KUL guest presentation Florin Stoican Norwegian University of - - PowerPoint PPT Presentation

KUL guest presentation

Florin Stoican

Norwegian University of Science and Technology (NTNU) - Department of Engineering Cybernetics

Tuesday 3rd July, 2012

Outline

1

Set theoretic elements

2

Fault tolerant control based on set-theoretic methods

3

Description of non-convex regions

4

Remarks upon the structure of explicit MPC

Outline

1

Set theoretic elements Families of sets Invariance notions Zonotope applications Other issues

2

Fault tolerant control based on set-theoretic methods

3

Description of non-convex regions

4

Remarks upon the structure of explicit MPC

Set theoretic elements Families of sets

Families of sets – generalities

Various families of sets in control: Issues to be considered: ellipsoids (Kurzhanski˘

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

flexibility of the representation numerical implementation

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 x1 x2

xTQx ≤ γ

−6 −4 −2 2 4 6 −6 −4 −2 2 4 6 x1 x2

Kern(S) = ∅

−0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 x1 x2

G(x) ≤ 0

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 1 / 38

Set theoretic elements Families of sets

Families of sets – generalities

Various families of sets in control: Issues to be considered: ellipsoids (Kurzhanski˘

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

flexibility of the representation numerical implementation

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 x1 x2

A0 +

• xiAi ≻ 0

−6 −4 −2 2 4 6 −6 −4 −2 2 4 6 x1 x2

Kern(S) = ∅

−0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 x1 x2

G(x) ≤ 0

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 1 / 38

Set theoretic elements Families of sets

Families of sets – polyhedral/zonotopic sets (more “structured”)

Best compromise: polytopic(zonotopic) sets Polyhedral sets: dual representation

half-space: hix ≤ ki, i = 1 . . . Nh vertex:

• i

αivi, αi ≥ 0,

• i

αi = 1, i = 1 . . . Nv

efficient algorithms for set containment problems (Gritzmann and Klee [1994])

can approximate any convex shape (Bronstein [2008])

−6 −4 −2 2 4 6 −6 −4 −2 2 4 6 x1 x2

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 2 / 38

−6 −4 −2 2 4 6 −6 −4 −2 2 4 6 x1 x2

Set theoretic elements Families of sets

Families of sets – polyhedral/zonotopic sets (more “structured”)

Best compromise: polytopic(zonotopic) sets Zonotopic sets:

• btained as

hypercube projection Minkowski sum of generators

additional representation

generator form:

• i

λigi, |λi| ≤ 1, i = 1 . . . Ng compact representation Fukuda: Nh = 2·

• Ng

n − 1

• , Nv = 2

d−1

• i=0
• Ng − 1

i

• limited to symmetric objects

−2 2 −4 −2 2 4 5 10

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1.5 −1 −0.5 0.5 1 1.5 x1 x2

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 2 / 38

Set theoretic elements Invariance notions

Invariance notions

Consider 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 be defined as: Ω∞ = f (f (. . . , ∆), ∆)

• ∞ iterations

= lim

k→∞ f (k)(0, ∆).

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 3 / 38

Set theoretic elements Invariance notions

Invariance notions

Consider 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 be defined as: Ω∞ =

∞

• i=0

AiB∆.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 3 / 38

Set theoretic elements Invariance notions

Invariance notions – exemplification

RPI set

−10 −8 −6 −4 −2 2 4 6 8 10 −8 −6 −4 −2 2 4 6 8 x1 x2

Ω AΩ ⊕ B∆

mRPI set

−10 −8 −6 −4 −2 2 4 6 8 10 −8 −6 −4 −2 2 4 6 8 x1 x2

Ω Ω∞ = AΩ∞ ⊕ B∆

AΩ ⊕ B∆ ⊆ Ω Ω∞ =

∞

• i=0

AiB∆

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 3 / 38

Set theoretic elements Zonotope applications

Ultimate bounds

Theorem (Ultimate bounds – Kofman et al. [2007]) For system x+ = Ax + Bδ with the Jordan decomposition A = V ΛV −1 and assuming that

• δ
• ≤ ¯

δ we have that the set ΩUB(ǫ) is RPI. Particularities: explicit linear formulations “good” approximation of the mRPI set can be extended to various degenerate cases (Haimovich et al.

[2008], Kofman et al. [2008])

−6 −5 −4 −3 −2 −1 1 2 3 4 5 6 −8 −6 −4 −2 2 4 6 8 x1 x2

|V −1x| ≤ b |x| ≤ |V |b

ΩUB(ǫ) =

• x : |V −1x| ≤ (I − |Λ|)−1|V −1B|¯

δ + ǫ

• Florin Stoican

KUL guest presentation Tuesday 3rd July, 2012 4 / 38

Set theoretic elements Zonotope applications

Ultimate bounds

Theorem (Ultimate bounds – Kofman et al. [2007]) For system x+ = Ax + Bδ with the Jordan decomposition A = V ΛV −1 and assuming that

• δ
• ≤ ¯

δ we have that the set ΩUB(ǫ) is RPI. δ1 ∈ ∆1 , |δ1| ≤ ¯ δ δ2 ∈ ∆2 , |δ2| ≤ ¯ δ ⇒

−4 −3 −2 −1 1 2 3 4 −3 −2 −1 1 2 3 x1 x2

Sets with the same bounding box will give the same UBI set for a given dynamic. Improvement (Stoican et al. [2011a]): use zonotopic sets for describing the

disturbance.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 4 / 38

Set theoretic elements Zonotope applications

QP problem with zonotopic bounds

min

u

1 2uTHu + xT

0 Fu

s.t. Gu ≤ W + Ex0. min

za,zb

1 2(za)THaza + xT

0 F aza

s.t. |V aza + V bzb| ≤ 1.

z x y

min

λ

1 2λT ˜ Hλ + xT

0 ˜

Fλ s.t. |λ| ≤ 1.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 5 / 38

Set theoretic elements Zonotope applications

QP problem with zonotopic bounds

min

u

1 2uTHu + xT

0 Fu

s.t. Gu ≤ W + Ex0. min

za,zb

1 2(za)THaza + xT

0 F aza

s.t. |V aza + V bzb| ≤ 1.

z x y

min

λ

1 2λT ˜ Hλ + xT

0 ˜

Fλ s.t. |λ| ≤ 1.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 5 / 38

Set theoretic elements Other issues

Other set theoretic topics

set separation between sets

through a separating hyperplane through a barrier function

−6 −4 −2 2 4 6 8 10 12 −2 −1 1 2 3 x1 x2

upper bound for the inclusion time

particular bounds for a given attractive set

−10 −8 −6 −4 −2 2 4 6 8 10 −10 −8 −6 −4 −2 2 4 6 8 10 x1 x2

RPI description for particular dynamics

switched/with delay cyclic invariance

−5 −4 −3 −2 −1 1 2 3 4 5 −1.5 −1 −0.5 0.5 1 1.5 x1 x2

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 6 / 38

Outline

1

Set theoretic elements

2

Fault tolerant control based on set-theoretic methods Problem statement FDI mechanism RC strategies Extensions

3

Description of non-convex regions

4

Remarks upon the structure of explicit MPC

Fault tolerant control based on set-theoretic methods

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 KUL guest presentation Tuesday 3rd July, 2012 7 / 38

Fault tolerant control based on set-theoretic methods

Fault tolerant control requirements

−4 −2 2 4 −5 5 10 15 20 25 30 35 40 45 50 −6 −4 −2 2 x1 t x2

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 8 / 38

Fault tolerant control based on set-theoretic methods

FTC generalities

Control (Reference) Governor Reconfigurable Feedforward Controller r Actuators u System w Sensors v z Fault Detection and Isolation (FDI) Reconfigurable Feedback Controller

• Reconfiguration

Mechanism Actuator Faults System Faults Sensor Faults u = inputs w = disturbances r = references v = noise z = tracking error Legend

FTC characterization FDI directions passive (robust control) active (adaptive control)

FDI and RC blocks link and reciprocal influences between FDI and RC

stochastic (Kalman filters, sensor fusion) artificial intelligence set theoretic methods

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 9 / 38

Fault tolerant control based on set-theoretic methods Problem statement

Multisensor scheme

uref + P u S1 S2 . . . SN . . . C1x C2x CNx + + + η1 η2 ηN + + + F1 y1 u F2 y2 u FN yN u xref − xref − xref − ˆ x1 + ˆ x2 + ˆ xN + ˆ z1 ˆ z2 ˆ zN v ∗ . . . − v ∗

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 10 / 38

Fault tolerant control based on set-theoretic methods Problem statement

Multisensor scheme – plant

uref + P u S1 S2 . . . SN . . . C1x C2x CNx + + + η1 η2 ηN + + + F1 y1 u F2 y2 u FN yN u xref − xref − xref − ˆ x1 + ˆ x2 + ˆ xN + ˆ z1 ˆ z2 ˆ zN v ∗ . . . − v ∗

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

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 10 / 38

Fault tolerant control based on set-theoretic methods Problem statement

Multisensor scheme – sensors

uref + P u S1 S2 . . . SN . . . C1x C2x CNx + + + η1 η2 ηN + + + F1 y1 u F2 y2 u FN yN u xref − xref − xref − ˆ x1 + ˆ x2 + ˆ xN + ˆ z1 ˆ z2 ˆ zN v ∗ . . . − v ∗

yi = Cix + ηi static and redundant sensors bounded noise: ηi ∈ Ni

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 10 / 38

Fault tolerant control based on set-theoretic methods Problem statement

Multisensor scheme – fault scenario

uref + P u S1 S2 . . . SN . . . C1x C2x CNx + + + η1 η2 ηN + + + F1 y1 u F2 y2 u FN yN u xref − xref − xref − ˆ x1 + ˆ x2 + ˆ xN + ˆ z1 ˆ z2 ˆ zN v ∗ . . . − v ∗

yi = Cix +ηi

FAULT

− − − − − − − ⇀ ↽ − − − − − − −

RECOVERY

yi = 0·x +ηF

i

bounded noise: ηF

i ∈ NF i

abrupt faults known model of the fault

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 10 / 38

Fault tolerant control based on set-theoretic methods Problem statement

Multisensor scheme – estimates

uref + P u S1 S2 . . . SN . . . C1x C2x CNx + + + η1 η2 ηN + + + F1 y1 u F2 y2 u FN yN u xref − xref − xref − ˆ x1 + ˆ x2 + ˆ xN + ˆ z1 ˆ z2 ˆ zN v ∗ . . . − v ∗

ˆ x+

i

= Aˆ xi + Bu + Li (yi − Ciˆ xi) LTI estimators

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 10 / 38

Fault tolerant control based on set-theoretic methods Problem statement

Multisensor scheme – tracking error

uref + P u S1 S2 . . . SN . . . C1x C2x CNx + + + η1 η2 ηN + + + F1 y1 u F2 y2 u FN yN u xref − xref − xref − ˆ x1 + ˆ x2 + ˆ xN + ˆ z1 ˆ z2 ˆ zN v ∗ . . . − v ∗

ˆ zi = ˆ xi − xref minimize tracking error

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 10 / 38

Fault tolerant control based on set-theoretic methods Problem statement

Multisensor scheme – controller

uref + P u S1 S2 . . . SN . . . C1x C2x CNx + + + η1 η2 ηN + + + F1 y1 u F2 y2 u FN yN u xref − xref − xref − ˆ x1 + ˆ x2 + ˆ xN + ˆ z1 ˆ z2 ˆ zN v ∗ . . . − v ∗

u = uref + v switch (and not fusion) fixed gain + reference governor MPC strategies

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 10 / 38

Fault tolerant control based on set-theoretic methods Problem statement

Modeling equations

plant dynamics x+ = Ax + Bu + Ew reference signal x+

ref = Axref + Buref

plant tracking error z+ = x − xref = Az + B (u − uref )

• v

+Ew estimations of the state ˆ x+

i

= (A − LiCi) ˆ xi + Bu + Li (yi − Ciˆ xi) estimations of the tracking error ˆ zi = ˆ xi − xref

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 11 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Set separation conditions

Reminder: z = x − xref yi = Cix + ηi

FAULT

− − − − − − ⇀ ↽ − − − − − − −

RECOVERY

yi = 0 · x + ηF

i

ηi ∈ Ni, ηF

i ∈ NF i

Consider the residual signal ri = yi − Cixref ,

• r H

i

= Ciz + ηi r F

i

= −Cixref + ηF

i

Set separation condition: ({Ciz} ⊕ Ni) ∩

• {−Cixref } ⊕ NF

i

• = ∅

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 12 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Set separation conditions

Reminder: z = x − xref yi = Cix + ηi

FAULT

− − − − − − ⇀ ↽ − − − − − − −

RECOVERY

yi = 0 · x + ηF

i

ηi ∈ Ni, ηF

i ∈ NF i

Consider the residual signal ri = yi − Cixref ,

• r H

i

∈ RH

i

= CiSz ⊕ Ni r F

i

∈ RF

i = −CiXref ⊕ NF i

Set separation condition:

• Ci Sz ⊕ Ni
• ∩
• −Ci Xref ⊕ NF

i

• = ∅

Assume that: z ∈ Sz xref ∈ Xref

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 12 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Set separation conditions

Reminder: z = x − xref yi = Cix + ηi

FAULT

− − − − − − ⇀ ↽ − − − − − − −

RECOVERY

yi = 0 · x + ηF

i

ηi ∈ Ni, ηF

i ∈ NF i

Consider the residual signal ri = yi − Cixref ,

• r H

i

∈ RH

i

= CiSz ⊕ Ni r F

i

∈ RF

i = −CiXref ⊕ NF i

Set separation condition:

• Ci Sz ⊕ Ni
• ∩
• −Ci Xref ⊕ NF

i

• = ∅

Assume that: z ∈ Sz xref ∈ Xref RH

i ∩ RF i = ∅ −

→

• ri ∈ RH

i

↔ yi = Cix + ηi ri ∈ RF

i ↔ yi = 0 · x + ηF i

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 12 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Sensor partitioning

IH =

• i ∈ I−

H : ri ∈ RH i

• ∪
• i ∈ I−

R : SR i ⊆ ˜

Si, ri ∈ RH

i

• IF =
• i ∈ I : ri /

∈ RH

i

• IR = I \ (IH ∪ IF).

I = IH ∪ IF ∪ IR

IH IF IR

IH IF IR ˜ xi ∈ ˜ Si

• —

X

ri ∈ RH

i

• X
• Florin Stoican

KUL guest presentation Tuesday 3rd July, 2012 13 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Sensor partitioning

IH =

• i ∈ I−

H : ri ∈ RH i

• ∪
• i ∈ I−

R : SR i ⊆ ˜

Si, ri ∈ RH

i

• IF =
• i ∈ I : ri /

∈ RH

i

• IR = I \ (IH ∪ IF).

I = IH ∪ IF ∪ IR

IH IF IR

ri ∈ RH

i

− → ri / ∈ RH

i

IH IF IR ˜ xi ∈ ˜ Si

• —

X

ri ∈ RH

i

• X
• Florin Stoican

KUL guest presentation Tuesday 3rd July, 2012 13 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Sensor partitioning

IH =

• i ∈ I−

H : ri ∈ RH i

• ∪
• i ∈ I−

R : SR i ⊆ ˜

Si, ri ∈ RH

i

• IF =
• i ∈ I : ri /

∈ RH

i

• IR = I \ (IH ∪ IF).

I = IH ∪ IF ∪ IR

IH IF IR

ri / ∈ RH

i

− → ri ∈ RH

i

IH IF IR ˜ xi ∈ ˜ Si

• —

X

ri ∈ RH

i

• X
• Florin Stoican

KUL guest presentation Tuesday 3rd July, 2012 13 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Sensor partitioning

IH =

• i ∈ I−

H : ri ∈ RH i

• ∪
• i ∈ I−

R : SR i ⊆ ˜

Si, ri ∈ RH

i

• IF =
• i ∈ I : ri /

∈ RH

i

• IR = I \ (IH ∪ IF).

I = IH ∪ IF ∪ IR

IH IF IR

˜ xi / ∈ ˜ Si − → ˜ xi ∈ ˜ Si IH IF IR ˜ xi ∈ ˜ Si

• —

X

ri ∈ RH

i

• X
• Florin Stoican

KUL guest presentation Tuesday 3rd July, 2012 13 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Recovery – preliminaries

Conditions for recovery acknowledgment (IR → IH) ri ∈ RH

i

– residual ˜ xi ∈ ˜ Si – estimation error ˜ xi = x − ˆ xi is not measurable but we construct SR

i

such that ˜ xi ∈ SR

i

Strategies: necessary conditions sufficient conditions

−10 −8 −6 −4 −2 2 4 6 8 10 −10 −8 −6 −4 −2 2 4 6 8 10

SR

i

~ Si

−10 −8 −6 −4 −2 2 4 6 8 10 −10 −8 −6 −4 −2 2 4 6 8 10

SR

i

~ Si

˜ xi ∈ SR

i , a necessary condition for ˜

xi ∈ ˜ Si is SR

i ∩ ˜

Si = ∅ ˜ xi ∈ SR

i , a sufficient condition for ˜

xi ∈ ˜ Si is SR

i ⊆ ˜

Si

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 14 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Recovery – validation

IR

i

− → IH : (i ∈ I−

R ) ∧ (SR i ⊆ ˜

Si) ∧ (ri ∈ RH

i )

Issues: gap time inclusion validation Strategies (during faulty functioning): gap time

keep the original dynamics of the estimator (Olaru et al. [2009]) change the dynamics of the estimator (Stoican et al. [2010b]) reset the estimation (ˆ x ◦

i = xref or ˆ

x ◦

i = ˆ

xl)

inclusion validation

wait for the validation of the inclusion compute the reachable set of SR

i

and observe when the inclusion is validated

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 15 / 38

−4 −3 −2 −1 1 2 3 4 5 6 7 8 −12 −10 −8 −6 −4 −2 2 4 6 8 10 12 x1 x2

Fault tolerant control based on set-theoretic methods FDI mechanism

Recovery – validation

IR

i

− → IH : (i ∈ I−

R ) ∧ (SR i ⊆ ˜

Si) ∧ (ri ∈ RH

i )

Issues: gap time inclusion validation Strategies (during faulty functioning):

−0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 −4 −3 −2 −1 1 2 3 4 x1 x2

gap time

keep the original dynamics of the estimator (Olaru et al. [2009]) change the dynamics of the estimator (Stoican et al. [2010b]) reset the estimation (ˆ x ◦

i = xref or ˆ

x ◦

i = ˆ

xl)

inclusion validation

wait for the validation of the inclusion compute the reachable set of SR

i

and observe when the inclusion is validated

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 15 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Illustrative example

Consider the interdistance example with dynamics x+ =

• 1

0.1 1

• A

x + 0.5

• B

u + 0.1

• E

w with W = {w : |w| ≤ 0.2}. C1 = 0.35 0.25 , |η1| ≤ 0.15, |ηF

1 | ≤ 1

C2 = 0.30 0.80 , |η2| ≤ 0.1, |ηF

2 | ≤ 1

C3 = 0.35 0.25 , |η3| ≤ 0.1, |ηF

3 | ≤ 0.3.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 16 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Illustrative example – FDI validation

Consider the interdistance example with dynamics x+ =

• 1

0.1 1

• A

x + 0.5

• B

u + 0.1

• E

w with W = {w : |w| ≤ 0.2}. RH

1 = {r1 : −22.9 ≤ r1 ≤ 22.9},

RH

2 = {r2 : −19.8 ≤ r1 ≤ 19.8},

RH

3 = {r3 : −22.9 ≤ r1 ≤ 22.9}.

RF

1 = {r1 : −58.9 ≤ r1 ≤ −49.8},

RF

2 = {r2 : −53.9 ≤ r1 ≤ −39.2},

RF

3 = {r3 : −58.1 ≤ r1 ≤ −50.5}.

−60 −50 −40 −30 −20 −10 10 20 30 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 x1

RH

1

RF

1

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 16 / 38

Fault tolerant control based on set-theoretic methods FDI mechanism

Illustrative example – recovery validation

Sensors estimations for test case when 3th sensor fails twice at f1 and f3 respectively:

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 1st component of the state

f1 = 6s f2 = 9s f3 = 14s f4 = 16s f5 = 26.5s t1 = 13.1s t2 = 25.5s t3 = 30.9s

set transitions for sensor with index 3

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 16 / 38

IH IF IR

f1 f2 f3 f4 t3

Fault tolerant control based on set-theoretic methods RC strategies

Control strategies

For all control strategies we use the separation condition as a design constraint: ({Ciz} ⊕ Ni) ∩

• {−Cixref } ⊕ NF

i

• = ∅

to assure exact FDI. Control strategies: z fixed and xref a decision variable: fixed gain feedback + reference governor z a decision variable and xref fixed: MPC strategy for the feedback action both z and xref as decision variables: MPC strategy involving both the reference and the feedback action

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 17 / 38

Fault tolerant control based on set-theoretic methods RC strategies

Fixed gain feedback

Assume a fixed feedback gain: such that a cost function is minimized: v = −Kˆ zl l = arg min

i∈IH J (ˆ

zi) and an index is selected, using current information: v∗ = −Kˆ zl l = arg min

i∈IH {||ˆ

zi||Q + ||v||R} ,

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 18 / 38

Fault tolerant control based on set-theoretic methods RC strategies

Fixed gain feedback

Assume a fixed feedback gain: such that a cost function is minimized: v = −Kˆ zl l = arg min

i∈IH J (ˆ

zi) and an index is selected, using current information: v∗ = −Kˆ zl l = arg min

i∈IH {||ˆ

zi||Q + ||v||R} , Use information over a prediction horizon to select the best index individual merit: keep the same sensor during the prediction horizon relay race: check the sensor index at each iteration collaborative scenario: consider a convex sum of the sensors (at least in the terminal step)

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 18 / 38

Fault tolerant control based on set-theoretic methods RC strategies

Fixed gain feedback

Assume a fixed feedback gain: such that a cost function is minimized: v = −Kˆ zl l = arg min

i∈IH J (ˆ

zi) and an index is selected, using current information: v∗ = −Kˆ zl l = arg min

i∈IH {||ˆ

zi||Q + ||v||R} , Take Sz = {z : Hz ≤ K} ⊆ Dz and enforce its invariance as a parameter after K (Stoican et al. [2010a]):

z+ = (A − B K )z+ E B K w ˜ xl

• ǫ∗ = max

l

min

K,H,ǫ ǫ≥0 HFz =Fz (A−BK) Hθz +Fz Bz,l δz,l ≤ǫθz δz,l ∈∆z,l

ǫ if ǫ∗ ≤ 1 the solution is feasible

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 18 / 38

Fault tolerant control based on set-theoretic methods RC strategies

FDI adjusted reference governor

Fix z (z ∈ Sz) and let xref be the decision variable: Dxref

• xref :
• {−Cixref } ⊕ NF

i

• ∩ (CiSz ⊕ Ni) = ∅, i = 1 . . . N
• .

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

u∗

ref [0,τ−1] = arg min uref [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 gain flexible reference

−12 −10 −8 −6 −4 −2 2 4 6 8 10 12 −12 −10 −8 −6 −4 −2 2 4 6 8 10 12 x1 x2

r xref Dxref

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 19 / 38

Fault tolerant control based on set-theoretic methods RC strategies

MPC with FDI feasibility guarantees

Fix xref (xref ∈ 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] = arg min v[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 guarantees numerical complexity (reachable sets)

−14 −12 −10 −8 −6 −4 −2 2 4 6 8 10 12 14 −15 −10 −5 5 10 15 x1 x2

z znom Dz

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 20 / 38

Fault tolerant control based on set-theoretic methods RC strategies

MPC with FDI feasibility guarantees

Fix xref (xref ∈ 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] = arg min vnom[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 guarantees numerical complexity (reachable sets)

−14 −12 −10 −8 −6 −4 −2 2 4 6 8 10 12 14 −15 −10 −5 5 10 15 x1 x2

z znom Dz

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 20 / 38

Fault tolerant control based on set-theoretic methods Extensions

The estimation error as residual signal

Consider the residual signal as ri = ˆ zi The residual sets for healthy to faulty transitions are: RH

i

= ˆ SH

i

(the invariant set of dynamics ˆ zi under healthy functioning) RF

i = ˆ

SH→F

i

(the one-step reachable set of ˆ SH

i

under faulty functioning for ˆ zi) Particularities: requires persistent faults recovers the entire information permits passive FTC has filter behavior

−50 50 100 150 200 250 300 350 400 450 −30 −20 −10 10 20 30 40 50 60 x1 x2

ˆ SH

i

ˆ SF

i

ˆ SH→F

i

ˆ SF→H

i Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 21 / 38

Fault tolerant control based on set-theoretic methods Extensions

The estimation error as residual signal

Consider the residual signal as ri = ˆ zi The residual sets for faulty to healthy transitions are: RH

i

= ˆ SF

i

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

i = ˆ

SF→H

i

(the one-step reachable set of ˆ SF

i

under healthy functioning for ˆ zi) Particularities: requires persistent faults recovers the entire information permits passive FTC has filter behavior

−50 50 100 150 200 250 300 350 400 450 −30 −20 −10 10 20 30 40 50 60 x1 x2

ˆ SH

i

ˆ SF

i

ˆ SH→F

i

ˆ SF→H

i Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 21 / 38

Fault tolerant control based on set-theoretic methods Extensions

Passive FTC implementation

For a cost function J(·) passive FTC is possible if: max

i∈IH J(ˆ

zi) < min

i∈I\IH J(ˆ

zi) quadratic function

−300 −250 −200 −150 −100 −50 50 100 −200 −150 −100 −50 50 100 150 200 x1 x2

max

i∈IH J(ˆ

zi) min

i∈I\IH J(ˆ

zi)

gauge function

−6 −4 −2 2 4 6 8 10 12 −2 −1 1 2 3 x1 x2

J(ˆ zi) = ˆ zT

i Pˆ

zi J(ˆ zi) = J∗ {⌈ρH(ˆ zi)⌉ − 1} + J∗ ⌈ρH(ˆ zl)⌉

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 22 / 38

Fault tolerant control based on set-theoretic methods Extensions

Passive FTC implementation

For a cost function J(·) passive FTC is possible if: max

i∈IH J(ˆ

zi) < min

i∈I\IH J(ˆ

zi)

−50 −40 −30 −20 −10 10 20 30 40 50 −100 −80 −60 −40 −20 20 40 60 x1 x2

Not always possible!

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 22 / 38

Fault tolerant control based on set-theoretic methods Extensions

Extended residual

Consider a receding observation horizon of length τ with extended residual ri = yi[−τ,0] − Ci,τxref [−τ,0] − Γi,τv[−τ,0] which leads to: r H

i

= Θi,τz[−τ] + Φi,τw[−τ,0] + ηi[−τ,0] r F

i = −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

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 23 / 38

Fault tolerant control based on set-theoretic methods Extensions

Extended residual

Consider a receding observation horizon of length τ with extended residual ri = yi[−τ,0] − Ci,τxref [−τ,0] − Γi,τv[−τ,0] which leads to: r H

i

= Θi,τz[−τ] + Φi,τw[−τ,0] + ηi[−τ,0] r F

i = −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 KUL guest presentation Tuesday 3rd July, 2012 23 / 38

Fault tolerant control based on set-theoretic methods Extensions

Extended residual (II)

Particularities: requires persistent faults (only for τ instants) recovers the entire information enhances the separation conditions adds delay in the control design

stability harder to enforce maximizes FDI admissible space

10 2 4 6 −10 x1 t x2

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 24 / 38

−8 −6 −4 −2 2 4 6 8 10 12 14 16 18 −15 −10 −5 5 10 15 x1 x2

t t − 1 t − 2

RH

i

RF

i

Fault tolerant control based on set-theoretic methods Extensions

Influences of extended residuals in RC design

General 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] = −Kˆ zi[−2τ,−τ]) leads to condition: −Θi,τxref [−τ] − Γi,τuref [−τ,0] / ∈ Pi ⊖

• −KSz[−2τ,−τ]
• ⊖ Sz

to be used in a reference governor. MPC formulation: (u∗

ref , v ∗) =

arg min

uref [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 KUL guest presentation Tuesday 3rd July, 2012 25 / 38

Fault tolerant control based on set-theoretic methods Extensions

From multisensor to multiple loops

x+ = Ax + u + Ew uref + u S1 E1 C1x y1 u + ˆ x1 xref − S2 E2 C2x y2 u + ˆ x2 xref − SNs ENs CNsx yNs u + ˆ xNs xref − KNg BNa ˆ zNs K2 B2 ˆ z2 K1 B1 ˆ z1 SW v

. . . . . . . . . . . . −50 −40 −30 −20 −10 10 20 30 40 50 −20 −15 −10 −5 5 10 15 20 x1 x2

S2

z

S1

z

S3

z τ12 = 8 τ21 = 8 τ13 = 2 τ31 = 7 τ23 = 7 τ32 = 8

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

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

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 26 / 38

Outline

1

Set theoretic elements

2

Fault tolerant control based on set-theoretic methods

3

Description of non-convex regions

4

Remarks upon the structure of explicit MPC

Description of non-convex regions

MIP – Preliminaries

Set separation problems usually lead to nonconvex feasible regions for

• ptimization problems (usually, the complement of a polyhedral set):

x∗ = arg min

x / ∈P J(x)

where P = {x : hix ≤ ki, i = 1 . . . N} . The goal is to reduce the number of binary variables in the extended representation.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 27 / 38

Description of non-convex regions

MIP – Basic idea

Linear extended representation: −hix ≤ −ki + Mαi, i = 1 : N

i=N

• i=1

αi ≤ N − 1 with (α1, . . . , αN) ∈ {0, 1} N

−10 −8 −6 −4 −2 2 4 6 8 10 −8 −6 −4 −2 2 4 6 8

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 28 / 38

Description of non-convex regions

MIP – Basic idea

Linear extended representation: −hix ≤ −ki + Mαi, i = 1 : N

i=N

• i=1

αi ≤ N − 1 with (α1, . . . , αN) ∈ {0, 1} N

−10 −8 −6 −4 −2 2 4 6 8 10 −8 −6 −4 −2 2 4 6 8

R−(Hi) P

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

• f binary variables

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

• i

, 1, . . . , 1)

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 28 / 38

Description of non-convex regions

MIP – Basic idea

Linear extended representation: −hix ≤ −ki + Mαi(λ), i = 1 : N 0 ≤ βl(λ) with αi(λ) : {0, 1}N0 → {0} ∪ [1, ∞) and N0 = ⌈log2 N ⌉

−10 −8 −6 −4 −2 2 4 6 8 10 −8 −6 −4 −2 2 4 6 8

R−(Hi) P

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

• f binary variables (Stoican et al. [2011b])

R−(Hi) ← → (λ1, . . . , λN0)i

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 28 / 38

Description of non-convex regions

MIP – Basic idea

Linear extended representation: −hix ≤ −ki + Mαi(λ), i = 1 : N 0 ≤ βl(λ) with αi(λ) : {0, 1}N0 → {0} ∪ [1, ∞) and N0 = ⌈log2 N ⌉

λ1 λ2 λ3 (1, 1, 0) (1, 1, 1) (0, 1, 0)

For any λ ∈ {0, 1}N0 unallocated to a region R−(Hi), the MI representation degenerates to the entire space Rn. Solution: add constraints that make the unallocated tuples infeasible

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 28 / 38

Description of non-convex regions

Exemplification of the approach

Consider a polytope P ⊂ R2 given by     −1 1 −1 1     x ≤     1 1 1 1    

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 29 / 38

Description of non-convex regions

Exemplification of the approach

and its complement C(P) by     1 −1 1 −1     x ≤     −1 + Mα1 −1 + Mα2 −1 + Mα3 −1 + Mα4     in the classical MI formulation.

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 29 / 38

Description of non-convex regions

Exemplification of the approach

and its complement C(P) by     1 −1 1 −1     x ≤     −1 + M( λ1 + λ2) −1 + M(1 − λ1 + λ2) −1 + M(1 + λ1 − λ2) −1 + M(2 − λ1 − λ2)     in the reduced MI formulation.

−1 −0.5 0.5 1 1.5 2 2.5 3 3.5 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5

R−(H2) :

• −1
• x ≤ −1

(λ2

1, λ2 2) = (0, 1)

In the reduced representation only N0 = ⌈log2 4⌉ = 2 binary variables are needed. For region R−(H2) associate tuple (λ2

1, λ2 2) = (0, 1) which leads to the

mapping α2 = 1 + λ1 − λ2

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 29 / 38

Description of non-convex regions

MIP – Non-connected regions

Consider the complement C(P) = cl(Rn \ P) of a union of polyhedral sets P =

j

Pj. A(H) =

• l=1,...,γ(N)

N

• i=1

Rσl(i)(Hi)

• Al

−10 −8 −6 −4 −2 2 4 6 8 10 −10 −8 −6 −4 −2 2 4 6 8 10 x1 x2

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

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 30 / 38

Description of non-convex regions

MIP – Non-connected regions

Consider the complement C(P) = cl(Rn \ P) of a union of polyhedral sets P =

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 5 10 15 −15 −10 −5 5 10 15 x1 x2

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

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 30 / 38

Description of non-convex regions

MIP – Non-connected regions

Consider the complement C(P) = cl(Rn \ P) of a union of polyhedral sets P =

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 5 10 15 −15 −10 −5 5 10 15 x1 x2

The number of cells can be reduced through merging procedures (Karnaugh maps, espresso heuristic minimizer).

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 30 / 38

Outline

1

Set theoretic elements

2

Fault tolerant control based on set-theoretic methods

3

Description of non-convex regions

4

Remarks upon the structure of explicit MPC Preliminaries Improved bound for active constraints Partial recursive explicit MPC

Remarks upon the structure of explicit MPC Preliminaries

Preliminaries

Let us consider the MPC formulation: min

u0,...,uN−1 N−1

• k=0
• uT

k Ruk + xT k Qxk

• + xT

N Qf xN

s.t. Gkxk + Hkuk ≤ bk, xk+1 = Axk + Buk, k = 0 min

u

1 2uTHu + xT

0 Fu

s.t. G u ≤ W + Ex0 where N is the prediction horizon length, Q, R, P are the cost matrices and there are q constraints.

⇔

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 31 / 38

Remarks upon the structure of explicit MPC Preliminaries

Preliminaries

Let us consider the MPC formulation: min

u0,...,uN−1 N−1

• k=0
• uT

k Ruk + xT k Qxk

• + xT

N Qf xN

s.t. Gkxk + Hkuk ≤ bk, xk+1 = Axk + Buk, k = 0 min

u

1 2uTHu + xT

0 Fu

s.t. G u ≤ W + Ex0 where N is the prediction horizon length, Q, R, P are the cost matrices and there are q constraints. Use the structure of matrix G to improve the bound limiting the number of sets of active constraints provide a recursive partial description of the explicit MPC

⇔

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 31 / 38

Remarks upon the structure of explicit MPC Preliminaries

KKT formulation and LICQ

KKT formulation: min

u

1 2uTHu + xT

0 Fu

s.t. Gu ≤ W + Ex0 Hu + F Tx0 + GTλ = 0 λ ≥ 0 Gu − W − Ex0 ≤ 0 λ × (Gu − W − Ex0) = 0 Definition (Tøndel et al. [2003]) For an active set of constraints, the linear independence constraint qualification (LICQ) holds if the set of active constraint gradients are linearly independent. First bound for the sets of LICQ active constraints: #I◦

N = N·m

• j=0

q j

• KKT

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 32 / 38

Remarks upon the structure of explicit MPC Preliminaries

Structure of MPC constraints

Up to instant k only the first {u1, . . . , uk} inputs appear in the constraint description → matrix G is lower-block triangular: R1 R2 . . . RN G = q1 × m q2 × 2m qN × Nm where each block Rk describing the kth order constraints has qk rows and k · m columns.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 33 / 38

Remarks upon the structure of explicit MPC Improved bound for active constraints

Improved bound for active constraints

Use the structure of matrix G to note that: at most min(qk, k · m) independent rows in block Rk at most k · m independent rows in the first R1, . . . , Rk blocks Let ik denote the number of constraints selected from each block Rk. Then the conditions 0 ≤ ik ≤ min(qk, k · m),

k

• j=0

ij ≤ k · m, ∀k = 1, . . . , N define all the selections of constraints which can be LICQ and the number of sets of active constraints has at most #I◦

N =

• (i1,...,iN) verifies selection

N

• j=1

qj ij

• elements.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 34 / 38

Remarks upon the structure of explicit MPC Improved bound for active constraints

Illustrative example

For N = 2, m = 1 and q1 = q2 = 4 we have the selection conditions: 0≤ i1 ≤ 1 0≤ i2 ≤ 2 0≤ i1 + i2 ≤ 2

(0, 0) (0, 1) (1, 1) (0, 2) (1, 0)

with the total number of candidate active sets being: #I◦

N = 31

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 35 / 38

Remarks upon the structure of explicit MPC Improved bound for active constraints

Application to bi-level optimization

Consider a bi-level optimization problem: min

y VU(y, z)

subject to GUI ≤ 0 GUE(y, z) = 0 z = arg min

z

VL(y, z) subject to GLI(y, z) ≤ 0 GLE(y, z) = 0

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 36 / 38

Remarks upon the structure of explicit MPC Improved bound for active constraints

Application to bi-level optimization

Consider a bi-level optimization problem with the lower-level problem in KKT form [Hovd, 2011]: min

y VU(y, z)

subject to GUI ≤ 0 GUE(y, z) = 0 λ ≥ 0 GLI(y, z) ≤ 0 GLE(y, z) = 0 λ × GLI(y, z) = 0 ∇zL(y, z, λ, ν) = 0

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 36 / 38

Remarks upon the structure of explicit MPC Improved bound for active constraints

Application to bi-level optimization

Consider a bi-level optimization problem with the lower-level problem in KKT form [Hovd, 2011]: min

y VU(y, z)

subject to GUI ≤ 0 GUE(y, z) = 0 λ ≥ 0 GLI(y, z) ≤ 0 GLE(y, z) = 0 λ × GLI(y, z) = 0 ∇zL(y, z, λ, ν) = 0 Hu + F Tx0 + GTλ = 0 λ ≥ 0 λ ≤ Mλs Gu − W − Ex0 ≤ 0 Gu − W − Ex0 ≥ −Mu(1 − s)

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 36 / 38

Remarks upon the structure of explicit MPC Improved bound for active constraints

Application to bi-level optimization

Consider a bi-level optimization problem with the lower-level problem in KKT form [Hovd, 2011]: min

y VU(y, z)

subject to GUI ≤ 0 GUE(y, z) = 0 λ ≥ 0 GLI(y, z) ≤ 0 GLE(y, z) = 0 λ × GLI(y, z) = 0 ∇zL(y, z, λ, ν) = 0 Hu + F Tx0 + GTλ = 0 λ ≥ 0 λ ≤ Mλs Gu − W − Ex0 ≤ 0 Gu − W − Ex0 ≥ −Mu(1 − s) 0 ≤

qk

• i=qk−1+1

si ≤ min(qk, k · m),

q1+···+qk

• i=1

si ≤ k · m The additional conditions permit only LICQ selections.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 36 / 38

Remarks upon the structure of explicit MPC Partial recursive explicit MPC

Partial recursive explicit MPC

Assume that τ · m constraints from R1, . . . , Rτ blocks are active, then u[0,τ−1] =

• uT

. . . uT

τ

T are uniquely determined: u[0,τ−1] = Ψx0 + φ If the (N − τ)-order explicit MPC is available, then a subset of the N-order explicit MPC can be determined: UN(x0) =

• Ψ

F ji

N−τ (Aτ + BτΨ)

• x0 +
• φ

F ji

N−τBτφ + gji N−τ

• , x0 ∈ Di

N.

where FN−τ, gN−τ characterize the (N − τ)-order solution.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 37 / 38

Remarks upon the structure of explicit MPC Partial recursive explicit MPC

Illustrative example

Let there be the LTI dynamics x+ = 1 1 1

• x +

1 0.3

• u

with constraints −5 ≤ uk ≤ 5,

• −25

−25

• ≤ xk ≤
• 25

25

• −30

−25 −20 −15 −10 −5 5 10 15 20 25 30 −20 −15 −10 −5 5 10 15 20 x1 x2

u0 = 5 u0 = −5

For N = 12 we have 51 regions, 32 of them can be obtained from the N = 11 order problem.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 38 / 38

Remarks upon the structure of explicit MPC Partial recursive explicit MPC

Illustrative example

Let there be the LTI dynamics x+ = 1 1 1

• x +

1 0.3

• u

with constraints −5 ≤ uk ≤ 5,

• −25

−25

• ≤ xk ≤
• 25

25

• −30

−25 −20 −15 −10 −5 5 10 15 20 25 30 −20 −15 −10 −5 5 10 15 20 x1 x2

Critical regions with the same first τ inputs can be merged together (if we care only about the first τ inputs). Particular case: for τ = 1, the 32 regions are merged into 14.

Florin Stoican KUL guest presentation Tuesday 3rd July, 2012 38 / 38

References

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

• K. Fukuda. Polytope examples. URL http://roso.epfl.ch/fukuda/lect/polyex_handout/expoly3.pdf.

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 Systematic Componentwise Ultimate-bound Computation Method. In Proccedings of the 17th World Congress IFAC, 2008.

• M. Hovd. Multi-level Programming for Designing Penalty Functions for MPC Controllers. In Proceedings of the 18th IFAC World

Congress, pages 6098–6103, Milano, Italy, 28 August-2 September 2011. 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 for feedback linearizable systems. In Aceptado en IFAC World Congress, 2008. AB Kurzhanski˘ ı and I. Vályi. Ellipsoidal calculus for estimation and control. Iiasa Research Center, 1997. 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.

Sorin Olaru, Florin Stoican, José A. De Doná, and María M. Seron. Necessary and sufficient conditions for sensor recovery in a multisensor control scheme. In Proceedings 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 Programming Study, 29:175–202, 1986. Florin Stoican, Sorin Olaru, and George Bitsoris. A fault detection scheme based on controlled invariant sets for multisensor systems. In Proceedings 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 multisensor control scheme. In Proceedings of the 29th American Control Conference, pages 4052–4057, Baltimore, Maryland, USA, 30 June-2 July 2010b. Florin Stoican, Sorin Olaru, María M. Seron, and José A. De Doná. Reference governor for tracking with fault detection capabilities. In Proceedings of the 2010 Conference on Control and Fault Tolerant Systems, pages 546–551, Nice, France, 6-8 October 2010c. Florin Stoican, Sorin Olaru, José A. De Doná, and María M. Seron. Zonotopic ultimate bounds for linear systems with bounded

• disturbances. In Proceedings of the 18th IFAC World Congress, pages 9224–9229, Milano, Italy, 28 August-2 September 2011a.

Florin Stoican, Ionela Prodan, and Sorin Olaru. On the hyperplanes arrangements in mixed-integer techniques. In Proceedings of the 30th American Control Conference, pages 1898–1903, San Francisco, California, USA, 29 June-1 July 2011b. Florin Stoican, Ionela Prodan, and Sorin Olaru. Enhancements on the hyperplane arrangements in mixed integer techniques. In Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, pages 3986–3991, Orlando, Florida, USA, 12-15 December 2011c.

• P. Tøndel, T.A. Johansen, and A. Bemporad. An algorithm for multi-parametric quadratic programming and explicit MPC solutions.

Automatica, 39(3):489–497, 2003.

Thank you!

Questions ?