Vi a

where and 6 = Hi — Ho is change magnitude which is known in this case. The jump time q is not known. Consequently, q in likelihood ratio (Eq. 8.62) and log-likelihood ratio (Eq. 8.64) should be replaced by its maximum likelihood estimate qk under hypothesis Ti\:

The resulting change detector with a threshold r is gk = A k{qk) = maxS*(Ho,5) (8.67) i where

Hence, the detector detects a jump of magnitude <5 in the mean at the first time where gk = Sk1(Ho,5) ~ min S\(ho,5)>t (8.69)

1<i <k which is called the Page-Hinkley Stopping Rule or the cumulative sum algorithm that may be computed recursively [44].

If the jump magnitude is unknown (ho is known but not /ii), one approach is to select a minimum jump magnitude in the mean that is desired to be detected and run two tests (increase and decrease in the mean). A second approach is to replace the unknown magnitude 5 by its maximum likelihood estimate (MLE) and then run the likelihood ratio test. In this case, gSfk = max maxS^(¿¿o,<5) (8.70)

i=q which reduces the double maximization in Eq. (8.70) to a single maximization [44]. The likelihood ratio test becomes

These maximum likelihood tests for composite hypothesis testing problems (such as finding MLE of S and the jump time q) are called generalized likelihood ratio (GLR) tests [45]. Likelihood ratio tests for more complex changes or models can be found in [45].

Maximum Likelihood and GLR Tests for Fault Diagnosis. When faulty operation is detected, the fault needs to be diagnosed. Diagnosis can be implemented by several approaches. Estimation of maximum likelihoods (ML) and the GLR test are popular techniques for model-based fault diagnosis.

Rewrite the joint density function in Eq. (8.51) in a generic form f(z(t),(*z(t)) = K exp

where K denotes the constant term preceding the exponential part (it remains the same for all hypotheses tested), z stands for the type of observation used (r,r,or r), and /¿z(£) the corresponding mean. The simplified log-likelihood function L[-] is defined as log L[(z(t),nz(t))} = -l[z(t) - Ma(t)]TE-1[z(t) - ^(t)] (8.74)

where log K is omitted because it will cancel out when the likelihood ratio is defined.

The ML test consists of the following procedure:

1. Compute the maximum likelihood estimates of the residual mean from observations under various hypotheses Jif

Azj{t) = arg max log L[(z(t),£„(«)) | Tij] j = 1, ■ • ■ , / (8.75)

where Hj are the hypotheses about various possible faults that impose constraints on the estimates of the mean and / is the number of faults. The hypotheses are a function of the properties of the residuals generators such as directional residuals or structured residuals discussed below.

2. Compute the conditional likelihood functions using the observations and conditional estimates log Lj(t) = log L[(z(t), £„•(*))] j = 1, ■ • • , / (8.76)

The most likely fault is the fault that yields the highest log-likelihood value. Extensions of ML approach with additional checks to account for the uncertainty in the decision because of signal noise are discussed in [189].

The basic idea is to check the parity (consistency) of the mathematical equations describing the process by using process measurement information. For illustration, consider a simple system with redundant information described by [162]

where y is the measurement vector of length q, C the q x n measurement matrix of rank n, x the unknown true measurements, and <5y the error vector. If Syl > r, where t1 is the error threshold for variable i. there is a faulty operation indicated by the it,h measured variable. Define a parity vector of dimension q — n such that p = Wy (8.78)

The projection matrix W is of dimension (q - n) x q is determined such that the parity vector is only a function of 5y. To achieve this, W is determined such that

These conditions assure that the rows of W are orthogonal and W is the null space of C. Consequently, p = Wiy (8.80)

Hence, parity equations are independent of x and contain only the errors 5y caused by faults. Furthermore, the columns of W define q distinct fault directions, each associated with only one of the measurements. If there is significant increase in the ith direction of p, it indicates faulty measurement Vi-

The residual vector r = y — Cx is related to p as r = WTp where x = (CTC)~1CTy is the least squares estimate of x. The FDD problem can be stated as a two-step procedure: (1) Find x and compute r; (2) Detect and diagnose the faulty measurements by parity checks. This concept is extended during the last three decades to handle more complex cases involving faults, disturbances, and noise. A short discussion of the formulation of residual generator and parity equations is given below.

Residual Generators. A residual generator is a linear discrete dynamic algorithm acting on observable variables [189]

where r(t) is the vector of residuals, and V(g) and W(q) are TFMs. Noting that r(t) must be zero when all inputs u(t) and y(t) are zero, and substituting y(t) = G(q)u(t) into Eq. (8.81) yields (V(g) + W(g)G(g))u(i) = 0. Hence, Eq. (8.81), the computational form of the residual generator can be written as

The term in brackets in Eq. (8.82) can be substituted using Eq. (8.48) to yield the internal form of the residual generator

Ideally, residuals r(t) should only be affected by faults. If specific unique residuals patterns for each fault could be generated, fault detection and isolation would reduce to checking the violation of limits of residuals and recognizing the patterns. However, disturbances, noise and modeling errors (nuisance inputs) contribute to residuals as well and interfere with FDD. The residual generator should be designed such that the effects of these nuisance inputs on the residuals are as small as possible, leading to robust residual generators. The differences in the properties of these three nuisance inputs determine the approach used in marginalizing them. Additive disturbances and modeling errors have similar temporal behavior to additive faults. Explicit decoupling of residuals from disturbances and modeling errors is necessary to improve the detection and diagnosis capability of the residuals.

Noises usually have much higher frequencies than fault signals and zero mean values. Therefore, filtering the residuals signals with low-pass filters reduces the effects of noise without affecting the fault signals significantly. In addition, testing the residuals against some threshold value as opposed to testing them for nonzero values reduces false alarms caused by noise. There is a tradeoff between the number of false alarms and the number of missed alarms which is affected by the level of thresholds selected (Type I and Type II errors).

The residual generator should be designed to improve fault isolation. The residual set should have different patterns for particular faults. Residual sets designed with the isolation objective are called enhanced residuals. There are two enhancement approaches, structured and directional. In structured residuals, each residual responds to a different set of faults and is insensitive to others. Threshold tests are applied to each element of the residual vector and the test results are converted to a fault code vector s(t) of binary digits. Defining a residual threshold vector (t), Si(t) = 1 if |ri(i)| > Ti\ Si(t) = 0 otherwise. The pattern of the fault code vector (a r(t) = W(q)[y}t)-G(q)u(t)}

r(t) = W(q)[SF(q)f(t) + SD(q)d(t) + SN(qMt) + Np(t)<fiP + NM(t)</>M]

binary string) is matched against the library of fault signatures for diagnosis. Directional residuals generate fault-specific vector directions ¡3 and the scalar transfer function 7(q) in that direction indicates the dynamics of the fault [189]

where 0j is the direction of the jth fault. Fault diagnosis is based on associating r(i|f) with the closest fault direction in the fault library.

The implementation of the residual generator may be done either in input-output form (Eq. (8.46)) or in the equivalent state-space form. (Note the conventional use of G in state-space representation which is different than its as a TFM G(q) and the difference between the conventional use of F in state-space representation and Fjr, F/>, and F ,v ■ )

xfc+1 = Fxfc + Gufc + E^ffe + EDdk + ENi/k yk = Cxfc + Dufc + FFffe + FDdfe + F Nvk (8.85)

The residual responses are specified such that detection and diagnosis are enhanced. For additive faults and disturbances (noise and multiplicative faults are neglected) define the specifications as r (t) = ZF(q){(t) + ZD(q)d(t) (8.86)

Comparing the internal residual expression Eq. (8.83) (ignoring the noise term) and the specification in Eq. (8.86), one can deduce that

The residual generator is obtained by solving Eq. (8.87) for W(q). Detailed examples in [189] illustrate the technique and its extensions with multiplicative faults and disturbances. Other extensions include integration of parity relation design and residual evaluation with GLR test and whitening filters for FDD of dynamic stochastic processes [464]. An implementation of this approach to continuous pasteurization systems and comparison of parity space approach with a statistical approach that combines T2 and SPE tests with contribution plots illustrates the strengths and limitations of both techniques [292].

The basic concept is to generate residuals for FDD by comparing measurements and their estimates computed using Kalman filters or observers. The estimation errors of observers or innovations of Kalman filters are used as residuals. Consider an observer for the deterministic process without faults and disturbances, the Kalman filter being used for the stochastic case that includes noise

The observer with a gain matrix K0b has a structure similar to Kalman filters discussed in Section 4.3.2, viz.,

The relations for the state estimation error e = x — x and the output estimation error e = y — y for the system with faults and disturbances become ek+1 = (F - Ko6C)efe + Guk + EFffc + EDdk - KobFFfk - KobFDdk when Eq. (8.88) is augmented with faults and disturbances by adding Epffc+E^dfc to the state equation and Fpffc+F^d/; to the output equation. Note that the term Gufc drops out because of the subtraction in deriving e, and the estimation errors are independent of deterministic inputs Ufc. The output estimation error e can be used as the residual r for FDD. In the absence of faults (f = 0), r is influenced only by unknown inputs d and noise that is not included in the process model. Faults can be detected by setting up threshold values for r (greater than zero to avoid false alarms due to noise and small disturbances) and developing some FDD logic.

Various observer and Kalman filter configurations have been considered to detect and diagnose multiple faults. One configuration is based on multiple hypotheses testing where a bank of estimators are designed such that each estimator is designed for a different fault hypothesis. For example, H.q would be no faults, Ti\, bias in sensor 1, 7i2 zero output in sensor 1, etc. The hypotheses are tested in terms of likelihood functions. The dedicated observer configuration has multiple estimators where each estimator is driven by a different single sensor output to estimate as many components of the output vector y as possible. When a certain sensor fails, the output estimate given by the corresponding estimator will be erroneous. FDD of multiple simultaneous faults is carried out by checking values of structured sets of estimation errors [106]. The generalized observer approach uses a bank of estimators where each estimator is dedicated to a certain sensor. Each estimator receives process information from all other sensors except

the sensor whose reading is being estimated. The residuals are checked using threshold logic to diagnose a faulty sensor. Reduced-order or nonlinear estimators can also be used to develop FDD systems with Kalman filters and diagnostic observers. The equivalence between parity relation based and diagnostic observer based FDD has been shown [162, 188].

Deterministic observers and filters were used in the previous section to estimate state variables and outputs. The effect of disturbances and noise were accounted for by using nonzero threshold limits for residuals. Robust observers can be designed by including disturbances [163] and both disturbances and noise [427]. To illustrate the methodology and design challenges, robust residuals generation using unknown deterministic input (disturbance) observers [163] are discussed. Consider the process model xfc+i = Fx*, + Gufc + EFffc 4- EDdfc yfc = Cxfc + FFffc + FDdk (8.91)

Define a linear transformation zfc - Txfc (8.92) for the fault free system and the robust unknown input observer zfc+1 = Rzfc + Syk + Jufc (8.93)

with the residual

such that if fk — 0 then lim/^oo r*, = 0 for all u and d, and for all initial conditions xo and zq. If ffc / 0, then rk 0. The estimation error equation for the observer is efc+i = zfc+1 - Txfc+i (8.95)

= Rzfe + Syfc + Jufc - TFxfe - TGufc - TEFffc - TEDdfe

Substituting for xk and yk, and imposing that the error should be independent of state variables, control inputs and disturbances, the following equations are established:

tf -rt |
= |
sc |

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