Source-linked AI summary
Fuzzy-based Robust Precision Consensus Tracking for Uncertain Networked Systems with Cooperative-Antagonistic Interactions
Amorey Lewis
TL;DR
Stochastic disturbances and actuator faults complicate precise finite-time bipartite consensus tracking for nonlinear multiagent systems. The paper develops an improved FTPF within a fuzzy fault-tolerant distributed cooperative controller, showing probabilistic boundedness and arbitrary-precision tracking within a predefined time.
Problem
Finite-time BCT for stochastic nonlinear multiagent systems is challenged by stochastic disturbances, actuator faults, and tight FTPF parameter-selection ranges.
Method
The paper designs an improved FTPF and combines it with fuzzy fault-tolerant distributed cooperative control for stochastic nonlinear multiagent systems.
Results
All system signals are semi-global uniformly ultimately bounded in probability, while bipartite consensus errors achieve arbitrary precision with probability in the predefined time.
Takeaways & Limitations
The relaxed FTPF parameter range supports better transient performance while the controller addresses actuator faults and assorted uncertainties without over-parameterization.
Abstract
from arXiv · showhide
In bipartite consensus tracking (BCT) tasks for nonlinear multiagent systems, stochastic disturbances and actuator faults are regarded as essential factors that hamper effective controller formulation and tracking precision improvement. To address these difficulties, we design an improved finitetime performance function (FTPF) for a fuzzy fault-tolerant distributed cooperative control scheme to achieve finite-time robust precision BCT tasks for nonlinear multiagent systems. The parameter selection range of the improved FTPF is relaxed, which renders systems to achieve better transient performance. Benefitting from stochastic Lyapunov stability theory, it is shown that all signals of systems are semi-global uniformly ultimately bounded in probability, and bipartite consensus errors can satisfy the arbitrary precision with probability in the predefined time. Finally, to verify its effectiveness, the proposed control scheme is applied to BCT tasks of a group of vehicles, which manifests anticipated control performance under various uncertainties.
I. INTRODUCTION
The introduction motivates finite-time robust bipartite consensus tracking for stochastic nonlinear multiagent systems with actuator faults. It proposes an improved FTPF and fuzzy fault-tolerant distributed control scheme to relax parameter restrictions and improve transient performance.
- Bipartite consensus tracking addresses cooperative and antagonistic interactions among agents, extending consensus beyond non-negative communication weights.
- Conventional prescribed performance control mainly targets steady-state behavior, whereas FTPFs address convergence-time requirements in practical tracking tasks.
- Existing FTPFs have tight parameter-selection ranges that hinder application and make desired performance difficult to ensure.
- The paper studies finite-time robust precision BCT for stochastic nonlinear multiagent systems subject to actuator faults.
- Its improved FTPF makes prescribed settling time independent of controller parameters and initial states, enabling direct task-based time selection.
- The proposed fault-tolerant scheme suppresses actuator-fault and uncertainty effects while avoiding adaptive-control over-parameterization.
A. Graph Theory
The graph-theoretic model represents cooperative and antagonistic communication using signed edge weights. Structural balance and a leader-rooted spanning tree provide the topology assumptions used in the analysis.
- A signed digraph models agent communication, with nonzero weighted edges indicating information flow between agents.
- Positive edge weights denote cooperative relationships, whereas negative edge weights denote antagonistic relationships.
- The graph Laplacian is formed from the signed adjacency matrix and an in-degree matrix based on absolute edge weights.
- A spanning tree exists when a node can reach every node through directed paths, and that node is the tree root.
- The leader communicates unidirectionally with followers, and bounded leader signals and first derivatives are assumed receivable by followers.
- Structural balance requires a partition of nodes in which within-group edges are positive and between-group edges are negative.
- The communication topology is assumed to be structurally balanced and to contain a leader-rooted spanning tree; under this assumption, the stated matrix is nonsingular.
B. Stochastic Stability Theorem
The stochastic stability section models system dynamics with an Itô stochastic differential equation and uses Lyapunov-generator conditions to characterize semi-global uniformly ultimately bounded behavior in probability.
- The stochastic system is modeled by an Itô-type differential equation with drift and diffusion functions.
- The state lies in R^n, the noise is an independent standard Brownian motion, and the drift and diffusion functions satisfy local Lipschitz conditions with zero values at the origin.
- For a twice-continuously differentiable Lyapunov function, the infinitesimal generator incorporates the diffusion covariance and its trace.
- The stability lemma requires the Lyapunov function to be bounded above and below by class K∞ functions of the state norm.
- Under these Lyapunov conditions, the stochastic system is semi-global uniformly ultimately bounded in probability.
- The expected Lyapunov value is bounded by an exponentially decaying initial-state term plus a constant ultimate-bound term.
C. Finite-Time Performance Function
The FTPF is designed as a smooth finite-time performance boundary that reaches a constant ultimate value after a prescribed settling time. The improved construction relaxes parameter selection compared with earlier FTPFs and supports better transient performance.
- An FTPF is defined as a smooth function that remains positive, nonincreasing, and reaches a constant value after its settling time.
- The segmentation function uses an exponentially varying transient term before Ts and a constant ultimate boundary afterward.
- Ts denotes the settling time, while the asymptotic boundary can be arbitrarily small and represents the FTPF's ultimate boundary.
- The construction verifies continuity and differentiability properties needed for the FTPF definition across the finite-time transition.
- Derivative analysis establishes higher-order smoothness of the performance function under the selected parameter conditions.
- The improved FTPF relaxes the tight parameter-selection range of earlier designs, supporting better transient performance.
D.Fuzzy Logie System
The paper introduces fuzzy logic systems to approximate unknown nonlinear functions within a stochastic multiagent model subject to actuator faults. It specifies fault modes, control objectives, and assumptions supporting finite-time bipartite consensus tracking.
- Fuzzy logic approximation: Fuzzy logic systems approximate unknown nonlinear functions using rule-based inputs, fuzzy sets, membership functions, and weighted outputs.The fuzzy basis-function representation rewrites the FLS output in parameterized form.
- Multiagent model: The i-th follower has state variables and output governed by unknown locally Lipschitz nonlinear functions with zero values at the origin.The model also includes stochastic terms and unknown parameters.
- Actuator faults: Actuator faults are modeled through normal operation, partial loss of control effectiveness, and total loss of effectiveness.The fault description includes unknown effectiveness parameters and fault onset and termination times.
- Control objectives and assumptions: The control objectives require all system signals to be semi-globally uniformly ultimately bounded in probability and consensus errors to attain arbitrary precision by a predefined settling time.The framework assumes at most M - 1 actuators operate in total loss-of-effectiveness mode and known actuator-control signs.
A. Error Transfonnation
The error transformation converts bipartite consensus errors into transformed errors so that bounded transformed errors enforce prescribed finite-time error bounds. This mechanism supports controller construction after the transformation.
- Error definition: The method defines bipartite consensus errors relative to the leader signal before applying the error transformation.The transformed-error construction is introduced specifically to achieve the finite-time precision objective.
- Transformation mechanism: The transformation uses μ(e_i1) = arctan(e_i1) and replaces the original consensus error with the transformed error in subsequent controller design.The transformed error is the variable used in later coordinate transformations and controller construction.
- Finite-time error bound: For t ≥ T_s, the transformed-error relation guarantees that the bipartite consensus error remains within a predefined region in finite time.The paper explains that bounded transformed errors ensure the required consensus-error inequality.
B. Fault-Tolerant Controller Design
The fault-tolerant controller design begins with coordinate transformations, command-filter compensation, and Lyapunov functions. Compensation signals are introduced to offset filtering errors during recursive controller construction.
- Coordinate transformation: Coordinate transformations are provided before designing the fault-tolerant controller.These transformations prepare the system variables for the recursive control design.
- Command filtering: A first-order command filter generates filtered virtual-controller outputs, with an associated positive design parameter and matched initial values.The filter output is related to the virtual controller through the stated filter dynamics.
- Lyapunov design: Step 1 selects a Lyapunov function using a compensated error and an estimated unknown bound.A compensation signal is designed to compensate the filtering error.
- Stochastic stability analysis: The infinitesimal generator of the first Lyapunov function is derived to support stability analysis of the stochastic controller.The derivation is based on the preceding Lyapunov function and compensation construction.
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The recursive design bounds unknown nonlinear terms with fuzzy approximators, constructs virtual controllers, and develops a second Lyapunov step with compensation for filtering errors.
- Unknown-term grouping: Unknown functions in the Lyapunov-generator inequality are grouped into a combined term before fuzzy approximation.The grouped expression collects terms containing the unknown functions.
- Fuzzy approximation: Fuzzy logic systems approximate the unknown functions, while their approximation errors are bounded by positive constants.The construction introduces fuzzy basis vectors and explicit approximation-error bounds.
- Generator bounds: Lemma-based inequalities transform the generator bounds into expressions containing designed positive parameters and error norms.These bounds are then used in the recursive controller derivation.
- Virtual-controller construction: Virtual controllers are constructed by substituting the derived bounds into the Lyapunov inequalities.The design proceeds through successive virtual-controller and generator expressions.
- Second recursive step: A second Lyapunov function and compensation signal address the filtering error in the next recursive step.The associated design parameters are selected as positive values.
By substituting Eq. (26) to inequality (25), it obtains
The section develops a fault-tolerant distributed controller through recursive Lyapunov analysis, compensation signals, adaptive laws, and fuzzy approximations for unknown nonlinear dynamics.
- Recursive stability design: The design recursively selects Lyapunov functions and virtual controllers to establish inequalities for the system's stability analysis.The construction proceeds through successive steps and substitutions among inequalities and controller expressions.
- Compensation and fault tolerance: A compensation signal is introduced for subsequent stability analysis, with positive design parameters governing its construction.The compensation signal is used together with compensated errors and filtering-error elimination in the recursive design.
- Fault-tolerant controller: The fault-tolerant controller combines adaptive laws, intermediate control variables, and sign-based terms to handle uncertain dynamics and actuator faults.The controller and adaptive laws are explicitly constructed after bounding the associated Lyapunov-generator terms.
- Fuzzy approximation: Two fuzzy logic systems approximate unknown nonlinearities in the drift and diffusion terms, while a reduced-parameter approach avoids adaptive-law growth with system order.The stated motivation is to reduce over-parameterization by making the number of adaptive laws independent of the agent order.