George M. Coghill

The Morven Framework

Motivation

• To provide properly constructive, constraint based qualitative simulation

• Retain QR ethos• To alleviate the problem of spurious behaviours• General purpose QR• Why a “Framework”

– No system is suitable for all situations– Permits testing and comparison of approaches– Consists in modular constituents

Context

PredictiveAlgorithm

Vector Envisionment

FuSim

Qualitative

Reasoning

P.A. V.E.

QSIM

TQA & TCP

Morven

Constituents

• Predecessors– Variables are represented as vectors– Models are distributed over differential planes– Fuzzy quantity spaces are utilised– Empirical knowledge can be incorporated.

• Specific to Morven– Transitions only generated for state variables– Constructive (assynchronous) simulation– Fuzzy Vector Envisionment– Different approach to prioritisation– Discrete time (synchronous) simulation

Constiuents (2)

• Permits multi-dimensional comparisons• Constructive & Non-constructive

• Simulation & Envisionment

• Synchronous & Assynchronous

The Morven Framework

ConstructiveNon-constructive

Simulation

Envisionment

Synchronous

Asynchronous

Fuzzy Qualitative Reasoning

• Motivation

• Integration of qualitative and vague quantitative information - captured in the nature of fuzzy sets

• Ability to utilise and calculate temporal information in a qualitative simulator

• To include empirically derived information into a qualitative simulator

• 4-tuple fuzzy numbers (a, b, )

• precise and approximate

• useful for computation

μ

a x

x a

x a x a a

x a b

b x x b b

x b

( )

( ) [ ]

[ ]

( ) [ ]

=

< −

− + ∈ −

∈

+ − ∈ +

> +

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

−

−

0

1

0

1

1

x

μA(x)

1

0 a x

(a)

μA(x)

1

0 a b x

(b)μA(x)

1

0 a- a xa+

(c)

μA(x)

1

0 a- b+a b

(d)

A convenient fuzzy representation

Fuzzy Quantity Spaces

μA

(x)

10 x-1 0.2 0.4 0.6 0.8-0.8 -0.6 -0.4 -0.2

n-top n-large n-medium n-small zero p-small p-medium p-large p-top

μA

(x)

10 x-1 0.2 0.4 0.6 0.8-0.8 -0.6 -0.4 -0.2

0

Curve Shapes

+ 0

0

+

_

_

d1d2

Transition Rules• Intermediate Value Theorem (IVT)

– States that for a continuous system, a function joining two points of opposite sign must pass through zero.

• Mean Value Theorem (MVT)– Defines the direction of change of a variable between two points.

[++] [+o] [+-]

[o+] [oo] [o-]

[-+] [-o] [- -]

Single Tank System

V

qi

qo

plane 0qO = kVV’ = qi - qO

plane 1q’O = kV’V’’ = q’i - q’O

plane 2q’’O = kV’’V’’’ = q’’i - q’’O

Single Compartment System

plane 0k10x1 = k10.x1x1’ = u - k10x1

plane 1k10x1’ = k10.x1’x1’’ = u’ - k10x1’

plane 2k10x1’’ = k10.x1’’x1’’’ = u’’ - k10x1’’

1

u

k10.x1

Models in Morven

(define-fuzzy-model <model_name>

(short-name <short_name_of_model>)

(variables <list-of [variable_name, bounds, quantity-space]>)

(auxiliary-variables <list-of auxiliary_variable_names>)

(input <list-of [input_name, bounds, quantity-space]>)

(constraints <list-of [differential_planes (list-of constraints)]>

(print <list-of variable_names>)

)

A JMorven Modelmodel-name: single-tankshort-name: fst

NumSystemVariables: 2variable: qo range: zero p-max NumDerivatives: 1 qspaces: tanks-quantity-spacevariable: V range: zero p-max NumDerivatives: 2 qsapces: tanks-quantity-space tanks-quantity-space2

NumExogenousVariables: 1variable: qi range: zero p-max NumDerivatives: 1 qspaces: tanks-quantity-space

Constraints:NumDiffPlanes: 2

Plane: 0 NumConstraints: 2Constraint: func (dt 0 qo) (dt 0 V) NumMappings: 9Mappings:

n-max n-maxn-large n-largen-medium n-mediumn-small n-smallzero zerop-small p-smallp-medium p-mediump-large p-largep-max p-max

Constraint: sub (dt 1 V) (dt 0 qi) (dt 0 qo)

NumVarsToPrint: 3 VarsToPrint: V qi qo

A JMorven Quantity Space NumQSpaces: 2

QSpaceName: tanks-quantity-spaceNumQuantities: 9

n-max -1 -1 0 0.1n-large -0.9 -0.75 0.05 0.15n-medium -0.6 -0.4 0.1 0.1n-small -0.25 -0.15 0.1 0.15zero 0 0 0 0p-small 0.15 0.25 0.15 0.1p-medium 0.4 0.6 0.1 0.1p-large 0.75 0.9 0.15 0.05p-max 1 1 0.1 0

QSpaceName: tanks-quantity-space2NumQuantities: 5

nl-dash -1 -0.75 0 0.15ns-dash -0.6 -0.15 0.1 0.15zero 0 0 0 0ps-dash 0.15 0.6 0.15 0.1pl-dash 0.75 1 0.15 0

Possible States

state vector state vector1 + + + + 22 + - o +2 + + + o 23 + - o o3 + + + - 24 + - o -4 + + o + 25 + - - +5 + + o o 26 + - - o6 + + o - 27 + - - -7 + + - + 28 o + + +8 + + - o 29 o + + o9 + + - - 30 o + + -10 + o + + 31 o + o +11 + o + o 32 o + o o12 + o + - 33 o + o -13 + o o + 34 o + - +14 + o o o 35 o + - o15 + o o - 36 o + - -16 + o - + 37 o o + +17 + o - o 38 o o + o18 + o - - 39 o o + -19 + - + + 40 o o o +20 + - + o 41 o o o o21 + - + -

Step Response

t

V

Solution Space

21

147

30V

qi

Soundness and Completeness

• Sound– Guarantees to find all possible behaviours of system

• Incomplete– Unfortunately also finds non-existent (spurious) behaviours

• Still useful for ascertaining that a dangerous state cannot be reached.

• Large research effort to remove spurious behaviours– we will skim the surfarce of the surface!

Single Tank System: Ramp Input

V

qi

qo

t

qi

•Input: Stepped Ramp

plane 0qO = kVV’ = qi - qO

plane 1q’O = kV’V’’ = q’i - q’O

plane 2q’’O = kV’’V’’’ = q’’i - q’’O

2 Element Vector Envisionment

State Vector21 + - +12 + 0 +3 + + +5 + + 07 + + -

30 0 + +32 0 + 034 0 + -

21 12 3

30

5

32

7

34

3 Element Vector EnvisionmentState Vector21 + - + -12 + 0 + -3 + + + -5 + + 0 07 + + - +

30 0 + + -32 0 + 0 034 0 + - +

21 12 3

30

5

32

7

34

Distinct Behaviours

t

V 21

12

3

7

3

5

32

34

30

Solution Space

21

7

30

V

qi

123

5

32

34

Total Solution Space: Single Compartment

u 0

u 1

d

1

,d

3

,d

2

x(0)

Cascaded Systems

plane 0qx = k1.h1qo = k2.h2h1’ = qi - qxh2’ = qx - qo

plane 1qx’ = k1.h1’qo’ = k2.h2’h1’’ = qi’ - qx’h2’’ = qx’ - qo’

plane 2qx’’ = k1.h1’’qo’’ = k2.h2’’h1’’’ = qi’’ - qx’’h2’’’ = qx’’ - qo’’

Tank A

Tank B

1 2

u

k12.x1

k20.x2

h1

h2

qi

qx

qo

Cascaded Systems Envisionment

1

11

12

6 2

0 10 13 9

8

7

5

3

4

State h1 h2 qx qo

0 [0 +] [0 0] [0 +] [0 0]1 [0 +] [+ -] [0 +] [+ -]2 [+ -] [0 +] [+ -] [0 +]3 [+ -] [+ -] [+ -] [+ -]4 [+ -] [+ 0] [+ -] [+ 0]5 [+ -] [+ +] [+ -] [+ +]6 [+ 0] [0 +] [+ 0] [0 +]7 [+ 0] [+ -] [+ 0] [+ -]8 [+ 0] [+ 0] [+ 0] [+ 0]9 [+ 0] [+ +] [+ 0] [+ +]10 [+ +] [0 +] [+ +] [0 +]11 [+ +] [+ -] [+ +] [+ -]12 [+ +] [+ 0] [+ +] [+ 0]13 [+ +] [+ +] [+ +] [+ +]

Cascaded Systems Solution Space

h2

h1

h1’=0

h1’=0

111

12

6 2010

13 9

8

7

5

3

4

Complete Solution Space: Cascaded Compartments

u 0

x 1

x 2

d11,d12,d13

d21

d22

d23

Categorisation of Behaviours

Behaviours

SpuriousReal

Non-chatteringChatteringPotentialActual

Fuzzy Set Theory and FQR

• Two main concepts: the cut and the Approximation principle

• The cutA = [p1, p2, p3, p4]

A= [p1+p3(p2+p4(1-

x

μ A ( )x

(a)

x

μ A (x)

α

(b)

Representational Primitives

Arithmetic primitives

Operation Result Conditions

-n (-d, -c, δ,γ) alln

n

d c d d c c

, ,

( )

,

( )

δ

δ

γ

γ+ −

⎛

⎝

⎜

⎞

⎠

⎟

n >0

0 ,n <0

0

m+n (a+ ,cb+ ,dτ+γ,+δ) allm,nm-n (a- ,d -b c,τ+δ,+γ) allm,nm n× (ac,bd,aγ+cτ−τγ,bδ+d+δ) m n> >

0 0

0 0,

(ad,bc,dτ−aδ+τδ,−bγ+c−γ) m n< >0 0

0 0,

(bc,ad,bγ−c+γ,−dτ+aδ−τδ) m n> <0 0

0 0,

(bd,ac,-bδ−d−δ,−aγ−cτ+τγ) m n< <0 0

0 0,

m=[a,b,τ,],n=[c,d,γ,δ]

Representational Primitives (2)

• Functional primitives– More specific than M+/- relations, though still

incomplete– Compiled (tabular) set of fuzzy if-then rules -

permits incusion of empirical information

• Derivative primitive

The Approximation Principle

The Approximation principle facilitates the mapping of the result of a fuzzy operation onto the values in the quantity space of the result variable.A measure of the Goodness of Approximation is

achieved by means of a Distance Metric

d(A, A’) = [(power(A)-power(A’))2+(centre(A)-(centre(A’))2]0.5

power([a,b, = 0.5[2(a+b) + centre([a,b, = 0.5[a+b]

Approximation Principle (2)

Propagated value

4 76 8 11 15

Predicted values

10 12 a

Transition RulesGeneration Rules

(G1) If there exists a derivative,dtmk, such that dtmk is the first non-zero derivative which is a higher derivative than dtjk, then If dtmk>0(dtmk<0) Ifqi∈ℜ,then dtjk+=qi+(dtjk+=qi−) else dtjk+=[ ]qqi i∨+(dtjk+=[ ]qqi i∨−)( 2)G Ifallderivativeshigherthandtjk ,are zero orifj=,nthen dtjk+=[ ]qqqi i i− +∨∨ FilteringRules( )F Iflili>+<−( ),the n¬ok( the adjacencyfilte )r ( 2)F Ifthereexists a derivative,dtmk, sucht hatdtmk is thefirstnon- zero derivativewhichis a higherderivativet handtjk,andmn<,then Ifdtmk>0(dtmk<0), then Ifqi∈ℜ,then Iflili≤≥()then¬ok else Iflili<>()then¬ok

Temporal CalculationsPersistence time, ΔTp:

If0∉+dn,then Δ T

W d

d

p

n

n

∈

+

( )

• calculates the end points of an interval

Relative Arrival time, ΔTa:

If 01∉+dnα, then [ ] [ ]

Δ T

L d U d

d

a

j

n

j

n

n

∈

−+

+

1

1

α α

α

Absolute Arrival time, TA:

TA

n

= Δ Tp

i

( d

n

)

i = 0

n − 1

∑+ Δ T

aj

( d

n

)

j = 1

n

∑

Absolute Departure time, TD:

TD

n

= TA

n

+ Δ Tp

n

TD

n + 1

= TD

n

® Δ Tp

n

+ Δ Tp

n , n + 1

Fuzzy Vector Envisionment

T.A.Q.A.

Generate Filter

Experimental Test

h

qi

qo

Fuzzy Vector State Labels

State Fuzzy Vector State Fuzzy Vector State Fuzzy Vector

1 pt z z z 8 pl nm pm nm 14 pm nm pm nm

2 pt ns ps ns 9 pl nl pl nl 15 ps pl nl pl

3 pt nm pm nm 10 pm pm nm pm 16 ps pm nm pm

4 pt nl pl nl 11 pm ps ns ps 17 ps ps ns ps

5 pl ps ns ps 12 pm z z z 18 ps z z z

6 pl z z z 13 pm ns ps ns 19 ps ns ps ns

7 pl ns ps ns

FVE Graph for a Step Input

34

9 8

14 13

171819

23

8 7

17 16

111213p-medium

p-small

p-large

p-max

p-mediump-small

2

11 10

16 15

567

1

11 10

16 15

5

p-maxp-largeqi

h

Fuzzy Qualitative Behaviours

h

t

72

6

510

16p-small

p-medium

p-large

p-max

Cascaded SystemS

mal

l

Small

Lar

ge

Large

h2

h1

Sm

all

Small

Med

ium

Medium Large

Lar

geH

uge

Huge

h2

h1