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OPTIMAL SENSOR PLACEMENT FOR VIBRATION-BASED STRUCTURAL
HEALTH MONITORING OBTAINED VIA VALUE OF INFORMATION
ANALYSIS AS PART OF A DIGITAL STRUCTURAL INTEGRITY
MANAGEMENT OF OFFSHORE STRUCTURES
L. Eichner, R. Schneider, P. Simon & M. Baeßler, Bundesanstalt für Materialforschung und -prüfung
(BAM), Germany
ABSTRACT
A digital structural integrity management of offshore structures enables an optimized planning of inspec-
tions and repairs with risk-based methods. In a risk-based approach, the inspection and repair strategy that
minimizes the expected lifetime costs consisting of the expected inspection, repair and failure costs is de-
termined. In addition to inspections, information on the structural condition can be continuously obtained
by monitoring the vibration response of the structural system. Changes in the vibration characteristics quan-
tified in terms of modal properties can be an indication of structural damage. In risk -based inspection and
repair planning, the effect of monitoring results is determined via Bayesian updating of the structural con-
dition and reliability. This information should be applied to inform decisions on inspections and may result
in a reduced inspection effort. The benefit of continuously monitoring the structural health can be quantified
in terms of the value of information, which corresponds to the difference between the expected lifetime
costs with and without monitoring. In this work, we demonstrate in a numerical example how an optimized
sensor placement for a vibration-based structural health monitoring system can be determined by maxi-
mizing the value of information.
1. INTRODUCTION
The optimization of inspection and maintenance ac-
tions for support structures in offshore wind farms is
a very important but complex task. Inspections are
performed to obtain information on the structural
condition and guide decisions on maintenance ac-
tions. Inspection and maintenance activities are,
however, associated with large costs [1] and it is
thus comprehensible that wind farm operators seek
to optimize them.
An essential key for optimizing inspection and
maintenance efforts for support structures in off-
shore wind farms is the implementation of system-
atic digital workflows. Within this context, the
DiMoWind-Inspect project develops concepts for a
digital structural integrity management for offshore
wind turbine structures with the aim of enabling tar-
geted predictive inspections and maintenance activ-
ities based on the digital management of inspection
and maintenance information [2]. This includes the
automated recording and management of the rele-
vant information related to the structural condition
and the integration of this information in risk-in-
formed inspection and maintenance planning proce-
dures.
In addition to inspections, information on the struc-
tural condition of offshore structures can be ob-
tained by means of continuous structural health
monitoring (SHM). Before the installation of an
SHM system, the question arises whether it is bene-
ficial. Monitoring a structural system subject to
slowly progressing deterioration processes such as
fatigue is beneficial if the SHM information can be
used to reduce the inspection and maintenance effort
while still ensuring that the requirements regarding
the structural reliability are fulfilled throughout the
structure’s lifetime.
The benefit of an SHM system can be quantified us-
ing value of information (VoI) analysis [3, 4, 5, 6,
7]. In this contribution, we adopt these approaches
to optimize the sensor setup of a vibration-based
SHM system before it is installed.
2. INSPECTION AND MAINTENANCE
PLANNING
2.1 BAYESIAN SYSTEM IDENTIFICATION
BASED ON INSPECTION RESULTS
Inspection and maintenance plans for deteriorating
structural systems can be derived using the heuristic
approach proposed in [8, 9]. In this approach, a
deteriorating structural system is described by a
physics-based model of the deterioration processes
and structural behavior as indicated in Figure 1. To
quantify the uncertainties in the structural model
(model and parameter uncertainties), it is combined
with a probabilistic model of its model param-
eters . The (prior) probabilistic structural model is
derived based on design information, expert
knowledge, and engineering experience. Based on
the probabilistic structural model, the time-depen-
dent reliability and the lifetime risk associated with
structural failure can be estimated [10].
As indicated in Figure 1, the probabilistic structural
model is updated with inspection results using
methods of Bayesian system identification [11].
This process can be thought of as identifying the
probabilistic model which best explains the obtained
inspection results.
Mathematically, Bayesian system identification is
expressed as
(1)
where is the updated (posterior) probabil-
ity density function of the model parameters con-
ditional on the available inspection results and
is the likelihood function, which quantifies
the plausibility of a realization of the model param-
eters in function of the obtained inspection results
. As an example, the likelihood function describing
inspection results detection/no detection of damages
is formulated based on the relevant probability of
detection (PoD) curves (see, for example, [12]).
Once the updated probabilistic model of the model
parameters is available, an updated esti-
mate of the lifetime reliability and risk can be deter-
mined [10].
2.2 OPTIMAL LIFETIME INSPECTION
AND MAINTENANCE PLANNING
Let denote the expected total lifetime
costs of an inspection and maintenance (or repair)
strategy . As indicated in Figure 2, the optimal
inspection and repair strategy is determined by min-
imizing :
(2)
Strategy
optimally balances the costs for con-
trolling deterioration with the achieved risk reduc-
tion.
The expected total lifetime costs of a strategy (for
simplicity’s sake, it is denoted by in the following)
are calculated as
(3)
where are the expected total life-
time costs conditional on all inspection results and
Figure 1: Bayesian system identification of a probabilistic structural model describing a deteriorating structural system based
on inspection and monitoring results obtained from the real structural system
the corresponding repairs, is the probability
distribution of the probabilistic inspection results ,
and is the domain of possible inspection and
repair histories.
is composed of the (conditional)
expected lifetime costs of inspection campaigns
, the expected lifetime costs of in-
spections themselves , the expected
lifetime costs of repairs , and the ex-
pected lifetime costs of structural failure (risk)
:
The conditional expected lifetime costs of inspec-
tion campaigns are calculated by
where is the cost of an inspection campaign,
is the number of inspection campaigns
launched in year , is the discount rate, and
is the probability of failure up to
year conditional on the inspection outcomes
and corresponding repairs. Similarly, the expected
lifetime costs of inspections and repairs are esti-
mated [9].
The conditional expected lifetime risk is estimated
as
(6)
wherein is the failure cost factor and
is the conditional annual system
failure probability in year .
The integral in Equation (3) is approximated by ap-
plying a Monte Carlo approach [8, 9]:
(7)
2.3 HEURISTIC INSPECTION AND
MAINTENANCE STRATEGIES
An inspection and maintenance strategy is a set
of policies adopted in each year of the structure’s
lifetime , i.e., [8, 9]. Each pol-
icy is a set of rules which prescribe the actions that
should be taken conditional on the available inspec-
tion results and corresponding repairs.
In the following, parameterized heuristics are intro-
duced to (a) reduce the solution space of all possible
(4)
(5)
Figure 2: Optimal decision minimizing the expected total lifetime costs (adapted from [13])
strategies and (b) express the inspection and repair
strategies in terms of rules which can be intuitively
understood by all stakeholders of the planning pro-
cess [8, 9]. The parameters of the heuristics are de-
noted by and a heuristic inspection and repair
strategy is denoted by . The optimal strategy
in Equation (2) can thus be approximated by
where is determined as:
(8)
3. INTEGRATION OF VIBRATION
MONITORING RESULTS
3.1 BAYESIAN SYSTEM IDENTIFICATION
BASED ON MONITORING RESULTS
In addition to inspection results, information from
vibration-based SHM can be included in the system
identification (see Figure 1). The measured vibra-
tion data can be processed using operational modal
analysis (OMA) to continuously identify the modal
properties of the structure in terms of mode shapes
and modal eigenvalues
, which can be trans-
formed into eigenfrequencies via .
Changes in these quantities may indicate structural
damages.
Based on the identified modal properties
and
and the model-predicted modal properties and
, a likelihood function can be formulated
which links the observed modal properties
and
with the model parameters (see, for example,
[14]). Subsequently, the monitoring results can be
included in the system identification in the same
way as the inspection results (see Section 2.1).
3.2 MONITORING-INFORMED LIFETIME
INSPECTION AND MAINTENANCE
PLANNING
In Section 2.3, the concept of heuristic inspection
and maintenance strategies is introduced. Continu-
ous vibration-based SHM is integrated into the in-
spection and maintenance planning by defining ad-
ditional parameterized heuristics, which (a) define
the monitoring setup (i.e., the number and position
of the sensors) and (b) guide decisions regarding in-
spections based on the available inspection and
monitoring results. The additional heuristic
parameters are included in the parameter vector .
SHM is associated with costs. The additional condi-
tional expected lifetime costs for monitoring
are a function of the initial
costs of the monitoring system , the annual mon-
itoring cost factor , the discount rate , and the
conditional time-variant system failure probability
(for simplicity’s sake,
is denoted by in the following):
(9)
4. VALUE OF INFORMATION
ANALYSIS
By continuously monitoring the vibration response
of a deteriorating structural system, it may be possi-
ble to reduce the inspection effort, which in turn
could lead to a reduction in the expected operating
costs. In the following, the expected total lifetime
costs (composed of expected campaign, inspection
and repair costs) associated with a monitoring-in-
formed inspection and repair strategy are de-
noted by . These costs are computed
in accordance with Section 2 taking into account the
additional information provided by the monitoring
system.
The difference between the expected total lifetime
costs associated with the optimal inspection and re-
pair strategy without SHM
and the ex-
pected total lifetime costs of the monitoring-in-
formed inspection and repair strategy
corresponds to the value of SHM [7, 15]:
(10)
It is determined without considering the expected
lifetime costs for installing and operating the moni-
toring system . By including
, the net value of SHM is obtained
[15]:
(11)
The monitoring-informed inspection and repair
strategy is beneficial if the is pos-
itive. The optimal monitoring-informed inspection
and repair strategy
and thus the optimal sen-
sor setup maximizes the :
(12)
This distinguishes this optimization approach from
most optimal sensor placement (OSP) methods,
where the optimality criterion often is information-
based, e.g., in [16, 17].
5. NUMERICAL EXAMPLE
5.1 STRUCTURAL MODELING
In the following numerical example, the framework
for VoI-based optimization of monitoring-infor-
mation inspection and maintenance strategies is ap-
plied to optimize the sensor setup of a vibration-
based SHM system for the frame shown in Figure 3.
This frame has been studied in numerous case stud-
ies before (see, for example, [9, 18, 19]).
The two-dimensional steel frame consists of welded
tubular steel members and has properties similar to
a jacket support structure of an offshore wind tur-
bine. The welded connections associated with the 13
braces are subject to fatigue. The frame contains a
total of 22 critical fatigue hotspots (see Figure 3).
Fatigue deterioration of each hotspot is modeled by
a probabilistic crack growth model based on Paris'
Law [20]. The parameters of the crack growth mod-
els are calibrated based on the available design in-
formation (i.e., design fatigue lives as defined in
Figure 3, S-N curves, partial safety factors) (see, for
example, [21]).
The frame has a lifetime of 25 years and is subject
to a time-varying lateral load, which is modeled by
its annual maximum . The annual reliability
index of the undamaged frame is . A more
detailed description of the deteriorating frame and
the applied inspection and repair models can be
found in [19, 22].
In this example, two structural models of the frame
are applied. First, a non-linear finite element (FE)
beam model is applied to evaluate the maximum ca-
pacity of the damaged frame subject to the horizon-
tal load. This model in combination with the system
fatigue deterioration model forms the basis for
Figure 3: Zayas frame with its properties, braces, and hotspots, based on [23]
estimating the time-variant failure probability
and the annual failure rate of the
frame conditional on a given inspection, monitoring
and repair history (see [19, 22] for more details).
Second, a linear-elastic dynamic FE beam model is
applied to determine the model-predicted modal ei-
genvalues and mode shapes of the frame
using numerical modal analysis. This model is in-
cluded in the likelihood function describing the
monitoring results (see Section 3.1).
5.2 GENERATING SYNTHETIC
INSPECTION, MONITORING AND
REPAIR OUTCOMES
The optimization of the sensor setup is performed at
the beginning of the frame’s lifetime before the sys-
tem is installed. Therefore, the inspection, monitor-
ing and repair histories must be generated syntheti-
cally based on the prior probabilistic model of the
frame. In particular, the first step in simulating mon-
itoring results at a given time is the stochastic sim-
ulation of the deterioration state based on the prior
probabilistic system fatigue model taking into ac-
count the inspection, monitoring and repair out-
comes available up to that time (see also [24]). In the
next step, the linear-elastic FE beam model, which
reflects the simulated deterioration state, is subject
to a horizontal white noise excitation at the supports.
During this time-domain analysis, the horizontal ac-
celerations at the sensor positions are recorded and
subsequently perturbed with a gaussian white noise
to account for measurement errors. The recorded
signals are input to an OMA using the covariance-
driven stochastic subspace identification algorithm
(SSI-COV) [25, 26]. This analysis results in the
(synthetically) identified modal properties
and
of the possibly damaged frame at time .
5.3 EXAMINED INSPECTION AND
MAINTENANCE STRATEGIES
The parameterized heuristics defining an inspection
and repair strategy are as follows [18]:
• The structure is inspected at fixed intervals
.
• In each inspection campaign, hotspots are
inspected.
• The hotpots are prioritized for inspection
according to a prioritization index which is a
function of parameter (see [18] for more
details).
• If the predicted annual system failure rate ex-
ceeds a threshold , an additional inspec-
tion campaign is performed.
• If the depth of a detected fatigue crack ex-
ceeds a threshold , the corresponding
hotspot is repaired.
The heuristic inspection and repair strategy is
thus defined in terms of the following parameters:
.
(13)
In this example, the heuristics defining a monitor-
ing-informed inspection and repair strategy are as
follows:
• The frame is continuously monitored by a vi-
bration-based SHM system. The number and
position of the horizontal accelerometers
(i.e., the sensor setup) are identified by a pa-
rameter .
• The vibration-data are processed once a
year. This approach is reasonable for highly
reliable structural systems subject to slowly
progressing deterioration processes.
• If the predicted annual system failure rate
exceeds a threshold , an inspection cam-
paign is launched.
• In each inspection campaign, all hotspots are
inspected.
• If the depth of a detected fatigue crack ex-
ceeds a threshold , the hotspot is repaired.
The monitoring-informed inspection and repair
strategy is thus defined by the following
heuristic parameters
.
(14)
As a baseline for the investigations into possible im-
provements of the inspection and repair effort by
means of a vibration-based SHM system, the opti-
mal strategy without monitoring is determined first
as described in Section 2. The optimization consid-
ers the following parameter ranges:
, , ,
and .
In the following, the optimal inspection and repair
strategy is denoted as . It is characterized by the
following optimal parameter vector
(15)
As shown in Figure 4, in addition to the inspection
and repair strategy without monitoring, three differ-
ent monitoring-informed inspection and repair strat-
egies are considered in this study. The heuristic pa-
rameters defining these strategies build on the pa-
rameters defining the optimal inspection and repair
strategy without monitoring . The monitoring-in-
formed inspection and repair strategies , and
differ from each other only in the sensor setup.
Strategy has a full setup with sensors in all pos-
sible positions (except at the supports). It is defined
by the following parameter vector:
12346789
(16)
For strategy , acceleration sensors are placed
along the left leg of the frame. It is defined by the
following parameters:
1234
(17)
In strategy , the sensors are installed alternatingly
along both legs as shown in Figure 4. The parame-
ters defining this strategy are:
1
(18)
5.4 NORMALIZED COST MODEL
This study’s analysis of the expected lifetime costs
is based on a normalized cost model introduced in
Sections 2.2 and 3.2. The discount rate is set to
2%. The cost factors are summarized in Table 1,
where is the unit cost for launching an inspection
campaign, is the unit cost for inspecting a compo-
nent, is the unit cost of repairing a component,
is the unit system failure cost, and is the annual
monitoring cost factor.
Table 1: Cost factors
1.00
0.10
0.30
1000
0.050
0.025
0.050
0.100
The calculation of the initial monitoring costs are
determined as follows (see also [27]):
(19)
wherein is the number of sensors above water,
is the number of sensors in the splash zone, and
is the number of sensors under water. The cor-
responding initial monitoring cost factors are ,
and (see Table 1). In this example, sen-
sors at positions 4 and 9 are located above water,
sensors at positions 3 and 8 are in the splash-zone,
and sensors at positions 1, , and 7 are located un-
der water (see Figure 4).
Figure 4: Setups of horizontal accelerometers considered as part of (monitoring-informed)
inspection and repair strategies , , and
5.5 RESULTS
The results of the analysis are summarized in Figure
5 and Table 2. Figure 5 shows that the
for strategies , and is positive.
This means that all three monitoring-informed in-
spection and repair strategies should be preferred
over the inspection-only strategy . Specifically,
the greatest reductions are obtained in the expected
lifetime costs for launching the inspection campaign
and performing the individual hotspot inspections.
Table 2 shows that – when including the information
of the vibration-based SHM system – the expected
lifetime campaign costs reduce to approxi-
mately 10% and the expected lifetime inspection
costs reduce to approximately 30% of the
corresponding costs associated with strategy
(without monitoring). This is mainly because in-
spection campaigns are performed on an ad-hoc ba-
sis when considering monitoring information in the
inspection planning.
The results also show that continuous monitoring
has a significant effect on the expected lifetime fail-
ure costs (expected lifetime risk). The results for all
three monitoring-informed strategies show that the
expected lifetime risk is approximately halved when
compared to strategy . This can be intuitively un-
derstood: On the one hand, SHM which continu-
ously indicates that the system is undamaged (i.e.,
no changes in the modal properties) results in lower
estimates of the failure probability since this infor-
mation confirms that deterioration progresses
slowly (see also [22]).
Table 2: Expected lifetime costs of (monitoring-informed)
inspection and repair strategies , , and
3.19
2.93
2.05
0.18
8.36
—
—
1.58
0.29
0.64
0.17
2.69
1.50
4.17
1.67
0.30
0.65
0.19
2.81
1.22
4.33
1.62
0.30
0.65
0.16
2.73
1.22
4.41
Figure 5: Expected lifetime costs of (monitoring-informed)
inspection and repair strategies , , and
On the other hand, continuous monitoring enables
launching an inspection and repair campaign in a
timely fashion as soon as it indicates damages (i.e.,
changes in the modal properties).
All the cost reductions surpass the additional costs
for monitoring . This leads to expected total
lifetime costs that are roughly divided by
three with monitoring-informed strategies in com-
parison to the inspection-only strategy .
When comparing strategies , and , it can be
seen that there are only minor differences in the ex-
pected lifetime campaign, inspection, repair and
failure costs. Albeit has the lowest expected total
lifetime costs , it is not the optimal strat-
egy. This is due to the expected lifetime monitoring
costs , which are higher than for strategies
with only four sensors ( and ). Since has the
highest , it is the optimal strategy.
6. CONCLUDING REMARKS
In this contribution, a heuristic approach to optimiz-
ing inspection and maintenance strategies for dete-
riorating structural system proposed in [8, 9] was
extended to enable the integration of vibration-
based SHM results in the inspection and mainte-
nance planning for deteriorating structural systems.
This in turn enabled the quantification of the value
of (vibration-based) SHM. In a numerical example
considering a steel frame subject to fatigue, the
framework was applied to optimize the sensor setup
of a vibration-based SHM system.
In the example, only a limited number of sensor set-
ups were considered. In the future, a suitable opti-
mization algorithm should be identified to enable an
efficient search within a larger set of possible sensor
setups. In addition – to allow transferring these ideas
into practice – they should be applied to more real-
istic (complex) structural systems such as support
structures of offshore wind turbines incorporating
variable environmental and operating conditions.
Further research should also focus on adapting the
described concepts to enable monitoring-based life-
time extensions of support structures in offshore
wind farms. Such lifetime extensions require a sys-
tematic digital management of all relevant data
obtained and information collected during the struc-
tures’ lifetime including design, fabrication and in-
stallation information as well as information on in-
spection, monitoring and maintenance actions. Con-
cepts for digitally managing such data as the basis
for a digital structural integrity management and
lifetime extensions are currently developed as part
of the DiMoWind-Inspect project [2].
ACKNOWLEDGEMENTS
This work was supported by the German Federal
Ministry for Economic Affairs and Climate Action
(BMWK) through grant 03EE3039A and Projektträ-
ger Jülich (PtJ).
REFERENCES
1. C. Dao, B. Kazemtabrizi and C. Crabtree, Wind
turbine reliability data review and impacts on
levelised cost of energy. Wind Energy, 2019. 22:
p. 1848-1871.
2. L. Eichner, P. Gerards, R. Herrmann, R.
Schneider, F. Hille and M. Baeßler. A
framework for data and structural integrity
management for support structures in offshore
wind farms based on building information
modelling (accepted for publishing). in 8th
International Symposium on Reliability and
Risk Management (ISRERM 2022). 2022.
3. D. Straub, Value of information analysis with
structural reliability methods. Structural Safety,
2014. 49: p. 75-85.
4. D. Zonta, B. Glisic and S. Adriaenssens, Value
of information: impact of monitoring on
decision-making. Structural Control and Health
Monitoring, 2014. 21(7): p. 1043-1056.
5. M. Pozzi and A. Der Kiureghian. Assessing the
value of information for long-term structural
health monitoring. in SPIE - The International
Society for Optical Engineering. 2011.
6. S. Thöns, On the Value of Monitoring
Information for the Structural Integrity and Risk
Management. Computer-Aided Civil and
Infrastructure Engineering, 2018. 33: p. 79–94.
7. A. Kamariotis, E. Chatzi and D. Straub,
[Preprint] A framework for quantifying the
value of vibration-based structural health
monitoring. 2022. Available from:
https://arxiv.org/abs/2202.01859.
8. J. Luque and D. Straub, Risk-based optimal
inspection strategies for structural systems
using dynamic Bayesian networks. Structural
Safety, 2019. 76: p. 68-80.
9. E. Bismut and D. Straub, Optimal adaptive
inspection and maintenance planning for
deteriorating structural systems. Reliability
Engineering & System Safety, 2021. 215.
10. D. Straub, R. Schneider, E. Bismut and H.-J.
Kim, Reliability analysis of deteriorating
structural systems. Structural Safety, 2020. 82.
11. D. Straub and I. Papaioannou, Bayesian
Updating with Structural Reliability Methods.
Journal of Engineering Mechanics, 2014.
141(3).
12. E. Bismut and D. Straub, A unifying review of
NDE models towards optimal decision support.
Structural Safety, 2022. 97.
13. D. Straub, Engineering Risk Assessment, in Risk
– A Multidisciplinary Introduction, C.
Klüppelberg, Straub, D. and Welpe, I.M.,
Editors. 2014, Springer International
Publishing.
14. M. W. Vanik, J. L. Beck and S. K. Au, Bayesian
Probabilistic Approach to Structural Health
Monitoring. Journal of Engineering Mechanics,
2000. 126(7): p. 738-745.
15. D. Straub, E. Chatzi, E. Bismut, W. Courage, M.
Döhler, M. H. Faber, J. Köhler, G. Lombaert, P.
Omenzetter, M. Pozzi, S. Thöns, H. Wenzel and
D. Zonta, Value of information: A roadmap to
quantifying the benefit of structural health
monitoring, in ICOSSAR - 12th International
Conference on Structural Safety & Reliability.
2017.
16. A. Mendler, M. Döhler and C. E. Ventura,
Sensor placement with optimal damage
detectability for statistical damage detection.
Mechanical Systems and Signal Processing,
2022. 170.
17. C. Papadimitriou, Optimal sensor placement
methodology for parametric identification of
structural systems. Journal of Sound and
Vibration, 2004. 278(4-5): p. 923-947.
18. E. Bismut, J. Luque and D. Straub, Optimal
prioritization of inspections in structural
systems considering component interactions
and interdependence, in 12th International
Conference on Structural Safety & Reliability
(ICOSSAR). 2017.
19. R. Schneider. Effect of repair models on risk-
based optimal inspection strategies for support
structures of offshore wind turbines. in
Proceedings of the 5th Conference on Smart
Monitoring, Assessment und Rehabilitation of
Civil Structures (SMAR 2019). 2019.
20. P. C. Paris and F. Erdogan, A critical analysis of
crack propagation laws. Journal of Basic
Engineering, 1963. 85(4): p. 528-533.
21. D. Straub, Generic Approaches to Risk Based
Inspection Planning for Steel Structures. 2004:
Swiss Federal Institute of Technology (ETH
Zürich).
22. R. Schneider, Time-variant reliability of
deteriorating structural systems conditional on
inspection and monitoring data. 2020:
Bundesanstalt für Materialforschung und -
prüfung (BAM).
23. R. Schneider, S. Thöns and D. Straub,
Reliability analysis and updating of
deteriorating systems with subset simulation.
Structural Safety, 2017. 64: p. 20-36.
24. A. Kamariotis, E. Chatzi and D. Straub, Value of
information from vibration-based structural
health monitoring extracted via Bayesian model
updating. Mechanical Systems and Signal
Processing, 2022. 166.
25. B. Peeters and G. de Roeck, Stochastic System
Identification for Operational Modal Analysis:
A Review. Journal of Dynamic Systems,
Measurement and Control, 2001. 123(4): p. 659-
667.
26. P. van Overschee and B. L. R. de Moor,
Subspace Identification for Linear Systems.
1996: Kluwer Academic Publishers Group.
27. A. Mehrjoo, M. Song, B. Moaveni, C.
Papadimitriou and E. Hines, Optimal sensor
placement for parameter estimation and virtual
sensing of strains on an offshore wind turbine
considering sensor installation cost.
Mechanical Systems and Signal Processing,
2022. 169.
