# Wavelet Adaptive Algorithm and Its Application to MRE Noise Control System

Authors: Zhang Yulin; Zhao Xiuyang
Journal: Shock and Vibration
 (2015)

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## Abstract

To address the limitation of conventional adaptive algorithm used for active noise control (ANC) system, this paper proposed and studied two adaptive algorithms based on Wavelet. The twos are applied to a noise control system including magnetorheological elastomers (MRE), which is a smart viscoelastic material characterized by a complex modulus dependent on vibration frequency and controllable by external magnetic fields. Simulation results reveal that the Decomposition LMS algorithm (D-LMS) and Decomposition and Reconstruction LMS algorithm (DR-LMS) based on Wavelet can significantly improve the noise reduction performance of MRE control system compared with traditional LMS algorithm.


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## Body

## 1. Introduction

The most popular algorithm used to adapt FIR filters is the Widrow-Hoff LMS [1], which is shown in Figure 1. Its popularity is due to its low computational complexity and robustness to implementation errors.Figure 1As in Figure1, x(n), d(n), e(n), and y(n) denote the reference of system, disturbance, control error, and control signal, respectively. The system control signal can be expressed as (1)yn=XTnWn,where W(n)=[w0,w1,…,wN-1]T is the weight vector of the N order control filter, XT(n)=x(n)∑i=0N-1z-i, and T denotes the vectors transpose. The goal is to minimize the output error: (2)en=dn-yn.According to the Widrow-Hoff LMS algorithm [1], the weight of control filter can be adjusted by (3)Wn+1=Wn+2μXnen,where μ is the step-size parameter. The convergence rate of this algorithm depends on the condition numbers of the autocorrelation Rx of the reference signal. When the eigenvalues of Rx are widely spread, the excess mean square error produced by LMS algorithm is primarily determined by the largest eigenvalue, and the time taken by the average tap-weight vector E[W^(k)] to converge is limited by the smallest eigenvalue. However, the speed of convergence of the mean square error is affected by the spread of the eigenvalues of Rx.The speed of convergence of the LMS algorithm may slow down when the correlation matrix of the inputs is ill-conditioned, which implies that the control system with Widrow-Hoff LMS adaptive algorithm might become unstable when inputs change indefinitely. In order to enhance the performance of the algorithm, Wavelet Transform is proposed in this study due to its time-frequency localization [2, 3]. This paper presents two kinds of adaptive wavelet algorithm: Decomposition LMS algorithm (D-LMS) and Decomposition and Reconstruction LMS algorithm (DR-LMS).


## 2. Decomposition LMS Algorithm (D-LMS)

In D-LMS algorithm, the input signal is decomposed into various wavelet spaces according to various scales to form the input vector, and then the adaptive filter weight corresponding to a certain frequency component is adjusted. This means that the proportion of each frequency of input signal is altered so as to reconstruct the output signal to approximate the desired signal [4–6].


### 2.1. Structure of D-LMS

Schematic of the D-LMS is provided in Figure2. Here, x(n) is the input signal, X0=x(n)∑i=0N-1z-i represents the input vector of the D-LMS algorithm, and DE denotes the signal decomposition with MALLAT algorithm [7]. Dj=[dj,0,dj,1,…,dj,2-jN-1]T(j=1,…,J) is the jth level detail signal of original signal, XJ=[x0,x1,…,x2-jN-1]T is the Jth level approximation signal of original signal, Wj=[wj,0,wj,1,…,wj,2-jN-1]T(j=1,…,J) is the weight vector of jth level details signal, and U=[u0,u1,…,u2-JN-1]T is weight vector of the Jth level approximation signal.Figure 2
Decomposition LMS algorithm.


### 2.2. Adaptation of Weight Vectors

As seen in Figure2,(4)Fj=DjTWjj=1,…,J,(5)F=XJTU,(6)en=dn-∑j=1JFj-F.Equation (6) also can be written as follows:(7)en=dn-∑j=1JDjTWj-XJU.According to the LMS algorithm, the equations of W and U are as follows: (8)Wj=Wj+μj-∇^jj=1,…,J,(9)U=U+μ-∇^J,where μj(j=1,…,J) and μ represent the step-size of the jth level detail signal and Jth level approximation signal, respectively. ∇^j and ∇^J, which denote the instantaneous estimation gradient vectors with e2(n) to Wj and U separately, can be expressed as follows:(10)∇^j=2e(n)∂en∂Wj(j=1,…,J)∇^J=2en∂en∂U.In this study, we can place the emphasis only on the adjustment of Wj as U resembles Wj in the adjustment way.First, rewrite (10) as follows:(11a)∇^j=2en∂en∂wj,0,∂en∂wj,1,…,∂en∂wj,2-jN-1T,(11b)∇^J=2en∂en∂u0,∂en∂u1,…,∂en∂u2-JN-1T.According to (4) and (5), the following equation is obtained:(12)∂e(n)∂wi,k=-∑j=0JDjT∂Wj∂wi,k-XJT∂U∂wi,k=-di,k.Then, the adaptation of W can be given by(13)wj,k=wj,k+2μjendj,k.Here, i=1,…,J, k=0,…,2-iN-1. Similarly, the adaptation of U can be expressed by(14)uk=uk+2uenxk.Here, 0≤k<2-JN.


## 2.1. Structure of D-LMS

Schematic of the D-LMS is provided in Figure2. Here, x(n) is the input signal, X0=x(n)∑i=0N-1z-i represents the input vector of the D-LMS algorithm, and DE denotes the signal decomposition with MALLAT algorithm [7]. Dj=[dj,0,dj,1,…,dj,2-jN-1]T(j=1,…,J) is the jth level detail signal of original signal, XJ=[x0,x1,…,x2-jN-1]T is the Jth level approximation signal of original signal, Wj=[wj,0,wj,1,…,wj,2-jN-1]T(j=1,…,J) is the weight vector of jth level details signal, and U=[u0,u1,…,u2-JN-1]T is weight vector of the Jth level approximation signal.Figure 2
Decomposition LMS algorithm.


## 2.2. Adaptation of Weight Vectors

As seen in Figure2,(4)Fj=DjTWjj=1,…,J,(5)F=XJTU,(6)en=dn-∑j=1JFj-F.Equation (6) also can be written as follows:(7)en=dn-∑j=1JDjTWj-XJU.According to the LMS algorithm, the equations of W and U are as follows: (8)Wj=Wj+μj-∇^jj=1,…,J,(9)U=U+μ-∇^J,where μj(j=1,…,J) and μ represent the step-size of the jth level detail signal and Jth level approximation signal, respectively. ∇^j and ∇^J, which denote the instantaneous estimation gradient vectors with e2(n) to Wj and U separately, can be expressed as follows:(10)∇^j=2e(n)∂en∂Wj(j=1,…,J)∇^J=2en∂en∂U.In this study, we can place the emphasis only on the adjustment of Wj as U resembles Wj in the adjustment way.First, rewrite (10) as follows:(11a)∇^j=2en∂en∂wj,0,∂en∂wj,1,…,∂en∂wj,2-jN-1T,(11b)∇^J=2en∂en∂u0,∂en∂u1,…,∂en∂u2-JN-1T.According to (4) and (5), the following equation is obtained:(12)∂e(n)∂wi,k=-∑j=0JDjT∂Wj∂wi,k-XJT∂U∂wi,k=-di,k.Then, the adaptation of W can be given by(13)wj,k=wj,k+2μjendj,k.Here, i=1,…,J, k=0,…,2-iN-1. Similarly, the adaptation of U can be expressed by(14)uk=uk+2uenxk.Here, 0≤k<2-JN.


## 3. Decomposition and Reconstruction LMS Algorithm (DR-LMS)

### 3.1. The Structure of DR-LMS

Schematic of the DR-LMS algorithm is shown in Figure3, in which the signal is not only decomposed, but also reconstructed.Figure 3
DR-LMS algorithm.WhereX0, DE, Dj(j=1,…,J), XJ, Wj, and U are the same meanings as in Figure 2. Vj(j=1,…,J-1) and YJ are the vectors, which can be expressed, respectively, by(15)Vj=Dj×^Wjj=1,…,J,YJ=XJ×^U.Here, ×^ denotes that each component of vector is multiplied by corresponding component of another vector. RE signifies the signal reconstruction. Y=[y0,y1,…,yN-1]T represents the results of reconstruction.It is obvious that the DR-LMS algorithm consists of three parts: firstly, the signal is decomposed into various levels, then each level detail signal and final approximation signal are multiplied by the corresponding weight vectors, and, finally, these signals are reconstructed by MALLAT algorithm to form the output of DR-LMS algorithm. In the process, the weight vectors are adapted to minimize thee(n); therefore, the law of the adaptation is crucial.


### 3.2. Adaptation of Weight Vector

As indicated in Figure3, the error e(n) is represented by(16)en=dn-YTT,where T=[1,1,…,1]1×NT. According to LMS algorithm, in order to minimize the error e(n), Wj and U can be adapted by (8) and (9). With the foregoing discussion, we only study the adaptation way of Wj and then can obtain the adaptation way of U in the same way.According to MALLAT algorithm and Figure3, the following equations are obtained:(17)Ym=2∑k2h(m-2k)Y1(k)+2∑kgm-2kV1k,Y1m=2∑kh(m-2k)Y2(k)+2∑kgm-2kV2k,Y2m=2∑kh(m-2k)Y3(k)+2∑kgm-2kV3k.Substitute (15) into (17):(18)Ym=2∑kh(m-2k)Y1(k)+2∑kgm-2kD1×^W1k,(19)Y1m=2∑kh(m-2k)Y2(k)+2∑kgm-2kD2×^W2k,(20)Y2m=∑kh(m-2k)Y3(k)+∑kgm-2kD3×^W3k.


#### 3.2.1.∂e(n)/∂wj,k and ∂e(n)/∂ui

According to (11a) and (11b) and (12), ∂e(n)/∂wj,k should be first given so as to calculate ∇^j. To address this problem, this study employs induction. Here, ∂e(n)/∂w1,i, ∂e(n)/∂w2,i, and ∂e(n)/∂w3,i are given as follows.(1)  ∂e(n)/∂w1,i. By (16), ∂e(n)/∂w1,i can be expressed as(21)∂e(n)∂w1,i=∑m=0N-1∂Y(m)∂w1,i.Substitute (18) into above equation:(22)∂e(n)∂w1,i=2∑m=0N-1∑k0h(m-2k0)∂Y1(k0)∂w1,i+2∑m=0N-1∑k0gm-2k0∂D1×^W1k0∂w1,i.It is clear that Y1 is not correlative with W1, so (22) can be rewritten as(23)∂e(n)∂w1,i=2∑m=0N-1g(m-2i)D1(i).(2)  ∂e(n)/∂w1,i. In the same way, ∂e(n)/∂w1,i can be expressed by(24)∂e(n)∂w2,i=∑m=0N-1∂Y(m)∂w2,i.Substitute (18) into (23), and (25)∂(D1×^W1)(k0)∂w2,i=0.Then (24) can be rewritten as(26)∂e(n)∂w2,i=2∑m=0N-1∑k0h(m-2k0)∂Y1(k0)∂w2,i.Substitute (19) into (26), and (27)∂Y2(k1)∂w2,i=0.So, the final expression of ∂e(n)/∂w1,i is(28)∂e(n)∂w1,i=22∑m=0N-1∑k0h(m-2k0)g(k0-2i)D1(i).(3)  ∂e(n)/∂w3,i. Similarly, ∂e(n)/∂w3,i can also be obtained; it is represented by(29)∂e(n)∂w3,i=23∑m=0N-1∑k0hm-2k0∑k1hk0-2k1gk1-2iD2i.From the above, ∂e(n)/∂w1,i, ∂e(n)/∂w1,i, and ∂e(n)/∂w3,i have been obtained. By comparing them with each other, the universal expression of ∂e(n)/∂wj,i can be induced: (30)Ifj=1∂e(n)∂w1,i=∑m=0N-1g(m-2i)D1(i),Ifj≠1∂e(n)wji=∑m=0N-1Zj(m,i)Dj(i),where j=1,…,J and i=0,…,2-jN-1. Consider(31)Zj(m,i)=2j∑k0∑k1⋯∑kj-1hm-2k0⋯hkj-3-2kj-2gkj-2-2i.As U is the same as Wj in the adaptation way, we can easily obtain the expression of ∂e(n)/∂ui:(32)∂e(n)∂ui=2J∑m=0N-1VJ(m,i)XJ(i),where i=0,…,2-JN-1. Consider(33)VJ(m,i)=2J∑k0∑k1⋯∑kj-1hm-2k0⋯hkJ-3-2kJ-2hkJ-2-2i.


#### 3.2.2. Adaptation ofWj and U

(1)  Wj(34)Ifj=1w1i=w1i+22μ1e(n)∑m=0N-1g(m-2i)D1(i)Ifj≠1wji=wji+2j+1μjen∑m=0N-1Zjm,iDji,where 1≤j≤J, 0≤i<2-jN.Z
j
(
m
,
i
) is the same meaning as expression above, and μj is the step-size of jth level detail signal.(2)  U(35)ui=ui+2J+1μJen∑m=0N-1VJm,iXJi,where VJ(m,i) is also the same meaning as the expression above and μJ is the step-size of final approximation signal.


## 3.1. The Structure of DR-LMS

Schematic of the DR-LMS algorithm is shown in Figure3, in which the signal is not only decomposed, but also reconstructed.Figure 3
DR-LMS algorithm.WhereX0, DE, Dj(j=1,…,J), XJ, Wj, and U are the same meanings as in Figure 2. Vj(j=1,…,J-1) and YJ are the vectors, which can be expressed, respectively, by(15)Vj=Dj×^Wjj=1,…,J,YJ=XJ×^U.Here, ×^ denotes that each component of vector is multiplied by corresponding component of another vector. RE signifies the signal reconstruction. Y=[y0,y1,…,yN-1]T represents the results of reconstruction.It is obvious that the DR-LMS algorithm consists of three parts: firstly, the signal is decomposed into various levels, then each level detail signal and final approximation signal are multiplied by the corresponding weight vectors, and, finally, these signals are reconstructed by MALLAT algorithm to form the output of DR-LMS algorithm. In the process, the weight vectors are adapted to minimize thee(n); therefore, the law of the adaptation is crucial.


## 3.2. Adaptation of Weight Vector

As indicated in Figure3, the error e(n) is represented by(16)en=dn-YTT,where T=[1,1,…,1]1×NT. According to LMS algorithm, in order to minimize the error e(n), Wj and U can be adapted by (8) and (9). With the foregoing discussion, we only study the adaptation way of Wj and then can obtain the adaptation way of U in the same way.According to MALLAT algorithm and Figure3, the following equations are obtained:(17)Ym=2∑k2h(m-2k)Y1(k)+2∑kgm-2kV1k,Y1m=2∑kh(m-2k)Y2(k)+2∑kgm-2kV2k,Y2m=2∑kh(m-2k)Y3(k)+2∑kgm-2kV3k.Substitute (15) into (17):(18)Ym=2∑kh(m-2k)Y1(k)+2∑kgm-2kD1×^W1k,(19)Y1m=2∑kh(m-2k)Y2(k)+2∑kgm-2kD2×^W2k,(20)Y2m=∑kh(m-2k)Y3(k)+∑kgm-2kD3×^W3k.


### 3.2.1.∂e(n)/∂wj,k and ∂e(n)/∂ui

According to (11a) and (11b) and (12), ∂e(n)/∂wj,k should be first given so as to calculate ∇^j. To address this problem, this study employs induction. Here, ∂e(n)/∂w1,i, ∂e(n)/∂w2,i, and ∂e(n)/∂w3,i are given as follows.(1)  ∂e(n)/∂w1,i. By (16), ∂e(n)/∂w1,i can be expressed as(21)∂e(n)∂w1,i=∑m=0N-1∂Y(m)∂w1,i.Substitute (18) into above equation:(22)∂e(n)∂w1,i=2∑m=0N-1∑k0h(m-2k0)∂Y1(k0)∂w1,i+2∑m=0N-1∑k0gm-2k0∂D1×^W1k0∂w1,i.It is clear that Y1 is not correlative with W1, so (22) can be rewritten as(23)∂e(n)∂w1,i=2∑m=0N-1g(m-2i)D1(i).(2)  ∂e(n)/∂w1,i. In the same way, ∂e(n)/∂w1,i can be expressed by(24)∂e(n)∂w2,i=∑m=0N-1∂Y(m)∂w2,i.Substitute (18) into (23), and (25)∂(D1×^W1)(k0)∂w2,i=0.Then (24) can be rewritten as(26)∂e(n)∂w2,i=2∑m=0N-1∑k0h(m-2k0)∂Y1(k0)∂w2,i.Substitute (19) into (26), and (27)∂Y2(k1)∂w2,i=0.So, the final expression of ∂e(n)/∂w1,i is(28)∂e(n)∂w1,i=22∑m=0N-1∑k0h(m-2k0)g(k0-2i)D1(i).(3)  ∂e(n)/∂w3,i. Similarly, ∂e(n)/∂w3,i can also be obtained; it is represented by(29)∂e(n)∂w3,i=23∑m=0N-1∑k0hm-2k0∑k1hk0-2k1gk1-2iD2i.From the above, ∂e(n)/∂w1,i, ∂e(n)/∂w1,i, and ∂e(n)/∂w3,i have been obtained. By comparing them with each other, the universal expression of ∂e(n)/∂wj,i can be induced: (30)Ifj=1∂e(n)∂w1,i=∑m=0N-1g(m-2i)D1(i),Ifj≠1∂e(n)wji=∑m=0N-1Zj(m,i)Dj(i),where j=1,…,J and i=0,…,2-jN-1. Consider(31)Zj(m,i)=2j∑k0∑k1⋯∑kj-1hm-2k0⋯hkj-3-2kj-2gkj-2-2i.As U is the same as Wj in the adaptation way, we can easily obtain the expression of ∂e(n)/∂ui:(32)∂e(n)∂ui=2J∑m=0N-1VJ(m,i)XJ(i),where i=0,…,2-JN-1. Consider(33)VJ(m,i)=2J∑k0∑k1⋯∑kj-1hm-2k0⋯hkJ-3-2kJ-2hkJ-2-2i.


### 3.2.2. Adaptation ofWj and U

(1)  Wj(34)Ifj=1w1i=w1i+22μ1e(n)∑m=0N-1g(m-2i)D1(i)Ifj≠1wji=wji+2j+1μjen∑m=0N-1Zjm,iDji,where 1≤j≤J, 0≤i<2-jN.Z
j
(
m
,
i
) is the same meaning as expression above, and μj is the step-size of jth level detail signal.(2)  U(35)ui=ui+2J+1μJen∑m=0N-1VJm,iXJi,where VJ(m,i) is also the same meaning as the expression above and μJ is the step-size of final approximation signal.


## 3.2.1.∂e(n)/∂wj,k and ∂e(n)/∂ui

According to (11a) and (11b) and (12), ∂e(n)/∂wj,k should be first given so as to calculate ∇^j. To address this problem, this study employs induction. Here, ∂e(n)/∂w1,i, ∂e(n)/∂w2,i, and ∂e(n)/∂w3,i are given as follows.(1)  ∂e(n)/∂w1,i. By (16), ∂e(n)/∂w1,i can be expressed as(21)∂e(n)∂w1,i=∑m=0N-1∂Y(m)∂w1,i.Substitute (18) into above equation:(22)∂e(n)∂w1,i=2∑m=0N-1∑k0h(m-2k0)∂Y1(k0)∂w1,i+2∑m=0N-1∑k0gm-2k0∂D1×^W1k0∂w1,i.It is clear that Y1 is not correlative with W1, so (22) can be rewritten as(23)∂e(n)∂w1,i=2∑m=0N-1g(m-2i)D1(i).(2)  ∂e(n)/∂w1,i. In the same way, ∂e(n)/∂w1,i can be expressed by(24)∂e(n)∂w2,i=∑m=0N-1∂Y(m)∂w2,i.Substitute (18) into (23), and (25)∂(D1×^W1)(k0)∂w2,i=0.Then (24) can be rewritten as(26)∂e(n)∂w2,i=2∑m=0N-1∑k0h(m-2k0)∂Y1(k0)∂w2,i.Substitute (19) into (26), and (27)∂Y2(k1)∂w2,i=0.So, the final expression of ∂e(n)/∂w1,i is(28)∂e(n)∂w1,i=22∑m=0N-1∑k0h(m-2k0)g(k0-2i)D1(i).(3)  ∂e(n)/∂w3,i. Similarly, ∂e(n)/∂w3,i can also be obtained; it is represented by(29)∂e(n)∂w3,i=23∑m=0N-1∑k0hm-2k0∑k1hk0-2k1gk1-2iD2i.From the above, ∂e(n)/∂w1,i, ∂e(n)/∂w1,i, and ∂e(n)/∂w3,i have been obtained. By comparing them with each other, the universal expression of ∂e(n)/∂wj,i can be induced: (30)Ifj=1∂e(n)∂w1,i=∑m=0N-1g(m-2i)D1(i),Ifj≠1∂e(n)wji=∑m=0N-1Zj(m,i)Dj(i),where j=1,…,J and i=0,…,2-jN-1. Consider(31)Zj(m,i)=2j∑k0∑k1⋯∑kj-1hm-2k0⋯hkj-3-2kj-2gkj-2-2i.As U is the same as Wj in the adaptation way, we can easily obtain the expression of ∂e(n)/∂ui:(32)∂e(n)∂ui=2J∑m=0N-1VJ(m,i)XJ(i),where i=0,…,2-JN-1. Consider(33)VJ(m,i)=2J∑k0∑k1⋯∑kj-1hm-2k0⋯hkJ-3-2kJ-2hkJ-2-2i.


## 3.2.2. Adaptation ofWj and U

(1)  Wj(34)Ifj=1w1i=w1i+22μ1e(n)∑m=0N-1g(m-2i)D1(i)Ifj≠1wji=wji+2j+1μjen∑m=0N-1Zjm,iDji,where 1≤j≤J, 0≤i<2-jN.Z
j
(
m
,
i
) is the same meaning as expression above, and μj is the step-size of jth level detail signal.(2)  U(35)ui=ui+2J+1μJen∑m=0N-1VJm,iXJi,where VJ(m,i) is also the same meaning as the expression above and μJ is the step-size of final approximation signal.


## 4. MRE Noise Control System

### 4.1. Dynamics of MRE

Magneto-rheological elastomers (MREs) are promising smart materials, which consist of magnetically polarizable particles in nonmagnetic solid or gel-like medium [8, 9]. As a compound of smart magneto-rheological (MR) fluid and viscoelastic materials, the MRE combines the salient features of both materials, including controllable stiffness and frequency-dependent viscoelastic behavior, which can be reversibly controlled under external magnetic fields in milliseconds.The MRE is generally regarded as a viscoelastic material [10], so its dynamic behavior can be described with a complex modulus which depends on vibration frequency and be controllable by external magnetic fields. In order to analyze the mechanical properties of MRE in the magnetic field, the modal of MRE can be simplified to a general kelvin model of viscoelastic material in parallel with an adjustable spring which was controlled by the applied magnetic field, as shown in Figure 4. In the noise control system, as the strain caused by the noise is small, the modal of MRE can be treated as a linear modal, in which the magnetic-induced modulus is adjusted by the applied magnetic field.Figure 4
The viscoelastic model of MRE.As shown in Figure4, when the MRE is subjected to the applied alternate strain where(36)γt=γ0eiωt=γ0cos⁡ωt+sinωt,the corresponding stress will become(37)τt=G0γ+Gmγ+ηγ˙=τ0eiωt+δ.According to the definition of complex modulus and (1) and (2), the complex modulus of MRE can be expressed as below:(38)Giω=τtγt=GIiω+iGRiω,where γ and τ denote the shear strain and stress of the MRE, γ0 and τ0 represent amplitudes of strain and stress, respectively, and G0, Gm, and η express the initial modulus, variable modulus, and viscosity coefficient, respectively. ω is the vibration frequency; δ is the delayed phase between stress and strain.The real part of the complex modulusGR(iω) is storage modulus which is representing the viscoelastic stiffness of the MRE. The imaginary part GI(iω) is loss modulus, and the ratio of GI(iω) to GR(iω) refers to the loss factor which is representing the viscoelastic damping and it is shown as(39)tanδ=GI(iω)GR(iω).According to the corresponding references [11–13], experimental results showed that the storage modulus linearly increases with vibration frequency in most cases, and both the storage modulus and the loss modulus linearly increase with external magnetic field strength in a certain range. Therefore, the storage modulus GR(iω) and the loss factor tanδ can be approximately expressed by(40a)GRiω=α0+α1ω,(40b)tanδ=β0,where the coefficients α0, α1, and β0 depend only on the applied magnetic field strength.


### 4.2. MRE Noise Control System

To test the feasibility of the algorithm stated before, an MRE noise control system is constructed as shown in Figure5. The system included the MRE, an electromagnetic device (not given in the figure), a noise generation device, and a measurement system. The MRE is made of silicone rubber filled with randomly dispersed carbonyl iron particles. The external magnetic field applied to MRE is generated by a DC electrical source. The measurement system includes M and L sensors which are used to record the excitation and response signals, respectively.Figure 5
A sketch of the MRE noise control system.


### 4.3. Simulation Experiment

In order to verify the performance of the algorithm, simulations are investigated in this section. Here, we apply the D-LMS algorithm, DR-LMS algorithm, and LMS algorithm to system identification [6], as shown in Figure 6.Figure 6
System identification.In Figure6, x(n) is the system input, and H(z) is an unknown system. In this system, the output of adaptive filter is desired to track the output of H(z), e(n) is the identification error, and we also choose e(n) as the criterion to judge the performance of the algorithms.


#### 4.3.1. Linear System with Nonlinear Input

This system can be described by the difference equation as below:(41)yn=0.3yn-1-0.6yn-2+fxn.It is a linear dynamic system with a nonlinear input mapping x(n) as the exciting signal. The nonlinear function fx is shown as(42)fx=x3+0.3x2-0.4x.Here, (43)xn=sin(2πn/20)+sin(2πn/40)+sin(2πn/50)+sin(2πn/250)·4.0-1.Table 1 shows the parameters chosen in this simulation. The simulation results are shown in Figures 7~9. The number of the iterations used to update weight vectors is different for different algorithm, while it takes 50 iterations for LMS algorithm but 20 iterations for D-LMS and DR-LMS algorithm. As seen in the figure, there is instability in system identification with LMS algorithm. Table 2 is the mean square error of system identification with different algorithm.Table 1
Simulation parameters.
Algorithm
Length
Step-size
LMS
32
0.0003
D-LMS
32
0.1185
DR-LMS
32
0.09Table 2
Mean square error (1024 data).
LMS
D-LMS
DR-LMS
4
.
662
×
1
0
-
4
9
.
0604
×
1
0
-
5
9.216
×
1
0
-
5Figure 7
The simulation result with LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 8
The simulation result with D-LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 9
The simulation result with DR-LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)It is obvious that the MSE with D-LMS and DR-LMS algorithm are smaller for the same number of samples.


#### 4.3.2. Nonlinear System with Linear Input

This is an example of a nonlinear system which possesses nonlinear dynamics but linear input excitation. The system can be described by a nonlinear difference equation as(44)yn=yn-11+y2n-1+xn+wn,where w(n) is the noise with uniform distribution and(45)xn=sin(2πn/20)+sin(2πn/50)+sin(2πn/250)+sin(2πnn/400)·4-1.The simulation parameters are the same as the previous simulation in Table 1.As shown in Figure12, the convergence rate of identification with LMS algorithm is slow, which takes about 70 iterations to obtain stabilization. After about 100 iterations, the mean square error is 0.0628 for 1024 samples. Figures 10 and 11 show that the convergence rates with D-LMS and DR-LMS are much faster. After 15 iterations the system can reach stabile state. After about 100 iterations, the mean square errors with D-LMS and DR-LMS are 0.0365 and 0.0355, respectively, for 1024 samples.Figure 10
The simulation result with DLMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 11
The simulation result with DR-LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 12
The simulation result with LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)


## 4.1. Dynamics of MRE

Magneto-rheological elastomers (MREs) are promising smart materials, which consist of magnetically polarizable particles in nonmagnetic solid or gel-like medium [8, 9]. As a compound of smart magneto-rheological (MR) fluid and viscoelastic materials, the MRE combines the salient features of both materials, including controllable stiffness and frequency-dependent viscoelastic behavior, which can be reversibly controlled under external magnetic fields in milliseconds.The MRE is generally regarded as a viscoelastic material [10], so its dynamic behavior can be described with a complex modulus which depends on vibration frequency and be controllable by external magnetic fields. In order to analyze the mechanical properties of MRE in the magnetic field, the modal of MRE can be simplified to a general kelvin model of viscoelastic material in parallel with an adjustable spring which was controlled by the applied magnetic field, as shown in Figure 4. In the noise control system, as the strain caused by the noise is small, the modal of MRE can be treated as a linear modal, in which the magnetic-induced modulus is adjusted by the applied magnetic field.Figure 4
The viscoelastic model of MRE.As shown in Figure4, when the MRE is subjected to the applied alternate strain where(36)γt=γ0eiωt=γ0cos⁡ωt+sinωt,the corresponding stress will become(37)τt=G0γ+Gmγ+ηγ˙=τ0eiωt+δ.According to the definition of complex modulus and (1) and (2), the complex modulus of MRE can be expressed as below:(38)Giω=τtγt=GIiω+iGRiω,where γ and τ denote the shear strain and stress of the MRE, γ0 and τ0 represent amplitudes of strain and stress, respectively, and G0, Gm, and η express the initial modulus, variable modulus, and viscosity coefficient, respectively. ω is the vibration frequency; δ is the delayed phase between stress and strain.The real part of the complex modulusGR(iω) is storage modulus which is representing the viscoelastic stiffness of the MRE. The imaginary part GI(iω) is loss modulus, and the ratio of GI(iω) to GR(iω) refers to the loss factor which is representing the viscoelastic damping and it is shown as(39)tanδ=GI(iω)GR(iω).According to the corresponding references [11–13], experimental results showed that the storage modulus linearly increases with vibration frequency in most cases, and both the storage modulus and the loss modulus linearly increase with external magnetic field strength in a certain range. Therefore, the storage modulus GR(iω) and the loss factor tanδ can be approximately expressed by(40a)GRiω=α0+α1ω,(40b)tanδ=β0,where the coefficients α0, α1, and β0 depend only on the applied magnetic field strength.


## 4.2. MRE Noise Control System

To test the feasibility of the algorithm stated before, an MRE noise control system is constructed as shown in Figure5. The system included the MRE, an electromagnetic device (not given in the figure), a noise generation device, and a measurement system. The MRE is made of silicone rubber filled with randomly dispersed carbonyl iron particles. The external magnetic field applied to MRE is generated by a DC electrical source. The measurement system includes M and L sensors which are used to record the excitation and response signals, respectively.Figure 5
A sketch of the MRE noise control system.


## 4.3. Simulation Experiment

In order to verify the performance of the algorithm, simulations are investigated in this section. Here, we apply the D-LMS algorithm, DR-LMS algorithm, and LMS algorithm to system identification [6], as shown in Figure 6.Figure 6
System identification.In Figure6, x(n) is the system input, and H(z) is an unknown system. In this system, the output of adaptive filter is desired to track the output of H(z), e(n) is the identification error, and we also choose e(n) as the criterion to judge the performance of the algorithms.


### 4.3.1. Linear System with Nonlinear Input

This system can be described by the difference equation as below:(41)yn=0.3yn-1-0.6yn-2+fxn.It is a linear dynamic system with a nonlinear input mapping x(n) as the exciting signal. The nonlinear function fx is shown as(42)fx=x3+0.3x2-0.4x.Here, (43)xn=sin(2πn/20)+sin(2πn/40)+sin(2πn/50)+sin(2πn/250)·4.0-1.Table 1 shows the parameters chosen in this simulation. The simulation results are shown in Figures 7~9. The number of the iterations used to update weight vectors is different for different algorithm, while it takes 50 iterations for LMS algorithm but 20 iterations for D-LMS and DR-LMS algorithm. As seen in the figure, there is instability in system identification with LMS algorithm. Table 2 is the mean square error of system identification with different algorithm.Table 1
Simulation parameters.
Algorithm
Length
Step-size
LMS
32
0.0003
D-LMS
32
0.1185
DR-LMS
32
0.09Table 2
Mean square error (1024 data).
LMS
D-LMS
DR-LMS
4
.
662
×
1
0
-
4
9
.
0604
×
1
0
-
5
9.216
×
1
0
-
5Figure 7
The simulation result with LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 8
The simulation result with D-LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 9
The simulation result with DR-LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)It is obvious that the MSE with D-LMS and DR-LMS algorithm are smaller for the same number of samples.


### 4.3.2. Nonlinear System with Linear Input

This is an example of a nonlinear system which possesses nonlinear dynamics but linear input excitation. The system can be described by a nonlinear difference equation as(44)yn=yn-11+y2n-1+xn+wn,where w(n) is the noise with uniform distribution and(45)xn=sin(2πn/20)+sin(2πn/50)+sin(2πn/250)+sin(2πnn/400)·4-1.The simulation parameters are the same as the previous simulation in Table 1.As shown in Figure12, the convergence rate of identification with LMS algorithm is slow, which takes about 70 iterations to obtain stabilization. After about 100 iterations, the mean square error is 0.0628 for 1024 samples. Figures 10 and 11 show that the convergence rates with D-LMS and DR-LMS are much faster. After 15 iterations the system can reach stabile state. After about 100 iterations, the mean square errors with D-LMS and DR-LMS are 0.0365 and 0.0355, respectively, for 1024 samples.Figure 10
The simulation result with DLMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 11
The simulation result with DR-LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 12
The simulation result with LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)


## 4.3.1. Linear System with Nonlinear Input

This system can be described by the difference equation as below:(41)yn=0.3yn-1-0.6yn-2+fxn.It is a linear dynamic system with a nonlinear input mapping x(n) as the exciting signal. The nonlinear function fx is shown as(42)fx=x3+0.3x2-0.4x.Here, (43)xn=sin(2πn/20)+sin(2πn/40)+sin(2πn/50)+sin(2πn/250)·4.0-1.Table 1 shows the parameters chosen in this simulation. The simulation results are shown in Figures 7~9. The number of the iterations used to update weight vectors is different for different algorithm, while it takes 50 iterations for LMS algorithm but 20 iterations for D-LMS and DR-LMS algorithm. As seen in the figure, there is instability in system identification with LMS algorithm. Table 2 is the mean square error of system identification with different algorithm.Table 1
Simulation parameters.
Algorithm
Length
Step-size
LMS
32
0.0003
D-LMS
32
0.1185
DR-LMS
32
0.09Table 2
Mean square error (1024 data).
LMS
D-LMS
DR-LMS
4
.
662
×
1
0
-
4
9
.
0604
×
1
0
-
5
9.216
×
1
0
-
5Figure 7
The simulation result with LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 8
The simulation result with D-LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 9
The simulation result with DR-LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)It is obvious that the MSE with D-LMS and DR-LMS algorithm are smaller for the same number of samples.


## 4.3.2. Nonlinear System with Linear Input

This is an example of a nonlinear system which possesses nonlinear dynamics but linear input excitation. The system can be described by a nonlinear difference equation as(44)yn=yn-11+y2n-1+xn+wn,where w(n) is the noise with uniform distribution and(45)xn=sin(2πn/20)+sin(2πn/50)+sin(2πn/250)+sin(2πnn/400)·4-1.The simulation parameters are the same as the previous simulation in Table 1.As shown in Figure12, the convergence rate of identification with LMS algorithm is slow, which takes about 70 iterations to obtain stabilization. After about 100 iterations, the mean square error is 0.0628 for 1024 samples. Figures 10 and 11 show that the convergence rates with D-LMS and DR-LMS are much faster. After 15 iterations the system can reach stabile state. After about 100 iterations, the mean square errors with D-LMS and DR-LMS are 0.0365 and 0.0355, respectively, for 1024 samples.Figure 10
The simulation result with DLMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 11
The simulation result with DR-LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)Figure 12
The simulation result with LMS algorithm. (a) Process of identification. (b) Error of identification.
(a)
(b)


## 5. Summary

In this study, two adaptive algorithms based on Wavelet are proposed and studied. To investigate the performance of the algorithms, the simulation for their application to MRE noise control system is constructed and explored in details. The results indicate that the wavelet adaptive algorithm possesses a good control effectiveness though it requires a great deal of calculation. The implementation of MRE noise control system with DR-LMS algorithm can significantly improve the performance of the control system.



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*Source: 968082-2015-07-26.xml*