Proceedings of the 26th International Workshop on Principles of Diagnosis
LPV subspace identification for robust fault detection using a set-membership
approach: Application to the wind turbine benchmark
Chouiref. H, Boussaid. B, Abdelkrim. M.N 1 , Puig. V2 and Aubrun.C3
1
Research Unit of Modeling, Analysis and Control of Systems (MACS), Gabès University
e-mail: houda.chouiref@gmail.com, dr.boumedyen.boussaid@ieee.org,naceur.abdelkrim@enig.rnu.tn
2
Advanced Control Systems Group (SAC), Technical University of Catalonia
e-mail: vicenc.puig@upc.edu
3
Centre de Recherche en Automatique de Nancy (CRAN), Lorraine University
e-mail: christophe.aubrun@univ-lorraine.fr
Abstract
This paper focuses on robust fault detection for
Linear Parameter Varying (LPV) systems using a
set-membership approach. Since most of models
which represent real systems are subject to mod-
eling errors, standard fault detection (FD) LPV
methods should be extended to be robust against
model uncertainty. To solve this robust FD prob-
lem, a set-membership approach based on an in-
terval predictor is used considering a bounded de-
scription of the modeling uncertainty. Satisfac-
tory results of the proposed approach have been
obtained using several fault scenarios in the pitch Figure 1: Fault diagnosis with set estimator schema
subsystem considered in the wind turbine bench-
mark introduced in IFAC SAFEPROCESS 2009.
must be robust. When modeling uncertainty in a determin-
1 Introduction istic way, there are two robust estimation methods: the first
The fault diagnosis of industrial processes has become an method is the bounded error estimation that assumes the pa-
important topic because of its great influence on the opera- rameters are considered time invariant and there is only an
tional control of processes. Reliable diagnosis and early de- additive error [6]. On the other hand, the second approach
tection of incipient faults avoid harmful consequences. Typ- is the interval predictor that takes into account the variation
ically, faults in sensors and actuators and the process itself of parameters and which considers additive and multiplica-
are considered. In the case of the wind turbine benchmark, tive errors [7], [8]. Here, the interval predictor is combined
a set of pre-defined faults with different locations and types with existing nominal LPV identification presented by [9],
are proposed in [1] where the dynamic change in the pitch allowing to include robustness and minimizing false alarms
system is treated. The procedure of fault detection is based (see Fig. 1) [10]. Thus, this paper contributes with a new
either on the knowledge or on the model of the system [2]. set-membership estimator approach that combines the in-
Model-based fault detection is often necessary to obtain a terval predictor scheme with the LPV identification through
good performance in the diagnosis of faults. The methods subspace methods in one step. To illustrate the methodology
used in model-based diagnosis can be classified according proposed in this work, the pitch subsystem of wind turbine
if they are using state observers, parity equations and pa- system proposed as a benchmark in IFAC SAFEPROCESS
rameter estimation [3]. For linear time invariant systems 2009 will be used. First, this subsystem is modeled as an
(LTI), the FD task is largely solved by powerful tools. How- LPV model using the hydraulic pressure as the scheduling
ever, physical systems generally present nonlinear behav- variable. On the hypothesis that damping ratio and natural
iors. Using LTI models in many real applications is not frequency have an affine variation with hydraulic pressure,
sufficient for high performance design. In order to achieve this affine LPV model is estimated by means of the subspace
good performance while using linear like techniques, Lin- LPV estimation algorithm. Second, the residue is synthe-
ear Parameter Varying systems are recently received con- sized to take into account the robustness against the uncer-
siderable attention [4]. Recently, many model-based appli- tainties in the parameters. This work is organized as fol-
cations using such systems and the subspace identification lows: In Section 2, the LPV subspace estimation method is
method were published [5]. In model-based FD, a residual recalled. In Section 3, the interval predictor approach com-
vector is used to describe the consistency check between bined with the LPV subspace method is proposed as tool
the predicted and the real behavior of the monitored sys- for robust fault detection. In Section 4, the modeling of the
tem. Ideally, the residuals should only be affected by the pitch system as a LPV model is introduced. Section 5 deals
faults. However, the presence of disturbances, noises and with simulation experiments that illustrate the implementa-
modeling errors yields the residual to become non zero. To tion and performance of the proposed approach applied to
take into account these errors, the fault detection algorithm the robust fault detection of wind turbine pitch system. Fi-
261
Proceedings of the 26th International Workshop on Principles of Diagnosis
nally, Section 6 gives some concluding remarks.
Y = [yp+1 , ..., yN ] (5)
2 LPV Subspace Identification method
In the literature, there are two methods for LPV identifica- ]
tion: First, the ones based on global LPV estimation. Sec- Z = [N1p z̄1p , ..., NN
p p
(6)
−p+1 z̄N −p+1
ond, the ones based on the interpolation of local models
[11], However, those approaches could lead to unstable rep- the controllability matrix can be expressed as:
resentations of the LPV structure while the original system
is stable [12]. That is why in this paper, we propose to κp = [lp , ..., l1 ]
use a subspace identification algorithm proposed (see [9] with [ ]
and [13]) to identify LPV systems which does not require
l1 = B̄ (1) , ..., B̄ (m)
interpolation or identification of local models and avoid in-
stability problems. and [ ]
lj = Ã(1) lj−1 , ..., Ã(m) lj−1
2.1 Problem formulation
[ ]
In the model used in identification in [9], the system ma- If the matrix Z T , U T has full row rank, the matrix
trices depend linearly on the time varying scheduling vector Cκp and D can be estimated by solving the following linear
as follows: regression problem [14]:
∑m 2
xk+1 =
(i)
µk (A(i) xk + B (i) uk + K (i) ek ) (1) min ∥Y − Cκp Z − DU ∥F (7)
Cκp ,D
i=1
where ∥∥F represents the Frobenius norm. This problem
yk = Cxk + Duk + ek (2) can be solved by using traditional least square methods as
in the case of LTI identification for time varying systems.
with xk ∈ R , uk ∈ R , yk ∈ R are the state, input and
n r l
Moreover, the observability matrix for the first model is cal-
output vectors and ek denotes the zero mean white innova- culated as follows:
tion process and m is the number of local model or schedul-
ing parameters: C
[ C Ã(1)
(2)
µk = 1, µk , ..., µm
T .
k ] Γp =
.
Eqs.(1) and (2) can be written in the predictor form: .
(1) p−1
m
∑ C(Ã )
(i)
xk+1 = µk (Ã(i) xk + B̃ (i) uk + K (i) yk ) (3)
with
i=1
⌣ ⌣ ⌣
with κ̄kp = [φp−1,k+1 B k , ..., φ1,k+p−1 B k+p−2 , B k+p−1 ]
Ã(i) = A(i) − K (i) C
and
B̃ (i) = B (i) − K (i) D ⌣
B k = [B̃, Kk ]
2.2 Assumptions and notation Then, Eq.(3) can be transformed into:
[ ]T
Defining zk = uTk , ykT and using a data window of
length p to define the following vector: xk+p = φp,k xk + κ̄kp z̄kp
xk+p = φp,k xk + κp Nkp z̄kp
zk
zk+1 where
φp,k = ÃK+p−1 ...Ãk+1 Ãk
p .
z̄k =
. If the system (3) is uniformly exponentially stable the ap-
. proximation error can be made arbitrarily small then:
zk+p−1
xk+p ≈ κp Nkp z̄kp
and introducing the matrix obtained using the Kronecker
product ⊗: To calculate the observability matrix Γp times the state X,
we first calculate the matrix Γp κp :
Pp/k = µk+p−1 ⊗ .... ⊗ µk ⊗ Ir+l
Clp Clp−1 . . Cl1
we can define 0 CA(1) lp−1 . . C Ã(1) l1
pp/k . . . 0
p p
Γ κ = . .
. pp−1/k+1 . .
p−1
Nkp = . . 0 (1)
C(Ã ) l1
. .
0 p1/k+p−1 Then, using the following Singular Value Decomposition
(SVD):
Now, by defining the matrices U , Y and Z : [ ∑ ][ ]
\
p p n 0
∑ V
U = [up+1 , ..., uN ] (4) Γ κ Z = [ υ υσ⊥ ]
0 V⊥
262
Proceedings of the 26th International Workshop on Principles of Diagnosis
the state is estimated by: with
⌢ ∑ ∏
p−j
X= V (Z i,j )T Z 1,j = ( µTÑ +v+j−i µÑ +v+j−1 )(zÑ
T
z
+j−i Ñ +j−1
)
n v=0
p
∑
Finally, C and D matrix are estimated using output equa- ZT Z = (Z 1,j )Z 1,j
tion (2) and A and B are estimated using the state equation j=1
(1). This algorithm can be summarized as follows [9]: (10)
Finally, the estimate sequence is obtained by solving the
• Create the matrices U , Y and Z using (4),(5) and (6), original SVD problem.
• Solve the linear problems given in (7) , The kernel method can be summarized as follows [9]:
• Construct Γp times the state X, • Create the matrices U T U using (4) and Z T Z and
• Estimate the state sequence, (Z i,j )T (Z i,j ) using (10),
• With the estimated state, use the linear relations to ob- • Solve the linear problem given in (8),
tain the system matrices. • Construct Γ times the state X using (9)and (10),
In the case of a very small p, we have in general a biased • Estimate the state sequence,
estimate. However, when the bias is too large, it will be a • With the estimated state, use the linear relation to ob-
problem. That is why a large p would be chosen. In the tain the system matrices.
case of a very large p, this method suffers from the curse
of dimensionality [13] and the number of rows of Z grows 3 Interval predictor approach
exponentially with the size of the past window. In fact, the
number of rows is given by: To add robustness to the LPV subspace identification ap-
proach presented in the previous section, it will be combined
∑p with the interval predictor approach [16]. The interval pre-
ρZ = (r + ℓ) mj
j=1 dictor approach is an extension of classical system identifi-
To overcome this drawback, the kernel method will be cation methods in order to provide the nominal model plus
introduced in the next subsection [15]. the uncertainty bounds for parameters guaranteeing that all
collected data from the system in non-faulty scenarios will
2.3 Kernel method be included in the model prediction interval. This approach
The
[ Tequation
] (7) has a unique solution if the matrix considers separately the additive and multiplicative uncer-
Z U T has full row rank and is given by: tainties. Additive uncertainty is taken into account in the
[ ] [ ] additive error term e(k) and modeling uncertainty is con-
[ T ] Z [ T ] sidered to be located in the parameters that are represented
d b
Cκ D = Y Z
p U T
( Z U T )−1
U by a nominal value plus some uncertainty set around. In the
When this is not the case, that will occurs when p is large, literature, there are many approximation of the set uncertain
the solution is computed by using the SVD of the matrix: parameter Θ. In our case, this set is described by a zonotope
[ ] [ ∑ ][ T ] [10] :
Z m 0 V Θ = θ0 ⊕ HB n = {θ0 + Hz : z ∈ B n } (11)
= [ υ υ⊥ ]
U 0 0 V⊥T
where: θ is the nominal model (here obtained with the
0
Then, the solution of the minimum norm is given by: identification approach, H is matrix uncertainty shape, B n
[ ] ∑−1 is a unitary box composed of n unitary (B = [−1, 1]) inter-
dp D
Cκ b =YV υT
m val vectors and ⊕ denotes the Minkowski sum. A particu-
To avoid computations in a large dimensional space, the lar case of the parameter set is used that corresponds to the
minimum norm results in: case where the parameter set Θ is bounded by an interval
2
box [17]:
min ∥α∥F (8)
α Θ = [θ1 , θ1 ] × ...[θi , θi ] × ...[θnθ , θnθ ] (12)
with [ ]
Y − α ZT Z + U T U = 0 where θi = θi0 − λi and θi = θi0 + λi with λi ≥ 0 and
[ ] i = 1, ..., nθ . In particular, the interval box can be viewed as
where α are the Lagrange multipliers and Z T Z + U T U a zonotope with center θ0 and H equal to an nθ ×nθ diagonal
is referred as the kernel matrix. matrix:
The matrix Γ times the state X can be constructed as fol-
lows: θ1 + θ1 θ2 + θ2 θn + θn
θ0 = ( , , ..., ) (13)
∑
p 2 2 2
1,j T 1,j
α j=1 (Z ) Z H = diag(λ1 , λ2 , ..., λn ) (14)
∑ p
T
α 2,j
(Z ) Z 1,j For every output, a model can be extracted in the following
j=2 regressor form:
Γκp Z = .
(9)
. y(k) = φ(k)θ(k) + e(k) (15)
.
∑ p
p,j T 1,j where
α (Z ) Z
j=p
263
Proceedings of the 26th International Workshop on Principles of Diagnosis
• φ(k) is the regressor vector of dimension 1× nθ which with
can contain any function of inputs u(k) and outputs x1 = β, x2 = β̇, u = βr
y(k). which can be discretised using an Euler approximation.
Then, the following system is obtained:
• θ(k) ∈ Θ is the parameter vector of dimension nθ ×1. {
x(k + 1) = Ax(k) + Bu(k)
(22)
y(k) = Cx(k)
• Θ is the set that bounds parameter values.
with [ ]
1 Te
• e(k) is the additive error bounded by a constant where A=
−Te wn2 −2Te ξwn + 1
|e(k)| ≤ σ. [ ]
0
In the interval predictor approach, the set of uncertain pa- B=
Te wn2
rameters Θ should be obtained such that all measured data
in fault-free scenario will be covered by the interval pre- C=[ 1 0 ]
dicted output.
4.2 LPV Pitch system model
y(k) ∈ [ŷ(k) − σ, ŷ(k) + σ] (16) The pitch parameters wn and ξ are variable with hydraulic
pressure P [1] [22]. Then, the pitch model can be written
where as the following LPV model according to [23] using P as
ŷ(k) = yˆ0 (k) − ∥φ(k)H∥1 (17) the scheduling variable ϑ :
{
x(k + 1) = A(ϑ)x(k) + B(ϑ)u(k)
(23)
ŷ(k) = yˆ0 (k) + ∥φ(k)H∥1 (18) y(k) = Cx(k)
and yˆ0 (k) is the model output prediction with nominal pa- with [ ]
1 Te
rameters with θ0 =[θ1 , θ2 , ..., θnθ ]T obtained using the LPV A(ϑ) =
identification algorithm: −Te wn2 (P ) −2Te ξ(P )wn (P ) + 1
[ ]
0
yˆ0 (k) = φ(k)θ0 (k) (19) B=
Te wn2 (P )
Then, fault detection will be based on checking if (16) y(k) = x1 (k) = β(k)
is satisfied. In case that, it is not satisfied a fault can be
indicated. Otherwise, nothing can be said. 4.3 Regressor form pitch system model
The pitch model can be transformed to the following regres-
4 Case study: wind turbine benchmark sion form [24]:
system y(k) = φ(k)θ(k) (24)
In this work, a specific variable speed turbine is considered.
where, φ(k) is the regressor vector which can contain any
It is a three blade horizontal axis turbine with a full con-
function of inputs u(k) and outputs y(k). θ(k) ∈ Θ is the
verter. The energy conversion from wind energy to mechan-
parameter vector. Θ is the set that bounds parameter values.
ical energy can be controlled by changing the aerodynamics
of the turbine by pitching the blades or by controlling the
In particular
rotational speed of the turbine relative to the wind speed.
φ(k) = [ y(k − 2) y(k − 1) u(k − 2)]
The mechanical energy is converted to electrical energy by
a generator fully coupled to a converter. Between the ro- T
θ = [ θ1 θ2 θ3 ]
tor and the generator, a drive train is used to increase the
rotational speed from the rotor to the generator [18]. This θ1 = (−Te2 wn2 + (2wn ξTe − 1))
model can be decomposed into submodels: Aerodynamic,
Pitch, Drive train and Generator [19] [20]. In this paper, θ2 = −2wn ξTe + 2
we focus on faults in the pitch subsystem as explained in the
following subsection. θ3 = Te2 wn2
4.1 Pitch system model
5 Results
In the wind turbine benchmark model, the hydraulic pitch
is a piston servo mechanism which can be modeled by a The pitch systems, which in this case are hydraulic, could
second order transfer function [21] [1]: be affected by faults in any of the three blades. The con-
sidered faults in the hydraulic system can result in changed
β(s) ωn2 dynamics due to a drop in the main line pressure. This dy-
= 2 (20)
βr (s) s + 2ζωn s + ωn2 namic change induces a change in the system parameters:
the damping ratio between 0.6 rad/s and 0.9 rad/s and the
Notice that βr refers to reference values of pitch angles. frequency between 3.42 rad/s and 11.11 rad/s according
The pitch model can be written in the following state space: to [23]. In this work, a fault detection subspace estimator
{ is designed to determine the presence of a fault. To distin-
ẋ1 = x2 guish between fault and modeling errors, an interval predic-
(21) tor approach is applied and a residual generation is used for
ẋ2 = −2ξwn x2 − wn 2 x1 + wn 2 u
264
Proceedings of the 26th International Workshop on Principles of Diagnosis
16 Upper
0.635
Lower
14
12 0.63
10
0.625
minmaxoutput
damping ratio in case1
8
0.62
6
4 0.615
2
0.61
0
−2 0.605
1.58 1.6 1.62 1.64 1.66 1.68
Time(s) x 10
4
0.6
0 0.5 1 1.5 2 2.5
Time(s) x 10
4
Figure 2: Upper (red line) and lower (blue line) bounds
Figure 3: Damping ratio in non-faulty case
deciding if there is a fault. To illustrate the performance of
this robust fault detection approach:: ξ ∈[ 0.6 0.63 ] and
wn ∈[ 10.34 11.11 ] are considered. Then, a parameter
set Θ is bounded by an interval box: 11.3
(25)
11.2
Θ = [θ1 , θ1 ] × [θ2 , θ2 ] × [θ3 , θ3 ]
11.1
11
and for i = 1, · · · , 3
frequency in case1
10.9
10.8
θi − θi
λi = ( ) (26) 10.7
2
10.6
θi + θi 10.5
θi0 = ( ) (27)
2 10.4
using equations (17) and (18), the output bounds are calcu- 10.3
0 0.5 1 1.5 2 2.5
lated to be used in fault detection test which are given in Time(s) x 10
4
Fig. 2.yˆ0 (k) is obtained by the use of the identification ap-
proach described in Section 2. To validate this algorithm Figure 4: Frequency in non-faulty case
two cases are used:
- Case 1: In this case, the pressure varies after time 10000s
while parameters vary in the interval of parametric uncer-
tainties, that is, damping ratio varies between 0.6 rad/s
and 0.63 rad/s and the frequency between 10.34 rad/s
and 11.11 rad/s. These parameters are presented respec- 2.1
tively in Figures. 3 and 4. The pitch angle in this case is 2
given in Fig. 5 altogether with the prediction intervals.
pitch angle in non faulty case
1.9
For fault detection, the residual signal, based on the com-
parison between the measured pitch angle and the estimated 1.8
one at each sampling instance, is calculated and it is shown 1.7
in Fig. 6. For fault decision, a fault indicator signal is used 1.6
and the decision is taken in function of this indicator. If
the actual angle is not within the predicted interval given in 1.5
Eq.(16), the fault indicator is equal to 1 and the system is 1.4
faulty. Otherwise, it is equal to 0 and the system is fault- 1.3
free. The fault indicator signal given in Fig. 7 shows that
there is no fault despite the pressure variation. The parame- 1.2
1.6608 1.6608 1.6608 1.6608 1.6608 1.6608 1.6608
ters variation is considered as a modeling error. Time(s) 4
x 10
- Case 2: In this case, the pressure P varies between time
t = 10000s and t = 17000s outside its nominal value. In Figure 5: Pitch angle in non-faulty case
this time interval, the damping ratio varies between 0.63
rad/s and 0.72 rad/s and the frequency varies between
265
Proceedings of the 26th International Workshop on Principles of Diagnosis
0.72
0.7
damping ratio in case2
0.68
0.025 0.66
0.02
0.015 0.64
0.01
0.62
0.005
residue
0 0.6
0 0.5 1 1.5 2 2.5
−0.005 Time(s) 4
x 10
−0.01
−0.015 Figure 8: Damping ratio in faulty case
−0.02
−0.025
0 0.5 1 1.5 2 2.5 11.5
Time(s) x 10
4
11
Figure 6: Residual in non-faulty case
frequency in case2 10.5
10
9.5
9
8.5
8
0 0.5 1 1.5 2 2.5
Time(s) 4
x 10
Figure 9: Frequency in faulty case
1
0.8
8.03 rad/s and 10.34 rad/s. On the other hand, the damp-
ing ratio varies between 0.6 rad/s and 0.63 rad/s and the
0.6
natural frequency varies between 10.34 rad/s and 11.11
0.4 rad/s outside as shown in Figures 8 and 9. In this case,
Fault indicator in case1
0.2 the pitch angle is given in Fig. 10, while the residual and
fault indicator signals are presented in Fig. 11 and Fig. 12,
0
respectively.
−0.2 Fig. 12 shows that the fault indicator signal changes its
−0.4 signature between time 10000s and 17000s which induce
−0.6
that the parameters vary larger than the modeling range due
to actuator fault in wind turbine benchmark system between
−0.8
instants t = 10000s and 17000s.
−1
0 0.5 1 1.5 2 2.5
Time(s) x 10
4
6 Conclusions
The proposed approach is based on an LPV estimation ap-
Figure 7: Fault indicator in non-faulty case proach to generate a residual as the difference between the
real and the nominal behavior of the monitored system.
When a fault occurs, this residual goes out of the inter-
val which represents the uncertainty bounds in non faulty
case. These bounds are generated by means of an inter-
val predictor approach that adds robustness to this fault de-
tection method, by means of propagating the parameter un-
certainty to the residual or predicted output. The proposed
266
Proceedings of the 26th International Workshop on Principles of Diagnosis
approach is illustrated by implementing a robust fault de-
tection scheme for a pitch subsystem of the wind turbine
Mesaured benchmark. Simulations show satisfactory fault detection
2.2 Max
Min
performance despite model uncertainties.
2.1
2
References
pitch angle in faulty case
1.9
1.8
[1] P. Odgaard, J. Stoustrup, and M. Kinnaert. Fault toler-
ant control of wind turbines-a benchmark model. In
1.7
7th IFAC symposium on fault detection, supervision
1.6
and safety of technical processes, Barcelona, spain,
1.5 2009.
1.4
[2] R. Isermann. Fault diagnosis systems: an introduction
1.3
from fault detection to fault tolerance. 2006.
1.2
[3] M. Blanke, M. Kinnaert, J. Lunze, M. Staroswiecki,
1.6608 1.6608 1.6608 1.6608 1.6608 1.6608 1.6608
Time(s) x 10
4 and J Schröder. Diagnosis and fault-tolerant control.
2006.
Figure 10: Pitch angle in faulty case [4] G. Mercere, M. Lovera, and E Laroche. Identifi-
cation of a flexible robot manipulator using a linear
parameter-varying descriptor state-space structure. In
Proc. of the IEEE conference on decision and control,
Orlando, Florida,USA, 2011.
0.15
[5] J. Dong, B. Kulcsár, and M Verhaegen. Fault detection
and estimation based on closed-loop subspace identi-
0.1 fication for linear parameter varying systems. In DX,
Stockholm, 2009.
0.05 [6] J. Bravo, T. Alamo, and E.F. Camacho. Bounded
error identification of systems with time-varying pa-
residue
rameters. IEEE Transactions on Automatic Control,
0 51:1144 Ű 1150., 2006.
[7] D. Efimov, L. Fridman, T. Raissi, A. Zolghadri, and
−0.05 R. Seydou. Interval estimation for lpv systems apply-
ing high order sliding mode techniques. Automatica,
48:2365–2371, 2012.
−0.1
0 0.5 1
Time(s)
1.5 2 2.5
4
[8] D. Efimov, T. Raissi, and A. Zolghadri. Control of
x 10
nonlinear and lpv systems: interval observer-based
framework. IEEE Transactions on Automatic Control.,
Figure 11: Residual signal 2013.
[9] J. Van Willem and M Verhagen. Subspace identifica-
tion of bilinear and lpv systems for open-and closed-
loop data. Automatica, 45:371–381, 2009.
1
[10] J. Blesa, V. Puig, J Romera, and J Saludes. Fault di-
agnosis of wind turbines using a set-membership ap-
0.95 proach. In the 18th IFAC world congress, Milano,
Italy, 2011.
0.9 [11] H. Tanaka, Y Ohta, and Y Okimura. A local approach
fault indicator
to lpv-identification of a twin rotor mimo system. In in
0.85 proceedings of the 47th IEEE Conference on Decision
and Control Cancun, Mexico, 2008.
0.8
[12] R. Toth, F. Felici, P. Heuberger, and P Van den Hof.
Discrete time lpv i/o and state-space representations,
differences of behavior and pitfalls of interpolation.
0.75
In in proceedings of the European Control Conference
0 0.5 1 1.5 2 2.5 (ECC), Kos, Greece, 2007.
Time(s) x 10
4
[13] J. Van Willem and M Verhagen. Subspace identifica-
tion of mimo lpv systems: the pbsid approach. In in
Figure 12: Fault indicator
Proceedings of the 47th IEEE Conference on Decision
and Control Cancun, Mexico, 2008.
267
Proceedings of the 26th International Workshop on Principles of Diagnosis
[14] P. Gebraad, J. Van Wingerden, G. Van der Veen, and
M Verhaegen. Lpv subspace identification using a
novel nuclear norm regularization method. In Ameri-
can Control Conference on O’Farrell Street, San Fran-
cisco, CA, USA, 2011.
[15] V. Verdult and M Verhaegen. Kernel methods for sub-
space identification of multivariable lpv and bilinear
systems. Automatica, 41:1557–1565, 2005.
[16] J. Blesa, V. Puig, and J Saludes. Identification for pas-
sive robust fault detection using zonotope based set
membership appraches. International journal of adap-
tive control and signal processing, 25:788–812, 2011.
[17] P. Puig, V. Quevedo, T. Escobet, F. Nejjari, and
S De las Heras. Passive robust fault detection of dy-
namic processes using interval models. IEEE Transac-
tions on Control Systems Technology, 16:1083 –1089,
2008.
[18] B. Boussaid, C. Aubrun, and M.N Abdelkrim. Set-
point reconfiguration approach for the ftc of wind tur-
bines. In the 18th World Congress of the International
Federation of Automatic Control (IFAC), Milano, Italy,
2011.
[19] B. Boussaid, C. Aubrun, and M.N Abdelkrim. Two-
level active fault tolerant control approach. In The
Eighth International Multi-Conference on Systems,
Signals Devices (SSD’11),Sousse, Tunisia, 2011.
[20] B. Boussaid, C. Aubrun, and M.N Abdelkrim. Active
fault tolerant approach for wind turbines. In The In-
ternational Conference on Communications, Comput-
ing and Control Applications (CCCA’11), Hammamet,
Tunisia, 2011.
[21] P. Odgaard, J. Stoustrup, and M Kinnaert. Fault toler-
ant control of wind turbines-a benchmark model. IEEE
Transactions on control systems Technology, 21:1168–
1182, 2013.
[22] P. Odgaard and J Stoustrup. Results of a wind tur-
bine fdi competition. In 8th IFAC symposium on
fault detection ,supervision and safety of technical pro-
cesses,Mexico, 2012.
[23] C. Sloth, T. Esbensen, and J Stoustrup. Robust and
fault tolerant linear parameter varying control of wind
turbines. Mechatronics, 21:645–659, 2011.
[24] H. Chouiref, B. Boussaid, M.N Abdelkrim, V. Puig,
and C Aubrun. Lpv model-based fault detection:
Application to wind turbine benchmark. In Interna-
tional conference on electrical sciences and technolo-
gies (cistem’14), Tunis, 2014.
268