How to reduce the trial period for experimental investigations of sealing elements by using the Kalman filter — The Kalman filter can be used to economize experimental investigations of sealing elements. In these investigations, the emission characteristics of sealing elements are determined to describe the sealing behavior. One of the most significant emission parameters is the total leakage rate.
The statistical test methodology, combining the experiment planning, set-up and procedure as well as evaluation, is a useful tool in a wide variety of scientific fields. Hereby, the experimental procedure provides a framework for the set-up and execution of the trials. In combination, further methods can be used to minimize the required time for investigations. Therefore, various algorithms or “filters” are suitable, one of them is the so-called Kalman filter.
Two typical trials are presented as below in the examples: The total leakage rate is displayed on the vertical axis and the measurement time on the horizontal. In Figure 1 the total leakage rate starts at LT 1 10-9 mbar · l/s and rises rapidly up to TT = 50 min to LT 3 10-6 mbar · l/s. The measurement values were recorded up to a measurement time of 200 min. Basically, the trials could have been terminated after a measuring time of 100 min. However, to obtain a reliable result, the test could run longer. Figure 2 shows an almost “ideal” series of trials with a standard deviation of 10-7 mbar · l/s.
These figures represent an “ideal” series of measurements whereas Figure 1 shows a total leakage rate over the trial period that cannot be described as “ideal”. The measured values scatter strongly, which can also be reflected in the standard deviation of the trials ( 6,2 10-7 mbar · l/s). Yet, the two trials show different scatters.
“Ideal” trials are not the standard when carrying out tests. In the case of “Non-ideal” trials, the scattering makes it difficult to determine the total stationary leakage rate, but it can be improved by increasing the testing period.
The observed scatter is caused by test itself and therefore it may refer as “Background noise/Disturbance”. It consists of process disturbance, system disturbance and measurement disturbance. The measuring disturbance can be taken from the specification of the mass spectrometer. The remaining part of the “Background noise/Disturbance” is the process disturbance. Although this can be minimized in terms of testing technology/experimental procedure, which consumes significant time and material. Alternatively, it is possible to use an experimental and evaluation procedure in order to reduce process or system and measurement disturbance as well as shorten the trial period.
The Kalman Filter
The Kalman filter allows to estimate the underlying physical model from measured values. However, this requires a Gaussian-distributed process disturbance, system disturbance and measurement disturbance. It can be used for online (real time) or offline test evaluations (after the trials).
The Kalman filter can be defined as follows: Determination of an algorithm (filter), which estimates optimal values in such a manner that average estimation error is minimal with the known dynamics of system and movement as well as the information about disturbance signals from the measurements. Observations and states of a physical system can be described as time-continuous or time-discrete. A distinction is made between the equation of state and the measuring or observational equation. Both together are also called as state space models.
The statistical analysis of the measured values from the experimental investigations requires a description of the measured values by an equation or a model, which is defined as a state or system equation for the test evaluations by using the Kalman filter. For this purpose, the Ornstein-Uhlenbeck process or Mean-Reversion process can be used, which describes the physical process underlying the measurement series. The parameter of the estimation for Mean-Reversion-Process is done using Maximum Likelihood Estimator. The estimated values are determined by using a “Program subroutine” in Matlab, which will passed to the “Main program routine” afterwards.
Figure 3 shows the use of the Kalman filter in the test evaluation. The Kalman filter is used as an example for the trials considering measurement time TT = 250 min, in which the significant influence of “Background Disturbance” (process, system and measurement) can be clearly visible. The estimated trials using the Kalman filter (red) takes relatively quickly the stationary value of the total leakage rate of LT 3 10-6 mbar · l/s. As it can be seen, the process disturbance, system disturbance and measurement disturbance are filtered out as much as possible.
After filtering out the process or system and measurement disturbance with the help of the Kalman filter, in determining the steady value of the total leakage rate, a required measurement time TT 50 results for this experiment. This results in a reduction of the test time by factor 5 compared to the actual original measurement time TT 250 min.
The application of the Kalman filter for above trials shows that it is possible with this instrument to make statistically reliable estimates of the stationary value. This has become particularly clear with the “Non-ideal” trials, since these process, system and measurement disturbances are most significant. With “ideal” measurement series, the influence of the Kalman filter cannot be visually observed, as the process, system and measurement are filtered out in very lower amount. In order to carry out this in practice, the integration of Kalman filter in a software-supported online evaluation is required, which break the series of measurements when the stationary value is reached.
* * Kar is a specialist for cost estimates, time scheduling and artificial intelligence at Shell; Berz works as a Senior Cost Estimator at Jacobs Engineering.