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.
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.
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 is the weight vector of the order control filter, , 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 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 tap-weight vector 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).
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 Figure 2. Here, is the input signal, represents the input vector of the D-LMS algorithm, and denotes the signal decomposition with MALLAT algorithm [7]. is the level detail signal of original signal, is the level approximation signal of original signal, is the weight vector of level details signal, and is weight vector of the level approximation signal.
2.2. 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 and represent the step-size of the level detail signal and level approximation signal, respectively. and , which denote the instantaneous estimation gradient vectors with 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 byHere, , . Similarly, the adaptation of can be expressed byHere, .
3. 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.
Where , , , , , and are the same meanings as in Figure 2. and are the vectors, which can be expressed, respectively, byHere, denotes that each component of vector is multiplied by corresponding component of another vector. signifies the signal reconstruction. 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 the ; therefore, the law of the adaptation is crucial.
3.2. Adaptation of Weight Vector
As indicated in Figure 3, the error is represented bywhere . 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.
According to MALLAT algorithm and Figure 3, the following equations are obtained:
3.2.1. and
According to (11a) and (11b) and (12), should be first given so as to calculate . To address this problem, this study employs induction. Here, , , and are given as follows.
(1) . By (16), can be expressed asSubstitute (18) into above equation:It is clear that is not correlative with , so (22) can be rewritten as
(2) . In the same way, can be expressed bySubstitute (18) into (23), and Then (24) can be rewritten asSubstitute (19) into (26), and So, the final expression of is
(3) . Similarly, can also be obtained; it is represented byFrom the above, , , and have been obtained. By comparing them with each other, the universal expression of can be induced: where and . ConsiderAs is the same as in the adaptation way, we can easily obtain the expression of :where . Consider
3.2.2. Adaptation of and
(1) where , .
is the same meaning as expression above, and is the step-size of level detail signal.
(2) where is also the same meaning as the expression above and 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.
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, and represent amplitudes of strain and stress, respectively, and , , 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 [11–13], 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 can be approximately expressed bywhere the coefficients , , and 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 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.
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.
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.
4.3.1. Linear System with Nonlinear Input
This system can be described by the difference equation as below:It is a linear dynamic system with a nonlinear input mapping as the exciting signal. The nonlinear function is shown asHere, 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.
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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 aswhere is the noise with uniform distribution andThe 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.
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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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This paper was supported by the National Natural Science Foundation of China (61373054).