““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