Unmanned aerial vehicle robust trajectory tracking control method based on event triggering and disturbance observation

Abstract

The invention belongs to the technical field of quad-rotor unmanned aerial vehicle control, and provides a quad-rotor unmanned aerial vehicle robust trajectory tracking control method based on an event triggering mechanism and an interference observer, the method has good tracking performance and anti-interference capability, the calculation requirement is small, and the problem of quad-rotor unmanned aerial vehicle robust control can be solved. The invention discloses an unmanned aerial vehicle robust trajectory tracking control method based on event triggering and disturbance observation, and the method comprises the following steps: 1, building a strong coupling and nonlinear model of a four-rotor unmanned aerial vehicle; a second part, designing a quadrotor unmanned aerial vehicle trajectory tracking controller based on model prediction control: constructing a target function based on a position tracking error, and designing a robust model prediction controller based on a fixed time disturbance observer on the basis; and a third part, designing an event triggering condition, and executing optimization solution of model prediction control only when the condition is met. The four-rotor unmanned aerial vehicle is mainly applied to design and manufacturing occasions of four-rotor unmanned aerial vehicles.

Description

Technical Field

[0001] This invention belongs to the field of quadcopter unmanned aerial vehicle (UAV) control technology, specifically relating to a robust trajectory tracking control method for quadcopter UAVs based on an event-triggered mechanism and an interference observer. Background Technology

[0002] Unmanned aerial vehicles (UAVs) are aircraft capable of autonomous flight using radio remote control equipment and onboard programs, performing various tasks without human intervention. Their ability to operate independently and their relatively low cost have made them a hot topic in current applied research. UAVs can be divided into fixed-wing and rotary-wing types. Rotary-wing UAVs, represented by quadcopter UAVs, have advantages such as simple mechanical structure, high maneuverability, and vertical takeoff and landing capabilities. They are suitable for performing tasks in small, complex environments and are widely used in aerial photography, power line inspection, agricultural plant protection, and logistics delivery.

[0003] However, due to the underactuated, strongly coupled, nonlinear, and easily disturbed characteristics of quadrotor UAVs, robust trajectory tracking control has become a current research hotspot. First, the mechanical structure of a quadrotor UAV is simple, which can be simplified to a six-DOF rigid body model powered by four symmetrically arranged rotors connected by two links. Each rotor generates a vertically upward thrust through rotation, with two rotors rotating clockwise and two counterclockwise. The thrust is adjusted by changing the rotational speed of different rotors, thus controlling the movement of the quadrotor UAV. Therefore, a quadrotor UAV uses four driving forces (four rotors) to control six degrees of freedom (three translational and three rotational degrees of freedom), reflecting its underactuated and strongly coupled characteristics. Second, the dynamic model of a quadrotor UAV involves multiple complex factors such as rotor aerodynamics, inertial forces, and attitude stability, and these factors interact intricately. Therefore, quadrotor UAVs exhibit significant nonlinear characteristics. Finally, quadcopter drones are typically used in high-altitude flight scenarios where wind disturbances are severe. They also face challenges from load variations, which can affect their center of gravity and flight characteristics, further increasing their susceptibility to interference.

[0004] Currently, classical model predictive control methods proposed by researchers both domestically and internationally for robust trajectory tracking control of quadrotor UAVs suffer from drawbacks such as insufficient robustness and high computational requirements. Therefore, this invention addresses the robust trajectory tracking control problem of quadrotor UAVs under conditions of limited onboard computer performance by designing a method that combines an event-triggered mechanism with robust model predictive control based on a fixed-time disturbance observer. This method enables quadrotor UAVs to reduce the number of calculations while maintaining control accuracy, thereby lowering computational requirements. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, this invention aims to propose a robust trajectory tracking control method for quadrotor UAVs based on an event-triggered mechanism and an interference observer. First, this method exhibits excellent tracking performance, enabling precise tracking of the desired trajectory by the quadrotor UAV. Second, it demonstrates strong anti-interference capabilities, addressing the robust control problem of quadrotor UAVs under complex interference conditions. Furthermore, this method has low computational requirements, allowing for the resolution of robust control problems of quadrotor UAVs under complex interference conditions with limited computational resources. Therefore, the technical solution adopted by this invention is a robust trajectory tracking control method for UAVs based on event triggering and interference observation, comprising the following steps:

[0006] Part 1: Design of a fixed-time disturbance observer based on the double-limit homogeneity theory: A strongly coupled and nonlinear model of a quadrotor UAV is established. Considering the complex disturbances experienced by the quadrotor UAV, the position dynamics model of the quadrotor UAV is rewritten as a multivariable first-order system. For this system, a fixed-time disturbance observer is designed based on the double-limit homogeneity theory.

[0007] Part Two: Design of a Quadrotor UAV Trajectory Tracking Controller Based on Model Predictive Control: For the strongly coupled and nonlinear model of the quadrotor UAV, combined with the disturbance estimates of the disturbance observer mentioned above, the predictive model constraints in the optimization problem are constructed. Considering the requirements of the trajectory tracking task, an objective function based on the position tracking error is constructed. On this basis, a robust model predictive controller based on a fixed-time disturbance observer is designed to solve the robust control problem of the quadrotor UAV under the influence of complex disturbances.

[0008] Part Three: Design of Event Triggering Conditions Based on Predictive State Errors: For a robust model predictive controller that integrates a fixed-time disturbance observer, considering the error between its predicted state and the actual state, event triggering conditions are designed. The optimization solution of model predictive control is executed only when the condition is met, thereby reducing the computational requirements of model predictive control and enabling the solution of robust control problems of quadcopter UAVs with limited computational resources.

[0009] The detailed steps are as follows:

[0010] Part 1: Design of a Fixed-Time Disturbance Observer Based on the Double-Limit Homogeneity Theory: The position dynamics model of a quadrotor UAV is rewritten as a multivariable first-order system, and a fixed-time disturbance observer is designed based on the double-limit homogeneity theory.

[0011] Homogeneity: Consider nonlinear systems:

[0012]

[0013] Where s = [s1,...,s n ] T Represents the state vector. For vector functions, Let n be the n-dimensional real number field. If for any ε > 0, the following equation holds:

[0014]

[0015] Where i∈1,...,n, then the vector function G(s) is said to be r-homogeneous, and is called... Let G(s) be the homogeneity of the vector function, or simply the (r,k)-homogeneous vector function G(s), where r = (r1,…,rk) n ) is the weight vector;

[0016] Bilimit Homogeneity: If, near s = 0, a vector function G(s) is approximated by a (r0, k0)-homogeneous function G0, where r0 is the weight vector and k0 is the homogeneity, then G(s) is said to be homogeneous with the related vector (r0, k0, G0) in the 0-limit. Similarly, if, near s = ∞, a vector function G(s) can be approximated by a (r0, k0)-homogeneous function G0, then G(s) is homogeneous with the related vector (r0, k0, G0). ∞ If G(s) is approximated, then G(s) is said to be in the ∞-limit with respect to the related vector (r). ∞ ,k ∞ G ∞ If a vector function G(s) is homogeneous in both the 0-limit and the ∞-limit, it is called bilimit homogeneity, or simply bihomogeneity.

[0017] Fixed-time convergence: If system (1) is globally asymptotically stable, and for any Where s0 is the initial value of the system, all of which guarantee that the system's settling time t≤t max , where t max If the constant is a fixed positive constant, then the system described by formula (1) is said to be fixed-time stable;

[0018] Lemma 1: Consider a bilimit homogeneous vector function that is semi-continuous on the upper side. Its double-limit correlation vectors are (r0,k0,G0) and (r ∞ ,k ∞ G ∞ If system (1) satisfies global asymptotic stability, then there exists a positive definite function V. G (s) guarantees its derivative. It is negative definite and satisfies the homogeneity of the bilimit; furthermore, if k0 ≤ k ∞ And there exist real numbers m0 and m ∞ satisfy and Then there exist positive constants κ0>0 and κ ∞ >0, guaranteeing that for any inequality Heng is established;

[0019] First, two coordinate systems need to be introduced to describe the model of the quadcopter drone, including the world coordinate system. and body coordinate system The world coordinate system is fixed to the ground, and the orientation of its coordinate axes does not change. The body coordinate system is coincident with the body of the quadrotor UAV, with its origin at its center, and the orientation of its coordinate axes changes with the UAV's pose. Considering the quadrotor UAV as a six-degree-of-freedom rigid body model, its kinematics and dynamics models are established:

[0020]

[0021] Where, p = [p x ,p y ,p z ] T With v = [v x ,v y ,v z ] T These represent the position and velocity of the quadcopter drone, respectively, and m is the mass of the quadcopter drone. This represents the rotation matrix from the body coordinate system to the world coordinate system. Its definition is related to the attitude information. c The scalar value represents the control thrust. This represents the unit vector pointing in the z-axis direction of the body coordinate system, therefore This can be understood as first setting the scalar T... c Mapped to the z-axis direction of the machine body, i.e., [0,0,T] c ] T Next, left multiplication Project the vector onto the world coordinate system where variable v resides, g = [0, 0, 9.81]. T m / s 2 It is the acceleration due to gravity, f d Representing an unknown external disturbance force, q = [q w ,q x ,q y ,q z ] T Let ω be a unit quaternion representing the attitude of the quadcopter drone, where ω = [ω x ,ω y ,ω z ] T J represents the angular velocity in the body coordinate system, J = diag(J x J y J z ) is the rotational inertia matrix, τ c =[τ c,x ,τ c,y ,τ c,z ] TTo control the torque, τ d Represents unknown external torque disturbance, × represents vector cross product, given the attitude information of the quadcopter UAV q=[q w ,q x ,q y ,q z ] T Then, the above rotation matrix Defined as:

[0022]

[0023] In model (3), with the symbol The relevant equations are defined as follows:

[0024]

[0025] Condition 1: Regarding the interference f in equation (3) d Assume it is continuous and first-order differentiable, and that it has positive constants δ and δ. Guarantee ||f d ||≤δ and Heng is established;

[0026] To control the position, velocity, and attitude of a quadcopter UAV, only the first three formulas of equation (3) are considered, where the disturbance factors exist in the position dynamics model:

[0027]

[0028] The model can be written as a multivariable first-order system as follows:

[0029]

[0030] Where z1 = mv, This serves as a virtual control input. For this system, the following disturbance observer is designed to estimate the disturbance f. d :

[0031]

[0032] in The observed value of z1 is obtained by solving the differential equation (8). It is a composite interference f d The estimated values, L1 and L2 are gain parameters designed to ensure observer stability, and the functions φ1 and φ2 are defined as follows:

[0033]

[0034] Where d ∞ ∈(0,1), k i >0,k′i >0,k″ i >0, i=(1,2). According to the definition of homogeneity, functions φ1(e1) and φ2(e1) are homogeneous at the 0-limit with a homogeneity of -1, and are also homogeneous at the ∞-limit with a homogeneity of d. ∞ Therefore, it is deduced that the functions φ1(e1) and φ2(e1) have bilimit homogeneity;

[0035] Combining equations (7) and (8), we obtain the following error system:

[0036]

[0037] in This is due to observational errors caused by interference.

[0038] Under condition 1, the fixed-time interference observer (8) This allows for the implementation of f in the system described by formula (7). d An effective estimate is obtained, and the estimation error converges within a fixed time. This is summarized as follows:

[0039] Consider the following bilimit homogeneous Lyapunov function:

[0040] V0(e)=V1(e1,e2)+V2(e2), (12)

[0041] Where V1 and V2 are defined as:

[0042]

[0043] Where 0 < p ∞ ≤max{1,3(1-d ∞ ) / 2},1<p0≤2p ∞ / (1-d ∞ ), and β0,β ∞ ,β′0,β′ ∞ All are arbitrary positive real numbers, ξ is defined as Furthermore, based on Young's inequality, we know that V1(e1,e2)≥0, which guarantees the positive definiteness of the Lyapunov function V0(e) defined in (12);

[0044] The derivative of the Lyapunov function can be obtained by taking the derivative.

[0045]

[0046] in

[0047]

[0048] get We need to prove its negative definiteness, that is, prove the negative definiteness of W(e). To do this, consider the structure of W(e) that converges to the following hyperplane:

[0049] Z1 = {φ1(e1) = e2}. (19)

[0050] In the hyperplane Z1, the equation can be obtained. Therefore, based on formulas (16) and (17), g1,σ1∈{0} always holds true in the hyperplane Z1. Then, by substituting g1=σ1=0 into formula (15), we obtain the convergence to the hyperplane Z1.

[0051]

[0052] Considering the multivariable symbolic function x T The properties of sign(x) = ||x|| and the hyperplane Z1 definition Its form is written as:

[0053]

[0054] Therefore, considering the conditions in inequality (11) function It is negative definite, that is Guaranteed Negative qualitative;

[0055] also, Since the condition for a continuous single-valued bilimit homogeneous function described in Lemma 1 is satisfied, based on the conclusion of Lemma 1, there exists a positive parameter L1 that satisfies the following condition, guaranteeing the negative definiteness of W(e):

[0056]

[0057] Here, χ(e1,e2) is an upper semi-continuous homogeneous function that reaches its maximum value on the homogeneous sphere. Thus, the positive definiteness of the Lyapunov function V0(e) and its derivative are... The negative definiteness of the observation error system (10) has been proven, and the global asymptotic stability of the observation error system (10) has been guaranteed. Furthermore, based on the conclusion of Lemma 1, there exist positive constants η0, η ∞ Make satisfy:

[0058]

[0059] Therefore, based on the fixed-time convergence theory, the observation error system (10) will converge at a fixed time. Converging to zero, i.e., the interference observer (8) can be within a fixed time. Internal implementation of composite interference f d Effective estimation, fixed time satisfy:

[0060]

[0061] Furthermore, as can be seen from formula (24), a fixed time Regardless of the initial observation error, the fixed-time stability of the observation error system (10) is guaranteed;

[0062] Part Two: Design of a Quadrotor UAV Trajectory Tracking Controller Based on Model Predictive Control: This section builds upon the disturbance estimates from Part One. Given the information, construct a prediction model that includes the effects of interference:

[0063]

[0064] The model uses position p, velocity v, and attitude q as state variables to control thrust T. c With reference angular velocity ω c For control variables, i.e., state variables in the optimization problem, we take x = [p, v, q]. T Control the amount taken

[0065] Next, considering the trajectory tracking requirements of a quadcopter drone, the following objective function V is constructed:

[0066]

[0067] Where k represents the sampling time, Δx (k+i|k) =x (k+i|k) -x (k+i|k),r It is the state error, x (k+i|k),r x represents the reference state in the desired trajectory. (k+N|k) For the terminal state, Δu (k+i|k) =u (k+i|k) -u (k+i|k),r Indicates the control input error, u (k+i|k),r As a reference control input, u is usually taken. (k+i|k),r =[mg,0,0,0] T , and Let represent the non-negative weight matrices for state error and control error, respectively. This is the weight matrix of the terminal state error, where N is the prediction step size of the model predictive control (MDC). That is, the MDC predicts N sampling times forward each time it solves the problem. If the sampling interval is dt, then the prediction time domain of the MDC is T = N * dt. Therefore, the objective function V is defined as N+1 state errors Δx. (k+i|k) and N control errors Δu(k+i|k) The weighted quadratic summation, if the optimization objective is to minimize the objective function, then the control quantity will minimize the error between the predicted state and the reference state, that is, to track the reference state in the desired trajectory.

[0068] In summary, the optimization problem for a quadrotor UAV trajectory tracking controller based on model predictive control is defined as follows:

[0069]

[0070] The objective function V is defined in equation (26). For the prediction model defined in equation (25), x k The state value at the current sampling time k is used as the initial value for the optimization solution iteration process. and These are control constraints, state constraints, and terminal state constraints.

[0071] At sampling time k, solve the optimization problem (27) to obtain the optimal control sequence. And the optimal predicted state sequence is obtained recursively. Both sequences satisfy optimality; in this case, only the first component of the optimal control sequence is taken. The system is applied to the problem, and at the next sampling time k+1, the optimization problem is solved again under new conditions to obtain a new optimal control sequence. With the optimal predicted state sequence Repeat this process to ensure real-time optimality;

[0072] Part Three: Design of Event Triggering Conditions Based on Predicted State Errors

[0073] Consider the predictive model (25) in model predictive control, which can be expressed in the following form:

[0074]

[0075] The model satisfies the following inequality:

[0076]

[0077] Where L N Let ||·|| be the Lipschitz constant for this system, and ||·|| denote the vector magnitude.

[0078] For the prediction model (28), the actual state x can be obtained by the Lipschitz lemma. k With the optimal predicted state sequence Upper limit of the difference:

[0079]

[0080] in express The largest eigenvalue, P is the terminal cost matrix in the model predictive control optimization problem;

[0081] Let t be a certain time when the calculation is performed. k Note that this time is different from the sampling time k. The sampling operation continues, and even if no calculation is performed, the sampling time will still increase; at t k At time t, the optimal predicted state sequence of model predictive control is denoted as . Where t = t k +i, i∈[0,T], the actual state after applying the optimal control sequence at this time is denoted as . Then the prediction error ω can be defined. t for:

[0082]

[0083] To address this error, the event triggering conditions are designed as follows:

[0084]

[0085] Where inf represents the infimum. This means that at t k After time ω t The moment t when the value is first greater than or equal to μ 0<θ≤1, the optimization solution of model predictive control is executed only when condition (32) is met; otherwise, the corresponding component in the optimal control sequence obtained in the previous solution is used as the optimal control quantity at the current moment.

[0086] The features and beneficial effects of this invention are:

[0087] This invention designs a trajectory tracking controller for a quadrotor UAV based on model predictive control (MMC), solving the tracking control problem for strongly coupled and nonlinear models and achieving accurate tracking of the desired trajectory by the quadrotor UAV. Secondly, this invention incorporates a fixed-time disturbance observer within the MMC framework to observe disturbances and compensate for them in real time within the controller, effectively improving the controller's robustness. Finally, this invention designs an event-triggered condition for the optimal predicted state sequence of MMC, performing optimization calculations only when this condition is met, effectively reducing the computational requirements of the algorithm.

[0088] This paper comprehensively analyzes and demonstrates the robust trajectory tracking control method for quadrotor UAVs based on an event-triggered mechanism and an interference observer, focusing on three aspects: the fixed-time convergence characteristics of the interference observer, the strong robustness and high-precision tracking of the controller, and the reduction in computational requirements. Simulation and theoretical analysis results show that the proposed fixed-time interference observer can achieve accurate estimation of interference within a fixed time. The proposed robust trajectory tracking control method for quadrotor UAVs based on an event-triggered mechanism and an interference observer exhibits high control accuracy and strong robustness, and significantly reduces the number of calculations compared to traditional methods, effectively lowering computational requirements. Attached image description:

[0089] Appendix Figure 1 Block diagram of robust trajectory tracking control structure for quadrotor UAVs based on event triggering mechanism and interference observer.

[0090] Appendix Figure 2 Schematic diagram of the coordinate system of a quadcopter drone.

[0091] Appendix Figure 3 The trajectory tracking curve of a quadcopter UAV under complex interference in a simulated environment.

[0092] Appendix Figure 4 Position tracking curve of a quadcopter UAV under complex interference in a simulated environment.

[0093] Appendix Figure 5 A schematic diagram illustrating the calculation of trigger timing in a simulation environment.

[0094] Appendix Figure 6 Tracking curve of physical experiment.

[0095] Appendix Figure 7 Three-dimensional position tracking curve of physical experiment.

[0096] Appendix Figure 8 Position error curve of physical experiment.

[0097] Appendix Figure 9 A diagram illustrating the triggering time of a physical experiment event. Detailed Implementation

[0098] To address the shortcomings of existing control methods and the practical need for high-precision and robust control of quadrotor UAVs, this invention comprehensively considers the strong coupling and nonlinear characteristics of the quadrotor UAV model, the requirements for trajectory tracking, robust control, and high computational demands. It proposes for the first time a trajectory tracking control method based on model predictive control, integrating an event-triggered mechanism and a disturbance observer. This method designs a quadrotor UAV trajectory tracking controller based on model predictive control to achieve accurate tracking of the desired trajectory. Simultaneously, a fixed-time disturbance observer is introduced to estimate disturbances in real time, improving the system's robustness. Finally, an event-triggered mechanism is integrated to reduce computational requirements.

[0099] This invention relates to the field of flight control technology for quadrotor unmanned aerial vehicles (UAVs). Specifically, it first proposes a robust trajectory tracking control method for quadrotor UAVs based on an event-triggered mechanism and an interference observer, which differs from previous control methods. The method is then tested in simulation environments and real-world test scenarios, and compared with classical model predictive control (MMC) methods and event-triggered MMC methods, verifying the advantages of the proposed method in terms of tracking accuracy, interference resistance, and algorithm requirements.

[0100] This invention combines theoretical derivation and virtual simulation technology to propose a trajectory tracking control method for quadrotor UAVs that features high tracking accuracy, strong robustness, and low computational requirements. Finally, tests were conducted in a simulation environment and a real-world test scenario to verify the effectiveness of the proposed method. A comparative analysis was also performed with classical model predictive control methods and event-triggered model predictive control methods in terms of tracking accuracy, disturbance rejection capability, and algorithm requirements.

[0101] The technical solution adopted in this invention is a robust trajectory tracking control method for UAVs based on event triggering and interference observation, the steps of which are as follows:

[0102] Part 1: Design of a fixed-time disturbance observer based on the double-limit homogeneity theory: A strongly coupled and nonlinear model of a quadrotor UAV is established. Considering the complex disturbances experienced by the quadrotor UAV, the position dynamics model of the quadrotor UAV is rewritten as a multivariable first-order system. For this system, a fixed-time disturbance observer is designed based on the double-limit homogeneity theory.

[0103] Part Two: Design of a Quadrotor UAV Trajectory Tracking Controller Based on Model Predictive Control: For the strongly coupled and nonlinear model of the quadrotor UAV, combined with the disturbance estimates of the disturbance observer mentioned above, the predictive model constraints in the optimization problem are constructed. Considering the requirements of the trajectory tracking task, an objective function based on the position tracking error is constructed. On this basis, a robust model predictive controller based on a fixed-time disturbance observer is designed to solve the robust control problem of the quadrotor UAV under the influence of complex disturbances.

[0104] Part Three: Design of Event Triggering Conditions Based on Predictive State Errors: For a robust model predictive controller that integrates a fixed-time disturbance observer, considering the error between its predicted state and the actual state, event triggering conditions are designed. The optimization solution of model predictive control is executed only when the condition is met, thereby reducing the computational requirements of model predictive control and enabling the solution of robust control problems of quadcopter UAVs with limited computational resources.

[0105] The detailed steps are as follows:

[0106] Part 1: Design of a Fixed-Time Disturbance Observer Based on the Double-Limit Homogeneity Theory: The position dynamics model of a quadrotor UAV is rewritten as a multivariable first-order system, and a fixed-time disturbance observer is designed based on the double-limit homogeneity theory.

[0107] Homogeneity: Consider nonlinear systems:

[0108]

[0109] Where s = [s1, ..., s n ] T Represents the state vector. For vector functions, Let n be the n-dimensional real number field. If for any ε > 0, the following equation holds:

[0110]

[0111] Where i∈1,...,n, then the vector function G(s) is said to be r-homogeneous, and is called... Let G(s) be the homogeneity of the vector function, or simply the (r,k)-homogeneous vector function G(s), where r = (r1,...,rk) n ) is the weight vector;

[0112] Bilimit Homogeneity: If, near s = 0, a vector function G(s) is approximated by a (r0, k0)-homogeneous function G0, where r0 is the weight vector and k0 is the homogeneity, then G(s) is said to be homogeneous with the related vector (r0, k0, G0) in the 0-limit. Similarly, if, near s = ∞, a vector function G(s) can be approximated by a (r0, k0)-homogeneous function G0, then G(s) is homogeneous with the related vector (r0, k0, G0). ∞ If G(s) is approximated, then G(s) is said to be in the ∞-limit with respect to the related vector (r). ∞ ,k ∞ G ∞ If a vector function G(s) is homogeneous in both the 0-limit and the ∞-limit, it is called bilimit homogeneity, or simply bihomogeneity.

[0113] Fixed-time convergence: If system (1) is globally asymptotically stable, and for any Where s0 is the initial value of the system, all of which guarantee that the system's settling time t≤t max , where t max If the constant is a fixed positive constant, then the system described by formula (1) is said to be fixed-time stable;

[0114] Lemma 1: Consider a bilimit homogeneous vector function that is semi-continuous on the upper side. Its double-limit correlation vectors are (r0,k0,G0) and (r ∞ ,k ∞ G ∞ If system (1) satisfies global asymptotic stability, then there exists a positive definite function V. G (s) guarantees its derivative. It is negative definite and satisfies the homogeneity of the bilimit; furthermore, if k0 ≤ k ∞ And there exist real numbers m0 and m ∞ Satisfying m0>max{r i 0} and m0>max{r i ∞ Then there exist positive constants κ0>0 and κ ∞ >0, guaranteeing that for any inequality Heng is established;

[0115] First, two coordinate systems need to be introduced to describe the model of the quadcopter drone, including the world coordinate system. and body coordinate system The world coordinate system is fixed to the ground, and the orientation of its coordinate axes does not change. The body coordinate system is coincident with the body of the quadrotor UAV, with its origin at its center, and the orientation of its coordinate axes changes with the UAV's pose. Considering the quadrotor UAV as a six-degree-of-freedom rigid body model, its kinematics and dynamics models are established:

[0116]

[0117] Where, p = [p x ,p y ,p z ] T With v = [v x ,v y ,v z ] T These represent the position and velocity of the quadcopter drone, respectively, and m is the mass of the quadcopter drone. This represents the rotation matrix from the body coordinate system to the world coordinate system. Its definition is related to the attitude information. c The scalar value represents the control thrust. This represents the unit vector pointing in the z-axis direction of the body coordinate system, therefore This can be understood as first setting the scalar T... c Mapped to the z-axis direction of the machine body, i.e., [0,0,T] c ] T Next, left multiplication

[0118] Project the vector onto the world coordinate system where variable v resides, g = [0, 0, 9.81]. T m / s 2 It is the acceleration due to gravity, f d Representing an unknown external disturbance force, q = [q w ,q x ,q y ,q z ] T Let ω be a unit quaternion representing the attitude of the quadcopter drone, where ω = [ω x ,ω y ,ω z ] T J represents the angular velocity in the body coordinate system, J = diag(J x J y J z ) is the rotational inertia matrix, τ c =[τ c,x ,τ c,y ,τ c,z ] T To control the torque, τ d Represents unknown external torque disturbance, × represents vector cross product, given the attitude information of the quadcopter UAV q=[q w ,q x ,q y ,q z ] T Then, the above rotation matrix Defined as:

[0119]

[0120] In model (3), with the symbol The relevant equations are defined as follows:

[0121]

[0122] Condition 1: Regarding the interference f in equation (3) d Assume it is continuous and first-order differentiable, and that it has positive constants δ and δ. Guarantee ||f d ||≤δ and Heng is established;

[0123] To control the position, velocity, and attitude of a quadcopter UAV, only the first three formulas of equation (3) are considered, where the disturbance factors exist in the position dynamics model:

[0124]

[0125] The model can be written as a multivariable first-order system as follows:

[0126]

[0127] Where z1 = mv, This serves as a virtual control input. For this system, the following disturbance observer is designed to estimate the disturbance f. d :

[0128]

[0129] in The observed value of z1 is obtained by solving the differential equation (8). It is a composite interference f d The estimated values, L1 and L2 are gain parameters designed to ensure observer stability, and the functions φ1 and φ2 are defined as follows:

[0130]

[0131] Where d ∞ ∈(0,1), k i >0,k′ i >0,k″ i >0, i=(1,2). According to the definition of homogeneity, functions φ1(e1) and φ2(e1) are homogeneous at the 0-limit with a homogeneity of -1, and are also homogeneous at the ∞-limit with a homogeneity of d. ∞ Therefore, it is deduced that the functions φ1(e1) and φ2(e1) have bilimit homogeneity;

[0132] Combining equations (7) and (8), we obtain the following error system:

[0133]

[0134] in This is due to observational errors caused by interference.

[0135] Under condition 1, the fixed-time interference observer (8) This allows for the implementation of f in the system described by formula (7). d An effective estimate is obtained, and the estimation error converges within a fixed time. This design result is summarized by the following theorem:

[0136] Theorem 1: For a system described by formula (7) that satisfies condition 1, if we choose 0 < d ∞ <1 and any positive gain k i >0,k′ i >0,k″ i If >0, i=(1,2), then there exist gains L1 and L2 satisfying the following equation:

[0137]

[0138] This makes the observation error system (10) stable at a fixed time, that is, the interference observer (8) can be stable at a fixed time. The interference f is obtained by internal estimation. d That is, for any Heng was established.

[0139] Proof of Theorem 1: Consider the following bilimit homogeneous Lyapunov function:

[0140] V0(e)=V1(e1,e2)+V2(e2), (12)

[0141] Where V1 and V2 are defined as:

[0142]

[0143] Where 0 < p ∞ ≤max{1,3(1-d ∞ ) / 2},1<p0≤2p ∞ / (1-d ∞ ), and β0,β ∞ ,β′0,β′ ∞ All are arbitrary positive real numbers, ξ is defined as Furthermore, based on Young's inequality, V1(e1,e2)≥0, which guarantees the positive definiteness of the Lyapunov function V0(e) defined in (12).

[0144] The derivative of the Lyapunov function can be obtained by taking the derivative.

[0145]

[0146] in

[0147]

[0148] get We need to prove its negative definiteness, that is, prove the negative definiteness of W(e). To do this, consider the structure of W(e) that converges to the following hyperplane:

[0149] Z1 = {φ1(e1) = e2}. (19)

[0150] In the hyperplane Z1, the equation can be obtained. Therefore, based on formulas (16) and (17), it can be seen that g1,σ1∈{0} always holds true in the hyperplane Z1. Then, by substituting g1=σ1=0 into formula (15), we can obtain the convergence to the hyperplane Z1.

[0151]

[0152] Considering the multivariable symbolic function x T The properties of sign(x) = ||x|| and the hyperplane Z1 definition Its form can be written as:

[0153]

[0154] Therefore, considering the conditions in inequality (11) function It is negative definite, that is Guaranteed The negative qualitative property.

[0155] also, Since the continuous single-valued double-limit homogeneous function condition described in Lemma 1 is satisfied, based on the conclusion of Lemma 1, there exists a positive parameter L1 that satisfies the following condition to guarantee the negative definiteness of W(e).

[0156]

[0157] Here, χ(e1,e2) is an upper semi-continuous homogeneous function that reaches its maximum value on the homogeneous sphere. Thus, the positive definiteness of the Lyapunov function V0(e) and its derivative are thus established. The negative definiteness of the observation error system (10) has been proven, and the global asymptotic stability of the observation error system (10) is guaranteed. Furthermore, based on the conclusion of Lemma 1, there exist positive constants η0, η... ∞ Make satisfy:

[0158]

[0159] Therefore, based on the fixed-time convergence theory, the observation error system (10) will converge at a fixed time. Converging to zero, i.e., the interference observer (8) can be within a fixed time. Internal implementation of composite interference f d Effective estimation, fixed time satisfy:

[0160]

[0161] Furthermore, as can be seen from formula (24), a fixed time This is independent of the initial observation error. Thus, the fixed-time stability of the observation error system (10) is guaranteed, and Theorem 1 is proved.

[0162] Part Two: Design of a Quadrotor UAV Trajectory Tracking Controller Based on Model Predictive Control: This section builds upon the disturbance estimates from Part One. Given the information, construct a prediction model that includes the effects of interference:

[0163]

[0164] The model uses position p, velocity v, and attitude q as state variables to control thrust T. c With reference angular velocity ω c For control variables, i.e., state variables in the optimization problem, we take x = [p, v, q]. T The control quantity is u = [T] c ,ω c ] T =[T c ,ω x,c ,ω y,c ,ω z,c ] T ;

[0165] Next, considering the trajectory tracking requirements of a quadcopter drone, the following objective function V is constructed:

[0166]

[0167] Where k represents the sampling time, Δx (k+i|k) =x (k+i|k) -x (k+i|k),r It is the state error, x (k+i|k),r x represents the reference state in the desired trajectory. (k+N|k) For the terminal state, Δu (k+i|k) =u (k+i|k) -u (k+i|k),r Indicates the control input error, u (k+i|k),r As a reference control input, u is usually taken. (k+i|k),r =[mg,0,0,0] T , and Let represent the non-negative weight matrices for state error and control error, respectively. This is the weight matrix of the terminal state error, where N is the prediction step size of the model predictive control (MDC). That is, the MDC predicts N sampling times forward each time it solves the problem. If the sampling interval is dt, then the prediction time domain of the MDC is T = N * dt. Therefore, the objective function V is defined as N+1 state errors Δx.(k+i|k) and N control errors Δu (k+i|k) The weighted quadratic summation, if the optimization objective is to minimize the objective function, then the control quantity will minimize the error between the predicted state and the reference state, that is, to track the reference state in the desired trajectory.

[0168] In summary, the optimization problem for a quadrotor UAV trajectory tracking controller based on model predictive control is defined as follows:

[0169]

[0170] The objective function V is defined in equation (26). For the prediction model defined in equation (25), x k The state value at the current sampling time k is used as the initial value for the optimization solution iteration process. and These are control constraints, state constraints, and terminal state constraints.

[0171] At sampling time k, solve the optimization problem (27) to obtain the optimal control sequence. And the optimal predicted state sequence is obtained recursively. Both sequences satisfy optimality; in this case, only the first component of the optimal control sequence is taken. The system is applied to the problem, and at the next sampling time k+1, the optimization problem is solved again under new conditions to obtain a new optimal control sequence. With the optimal predicted state sequence Repeat this process to ensure real-time optimality;

[0172] Part Three: Design of Event Triggering Conditions Based on Predicted State Errors

[0173] Consider the predictive model (25) in model predictive control, which can be expressed in the following form:

[0174]

[0175] The model satisfies the following inequality:

[0176]

[0177] Where L N Let ||·|| be the Lipschitz constant for this system, and ||·|| denote the vector magnitude.

[0178] For the prediction model (28), the actual state x can be obtained by the Lipschitz lemma. k With the optimal predicted state sequence Upper limit of the difference:

[0179]

[0180] in express The largest eigenvalue, P is the terminal cost matrix in the model predictive control optimization problem;

[0181] Let t be a certain time when the calculation is performed. k Note that this time is different from the sampling time k. The sampling operation continues, and even if no calculation is performed, the sampling time will still increase; at t k At time t, the optimal predicted state sequence of model predictive control is denoted as . Where t = t k +i, i∈[0,T], the actual state after applying the optimal control sequence at this time is denoted as . Then the prediction error ω can be defined. t for:

[0182]

[0183] To address this error, the event triggering conditions are designed as follows:

[0184]

[0185] Where inf represents the infimum. This means that at t k After time ω t The moment t when the value is first greater than or equal to μ 0<θ≤1, the optimization solution of model predictive control is executed only when condition (32) is met; otherwise, the corresponding component in the optimal control sequence obtained in the previous solution is used as the optimal control quantity at the current moment.

[0186] The present invention will be further described below with reference to the accompanying drawings and specific examples.

[0187] The structure diagram of the robust trajectory tracking control method for quadrotor UAVs based on event triggering mechanism and interference observer is shown below. Figure 1 As shown, the robust trajectory tracking control method for quadrotor UAVs based on event triggering mechanism and interference observer proposed in this invention mainly includes the following three parts:

[0188] Part 1: Design of a Fixed-Time Disturbance Observer Based on the Double-Limit Homogeneity Theory: The position dynamics model of a quadrotor UAV is rewritten as a multivariable first-order system, and a fixed-time disturbance observer is designed based on the double-limit homogeneity theory.

[0189] Homogeneity: Consider nonlinear systems:

[0190]

[0191] Where s = [s1, ..., sn ] T Represents the state vector. For vector functions, Represents the n-dimensional real number field. If for any ε > 0, the following equation holds:

[0192]

[0193] Where i∈1,...,n, then the vector function G(s) is said to be r-homogeneous, and is called... Let G(s) be the homogeneity of the vector function, or simply the (r,k)-homogeneous vector function G(s), where r = (r1,...,rk) n ) is the weight vector;

[0194] Bilimit Homogeneity: If, near s = 0, a vector function G(s) is approximated by a (r0, k0)-homogeneous function G0, where r0 is the weight vector and k0 is the homogeneity, then G(s) is said to be homogeneous with the related vector (r0, k0, G0) in the 0-limit. Similarly, if, near s = ∞, a vector function G(s) can be approximated by a (r0, k0)-homogeneous function G0, then G(s) is homogeneous with the related vector (r0, k0, G0). ∞ If G(s) is approximated, then G(s) is said to be in the ∞-limit with respect to the related vector (r). ∞ ,k ∞ G ∞ If a vector function G(s) is homogeneous in both the 0-limit and the ∞-limit, it is called bilimit homogeneity, or simply bihomogeneity.

[0195] Fixed-time convergence: If system (1) is globally asymptotically stable, and for any Where s0 is the initial value of the system, all of which guarantee that the system's settling time t≤t max , where t max If the constant is a fixed positive constant, then the system described by formula (1) is said to be fixed-time stable;

[0196] Lemma 1: Consider a bilimit homogeneous vector function that is semi-continuous on the upper side. Its double-limit correlation vectors are (r0,k0,G0) and (r ∞ ,k ∞ G ∞ If system (1) satisfies global asymptotic stability, then there exists a positive definite function V. G (s) guarantees its derivative. It is negative definite and satisfies the homogeneity of the bilimit. Furthermore, if k0 ≤ k ∞ And there exist real numbers m0 and m ∞ Satisfying m0>max{r i 0} and m0>max{r i∞}, then there exist positive constants κ0>0 and κ ∞ >0, guaranteeing that for any inequality Heng was established.

[0197] First, two coordinate systems need to be introduced to describe the model of the quadcopter drone, including the world coordinate system. and body coordinate system The world coordinate system is fixed to the ground, and the orientation of its coordinate axes does not change. The body coordinate system is coincident with the body of the quadrotor UAV, with its origin at its center, and the orientation of its coordinate axes changes with the UAV's pose. Considering the quadrotor UAV as a six-degree-of-freedom rigid body model, its kinematics and dynamics models are established:

[0198]

[0199] Where, p = [p x ,p y ,p z ] T With v = [v x ,v y ,v z ] T These represent the position and velocity of the quadcopter drone, respectively, and m is the mass of the quadcopter drone. This represents the rotation matrix from the body coordinate system to the world coordinate system. Its definition is related to the attitude information. c The scalar value represents the control thrust. This represents the unit vector pointing in the z-axis direction of the body coordinate system, therefore This can be understood as first setting the scalar T... c Mapped to the z-axis direction of the machine body, i.e., [0,0,T] c ] T Next, left multiplication Project the vector onto the world coordinate system where variable v resides, g = [0, 0, 9.81]. T m / s 2 It is the acceleration due to gravity, f d Representing an unknown external disturbance force, q = [q w ,q x ,q y ,q z ] T Let ω be a unit quaternion representing the attitude of the quadcopter drone, where ω = [ω x ,ω y ,ω z ] T J represents the angular velocity in the body coordinate system, J = diag(J x J y J z) is the rotational inertia matrix, τ c =[τ c,x ,τ c,y ,τ c,z ] T To control the torque, τ d Represents unknown external torque disturbance, × represents vector cross product, given the attitude information of the quadcopter UAV q=[q w ,q x ,q y ,q z ] T Then, the above rotation matrix Defined as:

[0200]

[0201] In model (3), with the symbol The relevant equations are defined as follows:

[0202]

[0203] Condition 1: Regarding the interference f in equation (3) d Assume it is continuous and first-order differentiable, and that it has positive constants δ and δ. Guarantee ||f d ||≤δ and Heng is established;

[0204] To control the position, velocity, and attitude of a quadcopter UAV, only the first three formulas of equation (3) are considered, where the disturbance factors exist in the position dynamics model:

[0205]

[0206] The model can be written as a multivariable first-order system as follows:

[0207]

[0208] Where z1 = mv, This serves as a virtual control input. For this system, the following disturbance observer is designed to estimate the disturbance f. d :

[0209]

[0210] in Let z1 be the observed value, which can be obtained by solving the differential equation (8). It is a composite interference f d The estimated values, L1 and L2 are gain parameters designed to ensure observer stability, and the functions φ1 and φ2 are defined as follows:

[0211]

[0212] Where d ∞ ∈(0,1), k i >0,k′ i >0,k″ i >0, i=(1,2). According to the definition of homogeneity, functions φ1(e1) and φ2(e1) are homogeneous at the 0-limit with a homogeneity of -1, and are also homogeneous at the ∞-limit with a homogeneity of d. ∞ Therefore, it can be deduced that the functions φ1(e1) and φ2(e1) have bilimit homogeneity;

[0213] Combining equations (7) and (8), we obtain the following error system:

[0214]

[0215] in This is due to observational errors caused by interference.

[0216] Under condition 1, the fixed-time interference observer (8) This allows for the implementation of f in the system described by formula (7). d An effective estimate is obtained, and the estimation error converges within a fixed time. This design result is summarized by the following theorem:

[0217] Theorem 1: For a system described by formula (7) that satisfies condition 1, if we choose 0 < d ∞ <1 and any positive gain k i >0,k′ i >0,k″ i If >0, i=(1,2), then there exist gains L1 and L2 satisfying the following equation:

[0218]

[0219] This makes the observation error system (10) stable at a fixed time, that is, the interference observer (8) can be stable at a fixed time. The interference f is obtained by internal estimation. d That is, for any Heng was established.

[0220] Proof of Theorem 1: Consider the following bilimit homogeneous Lyapunov function:

[0221] V0(e)=V1(e1,e2)+V2(e2), (12)

[0222] Where V1 and V2 are defined as:

[0223]

[0224] Where 0 < p ∞ ≤max{1,3(1-d ∞ ) / 2},1<p0≤2p ∞ / (1-d ∞ ), and β0,β ∞ ,β′0,β′ ∞ All are arbitrary positive real numbers, ξ is defined as Furthermore, based on Young's inequality, V1(e1,e2)≥0, which guarantees the positive definiteness of the Lyapunov function V0(e) defined in (12).

[0225] The derivative of the Lyapunov function can be obtained by taking the derivative.

[0226]

[0227] in

[0228]

[0229] get We need to prove its negative definiteness, that is, prove the negative definiteness of W(e). To do this, consider the structure of W(e) that converges to the following hyperplane:

[0230] Z1 = {φ1(e1) = e2}. (19)

[0231] In the hyperplane Z1, the equation can be obtained. Therefore, based on formulas (16) and (17), it can be seen that g1,σ1∈{0} always holds true in the hyperplane Z1. Then, by substituting g1=σ1=0 into formula (15), we can obtain the convergence to the hyperplane Z1.

[0232]

[0233] Considering the multivariable symbolic function x T The properties of sign(x) = ||x|| and the hyperplane Z1 definition Its form can be written as:

[0234]

[0235] Therefore, considering the conditions in inequality (11) function It is negative definite, that is Guaranteed The negative qualitative property.

[0236] also, Since the continuous single-valued double-limit homogeneous function condition described in Lemma 1 is satisfied, based on the conclusion of Lemma 1, there exists a positive parameter L1 that satisfies the following condition to guarantee the negative definiteness of W(e).

[0237]

[0238] Here, χ(e1,e2) is an upper semi-continuous homogeneous function that reaches its maximum value on the homogeneous sphere. Thus, the positive definiteness of the Lyapunov function V0(e) and its derivative are thus established. The negative definiteness of the observation error system (10) has been proven, and the global asymptotic stability of the observation error system (10) is guaranteed. Furthermore, based on the conclusion of Lemma 1, there exist positive constants η0, η... ∞ Make satisfy:

[0239]

[0240] Therefore, based on the fixed-time convergence theory, the observation error system (10) will converge at a fixed time. Converging to zero, i.e., the interference observer (8) can be within a fixed time. Internal implementation of composite interference f d Effective estimation, fixed time satisfy:

[0241]

[0242] Furthermore, as can be seen from formula (24), a fixed time This is independent of the initial observation error. Thus, the fixed-time stability of the observation error system (10) is guaranteed, and Theorem 1 is proved.

[0243] Part Two: Design of a Quadrotor UAV Trajectory Tracking Controller Based on Model Predictive Control: This section builds upon the disturbance estimates from Part One. Given the information, construct a prediction model that includes the effects of interference:

[0244]

[0245] The model uses position p, velocity v, and attitude q as state variables to control thrust T. c With reference angular velocity ω c For control variables, i.e., state variables in the optimization problem, we take x = [p, v, q]. T The control quantity is u = [T] c ,ω c ] T =[T c ,ω x,c ,ω y,c ,ω z,c ]T ;

[0246] Next, considering the trajectory tracking requirements of a quadcopter drone, the following objective function V is constructed:

[0247]

[0248] Where k represents the sampling time, Δx (k+i|k) =x (k+i|k) -x (k+i|k),r It is the state error, x (k+i|k),r x represents the reference state in the desired trajectory. (k+N|k) For the terminal state, Δu (k+i|k) =u (k+i|k) -u (k+i|k),r Indicates the control input error, u (k+i|k),r As a reference control input, u is usually taken. (k+i|k),r =[mg,0,0,0] T , and Let represent the non-negative weight matrices for state error and control error, respectively. This is the weight matrix of the terminal state error, where N is the prediction step size of the model predictive control (MDC). That is, the MDC predicts N sampling times forward each time it solves the problem. If the sampling interval is dt, then the prediction time domain of the MDC is T = N * dt. Therefore, the objective function V is defined as N+1 state errors Δx. (k+i|k) and N control errors Δu (k+i|k) The weighted quadratic summation, if the optimization objective is to minimize the objective function, then the control quantity will minimize the error between the predicted state and the reference state, that is, to track the reference state in the desired trajectory.

[0249] In summary, the optimization problem for a quadrotor UAV trajectory tracking controller based on model predictive control is defined as follows:

[0250]

[0251] The objective function V is defined in equation (26). For the prediction model defined in equation (25), x k The state value at the current sampling time k is used as the initial value for the optimization solution iteration process. and These are control constraints, state constraints, and terminal state constraints.

[0252] At sampling time k, solve the optimization problem (27) to obtain the optimal control sequence. And the optimal predicted state sequence is obtained recursively. Both sequences satisfy optimality; in this case, only the first component of the optimal control sequence is taken. The system is applied to the problem, and at the next sampling time k+1, the optimization problem is solved again under new conditions to obtain a new optimal control sequence. With the optimal predicted state sequence Repeat this process to ensure real-time optimality;

[0253] Part Three: Design of Event Triggering Conditions Based on Predicted State Errors

[0254] Consider the predictive model (25) in model predictive control, which can be expressed in the following form:

[0255]

[0256] The model satisfies the following inequality:

[0257]

[0258] Where L N Let ||·|| be the Lipschitz constant for this system, and ||·|| denote the vector magnitude.

[0259] For the prediction model (28), the actual state x can be obtained by the Lipschitz lemma. k With the optimal predicted state sequence Upper limit of the difference:

[0260]

[0261] in express The largest eigenvalue, P is the terminal cost matrix in the model predictive control optimization problem;

[0262] Let t be a certain time when the calculation is performed. k Note that this time is different from the sampling time k. The sampling operation continues, and even if no calculation is performed, the sampling time will still increase; at t k At time t, the optimal predicted state sequence of model predictive control is denoted as . Where t = t k +i, i∈[0,T], the actual state after applying the optimal control sequence at this time is denoted as . Then the prediction error ω can be defined. t for:

[0263]

[0264] To address this error, the event triggering conditions are designed as follows:

[0265]

[0266] Where inf represents the infimum. This means that at t k After time ω t The moment t when the value is first greater than or equal to μ 0<θ≤1, the optimization solution of model predictive control is executed only when condition (32) is met; otherwise, the corresponding component in the optimal control sequence obtained in the previous solution is used as the optimal control quantity at the current moment.

[0267] Finally, to verify the effectiveness of the robust trajectory tracking control method for quadrotor UAVs based on event triggering mechanism and interference observer proposed in this invention, a quadrotor UAV trajectory tracking control system was built in a simulation environment and a physical test scenario. The system was then compared and analyzed with classical model predictive control methods and model predictive control methods based on event triggering mechanism to verify the advantages of the proposed method in terms of tracking accuracy, anti-interference capability, and algorithm requirements.

[0268] The robust trajectory tracking control method for quadrotor UAVs based on an event-triggered mechanism and an interference observer proposed in this invention has been integrated and verified in simulation environments and physical test scenarios. To verify the effectiveness of the proposed robust trajectory tracking control method for quadrotor UAVs based on an event-triggered mechanism and an interference observer, the aircraft parameters are set as follows:

[0269] 1) The desired trajectory of the quadcopter UAV in the simulation environment is set as follows: p ref =[2sint,2cost-2,-1-0.1t] T m; The expected trajectory in the physical testing scenario is set to: p ref = [1.5sin0.03t] 2 cos0.03t 2 -2.5cos0.03t 2 +0.5, -1.0, 0] T m.

[0270] 2) Physical parameters of the quadcopter UAV: ​​mass m = 1.0 kg, inertial parameter J x =2.64×10 -3 kgm 2 J y =2.64×10 -3 kgm 2 J z =4.96×10 -3 kgm 2 .

[0271] 3) Propose the parameters of the method: , P=diag(1500,1500,1500,400,400,400,500,500,500), R=diag(1,10,10,10), N=10, dt=20ms, k1=k2=2.0, k′1=k′2=0.6, k″1=k″2=3.0, d ∞ =1 / 3, L1=L2=1.0, L N =1.0, θ=0.5.

[0272] 4) Other parameter settings: During the simulation test and verification process, the operating frequency of the model predictive controller and the interference observer is 50Hz, and the uncertain interference is transmitted through the time-varying function f. d =[0.2sin(2πt / 7.5),0.2cos(2πt / 7.5),0.1sin(2πt / 7.5)] T N-simulation; In the actual test scenario, the drone was affected by wind interference with a wind speed of 4.0-6.5m / s.

[0273] To verify that the key feature of the algorithm proposed in this invention is that a quadcopter UAV can accurately track the desired trajectory under complex interference, and that the algorithm requires fewer computations than traditional methods, a comparative method is introduced for verification. The comparative methods are: the classical model predictive control method is denoted as MPC, the event-triggered model predictive control method is denoted as ETMPC, and the method proposed in this invention is denoted as FxTDO-ETMPC.

[0274] To verify the robustness and high-precision tracking control capability of the proposed method under complex disturbances, complex disturbances were applied in the simulation. And enable the quadcopter drone to track the desired trajectory Tracking effect as Figure 3 and Figure 4 As shown in the figure, the FxTDO-ETMPC method proposed in this invention exhibits the best trajectory tracking performance, with the actual trajectory most closely matching the desired trajectory. The root mean square errors (RMS) of trajectory tracking for the three methods are 0.067m (MPC), 0.072m (ETMPC), and 0.028m (FxTDO-MPC), respectively. The error level of the method proposed in this invention is significantly lower than that of the comparative methods. In summary, the method proposed in this invention enables quadcopter UAVs to accurately track the desired trajectory under complex interference conditions, validating the controller's robustness and high-precision tracking control performance.

[0275] Furthermore, to verify the advantages of the proposed method in terms of computational requirements, the number of calculations for each method was recorded in simulation tests. Figure 5A schematic diagram of the computation triggering is provided, and the number of computation triggers for the three methods are 2231 (MPC), 1127 (ETMPC), and 874 (FxTDO-MPC), respectively. It can be seen that the number of computation triggers for the method proposed in this invention is significantly reduced compared to the MPC and ETMPC methods, effectively lowering the computational requirements of the algorithm.

[0276] In a physical verification scenario, the trajectory tracking effect is as follows: Figure 6 and Figure 7 As shown, the position errors of the proposed method and FxTDO-MPC in tracking the desired trajectory are as follows: Figure 8 As shown, the event triggering time is illustrated in the diagram. Figure 9 As shown, the results are consistent with the simulation results. The number of event triggers proposed in this invention is significantly reduced compared with the traditional method, but the error level is only slightly increased.

[0277] In summary, the robust trajectory tracking control method for quadrotor UAVs based on an event-triggered mechanism and an interference observer proposed in this invention has been comprehensively demonstrated and analyzed from the aspects of strong controller robustness, high-precision tracking, and reduced computational requirements. Simulation results show that this method can solve the trajectory tracking control problem of quadrotor UAVs under complex interference with limited computational resources. The theoretical basis of this invention fills the gap in the field of quadrotor UAV trajectory tracking control with limited computational resources and subject to complex interference. The proposed algorithm has a significant effect on improving the trajectory tracking accuracy of quadrotor UAVs and enhancing the robustness of the system, while its own computational requirements are low.

[0278] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A robust trajectory tracking control method for unmanned aerial vehicles (UAVs) based on event triggering and disturbance observation, characterized by the following steps: as follows: Part 1: Design of a fixed-time disturbance observer based on the double-limit homogeneity theory: A strongly coupled and nonlinear model of a quadrotor UAV is established. Considering the complex disturbances experienced by the quadrotor UAV, the position dynamics model of the quadrotor UAV is rewritten as a multivariable first-order system. For this system, a fixed-time disturbance observer is designed based on the double-limit homogeneity theory. Part Two: Design of a Quadrotor UAV Trajectory Tracking Controller Based on Model Predictive Control: For the strongly coupled and nonlinear model of the quadrotor UAV, combined with the disturbance estimates of the disturbance observer mentioned above, the predictive model constraints in the optimization problem are constructed. Considering the requirements of the trajectory tracking task, an objective function based on the position tracking error is constructed. On this basis, a robust model predictive controller based on a fixed-time disturbance observer is designed to solve the robust control problem of the quadrotor UAV under the influence of complex disturbances. Part Three: Design of Event Triggering Conditions Based on Predictive State Errors: For a robust model predictive controller that integrates a fixed-time disturbance observer, considering the error between its predicted state and the actual state, event triggering conditions are designed. The optimization solution of model predictive control is executed only when the condition is met, thereby reducing the computational requirements of model predictive control and enabling the solution of robust control problems of quadcopter UAVs with limited computational resources.

2. The robust trajectory tracking control method for UAVs based on event triggering and interference observation as described in claim 1, characterized in that, The detailed steps are as follows: Part 1: Design of a Fixed-Time Disturbance Observer Based on the Double-Limit Homogeneity Theory: The position dynamics model of a quadrotor UAV is rewritten as a multivariable first-order system, and a fixed-time disturbance observer is designed based on the double-limit homogeneity theory. Homogeneity: Consider nonlinear systems: Where s = [s1,...,s n ] T Represents the state vector. For vector functions, Let n be the n-dimensional real number field. If for any ε > 0, the following equation holds: Where i∈1,...,n, then the vector function G(s) is said to be r-homogeneous, and is called... Let G(s) be the homogeneity of the vector function, or simply the (r,k)-homogeneous vector function G(s), where r = (r1,…,rk) n ) is the weight vector; Bilimit Homogeneity: If, near s = 0, a vector function G(s) is approximated by a (r0, k0)-homogeneous function G0, where r0 is the weight vector and k0 is the homogeneity, then G(s) is said to be homogeneous with the related vector (r0, k0, G0) in the 0-limit. Similarly, if, near s = ∞, a vector function G(s) can be approximated by a (r0, k0)-homogeneous function G0, then G(s) is homogeneous with the related vector (r0, k0, G0). ∞ If G(s) is approximated, then G(s) is said to be in the ∞-limit with respect to the related vector (r). ∞ ,k ∞ G ∞ If a vector function G(s) is homogeneous in both the 0-limit and the ∞-limit, it is called bilimit homogeneity, or simply bihomogeneity. Fixed-time convergence: If system (1) is globally asymptotically stable, and for any Where s0 is the initial value of the system, all of which guarantee that the system's settling time t≤t max , where t max If the constant is a fixed positive constant, then the system described by formula (1) is said to be fixed-time stable; Lemma 1: Consider a bilimit homogeneous vector function that is semi-continuous on the upper side. Its double-limit correlation vectors are (r0,k0,G0) and (r ∞ ,k ∞ G ∞ If system (1) satisfies global asymptotic stability, then there exists a positive definite function V. G (s) guarantees its derivative. It is negative definite and satisfies the homogeneity of the bilimit; furthermore, if k0 ≤ k ∞ And there exist real numbers m0 and m ∞ Satisfying m0>max{r i 0 } and m0>max{r i ∞ }, then there exist positive constants κ0>0 and κ ∞ >0, guaranteeing that for any inequality Heng is established; First, two coordinate systems need to be introduced to describe the model of the quadcopter drone, including the world coordinate system. and body coordinate system The world coordinate system is fixed to the ground, and the orientation of its coordinate axes does not change. The body coordinate system is coincident with the body of the quadcopter UAV, with its origin at its center, and the orientation of its coordinate axes changes with the UAV's pose. Considering the quadcopter UAV as a six-degree-of-freedom rigid body model, its kinematics and dynamics models are established: Where, p = [p x ,p y ,p z ] T With v = [v x ,v y ,v z ] T These represent the position and velocity of the quadcopter drone, respectively, and m is the mass of the quadcopter drone. This represents the rotation matrix from the body coordinate system to the world coordinate system. Its definition is related to the attitude information. c The scalar value represents the control thrust. This represents the unit vector pointing in the z-axis direction of the body coordinate system, therefore This can be understood as first setting the scalar T... c Mapped to the z-axis direction of the machine body, i.e., [0,0,T] c ] T Next, left multiplication Project the vector onto the world coordinate system where variable v resides, g = [0, 0, 9.81]. T m / s 2 It is the acceleration due to gravity, f d Representing an unknown external disturbance force, q = [q w ,q x ,q y ,q z ] T Let ω be a unit quaternion representing the attitude of the quadcopter drone, where ω = [ω x ,ω y ,ω z ] T J represents the angular velocity in the body coordinate system, J = diag(J x J y J z ) is the rotational inertia matrix, τ c =[τ c,x ,τ c,y ,τ c,z ] T To control the torque, τ d Represents unknown external torque disturbance, × represents vector cross product, given the attitude information of the quadcopter UAV q=[q w ,q x ,q y ,q z ] T Then, the above rotation matrix Defined as: In model (3), with the symbol The relevant equations are defined as follows: Condition 1: Regarding the interference f in equation (3) d Assume it is continuous and first-order differentiable, and that it has positive constants δ and δ. Guarantee ||f d ||≤δ and Heng is established; For controlling the position, velocity, and attitude of a quadcopter UAV, only the first three formulas of equation (3) are considered, where the disturbance factors exist in the position dynamics model: The model can be written as a multivariable first-order system as follows: Where z1 = mv, This serves as a virtual control input. For this system, the following disturbance observer is designed to estimate the disturbance f. d : in The observed value of z1 is obtained by solving the differential equation (8). It is a composite interference f d The estimated values, L1 and L2 are gain parameters designed to ensure the stability of the observer, and the functions φ1 and φ2 are defined as follows: Where d ∞ ∈(0,1), k i >0,k i ′>0,k i ″>0, i=(1,2). According to the definition of homogeneity, functions φ1(e1) and φ2(e1) are homogeneous at the 0-limit with a homogeneity of -1, and are also homogeneous at the ∞-limit with a homogeneity of d. ∞ Therefore, it is deduced that the functions φ1(e1) and φ2(e1) have bilimit homogeneity; Combining equations (7) and (8), we obtain the following error system: in This is due to observational errors caused by interference. Under condition 1, the fixed-time interference observer (8) This allows for the implementation of f in the system described by formula (7). d An effective estimate of the , and the estimation error converges within a fixed time, is summarized as follows: Consider the following bilimit homogeneous Lyapunov function: V0(e)=V1(e1,e2)+V2(e2), (12) Where V1 and V2 are defined as: Where 0 < p ∞ ≤max{1,3(1-d ∞ ) / 2},1<p0≤2p ∞ / (1-d ∞ ), and β0,β ∞ ,β′0,β′ ∞ All are arbitrary positive real numbers, ξ is defined as Furthermore, based on Young's inequality, V1(e1,e2)≥0, which guarantees the positive definiteness of the Lyapunov function V0(e) defined in (12); The derivative of the Lyapunov function can be obtained by taking the derivative. in get We need to prove its negative definiteness, that is, prove the negative definiteness of W(e). To do this, consider the structure of W(e) that converges to the following hyperplane: Z1 = {φ1(e1) = e2}. (19) In the hyperplane Z1, the equation can be obtained. Therefore, based on formulas (16) and (17), g1,σ1∈{0} always holds true in the hyperplane Z1. Then, by substituting g1=σ1=0 into formula (15), we obtain the convergence to the hyperplane Z1. Considering the multivariable symbolic function x T The properties of sign(x) = ||x|| and the hyperplane Z1 definition Its form is written as: Therefore, considering the conditions in inequality (11) function It is negative definite, that is Guaranteed Negative qualitative; also, Since the condition for a continuous single-valued bilimit homogeneous function described in Lemma 1 is satisfied, based on the conclusion of Lemma 1, there exists a positive parameter L1 that satisfies the following condition, guaranteeing the negative definiteness of W(e): Here, χ(e1,e2) is an upper semi-continuous homogeneous function that reaches its maximum value on the homogeneous sphere. Thus, the positive definiteness of the Lyapunov function V0(e) and its derivative are... The negative definiteness of the observation error system (10) has been proven, and the global asymptotic stability of the observation error system (10) has been guaranteed. Furthermore, based on the conclusion of Lemma 1, there exist positive constants η0, η ∞ Make satisfy: Therefore, based on the fixed-time convergence theory, the observation error system (10) will converge at a fixed time. Converging to zero, i.e., the interference observer (8) can be within a fixed time. Internal implementation of composite interference f d Effective estimation, fixed time satisfy: Furthermore, as can be seen from formula (24), a fixed time Regardless of the initial observation error, the fixed-time stability of the observation error system (10) is guaranteed; Part Two: Design of a Quadrotor UAV Trajectory Tracking Controller Based on Model Predictive Control: This section builds upon the disturbance estimates from Part One. Given the information, construct a prediction model that includes the effects of interference: The model uses position p, velocity v, and attitude q as state variables to control thrust T. c With reference angular velocity ω c For control variables, i.e., state variables in the optimization problem, we take x = [p, v, q]. T The control quantity is u = [T] c ,ω c ] T =[T c ,ω x,c ,ω y,c ,ω z,c ] T ; Next, considering the trajectory tracking requirements of a quadcopter drone, the following objective function V is constructed: Where k represents the sampling time, Δx (k+i|k) =x (k+i|k) -x (k+i|k),r It is the state error, x (k+i|k),r x represents the reference state in the desired trajectory. (k+N|k) For the terminal state, Δu (k+i|k) =u (k+i|k) -u (k+i|k),r Indicates the control input error, u (k+i|k),r As a reference control input, u is usually taken. (k+i|k),r =[mg,0,0,0] T , and Let represent the non-negative weight matrices for state error and control error, respectively. This is the weight matrix of the terminal state error, where N is the prediction step size of the model predictive control (MDC). That is, the MDC predicts N sampling times forward each time it solves the problem. If the sampling interval is dt, then the prediction time domain of the MDC is T = N * dt. Therefore, the objective function V is defined as N+1 state errors Δx. (k+i|k) and N control errors Δu (k+i|k) The weighted quadratic summation, if the optimization objective is to minimize the objective function, then the control quantity will minimize the error between the predicted state and the reference state, that is, to track the reference state in the desired trajectory. In summary, the optimization problem for a quadrotor UAV trajectory tracking controller based on model predictive control is defined as follows: The objective function V is defined in equation (26). For the prediction model defined in equation (25), x k The state value at the current sampling time k is used as the initial value for the optimization solution iteration process. and These are control constraints, state constraints, and terminal state constraints. At sampling time k, solve the optimization problem (27) to obtain the optimal control sequence. And the optimal predicted state sequence is obtained recursively. Both sequences satisfy optimality; in this case, only the first component of the optimal control sequence is taken. The system is applied to the problem, and at the next sampling time k+1, the optimization problem is solved again under new conditions to obtain a new optimal control sequence. With the optimal predicted state sequence Repeat this process to ensure real-time optimality; Part Three: Design of Event Triggering Conditions Based on Predicted State Errors Consider the predictive model (25) in model predictive control, which can be expressed in the following form: The model satisfies the following inequality: Where L N Let ||·|| be the Lipschitz constant for this system, and ||·|| denote the vector magnitude. For the prediction model (28), the actual state x can be obtained by the Lipschitz lemma. k With the optimal predicted state sequence Upper limit of the difference: in express Let t be the largest eigenvalue of , and P be the terminal cost matrix in the model predictive control optimization problem; let t be a certain time when the calculation is performed. k Note that this time is different from the sampling time k. The sampling operation continues, and even if no calculation is performed, the sampling time will still increase; at t k At time t, the optimal predicted state sequence of model predictive control is denoted as . Where t = t k +i, i∈[0,T], the actual state after applying the optimal control sequence at this time is denoted as . Then the prediction error ω can be defined. t for: To address this error, the event triggering conditions are designed as follows: Where inf represents the infimum. This means that at t k After time ω t The moment t when μ first becomes greater than or equal to μ The optimization solution of model predictive control is only executed when condition (32) is met; otherwise, the corresponding component in the optimal control sequence obtained in the previous solution is used as the optimal control quantity at the current moment.

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