A new narrowband filter architecture for line spectrum vibration control and its experiments

2.1. The derivation of Fx-Newton narrowband control algorithm

It is assumed that the system is a linear system, and the frequency components of the system are orthogonal to each other, so the single frequency case can be analyzed only.

The frequency domain update formula of Fx-Newton algorithm [1, 6] is:

The secondary channel $S_{kl}$ that corresponds to the $l$ route control signal to the $k$ error signal.

In practical applications, the vibration signals are usually superimposed by multiple frequency components, and the frequency domain algorithm can be independently controlled by the orthogonal characteristics of each frequency component. At present, there are many ways to extract line spectrum signals in time domain. In this paper, a narrow band pass filter is used to extract single frequency line spectrum signal. The advantage is that it doesn't need to estimate frequency, and it can ensure that the filtered line spectrum signal is consistent with the original signal frequency.

The frequency response of the narrowband filter is $B$ at $\omega$. The frequency spectrum at $\omega$ of reference signal, disturbance signal and error signal before filtering is $X^o$, ${\mathbf{D}}^o$ and ${\mathbf{E}}^o$ respectively, and the correlation between them is:

The performance function of the control algorithm is:

Then the complex gradient vector of the function $J$ relative to the controller $\mathbf{g}$, $\mathbf{g}$ and their first derivative relative to $\mathbf{W}$ is:

So, the narrow-band Fx-Newton algorithm in frequency domain iterative formula can be obtained:

In which:

We can get the optimal controller:

We can see that when the line spectrum to be controlled is at the edge of two band-pass filters, the influence of transition band cannot be neglected. The following analysis of the filter characteristics of the convergence process and the impact of the results.

2.2. Influence of narrowband bandpass filter on algorithm

By comparing Eqs. (1), (8) and (11), we can see that for the same adaptive system, the addition of the pre-filter does not cause the algorithm principle error, but it will change the convergence process of algorithm and the optimal controller coefficients. Suppose the frequency response of the narrow-band filter is $B=\left|B\right|e^{i\theta \left(\omega \right)}$. Taking a fourth-order elliptic band-pass filter for an example, when the line spectrum is at the edge of the band, the amplitude response of the filter is less than the amplitude response within the pass band and the phase will change drastically. Usually in the sub-band narrowband control algorithm, often use the filter group shown in Fig. 1 for signal extraction, if the filter pass-band spacing is too large (Fig. 1(a)), can effectively prevent the signal Aliasing, but there is a possibility that the signal will be missed or that the signal has been severely attenuated, resulting in no control effect or poor control effect. If the filter pass-band is immediately followed (Fig. 1(b)), the signal fluctuations will lead to the algorithm to judge the signal frequency band repeatedly, nor is it the result we expected; or both compromise (Fig. 1(c)), but this cannot solve the problem fundamentally.

Fig. 1Three narrowband filter bank structures

Fig. 2The improved filter bank in narrowband algorithm

More seriously, when the algorithm is at the edge of the band, it is often judged as the change of the frequency of the reference signal to trigger the restart of the algorithm, which is not conducive to the stability of the algorithm. In fact, the signal fluctuates within a small interval Will affect the secondary channel characteristics of the algorithm, without the need to make major adjustments to the algorithm.

First, let us analyze the impact of narrowband filters on the optimal controller. Substitute $B\mathrm{\text{'}}=\left|B\mathrm{\text{'}}\right|e^{i{\theta }^{\mathrm{\text{'}}}\left(\omega \right)}$ into Eq. (10), we can get:

It can be seen that the change of the filter characteristics will lead to the change of the optimal controller parameters. The magnitude of the filter influences the size of the optimal controller coefficients. The phase affects the distribution of the controller coefficients. In order to lose the generality, suppose the phase of the filter is constant, namely let $B\mathrm{\text{'}}=\alpha B,{\theta }^{\mathrm{\text{'}}}\left(\omega \right)=\theta \left(\omega \right)$, $\alpha$ is the amplitude factor. Into the Eq. (8), if the algorithm is from ${\mathbf{W}}_n$ we can obtain:

Eq. (13) shows that the optimal controller coefficient increases, especially when the reference signal frequency is at the edge of the band. In the convergent process Eq. (12), the first and second terms are the renewal formulas of the weight coefficients when the filter is $B$. Since we assume that the phases are invariant, the convergence direction of the weight coefficients should be the same in both cases. Therefore, the second term indicates the direction of convergence. It can be seen that the directions of the third term and the second term are the same, that is, the recursion convergence increases with the same step factor, but the increased ratio is better than the optimal control the multiplication factor of increasing the coefficient is small, so will reduce the speed of system convergence.

4.1. Experimental setup

The experiment is carried out on the main and passive hybrid vibration isolation test platform for diesel generator set. The acceleration signals near the 6 hybrid vibration isolators on the base are taken as the error signals. The objective function is the sum of squares of the error signals.

4.2. Comparison of Fx-Newton algorithm and filter modified Fx-Newton algorithm

The frequency of the reference signal is adjusted to 62 Hz, and the signal frequency at this time is at the edge of the band in the B₁ filter bank. Two algorithms are run in the same convergence step to compare the convergence.

Fig. 5The compare of convergence rate when using shaker excites at 62 Hz

Fig. 6Power spectrum density of the acceleration signals before and after control

As can be seen from the Fig. 5, when at the edge of the band, the improved algorithm has a faster convergence rate than the original algorithm under the condition of guaranteed stability, and can only converge completely in only 4 seconds.

4.3. Multiline spectrum control in diesel generators

The diesel generating set is operated to excite the vibration components superimposed by the broad spectrum and the multitude of spectrums. The acceleration of the upper deck is taken as a reference signal, and the Fx-Newton algorithm is used for active control at multiple speeds.

Fig. 6 shows the diesel engine power at different speeds and other conditions under control before and after the base error of the power point density measurement points. Under each operating condition, the active control system automatically selects the most prominent 10 to 11 line spectra in the frequency band of 30 to 180 Hz for control, and still has a good convergence effect when the line spectrum is at the edge of the frequency band.