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
2 Theoretical tools, concepts and constructions
3 A conceptual solution
4 Extensions
5 Applications
6 Conclusions and future directions
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 2 / 46
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 3 / 46
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 4 / 46
Outline
1 Motivation
2 Theoretical tools, concepts and constructionsSet theoretic elementsMixed integer programming elements
3 A conceptual solution
4 Extensions
5 Applications
6 Conclusions and future directions
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 5 / 46
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
−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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 5 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 6 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 6 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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,∆).
Florin Stoican FTC based on set-theoretic methods October 6, 2011 7 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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∆.
Florin Stoican FTC based on set-theoretic methods October 6, 2011 7 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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∆
Florin Stoican FTC based on set-theoretic methods October 6, 2011 7 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 8 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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.
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.
Florin Stoican FTC based on set-theoretic methods October 6, 2011 8 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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.
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.
Florin Stoican FTC based on set-theoretic methods October 6, 2011 8 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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.
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 8 / 46
Theoretical tools, concepts and constructions Set theoretic elements
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 9 / 46
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.
Florin Stoican FTC based on set-theoretic methods October 6, 2011 10 / 46
Theoretical tools, concepts and constructions Mixed integer programming elements
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 11 / 46
Theoretical tools, concepts and constructions Mixed integer programming elements
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
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)
Florin Stoican FTC based on set-theoretic methods October 6, 2011 11 / 46
Theoretical tools, concepts and constructions Mixed integer programming elements
MIP – Basic ideaLinear extended representation:
−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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 11 / 46
Theoretical tools, concepts and constructions Mixed integer programming elements
MIP – Basic ideaLinear extended representation:
−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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 11 / 46
Theoretical tools, concepts and constructions Mixed integer programming elements
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 .
Florin Stoican FTC based on set-theoretic methods October 6, 2011 12 / 46
Theoretical tools, concepts and constructions Mixed integer programming elements
MIP – Non-connected regionsConsider the complement C(P) = cl(Rn \ P) of a union of polyhedral setsP =
⋃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]).
Florin Stoican FTC based on set-theoretic methods October 6, 2011 12 / 46
Theoretical tools, concepts and constructions Mixed integer programming elements
MIP – Non-connected regionsConsider the complement C(P) = cl(Rn \ P) of a union of polyhedral setsP =
⋃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.
Florin Stoican FTC based on set-theoretic methods October 6, 2011 12 / 46
Outline
1 Motivation
2 Theoretical tools, concepts and constructions
3 A conceptual solutionProblem statementFDI implementationControl strategies
4 Extensions
5 Applications
6 Conclusions and future directions
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 13 / 46
A conceptual solution Problem statement
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 13 / 46
A conceptual solution Problem statement
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 13 / 46
A conceptual solution Problem statement
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 13 / 46
A conceptual solution Problem statement
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 13 / 46
A conceptual solution Problem statement
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 13 / 46
A conceptual solution Problem statement
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 13 / 46
A conceptual solution Problem statement
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 14 / 46
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 15 / 46
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 ∈ RH
i = CiSz ⊕ Ni
rFi ∈ RF
i = −CiXref ⊕ NFi
Set separation condition:(Ci Sz ⊕ Ni
)∩(−Ci Xref ⊕ NF
i
)= ∅
Assume that:z ∈ Sz
xref ∈ Xref
RHi ∩ RF
i = ∅ −→
ri ∈ RH
i ↔ yi = Cix + ηi
ri ∈ RFi ↔ yi = 0 · x + ηF
i
Florin Stoican FTC based on set-theoretic methods October 6, 2011 15 / 46
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 ∈ RH
i = CiSz ⊕ Ni
rFi ∈ RF
i = −CiXref ⊕ NFi
Set separation condition:(Ci Sz ⊕ Ni
)∩(−Ci Xref ⊕ NF
i
)= ∅
Assume that:z ∈ Sz
xref ∈ Xref
RHi ∩ RF
i = ∅ −→
ri ∈ RH
i ↔ yi = Cix + ηi
ri ∈ RFi ↔ yi = 0 · x + ηF
i
Florin Stoican FTC based on set-theoretic methods October 6, 2011 15 / 46
A conceptual solution FDI implementation
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
]
Florin Stoican FTC based on set-theoretic methods October 6, 2011 16 / 46
A conceptual solution FDI implementation
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
]
Florin Stoican FTC based on set-theoretic methods October 6, 2011 16 / 46
A conceptual solution FDI implementation
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
]
Florin Stoican FTC based on set-theoretic methods October 6, 2011 16 / 46
A conceptual solution FDI implementation
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 17 / 46
A conceptual solution FDI implementation
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 17 / 46
A conceptual solution FDI implementation
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 17 / 46
A conceptual solution FDI implementation
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
xi /∈ Si −→ xi ∈ Si
IH IF IR
xi ∈ Si X — Xri ∈ RH
i X X X
Florin Stoican FTC based on set-theoretic methods October 6, 2011 17 / 46
A conceptual solution FDI implementation
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 18 / 46
A conceptual solution FDI implementation
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 19 / 46
A conceptual solution FDI implementation
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 19 / 46
A conceptual solution FDI implementation
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 20 / 46
A conceptual solution FDI implementation
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 20 / 46
A conceptual solution FDI implementation
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 20 / 46
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.
Florin Stoican FTC based on set-theoretic methods October 6, 2011 21 / 46
A conceptual solution Control strategies
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 22 / 46
A conceptual solution Control strategies
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 23 / 46
A conceptual solution Control strategies
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 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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 23 / 46
A conceptual solution Control strategies
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 24 / 46
A conceptual solution Control strategies
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 ⊕ Si
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 24 / 46
A conceptual solution Control strategies
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 ⊕ Si
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 24 / 46
Outline
1 Motivation
2 Theoretical tools, concepts and constructions
3 A conceptual solution
4 ExtensionsAdditional residual formulationsFDI influences in FTC designFrom multisensor to multiple loops
5 Applications
6 Conclusions and future directions
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 25 / 46
Extensions Additional residual formulations
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 25 / 46
Extensions Additional residual formulations
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 26 / 46
Extensions Additional residual formulations
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!
Florin Stoican FTC based on set-theoretic methods October 6, 2011 26 / 46
Extensions Additional residual formulations
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 27 / 46
Extensions Additional residual formulations
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 27 / 46
Extensions Additional residual formulations
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 28 / 46
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]
Florin Stoican FTC based on set-theoretic methods October 6, 2011 29 / 46
Extensions FDI influences in FTC design
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 ,
z : (Ciz ⊕ Ni ) ∩(−CiXref ⊕ NF
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 30 / 46
Extensions FDI influences in FTC design
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 ,
z : (Ciz ⊕ Ni ) ∩(−CiXref ⊕ NF
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 30 / 46
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 31 / 46
Extensions From multisensor to multiple loops
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])
Florin Stoican FTC based on set-theoretic methods October 6, 2011 32 / 46
Extensions From multisensor to multiple loops
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])
Florin Stoican FTC based on set-theoretic methods October 6, 2011 32 / 46
Outline
1 Motivation
2 Theoretical tools, concepts and constructions
3 A conceptual solution
4 Extensions
5 ApplicationsPositioning systemLane control mechanismWind turbine benchmark
6 Conclusions and future directions
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]).
Florin Stoican FTC based on set-theoretic methods October 6, 2011 33 / 46
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 34 / 46
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]).
Florin Stoican FTC based on set-theoretic methods October 6, 2011 35 / 46
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∗
Florin Stoican FTC based on set-theoretic methods October 6, 2011 36 / 46
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 37 / 46
Applications Wind turbine benchmark
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 .
Florin Stoican FTC based on set-theoretic methods October 6, 2011 38 / 46
Applications Wind turbine benchmark
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 39 / 46
Outline
1 Motivation
2 Theoretical tools, concepts and constructions
3 A conceptual solution
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 40 / 46
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 41 / 46
Conclusions and future directions Future directions
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
Florin Stoican FTC based on set-theoretic methods October 6, 2011 42 / 46
Conclusions and future directions Future directions
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.
Florin Stoican FTC based on set-theoretic methods October 6, 2011 43 / 46
Conclusions and future directions Future directions
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.
Florin Stoican FTC based on set-theoretic methods October 6, 2011 44 / 46
Conclusions and future directions Future directions
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)
Florin Stoican FTC based on set-theoretic methods October 6, 2011 45 / 46
Conclusions and future directions Future directions
Thank you!
Florin Stoican FTC based on set-theoretic methods October 6, 2011 46 / 46
Conclusions and future directions Future directions
Questions ?
Florin Stoican FTC based on set-theoretic methods October 6, 2011 46 / 46
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
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−1
−0.5
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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
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0.5
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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