ACS6124 Multisensor and Decision Systems Part I: Multisensor Systems
Lecture 2: Static Decision Systems
George Konstantopoulos g.konstantopoulos@sheffield.ac.uk
Automatic Control and Systems Engineering The University of Sheffield
(lecture notes produced by Dr. Inaki Esnaola and Prof. Visakan Kadirkamanathan)
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Sensor Faults
The value of sensor data relies on the integrity of the measurement. This can be compromised by faults emerging in sensors.
Satelite sensing position values can suffer from offsets. Robot position sensors can suffer drifts.
Hydraulic valve position sensing can suffer intermittent stuck position faults.
Satelite Positioning Robot Position Drift
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Fault Signal Patterns
The fault signal patterns can be classified into three common types.
1
2
3
important for its detection and compensation. Abrupt fault pattern is the most commonly assumed.
Abrupt Fault: The fault signals take constant values of zero and some fault level before and after the fault on-set time.
Incipient Fault: The fault signal changes gradually from zero and changes slowly and continuously from the fault on-set time.
Intermittent Fault: The fault signal takes non-zero values over very short time durations, but is repeated regularly.
A priori information about the fault signal pattern or type to be detected is
where tf is the fault onset time
ACS6124 Multisensor and Decision Systems
G. Konstantopoulos
Fault Signal Patterns
Incipient fault pattern is difficult to detect and is linked to system prognosis.
Intermittent fault pattern is linked to anomalous event history.
An intermittent fault can be mapped to the abrupt fault by
∆θ dt or ∆θ t ∑t
where tf is the fault onset time
An incipient fault can be mapped to the
abrupt fault by d∆θt dt
In practice, no explicit integration or differentiation is performed on the signal, but is carried out implicitly.
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Measurement Model – Change in Mean
Measurement Model: Representation of relationship between measurements and system parameters or states.
Measurement Model
The sensor measurement is a random signal and is a direct measurement of a state / parameter with additive noise.
yt = xt + vt
yt is the sensor measurement at time t, xt is the variable being sensed and vt is the noise signal at time t.
Dynamic characteristics ignored and model is consistent with steady state assumption.
Additive Fault – Change in Mean
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Problem Fomulation – Change in Mean Detection
Consider a random variable x(t), a measurand of the sensor
θ0 ifsensoris“normal” xt = θ1 if sensor is “faulty”
Level of fault ∆ = θ1 −θ0 Noisy measurements yt
To determine if the sensor is normal or faulty based on the observation yt , we can formulate hypothesis test such that:
H0: xt = θ0 i.e. the sensor is normal
H1: xt = θ1 i.e. the sensor is faulty
The change detection problem is to identify which hypothesis is true.
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
MAP Decision Rule
Maximum a posteriori (MAP) Decision Rule
Hypothesis H0 is favoured over H1, if
P(H0 |yt)>P(H1 |yt)
The MAP decision rule for favouring H0 is equivalent to:
P (H0 | yt ) > 1, Let Prior Probabilities Ratio τ = P (H0 )
P(H1 |yt) From Bayes Law,
P(H1)
p(yt |H0) p(yt |H1)
P(H0 |yt) = P(H1 |yt)
p(y|H )P(H0) t 0 p(yt) =
p(yt|H1) P(H1) p(yt )
p(yt |H0) p(yt |H1)
P(H0) = P(H1)
τ
Likelihood Ratio Test (Neyman – Pearson Test)
Hypothesis H0 is favoured over H1, if the likelihood ratio,
S(yt)= p(yt |H0) > 1 p(yt |H1) τ
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Model, Noise and Likelihood
Measurement model:
yt = xt + vt
Assumption: Independent zero mean noise E[vt ] = 0. Further vt is Gaussian
distributed with variance σ2. p(vt)=√
The probability distributions p(vt ) and p (yt |xt ) are:
Wherevt∼N(0,σ2)and p(yt |x)∼N(x,σ2)
1 1 [vt ]2 exp{− 2 }
2πσ 2σ
The likelihood if H0 is true:
p(yt |H0) = p(yt |xt = θ0)
∼ N(θ0,σ02)
1 1[yt−θ0]2 =√ exp− 2
2πσ0 2 σ0
The likelihood if H1 is true:
p(yt |H1) = p(yt |xt = θ1) ∼ N(θ1,σ02)
=√
exp −
2 2 σ0
1 2πσ0
1[yt−θ1]2
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Reducing MAP Decision Rule
Likelihood ratio (S(yt )) is: p(yt |H0)
1[yt−θ0]2 exp −2 2
σ0 exp −1 [yt−θ1]
2 σ 02
S(yt)=exp −2σ02([yt −θ0]2−[yt −θ1]2) =exp −2σ02(2yt −θ0−θ1)(θ1−θ0) Substituting ∆ = θ1 −θ0, and hence θ1 +θ0 = 2θ0 +∆,
S(yt)= p(y |H ) = t 1
√ √ 1
exp −2
2 σ0
1 2πσ0
1[yt−θ0]2
2 = exp −1 [yt−θ1]
2
∆ ∆ S(yt)=exp − 2 yt−θ0−
2 σ 02
2 π σ 0 11
Decide in favour of H0 if:
σ0 2 ∆ ∆1
exp − 2 yt−θ0− > σ0 2τ
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Reducing MAP Decision Rule (contd)
By taking the natural log of exp − ∆ y −θ − ∆ > 1 and simplifying,
−
σ02 t 0 2 τ
∆ ∆
2 yt −θ0− >−ln(τ)
σ0 2 ∆2
∆yt−θ0−2 <σ0ln(τ) ∆2
|∆|sign∆yt−θ0−2 <σ0ln(τ) where ∆ = sign∆|∆|. Through further simplification,
|∆|2 |∆|sign∆[yt −θ0]− 2 < σ02 ln(τ)
|∆|2 |∆|sign∆[yt −θ0] < 2 +σ02 ln(τ)
sign∆ σ
< 2σ 00
yt − θ0
|∆|
σ0
+ |∆| ln(τ)
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Change in Mean Detection Rule
By Defining the following variables and the fault signal to noise ratio ∆ ̃ = ∆ , σ0
̃
∆ 1
δ= 2 + ̃ln(τ)
∆
if sign∆ ̃ ·st < δ
if sign∆ ̃ ·st ≥ δ
The rule can equivalently be given in the measurement space (assuming ∆ > 0):
st=
The MAP decision rule is thus:
,
y t − θ 0
where
Decide in favour of H0: if yt < δ ̃ Decide in favour of H1: if yt ≥ δ ̃
̃ θ0+θ1 σ02
δ = 2 + ∆ ln(τ)
σ
0
Decide in favour of H0: Decide in favour of H1:
The decision rule can be seen as partitioning or classifying the measurement space into a region of normal and a region of fault.
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Decision Errors
Type I Error (light grey area) – Probability of False Alarm Pf Type II Error (dark grey area) – Probability of Miss Detection Pm
Let the cumulative distribution function of the normalised Gaussian pdf (N(0,1)) be, ζ 1 v2
φ(ζ)= √ exp − dν with Q(ζ)=1−φ(ζ)=φ(−ζ) −∞2π 2
Then the errors are:
Pf =P(st ≥δ |H0)=Q(δ), Pm =P(st <δ |H1)=Q(∆ ̃−δ)
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Multiplicative Faults
Sensor or system faults can also manifest as non-additive changes, but this is rare. An example of non-additive change is the multiplicative fault type.
Machine tool wear increases vibration in manufacturing processes. Seismic signals increase in their amplitude during earthquakes.
Machining Process and Vibration Signal Earthquake Seismic Signal
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Model 1: Measurement Model
Fault Type 2: Multiplicative Fault (Simplified)
Model:-yt =xt +(1+∆)vt,
where xt = θ0 is a constant and ∆ is the level of fault and vt ∼ N(0,σ02) is the noise.
Representing wt = (1 + ∆)vt , its mean and variance are given by,
E[wt]=(1+∆)E[vt]=(1+∆).0=0 V[wt]=(1+∆)2V[vt]=(1+∆)2σ02
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Change in Variance Detection
To determine if the system is normal or faulty based on the observation yt , we can formulate a hypothesis test such that:
H0: vt ∼ N(0,σ02) i.e. the sensor is normal
H1: vt ∼ N(0,σ12) i.e. the sensor is faulty
As with the Change in Mean Detection, the MAP decision rule is:
P(H0 |yt) p(yt |H0) P(H0) P(H1|yt)= p(yt|H1) P(H1)>1
S(yt )
The hypotheses H0 and H1 are equivalent to vt ∼ N(0,σ2)
H0: σ = σ0
H1: σ = σ1
The likelihood ratio is given by,
S(yt ) =
1 1[yt−θ0]2
√2πσ exp −2 σ2 00
2
√1 exp −1[yt−θ0] 2 π σ 1 2 σ 12
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Simplifying MAP Decision Rule
The likelihood ratio is simplified as,
σ 1[y −θ ]2 1[y −θ ]2 S(yt)=1exp− t 0 + t 0
σ0 2σ02 2σ12 σ1 11
= 1exp − [yt−θ0]2 − σ0 2 σ02
σ12
σ 1y−θ2σ2−σ2
=1exp−t0 10 σ0 2σ0 σ12
With τ = P(H0 ) and st = yt −θ0 , the decision rule becomes P(H1 ) σ0
σ1 σ2−σ21 1exp−[st]2 1 0 >
σ0 2 σ12 τ
1 σ2−σ21σ
exp − [st]2 1 0 > 0 2 σ12 τσ1
By taking the natural log, we obtain
1 σ2−σ2 σ
−[st]2 1 0 >ln 0 2 σ 12 τ σ 1
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Change in Variance Detection Rule
By further simplifying the decision rule,
1 σ2−σ2 σ
[st]2 1 0
[st]2<2 1 ln τ 1
σ 12 − σ 02 With the previous definition of st , we define δ ′ as:
σ 0
y−θ σ2 σ 1+2∆+∆2
st=t 0, δ′=2 1 lnτ1=2
2 ∆ + ∆ 2
ln(τ(1+∆))
σ 0 σ 12 − σ 02 The decision rule finally becomes:
DecideinfavourofH0: DecideinfavourofH1:
We have a two-sided test:
σ 0
if [st]2 <δ′ if [st]2 ≥δ′
√√ DecideH:− δ′