AC-515 Model based Fault-Diagnosis for Linear Systems

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AC-515 Model based Fault-Diagnosis for Linear Systems

Ce travail a bénéficié d'une aide de l'Etat français au titre du programme d'Investissements d'Avenir, IRT Nanoelec, portant la réference ANR-10-AIRT-05

1

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Damien Koenig

❑ Research affiliation:

✓ GIPSA-lab, Grenoble Images Speech Signal and Control, is a joint research unit of CNRS and University of

Grenoble.

❑ Teaching affiliation:

✓ Grenoble-Inp ESISAR school of Embedded Systems

My main concerns in automatic control are devoted to state estimation, fault diagnosis and fault tolerant control of complex systems.

Ce travail a bénéficié d'une aide de l'Etat français au titre du programme d'Investissements d'Avenir, IRT Nanoelec, portant la réference ANR-10-AIRT-05 2

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❑ ^Functions

✓ Fault Diagnosis

▪ Determination of the kind, size, location and time of detection of a fault.

Follows fault detection. Include Fault detection and identification.

✓ Supervision

▪ Monitoring a physical and taking appropriate actions to maintain the operation in the case of fault.

Nomenclature

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❑ ^Functions

✓ Diagnosis

▪ Determination of the kind, location and time of detection of a fault. Follows fault detection.

Nomenclature

r ¹ r ²

r

^N

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Example of various sensor faults on system measurements

The figure depicts the effect of the different faults on system

measurements.

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❑ ^Models

✓ Quantitative model

▪ Use of static and dynamic relations among system variables and parameters in order to describe a system’s behavior in quantitative mathematical terms.

✓ Qualitative model

▪ Use of static and dynamic relations among system variables in order to describe a system’s behavior in qualitative terms such as causalities and IF- THEN rules.

Nomenclature

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Application Examples

❑ Real Processes

❑ FDI applications

DC motor Three tank system Vehicle system

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Usual Fault Diagnosis Decomposition

❑ The fault system diagnosis can be decomposed into three parts including

✓

sensors,

✓

actuators,

✓

and system dynamics.

❑ A reliable fault diagnosis system should be able to distinguish faults from system disturbances and measurement noise. More precisely, the fault diagnosis system must be robust to these uncertainties while remaining sensitive to faults.

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Fault diagnosis of three tank system

❑ Modelling of faults

▪ Three types of faults are considered :

✓ component faults : leaks in the three tanks, which can be modelled as additional mass flows out of tanks,

where  A1 ,  A2 and  A3 are unknown and depend on the size of the leaks

▪ component faults: plugging between two tanks

▪ sensor faults: three additive faults in the three sensors, denoted by f1, f2 and f3

▪ actuator faults: faults in pumps, denoted by f4 and f5

leak

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leak leak

Fault Model diagnosis of three tank

❑ They are modelled as follows

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Method of Residual Generation

❑ The most relevant analytical model-based residual generation methods developed have been divided into three categories:

✓ Parity space approach

✓ Observer-based approach

✓ Parameter estimation approach

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Residual Generator Structure

❑ Residual generation via system simulator

❑ Simulator is usually an Observer, KF or UIO

𝑟 𝑡 = 𝑦 𝑡 − ො 𝑦(𝑡)

ො

𝑦(𝑡) is the estimated plant output

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Residual Generator Structure

❑ Residual generator:

❑ Constraint conditions: design

𝑟 𝑧 = 𝐻 _𝑢∗ 𝑧 𝑢 ^∗ (z)+𝐻 _𝑦 𝑧 𝑦 (z) r(t)=0 if and only if f(t) = 0

y 𝑧 = 𝐺 _𝑦𝑓 𝑧 𝑓(z)+𝐺 _𝑦𝑢 ^∗ 𝑧 𝑢 ^∗ (z)

𝐻 _𝑢∗ 𝑧 + 𝐻 _𝑦 𝑧 𝐺 _𝑦𝑢 ^∗ 𝑧 =0

𝐻 _𝑦 𝑧 𝐺 _𝑦𝑓 𝑧 ≠0

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Change Detection (Example 2)

❑ Residual Evaluation

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❑ Observer-based approaches

❑ Parity (vector) relations

❑ Fault detection via parameter estimation

Residual Generation Techniques

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❑ Residual General Structure

✓ Plant model without fault

▪ ቊ ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢 𝑦 = 𝐶𝑥

✓ Observer model

▪ ቄ ሶො 𝑥 = 𝐴 ො 𝑥 + Bu + L(y-C 𝑥) ො

✓ Dynamic of the state estimation 𝑒 _𝑥 = 𝑥 − ො 𝑥

▪ 𝑒 _𝑥 ሶ =(A-LC) 𝑒 _𝑥

✓ State estimation error property

▪ lim

𝑡 → 𝑒 _𝑥 = 0 𝑓𝑜𝑟 𝑓𝑎𝑢𝑙𝑡 𝑓𝑟𝑒𝑒

✓ Residual

▪ r(t) = W(y(t)- 𝑦(𝑡)) = 𝑊𝐶𝑒 ො _𝑥 (𝑡)

Observer-based approaches

B u

ሶො𝑥 

𝐶

y

ො 𝑥

A

ො 𝑦 ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢

𝑦 = 𝐶𝑥

W r(t) L

+

+ -

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❑ Residual General Structure

✓

Plant model with fault

▪ ቊ ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢+𝐹 _𝑥 f 𝑦 = 𝐶𝑥 + 𝐹 _𝑠 f

✓

Observer model

▪ ቊ ሶො𝑥 = 𝐴 ො𝑥+ Bu+L(y−C 𝑥) ො

ො

𝑦 = 𝐶 ො 𝑥

✓

Dynamic of the state estimation 𝑒 _𝑥 = 𝑥 − ො 𝑥

▪ 𝑒 _𝑥 ሶ =(A-LC) 𝑒 _𝑥 + 𝐹 _𝑥 f −L𝐹 _𝑠 f

✓

Output state estimation error

▪ 𝑦(t)=y(t)- ෤ 𝑦(𝑡)=C𝑒 ො _𝑥 𝑡 + 𝐹 _𝑠 f(t)

✓

State estimation error property

▪ lim

𝑡 → 𝑒 _𝑥 ≠ 0 𝑤ℎ𝑒𝑛 𝑎 𝑓𝑎𝑢𝑙𝑡 𝑜𝑐𝑐𝑢𝑟

✓

Residual

▪ r(t) = W(y(t)- 𝑦(𝑡)) = 𝑊𝐶𝑒 ො _𝑥 (𝑡)+W𝐹 _𝑠 f(t)

▪ lim

𝑡 → 𝑟 𝑡 = −𝑊𝐶 𝐴 − 𝐿𝐶 ⁻¹ (𝐹 _𝑥 − 𝐿𝐹 _𝑠 )+W𝐹 _𝑠 𝑓(𝑡) ≠ 0 𝑤ℎ𝑒𝑛 𝑎 𝑓𝑎𝑢𝑙𝑡 𝑜𝑐𝑐𝑢𝑟

Observer-based approaches

B u

ሶො𝑥 

𝐶

y

ො 𝑥

A

ො 𝑦 ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢 ^∗

𝑦 ^∗ = 𝐶𝑥

W r(t) L

+

+ - 𝐹 _𝑥

f

𝐹 _𝑦

𝑢 ^∗ 𝑦 ^∗

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❑ In order to describe the BFDF, the simple fault model is used

▪ ቊ ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢+𝑏 _𝑖 𝑓 _𝑎𝑖 (𝑡) 𝑦 = 𝐶𝑥 + 𝐼 _𝑗 𝑓 _𝑠𝑗 (𝑡)

✓

Where

▪ 𝑏 _𝑖 𝑓 _𝑎𝑖 (𝑡) (i=1,2, …, r) denotes that a fault occurred in the 𝑖 _𝑡ℎ 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟, 𝑏 _𝑖 ∈ 𝑅 ^𝑛 is the 𝑖 _𝑡ℎ column of the input matrix B and 𝑓 _𝑎𝑖 (𝑡) is an unknown scalar time-varying function which represents the evolution of the fault.

▪ 𝐼 _𝑗 𝑓 _𝑠𝑗 (𝑡) (j=1,2, …, m) denotes that a fault occurs in the 𝑗 _𝑡ℎ 𝑠𝑒𝑛𝑠𝑜𝑟, 𝐼 _𝑗 ∈ 𝑅 ^𝑚 is a unit vector corresponding to a fault occurs in the 𝑗 _𝑡ℎ 𝑠𝑒𝑛𝑠𝑜𝑟.

✓

BFDF

▪ ቄ ሶො 𝑥 = 𝐴 ො 𝑥 + Bu + L(y-C 𝑥) ො

✓

State estimation error property

▪ 𝑒 _𝑥 ሶ =(A-LC) 𝑒 _𝑥 +𝑏 _𝑖 𝑓 _𝑎𝑖 (𝑡)−L𝐼 _𝑗 𝑓 _𝑠𝑗 (𝑡)

✓

Output state estimation error

▪ 𝑦(t)=C𝑒 ෤ _𝑥 𝑡 + 𝐼 _𝑗 𝑓 _𝑠𝑗 (𝑡)

✓

Residual

▪ lim

𝑡 → 𝑟 𝑡 ≠ 0 𝑤ℎ𝑒𝑛 𝑓𝑎𝑢𝑙𝑡 𝑠𝑒𝑛𝑠𝑜𝑟 𝑜𝑟 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟 𝑜𝑐𝑐𝑢𝑟

Basic principles of fault detection filters (BFDF)

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❑ The sensor fault model is described by

▪ ቊ ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢 𝑦 = 𝐶𝑥 + 𝑓 _𝑠 (𝑡)

✓

Where

▪ 𝑓 _𝑠 𝑡 denotes the faults sensor vector

❑ We build a bank of Observers, where each observer i is piloted by sensor 𝑦 _𝑖

✓

ቄ ሶො 𝑥 = 𝐴 ො 𝑥 + Bu + 𝐿 _𝑖 (𝑦 _𝑖 -𝐶 _𝑖 𝑥) ො

✓

State estimation error property

▪ 𝑒 _𝑥 ሶ =(A-𝐿 _𝑖 𝐶 _𝑖 ) 𝑒 _𝑥 −𝐿 _𝑖 𝑓 _𝑠𝑖 (𝑡)

✓

Output state estimation error

▪ 𝑟 _𝑖 (t)=𝑦 _𝑖 - 𝑦 ෝ _𝑖 = 𝐶 _𝑖 𝑒 _𝑥 𝑡 + 𝑓 _𝑠𝑖 (𝑡)

✓

Residual

▪ lim

𝑡 → 𝑟 _𝑖 𝑡 ≠ 0 𝑤ℎ𝑒𝑛 𝑓 _𝑠𝑖 (𝑡) hold true

▪ 𝑟 _𝑖 𝑡 

Case 1 : Simultaneous faults sensor

ቊ ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢

𝑦 _𝑖 = 𝐶 _𝑖 𝑥 + 𝑓 _𝑠𝑖 (𝑡)

𝑖 ^𝑡ℎ sensor fault model

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Case 1 : Multiple sensors faults isolation scheme

u input

Actuators 𝑢 _𝑅 Plant 𝑦 _𝑅 Sensors

y

Identity Obs 1 or KF 1

⋮

Signal Ungrouping 𝑦 ₁

output

FDI Logic 𝑟 ₁ (𝑡)

𝑦 _𝑗

Identity Obs j or KF j

Identity Obs m or KF m

𝑦 _𝑚 𝑟 _𝑗 (𝑡)

𝑟 _𝑚 (𝑡)

❑ Dedicated Observer Scheme (DOS)

The number of these observers (estimators) is equal to the number m of system outputs, and each device is driven by a single output and all the inputs of the system. In this case a fault on the 𝑖 _𝑡ℎ output affects only the residual function of the output observer or filter driven by the 𝑖 _𝑡ℎ output.

𝑓 _𝑠

⋮

⋮ 𝑎 ₁

𝑎 _𝑗

𝑎 _𝑚 - +

- +

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❑ A simple threshold logic can be used to make decision about the appearance of a specific fault by the logic decision according to:

✓ {‖ 𝑟 _𝑖 (t)‖≥  _𝑖 implies 𝑓 _𝑠𝑖 ≠0

❑ This isolable residual structure is very simple and all faults can be detected simultaneously, however it is difficult to design in practice.

Case 1 : A simple threshold logic for DOS

𝒓 _𝟏 𝒓 _𝟐 … 𝒓 _𝒎

𝑓 _𝑠1 1 0 … 0

𝑓 _𝑠2 0 1 0

⋮ ⋱ 0

𝑓 _𝑠𝑚 0 0 1

Table 2 : DOS

In the tables above, a "1" in 𝑖 _𝑡ℎ row and 𝑖 _𝑡ℎ column denotes that the residual 𝑟 _𝑖 is sensitive to the fault 𝑓 _𝑠𝑖 , whilst a "0" denotes insensitivity.

𝑎 ₁ =1 if 𝑓 _𝑠1 hold true

𝑎 _𝑚 =1 if 𝑓 _𝑠𝑚 hold true

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Case 2 : No simultaneous sensors faults

u input

Actuators 𝑢 _𝑅 Plant 𝑦 _𝑅 Sensors

y

Obs 1

Obs j

Obs m

⋮

Signal Grouping 𝑦 ¹

output

FDI Logic 𝑟 ¹ (𝑡)

𝑦 ^𝑗

𝑦 ^𝑚 +

+

𝑟 ^𝑗 (𝑡)

𝑟 ^𝑚 (𝑡) 𝐶 ¹ -

𝑧 ¹

𝐶 ^𝑗 - 𝑧 ^𝑗

𝐶 ^𝑚 - 𝑧 ^𝑚

⋮

Where 𝑟 ^𝑖 (𝑡) ∈  ^𝑚−1 i=1, 2 …m

𝑓 _𝑠

⋮

⋮ 𝑎 ₁

𝑎 _𝑗

𝑎 _𝑚

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❑ Analysis of the fault sensor isolability

Case 2 : Sensor Generalized Observer Scheme

𝒓 ^𝟏 𝒓 ^𝟐 … 𝒓 ^𝒋 … 𝒓 ^𝒎

𝑓 _𝑠1 0 1 1 … 1

𝑓 _𝑠2 1 0 1 … 1

⋮ ⋱ ⋮

𝑓 _𝑠𝑗 1 1 0 1

⋮ ⋱

𝑓 _𝑠𝑚 1 1 1 0

Table 1: Generalized Observer Scheme GOS

In the table above, a "1" in 𝑖 _𝑡ℎ row and 𝑖 _𝑡ℎ column denotes that the

residual 𝑟 ^𝑖 is sensitive to the fault 𝑓 _𝑠𝑖 ,whilst a "0" denotes insensitivity.

❑ Fault alarm

▪ 𝑎 ₁ = 𝐿( ҧ𝑟 ¹ )L( 𝑟 ² )L( 𝑟 ³ ) L( 𝑟 ^𝑚 ) where L( 𝑟 ^𝑖 )=1 if 𝑟 ^𝑖 ≥  ^𝑖 (L=binary logic function)

𝑎 ₁ =1 if 𝑓 _𝑠1 hold true

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❑ The main task of robust FDI is to generate a residual signal which is robust to the system uncertainty.

❑ According to the previous discussion, a system with possible sensor and actuator faults can be described as :

▪ ቊ ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢+𝐸𝑑 𝑡 + 𝐵𝑓

_𝑎

(𝑡) 𝑦 = 𝐶𝑥 + 𝑓

_𝑠

(𝑡)

✓

Where

▪ 𝑓

_𝑎

(𝑡) 

^𝑟

is the actuator faults

▪ 𝑓

_𝑠

(𝑡) 

^𝑚

is the sensor faults

▪ 𝑑

^𝑞

is the unknown input (or disturbance)

❑ To generate a robust residual an UIO is designed :

▪ ቊ ሶ𝑧(𝑡) = 𝐹𝑧(𝑡) + 𝑇𝐵𝑢(t)+𝐺𝑦(𝑡)

ො

𝑥(𝑡) = 𝑧(𝑡) + 𝑁𝑦(𝑡)

✓

Where 𝑥 ො ^𝑛 is the estimated state vector and z ^𝑛 is the state of the full order observer, and F, T G, N are matrices to be designed for achieving unknown input de- coupling and other design requirements.

✓

Residual

▪ r(t)=y(t)-C 𝑥(𝑡) ො

Robust FDI schemes based on UIOs

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❑ When the UIO-based residual is applied to the previous system, the residual and state estimation errors become:

✓

𝑒

_𝑥

(𝑡) = 𝑥(𝑡) − ො 𝑥 𝑡 =(I-NC)x- 𝑧 𝑡 − 𝑁𝑓

_𝑠

𝑡 = Tx(t) − z(t) − N 𝑓

_𝑠

𝑡

✓

r(t)=y(t)-C 𝑥 𝑡 = 𝐶𝑒 ො

_𝑥

+ 𝑓

_𝑠

(𝑡)

✓

Where T=I-NC

❑ Dynamic of the state estimation error

✓

ሶ𝑒

_𝑥

= T ሶ𝑥 t − ሶ𝑧 t − N ሶ 𝑓

_𝑠

𝑡 = 𝑇𝐴 − 𝐺𝐶 𝑥+T𝐸𝑑 𝑡 + 𝑇𝐵𝑓

_𝑎

𝑡 − 𝐹𝑧 𝑡 − 𝐺𝑓

_𝑠

𝑡 − N ሶ 𝑓

_𝑠

𝑡

✓

ሶ𝑒

_𝑥

= 𝑇𝐴 − 𝐺𝐶 𝑥+T𝐸𝑑 𝑡 + 𝑇𝐵𝑓

_𝑎

𝑡 − 𝐹(Tx(t) − N 𝑓

_𝑠

𝑡 − 𝑒

_𝑥

) − 𝐺𝑓

_𝑠

𝑡 − N ሶ 𝑓

_𝑠

𝑡

✓

ሶ𝑒

_𝑥

= F 𝑒

_𝑥

+ 𝑇𝐴 − 𝐺𝐶 − 𝐹𝑇 𝑥+T𝐸𝑑 𝑡 + 𝑇𝐵𝑓

_𝑎

𝑡 + (𝐹N −G)𝑓

_𝑠

𝑡 − N ሶ 𝑓

_𝑠

𝑡

❑ If one can make the following relations hold true

✓

𝑇𝐴 − 𝐺𝐶 − 𝐹𝑇=0

✓

T=I-NC

✓

K= -(𝐹N−G)=G−FN

✓

TE=0

❑ The state estimation error will then be:

✓

ሶ𝑒

_𝑥

= F 𝑒

_𝑥

+ 𝑇𝐵𝑓

_𝑎

𝑡 − 𝐾𝑓

_𝑠

𝑡 − N ሶ 𝑓

_𝑠

𝑡

✓

If all eigenvalues of F are stable, then

▪ lim

𝑡

→ 𝑟 𝑡 < Threshold for fault free case

▪ lim

𝑡

→ 𝑟 𝑡 ≥ Threshold for faulty case

UIO observer design

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✓

1) Check the UI decoupled condition

▪ Rank(CE)=Rank(E)=dim(d)

✓

2) Compute

▪ TE=0  (I-NC)E=0  E=NCE N=E(CE)

⁺

=E (𝐶𝐸)(𝐶𝐸)

^{𝑇 −1}

(𝐶𝐸)

^𝑇

▪ Deduce N and T=I-NC

✓

3) Solve the Syvester equation

▪ 𝑇𝐴 − 𝐺𝐶 − 𝐹𝑇=0 TA-GC-F(I-NC)=0F=TA-(G-FN)C F=TA-KC

▪ Check the observabilty (or detectability) of the pair(TA, C), if (TA, C) observable

(detectable), a UIO exists and K can be computed using pole placement or LQ method.

▪ Deduce K, F, G=K+FN

❑ The UIO-based residual is also reduced to:

▪ ሶ𝑒

_𝑥

= F 𝑒

_𝑥

+ 𝑇𝐵𝑓

_𝑎

𝑡 − 𝐾𝑓

_𝑠

𝑡 − N ሶ 𝑓

_𝑠

𝑡

▪ r(t)=𝐶𝑒

_𝑥

+ 𝑓

_𝑠

(𝑡)

✓

It can be seen that the disturbance effects have been de-coupled from the residual.

✓

To detect actuator faults, one has to make:

▪ TB≠0

✓

More specifically, the fault in the 𝑖 _𝑡ℎ actuator will affect the residual iff:

▪ T𝑏

_𝑖

≠0

✓

Where 𝑏 _𝑖 is the 𝑖 _𝑡ℎ column of the matrix B.

UIO design procedure

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❑ The fault isolation problem is to located the fault, i.e., to determine in which sensor (or actuator) fault has occurred.

❑ The sensitivity and insensitivity properties makes isolation possible.

❑ The ideal situation is to make each residual only sensitive to a

particular fault and insensitive to all other faults. However, this ideal situation is normally difficult to achieve.

Robust fault isolation schemes based on UIOs

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❑ To design robust sensor fault isolation schemes , all actuators are assumed to be fault-free and the system can be described as:

▪ ሶ𝑥 𝑡 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡 + 𝐸𝑑(𝑡) 𝑦 ^𝑗 𝑡 = 𝐶 ^𝑗 𝑥 𝑡 + 𝑓 ^𝑗

𝑠 (𝑡) 𝑦 _𝑗 𝑡 = 𝑐 _𝑗 𝑥 𝑡 + 𝑓 _𝑠𝑗 (𝑡)

for j=1, 2, …, m

✓

Where

▪ 𝑐 _𝑗 ∈  ^1×𝑛 is the 𝑗 _𝑡ℎ 𝑟𝑜𝑤 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝐶

▪ 𝐶 ^𝑗 ∈  ^{(𝑚−1)×𝑛} is obtained from the matrix C by deleting 𝑗 _𝑡ℎ 𝑟𝑜𝑤𝑐 _𝑗

▪ 𝑦 _𝑗 𝑡 is the 𝑗 _𝑡ℎ component of y

▪ 𝑦 ^𝑗 (𝑡) ∈  ^𝑚−1 is obtained from the vector y by deleting 𝑗 _𝑡ℎ 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑦 _𝑗

❑ Based on the this description, m UIO-based residual generator can be constructed as:

▪ ൞

ሶ𝑧 ^𝑗 𝑡 = 𝐹 ^𝑗 𝑧 ^𝑗 𝑡 + 𝑇 ^𝑗 𝐵𝑢 𝑡 + 𝐺 ^𝑗 𝑦 ^𝑗 (𝑡)

ො

𝑥 𝑡 = 𝑧 ^𝑗 𝑡 + 𝑁 ^𝑗 𝑦 ^𝑗 (𝑡)

𝑟 ^𝑗 𝑡 =𝑦 ^𝑗 𝑡 −𝐶 ^𝑗 𝑥 𝑡 = (𝐼 − 𝐶 ො ^𝑗 𝑁 ^𝑗 )𝑦 ^𝑗 𝑡 − 𝐶 ^𝑗 𝑧 ^𝑗 𝑡

for j=1, 2, …, m

✓

Where 𝑟 ^𝑗 (𝑡) ∈  ^𝑚−1

Robust sensor fault isolation schemes

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❑ In order to obtain an UIO, the parameter matrices must satisfy the following equalities:

✓ 𝑁 ^𝑗 =E(𝐶 ^𝑗 E) ⁺ =E (𝐶 ^𝑗 𝐸)(𝐶 ^𝑗 𝐸) ^{𝑇 −1} (𝐶 ^𝑗 𝐸) ^𝑇

✓ 𝑇 ^𝑗 =I- 𝑁 ^𝑗 𝐶 ^𝑗

✓ 𝐹 ^𝑗 =𝑇 ^𝑗 A- 𝐾 ^𝑗 𝐶 ^𝑗 to be stabilized

✓ 𝐺 ^𝑗 = 𝐾 ^𝑗 +𝐹 ^𝑗 𝑁 ^𝑗

✓ for j=1, 2, …, m

❑ When all actuators are fault free and a fault occurs in the 𝑗 _𝑡ℎ sensor, the residual will satisfy the following isolation logic:

✓ ൝ 𝑟 ^𝑗 (𝑡) <  ^𝑗

𝑟 ^𝑘 (𝑡) ≥  ^𝑘 for k=1, …, j-1, j+1, …,m

✓ Where  ^𝑗 (j=1, 2…,m) are isolation threshold.

Robust sensor fault isolation schemes

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A robust sensor fault isolation scheme

u input

Actuators 𝑢 _𝑅 Plant 𝑦 _𝑅 Sensors

y

UIO 1

UIO j

UIO m

⋮

(𝐼 − 𝐶 ¹ 𝑁 ¹ ) Signal Grouping 𝑦 ¹

output

FDI Logic 𝑟 ¹ (𝑡)

(𝐼 − 𝐶 ^𝑗 𝑁 ^𝑗 )

(𝐼 − 𝐶 ^𝑚 𝑁 ^𝑚 ) 𝑦 ^𝑗

𝑦 ^𝑚 +

+

𝑟 ^𝑗 (𝑡)

𝑟 ^𝑚 (𝑡) 𝐶 ¹ -

𝑧 ¹

𝐶 ^𝑗 - 𝑧 ^𝑗

𝐶 ^𝑚 - 𝑧 ^𝑚

⋮

Where 𝑟 ^𝑖 (𝑡) ∈  ^𝑚−1 i=1, 2 …m

𝑓 _𝑠

⋮

⋮ 𝑎 ₁

𝑎 _𝑗

𝑎 _𝑚

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❑ Analysis of the fault sensor isolability

Sensor Generalized Observer Scheme

𝒓 ^𝟏 𝒓 ^𝟐 … 𝒓 ^𝒋 … 𝒓 ^𝒎

𝑓 _𝑠1 0 1 1 … 1

𝑓 _𝑠2 1 0 1 … 1

⋮ ⋱ ⋮

𝑓 _𝑠𝑗 1 1 0 1

⋮ ⋱

𝑓 _𝑠𝑚 1 1 1 0

Table 1: Generalized Observer Scheme GOS

In the table above, a "1" in 𝑖 _𝑡ℎ row and 𝑖 _𝑡ℎ column denotes that the

residual 𝑟 ^𝑖 is sensitive to the fault 𝑓 _𝑠𝑖 ,whilst a "0" denotes insensitivity.

❑ Fault alarm

▪ 𝑎 ₁ = 𝐿( ҧ𝑟 ¹ )L( 𝑟 ² )L( 𝑟 ³ ) L( 𝑟 ^𝑚 ) where L( 𝑟 ^𝑖 )=1 if 𝑟 ^𝑖 ≥  ^𝑖 (L=binary logic function)

𝑎 ₁ =1 if 𝑓 _𝑠1 hold true

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❑ Existence condition : In order to satisfy the constraints

✓ 𝑇 ^𝑗 E=0 𝑐 ₁ : decoupled condition

✓ 𝐹 ^𝑗 =𝑇 ^𝑗 A- 𝐾 ^𝑗 𝐶 ^𝑗 stable 𝑐 ₂ : detectability condition the following rank conditions must be verified for each observer :

✓ 𝑐 ₁ : rank 𝐶 ^𝑗 E

𝐸 =rank 𝐶 ^𝑗 E =dim(d)

✓ 𝑐 ₂ : rank 𝑝𝐼 _𝑛 − 𝑇 ^𝑗 𝐴

𝐶 ^𝑗 =n  ((𝑇 ^𝑗 𝐴))≥0

✓ Proof of 𝑐 ₁ : 𝑇 ^𝑗 E=0  (I- 𝑁 ^𝑗 𝐶 ^𝑗 )E=0  𝑁 ^𝑗 𝐶 ^𝑗 E=E

▪ The solution of 𝑁 ^𝑗 𝐶 ^𝑗 E=E depends on the rank of matrix 𝐶 ^𝑗 E.

▪ A solution exists iff : rank 𝐶 ^𝑗 E

𝐸 =rank 𝐶 ^𝑗 E =dim(d) [Rao]

▪ C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications. New York: Wiley, 1971.

Robust sensor fault isolation schemes

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❑ To design robust actuator fault isolation schemes, all sensors are assumed to be fault-free and the system can be described as:

▪ ሶ𝑥 𝑡 = 𝐴𝑥 𝑡 + 𝐵

^𝑖

𝑢

^𝑖

𝑡 + 𝐵

^𝑖

𝑓

^𝑖_𝑎

𝑡 + 𝑏

_𝑖

(𝑢

_𝑖

𝑡 + 𝑓

_𝑎𝑖

𝑡 ) + 𝐸𝑑(𝑡)

= 𝐴𝑥 𝑡 + 𝐵

^𝑖

𝑢

^𝑖

𝑡 + 𝐵

^𝑖

𝑓

^𝑖_𝑎

𝑡 + 𝐸

^𝑖

𝑑

^𝑖

(𝑡) 𝑦 𝑡 = 𝐶𝑥 𝑡

𝑓𝑜𝑟 𝑖 = 1, 2, … , 𝑟

✓

Where

▪ 𝑏

_𝑖

∈ 

^𝑛

is the 𝑖

_𝑡ℎ

𝑐𝑜𝑙𝑢𝑚𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝐵

▪ 𝐵

^𝑖

∈ 

^𝑛



^(𝑟−1)

is obtained from the matrix B by deleting 𝑖

_𝑡ℎ

𝑐𝑜𝑙𝑢𝑚𝑛 𝐵

_𝑖

▪ 𝑢

_𝑖

𝑡 is the 𝑖

_𝑡ℎ

component of u

▪ 𝑢

^𝑖

(𝑡) ∈ 

^𝑟−1

is obtained from the vector u by deleting 𝑖

_𝑡ℎ

𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑢

_𝑖

▪ 𝐸

^𝑖

=[𝐸 𝑏

_𝑖

]

▪ 𝑑

^𝑖

(𝑡)= 𝑑(𝑡) 𝑢

_𝑖

𝑡 + 𝑓

_𝑎𝑖

𝑡

❑ Based on the this description, r UIO-based residual generator can be constructed as:

▪ ൞

ሶ𝑧

^𝑖

𝑡 = 𝐹

^𝑖

𝑧

^𝑖

𝑡 + 𝑇

^𝑖

𝐵

^𝑖

𝑢

^𝑖

𝑡 + 𝐺

^𝑖

𝑦(𝑡)

ො

𝑥 𝑡 = 𝑧

^𝑖

𝑡 + 𝑁

^𝑖

𝑦(𝑡)

𝑟

^𝑖

𝑡 =𝑦 𝑡 −𝐶

^𝑖

𝑥 𝑡 = 𝐼 − 𝐶𝑁 ො

^𝑖

𝑦 𝑡 − 𝐶𝑧

^𝑖

𝑡

for i=1, 2, …, r

✓

Where 𝑟 ^𝑖 (𝑡) ∈  ^𝑚

Robust actuator fault isolation schemes

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❑ In order to obtain an UIO, the parameter matrices must satisfy the following equalities:

✓ 𝑁 ^𝑖 =𝐸 ^𝑖 (𝐶𝐸 ^𝑖 ) ⁺

✓ 𝑇 ^𝑖 =I- 𝑁 ^𝑖 𝐶

✓ 𝐹 ^𝑖 =𝑇 ^𝑖 A- 𝐾 ^𝑖 𝐶 to be stabilized

✓ 𝐺 ^𝑖 = 𝐾 ^𝑖 +𝐹 ^𝑖 𝑁 ^𝑖

✓ for i=1, 2, …, r

❑ When all sensors are fault free and a fault occurs in the 𝑖 _𝑡ℎ actuator, the residual will satisfy the following isolation logic:

✓ ൝ 𝑟 ^𝑖 (𝑡) <  ^𝑖

𝑟 ^𝑘 (𝑡) ≥  ^𝑘 for k=1, …, i-1, i+1, …,m

✓ Where  ^𝑖 (i=1, 2…,r) are isolation threshold.

Robust actuator fault isolation schemes

൝ ሶ𝑧 ^𝑖 𝑡 = 𝐹 ^𝑖 𝑧 ^𝑖 𝑡 + 𝑇 ^𝑖 𝐵 ^𝑖 𝑢 ^𝑖 𝑡 + 𝐺 ^𝑖 𝑦(𝑡) 𝑟 ^𝑖 𝑡 = 𝐼 − 𝐶𝑁 ^𝑖 𝑦 𝑡 − 𝐶𝑧 ^𝑖 𝑡

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A robust actuator fault isolation scheme

u input

Actuators 𝑢 _𝑅 Plant 𝑦 _𝑅 Sensors

y

UIO 1

UIO i

⋮

(𝐼 − 𝐶𝑁 ¹ ) 𝑦

output

FDI Logic 𝑟 ¹ (𝑡)

(𝐼 − 𝐶𝑁 ^𝑖 )

(𝐼 − 𝐶𝑁 ^𝑟 )

𝑦

𝑦 +

+

𝑟 ^𝑖 (𝑡)

𝑟 ^𝑟 (𝑡) 𝐶 -

𝑧 ¹

C - 𝑧 ^𝑖

𝐶 - 𝑧 ^𝑟

⋮

Where 𝑟 ^𝑖 (𝑡) ∈  ^𝑚 i=1, 2 …r Signal

Grouping

𝑢 ¹

𝑢 ^𝑖

𝑢 ^𝑟

UIO r

⋮

⋮ 𝑎 ₁

𝑎 _𝑖

𝑎 _𝑟

𝑓 _𝑢

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❑ Analysis of the fault actuator isolability

Actuator Generalized Observer Scheme

𝒓 ^𝟏 𝒓 ^𝟐 … 𝒓 ^𝒊 … 𝒓 ^𝒓

𝑓 _𝑢1 0 1 1 … 1

𝑓 _𝑢2 1 0 1 … 1

⋮ ⋱ ⋮

𝑓 _𝑢𝑖 1 1 0 1

⋮ ⋱

𝑓 _𝑢𝑟 1 1 1 0

Table 1: Generalized Observer Scheme GOS

In the tables above, a "1" in 𝑖 _𝑡ℎ row and 𝑖 _𝑡ℎ column denotes that the residual 𝑟 _𝑖 is sensitive to the fault 𝑓 _𝑢𝑖 , whilst a "0" denotes insensitivity.

𝑎 ₁ =1 if 𝑓 _𝑢1 hold true

❑ Fault alarm

▪ 𝑎 ₁ = 𝐿( ҧ𝑟 ¹ )L( 𝑟 ² )L( 𝑟 ³ ) L( 𝑟 ^𝑚 ) where L( 𝑟 ^𝑖 )=1 if 𝑟 ^𝑖 ≥  ^𝑖

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❑ Existence condition : In order to satisfy the constraints

✓ 𝑇 ^𝑖 𝐸 ^𝑖 =0 𝑐 ₁ : decoupled condition

✓ 𝐹 ^𝑖 =𝑇 ^𝑖 A- 𝐾 ^𝑖 𝐶 stable 𝑐 ₂ : detectability condition the following rank conditions must be verified for each UIO:

✓ 𝑐 ₁ : rank 𝐶𝐸 ^𝑖

𝐸 ^𝑖 =rank 𝐶𝐸 ^𝑖 =rank(𝐶 [𝐸 𝑏 _𝑖 ])=dim(d)+1

✓ 𝑐 ₂ : rank 𝑝𝐼 _𝑛 − 𝑇 ^𝑖 𝐴

𝐶 =n  ((𝑇 ^𝑖 𝐴))≥0

Robust actuator fault isolation schemes

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❑ ^Remarks

✓ The isolation schemes presented in this previous time can only isolate a single fault in either a sensor or an actuator, at the same time.

✓ This is based on the fact that the probability for two or more faults to occurs at the same time is very small in a real situation.

✓ If simultaneous faults need to be isolated, the fault isolation scheme should be modified based on a regrouping of faults. Each residual will be designed to be sensitive to one group of faults and insensitive to another group of faults. See the following Dedicated Observer Scheme (DOS).

A robust sensor fault isolation scheme

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❑ Observer-based approaches

❑ Parity (vector) relations

❑ Fault detection via parameter estimation

Residual Generation Techniques

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❑ Parity space approach

✓

For static system

▪ Y=CX

▪ without UI

✓

For static constraint system

▪ AX=0

▪ Y=CX

▪ without UI

✓

For static system without perfect UI decoupled

▪ 𝑦 _𝑘 = C𝑥 _𝑘 + _𝑘 +F𝑑 _𝑘  𝑦 _𝑘 = C𝑥 _𝑘 + _𝑘 +𝐹 ⁻ 𝑑 ⁻ _𝑘 +𝐹 ⁺ 𝑑 ⁺ _𝑘

▪ where 𝑦 _𝑘  ^𝑚 , 𝑥 _𝑘  ^𝑛 ,  _𝑘  ^𝑚 , 𝑑 _𝑘  ^𝑝 are respectively, the measurement vector, state vector, noise vector and fault vector.

▪ The fault vector is decomposed of two term, 𝑑 ⁻ _𝑘  ^𝑝−1 denotes the fault vector to be undetected and 𝑑 ⁺ _𝑘  ¹ the fault to be detected

▪ Assumption : m > n

✓

Dynamic system

▪ 𝑥 _𝑘+1 = A𝑥 _𝑘 + B𝑢 _𝑘 + 𝐸 ¹ 𝑑 _𝑘 + 𝑅 ¹ 𝑓 _𝑘

▪ 𝑦 _𝑘 = C𝑥 _𝑘 + D𝑢 _𝑘 + 𝐸 ² 𝑑 _𝑘 + 𝑅 ² 𝑓 _𝑘

▪ Where 𝑓 _𝑘  ^𝑔 denotes a fault vector which may contain actuator, component or sensor faults, 𝑑 _𝑘  ^𝑞 denotes the UI (or disturbance) vector.

Parity relation for fault detection and isolation

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❑ To begin with this problem, let us consider a general problem of the measurement of an n-dimensional vector using m sensors.

❑ The algebraic measurement equation is described as:

✓ 𝑦 _𝑘 =C𝑥 _𝑘 + 𝑓 _𝑘

✓ Where y ^𝑚 is the measurement vector , x ^𝑛 is the state vector, 𝑓 _𝑘 is the vector of sensor fault and ^C is an ^m ^ ⁿ measurement matrix.

❑ Two case can be considered

✓ m<n (without redundancy)

✓ m≥n (with redundancy)

Parity Relation for fault detection

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❑ Case 1 : m<n, 𝑦 _𝑘 = 𝐶𝑥 _𝑘 and rank(C) = m , the direct redundancy relation does not exist however, we may construct redundancy relations by collecting sensor outputs over a time interval (data window):

▪ Y(k-s:k)=

𝑦 _𝑘−𝑠 𝑦 _{𝑘−𝑠+1}

𝑦 ⋮ _𝑘

✓ This is known as "temporal redundancy" or "serial redundancy". An example is done at the end of this section.

Parity Relation for fault detection

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❑ Case 2 : m≥n and rank(C) = n , this implies that the rows of C are linearly dependent, i.e., the outputs of the sensors are related by a static relation.

❑ ^{Example 1:}

✓

Isolate a regular of matrix C, as:

✓

Since C1 is invertible, the system becomes

✓

Which is equivalent to

✓

The last relation gives 2 redundancy relation and a third relation can be obtained

✓

Hence, if a fault occur at sensor 𝑦 ₂ , we obtain and the fault can be detected.

Parity Relation for fault detection

X Y



 









 







=

0 2

1 1

0 1

2 1

 

 



= 

 

 



= 

0 2

1 1 0

1 2 1

2

1

, C

C



 



=

 =

 



=

−

1 1 1 2 2

1 1 1 2

2 1 1

Y C C Y

X Y C X C Y

X C Y

( )

 



=

−

=

−

 +

 =



 



− ⁻ 

0 2

0 5

. 0 5

. 0 0

4 2

3 2 1

2 1 1

1 2 y y

y y y

Y I Y C

C

0 5

. 0 5

.

0 y ₁ − y ₃ + y ₄ =

 

 



= +

−



−



− +

0 5

. 0 5

. 0

0 2

0 5

. 0 5

. 0

4 3

1 4 2

3 2 1

y y

y y y

y

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❑ Example 2 : 𝑦 _𝑘 =C𝑥 _𝑘 + 𝑓 _𝑘

❑ The dimension of 𝑦 _𝑘 is larger than the dimension of 𝑥 _𝑘 , i.e. m> n and rank(C) = n. For such system configurations, the number of measurements is greater than the number of variables x.

❑ For FDI purposes, the vector 𝑦 _𝑘 can be combined into a set of linearly independent parity equations to generate the parity vector (residual):

▪ 𝑟 _𝑘 =V𝑦 _𝑘 =V(C𝑥 _𝑘 + 𝑓 _𝑘 )

✓

In order to obtain a good characteristic of the residual 𝑟 _𝑘 (zero-valued for the fault-free case and insensitive of the unknown x), the matrix V must satisfy the condition:

▪ VC=0

✓

When this condition holds true, the residual (parity vector) only contains information on the faults

▪ 𝑟 _𝑘 = 𝑉𝑦 _𝑘 =𝑉𝑓 _𝑘  𝑟 _𝑘 = 𝑣 ₁ 𝑓 ₁ (𝑘)+ 𝑣 ₂ 𝑓 ₂ (k)+… 𝑣 _𝑚 𝑓 _𝑚 (𝑘)

❑ ^{where 𝑣} _𝑖 is the column of V, 𝑓 _𝑖 is the 𝑖 ^𝑡ℎ element of f(k) which denotes the fault in the 𝑖 ^𝑡ℎ sensor.

Parity Relation for fault detection

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❑ Example 3 : 𝑦 _𝑘 = C𝑥 _𝑘 + _𝑘 +F𝑑 _𝑘  𝑦 _𝑘 = C𝑥 _𝑘 + _𝑘 +𝐹 ⁻ 𝑑 ⁻ _𝑘 +𝐹 ⁺ 𝑑 ⁺ _𝑘

✓ Where the fault vector d is decomposed of two term, 𝑑 ⁻ _𝑘  ^𝑝−1 denotes the fault vector to be undetected and 𝑑 ⁺ _𝑘  ¹ the fault to be detected.

❑ Ideal solution (exact decoupling)

✓ Rewrite the above system as

▪ 𝑦 _𝑘 = 𝐶 𝐹 ⁻ 𝑥 _𝑘

𝑑 ⁻ _𝑘 +  _𝑘 +𝐹 ⁺ 𝑑 ⁺ _𝑘

✓ and find the parity vector P such that

▪ P= 𝑦 _𝑘 =  𝐹 ⁺ 𝑑 ⁺ _𝑘 +  _𝑘

▪  𝐶 𝐹 ⁻ =0

▪  𝐹 ⁺ 0

✓ Therefore P is sensitive to 𝑑 ⁺ and insensitive to x and 𝑑 ⁻ .

❑ Optimization projection (optimal decoupling)

▪ In practice the constraints  𝐶 𝐹 ⁻ =0 and  𝐹 ⁺ 0 are unfeasible.

▪ One solution is to find an optimal matrix  such that : min   𝐶 𝐹 ⁻ and max

  𝐹 ⁺

Parity Relation for fault Isolation

Numerical application

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❑ Multi-objective optimization

✓ One of methods to solve the multi-objective optimization problem is to optimize a new cost function J which accounts for both 𝐽 ₁ = min

 ԡ ԡ

 𝐶 𝐹 ⁻ and 𝐽 ₂ = max

  𝐹 ⁺ . A solution for minimizing J cannot minimize 𝐽 ₁ at the same time as maximizing 𝐽 ₂ .

✓ However, it could lead to a reasonable solution for robust residual design.

✓ A sensible mixture of performance indices is their ratio, i.e.

▪ 𝐽 = min



 𝐶 𝐹

⁻ ²

 ^𝐹

^{+ 2}

✓ Hence, the robust residual design is achievable by minimizing J.

Parity Relation for fault Isolation

(47)

❑ The problem min max

▪ min

  𝐶 𝐹 ⁻ and max

  𝐹 ⁺

Is transformed as the following optimization problem

▪ min



 ^𝐶 ^𝐹

⁻ ²

 𝐹

⁺ ²

or ቐ

C=0 min 

 ^𝐹

^{− 2}

 𝐹

⁺ ²

or ൝ C=0

min  (  𝐹 ^{− 2} − 𝑘 ²  𝐹 ^{+ 2} )

❑ Since C is a full column matrix we can decomposed the matrix C as

▪ C=0  (  ₁ ) ^𝑇 (  ₂ ) ^𝑇 𝐶 ₁ 𝐶 ₂ =0

▪ 𝑤ℎ𝑒𝑟𝑒 𝐶 ₁ 𝑖𝑠 𝑎 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑚𝑎𝑡𝑟𝑖𝑥

✓

After some manipulation we obtain

▪ ( ₁ ) ^𝑇 =-( ₂ ) ^𝑇 𝐶 ₂ (𝐶 ₁ ) ⁻¹ which implies C=0

▪ And  = ( ₁ ) ^𝑇 ( ₂ ) ^𝑇 = ( ₂ ) ^𝑇 −𝐶 ₂ (𝐶 ₁ ) ⁻¹ 𝐼

✓

The constrain optimization problem is reduce to a simple optimization problem

▪ ቐ

C=0 min 

 𝐹

^{− 2}

 𝐹

⁺ ²

 min



₂

(

₂

)

^𝑇

− ^𝐶

₂

^(𝐶

₁

⁾

⁻¹

^{𝐼 𝐹}

⁻ ²

(

₂

⁾

^𝑇

− ^𝐶

2

(𝐶

₁

)

⁻¹

𝐼 𝐹

⁺ ²

 min



₂

(

₂

⁾

^𝑇

^𝑃𝐹

⁻

^(𝐹

⁻

⁾

^𝑇

^𝑃

^𝑇



₂

(

₂

)

^𝑇

𝑃𝐹

⁺

(𝐹

⁺

)

^𝑇

𝑃

^𝑇



₂

▪ Where P= −𝐶 ₂ (𝐶 ₁ ) ⁻¹ 𝐼

Parity Relation for fault Isolation

(48)

❑ The optimization problem

▪ min



2

(

₂)^𝑇𝑃𝐹⁻(𝐹⁻)^𝑇𝑃^𝑇



₂

(

₂⁾^𝑇^𝑃𝐹⁺^(𝐹⁺⁾^𝑇^𝑃^𝑇



₂

 min



2

(

₂)^𝑇𝐴



₂

(

₂⁾^𝑇^𝐵



₂

▪ Where A=𝑃𝐹

⁻

(𝐹

⁻

)

^𝑇

𝑃

^𝑇

and B= 𝑃𝐹

⁺

(𝐹

⁺

)

^𝑇

𝑃

^𝑇

✓

is reduced to find the minimum eigenvalue of the pair(A, B) and the corresponding eigenvector.

❑ ^Proof

▪ 𝐹

₁

∶ ൝ 

^𝑛

→

X→

^𝑋^𝑇^𝐴𝑋

𝑋^𝑇𝐵𝑋

= )

▪  is a minimum of 𝐹

₁

for X = 𝑋

^∗

iff

✓

^𝛿𝐹  _{𝑋 𝑋=𝑋}

¹

ቚ

^∗

=0  ^𝛿

𝑋∗𝑇 𝐴𝑋∗

𝑋∗𝑇𝐵𝑋∗

 𝑋

^∗

=0  ^2𝐴𝑋

^∗

^𝑋

^∗𝑇

^𝐵𝑋

^∗

^−𝑋

^∗𝑇

^𝐴𝑋

^∗

^2𝐵𝑋

^∗

𝑋

^∗𝑇

𝐵𝑋

^∗²

=0  ²

𝑋

^∗𝑇

𝐵𝑋

^∗

A− ^𝑋

^∗𝑇

^𝐴𝑋

^∗

𝑋

^∗𝑇

𝐵𝑋

^∗



𝐵 𝑋 ^∗ =0

✓

Which is equivalent to 𝐴 − 𝐵 𝑋 ^∗ =0 where 𝑋 ^∗ is called the generalized eigenvector associated of the generalized eigenvalue 

✓

The roots of det(𝐴 − 𝐵) denoted by ∶  < _𝑗 < …<  _𝑖 <  ҧ are called the generalized eigenvalues of the matrix 𝐴 − 𝐵 and  the corresponding minimum eigenvalue.

Parity Relation for fault Isolation

(49)

❑ Consider the discrete-time system with the following description :

✓ Where 𝑢 _𝑘 ∈  ^𝑟 is the known input vector, 𝑦 _𝑘 ∈  ^𝑚 is the known output vector and 𝑥 _𝑘 ∈  ^𝑛 is the unknown state vector, 𝑓 _𝑘 ∈  ^𝑔 denotes a unknown fault vector which may contain actuator, component or sensors faults.

✓ 𝐴, 𝐵, 𝑅 ₁ , 𝑅 ₂ , 𝐶 𝑎𝑛𝑑 𝐷 are known real matrices with appropriate dimensions.

❑ How we obtain the redundancy relations ?

Parity Relation for the dynamic systems

ቊ 𝑥 _𝑘+1 = 𝐴𝑥 _𝑘 + 𝐵𝑢 _𝑘 + 𝑅 ₁ 𝑓 _𝑘

𝑦 _𝑘 = 𝐶𝑥 _𝑘 + 𝐷𝑢 _𝑘 + 𝑅 ₂ 𝑓 _𝑘

(50)

❑ How we obtain the redundancy relations ?

✓ Combining together the above relation from time instant k-s to time instant k yields the following redundant relations:

✓ where

✓ And the matrix M is constructed by replacing 𝐷, 𝐵 with 𝑅 ₂ , 𝑅 ₁ in the matrix H.

✓ To simplify the notation, the above equation can be written as:

▪ 𝑌 _𝑘 − 𝐻𝑈 _𝑘 = 𝑊𝑥 _𝑘−𝑠 + 𝑀𝐹 _𝑘

Parity Relation for the dynamic systems

𝑦 _𝑘−𝑠 𝑦 _{𝑘−𝑠+1}

⋮ 𝑦 _𝑘

𝑌 _𝑘

− H

𝑢 _𝑘−𝑠 𝑢 _{𝑘−𝑠+1}

⋮ 𝑢 _𝑘

𝑈 _𝑘

=W𝑥 _𝑘−𝑠 + M

𝑓 _𝑘−𝑠 𝑓 _{𝑘−𝑠+1}

⋮ 𝑓 _𝑘

𝐹 _𝑘

H=

𝐷 0 ⋯ 0

𝐶𝐵 𝐷 ⋯ 0

⋮ 𝐶𝐴 ^𝑠−1 𝐵

⋮ 𝐶𝐴 ^𝑠−2 𝐵

⋱ ⋮

⋯ 𝐷

∈  ^{𝑠+1 𝑚}  ^(𝑠+1)𝑟 W=

𝐶 𝐶𝐴

⋮ 𝐶𝐴 ^𝑠

∈  ^{𝑠+1 𝑚}  𝑛

(51)

❑ From the following data relation:

▪ 𝑌 _𝑘 − 𝐻𝑈 _𝑘 = 𝑊𝑥 _𝑘−𝑠 + 𝑀𝐹 _𝑘

✓ a residual signal can be defined as:

▪ 𝑟 _𝑘 = 𝑉 𝑌 _𝑘 − 𝐻𝑈 _𝑘

✓ Where V ∈  ^𝑝  𝑠+1 𝑚 and p is the residual vector dimension. The degree s is called the order of the parity relation.

✓ In order to analyze the residual, substituting 𝑌 _𝑘 − 𝐻𝑈 _𝑘 = 𝑊𝑥 _𝑘−𝑠 + 𝑀𝐹 _𝑘 into 𝑟 _𝑘 , we obtain:

▪ 𝑟 _𝑘 = 𝑉𝑊𝑥 _𝑘−𝑠 + 𝑉𝑀𝐹 _𝑘

✓ In order to make the parity vector useful for FDI, one should make it insensitive to unknown states, i.e.

▪ VW=0

✓ To satisfy the fault detectability condition, the matrix V should also satisfy the following condition:

▪ VM0

Parity Relation for the dynamic systems

(52)

❑ The parity relation approach for residual generation of dynamic system is shown in the following figure.

Parity Relation for the dynamic systems

System

Memory

‘s’ Samples H Memory

‘s’ Samples

V

𝑢 _𝑘 𝑦 _𝑘

𝑌 _𝑘

𝑈 _𝑘

+

−

𝑟 _𝑘

Residual generator via temporal redundancy

(53)

❑ FDI using the auto-redundance relation approach

✓ Consider the following fault sensor model :

✓ and the decision table

✓ Where

▪ 𝑟 _𝑖𝑘 = 𝑉 _𝑖 𝑌 _1𝑘 − 𝐻 _𝑖 𝑈 _𝑘 = 𝑉 _𝑖 𝑊 _𝑖 𝑥 _𝑘−𝑠 + 𝑉 _𝑖 𝑀 _𝑖 𝐹 _𝑖𝑘 , i=1, 2, …. m

Parity Relation for the dynamic systems

𝒇 _𝟏 𝒇 _𝟐 … 𝒇 _𝒎

𝑟 ₁ 1 0 … 0

𝑟 ₂ 0 1 … 0

⋮ ⋮

𝑟 _𝑚 0 0 … 1

ቊ 𝑥 _𝑘+1 = 𝐴𝑥 _𝑘 + 𝐵𝑢 _𝑘 𝑦 _𝑘 = 𝐶𝑥 _𝑘 + 𝑓 _𝑘

𝑊 _𝑖 = 𝐶 _𝑖 𝐶 _𝑖 𝐴

⋮ 𝐶 _𝑖 𝐴 ^𝑠

H=

0 0 ⋯ 0

𝐶 _𝑖 𝐵 0 ⋯ 0

⋮ 𝐶 _𝑖 𝐴 ^𝑠−1 𝐵

⋮ 𝐶 _𝑖 𝐴 ^𝑠−2 𝐵

⋱ ⋮

⋯ 0

𝑓 _{𝑖𝑘−𝑠} 𝑓 _{𝑖𝑘−𝑠+1}

⋮ 𝑓 _𝑖𝑘

𝐹 _𝑖𝑘

Constraints :

𝑉 _𝑖 𝑊 _𝑖 = 0

𝑉 _𝑖 𝑀 _𝑖  0

(54)

❑ Consider the discrete-time system with the following description :

✓ Where 𝑢 _𝑘 ∈  ^𝑟 is the input vector, 𝑦 _𝑘 ∈  ^𝑚 is the output vector and 𝑥 _𝑘 ∈  ^𝑛 is the state vector, 𝑓 _𝑘 ∈  ^𝑔 denotes a fault vector which may contain actuator, component or sensors faults, 𝑑 _𝑘 ∈  ^𝑞 is the unknown input (disturbance) vector.

✓ 𝐴, 𝐵, 𝐸 ₁ , 𝐸 ₂ , 𝑅 ₁ , 𝑅 ₂ , 𝐶 𝑎𝑛𝑑 𝐷 are known system model matrices with appropriate dimensions.

❑ In order to detect and isolate the faults a DOS or GOS scheme could be used.

Parity Relation for the dynamic systems

ቊ 𝑥 _𝑘+1 = 𝐴𝑥 _𝑘 + 𝐵𝑢 _𝑘 + 𝐸 ₁ 𝑑 _𝑘 + 𝑅 ₁ 𝑓 _𝑘 𝑦 _𝑘 = 𝐶𝑥 _𝑘 + 𝐷𝑢 _𝑘 + 𝐸 ₂ 𝑑 _𝑘 + 𝑅 ₂ 𝑓 _𝑘

(55)

❑ Observer-based approaches

❑ Parity (vector) relations

❑ Fault detection via parameter estimation

Residual Generation Techniques

(56)

❑ The process parameters are not known at all, or they are not known exactly enough. They can be determined with parameter estimation methods

❑ The basic structure of the model has to be known

❑ Based on the assumption that the faults are reflected in the physical system parameters

❑ The parameters of the actual process are estimated on-line using well-known parameter estimations methods

❑ The results are thus compared with the parameters of the reference model obtained initially under fault-free assumptions

❑ Any discrepancy (écart) can indicate that a fault may have occurred

Parameter estimation for fault detection

(57)

First Step : Parameter estimation without fault

An approach for modelling the input-output behavior of the monitored system will be recalled and exploited for fault detection

❑ SISO model Z=H𝜃 +

▪ 𝜃 unknown parameter vector

▪  noise of the sensor  N(0,V)

▪ Z sensor

▪ H known matrix

❑ Criteria

▪ min

𝜃 𝐽( ) = min

𝜃 1

2 Z−H𝜃 _𝑉 2 −1  min

𝜃 1

2 (Z−H𝜃) ^𝑇 𝑉 ⁻¹ (Z−H𝜃)

▪  ^𝜕𝐽(  ⁾ ฬ

𝜕   =෡  =0 and ^𝜕𝐽(  ⁾

𝜕  ∗𝜕  ^𝑇 >0

❑ Solution

▪ 𝜃=(𝐻 መ ^𝑇 𝑉 ⁻¹ 𝐻) ⁻¹ 𝐻 ^𝑇 𝑉 ⁻¹ Z

▪ ^𝜕𝐽(  ⁾

𝜕  ∗𝜕  ^𝑇 =𝐻 ^𝑇 𝑉 ⁻¹ 𝐻>0

Example of a Calorimeter Y=β ₀ + β ₁ 𝑈 + β ₃ 𝑈 ³

 Z=H𝜃+

 Y= 1 𝑈 𝑈 ³

β ₀ β ₁ β ₃

+

If V=𝜎 ² I then 𝜃=(𝐻 መ ^𝑇 𝐻) ⁻¹ 𝐻 ^𝑇 Z=𝐻 ⁺ 𝑍

It’s the Simple LS method

(58)

❑ Input/output model

✓

Measured signals:

▪ y(t)=Y(t)-𝑌

₀₀

; u(t)=U(t)-𝑈

₀₀

(Linearized around 𝑌

₀₀

, 𝑈

₀₀

)

✓

Basic equation without faults

▪ y(t)+𝑎

₁

𝑦

¹

𝑡 + ⋯ + 𝑎

_𝑛

𝑦

^𝑛

𝑡 =𝑏

₀

𝑢 𝑡 +𝑏

₁

𝑢

¹

𝑡 + ⋯ +𝑏

_𝑚

𝑢

^𝑚

𝑡

▪ y(t)=ℎ

^𝑇

𝑡 

▪ ℎ

^𝑇

= −𝑦

¹

𝑡 … −𝑦

^𝑛

𝑡 𝑢 𝑡 … 𝑢

^𝑚

𝑡

▪ 

^𝑇

𝑡 = 𝑎

₁

… 𝑎

_𝑛

𝑏

₀

… 𝑏

_𝑚

✓

Additive faults

▪ 𝑓

_𝑢

input fault; 𝑓

_𝑦

output fault

✓

Multiplicative faults

▪ 𝑎

_𝑖

, 𝑏

_𝑗

𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑓𝑎𝑢𝑙𝑡𝑠

Parameter Estimation for dynamic system

𝐺 _𝑝 𝑠 = 𝐵(𝑠) 𝐴(𝑠)

𝑓 _𝑦 𝑓 _𝑢

u y

𝑎 _𝑖 𝑏 _𝑗

(59)

❑ Spate space model

✓ Basic equation

ሶ 𝑥(𝑡)=Ax(t)+bu(t) y(t)= 𝑐 ^𝑇 x(t)

A=

0 0 1

0 1 −𝑎 ₁ 1

⋮ 0

⋮

−𝑎 ₂

⋮ 𝑏 ^𝑇 = 𝑏 ₀ 𝑏 ₁ … 𝑐 ^𝑇 = 0 0 … 1

✓ Additive fault

▪ 𝑓 _𝑙 input or state variable fault

▪ 𝑓 _𝑚 output fault

✓ Multiplicative fault

▪ A, B, c parameter faults

Parameter Estimation for dynamic system

b

𝑏 _𝑖

u ሶ𝑥 

𝑐 ^𝑇

𝑐 _𝑗 m 𝑓 _𝑚

x y

A

𝑎 _𝑖 I

𝑓 _𝑙

(60)

❑ Minimization of equation error

✓

Loss function:

▪ V= 𝑒 ² (k)

✓

Method

▪ Non-recursive

▪  ෡ = 𝐻 ^𝑇 𝐻 ⁻¹ 𝐻 ^𝑇 y

▪ Recursive

▪  ෡ _𝑘+1 = ෡  _𝑘 +𝐾 _𝑘+1 (𝑦 _𝑘+1 -ℎ ^𝑇 (𝑘 + 1)෡  _𝑘 )

✓

AC-515 Model based Fault-Diagnosis for Linear Systems