DecisionandOptimization DenisArzelierdptinfo.ens-paris-saclay.fr/Visites/do.pdf · Diagnosis: - Design and analyze methods for state estimation, fault de-tection, diagnosis and prognosis

DO

Decision and Optimization

Denis Arzelier

Scientific Domains

Composition : 3 teams, 41 CNRS/AP/P (25 HDRs), 3 AM, 3 PostDoc,40 Ph. D.

- DISCO: DIagnosis, Supervision and COntrol

- ROC: Operations Research Combinatorial Optimization and Constraints

- MAC: Methods and Algorithms for Control

Academic Fields:

Automatic Control

Applied Mathematics

Operational Research

Artificial Intelligence

Applied fields:

Aerospace

Energy

Transport Systems

Networks

Biotechnologies

Healthcare

Diag.

DO

A.I.

O.R.

A.C.

Appli. Math.

Optim.

ms∂2w

∂t2+ ζ(w)

∂w

∂t+ Y Is∆

2w = ∆m

x(t) = f(x(t), u(t), d(t), p(t), ξ)

y(t) = g(x(t), w(t)), x0 ∈ [X0]

y1∨

y2∨

y3,

y1∨

¬y2∨

y3,

y1∨

y2∨

¬y3,

y1∨

¬y2∨

y1¬y3,

DISCO

ROC

MAC

Scientific Communities

Visite ENS 2015 Decision and Optimization 2

Applied Fields

Composition : 3 teams, 41 CNRS/AP/P (25 HDRs), 3 AM, 3 PostDoc,40 Ph. D.

- DISCO : DIagnosis, Supervision and COntrol

- ROC : Operations Research Combinatorial Optimization and Constraints

- MAC : Methods and Algorithms for Control

Academic Fields:

Automatic Control

Applied Mathematics

Operational Research

Artificial Intelligence

Applied fields:

Aerospace

Energy

Transport Systems

Networks

Biotechnologies

Healthcare

Diag.

A.I.

O.R.

A.C.

Appli.Math.

Optim.

DISCO

ROC

MAC

Applied Fields


Scientific Objectives

Diagnosis:

- Design and analyze methods for state estimation, fault de-

tection, diagnosis and prognosis

- Continuous, discrete event and hybrid dynamical systems

Optimization:

- Design and analyze methods and algorithms

- Combinatorial optimisation and constraint propagation for

tasks planning and scheduling, resource allocation, polynomial

optimization and moment approach

Control Theory:

- Design and analyze control and supervision systems

- Robust S/C, AWU systems, hybrid S/C, observers


Life, National and International Collaborations

Organization and life

- 3 autonomous teams (budget and science)

- Scientific council (DOsc) = 3 × 2 delegates + 3 team leaders:

coordination, setting priorities, decision making (monthly based)

- Every-two-years internal scientific seminar on specific cross-

cutting topics (robustness, optimization)

- DO seminars: P. Baptiste, C. Bessiere, E. Bradley, F. Dufour,

J.C. Geromel, P. Nyberg, P. Pollett, L. Rodriguez, R. Sirdey, W.

Tucker, J.M. Vincent

DO

MNBT

ROB

ROB

GE

RCIC

NII

National and International Collaborations


Decision and Optimization: Robustness

Robustness

Robust control/diagnosis/optimization

in uncertain environments

- Robust optimization/Stochastic optimiza-

tion

- Set membership estimation for diagnosis

- Worst-case analysis and synthesis for S/C

stability and performance−1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

Re(s)

Im(s)

[A][x] = [b]

minx

f0(x,Θ)

s.t. f(x,Θ) 0, ∀ Θ ∈ U

Systemu

d

y

Objective. Guaranteed certification (worst-case) of solutions for all

instances in the feasible uncertainty set U

minx

supΘ∈U

f0(x,Θ)

s.t. supΘ∈U

f(x,Θ) 0, ∀ Θ ∈ U

[

xtj+1

]

=[

xtj

]

+

k−1∑

i=1

hif[i](

[

xtj

]

) + hkf[k](

[

xtj

]

)

maxΘ∈GU

‖Σ(Θ,K)‖∗ ≤ maxΘ∈U

‖Σ(Θ,K)‖∗ ≤ γg


Robustness: Identifiability, Stability Analysis, Scheduling Set-Membership identifiability/µ-SM identifiability

SM-I/µ-SM Idiff. alg.−→ restricted partial injectivity/p.i. of φ ∈ R

n(p1, · · · , pp)

- Mincing

- Evaluating

- Regularizing

P

Pµ

Pcµ

Regularizing

Mincing

Evaluating

φ(p)

−1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

p1

p2

Bernoulli equation y(t) = −p21y(t) + p22y(t)2

z

y

w

u [1]

[2]

[ν]Σ

Σ

Σ

K

Robust performance analysis maxΘ∈GU

‖Σ(δ,K)‖∗ ≤ maxδ∈U

‖Σ(δ,K)‖∗ ≤ γg

maxδ∈∆

minP (δ)

‖Σ(δ,K)‖∗ P (δ) =

ν∑

i=1

δiP[i]

γg = minP (δ)∈P

maxδ∈∆

‖Σ(δ,K)‖∗

[

0 P [i]

P [i] 0

]

+ S

[

A[i]′

−1

]

[

F G

]

≺ 0

Robust optimization for resource-constrained project scheduling with uncertain activity durations

minShi,xij ,ρ

ρ

ρ ≥ Shn+1 − S

∗n+1(p

h) h = 1, · · · , |P|

Shj ≥ S

hi + p

hi − M(1 − xij), (i, j) ∈ V

2

Shi ≥ 0 ∀i ∈ V, ρ ≥ 0, X ∈ X

0

0

11

1

1

1

11

1

1

1

1

1

2

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

45

5

5

5

5

5

m1

m1

m3

m3

m2

m2


Decision and Optimization: Optimization

Continuous and Discrete Optimization

1− x1 x1 + x2 x1

x1 + x2 2− x2 0

x1 0 1 + x2

0

x2 − x3 + y2 + y4 + z3 − z4 = 0

Polynomial Optimization and Generalized Moment Problem

(GMP)

minµ∈M(K)

∫

f0dµ

s.t.∫

fjdµ ≧ bj

- Measure Theory

- Duality and SDP relaxations

- Putinar’s positivity certificates and SOS representations

- Applications to impulsive optimal control, inner approxima-

tions of robust stability and attraction regions

Combinatorial Optimization Problems (COP)

- Multi-objective, uncertainties, human factors

- MILP formulations, CSP, graphs

- Scheduling, transportation, resource allocation

S2

S12S1 + 3S2 ≥ 9


Polynomial and Multi-Objective Optimization

f∗ = minx∈K

f(x) K =

x ∈ Rn

: gj(x) ≥ 0, j = 1, · · ·m

⊂ Rn

compact

f, gj ∈ R[x] Lagrangian Relaxation

- SDP relaxations (γd)d∈N, n small

γdf.c.−−−→d↑

f∗, f et−gj Conv and∇

2f(x

∗) ≻ 0 ⇒ γ1 = f

∗, γd = f

∗

- LP relaxations (θd)d∈N, n high

θd −−−→d↑∞

f∗, conditioning ↓

- Hybrid relaxations (ρd)d∈N

ρdf.c.−−−→d↑

f∗, f,−gj SOS − Conv+Slater ⇒ ρ1 = f

∗,K ⊆ 0, 1 ⇒ finite convergence

−6−4

−20

24

6

−6

−4

−2

0

2

4

6

−500

0

500

1000

xy

f(x,y)

f(x, y) = minf1(x, y), f2(x, y)

Generic branch-and-cut method for multi-objective optimization problems

f1 = −5.09 0 (1.94, 4.92)

f1 = −3.24 1 (1, 3.5)

f1 = −1.28 2 (0, 2)

f1 = −3.92 0 (2, 3)

f1 = −2.92 1 (1, 3)

f1 = −1.28 2 (0, 2)

f1 = −5.09 0 (1.94, 4.92)

Infeasible

Infeasible

f1 = −4.56 0 (2, 4)

Infeasible

Infeasible

Infeasible

Infeasible

Infeasible

x2 ≤ 3

x1 ≤ 1

x2 ≥ 4

x1 ≥ 2

minx1,x2

−1.00x1 − 0.64x2

minx3

x3

50x1 + 31x2 ≤ 250

3x1 − 2x2 ≥ −4

x1 + x3 ≤ 2

x1, x2 ≥ 0 integer

x3 ∈ 0, 1, 2−5 −4 −3 −2 −1 1

−1

−0.5

0.5

1

1.5

2

2.5

3

f1

f2

0

0


Hybrid Systems

Decision and Optimization: Hybrid Systems

x0

x1

x2

x3

x4

x5

C

D

x+ ∈ G(x)

x ∈ F (x)

Continuous time (flow) and discrete time (jumps)

(differential, difference and logic equations)

hybrid S/C for linear and nonlinear systems

Hybrid formalism and systems with impacts

Hybrid formalism and limited information

quantified I/O, sampling

Mode automata: Continuous and discrete dynamics

Abstraction of the continuous dynamics for diagnosis

Aging laws for prognosis

Uncertain continuous dynamics

x(t, p2) = f2(x(t, p2), u(t), p2)

y(t, p2) = h2(x(t, p2), p2)

x(t0, p2) = x0 ∈ X0

p2 ∈ P2 ⊂ Up2 , t0 ≤ t ≤ T

x(t, p4) = f4(x(t, p4), u(t), p4)

y(t, p4) = h4(x(t, p4), p4)

x(t0, p4) = x0 ∈ X0

p4 ∈ P4 ⊂ Up4 , t0 ≤ t ≤ T


Global Exponential Stability via Hybrid S/C

Reset rules

x+ = Gx

y u

xc, τ

xc = Acxc + Bcy

τ = 1 − dz

(

τ

ρ

)

u = Ccxc + Dcy

˙xp = Aexp + Beu + Ly

yp = Cpxp + Dpu

x+c , τ+

xp

Hco

yp

x =[

x′p x′

c e′]′

x(t) = Ax(t)

τ(t) = 1 − dz

(

τ

ρ

)

if x ∈ F or τ ∈ [0, ρ]

x+ = Gx

τ+ = 0

if x ∈ J and τ ∈ [ρ, 2ρ]

A = 0 × [0, 2ρ], ρ ∈ (0, ρ∗] is GES

F and J defined by Lyapunov/(xp, xc)

I/O behaviour for different G

xc(0, 0) = 0, xp(0, 0) = −[

1 1]′

State-feedback/observer-based output feed-

back

Effects of the dynamics of the observer: Over-

shoot

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

Time

Pla

nt O

utpu

t

0 1 2 3 4 5 6 7 8 9 10

0

0.5

1

Time

Pla

nt In

put

LinearBeker et al. 2004Prieur et al. 2011Theorem 1Theorem 3


Active and Passive Diagnosis of Hybrid Models

S = M(Q,Σ, T, q0) + Ξ(ζ,Q,C, ζ0)

q1

o1

o2

o2

f1f1

uo1

q2

q3

q4

Automata with enriched behavior

- ARR Analytic Redundancy Relations

ρpici (n) = Ωpi

i Ypi(n)− Ωpi

i Lpii (Ai, Bi, Ci, Di)U

pi(n)

- Residuals generation

rij = Boolean consistency indicator

- Mode signature

Sig(qj) =[

STj/1, · · · , S

Tj/j , · · · , S

Tj/m

]T

q1

o1

R012R013

R032

R024

o2

o2

f1

f1

uo1

q2 q3

q4

q12

q32

q24

q13

o1

o2

o2

(q1, )

(q3, )

(q3, f1)

(q4, )

(q4, f1)

Diagnoser

- State machine based on obser-

vations

- Estimation of actual mode

- Test of diagnosability

- On-line diagnosis

- Identify occurred faults

- Active diagnosis

o1

o1

R012

R013

R032

R032

R024o2 o2

o2

(q2, f1)

(q1, )

(q3, )

(q4, )(q24, f1)(q4, f1)(q3, f1)


Constraint Satisfaction Problems

Demo: M.-J. Huguet, ”Efficient algorithms for multimodal path computation”

Propagation and Analysis of Conflicts for Sequencing Constraints

SAT solvers and constraints propagation

AtMostSeqCard constrainst

Sequencing constraint:

Not more than 2 air conditioning for a sub-sequence of 3 vehicules

(vehicules assembly)

Propagation of sequencing constraints

Linear optimal arc consistency algorithm (O(n)) AtMostSeqCard(2/5/4)

Accelerating methods for the tree search via conflicts learning

Explain non satisfiability of constraints and conflict-driven clause-

learning ¬x2 ∧ ¬x3 ∧ ¬x4


Meta-Diagnosis Reasoning

ALGs

SDs OBSs

Non-faulty

diagnoser

Ont.

true

Covering

Complete Correct

Ontologically

true

Model-based diagnosis

- Knowledge about the system SD (predicate logic) + Com-

ponents CPS + Observations OBS + Algorithm ALG

- Diagnosis: Health status (Ab(ci) or ¬Ab(ci)) of compo-

nents satisfying constraints DS ∧OBS with ALG

Introduction of the meta-diagnosis problem

- How the solution of (SD,OBS,CPS) with ALG could be

faulty?

- Analyze required properties: correction - completion - cov-

ering

Design and solution of a meta-diagnosis problem

- Choose assumptions, meta-components MeCPS

- MEDITO: MeDP = (MeSD,MeCPS,MeOBS)

design

- Solution of MeDP by successive CSPs oriented by

minimal conflicts

¬Ab(mc1) ¬Ab(mc2) ¬Ab(mc3)

¬Ab(mc1)

∧¬Ab(mc2)

¬Ab(mc2)

∧¬Ab(mc3)

¬Ab(mc1)

∧¬Ab(mc3)

¬Ab(mc1)

∧¬Ab(mc2)

∧¬Ab(mc3)

stop stop

stoppropagation

minimal

conflict


Decision and Optimization for Embedded Systems

Embedded Control, Diagnosis and Optimization

- Limited capacity (CPU, memory, format)

- Real-time and communication constraints

- Rigorous computing (symbolic-numeric ob-

jects)

- Algorithms with guaranteed conver-

gence/performance

- Stochastic modelling−1 −0.5 0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8

9

10

x

y

γ=3

γ=2.7

γ=2.5

γ=3.5

γ=2.6

γ=2.3

γ=2.2

Formation/fleet of vehicles

- Limited or non standard information struc-

ture for control and estimation systems

- Non standard information structure

- Distributed actuators and sensors

- Distributed functional organization

- Distributed Optimization

LP/LMI/SDP/NLP = embedded technology?


Impulsive Space RDV problem Some results and theoretical stakes

Terre

Target

orbitcible

Initial chaser

orbit

ChaserTarget

−→R1(t0)

−→R2(t

0)−→R2(tf)

−→ R1(t

f) −→V

1 (t0 )

−→V2 (t

0 )

−→V1(tf )

−→V2(tf )

−−→

∆V

1

−−→

∆V

2

−→V2(t2)

−→ρ (t0)

−→ρ (tf)

1

2

3

4

01

23

45

67

0.5

1

1.5

2

2.5

3

3.5

4

4.5

iterations

true anomaly

prim

er v

ecto

r no

rm

- Open-loop or closed-loop/direct or indirect Architecture

for guidance/control

- Direct methods

Robustness, optimality / uncertainties and GNC sys-

tems errors

Collision avoidance and stable orbits

- Indirect methods

Impulsive OC (Primer Vector), PMP

Orbital perturbations

Technological and numerical stakes

- Toolbox PolyRDV 1.0r

- Embedded algorithm (LEON - 32-bit - SPARC-V8 RISC)

- Certified optimal trajectories via rigorous computations

- Collaborations: CNES (MAC+ROC), ADS, Region, TAS

0

-1 -0.5 0.5 1

f → (Tn, ∆)


Some Facts and Figures: Industrial Partnerships

Space Systems

- ∼10 PhDs (CNES/*, CNRS/*, CIFRE)

- ∼10 direct scientific collaborations

Aeronautics

- AIRSYS, CORAC, OCKF, SIRASAS

- 1 CIFRE, 1 direct PhD grant

Transportation Systems

- ANR, OSEO

- 3 CIFRE (1 SNCF - 2 MOBIGIS)

- Demo: M.-J. Huguet, ”Efficient algorithms for multi-

modal path computation”

Healthcare

- ONCOMATE Project (IMRCP, LBB, INNOPSYS SA, MNBT)

- 1 ANR BIOSTEC INNODIAG (DENDRIS, ITAV, ICR, MNBT)


Internships for 2015

Diagnosis and hybrid models:

- Health Monitoring of a system based on adaptive models

- Learning of temporal patterns for complex systems: Application to mobile

robots

Control, space rendezvous, computer science:

- Convergence proof of an algorithm for space rendezvous planning

- Robust analysis of an impulsive control algorithm for space rendezvous

CreditESA

Scheduling and combinatorial optimization:

- Scheduling with multiple energy sources

- Design and development of a constraint model for optimization of

tests of planning

- Solution of a crossdocking problem

- Scheduling of an aircraft assembly chain

- Algorithms for alternate path computation in multimodal graphs

- S-coloration of graphs


Documents

DecisionandOptimization DenisArzelierdptinfo.ens-paris-saclay.fr/Visites/do.pdf · Diagnosis: - Design and analyze methods for state estimation, fault de-tection, diagnosis and prognosis