Fuzzy Set Systems

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Smart Adaptive Methods in Modelling and Simulation of Complex Systems

Esko Juuso

Control Engineering Group,

Faculty of Technology

University of Oulu

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

EUROSIM

Federation of European Simulation Societies

OULU

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

EUROSIM

Federation of European Simulation Societies

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Control Engineering Group

Competence Pyramid

Detection of operating conditions

- system adaptation

-fault diagnosis, condition monitoring, quality

Intelligent analysers

-sensor fusion

-software sensors

-trends

Intelligent control

-adaptation

-model-based

Measurements

-on-line analysers

-DSP

Intelligent actuators

- model-based

Dynamic simulation

- controller design, prediction

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Outline

• Background

Soft computing: fuzzy set systems

– Hard computing: statistical analysis

• Modelling & Simulation

– Data + Knowledge + Decomposition

• Linguistic equation (LE) systems

– Generalised moments and norms

Nonlinear scaling

– Genetic tuning

• Application examples

• Conclusions

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Detection of operating conditions

Symptom generation

limit values, parameter esimates

analytic, heuristic

condition monitoring

statistical process control (SPC)

Classification and reasoning

case-based reasoning (CBR), models

fault and event trees

cause-effect relationships

novelty detection

Soft sensors

data-collection

pre-processing

normalisation and scaling

interpolation

data quality, outliers

signal processing

feature extraction

sensor fusion

Classification and reasoning methodologies

rule-based, fuzzy, neural, support vector

artificial immune systems

qualitative models, search strategies

Nonlinear multivariable methodologies

steady-state & dynamic

decomposition, clustering, composite models

mixed models

development and tuning

statistical, fuzzy, neural, genetic

Nonlinear process control

feedback

fuzzy, neural, sliding mode

adaptation (on-line, predefined)

model-based (FF, IMC, MPC)

high-level

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Steady-state modelling: Data

Statistical analysis

• Interactions

– Linear, quadratic & interactive  Response surface methodology

(RMS)

• Reduce dimensions

– Principal component analysis (PCA)

– Partial least squares regression (PLS)

Artificial neural networks

• Linear networks

– Regression

– Recursive tuning

• Multilayer perceptron

– Nonlinear activation

• Learning

– Backpropagation

– Advanced optimisation

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Steady-state modelling: Knowledge

Fuzzy arithmetics

• Extension principle

• Interval arithmetics

• Horizontal systems

Rules and relations

• Linguistic fuzzy

• Takagi-Sugeno fuzzy

• Singleton

• Fuzzy relational models

Type-2 fuzzy sets

• Uncertainty about the membership functions

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Fuzzy Fuzzy arithmetics

Fuzzy set systems

Fuzzy rulebase

Fuzzy relations

Fuzzy

Fuzzy

Crisp

Fuzzification

Fuzzy aritmetics

Fuzzy

Fuzzy reasoning

Defuzzification Crisp

Fuzzy

Fuzzy Fuzzy inequalities

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Steady-state modelling: Decomposition

Modelling

• Subprocesses

• Hierachical

• Composite models

– Linear parameter varying (LPV)

– Piecewise affine (PWA)

– TS fuzzy models

– Ensemble of redundant neural networks

Clustering

• Hierarchical

• Partitioning: K-means

• Fuzzy

– Fuzzy c-means (FCM)

– Subtractive

• Neural:

SOM

• Shape

(Gustafson-Kessel)

• Robust

• Optimal number

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Data mining

Complex applications: Fuzzy set systems

Domain expertise

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

EXPERTISE

Rules

Fuzzy Set Systems

+ Handling of uncerainty

+ Natural compromises

+ Easy to build (small systems)

+ Explanations

- Tuning (complex systems)

- (Doubts about stability)

Expert Systems

+ Extracting expert knowledge

- Complexity

- Handling of uncertainty

- Testing

Linguistic Equations

+ Very compact

+ Combining knowledge

+ Generalisation

+ Adaptive tuning

+ Easier testing

- Structure Restrictions

Chaos Theory

•Risk Analysis

•Economical factors

Knowledge-base alternatives

Genetic Algorithms

+ Large search space

+ Global/local optimisation

+ Design

- Computer Time Consuming

- Not for Control (off-line)

Neuro-fuzzy

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Neural Networks

+ ”Automatic” Modelling

+ Black Box Modelling

+ Precision (small systems)

- Only for Fragments

- Explanations

- Safety

- Precision (complex systems)

DATA

NN Structures

Fuzzy set systems

Linguistic equation systems

Linear interactions

Smart adaptive applications

Modelling

- Control

- Diagnostics

Meaning

How to define??

Hard computing??

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Data

Adaptation of scaling functions

- Generalised norms and moments

- Constraints

- Case specific

Data selection

- Outliers

- Suspicious

Nonlinear scaling

- Feasible ranges

- Membership definitions

- Membership functions

Adaptation

- Manual

- Neural

- Genetic

Linguistic relations

- Selected and scaled data

Linguistic equation alternatives

- Linear regression

- Case specific

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Selected equations

Domain expertise

Variable grouping

- 3-5 variables

- Include/exclude

- Correlation

- Causality

Selected variable groups

Manually defined equations

Final variable groups

Statistical analysis: norms

• A generalised norm about the origin

M

 p

N p

 

N s

(

M

 p ) 1 / p 

1

(

N i

N 

1 x i

(

) p

) 1 / p , p is a real number which is the l p norm 

M

 p p

 x

(

) p

.

• Special cases

– absolute mean x

(

)

1

 x

(

 av

)

1

N i

N 

1 x i

(

)

,

– rms value x

(

)

2

 (

 x rms

) 

1

(

N i

N 

1 x i

(

)

2

)

1 / 2

,

• Positive and negative values

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Generalised norms

• equal sized sub-blocks 

K

S

M

 p p

1

K

S i

K  S

1

(

M

 p

) i

1 / p

 p

 1 /

 p

1

K

S i

K  S

1

(

M

 p

) i

1 / p

,

• A maximum from several samples max(

M

 p

)

 i max

1 ,..., K

S

(

M

 p

)

1 i

/ p

• Increasing

(

M

 p

)

1 / p 

(

M q

)

1 / q p

 q

Recursive analysis!

x

(

)

1

 i

N 

1

N

1 x i

(

)

,

… x

(

)

1

1

N i

N 

1 x i

(

)

,

… x

(

)

2

1

(

N i

N 

1 x i

(

)

2

)

1 / 2

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Generalised moments

Normalised moments

• Skewness k

E

X

– Positive 

3

0

– Symmetric

X

E (

  k

– Negative

3

0

X

3

)

 k

0

• Generalised moment

 

E

X

(

) k 

X

 k

M

 p p k

• Locally linear if possible

• Corrections for corner points

• Core [( c l

) j

, ( c h

)]

• Support [min( x j

), max( x j

)] k = 3 Skewness k = 4 Kurtosis

Central value

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

LE: nonlinear scaling

linear models (interactions)

Data

Meaning

Expertise

Knowledge-based information: labels to numbers

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Tuning

(1) Core

Second order polynomials

[( c l

) j

, ( c h

)]

(2) Ratios

(3) Support

 j

 



1

3

, 3



  j



1

3

, 3



[min( x j

), max( x j

)] c j

• Centre point

• Corner points

 min( x j

), ( c l

) j

, ( c h

) j

, max( x j

)

• Calculation a

 j

1

2

( 1

  j

)

 c j

, b

 j

1

2

( 3

  j

)

 c j

, a

 j

1

2

(

 j

 

1 )

 c

 j

, b

 j

1

2

( 3

  j

)

 c

 j

X j

 b j

 b j

2 with x j

 max( x j

) b j

 2 

2 a

 j

4 a

 j

( c j

 x j

)

2 with c j

 x j

 max( x j

) b

 j

2 

4 a

 j

( c j

 x j

)

2

2 a j

2 with x j with

 min( x j

) min( x j

)

 x j

 c j

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

LE models: Dynamic simulator

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Genetic tuning

• Membership definitions

– Parameters

– No penalties

• Normalised interactions

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Lag phase

Exp.

phase

Decision system

X

X

Fuzzy weighting

+

Integration

Prediction

Steady state

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

X

Submodels

Fuzzy LE blocks

Measurements

CO

2 forecast

Volumetric mass transfer

Coefficient, k

L a

OTR forecast

DO forecast

Note: 3 phases & 3 models / phase  9 interactive dynamic models!

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

LE Application examples: Control

• Energy:

– Solar power plant

• Environment:

– Water circulation & wastewater treatment

~ 4 m

• Pulp&Paper:

– Lime kilns

Length > 100 m

Slow rotation: rotation time 42-45 s

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Solar thermal power plant

• Setpoint tracking

Principle: lower irradiation  lower temperatures

Operator can choose the risk level: smooth … fast

• Cloudy conditions www.psa.es

Clouds

High temperature are risky

Cloudy conditions are detected from fluctuations of irradiation  Working point is limited  Further limitations for the setpoint

• Optimisation

Constrained optimisation:

-Temperature (< 300 o C)

- Temperature increase (< 90 o C)

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Solar thermal power plant

• Intelligent control

– Adaptation, braking, asymmetrical action

– Automatic smart actions

– Disturbances are handled well if the working point is on a good level

• Intelligent indices

– react well to disturbances (clouds, load, …)

• Model-based limits for the working point

 Better adaptation

 Smooth adjustable operation

 A good basis for optimised operation within a Smart Grid

MODEL-BASED

CONTROL

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

LE Application examples: Diagnostics

• Stress indices

– Cavitation

• Condition indices

– Lime kiln

• Fatigue

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

Conclusions

• Soft computing

– Expertise

– Fuzzy reasoning

Complex systems

• Interactions

– Fuzzy set systems

– Linguistic equations

• Hard computing

– Data

– Statistical analysis

• Meaning

– Membership definitions

 Membership functions

• Generalised norms and moments

• Nonlinear scaling

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

EUROSIM

Federation of European Simulation Societies

34th Board Meeting in Vienna, February 2012,

NSS became an observer member of EUROSIM

COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

EUROSIM 2016

September 13-16, 2016, Oulu, Finland

The 9 th EUROSIM Congress on Modelling and Simulation

Oulu City Theatre

30 COMOD 2014

St. Petersburg, Russia, 2-4 July 2014

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