Sensor data validation using gray models

ISA TRANSACTIONS® ISA Transactions 42 共2003兲 9–17

Sensor data validation using gray models K. M. Tsang* Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

共Received 1 October 2001; accepted 16 April 2002兲

Abstract A new method based on the gray model is described for the online validation of measurements. A gray model is a differential equation describing the behavior of an accumulate generating operation 共AGO兲 data sequence. First-order gray models are fitted to measuring data records using the recursive orthogonal least-squares algorithm. Predictions derived from the fitted gray model are then compared with the actual measurements to generate a prediction error sequence. The quality of the measured value is determined by the prediction errors and variance of the prediction error sequence. Experimental results for detecting the quality of measurements from a thermistor are presented. © 2003 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Gray model; Data validation

1. Introduction Sensors provide online data which are prerequisites to monitor and control of technical processes. Sensors have been viewed as simple generator, their data have been assumed to be ‘‘correct.’’ However, many reported malfunction in a plant are found to stem from the lack of credibility of sensor data. Conventional contact between a sensor and the controller has been the unidirectional flow of measurement based upon the 4 –20-mA standard. This single stream of information has been used for monitoring the process, for feedback control, and for ensuring plant safety. Hence decision and control based on this single flow of information may lead to a sustained period of poor quality production or disaster if the quality of sensors or measurements are poor. There has been considerable effort devoted to the development of intelligent sensors in recent *Fax: 共852兲23301544. E-mail address: [email protected]

years 关1–5兴. The current trend towards intelligent sensors is to integrate a sensing element that can be made in a standard process with electronic circuits to fully and periodically calibrate and compensate the sensor, and circuits for generating a bus-compatible output with not only the sensing data but also the sensor health status. Yung and Clarke 关1兴 described a method of providing local sensor validation scheme which comprised of two stages. An autoregressive moving average 共ARMA兲 model of the sensor output was developed during a fault-free period of operation. The resultant model was then used to generate a prediction error sequence for determining the health of the sensor. Further signal processing is then required to determine the reason for any abnormal behavior and also generate fault indications. The technique was demonstrated on a simulation of a thermocouple system involving an increase in the variance of a noise source. Further work by Yang and Clarke 关3兴 led to the development of a selfvalidating thermocouple capable of determining several types of faults. Particular attention was

0019-0578/2003/$ - see front matter © 2003 ISA—The Instrumentation, Systems, and Automation Society.

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K. M. Tsang / ISA Transactions 42 (2003) 9–17

paid to a characteristic thermocouple fault resulting from a loss of contact with the process being measured. In the ARMA approach for data validation, a validated ARMA model is first fitted to the sensor during a fault-free period. The ARMA model is reapplied for the online validation of the measurements based on the prediction errors generated. Assumption has been made that both the signal and the prediction errors are stationary. The drawback of this approach is that a correct ARMA model has to be identified first and the signal itself may not be stationary. This limits the applicability of the algorithm. Gray system theory 关6兴 has been applied to various fields of data processing, modeling, control, prediction, system analysis, and decision making. The gray system modeling techniques have the advantages of 共i兲 assumption in probability distribution of data is not required; 共ii兲 as little as three data points are required for the modeling; 共iii兲 computation effort required is small for the construction of the model; and 共iv兲 highly adaptive to system dynamic behavior. The gray model has been found to be noise resistive and a large number of signals can be modeled by the first-order gray model. The first-order gray model could be applied in the processing of noisy measurements and this will eliminate the determination of the system order and validation of the system model during the learning stage. The simple structure of the first-order gray model also allows the model to be estimated online so that even nonstationary signals can be processed. This paper will investigate the feasibility of incorporating a gray model in online data validation scheme using the recursive orthogonal least-squares estimation algorithm 关7兴. Experimental results are included to demonstrate the effectiveness of the proposed approach.

2. Gray model A system which lacks complete information about its structure, operation mechanism, and behavior criteria is called a gray system 关6兴. The accumulate generating operation 共AGO兲 is aimed at transforming an irregular, scattered series of data into a smooth and monoincreasing series to lessen the effects of random characteristics and noise content in the data. Assume the original series of data with n samples is expressed as

Fig. 1. Schematic diagram showing the effect of AGO.

X 共 0 兲 ⫽ 兵 x 共 0 兲 共 T 兲 ,x 共 0 兲 共 2T 兲 , . . . ,x 共 0 兲 共 nT 兲 其 , 共1兲 where T is the sampling time and x ( 0 ) ( kT ) ⭓0 for k⫽1,2, . . . ,n. The first-order AGO of x ( 0 ) ( k ) is defined as

X 共 1 兲 ⫽ 兵 x 共 1 兲 共 T 兲 ,x 共 1 兲 共 2T 兲 , . . . ,x 共 1 兲 共 nT 兲 其 , 共2兲 k where x ( 1 ) ( kT ) ⫽ 兺 i⫽1 x ( 0 ) ( iT ) , for k⫽1, 2, . . . ,n. The purpose of AGO is to smooth out the scatter or randomness of the original data series and the impacts of the AGO in Eq. 共2兲 on the original data series is schematically shown in Fig. 1. If the firstorder AGO series is not smooth enough, reapplying the AGO on the AGO series can be conducted to reach the rth-order AGO series:

X 共 r 兲 ⫽ 兵 x 共 r 兲 共 T 兲 ,x 共 r 兲 共 2T 兲 , . . . ,x 共 r 兲 共 nT 兲 其 , 共3兲 k where x ( r ) ( kT ) ⫽ 兺 i⫽1 x ( r⫺1 ) ( iT ) , for k⫽1, 2, . . . ,n and r⫽2,3, . . . . The data series after the AGO smoothing can be modeled by a simple first-order differential equation to give a gray system model GM( r,m ) . The parameters r and m in GM( r,m ) actually denote the order of the operation of the AGO series and the number of variables in the differential equation, respectively. Among gray models, the firstorder gray model GM( 1,m ) is the most popular one 关8,9兴 employed in general applications. The differential equation relating the first-order AGO series X ( 1 ) is denoted by

dx 共 1 兲 共 t 兲 dt

⫹ax 共 1 兲 共 t 兲 ⫽b,

共4兲

K. M. Tsang / ISA Transactions 42 (2003) 9–17

where a and b are constants. Expressing dx ( 1 ) ( t ) /dt as

dx 共 1 兲 dt

共t兲

⫽ lim

x共1兲

共t兲

⫺x 共 1 兲

共 t⫺T 兲

T

T→0

共5兲

.

x 共 0 兲 共 k 兲 ⫽g 1 w 1 共 k 兲 ⫹g 2 w 2 共 k 兲 ⫹ ␰ 共 k 兲 ,

dt

冏

⫽x 共 1 兲

共k兲

⫺x 共 1 兲

共 k⫺1 兲

⫽x 共 0 兲

共k兲

t⫽kT

x 共 0 兲 共 k 兲 ⫹az 共 k 兲 ⫽b,

x 共 0 兲 共 k⫹1 兲 ⫽b⫺0.5a 关 x 共 1 兲 共 k⫹1 兲 ⫹x 共 1 兲 共 k 兲兴 ⫽b⫺0.5a 关

x共1兲

共k兲

⫹x 共 0 兲

⫹x 共 1 兲 共 k 兲兴

1⫹0.5a

.

共13兲

and n

1/n

兺

w 1共 k 兲 z 共 k 兲

k⫽1

␣⫽

n

兺

1/n

k⫽1

w 21 共 k 兲

1 ⫽ n

n

兺

z 共 k 兲 . 共14兲

k⫽1

The orthogonal parameters can be obtained by combining Eqs. 共11兲 and 共12兲 关7兴 to give

共 k⫹1 兲

共9兲

n

1 g 1⫽ n

兺

k⫽1

3. Recursive orthogonal estimation algorithm for gray models Consider the first-order gray model:

共15兲

兺

1/n

k⫽1

g 2⫽

x 共 0 兲共 k 兲 w 2 共 k 兲 ,

n

兺

k⫽1

w 22 共 k 兲

and the estimates of a and b are defined as 关7兴

aˆ ⫽g 2 , bˆ ⫽g 1 ⫺ ␣ aˆ .

共16兲

Eqs. 共13兲–共16兲 provide an offline calculation of the gray model parameters a and b for n data records. At time step k with n new measurements where k⬎n, define

1 Z k⫽ n

共10兲

where aˆ , bˆ , and ␰ ( k ) represent the two unknown parameters and the modeling error, respectively. The objective of the orthogonal estimation algorithm is to estimate the unknown parameters aˆ and bˆ in first transforming Eq. 共10兲 to an auxiliary equation:

x 共 0 兲共 k 兲 ,

n

1/n

With the availability of the measured data records x ( 0 ) ( k ) and z ( k ) , the numerically more stable orthogonal least-squares algorithm 关7兴 can be applied to Eq. 共7兲 to obtain estimates of the parameters a and b.

x 共 0 兲 共 k 兲 ⫽bˆ ⫺aˆ z 共 k 兲 ⫹ ␰ 共 k 兲 ,

共12兲

w 2 共 k 兲 ⫽z 共 k 兲 ⫺ ␣ w 1 共 k 兲 ⫽z 共 k 兲 ⫺ ␣ ,

and from Eq. 共8兲 the new predicted value xˆ ( 0 ) ( k ⫹1 ) can be obtained from past measurements as

xˆ 共 0 兲 共 k⫹1 兲 ⫽

w 1 共 k 兲 w 2 共 k 兲 ⫽0.

w 1 共 k 兲 ⫽1,

共8兲

b⫺ax 共 1 兲 共 k 兲

兺

k⫽1

The orthogonal data records can be constructed by defining 关7兴

共7兲

where the value of x ( 1 ) ( t ) is sampled and taken as the midpoint of the values x ( 1 ) ( k ) and x ( 1 ) ( k ⫺1 ) or z ( k ) ⫽0.5x ( 1 ) ( k ) ⫹0.5x ( 1 ) ( k⫺1 ) . Hence

n

1 n

共6兲

and the differential equation of Eq. 共4兲 can be reformulated to give

共11兲

where g 1 and g 2 are some constant coefficients, and w 1 ( k ) and w 2 ( k ) are constructed to be orthogonal over the n data records such that

If the sampling time T is set to 1, the derivative can be approximated by

dx 共 1 兲 共 t 兲

11

X k⫽ XZ k ⫽

1 n

1 n

n

兺

l⫽1

z 共 k⫺n⫹l 兲 ,

n

兺

l⫽1

x 共 0 兲 共 k⫺n⫹l 兲 ,

n

兺

l⫽1

x 共 0 兲 共 k⫺n⫹l 兲 z 共 k⫺n⫹l 兲 , 共17兲

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K. M. Tsang / ISA Transactions 42 (2003) 9–17

1 ZS k ⫽ n 1 XS k ⫽ n

␧RR 共 k⫹1 兲

n

兺 z 2共 k⫺n⫹l 兲 , l⫽1

⫽

n

兺

l⫽1

x 共 0 兲 共 k⫺n⫹l 兲 x 共 0 兲 共 k⫺n⫹l 兲 .

2 g 21 共 k⫹1 兲 ⫹g 22 共 k⫹1 兲共 ZS k⫹1 ⫺Z k⫹1 兲

XS k⫹1

⫻100 共20兲

Assuming the data size n for the estimation of the gray model is fixed, at time step ( k⫹1 ) when there is a new measurement x ( 0 ) ( k⫹1 ) the values of Z, X, XZ, ZS, and XS at time step ( k⫹1 ) can be derived from the values at time step k as

1 Z k⫹1 ⫽Z k ⫹ 关 z 共 k⫹1 兲 ⫺z 共 k⫺n⫹1 兲兴 , n 1 X k⫹1 ⫽X k ⫹ 关 x 共 0 兲 共 k⫹1 兲 ⫺x 共 0 兲 共 k⫺n⫹1 兲兴 , n 1 XZ k⫹1 ⫽XZ k ⫹ 关 x 共 0 兲 共 k⫹1 兲 z 共 k⫹1 兲 n ⫺x 共 0 兲 共 k⫺n⫹1 兲 z 共 k⫺n⫹1 兲兴 ,

which is an indicator that illustrates the percentage of the square of the output captured by the fitted model. If the value deviates from the value obtained in the fault-free period, it is a good indication that there is a change in the measured data. To summarize, the recursive orthogonal estimation algorithm for the online estimation of the gray model can be carried out in the following steps: 共i兲 Initialize Z, X, XZ, ZS, and XS to 0. 共ii兲 When there is a new measurement, apply Eq. 共18兲 to obtain new estimates of Z, X, XZ, ZS, and XS. 共iii兲 Apply Eq. 共19兲 to obtain the gray model parameters a and b. 共iv兲 Repeat procedures 共ii兲 and 共iii兲 when there is a new measurement to obtain new estimates of a and b for the gray model.

共18兲 1 ZS k⫹1 ⫽ZS k ⫹ 关 z 2 共 k⫹1 兲 ⫺z 2 共 k⫺n⫹1 兲兴 , n 1 XS k⫹1 ⫽XS k ⫹ 关 x 共 0 兲 共 k⫹1 兲 x 共 0 兲 共 k⫹1 兲 n ⫺x 共 0 兲 共 k⫺n⫹1 兲 x 共 0 兲 共 k⫺n⫹1 兲兴 , and from Eqs. 共14兲–共16兲 new estimates of ␣, g 1 , g 2 , aˆ , and bˆ at time step ( k⫹1 ) are then given by

␣ 共 k⫹1 兲 ⫽Z k⫹1 , g 1 共 k⫹1 兲 ⫽X k⫹1 , g 2 共 k⫹1 兲 ⫽

XZ k⫹1 ⫺ ␣ 共 k⫹1 兲 X k⫹1 2 ZS k⫹1 ⫺Z k⫹1

,

共19兲

aˆ 共 k⫹1 兲 ⫽g 2 共 k⫹1 兲 , bˆ 共 k⫹1 兲 ⫽g 1 共 k⫹1 兲 ⫺ ␣ 共 k⫹1 兲 aˆ 共 k⫹1 兲 . One ‘‘by-product’’ of the orthogonal estimation algorithm is the sum of the error reduction ratio test:

4. Sensor data validation The term ‘‘sensor’’ is often used to represent the total measurement system which comprises the sensor device, the transducer, the signal conditioner, and the necessary interface components. The input to the sensor is the measurand x ( t ) and the output is the measurement x ( 0 ) ( t ) . A sensor is declared faulty if the output measurement x ( 0 ) ( t ) gives an incorrect representation of the measurand x ( t ) . A conservative and common definition of sensor failure is that the output of the sensor x ( 0 ) ( t ) exceeds some threshold set down by the manufacturer. This is often called a ‘‘hard’’ failure. However, this definition is very restrictive and is insufficient for high performance and critical processes. A more general definition of sensor failure is given by Iserman 关8兴 as ‘‘a nonpermitted deviation from a characteristic property.’’ This includes ‘‘soft’’ failures such as an excessive drift as well as an abnormal increase or decrease in the noise level. Sensor validation is primarily based on the prediction error sequence 关1兴:

␧ 共 k 兲 ⫽x 共 0 兲 共 k 兲 ⫺xˆ 共 0 兲 共 k 兲 .

共21兲

K. M. Tsang / ISA Transactions 42 (2003) 9–17

Under normal fault-free operating conditions a number of statistics such as mean and variance can be determined for the prediction error sequence. Any subsequent changes to the sensor will be reflected in this sequence and hence in the derived statistics. Based on these signal patterns, a set of primary failure indicators are computed: the limit indicator, the jump indicator, and the noise indicator.

4.1. The limit indicator This indicator merely entails checking whether the signal output and its rate of change have violated some prescribed bounds. The threshold values are determined from a combination of the knowledge of the sensor specification and the operating environment. The limit indicator for the signal output has three possible states:

I lx 共 k 兲 ⫽

再

0兲 x 共 0 兲 共 k 兲 ⬍x 共min

⫺1, 0,

0兲 0兲 x 共min ⭐x 共 0 兲 共 k 兲 ⭐x 共max

1,

x共0兲

共22兲

0兲 , 共 k 兲 ⬎x 共max

(0) ( 0) where x min and x max are the minimum and maximum bounds on the signal output, respectively, and the limit indicator for the rate of change of signal can be reflected by the parameter a in the fitted gray model as

I la 共 k 兲 ⫽

再

13

tor I j ( k ) is highlighted if ␧ ( k ) exceeds a specified threshold, say ␭ ␴ ␧ :

再

I j 共 k 兲 ⫽ 0, 1,

兩 aˆ 共 k 兲 兩 ⭐a max

1,

兩 aˆ 共 k 兲 兩 ⬎a max,

共23兲

where a max is the bound on the rate of change of the signal.

4.2. The jump indicator The jump indicator detects a sudden displacement in the prediction error, and thus in the sensor output. The displacement may be due to a spike failure. If ␧ ( k ) is assumed to be a zero mean Gaussian noise with variance ␴ 2␧ , the jump indica-

⫺␭ ␴ ␧ ⭐␧ 共 k 兲 ⭐␭ ␴ ␧

共24兲

␧ 共 k 兲 ⬎␭ ␴ ␧ .

As an example, for the Gaussian assumption, ␭⫽3.8906 corresponds to a 99.99% confidence level. 4.3. The noise indicator The noise indicator identifies a change in the prediction error variance, which is a reflection of a deviation of sensor output variance. Stuck failure, erratic failure, and cyclic failure will cause permanent alternation in the variance. A useful test for comparing equality of variances from two independent samples is the F-ratio test 关10兴. Suppose 2 2 ␴ˆ ␧1 ( k ) and ␴ˆ ␧2 are the variances of the prediction error at the testing and learning stages, respectively. Under normal conditions, the variance of the prediction error should remain the same. Therefore the null hypothesis to be tested is 2 2 ␴ˆ ␧1 ( k ) ⫽ ␴ˆ ␧2 and the alternative hypothesis is 2 2 ␴ˆ ␧1 ( k ) ⫽ ␴ˆ ␧2 . At ␥ % significance level, the null hypothesis is accepted if

1 F ␥ %/2共 ␮ 2 , ␮ 1 兲

0,

␧ 共 k 兲 ⬍⫺␭ ␴ ␧

⫺1,

⭐F 共 k 兲 ⭐F ␭%/2共 ␮ 1 , ␮ 2 兲 , 共25兲

where

F共 k 兲⫽

␴ˆ 2␧1 共 k 兲 . ␴ˆ 2␧2

共26兲

2 ␴ˆ ␧1 ( k ) is the recursive estimate of the variance in 2 is the reference variance obthe testing stage, ␴ˆ ␧2 tained from N 2 samples in the learning stage, ␮ 1 ⫽N 1 ⫺1, and ␮ 2 ⫽N 2 ⫺1 where N 1 and N 2 are 2 ( k ) and the number of samples in computing ␴ˆ ␧1 2 ˆ␴ ␧2 , respectively. The noise indicator I n ( k ) is set according to the value of F ( k ) :

14

I n共 k 兲 ⫽

K. M. Tsang / ISA Transactions 42 (2003) 9–17

冦

⫺1, F 共 k 兲 ⬍ 0,

1 F ␥ %/2共 ␮ 2 , ␮ 1 兲

1 F ␥ %/2共 ␮ 2 , ␮ 1 兲

⭐F 共 k 兲 ⭐F ␭%/2共 ␮ 1 , ␮ 2 兲

共27兲

1, F 共 k 兲 ⬎F ␭%/2共 ␮ 1 , ␮ 2 兲 .

As an example, if the significance level is set to 5%, N 1 ⫽6, and N 2 ⫽21, then from the statistics tables 关10兴 F 2.5% ( 5,20) ⫽3.29 and 1/F 2.5% ( 20,5) ⫽1/6.33⫽0.158. 5. Experimental studies The temperature measuring unit attached to the process trainer PT326 关11兴 from Feedback Instruments Ltd. was used. The measuring unit composes of a bead thermistor connected in parallel with a resistor forming one arm of a bridge. The error voltage from the bridge is fed to an amplifier to produce the output measurement voltage. A 12bit A/D converter operated at 400 Hz was used to collect temperature measurements from the amplifier output. 5.1. Learning stage 4000 data points were collected from PT326 to obtain the characteristics of the fault-free measuring system. Fig. 2 shows the 4000 data points collected during the fault-free period. The recursive estimation algorithm of Eqs. 共17兲–共19兲 was applied to obtain new estimates of the gray model parameters a and b iteratively. New predicted out-

Fig. 2. Measured data records from PT326.

Fig. 3. 共a兲 The profiles for the fitted gray model parameters. 共b兲 The prediction error sequence.

puts and the prediction errors were obtained using Eqs. 共9兲 and 共21兲. For the fitting of the gray model, the data size n was set to 50 and Fig. 3 shows the profiles of the fitted gray model parameters and the prediction error sequence. The standard deviation ␴ˆ ␧2 of the estimation error was 6.25⫻10⫺3 and the mean of the estimation error was 5.09 ⫻10⫺7 . 5.2. Change of the input throttle value Fig. 4 shows 4000 data records collected from the PT326 when the input throttle valve was varying between positions 2 and 4 causing a variation in the temperature readings. The data window size

Fig. 4. Measured data records from PT326.

K. M. Tsang / ISA Transactions 42 (2003) 9–17

Fig. 5. 共a兲 The profiles for the fitted gray model parameters. 共b兲 The prediction error sequence.

n was fixed at 50. Fig. 5 shows the profiles of the fitted gray model parameters and the prediction error sequence, and Fig. 6 shows the profiles of the different fault indicators. Note that 99.99% confidence interval was used for the detection of the jump and noise indicators and a data window size of 2000 were used for the estimation of the noise variance. A larger window size was selected such that the estimated noise variance was more consistent. The maximum and minimum limits set on the readings were 10 and 0, respectively, and the maximum value set on the rate of change of

Fig. 6. Fault indicators: 共a兲 measurement output; 共b兲 rate of change of reading; 共c兲 jump indicator; 共d兲 noise indicator.

15

Fig. 7. Measured data records from PT326.

parameter was 0.0025. Even if there was a temperature change, there were no sign of any faults indicating that the measurements were of good quality. 5.3. Stuck failure A plastic tape was stuck to the thermistor to simulate the stuck failure. Fig. 7 shows the 4000 collected data records. Fig. 8 shows the profiles of the fitted gray model parameters and the prediction error sequence. Fig. 9 shows the profiles of different fault indicators. Clearly the variance of the prediction error sequence was much smaller

Fig. 8. 共a兲 The profiles for the fitted gray model parameters. 共b兲 The prediction error sequence.

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K. M. Tsang / ISA Transactions 42 (2003) 9–17

Fig. 9. Fault indicators: 共a兲 measurement output; 共b兲 rate of change of reading; 共c兲 jump indicator; 共d兲 noise indicator.

than the fault-free case which is clearly indicated by the noise fault indicator of Fig. 9共d兲. Also at approximately 1 sec, the jump fault indicator also showed that there were spikes in the measured data records. 5.4. Change of the bridge balance The balance of the measurement bridge was ramping up and down. Fig. 10 shows the 4000 collected data records as the balance of the bridge was ramping down. Fig. 11 shows the profiles of the fitted gray model parameters and the prediction error sequence. Fig. 12 shows the profiles of

Fig. 10. Measured data records from PT326.

Fig. 11. 共a兲 The profiles for the fitted gray model parameters. 共b兲 The prediction error sequence.

different fault indicators. The noise indicator clearly shows an increase in the noise variance and the jump fault indicator shows that there were spikes in the measured data records. 6. Conclusions The recursive orthogonal least-squares algorithm has been successfully implemented online for the estimation of gray model parameters. Experimental studies demonstrate that the prediction error sequence derived from the fitted gray model can be used to detect faults such as spikes and a

Fig. 12. Fault indicators: 共a兲 measurement output; 共b兲 rate of change of reading; 共c兲 jump indicator; 共d兲 noise indicator.

K. M. Tsang / ISA Transactions 42 (2003) 9–17

change in the noise variance. These fault indicators can accompany the raw measurements to enhance the quality of sensor measurement used for processes, feedback control, and thus ensuring greater plant safety. Acknowledgment The author gratefully acknowledges the support of the Hong Kong Polytechnic University through grant A-PD63. References 关1兴 Yung, S. K. and Clarke, D. W., Local sensor validation. Meas. Control 22, 132–141 共1989兲. 关2兴 Henry, M. P. and Clarke, D. W., The self-validating sensor: rationale, definitions and examples. Control Eng. Pract. 1, 585– 610 共1993兲. 关3兴 Yang, J. C. Y. and Clarke, D. W., A self-validating thermocouple. IEEE Trans. Control Syst. Technol. 5, 239–253 共1997兲. 关4兴 Qin, S. J., Yue, H., and Dunia, R., A self-validating inferential sensor for emission monitoring. Proceedings of the American Control Conference, 1997, pp. 473– 477. 关5兴 Moran, A. W., O’Reilly, P. G., and Irwin, G. W., A case study in on-line intelligent sensing. Proceedings of the American Control Conference, 2000, pp. 3949– 3953.

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关6兴 Deng, J. L., Control problems of gray systems. Syst. Control Lett. 1, 288 –294 共1982兲. 关7兴 Tsang, K. M. and Billings, S. A., Orthogonal estimation algorithm for complex number systems. Int. J. Syst. Sci. 23, 1011–1018 共1992兲. 关8兴 Huang, Y. P. and Chang, C. C., The integration and application of fuzzy and grey modeling methods. Fuzzy Sets Syst. 78, 107–119 共1996兲. 关9兴 Zhang, X. T., Zhu, H. Y., and Zhang, H. B., Robust gray model based on genetic algorithms and its application to prediction for chromotographic retention. Chemom. Intell. Lab. Syst. 44, 197–203 共1998兲. 关10兴 DeGroot, M. H., Probability and Statistics. AddisonWesley, Reading, MA, 1975. 关11兴 Feedback Instruments Ltd., Process Trainer PT326, FI Ltd., Crowborough, England.

K. M. Tsang received his B.Eng. and Ph.D. degrees in control engineering from the University of Sheffield, U.K., in 1985 and 1988, respectively. He is currently an Associate Professor in the Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong. His research interests include system identification, intelligent control, and pattern recognition.