Fault isolation based on transfer-function models using an MPC algorithm

Abstract

This work studies model-based fault isolation. A method of fault isolation filter design is developed. The fault to output transfer function model is obtained using system identification. The basic idea is to formulate the isolation problem as a dual decoupling control problem then use a MPC controller as the decoupler. In this way an efficient algorithm can be developed to obtain the fault isolation filter; time delays and right-half-plane (RHP) zeros can be handled. Model errors that occur during system identification are considered in designing the isolation filter. Optimal fault isolation filters that suppress disturbances are also developed. The method is demonstrated using a simulated 600MW supercritical power generation unit and the Tennessee Eastman process (TEP).

Full text

Fault isolation based on transfer-function models using an MPC algorithm
Jinming Zhoua, Yucai Zhua,∗
aState Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China
Abstract
This work studies model-based fault isolation. A method of fault isolation filter design is developed. The fault to output transfer
function model is obtained using system identification. The basic idea is to formulate the isolation problem as a dual decoupling
control problem then use a MPC controller as the decoupler. In this way an efficient algorithm can be developed to obtain the fault
isolation filter; time delays and right-half-plane (RHP) zeros can be handled. Model errors that occur during system identification
are considered in designing the isolation filter. Optimal fault isolation filters that suppress disturbances are also developed. The
method is demonstrated using a simulated 600MW supercritical power generation unit and the Tennessee Eastman process (TEP).
Keywords: Fault isolation, Model predictive control, System identification, Filter design, Tennessee Eastman process
1. Introduction1
Recent decades witness an increased focus on fault diagno-2
sis in modern industrial processes [1, 2, 3, 4, 5]. Fault refers to3
an unpermitted deviation of at least one characteristic or vari-4
able in a system [6]. In a broader context, faults can also re-5
fer to the slight changes in signals [7], which can be deemed6
as precursors before faults occur. A well-designed fault di-7
agnosis system can not only provide early detection of un-8
expected malfunctions or failures in process, give guidelines9
for equipment overhaul and maintenance, but improves prod-10
uct quality and brings economic benefits. Among the various11
branches of fault diagnosis methods, model-based fault diagno-12
sis is the most promising one and has been widely investigated13
through decades. Generally speaking, model-based methods14
contain two steps [2]: residual generation and residual eval-15
uation. Fig. 1 presents a typical model-based fault diagnosis16
scheme. In residual generation step, the consistency between17
process output and model output of is checked, according to18
which faults will be detected. The residual evaluation step can19
be viewed as a signal processing or filtering procedure, in which20
useful information about the faults will be extracted. This paper21
focuses on fault isolation (FI) problem, which aims at determin-22
ing the locations of the detected faults.23
As discussed in [2], the existing schemes for FI can be di-24
vided into three categories: using unknown input decoupling25
strategy; solving FI by means of a bank of residual generators;26
formulating FI as a dual problem of designing a decoupling27
controller. Solving FI by means of a bank of residual generators28
mainly refers to the dedicated observer scheme (DOS) [8] and29
generalized observer scheme (GOS) [9]. In DOS, each residual30
?This work is supported by the National Natural Science Foundation of
China, Grant Numbers: U1809207, 61673343.
∗Corresponding author
Email addresses: zhoujinming@zju.edu.cn (Jinming Zhou),
zhuyucai@zju.edu.cn (Yucai Zhu)
is only related to one output, which has very clear structure and 31
working principle, but its application is limited to sensor fault 32
isolation; in GOS, each residual is sensitive to all but one faults, 33
and the decision logic is complementary to DOS. 34
Unknown input decoupling technique aims at making the 35
generated signals independent of the disturbance (unknown in- 36
put) of no interest but preserving the sensitivity to the parts 37
to be detected. In fault detection (FD) problem, unknown in- 38
put decoupling technique is widely used to reduce the pro- 39
cess disturbance effect and improve the detection accuracy 40
[10, 11, 12, 13, 14]. FI problem can also be formulated as a 41
number of unknown input decoupling problems where faults of 42
no interest are handled as unknown input [2, 15, 16], then faults 43
can be clustered into some groups and FI is achieved according 44
to some decision logic. 45
Many model-based fault diagnosis design problems have 46
close relationships to control theory. For instance, both in con- 47
trol and fault diagnosis applications, model mismatch or model 48
uncertainty is a challenging problem. From around 1990, in- 49
spired by the development of H∞and robust control theory 50
which provide systematic analysis tools to handle model un- 51
certainties, many FD design methods based on H∞and linear 52
matrix inequality (LMI) have been developed [17, 18, 19, 20]. 53
FI is a problem that is closely related to decoupling control. 54
Decoupling is an important concept in multi-input multi-output 55
(MIMO) control systems. After the decoupling procedure each 56
output are only affected by one input thus the MIMO system is 57
reduced to several single-input single-output (SISO) systems. 58
Similarly, in FI, after processing of the original residuals, each 59
residual should only be affected by one or a group of faults. 60
Based on the similarities, [21] proposed an FI method utiliz- 61
ing the duality between FI and the state feedback decoupling; 62
in [2], this method was extended to handle more general cases. 63
However, several problems may occur when developing decou- 64
pling algorithms. In [22], it is pointed out that the realizability, 65
stability and robustness of the decouplers are three frequently 66
Preprint submitted to Computers &Chemical Engineering April 4, 2022

process
Process input Process output
process
model residual
processing Decision
logic
-residual
Model-based fault diagnosis system
information
of faults
residual generation
fault detection residual evaluation
fault isolation, identification, analysis, ...
process
Process input Process output
process
model residual
processing Decision
logic
-residual
Model-based fault diagnosis system
information
of faults
residual generation
fault detection residual evaluation
fault isolation, identification, analysis, ...
Figure 1: A diagram of model-based fault diagnosis system.
encountered problems in decoupling control. The reason lies in67
that for ideal decoupling [23], the inversion of the plant transfer68
function matrix is required, which can become improper, or un-69
stable when time delays and RHP zeros occur. In fact, these are70
common problems when plant model inversion is involved, see71
also [24]. In order to handle the problems of time delays and72
RHP zeros more complex algorithms are required [25, 26, 22],73
the result of which may become unreliable due to numerical74
issues when system dimension increases.75
Industrial processes are essentially dynamic and there are ca-76
sual relationships between various variables. These two fea-77
tures can be captured by state-space models or by transfer-78
function models. Model-based method is superior to data-79
driven and signal-based methods because models can reveal80
deeper information about processes than signals. Bottleneck81
of the model-based methods is the difficulty in model building,82
especially in process industries. In [27], we have studied the83
identification based residual generation and FD problem. Fol-84
lowing the same line, this work aims at developing an efficient85
FI algorithm based on the identified model. Model predictive86
control (MPC) is nowadays the most successful MIMO control87
strategy in process control [28, 29], which offers a powerful tool88
to handle large-scale interacting loops with constraints. Unlike89
traditional decoupling control that uses parametric controllers,90
MPC algorithms can handle time delays and RHP zeros inher-91
ently without special treatment. Based on the fact that FI can92
be solved by a dual problem of decoupling control, this work93
tries to solve FI problem by combining system identification94
and MPC technique.95
Contributions and outline96
The core of FI is to decouple the interactions between faults97
and residuals. In control applications, it turns out that MPC can98
inherently handle the interactions between manipulated vari-99
ables and controlled variables of large-scale plants, implying100
that the MPC controller also serves as a decoupler. Inspired by101
this nice feature of the MPC algorithm, this paper reveals that102
with some transformation and modifications, the unconstrained103
MPC controller can be used to address the couplings between104
faults, then to solve FI problem.105
The paper is divided into two main parts. In the first part,106
the relation between the unconstrained MPC controller and the107
isolation filter is analyzed. Then an algorithm is developed to108
obtain a parametric MPC controller and simultaneously an iso-109
lation filter. Considering that the algorithm is based on a model110
with errors, a method to validate the isolation filter is devel- 111
oped that considers errors in system identification. The pro- 112
posed method can handle large-scale systems containing time 113
delays and RHP zeros and is numerically reliable. The sec- 114
ond part deals with the effect of the nuisance factors containing 115
model error and unmeasured disturbance which degrade FI per- 116
formance. Based on a performance index and the related post- 117
filters, the FI performance can be enhanced. Further, an identi- 118
fication based fault detection and isolation (FDI) framework is 119
established combining the fault detection scheme in [27]. 120
The rest of the paper is organized as follows: Section 2 con- 121
tains preliminaries about the underlying system and identifica- 122
tion method, and formulates the FI problem to be addressed; 123
Section 3 presents details of the FI approach; Section 4 deals 124
with the FI performance and proposes the FDI framework; in 125
Section 5 the proposed method is validated in an example of a 126
600MW power generation unit; Section 6 presents the applica- 127
tion of the method in TEP. Summary and conclusion are given 128
in Section 7. 129
Notations 130
Vectors and matrices are in bold-face. Rndenotes the real n131
dimensional space, Rn×mdenotes the set of n×mreal matrices. 132
ATdenotes transpose of A,Ai j denotes ith raw jth column el- 133
ement of A. Kronecker product is denoted as ⊗. The identity 134
matrix of size n×nis denoted In.0n×pis the n×pmatrix full of 135
zero. diag{· · · } represents a diagonal or block-diagonal matrix. 136
For a vector x,xidenotes its ith element. kxkdenotes Euclidean 137
norm of x, and kxkQis the weighted norm. E[·], var[·], cov[·]138
denote mathematical expectation, variance and covariance op- 139
erators. |z|is the modulus of a complex number z.q−1is the 140
time-shift operator: q−1u(t)=u(t−1). Gaussian distribution and 141
chi-square distribution are denoted as N(·,·) and χ2(·). Φξ(ω)142
denotes the power spectrum of a signal sequence {ξ(t)}, when 143
this notation is used it is implied that the spectrum exists. 144
denotes the logical exclusive nor operator. 145
2. Preliminaries and problem statement 146
2.1. System description 147
This work considers MIMO linear time-invariant (LTI) sys- 148
tems. The true system without faults can be written as: 149
S:y(t)=G(q)u(t)+v(t),v(t)=H(q)e(t) (1) 150
where G(q) is a stable transfer matrix of dimension p×m,H(q) a 151
transfer matrix of dimension p×psatisfying: H(q) and H−1(q)152
are stable, H(q=∞)=Ip.u(t)∈Rmand y(t)∈Rpare 153
input and output signals, v(t)∈Rpdenotes the disturbance. 154
e(t)∈Rpis zero mean white noise with covariance matrix Λ,155
i.e. E{e(t)eT(t)}=Λ.156
Several ways exist to describe the true system S, for instance 157
state-space model [30, 31], input-output model [32, 33]. In this 158
work, we focus on input-output model. A parameterized model 159
structure Mis introduced as: 160
M:y(t)=G(q,θ)u(t)+v(t),v(t)=H(q,θ)e(t) (2) 161
2

where θ∈ Dθ∈Rdis a parameter vector, Dθrestricts θto162
the values that G(q,θ) is stable, H(q,θ) is stable and inversely163
stable.164
Prediction error method (PEM) is a standard method to165
identify θ, in which a quadratic cost function of the pre-166
diction error (PE) is minimized based on test data ZN=167
{u(1),y(1),· · · ,u(N),y(N)}:168
ˆ
θN=arg min
θ
1
N
N
X
t=1
(y(t)−ˆ
y(t,θ))TΛ−1(y(t)−ˆ
y(t,θ)),(3)169
where the predictor ˆ
y(t,θ) is given by the stable filter:170
ˆ
y(t,θ)=H−1(q,θ)G(q,θ)u(t)+[I−H−1(q,θ)]y(t).(4)171
For details, see [33].172
An important property of PEM is that, when the model struc-173
ture Mis flexible enough to capture the true system, i.e. there174
exists a θothat G(q)=G(q,θo), H(q)=H(q,θo) and assume175
that the white noise e(t) is Gaussian, the estimate ˆ
θNasymp-176
totically (N→ ∞) subjects to Gaussian distribution around the177
true parameter θo:178
ˆ
θN∼ N(θo,1
NP(θo)) (5)179
where180
P−1(θo)=E{Ψ(t,θo)Λ−1ΨT(t,θo)},(6)181
Ψ(t,θ) is defined as: d
dθˆ
y(t,θ). We refer to [34] for more explicit182
type of (5). This conclusion allows one to determine uncertainty183
regions for ˆ
θN, or in other words, provide an efficient way to184
quantify estimate error, which has been proved to be useful in185
robust controller design based on identified model [35, 36, 37],186
model quality validation [32] and input design [38, 34]. Sim-187
ilarly, this conclusion will serve as a tool to deal with model188
error in our subsequent study. Notice that in practice the true189
parameter θowill be replaced by the estimate ˆ
θN.190
Remark 1. This work focuses on input-output model of the191
system based on which FI and relevant issues will be investi-192
gated. Besides using PEM to build the model, one can also per-193
form subspace identification [30, 31] or first-principle modeling194
to get a state-space model then transform it to transfer-function195
type. Remember that, while PEM can provide model uncer-196
tainty description mentioned above, how to describe model197
errors in subspace identification method and in first-principle198
modeling is still an open question.199
2.2. Fault detection and fault isolation200
When faults occur, (1) becomes:201
y(t)=G(q)u(t)+v(t)+Gy f (q)f(t),(7)202
f(t)∈Rl,Gy f (q) is a p×ltransfer matrix. According to [2],203
both additive and multiplicative faults can be modelled in this204
form. For additive faults representative of offsets or drifts in205
actuators and sensors, if fkis a sensor (output) fault measuring206
y¯
i,207
Gy f
¯
ik =1,Gy f
ik =0 (i,¯
i); (8)208
if fkis an actuator (input) fault corresponding to uj,209
Gy f
ik (q)=Gi j(q).(9) 210
For multiplicative faults representative of process faults, we 211
have 212
Gy f (q)=δG(q),f(t)=u(t) (10) 213
where δG(q) denotes the change caused by process faults, and 214
typically this term is unknown. Moreover, it is augured that 215
multiplicative faults can affect the system stability [39]. In this 216
work we confine ourselves to isolation of additive faults. For 217
such faults, an estimated Gy f (q) can be obtained using system 218
model (through e.g. identification). Subsequently, the term ad- 219
ditive will be omitted when there is no ambiguity. 220
In practice, the faults to be considered should be chosen re- 221
lated to the process variables that is important for safety and 222
product quality. For additive faults, sensors and actuators in 223
the loops that is key for safe operations and control require- 224
ments should be taken into account. Prior knowledge obtained 225
by analysing data during the fault events and maintenance peri- 226
ods, or through insights of the devices can also be useful. 227
Based on the system model, the consistency between the 228
measured outputs and the model outputs can be checked, which 229
forms the basic idea of FD. In [27], it has been proved that out- 230
put error (OE) is more suitable for FD than PE. Denote the esti- 231
mated model of Sas ˆ
G(q) :=G(q,ˆ
θN), then OE can be written 232
as: 233
r(t)=y(t)−ˆ
G(q)u(t),(11) 234
which will be used as the basic residual in this work. From 235
(7), one can see that faults will cause deviations, or changes 236
in r(t). By combining components of r(t) into some statistics 237
J(r(t)) and choosing proper threshold Jth , faults can be detected 238
according to the following decision logic: 239







J(r(t)) >Jth ⇒alarm
J(r(t)) ≤Jth ⇒no alarm .(12) 240
Denote ∆G(q) :=G(q)−ˆ
G(q), r(t) can further be written as: 241
r(t)= ∆G(q)u(t)+v(t)
| {z }
¯
v(t)
+Gy f (q)f(t) (13) 242
where ¯
v(t) is a general disturbance term containing the nuisance 243
factors that can degrade the detection ability. In order to eval- 244
uate the detection ability, the following performance index is 245
introduced: 246
Definition 1. Let ε(t) be a scalar residual, if we denote εnas 247
the residual at the normal condition, εfas that at the faulty con- 248
dition when f(t) occurs, then the fault detection performance 249
index is defined as 250
J=
Enεf(t)o2
E{εn(t)}2.(14) 251
Let ε(t)=ri(t), and consider one fault fk, the detection per- 252
formance of rito fkcan be achieved, Jik :=E{rfk
i(t)}2
E{rn
i(t)}2. A larger 253
3

Jimplies a better performance of εto detect fin a statistical254
sense.255
To illustrate the FI problem, assume that ∆G=0,v=0, then256
FI problem can be formulated as:257
Problem 1. Find a stable filter L(q) of dimension l×psuch258
that259
L(q)r(t)=L(q)Gy f (q)f(t)=Γ(q)f(t) (15)260
where Γ(q)=diag{γ1(q),· · · , γl(q)}.261
In (15), each element in the new residual only responses to one262
fault, this is called perfect fault isolation (PFI) [2] and we call263
L(q) an isolation filter. From a pragmatic viewpoint, if the non-264
diagonal elements of Γ(q) is nonzero but tends to zero quickly265
with small dynamic fluctuations, it can be deemed that PFI is266
solved approximately with dynamic interactions. In this work,267
we extend the definition of PFI to encompass such approxima-268
tions.269
Concerning the solvability of Problem 1, the following result270
is important:271
Lemma 1. For additive fault f(t)∈Rland system (7), Problem272
1 is solvable if and only if273
rank(Gy f (q)) =l.(16)274
The proof of Lemma 1 can be found in [40, 2]. Before de-275
signing the fault diagnosis system, according to the considered276
faults, lcan be determined. Lemma 1 implies for a poutputs277
MIMO system, one can at most isolate pfaults. Hence if l≤p,278
isolation filter aiming at solving PFI can be developed; if l>p,279
more sensors are required for more available information, one280
can also consider isolating faults into groups as an alternative,281
see [15, 16].282
In this paper, we only focus on the occasions that l≤pwhere283
a PFI can be obtained. Subsequent sections aim at addressing284
several problems: How to develop efficient PFI algorithm using285
MPC controller? How to validate the designed isolation filter286
when model error exists? How to deal with disturbance and287
enhance isolation performance?288
3. Fault isolation using MPC controller289
In this section, it will be demonstrated that the transpose of290
an unconstraint MPC controller can serve as the isolation filter.291
According to the plant model, the fault to output model can292
be built, based on which an efficient algorithm to obtain the293
isolation filter is developed. Finally, fault isolation with model294
errors is considered.295
3.1. MPC controller and isolation filter296
Some brief review on MPC will be given first. Take dynamic297
matrix control (DMC) for instance, at each control interval, the298
following performance index is minimized:299
Jc=kw(t)−ˆ
yP(t)k2
Q+k∆uM(t)k2
R(17)300
where wis the reference trajectory, ˆ
yPdenotes predicted output301
over the prediction horizon P(for explicit expression, see e.g.302
controller plant
model
-+
+
-+
Figure 2: IMC structure.
[41]), and ∆uMdenotes the future control move over control 303
horizon M, which is optimized in (17). Qand Rare weighting 304
matrices. The first input in the optimal sequence is then sent 305
into the plant, and the entire calculation is repeated at subse- 306
quent control intervals [42]. For unconstraint MPC, the solution 307
of (17) is explicit. The control move at time tcan be expressed 308
as: 309
∆u(t)=L(ATQA +R)−1ATQ[w(t)−ˆ
yP(t)] (18) 310
where Acontains internal model parameters of DMC which are 311
formed by the coefficients of model step responses, 312
L=
1 0 0 0
...
0 1 0 0
.(19) 313
(18) can be viewed as a nonparametric model between w(t) and 314
u(t). 315
The unconstraint linear MPC has close connections to some 316
other control strategies. For instance, it can be transformed to 317
a IMC structure [43, 44]; when Pand Mapproach infinity, un- 318
constraint MPC becomes a standard linear quadratic regulator 319
(LQR) problem [45]. The role of MPC controller as a decou- 320
pler can hardly be seen from Jcand ∆u(t). Using the equiv- 321
alence between unconstraint MPC and IMC, and investigating 322
IMC structure, the decoupling function will emerge. The struc- 323
ture of IMC is shown in Fig. 2, the closed-loop transfer function 324
from wto yis: 325
T(q)=G(q)[Im+K(q)∆G(q)]−1K(q).(20) 326
A desirable controller design must satisfy T(1) =Ipto avoid 327
steady-state offset, which implies a decoupling with dynamic 328
interactions [25]. Notice that it differs from static decoupling 329
[22] where the static decoupler is designed simply based on 330
steady-state gains. A complete decoupling can be realized when 331
∆G=0[25]: factorize ˆ
G(q) as 332
ˆ
G(q)=ˆ
G+(q)ˆ
G−(q),ˆ
G+(1) =Ip(21) 333
where ˆ
G+(q) contains time delays and RHP zeros of ˆ
G(q) and 334
ˆ
G−(q) has a stable and realizable inverse. Then use 335
K(q)=ˆ
G−1
−(q)F(q) (22) 336
4

Algorithm 1: Designing isolation filter by identifying the MPC controller
Input: Plant test data ZN={u(1),y(1),· · · ,u(N),y(N)}, desired response Γ(q)
Output: Fault isolation filter L(q)
/* Step 1-3: tune MPC parameters to get the desired response. */
1Use system identification to build plant model based on ZNand get d
Gy f (q);
2Set d
Gy f (q)T
as the plant and internal model in MPC;
3Tune the MPC parameters Q,R,P,Mto get the desired closed-loop response;
/* Step 4-16: use system identification to get a parametric MPC contorller as well as the
isolation filter. */
4(Identification settings): set the length of identification experiment NK, controller order nKand design test signals
η1,· · · , ηl;
5for j←1to ldo
6Generate NK-sample test signal nηj(1),· · · , η j(NK)o, set {W(1),· · · ,W(NK)}according to (26);
7Run MPC simulation for NKsample time;
8for i←1to pdo
9Set the data set ZNK
i j ={W j(1),Ui(1),· · · ,Wj(NK),Ui(NK)};
10 Get ˆ
Ki j(q) using LS method;
11 Set Lji(q)=ˆ
Ki j(q);
12 end
13 end
14 if L(q)d
Gy f (q),Γ(q)then
15 Go back to Step 4 to adjust the identification settings;
16 end
as a controller delivers337
T(q)=ˆ
G+(q)F(q).(23)338
If ˆ
G+(q) and F(q) are chosen to be diagonal, T(q) will also have339
a diagonal structure.340
Now we are ready to demonstrate that the transpose of the341
MPC or IMC controller can be used as the isolation filter. If342
G(q)=ˆ
G(q)=hGy f (q)iT,343
T(q)=hGy f (q)iTK(q).(24)344
Transpose both sides of (24) yields345
KT(q)Gy f (q)=TT(q).(25)346
Factorize hGy f (q)iTaccording to (21), set a controller accord-347
ing to (22), and choose diagonal hGy f (q)iT
+and F(q), TT(q) be-348
comes diagonal, implying that KT(q) solves PFI. Analogously,349
when K(q) is a MPC or IMC controller that leads to a T(q) such350
that T(1) =Ip, PFI can be realized with some dynamic interac-351
tions.352
To obtain such model-based controllers, first the fault trans-353
fer matrix should be estimated based on the identified system354
model. From above discussion, one can see that the transpose355
of the ideal IMC controller can exactly solve PFI, whereas one356
must perform transfer matrix factorization and inversion as in357
(21) and (22), which can be numerically difficult when system358
Test signals
identification
input data output data
MPC
controller
Figure 3: Illustration of Problem 2.
dimension is large [25, 46]. For unconstrained MPC, after the 359
tuneable parameters are determined the control move can be 360
explicitly given as in (18), hence one possible alternative is to 361
design an unconstrained MPC with a very diagonal T(q), and 362
use the transpose of the MPC controller to realize FI. 363
3.2. Isolation filter identification 364
For FI purpose, the expression (18) is of little help and a 365
parametric (filter-type) model of the MPC controller is required. 366
For unconstrained linear MPC, the equivalent controller K(q) is 367
a linear filter which can be obtained by solving the following 368
black-box identification problem: 369
Problem 2. Design identification test using test signal vec- 370
tor W, use it as the reference trajectory of the de- 371
signed MPC, identify the transfer function matrix of the 372
MPC controller K(q) from simulation data set ZNK=373
5

{W(1),U(1),· · · ,W(NK),U(NK)}, where Wis used as in-374
put and Uis used as output.375
Problem 2 is illustrated by Fig. 3. Here Wand Udenote con-376
troller input and output, NKdenotes data length for identifying377
controller K(q). Problem 2 is an identification problem with the378
following features: 1) it is an open-loop identification problem;379
2) it is noise-free; 3) the order of K(q) to be identified is typi-380
cally high, because the ideal controller of IMC, or MPC is the381
(approximate) inverse of the plant model that is non-parametric382
[24, 46, 44]. Bearing the above points in mind, we recommend383
to perform lidentification tests (simulation runs), at each time384
only add a test signal to one component of W. Denote the test385
signal in jth test as ηj, we have:386
W(t)=[01×j−1, η j(t),01×l−j]T,at jth experiment. (26)387
ηjmust satisfy the persistent excitation condition [33], for in-
stance, it could be generalized binary noise (GBN) signal [47].
In each experiment, by using data set
ZNK
i j ={W j(1),Ui(1),· · · ,Wj(NK),Ui(NK)},1≤i≤p
to identify Ki j respectively, one can get the jth column of K(q).388
The dynamic relation between Wj(t) and Ui(t) can be ex-389
pressed as:390
Ui(t)=ϕT
i j(t)ϑi j (27)391
where392
ϕT
i j(t)=[−Ui(t−1),· · · ,−Ui(t−na),Wj(t),· · · ,Wj(t−nb)],
(28)
393
ϑT
i j =[a1
i j,· · · ,ana
i j ,b0
i j,· · · ,bnb
i j ].(29)394
395
ϑi j contains the parameters in Ki j, i.e.396
Ki j(q)=Bi j (q)
Ai j(q)=
b0
i j +b1
i jq−1+· · · +bnb
i j q−nb
1+a1
i jq−1+· · · +ana
i j q−na
.(30)397
In practice it is common to set na=nb=nK. Notice that (27) is398
a linear regression such that ϑi j can be obtained using the least-399
square (LS) method. After doing lidentification experiments400
and identifying l·pmodels, a MIMO model of K(q) whose401
elements are independently parameterized is obtained.402
Test signals may be simultaneously added in a single test and403
a MIMO model is identified using some MIMO identification404
methods. For our purpose MIMO test is not really necessary be-405
cause the “tests” here are simulations which do not have much406
economic cost. Moreover, using MIMO identification approach407
here will need to use very high orders, which can cause numer-408
ical problems.409
The isolation filter L(q) can be obtained by transposing the410
identified MPC controller. For checking whether it has decou-411
pled d
Gy f (q), one can investigate the step or frequency responses412
of L(q)d
Gy f (q). The above identification method are summa-413
rized in Algorithm 1 where the identification of plant model is414
also included.415
Some comments for the algorithm are given below:416
1) Data set ZNfor plant identification: Because Gyf (q)417
shares some same elements with G(q), it is necessary to 418
carry out identification tests using test signals (excitation) 419
and generate informative data, in order to guarantee a 420
high-quality fault to output model. 421
2) Γ(q)and related Q,R: To better approximate PFI and to 422
ensure a desirable detection speed, a high Qto Rratio set- 423
ting is suggested. The diagonal elements of Γ(q) should 424
be tuned to have fast dynamics while the nondiagonal ele- 425
ments should only pose some small fluctuations then tend 426
to zero quickly. 427
3) Test signals ηiand simulation time NK:NKmust be chosen 428
sufficient large while the design of the persistent excitation 429
signal ηican be found in [32]. 430
4) Controller order nK: Before identification, the order of 431
controller can be roughly estimated by investigating the or- 432
der of the inverse of d
Gy f (q). When the actual order is very 433
high, the above identification procedure actually plays the 434
role of model reduction. Note that despite the noise-free 435
feature of this identification problem (no variance error), 436
the bias error induced by model reduction must be care- 437
fully treated. The final determination of nKmay become 438
a trail and error procedure. It is suggested that check the 439
gain matrix, or step responses of the delivered Γto see 440
whether the identified controller model serves for the FI 441
purpose, some illustrations are given in Section 5.2. 442
Though in Section 3.1 we begin the discussion with DMC, 443
in practice other forms of MPC can also be used. Moreover, 444
the method to identify the MPC controller developed in Algo- 445
rithm 1 can also be used in other occasions where one needs to 446
decouple some variables, or requires to invert a system model. 447
The proposed algorithm has following advantages that makes it 448
very applicable: 449
1) it is numerically reliable because all the calculations are 450
explicit; 451
2) it requires no special treatment for time delays and RHP 452
zeros; 453
3) it can handle large-scale systems, which will be demon- 454
strated using the TE process. 455
3.3. Fault isolation with model errors 456
In Algorithm 1, the fault isolation filter L(q) is derived based 457
on estimated d
Gy f (q) without considering model errors. In order 458
to make the method more applicable, model errors are incorpo- 459
rated here. 460
Denote L(q)r(t) as rFI , when ∆G(q),0, ∆Gy f (q) :=Gy f (q)−461
d
Gy f (q),0, the transfer matrix from fto rFI becomes: 462
Γo(q)=L(q)Gy f (q)
=L(q)d
Gy f (q)+ ∆Gy f (q)=Γ(q)+L(q)∆Gy f (q)(31) 463
6

One can see that the diagonal structure of Γ(q) may be damaged464
by the additional term L(q)∆Gy f (q). Denote this term as ∆Γ(q).465
Obviously, when ∆Gy f (q) becomes significant, FI will fail.466
In system identification for model-based control, uncertainty467
regions of the estimated model are often developed for test-468
ing whether the achieved model is qualified for control purpose469
[32]. Due to that ∆Gy f (q) is closely related to plant model (as470
mentioned in Section 2.2), follow an analogous line as in iden-471
tification for control, uncertainty regions of ∆Gy f (q) can be de-472
veloped and used to validate whether L(q) is sufficiently accu-473
rate for the FI. One way is to introduce uncertainty into the step474
responses of Γo(q). By investigating step responses one can475
evaluate the severity of interactions. As Γ(q) is already known476
hence the remaining task is to derive the uncertainty region of477
∆Γ(q)f(t).478
When PEM is used to build the model, parameter uncertainty479
description in (5) can be used for the subsequent study. Follow-480
ing assumption is needed to establish (5):481
Assumption 1. Assume that S∈Msuch that there exists a θo
482
that G(q)=G(q,θo), H(q)=H(q,θo), the white noise e(t) is483
Gaussian; the identification are based on an informative data set484
ZN[33] of sufficient large N, and correct system order is used.485
With no loss of generality, denote the fault vector as fT(t)=486
[( fs(t))T,(fa(t))T], where fs(t)∈Rnsdenotes output faults and487
fa(t)∈Rnadenotes input faults, na+ns=l. Denote Ga(q,θo)488
as the real transfer matrix from fa(t) to y(t). Notice that489
∆Gy f (q)=h0p×ns∆Ga(q)i(32)490
where491
∆Ga(q) :=Ga(q,θo)−Ga(q,ˆ
θN) (33)492
contains elements of ∆G(q). According to (32)493
L(q)∆Gy f (q)f(t)=L(q)∆Ga(q)fa(t).(34)494
Denote ∆θ:=ˆ
θN−θo, then using first-order Taylor expansion495
∆Ga(q)fa(t)=
Pna
i=1∆Ga
1ifa
i
.
.
.
Pna
i=1∆Ga
pi fa
i

=
Pna
i=1
dGa
1i
dθT∆θfa
i
.
.
.
Pna
i=1
dGa
pi
dθT∆θfa
i

=[Ga(q,θo)]0fa(t)⊗Id∆θ
(35)
496
where [Ga(q,θo)]0is a block matrix whose (i,j) block equals497
d
dθTGa
i j(q,θo). Now (34) writes:498
L(q)∆Gy f (q)f(t)=L(q)[Ga(q,θo)]0fa(t)⊗Id
| {z }
Υ(t)
∆θ(36)499
where Υ(t)∈Rl×d. Under Assumption 1, (5) holds such500
that ∆θsubjects to Gaussian distribution, cov(∆θ)=1
NP(θo).501
Hence L(q)∆Gy f (q)f(t) also subjects to Gaussian distribution502
and Gauss’ approximation formula [33] directly gives:503
cov hL(q)∆Gy f (q)f(t)i=1
NΥ(t)P(θo)ΥT(t).(37)504
To summarize: 505
∆Γ(q)f(t)∼ N(0,1
NΥ(t)P(θo)ΥT(t)).(38) 506
Moreover, denote P(t) as a vector containing the diagonal ele- 507
ments of 1
NΥ(t)P(θo)ΥT(t), we have 508
∆Γ(q)f(t)i∼ N(0,Pi(t)).(39) 509
(39) implies that: 510
∆Γ(q)f(t)i≤ U β
2pPi(t),w.p.1−β(40) 511
where −U β
2
,Uβ
2specifies a confidence interval of the stan- 512
dard Gaussian distribution with confidence level 1−β,βis some 513
probability. Using (40), the uncertainty region of each compo- 514
nents of ∆Γ(q)f(t) can be determined. 515
Now by setting f(t) properly we can derive the step responses 516
of Γ(q) with error bounds which will be called step response 517
bands. Under Assumption 1, Γo(q) will fall into this step re- 518
sponse bands. Define a Boolean matrix Vas 519
Vi j =










1step response bands of Γi j intersects with 1
at steady state
0 else
.
(41) 520
Notice that Vii =1 and according to (31), when ∆Gy f is not 521
significant, the step response band will be narrow and Vi j =0522
while for large ∆Gy f , some Vi j may becomes 1. 523
Assume that faults do not occur simultaneously, using Vit 524
can be inferred how Γo(q) becomes when model error exists. 525
Several situations may occur: 526
1) Vis diagonal: It means that in the presence of model un- 527
certainty, the fault still excites most its corresponding com- 528
ponent in rFI . Hence one can simply decide which fault 529
occurs according to that which component of the residual 530
exceeds the threshold; 531
2) Vis nondiagonal but rank(V)=l: Due to model uncer- 532
tainty, the fault excite several components of rFI , to a in- 533
distinguishable extent. However, rank(V)=limplies that 534
different combinations of components will be excited (dif- 535
ferent patterns) and by matching the pattern FI can be real- 536
ized; for example, consider a Gy f with dimension 2 ×2. If 537
the step response bands of Γ21 intersects with 1 at steady 538
state while Γ12 does not, then 539
V="1 0
1 1#(42) 540
is of full rank. In this case f1excites rFI
1and rFI
2whereas f2541
only excites rFI
2, by investigating different patterns caused 542
by f1and f2, fault isolation can be realized; 543
3) rank(V)<l: This implies that many step response bands 544
of Γ(q) are wide and the fault model is poor. In this sit- 545
uation, l−rank(V) faults cannot be isolated. Model re- 546
identification with improved plant tests are recommended 547
here. 548
7

Remark 2. Typically, the model error contains the noise in-549
duced variance error and structure defect induced bias error.550
Notice that (5) only considers variance error. However, when551
the true system satisfies (1), it is arguable that the total error in552
any identified model is dominated by variance error [48].553
In situations 1) and 2) above, the matrix Vcan be used in fault554
isolation as follows. Define a Boolean vector Ξwith dimension555
l. If rFI
ialarms, set Ξi=1 else set Ξi=0. In the ideal case,556
a single column in Vequals Ξ, implying that the patterns are557
perfectly matched; then the fault is at this column. Otherwise,558
the fault number k∗can be determined by559
k∗=arg max
k
l
X
i=1
Vik Ξi,(43)560
where denotes the logical exclusive nor operator. This im-561
plies that the fault number is determined when two patterns are562
most closely matched. It should be reminded that due to incon-563
sistency between real systems and our assumptions, occasions564
may occur that (43) delivers nonunique k∗. We suggest that then565
choose the component that alarms, i.e. exceeds the threshold566
most intensively as the most possible candidate.567
4. Optimal FI filtering568
After addressing model errors in FI, the issue of disturbance569
will be studied. Unmeasured disturbances (referring ¯
v(t) in570
(13)) affect the isolation performance. That is to say, despite571
that in rFI (t) faults can be well structured, it may not cause572
alarms in rFI (t) if the detection performance of rFI (t) is poor,573
which means that the FI will fail. A situation might be encoun-574
tered is that a fault which could originally be detected by the575
basic residual rmay not be detected, causing miss alarms in576
the new residual rFI. In model-based fault diagnosis methods,577
one typical way to improve the detection ability of a certain578
residual is to use post-filters that maximize some performance579
indices [2]. For our purpose to enhance FI performance, the580
filter related to rFI
k(t) and fkis chosen such that:581
QFI
opt,k(q)=arg max
Q
EnQ(q)rFI,fk
k(t)o2
EnQ(q)rFI,n
k(t)o2.(44)582
We call QFI
opt,k(q) optimal filter due to that it maximizes a spe-583
cific performance index. According to [27], QFI
opt,k(q) can be584
obtained by investigating spectrum information, which is a fre-585
quency selection filter selecting the frequency586
ωopt,k=arg max
ω
ΦrFI,fk
k
(ω)
ΦrFI,n
k(ω).(45)587
In practice, considering realizability and phase delay, this opti-588
mal filter can be approximated by bandpass or lowpass filters.589
After a series of such optimal filters are designed, set590
QFI
opt(q)=diag nQFI
opt,1,· · · ,QFI
opt,ko(46)591
the filtered rFI becomes 592
rFI
opt(t)=QFI
opt(q)L(q)r(t).(47) 593
Notice that after being filtered by the diagonal QFI
opt(q), the de- 594
coupled structure is maintained in the residual. 595
Now combining the FD scheme developed in [27], we pro- 596
pose a FDI scheme in Fig. 4. In the FD block, rFD
opt which typi- 597
cally has the highest fault detection ability, is used; when faults 598
are detected, the isolation filter is used to make the faults struc- 599
tured and another optimal filter QFI
opt is used to improve the de- 600
tection ability of rFI . There are three filters in the FDI scheme 601
playing important roles, one isolation filter and two optimal fil- 602
ters aiming at enhancing abilities to detect faults. The two op- 603
timal filters are generally different and should be designed ac- 604
cording to different residuals. However for special cases such 605
as step and drift faults, they are both low-pass filters. 606
5. A 600MW power generation unit example 607
In this section we study fault isolation of a 600 MW super- 608
critical power generating unit. A coordinate control system 609
(CSS) is required to maintain the stable operation of the unit 610
while meeting the power demand from the power grid. In CCS, 611
three main controlled variables are steam flow (proportional to 612
output power at steady working points), main steam pressure 613
and main steam temperature; three main manipulated variables 614
are coal feed rate, main steam valve (steam turbine valve) and 615
feed water flow. The variables of the unit are coupled. This 616
section will use a 3 ×3 unit model identified from real data 617
(with asymptotic method [32], and at 100% load) to simulate 618
the dynamics of the unit. This model has been proved to be 619
high-quality, and used in a MPC system. 620
The continuous-time transfer matrix of the model is given 621
in (48). It can be verified that the system has time delays and 622
RHP zeros. The input-output variable names are given in Ta- 623
ble 1, in which boiler control command directly affect coal 624
feed rate; middle point temperature is an alternative of the main 625
steam temperature which can indicate the temperature changes 626
more sensitively; desuperheating water valve regulates the mid- 627
dle point temperature below some upper limit. We introduce 628
output disturbance to each output with 10% noise-to-signal ra- 629
tio, as given in (49) its discrete form and use three signals with 630
low-pass spectrums to simulate the input variations at the steady 631
working point: 632
v1(t)=1−0.1q−1
1−0.92q−1¯e1(t),
v2(t)=1−0.2q−1
1−0.89q−1¯e2(t),
v3(t)=1−0.29q−1
1−0.994q−1¯e3(t),
(50) 633
where ¯e1(t),¯e2(t),¯e3(t) are uncorrelated Gaussian white noise 634
with zero mean and unit variance. In simulation, the sampling 635
interval is chosen as 10s. The faults in the three inputs will be 636
8

process
Process input
process
model -
Process output No
Decision log ic
(for FI)
Improve FD
performace
Decision log ic
(for FD)
Decision log ic
(for FD)
Which faul t(s) occurs?
No
Yes
Improve FI
performace
Low FI perfomance
Yes
Alarm
FD block FI block
Continue monitoring Yes No
Redesign filter
Figure 4: Block diagram of the proposed FDI method.
G(s)=
0.0183(s+0.0609)(s+0.0118)e−40 s
(s+0.2586)(s+0.00541)(s+0.00406)
0.00489(s+0.175)(s+0.000910)e−40 s
(s+0.00267)(s2+0.0309s+0.00645)
0.000260(s+0.350)(s+0.00310)e−40 s
(s+0.0799)(s+0.00570)(s+0.00357)
0.000263(s2+0.0275s+0.00117)e−20 s
(s+0.00110)(s2+0.0876s+0.00280)
0.00137(s−0.0351)(s+0.0147)e−20 s
(s+0.302)(s+0.0385)(s+0.00648)
1.526×10−5(s+0.277)(s+0.00476)e−20s
(s+0.03283)(s+0.0148)(s+0.00339)
−0.00190(s+0.0134)(s−0.00270)e−130 s
(s+0.00111)(s2+0.0655s+0.000175)
−0.00157(s+0.0102)(s+0.00276)e−130 s
(s+0.00116)(s2+0.0133s+0.000264)
−0.000164(s2+0.0231s+0.000366)e−130 s
(s+0.00105)(s2+0.00857s+0.000164)

(48)
H(q)=diag (1−0.7413q−1+0.01897q−2
1−1.905q−1+0.9063q−2,1−0.42q−1
1−0.98q−1,1−0.6q−1
1−0.92q−1)(49)
considered, which, from above discussion, will have significant637
effect on CCS of the unit.638
Table 1: Input and output variables of the numerical example.
Input variables Output variables
u1Boiler main control command y1Output power
u2Main steam valve y2Steam pressure
u3Desuperheating water valve y3Middle point temperature
5.1. Plant identification639
To illustrate the proposed method, first a model is re-640
identified with model error quantified. Three uncorrelated GBN641
signal are added to the inputs, the experiment time N=8000642
samples. Three MISO BJ models are identified where cor-643
rect orders and delays are used, the result is shown in Fig. 5,644
from which one could see that the identified model can capture645
the dynamic characteristics of the real system and model error646
exists due to the disturbance (49). Notice that when decom-647
posing the system to pMISO subsystems, the corresponding648
estimated parameter vector ˆ
θ1,· · · ,ˆ
θpis uncorrelated, hence649
P(ˆ
θ)=diag{P(ˆ
θ1),· · · ,P(ˆ
θp)}.650
5.2. Isolation filter design and validation651
Based on the identified model, an MPC controller for iso-652
lation filter is designed and identified using Algorithm 1,653
the parameters used are P=400, M=200, Q=654
diag{7000,6500,2500},R=diag{1,1,1}. The simulation time 655
NK=8000 samples. As an illustration of determining the iden- 656
tified controller order nK, different orders are chosen, the rela- 657
tive error (RE) in identification of each element of Kand the 658
final gain of L(q)ˆ
G(q) is shown in Table 2. RE is defined as: 659
RE =100 ×var(y−ˆy)
var(y),(51) 660
ydenotes real output and ˆydenotes simulation output. RE is a 661
commonly used criteria to evaluate the fitness to real data of the 662
identified model. 663
From Table 2, it can be seen that for controller identifica- 664
tion, REs are typically small compared to plant identification 665
problem due to its noise-free feature. When nK=15, REs are 666
all small except for the one of K12, whereas the gain matrix is 667
very unsatisfactory because gains of Γ21,Γ23,Γ31 are large es- 668
pecially for Γ21. Compared nK=15 and nK=20, REs are 669
slightly reduced while the gains of the nondiagonal elements of 670
Γare reduced considerably. When nKincreases to 30, the REs 671
are almost unchanged compared to nK=20 however the gain 672
matrix is much improved, which will lead to a better FI. No sig- 673
nificant improvement occurs when increasing the order further, 674
so nK=30 will be used in this section. It can be concluded that 675
for noise-free identification, RE is not sensitive to the bias error 676
caused by model reduction. To check whether nKis proper, one 677
should pay attention to the final gain of Γ.678
Then, based on Section 3.3, the step response bands of Γ(q)679
9

0 2000 4000 6000
Time [s]
0
1
2
0 2000 4000 6000
Time [s]
0
0.1
0.2
0 2000 4000 6000
Time [s]
0
0.1
0.2
0 2000 4000 6000
Time [s]
0
0.05
0.1
0 2000 4000 6000
Time [s]
-10
-5
0
10-3
0 2000 4000 6000
Time [s]
0
0.005
0.01
0 2000 4000 6000
Time [s]
0
0.2
0.4
0 2000 4000 6000
Time [s]
-0.15
-0.1
-0.05
0
0 2000 4000 6000
Time [s]
-0.3
-0.2
-0.1
0
Boiler main control command Main steam valve Desuperheating water valve
2XWSXW
SRZHU
6WHDP
SUHVVXUH
0LGGOH
SRLQW
WHPSHUature
Figure 5: Step responses of real plant and of the identified model.
Table 2: REs and gain matrices corresponding to the identification results using
different orders.
Model order RE(%) Gain
15 
0.64 5.47 1.07
0.68 0.63 0.97
0.70 0.80 0.89

0.9637 0.0014 −0.0029
−1.8907 0.9376 −0.1558
0.2634 −0.0019 1.0171

20 
0.62 0.61 0.87
0.67 0.57 0.85
0.70 0.80 0.89

0.9907 0.0004 −0.0003
0.2298 0.9881 0.0264
−0.1834 0.0261 0.9812

30 
0.62 0.61 0.87
0.67 0.56 0.84
0.70 0.80 0.89

1.0012 −0.0001 0.0001
−0.0655 1.0045 0.0072
−0.0004 −0.042 0.9867

can be calculated and plotted, as shown in Fig. 6. From this fig-680
ure, we can see that step responses of Γ0(q) falls into the step681
response bands of Γ(q), and the derived isolation filter can be682
used for FI because Vis diagonal and has full rank, satisfying683
situation 1) in Section 3.3. By comparing Fig. 5 and Fig. 6, it684
can also be found that the dynamic characteristics of the faults685
to residuals transfer function are accelerated because in design-686
ing procedure of L(q), large Qto Rratio is used.687
5.3. Fault isolation implementation688
Faults are assumed to be step-type and three different simu-689
lations are carried out and each time only one fault occurs at690
7000s. The amplitude of the faults are set small compared to691
normal variations of the input signals. The results are shown692
in Fig. 7, where Hotelling’s T2statistic of each component of693
the residual is used and confidence level of the threshold is set694
to F, for details about this statistic and threshold setting see e.g.695
[27, 49]. For the reason that the three faults are all minor faults696
the detection performance of rFI is low, in Fig. 7 one can find697
that rFI cannot detect the faults. Then, by using optimal filters698
to components of rFI we obtain rFI
opt which can detect the faults699
and the three different faults can be isolated. The optimal fil-700
ters used are low-pass Butterworth filter with 0.05rad/s cutoff701
frequency.702
Table 3: Model inputs and outputs.
Block name Variable name Variable number
D feed flow XMV(1)
E feed flow XMV(2)
A feed flow XMV(3)
A and C feed flow XMV(4)
Compressor recycle valve XMV(5)
Model input Purge valve XMV(6)
Separator pot liquid flow XMV(7)
Stripper liquid product flow XMV(8)
Stripper steam valve XMV(9)
Reactor cooling water flow XMV(10)
Condenser Reactor cooling water flow XMV(11)
Reactor feed rate XMEAS(6)
Reactor pressure XMEAS(7)
Reactor temperature XMEAS(9)
Model output Separator pressure XMEAS(13)
Stripper pressure XMEAS(16)
Compressor work XMEAS(20)
Reactor cooling water outlet temperature XMEAS(21)
Component A XMEAS(23)
6. Application to Tennessee Eastman process 703
This section is dedicated to show the ability of the proposed 704
method to handle large-scale system with unknown model 705
structure. The well-known TEP benchmark is used for the pur- 706
pose. TEP was developed to provide a realistic simulation of 707
an industrial process for the evaluation of monitoring methods 708
[51]. Fig. 8 shows the flow diagram of TEP with 5 major units: 709
Table 4: Eight newly defined faults in TEP.
Variable number Process variable Type Amplitude
FAULT(1) Reactor feed rate sensor Step 0.2
FAULT(2) Separator pressure sensor Step 0.5
FAULT(3) Stripper pressure sensor Step 0.5
FAULT(4) D feed loss Step 0.3
FAULT(6) Separator pot liquid leakage Step 2
FAULT(5) A feed loss Step 3
FAULT(7) Reactor cooling water valve Step 0.5
FAULT(8) Condenser Reactor cooling water valve Step 4
10

Fault 1 Fault 2Fault 3
rFI-component 1
rFI-component 2
rFI-component 3
Figure 6: Step responses of Γo(q) and step response bands of Γ(q), the confidence level is 95%.
0 5000 10000 15000
0
20
40
60
T2 of component 1
0 5000 10000 15000
0
5
10
15
T2 of component 2
0 5000 10000 15000
Time [s]
0
5
10
15
T2 of component 3
FI FI
(a) u1fault
0 5000 10000 15000
0
5
10
15
T2 of component 1
0 5000 10000 15000
0
20
40
T2 of component 2
0 5000 10000 15000
Time [s]
0
5
10
15
T2 of component 3
), ),
(b) u2fault
0 5000 10000 15000
0
10
20
T2 of component 1
0 5000 10000 15000
0
5
10
15 T2 of component 2
0 5000 10000 15000
Time [s]
0
50
100
T2 of component 3
), ),
(c) u3fault
Figure 7: FI of three actuator faults, different components of rFI and rFI
opt are shown for comparison. The red dashed line denotes the 99.9% threshold.
11

Figure 8: A process flowsheet of Tennessee Eastman process given in [50].
reactor, condenser, compressor, separator and stripper. The pro-710
cess has two products from four reactants. Additionally, an inert711
and a by-product are also present making a total of 8 compo-712
nents denoted as A, B, C, D, E, F, G and H [49]. The process713
allows total 52 measurements out of which 41 are of process714
variables and 11 are manipulated variables; and the complete715
description of these variables can be found in [49]. A brief ver-716
sion covering the variables which will be used in this section is717
shown in Table 3.718
There are two alternatives to do fault diagnosis research with719
TEP, one is the data set given in [52], another is the Simulink720
code provided by Ricker [53] which is available to simulate the721
plant’s closed-loop behavior. In [52], 21 documented faults are722
defined for fault diagnosis study, but the they are mainly mul-723
tiplicative (process) faults. In order to validate the proposed724
method, we use the simulators offered by [53] and define eight725
additive faults as listed in Table 4. The fault to output transfer726
functions can be obtained using system identification. Among727
these faults, FAULT(1-3) are output faults and FAULT(4-8) are728
input faults. The sampling interval in this section is chosen as729
3 min.730
6.1. TEP identification731
Similarly to Section 5, a system identification is needed be-732
fore we perform FI. Based on the defined faults, we choose733
MV(1-11) as inputs and XMEAS(6,7,9,13,16,20,21,23) as out-734
puts to build a model for residual generation. TEP is operated735
in closed-loop so 11 uncorrelated GBN signals are added to736
the initial setpoints corresponding to the chosen outputs, then737
20000 samples are generated, among which 17000 for parame- 738
ter estimation and 3000 for model validation. Because 8 faults 739
are used in the study, it is sufficient to choose 8 outputs. The 740
comparison between real output and model simulation output 741
of the validation data is shown in Fig. 9. All the models we 742
used are 2nd-order ARMAX model and for each MISO system, 743
the orders of transfer functions from different inputs are set the 744
same. The identified model contains no time delays but have 745
RHP zeros. Based on the identified model, eight OE residuals 746
can be built and combined into the basic residual r.747
6.2. Isolation filter design 748
To get the isolation filter using Algorithm 1, first the fault 749
signal vector and fault transfer function should be determined. 750
Let 751
f(t)=[FAULT(1),FAULT(2),· · · ,FAULT(8)]T,(52) 752
then 753
Gy f (q)=

100G11 G13 G17 G1(10) G1(11)
000G21 G23 G27 G2(10) G2(11)
0 0
1 0
.
.
.0 1 .
.
..
.
..
.
..
.
..
.
.
.
.
.0.
.
..
.
..
.
..
.
..
.
.
.
.
.0
.
.
.
000G81 G83 G87 G8(10) G8(11)

,
(53) 754
12

0 1000 2000 3000
Sample
-1
0
1
XMEAS(6): RE= 17.58%
0 1000 2000 3000
Sample
-20
0
20
XMEAS(7): RE= 23.81%
0 1000 2000 3000
Sample
-1
-0.5
0
0.5
XMEAS(9): RE= 24.66%
0 1000 2000 3000
Sample
-20
0
20
XMEAS(13): RE= 26.62 %
0 1000 2000 3000
Sample
-20
0
20
40
XMEAS(16): RE= 23.65 %
0 1000 2000 3000
Sample
-10
0
10
XMEAS(20): RE= 2.326 %
0 1000 2000 3000
Sample
-1
0
1
XMEAS(21): RE= 21.84 %
0 1000 2000 3000
Sample
-2
0
2
XMEAS(23): RE= 9.202 %
Real output Simulation output
Figure 9: Identification result of TEP.
in which the left block corresponds to output faults FAULT(1-755
3), the right block corresponds to input faults FAULT(4-7). The756
step responses of d
Gy f (q) is shown in Fig. 10. The final parame-757
ters we used are Q=250·[1 1 1 1 2 1 1 1], R=[1 1 1 1 1 1 1 1],758
P=120, M=60, n=60. The step response bands of the759
achieved Γ(q) is shown in Fig. 11, from which one can see that760
the identified MPC controller eliminates the couplings in d
Gy f (q)761
to a great extent. According to Section 3.3762
V=

1 00000
1 00010
100000
10000
01000
05×300100
00010
00011

,(54)763
which has full rank and is in accordance with situation 2), hence764
the designed isolation filter can be used for FI purpose despite765
the model error.766
6.3. FI implementation767
As has mentioned and suggested in Section 4, to perform768
fault diagnosis, we recommend do FD based on rand rFD
opt and769
perform FI using the isolation filter. In [27], we have discussed770
the FD problem in detail and by using OEs and optimal filters,771
all the 21 documented faults in [52] can be well detected, based772
on the benchmark data set. Here, based on the simulator the773
8 newly defined faults can also be well detected following the 774
same method, and in order to avoid repetition FD step will be 775
omitted, only the FI step will be presented. 776
Similar to Section 5, each time only one fault occurs and 777
thus the change should occur only in one component of rFI .778
Faults occur at 200 sample time. The confidence interval of 779
the T2statistics are all set to 99.9%. In order to improve the FI 780
performance, QFI
opt(q) will be used as an optimal filter, which has 781
a diagonal structure and each diagonal element is a Butterworth 782
low-pass filter. The results are shown in Fig. 12, we take (d)(g) 783
for instance: for (d), Ξ=[00010000]T. According to (43), 784
k∗=4 hence one could conclude that FAULT(4) occurs; for (g), 785
k∗is not unique however due to the fact that only component 786
7 alarms, we conclude that FAULT(7) occurs. Similarly, the 787
8 faults can all be isolated. If faults occur sequentially, one 788
new fault will cause alarm in one or several new components 789
of rFI
opt, then decision should be made based on these new alarm 790
information. 791
Though TEP has become a benchmark for fault diagnosis 792
study, few model-based FD and FI methods has been validated 793
for TEP. One reason is that model-based methods are nearly all 794
based on first-principle state-space model and the exact model 795
of TEP is unknown. Recently, some identification based meth- 796
ods are proposed and validated in TEP, see e.g. [2, 49, 54, 27], 797
however they only focus on FD and the validations of FI meth- 798
ods are rare. In this paper, based on an identified model, the 799
proposed method is validated using TEP and the results can be 800
considered promising. 801
13

20
0
0.5
1
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 100 200
-0.4
-0.2
0
0 100 200
0
0.04
0.08
0 100 200
-0.2
-0.1
0
0 100 200
0
0.5
1
0 200
-0.02
0
0.02
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 20
-3
-2
-1
0
0 20
0
0.04
0.08
0 20
-0.1
-0.05
0
0 20
0
0.5
1
0 20
-0.08
-0.04
0
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 50
0
0.2
0.4
0 50 100
0
0.04
0.08
0 50
-0.08
-0.04
0
0 20 40
-1
-0.5
0
0 50
-0.15
-0.1
-0.05
0
0 1
-0.02
0
0.02
10 20
0
0.5
1
0 1
-0.02
0
0.02
0 20
-4
-2
0
0 20
0
0.04
0.08
0 20 40
0
0.05
0.1
0 20
0
1
2
3
0 20
-0.2
-0.1
0
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
10 20
0
0.5
1
0 20
-2
-1
0
0 20
0
0.05
0.1
0 20
0
0.05
0.1
0 20
-0.2
0
0.2
0.4
0.6
0 20
-0.02
0
0.02
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 500
-10
-5
0
0 500
0
1
2
0 500
-6
-4
-2
0
0 500
0
2
4
6
8
0 500
0
0.2
0.4
0.6
0.8
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 50
0
0.1
0.2
0 50 100
-0.02
0.02
0.06
0 50 100
-0.06
-0.04
-0.02
0
0 50
-1
-0.5
0
0 50 100
-0.1
-0.05
0
0 1 Sample
-0.02
0
0.02
0 1
-0.02
0
0.02
0 1
-0.02
0
0.02
0 1900
-3
-2
-1
0
0 1900
0
0.5
1
1.5
0 1900
-1
-0.5
0
0 1900
0
0.5
1
0 1900
0
0.2
0.4
0.6
0.8
FAULT(1)
XMEAS(6) FAULT(2)
XMEAS(13) FAULT(3)
XMEAS(16) FAULT(4)
XMV(1) FAULT(6)
XMV(7)
FAULT(5)
XMV(3) FAULT(7)
XMV(10) FAULT(8)
XMV(11)
100
0
0
Figure 10: Step responses of d
Gy f (q). Subfigures marked as red corresponds to d
Gy f
i j containing RHP zeros.
14

0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-1
0
1
0 50
-1
0
1
0 50
-0.5
0
0.5
1
1.5
0 50
-10
0
10
0 50
-5
0
5
0 50
-0.5
0
0.5
1
0 50
-2
0
2
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-1
0
1
2
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-2
0
2
0 50
-0.5
0
0.5
1
1.5
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-1
0
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-4
-2
0
2
4
0 50
-1
0
1
0 50
-0.5
0
0.5
1
0 50
-2
0
2
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-1
0
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
0 50
-0.5
0
0.5
1
Figure 11: Step response bands of Γ(q). In columns 4-8, the red dashed lines denote the unity lines.
15

0 200 400 600 800 1000
0
100
T2 of component 1
0 200 400 600 800 1000
0
5
10
T2 of component 2
0 200 400 600 800 1000
0
5
10
T2 of component 3
0 200 400 600 800 1000
0
5
10
T2 of component 4
0 200 400 600 800 1000
0
5
10
T2 of component 5
0 200 400 600 800 1000
0
5
10
T2 of component 6
0 200 400 600 800 1000
0
5
10
T2 of component 7
0 200 400 600 800 1000
Samples
0
5
10
T2 of component 8
(a) FAULT(1)
0 200 400 600 800 1000
0
5
10
T2 of component 1
0 200 400 600 800 1000
0
50
100 T2 of component 2
0 200 400 600 800 1000
0
5
10
T2 of component 3
0 200 400 600 800 1000
0
5
10
T2 of component 4
0 200 400 600 800 1000
0
5
10
T2 of component 5
0 200 400 600 800 1000
0
5
10
T2 of component 6
0 200 400 600 800 1000
0
5
10
T2 of component 7
0 200 400 600 800 1000
Samples
0
5
10
T2 of component 8
(b) FAULT(2)
0 200 400 600 800 1000
0
5
10
T2 of component 1
0 200 400 600 800 1000
0
5
10
T2 of component 2
0 200 400 600 800 1000
0
50
T2 of component 3
0 200 400 600 800 1000
0
5
10
T2 of component 4
0 200 400 600 800 1000
0
5
10
T2 of component 5
0 200 400 600 800 1000
0
5
10
T2 of component 6
0 200 400 600 800 1000
0
5
10
T2 of component 7
0 200 400 600 800 1000
Samples
0
5
10
T2 of component 8
(c) FAULT(3)
0 200 400 600 800 1000
0
5
10
T2 of component 1
0 200 400 600 800 1000
0
5
10
T2 of component 2
0 200 400 600 800 1000
0
5
10
T2 of component 3
0 200 400 600 800 1000
0
1
2104T2 of component 4
0 200 400 600 800 1000
0
5
10
T2 of component 5
0 200 400 600 800 1000
0
5
10
T2 of component 6
0 200 400 600 800 1000
0
5
10
T2 of component 7
0 200 400 600 800 1000
Samples
0
5
10
T2 of component 8
(d) FAULT(4)
0 200 400 600 800 1000
0
5
10
T2 of component 1
0 200 400 600 800 1000
0
5
10
T2 of component 2
0 200 400 600 800 1000
0
5
10
T2 of component 3
0 200 400 600 800 1000
0
5
10
T2 of component 4
0 200 400 600 800 1000
0
50
T2 of component 5
0 200 400 600 800 1000
0
5
10
T2 of component 6
0 200 400 600 800 1000
0
5
10
T2 of component 7
0 200 400 600 800 1000
Samples
0
5
10
T2 of component 8
(e) FAULT(5)
0 200 400 600 800 1000
0
5
10
T2 of component 1
0 200 400 600 800 1000
0
5
10
T2 of component 2
0 200 400 600 800 1000
0
5
10
T2 of component 3
0 200 400 600 800 1000
0
5
10
T2 of component 4
0 200 400 600 800 1000
0
5
10
T2 of component 5
0 200 400 600 800 1000
0
100
200
T2 of component 6
0 200 400 600 800 1000
0
5
10
T2 of component 7
0 200 400 600 800 1000
Samples
0
5
10
T2 of component 8
(f) FAULT(6)
0 200 400 600 800 1000
0
5
10
T2 of component 1
0 200 400 600 800 1000
0
5
10
T2 of component 2
0 200 400 600 800 1000
0
5
10
T2 of component 3
0 200 400 600 800 1000
0
5
10
T2 of component 4
0 200 400 600 800 1000
0
5
10
T2 of component 5
0 200 400 600 800 1000
0
5
10
T2 of component 6
0 200 400 600 800 1000
0
500
T2 of component 7
0 200 400 600 800 1000
Samples
0
5
10
T2 of component 8
(g) FAULT(7)
0 200 400 600 800 1000
0
5
10
T2 of component 1
0 200 400 600 800 1000
0
5
10
T2 of component 2
0 200 400 600 800 1000
0
5
10
T2 of component 3
0 200 400 600 800 1000
0
5
10
T2 of component 4
0 200 400 600 800 1000
0
5
10
T2 of component 5
0 200 400 600 800 1000
0
5
10
T2 of component 6
0 200 400 600 800 1000
0
5
10
T2 of component 7
0 200 400 600 800 1000
Samples
0
50
100
T2 of component 8
(h) FAULT(8)
Figure 12: FI of 8 different faults in TEP. In the figure, the blue line is component of rFI
opt while the red dashed line denotes the 99.9% threshold.
16

7. Conclusion802
A model based fault isolation method is developed. Using803
the identified plant transfer function model, the output errors804
are used as basic residuals. Then based on the faults to out-805
puts transfer function matrix, an isolation filter is designed by806
combining an MPC algorithm and an identification algorithm.807
A method to validate the isolation filter under model errors is808
developed; and an optimal isolation filter is proposed that sup-809
presses the disturbances. The method is straightforward that810
can handle large-scale processes, time delays and RHP zeros.811
The effectiveness of the method is verified using two case stud-812
ies including the well-known TE process. The ill-conditioning813
of the plant may pose a problem for fault isolation filter de-814
sign because decoupler is involved, which will be investigated815
in future work. Further research concerning identification based816
FDI can be detection and isolation of multiplicative faults and817
extending the developed methods to nonlinear systems.818
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