3.1. Determine the Optimal Algorithm Parameters
The standard wideband signal
is obtained as follows:
where
is the starting frequency and
is the rate of frequency change over time. Setting
to 10 Hz,
to 500, and the sampling period to 0.001 s, the standard LMS algorithm and the variable-step normalization LMS (VSNLMS) algorithm are used to filter the noisy signal
. The filtered signals
and
are compared for their errors.
Figure 3a shows the time domain of the original signal
and the noisy signal
with white noise, and
Figure 3b shows the Hilbert time-frequency energy.
As shown in
Figure 4, the output error diagram of a filter pair based on the LMS algorithm and VSNLMS algorithm is shown.
Obviously, the output error of the VSNLMS algorithm is significantly smaller than that of the LMS algorithm, indicating that the VSNLMS outperforms the LMS in filtering. The Pearson correlation coefficient
R between
and
is calculated to evaluate the filtering performance of different algorithms. Moreover, to quantify the impact of parameter changes on the filtering performance of the VSNLMS algorithm, the control variable method is used to obtain the
R value between
and
when the parameters
α,
β, and
γ range from 0.1 to 0.9. Finally, the optimal filtering parameters for the VSNLMS algorithm are determined. The formula for calculating the Pearson correlation coefficient is obtained as follows:
where
,
,
, and
represent the sample points and the corresponding means for the two datasets.
Table 1 shows the R values obtained based on the LMS and VSNLMS algorithms.
According to the above quantitative analysis, the optimal parameters for the VSNLMS algorithm are as follows:
,
, and
. On this basis, the impact of
θ on the M-VSNLMS algorithm is further analyzed.
Table 2 reports the R values obtained by the LMS and M-VSNLMS algorithms. By adjusting the value of
θ, the variation trend of
R with
θ is investigated based on the conclusions in
Table 1.
It can be observed from
Table 1 and
Table 2 that as the parameters
α,
β, and
θ increase, the correlation coefficient
R between the filter output sequence and the original sequence also increases, and the algorithm diverges when
θ is set to 0.8 or 0.9. Nevertheless, the correlation coefficient
R declines as the parameter
γ increases.
Figure 5 shows the relationship between the Pearson correlation coefficient
R and values of the parameters
α,
β,
γ, and
θ obtained via the control variable method.
From
Figure 5, it can be observed that the correlation coefficient between the output sequence of the M-VSNLMS algorithm and the original signal is much higher than the result of the LMS algorithm. Additionally, parameters
α and
β have the most significant impact on the algorithm, and the momentum term coefficient
θ further improves the prediction accuracy of the algorithm. Although the increase in
γ results in a decrease in prediction accuracy, this impact is slight.
The above analysis shows that the optimal parameters for the M-VSNLMS algorithm are as follows:
,
,
, and
. On the basis of the above conclusions, the LMS and M-VSNLMS algorithms are used to filter the noisy signal
.
Figure 6 illustrates the convergence process of the weight values for both algorithms. For the LMS algorithm, since its weight increment is fixed-step, the weight convergence curve is smooth, but the convergence rate is obviously slow. And at the sampling time of 15 s, the weights are still in a slow iteration process. For the M-VSNLMS algorithm, since its weight increment is variable-step, the weight convergence curve is erratic, but the convergence rate is fast. At t1 = 1.5 s, the weight of each tap quickly converges, and at t1 = 4 s, the weight of each tap roughly converges to a stable state. These demonstrate that the M-VSNLMS algorithm not only has significantly smaller convergence errors compared to those of the LMS algorithm but also has a distinct advantage in terms of the convergence rate.
Figure 7 presents the performance of the LMS, VSNLMS, and M-VSNLMS algorithms in filtering the signal
. The results demonstrate that the M-VSNLMS algorithm has the best performance on original signal reconstruction and noise removal in the time domain, frequency domain, and energy distribution.
Figure 7 shows the Hilbert time-frequency of the
filtering results using different algorithms.
3.2. Example Validation
To verify the convergence of the proposed algorithm and the optimization effect of the above conclusions on the algorithm’s learning capability, the M-VSNLMS algorithm with
,
,
, and
is used to identify the parameters of a dynamic system model.
Among them, when the model parameters are
,
,
,
,
,
is Gaussian white noise with a mean value of 0 and variance of 0.01. The input signal
u(
t) adopts the white noise sequence. In
Figure 8, the predicted results of the model are shown.
As shown in
Table 3, the true values of the estimated parameters are compared with the estimated values.
As shown in
Table 3, the identification accuracy of system model parameters is above 97%, while the identification accuracy of noise is only 0.55%, indicating a good ability to suppress noise, demonstrating the excellent system identification performance of the M-VSNLMS algorithm. In addition, the algorithm has poor performance in terms of noise model identification, indicating its good noise suppression ability.
3.3. Experimental Steps and Results
In order to verify whether the proposed algorithm has good application effects in engineering, instruments and equipment were built in the aerodynamic elasticity laboratory of CARDC, as shown in
Figure 9. By applying the driving voltage to the actuator, the aircraft model is vibrated, and the vibration signals are collected in real-time. In
Figure 9a,b, the schematic diagram of the experimental device and the schematic diagram of the measurement and control system are shown respectively.
In
Table 4, information on the selected measurement and control instruments is shown. The signal processing module uses a series of products from National Instruments (NI) in the United States, the controller is a product from dSPACE in Germany, and the actuator uses piezoelectric stacking materials from Physical Instruments (PI) in Germany.
As shown in
Figure 10, the real-time data fitting results based on the traditional ARMAX algorithm and N4SID algorithm are as follows:
As shown in
Figure 11, the real-time data fitting results of the data-driven self-learning algorithms LMS and M-VSNLMS are as follows:
Calculate the correlation coefficient
and mean square error
MSE between the output data of the above four algorithms and the expected data.
Among them, the correlation coefficient
is the ratio of the covariance of the output data and the expected data to the product of the expected values of the two.
,
,
, and
respectively represent the sampling and average values of the output data and the expected data, and
represents the sample data capacity. In
Table 5, the experimental results of the four algorithms are compared.
From
Table 5, it can be seen that the data fitting effect of the data-driven self-learning algorithm is superior to that of traditional algorithms, and the data fitting accuracy of the M-VSNLMS algorithm is higher than that of the LMS algorithm. This is consistent with the simulation results and also indicates that the M-VSNLMS algorithm has a stronger real-time identification ability for physical systems than the LMS algorithm. The mathematical model of traditional algorithms has a higher order, which makes the mathematical model more complex, and the correlation between the algorithm output data and the expected data is less than 0.9. The correlation between the output data based on self-learning algorithms and the expected data is above 0.95. The algorithm output data are obtained through the linear convolution operation, and the mathematical model structure of the algorithm is simple. Through the above experiments, it is shown that the algorithm proposed in this article has value for engineering applications.