A.3.1 Formulation of Unknown Stochastic Faults
Assuming a system subject to stochastic parametric input faults can be described by a set of nonlinear ordinary differential equations (ODEs) as following:
ẋ = f (t, x, u; g) (A.1)
0 ≤ t ≤ tf , x(0) = x0
where the vector x ϵ Rn represents the system states (measured quantities) with initial conditions x
0 ϵ Rn over time
domain [0, tf], and u denotes the known (measurable) inputs of the system. The vector g ϵ Rng is the unknown
(unmeasured) stochastic time varying input faults of interest, which has to be detected by a FDD algorithm. The function f is assumed to be a fundamental model of the process that can be developed from first principles. The input faults g considered in this current work consist of stochastic perturbation around a specific set of mean values as described in Fig.A.1 (a).
Figure A.1 Fault profile representing an intermittent stochastic input fault and resulting measured variable
It can be mathematically described as:
gi = ḡi + ∆gi (i = 1, …, ng) (A.2)
where {ḡi} are a set of constant mean values (operating modes), {∆gi} are stochastic variations around each mean
value. The statistical distribution of ∆gi is assumed to be a priori and time invariant, which can be estimated from
an offline model calibration algorithm. It is also assumed that the mean values {ḡi} of faults remain constant. The Parametric faults time profile
A m p li tu d e o f s to c h a s ti c f a u lt s ML-PRS Mean Mean # 2 Mean # 1 Mean # 3
Measured quantity time profile
A m p li tu d e o f m e a s u re d q u a n ti
ty Output for ML-PRS inputMean
(a) (b) A
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constancy of {ḡi} can be experimentally inferred from the constancy of the measured quantities, such as the
manipulated and/or controlled variables, through the steady state tests.As seen in Fig.A.1, the changes in the mean values of {ḡi} follow a Multilevel Pseudo Random Signal (ML-PRS) (Ljung, 1999). The inputs described by Eq.
A.2 are typical in chemical processes that experience both changes in means of operating variables but also in additional continuous random perturbations in time t. Then, the FDD problem can be defined as detecting a change in the unknown input mean ḡi and diagnosing around which particular ḡi the system is being operated. Each
particular mean ḡi will be referred heretofore as to an operating mode, thus the goal in the current work is to
diagnose the operating mode ḡi at a given time instant t.
A.3.2 Generalized Polynomial Chaos Expansion
The generalized polynomial chaos (gPC) expansion approximates a random variable as a polynomial series of another random variable following a standard distribution (Xiu D. , 2010). For the nonlinear chemical process defined by Eq. A.1, the gPC expansion can be used to quantify and propagate the effect of stochastic parametric inputs faults g onto the measured quantities x. The first step is to re-write each of the unknown input gi (i = 1,2,…,
ng) in g as a function of a set of random variables ξ = {ξi}:
gi = gi(ξi) (A.3)
where ξi is the ith random variable. The random variables (ξ = {ξi}) are further assumed to be independent and
identically distributed for simplicity. Using the gPC expansion, the unknown stochastic faults g(ξ) and system states x(t, ξ) can be approximated in terms of orthogonal polynomial basis functions Φk(ξ):
g(ξ)
=∑
gk𝛷k(ξ) ∞ k=0 (A.4) x(t, ξ)= ∑ xk(t)𝛷k(ξ) ∞ k=0 (A.5)where xk and gk are the gPC coefficients of measured quantities and faults at each time instant t, Φk(ξ) are multi-
dimensional orthogonal basis functions of ξ. If the faults (g) can be measured or estimated, the coefficients, i.e., {gk} in Eq. A.4, can be computed such that Eq. A.3 follows a priori probability density function. Then, the gPCs
coefficients, representing the responses of measured quantities (x) resulting from the stochastic faults (g), can be calculated using the first principle models of process in combination with a Galerkin projection (Ghanem & Spanos, 1991).
Using Galerkin projection, it is possible to calculate the gPC coefficients of the measured quantities {xk(t)} by
substituting Eq. A.4 and Eq. A.5 into Eq. A.1, and then projecting Eq. A.1 onto each one of the polynomial chaos basis functions {Φk(ξ)} as defined in Eq. A.6:
For practical implementation, Eq. A.4 is often truncated to a finite number of terms such as p, which is defined as the polynomial order. Hence, the total number of terms of measured quantities P in Eq. A.5 can be calculated as following:
P = ((ng + p)!/(ng!p!)) - 1 (A.7)
where p is the necessary terms used to approximate an a priori known distribution of g, and ng is the number of
faults of interest defined in Eq. A.2. From Eq. A.7, the number of the gPC expansion terms for the measured variables in Eq. A.5 increases as the polynomial order p and/or the number of unknown inputs ng increase. The
inner product in Eq. A.6 between two vectors can be computed with:
〈ψ(ξ),ψ'(ξ) 〉
=
∫ ψ(ξ)ψ'(ξ)W(ξ)dξ (A.8) where the integration is conducted over the entire event domain generated by the random variables ξ, and W(ξ)is the weighting function, which is the probability function of random variables and has to be chosen with respect to the polynomial basis function used to represent ξ so as the result of Eq. A.8 is one or zero (Xiu D. , 2010). To obtain orthogonality the basis functions {Φk(ξ)} have to be selected according to the choice of the distribution of ξ. For example, Hermite polynomials are chosen as basis functions for normally distributed ξ. Once the gPCcoefficients of the measured quantities x in Eq. A.5 are available, it is possible to compute statistical moments for the measured variables at a given time instant t with Eq. A.9 and Eq. A.10 as following (Xiu D. , 2010):
E(x(t)) = Ε [∑ xi(t)𝛷i P i=0 ] = x0(t)Ε[𝛷0] + ∑ Ε[𝛷k] P i=1 = x0(t) (A.9) Var(x(t)) = Ε [(x(t) - Ε(x(t)))2] = Ε [(∑ xi(t)𝛷i P i=0 - x(i= 0)(t)) 2 ] = Ε [(∑ xi(t)𝛷i P i=1 ) 2 ] = ∑ xi(t)2Ε(𝛷i2) P i=1 (A.10)
In addition, the probability density functions (PDFs) for measured variables, x(t), can be approximated by sampling from the distribution of ξ and substituting the samples into Eq. A.5. The ability of analytical formulae for calculating statistical moments as per Eq. A.9 and Eq. A.10 and to rapidly calculate the PDF profiles of the measured variables are the main rationale for using the gPC. It can reduce the computational effort required to approximate the PDF profiles, which are further used for the detection of faults and for the evaluation of fault detectability.
The first principle models based fault detection procedure used in this current work consists of the inverse of the procedures explained in this section, i.e., the distribution of the stochastic parametric faults (inputs) g is to be inferred from measurements of the process measured variables x.
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A.3.3 Gaussian Process
The Gaussian process (GP) extends multivariate Gaussian distributions to infinite dimensionality [12]. It can be used to generate a surrogate metamodel with the measurements x in Eq. A.1 to provide a prediction of how the process is behaving without knowing the true generative system, i.e., the value of g in this work. Assuming Ɗ = {(xi, gi)} (i = 1,…, N) is N pairs of observations, then the GP regression model can be formulated as follows:
gi = Ϭ(xi) + εi (A.11)
εi ~ N(0, σg2) (A.12)
where Ϭ denotes the GP metamodel and εi is a bias term. In other words, gi is related to xi nonlinearly through an
unknown function Ϭ that can be approximated with a GP. Moreover, each observation inside X = {xi} is related
to another with the covariance function k(xi, xj). A popular choice of the covariance function k is the squared
exponential kernel function (Shi & Choi, 2011) that can be defined as:
kij = k(xi, xj) = σG2exp(-2l12(xi-xj)2) (A.13)
where (σG, l) are unknown parameters and heretofore referred as hyper-parameters. For the given observations,
the covariance function k among all possible combinations of these N points can be computed with Eq. A.13. Let
K be the covariance matrix at all points of the N training observations, i.e., K = {kij} and 1 ≤ i, j ≤ N. It can be proved that the marginal distribution of G = {gi} follows a multivariate normal distribution (Rasmussen &
Williams, 2006; Shi & Choi, 2011):
{gi} ~ N(0, Kg) with Kg = K + σg2I (A.14)
where Kg is the N×N covariance matrix and each element (i, j)th inside Kg can be defined as:
{Kg}ij = cov(gi, gj) = k(xi, xj) + σg2δij (A.15)
where δij is the Kronecker delta function. Training of the GP involves the determination of the values for the
unknown parameters in Eq. A.13 and Eq. A.15, i.e., θ = {σG, l, σg}, based on the given observations Ɗ. This can
be solved with Empirical Bayes estimation algorithm by maximizing log p(g|x, θ), which can be given as (Shi & Choi, 2011):
arg max log p(g|x, θ) = -12Nlog(2π) -12log|Kg| -1
2g
T(K
g)-1g (A.16)
Based on the training results, the GP model can estimate the prediction g* for a new set of observations x*, which has the mean and variance as below:
E(g*|Ɗ, θ
opt) = k*TKg -1G (A.17)
var(g*|Ɗ, θ
opt) = k(x*, x*) - k*TKg -1k* (A.18) where k* = (k(x*, x
1), …, k(x*, xN))T is the vector of covariance between the new measured quantities x* and the