1.
1
Honeywell Technology Solutions I.I.T. Bombay, India
Decomposition Paradigms for Large Scale Systems
Department of Chemical
Engineering,
IIT Bombay, India.
Consultant – Research
Honeywell Technology
Solutions, Bangalore.
Dr. Ravi Gudi
ACM Technology talk
2.
Honeywell Technology Solutions I.I.T. Bombay, India
Talk Outline
Overview of general decomposition strategies
Approaches to Decomposition – brief preliminaries
Decomposition paradigms
Model coordination
Goal coordination
PSE applications: Optimization, Identification & Control
Illustrative examples & case studies
Concluding remarks.
3.
Honeywell Technology Solutions I.I.T. Bombay, India
Decomposition based problem solving
Systems engineering is posed with lots of challenging
problems from analysis, optimization, and control viewpoints.
A number of elegant solutions to the above class of problems
have been proposed
Generally successful for small to medium scale problems.
Require additional effort for tailoring to large scale applications
Complexity introduced by large scale systems needs to be
analyzed and decomposed for solvability.
Nature of complexity and the application requirements
influences the choice of the decomposition methodology.
4.
Honeywell Technology Solutions I.I.T. Bombay, India
Complexity ⇔ Decomposition
Complexity could be distributed across timescales, spatial
directions, combinatorial nature, etc.
Decompositions could be {hierarchical, spatial and coordinated},
{strategic, tactical, operational}.
Typical applications:
Modeling and Simulation: partitioning
Identification: segregation and composition
Optimization: relaxation and cooperation
Control: Optimizing control, communicationbased
Fault Detection and Diagnosis: discrimination / classification
5.
Honeywell Technology Solutions I.I.T. Bombay, India
Motivation for decomposition
Complex Systems: Challenges offered*
Dimensionality
Computation intensity grows faster than size
Information Structure Constraints
Distributed sources of data
Uncertainty
Interconnections between subsystems; Local relationships can be
modeled accurately.
Typical Applications: Manufacturing systems, Power networks,
Traffic networks, Digital communication networks, ...
*
Siljak (1996), Backx et al. (1998), Lu, (2000)
6.
Honeywell Technology Solutions I.I.T. Bombay, India
System description
System
Causes
(deterministic)
Effect
(measured)
Disturbances/ drifts
Causeeffect relationships could be complex (nonlinear and dynamic) and time varying (normal
versus abnormal situations, parameter shifts etc.).
Modeling & Simulation
Given a cause profiles, predict the effect profile
Optimization
Design the system (parameters) operation to maximize profit
Identification
Determine in an empirical manner the causeandeffect relationship
Control
Facilitate a cause to regulate the effect in the presence of disturbances
Fault detection and diagnosis
Mine the data to reveal data dependencies
7.
Honeywell Technology Solutions I.I.T. Bombay, India
Approaches to decomposition
System
Causes
(deterministic)
Effect
(measured)
Disturbances/ drifts
Represent the overall system in terms of smaller subsystems that are
relatively easily solvable
Issues of efficient partitioning that facilitates coexistence & solution ease
Union of these solutions does not necessarily represent the overall
system solution
Issues of interaction and solution degradation exist.
Coordinate so as integrate the local solutions such that it is optimal for
the entire problem.
8.
Honeywell Technology Solutions I.I.T. Bombay, India
Illustrative example: Control
Slurry
LCO
Gasoline
LPG
Tail Gas
Reactor
Regenerator
Catalyst/
coke
Catalyst
Air
Steam/ Oil feed
Slurry recycle
Main
Column
and Gas
Plant
9.
Honeywell Technology Solutions I.I.T. Bombay, India
Illustrative example: Control
Loop 1
Loop 2
Noise and
unmeasured
disturbancesMVC2 G2
Gd2
y3yd3
MVC1 G1
Gd
yYd
Gd1
u2u1
u3
+

+
+
+
++

Need to evolve a strategy to ‘Think globally but act locally’
10.
Honeywell Technology Solutions I.I.T. Bombay, India
Issues in Decentralized Control
Objective: Decentralize but seek centralized performance
through coordination*1
Decomposition
Controllability and Observability aspects
Vertical or Horizontal decomposition
Decentralized Controller Design*2
: Design independently on the
basis of local subsystem dynamics and the nature of the
interconnections.
*1
Marquardt, CPCVI, (2002), *2
Siljak (1996)
11.
Honeywell Technology Solutions I.I.T. Bombay, India
Coordination based control
MVC 1
MVC 2
MVC 3
MVC 4
Each node receives a plan of the other nodes moves and based on the
interacting dynamics, the node decides on its moves towards optimizing a
global cost.
12.
Honeywell Technology Solutions I.I.T. Bombay, India
Broad paradigms for decomposition*
G1 (m1
,y1
,x1
,x2
) = 0 G2 (m2
,y2
,x1
,x2
) = 0
m1
y1
m2
y2
x1
x2
Model coordination method
*1
Wismer, “Optimization methods for large scale systems
0x)y,G(m,..
),,(
,,
=ts
xymPMin
xym
13.
Honeywell Technology Solutions I.I.T. Bombay, India
Model coordination method
First level
Choose z to minimize
H(z) = H1(z) + H2(z)
min
m1
,y1
P1(m1
,y1
,z1
)H1(z) =
subjected to
G1(m1
,y1
,z1
,z2
) = 0
Determine
min
m2
,y2
P2(m2
,y2
,z2
)H2(z) =
Determine
subjected to
G2(m2
,y2
,z1
,z2
) = 0
m2
,y2
z
Second Level
Multilevel solution using model coordination
z
m1
,y1
14.
Honeywell Technology Solutions I.I.T. Bombay, India
Flow shop scheduling problem
A1
A2
A3
…………
…………
AnA
Platform A
b1
b2
…………bnb
Platform B
C1
C2
C3
Cnc
Platform C
D1
D2
D3
…………
Dnd
Platform D
nA – number of A lines; nB – number of B lines;
nc – number of C lines; nD– number of D lines
…………
15.
Honeywell Technology Solutions I.I.T. Bombay, India
Collaborative problem solving
Platform A Platform B Platform D
Individual formulations are simpler and intuitive when compared with a “monolith”
structure.
May perhaps be easier to solve to optimality at the individual steps.
Specialized solvers depending on nature of the problem can be used.
Often times, “interaction elements” are rather sparse – related to connectivity
Each platform has its individual formulation (constraints and solution method)
but updates the constraint bounds on other platform elements with which it
interacts.
16.
Honeywell Technology Solutions I.I.T. Bombay, India
Collaborative problem solving
PlatformA
Optimizer
PlatformB
Optimizer
,Exit if common cons traints s atis fied
Initialize
Optimizer 1 Optimizer 2
Decomposed
17.
Honeywell Technology Solutions I.I.T. Bombay, India
Some results: Flow shop scheduling problem
Scheduling for the lines in Platform A and B was solved using cooperative problem
solving for two scenarios:
• Cost functions were exactly the same using both approaches for each case.
• Decomposition and cooperation based solving is seen to be vastly superior to
monolith approach.
• Cooperative approach is definitely more scalable.
IterationsTimeProblem Type
940312Cooperative
3370268Monolith
IterationsTimeProblem Type
923412.2Cooperative
3436771Monolith
Scenario1 Scenario2
18.
Honeywell Technology Solutions I.I.T. Bombay, India
Lagrangian Relaxation methods
Broad philosophy:
Relax the constraint space of the problem by augmenting the objective function
with the difficult constraint(s) and solve the relaxed problem
A solution to the less constrained problem is as good as or better than the
constrained solution. For a minimization (maximization) problem therefore, this
relaxation gives a lower (upper) bound to the true solution.
bxh
xgts
xfMin
x
≤
≤
)(
0)(..
)(
Difficult constraints
Problem relaxation
0)(..
])([)(
≤
−+
xgts
bxhxfMin
x
λ
Relaxed problem easy
to solve
19.
Honeywell Technology Solutions I.I.T. Bombay, India
Lagrangian Relaxation methods
Tighten the relaxation
0)(..
])([)(
≤
−+
xgts
bxhxfMin
x
λ
λ
Max
For convex problems, the solution of the above relaxed problem is the same
as that of the original problem.
20.
Honeywell Technology Solutions I.I.T. Bombay, India
Goal coordination method
x1
z1
m1
y1
G1 (m1
,y1
,x1
,z2
) = 0 G2 (m2
,y2
,x2
,z1
) = 0
m2
y2
x2
z2
x1
z1
Interaction balance principle : Require xi
= zi
as a result of goal
coordination
22.
Honeywell Technology Solutions I.I.T. Bombay, India
Combinatorial Complexities: Sensor location in steam metering
flowsheet of methanol plant
5
7
11
9
8
4
3 1
62 10
1 3 15 24
2512
2769
13
4 17
28
14
7
8
20
21
26
18
19
10 11 16
22
2325
Objective: Determine
Sensor locations that
minimize failure rate
subject to cost constraint
*
Serth and Heenan, AIChE (1986)
Problem features:
11 balance equations involving 28
variables.
This flowsheet has a total of 21,474,180 sensor
combinations.Of these, 1,243,845 combinations
form an observable network.
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Honeywell Technology Solutions I.I.T. Bombay, India
Modeling failure rates
Measured Variable
Equal to the failure rate of the sensor measuring the variable
Unmeasured Variable
Sum of the failure rate of the sensors used for estimating the variable
( ) ( )
j
k k
C
j j j i i i
k 1 i C i C
i j j i
ˆ 1 x x 1 x j 1..nλ λ λ
= ∈ ∈
≠ ≠
= − + − ∀ =
∑ ∑ ∏
( )
( )
( ) ( )
j
k k
j
j N
*
j j
j N
i
i E
i
i N
C
j j j i i i
k 1 i C i C
i j j i
ˆMin max
s.t c 1 x C
x S 1, S V
1 x n m
ˆ 1 x x 1 x j 1..n
λ
λ λ λ
∀ ∈
∈
′∈
∈
= ∈ ∈
≠ ≠
− ≤
≥ − ∀ ⊆
− = −
= − + − ∀ =
∑
∑
∑
∑ ∑ ∏
Optimization formulation
Failure rate expression
24.
Honeywell Technology Solutions I.I.T. Bombay, India
Optimization Approaches
Brute Force enumeration
Time Consuming
Greedy Search Algorithms
Robust but do not guarantee optimality
Mathematical programming Techniques
Do not guarantee Optimality for MINLP
Needs an explicit optimization formulation
Constraint Programming
Needs an explicit optimization formulation
Guaranteed global optima and realizations
Easy to generate pareto fronts
25.
Honeywell Technology Solutions I.I.T. Bombay, India
Constraint programming – an illustration
Initial Constraint Propagation
{ }
1
, , 1,2,3
Solve y z
x y
x z
x y z
<
− =
≠
∈
Choice Point & Failure Choice Point & Solution
26.
Honeywell Technology Solutions I.I.T. Bombay, India
Constraint programming – Results on steam metering
500 secsConstraint Programming
2.5 hoursBrute Force Enumeration
50 secs*
MINLP SBB
Time TakenApproach
*
No guarantee of global optimality
27.
Honeywell Technology Solutions I.I.T. Bombay, India
Hierarchical decomposition : Flowshop facility
Line
1
Line
2
Stage
3
A A
A
B
B
B B
C C
C
Tanks Tanks
D
D D
Stage
2
A
B
C
D
Tanks
E
B
A
Illustration
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Honeywell Technology Solutions I.I.T. Bombay, India
Functional decomposition
Planning over a multiperiod horizon:
Order Redistribution
Detailed scheduling in each period:
Overall Inventory Profiles
Operator level inventory scheduling :
Individual Tank Assignments
Level1
Level2
Level3
29.
Honeywell Technology Solutions I.I.T. Bombay, India
Model granularity
Upper bounds on processing times:
Abstraction of total inventory
Upper bounds on total inventory :
Abstraction of total available compatible
tank volumes
Operator level inventory scheduling :
Individual Tank Assignments
Level1
Level2
Level3
Increasing model granularity
Specialized solvers could be used at each levels to fulfil goals at that level
30.
Honeywell Technology Solutions I.I.T. Bombay, India
Spatial decomposition: Model Identification for Control
Plantcontroller
yd
+

y
u
disturbance
d
+
+
Plant
u y Model 1u y
Model 2
Nonlinear plant
Locally linear models
31.
Honeywell Technology Solutions I.I.T. Bombay, India
Case study: high purity distillation
Local Model
y(t)=au(t) +
by(t) +
cy(t)u(t)
Model
Parameters
Gain Time
Constant
1 a 0.0030
b 0.9842
0.19 62.5
2 a 0.0053
b 0.9502
0.1064 18.75
3 a 0.004
b 0.9986
c 0.3424
 
4 a 0.0096
b 0.9963
2.59 260
32.
Honeywell Technology Solutions I.I.T. Bombay, India
Case study: high purity distillation
Switching
Function
33.
Honeywell Technology Solutions I.I.T. Bombay, India
Conclusions
Complexity introduced due to combinatoriality can be reduced using
intelligent enumeration via constraint programming.
Typical applications: problems involving large number of integer/ binary
decision making
Partitioning of large scale problems using collaborative /
communicative approaches simplifies solution procedures without
compromising solution rigor.
Typical application: large scale optimization and control problems.
Lagrangian relaxation methods help to work around difficult
constraints and gradually progress towards the optimal via bounding
and relaxation.
Typical applications: integer programming problems and those
bound by nonlinear constraints.
34.
Honeywell Technology Solutions I.I.T. Bombay, India
Thanks for your attention, Questions ?
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