IN THE
OF
C. G. Cassandras
Boston University
cgc@bu.edu,
Christos G. Cassandras
http://vita.bu.edu/cgc
CODES Lab. - Boston University
OUTLINE
¾ COMPLEXITY – FUNDAMENTAL LIMITS
¾ BARGAIN HUNTING IDEAS – AND PITFALLS…
- SAMPLE PATH ANALYSIS
- DECOMPOSITION
- ABSTRACTION
- SURROGATE PROBLEMS
- HIGH PROBABILITY v CERTAINTY
¾ THOUGHTS ON MANAGING COMPLEXITY
Christos G. Cassandras
CODES Lab. - Boston University
COMPLEXITY
PHYSICAL
COMPLEXITY
OPERATIONAL
COMPLEXITY
+
STOCHASTIC
COMPLEXITY
Christos G. Cassandras
NUMERICAL
COMPLEXITY
CODES Lab. - Boston University
THREE FUNDAMENTAL COMPLEXITY LIMITS
1/T1/2
LIMIT
NP-HARD
LIMIT
one order increase
in estimation
Tradeoff
between ACCURACY
INFO.
requires
GENERALITY
and
EFFICIENCY
SPACE
STRATEGY
DECISION
two orders
increase
in
learning
effort
=
of
an
algorithm
SPACE
SPACE
(e.g., [Wolpert
SIMULATION
LENGTH
and Macready,
IEEET)
TEC, 1997]
NO-FREE-LUNCH
LIMIT
Christos G. Cassandras
CODES Lab. - Boston University
THREE FUNDAMENTAL COMPLEXITY LIMITS
1/T1/2
LIMIT
NP-HARD
LIMIT
Effect is
MULTIPLICATIVE!
NO-FREE-LUNCH
LIMIT
Christos G. Cassandras
CODES Lab. - Boston University
A “BARGAIN” EXAMPLE
USING
SAMPLE PATH ANALYSIS
Christos G. Cassandras
CODES Lab. - Boston University
A COMPLEX SYSTEM IN [0, θ]
x(t)
θ
x& = f (u , t ;θ )
UNKNOWN
ff UNKNOWN
Christos G. Cassandras
CODES Lab. - Boston University
A COMPLEX SYSTEM IN [0, θ]
CONTINUED
PROBLEM: Determine θ to minimize:
T
J T (θ ) = ∫ L( x(t );θ )dt
x(t)
0
subject to x& = f (u , t ;θ )
θ
Random ff
Random
Christos G. Cassandras
Random
Random
Events
Events
CODES Lab. - Boston University
A COMPLEX SYSTEM IN [0, θ]
θ
β(t)
α(t)
LOSS
CONTINUED
0
x(t ) = 0, α (t ) − β (t ) ≤ 0
⎧
⎪
x& = ⎨
0
x(t ) = θ , α (t ) − β (t ) ≥ 0
⎪α (t ) − β (t )
otherwise
⎩
x(t)
LOSS
LOSS
θ
Increaseθθ
Increase
keeplow
low
totokeep
WORKLOAD
WORKLOAD
Christos G. Cassandras
Decreaseθθ
Decrease
keeplow
low
totokeep
CODES Lab. - Boston University
A COMPLEX SYSTEM IN [0, θ]
CONTINUED
PROBLEM: Determine θ to trade off LOSS vs WORKLOAD:
⎡ NT
⎤
=E
E
LT[L(θT ()θ=)]∑
dx
⎢∑ ∫ dx ⎥
∫
i =1 Fi ⎢
⎣ i =1 Fi ⎥⎦
NT
T
⎡
⎤
]
=
EQ[Q
(
θ
)
E
xdt
(
θ
)
=
xdt
T T
∫0 ⎢⎣∫0 ⎥⎦
T
LOSS
LOSS
FF11
FF22
WORKLOAD
WORKLOAD
Christos G. Cassandras
CODES Lab. - Boston University
SENSITIVITY ANALYSIS
Try and get :
dLT
LOSS Sensitivity:
dθ
WORKLOAD Sensitivity:
Christos G. Cassandras
dQT
dθ
CODES Lab. - Boston University
SENSITIVITY ANALYSIS - RESULTS
¾ Φ(θ) = set of periods with at least one overflow interval
¾ LOSS Sensitivity:
dLT
= − Φ (θ )
dθ
⎡ dL ⎤ d
E ⎢ T ⎥ = E [LT ]
ESTIMATE
⎣ dθ ⎦ dt
UNBIASED
¾ τk = time between first overflow and end of period, k∈Φ(θ)
dQT
= ∑τ k
¾ WORKLOAD Sensitivity:
dθ k∈Φ (θ )
ge of
No knowled cs,
nami
y
d
d
e
l
i
a
t
e
d
ristics,
e
t
c
a
r
a
h
c
c
stochasti
meters
a
r
a
p
l
e
d
o
or even m
⎡ dQ ⎤ d
E ⎢ T ⎥ = E [QT ]
ESTIMATE
⎣ dθ ⎦ dt
UNBIASED
LOSS
LOSS
FF11
FF22
WORKLOAD
WORKLOAD
τ
Christos G. Cassandras
CODES Lab. - Boston University
THE MORAL…
¾ Often, partial knowledge of system dynamics is adequate
to allow useful inferences from observed state trajectory
data; in particular: SENSITIVITY INFORMATION
¾ This “bargain” applies to a large (but not universal) class
of problems; otherwise, the NFL limit gets you!
¾ What about this system with x& = f ( x, u , t ;θ ) ?
Similar results, but more info.
needed regarding model
parameters
Christos G. Cassandras
Feedback
Feedback
CODES Lab. - Boston University
DECOMPOSITION
AND
ABSTRACTION
Christos G. Cassandras
CODES Lab. - Boston University
HIERARCHICAL DECOMPOSITION
???
FLIGHT PLAN
COMMANDS,
RANDOM
EVENTS
PLANNING
DISCRETE-EVENT
PROCESSES
AIRCRAFT
FLIGHT
DYNAMICS
PHYSICAL
PROCESSES
Christos G. Cassandras
CODES Lab. - Boston University
HIEARARCHICAL DECOMPOSITION
CONTINUED
MODEL
MODEL
TIME
TIMESCALE
SCALE
???
Weeks - Months
Minutes - Weeks
msec - Hours
Christos G. Cassandras
Diff. Eq’s, Flows, LP
PLANNING
DISCRETE-EVENT
PROCESSES
PHYSICAL
PROCESSES
Automata, Petri nets,
Queueing, Simulation
Diff. Eq’s,
Detailed Simulation
CODES Lab. - Boston University
HYBRID CONTROL SYSTEM
What exactly
does that mean?
DISCRETE-EVENT
PROCESSES
CONTROL
CONTROL
PHYSICAL
PROCESSES
Christos G. Cassandras
CODES Lab. - Boston University
WHAT’S A HYBRID SYSTEM?
TIME-DRIVEN
DYNAMICS
z&2 = g 2 ( z2 , u2 , t )
z&1 = g1 ( z1, u1, t )
TIME
x1
x0
x2
x1
x1 = f1 ( x0 , z1, u1, t )
x2 = f 2 ( x1, z2 , u2 , t )
DE
O
“M
”
EVENT-DRIVEN
DYNAMICS
Moreon
onmodeling
modelingframeworks,
frameworks,open
openproblems,
problems,etc:
etc:[Proc.
[Proc.
Wof IEEE Special Issue (Antsaklis, Ed.), 2000]
More
NE of IEEE Special Issue (Antsaklis, Ed.), 2000]
Christos G. Cassandras
CODES Lab. - Boston University
HYBRID SYSTEMS IN COOPERATIVE CONTROL
TARGET
EVENT: threat sensed
THREATS
TIME-DRIVEN
DYNAMICS
BASE
Christos G. Cassandras
EVENT: info. communicated by team member
CODES Lab. - Boston University
HYBRID SYSTEM IN MANUFACTURING
Key questions facing manufacturing system integrators:
• How to integrate ‘process control’ with ‘operations control’ ?
• How to improve product QUALITY within reasonable TIME ?
PROCESS CONTROL
• Physicists
• Material Scientists
• Chemical Engineers
• ...
Christos G. Cassandras
OPERATIONS CONTROL
• Industrial Engineers, OR
• Schedulers
• Inventory Control
• ...
CODES Lab. - Boston University
HYBRID SYSTEM IN MANUFACTURING
CONTINUED
Throughout a manuf. process, each part is characterized by
• A PHYSICAL state (e.g., size, temperature, strain)
• A TEMPORAL state (e.g., total time in system, total time to due-date)
PHYSICAL
STATE
Time-driven
Dynamics
NEW
PHYSICAL
STATE
OPERATION
OPERATION
TEMPORAL
STATE
Christos G. Cassandras
Event-driven
Dynamics
NEW
TEMPORAL
STATE
CODES Lab. - Boston University
DECOMPOSITION
Christos G. Cassandras
CODES Lab. - Boston University
LESS COMPLEX
MORE COMPLEX
TIME-DRIVEN
SYSTEM
What exactly
does that mean?
HYBRID
SYSTEM
EVENT-DRIVEN
SYSTEM
LESS COMPLEX
Christos G. Cassandras
DECOMPOSITION
CODES Lab. - Boston University
ABSTRACTION
(AGGREGATION)
Christos G. Cassandras
CODES Lab. - Boston University
MORE COMPLEX
LESS COMPLEX
TIME-DRIVEN
SYSTEM
ZOOM OUT
HYBRID
SYSTEM
LESS COMPLEX
Christos G. Cassandras
EVENT-DRIVEN
SYSTEM
ABSTRACTION
(AGGREGATION)
CODES Lab. - Boston University
MORECOMPLEX
LESS COMPLEX
TIME-DRIVEN
SYSTEM
HYBRID
SYSTEM
ABSTRACTION
(AGGREGATION)
Christos G. Cassandras
HYBRID
SYSTEM
EVENT-DRIVEN
SYSTEM
DECOMPOSITION
CODES Lab. - Boston University
WHAT IS THE RIGHT ABSTRACTION LEVEL ?
TOO FAR…
model not
detailed enough
TOO CLOSE…
too much
undesirable
detail
JUST RIGHT…
good model
Christos G. Cassandras
CREDIT: W.B. Gong
CODES Lab. - Boston University
DECOMPOSITION IN
OPTIMAL CONTROL
Christos G. Cassandras
CODES Lab. - Boston University
OPTIMAL CONTROL PROBLEMS
Physical State, z
zN
Temporal State, x
xN
• Get to desired final physical state zN in minimum time xN, subject to N-1 switching events
• Minimize: - deviations from N desired physical states (zi - qi)2
x2
1
- deviations xfrom
target
desired times (xi - τi)2
Christos G. Cassandras
CODES Lab. - Boston University
OPTIMAL CONTROL PROBLEMS
Time-driven
Dynamics
Temporal state
N
min ∑ ∫ Li ( zi (t ), ui (t ))dt
u
i =1
xi
xi −1
z&i = gi ( zi , ui , t )
s.t.
xi+1 = fi(xi,ui,t)
Physical state
Event-driven
Dynamics
Christos G. Cassandras
CODES Lab. - Boston University
OPTIMAL CONTROL PROBLEMS
N
CONTINUED
min ∑ ⎡ ∫ Li ( zi (t ), ui (t ))dt + ψ i ( xi )⎤
⎢
⎥
xi −1
u
⎣
⎦
i =1
xi
φi ( xi , xi −1 )
Cost under ui(t)
over [xi-1,xi]
Cost of switching time xi
N
min ∑ [φi ( xi , xi −1 ) + ψ i ( xi )]
u
Let:
i =1
si = xi - xi-1
Assume:
Time spent at ith operating mode
φi ( xi , xi −1 ) = φi ( si )
Christos G. Cassandras
CODES Lab. - Boston University
HIERARCHICAL DECOMPOSITION
z&i = gi ( zi , ui , t )
N
min ∑ [φi ( si ) + ψ i ( xi )]
u
xi+1 = fi(xi,ui,t)
i =1
HIGHER
LEVEL
PROBLEM:
LOWER
LEVEL
PROBLEMS:
s.t.
N
[
min ∑ φi ( si ) + ψ i ( xi )
s
i =1
*
si
]
min φi ( si ) = ∫ Li ( zi (t ), ui (t ))dt
0
ui
s.t.
xi+1 = fi(xi,si,t)
s.t.
z&i = gi ( zi , ui , t )
FIXED si
Christos G. Cassandras
CODES Lab. - Boston University
HIERARCHICAL DECOMPOSITION
zi0
zif
zif+1
zi0+1
zi0+ 2
si
si+1
xi-1
xi+1
xi
min
φi ( si )
f
0
min
f
0
u +1i ( zi +1 , zi +1 , si +1 )
u i ( z i , z i , si )
*
i
0
i
…
θ i ( z , zi , six) = min φi ( zi , ui , si )
f
1
ux2i
REALLY
REALLY
CHALLENGING
CHALLENGING
PROBLEM!
PROBLEM!
Christos G. Cassandras
φi +1 ( si +1 )
“ROUTINE”
“ROUTINE”
OPTIMAL
OPTIMALCONTROL
CONTROL
PROBLEM!
PROBLEM!
f
u ( z , zi , si )
0
i
CONTINUED
[
]
s.t.
min
θ i ( z , zi , si ) + ψ i ( xi )
∑
0 f
xi+1 = fi(xi,ui,t)
z ,z , s
i =1
N
0
i
Typical example:
f
x i + 1 = max( x i , a i + 1 ) + s i ( z i , u i )
CODES Lab. - Boston University
HIERARCHICAL DECOMPOSITION
CONTINUED
Decomposition works well in this case…
…but we still have to solve
all LOWER-LEVEL problems
and a HIGHER-LEVEL problem
[Gokbayrak
[Gokbayrakand
andCassandras,
Cassandras,2000]
2000]
x1
[Xu
[Xuand
andAntsaklis,
Antsaklis,2000]
2000]
…and there is also the issue of
selecting among many possible
modes to switch to
x2
[Bemporad
[Bemporadetetal,
al,2000]
2000]
Christos G. Cassandras
CODES Lab. - Boston University
ABSTRACTION
Christos G. Cassandras
CODES Lab. - Boston University
ABSTRACTION OF A DISCRETE-EVENT SYSTEM
DISCRETE-EVENT
SYSTEM
Christos G. Cassandras
CODES Lab. - Boston University
ABSTRACTION OF A DISCRETE-EVENT SYSTEM
DISCRETE-EVENT
SYSTEM
TIME-DRIVEN
FLOW RATE DYNAMICS
EVENTS
HYBRID
SYSTEM
Christos G. Cassandras
CODES Lab. - Boston University
STOCHASTIC FLOW MODELS (SFM)
θ
CONTROLLER
α(t)
α(t) – p(x)
β(t)
γ(t)
x(t)
(t),ββ(t):
(t):arbitrary
arbitrarystochastic
stochasticprocesses
processes
αα(t),
(piecewisecontinuously
continuouslydifferentiable)
differentiable)
(piecewise
0
x(t ) = 0, λ (t ) − p (0) ≤ 0
⎧
dx ⎪
=⎨
0
x(t ) = θ , λ (t ) − p (θ ) ≥ 0
dt ⎪
otherwise
⎩λ (t ) − p ( x(t ))
λλ(t)(t)==αα(t)(t)--ββ(t)(t)
Christos G. Cassandras
feedback
feedback
CODES Lab. - Boston University
WHY SFM?
¾ “Lower resolution” model of “real” system intended to
capture just enough info. on system dynamics
¾ Aggregates many events into simple continuous dynamics,
preserves only events that cause drastic change
⇒ computationally efficient
(e.g., orders of magnitude faster simulation)
¾ If used for CONTROL purposes,
another “BARGAIN” opportunity arises…
Christos G. Cassandras
CODES Lab. - Boston University
AN EXAMPLE OF A
“BARGAIN”
USING A
“SURROGATE” PROBLEM
Christos G. Cassandras
CODES Lab. - Boston University
THRESHOLD BASED BUFFER CONTROL
“REAL” SYSTEM
K
L(K): Loss Rate
ARRIVAL
PROCESS
LOSS
Q(K): Mean
Queue Lenth
x(t)
PROBLEM: Determine Κ to minimize [Q(K) + R·L(K)]
SFM
θ
β(t)
α(t)
RANDOM
PROCESS
γ(t)
x(t)
RANDOM
PROCESS
SURROGATE PROBLEM: Determine θ to minimize [QSFM(θ) + R·LSFM(θ)]
Christos G. Cassandras
CODES Lab. - Boston University
THRESHOLD BASED BUFFER CONTROL
CONTINUED
25
“Real” System
23
J(K)
21
SFM
19
Optim. Algorithm
using SFM-based
gradient estimates
17
DES
SFM
Opt. A lgo
15
0
5
10
15
20
25
30
35
40
45
K
Christos G. Cassandras
CODES Lab. - Boston University
“SURROGATE” PROBLEM IDEA
DISCRETE SET
“SURROGATE”
CONTINUOUS SET
Observe and estimate
Thisisisnot
not“real”…
“real”…
…but2.
thisObserve
“real”…and estimate
This
…but
this
isis“real”…
(e.g., gradient)
ρn -1
H(rn)
3. Update
ρn
rn
rn = f(ρn)
ρ n +1 = ρ n + ηn H (rn )ρ
1. Transform
ρ∗
r∗
r* = f(ρ*) ???
Christos G. Cassandras
CODES Lab. - Boston University
n+1
WHEN DOES THIS PROVABLY WORK?
¾ Need some structural properties;
otherwise, the NFL limit gets you!
¾ Similarities to Ordinal Optimization
[Ho
[Hoetetal,
al,JDEDS
JDEDS1992]
1992]
¾ Resource allocation problems
[Gokbayrak
[Gokbayrakand
andCassandras,
Cassandras,JOTA
JOTA2002]
2002]
¾ Cooperative control problems – see Session FrA06
Christos G. Cassandras
CODES Lab. - Boston University
BYPASSING COMPLEXITY IN
COOPERATIVE CONTROL
CONTINUED
V2
V3
V4
V1
V5
Christos G. Cassandras
CODES Lab. - Boston University
BYPASSING COMPLEXITY IN
COOPERATIVE CONTROL
V2
CONTINUED
Current Control Horizon
V3
V4
V1
V6
Optimal heading
Over
Event-Driven
Receding Horizon
V5
Christos G. Cassandras
CODES Lab. - Boston University
BYPASSING COMPLEXITY IN
COOPERATIVE CONTROL CONTINUED
MAIN IDEA:
Replace complex Discrete Stochastic Optimization problem
by a sequence of simpler Continuous Optimization problems
But how do we guarantee that vehicles actually
head for desired DISCRETE POINTS?
It turns out they do!
Can replace HARD problem by
several SIMPLER ones…
Christos G. Cassandras
CODES Lab. - Boston University
DANGERS OF DECOMPOSITION, ABSTRACTION
What exactly
does that mean?
TIME-DRIVEN
SYSTEM
EVENT-DRIVEN
SYSTEM
Christos G. Cassandras
CODES Lab. - Boston University
DANGERS OF DECOMPOSITION, ABSTRACTION
a
LOW-RESOLUTION
MODEL
.
e.g., x = f(x,a)
AVERAGING
A(u,
A(u,ωωNN))
aa == E[A(u,
E[A(u,ωω)])]
OUTPUT:
x(a)
A(u,ωω ))
... A(u,
11
INPUT u
RANDOMNESS ω
Christos G. Cassandras
HIGH-RESOLUTION
SIMULATOR
OUTPUT:
Data from
N simulated
scenarios
CODES Lab. - Boston University
WHY THIS FAILS...
a
LOW-RESOLUTION
MODEL
.
e.g., x = f(x,a)
OUTPUT:
x(a)
Averagecorresponds
correspondsto
tounlikely
unlikelyscenario
scenario ⇒
⇒ x(a)
x(a)isisway
wayoff...
off...
Average
MEAN a
Prob.Density
DensityFunction
Function
Prob.
A
ofofA
obtainedfrom
from
obtained
HighResolution
Resolutionmodel
model
High
x
Christos G. Cassandras
CODES Lab. - Boston University
WHY THIS FAILS...
f(E[A])
≠
¾ SIMPLE AVERAGING:
¾ SAMPLE, THEN AVERAGE:
E[f(A)]
If ultimate OUTPUT is x(a) = 0 or 1
this can result in 0 instead of 1
⇒ completely wrong conclusion !
Christos G. Cassandras
CODES Lab. - Boston University
WHAT’S THE WAY AROUND THIS?
QUESTION:
To average or not to average ?
ANSWER:
Average “just enough”
Replace AVERAGE by CONDITIONAL AVERAGES,
one for each CLUSTER
CLUSTER = group of “similar” scenarios from
High Resolution model
CLUSTER ANALYSIS
Christos G. Cassandras
CODES Lab. - Boston University
CLUSTERING
am
LOW-RESOLUTION
a1
...
MODEL
LOW-RESOLUTION
AVERAGING
xx == E[f(a)]
E[f(a)]
MODEL
...
CLUSTERING
aa11
...
A(u,
A(u,ωωNN))
aamm
A(u,ωω ))
... A(u,
11
INPUT u
RANOMNESS
Christos G. Cassandras
ω
HIGH-RESOLUTION
SIMULATOR
CODES Lab. - Boston University
HIGH PROBABLILITY
v
CERTAINTY
Christos G. Cassandras
CODES Lab. - Boston University
HIGH PROBABILITY v CERTAINTY
Birthday Paradox
1
1
51
101 151 201 251 301 351
1
Prob. of Success
0.9
0.8
0.7
0.6
0.1
0.5
0.4
CERTAINTY v 99% CONFIDENCE
0.3
0.01
0.2
0.1
366 v 60
66
51
56
61
41
46
0.001
6
11
16
21
26
31
36
1
0
No. of samples
Christos G. Cassandras
CODES Lab. - Boston University
A BRAVE NEW COMPLEX WORLD…
Christos G. Cassandras
CODES Lab. - Boston University
A BRAVE NEW COMPLEX WORLD…
Christos G. Cassandras
CODES Lab. - Boston University
MANAGING COMPLEXITY
¾ Better HARDWARE and SOFTWARE help…
¾ System ARCHITECTURE v OPERATIONAL CONTROL
¾ HIGH v LOW RESOLUTION models
(Too much detail can hurt)
¾ Know what PROBLEM needs to be solved,
then develop METHODOLOGIES (otherwise, NFL limit gets you!)
¾ MODEL-DRIVEN v DATA-DRIVEN approaches
(Embrace DATA -- and the NETWORK that gets data to you)
Christos G. Cassandras
CODES Lab. - Boston University
ACKNOWLEDGEMENTS
Thanks to several current and former students
whose contributions are
reflected in this talk…
+ Colleagues…
Y.C. (Larry) Ho
Wei-Bo Gong
Yorai Wardi
Ben Melamed
Young Cho
Liyi Dai
Xiren Cao
David Castanon
Jerry Wohletz
Yannis Paschalidis
+ Sponsors…
NSF, AFOSR, AFRL, DARPA, NASA,
EPRI, ARO, UT, ALCOA, NOKIA, GE