Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.
Transcript of Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.
Dr. Miguel Bagajewicz
Sanjay Kumar
DuyQuang Nguyen
Novel methods for Sensor Network Design
Minimize cost of instrumentation while satisfying the constraints on attributes like
• Accuracy• Precision• Reliability• Residual Accuracy
etc…
The Sensor Network Design Problem
Minimize Cost of instrumentation
such that accuracy of
S3= 7%S7= 8%
Similarly we can have constraints on residual accuracy, reliability,
precision etc..
The Sensor Network Design Problem
Tree Enumeration Procedure
• At each node calculate accuracy (and other attributes mandated by the constraints) compare with thresholds.
• If node is feasible, stop; explore sister nodes.
• If infeasible, go down.
How to find optimal solution?
• The Tree enumeration procedure can be made computationally effective by using cutsets instead of streams (Bagajewicz and Gala, 2006(a)).
• The efficiency is further more increased by decomposing the graph into subgraphs, (Bagajewicz and Gala, 2006(b))
• Gala M and M. Bagajewicz. (2006b). “Rigorous Methodology for the Design and Upgrade of Sensor Networks using Cutsets. Industrial and Engineering Chemistry Research”. Vol 45, No 21, pp. 6679-6686.
• Gala M and M. Bagajewicz. (2006b) “Efficient Procedure for the Design and Upgrade of Sensor Networks using Cutsets and Rigorous Decomposition”. Industrial and Engineering Chemistry Research, Vol 45, No 21, pp. 6687-6697.
Modified Tree Enumeration Procedure
•Accuracy has been conventionally defined as the sum of absolute value of the systematic error and the standard deviation of the meter (Miller, 1996).
•Since the above definition is of very less practical value, accuracy of a stream can defined as the sum of the precision and the maximum induced bias in the respective stream, Bagajewicz (2005).
Software Accuracy
-Software Accuracy -Precision -Maximum induced bias
• The maximum induced bias in a stream ‘i’ due a gross error in ‘s’ is given by, (using maximum power measurement test)
Where,
‘A’ is the incidence matrix and ‘S’ is the variance covariance matrix of measurements
Software Accuracyiiia ˆˆˆ
ia
i
i
ss
is
W
SWI
s
pcrit
psicrit
s
pi MaxZMax ])[()()(
,,)1,( ˆˆ
AASAAW TT 1)(
• In the presence of nT gross errors in positions given by a set T, the corresponding induced bias in variable ‘i’ is
• We have to explore all the possible combinations of locations of gross errors. Thus the problem can be stated using a binary vector as
Software accuracy in the presence of ‘nt’ gross errors
)(,
)(,
)()( )(]][[ˆ pscritis
Ts
picriti
pcrit
pi SWSWI
k
q
ZqW
W
ts
qSWqMaxT
kcritkT
pcritsTscrit
kk
ks
s
sTscritiss
iTicritp
i
0
.
)({)(ˆ
,,
)(,,,
,,,,,)(
•When there is more than one gross error, two gross errors may be equal in magnitude but opposite in sign which tend to cancel each other.
Gross Error Equivalency
S1
S2
S3
• Residual Accuracy of order ‘k’ is the software accuracy when ‘k’ gross errors have been found out and the measurements have been eliminated.
Residual Accuracy
Probability with which a variable ‘i’ can be estimated using its own measurement or through material balance equations in the time interval [0, t].
Estimation Reliability
Cutset is the set of edges (streams) when eliminated, separates the graph into two disjoint subgraphs. Deletion of a subset of the edges in cutset does not separate the graph into two subgraphs.
Streams 8, 6, 2 is a cutset. Streams 2, 3 is another cutset. There are several others.
Cutset
xm = [1, 2, 3]; xm is also a cutsetP{S1}= P{S2}= P{S3}= 0.9
• Probability of estimating S1= Probability of S1 working or Probability of S2, S3 working simultaneously.
Calculation of Estimation Reliability- Example
RS1= P{S1} υ [P{S2}∩P{S3}]
RS1= P{S1} υ [P{S2}×P{S3}]
RS1= P{S1}+ [P{S2}∩P{S3}]- [P{S1}×P{S2}×P{S3}]
RS1= 0.9+0.81-0.9×0.81
When xm = [2, 3]; S1 becomes non redundant and so it can be estimated
only by its material balance relations. Thus, RS1= P{S1} .P{S2}= 0.81
S2 S5
1
2
3S1
S3S4
4
S6
• If the variable is measured, then its estimation is directly the service reliability of the sensor measuring it.
• If the variable is not measured,
Estimation Reliability for Non Redundant Variable
smssv RRRR ........... 21
• Generate all the cutsets that has the variable of interest ‘i’.
• Removing the variable ‘i’ from those yields the reduced cutsets.
Estimation Reliability for Redundant Variable
S2
S5
1
2
3S1
S3
S44
S6
xm = [1, 2, 3]; Since the variable of interest is S1, the reduced cutset would be [2,3]. Let this be denoted by Zj(i), where ‘i’ is the variable of interest- here it is S1.
xm = [1, 2, 3, 4, 5]; [1, 2, 3], [1, 4, 5] are two cutsets.
[2, 3] and [4,5] are reduced cutsets.
P{S1}= P{S2}= P{S3}= P{S4}= P{S5}= 0.9
• Probability of estimating S1= Probability of S1 working or Probability of S2, S3 working simultaneously or P { S4 and S5} working simultaneously
Calculation of Estimation Reliability- Example
RS1= P{S1} υ [P{S2}∩P{S3}] υ [P{S4}∩P{S5}]
RS1= P{S1} υ [P{S2}×P{S3}] υ [P{S4}×P{S5}]
RS1= [P{S1}+ [P{S2}∩P{S3}]- [P{S1}×P{S2}×P{S3}] ] υ [P{S4}×P{S5}]
RS1= 0.981+0.81- 0.981×0.81
When xm = [1, 3]; S1 becomes non redundant and so it can be estimated only
by its direct measurement. Thus, RS1= P{S1} = 0.9
Z1(1)- reduced cutset
Z2(1)
S2 S5
1
2
3S1
S3S4
4
S6
ENV
• For a measured variable,
•For a unmeasured variable,
Estimation Reliability for Redundant Variable
)}().........()({)( 21 iZiZiZSPtR nkivi
)}(1){()()(
)}().........()({)( 21
tRtRtRtR
iZiZiZPtRsi
ui
si
vi
nkui
Estimation Reliability for Redundant Variable
smssss
nksink
ljs
ink
jl
ink
sj
ink
s
js
ink
sj
ink
ss
ink
s
ui
nkui
RRRRiZP
Also
iZiZP
iZiZiZP
iZiZPiZPtR
iZiZiZPtR
......)}({
,
)}(.......)({)1(
)}.....()()({
)}()({)}({)(
)}().........()({)(
321
1)(
)()()(
1
)()(
1
)(
1
21
Computation of estimation reliability of unmeasured variable- Sum of disjoint products
)}().........()({)( 21 iZiZiZPtR nkui
It can be proved that,
}1}....{1}.{1{)}({
)}({)}.({)}...({)}.({
)()(.....)()()(
,
)}({)}....({)}({)}().........()({
21
121
121*
**2121
sjm
sj
sjj
jj
jjj
nknk
RRRiZP
iZPiZPiZPiZP
iZiZiZiZiZ
where
iZPiZPiZPiZiZiZP
Input Data:1. Binary vector of measured streams at each node.2. Service reliability of sensors.3. Variables of interest.
Steps to be performed:4. Generate all the cutsets that has the variable of interest.
5. Choose only those reduced cutsets that have measured streams for reliability calculation. Other cutsets are useless as they do not make the variable of interest observable.
6. If no such cutset for unmeasured variable exist, then node is infeasible.
Implementation in the Program
Check if the variables of interest are non redundant. If so we got three cases.
• Case 1:The variable is measured, then estimation reliability is the sensor service reliability itself.
•Case 2:The variable is not measured, the estimation reliability is product of service reliabilities of sensors in the reduced cutset.
•Case 3:The variable of interest is not observable, then node is infeasible, go down the tree.
Implementation in the Program Non redundant variable
smssv RRRR ........... 21
Case 1: variable is measured too.
Case 2: unmeasured variable.
We have already discussed the computational method for above equations.
Implementation in the ProgramRedundant variable
)}().........()({)( 21 iZiZiZSPtR nkivi
)}().........()({)( 21 iZiZiZPtR nkui
Compare the obtained reliability with the specifications/ requirements/ thresholds,
• If node is feasible, transfer control to appropriate statement, which explores sister nodes
• If infeasible, go down the tree.
Implementation in the ProgramComparison with threshold
• Let there be ‘n’ sensors when calculating reliability. Assume one of the sensors has malfuntioned and the measurement eliminated, the estimation reliability we now have is “Residual Reliability of order one”
• The sensor that has a gross error or the malfunctioned sensor can be identified. This helps to know which measurement is eliminated.
Residual Reliability
Input Data:1. Binary vector of measured streams at a node. Say
‘ns’ streams are measured.
2. Sensor Service Reliability
3. Reduced Cutset Information
Calculation of Residual Reliability
Steps Involved:1. Choose reduced cutsets from already available
information. Eliminate those who have streams with the malfunctioning sensor.
2. Calculate Reliability the same way.
Calculation of Residual Reliability- Order One.
Example
1 2 3 4 5
6 7 10 11 8 9
5
11
16
21
22 23 24 12 14
13
19 2017
15
6 7 9 4
18 103
1
2
8
Madron and Veverka (1992)
Instrumentation Details- Madron and Veverka (1992)
Stream Flow Sensor cost Sensor Precision
(%)
Stream Flow Sensor Cost Sensor
Precision
1 140 19 2.5 13 10 12 2.5
2 20 17 2.5 14 10 12 2.5
3 130 13 2.5 15 90 17 2.5
4 40 12 2.5 16 100 19 2.5
5 10 25 2.5 17 5 17 2.5
6 45 10 2.5 18 135 18 2.5
7 15 7 2.5 19 45 17 2.5
8 10 6 2.5 20 30 15 2.5
9 10 5 2.5 21 80 15 2.5
10 100 13 2.5 22 10 13 2.5
11 80 17 2.5 23 5 13 2.5
12 40 13 2.5 24 45 13 2.5
Software Accuracy when all streams are measured
Stream Software Accuracy
1 6.9201
2 27.8008
3 5.6573
4 15.1941
5 52.7085
6 15.1186
7 36.4044
8 52.6420
9 52.6686
10 8.1288
11 7.4660
12 14.2693
Software Accuracy when all streams are measured
13 51.7501
14 51.7501
15 6.6338
16 5.9791
17 101.7665
18 5.4671
19 12.7847
20 18.6534
21 7.4659
22 52.6091
23 101.7665
24 12.7847
Requested Software Accuracy
• The software accuracy requested were.
•Three gross errors were allowed and no feasible nodes were found.
• Computed Reliability values were 90% for all streams when two gross errors are allowed.
Stream Threshold Accuracy
10 10%
16 8%
18 10%
19 15%
24 19%
Solution of Madron and Veverka (1992)
Cost of the Node Streams MeasuredValues of Accuracy of requested streams in percentage
185 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 19, 20, 23, 24
S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85
202 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 19, 20, 23, 24
S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85
220 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 23, 24
S10= 7.63S16= 5.48S18=5.49S19=12.91S24=12.91
227 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 23, 24
(node- a)
S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85
227 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 19, 20, 23, 24
(node- b)
S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85
239 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 23, 24
S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85
257 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24
S10= 7.64S16= 5.48S18=5.49S19=12.91S24=12.91
275 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 23, 24
S10= 8.91S16= 5.48S18=7.00S19=13.20S24=13.20
293 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24
S10= 7.62S16= 5.48S18=5.49S19=12.20S24=12.20
Thank You