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

Abstract

The method aims at the fault-tolerant formation control problem of the fuzzy multi-pilot unmanned aerial vehicle system. The invention provides a fuzzy unmanned aerial vehicle system fault-tolerant formation control method based on a reduced-order observer, and the method comprises the steps: carrying out the modeling of a multi-pilot unmanned aerial vehicle system through a T-S fuzzy model; an original system is firstly converted into a reduced-order form, and then an augmentation form is exported; designing a reduced-order fault observer for the reduced-order augmented system based on the intermediate variable, and estimating the state, process fault and system uncertainty of the following unmanned aerial vehicle; designing a tracking fault-tolerant formation controller by using relative state information, and taking estimation of process faults and system uncertainty as compensation items; self-adaptive parameters are designed 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; according to the method, nonlinear and uncertain control problems can be processed more prominently, meanwhile, the calculation amount is reduced, and the distributed decision-making efficiency is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of dynamic system monitoring and fault-tolerant control, and particularly to a fault-tolerant formation control method for a fuzzy UAV system based on a reduced-order observer. Background Art

[0002] In the past few decades, the formation tracking problem of multi-agent systems has attracted extensive attention, mainly due to its wide applications in fields such as distributed sensor networks, robotic systems, and formation tracking itself. Specifically, the formation tracking technology of multi-agent systems has been used in scenarios such as target surrounding, environmental monitoring, and intelligent traffic management, attracting continuous exploration by researchers. At the same time, the fuzzy control theory has also made great progress in recent years. It is worth noting that the Takagi-Sugeno (T-S) fuzzy model has been proven to have high-precision approximation ability for smooth nonlinear systems within a compact set. By integrating multiple local linear subsystems and using membership functions for interpolation, this model can effectively characterize complex nonlinear dynamic characteristics. Based on this, the research on the analysis and design of T-S fuzzy systems has become a hot direction, and many important research results have emerged in recent years.

[0003] Modern technological systems are characterized by their complexity and extensive interconnectivity, which makes them vulnerable to faults. In multi-agent systems, the frequent interaction between agents may cause the faults of a single agent to quickly spread and amplify throughout the agent network. Such faults will significantly reduce the system performance and may even lead to a complete system breakdown and costly downtime. In addition, external disturbances, parameter uncertainties caused by aging and wear, and fluctuating operating conditions may further affect the effectiveness and reliability of dynamic systems, thus triggering instability. Therefore, in recent decades, the research focus of many researchers has been concentrated on developing fault-tolerant systems to ensure safety, reliability, and availability.

[0004] Fault-tolerant control refers to a control strategy that can still ensure the performance of the system meets the requirements when faults exist in the system. Fault-tolerant control technologies are usually 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 resist specific faults by designing inherent robustness without an explicit fault detection mechanism. The present invention requires the control system to have high fault-tolerant ability to meet the formation target and improve the applicability of the algorithm.

[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 of the existing observer designs require the satisfaction of the observer matching condition, which is a restrictive assumption that is often difficult to hold in practical systems. In contrast, by introducing intermediate variables, the joint estimation of system states and faults has been achieved in nonlinear systems without the need to satisfy the strict positive real condition or the observer matching condition. This breakthrough has made the fault estimation method based on intermediate variables widely concerned in recent years. Nevertheless, how to use intermediate variables to achieve the collaborative research of fault estimation and formation tracking is still in the exploratory stage. Summary of the Invention

[0006] Aiming at the deficiencies of the existing technology, the present invention is based on a reduced-order intermediate variable observer to solve the problem that the traditional observer has a large dependence on the observer matching condition, resulting in the difficulty of fault estimation and formation control of multi-UAV formation systems in practical systems. The introduction of the reduced-order method will greatly reduce the computational cost of the reduced-order observer compared with the full-order one when the number of UAVs is large. The present invention mainly derives the reduced-order augmented system through coordinate transformation methods. On this basis, a fault observer based on intermediate variables is constructed to realize the joint reconstruction of system states, faults, and uncertainties without the need to satisfy the observer matching condition. Using the reconstructed faults and uncertainties, a fault-tolerant control strategy is designed for dynamic compensation to maintain system stability. Then, the formation tracking error is recalculated based on the reconstructed state variables, and the controller gain is determined by combining adaptive technology and Lyapunov's theorem. Compared with the previous UAV formation control methods, the fault-tolerant formation controller designed based on the above technology takes into account the influence of system faults and system uncertainties, and uses a fuzzy system to model the UAV system, which is expected to reduce the accuracy requirements for modeling the actual system and improve the limitations of previous research in this field.

[0007] To achieve the above technical objectives, the present invention provides the following technical solutions:

[0008] A fuzzy UAV system fault-tolerant formation control method based on a reduced-order observer, which specifically includes the following steps:

[0009] S1. Use the T-S fuzzy model to model the multi-leader UAV system, and the multi-leader UAV system includes leader UAVs and follower UAVs;

[0010] S2. By means of splitting the system states and coordinate transformation, transform the original system 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 based on the intermediate variable for the reduced-order augmented system to estimate the states, process faults, and system uncertainties of the following UAVs;

[0013] S5. Design a tracking fault-tolerant formation controller using the relative state information between UAVs, and use the estimates of process faults 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 to verify whether the formation tracking objective of the multi-leader fuzzy UAV system can be achieved.

[0016] Further, step S1 is specifically as follows:

[0017] Consider a multi-leader UAV system composed of N airframes, where M airframes are following UAVs, and the remaining N - M airframes are leader UAVs; use the T-S model to perform dynamic modeling on each leader UAV and following UAV respectively;

[0018] The dynamic model of the following UAV is:

[0019]

[0020] where, represents the rate of change of the state of the following UAV i, that is, the dynamic expression of the state; is the state of the following UAV, is the output of the following UAV, is the control input; are the process fault and system uncertainty respectively; q is the number of fuzzy rules, is the fuzzy weight, satisfying and i is the number of the following UAV, j is the number of the fuzzy rule, i ∈ {1,..., M}, j ∈ {1,..., q};

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

[0022] The dynamic model of the leader UAV is:

[0023]

[0024] where represents the rate of change of the state of the leader UAV, is the state of the leader UAV, is the output of the leader UAV; m is the number of the leader UAV, m ∈ {M + 1,..., N}.

[0025] Furthermore, step S2 is specifically as follows:

[0026] Define a matrix Left-multiply the inverse matrix of Y by the state x i (t) of the follower UAV to obtain:

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

[0028] where η i (t) is defined as the transformed state vector of the state x i (t), is an arbitrary orthogonal basis of the null space of matrix C, satisfying rank(C0) = n x - n y , and CC0 = 0;

[0029] Substitute η i (t) into the dynamic model of the follower UAV to obtain:

[0030]

[0031] where represents the rate of change of η i (t), that is, the dynamic expression of η i (t); then decompose η i (t) into Then there is:

[0032]

[0033] where is η i1(t), η i2 Dynamic representation of (t); η i1 (t) and η i2 (t) is the transformed state sub - vector, and η i1 (t) is used as the reduced - order state following the UAV;

[0034]

[0035] And it satisfies

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

[0037]

[0038] Finally, the reduced - order dynamics model following the UAV is obtained:

[0039]

[0040] Furthermore, step S3 is specifically as follows:

[0041] Let the augmented state of the UAV following be Convert the reduced - order dynamics model of the UAV following into an augmented form, and its formula is expressed as:

[0042]

[0043] Among them, the matrix is the system matrix of the augmented reduced - order dynamics model of the UAV following; is the dynamic expression of the augmented state.

[0044] Furthermore, the expression of the reduced - order fault observer designed in step S4 is specifically:

[0045]

[0046] Then, from the observed value of the observer and the reduced - order augmented system of the UAV following, it is derived:

[0047]

[0048] Among them, is the estimated value of the augmented state of the UAV following ; are respectively the estimated values of the uncertainty θ i (t), the process fault f i (t), and the state x i (t) of the UAV following system; L jis the designed observer gain; is the designed intermediate variable, which is expressed as where ζ is a selected scalar, is the intermediate variable the estimated value of; β i (t) is the designed new variable, and its expression is are respectively the estimated values of η i (t), η i1 (t).

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

[0050]

[0051] where, satisfies I represents the identity matrix, that is is the generalized inverse matrix of B j , and it is the controller compensation term matrix; the controller compensation term is consists of the estimation of the process state and the estimation of the uncertainty ; q is the number of fuzzy rules, is the fuzzy weight, which satisfies and ν i (t) is the compensation for the system redundancy term; are respectively the estimated values of s i (t), , s i (t) represents the relative state information between the i-th following UAV and other following UAVs, represents the relative state information between the i-th following UAV and the leader UAV, s i (t), The formulas are respectively expressed as:

[0052]

[0053]

[0054] where, ρ i (t), ρ l (t) respectively represent the desired time-varying queue information of the i-th and l-th following UAVs, x i (t), x l (t) are respectively the states of the i-th and l-th following UAVs; x m (t) is the state of the leader UAV, is x i (t), x l (t); a il is the element in the i-th row and l-th column of the adjacency matrix of the topological structure diagram of the multi-pilot UAV system, a im is the element in the i-th row and m-th column of the adjacency matrix of the topological structure diagram; is the controller gain matrix to be designed, M is the number of follower UAVs, and N represents the total number of UAVs.

[0055] Furthermore, step S6 is specifically as follows:

[0056] Design the parameters of the reduced-order fault observer and the tracking fault-tolerant formation controller based on the Lyapunov theorem and adaptive technology, specifically including:

[0057] If there exist scalars ζ > 0, ∈ > 0 and matrices Ω1 > 0, Ω2 > 0 such that the following linear matrix inequalities hold, and the reduced-order fault observer gain L j is selected to satisfy is a Hurwitz matrix, the error between the observed value and the actual value can asymptotically converge; the linear matrix inequalities are expressed as:

[0058]

[0059] where,

[0060]

[0061] I is the identity matrix; the matrices are respectively the state matrix, uncertainty distribution matrix, fault distribution matrix, and output matrix of the augmented reduced-order dynamics model of the follower UAV; H k2 is obtained from and H k is the uncertainty distribution matrix of the original follower UAV system;

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

[0063] When the controller gain in the tracking fault-tolerant formation controller and the compensation ν i (t) for the system redundancy term satisfy the following equations, the control objective is achieved:

[0064]

[0065] where, is the formation tracking error, expressed as is 's estimated value, Ω3 and Θ are symmetric positive definite matrices, Θ -1is the inverse matrix of matrix Θ, represents the value of the controller gain at the initial moment.

[0066] Furthermore, step S7 specifically includes:

[0067] S71. Select an appropriate Lyapunov function V(t) according to the estimated error systems of all follower drones, such as derived then the effectiveness of the reduced-order fault observer is verified;

[0068] S72. According to the formation tracking error of the follower drones select an appropriate Lyapunov function for stability judgment, then it is proved that under the tracking fault-tolerant formation controller, the formation tracking error of the follower drones converges to 0, and all follower drones can track the convex combination of multiple leaders and achieve a time-varying formation.

[0069] By means of the above technical solutions, the present invention provides a fault-tolerant formation control method for a fuzzy drone system based on a reduced-order observer, which has at least the following beneficial effects:

[0070] 1. The present invention uses a T-S fuzzy model to model the drone system and studies the fault-tolerant formation control problem of multiple leaders based on this model. The T-S fuzzy model has significant advantages in the modeling and control of complex systems, especially in dealing with nonlinearity and uncertainty. Therefore, the present invention has great advantages in modeling and dealing with system faults and uncertainties.

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

[0072] 3. Based on the faults and uncertainties reconstructed by the observer, corresponding active fault-tolerant control strategies are developed to compensate for the system faults and uncertainties. Since the actual state of the drone is not easy to measure, the reconstructed state information is used to replace the actual state information for formation control design, which improves the practicality of the algorithm. Description of the Drawings

[0073] The drawings described herein are used to provide a further understanding of the present application, form a part of the present application, and the illustrative embodiments and descriptions thereof of the present application are used to explain the present application and do not constitute an improper limitation to the present application. In the drawings:

[0074] Figure 1 is the overall flowchart of the method proposed by the present invention;

[0075] Figure 2Schematic diagram of the communication topology structure of the multi-leader UAV system in the present invention;

[0076] Figure 3 Estimated values of process fault signals and process faults of UAVs 1 to 4 in the present invention;

[0077] Figure 4 System uncertainty signals of UAVs 1 and 2 in the present invention and their estimated values;

[0078] Figure 5 Formation tracking errors of all follower UAVs in the present invention;

[0079] Figure 6 Schematic diagrams of the states of all follower UAVs and the leader UAV at each time period in the present invention. Specific implementation manners

[0080] To make the above objects, features, and advantages of the present invention more obvious and understandable, the following combines the attached Figure 1-6 and specific implementation manners to further elaborate on the present invention in detail. Thereby, a full understanding of how the present application uses technical means to solve technical problems and achieve the realization process of technical effects can be obtained and implemented accordingly.

[0081] Those of ordinary skill in the art can understand that all or part of the steps in implementing the methods of the above embodiments can be completed by instructing relevant hardware through a program. Therefore, the present application can adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk memories, CD-ROMs, optical memories, etc.) containing computer-usable program codes.

[0082] Please refer to Figure 1 - Figure 6 , which shows a specific implementation manner of this embodiment. This embodiment ensures the achievement of the formation control objective for the multi-leader UAV system in the presence of faults and uncertainties. mainly through coordinate transformation methods, a reduced-order augmented system is derived. On this basis, a fault observer based on intermediate variables is constructed to realize the joint reconstruction of system states, faults, and uncertainties, and there is no need to meet the observer matching conditions. Using the reconstructed faults and uncertainties, a fault-tolerant control strategy is designed for dynamic compensation to maintain system stability. Then, based on the reconstructed state variables, the formation tracking error is recalculated, and the controller gain is determined by combining adaptive technology and Lyapunov's theorem. The fault-tolerant formation controller designed based on the above technologies, compared with previous UAV formation control methods, takes into account the effects of system faults and system uncertainties, and uses a fuzzy system to model the UAV system, reducing the accuracy requirements for modeling the actual system and improving the limitations of previous research in this field.

[0083] As shown Figure 1 in the figure, the fault-tolerant formation control method of the fuzzy UAV system based on the reduced-order observer proposed by the present invention is shown, which specifically includes the following steps:

[0084] S1. Use the T-S fuzzy model to model the multi-leader UAV system, and the multi-leader UAV system includes leader UAVs and follower UAVs;

[0085] As a preferred embodiment, step S1 is specifically:

[0086] Consider a multi-leader UAV system composed of N airframes, where M airframes are follower UAVs, and the remaining N-M airframes are leader UAVs; use the T-S model to perform dynamic modeling on each leader UAV and follower UAV respectively;

[0087] The dynamic model of the follower UAV is:

[0088]

[0089] Among them, represents the state change rate of the follower UAV i, that is, the dynamic expression of the state; is the state of the follower UAV, is the output of the follower UAV, is the control input; are the process fault and system uncertainty respectively; q is the number of fuzzy rules, is the fuzzy weight, satisfying and i is the number of the follower UAV, j is the number of the fuzzy rule, i∈{1,...,M}, j∈{1,...,q};

[0090] In this application, comprehensively considers factors such as control input, process fault and uncertainty, and reflects the state change of the agent under the influence of these factors; while x i (t) is the state of the agent i itself, and does not directly include the influence of the above factors; the weight function dynamically adjusts the contributions of each subsystem through fuzzy rules, so that the overall UAV system can express approximately complex nonlinear dynamics.

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

[0092] The dynamic model of the leader UAV is:

[0093]

[0094] where, represents the rate of change of the state of the leader UAV, is the state of the leader UAV, is the output of the leader UAV; m is the number of the leader UAV, m ∈ {M + 1,..., N};

[0095] S2. By splitting the system state and coordinate transformation methods, the original system is transformed into a reduced - order form;

[0096] As a preferred implementation, step S2 is specifically:

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

[0098]

[0099] From this, it can be obtained that the matrix is the left inverse matrix of Y;

[0100] Therefore, left - multiplying the inverse matrix of Y by the state x i (t) of the follower UAV can obtain:

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

[0102] η i (t) is defined as the transformed state vector of the state x i (t); Let η iSubstituting (t) into the dynamic model of the follower UAV gives:

[0103]

[0104]

[0105] Left-multiplying both sides of the above equation by Y -1 , we can obtain:

[0106]

[0107] where represents the rate of change of η i (t), that is, the dynamic expression of η i (t); then decomposing η i (t) into and substituting it into the above equation, we have:

[0108]

[0109] Furthermore, the dynamic expressions of η i1 (t) and η i2 (t) are obtained:

[0110]

[0111] where is the dynamic expression of η i1 (t) and η i2 (t); η i1 (t) and η i2 (t) are transformation state sub-vectors, and η i1 (t) is taken as the reduced-order state of the follower UAV;

[0112]

[0113] and it satisfies

[0114] Then, according to defining the virtual output γ i (t), its formula expression is:

[0115]

[0116] Finally, the reduced-order dynamic model of the follower UAV is obtained:

[0117]

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

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

[0120] Let the augmented state of the follower UAV be The reduced-order dynamic model of the follower UAV is transformed into an augmented form, and the formula is expressed as:

[0121]

[0122] where the matrix is the system matrix of the augmented reduced-order dynamic model of the follower UAV; is the dynamic expression of the augmented state.

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

[0124] S4. Design a reduced-order fault observer based on the intermediate variable for the reduced-order augmented system to estimate the state, process fault, and system uncertainty of the follower UAV;

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

[0126]

[0127] Furthermore, it is derived from the observed value of the observer and the reduced-order augmented system of the follower UAV:

[0128]

[0129] where is the estimated value of the augmented state of the follower UAV; are the estimated values of the uncertainty θ i (t), the process fault f i (t), and the state x i (t) of the follower UAV system, respectively; L j is the designed observer gain; is the designed intermediate variable, which is expressed as where ζ is a selected scalar, is the estimated value of the intermediate variable ; β i (t) is the designed new variable, and its expression is are the estimated values of η i (t) and η i1 (t), respectively.

[0130] S5. Design a tracking fault-tolerant formation controller using the relative state information between unmanned aerial vehicles (UAVs), and use the estimation of process faults and system uncertainties as compensation terms;

[0131] As a preferred implementation manner, the expression of the tracking fault-tolerant formation controller in step S5 is specifically:

[0132]

[0133] Wherein, satisfies I represents the identity matrix, that is, is the generalized inverse matrix of B j , and is the controller compensation term matrix; the controller compensation term is composed of the estimation of the process state and the estimation of uncertainties ; q is the number of fuzzy rules, is the fuzzy weight, satisfying and ν i (t) is the compensation for the system redundancy term; are respectively the estimated values of s i (t), , s i (t) represents the relative state information between the i-th following UAV and other following UAVs, represents the relative state information between the i-th following UAV and the leading UAV,

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

[0135]

[0136] Wherein, ρ i (t), ρ l (t) respectively represent the expected time-varying queue information of the i-th and l-th following UAVs, x i (t), x l (t) are respectively the states of the i-th and l-th following UAVs; x m (t) is the state of the leading UAV, is the estimated value of x i (t), x l (t); a il is the element in the i-th row and l-th column of the adjacency matrix of the topological structure diagram of the multi-leading-UAV system, a im is the element in the i-th row and m-th column of the adjacency matrix of the topological structure diagram; is the controller gain matrix to be designed, M is the number of following UAVs, and N represents the total number of UAVs.

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

[0138] As a preferred implementation, step S6 is specifically as follows:

[0139] Design the parameters of the reduced-order fault observer and the tracking fault-tolerant formation controller based on Lyapunov's theorem and adaptive technology, specifically including:

[0140] If there exist scalars ζ > 0, ∈ > 0 and matrices Ω1 > 0, Ω2 > 0 such that the following linear matrix inequalities hold, and the reduced-order fault observer gain L j is selected to satisfy is a Hurwitz matrix, the error between the observed value and the actual value can asymptotically converge; the linear matrix inequalities are expressed as:

[0141]

[0142] where,

[0143]

[0144] I is the identity matrix; the matrices are respectively the state matrix, the uncertainty distribution matrix, the fault distribution matrix, and the output matrix of the augmented reduced-order dynamics model of the following UAVs; H k2 is obtained from and H k is the uncertainty distribution matrix of the original following UAV system;

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

[0146] When the controller gain in the tracking fault-tolerant formation controller and the compensation ν i (t) for the system redundancy term satisfy the following equations, the control objective is achieved:

[0147]

[0148] where, is the formation tracking error, expressed as is the estimated value of, Ω3 and Θ are symmetric positive definite matrices, and Θ -1 is the inverse matrix of matrix Θ, 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 the formation tracking objective of the multi-leader fuzzy UAV system can be achieved;

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

[0151] S71. Select an appropriate Lyapunov function V(t) according to the estimated error systems of all follower UAVs. If it is derived that then the effectiveness of the reduced-order fault observer is verified;

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

[0153]

[0154] Among them, ρ i (t) represents the desired time-varying queue information of the i-th follower UAV, and α k is a constant that satisfies x k (t) represents the state of the k-th UAV;

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

[0156] Assumption 1: The fault signal f i (t) and the system uncertainty θ i (t) are both differentiable and have bounded derivatives, that is, they satisfy and where μ1 > 0 and μ2 > 0.

[0157] Assumption 2: All follower UAVs can be divided into two categories: informed follower UAVs and uninformed follower UAVs. An informed follower UAV means that its neighbor set contains all leader UAVs, and an uninformed follower UAV means that its neighbor set does not contain leader UAVs. For each uninformed follower UAV, there is at least one direct undirected path leading to an informed follower UAV.

[0158] According to Assumption 2, we can obtain the Laplacian matrix of this multi-leader UAV system which can be expressed as where

[0159] Next, several lemmas will be introduced to facilitate the proof of the method proposed in the present invention later:

[0160] Lemma 1: If for each follower UAV, there is at least one leader with a directed path to this follower UAV, then all the real parts of the eigenvalues of the Laplacian matrix sub-block are positive. The matrix Each element in is non - negative, and the sum of elements in each row is equal to 1. Additionally, when Assumption 2 holds, each row of elements is the same.

[0161] Lemma 2: Assume that matrices Y1 and Y2 are dimension - compatible, then the following inequality holds:

[0162]

[0163] where ∈ is a positive real number;

[0164] Next, it will be specifically analyzed that under the parameter design conditions of S6, the reduced - order fault observer designed by the present invention based on the intermediate variable can complete the estimation tasks of the system state, fault, and uncertainty;

[0165] Define the estimation error: is the state estimation error of the augmented system corresponding to the i - th following UAV; is the estimation error of the intermediate variable of the observer corresponding to the i - th following UAV; is the estimation error of the system uncertainty of the i - th following UAV; is the estimation error of the system process fault of the i - th following UAV. Since the intermediate variable is defined as so it can be obtained that From the previous form of the observer, the dynamic form of the estimation error system can be obtained as:

[0166]

[0167] Write the above formula in a compact form

[0168]

[0169] where,

[0170]

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

[0172] Select the following Lyapunov function:

[0173]

[0174] where, Ω1, Ω2 are symmetric positive - definite, is The transpose of is The transpose of. From the dynamic equation of the previous error system, the dynamic equation of V(t) can be obtained as follows:

[0175]

[0176] According to Assumption 1, there exists an uncertain scalar value μ f and μ θ satisfying and Therefore, based on Lemma 2, the following inequality can be deduced to hold:

[0177]

[0178] where ε is a real constant;

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

[0180]

[0181] Define the matrix At this time, the above equation can be transformed into:

[0182]

[0183] where

[0184]

[0185] From the properties of the Lyapunov function, by scaling V(t), we can get:

[0186]

[0187] Here, λ max (Ω1) represents the largest eigenvalue of the matrix Ω1, and λ max (Ω2) is the same; represents half of the second norm of and the same for. Define the matrix So when Σ jk < 0, From the above equation and the inequality of can be obtained:

[0188]

[0189] where

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

[0191]

[0192] where λ min (Ω1) represents the minimum eigenvalue of matrix Ω1, and λ min (Ω2) is the same. Assume that is the complement of Φ. If then we can obtain:

[0193]

[0194] From the above equation and we can obtain:

[0195] If the matrix inequality condition Σ < 0 is satisfied, then by the Schur complement theorem, Σ < 0 can be transformed into the linear matrix inequality in step S4. Therefore, if According to the Lyapunov stability theory, the error pair will achieve uniform boundedness, thus ensuring the stability of the system under the specified conditions. The proof is completed, and thus the effectiveness of the reduced-order fault observer is verified;

[0196] S72. According to the formation tracking error of the following UAV select an appropriate Lyapunov function to make a stability judgment, then it is proved that under the tracking fault-tolerant formation controller, the formation tracking error of the following UAV converges to 0, and all following UAVs can track the convex combination of multiple leaders and achieve a time-varying formation;

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

[0198] First, define the vector as the formation state, where ρ i (t) represents the expected time-varying queue information of the i-th following UAV. Combining the state equation of the following UAV system, the designed controller u i (t) and the defined intermediate variable we can deduce that the dynamic equation of is:

[0199]

[0200] where q is the number of fuzzy rules, is the fuzzy weight, satisfying and ζ is a selected scalar, and the matrices A j , B j , H j , E j are known system matrices. is the controller gain to be designed.

[0201] Assume that ν i (t) satisfies Then the above equation 's dynamic equation can be transformed into:

[0202]

[0203] Rewrite the above equation in the compact form for all follower UAVs as:

[0204]

[0205] where I M represents the identity matrix of dimension M, represents the Kronecker product operation,

[0206]

[0207] Define the vector as the formation error of follower UAV i. Substitute the expression of s i (t), to obtain:

[0208]

[0209] Write it in the compact form for all follower UAVs as:

[0210]

[0211] where is the Jordan block of the Laplacian matrix Let represent the formation error of the entire system, which is also our main control variable. 's dynamic change can be obtained from 's dynamic equation and the state equation of the leader

[0212]

[0213] From the form of the reduced-order observer, it can be seen that So it can be obtained that

[0214] Therefore, when converges, will also converge. From the verification steps of S71, it can be seen that if the observer parameters satisfy the conditions of S61 step, the observation error will asymptotically converge to zero. Thus, as time t approaches infinity, e f (t) will tend to zero. Hence, the above equation is equivalent to:

[0215]

[0216] Select the following Lyapunov equation:

[0217]

[0218] where Ω3, Θ are symmetric positive definite matrices, is the transpose of, and tr(·) represents the trace of the matrix · inside the parentheses. From 's dynamic equation, the dynamic equation of V o (t) can be obtained as:

[0219]

[0220] Assume

[0221]

[0222] Simple transformation of the above equation gives:

[0223]

[0224] Thus, through the above equation, the expression of the controller gain can be written as:

[0225]

[0226] The above equation is the designed controller gain; so when the controller gain is the above equation, is re-expressed as:

[0227]

[0228] Define the matrix such that where Λ L1 is a diagonal matrix, and its diagonal elements correspond to the eigenvalues of the matrix i.e., Λ L1 = diag{λ1, λ2, …, λ M}}. In this case, define where Therefore, can be converted into the following form:

[0229]

[0230] where the matrix U jis symmetric positive definite and satisfies the Lyapunov equation in the form of From the form of V o (t), we can deduce that:

[0231]

[0232] Therefore, when time t approaches infinity, the formation error can converge to zero. From the expression of , we can obtain:

[0233]

[0234] Multiplying both sides of the above equation by yields:

[0235]

[0236] Here,

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

[0238]

[0239] where c il represents the element in the i-th row and l-th column of. According to Lemma 1 in Step S71, the sum of the elements in each row of is 1. Specifically, So the above equation is consistent with the control objective described in Step S71. Therefore, under the designed controller u i (t) and the controller gain designed according to S6, the multi-leader UAV system can achieve formation tracking, thus completing the effect verification.

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

[0241] The specific information of 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 Server VM mixed mode

[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 unmanned aerial vehicles (UAVs), including six follower UAVs and three leader UAVs, whose topological structure is as Figure 2 shown. Assume that the number of fuzzy rules q = 2, and the system matrix is as follows:

[0248]

[0249] According to Figure 2 the topological structure shown in the Laplacian matrix

[0250]

[0251] is determined as follows: Assume that process faults only occur in agents 1 to 4, and the dynamics of process faults f i (t), i = 1, 2,..., 6 are respectively 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:

[0256]

[0257] where y i1 (t) refers to the first element of the output y i (t) of the i-th follower UAV.

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

[0259]

[0260] Assume that 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 is a Hurwitz matrix, and the observer gain is set as:

[0263]

[0264] Figure 3 shows the fault estimation results of UAVs No. 1 to No. 4, while Figure 4 shows the uncertainty estimation of UAVs No. 1 and No. 2. It can be seen from the figure that the curves of the actual value and the estimated value approximately coincide, indicating that the observation error is very small, which shows that the estimation performance meets the expected requirements.

[0265] Figure 5 shows the formation tracking error The change of is shown in the figure. As shown in the figure, quickly converges to a small range near zero, indicating that the formation tracking target of the system has been effectively achieved. Figure 6 shows the position states of all UAVs at different time intervals. The square represents the leader, while the other shapes represent the followers. It can be seen from the figure that the followers continuously move around the leader in a rotating pattern. Therefore, the fault-tolerant tracking control (FTFC) scheme proposed in this paper successfully overcomes the influence of process faults and system uncertainties, ensuring that all follower UAVs can track multiple leaders according to the specified time-varying formation.

[0266] This embodiment aims to solve the problems of fault tolerance of fuzzy UAV systems and formation control of multiple leader UAVs. Through coordinate transformation, the original system is transformed into a reduced-order form, and then the fault is regarded as a special state. The reduced-order augmented system is derived, and a reduced-order fault observer based on intermediate variables is constructed to reconstruct the system state, fault, and uncertainty without the need to satisfy the observer matching condition. Using the reconstructed fault and uncertainty data, a fault-tolerant control strategy is developed to compensate for these problems and maintain system stability. Subsequently, the parameters of the fault observer and controller are designed based on adaptive technology and Lyapunov theorem, and then it is verified that the controller can achieve the tracking target of the multiple leader UAV system under the designed parameter conditions. Finally, the effectiveness of the method is verified through simulation experiments.

[0267] In the description of this specification, the descriptions referring to terms such as "one embodiment", "some embodiments", "example", "specific example", or "some examples" mean that the specific features, structures, materials, or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present application. Moreover, the specific features, structures, materials, or characteristics described can be combined in a suitable manner in any one or more embodiments or examples. In addition, without contradiction, those skilled in the art can combine and combine the different embodiments or examples described in this specification and the features of 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 definite sequence list of executable instructions for implementing logical functions, and can be specifically implemented in any computer-readable medium for use by an instruction execution system, apparatus, or device (such as a computer-based system, a system including a processor, or other systems that can fetch and execute instructions from the instruction execution system, apparatus, or device), or used in combination with these instruction execution systems, apparatuses, or devices.

[0269] The above embodiments have introduced the present invention in detail. Specific examples are used herein to illustrate the principle and implementation of the present invention. The description of the above embodiments is only for helping to understand the method and its core idea of the present invention; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation and application scope. In summary, 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 a fuzzy UAV system based on a reduced-order observer, characterized in that Specifically, it includes the following steps: S1. Use the T-S fuzzy model to model the multi-leader UAV system, which includes leader UAVs and follower UAVs; S2. By means of system state splitting and coordinate transformation, transform the original system 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 based on the intermediate variable for the reduced-order augmented system to estimate the state, process faults and system uncertainties of the follower UAVs; S5. Design a tracking fault-tolerant formation controller using the relative state information between UAVs, and use the estimation of process faults 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 to verify whether the formation tracking goal of the multi-leader fuzzy UAV system can be achieved.

2. The fault-tolerant formation control method for the fuzzy UAV system based on the reduced-order observer according to claim 1, characterized in that Step S1 is specifically as follows: Consider a multi-leader UAV system composed of N airframes, where M airframes are follower UAVs and the remaining N - M airframes are leader UAVs; use the T-S model to perform dynamic modeling on each leader UAV and follower UAV respectively; The dynamic model of the follower UAV is: Among them, represents the rate of change of the state following UAV i, that is, the dynamic expression of the state; is the state of the following UAV, is the output of the following UAV, is the control input; are the process fault and system uncertainty respectively; q is the number of fuzzy rules, is the fuzzy weight, satisfying and i is the number of the following UAV, j is the number of the fuzzy rule, i ∈ {1,..., M}, j ∈ {1,..., q}; A j , B j , H j , E j , and C are system matrices, where is the state matrix, is the control input matrix, is the uncertainty distribution matrix, is the output matrix, is the fault distribution matrix, n x , n y , n u , n f , n θ represent the number of elements in each matrix; all matrices are known and satisfy (A j , B j ) is controllable, (A j , C) is observable, and C has full row rank; The dynamic model of the leader UAV is: Among them, represents the state change rate of the leading UAV, is the state of the leading UAV, is the output of the leading UAV; m is the number of the leading UAV, m ∈ {M + 1,..., N}.

3. The fault-tolerant formation control method for a fuzzy UAV system based on a reduced-order observer according to claim 2, characterized in that, Step S2 is specifically as follows: Define a matrix Left multiply the state x i (t) of the following drone by the inverse matrix of Y to obtain the variable η i (t), and the formula is expressed as: η i (t) = Y -1 x i (t), i ∈ {1, ..., M}; where, η i (t) is defined as the transformed state vector of state x i (t), is an arbitrary orthogonal basis of the null space of matrix C, satisfying rank(C0) = n x - n y , and CC0 = 0; Substitute η i (t) into the dynamic model following the drone to obtain: Among them, represents the change rate of η i (t), that is, the dynamic expression of η i (t); then decompose η i (t) into Then there is: Among them, is η i1 (t), η i2 (t) of the dynamic representation; η i1 (t) and η i2 (t) are the transformed state sub-vectors, and η i1 (t) is taken as the reduced-order state following the UAV; System matrix and satisfy Then, according to Define the virtual output γ i (t), and its formula is expressed as: Finally, the reduced-order dynamic model of the follower UAV is obtained:

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

5. The fault-tolerant formation control method for a fuzzy UAV system based on a reduced-order observer according to claim 4, characterized in that, The expression of the reduced-order fault observer designed in step S4 is specifically: Furthermore, it is derived from the observed values of the observer and the reduced-order augmented system of the follower UAV: Among them, is the estimated value of the augmented state following the UAV ; are respectively the estimated values of the uncertainty θ i (t), process fault f i (t), and state x i (t) of the UAV system; L j is the designed observer gain; is the designed intermediate variable, which is expressed as where ζ is a selected scalar, is the estimated value of the intermediate variable ; β i (t) is the designed new variable, and its expression is are respectively the estimated values of η i (t), η i1 (t).

6. The fault-tolerant formation control method for a fuzzy UAV system based on a reduced-order observer according to claim 4, characterized in that The expression of the tracking fault-tolerant formation controller in step S5 is specifically: Among them, Satisfy I represents the identity matrix, that is Is the generalized inverse matrix of B j For the controller compensation term matrix; the controller compensation term is Consisted of the estimation of the process state And the estimation of uncertainty Constitute; q is the number of fuzzy rules, Is the fuzzy weight, satisfy And ν i (t) is the compensation for the system redundancy term; Are respectively s i (t), The estimated values of, s i (t) represents the relative state information between the i-th following UAV and other following UAVs, Represents the relative state information between the i-th following UAV and the leading UAV, s i (t), The formulas are respectively expressed as: where, ρ i (t) and ρ l (t) respectively represent the expected time-varying queue information of the \(i\)-th and \(l\)-th follower drones, x i (t) and x l (t) are the states of the \(i\)-th and \(l\)-th follower drones respectively; x m (t) is the state of the leader drone, is the estimated value of x i (t) and x l (t); a il is the element in the \(i\)-th row and \(l\)-th column of the adjacency matrix of the topology structure graph of the multi-leader drone system, a im is the element in the \(i\)-th row and \(m\)-th column of the adjacency matrix of the topology structure graph; is the controller gain matrix to be designed, M is the number of follower drones, and N represents the total number of drones.

7. The fault-tolerant formation control method for a fuzzy UAV system based on a reduced-order observer according to claim 6, characterized in that Step S6 is specifically as follows: Design the parameters of the reduced-order class fault observer and the tracking fault-tolerant formation controller based on the Lyapunov theorem and adaptive technology, specifically including: If there exist scalars ζ > 0, ∈ > 0 and matrices Ω1 > 0, Ω2 > 0 such that the following linear matrix inequalities hold, and the reduced-order fault observer gain L j is selected to satisfy is a Hurwitz matrix, the error between the observed value and the actual value can asymptotically converge; the linear matrix inequalities are expressed as: Among them, I is the identity matrix; the matrix is respectively the state matrix, the uncertainty distribution matrix, the fault distribution matrix, and the output matrix of the augmented reduced-order dynamics model following the UAV; H k2 is obtained from where H k is the uncertainty distribution matrix of the original UAV following system; Then, the parameters of the tracking fault-tolerant formation controller are designed as follows: When tracking the controller gains in the fault-tolerant formation controller and the compensation ν i (t) for the system redundancy terms satisfies the following equation, the control objective is achieved: wherein, is the formation tracking error, expressed as is the estimated value of, Ω3 and Θ are symmetric positive definite matrices, Θ -1 is the inverse matrix of matrix Θ, represents the value of the controller gain at the initial moment.

8. The fault-tolerant formation control method for a fuzzy UAV system based on a reduced-order observer according to claim 1, characterized in that Step S7 is specifically as follows: S71. Select an appropriate Lyapunov function V(t) according to the estimated error system of all drones following the drone, as derived Then the effectiveness of the reduced-order fault observer is verified; S72. According to the formation tracking error of the follower UAVs Select an appropriate Lyapunov function for stability judgment, then it is proved that under the tracking fault-tolerant formation controller, the formation tracking error of the follower UAVs converges to 0, and all follower UAVs can track the convex combination of multiple leaders and achieve a time-varying formation.

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