Model prediction control method and device based on online Gaussian process regression for precise air-drop system

Abstract

The invention discloses an online Gaussian process regression-based model prediction control method and device for a precise air-drop system, and the method comprises the steps: processing a dynamic model of a parafoil system into a transverse lateral motion linear time-invariant nominal model through the aerodynamic analysis of the parafoil system, and considering the flexibility of the parafoil and a residual part of the nominal model of wind disturbance; designing an online Gaussian process regression algorithm to learn a (potential) time-varying nominal model residual part; a new performance functional is designed for tracking error weighting based on posterior prediction variance of a Gaussian process regression algorithm, a random optimal control problem is constructed and solved in two parts, one part is used for controlling a nominal model, a suboptimal strategy of an off-line solving Riccati equation is adopted, and the optimal control problem is solved. And the other part is used for controlling the residual error and disturbance of the model obtained by online Gaussian process regression learning, and solving to obtain an optimal control sequence so as to obtain an optimal control law. The invention provides a novel and efficient solution for accurate homing control of the air-drop system.

Description

Technical Field

[0001] The present invention belongs to the field of navigation, guidance and control, and specifically proposes a model predictive control method and device based on online Gaussian process regression for a precision airdrop system. Background Art

[0002] Precision airdrop systems play an important role in military, rescue, and spacecraft recovery applications. Traditional round parachute airdrop systems are difficult to perform long-distance and high-precision airdrop tasks due to their uncontrollability and wind sensitivity. Precision airdrop systems based on parafoils have become a new hotspot in airdrop technology due to their excellent characteristics such as high glide ratio and high maneuverability. However, parafoil-based systems face a series of challenges in homing control:

[0003] 1. As a large flexible membrane structure, the aerodynamic shape of the parafoil is difficult to strictly maintain during flight, and the aerodynamic parameters are unsteady. The system model exhibits time-varying characteristics and large uncertainties, making it difficult to accurately model through physical mechanisms; 2. The parafoil has a large reference area and a relatively light system weight, resulting in high sensitivity of the system to the wind environment, and there is a coupling relationship between the system aerodynamic characteristics and the wind environment. The uncertainty caused by external disturbances has a large impact; 3. As a disposable device, the airdrop system is subject to cost constraints, and the insufficient sensing accuracy introduces random uncertainties into the feedback control of the system. 4. Model predictive control (MPC) is currently the most suitable method in the field of trajectory tracking control. However, the performance of MPC highly depends on the model, and model mismatch will greatly deteriorate the tracking accuracy, even leading to trajectory divergence, system out-of-control. Since it is difficult to model the precision airdrop system through physical mechanisms, general MPC methods are difficult to perform homing tasks in real complex environments.

[0004] The current research on the homing control of precision airdrop systems mainly focuses on a simplified rigid body six-degree-of-freedom model, with less consideration of wind disturbances and complex couplings between models. At the same time, the treatment of model noise and sensing noise is relatively limited. Also, the existing MPC methods based on model learning are not suitable for direct application in precision airdrop systems, mainly for two reasons: 1. Most of the existing model learning methods adopt parametric learning methods (such as neural network-based learning methods), whose fixed parametric structure limits the flexibility of their representation (especially when test data cannot be well represented in the test set), and neural networks usually require a large amount of data for offline training, making it difficult to effectively handle the problem of model time-variation online; 2. Most of the existing model learning methods are deterministic learning methods, that is, they can only give predicted values but cannot effectively evaluate the accuracy of the predicted values. The data collected in airdrop tasks in complex environments is limited, and it is actually difficult to ensure the accuracy of the predicted values. Therefore, it is more suitable to use stochastic learning methods to learn the model, give the random distribution of the predicted values in the form of probability, and incorporate the statistical characteristics into the controller design in an effective form, so that even when the accuracy of the learning result is not ideal, the system can adopt a more cautious control strategy to achieve better performance in trajectory tracking with robustness and stability. Summary of the Invention

[0005] The purpose of the present invention is to address the deficiencies of the prior art. This application proposes a model predictive control method based on online Gaussian process regression for precision airdrop systems. Generally speaking, Gaussian process regression (GPR) is used to perform online learning on the system model, and then a model predictive control (MPC) method is designed based on the learned model. The Gaussian process-based model predictive control (GP-MPC) proposed in this application verifies the effectiveness and robustness of this method through theoretical analysis and simulation experiments, providing a novel and efficient solution for solving the homing control of precision airdrop systems.

[0006] The purpose of the present invention is achieved through the following technical solutions: A model predictive control method based on online Gaussian process regression for precision airdrop systems, characterized in that the method includes the following steps:

[0007] S1: By analyzing the aerodynamics of the parafoil system, its dynamic model is processed into a lateral motion linear time-invariant nominal model, and a non-linear (potential) time-varying nominal model residual part considering the flexibility of the parafoil and wind disturbances;

[0008] S2: Predict the non-linear time-varying nominal model residual part based on the online Gaussian process regression algorithm;

[0009] S3: Design a performance functional by weighting the tracking error with the posterior predictive variance of the Gaussian process regression algorithm, construct a stochastic optimal control problem, and solve it in two parts. One part is used to control the nominal model, and a suboptimal strategy for solving the Riccati equation is obtained offline. The other part is used to control the model residuals and disturbances learned by the online Gaussian process regression. Solve the optimal control problem online to obtain the optimal control sequence, and then obtain the optimal control law.

[0010] Further, in step S1, the lateral state of the parafoil system is where φ is the roll angle, ψ is the yaw angle, p is the roll rate, and r is the yaw rate; establish a discrete parafoil system model through system identification methods as the nominal model, which is expressed as follows:

[0011]

[0012] where, a ij and b i are constants to be identified, where a ij and b i both represent elements in the matrix, ij represents the rows and columns of the matrix, T is the sampling period of the discrete system, u is the control input of the system, the subscript k represents the variable corresponding to the k-th moment, and I 4 represents the 4-dimensional identity matrix.

[0013] Based on the flight mechanics characteristics of the airdrop system with a parafoil, establish a dynamic equation, neglect the longitudinal motion mode, and for the lateral motion mode, establish a functional relationship between the control input and the yaw angle of the system, and its expression is the yaw rate is a function of the roll angle φ, the body-axis roll rate p, the body-axis yaw rate r, and the control input u; introduce noise and discretize the functional relationship as follows:

[0014]

[0015] where, ψ k+1 represents the yaw angle at the (k + 1)-th time step, are the estimated values of the roll angle, yaw angle, body-axis roll rate, and body-axis yaw rate at the k-th time step respectively, u k is the control input at the k-th time step, ∈ k is the noise set at the k-th time step, including the measurement noise of the navigation system and the process noise introduced by the random part in aerodynamics.

[0016] Further, construct the state vector x = [ψ, φ, p, r] T , and the noisy measurement y k as follows:

[0017]

[0018] Among them, is the nominal model part, and f(x k , u k ) represents the model residual part, and σ represents the standard deviation of the measurement error; an iterative solution for the prediction of the yaw angle is performed based on the learning algorithm of Gaussian process regression.

[0019] Furthermore, when using Gaussian process regression for prediction, the data set is updated online in the following manner:

[0020] (1) Through the kernel linear independence test KIT, judge the residual of the new data point represented by the original data set; compare the obtained residual with the residual threshold, and when it is less than the parameter threshold, the new data point is included in the data set;

[0021] (2) Delete the data point with the least information in the data set, that is, delete the data point with the largest KL divergence.

[0022] Furthermore, in step S3, the optimal index function is constructed as follows:

[0023]

[0024] Among them, Q and R are constant weighting matrices, and ∑ k is the posterior covariance matrix corresponding to the k-step prediction value output by Gaussian process regression, e k is the tracking error of the system at the k-th step, N represents the maximum prediction step of the system, represents the mathematical expectation.

[0025] Furthermore, for the control of the nominal model, a sub-optimal strategy is adopted, and the optimal gain matrix at the steady state of the system is obtained by solving the Riccati equation regarding the system equation and performance index. By solving the optimal gain matrix offline, a sub-optimal control is formed for part of the nominal model.

[0026] Furthermore, for the control of the model residual, a nonlinear optimization method is used to optimize the optimization index, and the optimal control problem is reconstructed as follows:

[0027]

[0028] Among them, x dk is the expected value of the system state, and u M is the maximum limit value of the wing parachute input rudder deflection; the optimal control problem is solved iteratively to obtain an optimal control sequence for the model residual

[0029] In a second aspect, the present invention further provides a model predictive control device based on online Gaussian process regression for a precision airdrop system, including a memory and one or more processors. An executable code is stored in the memory. When the processor executes the executable code, the model predictive control method based on online Gaussian process regression for a precision airdrop system as described above is implemented.

[0030] In a third aspect, the present invention further provides a computer-readable storage medium, on which a program is stored. The program is characterized in that when it is executed by a processor, the model predictive control method based on online Gaussian process regression for a precision airdrop system as described above is implemented.

[0031] The model predictive control method based on online Gaussian process regression for a precision airdrop system proposed in this application has the following main advantages compared with general methods:

[0032] 1. The non-parametric nature, small sample learning ability, and online learning framework of the adopted learner can effectively cope with the time-varying and large uncertainties brought by the flexibility of the parafoil, wind disturbance, and sensing noise, and effectively predict the system model.

[0033] 2. As a stochastic prediction method, Gaussian process regression gives the prediction of the system model in the form of a Gaussian random distribution, enabling the learner to effectively evaluate the accuracy of the predicted value and perform risk assessment with variance information in a large uncertainty environment and when the model / perturbation mutates, and informing the controller.

[0034] 3. By considering the uncertainty of the learner in the optimization index of the optimal controller, a more cautious control strategy can be achieved, and better robustness can be obtained when extreme perturbations occur.

[0035] 4. By using a nominal model and adopting a framework that combines learning the nominal model residual with nominal sub-optimal control, the problem of insufficient numerical solution accuracy of the learner and the controller can be effectively addressed, and the stability and robustness of the algorithm when the solution accuracy is limited can be improved. Description of the Drawings

[0036] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0037] Figure 1 It is a schematic diagram of the offline flight trajectory of the airdrop system;

[0038] Figure 2 It is a schematic diagram of the online learning curve under ideal conditions;

[0039] Figure 3 Schematic diagram of tracking control curve under ideal conditions;

[0040] Figure 4 Schematic diagram of input curve under ideal conditions;

[0041] Figure 5 Schematic diagram of online learning curve without nominal model;

[0042] Figure 6 Schematic diagram of tracking curve without nominal model;

[0043] Figure 7 Schematic diagram of model learning curve under sine interference;

[0044] Figure 8 Schematic diagram of tracking curve under sine interference;

[0045] Figure 9 Schematic diagram of online learning effect under model mutation;

[0046] Figure 10 Schematic diagram of tracking control effect under model mutation;

[0047] Figure 11 Schematic diagram of full - process simulation of airdrop system.

[0048] Figure 12 Structural diagram of a model predictive control device for a precision airdrop system based on online Gaussian process regression according to the present invention. Detailed implementation manners

[0049] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described below with reference to the accompanying drawings and implementation cases. It should be understood that the specific implementation cases described herein are only used to explain the present invention and are not used to limit the present invention.

[0050] As Figure 1 shown, a model predictive control method for a precision airdrop system based on online Gaussian process regression provided by the present invention includes the following steps:

[0051] Step S1, design a system identification algorithm: According to the offline flight data of the airdrop system collected in advance, preliminarily identify the system model for use as the nominal model of the learner and the controller;

[0052] Step S2, design a learner: According to the physical characteristics of the parafoil, design an online learning algorithm based on Gaussian process regression to learn the dynamic model residuals of the airdrop system, mainly including the following 3 sub - steps:

[0053] Step S21: Analyze the physical characteristics of the parafoil according to flight mechanics, design a learning algorithm framework that conforms to the dynamic laws of the airdrop system, and determine the input-output relationship of the learning algorithm;

[0054] Step S22: Design the prior and parameter training algorithms for Gaussian process regression to obtain appropriate kernel functions and hyperparameters;

[0055] Step S23: Design an online Gaussian process regression algorithm: During online operation, screen the optimal information points in real time and design the corresponding recursive algorithm to meet the requirements of online efficient calculation;

[0056] Step S3: Design a controller: Incorporate the model learned online by the learner in Step S2 into the model predictive controller, and calculate the optimal control law online in real time according to the reference signal:

[0057] Step S31: Design an optimization index function that considers the uncertainty of the learner;

[0058] Step S32: Design constraints, and adopt different processing strategies for the nominal model and the residuals of the GP-based learning model to construct an optimal control problem. Use the nonlinear optimization solver in MATLAB and the strategy of rolling horizon optimization to numerically solve the constructed optimal control problem.

[0059] Step S4: Implementation steps: According to the design steps of S1 - S3, Step S4 gives the steps in actual application, which are mainly divided into two parts: offline and online use:

[0060] Step S41: Offline flight: To obtain offline flight data, first conduct a low-altitude airdrop, use the nominal controller / open-loop flight to obtain a small arbitrary flight trajectory, and store the flight data (including navigation data and input rudder deflection signals) for nominal model identification and subsequent offline training of the learner.

[0061] Step S42: Online application: Select appropriate update steps for the learner and the controller, and execute the GP-MPC algorithm after the airdrop to track the desired trajectory.

[0062] The specific implementation process of each step of the present invention is described as follows:

[0063] 1. Step S1 establishes a linear time-invariant (LTI) nominal model for the lateral motion of the airdrop system through a simple system identification.

[0064] Note that different from traditional control methods that rely entirely on system identification or mechanism modeling, in the present invention, the result of system identification is only used as a part of the model finally adopted by the controller to relieve the learning pressure of the learner and the solution pressure of the controller. In control algorithms that highly rely on system identification, complex model classes are often required to perform regression on flight data, and strict verification of the identification results is needed to ensure the accuracy of the model. However, in the method proposed in the present invention, since a learner is used to perform online learning on the model residuals, only the simplest model structure needs to be selected to establish a rough model for the system.

[0065] Theoretically, the learner proposed in the present invention can arbitrarily approximate the actual model of the airdrop system. However, considering that in actual use, due to limited numerical solution accuracy, such as matrix inversion and angle operations involved in the learner, and numerical solution problems of nonlinear optimization involved in the controller, if the entire model is handed over to the learner for learning and then the controller performs calculations based on the results of the learner, compared with the method using the nominal model, the solution error of the computer may have a greater impact on the control performance.

[0066] The advantages of establishing a nominal model using an identification algorithm in advance are mainly in two aspects: 1. Reducing the influence of the learning error of the learner, enabling the learner to only learn the model residuals; 2. Reducing the influence of the optimization solution error of the controller. The model of model predictive control is divided into two parts for processing. For the nominal model part, the present invention adopts a suboptimal strategy, offline solves the Riccati matrix for the nominal LTI model, and uses numerical methods to solve the nonlinear optimization problem for the model residuals learned by the learner. This reduces the pressure on the online numerical optimization solver and the influence of numerical errors. It should be noted that in the case of serious mismatch of the identification algorithm, the method proposed in the present invention still has excellent performance. In the embodiments, the present invention will show the control effects without using the nominal model and using a seriously mismatched nominal model in the form of simulation results.

[0067] The present invention uses a simple lateral small perturbation linearized model of the parafoil system as the model class for identification, and takes the lateral states as the roll angle φ, yaw angle ψ, roll angular velocity p, and yaw angular velocity r respectively. The present invention only focuses on the structure and form of the system matrix and input matrix of the small perturbation model (such as matrix dimension, non-zero element position, and whether it is a constant) to obtain the following discrete system, and no explanation is made for specific variables. With knowledge of flight mechanics and control systems, it can be analyzed that this model is a linear time-invariant system, and each element in the matrix is a constant at steady state. The discrete system is denoted as:

[0068]

[0069] where a ij and b iis a constant to be identified, where a ij and b i both represent elements in the matrix, ij represents the rows and columns in the matrix, T is the sampling period of the discrete system, u is the control input of the system, and the subscript k represents the variable corresponding to the k-th moment. I 4 represents a 4-dimensional identity matrix. Then, the constants a ij and b i in the above formula need to be determined through statistical algorithms.

[0070] The weighted recursive least squares method is adopted:

[0071] Let the parameter to be identified be Let the measurement data be n i represents the measurement noise with zero mean, and the subscript i represents the i-th measurement. Then, the identification equation can be written as:

[0072]

[0073] where

[0074]

[0075] The recursive weighted least squares estimation is as follows:

[0076]

[0077] where P i is the error covariance, Q is the covariance of the measurement noise, represents the estimated value of the parameter to be identified after using the i-th measurement data.

[0078] By substituting the measurement data and recursively operating the above algorithm, the final Substituting it into the discrete system, it can be used as the nominal model for subsequent GP-MPC.

[0079] 2. Step S2 is to design a learner, which is divided into the following 3 sub-steps:

[0080] (1) Step S21: First, analyze the flight mechanics characteristics of the airdrop system based on the parafoil. Establish a 6-degree-of-freedom model for the parafoil system as follows:

[0081] The kinematic equations are as follows:

[0082]

[0083] Among them, (x, y, z) are the position coordinates of the center of mass of the airdrop system in the inertial system, (u, v, w) are the velocity components of the system's center of mass in the body system, φ, θ, ψ are the three Euler angles of the system, namely the roll angle, pitch angle, and yaw angle, and (p, q, r) are the angular velocities of the system in the body system, s x , c x , t x are the common abbreviations of sinx, cosx, and tanx respectively;

[0084] The matrix T represents the rotation matrix from the inertial system to the body system:

[0085]

[0086] The dynamic equations are as follows:

[0087]

[0088] Among them, m T is the total mass of the airdrop system, I T is the total inertia of the airdrop system about its center of mass, F A is the aerodynamic force received by the airdrop system in the body system, F W is the total gravity received by the airdrop system in the body system. In addition, denote:

[0089]

[0090] Among them, I T represents the moment of inertia of the parafoil system, I ** and I **I represent the corresponding elements in the matrix I T and its inverse matrix respectively;

[0091] The parafoil system is a system with good longitudinal aerodynamic stability. During flight, the longitudinal flight characteristics of the airdrop system remain almost unchanged after entering the steady state and are not affected by the flight state and control input. That is, make the following assumptions: after entering the steady state, θ remains unchanged and q = 0. Therefore, when the airdrop system performs trajectory tracking, the control target is only for the lateral and directional motion modes. Specifically, it is to achieve yaw angle tracking. To achieve this control purpose, the present invention expects to obtain the functional relationship between the control input and the system yaw angle, and calls this equation the dynamic model of the airdrop system trajectory tracking.

[0092] Decouple the yaw angle from the kinematic equation to obtain:

[0093]

[0094] From the previous analysis, it is considered that q = 0 and θ is a constant. It can be seen that the change rate of the yaw angle It is a function of the yaw angular velocity r and the roll angle φ. Since the roll angle φ can be obtained through the navigation system, to investigate the transfer relationship from the input rudder deflection u to the yaw angle change rate it is necessary to analyze the yaw angular velocity r and the dynamic equations of rotation. First, according to the relevant information on parafoil aerodynamics, the aerodynamic moment function relationship is analyzed:

[0095]

[0096] where Q is the free-stream dynamic pressure, S is the reference area of the parafoil, and Cl, Cm, Cn are the aerodynamic moment coefficients of the parafoil, which are given in the form of functions. The longitudinal moment coefficient is not considered in the present invention, so the specific independent variables are not given.

[0097] So far, the dynamic model of the airdrop system has been established. Next, the present invention analyzes the uncertainties brought by the parafoil flexibility and wind disturbance to the model.

[0098] First of all, the parafoil flexibility will have a significant impact on the aerodynamic parameters by affecting the aerodynamic shape and structural deformation of the parafoil, and ultimately affect the aerodynamic force. Specifically, due to its flexibility, the parafoil is different from the ideal rigid body assumption, and it will generate deformation and deflection during flight. Since the aerodynamic parameters of the aircraft (such as lift coefficient, drag coefficient, lateral aerodynamic parameters, etc.) are usually directly related to the geometric shape, the deformation caused by flexibility will directly affect the aerodynamic shape of the parafoil, and then affect the aerodynamic parameters. For large-flexibility textiles, their deformation often shows strong random characteristics when subjected to external force disturbances. Therefore, the present invention describes the changes in aerodynamic parameters caused by deformation as two parts: deterministic and random. In addition, the change in the aerodynamic shape may also have a small impact on the reference area of the parafoil. The general expression of the aerodynamic force is F A =Q·S·C A , where Q is the dynamic pressure, F A is the aerodynamic force, S is the reference area of the aircraft in the airflow, and C A is the aerodynamic coefficient. Therefore, the hypothesis is obtained: The aerodynamic force considering the influence of parafoil flexibility is expressed as:

[0099]

[0100] where F A (x,t) is the aerodynamic force considering the influence of flexibility, is the nominal part of the aerodynamic parameter, ΔC A (x,t) is the deterministic part of the aerodynamic coefficient caused by deformation, w C (t) is a zero-mean random process, representing the random part of the aerodynamic coefficient caused by deformation. The influence of flexibility on the aerodynamic moment is the same.

[0101] The present invention represents the influence of external wind disturbances through the change of aerodynamic force, and considers that the aerodynamic force brought by wind disturbances is expressed as F d (t)+w F (t), where F d (t) represents the deterministic time-varying part of wind disturbances, and w F (t) is a zero-mean random process representing the random part of wind disturbances. The moment brought by wind disturbances is the same.

[0102] According to the above analysis, the parafoil model considering the influence of flexibility and wind disturbances is described, that is, the expression of the aerodynamic force (moment) of the parafoil considering the influence of flexibility and wind disturbances:

[0103]

[0104] Substitute the expression of the aerodynamic moment M A into the rotational dynamics equation. After analysis, the following conclusion can be finally obtained: for the parafoil, the yaw angle rate is a function of the roll angle φ, the body-axis roll angular velocity p, the body-axis yaw angular velocity r, and the control input u.

[0105] Considering that the actual input used by the learner is the measured value of navigation data, the superscript "^" is used here to represent the estimated value. For the convenience of numerical integration by the computer control system, the present invention gives the functional relationship in discrete form, and there is

[0106]

[0107] where the subscript k represents that the variable takes the value at the k-th step of time. The present invention denotes the set of noises as ∈, including the total random term considering the measurement noise of the navigation system and the process noise introduced by the random part in aerodynamics.

[0108] Here, an assumption is made: for the convenience of the derivation of the learner, it is considered that the total noise ∈ follows a Gaussian distribution. Due to the existence of the central limit theorem, this assumption is often approximately satisfied in engineering practice; in addition, since the learning algorithm adopted by the present invention has good distribution robustness, if the noise does not satisfy the Gaussian distribution in actual use, the impact on the prediction accuracy of the learner is not significant. The assumption of Gaussian distribution here is mainly for the rigor of subsequent theoretical derivation.

[0109] The learner proposed by the present invention is for the subsequent prediction of the controller. During the use of the model predictive controller, it is necessary to predict the time domain H pThe variables within a step are continuously predicted. Therefore, within this framework, the learner also needs to predict the independent variables. According to the above aerodynamic formula, it can be known that under the assumption of longitudinal stability, the lateral and directional motion of the airdrop system is operationally closed, that is, the values of the four lateral and directional state variables ψ, φ, p, r at the next step are determined by the values of the four state variables at the current moment, that is, a state vector x = [ψ, φ, p, r]. T , and the noisy measurement y k , there is

[0110]

[0111] wherein, and are respectively the coefficient matrices of the simplified discrete system. Thus, the learning process of the learner is given as follows:

[0112] During one-step prediction (i.e., predicting the value at time step k + 1 at time step k): When each new time step arrives, obtain the roll angle φ of the navigation system, the roll angular velocity p of the airframe system, and the yaw angular velocity r of the airframe system from the navigation system, and record the value of the control input u calculated by the current controller. Combine these four numerical values into a vector as the input of the learner to solve the predicted value of the yaw angle at the next moment.

[0113] During multi-step prediction (i.e., predicting the value at time step k + i (i ≥ 2) at time step k): Use the predicted value at time step k + i - 1 k+i-1 and the control law u calculated by the controller

[0114] (2) In step S22: First, without proof, give the learning algorithm based on Gaussian process regression:

[0115] For the vector-valued function f(x), the noisy measurement where the noise obeys a joint Gaussian distribution with a mean of 0 and a standard deviation of . Define the training data set where the test input is the input to be predicted. From the mean and variance of the joint conditional Gaussian distribution, we can obtain:

[0116]

[0117] where the covariance function is used to measure the correlation between (x, x′), and is also called the kernel function in Gaussian process regression. The matrix function is called the covariance matrix, and each element of it Used to calculate the correlation between elements in the training dataset X, and each element in the vector-valued function calculates the correlation between the test input x* and the corresponding elements of the training input X.

[0118] The output of a single GP is a scalar. For an n-dimensional function f(x), n GPs can be used to predict each output respectively. The posterior mean and variance of the obtained vector value are denoted as:

[0119]

[0120] The following gives an algorithm based on Cholesky matrix decomposition that is more convenient for computer code implementation:

[0121]

[0122] α = L T \(L(Y - μ(X)1 τ ))

[0123] μ(x * ) = α T k Φ (X, x * )

[0124] υ := L\k Φ (X, x * )

[0125] ∑(x * ) := k(x * , x * ) - υ T υ

[0126] where L, α, υ are intermediate variables, 1 τ is a vector of all 1s, k Φ (X, x * ) is the kernel vector between the test data point x* and the dataset X (the calculation method is given later), K Φ (X, X) is the kernel matrix between datasets, μ(x * ), ∑(x * ) are the posterior mean and variance of the prediction for the test data point x* respectively.

[0127] The core of GP prediction lies in describing the correlation between the test input x* and the training input X through the covariance function. The following introduces the selection of the covariance function (kernel function):

[0128] The most commonly selected kernel function for modeling physical processes is the squared exponential kernel (also known as the RBF kernel). Since the exponential function has the property of being differentiable to an infinite order, the GP constructed based on this kernel is smooth enough and has good mathematical properties. Its expression is:

[0129]

[0130] where A, l i are the hyperparameters of the kernel function, which are undetermined coefficients. Denote their set as θ H , x[i] represents the i-th component of x. It is necessary to select appropriate hyperparameters to enable the GP to have good approximation ability for specific functions. The present invention adopts the maximum marginal likelihood method. First, give the likelihood function of the GP:

[0131]

[0132] Log-likelihood:

[0133]

[0134] Then the optimal hyperparameters are obtained by solving the following optimization problem:

[0135]

[0136] Any optimization solver provided by commercial software can be used to solve this problem to obtain the optimal hyperparameters, and then substitute them into the kernel function of the GP to predict the distribution of the output corresponding to x * . The prediction result is given in the form of the mean and covariance of a Gaussian distribution.

[0137] (3) In step S23, a method for selecting the dataset during online operation is given.

[0138] Since the GP needs to perform matrix inversion operations during prediction, its computational burden is highly correlated with the dimension of the matrix. As can be seen from step S12, the dimension of the matrix to be inverted is determined by the number of points in the training dataset. Therefore, it is desired to use as small a dataset as possible to predict the test points. Here, a method for online updating the dataset is given. The core principle is that when new data pairs arrive, it is decided whether to include the point in the dataset by judging whether the new input point can be well represented by the original dataset: if the new point can be well represented, the dataset is not updated; otherwise, the new point is added to the training set and a point with the least amount of information is deleted to ensure that the size of the dataset remains unchanged.

[0139] There are two key problems in implementing this mechanism: ① How to judge whether a new data point can be well represented? ② How to judge which point in the original data contains the least amount of information?

[0140] Now, algorithms are proposed respectively to solve the above problems: First, for ①, through the Kernel Independence Test (KIT), the residual of the new point represented by the original data set is judged, that is, calculate

[0141] β = k(x * , x * ) - k(X, x * ) T α τ

[0142] where α τ is calculated in the same way as α in the previous text. Here, the subscript τ is used to emphasize that the vector α is still τ-dimensional during online operation (it becomes α τ +1 when a new data point is introduced, and its expression will be given specifically later);

[0143] A residual threshold can be defined When the actual residual β is greater than the preset threshold , the new data point is incorporated into the data set.

[0144] Next, the solution to problem ② is discussed. Here, without derivation, the definitions and calculation methods of several parameters are given. By executing the following formula according to the process, this algorithm efficiently approximates the Kullback-Leibler (KL) divergence between the current Gaussian process and each alternative Gaussian process after deleting a data point, and then deletes the data point with the largest KL divergence. The formula is as follows:

[0145]

[0146] where are abbreviations of k Φ (X, x * ) and k(x * , x * ) respectively, C τ = [K(X, X) + σ 2 I] -1 , where

[0147] α τ+1 = T τ+1 (α τ ) + q (τ+1) s τ+1 ,

[0148]

[0149] where o τ+1 is defined as o τ+1 (τ + 1) = 1, o τ+1 (t) = 0 for all t ≠ τ + 1. T τ+1 (·), U τ+1(·) are respectively defined as expanding a τ - dimensional vector and matrix to a τ + 1 - dimensional one, and setting the newly added elements to 0;

[0150]

[0151] The specific implementation steps are as follows: Each time new measurement data is available, q, b, k Φ (X, x * ), and β. If β is less than the defined threshold, the data set is not updated, and only new is calculated and α and C are updated; otherwise, x* is incorporated into the data set, the inverse of matrix Q is calculated, and the after deleting the corresponding point is calculated. The above steps are executed cyclically until no new measurement data is available.

[0152] 3. Step S3, design the controller:

[0153] (1) Step S31, design an optimal index function considering the uncertainty of the learner:

[0154] The GP - MPC proposed by the present invention makes better use of the results given by the GP learner. According to Step S2, the GP learner will give the predicted value in the form of a Gaussian distribution during operation, that is, the mean μ and covariance ∑ of the predicted value are given. Currently, most technologies only utilize the information related to the mean μ of the predicted value in the application of GP, while ignoring or misusing the information about the covariance ∑.

[0155] The covariance ∑ of the predicted value reflects the possibility that the true value of the predicted value may deviate from the mean μ. Therefore, in essence, it describes a kind of uncertainty of prediction accuracy. When controlling in an environment with large uncertainty, it is more desirable to achieve such a goal: when the prediction covariance ∑ is small, the controller fully believes in the prediction result of the learner and hopes to assign a higher weight to the control error brought by its learning result in some way; on the contrary, when the prediction covariance ∑ is large, this means that the true value has a greater possibility of deviating from the mean μ of the predicted value. Over - believing the predicted value of the controller may lead to serious mistakes of the controller and even may cause the system to become unstable. Therefore, at this time, the present invention hopes that the controller adopts a more cautious control strategy, that is, constructs a certain weighting method to punish the contribution of the learner in the controller.

[0156] Stochastic MPC is a stochastic optimal control strategy based on rolling - horizon optimization. By constructing a certain cost function and continuously solving the optimization problem online within a certain time window to find the optimal control strategy in a given sense. This method aims to achieve the optimal regulation of the system state, and the online iterative structure enables the controller to have better robustness in dealing with uncertainties. Generally, for the tracking problem of a stochastic system, the following optimal index is designed:

[0157]

[0158] where \(e(k)\) (abbreviated as \(e\) k ) is the tracking error of the system at the \(k\)-th step, \(N\) represents the prediction horizon of the system (i.e., the maximum number of prediction steps), \(P\), \(Q\), \(R\) are weighting coefficients, and \(\|\cdot\|\) P represents the weighted ellipse norm of \(P\), represents the mathematical expectation,

[0159] Since it is emphasized that the system is subject to random disturbances, the system state \(x\) is actually a random variable. Therefore, it is necessary to take the expectation of the entire quadratic form to make \(J\) a scalar. At the same time, due to the online rolling solution characteristics of MPC, sometimes the terminal constraints are ignored

[0160] As mentioned above, the purpose of the present invention is to design an optimal index. When the prediction covariance \(\sum\) is large, a certain weighting method is constructed to punish the contribution of the learner in the controller, so that the system adopts a more cautious strategy. Here, the present invention adopts a new weighting method for the tracking error, that is, the Mahalanobis distance, also known as the Mahalanobis distance. For the distance between coordinates \(x\) and \(y\), the expression of the Mahalanobis distance is:

[0161]

[0162] where \(S\) -1 is the estimated value of the covariance matrix of the sample data.

[0163] The traditional method constructs the optimal index by the way of constant weighting, which is essentially the Euclidean distance. The Mahalanobis distance weights the distance by the covariance structure, and compared with the Euclidean distance, it considers the punishment of the error contribution due to the uncertainty of the random distribution. Since GP can effectively give the covariance \(\sum\) of the predicted value, inspired by this method, the present invention constructs the following optimal index function:

[0164]

[0165] where \(R\) is a constant weighting matrix, and \(\sum\) k is the covariance matrix corresponding to the predicted value at the \(k\)-th step output by GP.

[0166] (2) Step S32 mainly constructs the optimal control problem.

[0167] The controlled system in the present invention is as follows:

[0168]

[0169] where \(w\) k = \(d\) k+∈ k is the sum of various disturbances received by the system, including the external disturbance d k and the random term ∈ k .

[0170] Next, different processing methods will be adopted for the nominal model and the model residual part. For ease of representation, the model is decomposed as follows:

[0171]

[0172] The x at the next moment k+1 is divided into the nominal model and the model residual as well as the total disturbance w k into two parts. Correspondingly, the control law of the present invention is decomposed as follows:

[0173]

[0174] The control law is divided into two parts for separate solution. One part is used to control the part corresponding to the nominal model, while the other part is used to control the part of the model residual plus disturbance learned by the learner.

[0175] For the nominal model part, a sub-optimal strategy is adopted. From the knowledge of optimal control, the optimal gain matrix K at the steady state of the system is obtained by solving the Riccati equation for the system equation and the performance index. For a discrete linear time-invariant system,

[0176] A T KA-(A T KB)(R+B T KB) -1 (B T KA)+Q=0

[0177] where K is the steady-state solution, A is the state transition matrix, B is the input matrix, and Q and R are the error weight matrix and the input weight matrix in the quadratic form index respectively.

[0178] In the present invention, the selected nominal model and system indicators are fixed. Therefore, the optimal gain matrix K can be obtained by offline solution, so that the system forms a sub-optimal control for the part of the nominal model, that is

[0179]

[0180] For the controller part, the present invention uses a non-linear optimization method to optimize the optimization index. Next, the optimal control problem (OCP) in this framework is reconstructed.

[0181]

[0182] where x dk is the expected value of the system state, and u M is the maximum limit value of the wing parachute input rudder deflection.

[0183] By rolling and solving the above OCP, an optimal control sequence for the model residual can be obtained Generally, only the first element in the control sequence is used to act on the system in MPC, and a new optimal control sequence is continuously obtained by rolling and solving the OCP. However, due to the use of the controller to learn the model and disturbances in the method proposed by the present invention, and in order to avoid excessive solution burden, the restriction of only using the first element can be relaxed, and the first N elements of the control sequence are adopted, so that the controller can solve the OCP problem every N u time steps. N u is selected through the trade-off between the computational burden and performance in the experiment. u u

[0184] 4. Step S4 is the application step during actual use. Based on the design steps of S1-S3, the operation process is given, which is mainly divided into two parts: offline and online use:

[0185] (1) Step S41, Offline flight:

[0186] To obtain offline flight data, a low-altitude airdrop is first carried out. After the parachute is opened, the following methods can be used to fly to obtain a flight trajectory and store the flight data (including navigation data and input rudder deflection signal):

[0187] Method 1: Adopt the remote control mode. Through the remote controller / handle, the control rudder deflection is artificially given to manipulate the airdrop system to fly around an arbitrary trajectory. Note that to ensure that the system can be continuously excited, it is required that the control input be as diverse as possible, including various flight actions, different flight altitudes and speeds, etc.

[0188] Method 2: Fly with the nominal controller: The control algorithm obtained in the simulation environment can be used for flight to collect the trajectory.

[0189] It should be noted that for the trajectory tracking problem of the airdrop system targeted by this application, the flight data to be collected includes 5 variables: 4 state variables ψ, φ, p, r in the lateral direction, and the control input u.

[0190] After obtaining the flight data, the nominal model is obtained by using the identification algorithm in step S1;

[0191] Then, the optimal hyperparameters of GP are obtained by using the offline GP training algorithm in step S22.

[0192] (2) Step S42, Online Application:

[0193] To execute the GP-MPC algorithm, first select appropriate parameters, including: the update step size T of the learner and the controller, the dataset capacity N of the learner m , the threshold of the residual of the online GP for judging new points represented by the original dataset The prediction horizon Hp of MPC, the weights Q and R in the MPC optimal index. These coefficients are generally selected by trial and error, for example, weighing the computational burden and computational accuracy.

[0194] After selecting appropriate coefficients, when new data points arrive in the airdrop system, first execute step S23 to judge the new points and obtain a new dataset.

[0195] Using the dataset obtained by S23, execute step S22 to obtain the distribution of the yaw angle prediction for the next time step.

[0196] Substitute the predicted values in S22 into the optimal control problem reconstructed in step S32, and use an optimization solver in commercial software (such as MATLAB) to solve it, and an optimal control sequence facing the model residual can be obtained Let the final control law Act on the system.

[0197] Repeatedly execute the above three steps until the airdrop system reaches the specified position.

[0198] Example:

[0199] Taking a medium-sized parafoil with a span of 7.5m as an example, the specific implementation method of this method is illustrated by means of simulation verification.

[0200] The specific parameters of the parafoil used in the simulation are shown in Tables 1 and 2 below, mainly including geometric parameters and aerodynamic parameters:

[0201] Table 1

[0202]

[0203] Table 2

[0204]

[0205] In addition, the lift coefficient C l 、drag coefficient C D and the pitching moment coefficient at the quarter chord length are given in the form of Table 3 as a function of the parafoil angle of attack α for easy interpolation in MATLAB.

[0206] Table 3

[0207]

[0208] Steps S1 - S3 are design steps, which are general methods for the precision airdrop system and do not change with the change of the specific model of the parafoil. This embodiment only shows the implementation steps of Step S4.

[0209] From Step S41, offline flight: To obtain offline flight data, first simulate a small arbitrary flight trajectory using simulation and store 4 state variables v, φ, p, r in the lateral direction of the flight data, as well as the control input u. The trajectory is as Figure 1 shown

[0210] where the rudder deflection signal uses a sinusoidal sweep signal u(k) = 0.25 * sin(1 / 1000 * k * k * h), the simulation time is 200 seconds, and the simulation step size is 0.02.

[0211] Using this trajectory data, according to Steps S1 and S2, identify the nominal model of the system and train the hyperparameters of the GP.

[0212] Next, the performance during the online application of Step S42 is given; to verify the robustness and adaptability of the method proposed in this application, the performance of this method will be verified and demonstrated under different conditions below:

[0213] (I) Ideal situation: Using the identified nominal model, under the condition of zero - mean random disturbance and measurement noise, use GP - MPC for tracking. The online learning effect and tracking effect are as Figures 2 - 4 shown:

[0214] It can be seen that in the ideal situation, GP has quite ideal performance in both the online learning accuracy of the model and the control tracking error.

[0215] (II) Without nominal model: Without using the nominal model, directly use GP - MPC to online learn the entire model, and under the condition of zero - mean random disturbance and measurement noise, use GP - MPC for tracking.. The online learning effect and tracking effect are as Figure 5 and Figure 6 shown:

[0216] When not using the nominal model, the online learning and tracking curves still have good performance. However, it can be seen that affected by the numerical solution accuracy, both the learning error and the tracking error have relatively small errors, but the overall performance is within an acceptable range. This phenomenon is consistent with our analysis and shows that the GP - MPC based on the nominal model proposed by us has better control accuracy and robustness against numerical solution errors.

[0217] (3) Sine external disturbance condition: To simulate the anti-disturbance ability of the controller under sine wind, a sine wind disturbance is added in the simulation. Other conditions are the same as in experimental group (1) to verify the tracking situation of the controller. As Figure 7 and Figure 8 shown; the modulus value of the added sine wind disturbance is 0.8*sin(0.3*t).

[0218] The sine disturbance has a relatively small impact on the learner and the controller, especially in the initial stage. This is mainly because the dataset used for offline training in the initial stage does not contain disturbances, and there is a certain deviation from the situation during online use. In addition, the controller has not converged in the first few seconds, so there are certain learning and control errors in the first approximately 3 seconds. After 3 seconds, the prediction effect of the learner occasionally is not ideal because the external is not a function of the state and its variation law is unknown. Theoretically, it is impossible to completely predict through learning. However, due to the controller's ability to eliminate deviations, the tracking control performance under sine disturbance is good.

[0219] (4) Model mutation: At the 10th second, all key aerodynamic parameters of the parafoil system model are suddenly changed to 1.5 times the original. Other conditions are the same as in experimental group (1), and the online learning effect and tracking effect are observed. As Figure 9 and Figure 10 shown;

[0220] It can be seen that at the 10th second when the model mutates, it has an impact on the learner and the controller, resulting in a certain deviation between the instantaneous model prediction value and the tracking quantity of the controller. However, since the online GP can update the dataset in real time, this deviation is eliminated in about 2.5 seconds and then enters the steady state again, and there are no more prediction errors and control errors.

[0221] It should be noted that in this embodiment, the simulation step size is set to 0.5 s. If a smaller simulation step size is designed, it can be expected to eliminate the influence of the mutation disturbance in a shorter time.

[0222] Finally, the full-process simulation of the airdrop system based on the control method proposed in this application is given: In this embodiment, the nominal model is 0.6 times the identification model, considering random interference and measurement noise, and applying a certain slow-varying sine wind disturbance. The reference trajectory is generated by an optimization algorithm. The tracking effect is as Figure 11 shown.

[0223] Corresponding to the foregoing embodiment of a model predictive control method based on online Gaussian process regression for a precision airdrop system, the present invention also provides an embodiment of a model predictive control device based on online Gaussian process regression for a precision airdrop system.

[0224] See Figure 12, an apparatus for model predictive control based on online Gaussian process regression for a precision airdrop system provided by an embodiment of the present invention includes a memory and one or more processors. Executable code is stored in the memory. When the processor executes the executable code, it is used to implement a method for model predictive control based on online Gaussian process regression for a precision airdrop system in the above embodiment.

[0225] An embodiment of an apparatus for model predictive control based on online Gaussian process regression for a precision airdrop system provided by the present invention can be applied to any device with data processing capabilities. Such a device with data processing capabilities can be a device or apparatus such as a computer. The apparatus embodiment can be implemented by software, or by hardware, or by a combination of software and hardware. Taking software implementation as an example, as a logically defined apparatus, it is formed by a processor of any device with data processing capabilities reading corresponding computer program instructions in a non-volatile memory into the memory for running. From a hardware perspective, as Figure 12 shown, it is a hardware structure diagram of any device with data processing capabilities where an apparatus for model predictive control based on online Gaussian process regression for a precision airdrop system provided by the present invention is located. In addition to Figure 12 the shown processor, memory, network interface, and non-volatile memory, an apparatus in an embodiment is usually based on the actual functions of any device with data processing capabilities where the apparatus is located and may also include other hardware, which will not be elaborated here.

[0226] The specific implementation process of the functions and roles of each unit in the above apparatus can be specifically seen in the implementation process of the corresponding steps in the above method, which will not be elaborated here.

[0227] For the apparatus embodiment, since it basically corresponds to the method embodiment, relevant parts can refer to the partial description of the method embodiment. The apparatus embodiments described above are merely illustrative. The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place, or may be distributed to multiple network units. Some or all of the modules can be selected according to actual needs to achieve the purpose of the present invention solution. Those of ordinary skill in the art can understand and implement it without creative efforts.

[0228] An embodiment of the present invention also provides a computer-readable storage medium, on which a program is stored. When the program is executed by a processor, it implements a method for model predictive control based on online Gaussian process regression for a precision airdrop system in the above embodiment.

[0229] The computer-readable storage medium may be an internal storage unit of any device with data processing capabilities described in any of the foregoing embodiments, such as a hard disk or a memory. The computer-readable storage medium may also be an external storage device of any device with data processing capabilities, such as a plug-in hard disk, a Smart Media Card (SMC), an SD card, a Flash Card, etc. equipped on the device. Further, the computer-readable storage medium may also include both an internal storage unit and an external storage device of any device with data processing capabilities. The computer-readable storage medium is used to store the computer program and other programs and data required by any device with data processing capabilities, and may also be used to temporarily store the data that has been output or is to be output.

[0230] The above embodiments are used to explain the present invention, rather than to limit the present invention. Any modifications and changes made to the present invention within the spirit and scope of the claims of the present invention fall within the protection scope of the present invention.

Claims

1. A model predictive control method based on online Gaussian process regression for a precision airdrop system, characterized in that: The method comprises the following steps: S1: By analyzing the aerodynamics of the parafoil system, its dynamic model is processed into a linear time-invariant nominal model for lateral motion and a residual part of a nonlinear time-varying nominal model considering the parafoil flexibility and wind disturbance; S2: Predict the residual part of the nonlinear time-varying nominal model based on the online Gaussian process regression algorithm; S3: Based on the posterior prediction variance of the Gaussian process regression algorithm, a performance functional is designed for the tracking error weighting. A stochastic optimal control problem is constructed and solved in two parts. One part is used to control the nominal model, taking a suboptimal strategy of obtaining the Riccati equation offline. The other part is used to control the model residuals and disturbances learned by online Gaussian process regression. The optimal control problem is solved online to obtain the optimal control sequence, and then the optimal control law is obtained.

2. The model predictive control method based on online Gaussian process regression for a precision airdrop system according to claim 1, characterized in that: In step S1, the lateral state of the parafoil system is Where φ is the roll angle, ψ is the yaw angle, p is the roll angular velocity, and r is the yaw angular velocity. The discrete parafoil system model is established by the system identification method as a nominal model, which is expressed as follows: Among them, a ij and b i is the constant to be identified, where a ij and b i They all represent elements in the matrix, ij represents rows and columns in the matrix, T is the sampling period of the discrete system, u is the control input of the system, the subscript k represents the variable corresponding to time k, and I4 represents the 4-dimensional unit matrix. Based on the flight mechanics characteristics of the parafoil airdrop system, the dynamic equation is established, the longitudinal motion mode is ignored, and the functional relationship between the control input and the system yaw angle is established for the lateral motion mode, which is expressed as the yaw angle change rate. It is a function of the roll angle φ, the roll angular velocity p of the aircraft system, the yaw angular velocity r of the aircraft system and the control input u; the noise is introduced and the functional relationship is discretized as follows: Among them, ψ k+1 represents the yaw angle at the k+1th time step, are the estimated values ​​of the roll angle, yaw angle, roll angular velocity of the aircraft system and yaw angular velocity of the aircraft system at the kth step, u k is the control input at the kth step, ∈ k is the noise set at the k-th time step, including the measurement noise of the navigation system and the process noise introduced by the random part in aerodynamics.

3. The model predictive control method based on online Gaussian process regression for a precision airdrop system according to claim 2, characterized in that: Construct the state vector x = [ψ, φ, p, r] T , and the noisy measurement y k as follows: in, is the nominal model part, f(x k ,u k ) represents the residual part of the model, σ represents the standard deviation of the measurement error; the learning algorithm based on Gaussian process regression is used to iteratively solve the prediction of the yaw angle.

4. The model predictive control method based on online Gaussian process regression for a precision airdrop system according to claim 1, characterized in that: When using Gaussian process regression for prediction, the dataset is updated online in the following way: (1) Through the kernel linear independence test KIT, the residual of the new data point represented by the original data set is determined; the obtained residual is compared with the residual threshold, and when it is less than the parameter threshold, the new data point is included in the data set; (2) Delete the data point with the least information in the data set, that is, delete the data point with the largest KL divergence.

5. The model predictive control method based on online Gaussian process regression for a precision airdrop system according to claim 2, characterized in that: In step S3, the optimal index function is constructed as follows: Among them, Q and R are constant weight matrices, ∑ k is the posterior covariance matrix corresponding to the k-step prediction value output by Gaussian process regression, e k is the tracking error of the system at the kth step, N represents the maximum number of prediction steps of the system, represents mathematical expectation.

6. The model predictive control method based on online Gaussian process regression for a precision airdrop system according to claim 1, characterized in that: For the control of the nominal model, a suboptimal strategy is adopted. The optimal gain matrix of the system in steady state is obtained by solving the Riccati equation about the system equations and performance indicators. By solving the optimal gain matrix offline, suboptimal control is formed for the part of the nominal model.

7. The model predictive control method based on online Gaussian process regression for a precision airdrop system according to claim 5, characterized in that: For the control of model residuals, nonlinear optimization method is used to optimize the optimization index, and the optimal control problem is reconstructed as follows: where x dk is the expected value of the system state, u M Enter the maximum limit value of the parafoil rudder deviation; solve the optimal control problem in a rolling manner to obtain an optimal control sequence facing the model residual 8. A model prediction control device based on online Gaussian process regression for a precision airdrop system, comprising a memory and one or more processors, wherein the memory stores executable code, characterized in that: When the processor executes the executable code, a model predictive control method based on online Gaussian process regression for a precision airdrop system as described in any one of claims 1 to 7 is implemented.

9. A computer-readable storage medium having a program stored thereon, characterized in that: When the program is executed by the processor, a model predictive control method based on online Gaussian process regression for a precision airdrop system as described in any one of claims 1 to 7 is implemented.

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