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

Abstract

The invention provides a fuzzy multi-unmanned-aerial-vehicle fault-tolerant formation control method based on a reduced-order intermediate observer, and the method comprises the steps: constructing a multi-unmanned-aerial-vehicle system model containing a follower and a leader, the follower employing a T-S fuzzy model to represent a dynamic state, including system state output control input actuator fault and parameter uncertainty, and the leader employing a T-S fuzzy model to represent a dynamic state; the leader model does not contain faults and uncertainty; defining a time-varying formation tracking condition as follows: the state of a follower is equal to expected formation offset in a convex hull of a leader, and deducing a follower order reduction dynamic model; a reduced-order intermediate observer is designed to reconstruct system state, fault and uncertainty information, a distributed control protocol is combined, state error feedback, fault compensation and adaptive gain adjustment are fused, and the robustness of time-varying formation tracking control is ensured. Finally, the fault-tolerant capability and formation stability of the multiple unmanned aerial vehicles in a complex scene are remarkably improved, and a reliable guarantee is provided for successful execution of a task.

Description

Technical Field

[0001] This invention relates to the field of intelligent control technology, specifically to a fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer. Background Technology

[0002] Given the complexity of the system architecture and network interconnection, multi-UAV systems are highly susceptible to failure. In such systems, the information exchange between UAVs can cause a failure in one UAV to propagate rapidly throughout the network, degrading system performance and potentially leading to complete system collapse. Furthermore, aging, wear and tear, and external interference and uncertainties caused by diverse operations can further impact the system's stability and reliability.

[0003] Existing research on multi-UAV system control largely focuses on single-leader scenarios, while multi-leader situations are more common in practical applications. Furthermore, although the Takagi-Sugeno (TS) fuzzy model is a powerful tool for handling nonlinear systems, its application in multi-UAV systems is still insufficient. Regarding system safety and reliability, traditional fault estimation and fault-tolerant control methods (such as unknown input observers and adaptive observers) typically rely on observer matching conditions, which are often difficult to meet in practical engineering, limiting their application. While intermediate variable observer methods have emerged in recent years to circumvent these stringent conditions, existing designs are mostly full-order observers with high computational complexity, and research on using such observers to solve formation control problems in multi-UAV systems remains lacking. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this paper solves the formation control problem of fuzzy multi-UAV systems affected by actuator failures and parameter uncertainties. When a multi-UAV system described by a TS fuzzy model simultaneously suffers from actuator failures, parameter uncertainties, and multiple leaders, an effective control strategy is designed to ensure that the follower UAV swarm maintains the desired time-varying formation and converges to the convex hull formed by the states of multiple leaders.

[0005] To address this, a fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate variable observer is proposed. This method reconstructs the UAV state, actuator faults, and parameter uncertainties, achieving high-precision and high-reliability formation control even when the full system state cannot be directly obtained and faults and uncertainties occur in real time.

[0006] To achieve the above objectives, this invention provides the following technical solution: A multi-UAV system model containing followers and a leader is constructed, where the followers are dynamically represented using a TS fuzzy model, including system state output, control input, actuator faults, and parameter uncertainties; the leader model does not contain faults or uncertainties. A time-varying formation tracking condition is defined as the follower state equal to the leader's expected formation offset within the convex hull, and a reduced-order dynamic model for the followers is derived. A virtual output is defined using a left inverse matrix and a null space orthogonal basis, forming an augmented reduced-order system with derivatives for state fault uncertainty. A reduced-order intermediate observer is designed, defining intermediate variables to eliminate virtual output derivatives. The observer is constructed with the control input measurable output intermediate variables as inputs, augmented state estimation, and uncertainty auxiliary estimation as its core. The system state fault uncertainty is reconstructed by solving the observation gain using linear matrix inequalities. Based on the reconstructed information, a distributed formation control protocol is designed, defining the formation consistency error among followers, the follower-leader tracking error, and the overall formation error. The control input is fused to estimate the state error, feedback the fault uncertainty, and compensate for the adaptive gain. The gain update law is derived using Lyapunov functions, and TS fuzzy weights are fused to form the final input.

[0007] Preferably, the TS fuzzy model includes: determining a system matrix and fuzzy weights for each fuzzy rule, wherein the fuzzy weights satisfy normalization and summability conditions, and the system matrix is ​​controllable and observable; constructing a Laplace matrix to represent the topology of the multi-UAV system, wherein followers are divided into information-complete UAVs and information-incomplete UAVs, and each information-incomplete UAV has a direct undirected path with an information-complete UAV; assuming that the fault signal and parameter uncertainty are differentiable and bounded signals, satisfying the derivative bounded condition.

[0008] Preferably, the reduced-order dynamic model includes: determining the matrix and null space orthogonal basis that satisfy the left null space condition since the output matrix is ​​full rank; defining the virtual output combined with the reduced-order model to obtain the initial follower dynamic transformation into a reduced-order form; defining the augmented state variable as the system state fault uncertainty derivative to form the i-th follower augmented reduced-order system.

[0009] Preferably, the specific process for constructing the intermediate observer is as follows: a. Prerequisite preparation: Based on the reduced-order augmented system, the derivatives of faults and parameter uncertainties are incorporated into the augmented state based on the reduced-order dynamic model derived in the system model, forming an integrated augmented system of "state-fault-uncertainty" to ensure that the observer can estimate the three types of core information at the same time. b. Construct intermediate variables: Eliminate unmeasurable output derivatives and transform the "unmeasurable output derivatives" in the augmented system into "derivable terms containing fault / uncertainty information" to avoid introducing unmeasurable signals into the observer; c. Construct a dynamic structure for the observer: correlate the augmented state estimate with the uncertainty auxiliary estimate to ensure the continuity of the uncertainty estimate.

[0010] Preferably, the reduced-order intermediate observer includes: defining the intermediate variable as the product of the observer output estimate and a selected scalar, with the first derivative of the intermediate variable expressed by the intermediate coefficient matrix under fuzzy rules; constructing the observer dynamic equation to eliminate the virtual output derivative, and introducing new variables to correlate the augmented state estimation uncertainty to assist in estimation; defining the observation error to obtain the compact form of the observer error system, wherein the observation gain Lj satisfies the Hermite stability of the augmented matrix minus the observation term under the combination of fuzzy rules.

[0011] Preferably, the linear matrix inequality includes: the existence of a scalar and Positive definite matrices Ω1 and Ω2 satisfy inequality conditions, ensuring that the derivative of the Lyapunov function is negative definite; based on the boundedness assumption and lemma, an inequality chain is established, and the error system is proved to be uniformly bounded by the Schul complement formula.

[0012] Preferably, the distributed formation control protocol includes: defining the formation consistency error as the product of the adjacency matrix elements and the desired formation offset vector, and the tracking error as the product of the communication adjacency elements of the follower and leader; integrating the total formation error, and calculating negative feedback based on the estimated state for the error feedback term; using the pseudo-inverse of the fault distribution matrix to offset the reconstruction of fault uncertainty for the compensation term, and updating the adaptive gain with the integral of the total formation error.

[0013] Preferably, the Lyapunov function includes: selecting a Lyapunov function containing a gain quadratic form term, deriving the derivative based on the positive sum of the real parts of the eigenvalues ​​of the Laplace matrix and row consistency; defining the diagonal elements of the diagonal matrix as the eigenvalues ​​of the topological matrix, and converting it into a symmetric positive definite matrix form to ensure the convergence of the overall formation error.

[0014] Preferably, the control input includes: the final input fused TS fuzzy weight coefficients, the system dynamic equation is transformed into a follower integrated form; when the observation error converges and the total formation error approaches zero, the follower state is equal to the expected formation offset within the leader's convex hull.

[0015] Compared with existing technologies, this invention provides a fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer, which has the following beneficial effects: This invention addresses the interconnected challenges of equipment failures, parameter uncertainties, and dynamic environmental interference encountered by multiple UAVs in formation missions under complex environments. These problems often occur simultaneously in real-world missions, such as actuator failures leading to thrust loss, environmental interference causing attitude deviations, and information delays due to communication limitations, all of which collectively affect formation stability and mission execution efficiency. This invention accurately describes nonlinear dynamics by constructing a fuzzy model, designs a reduced-order intermediate observer to reconstruct system state, fault, and uncertainty information, and integrates a distributed control protocol, fusing state error feedback, fault compensation, and adaptive gain adjustment to ensure the robustness of time-varying formation tracking control. Ultimately, this invention significantly improves the fault tolerance and formation stability of multiple UAVs in complex scenarios, providing a reliable guarantee for successful mission execution. Attached Figure Description

[0016] Figure 1 This is a system topology diagram of the present invention; Figure 2 The fault estimation results for the UAV of this invention; Figure 3 The uncertainty estimation results for the UAV of this invention; Figure 4 This illustrates the variation of the formation tracking error φ(t) in this invention. Figure 5 These are snapshots of the motion trajectory status of all UAVs in this invention at different times. Detailed Implementation

[0017] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0018] In one embodiment, the method provided by this invention is applicable to formation control scenarios in multi-UAV systems. For example, when a UAV swarm performs a task, multiple UAVs need to follow a leader to form a specific formation while simultaneously dealing with potential equipment failures or environmental interference. By constructing a fuzzy model and a reduced-order observer, combined with a distributed control protocol, this method can effectively improve the robustness and stability of formation control. Specific implementation steps will be described in detail below.

[0019] Step S1 involves constructing a multi-UAV system model containing followers and a leader. The dynamics of the followers are represented using a fuzzy model, including system state, output, control input, actuator failures, and parameter uncertainties. The leader's model, however, does not consider failures and uncertainties. It should be noted that a multi-UAV system typically consists of multiple autonomous decision-making entities that collaborate through a communication network to complete tasks. In this embodiment, a follower refers to a UAV that needs to follow the leader's trajectory or formation, while the leader is responsible for guiding the entire formation's movement.

[0020] For example, in a drone swarm formation mission, suppose there are 5 follower drones and 1 leader drone. The leader drone flies along a preset path, while the follower drones need to maintain their relative positions to the leader, forming a triangular formation. When building the model, the dynamic model of each follower drone needs to consider the potential thrust reduction problem caused by its motors, and the possibility that external wind interference may cause changes in its flight parameters. Using a fuzzy model, these nonlinear effects can be decomposed into multiple linear subsystems and combined using fuzzy weights, thus more accurately describing the behavioral characteristics of each drone.

[0021] Step S11 involves determining the number and form of fuzzy rules when constructing the follower model. Assume that the dynamics of each follower can be described by three fuzzy rules, each corresponding to a different operating state, such as normal, low-speed, and high-speed states. For each rule, a corresponding system matrix and output matrix are defined, reflecting the dynamic response characteristics of the follower under different states. Simultaneously, actuator failure is modeled as an additional term affecting the effectiveness of the control input, while parameter uncertainty is represented as a perturbation matrix within the system matrix.

[0022] Step S12 involves constructing the leader model, assuming its dynamics are a known linear system with state variables including position and velocity, and control input consisting of a preset trajectory command. Since the leader model does not consider faults and uncertainties, its main function is to generate a stable reference signal for followers to track. In practical applications, the leader can be a core UAV in a cluster or a virtual reference point, and its state information is broadcast to all or some followers via a communication network.

[0023] It's important to note that fuzzy models have the advantage of handling complex nonlinear dynamics, especially in multi-UAV systems where the behavior of each UAV can be influenced by a combination of internal and external factors. By decomposing a nonlinear system into a combination of multiple linear subsystems, the complexity of control design can be significantly reduced, while improving the system's adaptability. In the UAV swarm example, this modeling approach helps the system maintain formation stability under varying flight conditions.

[0024] Step S2 defines the time-varying formation tracking condition as the follower state equal to the expected formation offset within the leader's convex hull, and derives the reduced-order dynamic model of the follower. A virtual output is defined using the left inverse matrix and a null space orthogonal basis, forming an augmented reduced-order system of state, fault, and uncertainty derivatives. Specifically, the time-varying formation tracking condition means that the follower's state needs to form a specific geometric formation with the leader, and this formation can change over time, for example, switching from a triangular formation to a straight-line formation.

[0025] Step S21, regarding the derivation of the reduced-order dynamic model for the follower, assuming the output matrix has full row rank, the original system model can be transformed into a reduced-order form through mathematical transformations. Specifically, by constructing a left inverse matrix, the system state is decomposed into observable and unobservable parts, and the model structure is further simplified using a null space orthogonal basis. The purpose of this reduction is to reduce the system dimensionality, thereby reducing the computational complexity of subsequent observer and controller design.

[0026] Step S22: Based on the reduced-order model, a virtual output is defined to replace some of the state variables of the original system. The virtual output can be obtained by linearly combining the measurable outputs, and its function is to simplify the expression of the system dynamics. By introducing the virtual output into the reduced-order model, the original follower dynamics can be transformed into an equivalent low-order system, thus laying the foundation for subsequent observer design.

[0027] Step S23 further defines augmented state variables, integrating the derivatives of system state, actuator faults, and parameter uncertainties into a unified vector, forming an augmented reduced-order system for each follower. This method of constructing augmented systems can explicitly incorporate faults and uncertainties into the system dynamics, thereby facilitating subsequent fault reconfiguration and control compensation.

[0028] It should be noted that the derivation of the reduced-order dynamic model is a key innovation of this method. By transforming a high-dimensional system into a low-dimensional system, not only can the computational resource requirements be reduced, but the real-time response capability of the system can also be improved. This reduction process is particularly important in the application of UAV swarms, because UAVs typically need to complete complex formation tasks under limited computing power and communication bandwidth. By defining virtual outputs and augmented state variables, the design process of subsequent observers can be further simplified, laying the foundation for achieving fault-tolerant control.

[0029] Step S3 involves designing a reduced-order intermediate observer. Intermediate variables are defined to eliminate the virtual output derivative. The observer is constructed with control input, measurable output, and intermediate variables as inputs, focusing on augmented state estimation and uncertainty-aided estimation. The observation gain is solved using linear matrix inequalities to reconstruct the system state, faults, and uncertainties. Specifically, the design of the reduced-order intermediate observer aims to solve the problem of directly measuring the virtual output derivative while simultaneously achieving accurate estimation of system state and fault information.

[0030] In one possible implementation, the design of a reduced-order intermediate observer first requires defining an intermediate variable. This intermediate variable is the product of the observer output estimate and a selected scalar, serving to replace the derivative term of the dummy output, thereby avoiding the introduction of cumbersome derivative calculations into the observer dynamics. By introducing the intermediate variable into the observer structure, the observer's input can be simplified to a control input and a measurable output, while maintaining the ability to estimate the augmented state.

[0031] Step S31: For the definition of intermediate variables, assume they are in the form of first derivatives and express them using the intermediate coefficient matrix under fuzzy rules. Specifically, intermediate variables can be obtained by combining the observer output with a preset constant, and their first derivatives are associated with certain matrices in the system dynamics. In this way, the influence of the virtual output derivative can be eliminated from the observer dynamics, thereby simplifying the design process.

[0032] In step S32, when constructing the observer's dynamic equations, a new variable is introduced to correlate the augmented state estimate and the uncertainty auxiliary estimate. The core state of the observer includes the estimated value of the augmented state and the auxiliary estimate of the uncertainty, while its inputs are the follower's control input, measurable output, and intermediate variables. By rationally designing the observer's structure, it can be ensured that its dynamic equations accurately reflect changes in the system state, while effectively estimating faults and uncertainties.

[0033] Step S33: Define the observation error as the difference between the actual augmented state and the estimated value, and derive the compact form of the observer error system. To ensure the convergence of the observation error, the observation gain needs to be designed so that, under the combination of fuzzy rules, the difference between the augmented matrix and the observation term satisfies the stability condition. This stability condition can be solved using linear matrix inequalities to obtain a suitable observation gain value.

[0034] In one embodiment, a formation flight mission of a UAV swarm in a complex environment is considered. It is assumed that strong wind interference exists in the mission scenario, causing deviations in the position sensor data of some UAVs. To address this, when designing a reduced-order intermediate observer, the intermediate variable is first defined as the product of the position estimate and a scalar, and its derivative is calculated using a coefficient matrix under fuzzy rules. The observer's dynamic equations take the UAV's thrust control commands, position output, and intermediate variables as inputs, and associate the augmented state estimate and the auxiliary estimate of wind interference through a new variable. The observation error is defined as the difference between the actual state and the estimated state. The observation gain is obtained by solving linear matrix inequalities, ensuring that the error system remains stable under interference. This design effectively improves the state estimation accuracy of the UAV swarm in complex environments, providing reliable data support for subsequent formation control.

[0035] In one embodiment, a dynamic adjustment scenario is considered when a drone swarm performs formation tasks. Assume the task requires the drones to dynamically adjust their formation according to environmental changes during flight, such as switching from a triangular formation to a straight formation when encountering obstacles. For this situation, the design of a reduced-order intermediate observer needs to consider the state estimation requirements under different formations. The intermediate variable is defined as the product of the position estimate and a dynamic adjustment constant. The observer's dynamic equation takes control input, position output, and the intermediate variable as inputs and outputs an augmented state estimate. By solving linear matrix inequalities, observation gain values ​​applicable to different formation switching scenarios are obtained, ensuring that the observation error remains within a small range during formation adjustment. This design effectively supports drone swarm formation tasks in dynamic environments.

[0036] Consider the application scenario of UAV swarms performing large-scale formation tasks. Assume the task involves 20 follower UAVs and 1 leader UAV, and the system needs to handle the state estimation requirements of multiple UAVs simultaneously. For this situation, a reduced-order intermediate observer can be designed in a modular manner. The intermediate variable is defined as the product of the position estimate and a fixed constant. The observer's dynamic equation takes control input, position output, and the intermediate variable as inputs and outputs an augmented state estimate. The system decomposes the observer function into three modules: state estimation, fault reconstruction, and uncertainty compensation. Through the collaborative work of these modules, accurate estimation of the state and fault information of each UAV is achieved. This design effectively improves the state estimation capability of the UAV swarm in large-scale tasks, thereby ensuring the overall stability of the formation and the successful completion of the task.

[0037] Based on the reconstruction information obtained from the aforementioned reduced-order intermediate observer, in step S4, a distributed formation control protocol is designed based on the reconstruction information. The formation consistency error among followers and the tracking error between followers and leaders are defined and integrated into the total formation error. The control input is fused with the estimated state error feedback, fault uncertainty compensation, and adaptive gain. The gain update law is derived through the Lyapunov function and fused with the TS fuzzy weights to form the final input.

[0038] Step S41 defines the formation consistency error as a combination of the products of the adjacency matrix elements and the desired formation offset vector. Specifically, for the i-th follower, its formation consistency error can be understood as the weighted sum of its expected relative position deviations from those of all its communicating neighbors, where the weights are determined by the adjacency relationships of the communication topology. The desired formation offset vector is a pre-designed sequence of vectors that may change slowly over time, used to describe the relative position each follower should be within the leader's convex hull at the current moment.

[0039] Step S42 defines the tracking error as the weighted state deviation between followers and the leader through communication adjacency elements. Typically, only some followers can directly or indirectly receive the leader's state broadcast information; therefore, the tracking error term is attenuated and weighted according to the communication level. When integrating the total formation error, the previously obtained formation consistency error and tracking error are superimposed in a certain proportion to form a comprehensive vector. This vector reflects both the relative coordination within the formation and the overall deviation from the reference trajectory.

[0040] Step S43: Construct the three main components of the control input: the first part is a negative feedback term based on the estimated state error, which is used to drive the system state to approach the desired value; the second part is an active compensation term using the fault and uncertainty information obtained from the reconstruction; and the third part is an adaptive gain term, which is used to dynamically adjust the compensation intensity to cope with different degrees of uncertainty and fault.

[0041] Step S44 further introduces the TS fuzzy weights into the synthesis process of the final control input. Specifically, the final control input is no longer a single linear form, but a global control quantity obtained by weighted fusion of multiple local control laws according to the weights of the current fuzzy membership function.

[0042] Step S45: Construct a suitable Lyapunov function to analyze the stability of the entire closed-loop system and derive the update law of the adaptive gain. The selected Lyapunov function typically includes the following components: a quadratic form of the total formation error, a quadratic form of the observation error, a quadratic form of the adaptive gain estimation error, and some cross-coupling terms related to the fuzzy weights.

[0043] Specifically, the quadratic term of the total formation error reflects the degree of formation convergence, and its weight matrix is ​​usually related to the lower bound of the eigenvalues ​​of the Laplace matrix of the communication topology; the quadratic term of the observation error is used to ensure the reliability of the reconstructed information; and the adaptive gain estimation error term is used to constrain excessive gain growth. By differentiating this function along the closed-loop system trajectory, and combining the symmetric positive definite property of the Laplace matrix and the characteristics of the convex combination of the fuzzy system, we can obtain a derivative expression containing negative definite terms and cancelable cross terms.

[0044] In step S46, during the derivation, the real part of the smallest non-zero eigenvalue of the Laplace matrix is ​​positive, and the error term is transformed into a form that can be suppressed by negative feedback. At the same time, by introducing appropriate Young's inequalities or Schur complement techniques, the cross-coupling terms are absorbed into the negative definite terms, so that the derivative of the Lyapunov function takes the form of negative definite or negative semi-definite plus integrable terms.

[0045] Step S47: Further define a diagonal matrix whose diagonal elements correspond to the eigenvalue sequence of the communication topology. Through similarity transformation, the influence of the asymmetric Laplace matrix is ​​transformed into a quadratic form of a symmetric positive definite matrix, which facilitates the stability analysis.

[0046] The following section, in conjunction with the accompanying drawings and the formula derivation of this application, elaborates on the mathematical implementation principles of this application. Please refer to the appendix. Figure 1-5 The derivation process of the formula in this application is as follows: I. Constructing a System Model Consider a multi-UAV system with N UAVs, where M UAVs are followers and the remaining N-M UAVs are leaders. Each UAV is represented by a nonlinear TS fuzzy model. The dynamic model of the follower UAVs is as follows: (1) in They are the first The system status, outputs, and control inputs of a follower drone. This represents faults and uncertainties. q represents the number of fuzzy rules. Represents fuzzy weights, satisfying System matrix as well as It is given. It is controllable. It is observable.

[0047] The dynamic model of the leader drone is as follows: (2) in, These are the state and output of the m-th leader drone, respectively. This represents the position coordinates, speed, and flight attitude data of the m-th leader. This represents the real-time location, speed, and flight attitude information of the m-th leader.

[0048] It should be noted that the present invention has fully constructed the leader model using formula (2). Unlike the followers, the leader, as the benchmark for formation tracking, does not contain fault terms or uncertainty terms in its model. Therefore, it does not require order reduction processing or the design of an intermediate variable observer for order reduction. The subsequent order reduction dynamic model of the followers and the fault-tolerant formation control protocol are all based on the leader model constructed in this invention as the core benchmark, achieving the goal of 'followers maintaining formation and tracking the leader'.

[0049] Definition 1: A multi-leader multi-UAV system (Equations (1) and (2)) can be ensured to achieve time-varying formation tracking if and only if the following conditions are met: (3) in Let represent the desired formation information of the i-th following element, which is a continuous offset vector. And assume... .

[0050] The Laplace matrix of this multi-UAV system can be expressed as: ,in , .

[0051] Assumption 1: Fault signal With system uncertainty All are differentiable and bounded signals, i.e., satisfying and ,in .

[0052] Assumption 2: All followers can be divided into two categories: drones with complete information and drones with incomplete information. Furthermore, for each drone with incomplete information, there exists at least one direct undirected path connecting it to a drone with complete information.

[0053] Lemma 1: If every follower has a direct connection to the leader, then All eigenvalues ​​of the matrix have positive real parts. Each element is non-negative, and the sum of the elements in each row is 1. Furthermore, when assumption 2 holds, the matrix... Each row is consistent.

[0054] Lemma 2: Let matrix and Compatible with any dimension (i.e.) The number of columns equals If a general matrix variable is defined with respect to the number of rows of the matrix, then the following inequalities hold: (4) in It is a positive real number.

[0055] Next, to design a reduced-order intermediate variable observer, it is necessary to derive the reduced-order form of system formula (1). Since... A matrix with full row rank exists. satisfy and ,in for An arbitrary orthogonal basis of the null space. Define the matrix. Then we have: (5) therefore, Construct a matrix The left inverse matrix. Define variables. We can obtain: (6) (7) Left multiplication of both ends We can obtain: (8) Assumption ,in According to equations (7) and (8), we can obtain: (9) in .

[0056] According to formula (7), we can obtain: (10) (11) By defining a virtual output and combining it with equation (11), we can obtain: (12) Therefore, the initial follower drone dynamic model can be transformed into a descending-order dynamic model as follows: (13) Define the state variables of the augmented system as Subsequently, the reduced-order dynamic model of the i-th follower drone can be expressed in the following augmented form: (14) in: , .

[0057] II. Fault Observer Design (I) Core Function Positioning The reduced-order intermediate variable observer designed in this invention is the "sensing core" of fault-tolerant control. Its core function is to simultaneously and accurately estimate three types of key information—1. the system state of the follower UAV—without relying on the "matching conditions" of traditional observers, in scenarios where it is impossible to directly measure all system states, actuator failures, and parameter uncertainties occur in real time. This is achieved through mathematical transformation and signal reconstruction. 2. Actuator fault signal 3. System parameter uncertainty This provides a quantitative basis for fault / uncertainty compensation for subsequent distributed control protocols, while reducing computational complexity through a down-order design.

[0058] (II) Observer Construction Steps 1. Prerequisites: Based on the reduced-order augmented system, and using the reduced-order dynamic model derived in "I. Building the System Model" as a foundation, the fault... Parameter uncertainty The derivative is included in the augmented state This forms an integrated augmented system of "state-fault-uncertainty" to ensure that the observer can simultaneously estimate the three types of core information.

[0059] 2. Construct intermediate variables: Eliminate unmeasurable output derivatives and define intermediate variables. The core objective is to augment the system by addressing the "unmeasurable output derivative". The term is transformed into a "derivable term containing fault / uncertainty information" to avoid introducing unmeasurable signals into the observer.

[0060] 3. Construct the dynamic structure of the observer based on intermediate variables. With measurable output Construct the observer equation: Observer input: Follower control input Measurable output intermediate variables ; Observer core state: augmented state estimate Uncertainty Auxiliary Estimator ; Key association: through By linking the two types of estimators, the continuity of uncertainty estimation is ensured.

[0061] 4. Solve for the observer gain To ensure asymptotic convergence of observation errors, the observer gain needs to be solved using the linear matrix inequality (LMI). .

[0062] 5. Reconstruct target observation information: from the observer output In this process, three types of core information are reconstructed through matrix mapping.

[0063] (III) Specific Formula Derivation To design a fault estimation observer that meets performance requirements, the following intermediate variables are defined: (15) in, It is a selected scalar. According to (14) and (15), The first derivative is given by the following equation: (16) Note 2: Let be the intermediate coefficient matrix (j∈{1,...,q}) under the j-th fuzzy rule, derived from the reduced-order augmented model formula (14). , It is composed of known matrices and is used to simplify the derivation of expressions.

[0064] For system (14), the following observer is designed based on (15) and (16): (17) Since y is a virtual output and cannot be directly measured, we can obtain the following from equations (12) and (17): (18) From equation (9), we can see that Therefore, the first derivative of y in equation (16) is unacceptable and must be eliminated. This can be achieved by defining a new variable. Combining equation (17), we can obtain: (19) Note 3: For the augmented matrix under the k-th fuzzy rule ( The block submatrix of equation (14) is used to simplify the derivation of the reduced-order model.

[0065] Therefore, a descending-order intermediate observer can be constructed as follows: (20) (twenty one) Where matrix Lⱼ represents the observer gain. The observation error is defined. according to achievable Based on equations (14), (16), and (17), the observer error system is: (twenty two) According to equation (22), the compact form of the observer error system can be obtained as follows: (twenty three) (twenty four) in .

[0066] Theorem 1: If there exist scalars ζ, ϵ > 0 and matrices Ω1, Ω2 > 0 satisfying the following linear matrix inequality, and the observer gain Lⱼ satisfies Āⱼ - LⱼC̄ k If (j, k = {1, ..., q}) is Hurwitz stable, then the observation error systems (23) and (24) converge asymptotically.

[0067] (25)

[0068] Proof 1: Choose the following Lyapunov function (26) According to equations (23) and (24), the first derivative of V(t) with respect to time t can be expressed as: (27) Based on hypothesis 1, there exists an undetermined scalar value. and satisfy and Therefore, according to Lemma 2, the following inequality relations can be established.

[0069] (28) (29) (30) Substituting equations (28), (29), and (30) into equation (27), we get: (31) Define a matrix that satisfies Then equation (31) is equivalent to: (32) in and ; Based on the equation, we can deduce that: (33) Define matrix Then when Available in time Based on equation (33), the following inequalities can be derived: (34) in ; Define a set as follows: (35) Assumption yes The complement, if Then we can obtain the following results: (36) Based on (33) and (35), we can conclude that: (37) If the linear matrix inequality condition (25) holds, it can be proved using the Schul complement formula. Therefore when At that time, according to Lyapunov's stability theory, systematic error It has consistent boundedness, thus ensuring the stability of the system under specific conditions.

[0070] Note 4: In traditional fault estimation methods (such as adaptive observers (AO) and unknown input observers (UIO), if the observer matching condition is not met, the matrix group (A, C, E) cannot be transformed into intermediate canonical form. This invention utilizes the characteristics of the fault distribution matrix to construct a unique intermediate variable, and builds the observer dynamic equation based on this intermediate variable. Through signal reconstruction and error optimization, reliable estimates of faults and uncertainties are finally obtained, thereby designing a dedicated observer. By establishing sufficient conditions to ensure the consistent eventual boundedness of the error system, the limitations of traditional methods on the observer matching condition are effectively overcome.

[0071] III. Distributed Formation Control Protocol (I) Core Function Positioning and Design Concept As the "decision-making and execution core" of the entire fault-tolerant formation control scheme, the distributed formation control protocol constructed in this invention aims to solve the problem of "time-varying formation with multiple leaders and followers under fault and uncertainty interference". Its design concept can be summarized as follows: based on the state, fault and uncertainty estimates output by the fault observer, and utilizing only the local communication information between UAVs (without requiring the global leader state), through the collaborative design of "error feedback adjustment + real-time fault compensation + adaptive gain optimization", it ensures that the follower group can accurately maintain the preset time-varying formation, and also achieves the convergence of the entire follower group to the convex hull formed by multiple leader states, ultimately achieving the dual control objectives of fault tolerance and formation.

[0072] (II) Protocol Construction Steps The core logic of protocol construction is "first clarify the control objectives, then design the control architecture, and finally optimize the adjustment mechanism and integrate verification," with the specific process as follows: The first step is to accurately define multi-layered formation errors and clarify the control direction. Error is the core feedback basis of the control protocol. Considering the dual requirements of "formation maintenance" and "leader tracking," error indicators need to be designed in two layers. The first is the formation consistency error among followers, whose core function is to ensure that the formation within the follower group does not diverge. It is defined as... .in, This is the time-varying expected formation offset vector, used to define the relative position reference of each follower in the formation; These are the adjacency matrix elements of the communication topology. This indicates that follower i and follower l have direct communication. There is no direct communication, making this design naturally suited for distributed communication scenarios.

[0073] Secondly, there is the tracking error between followers and leaders, which is used to guide the follower group towards the leader's goal and is defined as follows: ,in For the communication adjacency elements of follower i and leader m, logical AND... Consistent. To simplify subsequent control derivation, the two levels of error are integrated into a total formation error. The subsequent control objective can be simplified to: minimizing the total error. Asymptotically convergent to zero, at which point both "form preservation" and "convex hull convergence" can be achieved simultaneously.

[0074] The second step is to design the core architecture of the control protocol to achieve "feedback + compensation" coordination. Combining error indicators and system interference characteristics, the control protocol adopts a three-part architecture of "error feedback + fault / uncertainty compensation + adaptive gain," with each part complementing and cooperating. The error feedback component is the basic control unit, designed as... Here, the error calculated based on the observer's estimated state is used instead of the actual state error. This is mainly to avoid the control effect being affected by the actual state contaminated by faults. The error negative feedback directly pushes the follower to move closer to the target state.

[0075] The fault / uncertainty compensation term is the core unit of fault-tolerant control and is designed as follows: .in For matrix The Moore-Penrose inverse matrix, its core function is to utilize the fault estimate output by the observer. and uncertainty estimates Inverse adjustment of control input—when the actuator malfunctions and causes output attenuation, the compensation term will increase the control input accordingly to compensate for the attenuation; when there are parameter fluctuations in the system, the compensation term can also offset their interference with the formation effect in real time, thereby improving the fault tolerance of the system from the root.

[0076] Adaptive gain term It is a dynamic adjustment unit, whose core value lies in dynamically adjusting the feedback strength according to the error. Unlike fixed gain, it is time-varying. It can achieve dynamic adaptation of "the greater the error, the stronger the feedback": when the system is affected by faults / uncertainties and the error increases, the gain is updated quickly to enhance the feedback adjustment capability and improve the response speed; when the error gradually decreases and tends to stabilize, the gain also stabilizes, avoiding system oscillation and ensuring the smoothness of the formation process.

[0077] The third step is to derive the adaptive gain update law to ensure gain convergence. Gain updates must strictly adhere to stability constraints; otherwise, gain divergence may occur, compromising the stability of the entire control system. Based on Lyapunov stability theory, this is achieved by constructing a Lyapunov function containing a quadratic gain term. The gain update law is derived as follows: .

[0078] In engineering implementation, the initial value of the gain It is recommended to set it to a zero matrix or a small positive definite matrix to ensure the smoothness of the control input in the initial stage. The core logic of this update law is that the gain increases with the total error. The integral accumulation update enables real-time linkage between gain adjustment and error status, ensuring both adjustment effectiveness and avoiding sudden gain changes.

[0079] Finally, the final control protocol is formed through integration. Considering the nonlinear characteristics of the system, the weighting coefficients of the TS fuzzy model need to be incorporated. The three core units mentioned above are integrated into the final control input of follower i, ensuring that the protocol can adapt to the dynamic characteristics of nonlinear systems. The specific expression is as follows: .

[0080] (III) Detailed Derivation Process In the design of the following formation control protocol, the key control variable is the system's formation error. This error includes the formation error between follower UAVs and the error between each follower and the leader, specifically defined as follows: (38) (39) This invention addresses the fault-tolerant formation control problem by aiming to compensate for the impact of process faults and uncertainties within the system on unmanned aerial vehicles (UAVs). By integrating compensation terms into the control system to reduce the effects of these faults and uncertainties, the follower UAV can successfully perform formation tracking tasks. The fault-tolerant formation controller is designed as follows: (40) in express The Moore-Penrose inverse matrix, , . The time-varying gain of the controller to be designed, Used for compensation and its first derivative.

[0081] Define variables Based on this, combined with equations (1), (15), and (40), it can be deduced that... The dynamic equations are as follows: (41) Assumption satisfy Then equation (41) can be transformed into: (42) Expression (42) can be rewritten in the integrated form of the entire follower system as follows: (43) in Define variables We can obtain: (44) in , .make This represents the formation error of the entire system, which is also our main control variable. Based on equations (2) and (44), we can derive... The dynamic change equation is as follows: (45) From equation (20), we can see that Therefore when During convergence, It will also converge. According to Theorem 1, if the conditions of Theorem 1 are met, the observation error... It will gradually converge to zero. Therefore, when hour, It will approach zero. At this point, equation (41) is equivalent to: (46) Theorem 2: Under the conditions of Assumption 2 and below, if there exists a matrix and If the following conditions are met, then under the control protocol formula (40), the multi-UAV systems (1) and (2) can achieve inclusive formation control.

[0082] (47) (48) Proof 2: Choose the following Lyapunov function: (49) Among them, matrix and It is symmetrical. Based on (43) and (44), we can conclude: (50) set up: (51) Therefore, we can conclude that: (52) Therefore, the adaptive controller gain can be obtained. It can be represented as: (53) Therefore, when the controller gain satisfies the condition specified in equation (53), equation (50) can be rewritten as: (54) Define matrix satisfy ,in It is a diagonal matrix, whose diagonal elements are matrix elements. eigenvalues, i.e. Under these conditions, define ,in .

[0083] Therefore, equation (53) can be transformed into the following form: (55) in It is a symmetric positive definite matrix that satisfies the Lyapunov equation. From the form of equation (49), we can derive: (56) Therefore, as time t approaches infinity, the formation error... It will converge to zero. From equation (44), we can obtain: (57) Multiply both sides of equation (57) We can obtain: (58) in .

[0084] Furthermore, from equation (58), it can be derived that: (59) in, Representation matrix The elements. According to Lemma 1, The sum of all elements in each row is 1, meaning that for any row, there exists a sum of 1. This is consistent with the control objective described in Definition 1. Therefore, using the control protocol specified in Equation (40), the multi-leader system can achieve time-varying formation tracking.

[0085] Note 5: Based on the control protocol in equation (40), it can be seen that not all followers can obtain the leader's state information. Only when there is a direct path between the i-th follower and the m-th leader can the follower obtain the corresponding leader's state information.

[0086] IV. Simulation Verification The effectiveness of the proposed method is verified through numerical simulation. Consider a multi-UAV system comprising nine aircraft, including six follower aircraft and three leader aircraft, with the following communication topology: Figure 1 As shown. Assuming the number of fuzzy rules is q=2, the system matrix is ​​given as follows: ; ; .

[0087] according to Figure 1The Laplace matrix L of the topological structure shown can be determined as follows: .

[0088] Assuming the fault only occurs in drones 1-4, regarding the fault (in The dynamic construction of ) is as follows: ; ; ; ; The fuzzy membership function is selected as: .

[0089] Consider the following time-varying formation: ; Assume the uncertainty of the system is ,in ; To ensure the matrix It is Hurwitz stable, and the observer gain is set as follows: ; Figure 2 The fault estimation results for UAVs 1 through 4 are presented. Figure 2 In the text, "ad" represents drones 1-4 respectively, while... Figure 3 This shows the uncertainty estimates for UAVs 1 and 2. In this case, the system uncertainty estimates are only shown for UAVs 1 and 2. The graphical results indicate that the estimation performance meets the expected requirements.

[0090] Figure 4 The figure illustrates the variation of the formation tracking error φ(t). As shown, φ(t) converges quickly to a small neighborhood near zero, indicating that the system has effectively achieved formation tracking of the target.

[0091] Figure 5 The image shows the movement trajectories of all UAVs at different times. Red squares represent leaders, while the other colored squares represent followers. It is clear from the image that the followers continuously rotate around the leader. This demonstrates that the fault-tolerant formation control scheme proposed in this study can effectively suppress the effects of process failures and system uncertainties, ensuring that all followers successfully track multiple leaders under a specified time-varying formation.

[0092] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer, characterized in that, Includes the following steps: S1: Construct a multi-UAV system model containing followers and leaders. The followers are represented dynamically using a TS fuzzy model, including system state, output, control input, actuator failure, and parameter uncertainty. The leader model does not contain failures or uncertainties. S2: Define the time-varying formation tracking condition as the follower state equals the expected formation offset within the leader's convex hull, and derive the reduced-order dynamic model of the follower. Define the virtual output through the left inverse matrix and the null space orthogonal basis to form an augmented reduced-order system for state failure uncertainty derivatives. S3: Design a reduced-order intermediate observer, define intermediate variables to eliminate virtual output derivatives, build the dynamic structure of the observer, construct the observer equation based on the intermediate variables and measurable output, and solve the observation gain to reconstruct the uncertainty of system state faults through linear matrix inequalities; S4: Based on the reconstruction information, a distributed formation control protocol is designed, defining the formation consistency error among followers and the tracking error between followers and the leader. The total formation error is integrated, and the control input is fused to estimate the state error feedback failure uncertainty. The adaptive gain is compensated, and the gain update law is derived through the Lyapunov function. The TS fuzzy weights are fused to form the final input.

2. The fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer according to claim 1, characterized in that, In step S1, for each TS fuzzy model, the fuzzy rules are determined, and the system matrix and fuzzy weights are determined. The fuzzy weights satisfy the normalization and summability conditions, and the system matrix is ​​controllable and observable. Construct a Laplace matrix to represent the topology of a multi-UAV system, where followers are divided into information-complete UAVs and information-incomplete UAVs, and each information-incomplete UAV has a direct undirected path with an information-complete UAV. The fault signal and parameter uncertainty are differentiable and bounded signals, satisfying the condition of bounded derivative.

3. The fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer according to claim 2, characterized in that, In step S2, the output matrix of the dynamic model of the follower drone is of full row rank, and the matrix and null space orthogonal basis that satisfy the left null space condition are determined. By defining a virtual output and combining it with a reduced-order model, the initial follower is dynamically transformed into a reduced-order form. Define the augmented state variable as the derivative of the system state failure uncertainty, and form the i-th follower augmented reduced-order system.

4. The fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer according to claim 3, characterized in that, The specific process of constructing the intermediate observer in step S3 is as follows: a. Prerequisite preparation: Based on the reduced-order augmented system, the derivatives of faults and parameter uncertainties are incorporated into the augmented state based on the reduced-order dynamic model derived in the system model, forming an integrated augmented system of "state-fault-uncertainty" to ensure that the observer can estimate the three types of core information simultaneously. b. Construct intermediate variables: Eliminate unmeasurable output derivatives and transform the "unmeasurable output derivatives" in the augmented system into "derivable terms containing fault / uncertainty information" to avoid introducing unmeasurable signals into the observer; c. Construct a dynamic structure for the observer: correlate the augmented state estimate with the uncertainty auxiliary estimate to ensure the continuity of the uncertainty estimate.

5. The fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer according to claim 1, characterized in that, In step S3, the intermediate variable is defined as the product of the observer output estimate and the selected scalar. The first derivative of the intermediate variable is expressed by combining the intermediate coefficient matrix under the fuzzy rules. The observer dynamic equation is constructed to eliminate the virtual output derivative, and new variables are introduced to associate augmented state estimation and uncertainty-aided estimation. By defining the observation error, we obtain a compact form of the observer error system, where the observer gain satisfies the asymptotic convergence of the error under a combination of fuzzy rules.

6. The fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer according to claim 1, characterized in that, In step S4, the formation consistency error is defined as the product of the adjacency matrix elements and the desired formation offset vector, and the tracking error is the product of the communication adjacency elements of the follower and leader. The overall formation error is integrated, and the error feedback term is calculated based on the estimated state to provide negative feedback. The compensation term uses the pseudo-inverse of the fault distribution matrix to offset the uncertainty of the reconstructed fault, and the adaptive gain is updated with the integral of the total formation error.

7. The fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer as described in claim 1, wherein in step S4, a Lyapunov function containing a gain quadratic term is selected, and the derivative is derived based on the positive sum row consistency of the real part of the eigenvalues ​​of the Laplace matrix; Define the diagonal elements of the diagonal matrix as the eigenvalues ​​of the topological matrix, and transform it into a symmetric positive definite matrix form to ensure the convergence of the overall formation error.

8. In the fuzzy multi-UAV fault-tolerant formation control method based on a reduced-order intermediate observer as described in claim 1, the final input in step S4 is the fused TS fuzzy weight coefficient, and the system dynamic equation is transformed into a follower integrated form; When the observation error converges and the total formation error approaches zero, the follower state is equal to the expected formation offset within the leader's convex hull.

Images