Racing Control Method Based on Gaussian Process Regression Enhanced Model Predictive Control

Abstract

The present invention discloses a racing control method based on Gaussian process regression enhanced model predictive control, belonging to the field of trajectory planning and tracking control of unmanned driving systems. This method mainly includes: First, construct a stochastic optimal control problem model for racing that includes a state equation, a measurement equation, and probabilistic constraints. Secondly, use Gaussian process regression models to model system uncertainties and measurement links respectively. Subsequently, design a cautious model predictive control for racing and an extended Kalman filter GP-EKF based on Gaussian process regression to achieve high-precision state estimation. Finally, fuse GP-EKF and cautious model predictive control to construct a learning model predictive control GP-EKF-LMPC enhanced by GP-EKF, feedback the posterior state estimation to the optimization problem, and solve the optimal control command for racing in real time. Experiments show that the present invention can significantly improve the control accuracy and robustness in complex scenarios.

Description

Technical Field

[0001] The present invention belongs to the field of trajectory planning and tracking control of unmanned driving systems, and particularly relates to a racing car speed control method based on a Gaussian process regression enhanced model predictive control. Background Art

[0002] With the progress of technology, unmanned driving technology has become a research hotspot in the current automotive field. Among them, unmanned racing car speed, that is, autonomous driving racing car technology, integrates artificial intelligence, sensors and automatic control technology to enable the racing car to drive autonomously without a human driver. This technology not only improves the performance and safety of the racing car, but also greatly reduces human errors. Unmanned racing car speed is gradually becoming a new trend in motorsports, which not only promotes the innovation of racing car technology, but also brings a new viewing experience and competition mode to motorsports.

[0003] Benefiting from the continuously enhanced computing power, Model Predictive Control (MPC) can already be used for path tracking of autonomous vehicles and racing car speed. MPC optimizes the motion control of the vehicle within a finite time range based on a mathematical model. On the one hand, this model should fully represent the vehicle attributes and dynamics, and on the other hand, it should be suitable for use in an online optimization framework. Especially in autonomous racing cars, the vehicle operates at its performance limit, which is a challenging trade-off. In the disclosed control methods, the mean estimation of Gaussian Process Regression (GPR) is used as the dynamic model required by MPC. However, system states (such as speed, position) often cannot be directly obtained or can only be obtained with noise, which may lead to potential uncertainties and thus affect the actual control performance. Although some also have publicly available literature using the Extended Kalman Filter (EKF) to obtain state estimates, it is required that the measurement model of the unmanned racing car can be obtained in advance and is continuously differentiable. To address the above problems, the present invention proposes an enhanced learning model predictive control method combining Gaussian process regression and a filter to improve the control performance of unmanned racing cars under extreme conditions. Summary of the Invention

[0004] The purpose of the present invention is to provide a racing car speed control method based on a Gaussian process regression enhanced model predictive control, to solve the problem that the existing model predictive control MPC fails to control due to model deviation, inaccurate state estimation and non-continuous differentiability of the measurement model in racing car speed, so as to improve the control quality, stability and efficiency.

[0005] To achieve the above purpose, the technical solution adopted by the present invention is:

[0006] A racing car speed control method based on a Gaussian process regression enhanced model predictive control, comprising the following steps:

[0007] S1. Construct a stochastic optimal control problem for racing car speed, establish a state equation and a measurement equation for the discrete-time system of racing car speed. The state equation includes a nominal model and an uncertainty model containing a state-dependent residual model and an independent and identically distributed process noise, so as to obtain a stochastic optimal control problem model with the goal of minimizing the expected cost and satisfying the probability constraints of the system state and control input;

[0008] S2. Use Gaussian process regression models to model the system uncertainty and the measurement link respectively, to obtain a system uncertainty GPR model and a measurement link GPR model. The system uncertainty GPR model takes the state-control tuple as the input and the uncertainty as the output, and the measurement link GPR model takes the system state as the input and the measurement output as the output;

[0009] S3. Design a cautious model predictive control for racing car speed, make up the nominal model by using the Gaussian process regression model to predict the mean and variance of the uncertainty, and construct a probability reachable set based on the variance of the predicted uncertainty to convert the probability constraint into a tightened deterministic constraint, and adjust the tightening amplitude of the probability constraint in real time, thereby constructing the stochastic optimal control problem model described in step S1 into an approximate deterministic optimal control problem model;

[0010] S4. Design an extended Kalman filter GP-EKF based on Gaussian process regression, and iteratively update the system state estimation in a non-continuously differentiable scenario through a prediction step and a correction step. The prediction step predicts the state mean and variance based on the nominal model and the system uncertainty GPR model, and the correction step calculates the Kalman gain by linearizing the measurement link GPR model and updates the posterior state mean and variance;

[0011] S5. Integrate the GP-EKF and the cautious model predictive control, construct a learning model predictive control GP-EKF-LMPC enhanced by GP-EKF, take the posterior state estimation in step S4 as the input, feedback it to the approximate deterministic optimal control problem model in step S3, and generate an optimal control instruction in real time to complete the racing car speed task.

[0012] Further, in step S1, the stochastic optimal control problem model is as follows:

[0013]

[0014]

[0015]

[0016]

[0017]

[0018]

[0019] Wherein, E represents the expected value, 、 、 are the system state, control input, and measurement output at time k, respectively, represents the system state at time k and the control input corresponding cost function, represents the cost function of the system state at the terminal time N, is the nominal model, is the state-dependent residual model, is the independent and identically distributed process noise, , is the measurement function, is the measurement noise, , represents the control strategy, represents the probability.

[0020] Furthermore, in step S2, in the uncertainty GPR model, the state-control tuple is the input, and the corresponding uncertainty is the output. The uncertainty at time k is calculated by the following formula:

[0021]

[0022] Wherein, represents the uncertainty at time k, is the pseudo-inverse matrix.

[0023] Furthermore, the specific implementation process of step S2 includes: calculating the mean of the posterior distribution of the test data points by combining the similarity between the test data points and the observation data points, the covariance relationship between the observation data points, and the observed uncertainty, so as to predict the measurement output value of the test data points.

[0024] Furthermore, the specific implementation process of step S3 includes:

[0025] (1) Design the control strategy as linear state feedback, and the control strategy formula is as follows:

[0026]

[0027] Wherein, is the predicted state mean, is the predicted mean of the control input, is the system state at time k, is the feedback gain;

[0028] Define the state deviation and the control input deviation as:

[0029]

[0030]

[0031] wherein, is the state deviation, is the control input deviation;

[0032] (2) Using the first-order Taylor approximation of the nominal model and the mean function of the system uncertainty GPR model, obtain the system state mean and the system state variance:

[0033]

[0034]

[0035] wherein, represents the system state mean at time k + 1, represents the nominal model, represents the mean function of the uncertainty at time k obtained based on the Gaussian process regression model, 、 respectively represent the variances of the system state at time k + 1 and time k, , is the covariance function of the uncertainty at time k obtained based on the Gaussian process regression model, and T represents the transpose;

[0036] Combined with the control strategy formula, the control input variance is obtained as: , wherein, represents the control input variance at time k.

[0037] Furthermore, converting the probabilistic constraint into a tightened deterministic constraint in step S3 includes:

[0038] (1) Based on the system state variance and the control input variance, construct the k-step probabilistic reachable set of the state deviation with probability and the k-step probabilistic reachable set of the input deviation, and obtain:

[0039]

[0040]

[0041] (2) With the help of the probability reachable set, the probability constraints of the system state and control input at the k-th step are converted into tightened deterministic constraints as follows:

[0042]

[0043]

[0044] In the formula, and represent the deterministic constraint sets of the system state and control input after k-step conversion respectively, and represent the probability constraint sets of the original system state and control input respectively, represents the Minkowski difference operation.

[0045] Furthermore, in step S3, the mean of the predicted state and the mean of the predicted control input are used to approximate the expected cost function, and an approximate deterministic optimal control problem model is obtained:

[0046]

[0047]

[0048]

[0049]

[0050]

[0051]

[0052] In the formula, represents satisfying the deterministic constraint set of the system state , represents satisfying the deterministic constraint set of the control input, represents the initial state being , represents that the initial state covariance is 0.

[0053] Furthermore, in step S4, the prediction model of the prediction step is:

[0054]

[0055] In the formula, represents a Gaussian distribution with mean and covariance ;

[0056] Perform a first-order Taylor expansion on the prediction model at time t-1. The prior state mean and variance of the first-step prediction at this time are as follows:

[0057]

[0058]

[0059] In the formula, and respectively represent the prior state mean and variance of the first-step prediction at time t-1. and respectively represent the posterior state mean and variance at time t-1. represents the control input at time t-1. .

[0060] Furthermore, in step S4, the correction step specifically includes:

[0061] Linearize the measurement link GPR model at the mean through a first-order Taylor expansion to obtain the measurement matrix , and obtain the Kalman gain as:

[0062]

[0063] In the formula, represents the Kalman gain at time t, represents the measurement matrix, represents the covariance of the measurement noise given the prior state mean ;

[0064] Then the mean and variance of the posterior state are updated as follows:

[0065]

[0066]

[0067] In the formula, and respectively represent the updated posterior state mean and variance at time t, is the actual measurement value of the system at time t, represents the predicted measurement mean based on the prior state mean, and I is the identity matrix.

[0068] Furthermore, the control instructions in step S5 include acceleration and steering angle, and the goal is to minimize the lap time under the condition of satisfying the safety constraints of the track boundary.

[0069] The beneficial effects of the above solution are:

[0070] The present invention designs a predictive control method for enhancing learning models based on Gaussian process regression filters to solve problems such as the difficulty in directly observing system states in a racing scenario and the unsatisfactory control effects caused by inaccurate models in traditional model predictive control techniques. By adjusting control commands in real time, this method can handle various complex situations such as track changes and vehicle dynamics, thereby ensuring that the racing car completes the race in the safest state and the shortest time. Compared with traditional model predictive control techniques, the present invention demonstrates significant advantages in terms of control quality and stability. Description of the Drawings

[0071] Figure 1 It is the driving trajectory and speed graph of a racing car using the traditional MPC control method;

[0072] Figure 2 It is the driving trajectory and speed graph of a racing car using the learning MPC control method;

[0073] Figure 3 It is the driving trajectory and speed graph of a racing car using the cautious MPC control method;

[0074] Figure 4 It is the driving trajectory and speed graph of a racing car using the GP-EKF-LMPC control method of the present invention. Detailed Embodiment

[0075] The following further elaborates on the present invention in detail in conjunction with the drawings and specific embodiments.

[0076] It should be noted that unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which this application belongs.

[0077] To solve the problem that system states cannot be directly obtained and existing measurement models are not continuously differentiable in racing, the present invention designs a learning-based model predictive control (LMPC) that integrates the nominal model of the system and the Gaussian process regression model of residual uncertainty, estimates the future position and speed of the racing car in real time according to road conditions and the state of the racing car, and gives optimal acceleration and steering control commands.

[0078] A racing control method based on enhanced model predictive control with Gaussian process regression includes the following steps:

[0079] S1. Construct a stochastic optimal control problem for racing car speed competition, establish the state equation and measurement equation of the discrete-time system for racing car speed competition. The state equation includes a nominal model and an uncertainty model containing a state-dependent residual model and independent and identically distributed process noise, and obtain a stochastic optimal control problem model with the goal of minimizing the expected cost and satisfying the probability constraints of the system state and control input;

[0080] S2. Use Gaussian process regression models to model the system uncertainty and measurement link respectively, and obtain a system uncertainty GPR model and a measurement link GPR model. The system uncertainty GPR model takes the state-control tuple as the input and the uncertainty as the output, and the measurement link GPR model takes the system state as the input and the measurement output as the output;

[0081] S3. Design a cautious model predictive control for racing car speed competition. Compensate the nominal model by using the Gaussian process regression model to predict the mean and variance of the uncertainty, and construct a probabilistic reachable set based on the variance of the predicted uncertainty to convert the probability constraint into a tightened deterministic constraint. Adjust the tightening amplitude of the probability constraint in real time, and then construct the stochastic optimal control problem model described in step S1 into an approximate deterministic optimal control problem model;

[0082] S4. Design an extended Kalman filter GP-EKF based on Gaussian process regression. Iteratively update the system state estimation in the non-continuously differentiable scenario through the prediction step and the correction step. The prediction step predicts the state mean and variance based on the nominal model and the system uncertainty GPR model, and the correction step calculates the Kalman gain after linearizing the measurement link GPR model and updates the posterior state mean and variance;

[0083] S5. Integrate the GP-EKF and the cautious model predictive control to construct a learning model predictive control GP-EKF-LMPC enhanced by GP-EKF. Use the posterior state estimation in step S4 as the input and feedback it to the approximate deterministic optimal control problem model in step S3 to generate the optimal control command in real time and complete the racing car speed competition task.

[0084] The following is a detailed description of each step of the present invention:

[0085] Step S1. Construct a stochastic optimal control problem model for racing car speed competition.

[0086] For a discrete-time system of racing car speed competition, its state equation and measurement equation are respectively:

[0087] (1a)

[0088] (1b)

[0089] Among them, , , are the system state, control input, and measurement output at time step k, respectively. is the known part of the dynamic model and is the nominal model. The unknown part of the system is called the uncertainty, denoted as , which includes the state-dependent residual model and the independent and identically distributed process noise . . is the measurement function, is the measurement noise, . f and g are continuously differentiable functions. In addition, the state and input are subject to probabilistic constraints, allowing the probability of constraint violation to be no greater than . The stochastic optimal control problem model for the racing competition is obtained as follows:

[0090] (2a)

[0091] (2b)

[0092] (2c)

[0093] (2d)

[0094] (2e)

[0095] (2f)

[0096] where E represents the expected value, denotes the system state at time step k and the control input corresponding cost function, denotes the cost function of the system state at the terminal time step N, denotes the control strategy, denotes the probability.

[0097] Step S2: Design the Gaussian process regression models for the uncertainty and measurement link of the racing competition system.

[0098] Use GPR to model the uncertainty of the racing competition system in (1a) and the measurement link in (1b). In the uncertainty GPR model, the state-control tuple is the input, and the corresponding uncertainty is the output. The uncertainty at time step k is:

[0099] (3)

[0100] In the formula, represents the uncertainty at time k, is the pseudo-inverse matrix.

[0101] The set of M observed data is denoted as . Given a test data point , based on this data set, the posterior distribution of is still a Gaussian distribution:

[0102] (4)

[0103] The mean and variance of the posterior distribution of the test data point are respectively:

[0104] (5a)

[0105] (5b)

[0106] Here,

[0107] is

[0108] 's covariance matrix.

[0109] In addition, since the traditional measurement model of the racing system is not continuously differentiable, the present invention simultaneously uses GPR to model the observed dynamics. The system state and the measurement output are respectively used as the input and output of the GPR model. The data set composed of the observed data by is denoted as . Based on the data set, the GPR model of the measurement link is trained, and its mean and variance are similar to formulas (5a) and (5b).

[0110] It should be noted that the above-mentioned observed data sets are all measured by system sensors. The present invention combines the similarity between the test data point and the observed data points, the covariance relationship between the observed data points, and the observed uncertainty to calculate the mean of the posterior distribution of the test data point, so as to predict the measurement output value of the test data point.

[0111] Step S3, design a cautious model predictive control for the racing competition.

[0112] Assume that the state can be fully measured, design a cautious model predictive control for the racing competition, use the GPR model to predict the mean and variance of the uncertainty to make up for the inaccurate nominal model, and adjust the tightening amplitude of the probability constraint in real time according to the variance of the predicted uncertainty to achieve cautious control.

[0113] (1) Design the control strategy as linear state feedback:

[0114] (6)

[0115] In the formula, is the predicted state mean value, is the predicted control input mean value, is the system state at time k, is the feedback gain.

[0116] To improve the calculation efficiency, the feedback gain is preselected. The state deviation and control input deviation are defined as:

[0117] (7a)

[0118] (7b)

[0119] (2)Using the first-order Taylor approximation of the nominal model f and the mean function of the uncertainty GPR model, the system state mean value and system state variance are obtained:

[0120] (8a)

[0121] (8b)

[0122] In the formula, represents the system state mean value at time k + 1, represents the nominal model, represents the mean function of the uncertainty at time k obtained based on the Gaussian process regression model, , respectively represent the variances of the system state at time k + 1 and time k, , is the covariance function of the uncertainty at time k obtained based on the Gaussian process regression model, and T represents the transpose;

[0123] Combined with the control strategy formula (6), the control input variance is obtained as: , in the formula, represents the control input variance at time k.

[0124] (3)Define the n-step probabilistic reachable set: For the variable x, with the initial value , if , where n is an integer greater than 0, then the set is the n-step probabilistic reachable set (n-step PRS) of x.

[0125] Based on the system state variance and the control input variance , construct the probability as The k-step probabilistic reachable set of the state deviation and the k-step probabilistic reachable set of the input deviation , it can be known from the definition that:

[0126] (9a)

[0127] (9b)

[0128] Subsequently, with the help of PRS, the probabilistic constraints of the system state and control input at the k-th step are converted into tightened deterministic constraints as follows:

[0129] (10a)

[0130] (10b)

[0131] In the formula, and respectively represent the sets of deterministic constraints of the system state and control input after k-step conversion, and respectively represent the sets of probabilistic constraints of the original system state and control input, represents the Minkowski difference operation.

[0132] Then it can ensure that the state probability constraint in formula (2e) and the control input probability constraint in formula (2f) are satisfied. That is, if , then ;

[0133] if , then .

[0134] To reduce the computational complexity, the mean value of the predicted state and the mean value of the predicted control input are used to approximate the expected cost function. Therefore, in the case where the state is fully measurable, the stochastic optimal control problem model is constructed as an approximate deterministic optimal control problem model as follows:

[0135] (11)

[0136]

[0137]

[0138]

[0139]

[0140]

[0141] In the formula, denotes meeting the set of system state determinacy constraints , denotes meeting the set of control input determinacy constraints, denotes that the initial state is , denotes that the initial state covariance is 0. is the nominal input sequence.

[0142] Step S4: Design an extended Kalman filter GP-EKF based on Gaussian process regression.

[0143] When the system state cannot be directly or fully measured, a state estimation method must be adopted to ensure control performance. Considering that the measurement model of the racing car speed system is not continuously differentiable, the present invention designs an extended Kalman filter GP-EKF based on Gaussian process regression to fuse the above-mentioned cautious model predictive control to solve the stochastic optimal control problem of the racing car speed.

[0144] The GP-EKF process consists of two parts: prediction and correction. First, define the prior state mean and variance as and , and use and to denote the posterior state mean and variance.

[0145] The prediction step includes:

[0146] For the state equation (1a), its prediction model is expressed as:

[0147] (12)

[0148] In the formula, denotes a Gaussian distribution with a mean of and a covariance of .

[0149] The prediction model (12) includes a nominal model and the uncertainty modeled by GPR. Perform a first-order Taylor expansion approximation on the prediction model at time t-1, then the prior state mean and variance of the first-step prediction at this time are as follows:

[0150] (13a)

[0151] (13b)

[0152] In the formula, , respectively denote the prior state mean and variance of the first-step prediction at time t-1, , respectively represent the posterior state mean and variance at time t-1, represents the control input at time t-1, .

[0153] The correction step specifically includes:

[0154] Based on the GPR model of the system measurement link, given the state , then , whose mean and variance are similar to formulas (5a) and (5b). Linearize the GPR model of the measurement link at the mean through first-order Taylor approximation to obtain the measurement matrix . Then the Kalman gain is:

[0155] (14)

[0156] In the formula, represents the Kalman gain at time t, represents the measurement matrix, represents the covariance of the measurement noise under the given prior state mean .

[0157] Then the posterior state mean and variance are updated as follows:

[0158] (15a)

[0159] (15b)

[0160] In the formula, , respectively represent the updated posterior state mean and variance at time t, is the actual measurement value of the system at time t, represents the predicted measurement mean based on the prior state mean, and I is the identity matrix.

[0161] Step S5, construct a learning model predictive control GP-EKF-LMPC enhanced based on GP-EKF.

[0162] When the state cannot be directly or fully measured and the existing measurement model is not continuously differentiable, the present invention combines GP-EKF with cautious model predictive control to obtain an approximate deterministic optimal control problem for the racing random optimal control problem model. The learning model predictive control GP-EKF-LMPC model enhanced based on GP-EKF is as follows:

[0163] (16)

[0164]

[0165]

[0166]

[0167]

[0168]

[0169]

[0170] In the formula, and and respectively represent the prior state means predicted at the k-th step, the first step, and the N-th step at time t, and respectively represent the prior state variances predicted at the first step and the (k + 1)-th step at time t, represents the actual control input at time t, represents the control input predicted at the k-th step at time t, represents the updated posterior state mean at time t, , , represents the state certainty constraint set, represents the control input certainty constraint set.

[0171] During the process of solving the optimization problem, the posterior state mean and variance are updated according to formulas (15a) and (15b).

[0172] To intuitively demonstrate the effectiveness and superiority of the present invention, a traditional MPC, a learning MPC, a cautious MPC, and the GP-EKF-LMPC designed by the present invention are respectively used to control a racing car for speed racing, Figures 1 to 4 respectively showing the driving trajectories and speeds of the racing car under the control of the above four algorithms. Figures 1 to 4 In Figure 1 The traditional MPC does not consider system uncertainty. Figures 2 to 4 The model predictive control adopted considers system uncertainty, so the farther the prediction step is, the greater the prediction deviation. The solid dots on the prediction trajectory represent the prediction mean, and the shaded area represents the prediction variance. The competition rules require the racing car to complete 10 laps along the track at the fastest speed under the condition of safe driving (without going out of the track).

[0173] From Figures 1 to 4It can be seen that: (1) Since traditional MPC ignores model deviation, its control quality is the worst. It went off the track in the 6th lap and did not complete the race, while the other three methods all completed the race. (2) As can be seen from Figure 2 it, learning MPC fuses the model uncertainty learned by GPR with the nominal dynamics, making the system model more accurate, and thus obtaining better performance. However, due to the relatively high speed of the racing car, the racing car still frequently cuts the track boundary, especially at sharp turns. (3) Figure 3 describes the driving trajectory of the car under cautious MPC control. To avoid boundary violations, the cautious MPC scheme tightens the constraints according to the predicted variance, resulting in a relatively cautious control output and maintaining a safe distance from the boundary. (4) In addition, in the case of measurement uncertainty, the learning model predictive control GP-EKF-LMPC designed in the present invention makes the estimated state more accurate and weakens the influence of uncertainty online. Comparing Figure 3 and Figure 4 , the driving trajectory of the racing car is more consistent than that under only cautious MPC control, improving driving safety and efficiency.

[0174] Therefore, the main advantages of the present invention are:

[0175] (1) Effectively improve control accuracy and stability. By accurately estimating the system state through GP-EKF and combining with the dynamic model optimization of LMPC, high-precision control instructions can still be generated in extreme scenarios such as sharp turns and straight-line acceleration of the racing car, avoiding control failure caused by model deviation of traditional MPC.

[0176] (2) Enhance system robustness. Use Gaussian process regression models to model system uncertainty and measurement links respectively, enabling the system to adapt to changes in track conditions and vehicle dynamic disturbances; the dynamic constraint adjustment mechanism of the probabilistic reachable set ensures that the system always meets safety constraints within the allowable violation probability.

[0177] (3) Achieve a balance between efficiency and safety. GP-EKF-LMPC takes both safety and racing efficiency into account in the generation of control instructions. Experiments show that the time for it to complete 10 laps of the race is significantly shorter than that of cautious MPC, and there is no track overstepping ( Figure 4 ). By fusing state estimation and model prediction in real time, the system can still pass through sharp turns at a speed close to the limit, while traditional methods need to reduce speed to ensure safety.

[0178] (4) Reduce the dependence on the measurement model. The present invention does not require the measurement model to be continuously differentiable, and a nonlinear observation dynamics can be constructed only through historical data, solving the limitation of strong assumptions on the measurement model in existing methods and expanding the application scenarios.

[0179] In summary, through the GP-EKF-LMPC model, the present invention deeply integrates Gaussian process regression, state estimation, and model predictive control, achieving high-precision and high-robustness autonomous control in the racing scene. On the one hand, it solves the bottlenecks of traditional methods in state estimation, model deviation, and non-differentiability of the measurement model. At the same time, it provides a safe and efficient racing control scheme for unmanned racing cars and can be extended to the real-time optimal control of other nonlinear systems. The experimental results verify that this method has significant advantages in control quality, stability, and efficiency.

[0180] Finally, it should be noted that the parts not detailed in the present invention are all prior arts. Those of ordinary skill in the art can understand that the above description is only a preferred example of the invention and is not used to limit the invention. Although the invention has been described in detail with reference to the foregoing examples, for those skilled in the art, they can still modify the technical solutions described in the foregoing examples or perform equivalent replacements for some of the technical features. Any modifications, equivalent replacements, etc. made within the spirit and principle of the invention shall be included within the protection scope of the invention.

Claims

1. A racing car speed control method based on Gaussian process regression enhanced model predictive control, characterized in that: The following steps are involved: S1. Construct a random optimal control problem for a racing car, establish the state equation and measurement equation of the racing car discrete time system, the state equation includes a nominal model and an uncertainty model including a state-related residual model and an independent and identically distributed process noise, and obtain a random optimal control problem model with the goal of minimizing the expected cost and satisfying the system state and control input probability constraints; S2. Use Gaussian process regression model to model system uncertainty and measurement link respectively, and obtain system uncertainty GPR model and measurement link GPR model, wherein the system uncertainty GPR model takes state-control tuple as input and uncertainty as output, and the measurement link GPR model takes system state as input and measurement output as output; S3, designing a prudent model predictive control for racing cars, compensating the nominal model by using the Gaussian process regression model to predict the mean and variance of uncertainty, and constructing a probabilistic reachable set based on the variance of the predicted uncertainty to convert the probabilistic constraint into a tightened deterministic constraint, adjusting the tightening amplitude of the probabilistic constraint in real time, and then constructing the stochastic optimal control problem model described in step S1 into an approximate deterministic optimal control problem model; S4. Design an extended Kalman filter GP-EKF based on Gaussian process regression, and iteratively update the system state estimation in the non-continuous differentiable scenario through prediction steps and correction steps. The prediction step predicts the state mean and variance based on the nominal model and the system uncertainty GPR model. The correction step calculates the Kalman gain after linearizing the measurement link GPR model and updates the posterior state mean and variance. S5. Integrate the GP-EKF and prudent model predictive control to construct the GP-EKF-LMPC based on GP-EKF enhancement learning model predictive control. The posterior state estimation of step S4 is used as input and fed back to the approximate deterministic optimal control problem model of step S3 to generate optimal control instructions in real time to complete the racing task.

2. The racing speed control method based on Gaussian process regression enhanced model predictive control according to claim 1, characterized in that: In step S1, the stochastic optimal control problem model is as follows: In the formula, E represents the expected value, , , are the system state, control input and measurement output at time k, represents the system state at time k and control input The corresponding cost function is, Indicates the system status of the terminal at the moment N The cost function is is the nominal model, is the state-dependent residual model, is independent and identically distributed process noise, , is the measurement function, To measure noise, , represents the control strategy, Represents probability.

3. The racing speed control method based on Gaussian process regression enhanced model predictive control according to claim 2 is characterized in that: In step S2, in the uncertain GPR model, the state control tuple is the input, and the corresponding uncertainty The uncertainty at time k is calculated as follows: In the formula, represents the uncertainty at time k, is the pseudo-inverse matrix.

4. The racing speed control method based on Gaussian process regression enhanced model predictive control according to claim 3 is characterized in that: The specific implementation process of step S2 includes: combining the similarity between the test data points and the observed data points, the covariance relationship between the observed data points and the observed uncertainty, calculating the mean of the posterior distribution of the test data points, and thus predicting the measured output value of the test data points.

5. The racing speed control method based on Gaussian process regression enhanced model predictive control according to claim 2, characterized in that: The specific implementation process of step S3 includes: (1) The design control strategy is linear state feedback. The control strategy formula is as follows: In the formula, is the predicted state mean, is the predicted control input mean, is the system state at time k, is the feedback gain; Define the state deviation and control input deviation as: In the formula, is the state deviation, To control input deviation; (2) Using the first-order Taylor approximation of the nominal model and the mean function of the system uncertainty GPR model, the system state mean and system state variance are obtained: In the formula, represents the mean value of the system state at time k+1, represents the nominal model, represents the mean function of uncertainty at time k obtained based on the Gaussian process regression model, , They represent the variance of the system state at time k+1 and time k respectively, , is the covariance function of uncertainty at time k obtained based on the Gaussian process regression model, and T represents the transpose; Combining the control strategy formula, the control input variance is obtained as: , where Indicates the control input variance at time k.

6. The racing speed control method based on Gaussian process regression enhanced model predictive control according to claim 5, characterized in that: The conversion of the probabilistic constraints into tightened deterministic constraints in step S3 includes: (1) Based on the system state variance and control input variance, the constructed probability is The k-step probability reachable set of state deviations and the k-step probability reachable set of input deviations ,get: (2) With the help of probabilistic reachable sets, the probabilistic constraints of the system state and control input at step k are converted into tightened deterministic constraints, as follows: In the formula, , They represent the system state after k-step transformation and the deterministic constraint set of control input, respectively. , They represent the original system state and the probability constraint set of the control input respectively. Represents the Minkowski difference operation.

7. The racing speed control method based on Gaussian process regression enhanced model predictive control according to claim 6, characterized in that: In step S3, the expected cost function is approximated by the mean of the predicted state and the mean of the predicted control input to obtain an approximate deterministic optimal control problem model: In the formula, express Satisfy the system state deterministic constraints set , express Satisfy the set of control input deterministic constraints, Indicates that the initial state is , Indicates that the initial state covariance is 0.

8. The racing car speed control method based on Gaussian process regression enhanced model predictive control according to claim 7, characterized in that: In step S4, the prediction model of the prediction step is: In the formula, Indicates the mean , the covariance is Gaussian distribution of Performing a first-order Taylor expansion on the prediction model at time t-1, the mean and variance of the prior state of the first step prediction at that time are as follows: In the formula, , They represent the prior state mean and variance of the first step prediction at time t-1, , They represent the mean and variance of the posterior state at time t-1 respectively, represents the control input at time t-1, .

9. The racing speed control method based on Gaussian process regression enhanced model predictive control according to claim 7, characterized in that: In step S4, the correction step specifically includes: The measurement link GPR model is linearized at the mean through the first-order Taylor expansion to obtain the measurement matrix , the Kalman gain is: In the formula, represents the Kalman gain at time t, represents the measurement matrix, Indicates that the mean value of the given prior state The covariance of the measurement noise; Then the mean and variance of the posterior state are updated as follows: In the formula, , They represent the mean and variance of the updated posterior state at time t, respectively. is the actual measured value of the system at time t, represents the predicted measurement mean based on the prior state mean, and I is the identity matrix.

10. The racing speed control method based on Gaussian process regression enhanced model predictive control according to claim 1, characterized in that: The control instructions in step S5 include acceleration and steering angle, and the goal is to minimize the lap time while satisfying the track boundary safety constraints.

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