Coaxial multi-rotor unmanned aerial vehicle control method based on Koopman operator

Abstract

The invention relates to a coaxial multi-rotor unmanned aerial vehicle control method based on a Koopman operator, and the method comprises the steps: constructing a deep neural network based on the Koopman operator, and mapping a state variable to a high-dimensional observable space through an encoder; in a high-dimensional observable space, state variables are predicted through a linear connection layer based on dimension rising state variables and rotor rotating speeds; the decoder restores the prediction result to an original state space; training a deep neural network by using historical flight data to obtain codec parameters and a weight of a linear connection layer, and constructing a linear dynamic model of the observation function and the unmanned aerial vehicle; and the prediction controller predicts state variables of future multiple steps of the unmanned aerial vehicle by using a linear dynamics model, and enables the unmanned aerial vehicle to track a given trajectory by minimizing a cost function to obtain optimal control input. Compared with the prior art, the method has the advantages that complete data-driven modeling is realized, and the method can adapt to unmanned aerial vehicles with different sizes, loads and flight conditions.

Description

Technical Field

[0001] The present invention relates to the technical field of unmanned aerial vehicle (UAV) control, and in particular to a coaxial multi-rotor UAV attitude dynamics modeling and control method based on a Koopman operator. Background Art

[0002] Coaxial multi-rotor drones (UAVs) have been widely used in both military and civilian applications due to their significant advantages, including high efficiency, high payload capacity, efficient hovering, and extended flight time. However, the dynamics of coaxial UAVs are characterized by nonlinearity, strong coupling, and multiple uncertainties. These include the aerodynamic coupling effect of the upper rotor on the lower rotor, transonic shock waves generated at the blade tips during high-speed rotation, and external disturbances such as wind speed fluctuations and turbulence during flight, all of which further complicate UAV dynamics modeling.

[0003] Existing research on coaxial UAV dynamics primarily focuses on first-principles modeling based on the laws of physics and aerodynamics. For example, some utilize rigid-body kinematics, rotor aerodynamic theory, and wind tunnel experiments to derive torque expressions using a six-component balance on a rotor system test bench. Others employ blade unit theory, nonuniform inflow methods, and empirical functions to model coaxial rotor designs. Other studies also apply rigid-body kinematics theory to analyze the coupling between the position and attitude dynamics of coaxial UAVs. However, these first-principles modeling methods are often complex and computationally intensive, and traditional system identification methods are more applicable to specific operating points and may overlook the inherent nonlinear dynamics of the rotor system. Summary of the Invention

[0004] The purpose of the present invention is to address the shortcomings of the first principles modeling method, overcome the problems of strong nonlinearity, complex coupling and low modeling efficiency in the dynamics modeling of coaxial multi-rotor UAVs, and provide a UAV attitude dynamics modeling and automated hyperparameter optimization method based on the Koopman operator.

[0005] The purpose of the present invention can be achieved by the following technical solutions:

[0006] A coaxial multi-rotor UAV control method based on Koopman operator, comprising the following steps:

[0007] A deep neural network with an encoder-decoder structure based on the Koopman operator is constructed. The deep neural network uses rotor speed as the control input and attitude angle as the state variable. The encoder maps the state variable to a high-dimensional observable space and concatenates it with the original state variable to obtain an increased-dimensional state variable. In the high-dimensional observable space, the increased-dimensional state variable and the rotor speed are input into a linear connection layer to predict the next increased-dimensional state variable. The decoder restores the predicted increased-dimensional state variable to the original state space.

[0008] The deep neural network is trained using historical flight data of a coaxial multi-rotor UAV to obtain the encoder and decoder parameters and the weights of the linear connection layer, and to construct a linear dynamic model of the observation function and the UAV.

[0009] A model predictive controller (MPC) is constructed. The MPC uses a linear dynamics model to perform multi-step predictions on the future state variables of a coaxial multi-rotor UAV. The optimal control input of the coaxial multi-rotor UAV is obtained by minimizing a quadratic cost function so that the desired attitude angle of the UAV tracks a given trajectory.

[0010] As an optimal technical solution, the Koopman operator trains the encoder and decoder parameters and the linear matrix of the linear connection layer based on the historical state data of the drone at multiple times and the corresponding historical speed. The evolutionary loss terms of the Koopman operator include: encoder and decoder training loss, multi-step prediction loss, reconstruction error loss, physical loss, stability loss and L2 regularization term.

[0011] As a preferred technical solution, the encoder and decoder training loss functions are specifically as follows:

[0012]

[0013] Among them, L lift It represents the error between the dimension-raising state variable at the previous moment and the state variable at the next moment after it evolves to the next moment under the action of the Koopman operator; L pre represents the error between the increased-dimensional state variable and the low-dimensional state variable after it evolves and is decoded under the action of the Koopman operator; x k represents the state variable at time k; θ e represents the encoder parameters; θ d Denotes decoder parameters; A and B are linear matrices; ψ(x k ) is the dimension-raising state variable; u k is the rotor speed.

[0014] As a preferred technical solution, the multi-step prediction loss function is specifically as follows:

[0015]

[0016] in, n lift The cumulative error of step-up dimension, n pre The cumulative error of the step prediction, K i ψ(x k ) represents the multi-step evolution of the Koopman operator, specifically:

[0017]

[0018] As an optimal technical solution, the reconstruction error L rec It is used to reflect the accuracy of the original state variable being decoded again after being mapped to a high dimension by the dimensionality-raising function:

[0019]

[0020] As a preferred technical solution, the physical losses are specifically as follows:

[0021]

[0022] Among them, L phy n phy The total physical loss of the step state, represents the posture angle of the i-th step predicted by the deep neural network, T3 is the rotation matrix, ω i is the angular velocity of the drone at step i in the body coordinate system, λ ω is the residual coefficient, r ω Defined as:

[0023]

[0024] Among them, r ω represents the residual term of the angular velocity loop, which is used to reflect the error between the dimensionality-raising function mapping and the calculation of the real physical equation; Δt is the time interval between the previous and next moments; are the predicted torques of the UAV on the x, y, and z axes respectively; J x 、J y 、J z are the moments of inertia of the UAV on the x, y, and z axes respectively; p, q, and r are the angular velocities of the UAV around the x, y, and z axes in the body coordinate system respectively.

[0025] As a preferred technical solution, the stability loss is defined as:

[0026] L stab =RELU(ρ(A)-γ)

[0027] Among them, ρ(A) is the spectral radius of the state transfer matrix, γ is the set maximum value of the matrix eigenvalue modulus, RELU is the rectified linear unit, when the spectral radius violates the constraint, a penalty is imposed, otherwise, the output is 0.

[0028] As a preferred technical solution, the deep neural network uses differential evolution-ultra-bandwidth algorithm to optimize the optimal hyperparameter combination. The specific process is as follows:

[0029] Define the maximum and minimum budgets and the reduction rate η, and calculate the maximum number of scheduling times; randomly initialize and generate a subpopulation for differential evolution, where each individual in the subpopulation is a hyperparameter combination of a deep neural network;

[0030] Use differential evolution strategy to mutate individuals. After the mutation phase is completed, perform uniform crossover on the mutated individuals and the existing target individuals to generate experimental individuals. Select the better ones from the target individuals and experimental individuals as the next generation to update the population.

[0031] In the selection phase, the fitness of the new individual is compared with that of the corresponding parent individual, and the individual with higher fitness is selected;

[0032] The ultra-bandwidth idea is used to eliminate configurations with poor performance in advance. All individuals are trained with lower resources in the initial stage. After each round of iteration, inferior solutions are eliminated and only the top 1 / η ​​hyperparameter combinations are retained for higher budget training in the next stage.

[0033] As a preferred technical solution, the objective function of the hyperparameter optimization is defined as:

[0034] f object =L accuracy +L control

[0035] Among them, L accuracy is an accuracy indicator, representing the error after the neural network trained with a certain set of hyperparameter combinations performs a set number of step predictions, L control It is a controllability index, representing the difference between the dimension of the dimensionality-raising state and the rank of the controllability matrix. If it is not 0, the corresponding hyperparameter combination will be eliminated.

[0036] As a preferred technical solution, the model predictive controller uses the trained decoder as the observation function to increase the dimension of the current drone state variable and predicts the next increased dimension state variable through the drone's linearized dynamic model. By minimizing a quadratic cost function subject to linear constraints, the reference attitude angle in a given state is tracked and the optimal control input is obtained by solving the following optimal control problem:

[0037]

[0038] subjectto:ψ(x i+1 )=Aψ(x i )++Bu i

[0039] u min ≤u i ≤u max

[0040]

[0041] Among them, ψ(x k+i ) is the dimension-increasing state variable predicted by the linearized dynamics model; ψ(x ref ) is the reference attitude angle of the ascending dimension; u k is the decision variable of the kth step; Q and R are positive definite matrices, which penalize the tracking error and control increment respectively, u max ,u min Control the upper and lower limits of the drone's speed.

[0042] Compared with the prior art, the present invention has the following beneficial effects:

[0043] 1) The UAV attitude dynamics modeling and automated hyperparameter optimization method proposed in this invention integrates physical information and deep learning architecture to construct a physically interpretable neural network model for automatically learning the observation function of the Koopman operator. By embedding physical constraints in the network structure, the neural network is effectively guided to extract nonlinear system behaviors that conform to real physical characteristics in the high-dimensional observable space, thereby ensuring the stability and prediction accuracy of the model during the training process. In addition, a number of loss function collaborative optimization mechanisms are introduced into the neural network structure, covering multiple dimensions such as dimensionality-increasing space consistency, multi-step state prediction accuracy, and state reconstruction capabilities to enhance the model's ability to fit the global behavior of the dynamic system. The differential evolution-hyperband algorithm DEHB is used to efficiently and automatically optimize the key hyperparameters of the neural network, which can quickly screen high-quality model configurations under a multi-fidelity resource allocation strategy, greatly reducing the search space and training time, and improving modeling efficiency and convergence speed.

[0044] 2) The modeling of this invention is completely data-driven: Traditional UAV dynamics modeling relies heavily on rigid-body dynamics theory, aerodynamic models, and wind tunnel experiments, resulting in complex model structures and poor adaptability. This invention, based on Koopman operator theory, uses a deep neural network to automatically learn the system's observation function. Incorporating physical prior information, this model ensures that the learned model not only possesses data-driven fitting capabilities but also strikes a balance between physical consistency and robustness. This enhances the system's deployability in real-world flight environments, significantly lowers the modeling threshold, and improves versatility.

[0045] 3) This invention introduces DEHB to achieve automatic hyperparameter tuning: Differentiated from manual parameter tuning or traditional grid search and Bayesian optimization methods, this invention combines differential evolution with ultra-wideband multi-fidelity mechanisms to construct an efficient search framework suitable for high-dimensional mixed parameter spaces. While maintaining modeling accuracy, it significantly reduces parameter tuning time and greatly improves modeling efficiency. For coaxial drone structures of different sizes and loads, linearized models can be quickly trained even under flight conditions such as large attitude angle changes and intense disturbances, making this invention highly applicable to engineering applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 This is a flow chart of the attitude modeling and control method of a coaxial multi-rotor UAV based on the Koopman operator of the present invention;

[0047] Figure 2 This is the force analysis diagram of the coaxial octarotor UAV;

[0048] Figure 3 Schematic diagram of the neural network architecture with physical information constraints designed for the present invention. DETAILED DESCRIPTION

[0049] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0050] Example 1

[0051] In order to overcome the limitations of traditional modeling methods, the present invention proposes a coaxial multi-rotor UAV attitude modeling and control method based on the Koopman operator. The deep neural network with physical information fusion is used to automatically learn the Koopman observation function, and the differential evolution-hyperband algorithm (DEHB) is used to realize automatic optimization of hyperparameters, thereby improving the accuracy and generalization ability of the model, while reducing the manual design cost and computational burden. As a data-driven method, the Koopman operator theory can elevate nonlinear systems to the Hilbert space and obtain linear representations only by relying on input-output data. Constructing the observable subspace of the Koopman operator through deep neural networks (DNNs) can provide an intrinsic linear representation for strongly nonlinear systems and effectively construct the complex nonlinear attitude dynamics of UAVs. The specific steps are as follows: Figure 1 As shown:

[0052] First, the nonlinear dynamic equations of the drone are used to design a variety of control input excitation signals to generate multiple sets of state sequence data containing the drone's attitude angles and their rates of change. To ensure the richness and generalization of the dataset, periodic control signals of varying frequencies and amplitudes are used to simulate rotor speed inputs, comprehensively covering the various attitude angle changes that a drone may encounter during actual flight, including small-scale attitude adjustments and large-angle maneuvers. Specifically, the sampling period is set to 0.05s, the total length is 4000 steps, and the entire dataset is divided into 70%, 20%, and 10% proportions to form training, validation, and test sets.

[0053] After obtaining the training dataset, the differential evolution-ultra-bandwidth algorithm is introduced to automatically optimize the neural network hyperparameters. This process first sets the maximum search level based on the preset maximum and minimum budgets and reduction rates, and randomly initializes multiple populations, with each individual representing a set of hyperparameter combinations to be optimized. Subsequently, candidate solutions are generated through differential evolution processes such as mutation and crossover. During the optimization process, the DEHB algorithm utilizes an early stopping strategy and a multi-fidelity resource allocation mechanism to hierarchically eliminate individuals with poor performance, promoting only high-performing individuals to the next stage of higher-budget training. This allows for the rapid elimination of ineffective individuals at low cost, concentrating computing resources on the most promising solutions. It can efficiently screen the optimal hyperparameter combination in a high-dimensional mixed parameter space, and the final output includes configurations such as the optimal learning rate, number of hidden layers, number of nodes, dimension of the dimensionality-increasing space, and batch size.

[0054] The optimized hyperparameters are used to construct a deep neural network with an encoder-decoder structure. The encoder corresponds to the dimensionality-raising network (encoder), which maps the state variables (attitude angles) of the drone to the high-dimensional observable space ψ1(x k ), and compare the mapped state with the original state variable x k The dimensional state variable ψ(x k )=[x k ;ψ1(x k )]; In the high-dimensional observable space, a linear dynamics model is established through linear matrices A and B. The decoder corresponds to a dimensionality reduction network (decoder), which restores the prediction results of the linear dynamics model to the original state space.

[0055] A dataset is constructed based on the historical state data and corresponding historical rotation speed of the drone at multiple moments. The deep neural network is trained according to the preset loss function, and the encoder and decoder parameters and linear matrices are output to obtain the observation function and the linear dynamic model of the drone.

[0056] Finally, a model predictive controller (MPC) is designed using the trained linear dynamics model. The MPC controller uses the linear dynamics model learned by Koopman to make multi-step predictions of the drone's future attitude angles. It then obtains the optimal control input by minimizing a linearly constrained quadratic cost function. Tracking errors and control inputs are penalized based on the predictions and a pre-set cost function. This linearly constrained cost function is then transformed into a quadratic programming problem and solved, enabling precise trajectory tracking and stable control of the drone's desired attitude angle.

[0057] This embodiment is as follows Figure 2Taking the coaxial octocopters shown in the figure as an example, in the dynamic modeling process, the rotor motor speed is collected as input, and the attitude angle of the drone is used as the state variable. The specific steps include:

[0058] Consider the six-degree-of-freedom attitude dynamics model of a coaxial multi-rotor drone. Its state variables include linear velocity, angular velocity, and attitude angle. The dynamic equation is as follows:

[0059] ω=J -1 (M-ω×Jω)

[0060] Ω=T3ω (1)

[0061] in, Represents the angular velocity of the drone around the x, y, and z axes in the body coordinate system; are attitude angles, representing pitch, roll, and yaw angles respectively; represents the torque in three directions, is the UAV’s moment of inertia matrix, is the rotation matrix, and the corresponding relationship between the rotor speed and the torque it generates is:

[0062]

[0063] Where d is the distance from each rotor to the center of the fuselage, K T , K D is the tensile coefficient, n i , i = 1 to 8 is the rotational speed of each rotor. For coaxial multi-rotor drones, the main source of potential interference is the flapping effect. The source of uncertainty is the airflow coupling between the coaxial propellers. The downwash of the upper rotor often affects the operating efficiency of the lower rotor, causing the ideal model to fail under extreme operating conditions. Therefore, a data-driven modeling method is needed to compensate for the shortcomings of mechanism modeling.

[0064] The modeling process mainly includes the following parts: constructing a data set based on the historical state data and corresponding historical rotation speed of the drone at multiple times, constructing a linearized model of attitude dynamics based on the dimensionality-raising function, which includes a network model used to map the original state to a high-dimensional observable space, and training the Koopman linearized model according to the preset loss function to obtain the dynamic model of the drone.

[0065] The neural network used to fit the dimensionality-raising function mainly includes encoder, decoder and linear connection layer, such as Figure 3 As shown, encoder N e (x k |θ e ) promotes the original state to a higher-dimensional observable space, the decoder N d (xk |θ d ) returns the high-dimensional state to the original space. Both the encoder and decoder are composed of fully connected layers. The loss function of the training process is shown in Equations (3) and (4):

[0066]

[0067] Among them, x k represents the state variable at time k; θ e represents the encoder parameters; θ d Denotes decoder parameters; A and B are linear matrices; ψ(x k ) is the dimension-raising state variable; u k is the control input, i.e. the rotor speed; L lift It represents the error between the dimensionality-increasing state at the previous moment evolving to the next moment under the action of the Koopman operator and the state at the next moment directly increasing in dimension; L pre It represents the error between the higher-dimensional state and the lower-dimensional state after it evolves and is decoded under the action of the Koopman operator. In order to enhance the multi-step prediction effect of the model and prevent adjacent mutations or abnormal data from affecting the training accuracy, a multi-step prediction loss function is introduced here:

[0068]

[0069] in, is the cumulative error of multi-step dimensionality increase, is the cumulative error of multi-step prediction, and the multi-step evolution of the Koopman operator is expressed as

[0070] Reconstruction error L rec It reflects the accuracy of the original state after being re-decoded after being mapped to a high dimension by the dimensionality-raising function:

[0071]

[0072] In order to enhance the learning stability of deep neural networks, the proposed method designs ψ(x k ) contains the original state, so L rec The weight in the total loss is relatively small.

[0073] For the angular velocity loop of the UAV, its mechanism equation is determined by formula (1). Therefore, the present invention explicitly introduces physical equation constraints so that the designed deep neural network will learn to follow and simulate the physical motion laws of the UAV in continuous time. These laws are described by specific ordinary differential equations (Formula 1):

[0074]

[0075] Among them, Lphy n phy The total physical loss of the step state, represents the attitude angle state estimated by the deep neural network, T3 is the rotation matrix, ω i is the angular velocity of the UAV in the i-th step state in the body coordinate system, λ ω is the residual coefficient, r ω Defined as:

[0076]

[0077] Among them, r ω represents the residual term of the angular velocity loop, which is used to reflect the error between the dimensionality-raising function mapping and the calculation of the real physical equation; Δt is the time interval between the previous and next moments; are the predicted torques of the UAV in the x, y, and z directions, respectively. These torques can be obtained from the pre-fitted polynomial relationship between the rotor speed and the corresponding torque.

[0078] Since there is a large deviation between the predicted state and the true value at the beginning of training, the derivation will lead to an excessively large physical loss value, deviating from the actual direction of the constraint. Therefore, physical errors can be gradually introduced. After n rounds of iteration until convergence, physical losses are gradually added to ensure the accuracy of physical information.

[0079] In order to further guarantee the strict stability of the learning dynamics, a spectral radius regularization term is added, and the stability loss is defined as:

[0080] L stab =RELU(ρ(A)-γ) (10)

[0081] Among them, ρ(A) is the spectral radius of the state transfer matrix, γ is the artificially set maximum value of the matrix eigenvalue modulus, and RELU is the rectified linear unit. When the spectral radius violates the constraint, a penalty is imposed, otherwise the output is 0, thereby ensuring that the eigenvalue of the state transfer matrix remains within the unit circle, thereby promoting the existence of a stable feedback gain and facilitating the design of subsequent controllers.

[0082] Therefore, the total Koopman operator evolution loss term can be expressed as:

[0083]

[0084] By minimizing the total error, the linearized model of the UAV attitude dynamics can be approximated as: ψ(x k+1 )=Aψ(x k )+Bu k The training of the entire encoding and decoding network is carried out synchronously, and the weights of the linear connection layer are A and B. In order to prevent overfitting, the L2 regularization term is introduced and

[0085] Since the UAV has many dynamic state variables, the neural network used needs to have a complex structural design. To solve the problem of hyperparameter selection in deep neural network modeling, this paper uses the differential evolution-hyperband algorithm (DEHB) for automatic tuning. The process includes the following four steps: initialization, mutation, crossover and selection. During initialization, first, define the maximum and minimum budgets (b min ,b max ) and the reduction rate η, the maximum number of scheduling times is calculated based on these parameters:

[0086]

[0087] Then, a random initialization method is used to randomly generate N differential evolution subpopulations. Each individual in the population is considered to be a hyperparameter combination of a deep neural network, including learning rate, number of network layers, number of nodes, dimension of dimension increase, dropout ratio, batch size, etc.

[0088]

[0089] In formula (13), g is the number of generations, and D is the dimension of the problem to be solved; Denotes the i-th subpopulation in the g-th generation. Use differential evolution strategy to perform individual mutation and crossover to update the population. The mutation strategy in this invention adopts the classic mutation operator rand / 1 and randomly selects three individuals as parents, which are recorded as

[0090]

[0091] In formula (14), v i is the newly generated vector, and F is the scaling factor, which is usually in the range of (0, 1]. After the mutation phase is completed, the mutant individuals and the existing target individuals are uniformly (binomial) crossovered to generate experimental individuals. Finally, the better ones are selected from the target individuals and experimental individuals as the next generation:

[0092]

[0093] In formula (15), p is the crossover rate, and j is a random positive integer in [1, D] to ensure that at least one dimension is changed. This allows individuals (not necessarily the best individuals) to have the opportunity to guide the mutation process, effectively reducing the probability of the DE algorithm falling into local optimization. In order to evaluate the performance indicators of the offspring parameter combination, the objective function of the hyperparameter optimization algorithm is defined as:

[0094] f object =L accuracy +L control(16)

[0095] Among them, L accuracy is an accuracy indicator, representing the error after 1000-step prediction of a neural network trained with a certain set of hyperparameters, L control is the controllability index, representing the difference between the dimension of the dimensional state and the rank of the controllability matrix. If it is not 0, it proves that the obtained model has uncontrollable parts and should be eliminated. In the selection phase, the fitness of the new individual is compared with the corresponding parent individual, and the individual with higher fitness is selected:

[0096]

[0097] Because high-dimensional mixed-type parameter spaces (including continuous, discrete, and categorical variables) are prone to optimization challenges such as local optima and the curse of dimensionality, efficient solutions require a combination of global search and early stopping. Therefore, this paper leverages ultra-wideband technology to preemptively eliminate poorly performing configurations, accelerating optimization. Initially, all individuals are trained with low resources. After each iteration, inferior solutions are eliminated, retaining only the top 1 / η ​​configurations for the next, higher-budget training phase. This allows for rapid elimination of ineffective individuals at a low cost, concentrating resources on promising solutions.

[0098] After obtaining the linearized dynamic model, the model can be used to design the MPC controller for the UAV dynamics. The decoder obtained by training is used as the observation function ψ(x k )=N e (x k ∣θ e ) for the current UAV state variables to increase the dimension, and through the UAV linear dynamics model ψ(x k+1 )=Aψ(x k )+Bu k , predicting the next step of the increased-dimensional state variable. The goal of MPC is to minimize a quadratic cost function subject to linear constraints, resulting in a solvable convex optimization problem. To track the reference attitude angle at a given state, the optimal control input can be obtained by solving the following optimal control problem:

[0099]

[0100] Among them, ψ(x ref ) is the reference attitude angle of the ascending dimension; u k is the decision variable; and is a positive definite matrix, which penalizes the tracking error and control increment respectively. Equation (18) represents the dynamic constraints of the system. Considering the state and control constraints of the UAV in actual flight, the UAV speed is controlled at (u min ,u max ).

[0101] The MPC controller is described in further detail below:

[0102] Since the cost function is in quadratic form and the constraints are in linear form, the above optimization problem can be converted into a standard quadratic programming form (QP), u k As a decision variable, it is convenient to use the existing efficient standard quadratic programming QP solver to solve. In order to simplify the notation z k =ψ(x k ), stack all variables in the forecast time domain into vector form:

[0103]

[0104] The system constraints in Equation (18) are expressed as an overall form:

[0105] Z=A qp z k +B qp U

[0106] in:

[0107]

[0108] Substituting the cost function, the optimization problem can be written as a standard QP form:

[0109]

[0110] in, Q blk and R blk is the block diagonal matrix of state and control terms in the cost function, Z ref is the concatenation vector of the dimensional reference trajectory. The original optimal control problem can then be solved by efficient solvers such as qpOASES (C++).

[0111] The above describes in detail the preferred embodiments of the present invention. It should be understood that those skilled in the art can make numerous modifications and variations based on the concepts of the present invention without inventive effort. Therefore, any technical solutions that can be derived by those skilled in the art through logical analysis, reasoning, or limited experimentation based on the concepts of the present invention and the prior art should be within the scope of protection defined by the claims.

Claims

1. A coaxial multi-rotor UAV control method based on Koopman operator, characterized in that the steps include: A deep neural network with an encoder-decoder structure based on the Koopman operator is constructed. The deep neural network uses rotor speed as the control input and attitude angle as the state variable. The encoder maps the state variable to a high-dimensional observable space and concatenates it with the original state variable to obtain an increased-dimensional state variable. In the high-dimensional observable space, the increased-dimensional state variable and the rotor speed are input into a linear connection layer to predict the next increased-dimensional state variable. The decoder restores the predicted increased-dimensional state variable to the original state space. The deep neural network is trained using historical flight data of a coaxial multi-rotor UAV to obtain the encoder and decoder parameters and the weights of the linear connection layer, and to construct a linear dynamic model of the observation function and the UAV. A model predictive controller (MPC) is constructed. The MPC uses a linear dynamics model to perform multi-step predictions on the future state variables of a coaxial multi-rotor UAV. The optimal control input of the coaxial multi-rotor UAV is obtained by minimizing a quadratic cost function so that the desired attitude angle of the UAV tracks a given trajectory.

2. The method for controlling a coaxial multi-rotor UAV based on a Koopman operator according to claim 1, characterized in that: The Koopman operator trains the encoder and decoder parameters and the linear matrix of the linear connection layer based on the historical state data of the drone at multiple times and the corresponding historical speed. The evolutionary loss terms of the Koopman operator include: encoder and decoder training loss, multi-step prediction loss, reconstruction error loss, physical loss, stability loss and L2 regularization term.

3. The coaxial multi-rotor UAV control method based on Koopman operator according to claim 2 is characterized in that: The encoder and decoder training loss functions are as follows: Among them, L lift It represents the error between the dimension-raising state variable at the previous moment and the state variable at the next moment after it evolves to the next moment under the action of the Koopman operator; L pre represents the error between the increased-dimensional state variable and the low-dimensional state variable after it evolves and is decoded under the action of the Koopman operator; x k represents the state variable at time k; θ e represents the encoder parameters; θ d Denotes decoder parameters; A and B are linear matrices; ψ(x k ) is the dimension-raising state variable; u k is the rotor speed.

4. The method for controlling a coaxial multi-rotor UAV based on a Koopman operator according to claim 3, characterized in that: The multi-step prediction loss function is as follows: in, n lift The cumulative error of step-up dimension, n pre The cumulative error of the step prediction, K i ψ(x k ) represents the multi-step evolution of the Koopman operator, specifically:

5. The coaxial multi-rotor UAV control method based on Koopman operator according to claim 2 is characterized in that: The reconstruction error L rec It is used to reflect the accuracy of the original state variable being decoded again after being mapped to a high dimension by the dimensionality-raising function:

6. The method for controlling a coaxial multi-rotor UAV based on Koopman operator according to claim 2, characterized in that: The physical losses are as follows: Among them, L phy n phy The total physical loss of the step state, represents the posture angle of the i-th step predicted by the deep neural network, T3 is the rotation matrix, ω i is the angular velocity of the drone at step i in the body coordinate system, λ ω is the residual coefficient, r ω Defined as: Among them, r ω represents the residual term of the angular velocity loop, which is used to reflect the error between the dimensionality-raising function mapping and the calculation of the real physical equation; Δt is the time interval between the previous and next moments; are the predicted torques of the UAV on the x, y, and z axes respectively; J x 、J y 、J z are the moments of inertia of the UAV on the x, y, and z axes respectively; p, q, and r are the angular velocities of the UAV around the x, y, and z axes in the body coordinate system respectively.

7. The method for controlling a coaxial multi-rotor UAV based on Koopman operator according to claim 2, characterized in that: The stability loss is defined as: L stab =RELU(ρ(A)-γ) Among them, ρ(A) is the spectral radius of the state transfer matrix, γ is the set maximum value of the matrix eigenvalue modulus, RELU is the rectified linear unit, when the spectral radius violates the constraint, a penalty is imposed, otherwise, the output is 0.

8. The method for controlling a coaxial multi-rotor UAV based on Koopman operator according to claim 1, characterized in that: The deep neural network is optimized using the differential evolution-ultra-bandwidth algorithm to obtain the optimal hyperparameter combination. The specific process is as follows: Define the maximum and minimum budgets and the reduction rate η, and calculate the maximum number of scheduling times; randomly initialize and generate a subpopulation for differential evolution, where each individual in the subpopulation is a hyperparameter combination of a deep neural network; Use differential evolution strategy to mutate individuals. After the mutation phase is completed, perform uniform crossover on the mutated individuals and the existing target individuals to generate experimental individuals. Select the better ones from the target individuals and experimental individuals as the next generation to update the population. In the selection phase, the fitness of the new individual is compared with that of the corresponding parent individual, and the individual with higher fitness is selected; The ultra-bandwidth idea is used to eliminate configurations with poor performance in advance. All individuals are trained with lower resources in the initial stage. After each round of iteration, inferior solutions are eliminated and only the top 1 / η ​​hyperparameter combinations are retained for higher budget training in the next stage.

9. The method for controlling a coaxial multi-rotor UAV based on Koopman operator according to claim 8, characterized in that: The objective function of the hyperparameter optimization is defined as: in object =L accuracy +L control Among them, L accuracy is an accuracy indicator, representing the error after the neural network trained with a certain set of hyperparameter combinations performs a set number of step predictions, L control It is a controllability index, representing the difference between the dimension of the dimensionality-raising state and the rank of the controllability matrix. If it is not 0, the corresponding hyperparameter combination will be eliminated.

10. The method for controlling a coaxial multi-rotor UAV based on Koopman operator according to claim 1, characterized in that: The model predictive controller uses the trained decoder as the observation function to increase the dimension of the current UAV state variable and predicts the next increased dimension state variable through the UAV linearized dynamic model. It tracks the reference attitude angle in a given state by minimizing a quadratic cost function subject to linear constraints and obtains the optimal control input by solving the following optimal control problem: subject to:ψ(x i+1 )=Aψ(x i )++Bu i in min in i in max Among them, ψ(x k+i ) is the dimension-increasing state variable predicted by the linearized dynamics model; ψ(x ref ) is the reference attitude angle of the ascending dimension; u k is the decision variable of the kth step; Q and R are positive definite matrices, which penalize the tracking error and control increment respectively, u max ,u min Control the upper and lower limits of the drone's speed.

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