=Paper= {{Paper |id=Vol-1507/dx15paper33 |storemode=property |title=Faults Isolation and Identification of Heat-Exchanger/ Reactor with Parameter Uncertainties |pdfUrl=https://ceur-ws.org/Vol-1507/dx15paper33.pdf |volume=Vol-1507 |dblpUrl=https://dblp.org/rec/conf/safeprocess/ZhangDCL15 }} ==Faults Isolation and Identification of Heat-Exchanger/ Reactor with Parameter Uncertainties== https://ceur-ws.org/Vol-1507/dx15paper33.pdf

Proceedings of the 26th International Workshop on Principles of Diagnosis

Faults isolation and identification of Heat-exchanger/ Reactor
with parameter uncertainties

Mei ZHANG1,4,5 , Boutaïeb DAHHOU2,3 Michel CABASSUD 4,5 Ze-tao LI1
1
Guizhou University
gzgylzt@163.com
2
CNRS LAAS, Toulouse, France
boutaib.dahhou@laas.fr
3
Université de Toulouse, UPS, LAAS, Toulouse, France
4
Université de Toulouse, UPS, Laboratoire de Génie Chimique
michel.cabassud@ensiacet.fr
5
CNRS, Laboratoire de Génie Chimique

Abstract Supervision studies in chemical reactors have been reported
in the literature concerning process monitoring, fouling de-
This paper deals with sensor and process fault de- tection, fault detection and isolation. Existing approaches
tection, isolation (FDI) and identification of an in-
can be roughly divided into data based method as in [3],
tensified heat-exchanger/reactor. Extended high
neural networks as in [4] and model based method as in
gain observers are adopted for identifying sensor [5,6,7,8,9]. Among the model based approach, observer
faults and guaranteeing accurate dynamics since
based methods are said to be the most capable
they can simultaneously estimate both states and
[10,11,12,13,14] if analytical models are available.
uncertain parameters. Uncertain parameters in- Most of previous approaches focus on a particular class of
volve overall heat transfer coefficient in this paper.
failures. This paper deals with integrated fault diagnosis for
Meanwhile, in the proposed algorithm, an ex-
both sensor and process failures. Using temperature meas-
tended high gain observer is fed by only one meas- urements, together with state observers, an integrated diag-
urement. In this way, observers are allowed to act
nosis scheme is proposed to detect, isolate and identify
as soft sensors to yield healthy virtual measures for
faults. As for sensor faults, a FDI framework is proposed
faulty physical sensors. Then, healthy measure- based on the extended observer developed in [15]. Extended
ments, together with a bank of parameter interval
high gain observers are adopted in this paper due to its ca-
filters are processed, aimed at isolating process
pability of simultaneous estimation of both states and pa-
faults and identifying faulty values. Effectiveness rameters, resulting in more accurate system dynamics. The
of the proposed approach is demonstrated on an in-
estimates information provided by the observers and the
tensified heat-exchanger/ reactor developed by the
sensors measurements are processed so as to recognize the
Laboratoire de Génie Chimique, Toulouse, France. faulty physical sensors, thus achieving sensor FDI. Moreo-
ver, the extended high gain observers will work as soft sen-
1 Introduction sors to output healthy virtual measurements once there are
Nowadays, safety is a priority in the design and develop- sensor faults occurred. Then, the healthy measures are uti-
ment of chemical processes. Large research efforts contrib- lized to feed a bank of parameter intervals filters developed
uted to the improvement of new safety tools and methodol- in [11] to generate a bank of residuals. These residuals are
ogy. Process intensification can be considered as an inher- processed for isolating and identifying process faults which
ently safer design such as intensified heat exchangers involves jumps in overall heat transfer coefficient in this
(HEX) reactors in [1], the prospects are a drastic reduction work.
of unit size and solvent consumption while safety is in- It should be pointed out that the contribution of this work
creased due to their remarkable heat transfer capabilities. does not lie with the soft sensor design or the parameter in-
However, risk assessment presented in [2] shows that po- terval filter design as either part has individually already
tential risk of thermal runaway exists in such intensified been addressed in the existing literature. However, the au-
process. Further, several kinds of failures may compromise thors are not aware of any studies where both tasks are com-
safety and productivity: actuator failures (e.g., pump fail- bined for integrated FDI, besides, there is no report whereby
ures, valves failures), process failures (e.g., abrupt varia- parameter estimation capacity of the extended high gain ob-
tions of some process parameters) and sensor failures. server is used to adapt the coefficient, rather than parameter
Therefore, supervision like FDI is required prior to the im- FDI, thus together with sensor FDI framework forms the
plementation of an intensified process. contribution of this work.
For complex systems (e.g. heat-exchanger/reactors), fault
detection and isolation are more complicated for the reason 2 System modelling
that some sensors cannot be placed in a desirable place, and
for some variables (concentrations), no sensor exists. In ad- 2.1 Process description
dition, complete state and parameters measurements (i.e. The key feature of the studied intensified continuous heat-
overall heat transfer coefficient) are usually not available. exchanger/reactor is an integrated plate heat-exchanger

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Proceedings of the 26th International Workshop on Principles of Diagnosis

technology which allows for the thermal integration of sev- channel volume and cannot be a failure leads to fatal acci-
eral functions in a single device. Indeed, by combining a re- dent normally, but it may influence the dynamic of the pro-
actor and a heat exchanger in only one unit, the heat gener- cess and it is rather difficult to calculate the changes online.
ated (or absorbed) by the reaction is removed (or supplied) In this paper, we treat the parameter uncertainty as an un-
much more rapidly than in a classical batch reactor. As a measured state, and employ an observer as soft sensor to
consequence, heat exchanger/reactors may offer better estimate it, unlike other literature, the estimation here is not
safety (by a better thermal control of the reaction), better for fouling detection but for more accurate model dynamics,
selectivity (by a more controlled operating temperature). and to ensure the value of the variable is within acceptable
parameter, (e.g., upper and lower bounds of the process var-
2.2 Dynamic model iable value).
Supervision like FDI study can be much more efficient if a To rewrite the whole model in the form of state equations,
dynamic model of the system under consideration is availa- due to the assumption that every element behaves like a per-
ble to evaluate the consequences of variables deviations and fectly stirred tank, we suppose that one cell can keep the
the efficiency of the proposed FDI scheme. main feature of the qualitative behavior of the reactor. For
Generally speaking, intensified continuous heat-exchanger/ the sake of simplicity, only one cell has been considered.
reactor is treated as similar to a continuous reactor [16,17], Let us delete the subscript k for a given cell.
then flow modelling is therefore based on the same hypoth- Define the state vector as x1 T = [x11 , x12 ]T = [Tp , Tu ]T , un-
dhp
esis as the one used for the modelling of real continuous re- measured state x2 T = [x21 , x22 ]T = [hu , hp ]T , =
dt
actors, represented by a series of N perfectly stirred tank re- dhu
actors (cells). According to [18] , the number of cells N = ε(t) , ε(t) is an unknown but bounded function refers
dt
should be greater than the number of heat transfer units, and to variation of h, the control input u = Tui , the output vector
the heat transfer units is related with heat capacity flowrate. T
of measurable variables y T = [y1 , y2 ]T = [Tp , Tu ] , then
The modelling of a cell is based on the expression of bal- the equation (1) and (2) can be rewritten in the following
ances (mass and energy) which describes the evolution of state-space form:
the characteristic values: temperature, mass, composition,
ẋ 1 = F1 (x1 )x2 + g1 (x1 , u)
pressure, etc. Given the specific geometry of the heat-ex-
changer/reactor, two main parts are distinguished. The first { ẋ 2 = ε(t) (3)
part is associated with the reaction and the second part en- y = x1
compasses heat transfer aspect. Without reaction, the basic
A
mass balance expression for a cell is written as: (Tp − Tu ) 0
ρp Cp Vp
{Rate of mass flow in – Rate of mass flow out = Rate of Where, F1 (x1 ) = ( p
),
A
change of mass within system} 0 (Tu − Tp )
ρu Cp Vu
The state and evolutions of the homogeneous medium cir- u
(Tpi −Tp )Fp
culating inside cell 𝑘 are described by the following bal-
Vp
ance: and g1 (x) = ( ) , Tpi , Tui is the output of previ-
(Tui −Tu )Fu
−1
2.2.1 Heat balance of the process fluid (J. s ) Vu
k ous cell, for the first cell, it is the inlet temperature of pro-
k dTp
ρkp Vpk Cp = hkp Ak (Tpk − Tuk ) + ρkp Fpk Cp k (Tpk−1 − Tpk ) (1) cess fluid and utility fluid.
p dt p
where ρkp is density of the process fluid in cell k (in In this case, the full state of the studied system is given as:
kg. m−3 ), Vpk is volume of the process fluid in cell k (in m3 ),
Cp k specific heat of the process fluid in cell k (in ẋ = F(x1 )x + G(x1 , u) + ε̅(t)
p { (4)
J. kg −1 . K −1 ) , hkp is the overall heat transfer coefficient (in y = Cx
J. m−2 . K −1 . s −1 ).
x1 0 F1 (x1 )
2.2.2 Heat balance of the utility fluid (J. s −1 ) Where x = [x ] , F(x1 ) = ( ) , G(x1 , u) =
2 0 0
dTku g (x, u) 0
ρku Vuk Cp k = hku Ak (Tuk − Thk ) + ρku Fuk Cp k (Tuk−1 − Tuk ) (2) ( 1 ) , C = (I 0), ε̅(t) = ( )
u dt u 0 ε(t)
whereρku is density of the utility fluid in cell k (in kg. m−3 ), 3 Fault detection and diagnose scheme
Vuk is volume of the utility fluid in cell k (in m3 ), Cp k specific
u
heat of the utility fluid in cell k (in J. kg −1 . K −1 ) , hku is 3.1 Observer design for sensor FDI
overall heat transfer coefficient (in J. m−2 . K −1 . s −1 ). The extended high gain observer proposed by [15] can be
The eq. (1) (2) represent the dynamic reactor comportment. used like an adaptive observer for estimation both states and
The two equations represent the evolution of two states (Tp : parameters simultaneously, in this paper, the latter capabil-
reactor temperature and Tu : utility fluid temperature).The ity is utilized to estimate incipient degradation of overall
heat transfer coefficient (h) is considered as a variable heat transfer coefficient (due to fouling), thus guaranteeing
which undergoes either an abrupt jumps (by an expected a more accurate approximation of the temperature. It is quite
fault in the process) or a gradual variation (essentially due useful in chemical processes since parameters are usually
to degradation). The degradation can be attributed to foul- with uncertainties and unable to be measured.
ing. Fouling in intensified process is tiny due to the micro

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Proceedings of the 26th International Workshop on Principles of Diagnosis

Consider a nonlinear system as the form: With this formulation, the faulty model becomes:
ẋ = F(x1 )x + G(x1 , u) ẋ = F(x1 )x + G(x1 , u) + ε̅(t)
{ (5) { (11)
y = Cx y = Cx + Fs fs
where x = (x1 , x2 )T ∈ ℛ 2n , x1 ∈ ℛ n is the state, x2 ∈ ℛ n 𝐹𝑠 is the fault distribution matrix and we consider that fault
is the unmeasured state, x2 = ϵ(t), u ∈ ℛ m , y ∈ ℛ p are in- vector 𝑓𝑠 ∈ ℛ 𝑝 (𝑓𝑠𝑗 is the 𝑗𝑡ℎ element of the vector) is also a
put and output, ϵ(t) is an unknown bounded function bounded signal. Notice that, a faulty sensor may lead to in-
which may depend on u(t), y(t), noise, etc., and correct estimation of parameter. That is why we emphasized
healthy measurement for parameter fault isolation as men-
0 F1 (x1 ) g (x, u)
F(x1 ) = ( ) , G(x1 , u) = ( 1 ), C(I 0), tioned above.
0 0 0
F1 (x1 ) is a nonlinear vector function, g1 (x, u) is a matrix 3.2.2 Fault detection and isolation scheme
function whose elements are nonlinear functions. The proposed sensor FDI framework is based on a bank of
Supposed that assumptions related boundedness of the observers, the number of observers is equal to the number
states, signals, functions etc. in [15] are satisfied, the ex- of sensors. Each observer use only one sensor output to es-
tended high gain observer for the system can be given by: timate all the states and parameters. First, assumed the sen-
sor used by ith observer is healthy, let yi denotes the ith
x̂̇ = F(x̂1 )x + G(x̂1 , u) − Λ−1 (x̂1 )Sθ−1 C T (ŷ − y) system output used by the ith observer. Then we form the
{ (6)
ŷ = Cx̂ observer as:
𝐼 0 x̂̇ i = F(x̂1i )x + G(x̂1i , u) + Hi (yi − ŷii )
Where: Λ(𝑥̂1 ) = [ ] 1 ≤ i ≤ p{ i
0 𝐹1 (𝑥̂1 ) (12)
ŷ = Cx̂ i
𝑆𝜃 is the unique symmetric positive definite matrix satisfy-
ing the following algebraic Lyapunov equation: Define eix = x̂ i − x, eiy = Ceix , eiyj = ŷji − yj , rji (t) = ‖ŷji −
yj ‖, μi = ‖rji (t)‖ ≔ sup‖rji (t)‖, for t ≥ 0.
θSθ + AT Sθ + Sθ A − C T C = 0 (7)
Where i denotes the ith observer, ŷii , ŷji denotes the ith, jth
0 I
Where A = [ ] , θ > 0 is a parameter define by [15] estimated system output generated by the ith observer, Hi is
0 0
and the solution of eq. (7) is: the gain of ith observer determined by the following equa-
1 1
tion :
I − 2I 2θi I
Sθ = [ θ θ
] (8) Hi = Λ−1 (x̂1 )Sθ−1 C T = Λ(x̂1 ) [ 2 −1 ]
− 2I
1 2
I
i θi F1 (x̂1 )
θ θ3 Then we get:
Then, the gain of estimator can be given by: Theorem 1:
If the lth sensor is faulty, then for system of form (4), the
2θI
H = Λ−1 (x̂1 )Sθ−1 C T = Λ(x̂1 ) [ 2 −1 ] (9) observer (12) has the following properties:
θ F1 (x̂1 )
For i ≠ l , ŷ i = y asymptotically
Notice that larger 𝜃 ensures small estimation error. How- For i = l, ŷ i ≠ y
ever, very large values of 𝜃 are to be avoided in practice due Proof: If the lth sensor is faulty, then:
to noise sensitiveness. Thus, the choice of 𝜃 is a compro- For i ≠ l, means that fsi = 0, yi = θsi , we have:
mise between fast convergence and sensitivity to noise.
lim eix = lim (x̂ i − x) = 0 (13)
t→∞ t→∞
3.2 Sensor fault detection and isolation scheme
Then the vector of the estimated output ŷ i generated by ith
The above observer could guarantee the heat-exchanger/re- observer guarantee ŷ i = y after a finite time.
actor dynamics ideally. Then, a bank of the proposed ob- For i = l, means that θsl = ylf = yl + fsl , fsl ≠ 0 , the ob-
servers, together with sensor measurements, are used to server is designed on the assumption that there is no fault
generate robust residuals for recognizing faulty sensor. occurs, because there is fault fsl exit, so the estimation error
Thus, we propose a FDI scheme to detect, meanwhile, iso-
elx = 0 asymptotically cannot be satisfied, then :
late and recovery the sensor fault.
lim (x̂ i − x) = lim (x̂ l − x) ≠ 0 (14)
t→∞ t→∞
3.2.1 Sensor faulty model we have:
ė lx = F(x̂1i , u) elx − Hi G(x̂1i , u, fsl ) elx (15)
A sensor fault can be modeled as an unknown additive term
i
in the output equation. Supposed θsj is the actual measured Then the vector of the estimated output ŷ generated by the
output from jth sensor, if jth sensor is healthy, θsj= yj , while ith observer is different from y, that is ŷ i ≠ y.⊡
if jth sensor is faulty, θsj = yjf = yj + fsj , (𝑓𝑠𝑗 is the fault), As mentioned above, the observers are deigned under the
for t ≥ t f and lim |yj − θsj | ≠ 0.That means yjf is the actual assumption that no fault occurs, furthermore, each observer
t→∞
output of the jth sensor when it is faulty, while yj is the ex- just subject to one sensor output. Residual rii is the differ-
pected output when it is healthy, that is:
ence between the ith output estimation ŷii determined by
yi ; jth sensor when it is faulty the ith observer and the ith system output yi , then Theorem
θsi = { f (10)
yi = yi + fsi ; jth sensor when it is faulty 2 formulates the fault detection and isolation scheme.

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Proceedings of the 26th International Workshop on Principles of Diagnosis

Theorem 2: all the intervals whether or not one of them contains the
If the lth sensor is faulty, then: faulty parameter value of the system, the faulty parameter
For i ≠ l, we have: value is found, the fault is therefore isolated and estimated.
fsi = 0, yi = θsi (16) The practical domain of each parameter is partitioned into a
thus ŷii converges to yi asymptotically, we get: certain number of intervals. For example, parameter hp is
rii = ‖ŷii − yi ‖ ≤ μi (17) partitioned into q intervals, their bounds are denoted
(0) (1) (i) (q)
For i = l, we have: by hp , hp , … , hp , … , hp . The bounds of ith interval are
fsl ≠ 0, θsl = ylf = yl + fsl ≠ yl , then ŷll could not track yl (i−1) (i)
hp and hp , are also noted as hbi ai
p and hp , and the nomi-
correctly: nal value for hp denotes by hp0 .
rll = ‖ŷll − yl ‖ ≥ μl (18) To verify if an interval contains the faulty parameter value
Therefore, in practice, we can check all the residuals rii , for of the post-fault system, a parameter filter is built for this
1 ≤ i ≤ p, if rii ≥ μi denotes that ith sensor is faulty, then interval. A parameter filter consists of two isolation observ-
the sensor fault detection and isolation is achieved. ers which correspond to two interval bounds, and each iso-
The residuals are designed to be sensitive to a fault that lation observer serves two neighboring intervals. An inter-
comes from a specific sensor and as insensitive as possible val which contains a parameter nominal value is unable to
to all the others sensor faults. This residual will permit us to contain the faulty parameter value, so a parameter filter will
treat not only with single faults but also with multiple and not be built for it.
simultaneous faults. Define Eq. (3) into a simple form as:
Let rsi denotes the fault signature of the ith sensor, define: ẋ = F1 (x1 )x2 + g1 (x1 , u) ẋ = f(x1 , hp , u)
{ 1 = { 1 (24)
1 if rii ≥ μi ; ith sensor is faulty y = x1 y = x1
rsi (t) = { (19)
0 if rii ≤ μi ; ith sensor is health The parameter filter for ith interval of hp is given below.
3.2.3 Fault identification and handling mechanism The isolation observers are:
1) Fault identification x̂̇ ai = f(x̂1 , hai ̂ ai )
p0 , u) + H(y − y

Supposed there are m healthy sensors and p − m faulty {ŷ̇ ai = cx̂̇ ai (25)
ones, then to identify the faulty size of ith sensor, use m ai
ε = y − cx̂ ̇ ai
estimated output ŷim generated by m observers which use
healthy measures, 1 ≤ m ≤ p − 1, m ≠ i , define f̂si as the x̂̇ bi = f(x̂1 , hbi ̂ bi )
p0 , u) + H(y − y
estimated faulty value of the ith sensor, then: {ŷ̇ bi = cx̂̇ bi (26)
∆
̂fsi = 1 ∑m |ŷ m − θsi | → fsi (20)
m i=1 i ε = y − hx̂̇
bi bi

2) Fault recovery Where:
As mentioned above, the extended high gain observer is hp0 , t < t f hp0 , t < t f
also worked as a software sensor to provide an adequate hai
p0 (t) = { (i) , hbi
p0 (t) = { (i−1) ,(27)
hp , t ≥ t f hp , t ≥ t f
estimation of the process output, thus replacing the meas-
urement given by faulty physical sensor. The isolation index of this parameter filter is calculated by:
θsi is the actual measured output from ith sensor:
yi νi (t) = sgn(εai )sgn(εbi ) (28)
θsi = { f (21) As soon as νi (t) = 1, the parameter filter sends the ’non-
yi = yi + fsi
Let m observers use healthy measurements as the soft sen- containing’ signal to indicate that this interval does not con-
sor for ith sensor, define: tain the faulty parameter value. And if the fault is in the ith
m interval. Let:
1 1
y̅i = ∑ ŷim (22) ĥA = (hai A + hbi A) (29)
m 2
i=1 to represent the faulty value, fault isolation and identifica-
If ith sensor is healthy, let the sensor actual output as θsi
tion is then achieved.
its output, while if it is faulty, let y̅i to replace θsi , that is:
θ , if ith sensor healthy 4 Numerical simulation
yi = { si (23)
y̅i , if ith sensor faulty A case study is developed to test the effectiveness of the
proposed scheme. The real data is from a laboratory pilot of
3.3 process fault diagnose a continuous intensified heat-exchanger/reactor. The pilot is
In order to achieve process FDD, healthy measurements are made of three process plates sandwiched between five util-
fed to a bank of parameter intervals filters developed in [11] ity plates, shown in Fig.1. More Relative information could
to generate a bank of residuals. These residuals are pro- be found in [2]. As previously said, the simulation model is
cessed for identifying parameter changes, which involves considered just for one cell which may lead to moderate in-
variation of overall heat transfer coefficient in this paper. accuracy of the dynamic behavior of the realistic reactor.
The main idea of the method is as follows. However, this point may not affect the application and
The practical domain of the value of each system parameter demonstration of the proposed FDD algorithm encouraging
is divided into a certain number of intervals. After verifying results are got.

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Proceedings of the 26th International Workshop on Principles of Diagnosis

4.3 Sensor FDI and recovery demonstration
In order to show effectiveness of the proposed method on
sensor FDI, multi faults and simultaneous faults in the tem-
Figure 1 (a) Reactive channel design; (b) utility channel de- perature sensors are considered in case 1 and case 2 respec-
sign; (c) the heat exchanger/reactor after assembly. tively. Besides, the pilot is suffered to parameter uncertain-
The constants and physical data used in the pilot are given ties caused by heat transfer coefficient decreases with ℎ =
in table1. (1 − 0.01𝑡)ℎ. Two extended high gain observers are de-
signed to generate a set of residuals achieving fault detec-
Table 1. Physical data used in the pilot tion and isolation in individual sensors. Observer 1 is fed by
Constant Value units output of sensor 𝑇𝑝 to estimate the whole states and param-
eter while observer 2 uses output of sensor 𝑇𝑢 . Advantages
hA 214.8 W. K −1
of the proposed FDI methodology drop on that if one sensor
A 4e−6 m3 is faulty, we can use the estimated value generated by the
Vp 2.685e−5 m3 healthy one to replace the faulty physical value, thus provid-
Vu 1.141e−4 m3 ing a healthy virtual measure.
ρp , ρu 1000 kg. m−3 Case 1: abrupt faults occur at output of sensor 𝑇𝑝 at t=80s,
cp , cp 4180 J. kg −1 . k −1 100s, with an amplitude of 0.3℃, 0.5℃ respectively.the re-
p u sults are reported in Fig.5-8.
4.1 operation conditions

The inlet fluid flow rate in utility fluid and process fluid are
𝐹𝑢 = 4.17𝑒 −6 𝑚3 , 𝐹𝑝 = 4.22𝑒 −5 𝑚3 𝑠 −1 .The inlet tempera-
ture in utility fluid is time-varying between 15.6℃ and
12.6℃, which is a classical disturbance in the studied sys-
tem, as shown in Fig.2. The inlet temperature in process
fluid is 76℃. Initial condition for all observers and models
are supposed to be T̂𝑝0 = T̂𝑢0 = 30℃, hA = 214.8 W. K −1 .

Fig. 5 output temperature of both fluid in case 1 by observer
1, red curve demonstrates the estimated value while black
one is the measured value.

It is obviously that since t=80s, 𝑇̂𝑢 (red curve) cannot track
𝑇𝑢 (black curve) correctly, while it needs about 0.2s for 𝑇̂𝑝
Fig.2 utility inlet temperature 𝑇𝑢𝑖 to track 𝑇𝑝 at t=80s and t=100s. It suggests that faults occur,
then the following task is to identify size and location of
4.2 High gain observer performance faulty sensors. Fig.6 and Fig.7 achieves the goal. It takes
0.1s and 0.3s for isolating the faults at 80s, 100s respec-
To prove the convergence of the observers and show their tively.
tracking capabilities, suppose the heat transfer coefficient
subjects to a decreasing of ℎ = (1 − 0.01𝑡)ℎ and followes
by a sudden jumps of 15 at 𝑡 = 100𝑠. These variations and
observer estimation results are reported in Fig.3.

Fig.3. simulation and estimation of heat transfer coefficient
variation.
Fig.6 isolation residual in case 1.
Black curve simulates the actual changes of the parameter
while the red one illustrates the estimation generated by the
proposed observer, it can be seen from Fig. 3 that the esti-
mation value tracks behavior of the real value with a good
accuracy, thus ensuring a good dynamics.

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Proceedings of the 26th International Workshop on Principles of Diagnosis

Fig.7b fault signature in case 1, obviously, faults only occur
at output of sensor 𝑇𝑝 . Fig. 9 isolation residual in case 2

For fault recovery, we can employ observer 2 as soft sensor
to generate a health value for faulty sensor 𝑇𝑝 . Observer 2
uses only measured 𝑇𝑢 to estimate all states and parameters.
Therefore, 𝑇̂𝑢 , 𝑇̂𝑝 generated by observer 2 are only decided
by 𝑇𝑢 . In case 1, faults occur only on sensor 𝑇𝑝 , sensor 𝑇𝑢 is
healthy, that is to say 𝑇̂𝑢 , 𝑇̂𝑝 generated by observer 2 will be
satisfied their expected values. As shown in Fig.8, we can
see that since 𝑇𝑢 is healthy, estimated value 𝑇̂𝑢 tracks meas-
ured 𝑇𝑢 perfectly, while estimated value 𝑇̂𝑝 (red curve) does
not track the faulty measured value 𝑇𝑝 (black curve), 𝑇̂𝑝 (red
curve) illustrates the expected value for sensor 𝑇𝑝 , we can
Fig 10. Fault signature in case 2
use estimate 𝑇̂𝑝 (red curve) to replace measured faulty value
𝑇𝑝 ( black curve) for fault recovery. 4.4 Fast process fault isolation and identification

Process fault is related to variation of overall heat transfer
coefficient (h). The heat transfer coefficient is considered as
variable which undergoes either an abrupt jumps (by an ex-
pected fault in the flow rate) or a gradual variation (essen-
tially due to fouling). For incipient variation, since fouling
in intensified heat-exchanger/reactor is tiny and only influ-
ence dynamics, we have employed extended high observers
to ensure the dynamic influenced by this slowly variation.
Therefore, the abrupt changes in heat transfer coefficient ℎ
can only be because of sudden changes in mass flow rate. It
Fig.8 fault recovery in case 1, red curve demonstrates the implies that the root cause of process fault is due to actuator
estimated value while black one is the measured value. fault in this system.
If there are faults occurred only on output of sensor 𝑇𝑢 , the Supposed an abrupt jumps in ℎ at t=40 from 214.8 to 167.
same results can be yield easily. For multi and simultaneous
faults on both sensors, we can still isolate the faults cor-
rectly. Case 2 will verify this point.

Case 2: simultaneous faults imposed to the outputs of sen-
sors 𝑇𝑝 as in case 1 and 𝑇𝑢 at t=80s with amplitude of 0.6℃.
Results are reported in Fig.9-10. Residuals are beyond their
threshold obviously at time 80s, 100s.
Fig.11 detection residual in process faulty case
It can be seen from Fig.9, Fig .10 that the proposed FDI
scheme can isolate faults correctly, and it takes 0.25s, 0.4s From Fig.11, at t=40s, unlike sensor fault cases, the residual
for isolating the faults in sensor 𝑇𝑝 at 80s, 100s and 0.2s for leaves zero and never goes back, this indicates that process
isolating that in sensor 𝑇𝑢 at t=80s respectively. Compared fault occurs. For fast fault isolation and identification, we
with Case 1, more times is needed in this Case 2. use the methodology of parameter interval filters developed
in [11]. In [2], heat transfer coefficient ℎ changes between
130.96 and 214.8, then ℎ is divided into 4 intervals as shown
in table 2 and simulation results are shown in Fig.12. It can
be seen at t=40s, only index for interval 150-170 goes to
zero rapidly, then there is a fault in this interval. The faulty
1 1
value is estimated by ℎ̂𝐴 = (ℎ𝑎 𝐴 + ℎ𝑏 𝐴) = (150 +
2 2
170) = 160. We can see it is closely to actual faulty value
167, and if more intervals are divided, the estimated value
may be closer to the actual faulty value.

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