passive network-redesign-ntua
DESCRIPTION
The paper examines the problem of systems redesign within the context of passive electrical networks and through analogies provides also the means of addressing issues of re-design of mechanical networks. The problem addressed here are special cases of the more general network redesign problem. Redesigning autonomous passive electric networks involves changing the network natural dynamics by modification of the types of elements, possibly their values, interconnection topology and possibly addition, or elimination of parts of the network. We investigate the modelling of systems, whose structure is not fixed but evolves during the system lifecycle. As such, this is a problem that differs considerably from a standard control problem, since it involves changing the system itself without control and aims to achieve the desirable system properties, as these may be expressed by the natural frequencies by system re-engineering. In fact, this problem involves the selection of alternative values for dynamic elements and non-dynamic elements within a fixed interconnection topology and/or alteration of the network interconnection topology and possible evolution of the cardinality of physical elements (increase of elements, branches). The aim of the paper is to define an appropriate representation framework that allows the deployment of control theoretic tools for the re-engineering of properties of a given network. We use impedance and admittance modelling for passive electrical networks and develop a systems framework that is capable of addressing “life-cycle design issues” of networks where the problems of alteration of existing topology and values of the elements, as well as issues of growth, or death of parts of the network are addressed. We use the Natural Impedance/ Admittance (NI-A) models and we establish a representation of the different types of transformations on such models. This representation provides the means for an appropriate formulation of natural frequencies assignment using the Determinantal Assignment Problem framework defined on appropriate structured transformations. The developed natural representation of transformations are expressed as additive structured transformations. For the simpler case of RL or RC networks it is shown that the single parameter variation problem (dynamic or non-dynamic) is equivalent to Root Locus problems. follow IEEE NTUA SB on facebook: https://www.facebook.com/IeeeNtuaSBTRANSCRIPT
““Structural Transformations of Passive Structural Transformations of Passive Electrical Networks and System Electrical Networks and System
PropertiesProperties””
Professor Nicos Karcanias
Athens, 17 January 2013NTUA
Systems & Control Research Centre,
EEIE DepartmentSchool of Engineering & Mathematical Sciences
General Problem :ThemeGeneral Problem :Theme
TRADITIONAL CONTROL: THE SYSTEM IS ASSUMED FIXED
INTEGRATED SYSTEMS DESIGN: THE SYSTEM EVOLVES THROUGH THE DIFFERENT DESIGN STAGES.
CONTROL AS INSTRUMENT IN GENERALISED DESIGN.
NEW PARADIGM:
“STRUCTURE EVOLVING SYSTEMS”
SPECIAL CASE: “Passive Networks Redesign”
Evolution in Engineering ProcessesEvolution in Engineering Processes
Ad Hoc Networks
Operational Embedding
Top-Down Bottom-Up
Design Stage
Evolution
EARLY STAGES
LATE STAGES
Design Time
Evolution
▲ Integration of Design / Operations of Technological Processes
▲ Power Generation & Distribution Systems under Market De-regulation
▲ Networks of SYSTEMS: Control of Communication/ Traffic Networks
▲ Re-Engineering of Technological Processes
Abstraction of Integrated DesignAbstraction of Integrated DesignCONCEPTUAL
DESIGN
PROCESS SYNTHESIS
OVERALL INSTRUMENTATION
CONTROL OF THE PROCESS
EARLY
LATE
INTEGRATED CONTROL DESIGN:
FORMS OF SYSTEM EVOLUTION:
CASCADE PROCESS EVOLUTION
EARLY / LATE DESIGN EVOLUTION
REDESIGN AND SYSTEMS GROWTH
PROCESSENGINEERING
INSTRUMENT.ENGINEERING
CONTROL ENGINEER
1 1s
1x
2x
-11
1 1s
1R
2R
1
1
1-1
2x
1x
1s
1s
1C
2C
1R
2R
1C
2C
-11
1
2x
1x1s
1s
Passive Network RedesignPassive Network Redesign
▲
GENERAL PROBLEM:
Given a passive network define the required changes in the nature of elements, values of physical parameters and network topology required to achieve a desirable dynamic behaviour.
Network Representation Problem: Admittance and
Impedance Operators (integral-Differential Operators)
Transformation Representation Problem: Describe transformations of Network Reengineering
Network Redesign Problem: Develop and solve structure assignment problems.
Classification of Lamped Classification of Lamped Physical ElementsPhysical Elements
•
A-TYPE ELEMENTS:
Translational mass; Inertia; Electrical capacitance; Fluid capacitance; Thermal capacitance
•
T-TYPE ELEMENTS:
Translational spring; Rotational spring; Inductance; Fluid inertance
•
D-TYPE ELEMENTS: Translational damper; Rotational damper; Electrical resistance; Fluid resistance; Thermal resistance
Network Example
Mechanical Network
Nodal Network Graph
Example: Loop Methodology
••
Impedance ModellingImpedance Modelling
1
1 1 1 2
3
1 1 2 1 1 2
1 1 2 2
1 1( ) ( ) 00
1 1 1 1 1( ) ( ) ( )
1 1 10 ( ) ( )
s f vb k b f
f fb b b m s m s m s
sm s m s m s k
( ) ( )sf s v sZ (s ) Impedance M r x: at iZ(s)
Example: Node Methodology
••
Admittance ModellingAdmittance Modelling
A dm ittance M atrix( ) : Y s
(b1 b2 k1
s) b2 0
b2 (m1s b2 k2
s) (k2
s)
0 (k2
s) (m2s
k2
s)
u1
u2
u3
k1
sv
0F
s( ) ( ) ( )Y s u s i s
The Vertex TopologyThe Vertex Topology
Components Vertex TopologyComponents Vertex Topology
•
Component Topologies
1 2 2
2 2
00
0 0 0
b b bb b
D = :Y D-type
1
2 2
2 2
0 000 -
kk kk k
T= Y : T-type ( 1/s)1
2
0 0 00 00 0
mm
A = Y : A-type (s)
Boolean Symmetric matrices Boolean Symmetric matrices defining the location of the defining the location of the physical elements in the Vertex physical elements in the Vertex Topology.Topology.
A A-Vertex TopologyY : A-typ A-Vertexe h Grap
T T-Vertex TopologyY : T-typ T-Vertex e h Grap
D D-Vertex TopologyY : D-typ D-Vertex e h Grap
The Loop TopologyThe Loop Topology
Components Loop TopologyComponents Loop Topology•
Component Topologies
1 1
1 1 2
1/ 1/ 01/ 1/ 1/ 00 0 0
b bb b b
D= Z : D-type
1
2
1/ 0 00 0 00 0 1/
k
k
T= Z : T-type (s)1 1
1 1 2
0 0 00 1/ 1/0 1/ 1/ 1/
m mm m m
A= Z : A-type (1/s)
Boolean Symmetric matrices Boolean Symmetric matrices defining the location of the defining the location of the physical elements in the Loop physical elements in the Loop TopologyTopology
A A-Loop TopologyZ : A-ty A-Loope p Graph
T T-Loop TopologyZ : T-ty T-Loo p p phe Gra
D D-Loop TopologyZ : D-ty D-Loo p p phe Gra
Admittance ModellingAdmittance Modelling•
is the sum of admittances in loop I
•
is the sum of admittances commonbetween loops i and j
y11 y12 y13 ... y1n
y12 y22 y23 y2n
y13 y22 y33 y3n
y1q y2q y3q ynn
u1
u2
u3
un
is1
is2
is3
isn
yii (s)
yij (s)
Impedance ModellingImpedance Modelling•
is the sum of impedances in loop I
•
is the sum of impedances common
between•
loops i and j
11 12 13 1 1 1
12 22 23 2 2 2
13 22 33 3 3 3
1 2 3
... q s
q s
q s
q q q qq q sq
z z z z f vz z z z f vz z z z f v
z z z z f v
zii (s)
( )ijz s
System & Structural Aspects System & Structural Aspects of Network Modelsof Network Models
•
General Network Operator
1( )s ss
B CW D
Impedance, or Admittance OperatorImpedance, or Admittance Operator
W(s)is symmetric and the structure of B, C and D matricis symmetric and the structure of B, C and D matrices es characterizes the topology of Acharacterizes the topology of A--, T, T-- and Dand D-- type matrices type matrices associated with the network.associated with the network.
1( )s ss
TA DY Y Y Y1( )s ss
AT DZ Z Z Z
Impedance OperatorImpedance Operator
Admittance OperatorAdmittance Operator
AA--D & TD & T--D Networks, Matrix D Networks, Matrix Pencils & RePencils & Re--engineeringengineering
•
SPECIAL CASES: A-D and T-D Networks
1ˆ ˆ( ) ( )s s s ss
A TW W D =CD CB D
ˆA TW (s), W (s) are symmetric structured matrix penare symmetric structured matrix pencils and cils and the structure of B, C and D matrices characterizes the the structure of B, C and D matrices characterizes the topology of Atopology of A--, T, T-- and Dand D-- type matrices associated with the type matrices associated with the network. Furthermore, passivity of the network implies network. Furthermore, passivity of the network implies stabilitystability
PROBLEM:PROBLEM: Define the properties of the spectrum of theDefine the properties of the spectrum of the pencils under pencils under structural transformationsstructural transformations in in
the corresponding networkthe corresponding network( ), ( )s sTAW W
Basics of Network TopologyBasics of Network Topology•
Remark:
The presence of an element of A-, T-
or D-type is expressed by an entry in the corresponding matrix C, B or D, respectively. In particular,
•
(i)
if an element is present in the i-th
loop (node), then its value is added to the (i, i) position of the respective matrix.
•
(ii)
if an element is common to the i-th
and j-th
loop (node), then its value is added to the (i, i) and (j, j) positions, as well as subtracted from the (i, j) ) and (j, i) positions of the corresponding matrix.
•
Structured elementary matrices expressing addition, or elimination of network elements without altering the cardinality of the respective graph.
The RC, RL Matrix PencilThe RC, RL Matrix Pencil••
PROPERTIES OF THE RC, OR RL NETWORK OPERATORPROPERTIES OF THE RC, OR RL NETWORK OPERATOR
[ ]
s
kxk
t t
sF + G (sF + G) (sF + G) 0 F G 0
F = F 0, G = G 0
Given the network pencil we have the following properties :
(i) is regular, ie det and ker{ } ker{ } = { }
(ii) and all eigenvalues are real and non - posi
1 1 fk
}
}
:
kxk
1
1,
sF + G
sF + G (sF + G) F
F G 0 T 0
tive
(iii) All finite eigenvalues are real and index{
(iv) If index{ then deg{det } = rank{ }
(v) If ker{ } ker{ } = { }, and diag{ },
,...,
0{ }
fk k k
tT sF + G T = sI + sI sI block - diag{ }
Passive Networks Redesign Problem
PROBLEM: CHANGE PERFORMANCE OF PASSIVE NETWORKS BY REDESIGN
CHANGE VALUES OF COMPONENTS ALTER NATURE OF COMPONENTS MODIFY EXISTING TOPOLOGY BY REDUCING THE SYSTEM, EXISTING STRUCTURE AUGMENT THE SYSTEM BY ADDING SUBSYSTEMS AND EVOLVING EXISTING
TOPOLOGY
CASE (i): DETERMINE THE RESISTORS IN AN RL NETWORK
IMPEDANCE MATRIX: 1 2Q sL + R Q + DDiagonal design parameters
CASE (ii): DETERMINE THE RESISTORS IN AN RLC NETWORK
IMPEDANCE MATRIX: t t1 1 2 2
1Q sL + R + Q + Q D QsC
PROBLEMS ARE REDUCED TO DETERMINANTAL ASSIGNMENT
Example (1a) of RedesignExample (1a) of Redesign
1 11 111 1
1 1 1 11 1 2 3 3 21 1 2 2
1 13 3 4 32 2
1
0 0 0 00( )0 0
0 0 00
.
R RC CZ s s sR R R R R LC C C C
R R R LC C
s C D sB
Example (1b) of RedesignExample (1b) of Redesign
'3 3 2 2
3
1 , 0 1 0 ttC C b b b eC
1 1 1 1' 11 1 1 1
31 1 1 1 1 1 11 1 2 1 1 2 3
1 13 3
0 0 00 0 0 1 00 00 0 00 0 0 0
C C C CC CC C C C C C C
C C
For the A-type elements:
Example (1c) of RedesignExample (1c) of Redesign
For the D-type elements: '5 1 1 1 1, 1 0 0 ttD D R b b b e
1 1 5 1 5 1'
1 1 2 3 3 1 1 2 3 3
3 3 4 3 3 4
0 0 0 00 0 0
0 0 0 0 0
R R R R R RD
R R R R R R R R R RR R R R R R
For the T-type elements:'
4 12 12 12 1 2, tB B L b b b e e
1 4 4 1 4 4'4
2 4 4 4 2 4
3 3
1 1 0 0 0 0 01 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
L L L L L LB B L
L L L L L LL L
Example (2a) of RedesignExample (2a) of Redesign
Augmented Network
Example (2b) of RedesignExample (2b) of Redesign
1 1 11
1 1 2 2
4 4 5 5
3 5 3 5
11
2 4 4
4 4
3
1/ 1/ 0 0 0 0 0( ) 1/ 1/ 1/ 0 0 0 0 0
0 0 1/ 0 0 00 0 0 1/ 0 0
0 0 0 0 0
0 00 0 0
C C LZ s C C C s L s
C L L LC L L L
RR R R s C sB D
R RR
.
Augmented Impedance: Column-Row Expansion
AA--D & TD & T--D Networks: single D Networks: single Parameter PerturbationsParameter Perturbations
•
PROBLEM: [ ] s
kxk
'
sF + G
F F + F(x,b), G = G + G(x,b),
F(x,b), G(x,b) = xbb
Given investigate the effect of perturbations on the pencil of the type : where , ,
t i i j b = e b = e e
f(s,F, G, x,b) = det(s(F + F(x,b)) + G)f(s,F
or
Study the determinantal assignment problems , G, x,b) = det(sF + G + (G(x,b))
The Network Characteristic The Network Characteristic Polynomial Polynomial (a)(a)
•
Problem Formulation: Binet-Cauchy Theorem
( )
[ ]
i
i j
kk
- k b = eb = e e
f(s,F, G, x,b) = det s(F + F(x,b)) + G =I
= det sF + G, IsF(x,b)
(second order
(firs
variatio
node or loop graph a t order variation)nd or
n)
Assume :
2k k
2k1 k
x
[ ]
t
ktk k k
g(s,F, G) p(s, x,b)
I g(s,F, G) = C sF + G, I , p(s, x,b) = C
sF(x,b) g(s,F, G) :
p(s, x,b) : Grass
Grassmann vector of
mann vector of the
the netw
structu
ork
ral perturbation
The Network Characteristic The Network Characteristic Polynomial Polynomial (b)(b)
•
Lemma: Grassmann
vector of Structural Perturbation
, ,
( )= ( , ( )
i
sx
sign
k,2k
[ ],
) Q
( ) ( ) ( )
j
1,0,.., a 0,...,0 a
1,.., -1, +1,k,...k+
b = e
b =
p(s,x,b),
p s,x,e =
p(s,x,b) e e
first order variation
second order varia
has the structure:
tion
: ha-
:
1 2 3 4,
( ) , r= ( )
( )= ( , ( )= (
i
r
j j i j j j
r sx r
k,2k
[ ]
) Q
( ) , , ,j ( ) ( ) ( ) ( )
( )
1,0,.., a 0,...,0,a 0,...,0,a 0,...,0,a 0,...,0
a 1,2,3,4 1
1 1,2,.., -1, +1,...,k,k+ 2 1,2,.., -1, +1,...,k,k+
p s,x,e e =
s the structure:
-
,( )= ( , ( )= (i i i i j
k,2k
k,2k k,2k
) Q
) Q ) Q3 1,2,..,i -1, +1,...,k,k+ 4 1,2,.., -1, +1,...,k,k+
The Network Characteristic The Network Characteristic Polynomial Polynomial (c)(c)
•
LEMMA: GRASSMANN VECTORS OF STRUCTURAL PERTURBATIONS
, ,
( )= ( , ( )
i
sx
sign
k,2k
[ ],
) Q
( )
first order variation
second order
has
the structure:
variation ha:-
:
( ) ( )
j
1,0,.., a 0,...,0 a
1,.., -1, +1,k,...k+
p(s,x,b),
p s,x,e =
p(s,
b = e
b = e e x,b)
1 2 3 4,
( ) , r= ( )
( )= ( , ( )= (
i
r
j j i j j j
r sx r
k,2k
[ ]
) Q
( ) , , ,
s the structure:
- j ( ) ( ) ( ) ( )
( )
1,0,.., a 0,...,0,a 0,...,0,a 0,...,0,a 0,...,0
a 1,2,3,4 1
1 1,2,.., -1, +1,...,k,k+ 2 1,2,.., -1, +1,...,k,k+
p s,x,e e =
,( )= ( , ( )= (i i i i j
k,2k
k,2k k,2k
) Q
) Q ) Q3 1,2,..,i -1, +1,...,k,k+ 4 1,2,.., -1, +1,...,k,k+
RC, RL Single Variations as RC, RL Single Variations as Root Locus ProblemRoot Locus Problem
••
THEOREM:THEOREM:
ROOT LOCUS FOR RC, RL SINGLE PARAMETER VARIATIONS: The network characteristic polynomial is expressed as:
( ) ( )
+) ( ) (s s
ss
and is formed from the nonzero compon
first order variation
ents of
according to the rules
(i)
F,G F,G,b
F, F,G,bG
p det(sF + G) p(s,
b =
x,b)
e
f(s,F, G, x,b) =z
p xsz
4
1
( ) ( ) ( ), ( )
( )= (
( )
i
s s s
s
k,2k) Q
where is the
component of that corr
second order variation
esponds to
(ii : ) -
: F,G,b F,G, F,G,
F,G,bj F,G,
( )
1, .., - 1, + 1,k, ...k +
b = e e
p(s, x,b)
z z z a
z z
( )
( )
( )= ( , ( )= (j j i j j j
s
s
k,2k k) Q ) Q
where are the components of that corresponds to the sequences
characterising the nonzero elements ie F,G,
1 1, 2, .., - 1, + 1, ..., k, k + 2 1, 2, .., - 1, + 1, ..., k, k +
p(s, x,b) z
,( )= ( , ( )= (i i i i j
,2k
k,2k k,2k) Q ) Q3 1, 2, .., i - 1, + 1, ..., k, k + 4 1, 2, .., - 1, + 1, ..., k, k +
Root Locus & Fixed ModesRoot Locus & Fixed Modes•
CROLLARY: FIXED MODES OF THE SINGLE VARIATION PROBLEM:
( ) ( ) ( ), ( )
( )
s s s
s
The polynomial has a fixed mode iff :
(i) and
where is the component of that corr
first order variatio
esponds to
n : F,G F,G,b F,G,
F,G,
p det(sF +b = e G)
p(s, x,b)
f(s,F, G,x,b)z z
z= (
( )
( ), i s
s
k,2k) Q
second ord
have a nontrivial gcd.
(ii) and the set of
polynomials = which are the compon
er variati
ents o
on
f
:
- F,Gj
F,G,
1, .., - 1, + 1,k, ...k +
1,2,3,4
b = e e p det(sF +G)
p(s, xz
( )= ( ,( )= ( , ( )= ( , ( )= (
j j i
j j j i i
i i j
k,2k
k,2k k,2k
k,2k
) Q
) Q ) Q
) Q
corresponding to the sequences :
have a
1 1,2, .., - 1, + 1, ...,k,k +
2 1,2, .., - 1, + 1, ...,k,k + 3 1,2, ..,i - 1, + 1, ...,k,k +
4 1,2, .., - 1, + 1, ...,k,k +
,b)
nontrivial gcd.
Properties of Root LocusProperties of Root LocusROOT LOCUS STRUCTURE AND PROPERTIES
+ 0( ) ,( )
( ) :( ):
sss s
F,G,F,G
F,G
b
F,G,b
Characte xsristic Equation: psp
zzpole polynomial, zero polynomial
For RC, or RL networks with first, or second order parameter varPROP iatiOSITION: ons
the
p 0 2, p
1,
x > 0, x < 0 following properties hold true :
For all branches are on the Re- axis
(b) If a multiple pole with multiplicity then is a zero of multiplicity
at least ie the root
(a)
( ) ( ) s sF,G F,G,bp sz locus has fixed points
(c) If all cancellations are made between and then for the reduced
root locus we cannot have two poles next to each other.
Interlacing Properties of Root Interlacing Properties of Root LocusLocus
•
INTERLACING PROPERTY OF ROOT LOCUS
t
sF + Gb,
s(F + xbb ) + G x
: For an RC, or RL network described by and
with first, or second order parameter variations if
is regular,
then t
(Interlacing
he
Property) THEOREM
following properties hold true :
(a) All poles and zeros are located on the real axis.
(b) There are no poles and zeros of multiplicity higher than one.
(c) There cannot be two poles, or two zeros nex
z < 0, p, p < z
x < 0
t to each other
(d) a zero, such that for all poles
For all allpoles move to tPole Mo hvement : e l
Interlacing Property
x > 0
eft and for all
poles move to right
Future ResearchFuture Research♦
Provide a new dimension to system theory by exploring the properties of the network representations and network models
♦
Develop descriptions of system structure evolution
♦
Develop a framework for network redesign, by studying structure assignment problems
♦
Provide a formal way for describing issues of duality and analogy
♦
Develop a multi-parameter approach for re-engineering based on the DAP formulation.
APPENDIXAPPENDIX
Dimensional Variability of Dimensional Variability of Interconnection GraphInterconnection Graph
f
t4t5
eba
t6
c
dt1 t2
t3
t7
, , , , , :a b c d e f SCALARS OF VECTORS WITH DIFFERENT DIMENSIONS
1 2 3 4 5 6 7, , , , , , :t t t t t t t VECTOR SIGNALS (VERTICES) & VECTOR FUNCTIONS
EARLY STAGES:
SCALAR SIGNALS (VERTICES) & SCALAR FUNCTIONS (EDGES)LATE STAGES:
VECTOR SIGNALS (VERTICES) & VECTOR FUNCTIONS (EDGES)
PROBLEMS: REPRESENTATION OF DIMENSIONAL VARIABILITY SYSTEM PROPERTIES AND DIMENSIONAL VARIABILITY INVARIANCE
EVOLUTION
Life Cycle Evolution of Systems: Life Cycle Evolution of Systems:
Growth, DeathGrowth, Death
THEME: GRAPH STRUCTURE EVOLUTION PROBLEMS
t6
t9
t8 t4
g
c
d a b e
f
t1 t2 t3
t5
t7
PROBLEMS: GRAPH STRUCTURAL GROWTH PROBLEMGRAPH LOSS PROBLEMGRAPH PRESERVATION WITH EDGES REDESIGN
THE REPRESENTATION PROBLEM
(DEVELOP AN APPROPRIATE REPRESENTATION TO DESCRIBE GROWTH, DEATH)
Canonical ExampleCanonical Example
1 2
3 +
+
+ + +
-1w 1e 1y
2w
2e 2y
3y 3e3w
: , 1, 2, 3i i i i ii
i i i
x A x B eS i
y C x
AGGREGATE SYSTEM: , ,aS A B C1 1 1
2 2 2
3 3 3
0 0 0 0 0 00 0 , 0 0 , 0 00 0 0 0 0 0
A B CA A B B C C
A B C
INTERCONNECTION RULE:
0 0 , 0
0 0
Ie w F y F I I
I
COMPOSITE SYSTEM: , , , ,c aS A B C S A B C F
1 1 3
2 1 2 2 3
3 2 3
0, ,
0
A B CA A B FC B C A B C B B C C
B C A
Composition, Completeness & Equivalent Feedback Configuration
GIVEN:
(i)
TOPOLOGY SYSTEM INTERCONNECTIONS(ii)
FAMILY OF SUBSYSTEM MODELS
COMPOSITION ASSUMPTIONS:
ke
kG
,k jF,1 1kF z
,k j jF z
kwk
kz
jz
1,2,...,j : i i iG e z
EQUIVALENT FEEDBACKCONFIGURATION:
w ++
e G s
F
z
z G s ee w Fz
THE COMPLETENESS ASSUMPTION:(i)
SUBSYSTEM OUTPUT , (ii)
EXOGENOUS SUBSYSTEM INPUTS DEGREES OF FREEDOM :, 1,2,...,k ky z k
kw kl
1dim colsp ;...; , 1, 2,...,k k kl F F k
Equivalent Feedback Configuration: Non-Complete Composite Systems
INTERCONNECTIONS
REMARK:
DEVIATIONS FROM COMPLETENESS CORRESPONDS TO INPUT-OUTPUT DECENTRALISED SQUARING DOWN
REMARK:
THE SYSTEM GRAPH BECOMES ESSENTIAL FOR THE STRUCTURAL PROPERTIES OF NONCOMPLETE SYSTEMS
- AGGREGATEA
INPUTSTRUCTURE
:L OUTPUTSTRUCTURE
:K
u 1L
L0
0
w1
0
0
z y1K
K0
0
COMPOSITE SYSTEM ; ; ; :C a L K
Design Time Dependent Evolution Design Time Dependent Evolution In Integrated DesignIn Integrated Design
ASSUMPTION: INTERCONNECTION TOPOLOGY FIXED, SUBPROCESS MODELS MAY HAVE VARIABLE COMPLEXITY
EARLY (simple)
LATE (complex)
EVOLUTION OF MODELS IN EARLY DESIGN
= 0
1
2
3
…
0
ISSUES: MODELLING IN EARLY –
LATE DESIGN: AN EVOLUTIONARY STRUCTURAL PROCESSES
VARIABLE DIMENSIONALITY MODELLING VARIABLE COMPLEXITY MODELLING
0
1
k
Variable Complexity ModellingASSUMPTION: FIXED DIMENSIONALITY OF VERTICES, BUT VARIABLE COMPLEXITY MODELLING FOR SUBSYSTEMS
= 0
1
2
3
…
0
KERNELMODEL
GRAPH STEADY-
STATEGRAPH + FIRST
ORDER DYNAMICSGRAPH + FULL
LINEAR MODELSGRAPH + NON-LINEAR MODEL
1 :k k k 1kNESTING: IS A STRUCTURED SIMPLIFICATION OF
PROBLEMS:STRUCTURED MODEL REDUCTION THAT PRESERVES INTERCONNECTION RULE.VARIABILITY AND INVARIANCE OF SYSTEM PROPERTIES UNDER STRUCTURED MODEL REDUCTION. PREDICTION OF PROPERTIES OF FULL MODEL FROM THE PROPERTIES OF SIMPLE (EARLY) MODELS.
INVARIANCE EVOLUTION
The Notion of Kernel ModelThe Notion of Kernel Model
: , : :i i i i iie w g e wSUBSYSTEMS:
, :i ie w INPUT, OUTPUT VERTICES
:ig INPUT -
OUTPUT OPERATOR
1,2,...., :a i
i AGGREGATE SYSTEM
1 2 1 2, ,..., , , ,...,e e e w w w
KERNEL GRAPH: A RELATION :C i.e. A SUBSET OF KERNEL FUNCTION:
, ,: :
, ,
i f
i f
j k C
C jk j k j k
j k C
w ef w e f
w e
KERNEL MODEL OF COMPOSITE SYSTEM:
1
* :CC k jk ja a
je f w