater than zero.

[0052] Similarly, define about Optimal cost function

[0053]

[0054] in, For definition in compact set The virtual control law on, and satisfies . For the utility function in the second step, This is the optimal virtual control law for the second step.

[0055] Optimal cost function Satisfying the following HJB equation:

[0056]

[0057] in, for The gradient is calculated. Therefore, the optimal virtual control law can be obtained.

[0058]

[0059] For ease of calculation, this invention sets the unknown function... And set

[0060]

[0061]

[0062] in, It is a constant greater than 0. To introduce a new unknown term. Because... and Since all are unknown, an identification neural network approximation similar to equation (15) is introduced. And introduce unknowns in the approximate optimal virtual control law of the evaluation neural network, similar to that in equation (16). .

[0063] Therefore, the approximate optimal virtual control law Can be rewritten as

[0064]

[0065] Define the Lyapunov function in the second step as follows:

[0066]

[0067] in, This represents the weight error of the recognition neural network. This represents the weight error of the evaluation neural network. This represents a positive definite matrix.

[0068] The rules for identifying neural network weights and evaluating their update are as follows:

[0069]

[0070]

[0071] in, It is a constant greater than zero. It is also a constant greater than zero.

[0072] Similar to the process described above, define about Optimal cost function

[0073]

[0074] in, For definition in compact set The virtual control law on, and satisfies . This is the optimal actual control law. Let be the utility function for the third step. The HJB equation for the third step is defined as follows:

[0075]

[0076] in, express The gradient is calculated. The optimal control law can be obtained.

[0077]

[0078] For ease of calculation, the present invention makes And set

[0079]

[0080]

[0081] in, It is a constant greater than 0. This introduces new unknowns. Because... and Since all are unknown, an identification neural network approximation similar to equation (15) is introduced. And introduce unknowns in the approximate optimal virtual control law of the evaluation neural network, similar to that in equation (16). .

[0082] Therefore, the approximate optimal virtual control law Can be rewritten as

[0083]

[0084] Define the Lyapunov function in the third step as follows:

[0085]

[0086] in, This represents the weight error of the recognition neural network. This represents the weight error of the evaluation neural network. This represents a positive definite matrix.

[0087] The rules for identifying neural network weights and evaluating their update are as follows:

[0088]

[0089]

[0090] in, It is a constant greater than zero. It is also a constant greater than zero. The optimal control law... When applied to an automated vehicle system, it enables the automated vehicle system to track an ideal trajectory. Attached Figure Description

[0091] Figure 1 Schematic diagram of an autonomous vehicle system

[0092] Figure 2 Optimal backstepping tracking control structure diagram of an automated vehicle system

[0093] Figure 3 Position tracking effect diagram

[0094] Figure 4 Tracking error

[0095] Figure 5 Control input

[0096] Figure 6 Cost function Detailed Implementation

[0097] This invention, based on MATLAB, conducts simulation experiments on an automated vehicle system to verify the control performance of the designed optimal backstepping tracking controller. The parameter values ​​of the algorithm proposed in this invention are selected as follows:

[0098] (1) The reference trajectory is The initial position is Gaussian function width Engine time constant

[0099] (2) Adaptive parameters , , ;

[0100] (3) The initial weights of the recognition neural network are set as follows: , , The initial weights of the evaluation neural network are set to... , , .

[0101] (4) The parameters related to the update of neural network weights are selected as follows: , , ; , , ; , , .

[0102] Figure 3 Describes the position of the automated vehicle system under the desired trajectory. As can be seen from the trajectory, the system output can accurately track the specified desired trajectory and exhibit a small tracking error, such as... Figure 4 As shown. Figure 5 Describes control input The convergence curve. Figure 6 The cost function convergence curve is presented, showing that the cost function can quickly converge to near the stable value. Figures 3-6 This verifies the effectiveness and rationality of the present invention.

[0103] This invention designs an optimal backstepping tracking control to achieve optimal tracking of the position, velocity, and acceleration of an automated vehicle system. By optimizing the neural network weight update mechanism, the dependence on Bellman residuals and continuous excitation conditions is eliminated, ensuring the stability of weight convergence. Simultaneously, by utilizing the neural network's learning ability for complex virtual control laws, direct calculation of higher-order derivatives is avoided, significantly reducing computational complexity. Compared with traditional backstepping methods, the proposed method not only enables the automated vehicle system to track the desired trajectory but also achieves relatively optimal system performance. Simulation analysis demonstrates the reasonable effectiveness of the proposed method.

Claims

1. An optimal backstepping tracking control method for a class of automated vehicle systems, characterized in that... Includes the following steps: Step 1: Establish the optimal tracking control problem for the automated vehicle system; Step 2: Design the optimal virtual control law and the optimal tracking controller; Step 1: Establish the optimal tracking control problem for the autonomous vehicle system; consider the longitudinal dynamics of the vehicle as described by the following nonlinear third-order model: in: , , These represent the vehicle's position, speed, and acceleration, respectively. , , These represent the first derivatives of the vehicle's position, velocity, and acceleration, respectively. , , This represents the unknowns caused by parameter variations and modeling errors; Input for the engine; This represents the disturbance caused by gusts of wind; It is the engine time constant, which is usually unknown; All forms represent The first derivative; in: This represents the mass of the car; Represents the air drag coefficient; Represents air quality; Represents the cross-sectional area; Mechanical resistance; engine input Designed for in: To control the input; as can be seen from the above analysis. Define the position tracking error as Speed ​​tracking error is Acceleration tracking error is Its expression is in: For position reference signal; and These are the optimal virtual control laws for the first and second steps, respectively. and They represent the optimal virtual control law respectively. and The estimated value.

2. The method according to claim 1, characterized in that, Step 2 is as follows: Define about Optimal cost function in, For definition in compact set The virtual control law on, and satisfies ; The optimal virtual control law for the first step; Let be the utility function for the first step; the Hamiltonian function is defined as follows: Among them, regarding Optimal cost function The gradient is ;about The optimal cost function satisfies the following HJB equation: Obtain the optimal virtual control law in the first step. Let the unknown function ; and set in, It is a constant greater than 0. To introduce new unknowns; because and Since all terms are unknown, a radial basis function neural network is used to approximate the unknowns; a recognition neural network is then introduced for further approximation. And introduce unknowns in the approximate optimal virtual control law of the evaluation neural network. ; in, Ideal recognition neural network weights and For the ideal evaluation of neural network weights, all weights are unknowns; and For activation functions; and This represents the approximation error of the neural network. and These are the number of neurons in the identification neural network and the number of neurons in the evaluation neural network, respectively, and their values ​​are given in the parameter section; Since the ideal weights are unknown, a recognition neural network and a judging neural network are constructed separately. in, and They are respectively and The estimated value; Therefore, the approximate optimal virtual control law Rewritten as Define the Lyapunov function for the first step as follows: in, This represents the weight error of the recognition neural network. This represents the weight error of the evaluation neural network. Represents a positive definite matrix; The rules for identifying neural network weights and evaluating their update are as follows: in, It is a constant greater than zero. It is also a constant greater than zero; Define about Optimal cost function in, For definition in compact set The virtual control law on, and satisfies ; For the utility function in the second step, This is the optimal virtual control law for the second step; Optimal cost function Satisfying the following HJB equation: in, for The gradient; by calculating Thus, the optimal virtual control law is obtained. Let the unknown function ; and set in, It is a constant greater than 0. To introduce new unknowns; because and Since all are unknown, an identification neural network approximation similar to equation (15) is introduced. And introduce unknowns in the approximate optimal virtual control law of the evaluation neural network, similar to that in equation (16). ; Therefore, the approximate optimal virtual control law Rewritten as Define the Lyapunov function in the second step as follows: in, This represents the weight error of the recognition neural network. This represents the weight error of the evaluation neural network. Represents a positive definite matrix; The rules for identifying neural network weights and evaluating their update are as follows: in, It is a constant greater than zero. It is also a constant greater than zero; Similar to the process described above, define about Optimal cost function in, For definition in compact set The virtual control law on, and satisfies ; This is the optimal actual control law; Let be the utility function for the third step; the HJB equation for the third step is defined as follows: in, express The gradient; by calculating Obtain the optimal control law For ease of calculation, let ; and set in, It is a constant greater than 0. To introduce new unknowns; because and Since all are unknown, an identification neural network approximation similar to equation (15) is introduced. And introduce unknowns in the approximate optimal virtual control law of the evaluation neural network, similar to that in equation (16). ; Therefore, the approximate optimal virtual control law Rewritten as Define the Lyapunov function in the third step as follows: in, This represents the weight error of the recognition neural network. This represents the weight error of the evaluation neural network. Represents a positive definite matrix; The rules for identifying neural network weights and evaluating their update are as follows: in, It is a constant greater than zero. It is also a constant greater than zero; the optimal control law It is applied to the autonomous vehicle system to enable the autonomous vehicle system to track the ideal trajectory.

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