Wavelet Adaptive Algorithm and Its Application to MRE Noise Control System

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.


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.
As in Figure 1, (), (), (), and () denote the reference of system, disturbance, control error, and control signal, respectively.The system control signal can be expressed as where () = [ 0 ,  1 , . . .,  −1 ]  is the weight vector of the  order control filter,   () = () ∑ −1 =0  − , and  denotes the vectors transpose.The goal is to minimize the output error:  () =  () −  () . ( According to the Widrow-Hoff LMS algorithm [1], the weight of control filter can be adjusted by  ( + 1) =  () + 2  ()  () , where  is the step-size parameter.The convergence rate of this algorithm depends on the condition numbers of the autocorrelation   of the reference signal.When the eigenvalues of   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 tapweight vector [ Ŵ()] 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   .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).

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][5][6].

Adaptation of Weight Vectors.
As seen in Figure 2, Equation ( 6) also can be written as follows: According to the LMS algorithm, the equations of  and  are as follows: where   ( = 1, . . ., ) and  represent the step-size of the th level detail signal and th level approximation signal, respectively.∇ and ∇ , which denote the instantaneous estimation gradient vectors with  2 () to   and  separately, can be expressed as follows: In this study, we can place the emphasis only on the adjustment of   as  resembles   in the adjustment way.First, rewrite (10) as follows: According to (4) and ( 5), the following equation is obtained: Then, the adaptation of  can be given by  , =  , + 2   ()  , .

Decomposition and Reconstruction LMS Algorithm (DR-LMS)
3.1.The Structure of DR-LMS.Schematic of the DR-LMS algorithm is shown in Figure 3, in which the signal is not only decomposed, but also reconstructed.
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 the (); therefore, the law of the adaptation is crucial.3, the error () is represented by

Adaptation of Weight Vector. As indicated in Figure
where  = [1, 1, . . ., 1]  1× .According to LMS algorithm, in order to minimize the error (),   and  can be adapted by (8) and (9).With the foregoing discussion, we only study the adaptation way of   and then can obtain the adaptation way of  in the same way.
(, ) is the same meaning as expression above, and   is the step-size of th level detail signal. (2) where   (, ) is also the same meaning as the expression above and   is the step-size of final approximation signal.

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 magnetorheological (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.
As shown in Figure 4, when the MRE is subjected to the applied alternate strain where the corresponding stress will become According to the definition of complex modulus and ( 1) and ( 2), the complex modulus of MRE can be expressed as below: where  and  denote the shear strain and stress of the MRE,  0 and  0 represent amplitudes of strain and stress, respectively, and  0 ,   , and  express the initial modulus, variable modulus, and viscosity coefficient, respectively. is the vibration frequency;  is the delayed phase between stress and strain.The real part of the complex modulus   () is storage modulus which is representing the viscoelastic stiffness of the MRE.The imaginary part   () is loss modulus, and the ratio of   () to   () refers to the loss factor which is representing the viscoelastic damping and it is shown as According to the corresponding references [11][12][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   () and the loss factor tan  can be approximately expressed by tan where the coefficients  0 ,  1 , and  0 depend only on the applied magnetic field strength.

MRE Noise Control System.
To test the feasibility of the algorithm stated before, an MRE noise control system is constructed as shown in Figure 5.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.

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.
In Figure 6, () is the system input, and () is an unknown system.In this system, the output of adaptive filter is desired to track the output of (), () is the identification error, and we also choose () as the criterion to judge the performance of the algorithms.
where () is the noise with uniform distribution and (45) The simulation parameters are the same as the previous simulation in Table 1.
As shown in Figure 12, 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.

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.

Figure 4 :
Figure 4: The viscoelastic model of MRE.

Table 2 :
Mean square error (1024 data).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