7.2.1.6.5. Vold-Kalman Filter

The Vold-Kalman Filter is a useful order tracking algorithm based on bandwidth pass filter. The noise and vibration are excited by the rotational frequency. Therefore, when the rotational frequency is used as a reference bandpass frequency of the Vold-Kalman Filter, then the solution has relatively smoothing and exact comparison to the FFT based order tracking algorithm. In the case of the FFT based order tracking (Campbell), the order line is a wavy shape signal.

The Vold-Kalman Filter is used for getting an order line signal in the RPM-Order Section View of the Campbell3D.

Algorithm of Vold-Kalman Filter

The second generation Vold-Kalman filter is also based on the Kalman filter.

The measurement equation is defined (7.1).

(7.1)\[\begin{split}\begin{aligned} & y(n)=x(n){{e}^{j\Theta (n)}}+\eta (n) \\ & \Theta (n)=\sum\limits_{i=0}^{n}{a\omega (i)\Delta t} \\ \end{aligned}\end{split}\]

Where,

\(n\) is an index of the signal.

\(y(n)\) is the target signal of the Order Tracking.

\(\eta (n)\) is an error.

\(\omega (n)\) is the reference rotational velocity (=frequency) of the rotational equipment.

\(a\) is a scalar and the Order value.

\(x(n)\) is the filter output signal.

The matrix form of (7.1) is defined (7.2).

(7.2)\[\begin{split}\begin{aligned} & \mathbf{y}-\mathbf{Cx}=\mathbf{\eta } \\ & \mathbf{C}=diag({{e}^{j\Theta (1)}},\ldots ,{{e}^{j\Theta (n)}}) \\ \end{aligned}\end{split}\]

The matrix \(\mathbf{C}\) is defined with the reference order frequency.

The process equation is defined as (7.3).

(7.3)\[\begin{split}\begin{aligned} & 2-Pole:\quad x(n)-2x(n-1)+x(n-2)=\varepsilon (n) \\ & 3-Pole:\quad x(n)-3x(n-1)+3x(n-2)-x(n-3)=\varepsilon (n) \\ \end{aligned}\end{split}\]

Where, \(\varepsilon (n)\) is also an error in process.

The matrix form of Eq. 3 is defined as (7.4).

(7.4)\[\mathbf{Ax}=\mathbf{\varepsilon }\]

The matrix \(\mathbf{A}\) is a rectangular shape. The matrix dimension of \(\mathbf{A}\) is ((N-2) x N) for 2-Pole and ((N-3) x N) for 3-Pole, respectively.

The solution of the second generation Vold-Kalman filter is defined as the minimization of the least squares of two errors.

(7.5)\[J={{r}^{2}}{{\mathbf{\varepsilon }}^{T}}\mathbf{\varepsilon }+{{\mathbf{\eta }}^{H}}\mathbf{\eta }\]

Where, \(J\) is an objective function for minimization.

\(r\) is the weighting factor of the Vold-Kalman Filter. This weighting factor is defined the bandwidth of filteration.

The solution is defined as following (7.6).

(7.6)\[\begin{split}\begin{aligned} & \frac{\partial J}{\partial \mathbf{x}}=\frac{\partial ({{r}^{2}}{{\mathbf{\varepsilon }}^{T}}\mathbf{\varepsilon }+{{\mathbf{\eta }}^{H}}\mathbf{\eta })}{\partial \mathbf{x}}=({{r}^{2}}{{\mathbf{A}}^{T}}\mathbf{A}+\mathbf{I})\mathbf{x}-{{\mathbf{C}}^{H}}\mathbf{y} \\ & \mathbf{x}={{({{r}^{2}}{{\mathbf{A}}^{T}}\mathbf{A}+\mathbf{I})}^{-1}}{{\mathbf{C}}^{H}}\mathbf{y} \\ \end{aligned}\end{split}\]

Where, the right sided matrix \(({{r}^{2}}{{\mathbf{A}}^{T}}\mathbf{A}+\mathbf{I})\) is a sparse real matrix.

\({{\mathbf{C}}^{H}}\) is calculated using the order value and the reference rotational velocity (=the reference bandwidth frequency).

\(\mathbf{y}\) is the target signal of Order Tracking signal.

In order to get the solution, the MKL sparse linear solver named DSS is used for the Vold-Kalman Filter of the Campbell3D function.

As a single computation, the multiple order tracking result can be calculated with two or more reference rotational velocities using the Vold-Kalman Multiorder Tracking Filter.

\[\begin{split}\begin{aligned} \begin{bmatrix} & r^2 {\bf{A}}^T{\bf{A}} + \bf{I} &{\bf{C}}_1^H{\bf{C}}_2 & ... &{\bf{C}}_1^H{\bf{C}}_n \\ & {\bf{C}}_2^H{\bf{C}}_1 & r^2 {\bf{A}}^T{\bf{A}} + \bf{I} & ... & {\bf{C}}_2^H{\bf{C}}_n \\ & ... & ... & ... & ...\\ & {\bf{C}}_n^H{\bf{C}}_1 & {\bf{C}}_n^H{\bf{C}}_2 & ... & r^2 {\bf{A}}^T{\bf{A}} + \bf{I} \\ \end{bmatrix} \begin{Bmatrix} {\bf{x}}_1 \\ {\bf{x}}_2 \\ ... \\ {\bf{x}}_n \end{Bmatrix} = \begin{Bmatrix} {\bf{C}}_1^H{\bf{y}} \\ {\bf{C}}_2^H{\bf{y}} \\ ... \\ {\bf{C}}_n^H{\bf{y}} \end{Bmatrix} \end{aligned}\end{split}\]

[Vold1995]

Vold, H., Leuridan, J., “High Resolution Order Tracking at Extreme Slew Rates, using Kalman Tracking Filters”, Shock and Vibration, Vol. 2, No. 6, pp. 507-515 (1995)

[Vold1997]

Vold, H., Herlufsen, H., Mains, M., Corwin-Renner, D., “Multi Axle Order Tracking with the Vold-Kalman Tracking Filter”, Sound and Vibration 31, May 1997, pp. 30-35

[Tuma]

Tuma, J., “Vold-Kalman Order Tracking Filtration”, http://homel.vsb.cz/~tum52/download/VoldKalman.pdf