New in Situ Sensor Modeling Approach to Measurement Validation

tify faults locally at the sensor level. Himmelblau and. Bhalodia2 propose sensor validation using statistical tests and signal modeling with artifici...
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New in Situ Sensor Modeling Approach to Measurement Validation† Deepak Srinivasagupta‡ and Babu Joseph*,‡ Process Control Systems Laboratory, Department of Chemical Engineering, Washington University, One Brookings Drive, St. Louis, Missouri 63130

Robert E. Morley Electronic Systems and Signals Research Laboratory, Department of Electrical Engineering, Washington University, One Brookings Drive, St. Louis, Missouri 63130

Measurement error due to sensor degradation (fouling, miscalibration, etc.) is more difficult to identify compared to catastrophic sensor failure. Passive methods previously proposed for sensorlevel monitoring are based on power spectrum or multiscale analysis of sensor data. These methods have limitations caused by not accounting for various noise sources and assumptions about sensor noise characteristics, thus resulting in false and missed alarms. In this paper, an online sensor fault detection scheme based on the identification of sensor response characteristics is proposed and evaluated. We develop both robust passive and active in situ techniques to identify sensor response characteristics that relate directly to its health. Using the identified sensor model, various kinds of sensor faults are quantified and mapped into the model parameters. A dynamic model-based estimator is proposed for data reconciliation. These ideas were experimentally validated using thermocouples, flowmeters, and resistance thermometric devices on laboratory-scale processes. The proposed approach was seen to accurately quantify the sensor model parameters and aid in measurement reconstruction. 1. Introduction Accurate measurements are necessary in providing controllers and operators with a view of the process status. Safe and profitable operations of a chemical plant rely heavily on the proper functioning of sensors. In process control, many (reportedly up to 60%) of the perceived malfunctions in a plant are believed to originate from the lack of credible measurements.1 Reliable measurement validation mechanisms are needed by operators and the control systems that use the information provided by the sensor for decision making. Sensor faults can be classified into two broad types: hard (catastrophic) sensor failure, which results in completely erroneous or no readings from the sensor, and sensor degradation (change in physical condition or calibration), which degenerates the performance of the sensor. In this work, we are primarily concerned with identifying and rectifying the latter type of sensor faults, though many of our methods are valid for the former as well. Sensor degradation can be due to a variety of circumstances such as fouling, mechanical damage, electrical malfunction, or incorrect calibration. Because sensors cannot be easily removed and tested for faults, it is desirable to develop in situ measurement validation schemes. There is inherent isolation from process-level faults because such schemes are based on the sensor measurement alone. Process-model-oriented sensor fault detection methods do not easily separate sensor faults from process problems. Furthermore, the † Based on a presentation at the AIChE Annual Meeting, Indianapolis, IN, Nov 2002. * To whom correspondence should be addressed. Tel.: (813) 974-0692. Fax: (813) 974-3651. E-mail: [email protected]. ‡ Present address: Department of Chemical Engineering, University of South Florida, 4202 E. Fowler Avenue, ENB 118, Tampa, FL 33260.

disparity in time scales between the process and the measurement system is detrimental to the efficacy of process-model-oriented methods. Sensor degradation affects the dynamic performance of the sensor and is difficult to detect during normal operating conditions. Measurement aberration detection denotes methods and algorithms that attempt to identify faults locally at the sensor level. Himmelblau and Bhalodia2 propose sensor validation using statistical tests and signal modeling with artificial neural networks. Time-series models have also been used.3,4 Luo et al.5 used multiscale (wavelet) filters to retain the midfrequency noise and applied various statistical tests to identify sensor degradation. Luo et al.6 used dynamic principle component analysis (PCA)7,8 on the wavelet decomposition of the signal to detect sensor degradation. Ying and Joseph9 proposed a sensor fault detection and identification (FDI) scheme based on the power spectral density (PSD) of measurement noise. All of these methods are passive, often require large amounts of data, and are prone to both type I (false alarms) and type II (undetected faults) errors. Because these methods mostly rely on validation tests using univariate or multivariate statistics, quantification of sensor degradation and measurement rectification is not easy. Nevertheless, these methods are relevant to us because they deal with sensor diagnosis at the lowest level of a fault detection hierarchy. This has become possible because of the advent of plantwide intelligent sensor networks that support self-validating (SEVA) systems10,11 and the availability of sufficient local computing power at the sensor level. Among proactive approaches, the loop current step response (LCSR) method, sometimes called the resistive heating method,12,13 was developed in the late 1970s14,15 for in situ response time testing of resistance thermometric devices (RTDs), thermocouples,1,16-18 and ther-

10.1021/ie0209834 CCC: $25.00 © 2003 American Chemical Society Published on Web 05/03/2003

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Figure 1. Closed-loop feedback control system with noise sources.

mistors. Henry and Clarke19 proposed a SEVA scheme to monitor the health of a sensor. These methods result in fewer false alarms. Such proactive means seem promising in the development of robust FDI schemes. However, they are limited to sensors that can be actively perturbed. All of the studies presented so far have been restricted to temperature sensors where resistive heating is applicable. In this work, we propose a new in-situ robust sensor modeling approach to measurement validation. A sensor’s health can be related to its response model, and it is often possible to devise a suitable compensator to provide an estimate of the true measurement if degradation is detected. The organization is as follows. First, an understanding of process measurement and measurement noise is developed by a review of the passive method recently developed by Ying and Joseph.9 Next, the theory behind in situ sensor modeling using robust passive methods is developed. We then develop the notion of active sensor perturbation (ASP) to build accurate sensor models. We extend the LCSR method to show that it can measure not only changes in the response time but also quantify calibration errors. We then overcome some of its drawbacks by developing a more robust frequency domain perturbation technique. Both active and passive methods are validated with laboratory experiments on thermocouples and RTDs. Finally, measurement reconciliation in the case of sensor degradation using a sensor model is described and experimentally validated. 2. Passive Sensor FDI Because most chemical process dynamics are slow and most sensors are designed to be fast responding, the lowfrequency content in the data streaming from a sensor is primarily related to process variations of operational interest. Any mid- or high-frequency content usually arises from process noise, such as turbulence, and instrument noise. The naturally occurring, random process noise is attenuated by the sensor, and we can analyze this attenuated noise to detect sensor faults. This forms the basis of passive techniques that examine sensor measurement at mid- and high-frequencies to detect sensor degradation. For continuous processes operating largely around a steady state, noise characteristics and sensor dynamics can be assumed to be time-invariant and relatively independent of the operating conditions. 2.1. Sensor Noise Characteristics. Consider the block diagram structure of a simple feedback control system shown in Figure 1. Measurement noise enters the system from various sources, but the two primary

sources of noise are (i) the process and (ii) the signal conditioning and transmission assembly. We regard the sensor as comprising two parts: a transducer (or sensing element), which is in contact with the process, and a signal conditioner, which processes the signal from the transducer. Process Noise. Process noise arises as a result of local fluctuations introduced by turbulence, Brownian motion, and other physical phenomena that exist over various lengths and time scales. Process noise involves real, local, time-varying changes in the sensed variable and is usually not of interest during normal operation. As this noise arises within the process, it is attenuated both by the sensing element and the signal-conditioning unit. The degree of this attenuation is thus assumed to be indicative of the health of the measurement system. Nonprocess Noise. We regard it as comprising two components: electrical noise, which is added on by the circuitry that conditions (e.g., amplification, level shifting) the transducer signal, and transmission noise when the measurement is sent to the control system. The electrical noise usually consists of harmonics of the 60 Hz supply voltage, ground noise, resistive noise (Thompson noise), and ac transients. The electrical noise is not attenuated by the transducer but could possibly be used to identify calibration errors such as gain variation because it passes through the signal conditioning unit. The transmission noise arises mainly from electromagnetic induction (radio signals, lighting, etc.) and is not attenuated by the measurement system. It can be avoided in digital control systems, and we will make that assumption for our study. Unless mentioned otherwise, the term nonprocess noise will be used synonymously with electrical noise. 2.2. Experimental Setup. A PC-based data acquisition (DAQ) system was used to acquire temperature measurements for our studies. A temperature sensor (Ktype thermocouple) was set up to measure the temperature of air from a hot air blower. The air heater power could be controlled from the PC. Data Translation’s DT2801 and National Instruments’ PC-516 DAQ cards were used for low- and high-speed data acquisitions, respectively. The thermocouple was covered with cottonfilled thin rubber tubing to simulate fouling. A further description of this air temperature control process is given by Srinivasagupta.20 2.3. Review of the Ying-Joseph Method. The online PSD-based sensor fault detection scheme devised by Ying and Joseph9 involves high-frequency sampling, outlier removal, detrending, PSD calculation, and reduction of variable space (due to cross-correlated power spectral trends at different frequencies) using PCA and estimating the Hotelling T 2 test statistic. To verify this scheme, the normal (Figure 2) and coated thermocouples (Figure 3) were sampled at a rate of 500 Hz. The measurement noise characteristics are dependent on the operating conditions. To monitor sensor health, we are particularly interested in the process noise generated by the thermohydraulics of the system. At increased temperatures, thermal excitation from the heater ensures adequate process noise. This process noise seems to have a smaller variance for the coated thermocouple because of increased attenuation as it passes through the sensing system. Z-score normalization was applied to the power trend data at each frequency, and a PCA model was trained on this PSD data (specifications in Table 1). We used

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Figure 2. High-speed measurement for a normal thermocouple.

Figure 4. Hotelling T 2 test applied to data from a coated thermocouple.

modeling methods in subsequent sections. In principle, in situ sensor modeling without access to the inputs is a strictly intractable problem. This is overcome in two ways: (a) for passive modeling methods, we make certain assumptions about the sensor’s input, and (b) for active modeling methods, we perturb the sensor directly from the output. 3. Passive Sensor Modeling

Figure 3. High-speed measurement from a coated thermocouple. Table 1. Algorithm Specifications for the Ying-Joseph9 Method specification

value

sampling frequency window length periodogram length FFT window type frequencies for the PCA model no. of principal components

500 Hz 5000 points 32 points 32-point Hanning (50 + 15.63k) Hz, k ) 1, ..., 10 2 (for 98% of variance)

the PLS Toolbox21 for Matlab to build the PCA model. The PSD calculation was repeated for data from the coated thermocouple with the heater and the blower fully on. The Hotelling T 2 statistic was used to compare the PSD data from the coated thermocouple with the PCA model. The Hotelling T 2 statistic stayed above the 95% confidence limit (dashed horizontal line in Figure 4) obtained from the training data set, for sustained periods of time, suggesting the presence of a fault, if somewhat unconvincingly. The Ying-Joseph9 method assumes that most of the measured noise is from the process and is stationary. In the absence of sufficient process noise, the method fails completely.20 An improved performance may, therefore, be expected by filtering out the nonprocess noise when used to detect sensor coating faults. Other drawbacks of the Ying-Joseph method include lack of sufficient guidelines in identifying and quantifying sensor fault types. To address all of these drawbacks, we develop passive and active in situ sensor

3.1. Categorization of Sensor Degradation. For the purpose of sensor modeling, we divide sensor degradation into two main categories, based on how they affect the dynamic performance of the sensor. These are described below. Transience Faults. By transience errors, we refer to errors arising as a result of changes in the dynamic response time of the sensor. Changes can take place as a result of fouling, primarily from the extended contact of the sensor with a dirty process fluid. Over a period of time, there may be inert coatings on the sensing element acting as a barrier, effectively decreasing the heat- or mass-transfer coefficient. Alternatively, there may be corrosion. Both of these affect the speed of the sensor response by altering its time constant, without introducing significant steady-state bias. Calibration Faults. Any errors in span or the zero reading will be termed calibration faults because both of them introduce a fixed steady-state bias in the measurement. In general, if yp(t) is the steady-state process output and y′(t) is the corresponding sensor output, linear calibration errors can be represented as a linear transformation y′(t) ) y0′ + K′[yp(t) - y0], where K′ denotes any (nonunity) gain brought about by an error in span. y0 and y0′ denote the correct and incorrect zero readings. Zero errors (y0′ * y0 and K′ ) 1) cannot be detected from a dynamic modeling test alone. In principle, by forcing a zero signal into the signal conditioning unit, we can obtain the zero reading and determine if there has been an error. Span errors (y0′ ) y0 and K′ * 1) manifest themselves as a steady-state gain in a sensor’s dynamic model. Cases of outright sensor failure, such as frozen readings, can be regarded as special cases of calibration faults (K′ ) 0). Even if sensors are initially well calibrated, mechanical damage over a period of time can change the gain (e.g., for strain gauges and RTDs) or introduce zero errors.

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3.2. Principles of Sensor Modeling. We propose to identify sensor faults by continuously identifying a sensor model and then detecting statistically significant changes in the model parameters. For example, eq 1 shows the energy balance on a temperature transducer (thermocouple, or RTD with negligible self-heating) where the transducer directly contacts the process fluid, modeled as

mCp

dT + hA(T - Tp) ) 0 dt

(1)

Here, Tp and T are the time-varying process and transducer temperatures, respectively. For the transducer element, m, Cp, h, and A denote the mass, specific heat, heat-transfer coefficient, and exposed surface area, respectively. The time constant of the transducer is τ ) mCp/hA. We have ignored the effect of any (usually small) temperature-dependent electromotive force generated internally, such as for a thermocouple. Many simple sensors can be similarly modeled using first principles as first-order lag processes [GT(s) ) K/(τs + 1)] with gain K ) 1, zero delay, and time constant τ. Sensors with overdamped high-order dynamics (e.g., a thermocouple inside a thermowell) can still be approximated as first order. Any change in the sensor model parameters is suggestive of a fault in the sensor, both hard sensor failure and sensor degradation. Sensor fouling usually increases its time constant. Calibration errors (such as changes in span), stuck readings (including electric disconnection), and erratic errors bring about a change in the gain in the sensor model. A faulty sensor can be similarly represented as G′T(s). In situ sensor model identification necessitates that reasonable assumptions be made about the sensor input. The passive in situ sensor modeling proposed here is valid provided the process noise is stationary. Even though we use linear Laplacian domain models for notational simplicity, the approach is general and extensible to higher-order models, as well as other linear model representations, both continuous and discrete. It must be noted that, for poorly chosen models, spurious drifts in model parameters are possible. 3.3. Passive Sensor Modeling. If Pss(Ω) is the PSD of a continuous signal and Pxx(ω) is the PSD of the sampled signal, the relation between them is22

1 ω Pxx(ω) ) Pss T T

()

where the angular frequency is |ω| < π, T is the sampling period, and Ω ) ω/T. In practice, PSD is usually computed using the Welch method23 or one of its variants. The Welch method provides an asymptotically biased estimate of the PSD. For a fixed data length N, there is a tradeoff between the bias and the variance of the PSD estimate. However, by an increase of the sampling rate to the maximum extent supported by the DAQ hardware and thus an increase of the data length N, both the bias and the variance can be reduced simultaneously. Besides, we are primarily interested in frequencies well above the closed-loop corner frequency of the process. Fast sampling, on the order of kilohertz and even megahertz, is supported on current DAQ systems. When operating around a steady state, eq 2 relates the sensor output y(s) to the process noise np(s) and the electrical noise ne(s) as

y(s) ) K

{τs 1+ 1[y (s) + n (s)] + n (s)} p

p

e

(2)

Let Ppxx(ω) and Pexx(ω) denote the respective PSD of these noise inputs. We consider only those frequencies that are well above the stop-band cutoff frequency of the process. For stationary noise, Ppxx(ω) and Pexx(ω) do not change with time. For identification of the model parameters, we propose a five-stage sensor modeling technique as described below. Healthy Sensor Modeling. At the time of installation, if possible, the healthy sensor model parameters K and τ are determined at close to standard operating conditions (from standard offline identification tests or manufacturer specifications). Stationary Noise Characterization. After the sensor has been installed, it is continuously sampled at the maximum supported rate. The stationary nonprocess noise and the sensor output noise signatures are determined in situ, under standard process operating conditions. In addition, the standard uncertainty (usually 95% confidence intervals) is also computed. Detection of the Calibration Fault. Under normal process operations, it may be reasonable to assume that span errors and transience errors do not occur simultaneously. On the basis of the attenuation of a signal through a first-order filter, it is possible to relate the PSD of the output from the normal [Pyy(ω)] and faulty [P′yy(ω)] sensors to their input.24 For span errors (τ ) τ′) alone, if ωt and ωp are the corner frequencies of the transducer and process, respectively, under conditions of stationary noise and for ωt > ω . ωp:

K′ ) K

x

P′yy(ω) Pyy(ω)

(3)

If the ratio [P′yy(ω)/Pyy(ω)]1/2 (statistically) significantly deviates from unity and is independent of the frequency, it can be concluded that a span error has occurred. Removal of Nonprocess Noise. Once span errors can be ruled out, the nonprocess noise ne(s) is then filtered out for increased robustness and quantifying transience errors. The filter pass bands are based on the nonprocess noise signature built during the noise characterization stage. Often the nonprocess noise exists at much higher frequencies compared to the process noise and can be easily removed using standard digital filters. Care must be taken to ensure the greatest selectivity in removing the nonprocess noise in the case of spectral overlap. We suggest the use of wavelet softthresholding filters in such cases because they can resolve in both time and frequency domains. The soft threshold allows a desired selectivity in the removal of nonprocess noise for optimal performance. Detection of the Transience Fault. For transience faults alone (K′ ) K ) 1), it can be seen that the PSD ratio varies with frequency (ω > ωt). As ω f ∞

τ ) τ′

x

P′yy(ω) Pyy(ω)

(4)

If the ratio [P′yy(ω)/Pyy(ω)]1/2 deviates from unity at high frequencies beyond the limits of uncertainty, it may be concluded that a transience fault has occurred. In this way, we can distinguish between the two main types: transience and calibration faults. Because the

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ratio [P′yy(ω)/Pyy(ω)]1/2 plays an important factor in our analysis, it will, henceforth, be termed as a sensor degradation factor (SDF), γ(ω). The superscripts r and f denote the raw and filtered SDFs as in eqs 3 and 4, respectively. Statistically significant nonunity values of the SDF indicate sensor faults. In addition, if the healthy sensor model parameters K and τ are known, we can determine K′ and τ′ from eqs 3 and 4. If ∆Pyy(ω) and ∆P′yy(ω) are the uncertainty (e.g., 95% confidence intervals) in Pyy(ω) and P′yy(ω), respectively, the uncertainty in the SDF is approximated as

∆γ(ω) )

x

1 2

[∆P′yy(ω)]2

Pyy(ω) P′yy(ω)

+

P′yy(ω) [∆Pyy(ω)]2 Pyy2(ω)

(5)

In the testing for fouling faults, the accurate removal of the nonprocess noise from the measurement is important. Filtering out the nonprocess noise when it does not spectrally overlap with process noise is a trivial exercise. For frequency bands where there is some overlap between the process and nonprocess noise, filtering must be done selectively so that we minimize the loss of process noise energies. Conventionally designed stop-band and pass-band filters attenuate all of a signal in specific frequency bands. On the other hand, wavelet decomposition (see Graps25 for a tutorial introduction) resolves the signal in both time and frequency domains up to the limits of Heisenberg uncertainty. Under the assumption of stationary noise in a process operating around a steady state and using soft thresholding filters, we can select threshold levels (based on the nonprocess noise signature) that eliminate the nonprocess noise and retain the process noise in an optimal sense. In addition, power spectral analysis assumes that the underlying noise has a Gaussian distribution and is fully described by second-order statistics. Using the Wavelet toolbox for Matlab, decomposition of the measurement with significant process noise (last 200 000 points of Figure 2) suggests that our observed noise is Gaussian (Figure 5), although it cannot be conclusively said so. The histograms have been plotted using 50 bins. In general, noise characteristics are local and specific to the particular measurement. The passive sensor modeling method was verified using the part of the data from Figure 3 that possessed significant process noise. Separate step tests on the normal and coated thermocouples resulted in time constants of τ ) 19 s and τ′ ) 30 s. A wavelet denoising filter was applied to eliminate the nonprocess noise to test for fouling faults. The signal was decomposed into four levels using the db1 wavelet as before. The denoising was performed on the detail coefficients using a soft thresholding. The soft thresholds at various detail levels (level 1, 0.005; level 2, 0.007; level 3, 0.008; level 4, 0.010) were determined from the measurement with only nonprocess noise (blower, heater off). In general, the nonprocess noise signature can be captured by shorting the input terminals to the signal conditioner or by using dummy wiring to capture only the electric noise. Even though sampling rates much beyond 500 Hz are desirable for passive in situ sensor modeling and are indeed supported by common DAQ systems, when using the temperature sensors, we were limited by the maximum write speed to the disk in our offline analysis. In practice, the streaming data can be processed online

Figure 5. Noise histograms at four levels with db1 wavelet decomposition.

Figure 6. Filtered SDF for a thermocouple.

in solid-state buffers, limited only by memory space and computational speed. Being able to isolate the process and nonprocess noise signatures leads to increased robustness and performance in PSD-based sensor FDI. The PSD calculation involved a 32-point discrete FFT using a Hanning window of the same size. We have seen that this window length offers the best tradeoff between bias and variance, when a data segment of around 200 000 points is used for PSD calculation after outlier removal and mean detrending. It may be noted that, as ω f 0, the SDF (shown in Figure 6 along with 95% confidence intervals) approaches 0.96 (not unity) possibly because of (i) differences in closed-loop process dynamics and (ii) a small offset arising from the steady-state temperature gradient across the inert coating. As ω f ∞, the SDF shows some signs of leveling out with a value of γf(ω) ) 0.65, close to the true value τ/τ′ ) 0.63. This technique is general and applicable for a wide variety of sensors in

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Figure 9. Interaction between electrical, thermal, and mechanical systems.

Figure 7. Raw SDF for a flow sensor.

Figure 8. Schematics of a “smart transmitter” based on active sensor modeling.

practice, provided the assumption of stationary process noise holds. Because process noise characteristics can vary between successive time windows, additional robustness can be ensured by looking for sustained model parameter deviations over a period of time. Figure 7 shows the raw SDF computed for high-speed measurement (10 s data window with a sampling rate of 20 kHz from a flowmeter (Omega FLR 1000 Series) after a span error of 10% was introduced. It can be seen that the SDF is fairly close to the true value of 0.9 over a wide range of frequencies, well within the confidence intervals of 95% as shown in the figure. The FFT computation involved a 32-point periodogram with a Hanning window of the same size and the mean detrended from each FFT window. 4. Active Sensor Modeling 4.1. Principles of Active Sensor Modeling. It is desirable to develop deterministic (rather than stochastic) methods that model the sensor in situ, thus having high reliability. Such techniques if available can lead to very few false or missed alarms. To obtain the sensor dynamics directly from the sensor, we look at techniques that perturb the sensor directly from the sensor output. Such proactive probing to observe the converse transduction is possible for several active transducers26 such as piezoelectric, thermoelectric, electromagnetic, magnetostriction, and inductive transducers. Figure 8 shows the schematic of a transmitter based on this ASP concept. In principle, we believe that ASP is applicable to linear (and possibly higher order) interaction processes. A simplified schematic used to commonly represent such processes27 is shown in Figure

9. Such interactions form the basis of several active transducers such as piezoelectric pressure sensors. The linear interaction coefficients are related through Onsager’s reciprocal relationships.28 The converse effect (e.g., Peltier effect in thermoelectricity) is often weaker than the primary transduction (Seebeck effect) and confounded by irreversible effects such as the Joule effect. In many cases, this may involve some amount of sensor redesign. For the special case of temperature transducers (whether active or passive), this problem can be overcome by resorting to resistive heating similar to the LCSR method. The perturbing signal may be chosen as a simple pulse or as a pseudorandom binary signal for excitation at several frequencies. Linear sensor models can be represented by transfer functions or as a frequency response. Storing the sensor model in the frequency domain keeps it nonparametric and enables easy filtering and estimation in subsequent stages. 4.2. Model Identification with LCSR. The LCSR technique used by Yang and Clarke1 among others consists of resistively heating temperature transducers with an electric current for a short period of time and observing the ensuing decay characteristics. This excitation can be applied periodically to monitor the sensor’s health. It must be noted that, because resistive heating results only in a temperature rise, the excitation electric signal must be such that the transducer can decay to the process temperature eventually or at least equilibrate at a small offset in excess of the process temperature. This consideration will determine the time period between successive excitations. A simple rule of thumb that can be applied is to wait for at least 4τ for transducers with first-order dynamics. 4.3. LCSR Applied to RTD. We extend the LCSR method to detect calibration faults. We used a Pt-100 RTD with a calibration error (30% of span) corresponding to K′ ) 0.70. It can be seen in Figure 10 that the response time remained the same, and we could successfully estimate the calibration error (estimated as K′ ) 0.61). It must be noted that the successful determination of the calibration fault is made by comparison of the initial overshoot and not from the final steadystate sensor output. 5. Robust in Situ Frequency Domain Identification Because LCSR is based on time-domain identification and involves excitation with a step input that perturbs mostly at low frequencies, there is a possibility that the sensor dynamics can be significantly confounded with process dynamics. The decay time scales (∼4τ) are not isolated from the process variations of interest (frequency bands of 0.1 Hz and lower). The LCSR test should thus be ideally performed when the process is

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Figure 10. Calibration fault detection in RTD using LCSR.

Figure 11. LCSR in the presence of process noise and dynamics.

operating stably at a steady state. The LCSR tests described in previous sections were conducted at steadystate conditions. Figure 11 shows the detrimental effect of this confounding when the LCSR was applied to thermocouples in the presence of significant process dynamics. Identification of the sensor time constants gave incorrect values of 17.1 and 16.5 s for the normal and coated thermocouples, respectively. To develop a more robust identification method that is well immune to closed-loop process dynamics, we propose a frequency-domain approach. Extending the temperature sensor model (eq 1) to include the effect of active perturbation, we have the following nonlinear ordinary differential equation (eq 6):

[

]

i2(t) R(T) dT T - Tp + )K dt τ mCp

parameter

value

50% response time in air (speed ) 1 m/s) length width height specific heat of platinum (Cp) density of platinum temperature coefficient of resistance (R)

4s 2.3 ( 0.2 mm 2.0 ( 0.2 mm 1.3 ( 0.2 mm 130 J/kg‚K 21450 kg/m3 0.00385 Ω/Ω‚°C

approximations. We use a linear form R(T) ) R0[1 + RT (°C)] where, the ice-point resistance is R0 ) 100 Ω for the standard Pt-100 RTD that we used (see Table 2). Also, the resistance of the connecting leads can be ignored compared to R(T). Independent of the nature of the identification tests (frequency response/step tests) being performed, the excitation current i(t) must be designed such that the temperature rise ∆T must not be confounded by the process and nonprocess noise (can be computed from the noise variance). The time period ∆t in which this temperature rise is attained must be small compared to the sampling rate/control system feedback rate. Also, it must be possible to determine ∆T within the resolution of the measurement system. Equation 6 suggests that a root-mean-square current of 1 mA through the Pt-100 RTD causes a steady-state temperature rise (also termed as the self-heating temperature rise) of 0.05 °C. Accordingly, the Omega CCT-20 signal conditioner drives a maximum base current of 1 mA (dc) through the RTD as a part of a Wheatstone bridge circuit. 5.1. Design of the Excitation Signal. Instead of LCSR, where a constant current flows through the temperature sensor for a short period of time, we vary the current to excite the sensor at frequencies of interest. Typically, we want our excitation frequency to be at least 10 times higher than the corner frequency of the process. Because the excitation current i(t) appears in a nonlinear fashion in eq 6, care must be taken in designing this signal to ensure that the transducer is perturbed at frequencies of interest. Two points are usually chosen: one on the low-frequency asymptote, to determine the gain; the other on the downward slope of the frequency response to determine the time constant. It must be noted that the RTD responses to a process temperature change [T(s)/Tp(s) ) K/(τs + 1)] and excitation power change [T(s)/P(s) ) Kτ/(τs + 1)] differ by a factor. Figure 12 shows the Bode plots for the healthy RTD (τ ) 5.772 s) and the closedloop air temperature control process20 under a PI controller, approximated as first-order with a corner frequency of 0.0316 Hz.9 For the laboratory-scale setup that we used, identification on the low-frequency as-

(6)

Here, R(T) is the temperature-dependent resistance, K denotes the effective gain arising from any span calibration error, and i(t) is the excitation current. The excitation power is, therefore, P(t) ≡ i2(t) R(T). Our goal is to identify K and τ using a suitably chosen excitation current i(t). For the purpose of estimating the necessary current across the transducer to ensure a satisfactory S/N ratio for identification, we made some simple

Figure 12. Bode plots of the air temperature control process and RTD sensor.

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Figure 13. Sine-wave excitation of RTD. Figure 14. Dynamic compensator for coating fault.

ymptote of the sensor is difficult under process dynamics because of the low process time constant and only steady-state identification is possible. For industrialscale processes with much lower process corner frequencies, this is easily achieved. Using an excitation voltage signal V(t) ) [0.1 + 0.1 sin(ωt)]1/2 V, such that we excite the RTD at 1 Hz frequency (ω ) 6.28 rad/s), we obtain the RTD temperature response as shown in Figure 13. We assume that a 100 Ω resistance is in series with the RTD as part of a voltage divider circuit. We see a temperature rise of 0.1 °C at steady state. The amplitude of the sinusoidal variations in the RTD temperature is well below the resolution (0.025 °C, for a span of 0-100 °C) of the 12bit DAQ board. Fortunately, this problem can be overcome in the Sigma-Delta converters (see the appendix), which are the basis of most commercial successive approximation A/D converters today, by adequately oversampling to achieve higher resolution.29 Because calibration faults are frequency-independent, sweeping a sine wave of varying frequency can be used to distinguish between calibration and transience faults as suggested previously. Frequency-domain identification using this simulated data (with uniform white noise added) resulted in an amplitude ratio of 0.06, close to the theoretical amplitude ratio of 0.044. Further parameter estimation can be done, or the sensor dynamics can be simply retained in the form of its frequency response, which will be conducive for easy processing by digital signal processing hardware during future estimation. Even though it seems conceivable that the small excitation amplitudes involved here can be confounded with process variations, it must be realized that accurate filtering should be possible because the exact frequency of interest is known a priori. Besides, because the process noise at these high excitation frequencies is expected to show stationary characteristics, it may be possible to evolve a hybrid active-passive technique by retaining that noise. Further investigation on these issues, including refining and validating this technique experimentally, is necessary and will form the basis of a future paper. 6. Dynamic Estimation For cases where an accurate invertible dynamic sensor model G′T(s) can be obtained (K′ * 0), it is possible to reconstruct the true measurement from the erroneous reading using inverse filtering. In cases of

outright sensor failure (K′ ) 0), reconstruction is possible only by using other measurements from a process model. After filtering out the nonprocess noise (using the denoising procedures previously described), the problem of dynamic estimation can be described as30 one of estimating the true sensor reading, y(s) ) GT(s) yp(s) from the faulty measurement y′(s) ) G′T(s)yp(s). In the simplest case, this can be accomplished using an estimator {G ˆ T(s) ) [G′T(s)]-1GT(s)}. The estimator was applied to the step-test response of the coated thermocouple (Figure 14). It can be seen that the dynamic estimator performs well in improving its response time and results in no steady-state offset when compared with the response of the normal thermocouple. It may be noted that the responses of the two thermocouples are from separately conducted step tests. The estimator in the form of a lead-lag term also suppresses some noise and smoothens out the outliers and the spikes. The confidence limits on the estimated measurement can be computed from the uncertainty in the sensor output and in the identified sensor model {e.g., for the passive FDI case, the model uncertainty is ∆G′T(ω) ) f[∆γ(ω)]}, provided the historic sensor model has been identified accurately. 7. Conclusions In this work, we propose an in situ sensor modeling approach to detection, identification, and compensation of sensor faults. Experimental studies under laboratory conditions demonstrate the validity and benefits of this approach. Both passive and active methods have been developed and compared in this regard. Passive approaches are universally applicable but under conditions of stationary noise. The active techniques do not have such assumptions and are more accurate but are limited to a limited class of sensors that can be perturbed in situ. The proposed methods look promising because they are simple, deal with measurement validation at the sensor level, and can be easily implemented in embedded hardware. It must be realized, however, that sensor modeling is dependent on the operating conditions. Sensor-level measurement validation has the benefit of not having to deal with faults at other levels, such as process faults, and can be available as off-the-shelf solutions applicable to any process. Plantwide instrumentation standards such as Fieldbus and Hartbus can support intelligent SEVA sensors.

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Figure 15. Schematic of sigma-delta converter. Table 3. Sample Calculations on the Sigma-Delta A/D Converter clock period 0 1 2 3 4

Vin

V1

V2

0.75 0 0 0.75 0.75 0.75 0.75 -0.25 0.5 0.75 -0.25 0.25 0.75 -0.25 0

V3 0 1 1 1 0

clock period 5 6 7 8

Vin

V1

V2

0.75 0.75 0.75 0.75 -0.25 0.5 0.75 -0.25 0.25 0.75 -0.25 0

V3 1 1 1 0

Wherever an accurate and invertible dynamic sensor model cannot be obtained, estimation using analytical redundancy in online process models using other measurements will be necessary. Alternatively, optimal or suboptimal inferential estimators using other process measurements30 following a design procedure analogous to that of the dynamic estimator can be used. For nonGaussian distributions of noise, higher order spectral analysis31 can be used in place of the power spectrum to extract additional information. While our interest lies primarily in in situ sensor validation, monitoring the sensors can also provide insights into the health of the process. Sensor fouling can be indicative of similar conditions in the rest of the process vessel. Acknowledgment Financial support from the Boeing-McDonnell Douglas Foundation Graduate Fellowship is gratefully acknowledged. D.S. acknowledges useful discussions with Mr. Karthikeyan Mariappan and Mr. Aniket Basu. We also thank the anonymous reviewers for their comments which helped to improve the quality of the manuscript. Appendix: Sigma-Delta A/D Converter Sigma-Delta A/D converter have formed the basis of most commercial A/D converters in the last 2 decades. The Sigma-Delta converter is a 1-bit A/D converter that uses oversampling to achieve higher resolution. A simple first-order converter is shown in Figure 15. For simplicity, we assume that the input voltage Vin ∈ (-1, 1). The initial nodal voltages V1, V2, and V3 are set to zero. The relations at each clock cycle are as follows: (1) At the differential amplifier, V1[k] ) Vin - V3[k 1]. (2) At the integrator, V2[k] ) V2[k - 1] + V1[k]. (3) At the comparator, V3[k] ) sgn(V2[k]) {i.e., V3[k] ) V2[k]/abs(V2[k])}. Table 3 shows a sample calculation when a steady signal of 0.75 V is encoded by the Sigma-Delta A/D converter. The nodal states (V1, V2, and V3) at some

point attain their starting values (e.g., clock periods 1 and 5 are identical) and thus become periodic. The average value of V3 in a period is 0.75. It can be seen that, for n additional bits of resolution from this primarily 1-bit A/D converter by the above simple averaging method, we need to oversample by a factor of 2n. Therefore, from a 12-bit DAQ board (where the 12-bit resolution is achieved internally by oversampling in the hardware), we can achieve 16-bit resolution by increasing the desired sampling rate by a factor of 16 and using simple boxcar averaging. More refined filtering techniques reduce the oversampling factor required. Literature Cited (1) Yang, J. C.-Y.; Clarke, D. W. A Self-Validating Thermocouple. IEEE Trans. Control Syst. Technol. 1997, 5, 239. (2) Himmelblau, D. M.; Bhalodia, M. On-line Sensor Validation of Single Sensors using Artificial Neural Networks. Proceedings of the 1995 American Control Conference, American Automatic Control Council: Seattle, WA, Jun 21-23, 1995; Vol. 1, Part 1 (of 6), p 766. (3) Upadhyaya, B. R.; Kerlin, T. W. Estimation of Response Time Characteristics of Platinum Resistance Thermometers by the Noise Analysis Technique. ISA Trans. 1978, 17 (4), 21. (4) Yung, S. K.; Clarke, D. W. Local Sensor Validation. Meas. Control 1989, 22 (5), 132. (5) Luo, R.; Misra, M.; Qin, S. J.; Barton, R.; Himmelblau, D. Sensor Fault Detection via Multiscale Analysis and Nonparametric Statistical Inference. Ind. Eng. Chem. Res. 1998, 37, 1024. (6) Luo, R.; Misra, M.; Himmelblau, D. Sensor Fault Detection via Multiscale Analysis and Dynamic PCA. Ind. Eng. Chem. Res. 1999, 38, 1489. (7) Bakshi, B. R.; Locher, G.; Stephanopoulos, G.; Stephanopoulos, G. Analysis of Operating Data for Evaluation, Diagnosis and Control of Batch Operations. J. Process Control 1994, 4, 179. (8) Bakshi, B. R. Multiscale PCA with Application to Multivariate Statistical Process Monitoring. AIChE J. 1998, 44, 1596. (9) Ying, C.-M.; Joseph, B. Sensor Fault Detection Using Noise Analysis. Ind. Eng. Chem. Res. 2000, 39, 396. (10) Henry, M. Automatic Sensor Validation. Control Instrum. 1995, 27 (9), 60. (11) Henry, M. Sensor Validation and Fieldbus. Comput. Control Eng. J. 1995, 6, 263. (12) Wu, S. M.; Hsu, M. C.; Chow, M. C. The Determination of Time Constants of Reactor Pressure and Temperature Sensors: The Dynamic Data System Method. Nucl. Sci. Eng. 1979, 72, 84. (13) Lockwood, F. C.; Moneib, H. A. A New On-line Pulsing Technique for Response Measurements of Thermocouple Wires. Combust. Sci. Technol. 1981, 26, 177. (14) Ballantyne, A.; Moss, J. B. Fine Wire Thermocouple Measurements of Fluctuating Temperature. Combust. Sci. Technol. 1977, 17, 63. (15) Carroll, R. M.; Shepard, R. L. Measurement of the transient response of thermocouples and resistance thermometers using an in situ method; Available NTIS Report ORNL/TM-4573; Oak Ridge National Laboratory: Oak Ridge, TN, 1977; pp 45. From: Energy Res. Abstr. 1977, 2 (22), Abstr. No. 54343.

Ind. Eng. Chem. Res., Vol. 42, No. 11, 2003 2333 (16) Robinson, J. C.; Allen, J. W.; Cartwright, W. New Techniques for RTD Time Response Measurement. Nucl. Plant J. 1994, 12, 70. (17) Hashemian, H. M.; Jones, C. N.; Shell, C. S.; Harkleroad, J. D. New Technology for Testing the Attachment of Sensors to Solid Materials. Proc. Int. Instrum. Symp. 1995, May, 581. (18) Jackowska-Strumillo, L.; Sankowski, D.; McGhee, J.; Henderson, I. A. Modelling and MBS Experimentation for Temperature Sensors. Measurement 1997, 20, 49. (19) Henry, M. P.; Clarke, D. W. The Self-Validating Sensor: Rationale, Definitions and Examples. Control Eng. Pract. 1993, 1, 585. (20) Srinivasagupta, D. Studies in Process Modeling, Design, Monitoring, and Control, with Applications to Polymer Composites Manufacturing. D.Sc. Dissertation, Washington University, St. Louis, MO, 2002. (21) Wise, B. M.; Gallagher, N. B. PLS Toolbox 2.0 for use with MATLAB; Eigenvector Research: Manson, WA, 1998. (22) Oppenheim, A. V.; Schafer, R. W.; Buck, J. R. DiscreteTime Signal Processing, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1999. (23) Welch, P. D. The use of Fast Fourier Transform for the Estimation of Power Spectra: A Method based on Time Averaging over Short, Modified Periodograms. IEEE Trans. Audio Electroacoust. 1967, AU-15 (2), 70.

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Received for review December 5, 2002 Revised manuscript received March 18, 2003 Accepted April 7, 2003 IE0209834