Fault tolerant control based on set-theoretic methods - PhD Thesis defenseacse.pub.ro/wp-content/uploads/2013/08/slides3.pdf · Motivation TheneedforFTCincontrolapplications Bhopalchemicalspill

Fault tolerant control based on set-theoreticmethods

PhD Thesis defense

Florin Stoican

Supervisor: Sorin Olaru

SUPELEC Systems Sciences (E3S) - Automatic Control Department

October 6, 2011

Outline

1 Motivation

2 Theoretical tools, concepts and constructions

3 A conceptual solution

4 Extensions

5 Applications

6 Conclusions and future directions

Outline

1 Motivation



4 Extensions

5 Applications


Motivation

The need for FTC in control applications

Bhopal chemical spill(~4000 casualties)

Flight 1862 crash(43 casualties)

Fukushima meltdown(~40 km exclusion zone)

BP oil spill(~60000 barrels/day)

Florin Stoican FTC based on set-theoretic methods October 6, 2011 1 / 46

Motivation

Practical justification

d

y1

y2

FTC schemeu

exploit the sensor redundancydetect and isolate the faultsrecover from a temporary failure

reconfiguration of the controlstability guaranteesperformance objectives


Motivation

Fault tolerant control requirements

−4−2

02

4 −5 0 5 10 15 20 25 30 35 40 45 50

−6

−4

−2

0

2

x1

t

x 2


Motivation

FTC generalities

Control(Reference)Governor

ReconfigurableFeedforwardController

rActuators

u System

w

Sensors

v

z

Fault Detectionand Isolation (FDI)

ReconfigurableFeedbackController

-ReconfigurationMechanism

ActuatorFaults

SystemFaults

SensorFaults

u = inputsw = disturbancesr = referencesv = noisez = tracking error

Legend

FTC characterization FDI directionspassive (robust control)active (adaptive control)

FDI and RC blockslink and reciprocal influencesbetween FDI and RC

stochastic (Kalman filters,sensor fusion)artificial intelligenceset theoretic methods


Outline

1 Motivation

2 Theoretical tools, concepts and constructionsSet theoretic elementsMixed integer programming elements


4 Extensions

5 Applications


Theoretical tools, concepts and constructions Set theoretic elements

Families of sets – generalitiesVarious families of sets in control: Issues to be considered:

ellipsoids (Kurzhanskiı and Vályi [1997])polytopes/zonotopes (Motzkin et al. [1959])(B)LMIs (Nesterov and Nemirovsky [1994])star-shaped sets (Rubinov and Yagubov [1986])

flexibility of therepresentationnumericalimplementation

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x1

x 2

xT Qx ≤ γ

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

x1

x 2

Kern(S) 6= ∅

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x1

x 2

G(x) ≤ 0



Families of sets – generalitiesVarious families of sets in control: Issues to be considered:

ellipsoids (Kurzhanskiı and Vályi [1997])polytopes/zonotopes (Motzkin et al. [1959])(B)LMIs (Nesterov and Nemirovsky [1994])star-shaped sets (Rubinov and Yagubov [1986])

flexibility of therepresentationnumericalimplementation

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x1

x 2

A0 +∑

xiAi 0

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

x1

x 2

Kern(S) 6= ∅

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x1

x 2

G(x) ≤ 0



Families of sets – polyhedral/zonotopic sets(more “structured”)Best compromise: polytopic(zonotopic) sets

Polyhedral sets:dual representation

half-space:

hi x ≤ ki , i = 1 . . .Nh

vertex:∑i

αi vi , αi ≥ 0,∑

i

αi = 1, i = 1 . . .Nv

efficient algorithms for set containmentproblems (Gritzmann and Klee [1994])can approximate any convex shape(Bronstein [2008])

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

x1

x 2



Families of sets – polyhedral/zonotopic sets(more “structured”)Best compromise: polytopic(zonotopic) sets

Zonotopic sets:obtained as

hypercube projectionMinkowski sum of generators

additional representationgenerator form:∑

i

λi gi , |λi | ≤ 1, i = 1 . . .Ng

compact representationlimited to symmetric objects

−20

2

−4−2024

5

10

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x1

x 2



Invariance notionsConsider a system in Rn

x+ = f (x , δ)

with disturbances bounded by the set ∆ ⊂ Rn.

Definition (RPI set)A set Ω is called robust positive invariant (RPI) iff

f (Ω,∆) ⊆ Ω.

The minimal RPI set (which is contained in all the RPI sets) can bedefined as:

Ω∞ = f (f (. . . ,∆),∆)︸︷︷︸∞ iterations

= limk→∞

f (k)(0,∆).



Invariance notionsConsider a LTI system in Rn

x+ = Ax + Bδ

with A a Schur matrix and disturbances bounded by the set ∆ ⊂ Rn.

Definition (RPI set)A set Ω is called robust positive invariant (RPI) iff

AΩ⊕ B∆ ⊆ Ω.

The minimal RPI set (which is contained in all the RPI sets) can bedefined as:

Ω∞ =∞⊕

i=0AiB∆.



Invariance notions – exemplification

RPI set

−10 −8 −6 −4 −2 0 2 4 6 8 10−8

−6

−4

−2

0

2

4

6

8

x1

x 2

Ω

AΩ⊕ B∆

mRPI set

−10 −8 −6 −4 −2 0 2 4 6 8 10−8

−6

−4

−2

0

2

4

6

8

x1x 2

Ω

Ω∞ = AΩ∞ ⊕ B∆

AΩ⊕ B∆ ⊆ Ω Ω∞ =∞⊕

i=0AiB∆



Ultimate bounds for zonotopic setsTheorem (Ultimate bounds – Kofman et al. [2007])

For sytem x+ = Ax + Bδ with the Jordan decomposition A = V ΛV−1and assuming that

∣∣δ∣∣ ≤ δ we have that the set ΩUB(ε) is RPI.

Particularities:explicit linear formulations“good” approximation of the mRPIsetcan be extended to variousdegenerate cases (Haimovich et al.[2008], Kofman et al. [2008])

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

−8

−6

−4

−2

0

2

4

6

8

x1

x 2

|V−1x | ≤ b

|x | ≤ |V |b

ΩUB(ε) =

x : |V−1x | ≤ (I − |Λ|)−1|V−1B|δ + ε






δ1 ∈ ∆1 , |δ1| ≤ δ

δ2 ∈ ∆2 , |δ2| ≤ δ

⇒

−4 −3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

x1

x 2Sets with the same bounding box will give the same UBI set for a givendynamic.Improvement (Stoican et al. [2010c]): use zonotopic sets for describing thedisturbance.






δ1 ∈ ∆1 , |δ1| ≤ δ

δ2 ∈ ∆2 , |δ2| ≤ δ

⇒

−4 −3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

x1

x 2Sets with the same bounding box will give the same UBI set for a givendynamic.Improvement (Stoican et al. [2010c]): use zonotopic sets for describing thedisturbance.






For a zonotopic perturbation

∆ = C Bm∞

the dynamics become

x+ = Ax + Bδ = Ax + BCw

and the UBI set becomes: −25 −20 −15 −10 −5 0 5 10 15 20 25−15

−10

−5

0

5

10

15

ΩUB

ΩUB

x1

x 2ΩUB(ε) =

x : |V−1x | ≤ (I − |Λ|)−1|V−1B C |1 + ε



Other set theoretic topics

set separation between setsthrough a separating hyperplanethrough a barrier function

−6 −4 −2 0 2 4 6 8 10 12

−2

−1

0

1

2

3

x1

x 2

upper bound for the inclusion timeparticular bounds for a given attractive set

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

x1

x 2

RPI description for particular dynamicsswitched/with delaycyclic invariance

−5 −4 −3 −2 −1 0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

1.5

x1

x 2


Theoretical tools, concepts and constructions Mixed integer programming elements

MIP – Preliminaries

Set separation problems usually lead to nonconvex feasible regions foroptimization problems (usually, the complement of a polyhedral set):

x∗ = argminx /∈P

J(x)

whereP = x : hix ≤ ki , i = 1 . . .N .

The goal is to reduce the number of binary variables in the extendedrepresentation.



MIP – Basic ideaLinear extended representation:

−hix ≤ −ki + Mαi , i = 1 : Ni=N∑i=1

αi ≤ N − 1

with (α1, . . . , αN) ∈ 0, 1 N

and

N0 = dlog2 N e

−10 −8 −6 −4 −2 0 2 4 6 8 10−8

−6

−4

−2

0

2

4

6

8




−hix ≤ −ki + Mαi , i = 1 : Ni=N∑i=1

αi ≤ N − 1

with (α1, . . . , αN) ∈ 0, 1 N

and

N0 = dlog2 N e

−10 −8 −6 −4 −2 0 2 4 6 8 10−8

−6

−4

−2

0

2

4

6

8

R−(Hi)

P

Any of the regions R−(Hi ) of C(P) can be obtained by a suitable choiceof binary variables

(Stoican et al. [2011b])

R−(Hi)←→ (α1, . . . , αN)i , (1, . . . , 1, 0︸︷︷︸

i

, 1, . . . , 1)




−hix ≤ −ki + Mαi (λ), i = 1 : N

0 ≤ βl (λ)

with αi (λ) : 0, 1N0 → 0 ∪ [1,∞)and

N0 = dlog2 N e−10 −8 −6 −4 −2 0 2 4 6 8 10−8

−6

−4

−2

0

2

4

6

8

R−(Hi)

P

Any of the regions R−(Hi ) of C(P) can be obtained by a suitable choiceof binary variables (Stoican et al. [2011b])

R−(Hi)←→ (λ1, . . . , λN0)i




−hix ≤ −ki + Mαi (λ), i = 1 : N

0 ≤ βl (λ)

with αi (λ) : 0, 1N0 → 0 ∪ [1,∞)and

N0 = dlog2 N eλ1

λ2

λ3

(1, 1, 0)

(1, 1, 1)

(0, 1, 0)

For any λ ∈ 0, 1N0 unallocated to a region R−(Hi ), the MIrepresentation degenerates to the entire space Rn.

Solution: add constraints that make the unallocated tuples infeasible



MIP – Non-connected regionsConsider the complement C(P) = cl(Rn \ P) of a union of polyhedral setsP =

⋃j

Pj .

A(H) =⋃

l=1,...,γ(N)

( N⋂i=1

Rσl (i)(Hi )

)︸︷︷︸

Al

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

x1

x 2Using the hyperplanes Hi we partition the space into disjoint cells Al .




⋃j

Pj .

. . .

Al

σl (1)h1x ≤ σl (1)k1 + Mαl (λ)

...σl (N)hNx ≤ σl (N)kN + Mαl (λ)

. . .

0 ≤ βl (λ) −15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

15

x1

x 2Using the same procedure we associate a linear combination of binaryvariables αl (λ) to each cell (Stoican et al. [2011c]).




⋃j

Pj .

. . .

Al

σl (1)h1x ≤ σl (1)k1 + Mαl (λ)

...σl (N)hNx ≤ σl (N)kN + Mαl (λ)

. . .

0 ≤ βl (λ) −15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

15

x1

x 2The number of cells can be reduced through merging procedures.


Outline

1 Motivation


3 A conceptual solutionProblem statementFDI implementationControl strategies

4 Extensions

5 Applications


A conceptual solution Problem statement

Multisensor scheme

uref+

Pu

S1

S2

...

SN

...

C1x

C2x

CNx

+

+

+

η1

η2

ηN

+

+

+

F1y1

u

F2y2

u

FNyN

u

xref

−

xref

−

xref

−

x1+

x2+

xN

+

z1

z2

zN

v∗

...

− v∗

x+ = Ax + Bu + Ew LTI systembounded noise: w ∈W



Multisensor scheme – plant

uref+

Pu

S1

S2

...

SN

...

C1x

C2x

CNx

+

+

+

η1

η2

ηN

+

+

+

F1y1

u

F2y2

u

FNyN

u

xref

−

xref

−

xref

−

x1+

x2+

xN

+

z1

z2

zN

v∗

...

− v∗

x+ = Ax + Bu + Ew LTI systembounded noise: w ∈W



Multisensor scheme – sensors

uref+

Pu

S1

S2

...

SN

...

C1x

C2x

CNx

+

+

+

η1

η2

ηN

+

+

+

F1y1

u

F2y2

u

FNyN

u

xref

−

xref

−

xref

−

x1+

x2+

xN

+

z1

z2

zN

v∗

...

− v∗

yi = Cix + ηistatic and redundant sensorsbounded noise: ηi ∈ Ni



Multisensor scheme – fault scenario

uref+

Pu

S1

S2

...

SN

...

C1x

C2x

CNx

+

+

+

η1

η2

ηN

+

+

+

F1y1

u

F2y2

u

FNyN

u

xref

−

xref

−

xref

−

x1+

x2+

xN

+

z1

z2

zN

v∗

...

− v∗

yi = Cix +ηiFAULT−−−−−−−−−−−−−−

RECOVERYyi = 0 · x +ηF

i

bounded noise: ηFi ∈ NF

i

abrupt faultsknown model of the fault



Multisensor scheme – estimates

uref+

Pu

S1

S2

...

SN

...

C1x

C2x

CNx

+

+

+

η1

η2

ηN

+

+

+

F1y1

u

F2y2

u

FNyN

u

xref

−

xref

−

xref

−

x1+

x2+

xN

+

z1

z2

zN

v∗

...

− v∗

x+i = Axi + Bu + Li (yi − Ci xi )

LTI estimators



Multisensor scheme – tracking error

uref+

Pu

S1

S2

...

SN

...

C1x

C2x

CNx

+

+

+

η1

η2

ηN

+

+

+

F1y1

u

F2y2

u

FNyN

u

xref

−

xref

−

xref

−

x1+

x2+

xN

+

z1

z2

zN

v∗

...

− v∗

zi = xi − xrefminimize tracking error



Multisensor scheme – controller

uref+

Pu

S1

S2

...

SN

...

C1x

C2x

CNx

+

+

+

η1

η2

ηN

+

+

+

F1y1

u

F2y2

u

FNyN

u

xref

−

xref

−

xref

−

x1+

x2+

xN

+

z1

z2

zN

v∗

...

− v∗

u = uref + v switch (and not fusion)fix gain + referencegovernorMPC strategies



Modeling equationsplant dynamics

x+ = Ax + Bu + Ew

reference signalx+

ref = Axref + Buref

plant tracking error

z+ = x − xref = Az + B (u − uref )︸︷︷︸v

+Ew

estimations of the state

x+i = (A− LiCi ) xi + Bu + Li (yi − Ci xi )

estimations of the tracking error

zi = xi − xref


A conceptual solution FDI implementation

Set separation conditionsReminder:

z = x − xref

yi = Cix + ηiFAULT−−−−−−−−−−−−−

RECOVERYyi = 0 · x + ηF

i

ηi ∈ Ni , ηFi ∈ NF

i

Consider the residual signal

ri = yi − Cixref ,

rHi = Ciz + ηi

rFi = −Cixref + ηF

i

Set separation condition:

(Ciz ⊕ Ni ) ∩(−Cixref ⊕ NF

i)

= ∅

Assume that:z ∈ Sz

xref ∈ Xref

RHi ∩ RF

i = ∅ −→

ri ∈ RH

i ↔ yi = Cix + ηi

ri ∈ RFi ↔ yi = 0 · x + ηF

i




z = x − xref

yi = Cix + ηiFAULT−−−−−−−−−−−−−


i


i



rHi ∈ RH

i = CiSz ⊕ Ni

rFi ∈ RF

i = −CiXref ⊕ NFi

Set separation condition:(Ci Sz ⊕ Ni

)∩(−Ci Xref ⊕ NF

i

)= ∅


xref ∈ Xref

RHi ∩ RF

i = ∅ −→

ri ∈ RH


ri ∈ RFi ↔ yi = 0 · x + ηF

i




z = x − xref

yi = Cix + ηiFAULT−−−−−−−−−−−−−


i


i



rHi ∈ RH

i = CiSz ⊕ Ni

rFi ∈ RF

i = −CiXref ⊕ NFi

Set separation condition:(Ci Sz ⊕ Ni

)∩(−Ci Xref ⊕ NF

i

)= ∅


xref ∈ Xref

RHi ∩ RF

i = ∅ −→

ri ∈ RH


ri ∈ RFi ↔ yi = 0 · x + ηF

i



Auxiliary setsboundedness assumptions: Ni , NF

i , WXref – set for the reference signalSi – invariant set for the state estimation errorSz – invariant set for the plant tracking error

State estimation error:

x+i = x+ − x+

i = (A− LiCi ) xi +[E −Li

] [wηi

]

Plant tracking error (for fix gain v = −Kzl):

z+ = (A− BK ) z +[E BK

] [wxl

]






x+i = x+ − x+


] [wηi

]


z+ = (A− BK ) z +[E BK

] [wxl

]






x+i = x+ − x+


] [wηi

]


z+ = (A− BK ) z +[E BK

] [wxl

]



Sensor partitioning

IH =

i ∈ I−H : ri ∈ RHi∪

i ∈ I−R : SRi ⊆ Si , ri ∈ RH

i

IF =

i ∈ I : ri /∈ RHi

IR = I \ (IH ∪ IF ).

I = IH ∪ IF ∪ IR

IH IF

IR

ri ∈ RHi −→ ri /∈ RH

i

IH IF IR

xi ∈ Si X — Xri ∈ RH

i X X X



Sensor partitioning

IH =



i

IF =


IR = I \ (IH ∪ IF ).


IH IF

IR

ri ∈ RHi −→ ri /∈ RH

i

IH IF IR


i X X X



Sensor partitioning

IH =



i

IF =


IR = I \ (IH ∪ IF ).


IH IF

IR

ri /∈ RHi −→ ri ∈ RH

i

IH IF IR


i X X X



Sensor partitioning

IH =



i

IF =


IR = I \ (IH ∪ IF ).


IH IF

IR

xi /∈ Si −→ xi ∈ Si

IH IF IR


i X X X



Recovery – preliminariesConditions for recovery acknowledgment (IR → IH)

ri ∈ RHi – residual

xi ∈ Si – estimation error

xi = x − xi is not measurable but we constructSR

i such that xi ∈ SRi

Strategies:necessary conditionssufficient conditions

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

SRi

~Si

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

SRi

~Si

xi ∈ SRi , a necessary condition for xi ∈ Si is SR

i ∩ Si 6= ∅xi ∈ SR

i , a sufficient condition for xi ∈ Si is SRi ⊆ Si



Recovery – validation

IRi−→ IH : (i ∈ I−R ) ∧ (SR

i ⊆ Si ) ∧ (ri ∈ RHi )

Issues:gap timeinclusion validation

Strategies (during faulty functioning):

gap timekeep the original dynamics of the estimator (Olaru et al. [2009])change the dynamics of the estimator (Stoican et al. [2010b])reset the estimation (xi = xref or xi = xl)

inclusion validationwait for the validation of the inclusioncompute the reachable set of SR

i and observe when the inclusion isvalidated



Recovery – validation

IRi−→ IH : (i ∈ I−R ) ∧ (SR

i ⊆ Si ) ∧ (ri ∈ RHi )

Issues:gap timeinclusion validation

Strategies (during faulty functioning):−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−4

−3

−2

−1

0

1

2

3

4

x1

x 2

gap timekeep the original dynamics of the estimator (Olaru et al. [2009])change the dynamics of the estimator (Stoican et al. [2010b])reset the estimation (xi = xref or xi = xl)

inclusion validationwait for the validation of the inclusioncompute the reachable set of SR

i and observe when the inclusion isvalidated



Illustrative example

Consider the interdistance example with dynamics

x+ =

[1 0.10 1

]︸︷︷︸

A

x +

[00.5

]︸︷︷︸

B

u +

[00.1

]︸︷︷︸

E

w

with W = w : |w | ≤ 0.2.

C1 =[0.35 0.25

], |η1| ≤ 0.15, |ηF

1 | ≤ 1C2 =

[0.30 0.80

], |η2| ≤ 0.1, |ηF

2 | ≤ 1C3 =

[0.35 0.25

], |η3| ≤ 0.1, |ηF

3 | ≤ 0.3.

−60 −50 −40 −30 −20 −10 0 10 20 30−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1

RH1

RF1



Illustrative example – FDI validation

Consider the interdistance example with dynamics

x+ =

[1 0.10 1

]︸︷︷︸

A

x +

[00.5

]︸︷︷︸

B

u +

[00.1

]︸︷︷︸

E

w

with W = w : |w | ≤ 0.2.

RH1 = r1 : −22.9 ≤ r1 ≤ 22.9,

RH2 = r2 : −19.8 ≤ r1 ≤ 19.8,

RH3 = r3 : −22.9 ≤ r1 ≤ 22.9.

RF1 = r1 : −58.9 ≤ r1 ≤ −49.8,

RF2 = r2 : −53.9 ≤ r1 ≤ −39.2,

RF3 = r3 : −58.1 ≤ r1 ≤ −50.5.

−60 −50 −40 −30 −20 −10 0 10 20 30−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1

RH1

RF1



Illustrative example – recovery validationSensors estimations for test case when 3th sensor fails twice at f1 and f3respectively:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

140

150

160

170

180

time

1stcompo

nent

ofthestate

f1 = 6s f2 = 9s f3 = 14sf4 = 16s

f5 = 26.5s

t1 = 13.1s t2 = 25.5s t3 = 30.9s

set transitions for sensorwith index 3


A conceptual solution Control strategies

Control strategies

Issues related to FTC:faults may impose control law reconstructionunder fault, the objectives may need to change

Control strategies:fix gain with reference governorMPC formulation

Both strategies use the separation condition

(Ciz ⊕ Ni ) ∩(−Cixref ⊕ NF

i)

= ∅

to assure exact FDI.



FDI adjusted reference governorFix z and let xref be the decision variable:

Dxref ,

xref :(−Cixref ⊕ NF

i)∩ (CiSz ⊕ Ni ) = ∅, i = 1 . . .N

.

Reference governor (Stoican et al. [2010e]):

u∗ref [0,τ−1] = argminuref [0,τ−1]

τ−1∑i=0

(||r[i] − xref [i]||Qr + ||uref [i]||Rr

)

subject to:

x+ref [i] = Axref [i] + Buref [i]

x+ref [i] ∈ Dxref

Characteristics:fix gainflexible reference

−12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12−12

−10

−8

−6

−4

−2

0

2

4

6

8

10

12

x1

x 2

rxref

Dxref



MPC with FDI feasibility guaranteesFix xref and let z be the decision variable:

Dz ,

z : (Ciz ⊕ Ni ) ∩(−CiXref ⊕ NF

i)

= ∅, i = 1 . . .N

into the MPC formulation:

v∗[0,τ−1] = argminv[0,τ−1]

τ−1∑i=0

(||z[i]||Q + ||v[i]||R

)+ ||z[τ ]||P

subject to:

z+[i] = Az[i] + Bv[i] + E w[i]

z+[i] ∈ Dz

Issues:stability guaranteesnumerical complexity(reachable sets) −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14

−15

−10

−5

0

5

10

15

x1

x 2

zznom

Dz



MPC with FDI feasibility guaranteesFix xref and let z be the decision variable:

Dz ,


i)

= ∅, i = 1 . . .N

into the tube-MPC formulation(z ∈ znom ⊕ Sz):

v∗nom[0,τ−1] = argminvnom[0,τ−1]

τ−1∑i=0

(||znom[i]||Q + ||vnom[i]||R

)+ ||znom[τ ]||P

subject to:

z+nom[i] = Aznom[i] + Bvnom[i]

z+nom[i] ∈ Dz Sz

Issues:stability guaranteesnumerical complexity(reachable sets) −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14

−15

−10

−5

0

5

10

15

x1

x 2

zznom

Dz



MPC with FDI feasibility guarantees (II)Reminder:

tracking error estimation: zi = xi − xref

tracking error: z = x − xref

estimation error: xi = x − xi

znom =?

which leads to:z = zi + xi

Several directions (i ∈ IH and j = 0 . . . τ − 1):individual meritkeep the same sensor during the prediction horizon: znom[j] = zi[j]

relay racecheck the sensor index at each iteration: znom[j] = zij [j]

collaborative scenarioconsider a convex sum of the sensors (at least in the terminal step):znom[τ ] = convzi[τ ]

Can be applied also for fix control laws v = −Kzi







znom =?

which leads to:z ∈ zi ⊕ Si











znom =?

which leads to:z ∈ zi ⊕ Si






Outline

1 Motivation



4 ExtensionsAdditional residual formulationsFDI influences in FTC designFrom multisensor to multiple loops

5 Applications


Extensions Additional residual formulations

The estimation error as residual signalConsider the residual signal as

ri = zi

The residual sets for healthy to faulty transitions are:

RHi = SH

i (the invariant set of dynamics zi under healthyfunctioning)RF

i = SH→Fi (the one-step reachable set of SH

i under faultyfunctioning for zi)

Particularities:requires persistent faultsrecovers the entire informationpermits passive FTChas filter behavior

−50 0 50 100 150 200 250 300 350 400 450−30

−20

−10

0

10

20

30

40

50

60

x1

x 2

SHi

SFi

SH→Fi

SF→Hi



The estimation error as residual signalConsider the residual signal as

ri = zi

The residual sets for faulty to healthy transitions are:

RHi = SF

i (the invariant set of dynamics zi under faulty functioning)RF

i = SF→Hi (the one-step reachable set of SF

i under healthyfunctioning for zi)

Particularities:requires persistent faultsrecovers the entire informationpermits passive FTChas filter behavior

−50 0 50 100 150 200 250 300 350 400 450−30

−20

−10

0

10

20

30

40

50

60

x1

x 2

SHi

SFi

SH→Fi

SF→Hi



Passive FTC implementationFor a cost function J(·) passive FTC is possible if:

maxi∈IH

J(zi ) < mini∈I\IH

J(zi )

quadratic function

−300 −250 −200 −150 −100 −50 0 50 100−200

−150

−100

−50

0

50

100

150

200

x1

x 2

maxi∈IH

J(zi)

mini∈I\IH

J(zi)

gauge function

−6 −4 −2 0 2 4 6 8 10 12

−2

−1

0

1

2

3

x1

x 2

J(zi ) = zTi Pzi J(zi ) = J∗ dρH(zi )e − 1+ J∗ dρH(zl )e



Passive FTC implementationFor a cost function J(·) passive FTC is possible if:

maxi∈IH

J(zi ) < mini∈I\IH

J(zi )

−50 −40 −30 −20 −10 0 10 20 30 40 50−100

−80

−60

−40

−20

0

20

40

60

x1

x 2

Not always possible!



Extended residual

Consider a receding observation horizon of length τ with extendedresidual

ri = yi[−τ,0] − Ci,τxref [−τ,0] − Γi,τv[−τ,0]

which leads to:

rHi = Θi,τ z[−τ ] + Φi,τw[−τ,0] + ηi[−τ,0]

rFi = −Θi,τxref [−τ ] − Γi,τ

(uref [−τ,0] + v[−τ,0]

)+ ηF

i[−τ,0]

Set separation guarantee for FDI:

−Θi,τ

(z + xref [−τ ]

)− Γi,τ

(uref [−τ,0] + v[−τ,0]

)/∈ Pi

All control parameters influence the capacity of fault detection



Extended residual

Consider a receding observation horizon of length τ with extendedresidual

ri = yi[−τ,0] − Ci,τxref [−τ,0] − Γi,τv[−τ,0]

which leads to:

rHi = Θi,τ z[−τ ] + Φi,τw[−τ,0] + ηi[−τ,0]

rFi = −Θi,τxref [−τ ] − Γi,τ

(uref [−τ,0] + v[−τ,0]

)+ ηF

i[−τ,0]

Set separation guarantee for FDI:

−Θi,τ

(z + xref [−τ ]

)− Γi,τ

(uref [−τ,0] + v[−τ,0]

)/∈ Pi

All control parameters influence the capacity of fault detection



Extended residual (II)

Particularities:requires persistent faults (only for τinstants)recovers the entire informationenhances the separation conditionsadds delay in the control design

stability harder to enforcemaximizes FDI admissible space

0100 2 4 6

−10

0

x1t

x 2


Extensions FDI influences in FTC design

Influences of extended residuals in RC designGeneral condition for FDI validation:

Dref ,−Θi,τ

(z + xref [−τ ]

)− Γi,τ

(uref [−τ,0] + v[−τ,0]

)/∈ Pi

Control strategies:

fix gain with delayed information (v[−τ,0] = −Kzi[−2τ,−τ ]) leads tocondition:

−Θi,τxref [−τ ] − Γi,τuref [−τ,0] /∈ Pi −KSz[−2τ,−τ ]

Sz

to be used in a reference governor.MPC formulation:

(u∗ref , v∗) = argminuref [0,σ],v[0,σ]

σ∑j=0

f(xref [j], z[j], uref [j], v[j]

)subject to:

x+ref [j] = Axref [j] + Buref [j]

z+[j] = Az[j] + Bv[j] + Ew[j](

xref [j−τ ], uref [j−τ,j], v[j−τ,j], z[j])∈ Dref [j]



FDI adjustment for fix gain controlControl strategy for fix gain feedback:

instead of computing the set invariant for a given dynamics we tryto determine the dynamics that make a given set invariantfor a bounded reference xref ∈ Xref the feasible tracking error regionis given by

Dz ,


i)

= ∅, i = 1 . . .N

Take Sz ⊆ Dz and enforce its invariance as a parameter after K (Stoicanet al. [2010a]):

Sz = z : Hz ≤ K ⊆ Dz

z+ = (A− B K )z+[E B K

] [wxl

]ε∗ = max

lmin

K ,H,εε≥0

HFz =Fz (A−BK)Hθz +Fz Bz,lδz,l≤εθz

δz,l∈∆z,l

ε

if ε∗ ≤ 1 the solution is feasible



FDI adjustment for fix gain controlControl strategy for fix gain feedback:

instead of computing the set invariant for a given dynamics we tryto determine the dynamics that make a given set invariantfor a bounded reference xref ∈ Xref the feasible tracking error regionis given by

Dz ,


i)

= ∅, i = 1 . . .N

Take Sz ⊆ Dz and enforce its invariance as a parameter after K (Stoicanet al. [2010a]):

Sz = z : Hz ≤ K ⊆ Dz

z+ = (A− B K )z+[E B K

] [wxl

]ε∗ = max

lmin

K ,H,εε≥0

HFz =Fz (A−BK)Hθz +Fz Bz,lδz,l≤εθz

δz,l∈∆z,l

ε

if ε∗ ≤ 1 the solution is feasible


Extensions From multisensor to multiple loops

From multisensor to multiple loops

x+ = Ax + u + Ew

uref

+ u

S1 E1C1x y1

u

+

x1

xref

−

S2 E2C2x y2

u

+

x2

xref

−

SNs ENs

CNs x yNs

u

+

xNs

xref

−

KNgBNa

zNs

K2B2z2

K1B1z1

SW

v

. . .. . .

. . .. . .

the same principles hold for actuator/subsystems faultsissues to be considered:

computations more difficult (star-shaped sets)the system becomes switched



Switched systems particularitiesNote (Branicky [1994]): A switched system may not be stable even if allits subsystems are stable:

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

x1

x 2

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

x1

x 2 −3,000−2,000−1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000−1,000

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

x1

x 2

then the system is globally stable for any switch occurring at momentsgreater or equal with T .Difficulty: RPI construction for switched systems (Stoican et al. [2010d])



Switched systems particularitiesTheorem (Geromel and Colaneri [2006])

Let there be the switched system x+ = Aix and assume that:

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

x1

x 2

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

x1

x 2

Pi > 0A′iPiAi + Pi ≤ 0A′i

T PjAiT < Pi ∀j 6= i

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

x1

x 2

then the system is globally stable for any switch occurring at momentsgreater or equal with T .Difficulty: RPI construction for switched systems (Stoican et al. [2010d])


Outline

1 Motivation



4 Extensions

5 ApplicationsPositioning systemLane control mechanismWind turbine benchmark


Applications Positioning system

Positioning system

uref+ +

v

+

d

+A M = Kv

s(τs+1)

umreducer

θm Sxx Vx

Ω

SΩ

VΩ

Characteristics:“real example” – laboratory experiment with computer generatedfeedback2 sensors (velocity and position)the goal is to follow a position of reference

Important because it shows that for systems with a small number ofsensors, any missed fault is significant (Stoican and Olaru [2010]).


Applications Positioning system

Positioning system (II)

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

−5

0

5

10

time

1stcompo

nent

ofthestate

0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300 1,400 1,500 1,600 1,700 1,800 1,900 2,000

0

5

10

15

time

1stcompo

nent

ofthestate

Simple FTC implementation:residual using only current informationrecovery mechanism with estimation reset and counter


Applications Lane control mechanism

Lane control mechanism

yCGL > 0

`s

yL > 0

ψL

yl

yr

ΨL

η1

η2

F1

F2

y1

y2

S1

S2

sensor

selec

tion

x1

x2controlsele

ction

u x∗

ud

uaK

FDI2

FDI1

Characteristics:practical application: Minoiu Enache et al. [2010]maintain the car inside a nominal(safety) striphybrid control law (driver+corrective mechanism)2 sensors (vision algorithms and GPS RTK)

Important because it highlights a case where the FTC applies away fromthe origin (Stoican et al. [2011a]).


Applications Lane control mechanism

Lane control mechanism (II)Stability of the scheme assured:

convergence into the nominalregion:

Ωm ⊆ S∗

inclusion into the safety region:

ΩM ⊆ S∗

Front wheels of the vehicle

0 2 4 6 8 10 12 14 16 18 20 22 24−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time[s]

y l[m

],y r

[m]

u =

ud , x∗ ∈ S∗

ua, x∗ ∈ S∗ \ S∗

FTC necessary when u = ua:separation condition

X ∈ S∗ \ S∗−10 −8 −6 −4 −2 0 2 4 6 8 10

−1

−0.5

0

0.5

1

yL[m]

ψL[]

ΩM

S∗X o

Ωm

S∗


Applications Wind turbine benchmark

Wind turbine benchmarkA benchmark proposed in Odgaard et al. [2009]and response given in Stoicanet al. [2010f]:

a tri-pale windturbine Simulink modelfaults at sensor, actuator and plant level

Blade &Pitch System Drive Train Generator &

Converter

Controller

vwτr

ωr

τg

ωg

ωr,m, ωg,m τg,m,Pgβm, τr,m

βr

τg,r

Pr

Implementation of a FDI mechanism which:detects in less than a predefined measure horizon a fault occurrenceis robust to noise and perturbations



Wind turbine benchmark (II)Fault f8 (bias in generator)

The dynamics of the generator are affected by a bias fault:τ+

g = Aτg τg + Bτg τg,r + (1− f8) bConsidering the fault we have the healthy and faulty residual sets:

RHf8 = Zτg ⊕ Nτg,m

RFf8 = Zτg ⊕ Nτg,m ⊕

(I − Aτg )−1b

where Zτg is the invariant set corresponding to the dynamics of τg .



Wind turbine benchmark (III)Fault f6 (changed dynamics in actuator)

Consider the dynamicsx+β2

= Aβ2,f6xβ2 + Bβ2,f6(βr + β2,f )

β2 = (1 + (1− f6)K ) Cβ2,f6xβ2

and associate a residual rf6 , xβ2,m1 − xβ2,nom where β2,m1 is an estimateof the state and xβ2,nom is the nominal system.

−0.2 −0.15 −0.1 − 5 · 10−2 0 5 · 10−2 0.1 0.15 0.2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2·10−2

x1

x 2

Tf = 50Ts

−5 −4 −3 −2 −1 0 1 2 3 4 5

·10−2

−4

−2

0

2

4

·10−3

x1

x 2

Tf = 10Ts


Outline

1 Motivation



4 Extensions

5 Applications

6 Conclusions and future directionsConclusionsFuture directions

Conclusions and future directions Conclusions

Conclusions

invariant sets offer a robust FTC approacha countable number of sensor fault scenarios can be arbitrary chosena global view in considering the effects of the FDI mechanismextensions to MPCgood balance between computational effort and precisionrobust fault detection


Conclusions and future directions Future directions

Future directions – set theoretic elements

A) “Wish list” for set theoretic constructions:invariance computations

explicit formula for the boundary of the mRPI setRPI sets for switched/with delay systems

optimization problem which returns an RPI set for given dynamics andconstraints (+ fix structure)faster algorithms for set operations (treat degenerate polyhedral cases)comprehensive framework for zonotopes

B) A broader perspective:bridge the gap with stochastic FTC by the use of probabilisticinvarianceremain under boundedness assumptions and reformulate FTC in theviability theory framework



Future directions – viability theory

Viability theory generalizes a series of geometrical notions:

set valued maps and differenceinclusionscontinuity/derivabilityset shapepositive (control) invariance(viability/invariance kernels)

−10 −8 −6 −4 −2 0 2 4 6 8 10−6

−4

−2

0

2

4

6

x ′(t) ∈ F (x(t), u(t), f (t))

u(t) ∈ U(x(t))

Rk(x) = u(x) ∈ U(x), F (x , u(x), f (t)) ⊆ TK (x)

Issues:numerical algorithmshard to applydense mathematicalframework



References IM.S. Branicky. Stability of switched and hybrid systems. In IEEE Conference on Decision and Control, volume 4, pages 3498–3498.

Institute of electrical engineers INC (IEE), 1994.

EM Bronstein. Approximation of convex sets by polytopes. Journal of Mathematical Sciences, 153(6):727–762, 2008.

J.A. De Dona, X.W. Zhuo, M. Seron, and S. Olaru. A fault tolerant multisensor switching scheme for state estimation. In FaultDetection, Supervision and Safety of Technical Processes, pages 704–709, 2009.

José A. De Doná, María M. Seron, and Alain Yetendje. Multisensor fusion fault-tolerant control with diagnosis via a set separationprinciple. In Proceedings of Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, pages7825–7830, Shanghai, P.R. China, 16-18 December 2009.

JC Geromel and P. Colaneri. Stability and stabilization of discrete time switched systems. International Journal of Control, 79(7):719–728,2006.

Peter Gritzmann and Victor Klee. On the complexity of some basic problems in computational convexity: I. Containment problems.Discrete Mathematics, 136(1-3):129–174, 1994.

Hernan Haimovich, Ernesto Kofman, María M. Seron, I. y Agrimensura, and R. de Rosario. Analysis and Improvements of a SystematicComponentwise Ultimate-bound Computation Method. In Proccedings of the 17th World Congress IFAC, 2008.

Ernesto Kofman, Hernan Haimovich, and María M. Seron. A systematic method to obtain ultimate bounds for perturbed systems.International Journal of Control, 80(2):167–178, 2007.

Ernesto Kofman, F. Fontenla, Hernan Haimovich, María M. Seron, and A. Rosario. Control design with guaranteed ultimate bound forfeedback linearizable systems. In Aceptado en IFAC World Congress, 2008.

AB Kurzhanskiı and I. Vályi. Ellipsoidal calculus for estimation and control. Iiasa Research Center, 1997.

Nicoleta Minoiu Enache, Said Mammar, Sebastien Glaser, and Benoit Lusetti. Driver assistance system for lane departure avoidance bysteering and differential braking. In 6th IFAC Symposium Advances in Automotive Control, 12 - 14 July, Munich, Germany, 2010.

TS Motzkin, H. Raiffa, GL Thompson, and RM Thrall. The double description method. Contributions to the theory of games, 2:51, 1959.

Y. Nesterov and A. Nemirovsky. Interior point polynomial methods in convex programming. Studies in applied mathematics, 13, 1994.

C. Ocampo-Martinez, J.A. De Doná, and MM Seron. Actuator fault-tolerant control based on set separation. International Journal ofAdaptive Control and Signal Processing, 24(12):1070–1090, 2010.

P.F. Odgaard, J. Stoustrup, and M. Kinnaert. Fault Tolerant Control of Wind Turbines–a benchmark model. In Proc. of the 7th IFACSymp. on Fault Detection, Supervision and Safety of Technical Processes, pages 155–160, Barcelona, Spain, 30 June-3 July 2009.



References IISorin Olaru, Florin Stoican, José A. De Doná, and María M. Seron. Necessary and sufficient conditions for sensor recovery in a multisensor

control scheme. In Proc. of the 7th IFAC Symp. on Fault Detection, Supervision and Safety of Technical Processes, pages 977–982,Barcelona, Spain, 30 June-3 July 2009.

AM Rubinov and AA Yagubov. The space of star-shaped sets and its applications in nonsmooth optimization. Mathematical ProgrammingStudy, 29:175–202, 1986.

M.M. Seron and J.A. De Dona. Actuator fault tolerant multi-controller scheme using set separation based diagnosis. International Journalof Control, 83(11):2328–2339, 2010.

Florin Stoican and Sorin Olaru. Fault tolerant positioning system for a multisensor control scheme. In Proceedings of the 19th IEEEInternational Conference on Control Applications, part of 2010 IEEE Multi-Conference on Systems and Control, pages 1051–1056,Yokohama, Japan, 8-10 September 2010.

Florin Stoican, Sorin Olaru, and George Bitsoris. A fault detection scheme based on controlled invariant sets for multisensor systems. InProceedings of the 2010 Conference on Control and Fault Tolerant Systems, pages 468–473, Nice, France, 6-8 October 2010a.

Florin Stoican, Sorin Olaru, José A. De Doná, and María M. Seron. Improvements in the sensor recovery mechanism for a multisensorcontrol scheme. In Proceedings of the 29th American Control Conference, pages 4052–4057, Baltimore, Maryland, USA, 30 June-2July 2010b.

Florin Stoican, Sorin Olaru, José A. De Doná, and María M. Seron. Zonotopic ultimate bounds for linear systems with boundeddisturbances. Accepted to the 18th IFAC World Congress, 2010c.

Florin Stoican, Sorin Olaru, María M. Seron, and José A. De Doná. A fault tolerant control scheme based on sensor switching and dwelltime. In Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, Georgia, USA, 15-17 December 2010d.

Florin Stoican, Sorin Olaru, María M. Seron, and José A. De Doná. Reference governor for tracking with fault detection capabilities. InProceedings of the 2010 Conference on Control and Fault Tolerant Systems, pages 546–551, Nice, France, 6-8 October 2010e.

Florin Stoican, Catalin-Florentin Raduinea, and Sorin Olaru. Adaptation of set theoretic methods to the fault detection of a wind turbinebenchmark. Accepted to the 18th IFAC World Congress, 2010f.

Florin Stoican, N. Minoiu Enache, and Sorin Olaru. A lane control mechanism with fault tolerant control capabilities. Submitted to the50th IEEE Conference on Decision and Control and European Control Conference, 2011a.

Florin Stoican, Ionela Prodan, and Sorin Olaru. On the hyperplanes arrangements in mixed-integer techniques. accepted to the 30thAmerican Control Conference, 2011b.

Florin Stoican, Ionela Prodan, and Sorin Olaru. Enhancements on the hyperplane arrangements in mixed integer techniques. Accepted tothe 50th IEEE Conference on Decision and Control and European Control Conference, 2011c.



Additional features related to the PhD thesispresentation

Research project financed by the Carnot C3S Institute:

Collaborations:theoretical

various conference/journal papers (Univ. of Newcastle, Australia)viability theory discussions (VIDAMES (J.P. Aubin))

practicalpositioning system (College of Electrical and Computer Engineering,Serbia – a Pavle Slavic project)lane control (Renault – N. Minoiu-Enache)windturbine benchmark (competition proposed at SafeProcess’09 andpresented at IFAC’11)



Thank you!



Questions ?


Additional remarks

set-theoretic FTC for actuator faults:the fault affects the scheme for at least one stepfault detection through a bank of observers (the estimation shouldstay in one set if the model of the observer corresponds with themodel of the fault)Ocampo-Martinez et al. [2010], Seron and De Dona [2010]

as control implementation, the switch between sensors is equivalentwith sensor fusion strategies:

the switch will have a “leveling” effect, if the noise levels arecomparable, there will be a continuous switch between sensorsthe same set constructions can be used for FDI even if the control isrealized through sensor fusion methodsDe Doná et al. [2009], De Dona et al. [2009]

Collaborative MPCindividual merit

−10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

−1

−0.5

0

0.5

1

time

inde

x

relay race

−10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

−1

−0.5

0

0.5

1

time

inde

x

collaborative scenario

−10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

−1

−0.5

0

0.5

1

time

inde

x

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