A variable step-size FxLMS algorithm for active vibration control of cantilever beam

Abstract

The conventional filtered-x least mean square (FxLMS) algorithm for active vibration control (AVC) is limited by the trade-off between convergence rate and steady-state error and by its sensitivity to abrupt input changes. To address these issues, a nonlinear step-size adjustment mechanism combining a bounded hyperbolic tangent mapping with a normalized variable step-size strategy is developed, based on which a variable step-size FxLMS algorithm, termed VSS-FxtanhLMS, is proposed. The convergence properties of the algorithm are analyzed in the mean and mean-square senses. For AVC of a flexible cantilever beam, the proposed algorithm is evaluated on a co-simulation platform established using ADAMS and Simulink under single-frequency, dual-frequency, and amplitude-step excitations, and is further validated on an experimental AVC platform. Co-simulation results show that VSS-FxtanhLMS achieves the smallest steady-state error, the lowest mean squared error, and the shortest convergence time under both single-frequency and dual-frequency excitations, while remaining convergent under amplitude-step excitation. In experiments, compared with FxasinhLMS, the strongest competing method, VSS-FxtanhLMS reduces the steady-state error by 17.48%, shortens the convergence time by 33.33%, and reduces the power spectral density by 1.72 dB/Hz. These results indicate that VSS-FxtanhLMS achieves improved steady-state accuracy, convergence behavior, and robustness under both simulation and practical conditions.

Keywords

variable step-size FxLMS hyperbolic tangent function active vibration control co-simulation experimental validation flexible cantilever beam

1 Introduction

With the rapid development of aerospace engineering and smart structural systems, lightweight flexible structures are increasingly used in modern spacecraft and precision engineering applications. To meet the requirements of lightweight design, high performance, and multifunctional integration, many aerospace components are intentionally designed with high flexibility and are often integrated with piezoelectric materials for sensing and actuation.¹ However, owing to their low stiffness and low intrinsic damping, these structures are highly susceptible to external disturbances and may exhibit persistent low-frequency vibrations, especially in vacuum or weakly damped environments.² Such vibrations can degrade structural stability and control accuracy and may further reduce the service life and reliability of critical components. Therefore, effective vibration suppression of flexible structures is of considerable importance in both theory and engineering practice.

To suppress such vibrations, passive vibration control has been widely studied. However, passive methods are often ineffective for low-frequency vibration suppression and lack sufficient adaptability under varying operating conditions. In contrast, AVC provides higher control flexibility and better suppression capability for low-frequency disturbances.³ Among various AVC approaches, adaptive-filtering-based methods have attracted significant attention because they do not require an accurate model of the controlled plant and can adapt to time-varying operating conditions. In particular, the filtered-x least mean square (FxLMS) algorithm has become one of the most widely used methods in active noise and vibration control due to its simple structure, low computational complexity, effective performance, and ease of implementation.⁴

Despite these advantages, the conventional FxLMS algorithm still suffers from two fundamental limitations. One is the inherent trade-off between convergence rate and steady-state error under a fixed step size. A large step size can accelerate convergence but usually increases steady-state misadjustment, whereas a small step size improves steady-state accuracy at the expense of convergence speed. The other is its limited capability to maintain stable and accurate adaptation under abrupt variations in the input signal. In practical vibration control systems, excitation signals are often nonstationary and may undergo sudden amplitude changes. Under such conditions, the conventional FxLMS algorithm may exhibit degraded convergence behavior or even instability. These limitations indicate that the fixed-step-size structure and insufficient nonlinear response to rapidly varying excitation still restrict the overall control performance of the conventional FxLMS algorithm.

To improve the performance of FxLMS, existing studies have mainly proceeded along two directions. One direction focuses on nonlinear error transformation and related cost-function modifications. Gu et al.⁵ proposed the FxatanLMS algorithm for impulsive-noise suppression, while Wu et al.^6,7 developed the FxlogLMS and Fair algorithms. These methods mainly improve robustness by reshaping the error response and reducing the influence of large-amplitude disturbances. In addition, Padhi et al.⁸ proposed FxwaLMS to improve tracking performance, whereas Cheer and Elliott⁹ developed LFxLMS to enhance numerical stability under specific operating conditions. The other direction focuses on variable step-size (VSS) strategies, in which the step size is adjusted online to balance convergence rate and steady-state error. Representative examples include NASFSxLMS proposed by Meng et al.,¹⁰ the constant forgetting factor scheme proposed by Kwong,¹¹ the inverse-cosine-based improvement proposed by Fang et al.,¹² and the normalized sinusoidal VSS strategy proposed by Lan et al.¹³ FxLMS-based AVC methods have also been investigated in piezoelectric structures and cantilever-type systems,^14–18 further demonstrating the engineering relevance of this class of algorithms. In summary, nonlinear-error-transformation methods enhance robustness but retain the fixed-step-size limitation, whereas VSS methods improve the trade-off between convergence speed and steady-state error but usually lack explicit handling of abrupt input-amplitude variations.

Recent studies have further shown that model uncertainty, secondary-path nonlinearity, actuator hysteresis, and parameter variations can noticeably affect the robustness of adaptive AVC systems. For example, Pu et al. investigated adaptive AVC of piezoelectric smart structures with online hysteresis identification and compensation.¹⁹ Jiang et al. applied a VSS-FxLMS algorithm to active vibration control of blades and verified its effectiveness through co-simulation and experiments.²⁰ Umar et al. studied MFC-actuated cantilever-beam vibration control using hysteresis modeling and a VSS-FxLMS control algorithm.²¹ These studies indicate that recent AVC research has increasingly emphasized variable-step-size adaptation and uncertainty effects. However, under abrupt input-amplitude variations, simultaneously improving convergence rate, steady-state accuracy, and robustness remains insufficiently addressed.

To address this issue, this paper proposes a variable step-size FxLMS algorithm, termed VSS-FxtanhLMS, for AVC of a flexible cantilever beam. Rather than relying solely on nonlinear error transformation, the proposed method combines a bounded hyperbolic tangent nonlinear mapping with a normalized VSS mechanism. Unlike the Fair, FxatanLMS, and FxasinhLMS algorithms, which mainly rely on nonlinear error transformation, the proposed algorithm jointly adjusts the step-size factor according to the reference input and the instantaneous error, so that both convergence performance and robustness can be improved under varying excitation conditions.

The main contributions of this paper are summarized as follows:

1.A VSS-FxtanhLMS algorithm is proposed by combining a hyperbolic tangent nonlinear transformation with a normalized VSS mechanism.

2.The convergence properties of the proposed algorithm are analyzed in the mean and mean-square senses.

3.A co-simulation and experimental framework for a flexible cantilever beam is established to evaluate the overall performance of the proposed algorithm in terms of convergence rate, steady-state error, and robustness.

2 Filtered-X least mean square (FxLMS) algorithm

The conventional least mean square (LMS) algorithm minimizes the instantaneous squared error by employing the steepest descent method,²² which endows it with excellent computational simplicity and real-time processing efficiency. However, in practical AVC systems, the secondary-path dynamics between the control actuator and the error sensor introduce phase distortion, which can seriously compromise the stability of the direct LMS gradient estimate. To compensate for this inherent delay and phase shift, the FxLMS algorithm incorporates a secondary path model to prefilter the reference signal, thereby correctly aligning the gradient estimation and ensuring stable convergence.²³ The overall workflow of the present study is shown in Figure 1, and the fundamental block diagram of the FxLMS algorithm is presented in Figure 1(a).

In Figure 1(a), $P (z)$ is the transfer function from the input signal to the error sensor, called the primary path transfer function; $S (z)$ is the secondary path transfer function, and $\hat{S} (z)$ is its estimate; $x (n)$ is the input signal; $d (n)$ is the desired signal; $x_{f} (n)$ is the filtered-x input signal obtained after $x (n)$ is filtered by $\hat{S} (z)$ ; $u (n)$ is the adaptive filter output signal; $y (n)$ is the output signal of $u (n)$ after the secondary path; and $e (n)$ is the corrected error signal. It is assumed that the transfer function coefficients of $S (z)$ and $\hat{S} (z)$ are exactly the same in the ideal state.

For the adaptive FIR filter, the control output at discrete time index $n$ is given by:

u (n) = w^{T} (n) x (n)

(1)

where

w (n)

denotes the adaptive-filter weight vector,

x (n) = {[x (n), x (n - 1), \dots, x (n - M + 1)]}^{T}

denotes the reference input vector,

M

is the adaptive-filter length, and

{(\cdot)}^{T}

epresents the transpose operation.

y (n) = s (n) * u (n)

(2)

where * denotes the convolution operation.

x_{f} (n) = \hat{s} (n) * x (n)

(3)

where

x_{f} (n)

denotes the scalar filtered reference signal. The corresponding filtered reference input vector used in the adaptive update is defined as

x_{f} (n) = {[x_{f} (n), x_{f} (n - 1), \dots, x_{f} (n - M + 1)]}^{T}

e (n) = d (n) - y (n)

(4)

By substituting Eqs. (1)-(3) into Eq. (4), the error signal $e (n)$ can be expressed as:

e (n) = d (n) - w^{T} (n) x_{f} (n)

(5)

The mean square error of the system, denoted as $ε (n)$ , is given by:

ε (n) = E [e^{2} (n)] = E {{[d (n) - w^{T} (n) x_{f (n)}]}^{2}}

(6)

From Eq. (6), it can be seen that the mean squared error (MSE) is a quadratic function²⁴ of the weight vector $w (n)$ , and the optimal weight vector is $w_{*}$ when the MSE is minimized.

w_{*} = {R_{f}}^{- 1} p_{f}

(7)

R_{f} = E [x_{f} (n) x_{f}^{T} (n)]

(8)

p_{f} = E [x_{f} (n) d (n)]

(9)

where

R_{f}

is the auto correlation matrix of the filtered reference input vector

x_{f} (n)

, and

p_{f}

is the cross-correlation vector between

x_{f} (n)

and primary disturbance signal

d (n)

Substituting Eq. (7) into Eq. (5) yields the optimal error signal $e_{*} (n)$ as:

e_{*} (n) = d (n) - {w_{*}}^{T} x_{f} (n)

(10)

As shown in Eq. (7), the optimal weight vector $w_{*}$ can be calculated directly from $R_{f}$ and $p_{f}$ . However, practical implementation requires a large amount of statistical data and computationally expensive matrix inversion operations, which is prohibitive for real-time AVC systems. To circumvent these challenges, the FxLMS algorithm adopts a gradient-based iterative weight update law, which is given by:

w (n + 1) = w (n) + μ e (n) x_{f} (n)

(11)

where

μ

denotes the step-size parameter that governs the convergence speed and stability of the adaptive algorithm.

3 Improved FxLMS algorithm

The weight update of the conventional FxLMS algorithm is strongly affected by the step-size parameter $μ$ and the error signal $e (n)$ . To improve the balance among convergence rate, steady-state error, and robustness, this paper introduces two modifications: a nonlinear transformation of the error signal $e (n)$ and a VSS strategy.

3.1 Error signal transformation

In this paper, we propose the FxtanhLMS algorithm, in which the hyperbolic tangent function is adopted to nonlinearly transform the error signal. The hyperbolic tangent function is a continuous, bounded, and monotonically increasing odd function whose range is (-1,1), In addition, $\tan h (x) \to 0$ as $x \to 0$ . Owing to these properties, the transformed error signal is confined to a bounded range. The transformation function is defined as follows:

ϕ [e (n)] = \frac{2 \tan h (e^{2} (n))}{e (n) + δ}

(12)

ϕ (e (n)) = {\begin{cases} ϕ (e (n)), & | ϕ (e (n)) | \leq b, \\ b s g n (ϕ (e (n))), & | ϕ (e (n)) | > b . \end{cases}

(13)

Where

\tan h

denotes the hyperbolic tangent function;

b

denotes the prescribed threshold; and

δ > 0

is a very small constant introduced to avoid division by zero.

3.2 VSS algorithm

In the FxLMS algorithm, the step-size parameter $μ$ directly determines the trade-off between convergence rate and steady-state error. A larger $μ$ accelerates convergence but may increase steady-state fluctuation or even cause instability, whereas a smaller $μ$ improves steady-state accuracy at the cost of slower convergence. This limitation becomes more pronounced when the input amplitude changes abruptly, because the energy of the filtered reference input may increase rapidly and enlarge the weight-update term. Therefore, a normalized variable step-size strategy is introduced to regulate the adaptation gain according to the transformed error signal while suppressing the influence of large filtered-reference input energy.The proposed normalized step-size function is defined as

μ (n) = \frac{α {(1 - \exp [- | ϕ (e (n)) |])}^{1 / p}}{β {‖ x_{f} (n) ‖}_{2}^{2} + σ}

(14)

Where

p

denotes the filter order;

α

controls the overall step-size magnitude; and

β

controls the normalization strength with respect to the filtered-reference input energy; and

σ > 0

is a very small constant introduced to avoid division by zero.In this study, p=8,

α

=5,

β

=0.3, and

σ

=0.001are used.

When the above VSS strategy is incorporated into the FxtanhLMS algorithm, the resulting method is termed the VSS-FxtanhLMS algorithm. The weight update equation of the proposed algorithm can be written as:

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

(15)

3.3 Convergence analysis of the VSS-FxtanhLMS algorithm

To clarify the convergence behavior of the proposed VSS-FxtanhLMS algorithm, its mean and mean-square convergence properties are analyzed in this section. The mean convergence analysis is used to examine whether the adaptive weight vector can approach its optimal value on average, while the mean-square analysis is used to evaluate whether the residual error energy remains bounded. Since the proposed step size $μ (n)$ varies with the transformed error signal and the filtered-reference input energy, it is assumed to vary more slowly than the instantaneous error and input signals. Under this commonly used independence assumption, $μ (n)$ , $e (n)$ , $x_{f} (n)$ , and $w (n)$ are treated as mutually independent.For FxLMS-based adaptation, the filtered reference input vector $x_{f} (n)$ is used in the weight update. Therefore, the auto correlation matrix of the filtered reference input vector is defined as

R_{f} = E [x_{f} (n) x_{f}^{T} (n)] = Q Λ Q^{T}

(16)

Where

Λ

is the diagonal matrix of eigenvalues, and

Q

is the corresponding eigenvector matrix of

R_{f}

3.3.1 Dynamic analysis in the mean sense

To investigate the convergence behavior of the step-size parameter, the tracking error of the adaptive filter weights is analyzed in the mean sense.

V^{'} (n + 1) = [I - μ (n) x^{'} (n) x^{'} T (n)] V^{'} (n) + μ (n) x^{'} (n)

(17)

V (n) = w (n) - w_{*}

(18)

V^{'} (n) = Q^{T} V (n)

(19)

x^{'} (n) = Q^{T} x (n)

(20)

Usually, the variation rate of the step size $μ (n)$ is much smaller than that of the error signal $e (n)$ and the input signal $x (n)$ . Therefore, it is assumed that the step size $μ (n)$ , the error signal $e (n)$ , the input signal $x (n)$ , and the adaptive filter weight vector $w (n)$ are mutually independent. Accordingly, the expectations of $V^{'} (n + 1)$ and $V (n + 1)$ can be expressed as follows:

E [V^{'} (n + 1)] = E {[I - μ (n) x^{'} (n) x'^{T} (n)] V^{'} (n)} + E [μ (n) x^{'} (n)]

(21)

E [V (n + 1)] = E {[I - μ (n) x (n) x^{T} (n)] V (n)}

(22)

For the stability of Eq. (17), the following conditions must hold:

\prod_{n = 0}^{N} {I - E [μ (n) R]} \to 0 n \to \infty

(23)

Therefore, a sufficient condition for the convergence of the average weight vector is:

0 < E [μ (n)] < \frac{2}{λ_{\max}}

(24)

Where

λ_{\max}

denotes the largest eigenvalue of matrix

R

Accordingly, Eq. (24) can be rewritten as:

{μ \frac{2}{λ_{\max}}}_{\max}

(25)

Where

μ_{\max}

denotes the maximum allowable value of the step size

μ (n)

3.3.2 Dynamic analysis in the mean square sense

The upper bound of the step size derived from the stability analysis in the mean sense is relatively conservative. To establish a more precise and practical stability criterion, further analysis in the mean-square sense is required. Accordingly, the MSE of the error signal can be expressed as:

ε (n) = E [e^{2} (n)] = E [e (n) e^{T} (n)]

(26)

From Eq. (18), the following relation holds at time instant

n + 1

V (n + 1) = w (n + 1) - w_{*}

(27)

Combining Eqs. (15) and (27), the weight-error recursion can be obtained as:

V (n + 1) = w (n + 1) - w_{*} = w (n) - w_{*} + μ (n) ϕ (e (n)) x_{f} (n) = V (n) + μ (n) ϕ (e (n)) x_{f} (n)

(28)

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

(29)

According to Eqs. (3) and (5), the error signal can be expanded as:

e (n) = d (n) - w^{T} (n) x_{f} (n)

(30)

Substituting Eqs. (3), (10) and (18) into Eq. (30) yields

e (n) = e_{*} (n) - V^{T} (n) x_{f} (n)

(31)

Substituting Eq. (31) into Eq. (26) yields

ε (n) = E [{(e_{*} (n) - V^{T} (n) x_{f} (n))}^{2}]

(32)

Then, Eq. (32) can be simplified as:

ε (n) = E [{e_{*}}^{2} (n)] + E [V^{T} (n) R f V (n)] = ε_{*} (n) + ε_{e x} (n)

(33)

Where

ε_{*} (n)

denotes the minimum MSE, and

ε_{e x} (n)

denotes the steady-state residual error of the algorithm, which influences the steady-state behavior of the error-signal MSE.

ε_{e x} (n + 1) = E [V^{T} (n + 1) R f V (n + 1)]

(34)

Substituting Eq. (28) into Eq. (34) yields

ε_{e x} (n + 1) = {E [(V (n) + μ (n) ϕ (e (n)) x f (n))}^{T} R f (V (n) + μ (n) ϕ (e (n)) x f (n))]

(35)

For Eq. (35), a sufficient condition to guarantee the convergence of the algorithm in the mean-square sense is:

0 < \frac{E [μ^{2} (\infty)]}{E [μ (\infty)]} < \frac{2}{3 t r (R)}

(36)

Where

E [μ^{2} (\infty)]

and

E [μ (\infty)]

denote the steady-state values of

E [μ^{2} (n)]

and

E [μ (n)]

, respectively, and

t r (R)

is the trace of the auto correlation matrix

R

, namely, the sum of its diagonal elements.

E [μ (\infty)] = E [\frac{α {(1 - e^{- | ϕ [e (\infty)] |})}^{\frac{1}{p}}}{β {x_{f}}^{2} + σ}] = \frac{α E [{(1 - e^{- | ϕ [e (\infty)] |})}^{\frac{1}{p}}]}{β E^{2} [x_{f}] + σ}

(37)

Let $A_{\infty} = {(1 - \exp [- | ϕ (e (\infty)) |])}^{1 / p}, B_{\infty} = β {‖ x_{f} (\infty) ‖}_{2}^{2} + σ,$ then the steady-state step size can be written as $μ (\infty) = α A_{\infty} / B_{\infty} .$

E [μ^{2} (\infty)] = E [{(\frac{α A_{\infty}}{B_{\infty}})}^{2}] = α^{2} E [\frac{A_{\infty}^{2}}{B_{\infty}^{2}}]

(38)

E [μ (\infty)] = E [\frac{α A_{\infty}}{B_{\infty}}] = α E [\frac{A_{\infty}}{B_{\infty}}]

(39)

The ratio between the second- and first-order moments of the steady-state step size is defined as

γ = \frac{E [μ^{2} (\infty)]}{E [μ (\infty)]} = α \frac{E [A_{\infty}^{2} / B_{\infty}^{2}]}{E [A_{\infty} / B_{\infty}]}

(40)

Therefore, to ensure mean-square convergence, the ratio

γ

should satisfy

0 < γ < \frac{2}{3 t r (R_{f})}

(41)

Since

0 < A \infty \leq 1

and

σ

is sufficiently small, a sufficient practical condition for parameter selection can be expressed as

0 < \frac{α}{β} < \frac{{2 | x_{f} (\infty) |}_{2}^{2}}{3 t r (R_{f})}

(42)

From Eq. (42), it can be seen that the two parameters $α$ and $β$ in the VSS-FxtanhLMS algorithm are related to the input signal. if these two parameters are properly chosen, the algorithm can achieve convergence.

4 Co-simulation system for AVC of a flexible cantilever beam

4.1 Construction of the cantilever beam model

Under external excitation, a cantilever beam system exhibits vibration deformation.²⁵ To suppress the resulting vibration, a control force is applied near the excitation source. In this paper, a flexible cantilever beam is investigated, and a finite element model is established in Adams, as shown in Figure 1(b). The corresponding material properties are given in Table 1.

Table 1.

Geometric and material parameters of the cantilever beams used in the co-simulation and experiments.

Parameter	Cantilever beam	Experimental beam
Length/width/thickness (mm)	900/25/1	500/70/1
Density (kg/m³)	7.80 × 10³	8.00 × 10³
Mass (kg)	—	0.14
Elastic modulus (GPa)	207	190
Yield strength (MPa)	—	206
Poisson’s ratio	0.29	0.29

Under the given initial conditions, the flexible cantilever beam is excited by an external force applied near its free end. Modal analysis indicates that a pronounced vibration response is observed near the middle section of the beam. Accordingly, a control force is introduced at this location to counteract the excitation and accelerate vibration suppression. In addition, five marker points are arranged along the beam span from the fixed end to the free end, and their displacements in the x-direction are recorded.

The co-simulation requires data exchange between Adams and Simulink. First, a control interface is established in Adams to define the system inputs and outputs. The two control forces applied at the free end and the middle section of the cantilever beam are specified as the input variables of Adams, while the displacements of the selected marker points along the x-axis are defined as the output variables. Second, the external force applied at the free end of the cantilever beam is replaced by a real-time input signal generated from Simulink. Finally, the data communication between Adams and Simulink is realized by importing the cantilever beam model into Simulink.

4.2 Construction of the Co-Simulation system

To verify the FxLMS-based AVC system in simulation, a co-simulation platform is constructed using ADAMS and Simulink. The ADAMS/Simulink flexible cantilever beam model serves as the primary path, while the estimated secondary path $\hat{S} (z)$ is used to generate the filtered reference signal for the FxLMS update. The estimated secondary path is represented by a discrete-time transfer-function model rather than an FIR filter. The adaptive controller is implemented using a MATLAB Function block, and the ADAMS model is imported into Simulink as the mechanical module. An enabled subsystem is introduced so that control is activated only after the vibration response reaches a stable state.

In the co-simulation, the filtered reference signal is used by the FxLMS, Fair, LFxLMS, FxatanLMS, NASFSxLMS, and proposed VSS-FxtanhLMS algorithms to generate the control force, which is applied at the designated control location to suppress the vibration response.

5 Simulation and experimental verification

To validate the proposed VSS-FxtanhLMS algorithm, co-simulation and experimental studies are conducted for AVC of the flexible cantilever beam. The co-simulation evaluates convergence, steady-state performance, and robustness under different excitation conditions, while the experiment verifies practical vibration suppression performance.

5.1 Simulation validation and results analysis

A co-simulation platform for AVC of the flexible cantilever beam is established using Simulink and ADAMS. Six algorithms are compared, namely FxLMS, Fair, LFxLMS, FxatanLMS, NASFSxLMS, and the proposed VSS-FxtanhLMS. In the co-simulation model, the ADAMS/Simulink flexible cantilever beam model serves as the primary path. The estimated secondary path $\hat{S} (z)$ , which is used to generate the filtered reference signal in the FxLMS update, is represented by a first-order discrete-time transfer-function model: $\hat{S} (z) = \frac{3.983}{1 - 0.916 z^{- 1}}$ .The adaptive input vector is constructed using the current sample and seven delayed samples; therefore, the adaptive filter order is set to $p = 8$ . The sampling frequency is 1 kHz. The threshold parameter of the Fair algorithm is set to $c = 10$ , and the parameters of the VSS-FxtanhLMS algorithm are chosen as $b = 0.2$ , $δ = 0.001$ , $p = 8$ , $α = 5 \times 1 0^{- 5}$ , $β = 0.3$ , and $σ = 0.001$ . The excitation frequencies are selected from the first two natural frequencies of the cantilever beam, and three excitation conditions are considered: single-frequency excitation, dual-frequency excitation, and amplitude-step excitation.

5.1.1 Single-frequency excitation

To evaluate the vibration suppression performance of the cantilever-beam active control co-simulation system under low-frequency excitation, a sinusoidal excitation signal is adopted. Specifically, the excitation frequency is set to the first-order natural frequency of the cantilever beam. Because the output range of the piezoelectric actuator is limited, the excitation amplitude is adjusted to an appropriate level. Accordingly, the input to the co-simulation system is set as a sinusoidal signal with an amplitude of 1 V and a frequency of 14 Hz. The system runs in an uncontrolled state for 50 s before control is activated, and the responses of the six algorithms are shown in Figure 2(a). The results indicate that the proposed VSS-FxtanhLMS algorithm achieves the fastest convergence rate.

In AVC, algorithm performance is commonly evaluated using two criteria: steady-state error and convergence rate. In this paper, the response data over the last 10 s of each simulation are extracted, and the mean of the corresponding peak values is taken as the steady-state error metric. The convergence rate is quantified by the time required for the system response to decrease from its initial state to 10% of the maximum observed response. To provide a more objective and comprehensive comparison, the MSE computed from the peak values of the sampled response is further adopted as an additional performance index.

Table 2 summarizes the steady-state error, MSE, and convergence time of the FxLMS, Fair, LFxLMS, FxatanLMS, NASFSxLMS, and proposed VSS-FxtanhLMS algorithms obtained from the co-simulation study. As shown in Table 2, the proposed VSS-FxtanhLMS algorithm achieves a steady-state error of 9.96×10⁻⁶, an MSE of 6.18×10⁻⁷, and a convergence time of 40 s. All three indices are lower than those of the other five algorithms. This indicates that, under single-frequency excitation, the proposed algorithm achieves both lower steady-state error and faster convergence than the other five algorithms.

Table 2.

Steady-state error, MSE and convergence time of six algorithms under single-frequency and dual-frequency input signal.

Types of algorithms	Single-frequency input signal			Dual-frequency input signal
Types of algorithms	Steady-state error(V)	Mean squared error(V)	Convergence time(s)	Steady-state error(V)	Mean squared error(V)	Convergence time(s)
FxLMS	1.83 × 10^-3	1.70 × 10^-2	160	1.10 ×10^-3	4.73 × 10^-2	180
Fair	1.39 × 10^-3	9.73 × 10^-3	135	1.04 × 10^-3	4.19 × 10^-2	165
LFxLMS	1.09 × 10^-3	5.95 × 10^-3	115	9.71 × 10^-4	3.68 × 10^-2	135
FxatanLMS	6.34 × 10^-4	2.03 × 10^-3	105	8.68 × 10^-4	2.94 × 10^-2	120
NASFSxLMS	5.78 × 10^-5	1.69 × 10^-5	45	1.71 × 10^-4	1.16 × 10^-2	70
VSS-FxtanhLMS	9.96 × 10^-6	6.18 × 10^-7	40	1.62 × 10^-4	1.05 × 10^-2	50

5.1.2 Dual-frequency excitation

Excitation in practical environments is often complex and commonly contains dual-frequency components. To evaluate vibration control performance under dual-frequency excitation, a composite excitation signal is adopted. Specifically, the vibration excitation is constructed as the superposition of two sinusoidal signals at the first-order and second-order natural frequencies of the cantilever beam. Because the output range of the piezoelectric actuator is limited, the excitation amplitude is adjusted to an appropriate level. Accordingly, the input to the co-simulation system is set as the superposition of two sinusoidal signals, each with an amplitude of 0.5 V and frequencies of 14 Hz and 87 Hz, respectively. The system runs in an uncontrolled state for 50 s before control is activated, and the responses of the six algorithms are shown in Figure 2(b). From these results, the steady-state error, MSE, and convergence time are calculated and summarized in Table 2.

As shown in Table 2, under dual-frequency excitation, the proposed VSS-FxtanhLMS algorithm achieves a steady-state error of 1.62×10⁻⁴, an MSE of 1.05×10⁻², and a convergence time of 50 s. Although the increased input complexity leads to larger residual error and longer convergence time than the single-frequency case, the proposed algorithm still achieves the smallest steady-state error, the lowest MSE, and the shortest convergence time among the compared methods. This result indicates that the proposed algorithm maintains stable convergence and low residual error under dual-frequency excitation.

5.1.3 Amplitude step excitation

In practical environments, vibration excitation is often complex, and abrupt changes in excitation amplitude may occur. When the excitation amplitude changes abruptly, the vibration period and phase remain unchanged, while the amplitude varies instantaneously. This disturbance alters the state of the co-simulation system and may degrade the control performance of the algorithm. To evaluate the robustness of the control algorithms under such conditions, abrupt amplitude changes are introduced into the vibration excitation signal during the simulation. Specifically, for both single-frequency and dual-frequency excitation scenarios, the sinusoidal input is subjected to an abrupt amplitude change within the first 100 s of the simulation. This amplitude-change scenario is used to assess the robustness of the algorithms throughout the vibration suppression process. Considering that the voltage range of the acquisition board is generally 0-10 V, the sinusoidal input amplitude is set to 10 V in this paper.

To clearly show the performance of the six algorithms, Figure 2(c) and 2(d) show the responses under 10 V amplitude-step excitation for single-frequency and dual-frequency inputs, respectively. It can be observed that the FxLMS, Fair, LFxLMS, and FxatanLMS algorithms fail to converge under amplitude-step excitation for both single-frequency and dual-frequency inputs. In contrast, the NASFSxLMS and VSS-FxtanhLMS algorithms are still able to converge under these excitation conditions. Table 3 presents the steady-state error, MSE, and convergence time of the NASFSxLMS and VSS-FxtanhLMS algorithms under amplitude-step excitation with different input signals. From Table 3, it can be seen that the VSS-FxtanhLMS algorithm outperforms the NASFSxLMS algorithm in terms of steady-state error, MSE, and convergence time. The VSS-FxtanhLMS algorithm exhibits effective vibration suppression under amplitude-step excitation in the single-frequency case. Under dual-frequency amplitude-step excitation, although the convergence time becomes longer, the algorithm remains stable and does not diverge. In general, the VSS-FxtanhLMS algorithm substantially enhances the robustness of the control algorithm.

Table 3.

Steady-state error, MSE, and convergence time of the NASFSxLMS algorithm and the VSS-FxtanhLMS algorithm under amplitude step excitations of different input signals.

Types of algorithms	Single-frequency input signal			Dual-frequency input signal
Types of algorithms	Steady-state error(V)	Mean squared error(V)	Convergence time(s)	Steady-state error(V)	Mean squared error(V)	Convergence time(s)
NASFSxLMS	1.76 × 10^-4	1.57 × 10^-4	100	1.63 × 10^-3	0.1063	120
VSS-FxtanhLMS	1.61 × 10^-6	1.30 × 10^-8	45	1.57 × 10^-3	0.1030	80

5.2 Experimental platform setup and experimental validation

To further evaluate the practical vibration suppression capability and robustness of the proposed VSS-FxtanhLMS algorithm, an experimental platform was developed based on the co-simulation framework.FxLMS, Fair, FxatanLMS, FxasinhLMS, and VSS-FxtanhLMS are compared experimentally. Among them, FxasinhLMS is implemented in the filtered-x structure based on the inverse hyperbolic sine function adaptive filtering algorithm reported in Ref. 26.The experimental study consists of platform construction, secondary-path model identification, and comparative evaluation of different control algorithms. Comparison between the experimental and co-simulation results further verifies the effectiveness, robustness, and engineering applicability of the proposed algorithms in practical engineering scenarios.

5.2.1 Experimental platform setup

The experimental platform for AVC of the flexible cantilever beam is illustrated in Figure 1(d). It mainly consists of a flexible stainless-steel cantilever beam, MFC sensors and actuators, charge amplifiers, voltage amplifiers, and an xPC real-time control system. The sensor signals are conditioned by the charge amplifiers and acquired by the data acquisition system, whereas the control signal computed by the real-time controller is output through the D/A channel, amplified by the voltage amplifier, and applied to the actuator. The host computer is used for controller development, parameter configuration, and monitoring, while the target computer executes the control algorithm in real time and handles data acquisition and control output. The main parameters of the beam and the piezoelectric devices are listed in Tables 1 and 4, respectively.

Table 4.

Physical parameters of the piezoelectric sensors and actuators.

Parameter	Sensor (m-0714-p2)	Actuator (m-5628-p2)
Effective length/width (mm)	7/14	56/28
Overall length/width (mm)	16/16	66/31
Static capacitance (nF)	7.89	113.06
Maximum displacement (μm)	—	-45
Maximum thrust (N)	—	-205

The experimental platform adopts a Simulink Real-Time host–target architecture. The host computer is used for controller development and monitoring, while the target computer executes the control algorithm and handles data acquisition and output in real time, reducing operating-system-induced delay.

5.2.2 Analysis of experimental results

In AVC experiments, the performance of FxLMS algorithms strongly depends on the accuracy of the secondary-path model. Therefore, secondary-path identification is conducted before closed-loop control experiments. Considering the dominant dynamics of the experimental platform and real-time implementation requirements, the secondary path is modeled as a second-order discrete-time transfer function. The identification input is the driving signal applied to the disturbance piezoelectric patch, and the output is the response measured by the error sensor under uncontrolled conditions. Based on the measured input–output data, three identification algorithms are employed for secondary-path modeling: recursive least squares (RLS), recursive maximum likelihood (RML), and differential evolution (DE). RLS is a recursive estimation method that updates the model parameters by minimizing a weighted least-squares cost function.²⁷ RML is based on the maximum-likelihood principle and is used for recursive parameter identification of dynamic systems.²⁸ DE is a population-based global optimization algorithm originally proposed by Storn and Price.²⁹The comparison framework and output errors are shown in Figure 3(a) and 3(b), respectively. The MSEs obtained by RLS, RML, and DE are 0.3276, 0.3224, and 0.1540, respectively. Since DE yields the smallest MSE, the DE-identified second-order secondary-path model is selected for the subsequent AVC experiments.

With the secondary-path model determined, five algorithms, namely FxLMS, Fair, FxatanLMS, FxasinhLMS, and VSS-FxtanhLMS, are selected for experimental comparison. Figure 4(a) shows the time-domain error responses. All five algorithms are capable of suppressing the vibration, while noticeable differences are observed in convergence behavior and steady-state error. Among the benchmark algorithms, FxasinhLMS exhibits the best performance and is therefore selected as the reference for quantitative comparison. As summarized in Table 5, compared with FxasinhLMS, VSS-FxtanhLMS reduces the steady-state error from 3.49×10^-1 V to 2.88×10^-1 V and shortens the convergence time from 60 s to 40 s, corresponding to reductions of 17.48% and 33.33%, respectively.

Table 5.

Experimental time-domain and frequency-domain performance of the five algorithms.

Algorithm	Steady-state error (V)	Convergence time(s)	PSD
FxLMS	3.74e^-1	60	-22.67
Fair	3.88e^-1	60	-22.36
FxatanLMS	3.56e^-1	60	-23.10
FxasinhLMS	3.49e^-1	60	-23.26
VSS-FxtanhLMS	2.88e^-1	40	-24.98

The frequency-domain performance is evaluated using the power spectral density (PSD) of the experimental error signals, as shown in Figure 4(b). The frequency range is set to 0–100 Hz to cover the dominant low-frequency vibration band. Table 5 shows that VSS-FxtanhLMS achieves the lowest PSD level among the five algorithms. Compared with FxasinhLMS, the PSD level is reduced from −23.26 dB/Hz to −24.98 dB/Hz, corresponding to a reduction of 1.72 dB/Hz. This result is consistent with the time-domain evaluation. Although the overall experimental performance is lower than that under ideal co-simulation conditions, the proposed VSS strategy still shows better convergence behavior and steady-state suppression under practical conditions.

6 Conclusions

This paper addresses the trade-off among convergence rate, steady-state error, and robustness to abrupt input variations in active vibration control of a flexible cantilever beam by proposing the VSS-FxtanhLMS algorithm and analyzing its mean and mean-square convergence properties. Co-simulation results show that, under both single-frequency and dual-frequency excitations, the proposed algorithm achieves lower steady-state error, lower mean-square error, and shorter convergence time than the compared algorithms, and remains convergent under amplitude-step excitation. Experimental results further confirm that the proposed algorithm provides improved steady-state accuracy, faster convergence, and better vibration suppression performance under practical operating conditions. These results indicate that the proposed algorithm improves adaptability to input variations while maintaining favorable steady-state and convergence performance. Parameter and model errors, especially secondary-path estimation errors and actuator/sensor variations, may affect the filtered-reference signal and should be considered by accurate identification and proper parameter selection. Future work will extend the proposed method to more complex flexible structures, multichannel control systems, and broader disturbance conditions.

Footnotes

ORCID iD

Xiaoyan Chen

Author contributions

Jinhua Jiang: Conceptualization, methodology, validation, investigation, writing – review and editing.

Xiaoyan Chen: Software, investigation, writing – original draft.

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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