A distributed fault detection system and method for a fractional order nonlinear multi-agent system

Abstract

The application discloses a kind of distributed fault detection systems and methods of fractional nonlinear multi-agent system, the system includes fractional K filter module, residual signal generation module, fractional noise suppression filter module, detection threshold calculation module and fault real-time judging module.The method first establishes the strict feedback dynamics model of fractional follower agent;Second, based on the nominal system design fractional K filter for state estimation;Then, through fractional noise suppression filter, measurement noise is suppressed, residual signal is generated and dynamic detection threshold is calculated;Finally, by comparing residual signal with dynamic detection threshold, fault real-time judging is realized.The application is aimed at fractional multi-agent system with time-varying actuator fault and bounded measurement noise, realizes fault-free scene zero false alarm and fault finite-time detection, and each agent only depends on local signal to complete detection independently, with low communication overhead and strong scalability.

Description

Technical Field

[0001] This invention relates to the field of fault detection technology for multi-agent systems, and in particular to a distributed fault detection method based on a fractional-order K-filter, applicable to fractional-order multi-agent systems with time-varying actuator faults and bounded measurement noise. Background Technology

[0002] With the development of fractional calculus theory, fractional-order systems, due to their ability to accurately describe real-world physical processes with memory and genetic properties, have found widespread application in fields such as multi-agent cooperative control, industrial process control, and intelligent robotics. In the actual operation of multi-agent systems, problems such as actuator failures, external disturbances, and measurement noise are unavoidable. These can range from reducing the accuracy of cooperative control to causing system instability and safety accidents. Therefore, achieving rapid and accurate fault detection in fractional-order multi-agent systems is a key technology for ensuring the safe and reliable operation of such systems.

[0003] Currently, scholars both domestically and internationally have conducted relevant research on fault detection in multi-agent systems, but the following problems still exist:

[0004] First, most existing multi-agent fault detection methods are designed for integer-order system architectures and do not consider the special dynamic characteristics of fractional-order systems. When directly applied to fractional-order multi-agent systems, the convergence of state estimation cannot be guaranteed, and the accuracy and reliability of fault detection are difficult to ensure.

[0005] Second, most existing fault detection solutions do not fully consider the interference of measurement noise. In actual noisy operation scenarios, they are prone to false alarms or missed alarms, and cannot simultaneously take into account detection sensitivity and anti-interference ability.

[0006] Third, most existing fault detection methods rely on global system information to achieve centralized detection, which is difficult to adapt to the distributed communication topology of multi-agent systems. Furthermore, they lack the ability to detect time-varying actuator faults within a limited time frame, failing to meet the real-time requirements of engineering scenarios. Summary of the Invention

[0007] The purpose of this invention is to provide a distributed fault detection system and method for fractional-order nonlinear multi-agent systems. It combines fractional-order K-filters with residual detection theory to achieve accurate detection of time-varying actuator faults within a finite time, while ensuring zero false alarms in fault-free scenarios and adapting to multi-agent distributed communication architectures.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] A distributed fault detection system for a fractional-order nonlinear multi-agent system includes a fractional-order K-filter module, a residual signal generation module, a fractional-order noise suppression filter module, a detection threshold calculation module, and a real-time fault determination module, wherein: The input of the fractional-order K-filter module is connected to the local control input and state measurement output of the follower agent, and the output of the fractional-order K-filter module is connected to the system state estimate. The input of the residual signal generation module is connected to the system state measurement value and the state estimate value output by the fractional-order K-filter module, and the output of the residual signal generation module is connected to the fault detection residual signal. The input of the fractional-order noise suppression filter module is connected to the nominal system state estimation error, and the output of the fractional-order noise suppression filter module is connected to the filtered nominal estimation error. The input of the detection threshold calculation module is connected to the upper bound of the nominal estimation error after filtering and the system design parameters, and the output of the detection threshold calculation module is connected to the dynamic fault detection threshold. The input of the real-time fault determination module is connected to the fault detection residual signal and the dynamic fault detection threshold, and the output of the real-time fault determination module is connected to the fault detection result signal.

[0010] A distributed fault detection method for fractional-order nonlinear multi-agent systems includes the following steps: Step A: Establish a rigorous feedback dynamic model for the i-th fractional-order follower agent. This model includes system state variables, control inputs, and system outputs, and contains known smooth nonlinear functions, unknown constant parameter vectors, time-varying actuator fault functions, and bounded measurement noise. Step B involves designing a fractional-order K-filter based on the nominal system to achieve high-precision estimation of the system state variables. This includes constructing a fault-free, disturbance-free, and noise-free nominal system model, determining the parameter matrix and update rules of the fractional-order K-filter for the nominal system, deriving the dynamic equation of the state estimation error, and analyzing the stability of the error system. Step C: Define a fractional-order stable noise suppression filter to filter the estimation error and suppress measurement noise interference; Step D: Generate residual signals based on the state estimation results, and define the residual signals as the difference between the actual output and the estimated output of the system. Step E: Determine the dynamic fault detection threshold based on the upper bound of the nominal estimation error after filtering and the system design parameters; Step F: Compare the residual signal with the dynamic fault detection threshold. When the residual signal exceeds the threshold, a fault is determined to have occurred.

[0011] Furthermore, in step A, the first The rigorous feedback dynamics model formula for a fractional-order follower agent is: (1) in, It is a Caputo-type fractional differential operator. ; It is a time variable, and ; , for An identity matrix of order 1; ; ; , , They are the first The state variables, control inputs, and outputs of each subsystem; , It is a known smooth nonlinear function; A vector of unknown constants; , For time-varying fault functions; The exact time of the unknown failure is unknown. Let this be a fault step function, used to indicate whether an actuator fault has occurred, satisfying: when hour, ;when hour, ; This indicates bounded measurement noise, meaning there exists a normal constant. satisfy And its fractional derivative It also has boundaries.

[0012] Furthermore, in step B, the nominal system model formula is: (2) in, For the first The state vector of a follower agent in the nominal system. For the corresponding nominal system output; subscript Indicates the nominal system; The design formula for a fractional-order K-filter is: (3) in, For input channel vectors / selection vectors, These are the internal state variables of the fractional-order K-filter; It is a Hurwitz matrix and for any given symmetric positive definite matrix There exists a symmetric positive definite matrix. Satisfying Lyapunov inequalities ; For design parameters; The system state variable estimation formula based on the fractional-order K-filter is: (4).

[0013] The nominal system state estimate, This is the estimated output value of the nominal system.

[0014] Furthermore, in step C, a fractional-order stable noise suppression filter is defined: (5) filter The condition that the input is bounded and the output is stable is satisfied. is the complex frequency variable in the Laplace transform domain; The filter order parameter; For fractional order, ;for , and The filter coefficients are non-negative constants; their impulse response It satisfies the Mittag-Leffler type monotonically decaying characteristic; it has a positive real constant. , This makes it possible for all ,have .

[0015] Furthermore, in step D, the formula for generating the residual signal is: (6) in, Let i be the residual signal of the i-th agent; The output error is the difference between the actual output and the estimated output; this is addressed by setting a corresponding detection threshold. ,when When the fault occurs, it is determined that a fault has occurred.

[0016] Furthermore, in step E, the residual signal satisfies: (7) in, The filtered noise, and satisfies , Noise measurement after filtering The upper bound; For the output matrix transpose; The filtered state estimation error, and , ,in The state estimation error after filtering The upper bound function, It is a known upper bound of the system's initial state norm; The fault detection threshold for a fractional-order multi-agent system is designed as follows: (8) in, It is generated by the following equation (9): (9) in, It is a bounded decay term; It is the Gamma function; Impulse response exist The value at time; This represents the positive parameter in the derivation of the upper bound of the error. It is a positive number.

[0017] Furthermore, in step F, the fault determination conditions are as follows: Before the failure occurred For all follower intelligent agents Hengyou This means that the system will not generate false alarms when there are no faults. If the fault is It occurs at any moment, and exists at a finite number of moments. This makes the following inequality hold: (10) The fault can be found in It is detected at all times, that is ; in, For a finite moment when the fault is detected; For integration variables; It is a two-parameter Mittag-Leffler function; Representation matrix The smallest eigenvalue; Representation matrix The largest eigenvalue; A symmetric positive definite matrix that satisfies Lyapunov's inequality; and They are respectively Time The dynamic residual signal and detection threshold of each agent. Beneficial effects

[0018] Compared with the prior art, the present invention has the following beneficial effects: First, it adapts to the dynamic characteristics of fractional-order systems. State estimation is achieved based on a fractional-order K-filter, ensuring that the estimation error meets the Mittag-Leffler convergence requirement. This solves the problem that traditional integer-order fault detection methods are not applicable to fractional-order multi-agent systems, thus providing greater assurance in detection accuracy and stability.

[0019] Second, it balances anti-interference capability with real-time detection. By suppressing measurement noise interference through fractional-order filtering and combining it with theoretically derived dynamic detection thresholds, it achieves zero false alarms in fault-free scenarios, and can trigger detection within a limited time after a fault occurs.

[0020] Third, a purely distributed detection architecture. Each agent can independently complete fault detection based solely on its own locally measurable signals, without the need for a global central node or interaction with neighboring agents to exchange detection information. This results in low communication overhead and stronger system scalability and fault tolerance.

[0021] Fourth, it has a wide range of applicable scenarios. It can be adapted to complex working conditions with unknown parameters, time-varying actuator failures, and bounded measurement noise, without needing to know the time and magnitude of the failure in advance, making it highly practical for engineering applications. Attached Figure Description

[0022] Figure 1 This is a schematic diagram of a fractional-order multi-agent fault detection system;

[0023] Figure 2 This is a diagram showing the state of i agents (Case 1).

[0024] Figure 3 This is a diagram showing the fault detection and judgment effect without a filter (Case 1).

[0025] Figure 4 This is a diagram showing the fault detection and judgment effect under the condition of having a filter (case 1).

[0026] Figure 5 This is a diagram showing the state of i agents (case 2).

[0027] Figure 6 This is a diagram showing the fault detection and judgment effect under the condition of no filter (case 2).

[0028] Figure 7This is a diagram showing the effect of fault detection and judgment under the condition of having a filter (case 2). Detailed Implementation

[0029] The invention will now be further explained with reference to the accompanying drawings.

[0030] Consider a fractional-order multi-agent system consisting of one leader and N followers. The i-th fractional-order follower agent adopts a dynamic model with strict feedback. The model includes system state variables, control inputs, and system outputs. It contains a known smooth nonlinear function, an unknown constant parameter vector, a time-varying actuator fault function, and bounded measurement noise. The time of fault occurrence is unknown. Before the fault occurs, the fault function is 0, and after the fault occurs, it exhibits time-varying characteristics.

[0031] This method is based on the following basic assumptions: the communication topology contains a directed spanning tree rooted at the virtual leader; the unknown parameters of the system are bounded; and the time-varying fault function satisfies the conditions of boundedness and decomposability.

[0032] A distributed fault detection system for fractional-order multi-agent systems includes the following modules: a fractional-order K-filter module, a residual signal generation module, a fractional-order noise suppression filter module, a detection threshold calculation module, and a real-time fault determination module.

[0033] The input of the fractional-order K-filter module is connected to the local control input and state measurement value of the follower agent, and the output of the fractional-order K-filter module is connected to the system state estimate value; the input of the residual signal generation module is connected to the system state measurement value and state estimate value, and the output of the residual signal generation module is connected to the fault detection residual signal; the input of the fractional-order noise suppression filter module is connected to the nominal system state estimation error, and the output of the fractional-order noise suppression filter module is connected to the filtered nominal estimation error; the input of the detection threshold calculation module is connected to the upper bound of the filtered nominal estimation error and the system design parameters, and the output of the detection threshold calculation module is connected to the dynamic fault detection threshold; the input of the real-time fault determination module is connected to the fault detection residual signal and the dynamic fault detection threshold, and the output of the real-time fault determination module is connected to the fault detection result signal.

[0034] The present invention discloses a distributed fault detection method for a fractional-order multi-agent system, comprising the following steps:

[0035] Step A, System Description

[0036] No. The rigorous feedback dynamics model formula for a fractional-order follower agent is:

[0037] (1)

[0038] in, It is a Caputo-type fractional differential operator. ;

[0039] It is a time variable, and ;

[0040] , for An identity matrix of order 1; ; ;

[0041] , , They are the first The state variables, control inputs, and outputs of each subsystem;

[0042] , It is a known smooth nonlinear function;

[0043] A vector of unknown constants;

[0044] , For time-varying fault functions;

[0045] The exact time of the unknown failure is unknown. Let this be a fault step function, used to indicate whether an actuator fault has occurred, satisfying: when hour, ;when hour, ;

[0046] This indicates bounded measurement noise, meaning there exists a normal constant. satisfy And its fractional derivative It also has boundaries.

[0047] Step B, Observer Design

[0048] Based on the nominal system, a fractional-order K-filter is designed to achieve high-precision estimation of system state variables. This includes: constructing a fault-free, disturbance-free, and noise-free nominal system model; designing a fractional-order K-filter for the nominal system and determining the filter's parameter matrix and update rules; deriving the dynamic equation of the state estimation error; selecting an appropriate Lyapunov function to analyze the stability of the error system; and verifying that the estimation error converges according to the Mittag-Leffler law.

[0049] The formula for the nominal system model is:

[0050] (2)

[0051] in, For the first The state vector of a follower agent in the nominal system. For the corresponding nominal system output; subscript Indicates the nominal system;

[0052] The design formula for a fractional-order K-filter is:

[0053] (3)

[0054] in, For input channel vectors / selection vectors, These are the internal state variables of the fractional-order K-filter;

[0055] It is a Hurwitz matrix and for any given symmetric positive definite matrix There exists a symmetric positive definite matrix. Satisfying Lyapunov inequalities ;

[0056] These are design parameters.

[0057] The system state variable estimation formula based on the fractional-order K-filter is:

[0058] (4)

[0059] in, The nominal system state estimate, This is the estimated output value of the nominal system.

[0060] Step C, fractional-order noise suppression filter

[0061] Define a fractional-order stable noise suppression filter:

[0062] (5)

[0063] filter The condition that the input is bounded and the output is stable is satisfied. is the complex frequency variable in the Laplace transform domain; The filter order parameter; For fractional order, ;for , and The filter coefficients are non-negative constants; their impulse response It satisfies the Mittag-Leffler type monotonically decaying characteristic; it has a positive real constant. , This makes it possible for all ,have .

[0064] Step D, define the residual signal

[0065] The residual signal is generated based on the state estimation result and serves as the core criterion for fault detection. This includes defining the residual signal as the difference between the actual output and the estimated output of the system, and preprocessing the residual signal with a fractional-order filter to enhance the residual signal's sensitivity to faults and its robustness to interference.

[0066] The formula for generating the residual signal is:

[0067] (6)

[0068] in, Let i be the residual signal of the i-th agent; The output error is the difference between the actual output and the estimated output; this is addressed by setting a corresponding detection threshold. ,when When the fault occurs, it is determined that a fault has occurred.

[0069] Step E: Determine the detection threshold

[0070] The residual signal satisfies:

[0071] (7)

[0072] in, The filtered noise, and satisfies , Noise measurement after filtering The upper bound; For the output matrix transpose; The filtered state estimation error, and , ,in The state estimation error after filtering The upper bound function, It is a known upper bound of the system's initial state norm;

[0073] The fault detection threshold for a fractional-order multi-agent system is designed as follows:

[0074] (8)

[0075] in, It is a bounded decay term; It is the Gamma function; Impulse response exist The value at time;

[0076] This represents the positive parameter in the derivation of the upper bound of the error. It is a positive number;

[0077] The filtered noise, and satisfies ,and , ;

[0078] It is generated by the following formula:

[0079] (9)

[0080] Step F, Real-time Fault Determination

[0081] In summary, for a fractional-order multi-agent system that satisfies the assumptions, based on the previously designed fractional-order K-filter, fractional-order noise suppression filter, and residual signal... With detection threshold The following conclusions can be drawn:

[0082] 1. Before the fault occurs For all follower intelligent agents Hengyou This means that the system will not generate false alarms when there are no faults.

[0083] 2. If the fault is in It occurs at any moment, and exists at a finite number of moments. This makes the following inequality hold:

[0084] (10)

[0085] The fault can be found in It is detected at all times, that is ;

[0086] in, For a finite moment when the fault is detected; For integration variables; It is a two-parameter Mittag-Leffler function; Representation matrix The smallest eigenvalue; Representation matrix The largest eigenvalue; A symmetric positive definite matrix that satisfies Lyapunov's inequality; and They are respectively Time The dynamic residual signal and detection threshold of each agent.

[0087] The present invention will be further described below with reference to the embodiments.

[0088] Example

[0089] The overall architecture of the fractional-order multi-agent system distributed fault detection system designed in this embodiment is as follows: Figure 1 As shown, the system consists of a fractional-order K-filter module, a residual signal generation module, a fractional-order noise suppression filter module, a detection threshold calculation module, and a real-time fault determination module, which are cascaded in sequence. Each follower agent is independently configured with a complete detection module, which can complete fault detection by relying only on local control input and state measurement values. It does not require a global central node or information interaction with neighboring agents, and has a purely distributed characteristic.

[0090] This embodiment selects a fractional-order multi-agent system with one leader and four followers as the verification object. The communication topology contains a directed spanning tree rooted at the virtual leader, satisfying the basic system assumptions. A Caputo-type fractional-order differential order is set. The total simulation time is 60 seconds, and the actuator failure occurs at the specified time. , respectively in , (Case 1) and , (Scenario 2) Verify the performance of the method under two working conditions.

[0091] The i-th fractional-order follower agent adopts a second-order strict feedback dynamic model, specifically expressed as:

[0092] (11)

[0093] Fault function .

[0094] Design a fractional-order K-filter for a fault-free, disturbance-free, and noise-free nominal system, specifically in the form of:

[0095] (12)

[0096] (13)

[0097] (14)

[0098] in , , , , , , , ,

[0099] The fractional-order noise suppression filter is designed as follows:

[0100] (15)

[0101] Residual signal without filtering:

[0102] (16)

[0103] Residual signal under filtered conditions:

[0104] (17)

[0105] The dynamic threshold is:

[0106] (18)

[0107] In the formula .

[0108] The design goal of this embodiment is to achieve zero false alarms in a fault-free scenario and finite-time detection after a fault occurs in a fractional-order multi-agent system with time-varying actuator faults and bounded measurement noise, while adopting a purely distributed architecture to reduce communication overhead. The total simulation time is set to 60 seconds, and the fault is injected into the system at time 20 seconds.

[0109] Scenario 1: Low noise condition (noise amplitude 0.05)

[0110] Figure 2 The figures show the state response curves of the four follower agents under low-noise conditions. During the fault-free period of 0-20s, the system state gradually converges from the initial value and tends to stabilize, with a smooth and regular trajectory, only superimposed with weak high-frequency noise fluctuations, and the operation is stable. After the actuator fault is injected at 20s, the state deviates from the original steady-state range under the action of the fault, but the system still remains stable and does not diverge. The steady-state deviation characteristics caused by the fault are clear and intuitive.

[0111] Figure 3 The diagram shows the fault detection performance in Case 1 without a fractional-order noise suppression filter. Due to the presence of measurement noise, the residual signal exhibits significant high-frequency fluctuations. During the fault-free period of 0-20 seconds, the residual amplitude repeatedly approaches the detection threshold, posing a high risk of false alarms. After the fault occurs at 20 seconds, the residual signal rises gradually and takes a considerable amount of time to break through the threshold, resulting in a significant fault detection delay and insufficient real-time performance.

[0112] Figure 4The image shows the fault detection effect after adding a fractional-order noise suppression filter in Case 1. After fractional-order filtering, the high-frequency noise components in the residual signal are significantly suppressed, and the overall waveform is smooth and stable. During the 0-20s fault-free operation phase, the residual amplitude remains below the dynamic detection threshold, and no false alarms occur, verifying the zero-false-alarm characteristic in a fault-free scenario. After the fault occurs at 20s, the residual signal rises rapidly, breaking through the detection threshold in a short time and successfully triggering the fault detection signal. This achieves rapid fault detection within a limited time, and both the detection sensitivity and real-time performance meet the design requirements.

[0113] Scenario 2: High noise condition (noise amplitude 2)

[0114] Figure 5 The figures show the state response curves of four follower agents under high noise conditions. The system itself remains stable, and the state eventually tends to the steady-state range. However, due to strong measurement noise, it is accompanied by violent high-frequency oscillations near the steady-state value throughout the process. During the fault-free period of 0-20s, the normal steady-state trend is largely masked by noise. After the fault injection at 20s, the steady-state deviation caused by the fault is submerged by strong noise, and it is impossible to distinguish between normal oscillations and fault anomalies based solely on the original trajectory.

[0115] Figure 6 This is a diagram showing the fault detection and judgment effect in Case 2 without a fractional-order noise suppression filter. It can be observed that the residual signal... Severely contaminated by high-intensity measurement noise, the residual amplitude frequently exceeds the detection threshold during the 0-20s fault-free period, resulting in a large number of false alarms and making it impossible to achieve stable and reliable fault detection. After the fault occurs at 20s, the residual increment caused by the fault is submerged by noise, making it impossible to effectively distinguish between normal noise fluctuations and fault anomalies. This results in serious problems of missed alarms and false alarms, and extremely poor detection reliability.

[0116] Figure 7 The image shows the fault detection performance after adding a fractional-order noise suppression filter in case 2. Through the filtering effect of the fractional-order noise suppression filter, high-frequency interference in the residual is effectively removed, and signal stability is significantly improved. During the 0-20s fault-free period, the residual amplitude remains below the detection threshold, maintaining the zero false alarm characteristic and demonstrating significant anti-interference capability. After the fault occurs at 20s, the residual steadily increases with the development of the fault and exceeds the detection threshold, accurately detecting the actuator fault within a limited time. Comparing the detection results with those without filtering shows that the proposed method maintains reliable detection performance even in strong noise environments, effectively balancing fault detection sensitivity and anti-interference capability.

[0117] Simulation results show that the distributed fault detection method based on fractional-order K-filter proposed in this invention can adapt to the special dynamic characteristics of fractional-order systems and ensure the convergence of Mittag-Leffler state estimation. Through fractional-order noise suppression filtering and dynamic threshold design, it can achieve fault-free, zero-false-alarm, and finite-time fault detection under different noise intensities. At the same time, it adopts a purely localized distributed architecture, where each agent can independently complete the detection based on its own measurement signal, resulting in low communication overhead, strong system scalability and fault tolerance, and adaptability to complex engineering conditions with unknown parameters, time-varying actuator faults, and bounded measurement noise.

[0118] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A distributed fault detection system for a fractional-order nonlinear multi-agent system, characterized in that, It includes a fractional-order K-filter module, a residual signal generation module, a fractional-order noise suppression filter module, a detection threshold calculation module, and a real-time fault determination module, wherein: The input of the fractional-order K-filter module is connected to the local control input and state measurement output of the follower agent, and the output of the fractional-order K-filter module is connected to the system state estimate. The input of the residual signal generation module is connected to the system state measurement value and the state estimate value output by the fractional-order K-filter module, and the output of the residual signal generation module is connected to the fault detection residual signal. The input of the fractional-order noise suppression filter module is connected to the nominal system state estimation error, and the output of the fractional-order noise suppression filter module is connected to the filtered nominal estimation error. The input of the detection threshold calculation module is connected to the upper bound of the nominal estimation error after filtering and the system design parameters, and the output of the detection threshold calculation module is connected to the dynamic fault detection threshold. The input of the real-time fault determination module is connected to the fault detection residual signal and the dynamic fault detection threshold, and the output of the real-time fault determination module is connected to the fault detection result signal.

2. A distributed fault detection method for a fractional-order nonlinear multi-agent system, characterized in that, Includes the following steps: Step A: Establish a rigorous feedback dynamic model for the i-th fractional-order follower agent. This model includes system state variables, control inputs, and system outputs, and contains known smooth nonlinear functions, unknown constant parameter vectors, time-varying actuator fault functions, and bounded measurement noise. Step B involves designing a fractional-order K-filter based on the nominal system to achieve high-precision estimation of the system state variables. This includes constructing a fault-free, disturbance-free, and noise-free nominal system model, determining the parameter matrix and update rules of the fractional-order K-filter for the nominal system, deriving the dynamic equation of the state estimation error, and analyzing the stability of the error system. Step C: Define a fractional-order stable noise suppression filter to filter the estimation error and suppress measurement noise interference; Step D: Generate residual signals based on the state estimation results, and define the residual signals as the difference between the actual output and the estimated output of the system. Step E: Determine the dynamic fault detection threshold based on the upper bound of the nominal estimation error after filtering and the system design parameters; Step F: Compare the residual signal with the dynamic fault detection threshold. When the residual signal exceeds the threshold, a fault is determined to have occurred.

3. The method according to claim 2, characterized in that, In step A, the first The rigorous feedback dynamics model formula for a fractional-order follower agent is: （1） in, It is a Caputo-type fractional differential operator. ; It is a time variable, and ; , for An identity matrix of order 1; ; ; , , They are the first The state variables, control inputs, and outputs of each subsystem; , It is a known smooth nonlinear function; A vector of unknown constants; , For time-varying fault functions; The exact time of the unknown failure is unknown. Let this be a fault step function, used to indicate whether an actuator fault has occurred, satisfying: when hour, ;when hour, ; This indicates bounded measurement noise, meaning there exists a normal constant. satisfy And its fractional derivative It also has boundaries.

4. The method according to claim 3, characterized in that, In step B, the nominal system model formula is: （2） in, For the first The state vector of a follower agent in the nominal system. For the corresponding nominal system output; subscript Indicates the nominal system; The design formula for a fractional-order K-filter is: （3） in, For input channel vectors / selection vectors, These are the internal state variables of a fractional-order K-filter; It is a Hurwitz matrix and for any given symmetric positive definite matrix There exists a symmetric positive definite matrix. Satisfying Lyapunov inequalities ; For design parameters; The system state variable estimation formula based on the fractional-order K-filter is: （4） in, The nominal system state estimate, This is the estimated output value of the nominal system.

5. The method according to claim 4, characterized in that, In step C, a fractional-order stable noise suppression filter is defined: （5） filter The condition that the input is bounded and the output is stable is satisfied. For the complex frequency variable in the Laplace transform domain; The filter order parameter; For fractional order, ;for , and The filter coefficients are non-negative constants; their impulse response It satisfies the Mittag-Leffler type monotonically decaying characteristic; it has a positive real constant. , This makes it possible for all ,have .

6. The method according to claim 5, characterized in that, In step D, the formula for generating the residual signal is: （6） in, Let i be the residual signal of the i-th agent; The output error is the difference between the actual output and the estimated output; this is addressed by setting a corresponding detection threshold. ,when When the fault occurs, it is determined that a fault has occurred.

7. The method according to claim 6, characterized in that, In step E, the residual signal satisfies: （7） in, The filtered noise, and satisfies , Noise measurement after filtering The upper bound; For the output matrix Transpose of; The filtered state estimation error, and , ,in The state estimation error after filtering The upper bound function, It is a known upper bound of the system's initial state norm; The fault detection threshold for a fractional-order multi-agent system is designed as follows: （8） in It is generated by the following equation (9): （9） in It is a bounded decay term; It is the Gamma function; Impulse response exist The value at time; This represents the positive parameter in the derivation of the upper bound of the error. It is a positive number.

8. The method according to claim 7, characterized in that, In step F, the fault determination conditions are as follows: Before the failure occurred For all follower intelligent agents Hengyou This means that the system will not generate false alarms when there are no faults. If the fault is It occurs at any moment, and exists at a finite number of moments. This makes the following inequality hold: （10） The fault can be found in It is detected at all times, that is ; in, For a finite moment when the fault is detected; For integration variables; It is a two-parameter Mittag-Leffler function; Representation matrix The smallest eigenvalue; Representation matrix The largest eigenvalue; A symmetric positive definite matrix that satisfies Lyapunov's inequality; and They are respectively Time The dynamic residual signal and detection threshold of each agent.

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