Unmanned aerial vehicle trajectory tracking control method based on model prediction and preset performance constraint

Abstract

The present application relates to a model prediction and preset performance constraint based unmanned aerial vehicle trajectory tracking control method, which introduces Lyapunov stability constraint in the model prediction control framework and integrates preset performance control mechanism, thereby realizing the unification of performance constraint and system stability analysis, including the following steps: establishing a nonlinear system model based on unmanned aerial vehicle dynamics; defining position error and attitude error and constructing preset performance function, converting error with performance constraint into unconstrained error through error normalization and nonlinear transformation; establishing model prediction optimization problem under the premise of considering input saturation and stability constraint; designing auxiliary control law based on transformed error to construct stability constraint; proving that the control strategy can guarantee that the error meets preset performance constraint and system local asymptotic stability. The present application can realize stable and reliable trajectory tracking control of unmanned aerial vehicle system, and has high tracking precision and good dynamic performance.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology, specifically to a method for unmanned aerial vehicle (UAV) trajectory tracking control based on model prediction and preset performance constraints. Background Technology

[0002] As high-tech equipment integrating advanced electronic and computer technologies, unmanned aerial vehicles (UAVs) can complete various flight missions through preset programs or autonomous decision-making systems without real-time human control. With the continuous iteration of sensor and automatic control technologies and the increasing maturity of manufacturing processes, UAVs are being applied in civilian fields such as smart agriculture, power line inspection, and road monitoring. Therefore, designing an effective control strategy to ensure high-precision tracking control of UAVs is an important and challenging research topic.

[0003] Conventional control methods can meet the trajectory tracking requirements of UAVs to a certain extent, but they usually lack systematic means when dealing with hard constraints related to UAVs, and it is difficult to guarantee the transient performance of the closed-loop system.

[0004] The DQN-based quadrotor UAV model predictive control method (CN120215553A), the multi-model-based quadrotor UAV model predictive control method (CN118092493A), and the hybrid strategy-driven UAV model predictive control method (CN115480487A) improve the predictive capability and control accuracy of UAVs from different perspectives. However, none of these methods can directly guarantee the transient performance of the closed-loop system. Directly using preset performance constraints as constraints for model prediction may lead to the model prediction optimization problem becoming unsolvable. In addition, the multi-quadrotor UAV preset performance trapping fault-tolerant control method (CN120065731A) and the UAV preset performance control method and system considering time-varying position constraints (CN119536343A) have achieved good results in guaranteeing tracking performance and handling time-varying constraints, but their preset performance frameworks are difficult to handle complex constraints such as input saturation and state constraints. Summary of the Invention

[0005] To address the problems existing in the prior art, the purpose of this invention is to provide a UAV trajectory tracking control method based on model prediction and preset performance constraints, which can ensure that the tracking error meets the preset performance constraints and ensure recursive feasibility under the condition that the auxiliary control law is bounded.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] A UAV trajectory tracking control method based on model prediction and preset performance constraints includes the following steps:

[0008] Based on the kinematic and dynamic relationship of the UAV, a nonlinear system model is established to obtain the state variables of the UAV;

[0009] Based on the obtained UAV state variables, position error and attitude error are defined, and the tracking error is obtained based on the position error and attitude error. A preset performance function is constructed, and the tracking error constrained by the preset performance function is normalized and transformed through error normalization and nonlinear transformation. Transform into an equivalent unconstrained error variable ;

[0010] Unconstrained error variables As the prediction error is introduced into the cost function, a model predictive control optimization problem is established under the premise of considering input saturation and stability constraints.

[0011] Based on unconstrained error variables Design auxiliary control laws;

[0012] The model predictive control optimization problem is solved based on the auxiliary control law to obtain the optimal control sequence. The control command corresponding to the first sampling period in the optimal control sequence is then used as the actual control input to the UAV to achieve trajectory tracking.

[0013] Furthermore, the nonlinear system model of the UAV is as follows:

[0014] ; ;

[0015] in, This indicates the position and yaw angle of the UAV in the inertial coordinate system. These are the linear velocity and angular velocity of the drone in its body coordinate system. Indicates control input, and It is a gain matrix described by a diagonal matrix. , , , This is the gain coefficient. , , , It is a time constant. This represents the transformation matrix that transforms a vector from the body coordinate system to the inertial coordinate system; the tracking error is defined as... , For reference trajectory.

[0016] Furthermore, to ensure tracking error To achieve the desired transient and steady-state performance, a pre-defined performance function matrix is ​​introduced. , It is a diagonal matrix composed of smooth, bounded, and strictly decreasing scalar functions, i.e. This ensures that each component of the unconstrained error variable satisfies:

[0017] , , ;

[0018] in, Representing the corresponding components of position and yaw angle, each boundary function is selected. for:

[0019] ;

[0020] in, , , Let these represent the initial error boundary, final steady-state boundary, and convergence rate for each component, respectively, and they must satisfy the following conditions: .

[0021] Furthermore, by utilizing a preset performance function matrix and tracking error... Define the normalized error variable:

[0022] ;

[0023] in, , and Transform each normalized error component:

[0024] ;

[0025] The above formula uses functions Transforming the normalized error into an unconstrained error, as long as... By maintaining boundaries, we can ensure that when hour, Establishment, Definition .

[0026] Furthermore, the cost function of the model predictive control optimization problem At sampling time Defined as:

[0027] ;

[0028] in, It is the prediction error. , To predict the time domain, Classified as step, It is the sampling period. It is a predicted state. It is a control sequence in the prediction time domain. It is the positive definite diagonal weighted matrix of the prediction error. It is the positive definite diagonal weighted matrix of the system input. express , express , Is The cost function at time step, Satisfy control input saturation constraints , This is the maximum input limit.

[0029] Furthermore, the model predictive control optimization problem introduces Lyapunov stability contraction constraints:

[0030] ;

[0031] in, Is Time-based auxiliary control law, It is a Lyapunov function. Represents the nonlinear model of the drone. Indicates in The initial state at a given moment.

[0032] Furthermore, for unconstrained error variables subscript Represents the corresponding components of position and yaw angle, and the time derivative of each component. It can be represented as:

[0033] ;

[0034] To simplify the expression, define

[0035] ,

[0036] Defined here and ,therefore It can be rewritten as

[0037] ;

[0038] To design the auxiliary control law, the following error variable is defined. ,in As a virtual control input vector, it is designed as

[0039] ;

[0040] in, It is a positive definite gain matrix. It is a virtual control input vector.

[0041] Furthermore, auxiliary control law The design employs a backstepping method, specifically in the following form:

[0042] ;

[0043] in, It is a positive definite gain matrix. For the error vector, The derivative of the transformation matrix is ​​represented. The second derivative of the reference trajectory is represented. for derivative, .

[0044] Furthermore, the UAV trajectory tracking control method meets the following requirements: Not less than the auxiliary control law It is recursively feasible when bounded to an infinite norm; this stability exists only within the defined region of attraction. The specific range of the attraction domain is constrained by the controller parameters.

[0045] An electronic device includes a processor and a memory, wherein a computer program is stored in the memory, and the processor executes the computer program to implement the above-described UAV trajectory tracking control method.

[0046] In summary, the present invention has the following advantages:

[0047] This invention introduces Lyapunov stability constraints and integrates a pre-defined performance control mechanism into the model predictive control framework. By utilizing error normalization and nonlinear transformation, it converts performance-constrained tracking errors into unconstrained error variables, achieving a unification of performance constraints and system stability analysis. This ensures that the UAV trajectory tracking error meets pre-defined transient and steady-state performance constraints while guaranteeing the recursive feasibility and local asymptotic stability of the system under input saturation conditions. This solves the technical problems of traditional model predictive control, which is prone to error overshoot due to its finite prediction time domain and optimization infeasibility under complex constraints, as well as the difficulty of pre-defined performance control in handling complex constraints. Compared with classical model predictive control, this invention significantly improves the transient performance of UAVs, reduces overshoot and weakens oscillations, and significantly improves the accuracy and dynamic performance of UAV trajectory tracking. Furthermore, this method theoretically achieves the organic integration of pre-defined performance control and model predictive control, expanding the application scope of both control strategies. Attached Figure Description

[0048] Figure 1This is a schematic diagram of the UAV tracking error curve in this embodiment;

[0049] Figure 2 This is a schematic diagram of the UAV control input curve in this embodiment;

[0050] Figure 3 This is a flowchart of the UAV trajectory tracking control method based on model prediction and preset performance constraints in this embodiment. Detailed Implementation

[0051] The present invention will now be described in further detail.

[0052] This embodiment provides a UAV trajectory tracking control method based on model prediction and preset performance constraints, which solves the problem of difficulty in balancing stability and performance constraints in traditional model predictive control for UAV trajectory tracking. The method includes the following steps:

[0053] Step 1: Based on the kinematic and dynamic relationship of the UAV, establish a nonlinear system model to obtain the state variables of the UAV;

[0054] Step 2: Based on the obtained state variables of the UAV, construct a preset performance function, and transform the performance-constrained error into an equivalent unconstrained error variable through error normalization and nonlinear transformation.

[0055] Step 3: Combining the unconstrained error variables after nonlinear transformation, establish a model predictive control optimization problem under the premise of considering input saturation and stability constraints;

[0056] Step 4: Based on the transformed unconstrained error variables, design an auxiliary control law based on the backstepping method to construct stability constraints.

[0057] Step 5: Analyze the constructed control framework from the aspects of recursive feasibility of model predictive control and system stability, and prove that the control strategy can ensure that the error meets the preset performance constraints and the local asymptotic stability of the system.

[0058] Step 6: Verify the effectiveness of the proposed control strategy through MATLAB simulation.

[0059] As an example, step 1, further, the nonlinear model of the UAV can be described by the following equation:

[0060] ;

[0061] (1)

[0062] in, This indicates the position and yaw angle of the UAV in the inertial coordinate system. These are the linear velocity and angular velocity of the drone in its body coordinate system. Indicates control input, and It is a gain matrix described by a diagonal matrix. , , , This is the gain coefficient. , , , It is a time constant. The transformation matrix that transforms a vector from the body coordinate system to the inertial coordinate system is expressed as follows:

[0063] (2)

[0064] To simplify the subsequent analysis, the system shown in equation (1) can be rewritten as:

[0065] (3)

[0066] in .

[0067] Step 2: Further, define the UAV reference trajectory as follows:

[0068] ;

[0069] (4)

[0070] in, , , , , , The arctangent function is defined in the fourth quadrant, and its range is [range missing]. The tracking error is defined as... To ensure accurate trajectory tracking by the drone, this tracking error must meet the following requirements. In addition, the control input must also meet the following requirements. .

[0071] To address the UAV trajectory tracking control problem based on model prediction and preset performance constraints, the following assumptions are proposed: Desired trajectory Its first and second derivatives are both bounded, i.e. , , .

[0072] To ensure tracking error To achieve the desired transient and steady-state performance, a pre-defined performance function matrix is ​​introduced. . It is a diagonal matrix composed of smooth, bounded, and strictly decreasing scalar functions, i.e. This ensures that each component of the tracking error satisfies:

[0073] , , (5)

[0074] in, This represents the corresponding components of position and yaw angle. Each boundary function is selected. for:

[0075] (6)

[0076] in, , , Let these represent the initial error boundary, final steady-state boundary, and convergence rate for each component, respectively, and they must satisfy the following conditions: To facilitate controller design, a normalized error variable is defined: (7)

[0077] in, , and Transform each normalized error component:

[0078] (8)

[0079] The above formula uses functions Transforming the normalized error into an unconstrained error, as long as... By maintaining boundaries, we can ensure that when hour, Established. Definition. .

[0080] Step 3: Furthermore, we propose a model predictive control method that combines preset performance control. Unlike classical model predictive control, this method introduces contraction constraints as stability constraints to ensure the stability of the system at each sampling time. Moreover, considering actuator constraints, this control framework can achieve accurate trajectory tracking while ensuring input feasibility and system closed-loop stability. Under the above constraints, the following optimization problem is proposed:

[0081] (9)

[0082] The following constraints must be met:

[0083] (10)

[0084] (11)

[0085] (12)

[0086] (13)

[0087] in, It is the prediction error. , To predict the time domain, Classified as step, It is the sampling period. It is a predicted state. It is a control sequence in the prediction time domain. It is the maximum input limit. Is Time-based auxiliary control law, It is a Lyapunov function, which will be given in the subsequent design of the auxiliary control law. Is The cost function at time step, It is the positive definite diagonal weighted matrix of the prediction error. It is the positive definite diagonal weighted matrix of the system input. express , express In equations (10)-(13), equation (10) represents the nonlinear model in equation (3) used to predict future states, and equation (11) provides the model predictive control... At the initial state at time step (12), the control input constraint is represented by equation (13), and the stability constraint is represented by equation (13). This constraint ensures that the derivative of the Lyapunov function generated by the model predictive controller at each sampling time step is not greater than that of the auxiliary control law. The resulting derivative ensures the closed-loop stability of the system. Let , Indicates in The optimal control input sequence obtained by solving the optimization problem at each time step is used only for the first sampling period. The control commands serve as the actual control inputs to the system. To construct constraints (13), auxiliary control laws need to be designed. Given the wide application of backstepping control in UAV systems, we employ the backstepping method to design the auxiliary control law. .

[0088] Step 4: Further, in order to construct the constraint (13), an auxiliary control law needs to be designed. Given the wide application of backstepping control in UAV systems, we employ the backstepping method to design the auxiliary control law. .

[0089] For the unconstrained error variable defined in (8) subscript Represents the corresponding components of position and yaw angle, and the time derivative of each component. It can be represented as:

[0090] (14)

[0091] To simplify the expression, define

[0092] , (15)

[0093] Defined here and Therefore, (15) can be rewritten as (16)

[0094] because and It can be concluded that Furthermore, as long as Strictly meet (5), It is bounded, therefore It is also bounded, therefore It also has boundaries. According to... From the expression, we can see that Both the numerator and denominator are bounded, and the denominator is not equal to zero, therefore Bounded. For ease of use later, we assume... , and The upper bounds are positive constants. , , .

[0095] To design this auxiliary control law, we define the following error vector: (17)

[0096] in It is a virtual control input vector.

[0097] Choose the following Lyapunov functions. (18)

[0098] right Taking the derivative, we get

[0099] (19)

[0100] in ,and It is a virtual control input vector. The virtual control input is designed as follows: (20)

[0101] in It is a positive definite gain matrix. It is a virtual control input vector. Substituting (20) into (19) yields...

[0102] (twenty one)

[0103] The above formula shows that if there are no overlapping terms , It is negative semi-definite, and this cross term will be eliminated in subsequent designs.

[0104] Therefore, the next step will be to construct a composite Lyapunov function. To introduce auxiliary variables And we complete the stability proof. We choose the Lyapunov function.

[0105] (twenty two)

[0106] according to , and in equation (1) and The expression, for Differentiation yields

[0107] (twenty three)

[0108] auxiliary controller The design is as follows:

[0109] (twenty four)

[0110] in It is a positive definite gain matrix. Substituting (24) into (23) and simplifying, we can obtain the Lyapunov derivative as follows:

[0111] (25)

[0112] because Ordinary and Since the system is negatively definite, according to Lyapunov stability theory, it is asymptotically stable. The recursive feasibility and system stability of the proposed model predictive control scheme will be analyzed next.

[0113] Step 5: Further analyze the recursive feasibility and system stability of the proposed model predictive control scheme. To ensure recursive feasibility, an auxiliary controller... Must meet .

[0114] because Negative definiteness can be obtained

[0115] (26)

[0116] Therefore, we have

[0117] (27)

[0118] Since both sides of the above inequality are the sum of two non-negative terms, we have

[0119] ;

[0120] (28)

[0121] make From equation (28), we can obtain and .so and It is bounded. Because in All variables in the expression , , , and They are all bounded, therefore It is also bounded, meaning that there exist positive real numbers. , making .right Differentiation yields ,because Each term consists of bounded variables, and thus It is also bounded, meaning that there exist positive real numbers. , making .

[0122] According to equation (1), we have

[0123] (29)

[0124] Depend on and , can be obtained

[0125] (30)

[0126] Then we can get

[0127] (31)

[0128] Considering that the system parameters of a drone are actually bounded, we assume that there exists , and Make , and .according to From the definition, we can obtain

[0129] (32)

[0130] Since the infinite norm of a matrix is ​​defined as the maximum sum of the absolute values ​​of the elements in each row of the matrix, and The sum of the absolute values ​​of the elements in each row does not exceed , can be obtained And because , can be obtained ,so

[0131] (33)

[0132] For practical auxiliary controllers , can be obtained

[0133] (34)

[0134] Equation (34) shows that the auxiliary controller is bounded, if If the upper bound of the auxiliary controller is not less than that of the auxiliary controller, the optimization problem will have at least one solution at each sampling time, namely the auxiliary controller itself. Therefore, the predictive control strategy of this model is recursively feasible. Equation (34) guarantees recursive feasibility, thereby limiting the range of selectable controller parameters. Once the controller parameters are determined, an attraction domain can be obtained. If and only if the initial state is located The system can only be stable within the internal attraction domain; stability is only guaranteed outside of it.

[0135] It should be noted that the backstepping controller (24) does not act directly on the actual system, but rather serves as an auxiliary control law to ensure the recursive feasibility and stability of the model predictive control problem. The actual control input is generated by the model predictive controller based on the auxiliary control law.

[0136] Consider the Lyapunov function of equation (25) The function is continuously differentiable, positive definite, and radially unbounded. According to the local Lyapunov inverse theorem, there exists a function in a neighborhood of the equilibrium point. Class function For i = 1, 2, 3, the following relationship is satisfied:

[0137] (35)

[0138] (36)

[0139] The optimal sequence obtained by solving the prediction problem of the optimization model , The middle corresponds to the first sampling period The control commands serve as the actual control inputs of the system. Since equation (13) guarantees that the actual control input is determined by... The obtained Lyapunov function at time t The derivative is less than or equal to that of the auxiliary controller. The derivative of the Lyapunov function is obtained. Therefore, we can obtain...

[0140] (37)

[0141] Based on the above equation and the existence of the attraction domain, it can be inferred that the model predictive controller guarantees the local asymptotic stability of the closed-loop system. The recursive feasibility of the proposed model predictive controller and the stability of the closed-loop system have been theoretically proven.

[0142] Step 6: In the simulation, define the reference trajectory of the UAV as... , and Set the control input limit to The sampling interval and prediction time domain were respectively selected as... and The initial state of the drone is selected as follows: The time constants related to the position and attitude of the UAV are as follows: , , , The corresponding gain coefficients are as follows: , , , Define the control gain matrix as , In the proposed model predictive control scheme design, the weight matrix in the cost function is selected as... and The following preset performance function is selected for the simulation: , , as well as .

[0143] Based on the above parameters, we performed a simulation using a model predictive controller and obtained the simulation results as follows: Figure 1 and Figure 2 As shown, Figure 1 This represents the drone tracking error curve. Figure 2 This is the input curve for drone control. Figure 1 Indicates tracking error , , , Throughout the simulation process, the proposed method (PPC-LMPC) was able to keep all errors within the preset boundaries compared to LMPC, thus verifying the effectiveness of the performance constraints. Figure 2 Indicates control input , , , The curve showed that all inputs remained within ±20, satisfying the feasibility requirements. Furthermore, the control signal was relatively smooth, confirming that the optimization process effectively balanced tracking accuracy and control variable variations.

[0144] The core synergistic effect of this invention lies in the organic integration of preset performance control, backstepping, and model predictive control: First, preset performance control imposes quantitative constraints on the transient and steady-state performance of the tracking error through time-varying boundary functions, and transforms the constrained error into an unconstrained variable through nonlinear transformation, providing performance assurance for high-precision tracking; Second, the backstepping auxiliary control law designed based on the transformed error and its corresponding Lyapunov function not only ensure asymptotic stability of the system, but more importantly, it is used to construct the stability contraction constraint of model predictive control, thereby transferring the preset performance requirements to the optimization problem; Finally, while receiving this stability constraint, model predictive control also handles complex constraints such as input saturation, and uses its rolling optimization capability to improve control performance, while the boundedness of the backstepping control law provides a theoretical guarantee for the recursive feasibility of model predictive control. Through the progressive coupling of performance constraints, stability assurance, and optimization solution, these three aspects solve the inherent defects of preset performance control in handling complex constraints, and traditional model predictive control in generating overshoot and being difficult to directly integrate performance constraints, achieving a unity of stability and high performance.

[0145] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A UAV trajectory tracking control method based on model prediction and preset performance constraints, characterized in that: Includes the following steps, Based on the kinematic and dynamic relationship of the UAV, a nonlinear system model is established to obtain the state variables of the UAV; Based on the obtained UAV state variables, position error and attitude error are defined, and the tracking error is obtained based on the position error and attitude error. A preset performance function is constructed, and the tracking error constrained by the preset performance function is normalized and transformed through error normalization and nonlinear transformation. Transform into an equivalent unconstrained error variable ; Unconstrained error variables As the prediction error is introduced into the cost function, a model predictive control optimization problem is established under the premise of considering input saturation and stability constraints. Based on unconstrained error variables Design auxiliary control laws; Based on the model predictive control optimization problem solved by the auxiliary control law, an optimized control sequence is obtained. The control command corresponding to the first sampling period in the optimized control sequence is then used as the actual control input to the UAV to achieve trajectory tracking. To ensure tracking error To achieve the desired transient and steady-state performance, a pre-defined performance function matrix is ​​introduced. , It is a diagonal matrix composed of smooth, bounded, and strictly decreasing scalar functions, i.e. This ensures that each component of the tracking error satisfies: ， ， ； in, This represents the corresponding components of position and yaw angle. for The corresponding components, select each boundary function for: ； in, , , Let these represent the initial error boundary, final steady-state boundary, and convergence rate for each component, respectively, and they must satisfy the following conditions: ; Cost function of model predictive control optimization problem At sampling time Defined as: in, It is the prediction error. , To predict the time domain, Classified as step, It is the sampling period. It is a predicted state. It is a control sequence in the prediction time domain. It is the positive definite diagonal weighted matrix of the prediction error. It is the positive definite diagonal weighted matrix of the system input. express , express , Is The cost function at time step, Satisfy control input saturation constraints , This is the maximum input limit; The model predictive control optimization problem introduces Lyapunov stability contraction constraints: ;in, Indicates the drone's status. Is Time-based auxiliary control law, It is a Lyapunov function. Represents the nonlinear model of the drone. Indicates in The initial state at a given moment.

2. The UAV trajectory tracking and control method according to claim 1, characterized in that: The nonlinear system model of the UAV is as follows: ； ； in, This indicates the position and yaw angle of the UAV in the inertial coordinate system. These are the linear velocity and angular velocity of the drone in its body coordinate system. Indicates control input, and It is a gain matrix described by a diagonal matrix. , , , This is the gain coefficient. , , , It is a time constant. This represents the transformation matrix that transforms a vector from the body coordinate system to the inertial coordinate system; the tracking error is defined as... , For reference trajectory.

3. The UAV trajectory tracking and control method according to claim 2, characterized in that: Using a preset performance function matrix and tracking error Define the normalized error variable: ； in, , and Transform each normalized error component: ； The above formula uses functions Transforming the normalized error into an unconstrained error, as long as... By maintaining boundaries, we can ensure that when hour, Establishment, Definition .

4. The UAV trajectory tracking and control method according to claim 3, characterized in that: For unconstrained error variables subscript Represents the corresponding components of position and yaw angle, and the time derivative of each component. It can be represented as: ； To simplify the expression, define ， ； Defined here and ,therefore It can be rewritten as ； To design the auxiliary control law, the following error variable is defined. ,in As a virtual control input vector, it is designed as ； in, It is a positive definite gain matrix. It is a virtual control input vector.

5. The UAV trajectory tracking and control method according to claim 4, characterized in that: Auxiliary control law The design employs a backstepping method, specifically in the following form: ； in, It is a positive definite gain matrix. For the error vector, The derivative of the transformation matrix is ​​represented. The second derivative of the reference trajectory is represented. for derivative, .

6. An electronic device, characterized in that, It includes a processor and a memory, the memory storing a computer program, and the processor executing the computer program to implement the UAV trajectory tracking control method as described in any one of claims 1-5.

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