Multipoint linearization model prediction trajectory tracking control method based on Gaussian process

Abstract

The invention discloses a multi-point linearization model prediction trajectory tracking control method based on a Gaussian process, and the method comprises the steps: building an ASV kinetic model for the trajectory tracking problem of an autonomous unmanned surface vehicle ASV; a GP prediction model is obtained by adopting unmodeled dynamics of a trajectory tracking system of Gaussian process GP approximation ASV; on the basis of dictionary scale limitation and a maximum likelihood method, updating criteria of dictionary data and hyper-parameters are established, and a GP prediction model is optimized; combining the optimized GP prediction model with the ASV kinetic model, using a learning-based model prediction control MPC method to construct a GP-MPC framework, and designing a terminal state set; on the basis of a multi-point linearization method, a local linear model predictive control LMPC method is combined, an algorithm for solving a GP-MPC framework optimization problem is designed, and real-time control over ASV trajectory tracking is achieved. According to the method, the calculation time is remarkably shortened while the ASV trajectory tracking precision is ensured.

Description

Technical Field

[0001] The invention belongs to the technical field of intelligent control and relates to a multi-point linearization model prediction trajectory tracking control method based on Gaussian process. Background Art

[0002] In recent years, autonomous surface vehicles (ASVs) have been widely used in marine environmental monitoring, resource exploration, rescue operations, and military fields. In these application scenarios, ASVs need to perform trajectory tracking tasks in complex, changeable, and uncertain marine environments, so extremely high requirements are placed on the robustness, accuracy, and real-time performance of their control systems. However, existing ASV control methods, such as classic proportional integral differential (PID) control, linear quadratic regulator (LQR) control, and standard model predictive control (MPC) methods, usually rely on accurate modeling of system dynamics, but in actual environments, ASV motion involves complex nonlinear hydrodynamics and unknown environmental disturbances (see Teng F, Han S. Nonlinear dynamics of marine vehicles under environmental disturbances. Journal of Marine Science and Technology, 2020: 1340-1350.), which makes it difficult for control methods based on precise models to achieve ideal results in dynamic, complex, and uncertain scenarios.

[0003] MPC is an advanced control method that has been widely used in the field of industrial and robotic control because it can generate multi-step prediction and control sequences through rolling optimization strategies (see Camacho EF, Bordons C. Model predictive control. Springer Science & Business Media, 2013.). A significant advantage of MPC is that it can handle system constraints and improve control accuracy through online optimization. However, the performance of MPC is heavily dependent on the accuracy of the system model. When faced with the complex nonlinear dynamic characteristics and uncertainties of ASV, traditional MPC methods are difficult to deal with effectively.

[0004] Although GP-MPC has shown significant potential in solving nonlinear control problems, it still faces challenges in reducing computational complexity as the size of the data set increases (see Quinonero-Candela J, Rasmussen CE. Aunifying view of sparse approximate Gaussian process regression. Journal of Machine Learning Research, 2005: 1939-1959.). However, the current theoretical analysis of the iterative feasibility and asymptotic stability of the GP-MPC method is still insufficient, which limits its widespread promotion in practical applications.

[0005] In addition, MPC can be divided into linear MPC (LMPC) and nonlinear MPC (NMPC) (see Mayne DQ, Rawlings JB. Model predictive control: Theory and design. Nob Hill Publishing, 2009.). Although NMPC is applicable to nonlinear dynamic models, its optimization problem is complex and it is difficult to achieve real-time control (see Diehl M, Bock HG. Efficient solution of nonlinear MPC problems. IEEE Transactions on Automatic Control, 2005: 255-269.). In contrast, LMPC uses a linear model to reduce the computational burden, but when the state deviates from the linearization point, the model accuracy decreases rapidly, resulting in reduced control performance (see Wang L. Model predictive control system design and implementation using MATLAB. Springer, 2009.). Summary of the invention

[0006] Purpose of the invention: The purpose of the present invention is to provide a Gaussian process-based multi-point linearized model predictive trajectory tracking control (GP-MP-LMPC) method that significantly reduces the calculation time while ensuring the ASV trajectory tracking accuracy.

[0007] Technical solution: The multi-point linearization model prediction trajectory tracking control method based on Gaussian process of the present invention comprises the following steps:

[0008] S1. Aiming at the trajectory tracking problem of autonomous surface vehicle (ASV), an ASV dynamics model is constructed.

[0009] S2, introduce the Gaussian process GP into the trajectory tracking system of ASV, approximate the unmodeled dynamics of the system, and obtain the GP prediction model;

[0010] S3. Based on dictionary size limitation and maximum likelihood method, establish the updating criteria of dictionary data and hyperparameters to optimize the GP prediction model;

[0011] S4. Combining the optimized GP prediction model with the ASV dynamics model, using the learning-based model predictive control MPC method, constructing the GP-MPC framework, and designing the terminal state set;

[0012] S5. Based on the multi-point linearization method, the state prediction model in the prediction time domain is derived. Combined with the local linear model predictive control LMPC method, an algorithm for solving the GP-MPC framework optimization problem is designed to achieve real-time control of ASV trajectory tracking.

[0013] Furthermore, the ASV kinetic model constructed in step S1 is expressed as:

[0014]

[0015] in, is the time derivative of the state variable x, x = [η T v T ] T =[xy ψ uvr] T , is a generalized coordinate vector, where x, y, and ψ are the horizontal coordinate, vertical coordinate, and heading angle of ASV in the earth coordinate system, respectively. is the velocity vector, u, v, and r are the forward velocity, roll velocity, and yaw velocity of the ASV in the hull coordinate system, respectively. T is the transpose of the matrix, u is the control input, and u = τ, τ = [τ u τ v τ r ] T is the torque of the control force, τ u is the torque in the direction of forward speed, τ v is the moment in the direction of the rolling velocity, τ r is the torque in the yaw angular velocity direction, R(η) is the rotation matrix, A(v)=-M -1 (C(v)+D(v)), B=M -1 , M -1 is the inverse matrix of M, M is the positive definite symmetric inertia matrix, C(v) is the Coriolis matrix, D(v) is the damping matrix, represents the unmodeled dynamics in the ASV trajectory tracking problem, e1(v) is the component of the unmodeled dynamics that affects the forward velocity, e2(v) is the component of the unmodeled dynamics that affects the roll velocity, and e3(v) is the component of the unmodeled dynamics that affects the yaw velocity.

[0016] Furthermore, when solving the ASV trajectory tracking problem in step S1, each state of the ASV trajectory tracking system has a corresponding reference state x r (t) = [x r (t), y r (t), ψ r (t),u r (t), v r (t), r r (t)] T , defined as follows:

[0017]

[0018] v r (t) = 0,

[0019]

[0020] Among them, x r (t) is the reference horizontal coordinate, y r (t) is the reference ordinate, ψ r (t) is the reference heading angle, u r (t) is the reference forward speed, v r (t) is the reference rolling velocity, r r (t) is the reference yaw angular velocity, For x r The derivative of (t) with respect to time, for y r The derivative of (t) with respect to time, for The derivative with respect to time, for With respect to the time derivative, atan2 represents the four-quadrant inverse tangent operator.

[0021] Furthermore, in step S2, the Gaussian process GP is introduced into the trajectory tracking system of the ASV to approximate the unmodeled dynamics of the system and obtain the GP prediction model, including:

[0022] (1) Apply GP to each dimension e of the unmodeled dynamics E(v) j (v) Modeling, j = 1, 2, 3, given a GP prior and a training set containing n sets of data z i For the training set The i-th group of input data, y i For the training set The GP posterior distribution d of the unmodeled dynamics in the jth dimension at the test point z for the i-th set of output data GP (z) is:

[0023]

[0024] Among them, e j (z) represents the value of the unmodeled dynamics in the jth dimension at the test point z, d GP The mean and covariance of (z) are:

[0025]

[0026] Among them, Z is the dictionary input data set, Y is the dictionary output data set, [K(z,Z)] 1i = k(z, z i ), [K(Z, Z)] ij = k(z i , z j ), K(z, Z) is the covariance matrix of z and Z, K(Z, Z) is the covariance matrix of Z and Z, is the covariance of GP noise, I is the unit matrix, K(z, z) is the covariance of z and z, and K(Z, z) is the transpose of K(z, Z); let the mean function m(z) = 0, and the covariance function is:

[0027]

[0028] in, is the output variance, z′ is the derivative of z, is the length scale covariance matrix, expressed as:

[0029]

[0030] Among them, l i ≥0, i=1,,...,n z , n z is the dimension of the test point z, and the hyperparameters of the kernel function are:

[0031]

[0032] (2) Using the explicit fourth-order Runge-Kutta method, the ASV dynamics model with additional disturbance is discretized as:

[0033]

[0034] in, t k +1 moment status, By discrete state variables and discrete control inputs The partial components of is the transformation matrix for the state, is the transformation matrix for the control input, f true represents the actual ASV dynamics model, f nom represents the nominal ASV kinetic model, is a multidimensional GP constructed by independently training GPs in each dimension to approximate the unmodeled dynamics E(v), The distribution of is:

[0035]

[0036] in, Indicates that unmodeled dynamics are at the test point The value vector at , represents the unmodeled dynamics of the jth dimension at the test point The value at The mean and variance Respectively expressed as:

[0037]

[0038] in, for Corresponding GP The mean value of for Corresponding GP The covariance at , d represents a Gaussian process;

[0039] The distribution of Gaussian process noise ω is is the covariance of the single-dimensional GP noise, the state input matrix B d Used to select states that are affected by model errors.

[0040] Furthermore, in step S3, based on the dictionary size limit and the maximum likelihood method, the update criteria for dictionary data and hyperparameters are established, including:

[0041] (1) Dictionary Design a maximum capacity n max , n max is a preset threshold. When the number of data points exceeds this preset threshold, this limit is maintained by removing the point with the lowest covariance function value;

[0042] (2) The maximum likelihood method is used to find the hyperparameter θ that makes the observed data most likely * ,θ * Obtained by the following formula:

[0043] θ * = argmax θ logp(Y|Z,θ)

[0044] The log-likelihood function is:

[0045]

[0046] Among them, Z is the dictionary input data set, Y is the dictionary output data set, K(Z, Z) is the covariance matrix of Z and Z, is the covariance of GP noise, I is the identity matrix, K y is the inverse matrix of , and θ is the hyperparameter vector of the kernel function.

[0047] Furthermore, the GP-MPC framework constructed in step S4 includes:

[0048] The objective function is:

[0049]

[0050] in, is the mean sequence of all states in the control time domain at time t, u 0：N-1|t is the sequence of all control inputs in the control domain at time t, is the state mean of the kth step in the control domain at time t, x r (k|t) is the reference state of the kth step in the control domain at time t, u k|t is the control input of the kth step in the control domain at time t, is the terminal state mean in the control time domain at time t, x r (N|t) is the terminal reference state in the control domain at time t, l is the stage cost function, l f is the terminal cost function, N is the control time domain;

[0051] The initial conditions are:

[0052]

[0053] in, is the initial state mean in the control time domain at time t, and xt is the real system state at time t;

[0054] The prediction equation is:

[0055]

[0056] in, is the state mean of the k+1th step in the control time domain at time t, is the mean of the Gaussian process of the kth step in the control time domain at time t, B d Input matrix for the state;

[0057] The state and control input constraints are:

[0058]

[0059] in, is the set of feasible states, A set of feasible control inputs;

[0060] The terminal state constraints are:

[0061]

[0062] in, is the terminal state set;

[0063] At each time instant, the optimization problem of the GP-MPC framework is solved to obtain the control input sequence is the optimal initial control input, is the optimal terminal control input, and Applied to the system; using a linearized approximation method to propagate the desired state, specifically:

[0064]

[0065] in, is the state mean at time k+1, is the state mean at time k, is the Gaussian process mean at time k, u k is the discrete control input.

[0066] Furthermore, the method for designing the terminal state set is:

[0067] Consider a terminal control law:

[0068]

[0069] in, is the control input obtained by updating the terminal control law, μ x is the mean of the states, is the terminal feedback gain, u r is the ideal reference control input; the terminal state set Designed to:

[0070]

[0071] in, It is obtained by substituting the calculation results of the terminal control law into the prediction equation.

[0072] Furthermore, the method for designing an algorithm for solving the GP-MPC framework optimization problem in step S5 is:

[0073] (1) At each time t, solve the MPC framework to obtain the predicted state sequence X t =[x 1|t , x 2|t , ..., x N|t ] T and the control input sequence U executed t =[u 0|t ,u 1|t , ..., u N-1|t ] T , where X 1|t is the predicted state at time t+1, x 2|t is the predicted state at time t+2, X N|t is the predicted state at time t+N, u 0|t is the control input executed at time t, u 1|t is the control input executed at time t+1, u N-1|t is the control input executed at time t+N-1; the multi-point linear model of ASV is based on the state sequence X predicted in the previous step t-1 and control input sequence U t-1 , which is expressed as:

[0074]

[0075] Among them, x k+1|t is the state of the k+1th step in the control time domain at time t, f est To predict the ASV kinetic function, x k+1|t-1 is the state of the k+1th step in the control time domain at time t-1, u k+1|t-1 is the control input of the k+1th step in the control domain at time t-1, is the gradient of the state, is the gradient of the control input, x k|t is the state of the kth step in the control time domain at time t, u k|t is the control input of the kth step in the control domain at time t, x is the state variable, u is the control input, and:

[0076]

[0077] (2) The mean and covariance matrix of the first-order derivative of GP are:

[0078]

[0079] Among them, d′ GP (z) is the first-order derivative of GP, μ d ′(z) is the mean of the first-order derivative of GP at z, ∑ d ′(z) is the covariance of the first-order derivative of GP at z, K(z, Z) is the covariance matrix of z and Z, K(Z, Z) is the covariance matrix of Z and Z, is the covariance of GP noise, I is the identity matrix, K(z, z) is the covariance of z and z;

[0080] (3) Based on the multi-point linearization model of ASV in step (1), the state prediction model in the prediction time domain is:

[0081] X t =P x x(t)+P u U t +P e D e

[0082] in:

[0083]

[0084] in,

[0085] (4) The cost function of the GP-based multi-point linearized MPC is reformulated as:

[0086]

[0087] Among them, J is the cost function, is the reference state at time t+1, is the reference state at time t+2, is the reference state at time t+N, and the matrix Q is the state weight matrix, Q f is the terminal state weight matrix, R is the weighted matrix. Substituting the state prediction model into the above formula, we get:

[0088]

[0089]

[0090] in, For U t The transposed matrix of

[0091] After omitting the tops that are not relevant to the control input sequence, the cost function is summarized in quadratic form:

[0092]

[0093] in, is the simplified cost function;

[0094] Through the multi-point linearization method, the GP-based nonlinear MPC optimization problem is transformed into a quadratic cost function in step (4) and a multi-point prediction model in step (3).

[0095] The system corresponding to the method includes:

[0096] A dynamic model building unit is used to build an ASV dynamic model for the trajectory tracking problem of the autonomous surface vehicle (ASV);

[0097] Unmodeled dynamics approximation unit, used to introduce Gaussian process GP into the trajectory tracking system of ASV, approximate the unmodeled dynamics of the system, and obtain the GP prediction model;

[0098] An updating criterion establishment unit is used to establish updating criteria for dictionary data and hyperparameters based on dictionary size restrictions and maximum likelihood method to optimize the GP prediction model;

[0099] The GP-MPC framework and terminal state set construction unit is used to combine the optimized GP prediction model with the ASV dynamics model, use the learning-based model predictive control MPC method, build the GP-MPC framework, and design the terminal state set;

[0100] The optimization control unit is used to derive the state prediction model in the prediction time domain based on the multi-point linearization method, and design an algorithm to solve the GP-MPC framework optimization problem in combination with the local linear model predictive control LMPC method to achieve real-time control of ASV trajectory tracking.

[0101] A computer-readable storage medium for storing and executing the method, wherein the computer-readable storage medium stores computer instructions, and when the computer instructions are called, they are used to execute the steps of the multi-point linearization model prediction trajectory tracking control method based on Gaussian process.

[0102] Beneficial effects: Compared with the prior art, the significant technical effects of the present invention are as follows: (1) Based on the multi-point linearization model of ASV, a corresponding local linear model predictive control (LMPC) method is proposed and combined with the GP-MPC framework; compared with other trajectory tracking control methods based on GP-MPC, the solution characteristics of MPC rolling time domain are combined, the utilization rate of historical data is fully improved, the calculation amount of MPC solution optimization problem is greatly reduced, and the requirements of real-time control can be met; (2) Compared with other trajectory tracking control methods based on GP-MPC, the present invention proposes a terminal control law and Based on this, a terminal state set is designed in the GP-MPC framework, which provides theoretical help to ensure the iterative feasibility and asymptotic stability of the GP-MPC framework; (3) Based on the dictionary size limit and the maximum likelihood method, the dictionary data and hyperparameter update criteria are established; by limiting the size of the dictionary data set, the computational complexity problem caused by the large data set is effectively prevented, and supplemented by the hyperparameter update criterion, the prediction accuracy of the GP model is guaranteed while reducing the amount of calculation; (4) It can be easily expanded to drone or unmanned vehicle systems, as well as other motion control problems, and has broad application prospects and practicality. BRIEF DESCRIPTION OF THE DRAWINGS

[0103] Figure 1 It is a schematic diagram of the steps of the multi-point linearization model prediction trajectory tracking control method based on Gaussian process of the present invention;

[0104] Figure 2 It is a diagram showing the position state and control input changes during the ASV trajectory tracking process of the simulation 1 of the present invention;

[0105] Figure 3 It is a schematic diagram of the ASV trajectory tracking results based on different control methods of the simulation 1 of the present invention;

[0106] Figure 4 It is a diagram showing the position state and control input changes during the ASV trajectory tracking process of the simulation 2 of the present invention;

[0107] Figure 5 It is a diagram showing the change of disturbance and prediction error during the ASV trajectory tracking process of the simulation 2 of the present invention;

[0108] Figure 6 It is a schematic diagram of the ASV trajectory tracking results based on different control methods of simulation 2 of the present invention. DETAILED DESCRIPTION

[0109] In order to make the purpose of the invention, technical solutions and beneficial effects of the invention more clear and definite, the purpose of the invention, technical solutions and beneficial effects of the invention of the invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the embodiments are only used to explain and understand the present invention and are not used to limit the present invention.

[0110] like Figure 1 As shown, the Gaussian process-based multi-point linearized model predictive trajectory tracking control (GP-MP-LMPC) method disclosed in the embodiment of the present invention first constructs a dynamic model for the autonomous surface vehicle (ASV) trajectory tracking problem; then applies the Gaussian process (GP) to the trajectory tracking control system to approximate the unmodeled dynamics of the system; then, based on the dictionary size limit and the maximum likelihood method, the dictionary data and hyperparameter update criteria are established; then, combined with the learning-based model predictive control (MPC) method, the GP-MPC framework is constructed, and the terminal state set is designed. Subsequently, based on the local linear MPC (LMPC) method of the multi-point linearized model, an algorithm for solving the trajectory tracking problem is designed, and finally, while ensuring the ASV trajectory tracking accuracy, the calculation time is significantly reduced.

[0111] Specifically, in an embodiment of the present invention, a multi-point linearized model prediction trajectory tracking control method based on Gaussian process mainly includes the following steps:

[0112] Step 1: Aiming at the trajectory tracking problem of autonomous surface vehicle (ASV), an ASV dynamics model is constructed. The specific process of this step is:

[0113] Step 1-1: The three-degree-of-freedom nonlinear dynamic model of the autonomous surface vehicle (ASV) describes its horizontal position in the earth coordinate system, where the horizontal coordinate is The vertical axis is And heading angle In the hull coordinate system, the forward speed is Roll speed and yaw rate Together they represent the motion state of the autonomous surface vehicle. The ASV dynamics model is as follows:

[0114]

[0115] in, is the generalized coordinate vector, is the time derivative of η, is the velocity vector, T is the transpose of the matrix, is the time derivative of v. The unmodeled dynamics due to gravity, buoyancy, and environmental forces and moments are Indicates that e1(v) is the component that affects the forward velocity in the unmodeled dynamics, e2(v) is the component that affects the roll velocity in the unmodeled dynamics, and e3(v) is the component that affects the yaw angular velocity in the unmodeled dynamics; the torque of the control force is τ = [τ u τ v τ r ] T , τ u is the torque in the direction of forward speed, τ v is the moment in the direction of the rolling velocity, τ r is the torque in the direction of the yaw angular velocity. R(η) is the rotation matrix, defined as:

[0116]

[0117] Where ψ is the heading angle.

[0118] M is a positive definite symmetric inertia matrix, expressed as:

[0119]

[0120] in, m is the mass of ASV, is the hydrodynamic coefficient related to the first-order derivative of u, is the hydrodynamic coefficient related to the first-order derivative of v, I z is the identity matrix, is the hydrodynamic coefficient related to the first-order derivative of r, x g are the relevant ASV model parameters, is the hydrodynamic coefficient related to the first-order derivative of r.

[0121] C(v) is the Coriolis matrix, defined as:

[0122]

[0123] Among them, C 13 (v)=-M 22 v s -M 23 r s , C 23 (v) = M 11 u s .

[0124] D(v) is the damping matrix, expressed as:

[0125]

[0126] The damping coefficient is:

[0127] D 11 (v) = -Xu -X |u|u |u|-X uuu u 2 ,

[0128] D 22 (v) = -Y v -Y |v|v |v|-Y |r|v |r|,

[0129] D 23 (v) = -Y r -Y |v| r|v|-Y |r|r |r|,

[0130] D 32 (v)=-N v -N |v|v |v|-N |r|v |r|,

[0131] D 33 (v)=-N r -N |v|r |v|-N |r|r |r|.

[0132] Among them, X (·) , Y (·) and N (·) are the hydrodynamic coefficients, X u , X |u|u , X uuu are the hydrodynamic coefficients related to u, Y v , Y |v|v 、N v 、N |v|v are the hydrodynamic coefficients related to v, Y |r|v , Y |v|r 、N |r|v 、N |v|r are the hydrodynamic coefficients related to r and v, Y r , Y |r|r 、N r 、N |r|r are all hydrodynamic coefficients related to r, u is the forward speed of ASV in the hull coordinate system, v is the rolling speed of ASV in the hull coordinate system, and r is the yaw angular velocity of ASV in the hull coordinate system.

[0133] Step 1-2: Set the state variable x = [η T v T ] T =[xy ψ uvr] T, control input u = τ, x is the horizontal coordinate of ASV in the earth coordinate system, y is the vertical coordinate of ASV in the earth coordinate system. The dynamic model of ASV can be rewritten from equation (1) as:

[0134]

[0135] in, is the derivative of x with respect to time, A(v)=-M -1 (C(v)+D(v)), B=M -1 , M -1 is the inverse matrix of M.

[0136] Steps 1-3: When solving the ASV trajectory tracking problem, each state of the ASV dynamics model (6) has a corresponding reference state x r (t) = [x r (t), y r (t), ψ r (t),u r (t), v r (t), r r (t)] T , defined as follows:

[0137]

[0138] Among them, x r (t) is the reference horizontal coordinate, y r (t) is the reference ordinate, ψ r (t) is the reference heading angle, u r (t) is the reference forward speed, v r (t) is the reference rolling velocity, r r (t) is the reference yaw angular velocity, For x r The derivative of (t) with respect to time, for y r The derivative of (t) with respect to time, for The derivative with respect to time, for With respect to the time derivative, atan2 represents the four-quadrant tangent inverse operator. The above formula ensures that x r (t) satisfies the first ASV kinetic equation in the ASV kinetic model (1). Under the condition that the unmodeled kinetics is zero (i.e., E(v) = 0), x r Substituting (t) into the second ASV dynamics equation in the ASV dynamics model (1) yields the ideal reference control input u r =τ r ,u ris the ideal reference control input, τ r The trajectory tracking problem of ASV is actually to design the control input u so that the actual state x closely follows the reference state x. r (t).

[0139] Step 2: Introduce Gaussian process (GP) into the ASV trajectory tracking control system to approximate the unmodeled dynamics of the system, obtain the GP prediction model, and supplement the ASV dynamics model. The specific process of this step is:

[0140] Step 2-1: Apply GP to each dimension e of the unmodeled dynamics E(v) j (v) Modeling, j = 1, 2, 3, given a GP prior and a training set containing n sets of data z i For the training set The i-th group of input data, y i For the training set The GP posterior distribution d of the unmodeled dynamics in the jth dimension at the test point z for the i-th set of output data GP (z) is:

[0141]

[0142] Among them, e j (z) represents the value of the unmodeled dynamics in the jth dimension at the test point z, d GP The mean and covariance of (z) are:

[0143]

[0144] Among them, Z is the dictionary input data set, Y is the dictionary output data set, [K(z,Z)] 1i = k(z, z i ), [K(Z, Z)] ij = k(z i , z j ), K(z, Z) is the covariance matrix of z and Z, K(Z, Z) is the covariance matrix of Z and Z, is the covariance of GP noise, I is the identity matrix, K(z, z) is the covariance of z and z, and K(Z, z) is the transpose of K(z, Z). Assume that the mean function m(z) = 0 and the covariance function k(z, z′) is:

[0145]

[0146] in, is the output variance, z′ is the derivative of z, is the length scale covariance matrix, expressed as:

[0147]

[0148] Among them, l i ≥0, i=1,,...,n z , n z is the dimension of the test point z, and the hyperparameters of the kernel function are:

[0149]

[0150] in, is the covariance of the GP noise ω.

[0151] Step 2-2: Apply the explicit fourth-order Runge-Kutta method to discretize the ASV dynamics model with additional disturbances as follows:

[0152]

[0153] in, t k +1 moment status, The discrete states of the ASV dynamics model and discrete control inputs The partial components of is the transformation matrix for the state, is the transformation matrix for the control input, and these components are assumed to be the factors that affect the model error. true represents the actual ASV dynamics model, f nom represents the nominal ASV kinetic model. It is a multi-dimensional GP approximation constructed by independently training GPs in each dimension. Since there is no correlation between different dimensions, the predicted mean and variance of each dimension are calculated independently. The distribution of is:

[0154]

[0155] in, Indicates that unmodeled dynamics are at the test point The value vector at , represents the unmodeled dynamics of the jth dimension at the test point The value at The mean and variance Respectively expressed as:

[0156]

[0157] in, for Corresponding GP The mean value of for Corresponding GP The covariance at , d represents a Gaussian process.

[0158] In addition, the distribution of the noise ω of the multidimensional Gaussian process is is the covariance of the single-dimensional GP noise, the state input matrix B d Used to select states that are affected by model errors.

[0159] Step 3: Based on the dictionary size limit and maximum likelihood method, establish the update criteria for dictionary data and hyperparameters to optimize the GP prediction model to ensure the prediction accuracy of GP and reduce the computational complexity of GP. The specific process of this step is:

[0160] Step 3-1: Dictionary Design a maximum capacity n max , when the number of data points exceeds this preset threshold (maximum capacity n max ), this restriction is maintained by removing the points with the lowest covariance function values, which reduces the computational complexity of GP regression by limiting the number of data points in the dictionary.

[0161] Step 3-2: Maximum likelihood method is used to find the hyperparameter θ that makes the observed data most likely * ,θ * It can be obtained by the following formula:

[0162] θ * = argmax θ logp(Y|Z,θ) (16)

[0163] The log-likelihood function is:

[0164]

[0165] in, K y is the inverse matrix of , and θ is the hyperparameter vector defined above.

[0166] Step 4: Combine the optimized GP prediction model with the ASV dynamics model, use the learning-based model predictive control MPC method, build the GP-MPC framework, and design the terminal state set. The specific process of this step is:

[0167] Step 4-1, the objective function of the GP-MPC framework is:

[0168]

[0169] in, is the mean sequence of all states in the control time domain at time t, u 0：N-1|t is the sequence of all control inputs in the control domain at time t, is the state mean of the kth step in the control domain at time t, x r (k|t) is the reference state of the kth step in the control domain at time t, u k|t is the control input of the kth step in the control domain at time t, is the terminal state mean in the control time domain at time t, x r (N|t) is the terminal reference state in the control domain at time t, l is the stage cost function, l f is the terminal cost function, N is the control time domain;

[0170] The initial conditions are:

[0171]

[0172] in, is the initial state mean in the control time domain at time t, and xt is the real system state at time t;

[0173] The prediction equation is:

[0174]

[0175] in, is the state mean of the k+1th step in the control time domain at time t, is the mean of the Gaussian process of the kth step in the control time domain at time t;

[0176] The state and control input constraints are:

[0177]

[0178] in, is the set of feasible states, A set of feasible control inputs;

[0179] The terminal state constraints are:

[0180]

[0181] Where N represents the control time domain, is the terminal state set, and the objective function is composed of the stage cost function l and the terminal cost function l f Composition, defined as follows:

[0182]

[0183] in, The representative state x and the reference state x r The deviation for The transposed matrix, u T is the transposed matrix of u. The weight matrices Q, R and Q f are all positive definite matrices.

[0184] At each time instant, the optimization problem of the GP-MPC framework is solved to obtain the control input sequence is the optimal initial control input, is the optimal terminal control input, and Applied to the system. To improve the computational efficiency of the prediction equation and maintain computational feasibility, a linearized approximation method is used to propagate the desired state, specifically:

[0185]

[0186] in, is the state mean at time k+1, is the state mean at time k, is the Gaussian process mean at time k;

[0187] Step 4-2, consider a terminal control law:

[0188]

[0189] in, is the control input obtained by updating the terminal control law, μ x is the mean of the states, is the terminal feedback gain. The terminal state set Designed to:

[0190]

[0191] Among them, the terminal prediction state It is obtained by substituting the calculation results of the terminal control law into the prediction equation.

[0192] Step 5: Based on the multi-point linearization method, derive the state prediction model in the prediction time domain, combine the local linear model predictive control LMPC method, design an algorithm to solve the GP-MPC framework optimization problem, and realize the real-time control of ASV trajectory tracking. The specific process of this step is:

[0193] Step 5-1: At each time t, solve the MPC framework to obtain the predicted state sequence X t =[x 1|t , x 2|t , ..., x N|t] T and the control input sequence U executed t =[u 0|t ,u 1|t , ..., u N-1|t ] T , where x 1|t is the predicted state at time t+1, x 2|t is the predicted state at time t+2, X N|t is the predicted state at time t+N, u 0|t is the control input executed at time t, u 1|t is the control input executed at time t+1, u N-1|t is the control input executed at time t+N-1. The multi-point linear model of ASV is based on the state sequence X predicted in the previous step. t-1 and control input sequence U t-1 , which is expressed as:

[0194]

[0195] Among them, x k+1|t is the state of the k+1th step in the control time domain at time t, f est To predict the ASV kinetic function, x k+1|t-1 is the state of the k+1th step in the control time domain at time t-1, u k+1|t-1 is the control input of the k+1th step in the control domain at time t-1, is the gradient of the state, is the gradient of the control input, X k|t is the state of the kth step in the control time domain at time t, u k|t is the control input of the kth step in the control domain at time t, f est (x,u)=f nom (x,u)+B d d GP (z) and:

[0196]

[0197] In addition, during the multi-point linearization process, it is assumed that the planning input of step N is the same as that of step N-1, that is, u N|t-1 =u N-1|t-1 .

[0198] Step 5-2: Calculate and When , we need to use the derivative of Gaussian process (GP). According to previous research, the derivative of GP is still GP. Consider the GP model above, the output vector d is known GP and the unknown output vector d ′ GPThe joint probability density function of also obeys Gaussian distribution. The mean and covariance matrix of the first-order derivative of GP are:

[0199]

[0200]

[0201] Among them, d′ GP (z) is the first-order derivative of GP, μ d ′(z) is the mean of the first-order derivative of GP at z, ∑ d ′(z) is the covariance of the first-order derivative of GP at z.

[0202] Step 5-3: Based on the multi-point linearization model of ASV in step 5-1, the state prediction model in the prediction time domain can be summarized as follows:

[0203] X t =P x x(t)+P u U t +P e D e (32)

[0204] in:

[0205]

[0206] in,

[0207]

[0208] Step 5-4: Re-express the cost function of the GP-based multi-point linearized MPC as:

[0209]

[0210] Among them, J is the cost function, is the reference state at time t+1, is the reference state at time t+2, is the reference state at time t+N, and the matrix Q is the state weight matrix, Q f is the terminal state weight matrix, Substituting the state prediction model into the above formula, we can get:

[0211]

[0212] in, For U t The transposed matrix of

[0213] After omitting the top that is not related to the control input sequence, the quadratic cost function is summarized as:

[0214]

[0215] in, is the simplified cost function.

[0216] Through the multi-point linearization method, the GP-based nonlinear MPC optimization problem is transformed into a quadratic cost function in step 5-4 and a state prediction model in the time domain in step 5-3.

[0217] The following is a simulation verification of the ASV trajectory tracking control of the Gaussian process-based multi-point linearized model predictive trajectory tracking control framework designed by the present invention for environmental disturbances.

[0218] For ASV dynamics, the multivariate GP approximation d GP The dimension is 3, that is, n d =3. GP Each dimension of corresponds to the unmodeled dynamics E(v) = [e1(v) e2(v) e3(v)] T A dimension of . Then the matrix B d Given by:

[0219]

[0220] The components of the unmodeled dynamics, e1(v), e2(v), and e3(v), represent gravity, buoyancy, and environmental forces and their moments, respectively. These components depend on the shape, displacement, and motion state of the ASV, affecting its trajectory tracking performance, such as its ability to maintain heading and speed. In the simulation experiment, the unmodeled dynamics are given as:

[0221] e1(v)=0.279uv 2 +0.342v 2 r

[0222] e2(v)=0.912u 2 v

[0223] e3(v)=0.156ur 2 +0.278urv 3

[0224] Based on the above equation, it can be expressed that in is a zero matrix, and:

[0225]

[0226] The remaining ASV model parameters are summarized in Table 1.

[0227] Table 1

[0228]

[0229] All subsequent calculations were performed using MATLAB R2020a on a computer equipped with a 2.60 GHz i9 processor and 16 GB of memory. The fmincon function in MATLAB was used with the interior point method, a maximum of 6 iterations, and 10 -3 The optimization tolerance setting is used to balance control performance and computational efficiency.

[0230] The ASV trajectory tracking experiment was conducted in two scenarios: straight trajectory and curved trajectory.

[0231] Simulation experiment 1:

[0232] The reference straight line trajectory used in simulation experiment 1 is generated by the following formula:

[0233] y r =x r -1

[0234] According to steps 1-3, the reference state x r The corresponding components are as follows:

[0235] x r (t) = t + 3, y r (t) = t + 2,

[0236] ψ r (t) = tan -1 (1)

[0237] v r (t) = 0, r r (t)=0.

[0238] The weighting matrix is ​​set to R = diag([0.1, 0.1, 0.1]), Q = diag([10 6 , 10 6 , 10 3 , 10 2 , 10 2 , 10 2 ]) and Q f =2Q, the prediction range of MPC is N=8. According to the actual needs of ASV, the upper limit of the control input is set to 200. The initial hyperparameter is set to σ f =[0.01,0.01,0.01]T, and σ n =diag(σ f / 3). These hyperparameters are then optimized from their initial values. The maximum number of points in the dictionary is set to n max =300. The simulation time is 12 seconds, and the time step is 0.02 seconds. The initial posture of the ASV is η0 = [6.0, 0.0, 0.0] T , the initial velocity is V0 = [0.0, 0.0, 0.0] T .

[0239] Under the GP-MP-LMPC framework, the changes in the position state (x and y) and the control input u during the ASV trajectory tracking process are shown in the following figure: Figure 2 As shown in the figure. Starting from the initial position, the GP model is used to make predictions immediately and its dictionary is continuously updated with new data points. After 1.5 seconds, the GP model continues to make predictions, but the dictionary is no longer updated. Figure 2 As can be seen in Figure 2, the ASV successfully tracks the reference trajectory after about 3 seconds and there is no deviation in the subsequent simulation. After the first 3 seconds of variation, the control input gradually stabilizes and approaches zero.

[0240] Figure 3 The trajectory tracking performance of the GP-MP-LMPC framework is compared with that of MP-LMPC and GP-NMPC. It can be clearly seen that all three methods successfully achieve the trajectory tracking of ASV, among which GP-NMPC and GP-MP-LMPC perform best, with a small performance gap between the two. MP-LMPC takes longer to achieve a satisfactory tracking effect, and there is greater uncertainty in the early stage of the trajectory. The comparison of some control performance indicators of the three control frameworks is shown in Table 2.

[0241] Table 2

[0242] index GP-MP-LMPC MP-LMPC GP-NMPC Prediction Error 0.00126 66.2696 0.00125 Calculation time (seconds) 10 7 5624

[0243] The prediction error is defined as the actual system dynamic model f true With the prediction model f nom +B d d GP The absolute value of the difference between , and only the first 3 seconds (150 time steps) are calculated. As can be seen from Table 2, the prediction errors of GP-MP-LMPC and GP-NMPC are significantly smaller than those of MP-LMPC. In addition, the computation time of GP-MP-LMPC is shorter, which indicates that the proposed GP-MP-LMPC framework has higher computational efficiency while achieving effective trajectory tracking. Further analysis shows that both GP-MP-LMPC and MP-LMPC frameworks support real-time control.

[0244] It is worth noting that in simulation experiment 1, since the reference trajectory is a straight line trajectory, ASV can track the reference trajectory after 3 seconds, so there is no need to make significant direction adjustments. The velocity components v and r in the state become very small, resulting in a small perturbation E value, which makes the prediction error after 3 seconds very small, and the learning effect of GP is not obvious. Therefore, we choose the total prediction error of the first 3 seconds for comparison.

[0245] Simulation experiment 2:

[0246] In order to better demonstrate the control performance of the framework proposed in this paper, in simulation experiment 2, we replaced the tracking trajectory of ASV with the following "8"-shaped trajectory:

[0247] x r (t) = 6.8sin(0.3t)-0.5

[0248] y r (t) = 9.6cos(0.15t) - 6

[0249] The initial posture of ASV is η0 = [-0.4, 0.0, 0.0] T , the initial speed is the same as that set in simulation experiment 1. In order to fully display the “8”-shaped trajectory, the simulation time is set to 42 seconds, and other parameter settings are consistent with simulation experiment 1.

[0250] Figure 4 The change of position state (x and y) and control input over time during ASV trajectory tracking using the GP-MP-LMPC framework is shown. It can be seen that ASV starts to track the reference trajectory after about 3 seconds, and the control input tends to be stable with small regular fluctuations. Figure 5 The change of disturbance E and the impact of the GP model on the prediction error are shown. In the first 3 seconds, the ASV has not reached the reference trajectory and needs to accelerate, which increases the disturbance E. However, the GP model effectively alleviates the impact of the disturbance. With GP, the prediction error is reduced to a very small level after 3 seconds, and the total prediction error is about ten times smaller than the disturbance.

[0251] Figure 6 The tracking effects of the three control methods, GP-MP-LMPC, MP-LMPC and GP-NMPC, on the “8” curve trajectory are shown. The values ​​of the control performance indicators (prediction error and calculation time) are shown in Table 3.

[0252] Table 3

[0253] index GP-MP-LMPC MP-LMPC GP-NMPC Prediction Error 0.78149 79.2069 0.72263 Calculation time (seconds) 40 34 14622

[0254] from Figure 6It is not difficult to see from Table 3 that GP-MP-LMPC and GP-NMPC can quickly guide the ASV to the reference trajectory, while MP-LMPC has poor control performance and large fluctuations in the early stage. In addition, the GP-MP-LMPC framework ensures a prediction error comparable to that of GP-NMPC while taking less time to compute, which is critical for practical applications. At the same time, similar to simulation experiment 1, only the GP-MP-LMPC and MP-LMPC frameworks support real-time control.

[0255] Based on the above two simulation experiments, the Gaussian process-based multi-point linearized model predictive trajectory tracking control (GP-MP-LMPC) framework design method designed in the present invention realizes the trajectory tracking control of the autonomous surface vehicle (ASV) under the condition of environmental disturbance with high uncertainty, and can meet the accuracy and real-time requirements.

[0256] It is worth noting that the above is only a more preferred embodiment of the present invention, which is only used to verify and explain the present invention, and does not constitute a limitation on the present invention. Various changes, equivalent substitutions and modifications made by relevant technical personnel without departing from the technical ideas of the present invention are all included in the protection scope of the present invention.

Claims

1. A multi-point linear model prediction trajectory tracking control method based on Gaussian process, characterized in that: The following steps are involved: S1. Aiming at the trajectory tracking problem of autonomous surface vehicle (ASV), an ASV dynamics model is constructed. S2, introduce the Gaussian process GP into the trajectory tracking system of ASV, approximate the unmodeled dynamics of the system, and obtain the GP prediction model; S3. Based on dictionary size limitation and maximum likelihood method, establish the updating criteria of dictionary data and hyperparameters to optimize the GP prediction model; S4. Combining the optimized GP prediction model with the ASV dynamics model, using the learning-based model predictive control MPC method, constructing the GP-MPC framework, and designing the terminal state set; S5. Based on the multi-point linearization method, the state prediction model in the prediction time domain is derived. Combined with the local linear model predictive control LMPC method, an algorithm for solving the GP-MPC framework optimization problem is designed to achieve real-time control of ASV trajectory tracking.

2. The multi-point linear model prediction trajectory tracking control method based on Gaussian process according to claim 1 is characterized in that: The ASV kinetic model expression constructed in step S1 is: in, is the time derivative of the state variable x, x = [η T v T ] T =[xy ψ uvr] T , is a generalized coordinate vector, where x, y, and ψ are the horizontal coordinate, vertical coordinate, and heading angle of ASV in the earth coordinate system, respectively. is the velocity vector, u, v, and r are the forward velocity, roll velocity, and yaw velocity of the ASV in the hull coordinate system, respectively. T is the transpose of the matrix, u is the control input, and u = τ, τ = [τ u τ v τ r ] T is the torque of the control force, τ u is the torque in the direction of forward speed, τ v is the moment in the direction of the rolling velocity, τ r is the torque in the yaw angular velocity direction, R(η) is the rotation matrix, A(v)=-M -1 (C(v)+D(v)), B=M -1 , M -1 is the inverse matrix of M, M is the positive definite symmetric inertia matrix, C(v) is the Coriolis matrix, D(v) is the damping matrix, represents the unmodeled dynamics in the ASV trajectory tracking problem, e1(v) is the component of the unmodeled dynamics that affects the forward velocity, e2(v) is the component of the unmodeled dynamics that affects the roll velocity, and e3(v) is the component of the unmodeled dynamics that affects the yaw velocity.

3. The multi-point linear model prediction trajectory tracking control method based on Gaussian process according to claim 2 is characterized in that: When solving the ASV trajectory tracking problem in step S1, each state of the ASV trajectory tracking system has a corresponding reference state x r (t) = [x r (t), y r (t), ψ r (t),u r (t), v r (t), r r (t)] T , defined as follows: v r (t)=0, Among them, x r (t) is the reference horizontal coordinate, y r (t) is the reference ordinate, ψ r (t) is the reference heading angle, u r (t) is the reference forward speed, v r (t) is the reference rolling velocity, r r (t) is the reference yaw angular velocity, For x r The derivative of (t) with respect to time, for y r The derivative of (t) with respect to time, for The derivative with respect to time, for With respect to the time derivative, atan2 represents the four-quadrant inverse tangent operator.

4. The multi-point linear model prediction trajectory tracking control method based on Gaussian process according to claim 1 is characterized in that: In step S2, the Gaussian process GP is introduced into the trajectory tracking system of the ASV to approximate the unmodeled dynamics of the system and obtain the GP prediction model, including: (1) Apply GP to each dimension e of the unmodeled dynamics E(v) j (v) Modeling, j = 1, 2, 3, given a GP prior and a training set containing n sets of data z i For the training set The i-th group of input data, y i For the training set The GP posterior distribution d of the unmodeled dynamics in the jth dimension at the test point z for the i-th set of output data GP (z) is: Among them, e j (z) represents the value of the unmodeled dynamics in the jth dimension at the test point z, d GP The mean and covariance of (z) are: Among them, Z is the dictionary input data set, Y is the dictionary output data set, [K(z,Z)] 1i = k(z, z i ), [K(Z, Z)] ij = k(z i , z j ), K(z, Z) is the covariance matrix of z and Z, K(Z, Z) is the covariance matrix of Z and Z, is the covariance of GP noise, I is the identity matrix, K(z, z) is the covariance of z and z, and K(Z, z) is the transpose of K(z, Z); let the mean function m(z) = 0, and the covariance function is: in, is the output variance, z′ is the derivative of z, is the length scale covariance matrix, expressed as: Among them, l i ≥0, i=1,,...,n z , n z is the dimension of the test point z, and the hyperparameters of the kernel function are: (2) Using the explicit fourth-order Runge-Kutta method, the ASV dynamics model with additional disturbance is discretized as: in, t k +1 moment status, By discrete state variables and discrete control inputs The partial components of is the transformation matrix for the state, is the transformation matrix for the control input, f true represents the actual ASV dynamics model, f nom represents the nominal ASV kinetic model, is a multidimensional GP constructed by independently training GPs in each dimension to approximate the unmodeled dynamics E(v), The distribution of is: in, Indicates that unmodeled dynamics are at the test point The value vector at , represents the unmodeled dynamics of the jth dimension at the test point The value at The mean and variance Respectively expressed as: in, for Corresponding GP The mean value of for Corresponding GP The covariance at , d represents a Gaussian process; The distribution of Gaussian process noise ω is is the covariance of the single-dimensional GP noise, the state input matrix B d Used to select states that are affected by model errors.

5. The multi-point linear model prediction trajectory tracking control method based on Gaussian process according to claim 1 is characterized in that: In step S3, based on the dictionary size limit and the maximum likelihood method, the update criteria for dictionary data and hyperparameters are established, including: (1) Dictionary Design a maximum capacity n max , n max is a preset threshold. When the number of data points exceeds this preset threshold, this limit is maintained by removing the point with the lowest covariance function value; (2) The maximum likelihood method is used to find the hyperparameter θ that makes the observed data most likely * ,θ * Obtained by the following formula: θ * =argmax θ logp(Y|Z,θ) The log-likelihood function is: Among them, Z is the dictionary input data set, Y is the dictionary output data set, is the covariance matrix of Z and Z, is the covariance of GP noise, I is the identity matrix, K y is the inverse matrix of , and θ is the hyperparameter vector of the kernel function.

6. The multi-point linear model prediction trajectory tracking control method based on Gaussian process according to claim 1 is characterized in that: The GP-MPC framework constructed in step S4 includes: The objective function is: in, is the mean sequence of all states in the control time domain at time t, u 0：N-1|t is the sequence of all control inputs in the control domain at time t, is the state mean of the kth step in the control domain at time t, x r (k|t) is the reference state of the kth step in the control domain at time t, u k|t is the control input of the kth step in the control domain at time t, is the terminal state mean in the control time domain at time t, x r (N|t) is the terminal reference state in the control domain at time t, l is the stage cost function, l f is the terminal cost function, N is the control time domain; The initial conditions are: in, is the initial state mean in the control time domain at time t, and xt is the real system state at time t; The prediction equation is: in, is the state mean of the k+1th step in the control time domain at time t, is the mean of the Gaussian process of the kth step in the control time domain at time t, B d Input matrix for the state; The state and control input constraints are: in, is the set of feasible states, A set of feasible control inputs; The terminal state constraints are: in, is the terminal state set; At each time instant, the optimization problem of the GP-MPC framework is solved to obtain the control input sequence is the optimal initial control input, is the optimal terminal control input, and Applied to the system; using a linearized approximation method to propagate the desired state, specifically: in, is the state mean at time k+1, is the state mean at time k, is the Gaussian process mean at time k, u k is the discrete control input.

7. The multi-point linear model prediction trajectory tracking control method based on Gaussian process according to claim 6 is characterized in that: The method for designing a terminal state set is: Consider a terminal control law: in, is the control input obtained by updating the terminal control law, μ x is the mean of the states, is the terminal feedback gain, u r is the ideal reference control input; the terminal state set Designed to: in, It is obtained by substituting the calculation results of the terminal control law into the prediction equation.

8. The multi-point linear model prediction trajectory tracking control method based on Gaussian process according to claim 1 is characterized in that: The method for designing an algorithm for solving the GP-MPC framework optimization problem in step S5 is: (1) At each time t, solve the MPC framework to obtain the predicted state sequence X t =[x 1|t , x 2|t , ..., x N|t ] T and the executed control input sequence U t =[u 0|t ,u 1|t , ..., u N-1|t ] T , where x 1|t is the predicted state at time t+1, x 2|t is the predicted state at time t+2, x N|t is the predicted state at time t+N, u 0|t is the control input executed at time t, u 1|t is the control input executed at time t+1, u N-1|t is the control input executed at time t+N-1; the multi-point linear model of ASV is based on the state sequence X predicted in the previous step t-1 and control input sequence U t-1 , which is expressed as: Among them, x k+1|t is the state of the k+1th step in the control time domain at time t, f est To predict the ASV kinetic function, x k+1|t-1 is the state of the k+1th step in the control time domain at time t-1, u k+1|t-1 is the control input of the k+1th step in the control domain at time t-1, is the gradient of the state, is the gradient of the control input, x k|t is the state of the kth step in the control time domain at time t, u k|t is the control input of the kth step in the control domain at time t, x is the state variable, u is the control input, and: (2) The mean and covariance matrix of the first-order derivative of GP are: Among them, d′ GP (z) is the first-order derivative of GP, μ d ′(z) is the mean of the first-order derivative of GP at z, ∑ d ′(z) is the covariance of the first-order derivative of GP at z, K(z, Z) is the covariance matrix of z and Z, K(Z, Z) is the covariance matrix of Z and Z, is the covariance of GP noise, I is the identity matrix, K(z, z) is the covariance of z and z; (3) Based on the multi-point linearization model of ASV in step (1), the state prediction model in the prediction time domain is: X t =P x x(t)+P u U t +P e D e in: in, (4) The cost function of the GP-based multi-point linearized MPC is reformulated as: Among them, J is the cost function, is the reference state at time t+1, is the reference state at time t+2, is the reference state at time t+N, and the matrix Q is the state weight matrix, Q f is the terminal state weight matrix, R is the weighted matrix. Substituting the state prediction model into the above formula, we get: in, For U g The transposed matrix of After omitting terms that are not relevant to the control input sequence, the cost function is summarized in quadratic form: in, is the simplified cost function; Through the multi-point linearization method, the GP-based nonlinear MPC optimization problem is transformed into a quadratic cost function in step (4) and a multi-point prediction model in step (3).

9. A multi-point linearized model prediction trajectory tracking control system based on Gaussian process, characterized in that: include: A dynamic model building unit is used to build an ASV dynamic model for the trajectory tracking problem of the autonomous surface vehicle (ASV); Unmodeled dynamics approximation unit, used to introduce Gaussian process GP into the trajectory tracking system of ASV, approximate the unmodeled dynamics of the system, and obtain the GP prediction model; An updating criterion establishment unit is used to establish updating criteria for dictionary data and hyperparameters based on dictionary size restrictions and maximum likelihood method, and optimize the GP prediction model; The GP-MPC framework and terminal state set construction unit is used to combine the optimized GP prediction model with the ASV dynamics model, use the learning-based model predictive control MPC method, build the GP-MPC framework, and design the terminal state set; The optimization control unit is used to derive the state prediction model in the prediction time domain based on the multi-point linearization method, and design an algorithm to solve the GP-MPC framework optimization problem in combination with the local linear model predictive control LMPC method to achieve real-time control of ASV trajectory tracking.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores computer instructions, which, when called, are used to execute the steps of the multi-point linearization model prediction trajectory tracking control method based on Gaussian process according to any one of claims 1 to 8.

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