Fault-tolerant control method for recovering preset performance in disturbed multi-agent systems

Abstract

This invention discloses a fault-tolerant control method for recoverable preset performance of a disturbed multi-agent system. By designing a dynamic recoverable performance boundary function based on the local coordination error affected by the attack, the attacked error is mapped to an unconstrained variable, ensuring that the error is forcibly captured within the nominal boundary within a preset time. A model parameter estimator and a lumped attack disturbance observer are constructed to compensate for FDI attack residuals and parameter uncertainties online. A smooth time-axis interval saturation operator with dynamic safety margin is constructed to smooth physical intermediate states such as velocities and accelerations at each order through a command filter. By introducing input constraints from physical actuators, a fault-tolerant control law for recoverable preset performance is obtained, which can be mathematically and smoothly constrained within a time-varying safety corridor. This solves the problem that existing methods cannot achieve initial performance recovery and cooperative control of multi-agent systems under conditions of additive FDI attacks at the communication end and multiple physical constraints.

Description

Technical Field

[0001] This invention relates to the fields of multi-agent cooperative control and network security technology, and in particular to a fault-tolerant control method for recoverable preset performance of a disturbed multi-agent system. Background Technology

[0002] Distributed cooperative control has become a core topic in networked control, widely applied in fields such as robot swarms, autonomous vehicles, and cyber-physical infrastructure. In these scenarios, agents need to achieve consistency through communication networks while simultaneously addressing nonlinear uncertainties, limited execution capabilities, and security requirements. Therefore, high-performance leader-follower consistency control for constrained nonlinear multi-agent systems has significant practical implications and technical challenges. Preset performance control (PPC) directly encodes transient and steady-state requirements through preset error boundaries, making it an effective control mechanism. However, most PPC results implicitly assume that the tracking error is initially within the preset performance boundary (PPB); once the initial conditions exceed the preset region, the associated error transformation may become ambiguous. Initial constraint violations are very common in practice. For leader-follower multi-agent systems, due to initialization mismatches, initial deviations caused by communication topology, and network attacks, the cooperative error is very likely to be outside the nominal preset boundary at the initial moment. Another challenge stems from communication security. Multi-agent systems rely on information exchange between neighbors, making the communication layer highly vulnerable to network attacks, with spoofing / FDI attacks being particularly severe, directly disrupting transmission measurements used in controller design. While existing resilient control research considers spoofing / FDI attacks, most schemes employ multiplicative or parameterized attack models, or require knowledge of the first step of the attack, and often focus on stable performance while failing to simultaneously address the issues of recoverable preset performance and time-varying safety constraints. Furthermore, maintaining the distributed implementation of the control scheme and satisfying strict state and input constraints when the original neighbor states are unavailable and only attacked information is available further complicates the problem. Therefore, how to achieve initial performance recovery and collaborative control of multi-agent systems under conditions of additive FDI attacks at the communication end and multiple physical constraints is a technical challenge that urgently needs to be solved in this field. Summary of the Invention

[0003] This invention provides a fault-tolerant control method for recoverable preset performance in a disturbed multi-agent system to overcome the above-mentioned technical problems.

[0004] To achieve the above objectives, the technical solution of the present invention is as follows: A fault-tolerant control method for recoverable preset performance in a disturbed multi-agent system, specifically including the following steps: S1: Establish a rigorous feedback nonlinear dynamic model for a multi-agent system that includes unknown model parameter terms, and simultaneously construct an FDI attack model for additive fake data injection at the communication end. S2: Based on the rigorous feedback nonlinear dynamics model and the FDI attack model, define the local coordination error affected by the attack; construct the dynamic recoverable performance boundary function of the agent, and define the boundary center function and height function based on the dynamic recoverable performance boundary function; based on the local coordination error, obtain the normalized error according to the boundary center function and height function; and convert the normalized error into an unconstrained transformation error. S3: Construct a model parameter estimator and a lumped attack perturbation observer based on the unconstrained transformation error; the model parameter estimator is used to estimate unknown model parameter terms and obtain an optimized rigorous feedback nonlinear dynamic model; the lumped attack perturbation observer is used to estimate the lumped attack perturbation caused by FDI attacks; S4: Based on the backstepping method and the optimized strict feedback nonlinear dynamic model, an unlimited original virtual control law is constructed by combining a lumped attack perturbation observer; for the internal state constraints of the multi-agent system, a state tightening interval of the multi-agent system is constructed by introducing a safety margin function; based on the state tightening interval, a smooth time axis interval saturation operator is constructed to constrain the unlimited original virtual control law, so as to obtain a constrained and protected virtual control law; S5: Based on a preset low-pass dynamic surface filter, the constrained virtual control law is filtered to obtain a filtered signal; the sliding mode error is obtained based on the filtered signal, and the original target thrust of the agent is obtained based on the sliding mode error; the original target thrust is subjected to smooth input saturation control constraints based on the input constraints of the physical actuator to obtain a recoverable preset performance fault-tolerant control law, thereby realizing the recoverable preset performance fault-tolerant control of the multi-agent system under FDI attack.

[0005] Furthermore, the rigorous feedback nonlinear dynamic model of S1, which includes unknown model parameter terms, is as follows:

[0006] In the formula: Indicates the first The first-order physical state variables of a follower; Indicates the first An unknown constant parameter vector in a multi-agent subsystem; Indicates the first The first follower Order of physical state variables; Indicates control input; Indicates the measurement output; This represents a known nonlinear regression function; Indicates the first An unknown constant parameter vector in a multi-agent subsystem; express The first derivative; Indicates the first The first follower Order of physical state variables; Indicates transpose; express The first derivative; Indicates the first The first follower Order of physical state variables; The FDI attack model is as follows:

[0007]

[0008] In the formula: express Time of the first The follower received from the first The first follower Attack status signal; express Time of the first The first follower received the first [item / item] from the navigator. Attack status signal; express Time of the first The first follower The true state signal of the first order; express The first time the navigator The true state signal of the first order; and express Attack signals injected at all times and .

[0009] Furthermore, step S2 specifically includes the following steps: S21: Based on the rigorous feedback nonlinear dynamics model and the FDI attack model, define the local coordination error affected by the attack. for:

[0010] In the formula: In the topology of a communication network, the first... The first follower and the first Decision variables where followers have adjacency relationships; Indicates the navigator and the first The communication connection weight of each follower; Indicates the relationship with the first Each follower has a set of neighboring nodes in the communication link; Indicates the first The follower received from the first The first follower Attack status signal; Indicates the first The first follower received the first [item / item] from the navigator. Attack status signal; S22: The boundary function for constructing the dynamic recoverable performance of the agent is:

[0011]

[0012]

[0013] In the formula: , Describes the boundary function of dynamic recoverable performance. , Represent the nominal upper and lower boundary functions in the form of exponential decay; Represents the local recovery margin function; Represents the initial local recovery margin constant; Indicates the preset maximum recovery time; Indicates a smooth decay power and ; S23: Define the boundary center function based on the dynamic recoverable performance boundary function. With height function for:

[0014] Based on the local coordination error, the normalized error obtained from the boundary center function and the height function is:

[0015] S24: Convert the normalization error into an unconstrained transformation error. for: .

[0016] Furthermore, the formula for constructing the model parameter estimator described in S3 is as follows:

[0017] In the formula: The positive definite diagonal gain matrix representing the adaptive law; This represents a nonlinear regression compensation vector obtained based on the local state and information about the attacked neighbor. This indicates the correction factor for the leakage term; Indicates the first The first follower Estimates of the order unknown constant parameter vector; express The first derivative; The formula for constructing the lumped attack perturbation observer is as follows:

[0018] In the formula: express The first derivative; This represents an adaptive estimate of the lumped attack disturbance caused by an FDI attack; This represents the adaptive positive definite gain parameter.

[0019] Furthermore, step S4 specifically includes the following steps: S41: Based on the backstepping method and the optimized rigorous feedback nonlinear dynamic model, combined with the lumped attack disturbance observer, the unlimited original virtual control law is constructed as follows:

[0020] In the formula: Indicates the first The sum of the in-degrees of the network topology of each follower; The smooth scaling buffer parameter represents the hyperbolic tangent function; The first step, derived by the backstepping method, represents the... The first-order unlimited original virtual control guidance command of a follower; The state feedback gain parameter represents the first-order unlimited original virtual control guidance command. Indicates the feedforward term; express The first derivative; express The abbreviated form; express The first derivative; express The abbreviated form; S42: Internal state constraints for multi-agent systems By introducing a safety margin function The state tightening interval for constructing a multi-agent system is:

[0021] In the formula: The multi-agent system represents the first Internal physical state variables of the order; Indicates the first Time-varying absolute physical lower boundary of the state constraint; Indicates the first Time-varying absolute physical boundary of order-state constraints; Indicates the first The dynamic safety buffer margin function preset for each state; This represents the lower bound of the virtual control command for the state tightening interval after the introduction of a safety margin. This represents the upper bound of the virtual control command in the state tightening interval after the introduction of a safety margin; S43: The smoothing time axis interval saturation operator corresponding to the original virtual control law used to constrain the unlimited amplitude, constructed based on the state tightening interval, is as follows:

[0022] In the formula: This indicates a saturation operator for smoothing the time axis interval; The smoothness steepness parameter represents the saturation operator; S44: Construct the corresponding local coordination error based on the smoothing time axis interval saturation operator. Constrained and protected virtual control laws for: .

[0023] Furthermore, S5 specifically includes the following steps: S51: Based on a preset first-order low-pass dynamic surface filter, the constrained virtual control law is filtered to obtain the filtered signal. The sliding mode errors of each order are obtained from the filtered signal as follows: , , ,......, ; Get the Step sliding mode error derivative for

[0024] In the formula: Indicates the corresponding number Step sliding mode error The virtual control law that is subject to constraints and protection Corresponding filtered signal The derivative; S52: Based on Stability theory, based on the first Step sliding mode error derivative Get the The first follower Unlimited raw virtual control boot instructions for:

[0025] In the formula: Indicates the first k State feedback gain parameters of the order; This represents the compensation signal used to eliminate the effects of filter delay. S53: Based on the first Unlimited raw virtual control boot instructions Combined with steps S42 to S44, obtain the corresponding first... Step sliding mode error Constrained and protected virtual control laws Obtain the corresponding virtual control law Corresponding filtered signal derivative To obtain the first The first follower corresponds to the agent's... k Original target thrust for:

[0026] In the formula: Indicates the first k The height function of order; Indicates the first State feedback gain parameters of the order; Indicates the first Step sliding mode error; Indicates the first Unconstrained transformation error of order; Indicates the first The normalized error of the order; S53: Based on the input constraints of the physical actuator, apply smooth input saturation control constraints to the original target thrust to obtain a recoverable preset performance fault-tolerant control law. for:

[0027] In the formula: This represents the maximum value of the physical actuator input constraint; Based on the recoverable preset performance fault-tolerant control law This enables recoverable, preset performance-tolerant fault control for multi-agent systems subjected to FDI attacks.

[0028] Beneficial effects: This invention provides a fault-tolerant control method for recoverable preset performance in a disturbed multi-agent system. First, by designing a dynamic recoverable performance boundary function for the agents based on the local coordination error affected by the attack, and combining it with asymmetric... The transformation maps the attacked error to an unconstrained variable, ensuring that the error is forcibly captured within the nominal boundary within a preset time. Secondly, a distributed model parameter estimator and a lumped attack disturbance observer are constructed to compensate for FDI attack residuals and parameter uncertainties online. At the same time, by constructing a smooth time axis interval saturation operator with dynamic safety margin, the traditional solution of extremely complex control obstacle functions or model predictive control online optimization schemes is replaced. This allows physical intermediate states such as velocities and accelerations to be smoothed by the instruction filter, while the input constraints of the physical actuator are introduced to obtain a recoverable preset performance fault-tolerant control law. This law can be mathematically strictly and smoothly constrained within the time-varying safety corridor and seamlessly compatible with the final hardware output limit of the actuator. This achieves high-precision, robust, safe, collaborative fault-tolerant tracking control of multi-agent systems under multiple constraints. Attached Figure Description

[0029] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0030] Figure 1 This is a flowchart of the fault-tolerant control method for recoverable preset performance in a disturbed multi-agent system according to the present invention; Figure 2 This is a schematic diagram of the communication topology of a multi-satellite formation system with FDI attack capabilities in this embodiment; Figure 3 Different preset capture times in this embodiment Comparison of entry capture characteristics of system attack coordination error under the following conditions; Figure 4 This is a diagram showing the error evolution process of follower 3 under the recoverable preset performance boundary and the nominal boundary in this embodiment; Figure 5 This is a diagram showing the evolution trajectory of the formation system in three-dimensional space in this embodiment; Figure 6 This is a graph showing the variation of the tracking error norm in the first state of the formation system in this embodiment; Figure 7 This is a diagram showing the first state evolution trajectory of the follower under time-varying position constraints in this embodiment; Figure 8This is a diagram showing the second state evolution trajectory of the follower under time-varying velocity constraints and safety margin allocation in this embodiment. Figure 9 This is a graph showing the control input command output curve of the follower under actuator saturation constraints in this embodiment; Figure 10 This is a diagram showing the norm distribution of the FDI attack signal acting on communication channels 0-1 and 1-2 in this embodiment; Figure 11 This is a comparison chart of the tracking performance of the method described in this embodiment and the traditional bounded pilot-follower protocol under disturbed conditions. Detailed Implementation

[0031] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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, 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.

[0032] To address the issues of existing strictly feedback nonlinear multi-agent systems, where initial coordination errors easily exceed preset performance boundaries leading to control failure when subjected to additive spurious data injection (FDI) attacks at the communication end, and where it is difficult to simultaneously satisfy dynamic state constraints and input constraints, this embodiment provides a fault-tolerant control method for disturbed multi-agent systems that can recover preset performance. This method can achieve safe "capture" even with extremely poor initial states, ensuring that the system state does not exceed limits and ultimately converges to within a set tracking accuracy. Figure 1 As shown, the specific steps include: S1: Establish a rigorous feedback nonlinear dynamic model for a multi-agent system that includes unknown model parameter terms, and simultaneously construct an FDI attack model for additive fake data injection at the communication end. Specifically, this embodiment considers a multi-agent system consisting of N followers and 1 leader, the first... The rigorous feedback nonlinear dynamics model of the followers, including unknown model parameter terms, is as follows:

[0033] In the formula: Indicates the first The first-order physical state variables of a follower; Indicates the first An unknown constant parameter vector in a multi-agent subsystem; Indicates the first The first follower Order of physical state variables; Indicates control input; Indicates the measurement output; This represents a known nonlinear regression function; Indicates the first An unknown constant parameter vector in a multi-agent subsystem; express The first derivative; Indicates the first The first follower Order of physical state variables; Indicates transpose; express The first derivative; Indicates the first The first follower Order of physical state variables; Based on the data packet tampering mechanism of multi-agent systems in open communication networks, an additive FDI attack model is defined on the communication channel. In this embodiment, for the channel... and The received attack signal is:

[0034]

[0035] In the formula: express Time of the first The follower received from the first The first follower Attack status signal; express Time of the first The first follower received the first [item / item] from the navigator. Attack status signal; express Time of the first The first follower The true state signal of the first order; express The first time the navigator The true state signal of the first order; and express Attack signals injected at all times and ; S2: Based on the rigorous feedback nonlinear dynamics model and the FDI attack model, define the local coordination error affected by the attack; construct the dynamic recoverable performance boundary function of the agent, and define the boundary center function and height function based on the dynamic recoverable performance boundary function; obtain the normalized error based on the local coordination error and the boundary center function and height function; and convert the normalized error into an unconstrained transformation error, specifically including the following steps: S21: Based on the rigorous feedback nonlinear dynamics model and the FDI attack model, define the local coordination error affected by the attack. for:

[0036] In the formula: In the topology of a communication network, the first... The first follower and the first Decision variables where followers have adjacency relationships; Indicates the navigator and the first The communication connection weight of each follower; Indicates the relationship with the first Each follower has a set of neighboring nodes in the communication link; Indicates the first The follower received from the first The first follower Attack status signal; Indicates the first The first follower received the first [item / item] from the navigator. Attack status signal; S22: Based on local coordination error The dynamic recoverable performance boundary function of the adaptive design agent, to ensure that the error is absolutely wrapped, is expressed as follows:

[0037]

[0038] Configure the nominal upper and lower boundary functions of exponential decay form globally in the system. , To define the steady-state convergence accuracy under the ideal state of no attack among multi-agent agents. The local recovery margin function is:

[0039] In the formula: , Describes the boundary function of dynamic recoverable performance. , Represent the nominal upper and lower boundary functions in the form of exponential decay; Represents the local recovery margin function; Represents the initial local recovery margin constant; Indicates the preset maximum recovery time; Indicates a smooth decay power and Furthermore, in Initial activation time, the The initial contamination error was collected by the follower. Adaptive calculation of the initial local recovery margin constant for crash prevention:

[0040] In the formula: This represents a preset strict positive buffer constant, used to ensure that the denominator is not zero and there are no singularities during the initial performance transformation; in this embodiment, a globally uniform forced capture time is set. and smooth decay power The synthesized local recovery margin function that decays over time is: ; S23: Utilizing asymmetry The transformation converts constrained errors into unconstrained variables, that is, it defines the boundary center function based on the dynamic recoverable performance boundary function. With height function for:

[0041] Based on the local coordination error, the normalized error obtained from the boundary center function and the height function is:

[0042] S24: Convert the normalization error into an unconstrained transformation error. for:

[0043] S3: Construct a model parameter estimator and a lumped attack perturbation observer based on the unconstrained transformation error; the model parameter estimator is used to estimate unknown model parameter terms and obtain an optimized rigorous feedback nonlinear dynamic model; the lumped attack perturbation observer is used to estimate the lumped attack perturbation caused by FDI attacks; Specifically, the formula for constructing the model parameter estimator is as follows:

[0044] In the formula: The positive definite diagonal gain matrix representing the adaptive law; This represents a nonlinear regression compensation vector obtained based on the local state and information about the attacked neighbor. This indicates the correction factor for the leakage term; Indicates the first The first follower Estimates of the order unknown constant parameter vector; express The first derivative; The lumped attack disturbance observer, used to hedge the unknown derivative terms of the FDI attack signal and the system modeling error, is constructed using the following formula:

[0045] In the formula: express The first derivative; This represents an adaptive estimate of the lumped attack disturbance caused by an FDI attack; This represents the adaptive positive definite gain parameter; S4: Based on the backstepping method and the optimized strict feedback nonlinear dynamic model, an unlimited original virtual control law is constructed by combining a lumped attack perturbation observer; for the internal state constraints of the multi-agent system, a state tightening interval of the multi-agent system is constructed by introducing a safety margin function; based on the state tightening interval, a smooth time axis interval saturation operator is constructed to constrain the unlimited original virtual control law, so as to obtain a constrained and protected virtual control law; The specific steps include: S41: Based on the backstepping method and the optimized rigorous feedback nonlinear dynamic model, combined with a lumped attack perturbation observer, i.e., combined with the feedforward term. Adaptive compensation terms and virtual conversion error The original virtual control law without amplitude limit is constructed as follows:

[0046] In the formula: Indicates the first The sum of the in-degrees of the network topology of each follower; The smooth scaling buffer parameter represents the hyperbolic tangent function; The first step, derived by the backstepping method, represents the... The first-order unlimited original virtual control guidance command of a follower; The state feedback gain parameter represents the first-order unlimited original virtual control guidance command. Indicates the feedforward term; express The first derivative; express The abbreviated form; express The first derivative; express The abbreviated form; S42: Internal state constraints for multi-agent systems By introducing a safety margin function The state tightening interval for constructing a multi-agent system is:

[0047] In the formula: The multi-agent system represents the first Internal physical state variables of the order; Indicates the first Time-varying absolute physical lower boundary of the state constraint; Indicates the first Time-varying absolute physical boundary of order-state constraints; Indicates the first The dynamic safety buffer margin function preset for each state; This represents the lower bound of the virtual control command for the state tightening interval after the introduction of a safety margin. This represents the upper bound of the virtual control command in the state tightening interval after the introduction of a safety margin; S43: The smoothing time axis interval saturation operator corresponding to the original virtual control law used to constrain the unlimited amplitude, constructed based on the state tightening interval, is as follows:

[0048] In the formula: This indicates a saturation operator for smoothing the time axis interval; The smoothness steepness parameter represents the saturation operator and is used to determine the smoothness of the amplitude limiting curve. S44: Construct the corresponding local coordination error based on the smoothing time axis interval saturation operator. Constrained and protected virtual control laws for:

[0049] S5: Based on a preset low-pass dynamic surface filter, the constrained virtual control law is filtered to obtain a filtered signal; the sliding mode error is obtained based on the filtered signal, and the original target thrust of the agent is obtained based on the sliding mode error; the original target thrust is smoothed by input saturation control constraint based on the input constraint of the physical actuator to obtain a recoverable preset performance fault-tolerant control law, thereby realizing the recoverable preset performance fault-tolerant control of the multi-agent system under FDI attack; The specific steps include: S51: Based on a preset first-order low-pass dynamic surface filter, the constrained virtual control law is filtered to obtain the filtered signal. The sliding mode errors of each order are obtained from the filtered signal as follows: , , ,......, ; Get the Step sliding mode error derivative for

[0050] In the formula: Indicates the corresponding number Step sliding mode error The virtual control law that is subject to constraints and protection Corresponding filtered signal The derivative; in this embodiment, the safety command after the amplitude limit is used. Feed time constant is In a low-pass dynamic surface filter (DSC), the filtering smoothing command is extracted. and derivative This eliminates the computational explosion problem caused by backstepping differentiation; S52: Based on Stability theory, based on the first Step sliding mode error derivative Get the The first follower Unlimited raw virtual control boot instructions for:

[0051] In the formula: Indicates the first k State feedback gain parameters of order; This represents the compensation signal used to eliminate the effects of filter delay; the formula is obtained through online estimation. Real-time cancellation of nonlinear dynamics ; S53: Based on the first Unlimited raw virtual control boot instructions Combined with steps S42 to S44, obtain the corresponding first... Step sliding mode error Constrained and protected virtual control laws Obtain the corresponding virtual control law Corresponding filtered signal derivative To obtain the first The first follower corresponds to the agent's... k Original target thrust for:

[0052] In the formula: Indicates the first k The height function of order; Indicates the first State feedback gain parameters of order; Indicates the first Step sliding mode error; Indicates the first Unconstrained transformation error of order; Indicates the first Normalization error of order; S53: Based on the input constraints of the physical actuator, apply smooth input saturation control constraints to the original target thrust to obtain a recoverable preset performance fault-tolerant control law. for:

[0053] In the formula: This represents the maximum value of the physical actuator input constraint; Based on the recoverable preset performance fault-tolerant control law This enables recoverable, preset performance-tolerant fault control for multi-agent systems subjected to FDI attacks.

[0054] To fully verify the effectiveness, superiority, and engineering feasibility of the method described in this embodiment, this example uses a planar satellite formation system subjected to a communication-end spurious data injection (FDI) attack and facing severe initial error exceeding limits and multiple time-varying physical constraints as the controlled object. It strictly follows the patent claims and theoretical derivation logic, and details the implementation process. The simulation environment is built based on MATLAB / Simulink. All derivation steps and formula applications correspond to the core designs in the invention, including recoverable boundary reconstruction, asymmetric spatial mapping, smooth dynamic limiting, and adaptive compensation, ensuring a high degree of consistency between theoretical derivation and engineering practice. The specific implementation process is as follows: This embodiment selects a space-plane satellite formation system consisting of one leader (numbered 0) and four followers (numbered 1, 2, 3, and 4). Each follower satellite is configured as follows: Onboard sensor array: Used for real-time measurement of the satellite's absolute position and velocity. Inter-satellite communication link: Topology connection including edges. The onboard main control computer incorporates the fault-tolerant control algorithm of this invention, with a computation cycle of [missing information]. Thrust actuator: The physical thrust limit is locked by hardware.

[0055] S1: Constructing a double-integral relative dynamics model and distortion error perception: S10: Perform model coordinate calculation; the onboard microprocessor is based on... Transformation, in axis Establish current followers That is, the first The dynamic control equations of a follower:

[0056]

[0057] In the formula: This indicates that the expected bias has been superimposed. The relative position error; Indicates relative velocity error. This represents the control thrust command to be solved; S11: Expected bias and initial state configuration, the expected leader center bias vector for each follower is:

[0058] The initial state of the multi-agent system is set to an extremely poor misaligned state (initial velocity value is 0):

[0059]

[0060] The time-varying trajectory parameters of the navigator are set as follows: and

[0061] S2: Construct the malicious attack perturbation function (FDI), set at... During the specified time, the communication link was subjected to a malicious additive false data injection (FDI) attack. The distorted position signal received by the communication module was... The distorted speed signal is The attack occurs within a window function. satisfy:

[0062] The injected location attack waveform specifically manifests as follows:

[0063]

[0064] At this time, the followers The received distortion coordination error is expressed as:

[0065] For followers who directly connect to the leader The received distortion coordination error is expressed as:

[0066] For followers , Indicates its parent node; S3: Parameter settings for full-state safety constraints: To prevent satellites from colliding or overstepping boundaries, an absolute time-varying safety range is written into the low-level control: Position state constraints:

[0067] Safety margin function:

[0068] State tightening interval of multi-agent systems:

[0069]

[0070] S4: Construction and spatial mapping calculation of recoverable preset performance boundary functions: S40: Nominal and Recovery Boundary Construction: Nominal upper and lower boundary functions , Set as:

[0071]

[0072] The system forces the capture time to be set to Seconds, decay index Each follower according to When the initial error exceeds the limit at time 1, the locally adaptive allocation recovery margin is:

[0073] Taking the follower 3, which deviates most severely by d, as an example, its local time-varying recoverable boundary is:

[0074]

[0075]

[0076] Using the central function With half-height functions The normalized attack error is:

[0077] Then, mapping it to an unconstrained space to obtain the virtual reference error is:

[0078] The variable It will serve as the core feedback driver for subsequent robust control.

[0079] S5: Solve fault-tolerant control instructions: S50: Online compensation for FDI attacks. To eliminate the unknown upper bound of residuals caused by FDI attacks, a local adaptive update law is run:

[0080] In the formula: the initial conditions are set as follows Combined with feedback gain and smooth scaling parameters Calculate the original virtual velocity command, combined with the known coupling feedforward term. The processor calculates the unlimited original velocity guiding law using the backstepping method. :

[0081] The applied steepness parameter is The smoothing time axis interval saturation operator, for Strict amplitude limiting was implemented to obtain safety instructions. for:

[0082] After the limit Feed time constant is First-order low-pass command filter ; Then calculate the sliding mode error in the second step. And so on, until the deduction reaches the [number]. The order is obtained And its derivative expression is:

[0083] according to Stability requirements, according to the first Step sliding mode error derivative Get the The first follower Unlimited raw virtual control boot instructions for:

[0084] Finally, calculate the original driving torque of the actuator. (where feedback parameters) ): Finally, to ensure that the hardware thrust limits of the satellite actuators are met. The input saturation process is performed as follows:

[0085] Therefore, this implementation of the multi-agent system is based on this. Drive the actuator to complete the full closed loop of multi-agent collaborative fault-tolerant control.

[0086] Compared with the prior art, the beneficial effects of the method described in this embodiment are as follows: (1) Fault-tolerant control with explicit entry-capture characteristics: The method described in this embodiment breaks the fate of traditional PPC algorithms that face collapse when the initial state is violated. The contract mechanism with a recoverable preset performance boundary function has the intelligent adaptive capability of "actively wrapping first and then forcibly shrinking", allowing the agent to automatically generate a tolerant safe channel when attacked or with extremely poor initialization parameters, and within the set... Within a short period of time, the multi-agent system is forcibly pulled into an ideal synchronization trajectory, which greatly improves the survival rate and robustness of actual engineering deployments.

[0087] (2) Robust Distributed Defense Without Prior Knowledge: When dealing with FDI attacks, the method described in this embodiment completely eliminates the stringent dependence of existing technologies on multiplicative attack models, known first derivatives of attack signals, or known global topological connectivity information. This is achieved through a designed asymmetric space mapping and a single-parameter perturbation adaptive law. This allows each microcontroller node, or intelligent agent, to independently compensate for and offset attack signals locally by simply listening to the communication messages of its contaminated neighbors, thus possessing extremely high resistance to destruction.

[0088] (3) Achieving high-fidelity satisfaction of full-state physical constraints with low computational cost: The method described in this embodiment innovatively adopts a smooth time-axis interval saturation operator with dynamic safety margin, replacing the traditional solution of extremely complex control obstacle functions or model predictive control online optimization schemes. It enables physical intermediate states such as velocities and accelerations of each order to be mathematically and smoothly constrained within the time-varying safety corridor while being smoothed by the instruction filter, and is seamlessly compatible with the final hardware output limit of the actuator, possessing excellent industrial chip portability.

[0089] like Figure 2The diagram illustrates the communication topology of a multi-satellite formation system subjected to an FDI attack, showing a directed communication network consisting of one leader (labeled 0) and four followers (labeled 1, 2, 3, and 4). The arrows indicate the direction of information flow. The leader only sends signals (pinned) to follower 1, while the other followers obtain information through neighbor cooperation. The yellow lightning bolt symbol indicates a False Data Injection (FDI) attack on this communication channel, where the attacker interferes with the formation configuration by tampering with the transmitted position and velocity messages.

[0090] Figure 3 Different preset capture times were displayed. The graph compares the entry capture characteristics of the system's attack coordination error under different conditions, detailing the dynamic performance of the "forced capture" mechanism, the core innovation of this invention, under various preset parameters. The graph sets the capture time for each parameter. For 3s, 5s, and 7s, the results clearly show that no matter how severe the coordination error deviation is at the initial moment outside the nominal boundary (black dashed line), the system can wrap it with adaptive expansion of the recoverable boundary and smoothly "suppress" it back into the high-precision funnel at the user-specified moment along the corresponding solid line trajectory. This proves that the method described in this embodiment can explicitly control the error convergence rhythm according to the urgency of the actual task.

[0091] Figure 4 The diagram illustrates the error evolution of representative followers under recoverable pre-defined performance boundaries (R-PPB) and nominal boundaries. Taking follower 3, which initially violated the rules most severely, as an example, the diagram compares the differences in the evolution of recoverable performance boundaries (R-PPB) and traditional nominal boundaries when handling out-of-bounds errors. Instantly, due to the severely excessive initial error, R-PPB expands upwards instantaneously through the local margin of the underlying solution to safely contain the error, thus completely avoiding the computational collapse of the traditional PPC algorithm at this moment; subsequently... During this period, the expanded boundary decays smoothly according to a preset power order, guiding the error trajectory to merge into the nominal trajectory precisely at 5s without any singular jumps, thus verifying the excellent numerical stability of the spatial mapping transformation combined with the boundary reconstruction mechanism.

[0092] Figures 5 to 6 This paper presents the evolution trajectory of a multi-agent formation system in three-dimensional space and the variation curve of the tracking error norm in the first state, comprehensively demonstrating the macroscopic configuration evolution and overall tracking performance indicators of the multi-agent formation system in three-dimensional space. Figure 5 The motion trajectories of the four following satellites in three-dimensional spacetime were depicted. Each celestial body successfully overcame the extremely disordered initial dispersion state and accurately converged towards the navigator target vertex with the desired bias, achieving regular configuration reorganization. Figure 6 The convergence curve of the error norm of the entire system is given. The combined error of all stars converges rapidly after the initial adjustment period and approaches zero. This quantitatively proves the semiglobal-practical ultimate bounded (UUB) stability of the entire system under harsh attack conditions.

[0093] Figure 7 The diagram illustrates the first-state evolution trajectory of a representative follower under time-varying position constraints. Through real-time comparison of the actual position trajectory with the time-varying boundary, the system's forward invariance security guarantee capability for the first state (position) is rigorously demonstrated. The black dashed line in the diagram represents the absolutely safe physical corridor that tightens exponentially over time. Although the system underwent significant transient adjustments during the initialization phase and subsequently suffered severe FDI attack interference, the real-time position trajectory of the representative celestial body remained strictly "locked" within the safe corridor, without any boundary breaching. This directly proves the extremely high reliability of the method described in this embodiment in safety-sensitive tasks such as collision avoidance.

[0094] Figure 8 The diagram illustrates the second-state evolution trajectory of a representative follower under time-varying velocity constraints and safety margin allocation, further demonstrating the technical details of the method described in this embodiment for implementing refined constraint protection for higher-order internal states (velocities). To reserve sufficient safety buffer for the underlying servo control, the algorithm utilizes dynamic safety margins to tighten the absolute physical boundary of velocity inwards. The smooth solid line trajectory in the diagram represents the actual velocity after being constrained by the smooth interval saturation operator. It not only maintains a safe distance from the dangerous physical red line but also exhibits an extremely smooth evolution process without overshoot, fully demonstrating the advantages of the method described in this embodiment in soft-limiting control when resolving conflicts in multi-order state constraints.

[0095] Figure 9 The diagram illustrates the control input command output waveforms of a representative follower under actuator saturation constraints, depicting the actual physical thrust waveforms of the satellite attitude and orbit control thrusters during mission execution. The black dashed line in the diagram rigidly defines the maximum thrust limit (±2.0) of the actuator hardware. During the initial forced acquisition phase and the sudden moment of an FDI attack, control demands rapidly increase. However, under the active intervention of the smooth input saturation operator described in this embodiment, the thrust commands are perfectly and smoothly intercepted at the maximum threshold, effectively preventing hardware damage caused by long-term thruster overload. Subsequently, with compensation from the adaptive hedging term, the thrust quickly returns to a stable level.

[0096] Figure 10The diagram shows the norm distribution of FDI attack signals acting on representative communication channels, serving as a test input background to verify anti-interference capabilities. It also demonstrates the time-frequency distribution characteristics of fake data injection attack signals acting on some representative communication links. This attack exhibits severe characteristics of high frequency and time-varying fluctuations within a set time window. Combined with the system's response curve, it can be seen that even with severe and completely unknown attack intensity, the adaptive observer running inside this controller can still accurately and in real-time capture and offset these injected energies, maintaining extremely strong steady-state anti-disturbance stability of the terminal tracking trajectory.

[0097] Figure 11 This paper presents a comparison of the tracking performance of the method described in this embodiment and the traditional bounded leader-follower protocol under disturbed conditions. The comparative experiments visually demonstrate the generational performance leap of the method described in this embodiment compared to existing technologies. Under the same initial out-of-bounds and FDI attack conditions, the traditional bounded leader-follower protocol not only fails to forcibly pull the error back into the performance funnel within the specified time, but also exhibits severe steady-state drift that cannot be eliminated when facing attacks. In contrast, this invention, through the deep coupling of the recoverable preset performance boundary (R-PPB) mechanism and a robust adaptive architecture, not only achieves faster lossless capture, but also firmly locks the error under disturbed conditions within a high-precision preset funnel, demonstrating extremely high fault-tolerant engineering application value. In summary, the method described in this embodiment aims to solve the problem of system computational singularities and cooperative loss of control that are prone to occur in modern multi-agent networks when facing the dual attacks of malicious network tampering and severe physical limits. To address the critical flaw of traditional Preset Performance Control (PPC) algorithms, which require initial errors to be strictly within safety boundaries, the method described in this embodiment innovatively constructs a recoverable preset performance boundary based on adaptive adjustment of the local initial state of nodes. Combined with asymmetric inverse hyperbolic tangent space mapping technology, this endows the control system with a powerful "forced entry capture" capability, allowing the agent to be safely, smoothly, and timely pulled back into a high-precision steady-state funnel even under extremely adverse initial position deviations. Simultaneously, facing unknown and covert communication-end spoofing (FDI) attacks, the method described in this embodiment designs a fully distributed adaptive disturbance observer, achieving online and precise hedging against locally maliciously tampered messages without needing to know any prior attack models or derivative signals. Furthermore, by introducing a smooth time-axis interval saturation operator and dynamic surface filtering technology into the underlying control architecture, the method described in this embodiment strictly locks the safety corridor of the final actuator thrust at each order of internal states of satellites or drones at the physical level, achieving absolute safety guarantees against collisions and hardware burnout.

[0098] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A fault-tolerant control method for a disturbed multi-agent system with recoverable preset performance, characterized in that, Specifically, the following steps are included: S1: Establish a rigorous feedback nonlinear dynamic model for a multi-agent system containing unknown model parameter terms, and simultaneously construct an FDI attack model for additive fake data injection at the communication end. S2: Based on the rigorous feedback nonlinear dynamics model and the FDI attack model, define the local coordination error affected by the attack; construct the dynamic recoverable performance boundary function of the agent, and define the boundary center function and height function based on the dynamic recoverable performance boundary function; based on the local coordination error, obtain the normalized error according to the boundary center function and height function; and convert the normalized error into an unconstrained transformation error. S3: Construct a model parameter estimator and a lumped attack perturbation observer based on the unconstrained transformation error; the model parameter estimator is used to estimate unknown model parameter terms and obtain an optimized rigorous feedback nonlinear dynamic model; the lumped attack perturbation observer is used to estimate the lumped attack perturbation caused by FDI attacks; S4: Based on the backstepping method and the optimized strict feedback nonlinear dynamic model, an unlimited original virtual control law is constructed by combining a lumped attack perturbation observer; for the internal state constraints of the multi-agent system, a state tightening interval of the multi-agent system is constructed by introducing a safety margin function; based on the state tightening interval, a smooth time axis interval saturation operator is constructed to constrain the unlimited original virtual control law, so as to obtain a constrained and protected virtual control law; S5: Based on a preset low-pass dynamic surface filter, the constrained virtual control law is filtered to obtain a filtered signal; the sliding mode error is obtained based on the filtered signal, and the original target thrust of the agent is obtained based on the sliding mode error; the original target thrust is subjected to smooth input saturation control constraints based on the input constraints of the physical actuator to obtain a recoverable preset performance fault-tolerant control law, thereby realizing the recoverable preset performance fault-tolerant control of the multi-agent system under FDI attack.

2. The method for recoverable preset performance fault-tolerant control of a disturbed multi-agent system according to claim 1, characterized in that, The rigorous feedback nonlinear dynamic model containing unknown model parameter terms described in S1 is: In the formula: Indicates the first The first-order physical state variables of a follower; Indicates the first An unknown constant parameter vector in a multi-agent subsystem; Indicates the first The first follower Order of physical state variables; Indicates control input; Indicates the measurement output; This represents a known nonlinear regression function; Indicates the first An unknown constant parameter vector in a multi-agent subsystem; express The first derivative; Indicates the first The first follower Order of physical state variables; Indicates transpose; express The first derivative; Indicates the first The first follower Order of physical state variables; The FDI attack model is as follows: In the formula: express Time of the first The follower received from the first The first follower Attack status signal; express Time of the first The first follower received the first [item / item] from the navigator. Attack status signal; express Time of the first The first follower The true state signal of the first order; express The first time the navigator The true state signal of the first order; and express Attack signals injected at all times and .

3. The method for recoverable preset performance fault-tolerant control of a disturbed multi-agent system according to claim 2, characterized in that, S2 specifically includes the following steps: S21: Based on the rigorous feedback nonlinear dynamics model and the FDI attack model, define the local coordination error affected by the attack. for: In the formula: In the topology of a communication network, the first... The first follower and the first Decision variables where followers have adjacency relationships; Indicates the navigator and the first The communication connection weight of each follower; Indicates the relationship with the first Each follower has a set of neighboring nodes in the communication link; Indicates the first The follower received from the first The first follower Attack status signal; Indicates the first The first follower received the first [item / item] from the navigator. Attack status signal; S22: The boundary function for constructing the dynamic recoverable performance of the agent is: In the formula: , Describes the boundary function of dynamic recoverable performance. , Represent the nominal upper and lower boundary functions in the form of exponential decay; Represents the local recovery margin function; Represents the initial local recovery margin constant; Indicates the preset maximum recovery time; Indicates a smooth decay power and ; S23: Define the boundary center function based on the dynamic recoverable performance boundary function. With height function for: Based on the local coordination error, the normalized error obtained from the boundary center function and the height function is: S24: Convert the normalization error into an unconstrained transformation error. for: 。 4. The method for recoverable preset performance fault-tolerant control of a disturbed multi-agent system according to claim 3, characterized in that, The formula for constructing the model parameter estimator described in S3 is: In the formula: The positive definite diagonal gain matrix representing the adaptive law; This represents a nonlinear regression compensation vector obtained based on the local state and information about the attacked neighbor. This indicates the correction factor for the leakage term; Indicates the first The first follower Estimates of the order unknown constant parameter vector; express The first derivative; The formula for constructing the lumped attack perturbation observer is as follows: In the formula: express The first derivative; This represents an adaptive estimate of the lumped attack disturbance caused by an FDI attack; This represents the adaptive positive definite gain parameter.

5. The fault-tolerant control method for recoverable preset performance of a disturbed multi-agent system according to claim 4, characterized in that, S4 specifically includes the following steps: S41: Based on the backstepping method and the optimized rigorous feedback nonlinear dynamic model, combined with the lumped attack disturbance observer, the unlimited original virtual control law is constructed as follows: In the formula: Indicates the first The sum of the in-degrees of the network topology of each follower; The smooth scaling buffer parameter represents the hyperbolic tangent function; The first step, derived by the backstepping method, represents the... The first-order unlimited original virtual control guidance command of a follower; The state feedback gain parameter represents the first-order unlimited original virtual control guidance command. Indicates the feedforward term; express The first derivative; express The abbreviated form; express The first derivative; express The abbreviated form; S42: Internal state constraints for multi-agent systems By introducing a safety margin function The state tightening interval for constructing a multi-agent system is: In the formula: The multi-agent system represents the first Internal physical state variables of the order; Indicates the first Time-varying absolute physical lower boundary of the state constraint; Indicates the first Time-varying absolute physical boundary of order-state constraints; Indicates the first The dynamic safety buffer margin function preset for each state; This represents the lower bound of the virtual control command for the state tightening interval after the introduction of a safety margin. This represents the upper bound of the virtual control command in the state tightening interval after the introduction of a safety margin; S43: The smoothing time axis interval saturation operator corresponding to the original virtual control law used to constrain the unlimited amplitude, constructed based on the state tightening interval, is as follows: In the formula: This indicates a saturation operator for smoothing the time axis interval; The smoothness steepness parameter represents the saturation operator; S44: Construct the corresponding local coordination error based on the smoothing time axis interval saturation operator. Constrained and protected virtual control laws for: 。 6. The fault-tolerant control method for recoverable preset performance of a disturbed multi-agent system according to claim 5, characterized in that, S5 specifically includes the following steps: S51: Based on a preset first-order low-pass dynamic surface filter, the constrained virtual control law is filtered to obtain the filtered signal. ; The sliding mode errors of each order are obtained from the filtered signal as follows: , , ,......, ; Get the Step sliding mode error derivative for In the formula: Indicates the corresponding number Step sliding mode error The virtual control law that is subject to constraints and protection Corresponding filtered signal The derivative; S52: Based on Stability theory, based on the first Step sliding mode error derivative Get the The first follower Unlimited raw virtual control boot instructions for: In the formula: Indicates the first k State feedback gain parameters of the order; This represents the compensation signal used to eliminate the effects of filter delay. S53: Based on the first Unlimited raw virtual control boot instructions Combined with steps S42 to S44, obtain the corresponding first... Step sliding mode error Constrained and protected virtual control laws ; Obtain the corresponding virtual control law Corresponding filtered signal derivative To obtain the first The first follower corresponds to the agent's... k Original target thrust for: In the formula: Indicates the first k The height function of order; Indicates the first State feedback gain parameters of the order; Indicates the first Step sliding mode error; Indicates the first Unconstrained transformation error of order; Indicates the first Normalization error of order; S53: Based on the input constraints of the physical actuator, apply smooth input saturation control constraints to the original target thrust to obtain a recoverable preset performance fault-tolerant control law. for: In the formula: This represents the maximum value of the physical actuator input constraint; Based on the recoverable preset performance fault-tolerant control law This enables recoverable, preset performance-tolerant fault control for multi-agent systems subjected to FDI attacks.

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