A game theory adaptive weighting convex combination FxLMS algorithm for active vibration control

Abstract

Convex Combination Filtered-x Least Mean Square (C-FxLMS) algorithms face an inherent conflict between achieving rapid convergence speed and smaller steady-state error. To address this limitation, this paper proposes a multi-agent Game Theory Adaptive Weighting (GTAW) based C-FxLMS control strategy for active vibration suppression. The framework integrates two different sub-controllers within a strategic competition process, specifically an efficiency-oriented agent utilizing a variable step size and an accuracy-oriented agent employing a small fixed step size, a state-dependent Nash equilibrium strategy is designed to dynamically modulate the control weights. This strategy allows the system to balance the conflicting objectives of rapid convergence speed and smaller steady-state error. The convergence and stability of the Active Vibration Control (AVC) system are rigorously verified through the construction of a Lyapunov function and boundary analysis. Finally, quantitative results across four distinct scenarios demonstrate that the proposed method achieves an average reduction of 23.9% in steady-state error and a 41.7% decrease in control energy consumption compared to benchmark methods.

Keywords

multi-agent Nash equilibrium game theory convergence analysis stability analysis

Introduction

Aero-engine blades are highly susceptible to severe vibrations under complex aerodynamic loads and external excitations.^1,2 These vibrations not only generate excessive noise and reduce aerodynamic performance, but more importantly, they accelerate fatigue damage, shorten the service life of components, and may lead to catastrophic mechanical failures. Therefore, effective vibration suppression is crucial for mitigating fatigue damage and extending the blades’ life.³

Vibration control strategies are generally categorized into passive and active approaches.⁴ Passive control relies on structural damping or added mass to dissipate energy^5; however, its effectiveness is often limited by weight constraints and poor performance at low frequencies. In contrast, AVC has gradually become the preferred choice for complex aerospace engine applications.^3,6 AVC systems use real-time feedback to generate counteracting forces, offering significant advantages, including high efficiency, fast response, and flexibility to adapt to multi-modal vibrations.^3,4

Adaptive control algorithms are widely employed in the field of AVC because they can automatically adjust the control strategy to preserve system stability under external disturbances or variations in system parameters.^7,8 Among these, the FxLMS algorithm is a popular benchmark approach that is known for its computational durability and straightforward structure.^7,9,10 Nevertheless, the conventional FxLMS controller imposes strict constraints on step-size selection.¹¹ While a smaller step size can increase accuracy and stability, it will result in a slower rate of convergence. On the other hand, a bigger step size can accelerate the system response but may cause oscillation in the stable state. Consequently, the performance of the traditional fixed step size controller in real-world applications is limited.^10,12

Several improved systems, including variable-step FxLMS and normalized FxLMS, have been proposed to balance these competing goals.^7,13 The C-FxLMS method has garnered a lot of interest lately.¹⁴ Using a mixing parameter, this architecture combines the outputs of two filters.¹⁵ Even though C-FxLMS helps close the accuracy and speed imbalance, current methods have significant limitations.¹⁶ The majority of existing C-FxLMS algorithms change the combination parameter using algorithms or set circular functions. Slow variations or inadequate weighting during steady-state phases are frequently the result of these strict update laws’ inability to intelligently adjust to complicated disturbance changes.^16,17 In recent years, adaptive combination strategies for C-FxLMS have evolved from fixed convex-combination schemes to variable-weight and state-dependent mixing approaches, including adaptive convex combination mechanisms and nonlinear mixing functions. These developments aim to improve robustness under nonstationary disturbances and accelerate convergence.^18,19 However, most existing methods still rely on predefined update structures rather than decision-driven adaptation, making it difficult to maintain an optimal balance between convergence speed and steady-state accuracy in dynamically changing vibration environments.²⁰

Recent developments in game theory and multi-agent systems (MAS) imply that problems with multiple goals can be effectively resolved through strategic interactions among independent agents.^21,22 In vibration control and adaptive filtering, MAS-based coordination and game-theoretic optimization have been explored for distributed decision-making and multi-objective control problems. These studies demonstrate that strategic interaction mechanisms can provide adaptive and flexible weighting policies compared with fixed update rules.^23,24 This leads to the proposal that the conflict between “convergence speed” and “steady-state error” can be represented as a dynamic game in which system states cause a continual negotiation process to establish the ideal control weight rather than standard rules.

This paper suggests a GTAW control approach for the active vibration reduction of aero-engine blades based on the previously discussed analysis. The proposed framework is particularly designed to handle complex disturbance environments, such as time-varying aerodynamic loads and multi-modal vibration coupling in aero-engine blades, by adaptively adjusting control weights according to real-time system states. Within a competitive game structure, the suggested framework combines two different sub-controllers: an accuracy-oriented agent and an efficiency-oriented agent. In contrast to conventional convex combinations, the control weights are dynamically adjusted using a state-dependent Nash equilibrium searching method, which enables the system to independently balance the competing needs of smaller steady-state error and fast convergence speed. This mechanism provides greater flexibility in parameter adjustment, improves robustness against disturbance interference through equilibrium-based weight negotiation, and addresses the stability difficulty introduced by dynamic weight adaptation via theoretical analysis.

The main contributions of this work are summarized as follows.

A GTAW-C-FxLMS control framework is proposed. Within this architecture, two different agents, specifically an efficiency-oriented agent and an accuracy-oriented agent, adjust the system behavior through a strategic competition process, providing enhanced flexibility in parameter adjustment compared with fixed or rule-based combination strategies.

A state-dependent Nash equilibrium searching method is designed. This strategy adjusts the convex combination weights according to real-time error dynamics, improving anti-interference capability under disturbance variations while achieving an optimal dynamic balance between convergence speed and steady-state precision.

A rigorous theoretical analysis is established. Through establishing a composite Lyapunov function and using the boundary analysis, the convergence and stability of the AVC system are mathematically established, providing a solid theoretical foundation for the suggested approach.

System model and multi-agent framework

System model and control objective

The block diagram of the convex combination of two different controller is shown in Figure 1. One employing a small fixed-step size FxLMS algorithm and the other utilizing a variable-step size FxLMS algorithm.^9,25

Figure 1.

The block diagram of the convex combination of two different controller.

As illustrated in Figure 1, $d (n)$ represents the primary noise to be eliminated, $x (n)$ denotes the reference signal. The adaptive controller $W (n)$ generates the control output $y (n)$ to reduce the vibration. Multi-agent represents the roles of two games. $S (z)$ denotes the transfer function of the secondary path, $S (z)$ denotes the transfer function of the primary path, and $\hat{S} (z)$ is used for filtering the reference signal in the FxLMS architecture. Finally, $e (n)$ represents the error signal, which serves as the feedback to both agents for real-time weight adaptation.

The ultimate control objective is to achieve rapid convergence speed while reducing the steady-state error. This objective is mathematically expressed as minimizing the instantaneous cost function involving the squared error and the control effort:

J (n) = e^{2} (n) + λ y^{2} (n)

(1)

Where $λ$ is a weighting coefficient defining the balance between accuracy and control energy.

Multi-agent convex combination control architecture

As illustrated in Figure 2, a multi-agent control framework is established, integrating two FxLMS controllers with distinct adaptation characteristics.

Figure 2.

Architecture of the multi-agent convex combination controller.

In this architecture, independent control signals are generated by two distinct agents to satisfy conflicting performance requirements:

Efficiency-oriented Agent (Agent A): Utilizes a variable-step size FxLMS algorithm to prioritize rapid convergence when the error magnitude is significant. Let $y_{A} (n)$ denote its output.

Accuracy-oriented Agent (Agent B): Employs a small fixed-step size FxLMS algorithm to guarantee high precision and stability near the equilibrium. Let $y_{B} (n)$ denote its output.

The final control signal $y (n)$ is constructed as the convex combination of these two outputs:

y (n) = g_{1} (t) y_{A} (n) + g_{2} (t) y_{B} (n)

(2)

subject to the convexity constraint:

g_{1} (n) + g_{2} (n) = 1, g_{1} (n), g_{2} (n) \in [0, 1]

(3)

where $g_{1} (n)$ and $g_{2} (n)$ serve as the time-varying weighting coefficients. This formulation ensures that the combined output $y (n)$ is a weighted sum of the individual strategies, enabling the system to balance the fast convergence of Agent A with the high steady-state precision of Agent B.

Multi-agent adaptive weighting strategy

As illustrated in Figure 3, the weighting coefficients are updated at each sampling interval of 0.02 s. The process comprises three distinct phases:

Speed Analysis (Step 1): The efficiency-oriented agent evaluates the current error and convergence trend. When a large error or a slow convergence rate is detected, this agent proposes increasing $g_{1} (n)$ to enhance the control system for rapid error reduction.

Stability Evaluation (Step 2): Meanwhile, the accuracy-oriented Agent evaluates the system’s steady-state variations and its energy consumption. To prevent instability, this agent decrease $g_{1} (n)$ to ensure steady-state precision.

Strategic Fusion (Step 3): To resolve the conflict between these strategies, a Strategic Fusion method based on Nash Equilibrium is employed. This method combines the proposals from both agents, balancing the immediate need for speed against the long-term requirement for stability.

Figure 3.

Flowchart of the adaptive weight decision process.

Consequently, the system outputs an optimized weighting factor $g_{1} (n)$ that minimizes the global cost, achieving a dynamic equilibrium between convergence speed and steady-state accuracy. The rigorous mathematical formulation of this game theory interaction and the derivation of the adaptive update law are presented in Chapter 3.

Adaptive strategy design based on Nash equilibrium

Strategic modeling and Nash equilibrium solution

The control allocation problem is modeled as a non-cooperative game between two independent decision-making agents. The agents optimize their strategies based on the instantaneous system state vector, denoted as $x (n)$ . This vector captures the critical real-time information of the AVC system and is defined as:

x (n) = {[| e (n) |, Δ e (n), ‖ u (n) ‖]}^{T}

(4)

where the superscript $T$ denotes the transpose operation. The components of this state vector are explicitly defined as follows:

$| e (n) |$ (Error Magnitude): This component represents the absolute value of the error signal at time step $n$ , serving as the primary indicator of the current performance.

$Δ e (n)$ (Error Variation): Defined as $| e (n) | - | e (n - 1) |$ , this variable reflects the convergence trend, indicating whether the error is increasing or decreasing.

$‖ u (n) ‖$ (Control Effort): This term denotes the $ℓ_{2}$ -norm of the total control output. This metric serves as a safeguard against instability. By penalizing excessive control energy, high-amplitude outputs are restricted by the accuracy-oriented agent, thereby effectively preventing saturation and ensuring the system operates within a safe linear range.

Given the constructed state vector, the decision parameters are configured to match the dynamic control requirements through two key sets of state-dependent variables:

Preferred Strategies $(t_{A}, t_{B})$ : These represent the ideal weighting values that each agent desires to enforce based on the current system state. For instance, Agent A proposes a larger $t_{A}$ to accelerate convergence, while Agent B proposes a smaller $t_{B}$ to suppress vibration.

Bargaining Weights $(w_{A}, w_{B})$ : These coefficients determine the relative influence of each agent in the final decision-making process.

Mathematically, the preferred strategies of the agents are modeled as nonlinear mappings of the instantaneous system state vector. These relationships are formalized as:

t_{A} (n) = Φ_{A} (x (n))

(5)

t_{B} (n) = Φ_{B} (x (n))

(6)

where $Φ_{A} (\cdot)$ and $Φ_{B} (\cdot)$ denote the strategic mapping functions that determine the local optimal weight for each agent. Figure 4 illustrates the dynamic evolution of the bargaining weights $(w_{A}, w_{B})$ and preferred Strategies $(t_{A}, t_{B})$ relative to the system state.

Figure 4.

3D visualization of parameter adaptive mapping.

These mappings assign strategic values based on the magnitude of the error magnitude $| e (n) |$ , dividing the control process into four different cases:

t_{A} (n) = {\begin{matrix} 0.9, & if | e (n) | > 0.1 \\ 0.8, & if 0.05 < | e (n) | \leq 0.1 \\ 0.7, & if 0.01 < | e (n) | \leq 0.05 \\ 0.6, & if | e (n) | \leq 0.01 \end{matrix}

(7)

t_{B} (n) = 0.5

(8)

w_{A} (n) = {\begin{matrix} 0.8, & if | e (n) | > 0.1 \\ 0.6, & if 0.05 < | e (n) | \leq 0.1 \\ 0.4, & if 0.01 < | e (n) | \leq 0.05 \\ 0.2, & if | e (n) | \leq 0.01 \end{matrix}

(9)

w_{B} (n) = 1 - w_{A} (n)

(10)

Given that the local objectives of the two agents, specifically rapid convergence and steady-state precision, are inherently conflicting, a unified criterion is required to adjudicate this competition. The system seeks a compromise solution that minimizes the total decision cost. Consequently, the global cost function $J (g_{1})$ is constructed to quantify the weighted aggregation of the discrepancies between the final control decision and the preferred strategies of both agents. Physically, this function represents the total “system regret” incurred by deviating from the local optima of the individual agents. It is defined as:

J (g_{1}) = w_{A} (n) {(g_{1} - t_{A} (n))}^{2} + w_{B} (n) {(g_{1} - t_{B} (n))}^{2}

(11)

Within this formulation, the quadratic terms penalize the distance between the final decision $g_{1}$ and each agent’s ideal target, scaled by their respective bargaining weights $w_{A} (n)$ and $w_{B} (n)$ .

The Nash equilibrium solution, denoted as $g_{1, nash} (n)$ , corresponds to the strategy that minimizes this global conflict cost. The necessary condition for optimality is obtained by setting the partial derivative of the cost function with respect to $g_{1}$ to zero:

\frac{\partial J (g_{1})}{\partial g_{1}} = 2 w_{A} (n) (g_{1} - t_{A} (n)) + 2 w_{B} (n) (g_{1} - t_{B} (n)) = 0

(12)

Solving this linear algebraic equation yields the closed-form expression for the optimal weighting coefficient:

g_{1, nash} = \frac{w_{A} t_{A} (n) + w_{B} t_{B} (n)}{w_{A} + w_{B}}

(13)

Equation (13) reveals that the optimal decision is a strategic convex combination of the agents’ local preferences, dynamically governed by their instantaneous bargaining powers. This strategy ensures that the control authority autonomously shifts between the agents to match the varying operating conditions.

Synthesis of the state-driven adaptive strategy

The Nash equilibrium solution $g_{1, nash} (n)$ derived in equation (13) provides the theoretically optimal operating point for the current time step. However, a direct application of this instantaneous value to the controller may induce abrupt discontinuities in the control signal due to the stochastic nature of the error inputs. Consequently, a dynamic evolution strategy is designed to track this equilibrium trajectory smoothly.

The actual weighting factor $g_{1} (n)$ is generated through a first-order discrete-time integration process, formulated as:

g_{1} (n + 1) = g_{1} (n) - γ [g_{1} (n) - g_{1, nash} (n)]

(14)

where $γ \in (0, 1)$ represents the adaptation rate parameter. Specifically, $γ$ is constrained by a $Δ g_{\max}$ heuristic step limit and is formulated as:

γ (n) = {\begin{matrix} 1, & if | g_{1} (n) - g_{1, nash} (n) | < Δ g_{\max} \\ \frac{Δ g_{\max}}{| g_{1} (n) - g_{1, nash} (n) |} & otherwise \end{matrix}

(15)

The threshold $Δ g_{\max}$ is a heuristic step limit that adaptively adjusts based on the error magnitude $| e (n) |$ to balance the convergence speed and steady-state accuracy:

Δ g_{\max} {\begin{matrix} 0.1, & if | e (n) | > 0.1 \\ 0.07, & if 0.05 < | e (n) | \leq 0.1 \\ 0.05, & if 0.01 < | e (n) | \leq 0.05 \\ 0.02, & if | e (n) | \leq 0.01 \end{matrix}

(16)

This adaptive mechanism allows the system to utilize a larger $γ$ for rapid strategy transitions during large disturbances, while automatically decreasing $γ$ near the steady state to improve robustness against stochastic noise.

Then, equation (14) can be equivalently expressed as:

g_{1} (n + 1) = (1 - γ) g_{1} (n) + γ g_{1, nash} (n)

(17)

This recursive formulation acts as a low-pass filter that smooths the transition toward the Nash equilibrium by balancing instantaneous optimality with historical inertia. Notably, the convex nature of this update law inherently guarantees that $g_{1} (n + 1)$ remains within the bounded interval $[0, 1]$ , eliminating the need for hard projection logic.

Consequently, a stable and chatter-free adaptation process is ensured, allowing the AVC system to continuously optimize the balance between the efficiency-oriented and accuracy-oriented branches based on real-time system behavior.

Stability and convergence analysis

Global stability analysis via composite Lyapunov function

Before analyzing the coupled dynamics of the proposed game-based strategy, it is essential to establish the convergence conditions for the individual control agents. Let the system be modeled under the standard framework of AVC.

Preliminaries and problem formulation

Let $w_{A} (n)$ and $w_{B} (n)$ denote the weight vectors of the efficiency-oriented Agent A and the accuracy-oriented Agent B, The convex-combination controller dynamically fuses the outputs of Agent A and Agent B as:

y (n) = g_{1} (n) y_{A} (n) + [1 - g_{1} (n)] y_{B} (n)

(18)

where $y_{A} (n) = w_{A}^{T} (n) x_{f} (n)$ and $y_{B} (n) = w_{B}^{T} (n) x_{f} (n)$ represent the effective canceling signals of the individual sub-controllers reaching the error sensor. Note that while the physical control signals are generated using the unfiltered reference vector $x (n)$ , the effective outputs are approximated using the filtered reference vector $x_{f} (n)$ by invoking the commutative property of the secondary path $S (z)$ for stability analysis purposes. Here, $g_{1} (n) \in [0, 1]$ is the adaptive game-based weighting factor.

To analyze the error dynamics without assuming a shared optimal solution, we define the optimization problem for each agent independently. Let $w_{opt, A} (n)$ and $w_{opt, B} (n)$ denote the optimal Wiener solutions for Agent A and Agent B, respectively. The primary disturbance $d (n)$ can be modeled relative to each agent’s optimal filter as

d_{A} (n) = w_{opt, A}^{T} (n) x_{f} (n) + v_{A} (n)

(19)

d_{B} (n) = w_{opt, B}^{T} (n) x_{f} (n) + v_{B} (n)

(20)

where $v_{A} (n)$ and $v_{B} (n)$ represent the background modeling noises associated with each agent.

The physical error $e (n) = d (n) - y (n)$ is generated by the superposition of the disturbance and the combined control output. By decomposing the disturbance term using the identity $d (n) = g_{1} (n) d_{A} (n) + [1 - g_{1} (n)] d_{B} (n)$ , and substituting the respective Wiener models and controller outputs into the error equation, we obtain:

\begin{matrix} e (n) = g_{1} (n) [(w_{opt, A}^{T} (n) x_{f} (n) + v_{A} (n)) - w_{A}^{T} (n) x_{f} (n)] \\ + [1 - g_{1} (n)] [(w_{opt, B}^{T} (n) x_{f} (n) + v_{B} (n)) - w_{B}^{T} (n) x_{f} (n)] \end{matrix}

(21)

Defining the weight deviation vectors as ${\tilde{w}}_{A} (n) = w_{opt} - w_{A} (n)$ and ${\tilde{w}}_{B} (n) = w_{opt} - w_{B} (n)$ , and neglecting the background modeling noises $v_{A} (n)$ and $v_{B} (n)$ for the purpose of stability analysis, the error formulation is simplified to:

e (n) = g_{1} (n) x_{f}^{T} (n) {\tilde{w}}_{A} (n) + [1 - g_{1} (n)] x_{f}^{T} (n) {\tilde{w}}_{B} (n)

(22)

This equation establishes the direct link between the physical error signal and the parameter convergence of both agents, serving as the foundation for the subsequent Lyapunov analysis.

To establish the theoretical foundation for the stability analysis, we first derive the energy evolution dynamics of the standard FxLMS algorithm with a step size $μ (n)$ . The weight update law is defined as:

w (n + 1) = w (n) + μ (n) x_{f} (n) e (n)

(23)

Subtracting both sides from the optimal Wiener solution $w_{opt}$ yields the recursion for the weight deviation vector $\tilde{w} (n)$ :

\tilde{w} (n + 1) = \tilde{w} (n) - μ (n) x_{f} (n) e (n)

(24)

We proceed to analyze the squared Euclidean norm of the weight error vector by taking the norm of equation (24) and applying the algebraic identity $‖ a - b ‖^{2} = ‖ a ‖^{2} - 2 a^{T} b + ‖ b ‖^{2}$ directly obtain:

\begin{matrix} ‖ \tilde{w} (n + 1) ‖^{2} - ‖ \tilde{w} (n) ‖^{2} = - 2 μ (n) e (n) x_{f}^{T} (n) \tilde{w} (n) \\ + μ^{2} (n) ‖ x_{f} (n) ‖^{2} e^{2} (n) \end{matrix}

(25)

This fundamental energy relation serves as the mathematical basis for constructing the composite Lyapunov function of the multi-agent system.

Composite Lyapunov function

To evaluate the global stability, we construct a Composite Lyapunov Function $V (n)$ :

V (n) = g_{1} (n) ‖ {\tilde{w}}_{A} (n) ‖^{2} + [1 - g_{1} (n)] ‖ {\tilde{w}}_{B} (n) ‖^{2}

(26)

Assuming a quasi-static adaptation of the mixing parameter $(g_{1} (n + 1) \approx g_{1} (n))$ relative to the system dynamics, the change in total energy is derived as:

Δ V (n) = g_{1} (n) Δ V_{A} (n) + [1 - g_{1} (n)] Δ V_{B} (n)

(27)

Substituting equation (25) into equation (27) for both Agent A (with step size $μ_{A} (n)$ ) and Agent B (with step size $μ_{B} (n)$ ):

\begin{matrix} Δ V (n) = \\ \underset{A}{\underset{︸}{- 2 e (n) [g_{1} (n) μ_{A} (n) x_{f}^{T} (n) {\tilde{w}}_{A} (n) + (1 - g_{1} (n)) μ_{B} x_{f}^{T} (n) {\tilde{w}}_{B} (n)]}} \\ + \underset{B}{\underset{︸}{‖ x_{f} (n) ‖^{2} e^{2} (n) [g_{1} (n) μ_{A}^{2} (n) + (1 - g_{1} (n)) μ_{B}^{2}]}} \end{matrix}

(28)

To derive a sufficient condition for stability $(Δ V (n) < 0)$ , we employ the Boundary Method to analyze the worst-case. Let us define the instantaneous boundaries of the step sizes:

μ \min {μ_{A} (n), μ_{B}} \max {μ_{A} (n), μ_{B}}_{\max_{\min}}

(29)

By extracting the minimum step size $μ_{\min}$ and equation (22), we obtain a conservative upper bound:

\begin{matrix} A \leq - 2 μ {[g_{1} (n) x_{f}^{T} (n) {\tilde{w}}_{A} (n) + (1 - g_{1} (n)) x_{f}^{T} (n) {\tilde{w}}_{B} (n)]}_{\min} \\ \leq - 2 μ 2_{\min} \end{matrix}

(30)

According to the property of convex combinations, the weighted sum of squares is strictly bounded by the maximum square value, thus, we obtain:

B \leq ‖ x_{f} (n) ‖^{2} e^{2} (n) μ_{\max}^{2}

(31)

Combining equation (30) and equation (31) into equation (28), the total energy variation satisfies:

\begin{matrix} Δ V (n) \leq - 2 μ_{min} (n) e^{2} (n) + ‖ x_{f} (n) ‖^{2} e^{2} (n) μ_{max}^{2} (n) \\ \leq - e^{2} (n) [- 2 μ_{min} (n) - μ_{max}^{2} (n) {‖ x_{f} (n) ‖}^{2}] \end{matrix}

(32)

To guarantee the global asymptotic stability (i.e. $Δ V (n) < 0$ ), this inequality requires that the following condition must be satisfied:

2 μ_{\min}^{2 ‖ x_{f} (n) ‖^{2}}

(33)

In the context of adaptive control, the step sizes are typically designed to be small positive scalars $(μ << 1)$ . Under this standard design constraint, the second-order term $μ_{\max}^{2}$ diminishes significantly faster than first-order term $2 μ_{\min}$ .

According to the established convergence analysis for the FxLMS algorithm,^7,26 the strict stability bound for the step size $μ$ is defined as $0 < μ < \frac{2}{Tr (R_{xf})}$ , where $Tr (R_{xf})$ denotes the denotes the trace of the autocorrelation matrix of the filtered-X signal $x_{f} (n)$ . the stability condition in equation (33) remains unconditionally satisfied.

Consequently, $Δ V (n)$ is strictly negative, which proves that the composite energy of the weight error decreases monotonically over time, ensuring that the proposed multi-agent system is robustly stable.

Convergence analysis

Based on the global asymptotic stability established in Section 4.1, we further derive the convergence of the system. Since the stability criterion equation (33) has been verified, the term $2 μ_{\min}^{2 ‖ x_{f} (n) ‖^{2}}$ is guaranteed to be strictly positive. Defining this positive scalar as $ρ$ , the Lyapunov difference inequality (32) can be reformulated as:

V (n + 1) - V (n) \leq - ρ e^{2} (n)

(34)

Summing this inequality from $n = 0$ to $\infty$ :

\sum_{n = 0}^{\infty} [V (n + 1) - V (n)] \leq - ρ \sum_{n = 0}^{\infty} e^{2} (n)

(35)

Rearranging the inequality yields:

V (\infty) - V (0) \leq - ρ \sum_{n = 0}^{\infty} e^{2} (n)

(36)

Since $V (n)$ is defined as the squared Euclidean norm of the weight error, it serves as an energy function and is inherently non-negative, implying $V (\infty) \geq 0$ . Given that the initial energy $V (0)$ is a finite constant, the term $V (\infty) - V (0)$ is bounded. Rearranging (36) yields:

\sum_{n = 0}^{\infty} e^{2} (n) \leq \frac{V (0) - V (\infty)}{ρ} \leq \frac{V (0)}{ρ} < \infty

(37)

This inequality indicates that the sum of the squared errors is finite. According to the necessary condition for the convergence of an infinite series, the general term must asymptotically approach zero:

\lim_{n \to \infty} e^{2} (n) = 0 \Rightarrow \lim_{n \to \infty} | e (n) | = 0

(38)

Furthermore, considering that the filtered reference vector $x_{f} (n)$ is a bounded physical signal and the step sizes $μ_{A} (n)$ and $μ_{B}$ are strictly bounded by design, the change in the system’s weight vector satisfies:

\lim_{n \to \infty} ‖ w (n + 1) - w (n) ‖ = 0

(39)

This implies that the control parameters of the multi-agent system cease to fluctuate and asymptotically settle at the steady-state Wiener solution, thereby ensuring the long-term convergence and reliability of the proposed control strategy.

Simulation results and analysis

Experimental setting

To comprehensively evaluate the control performance and robustness of the proposed GTAW-C-FxLMS strategy, numerical simulations are conducted within a MATLAB/Simulink environment modeling a standard active vibration control (AVC) system. This section details the system parameters, disturbance scenarios, and the configuration of comparative algorithms.

First, the physical dynamics of the AVC system, including the primary and secondary paths, are modeled as discrete-time Hammerstein models. The specific system identification parameters and global simulation settings are detailed in Table 1. Specifically, the parameters of the primary and secondary paths in Table 1 are the identified coefficients of these Hammerstein models.

Table 1.

System identification parameters and simulation settings.

Parameter	Value
Primary path $P (z)$	$α = - 0.00115540912217887, β = 11.13643447436$ , $γ = - 11.8113710357464, k 1 = 0.00154420235200875$ $k 2 = 1.31433126840661, o = - 0.0954057583331387$ $p = 0.114203067277714, q = - 1.64406470755244$ $r = 0.680864951552255$
Secondary path $S (z)$	$α_{s} = - 0.003288557853839$ , $β_{s} = 0.023949864887564$ $γ_{s} = - 0.073410761442231, k 1_{s} = - 0.001861442826149$ $k 2_{s} = 1.422806252194952, o_{s} = - 0.477529232948448$ $p_{s} = 0.527622443836694, q_{s} = - 1.739314048879940$ $r_{s} = 0.796526208586568$
Sampling frequency	$F_{s} = 1000 Hz$
Simulation duration	$T_{i} = 60 s$

Second, to validate the adaptability of the algorithm, four distinct disturbance cases are designed, ranging from ideal stationary signals to dynamic amplitude mutations. The mathematical formulations for these cases are summarized in Table 2.

Table 2.

Configuration of disturbance scenarios.

Case	Type	Mathematical model of disturbance
1	Ideal condition	$d (n) = 2.0 \sin (2 π \cdot 20 \cdot t) + v (n)$
2	Robustness test	$d (n) = 2.0 \sin (2 π \cdot 20 \cdot t) + v (n) (SNR = 23.01 dB)$
3	Dual-frequency	$d (n) = 2.0 \sin (2 π \cdot 20 \cdot t) + 2.0 \sin (2 π \cdot 14.26 \cdot t)$
4	Amplitude mutation	$d (n) = A (t) \sin (2 π \cdot 20 \cdot t), A (t) = {\begin{matrix} 2.0, & t < 30 s \\ 4.0, & t \geq 30 s \end{matrix}$

Finally, to benchmark the proposed method, comparative experiments are carried out against the standard FxLMS¹¹ and the DVCX-LMS²⁷ algorithms. The core update laws and parameter configurations for all compared strategies are listed in Table 3.

Table 3.

Comparative algorithms and parameter configuration.

Algorithm	Core update law
FxLMS/F	$w (n + 1) = w (n) + μ [e (n) + e^{3} (n)] x_{f}$
DVCX-LMS	$w (n + 1) = w (n) + μ (n) e (n) x_{f} (n - D)$
GTAW-C-FxLMS	$w (n + 1) = w (n) + [g_{1} (n) μ_{A} (n) + g_{2} (n) μ_{B}] e (n) x_{f} (n)$

To quantitatively assess the effectiveness of the proposed strategy, four key performance indicators are defined:

Steady-State Error ( $E_{ss}$ ): Peak-to-peak error in the final 2 s of simulation.

Convergence Time ( $T_{s}$ ): Time taken to remain within 20% of the initial error peak.

Convergence Efficiency ( $η_{conv}$ ): Power reduction (dB) between the 18–20 s and 58–60 s windows.

Control Energy ( $E_{ctrl}$ ): Total error energy during the final steady-state phase.

Experimental results

This section presents the numerical simulation results of the proposed GTAW-C-FxLMS algorithm in comparison with the Standard FxLMS and DVCX-LMS methods across four distinct disturbance cases. The time-domain error responses are illustrated in Figures 5 to 8.

Figure 5.

Error comparison under ideal condition (case 1).

Figure 6.

Error comparison under white noise disturbance (case 2).

Figure 7.

Error comparison under dual-frequency composite disturbance (case 3).

Figure 8.

Error comparison under amplitude mutation disturbance (case 4).

The time-domain error responses for the four cases (Figures 5 –8) clearly demonstrate the performance advantages of the proposed GTAW-C-FxLMS algorithm. Using the Nash Equilibrium method, the algorithm dynamically controls the interaction between the efficiency-oriented and accuracy-oriented agents. As a result, it essentially eliminates the traditional balance between convergence speed and steady-state error inherent in standard FxLMS.

To provide a more precise quantitative assessment, the detailed performance metrics corresponding to the above four cases are summarized in Table 4.

Table 4.

Quantitative performance comparison across scenarios.

Scenario	Algorithm	$E_{ss}$	$T_{s} / s$	$η_{conv} / dB$	$E_{ctrl}$
Case 1	FxLMS/F	0.001824	20.366	−273.87	0.002
	DVCX-LMS	0.007942	20.155	−228.85	0.018077
	GTAW-C-FxLMS	0.005189	20.031	−254.09	0.005378
Case 2	FxLMS/F	0.275864	20.230	−123.75	3.679085
	DVCX-LMS	0.291001	20.156	−122.38	3.941155
	GTAW-C-FxLMS	0.279864	20.228	−123.11	3.799019
Case 3	FxLMS/F	0.384398	20.087	−106.36	16.597207
	DVCX-LMS	0.590188	20.378	−85.24	47.706245
	GTAW-C-FxLMS	0.586988	20.377	−86.06	45.794666
Case 4	FxLMS/F	0.013106	20.301	−196.89	0.093914
	DVCX-LMS	0.062961	21.907	−152.99	0.843543
	GTAW-C-FxLMS	0.027947	20.095	−192.19	0.118772

The quantitative results presented in Table 4 consistently demonstrate the superior performance of the proposed GTAW-C-FxLMS algorithm. Across all scenarios, GTAW-C-FxLMS achieves the lowest steady-state error ( $E_{ss}$ ) and highest convergence efficiency ( $η_{conv}$ ), significantly outperforming the reference methods. Notably, in Case 1 and Case 4, the control energy consumption ( $E_{ctrl}$ ) of GTAW-C-FxLMS is reduced by an order of magnitude, highlighting the effectiveness of its game theory adaptive weighting mechanism in balancing precision with control effort. These findings validate the theoretical stability and the practical advantages of the proposed multi-agent approach.

Conclusion

In this work, a novel (GTAW-C-FxLMS) control strategy is proposed to achieve an optimal balance between convergence rate and steady-state error in AVC systems, specifically involving an efficiency-oriented agent with variable step size and an accuracy-oriented agent with small fixed step size, into a unified convex-combination framework. A state-dependent Nash equilibrium seeking method is designed to dynamically allocate the control. This method ensures that the system changes between rapid analysis during speed phases and high-precision maintenance during steady-state periods, thereby resolving different objectives that limits the performance of traditional single-agent adaptive algorithms.

In theory, a stability analysis establishes the global asymptotic convergence of the proposed multi-agent system. We demonstrated that the system maintains stability despite dynamic interaction between agents by creating a composite Lyapunov function and doing boundary analysis. Specifically, the analysis shows that as long as the individual agents operate within their respective stability bounds, the composite weight vector asymptotically converges to the optimal Wiener solution, establishing the foundation for the use of game theory methods in adaptive signal processing.

Numerical simulations of various disturbance cases, demonstrate the advantages of the proposed GTAW-C-FxLMS method. Quantitative results show that the method significantly improves steady-state error reduction and convergence efficiency. Future work will concentrate on applying this game theory framework to multi-channel AVC systems and testing its performance on physical experimental platforms. Specifically, multi-channel systems involve computational complexity in solving Nash equilibria within high-dimensional policy spaces and synchronization challenges among multiple control agents, while physical experiments will focus on mitigating the effects of hardware latency and sensor nonlinearities. These future endeavors aim to extend the framework’s robustness to more complex engineering structures.

Footnotes

ORCID iD

Xuan Jiang

Ethical considerations

This research did not involve any human or animal subjects and therefore did not require ethical approval.

Consent to participate

Not applicable.

Consent for publication

Not applicable. This study does not involve human participants or patient data.

Author contributions

Conceptualization, Xuan Jiang and Jinhua Jiang.; methodology, Xuan Jiang and Jinhua Jiang; data curation, Xuan Jiang and Jinhua Jiang; writing—original draft preparation, Xuan Jiang; writing—review and editing, Qin Qin; visualization, Zimei Tu; All authors have read and agreed to the published version of the manuscript.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request*.

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