New feedforward filtered-x least mean square algorithm with variable step size for active vibration control

Abstract

The step size of least mean square (LMS) algorithm is significant for its performance. To be specific, small step size can get small excess mean square error but results in slow convergence. However, large step size may cause instability. Many variable step size least mean square (VSSLMS) algorithms have been developed to enhance the control performance. In this paper, a new VSSLMS was proposed based on Kwong’s algorithm to evaluate the robustness. The approximate analysis of dynamic and steady-state performance of this developed VSSLMS algorithm was given. An active vibration control system of piezoelectric cantilever beam was established to verify the performance of the VSSLMS algorithms. By comparing with the current VSSLMS algorithms, the proposed method has better performance in active vibration control applications.

Introduction

In mechanical and aerospace engineering, an increasing number of flexible structures were adopted due to their good characteristics. ¹ However, mechanical flexible structures often incur unwanted vibration which can significantly degrade the performance and even result in catastrophic system failure. ² In the past two decades, many useful achievements have been made by scholars in the field of active control of structural vibration.^3–6 It shows that smart materials such as piezoelectric material can provide a feasible method to solve the active vibration control (AVC) of flexible mechanical structures.

Many flexible structures such as robot arms and aircraft wings can be modeled as a cantilever beam. Thus, it has become a research hotspot to suppress the vibration of piezoelectric cantilever beam effectively.^7,8 The control method to suppress the vibration of piezoelectric cantilever beam covers almost all the branches of modern control theory: fuzzy control, ⁹ H∞ control, ¹⁰ optimal control, ¹¹ adaptive control, ¹² etc. Least mean square (LMS) algorithm is commonly used due to its robustness and ease of use. ¹³ However, there is a contradiction between the convergence speed and the steady-state error during the selection of step size in traditional LMS algorithm. Thus, different kinds of variable step size LMS (VSSLMS) algorithms were proposed to improve the performance.

Generally, the VSSLMS algorithm may be categorized by the adjusting signal. ¹⁴ Among them, most research works choose residual error as the adjusting signal. For example: Shan’s algorithm, ¹⁵ Kwong’s algorithm, ¹⁶ Aboulnasr’s algorithm, ¹⁷ Gao’s algorithm, ¹⁸ Bruhn’s algorithm, ¹⁹ Haweel’s algorithm, ²⁰ Tortora’s algorithm, ²¹ Lan’s algorithm, ²² Zhao’s algorithm, ²³ Costa’s algorithm, ²⁴ Li’s algorithm, ²⁵ Yan’s algorithm ²⁶ and Huang’s algorithm. ²⁷ Among them, the VSSLMS algorithm proposed by Kwong and Johnston ¹⁶ has variable step size according to the square of instantaneous error $e\left(n\right)$ . Compared to conventional LMS algorithm, it improved the tracking capability, but compromised the robustness.

As there exists a secondary path (including the digital-to-analog converter, reconstruction filter, power amplifier, vibrational path from actuator to error sensor, error sensor, and analog-to-digital converter) in an AVC system, the convergence of LMS algorithm or VSSLMS algorithm depends on the phase of the secondary path filter, exhibiting ever-increasing oscillatory behavior as the phase increases and finally becomes unstable at 90°. ²⁸ Morgan proposed a solution to this problem by employing an inverse filter in the reference path, which is known as the “filtered-reference” method. Because the reference signal is commonly denoted by ‘x’, it is more popularly known as the filtered-x LMS algorithm. ²⁹

In this investigation, a VSSLMS algorithm based on Kwong’s algorithm ¹⁶ was proposed to improve the robustness. An approximate analysis of convergence and steady-state performance for zero-mean stationary Gaussian inputs were provided. Also, an AVC experimental system with flexible piezoelectric cantilever beam was set up to verify the effectiveness.

This paper is organized as follows. In the next section, three VSSLMS algorithms and the proposed VSSLMS algorithm are presented. Then, the mathematical analysis of dynamic and steady-state characteristics of the new algorithm are presented. An AVC experimental system with flexible cantilever beam was set up to verify the effectiveness of the VSSLMS algorithms that is discussed in the penultimate section. The final section provides some conclusions.

The proposed VSSLMS algorithm

In this paper, the conventional LMS algorithm and three existing VSSLMS algorithms are selected to compare the effect of the proposed algorithm. In Table 1, the step size update equations and complexity comparison of these VSSLMS algorithms are presented. For convenience, Kwong’s algorithm ¹⁶ was expressed as VSSLMS-A. In the same way, Aboulnasr’s algorithm ¹⁷ and Huang’s algorithm ²⁷ were expressed as VSSLMS-B and VSSLMS-C, respectively.

Table 1.

Comparison of complexity among the VSS-LMS algorithms.

VSSLMS	Step size update	Parameters	Mul.	Add.
VSSLMS-A 16	$\mu \left(n\right)=\xi \mu \left(n-1\right)+\eta e^2\left(n-1\right)$	$2\left(\xi ,\eta \right)$	3	1
VSSLMS-B 17	$\mu \left(n\right)=\xi \mu \left(n-1\right)+\eta p^2\left(n-1\right)$ $p\left(n\right)\text{=}\beta p\left(n-1\right)+\left(1-\beta \right)e\left(n\right)e\left(n-1\right)$	$3\left(\xi ,\eta ,\beta \right)$	6	2
VSSLMS-C 27	$\mu \left(n\right)=\xi \mu \left(n-1\right)+\eta e^2\left(n-1\right)e^2\left(n-2\right)$	$2\left(\xi ,\eta \right)$	5	1

VSSLMS: variable step size least mean square.

As presented in Table 1, the step size of VSSLMS-A ¹⁶ is updated directly from the instantaneous error. As the instantaneous error is oscillating and usually mixed with measurement noise, the algorithm VSSLMS-A ¹⁶ is inevitably sensitive to the noise. To improve the robustness, Aboulnasr and Mayyas ¹⁷ developed a VSSLMS algorithm updating the step size with the autocorrelation between $e\left(n\right)$ and $e(n-1)$ . Huang et al. ²⁷ simplified the Aboulnasr’s algorithm ¹⁷ to strengthen the advantage of calculation while maintaining its performance.

To obtain a better robustness, a VSSLMS algorithm was proposed based on VSSLMS-A. It was named VSSLMS-New in this paper. The updated equation of the step size of VSSLMS-New algorithm is as follows

$$\mu \left(n\right)=\xi \mu \left(n-1\right)+\eta \left(n-1\right)e^2\left(n-1\right)$$

(1)

$$\eta \left(n\right)=\alpha \cdot \mathrm{arccot}\left(\left|e\left(n\right)\right|\right)$$

(2)

In the new algorithm, a varying user parameter algorithm is used to refer to the fixed parameter in VSSLMS-A. With the property of the arc-cotangent function, the step size can be restrained in a reasonable range while the noise impacts on the error sensor. VSSLMS-New contains two user parameters, and it requires four multiplications in each interaction. It is comparable to its original version VSSLMS-A ¹⁶ in computational complexity. Extensive experiments in AVC were conducted to compare VSSLMS-New with the abovementioned three VSSLMS algorithms.

Performance analysis of VSSLMS-New algorithm

A block diagram illustrating the LMS algorithm in adaptive process is presented in Figure 1, where $w(n)$ is the adaptive filter, $x(n)$ denotes the input sequence, $d(n)$ represents the response of disturbance, $y(n)$ denotes the output of adaptive filter, $e(n)$ is the residual error signal, $\mu$ is the step size of the algorithm, $n$ represents the discrete time sequence. As shown in Figure 1, the adaptive process of LMS algorithm can be described as equations (3) to (5).

$$w\left(n+1\right)=w\left(n\right)\text{+}\mu e\left(n\right)x\left(n\right)$$

(3)

$$e\left(n\right)=d\left(n\right)-y\left(n\right)$$

(4)

$$y\left(n\right)\text{=}{x}^T\left(n\right)\ast w\left(n\right)$$

(5)

Figure 1.

Structure of adaptive filter with LMS algorithm.

In the adaptive process, the updated equation of adaptive filter $w(n)$ is as follows

$$w\left(n+1\right)=w\left(n\right)\text{+}\mu \left(n\right)e\left(n\right)x\left(n\right)$$

(6)where $\mu \left(n\right)$ is the varying step size.

The autocorrelation matrix of $x(n)$ is defined as $R\text{=}E\left\{x\left(n\right)x^T\left(n\right)\right\}$ . Furthermore, it can be expressed as $R\text{=}Q\Lambda Q^T$ , where $\Lambda$ is the matrix of eigenvalues and $Q$ is the modal matrix of $R$ .

Residual error $e(n)$ is obtained by the desired response $d(n)$ and the adaptive filter output response $y(n)$ , respectively, by

$$e\left(n\right)=d\left(n\right)-y\left(n\right)$$

(7)

$$d\left(n\right)\text{=}{x}^T\left(n\right)w\left(n\right)$$

(8)

$$y\left(n\right)=x^T\left(n\right)w^{\text{*}}\left(n\right)\text{+}\zeta \left(n\right)$$

(9)where $\zeta (n)$ is the measurement noise and is assumed to be zero-mean, white-Gaussian sequence. $w^{\text{*}}(n)$ is the optimal estimate of the weight vector $w(n)$ . It is modeled as

$$w^{*}\left(n\text{+}1\right)=w^{*}\left(n\right)+\nu \left(n\right)$$

(10)where $\nu (n)$ is the noise process of zero-mean which has the autocorrelation matrix ${\sigma }_n^{2}I$ .

Mean behavior of the weight vector

According the updated equation of the VSSLMS-New, submitting equations (7) to (10) in equation (6), results in

$$V^{\prime }\left(n\text{+}1\right)=\left[I-\mu \left(n\right)x^{\prime }\left(n\right){x^{\prime }}^T\left(n\right)\right]V^{\prime }\left(n\right)\text{+}\mu \left(n\right)\zeta \left(n\right)x^{\prime }\left(n\right)-{\nu }^{\prime }\left(n\right)$$

(11)where $V(n)=w(n)-w^{\ast }(n)$ , $V^{\prime }(n)=Q^{T}V(n)$ , ${\nu }^{\prime }(n)=Q^T\nu (n)$ , $x^{\prime }(n)=Q^{T}x(n)$ . Normally, $\mu (n)$ slowly varies when compared with $e(n)$ and $x(n)$ . Hence, the independent assumption of $\mu (n)$ with $e(n)$ , $w(n)$ and $x(n)$ is true.

Then, the estimation of $V^{\prime }(n\text{+}1)$ can be obtained as

$$E\left\{V^{\prime }\left(n+1\right)\right\}\text{=}E\left\{\left[I-\mu \left(n\right)x^{\prime }\left(n\right){x^{\prime }}^T\left(n\right)\right]V^{\prime }\left(n\right)\right\}+E\left\{\mu \left(n\right)\zeta \left(n\right)x^{\prime }\left(n\right)\right\}-E\left\{{\nu }^{\prime }\left(n\right)\right\}$$

(12)

The estimation of $V(n\text{+}1)$ can be obtained as equation (13).

$$E\left\{V\left(n+1\right)\right\}=E\left\{\text{[}I-\mu \left(n\right)x\left(n\right)x^T\left(n\right)\text{]}V\left(n\right)\right\}$$

(13)

Equation (13) is stable if and only if

$$\underset{n=0}{\overset{N}{\Pi }}\left[I-E\left\{\mu \left(n\right)R\right\}\right]\to 0.\text{as\hspace{1em}}n\to \infty$$

(14)

Accordingly, a sufficient condition for the convergence of the weight vector mean is

$$0<E\left\{\mu \left(n\right)\right\}<\frac{2}{{\lambda }_{\max }\left(R\right)}$$

(15)where ${\lambda }_{\max }(R)$ is the maximum eigenvalue of $R$ . However, the convergence of the mean weight vector cannot ensure the convergence of mean square error (MSE). Therefore, we need to determine the conditions that will ensure the convergence in the mean square sense.

MSE

The definition of MSE is given in Widrow et al. ³⁰

$$E\left\{e^2\left(n\right)\right\}={\epsilon }_{\min }+{\epsilon }_{ex}\left(n\right)={\epsilon }_{\min }+E\left\{{V^{\prime }}^T\left(n\right)\Lambda V^{\prime }\left(n\right)\right\}$$

(16)where ${\epsilon }_{\min }=E\{{\zeta }^2(n)\}$ is the minimum value of the MSE and ${\epsilon }_{\mathrm{e}\mathrm{x}}(n)$ is the excess MSE. Consequently, equation (16) shows that the stability of MSE is directly related to the stability of $E\{{V^{\prime }}^T(n)\Lambda V^{\prime }(n)\}$ . Then, post-multiplying both sides of equation (11) by ${V^{\prime }}^T(n+1)$ and taking the expected value

$$\begin{array}{l}E\left\{V^{\prime }\left(n+1\right){V^{\prime }}^T\left(n+1\right)\right\}=E\left\{V^{\prime }\left(n\right){V^{\prime }}^T\left(n\right)\right\}-E\left\{\mu \left(n\right)\right\}E\left\{V^{\prime }\left(n\right){V^{\prime }}^T\left(n\right)\right\}\Lambda \\ -E\left\{\mu \left(n\right)\right\}\Lambda E\left\{V^{\prime }\left(n\right){V^{\prime }}^T\left(n\right)\right\}+2E\left\{{\mu }^2\left(n\right)\right\}\Lambda E\left\{V^{\prime }\left(n\right){V^{\prime }}^T\left(n\right)\right\}\Lambda \\ +E\left\{{\mu }^2\left(n\right)\right\}\Lambda \text{tr}\left(\Lambda E\left\{V^{\prime }\left(n\right){V^{\prime }}^T\left(n\right)\right\}\right)+E\left\{{\mu }^2\left(n\right)\right\}{\epsilon }_{\min }\Lambda +{\sigma }_n^{2}I\end{array}$$

(17)

According to equations (4) and (5), the mean and mean-square behavior of the step-size $\mu (n)$ can be obtained as

$$E\left\{\mu \left(n+1\right)\right\}=\xi E\left\{\mu \left(n\right)\right\}+E\left\{\eta \left(n\right)e^2\left(n\right)\right\}\text{=}\xi E\left\{\mu \left(n\right)\right\}+\alpha E\left\{\mathrm{arccot}\left(\left|e\left(n\right)\right|\right)e^2\left(n\right)\right\}$$

(18)

$$\begin{array}{c}E\left\{{\mu }^2\left(n+1\right)\right\}={\xi }^{2}E\left\{{\mu }^2\left(n\right)\right\}+2\alpha \xi E\left\{\mathrm{arccot}\left(\left|e\left(n\right)\right|\right)e^2\left(n\right)\right\}+{\alpha }^{2}E\left\{{\mathrm{arccot}}^2\left(\left|e\left(n\right)\right|\right)e^4\left(n\right)\right\}\end{array}$$

(19)

Steady-state misadjustment

In this analysis, we assumed that the algorithm has converged. Therefore, the samples of the error $e(n)$ can be assumed uncorrelated. The steady-state misadjustment was used as the figure of merit to examine the proposed VSSLMS algorithm. As the same argument in Widrow et al., ³⁰ the misadjustment is defined as

$$M=\frac{{\epsilon }_{\text{ex}}\left(\infty \right)}{{\epsilon }_{\min }}$$

(20)where ${\epsilon }_{\text{ex}}(\infty )$ is the steady-state value of ${\epsilon }_{\text{ex}}(n)$ . Also, a sufficient condition that ensures the convergence of the MSE is given as ¹⁶

$$0<\frac{E\left\{{\mu }^2\left(\infty \right)\right\}}{E\left\{\mu \left(\infty \right)\right\}}\le \frac{2}{3\text{tr}\left(R\right)}$$

(21)where $E\{\mu (\infty )\}$ and $E\{{\mu }^2(\infty )\}$ is the steady-state values of $E\{\mu (n)\}$ and $E\{{\mu }^2(n)\}$ .

When the parameters satisfied equation (21), we can obtain equation (22) by using equations (16) and (17) in equation (20).

$$M=\frac{{\displaystyle {\sum }_{j=1}^N\frac{E\left\{{\mu }^2\left(\infty \right)\right\}{\lambda }_j}{2E\left\{\mu \left(\infty \right)\right\}-2E\left\{{\mu }^2\left(\infty \right)\right\}{\lambda }_j}+{\displaystyle {\sum }_{j=1}^N\frac{{\sigma }_n^2}{\left(2E\left\{\mu \left(\infty \right)\right\}-2E\left\{{\mu }^2\left(\infty \right)\right\}{\lambda }_j\right){\epsilon }_{\min }}}}}{1-{\displaystyle {\sum }_{j=1}^N\frac{E\left\{{\mu }^2\left(\infty \right)\right\}{\lambda }_j}{2E\left\{\mu \left(\infty \right)\right\}-2E\left\{{\mu }^2\left(\infty \right)\right\}{\lambda }_j}}}$$

(22)

Recall that

$$E\left\{e^2\left(\infty \right)\right\}\text{=}{\epsilon }_{\min }+{\epsilon }_{\text{ex}}\left(\infty \right)$$

(23)

$$E\left\{\mu \left(\infty \right)\right\}\text{=}\frac{E\left[\mathrm{arccot}\left(\left|e\left(\infty \right)\right|\right)\right]\alpha \left({\epsilon }_{\min }+{\epsilon }_{\text{ex}}\left(\infty \right)\right)}{1-\xi }$$

(24)

$$E\left\{{\mu }^2\left(\infty \right)\right\}\text{=}\frac{\left\{\begin{array}{c}2E\left[\mathrm{arccot}\left(\left|e\left(\infty \right)\right|\right)\right]\alpha \xi \left({\epsilon }_{\min }+{\epsilon }_{\text{ex}}\left(\infty \right)\right)\text{+}{\left\{E\left[\mathrm{arccot}\left(\left|e\left(\infty \right)\right|\right)\right]\right\}}^2{\alpha }^2{\left({\epsilon }_{\min }+{\epsilon }_{\text{ex}}\left(\infty \right)\right)}^2\end{array}\right\}}{1-{\xi }^2}$$

(25)

We can assume that ${\epsilon }_{\min }\gg {\epsilon }_{\text{ex}}(\infty )$ , when the algorithm has been converged.^16,31 Then from equations (24) and (25), we have

$$\frac{E\{{\mu }^2(\infty )\}}{E\{\mu (\infty )\}}\approx y$$

(26)

$$y=\frac{2\xi \text{+}\alpha k{\epsilon }_{\min }}{1\text{+}\xi }$$

(27)where $k=\mathrm{arccot}(\left|{\epsilon }_{\min }\right|)$ . Note that $0<\xi <1$ , $0<\alpha <1$ and $0<k<\frac{\pi }{2}$ , then

$$y<\frac{4\xi \text{+}\pi \alpha {\epsilon }_{\min }}{1\text{+}\xi }$$

(28)

From equations (21) and (26), a sufficient condition on parameters $\xi$ and $\alpha$ for convergence of the MSE is

$$0<\frac{4\xi \text{+}\pi \alpha {\epsilon }_{\min }}{1\text{+}\xi }\le \frac{1}{3\text{tr(}R\text{)}}$$

(29)

Substituting equation (26) in equation (22), we have

$$M=\frac{{\displaystyle {\sum }_{j=1}^N\frac{y{\lambda }_j}{2-2y{\lambda }_j}+{\displaystyle {\sum }_{j=1}^N\frac{{\sigma }_n^2}{E\left\{\mu \left(\infty \right)\right\}\left(2-2y{\lambda }_j\right){\epsilon }_{\min }}}}}{1-{\displaystyle {\sum }_{j=1}^N\frac{y{\lambda }_j}{2-2y{\lambda }_j}}}$$

(30)

In the event of small values of misadjustment so that ${\displaystyle \sum y{\lambda }_i}\ll 1$ .

$$M\approx \frac{y}{2}\text{tr}\left(R\right)\text{+}\frac{N{\sigma }_n^2}{2E\left\{\mu \left(\infty \right)\right\}{\epsilon }_{\min }}$$

(31)where

$$E\left\{\mu \left(\infty \right)\right\}\approx \frac{\pi \alpha {\epsilon }_{\min }}{2\left(1-\xi \right)}$$

(32)

In a stationary environment, ${\sigma }_n^2\text{=}0$ , and the misadjustment is given by

$$M\approx \frac{y}{2}\text{tr}\left(R\right)$$

(33)

Experimental system

In this paper, an AVC experimental system of flexible piezoelectric cantilever beam was established. To better illustrate the functionality of each component, a schematic of the experimental setup is presented in Figure 2. There are six piezoelectric patches on the cantilever beam. Two of them are used as sensors to measure the vibration. The one near the cantilevered base is called as reference sensor, and the one near the free end is called as error sensor. The other four are used as actuators. To strengthen the output of the actuators, every two piezoelectric patches are bonded on both sides of the beam at the same location. The group closer to the cantilevered base is called as the disturbance actuator and the other group is called as the suppression actuator.

Figure 2.

Block diagram of cantilever beam experimental system.

The sketch of flexible cantilever beam is shown in Figure 2, and the parameters of the beam are given in Table 2.

Table 2.

Parameters of the cantilever beam.

Characteristics	Values
Beam length	500 mm
Beam width	35 mm
Beam thickness	1 mm
Disturbance piezoelectric patches position	32 mm
Suppress piezoelectric patches position	200 mm
Modulus of elasticity	190 GPa
Poisson's ratio	0.29

By using the Real-Time Workshop Toolbox in SIMULINK, two industrial computers (ACP-4020, Advantech©) are combined as target PC and host PC, respectively. A DAQ card (PCI-6289, NI©) is used for data collection and implementation of control algorithms. The control signal with 0 V–10 V is amplified to 0 V–300 V by the piezoelectric power amplifier (E00.A4, XMT©). And the piezoelectric actuator is used to suppress the vibration. The sensor signals are amplified by the charge-amplifier (YE5852A, SINOCERA©). The picture of experimental system is shown in Figure 3.

Figure 3.

Picture of AVC experimental system.

Experimental results

A sine signal with a resonance frequency of 20.53 Hz and amplitude of 4 V was used as the disturbance to excite vibration of the beam. To compare the performance of VSSLMS algorithms confronting noise, the sine signals were embedded in a white noise with variances 10⁻² and 10⁻¹. The sampling frequency $f=1000\text{Hz}$ . The order of adaptive filter is set to 24 and the initial value is set to zero.

As shown in Figure 4, VSSLMS-New was implemented in the filtered-x structure AVC system. The estimation of secondary path $\widehat{S}(z)$ should be identified before the experiment. Many methods were proposed^4,32–34; the experiment modeling method and system identification toolbox (MATLAB 2016b) were employed here. All the user parameters were adjusted very carefully to let the VSSLMS algorithms have a fair competition. Moreover, the choice of these parameters was also guided by the recommended values in their papers.

Figure 4.

Block diagram of Fx-LMS algorithm in feed-forward AVC system.

It should be noted that the VSSLMS algorithms had to be readjusted when switching from the wideband noise variance 10⁻² to 10⁻¹. The values of user parameters are shown in Table 3. The initial step size of all the VSSLMS algorithms are the same as that of the conventional LMS algorithm, such that all algorithms present similar initial convergence. And the maximum and minimum of the step size of all the VSSLMS algorithms if they need are all set a same value.

Table 3.

The values of parameters in experiments.

Algorithm	Noise variance ${10}^{-2}$	Noise variance ${10}^{-1}$
LMS	$u=0.001$	$u=0.0008$
VSSLMS-A 16	$\xi \text{=}0.97$ ; $\eta =0.02$	$\xi \text{=}0.97$ ; $\eta =0.012$
VSSLMS-B 17	$\xi \text{=}0.97$ ; $\eta =4$ ; $\beta =0.2$	$\xi \text{=}0.97$ ; $\eta =4$ ; $\beta =0.2$
VSSLMS-C 27	$\xi \text{=}0.97$ ; $\eta =0.025$	$\xi \text{=}0.97$ ; $\eta =0.015$
VSSLMS-New	$\xi \text{=}0.97$ ; $\alpha =0.02$	$\xi \text{=}0.97$ ; $\alpha =0.02$

LMS: least mean square; VSSLMS: variable step size least mean square.

Performance comparison was made in terms of total MSE and final MSE. The total MSE is the sum of MSE during the whole experiment. It can manifest the overall performance of the algorithm, including the convergence speed and the resulting MSE. The final MSE is the MSE obtained during the steady state. The results of the experiments of AVC are presented in Table 4 after averaging 200 individual runs. Figure 5 presents the comparison of MSE of various VSSLMS algorithms in AVC experiments.

Table 4.

Results of active vibration control experiments.

Algorithm	Noise variance ${10}^{-2}$		Noise variance ${10}^{-1}$
	Total MSE	Final MSE	Total MSE	Final MSE
LMS	9.453e+2	2.298e-3	1.096e+3	2.946e-3
VSSLMS-A 16	4.859e+2	1.977e-3	5.607e+2	2.670e-3
VSSLMS-B 17	4.803e+2	2.086e-3	4.838e+2	2.124e-3
VSSLMS-C 27	4.637e+2	1.793e-3	5.290e+2	2.387e-3
VSSLMS-New	4.396e+2	1.570e-3	4.365e-2	1.545e-3

VSSLMS: variable step size least mean square; MSE: mean square error.

Figure 5.

Comparison of MSE among LMS, VSSLMS-A, VSSLMS-B, VSSLMS-C and VSSLMS-New for different noise levels. (a) Part1: small noise level ( ${\sigma }^2\text{=}0.01$ ); (b) Part2: small noise level ( ${\sigma }^2\text{=}0.01$ ); (c) Part1: large noise level ( ${\sigma }^2\text{=}0.1$ ); and (d) Part2: large noise level ( ${\sigma }^2\text{=}0.1$ ).

From Table 4, Figure 5, and numerous unpresented experimental results, it can be concluded that all the control algorithms can suppress the vibration effectively. All the VSSLMS algorithms have a better performance than conventional LMS algorithm both in small and large noise level at the expense of higher computational cost.

Overall, the VSSLMS-B and the VSSLMS-C present almost the same performance in small noise level ( ${\sigma }^2\text{=}0.01$ ) with their original form, namely the VSSLMS-A algorithm. While in the large noise level ( ${\sigma }^2\text{=}0.1$ ), VSSLMS-B obtains a faster convergence speed and a lower MSE in steady state than in VSSLMS-A and VSSLMS-C. However, it should be noted that VSSLMS-B requires more user parameters and calculations.

The VSSLMS-New algorithm provides slightly better MSE than the other VSSLMS algorithms in small additive noise environment. Fortunately, it obtained a faster convergence speed and an obviously lower MSE than other VSSLMS algorithms in large additive noise environment.

As the VSSLMS-New and VSSLMS-A have a similar form, two sets of the same user parameters were used in different noise levels for further comparison. Figure 6(a) and (b) shows the MSE among VSSLMS-A algorithm and VSSLMS-New in small noise level ( ${\sigma }^2\text{=}0.01$ ) with the same parameters. Figure 6(c) and (d) shows them in large noise level ( ${\sigma }^2\text{=}0.1$ ). It is clearly observed that the proposed VSSLMS-New algorithm has a lower MSE level. The proposed VSSLMS-New algorithm has better robustness when compared to noise VSSLMS-A algorithm in AVC application of this paper.

Figure 6.

Comparison of MSE among VSSLMS-A and VSSLMS-New with the same parameters in different noise levels (200 runs). (a) Noise ${\sigma }^2\text{=}0.01$ , $\xi \text{=}0.97$ , $\eta =\alpha \text{=}0.02$ ; (b) noise ${\sigma }^2\text{=}0.01$ , $\xi \text{=}0.96$ , $\eta =\alpha \text{=}0.01$ ; (c) noise ${\sigma }^2\text{=}0.1$ , $\xi \text{=}0.97$ , $\eta =\alpha \text{=}0.02$ ; and (d) noise ${\sigma }^2\text{=}0.1$ , $\xi \text{=}0.96$ , $\eta =\alpha \text{=}0.01$ .

Conclusion

A new VSSLMS algorithm was proposed for AVC of piezoelectric cantilever beam. The new algorithm has a user parameter which varies by the instantaneous error instead of a constant. As a result, the algorithm can effectively isolate the influence of noise on step size. Approximate theoretical analysis for the dynamic and steady-state behavior of the new algorithm was derived. By an AVC experimental system of the flexible beam, the performance of the proposed algorithm was compared with the conventional LMS algorithm ³⁵ as well as other VSSLMS algorithms^16,17,27 through experiments. The results show that the proposed algorithm has better robustness to noise both in high and low noise levels compared with the other VSSLMS algorithms.