The assessment of any technical object functioning stability is often limited by the monitoring of midrange constancy and monitored parameters dispersion. For that, the methods of multivariate statistical monitoring, used for the assessment of process stability, are offered. Midrange multivariate process monitoring is accomplished with the help of algorithms, based on Hotelling's chart statistics. While assessing the dispersion stability, one can use generalized variance based algorithms - covariance matrix determinant. The approaches described here to increase the efficiency of multivariate dispersion monitoring.
(xijt x j )( xikt xk ) , UCL |0| (b1 u1-/2 b2 ), LCL where u1-/2 is normal inverted distribution of order 1 – /2, is a confidence level (probability of false alert); the coefficients are computed aps per the following formulae : b1 (n 11) p (n j); (4) b2 1 j1p (n j)[p (n k 2) p (n k )] , (5) (n 1) 2 p j1 k 1 k 1 the assessment of destination generalized variance |0| is found as per the learning sample .If the lower control line LCL as per formula (3) is negative, zero value is taken.
Destabilization of the process is witnessed by at least one point getting beyond one of the control lines on the chart of the generalized variance , i.e. the process is steady when the in equation below is satisfied:
LCL< |St| <UCL, (6) where t is the number of monitored samples. For example, Fig.1 shows the chart of generalized variance: lower control line is zero, no points beyond the control line: the process is steady.
(3) Рис. 1.Картаобобщеннойдисперсии
The process is considered steady as per criteria of multivariate dispersion if on the chart of generalized variance there are no points beyond the control lines, i.e. the condition is followed (6). This condition is important, but very often insufficient to ensure the process stability. Sometimes on the chart there are special form structures, which testify process instability: these are the structures, the probability of which is commensurate with the probability of false alert. For example, several successive points increasing or decreasing indicate the trend of process monitored parameter. The specialists have no unanimous opinion regarding the structures to be used for stability assessment. Western Electric [4,5] four criteria are widely popular; one of them for example is as follows: at least eight successive points located on one side of the central line show the process instability. ISO distinguish eight criteria [6], six criteria are offered for Hotelling’s chart [7].
Generalized variance algorithm is based on normal inverted distribution, and as a rule, practical calculations are done on the basis of three sigma rule: in formula (3) we take u1-/2 = 3. To find the fault one can use the same specialized structures types as for Schewart’s chart. They are: 1) at least one point getting out beyond the control lines, 2) at least two out of three points located on one side of the central line, getting out beyond the twin sigma limits 3) at least four out of five successive points located on one side of the central line, getting out beyond one sigma limit 4) at least eight successive points located on one side of the central line , 5) six decreasing or increasing points in a row (trend), 6) fourteen in turn increasing and decreasing points ( cycles) etc.
The probability of eight points in a row on one side of the central line may be detected as follows. Firstly, we check the criterion fault (6) i.e. a point getting out beyond one of the control lines, probability of this event while using tree sigma rule is equal to 0,0027/2 = 0,00135. The probability of one point getting to one side of the central line is equal to 0,5. Then the probability of eight points on one side of the central line provided all the points are located within the control lines is equal to (0,5 – 0,00135)8 = 0,003823, this is commensurate with the probability of false alert 0,0027.
Plotting the control charts on PC, the search of specialized structures of any type on the charts is easily computerized, without any difficulties. But, take into account that the increase of criteria number will lead to decrease of observations number among false alerts. Using only structure 1 to detect the unsteady state this number is equal to 1/α ≈ 370 samples, structures 1 and 4 get 153 samples selecting four structures from the 1st to 4th lead to 92 samples. This value is acceptable, but the use of additional criteria may bring the number of observations among the false alerts to unacceptably small value.
To detect the step-wise increase in the dispersion exponentially weighted moving average algorithm for generalized varianceis seldom used; the corresponding values are determined as per formula Et = (1 –k) Et-1 + k|St|; where H is a parameter, which determines the location of the control lines; Et is a mean square deviation of Et values , determined as per formula :
The conducted investigation made it possible for us to offer the following way of object operation stability assessment: 1. Under the conditions of flawless stable operation the detectors readings are taken and main statistical characteristics are calculated: mean values vector and covariant matrix (characteristics of learning sample). 2. A set of all possible statistical tools is selected for further monitoring. Non correlated data are monitored by the tools based on Schewart’s chart. To monitor mean level of correlated parameters, Hotelling’s chart is used, to monitor multivariate dispersion, generalized variancechart is used. 3. When necessary the exponentially weighted moving average algorithm based on Hotelling statistics and generalized variance is used. 4. Constant monitoring of object operation is done in order to detect destabilization. Specialized structures are searched on the charts, which prove the possible fault in the process.
The computational investigation was done based on the example of hydraulic unit serviceability with the use of vibration dispersion stability criterion [12]. The detectors readings were correlated; simulated dispersion increase was captured by the exponentially weighted moving averages chart (Fig.2)
To assess the stability of object functioning stability as per the criteria of multivariate dispersion one may use the control charts of generalized variance. But these charts do not always detect the faults on time. There were offered methods for the charts sensitivity improvement: the search of non- random structures and the use of algorithmof exponentially weighted moving averages can significantly increase the monitoring efficiency.
The offered way of object functioning destabilization diagnostics, based on the process statistical control methods, enables timely detection of the operation faults , connected with its parameters dispersion alternation , and prevents an emergency when necessary.
The investigation is done with financial support of RFFI (РФФИ), projects №16-48-732002.
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