Fuzzy unmanned aerial vehicle system fault-tolerant formation control method based on reduced-order observer

Abstract

The application is directed to the problem of fault-tolerant formation control of fuzzy multi-leader UAV system. A fault-tolerant formation control method based on reduced-order observer is proposed, including: using T-S fuzzy model to model the multi-leader UAV system; transforming the original system into a reduced-order form, and then deriving an augmented form; designing a reduced-order fault observer based on the intermediate variable for the reduced-order augmented system to estimate the state of the follower UAV, process fault and system uncertainty; using the relative state information to design a tracking fault-tolerant formation controller, taking the estimation of process fault and system uncertainty as compensation term; designing adaptive parameters for the reduced-order fault observer and the tracking fault-tolerant formation controller; verifying the performance of the reduced-order fault observer and the tracking fault-tolerant formation controller; the application can more prominently deal with the nonlinear and uncertain control problem, while reducing the calculation amount and improving the efficiency of distributed decision-making.

Description

Technical Field

[0001] This invention relates to the field of dynamic system monitoring and fault-tolerant control technology, and in particular to a fault-tolerant formation control method for fuzzy unmanned aerial vehicle systems based on reduced-order observers. Background Technology

[0002] Over the past few decades, the formation tracking problem of multi-agent systems has attracted widespread attention, mainly due to its extensive applications in distributed sensor networks, robotic systems, and formation tracking itself. Specifically, formation tracking technology for multi-agent systems has been used in scenarios such as target circling, environmental monitoring, and intelligent traffic management, attracting continuous exploration by researchers. Meanwhile, fuzzy control theory has also made significant progress in recent years. Notably, the Takagi-Sugeno (TS) fuzzy model has been proven to have high-accuracy approximation capabilities for smooth nonlinear systems within compact sets. This model, by fusing multiple locally linear subsystems and using membership functions for interpolation, can effectively characterize complex nonlinear dynamic characteristics. Based on this, the analysis and design research of TS fuzzy systems has become a hot topic, with many important research results emerging in recent years.

[0003] Modern technological systems are characterized by their complexity and widespread interconnectivity, making them susceptible to failures. In multi-agent systems, frequent interactions between agents can cause a single agent's failure to propagate and amplify rapidly throughout the network. Such failures can significantly degrade system performance and may even lead to complete system failure and costly downtime. Furthermore, external disturbances, parameter uncertainties due to aging and wear, and fluctuating operating conditions can further affect the effectiveness and reliability of dynamic systems, thus inducing instability. Therefore, in recent decades, much research has focused on developing fault-tolerant systems to ensure security, reliability, and availability.

[0004] Fault-tolerant control refers to a control strategy that ensures the system's performance meets requirements even when faults exist. Fault-tolerant control techniques are generally divided into two categories: active fault-tolerant control and passive fault-tolerant control. Active fault-tolerant strategies rely on a fault estimation unit to explicitly detect and identify fault characteristics, while passive fault-tolerant strategies rely on inherent robustness to resist specific faults, without requiring an explicit fault detection mechanism. This invention requires the control system to have high fault tolerance to meet formation objectives and improve algorithm applicability.

[0005] In the field of fault estimation, various methods have been proposed and developed, including unknown input observers, robust observers, adaptive observers, and sliding mode observers. However, most existing observer designs require the satisfaction of observer matching conditions, a restrictive assumption that is often difficult to hold in practical systems. In contrast, by introducing intermediate variables, joint estimation of system state and fault has been achieved in nonlinear systems without satisfying strict positive real number conditions or observer matching conditions. This breakthrough has led to widespread attention being paid to fault estimation methods based on intermediate variables in recent years. Nevertheless, how to utilize intermediate variables to achieve synergistic research on fault estimation and formation tracking remains in the exploratory stage. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention utilizes a reduced-order intermediate variable observer to solve the problem that traditional observers, heavily reliant on observer matching conditions, are difficult to implement in real-world multi-UAV formation systems for fault estimation and formation control. The introduction of the reduced-order method significantly reduces computational costs compared to full-order observers when dealing with a large number of UAVs. This invention primarily derives a reduced-order augmented system through coordinate transformation. Based on this, a fault observer based on intermediate variables is constructed to achieve joint reconstruction of system state, faults, and uncertainties without requiring observer matching conditions. Using the reconstructed faults and uncertainties, a fault-tolerant control strategy is designed for dynamic compensation, thereby maintaining system stability. Then, the formation tracking error is recalculated based on the reconstructed state variables, and the controller gain is determined by combining adaptive techniques and Lyapunov's theorem. The fault-tolerant formation controller designed based on these techniques, compared to previous UAV formation control methods, considers the impact of system faults and uncertainties and uses fuzzy systems to model the UAV system, potentially reducing the accuracy requirements for modeling the actual system and overcoming the limitations of previous research in this field.

[0007] To achieve the above-mentioned technical objectives, the present invention provides the following technical solution:

[0008] The fault-tolerant formation control method for fuzzy unmanned aerial vehicle (UAV) systems based on reduced-order observers specifically includes the following steps:

[0009] S1. Model the multi-leader UAV system using the TS fuzzy model. The multi-leader UAV system includes a leader UAV and a follower UAV.

[0010] S2. By decomposing the system state and transforming the coordinates, the original system is transformed into a reduced-order form;

[0011] S3. Analyze the reduced-order system obtained in step S2 and derive the augmented form of the reduced-order system.

[0012] S4. Design a reduced-order fault observer for the reduced-order augmented system based on intermediate variables to estimate the state, process faults and system uncertainties of the following UAV.

[0013] S5. Design a tracking fault-tolerant formation controller using the relative state information between UAVs, and use the estimation of process failures and system uncertainties as compensation terms.

[0014] S6. Design adaptive parameters for the reduced-order fault observer and the tracking fault-tolerant formation controller;

[0015] S7. Evaluate the performance of the reduced-order fault observer and the tracking fault-tolerant formation controller through error analysis, and verify whether it can achieve formation tracking of targets for multi-leader fuzzy UAV systems.

[0016] Furthermore, step S1 specifically includes:

[0017] Consider a multi-leader UAV system consisting of N bodies, where M bodies are follower UAVs and the remaining NM bodies are leader UAVs; use the TS model to perform dynamic modeling for each leader UAV and follower UAV respectively;

[0018] The dynamic model for following the drone is as follows:

[0019]

[0020] in, This represents the rate of change of the state of the drone i, i.e., the dynamic expression of the state; To track the drone's status, To follow the output of the drone, For control input; These represent process failures and system uncertainties, respectively; q represents the number of fuzzy rules. For fuzzy weights, satisfying and i is the number of the drone being followed, and j is the number of the fuzzy rule, i∈{1,...,M}, j∈{1,...,q};

[0021] A j B j H j E j C is the system matrix, where, The state matrix, To control the input matrix, The uncertainty distribution matrix is... For the output matrix, Let n be the fault distribution matrix. x n y n u nf n θ Let A represent the number of elements in each matrix; all matrices are known and satisfy (A j B j (A) is controllable. j C) is observable, and C is full rank;

[0022] The dynamic model of the pilot drone is as follows:

[0023]

[0024] in, This indicates the rate of change of state of the pilot drone. The status of the navigation drone. is the output of the navigation drone; m is the number of the navigation drone, m∈{M+1,...,N}.

[0025] Furthermore, step S2 specifically includes:

[0026] Define a matrix The status x of the following drone i (t) Multiplying the inverse of Y by the left matrix yields:

[0027] η i (t)=Y -1 x i (t), i∈{1,...,M};

[0028] Where, η i (t) is defined as state x i The transformed state vector of (t), Let C be any orthogonal basis of the null space of matrix C, satisfying rank(C0) = n x -n y And CC0 = 0;

[0029] η i Substituting (t) into the dynamic model of the following UAV, we get:

[0030]

[0031] in, Indicates η i The rate of change of (t), i.e. η i The dynamic expression of (t); and then η i (t) is decomposed into Then we have:

[0032]

[0033] in, For η i1(t), η i2 The dynamic representation of (t); η i1 (t) and η i2 (t) is the transformed state subvector, and η i1 (t) represents the reduced-order state of the following drone;

[0034]

[0035] And satisfy

[0036] Then according to Define virtual output γ i (t), its formula is expressed as:

[0037]

[0038] Finally, a reduced-order dynamic model of the following drone is obtained:

[0039]

[0040] Furthermore, step S3 specifically includes:

[0041] Let the augmented state of the following drone be The reduced-order dynamics model of the drone is transformed into an augmented form, expressed by the following formula:

[0042]

[0043] Among them, matrix The system matrix for the augmented reduced-order dynamics model of the drone; For the dynamic representation of augmented states.

[0044] Furthermore, the expression for the reduced-order fault observer designed in step S4 is as follows:

[0045]

[0046] Then, the following is derived from the observer's observations and the reduced-order augmentation system following the UAV:

[0047]

[0048] in, It is an augmented state following the drone. The estimated value; These are the uncertainties θ of the drone system. i (t), process fault f i (t), state x i The estimated value of (t); L jIt is the observer gain of the design; It is an intermediate variable in the design, represented as Where ζ is a selected scalar, It is an intermediate variable The estimated value of β; i (t) is a new variable in the design, and its expression is: η i (t), η i1 The estimated value of (t).

[0049] Furthermore, the expression for the tracking fault-tolerant formation controller in step S5 is specifically as follows:

[0050]

[0051] in, satisfy I represents the identity matrix, i.e. It is B j The generalized inverse matrix is ​​the controller compensation term matrix; the controller compensation term is... Estimation of process state And estimation of uncertainty Composition; q is the number of fuzzy rules, For fuzzy weights, satisfying and ν i (t) represents the compensation for system redundancy; s i (t), The estimated value, s i (t) represents the relative state information between the i-th following drone and the other following drones. This represents the relative state information between the i-th following drone and the lead drone. s i (t), The formulas are expressed as follows:

[0052]

[0053]

[0054] Where, ρ i (t), ρ l (t) represents the expected time-varying queue information of the i-th and l-th following drones, respectively, x i (t), x l (t) represents the states of the i-th and l-th following drones, respectively; x m (t) represents the state of the lead drone. For x i (t), x l The estimated value of (t); a il Let a be the element in the i-th row and l-th column of the adjacency matrix of the topology graph of the multi-navigation unmanned aerial vehicle system. im Let i be the element in the i-th row and m-th column of the adjacency matrix of the topological graph; This is the controller gain matrix that needs to be designed, where M is the number of following drones and N represents the total number of drones.

[0055] Furthermore, step S6 specifically includes:

[0056] Based on Lyapunov's theorem and adaptive techniques, a reduced-order fault observer and a fault-tolerant formation controller parameter tracking method are designed, specifically including:

[0057] If there exist scalars ζ > 0, ∈ > 0 and matrices Ω1 > 0, Ω2 > 0 such that the following linear matrix inequality holds, and the reduced-order fault observer gain L j The selection satisfies When the matrix is ​​a Hurwitz matrix, the error between the observed and actual values ​​converges asymptotically; the linear matrix inequality is expressed as:

[0058]

[0059] in,

[0060]

[0061] I is the identity matrix; matrix These represent the state matrix, uncertainty distribution matrix, fault distribution matrix, and output matrix of the augmented reduced-order dynamics model of the drone; H k2 Depend on H was obtained. k It is the uncertainty distribution matrix of the original follow-up drone system;

[0062] Then, the parameters for the tracking fault-tolerant formation controller are designed as follows:

[0063] When tracking the controller gain in a fault-tolerant formation controller and compensation for system redundancy ν i (t) The control objective is achieved when the following equation is satisfied:

[0064]

[0065] in, The formation tracking error is denoted as... yes The estimated values, Ω3 and Θ are symmetric positive definite matrices, Θ -1It is the inverse of matrix Θ. This represents the value of the controller gain at the initial moment.

[0066] Furthermore, step S7 specifically includes:

[0067] S71. Based on the estimation error system of all following UAVs, select the appropriate Lyapunov function V(t), as shown in the derivation... This verifies the effectiveness of the reduced-order fault observer;

[0068] S72, Based on the formation tracking error of the following drone By selecting an appropriate Lyapunov function for stability assessment, it is proven that the formation tracking error of the UAV under the tracking fault-tolerant formation controller is... Converging to 0, all following drones can track convex combinations of multiple leaders and achieve time-varying formations.

[0069] By employing the above technical solution, the present invention provides a fault-tolerant formation control method for fuzzy unmanned aerial vehicle systems based on reduced-order observers, which has at least the following beneficial effects:

[0070] 1. This invention employs the TS fuzzy model to model unmanned aerial vehicle (UAV) systems and studies the fault-tolerant formation control problem with multiple leaders based on this model. The TS fuzzy model has significant advantages in the modeling and control of complex systems, especially in handling nonlinearity and uncertainty. Therefore, this invention has great advantages in modeling and handling system faults and uncertainties.

[0071] 2. This invention designs a reduced-order intermediate variable observer for reconstructing system states, faults, and uncertainties. The introduction of the reduced-order observer significantly reduces computational complexity, decreases local computational burden, and improves distributed decision-making efficiency.

[0072] 3. Based on the faults and uncertainties in the observer reconstruction, a corresponding active fault-tolerant control strategy was developed to compensate for the system's faults and uncertainties. Since the actual state of the UAV is not easily measured, the reconstructed state information is used to replace the actual state information for formation control design, improving the algorithm's practicality. Attached Figure Description

[0073] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings:

[0074] Figure 1 This is an overall flowchart of the method proposed in this invention;

[0075] Figure 2This is a schematic diagram of the communication topology of the multi-pilot unmanned aerial vehicle system in this invention;

[0076] Figure 3 These are the process fault signals and estimated values ​​of process faults for UAVs 1 to 4 in this invention;

[0077] Figure 4 This invention provides the system uncertainty signals of UAVs 1 and 2, and their estimated values.

[0078] Figure 5 This refers to the formation tracking error of all following drones in this invention;

[0079] Figure 6 This is a schematic diagram showing the status of all follower drones and lead drones in various time periods in this invention. Detailed Implementation

[0080] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the following description is provided in conjunction with the accompanying drawings. Figure 1-6 The present invention will be further described in detail below with reference to specific embodiments. This will allow for a full understanding of how the present application uses technical means to solve technical problems and achieve technical effects, and to facilitate its implementation.

[0081] Those skilled in the art will understand that all or part of the steps in the methods of the above embodiments can be implemented by a program instructing related hardware. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0082] Please refer to Figures 1-6 This illustration demonstrates a specific implementation of this embodiment. This embodiment ensures the achievement of formation control objectives for multi-leader UAV systems under fault and uncertainty conditions. It primarily derives a reduced-order augmented system through coordinate transformation. Based on this, a fault observer based on intermediate variables is constructed to achieve joint reconstruction of system state, faults, and uncertainties without requiring observer matching conditions. Using the reconstructed faults and uncertainties, a fault-tolerant control strategy is designed for dynamic compensation, thereby maintaining system stability. Then, the formation tracking error is recalculated based on the reconstructed state variables, and the controller gain is determined by combining adaptive techniques and Lyapunov's theorem. The fault-tolerant formation controller designed based on the above techniques, compared to previous UAV formation control methods, considers the impact of system faults and uncertainties, and uses a fuzzy system to model the UAV system, reducing the accuracy requirements for modeling the actual system and overcoming the limitations of previous research in this field.

[0083] like Figure 1 The diagram illustrates the fault-tolerant formation control method for fuzzy unmanned aerial vehicle (UAV) systems based on a reduced-order observer proposed in this invention, which specifically includes the following steps:

[0084] S1. Model the multi-leader UAV system using the TS fuzzy model. The multi-leader UAV system includes a leader UAV and a follower UAV.

[0085] In a preferred embodiment, step S1 specifically includes:

[0086] Consider a multi-leader UAV system consisting of N bodies, where M bodies are follower UAVs and the remaining NM bodies are leader UAVs; use the TS model to perform dynamic modeling for each leader UAV and follower UAV respectively;

[0087] The dynamic model for following the drone is as follows:

[0088]

[0089] in, This represents the rate of change of the state of the drone i, i.e., the dynamic expression of the state; To track the drone's status, To follow the output of the drone, For control input; These represent process failures and system uncertainties, respectively; q represents the number of fuzzy rules. For fuzzy weights, satisfying and i is the number of the drone being followed, and j is the number of the fuzzy rule, i∈{1,...,M}, j∈{1,...,q};

[0090] In this application, It comprehensively considers factors such as control input, process failures, and uncertainties, reflecting the state changes of the agent under the influence of these factors; while x i (t) represents the state of agent i itself, and does not directly include the influence of the above factors; weight function By dynamically adjusting the contributions of each subsystem using fuzzy rules, the overall UAV system can express approximately complex nonlinear dynamics.

[0091] A j B j H j E j C is the system matrix, where, The state matrix, To control the input matrix, The uncertainty distribution matrix is... For the output matrix, Let n be the fault distribution matrix. x n y n u n f n θ Let A represent the number of elements in each matrix; all matrices are known and satisfy (A j B j (A) is controllable. j C) is observable, and C is full rank;

[0092] The dynamic model of the pilot drone is as follows:

[0093]

[0094] in, This indicates the rate of change of state of the pilot drone. The status of the navigation drone. is the output of the navigation drone; m is the number of the navigation drone, m∈{M+1,...,N};

[0095] S2. By decomposing the system state and transforming the coordinates, the original system is transformed into a reduced-order form;

[0096] In a preferred embodiment, step S2 specifically includes:

[0097] Define a matrix Since the system matrix C is a full-rank row matrix, there exists a matrix... Satisfying rank(C0) = n x -n y And CC0 = 0, which means that C0 is any orthogonal basis of the null space of C. Therefore, we have:

[0098]

[0099] Therefore, the matrix It is the left inverse matrix of Y;

[0100] Therefore, the state x of the drone is... i (t) Multiplying the inverse of Y by its left side yields:

[0101] η i (t)=Y -1 x i (t), i∈{1,…,M};

[0102] η i (t) is defined as state x i The transformed state vector of (t); η iSubstituting (t) into the dynamic model of the following UAV, we get:

[0103]

[0104]

[0105] Multiply both sides of the above equation by Y on the left. -1 We can obtain:

[0106]

[0107] in, Indicates η i The rate of change of (t), i.e. η i The dynamic expression of (t); and then η i (t) is decomposed into Substituting into the above equation, we get:

[0108]

[0109] Furthermore, we obtain η i1 (t), η i2 Dynamic representation of (t):

[0110]

[0111] in, For η i1 (t), η i2 The dynamic representation of (t); η i1 (t) and η i2 (t) is the transformed state subvector, and η i1 (t) represents the reduced-order state of the following drone;

[0112]

[0113] And satisfy

[0114] Then according to Define virtual output γ i (t), its formula is expressed as:

[0115]

[0116] Finally, a reduced-order dynamic model of the following drone is obtained:

[0117]

[0118] S3. Analyze the reduced-order system obtained in step S2 and derive the augmented form of the reduced-order system.

[0119] In a preferred embodiment, step S3 specifically includes:

[0120] Let the augmented state of the following drone be The reduced-order dynamics model of the drone is transformed into an augmented form, expressed by the following formula:

[0121]

[0122] Among them, matrix The system matrix for the augmented reduced-order dynamics model of the drone; For the dynamic representation of augmented states.

[0123] In this application, both the observer and the controller are designed within the drone, so only the dynamic model of the drone needs to be reduced in order and augmented.

[0124] S4. Design a reduced-order fault observer for the reduced-order augmented system based on intermediate variables to estimate the state, process faults and system uncertainties of the following UAV.

[0125] As a preferred embodiment, the expression for the reduced-order fault observer designed in step S4 is as follows:

[0126]

[0127] Then, the following is derived from the observer's observations and the reduced-order augmentation system following the UAV:

[0128]

[0129] in, It is an augmented state following the drone. The estimated value; These are the uncertainties θ of the drone system. i (t), process fault f i (t), state x i The estimated value of (t); L j It is the observer gain of the design; It is an intermediate variable in the design, represented as Where ζ is a selected scalar, It is an intermediate variable The estimated value of β; i (t) is a new variable in the design, and its expression is: η i (t), η i1 The estimated value of (t).

[0130] S5. Design a tracking fault-tolerant formation controller using the relative state information between UAVs, and use the estimation of process failures and system uncertainties as compensation terms.

[0131] In a preferred embodiment, the expression for the tracking fault-tolerant formation controller in step S5 is specifically as follows:

[0132]

[0133] in, satisfy I represents the identity matrix, i.e. It is B j The generalized inverse matrix is ​​the controller compensation term matrix; the controller compensation term is... Estimation of process state And estimation of uncertainty Composition; q is the number of fuzzy rules, For fuzzy weights, satisfying and ν i (t) represents the compensation for system redundancy; s i (t), The estimated value, s i (t) represents the relative state information between the i-th following drone and the other following drones. This represents the relative state information between the i-th following drone and the lead drone.

[0134] s i (t), The formulas are expressed as follows:

[0135]

[0136] Where, ρ i (t), ρ l (t) represents the expected time-varying queue information of the i-th and l-th following drones, respectively, x i (t), x l (t) represents the states of the i-th and l-th following drones, respectively; x m (t) represents the state of the lead drone. For x i (t), x l The estimated value of (t); a il Let a be the element in the i-th row and l-th column of the adjacency matrix of the topology graph of the multi-navigation unmanned aerial vehicle system. im Let i be the element in the i-th row and m-th column of the adjacency matrix of the topological graph; This is the controller gain matrix that needs to be designed, where M is the number of following drones and N represents the total number of drones.

[0137] S6. Design adaptive parameters for the reduced-order fault observer and the tracking fault-tolerant formation controller;

[0138] In a preferred embodiment, step S6 specifically includes:

[0139] Based on Lyapunov's theorem and adaptive techniques, a reduced-order fault observer and a fault-tolerant formation controller parameter tracking method are designed, specifically including:

[0140] If there exist scalars ζ > 0, ∈ > 0 and matrices Ω1 > 0, Ω2 > 0 such that the following linear matrix inequality holds, and the reduced-order fault observer gain L j The selection satisfies When the matrix is ​​a Hurwitz matrix, the error between the observed and actual values ​​converges asymptotically; the linear matrix inequality is expressed as:

[0141]

[0142] in,

[0143]

[0144] I is the identity matrix; matrix These represent the state matrix, uncertainty distribution matrix, fault distribution matrix, and output matrix of the augmented reduced-order dynamics model of the drone; H k2 Depend on H was obtained. k It is the uncertainty distribution matrix of the original follow-up drone system;

[0145] Then, the parameters for the tracking fault-tolerant formation controller are designed as follows:

[0146] When tracking the controller gain in a fault-tolerant formation controller and compensation for system redundancy ν i (t) The control objective is achieved when the following equation is satisfied:

[0147]

[0148] in, The formation tracking error is denoted as... yes The estimated values, Ω3 and Θ are symmetric positive definite matrices, Θ -1 It is the inverse of matrix Θ. This represents the value of the controller gain at the initial moment.

[0149] S7. Evaluate the performance of the reduced-order fault observer and the tracking fault-tolerant formation controller through error analysis, and verify whether it can achieve formation tracking of the target of the multi-leader fuzzy UAV system.

[0150] In a preferred embodiment, step S7 specifically includes:

[0151] S71. Based on the estimation error system of all following UAVs, select the appropriate Lyapunov function V(t), as shown in the derivation... This verifies the effectiveness of the reduced-order fault observer;

[0152] In this embodiment, the formation control objective function that needs to be implemented is:

[0153]

[0154] Where, ρ i (t) represents the expected time-varying queue information of the i-th following drone, α k It is a constant that satisfies x k (t) represents the state of the kth drone;

[0155] To ensure the rationality of subsequent design, the following assumptions need to be made:

[0156] Assumption 1: Fault signal f i (t) and system uncertainty θ i All (t) are differentiable and have bounded derivatives, i.e., satisfying and Where μ1>0, μ2>0.

[0157] Assumption 2: All following drones can be divided into two categories: informed following drones and uninformed following drones. An informed following drone is one whose neighborhood set includes all the lead drones, while an uninformed following drone is one whose neighborhood set does not include the lead drones. For each uninformed following drone, there exists at least one direct undirected path leading to some informed following drone.

[0158] Based on hypothesis 2, we can obtain the Laplace matrix of this multi-navigation UAV system. It can be represented as in

[0159] The following lemmas will be introduced to facilitate the subsequent proof of the method proposed in this invention:

[0160] Lemma 1: If for each follower drone, there exists at least one leader with a directed path to that follower drone, then the Laplace matrix subblock... All eigenvalues ​​of the matrix have positive real parts. Each element in the array is non-negative, and the sum of the elements in each row is equal to 1. Furthermore, when assumption 2 is true, Each row contains the same elements.

[0161] Lemma 2: Assume that matrices Y1 and Y2 have compatible dimensions, then the following inequalities hold:

[0162]

[0163] Here, ∈ is a positive real number;

[0164] The following section will analyze in detail how, under the parameter design conditions of S6, the reduced-order fault observer based on intermediate variables designed in this invention can complete the estimation tasks of system state, faults and uncertainties.

[0165] Define the estimation error: It is the state estimation error of the augmented system corresponding to the i-th following UAV; It is the estimation error of the intermediate variable corresponding to the i-th follower UAV; It is the estimation error of the uncertainty of the i-th following UAV system; It is the estimation error of the process fault of the i-th following UAV system. Since the intermediate variable is defined as... Therefore, we can conclude that... From the observer form above, the dynamic form of the estimation error system can be obtained as follows:

[0166]

[0167] Write the above expression in compact form

[0168]

[0169] in,

[0170]

[0171] I M It is an identity matrix of dimension M. It is of dimension n θ The identity matrix, L represents the Kronecker product operation. j It is the observer gain.

[0172] Choose the following Lyapunov functions:

[0173]

[0174] Among them, Ω1 and Ω2 are symmetric positive definite. yes transpose, yes The transpose of . From the dynamic equation of the error system above, the dynamic equation of V(t) is:

[0175]

[0176] According to hypothesis 1, there exists an uncertain scalar value μ. f and μ θ satisfy and Therefore, based on Lemma 2, the following inequalities can be derived:

[0177]

[0178] Where ε is a real constant;

[0179] Substituting the above inequality into the dynamic equation of V(t), we get:

[0180]

[0181] Define matrix At this point, the above formula can be transformed into:

[0182]

[0183] in,

[0184]

[0185] By scaling V(t) according to the properties of the Lyapunov function, we can obtain:

[0186]

[0187] Here λ max (Ω1) represents the largest eigenvalue of matrix Ω1, λ max (Ω2) Similarly; express The bisection of the 2norm, Similarly, define a matrix. Therefore, when Σ jk When <0, From the above formula and The inequality can be obtained as follows:

[0188]

[0189] in,

[0190] Define a set Φ, which is represented as:

[0191]

[0192] Where, λ min (Ω1) represents the smallest eigenvalue of matrix Ω1, λ min (Ω2) Similarly. Assume... It is the complement of Φ, if That would yield:

[0193]

[0194] From the above formula and We can obtain:

[0195] If the matrix inequality condition Σ<0 is satisfied, then Σ<0 can be transformed into a linear matrix inequality in step S4 using the Schul complement theorem. Therefore, if According to Lyapunov's stability theory, the error pair This achieves consistent boundedness, thus guaranteeing the stability of the system under specified conditions. The proof is complete, thus verifying the effectiveness of the reduced-order fault observer.

[0196] S72, Based on the formation tracking error of the following drone By selecting an appropriate Lyapunov function for stability assessment, it is proven that the formation tracking error of the UAV under the tracking fault-tolerant formation controller is... Converging to 0, all following drones can track convex combinations of multiple leaders and achieve time-varying formations;

[0197] This embodiment also provides the following specific verification process:

[0198] First, define a vector. In formation state, where ρ i (t) represents the expected time-varying queue information of the i-th following drone. Combining the previous state equations of the following drone system and the designed controller u... i (t) and the defined intermediate variables It can be deduced The dynamic equation is:

[0199]

[0200] Where q represents the number of fuzzy rules. For fuzzy weights, satisfying and ζ is a chosen scalar, and matrix A j B j H j E j It is a known system matrix. The controller gain that needs to be designed.

[0201] Assume ν i (t) satisfies Then the above equation The dynamic equation can be transformed into:

[0202]

[0203] The above formula can then be written in a compact form for all following drones:

[0204]

[0205] in, I M This represents the identity matrix of dimension M. This represents the Kronecker product operation.

[0206]

[0207] Define vector To track the formation error of drone i, s i (t), Substituting the expression, we get:

[0208]

[0209] This can be written in a compact form as follows:

[0210]

[0211] in, It is the Laplace matrix The block of Jordan. Let This represents the formation error of the entire system, which is also our main control variable. The dynamic changes can be seen from The dynamic equations and the leader's state equations are obtained.

[0212]

[0213] From the form of the reduced-order observer, we can see that... Therefore, we can obtain

[0214] Therefore, when During convergence, It will also converge. As can be seen from the verification step in S71, if the observer parameters satisfy the conditions in step S61, the observation error... It will asymptotically converge to zero. Therefore, as time t approaches infinity, e f (t) will tend to zero. Therefore, the above equation is equivalent to:

[0215]

[0216] Choose from the following Lyapunov equations:

[0217]

[0218] Where Ω3,Θ is a symmetric positive definite matrix. yes The transpose of , tr(·) denotes the trace of the matrix · within the parentheses. The dynamic equation can be used to obtain V o The dynamic equation of (t) is:

[0219]

[0220] Assumption

[0221]

[0222] By making a simple transformation of the above equation, we can obtain:

[0223]

[0224] Therefore, the controller gain can be written from the above formula. The expression is:

[0225]

[0226] The above formula represents the designed controller gain; therefore, when the controller gain is as shown in the above formula, Reexpressed as:

[0227]

[0228] Define matrix Make Among them Λ L1 It is a diagonal matrix whose diagonal elements correspond to the matrix. The eigenvalue, i.e., Λ L1 =diag{λ1,λ2,…,λ M In this case, the definition is... in therefore, It can be converted into the following forms:

[0229]

[0230] Wherein, matrix U jIt is symmetric positive definite and satisfies the form: Lyapunov's equations. From V o From the form (t), we can derive:

[0231]

[0232] Therefore, as time t approaches infinity, the formation error... It converges to zero. (From...) From the expression, we can obtain:

[0233]

[0234] Multiply both sides of the above expression by... Afterwards, we can obtain:

[0235]

[0236] Here

[0237] Therefore, from the above equation, we can deduce that:

[0238]

[0239] Among them, c il express The element in the i-th row and l-th column. According to Lemma 1 of step S71, The sum of the elements in each row is 1. Specifically Therefore, the control objective described in the above formula is consistent with that in step S71. Thus, in the designed controller u... i (t) and under the action of the controller gain designed according to S6, the multi-navigation UAV system can achieve formation tracking, thereby completing the effect verification.

[0240] To enable researchers in the field to better understand the implementation of this invention, Matlab software is used for simulation in this embodiment.

[0241] The specific information about the simulation software is as follows:

[0242] Software Name: MATLAB

[0243] Version information: 9.11.0.1769968 (R2021b)

[0244] License Number: 968398

[0245] Operating System: Microsoft Windows 11 Home Chinese Edition, version 22H2 (KB5034467)

[0246] Java version: Java 1.8.0_202-b08 with Oracle Corporation Java HotSpot(TM)64-Bit ServerVMmixedmode

[0247] To verify the effectiveness of the fault-tolerant formation control method proposed in this patent, we considered a multi-agent system consisting of nine UAVs, including six follower UAVs and three lead UAVs, with the following topology: Figure 2 As shown. Assume the number of fuzzy rules is q = 2, and the system matrix is ​​as follows:

[0248]

[0249] according to Figure 2 The topological structure shown, the Laplace matrix The following is confirmed:

[0250]

[0251] Assuming the process failure only occurs in agents 1 through 4, the process failure f i The dynamics of (t), i = 1, 2, ..., 6 are represented as follows:

[0252] f1(t)=sin(0.2t),

[0253]

[0254] f5(t)=0, f6(t)=0

[0255] The fuzzy membership function is selected as follows:

[0256]

[0257] Among them, y i1 (t) refers to the output y of the i-th following drone. i The first element of (t).

[0258] Consider the following time-varying expected formation:

[0259]

[0260] Assume the uncertainty of the system is:

[0261] θ i (t)=cos(0.2t-3i)+(-1) i-1 sin(0.5t+1), i=1,2,…,6:

[0262] To ensure It is the Hurwitz matrix, and the observer gain is set to:

[0263]

[0264] Figure 3 The fault estimation results for UAVs 1 through 4 are shown, while Figure 4 The figure shows the uncertainty estimates for UAVs 1 and 2. As can be seen from the figure, the curves of the actual values ​​and the estimated values ​​approximately overlap, indicating that the observation error is very small, which shows that the estimation performance meets the expected requirements.

[0265] Figure 5 Demonstrates formation tracking error The changes are shown in the figure. The rapid convergence to a small range near zero indicates that the system's formation tracking of the target has been effectively achieved. Figure 6 The diagram illustrates the positional states of all UAVs at different time intervals. Squares represent leaders, while other shapes represent followers. As can be seen, the followers continuously move in a rotational pattern around the leader. Therefore, the fault-tolerant tracking control (FTFC) scheme proposed in this paper successfully overcomes the effects of process failures and system uncertainties, ensuring that all following UAVs can track multiple leaders in a specified time-varying formation.

[0266] This embodiment addresses the fault tolerance issues and formation control problems of fuzzy unmanned aerial vehicle (UAV) systems with multiple lead UAVs. Through coordinate transformation, the original system is converted into a reduced-order form. Faults are then treated as a special state, leading to the derivation of a reduced-order augmented system. A reduced-order fault observer based on intermediate variables is constructed to reconstruct system states, faults, and uncertainties without requiring observer matching conditions. Using the reconstructed fault and uncertainty data, a fault-tolerant control strategy is developed to compensate for these problems and maintain system stability. Subsequently, based on adaptive techniques and Lyapunov's theorem, fault observer and controller parameters are designed. The results demonstrate that, under the designed parameter conditions, the controller can achieve target tracking in a multi-leader UAV system. Finally, simulation experiments verify the effectiveness of the proposed method.

[0267] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of those different embodiments or examples.

[0268] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus or device (such as a computer-based system, a processor-included system or other system that can fetch and execute instructions from, an instruction execution system, apparatus or device).

[0269] The above embodiments provide a detailed description of the present invention. Specific examples have been used to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of the present invention. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A fault-tolerant formation control method for fuzzy unmanned aerial vehicle (UAV) systems based on reduced-order observers, characterized in that, Specifically, the steps include: S1. Model the multi-leader UAV system using the TS fuzzy model. The multi-leader UAV system includes a leader UAV and a follower UAV. S2. By decomposing the system state and transforming the coordinates, the original system is transformed into a reduced-order form; S3. Analyze the reduced-order system obtained in step S2 and derive the augmented form of the reduced-order system. S4. Design a reduced-order fault observer for the reduced-order augmented system based on intermediate variables to estimate the state, process faults and system uncertainties of the following UAV. S5. Design a tracking fault-tolerant formation controller using the relative state information between UAVs, and use the estimation of process failures and system uncertainties as compensation terms. S6. Design adaptive parameters for the reduced-order fault observer and the tracking fault-tolerant formation controller; S7. Evaluate the performance of the reduced-order fault observer and the tracking fault-tolerant formation controller through error analysis, and verify whether it can achieve formation tracking of targets for multi-leader fuzzy UAV systems.

2. The fault-tolerant formation control method for a fuzzy unmanned aerial vehicle system based on a reduced-order observer as described in claim 1, characterized in that, Step S1 is as follows: Consider a multi-leader UAV system consisting of N bodies, where M bodies are follower UAVs and the remaining NM bodies are leader UAVs; use the TS model to perform dynamic modeling for each leader UAV and follower UAV respectively; The dynamic model for following the drone is as follows: in, This represents the rate of change of the state of the drone i, i.e., the dynamic expression of the state; To track the drone's status, To follow the output of the drone, For control input; These represent process failures and system uncertainties, respectively; q represents the number of fuzzy rules. For fuzzy weights, satisfying and i is the number of the drone being followed, and j is the number of the fuzzy rule, i∈{1,...,M}, j∈{1,...,q}; A j B j H j E j C is the system matrix, where, The state matrix, To control the input matrix, The uncertainty distribution matrix is... For the output matrix, Let n be the fault distribution matrix. x n y n u n f n θ Let A represent the number of elements in each matrix; all matrices are known and satisfy (A j B j (A) is controllable. j C) is observable, and C is full rank; The dynamic model of the pilot drone is as follows: in, This indicates the rate of change of state of the pilot drone. The status of the navigation drone. is the output of the navigation drone; m is the number of the navigation drone, m∈{M+1,...,N}.

3. The fault-tolerant formation control method for a fuzzy unmanned aerial vehicle system based on a reduced-order observer according to claim 2, characterized in that, Step S2 is as follows: Define a matrix The status x of the following drone i (t) Multiplying the inverse matrix of Y by the left yields the variable η. i (t), expressed by the formula: η i (t)=Y -1 x i (t),i∈{1,...,M}; Where, η i (t) is defined as state x i The transformed state vector of (t), Let C be any orthogonal basis of the null space of matrix C, satisfying rank(C0) = n x -n y And CC0 = 0; η i Substituting (t) into the dynamic model of the following UAV, we get: in, Indicates η i The rate of change of (t), i.e. η i The dynamic expression of (t); and then η i (t) is decomposed into Then we have: in, For η i1 (t), η i2 The dynamic representation of (t); η i1 (t) and η i2 (t) is the transformed state sub-vector, and η i1 (t) represents the reduced-order state of the following drone; System Matrix And satisfy Then according to Define virtual output γ i (t), its formula is expressed as: Finally, a reduced-order dynamic model of the following drone is obtained:

4. The fault-tolerant formation control method for a fuzzy unmanned aerial vehicle system based on a reduced-order observer according to claim 3, characterized in that, Step S3 is as follows: Let the augmented state of the following drone be The reduced-order dynamics model of the drone is transformed into an augmented form, expressed by the following formula: Among them, matrix The system matrix for the augmented reduced-order dynamics model of the drone; For the dynamic representation of augmented states.

5. The fault-tolerant formation control method for a fuzzy unmanned aerial vehicle system based on a reduced-order observer according to claim 4, characterized in that, The expression for the reduced-order fault observer designed in step S4 is as follows: Then, the following is derived from the observer's observations and the reduced-order augmentation system following the UAV: in, It is an augmented state following the drone. The estimated value; These are the uncertainties θ of the drone system. i (t), process fault f i (t), state x i The estimated value of (t); L j It is the observer gain of the design; It is an intermediate variable in the design, represented as Where ζ is a selected scalar, It is an intermediate variable The estimated value of β; i (t) is a new variable in the design, and its expression is: η i (t), η i1 The estimated value of (t).

6. The fault-tolerant formation control method for a fuzzy unmanned aerial vehicle system based on a reduced-order observer according to claim 4, characterized in that, The specific expression for the tracking fault-tolerant formation controller in step S5 is as follows: in, satisfy I represents the identity matrix, i.e. It is B j The generalized inverse matrix is ​​the controller compensation term matrix; the controller compensation term is... Estimation of process state And estimation of uncertainty Composition; q is the number of fuzzy rules, For fuzzy weights, satisfying and ν i (t) represents the compensation for system redundancy; s i (t), The estimated value, s i (t) represents the relative state information between the i-th following drone and the other following drones. This represents the relative state information between the i-th following drone and the lead drone. s i (t), The formulas are expressed as follows: Where, ρ i (t), ρ l (t) represents the expected time-varying queue information of the i-th and l-th following drones, respectively, x i (t), x l (t) represents the states of the i-th and l-th following drones, respectively; x m (t) represents the state of the lead drone. For x i (t), x l The estimated value of (t); a il Let a be the element in the i-th row and l-th column of the adjacency matrix of the topology graph of the multi-navigation unmanned aerial vehicle system. im Let i be the element in the i-th row and m-th column of the adjacency matrix of the topological graph; This is the controller gain matrix that needs to be designed, where M is the number of following drones and N represents the total number of drones.

7. The fault-tolerant formation control method for a fuzzy unmanned aerial vehicle system based on a reduced-order observer according to claim 6, characterized in that, Step S6 is as follows: Based on Lyapunov's theorem and adaptive techniques, a reduced-order fault observer and a fault-tolerant formation controller parameter tracking method are designed, specifically including: If there exist scalars ζ > 0, ∈ > 0 and matrices Ω1 > 0, Ω2 > 0 such that the following linear matrix inequality holds, and the reduced-order fault observer gain L j The selection satisfies When the matrix is ​​a Hurwitz matrix, the error between the observed and actual values ​​converges asymptotically; the linear matrix inequality is expressed as: in, I is the identity matrix; matrix These represent the state matrix, uncertainty distribution matrix, fault distribution matrix, and output matrix of the augmented reduced-order dynamics model of the drone; H k2 Depend on H was obtained. k It is the uncertainty distribution matrix of the original follow-up drone system; Then, the parameters for the tracking fault-tolerant formation controller are designed as follows: When tracking the controller gain in a fault-tolerant formation controller and compensation for system redundancy ν i (t) The control objective is achieved when the following equation is satisfied: in, The formation tracking error is denoted as... yes The estimated values, Ω3 and Θ are symmetric positive definite matrices, Θ -1 It is the inverse of matrix Θ. This represents the value of the controller gain at the initial moment.

8. The fault-tolerant formation control method for a fuzzy unmanned aerial vehicle system based on a reduced-order observer according to claim 1, characterized in that, Step S7 is as follows: S71. Based on the estimation error system of all following UAVs, select the appropriate Lyapunov function V(t), as shown in the derivation... This verifies the effectiveness of the reduced-order fault observer; S72, Based on the formation tracking error of the following drone By selecting an appropriate Lyapunov function for stability assessment, it is proven that the formation tracking error of the UAV under the tracking fault-tolerant formation controller is... Converging to 0, all following drones can track convex combinations of multiple leaders and achieve time-varying formations.

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