UNIVERSITAT POLITÈCNICA DE CATALUNYAPrograma de doctorat: Tecnologies avançades de la producció

Qualitative Modelling of Complex Systems by Means of Fuzzy Inductive Reasoning. Variable Selection and

Search Space Reduction.

Josep Maria Mirats Tur

Directors:

Rafael Huber François E. Cellier

(Univ. Politècnica de Catalunya) (Univ. of Arizona)

Barcelona, Novembre 2001

Context and motivation

• To model and simulate the output or outputs of a system in order to control it

• To solve the modelling and simulation problem we can use Deductive and/or Inductive modelling approaches

Context and motivation

• Unfortunately FIR, in its previous state, could not deal with large-scale systems

•FIR, is a modelling and simulation methodology capable of generating an input-output model

•It qualitatively learns the behaviour of a system from its real past data. Interesting for ill-defined systems

Objectives

• Problem: To obtain a qualitative model of a (ill-defined) system with a large number of measurable variables

• Objective: To reduce the FIR model search space in order to solve the aforementioned problem

Index of subjects

• About the Fuzzy Inductive Reasoning methodology

• Main lines of research in the dissertation

• Developed methods– Sub-optimal mask search algorithm– Method based on spectral coherence functions– Subsystem decomposition methods

• Applications

• Conclusions and future work

About the FIR methodology

Fuzzification Module

(recoding)

Qualitative Modelling Engine

Defuzzification Module

(regeneration)

Qualitativedata

Qualitativepredictions

Qualitative Simulation Engine

Model(mask + behaviour)

FIR

Quantitativepredictions

Prediction error

4.22.58.61.0

3.14.12.31.4

2.51.14.07.2

1.84.31.02.1321 yxxx

real-valued trajectories from the system variables

42.2

33.1

27.5

98.7

y

About the FIR methodologyThe Qualitative modelling engine

1101

1111

1111

mcan

))(),(),(),2(()( 113 txttyttxttxfty

t

tt

tt

m

yxxx

2

1004

3002

0100321

x1 x2 x3 y1

0 1 2 1 3t 2 2 1 3 i1 i2 i3 i4 o1 i1 i2 i3 i4 o1

2t 1 1 3 2 1 2 3 1 2 1 1 2 1 13t 1 1 2 1 1 1 2 1 1 1 2 3 1 24t 2 2 3 1 3 1 1 2 1 2 2 1 2 15t 2 1 2 1 2 2 1 2 1 3 1 1 2 16t 1 2 1 3 3 2 1 1 3 3 2 1 1 3

Dynamic relations Static relations Sorted static relations(Data system) (Behaviour System)

121 c

nnn

Index of subjects

• About the Fuzzy Inductive Reasoning methodology

• Main lines of research in the dissertation

• Developed methods– Sub-optimal mask search algorithm– Method based on spectral coherence functions– Subsystem decomposition methods

• Applications

• Conclusions

Main lines of research

Methods that obtain a decomposition of

the system into subsystems

Sets of variables containing maximum information about the system

Sets of variables maximally related between them

Methods that directly simplify the

FIR mask search space

Sub-optimal mask search algorithms

Reducing the number of possible m-input variables to the FIR model

Index of subjects

• About the Fuzzy Inductive Reasoning methodology

• Main lines of research in the dissertation

• Developed methods– Sub-optimal mask search algorithm– Method based on spectral coherence functions– Subsystem decomposition methods

• Applications

• Conclusions

Developed methodsNew sub-optimal search algorithm

• A new approach for reducing the model search space of FIR is proposed

• The algorithm is based on proposing to FIR mask candidates of increasing depth

• It can deal with previously unmanageable large-scale MISO systems

Developed methodsNew sub-optimal search algorithm

Complexity, c Depth, d = 1,

Mcan = (-1 -1 …. -1 -1 +1)

Exhaustive search keeping information about Q of each mask

Find the highest quality mask Qbest, and compute relative

quality of all masks Qrel=Q/Qbest

Determine the good masks, with quality Qrel > s

Determine significant inputs as those used at least in the t% of all

good masks

c = 2

c < 5

c = c+1

Yes

Does the overall quality

increases?No

Yes

No

END

FIR sub-optimal models together with a value for parameter d are obtained

Elaborate matrix M where the rows r d are filled by -1 at the locations of significant m-inputs and 0 for insignificant m-inputs

d = d+1

Propose a new candidate matrix

MMcan

11...11

Developed methodsNew sub-optimal search algorithm

Application to a garbage incinerator plant

Number of visited models

10

1depthdepthmasksvisited

Number of models tovisit using a depth 10full candidate mask

Percent ofcomputationalleviation

1.362.501 64.704.850 97.89%

y(t) = f{y(t-1),y(t-4),x2(t-8)}

Qopt=0.6548

Q=0.6312

Qloss = 3.60%

Index of subjects

• About the Fuzzy Inductive Reasoning methodology

• Main lines of research in the dissertation

• Developed methods– Sub-optimal mask search algorithm– Method based on spectral coherence functions– Subsystem decomposition methods

• Applications

• Conclusions

Developed methodsMethod based on spectral coherence functions

•Computing the energy of the signals it can be determined at which delays each input variable has more energy related to the output

•To propose FIR a unique sparse candidate matrix to obtain a dynamic qualitative model of a large-scale system

•Each variable trajectory is seen as the collection of values measuring a desired physical characteristic

Developed methodsMethod based on spectral coherence functions

Start

Compute the cross-coherence function, Cxy and significant peaks for the pair

<xi,y>

Obtain the significant delays for which each xi is most related to the output in energy

terms.

Decide a mask depth, d

Form a mask candidate matrix with

information from delays 2 up to d.

Fill rows 0 and 1 of the candidate matrix,for example using the suboptimal algorithm

of Section 2.4.1

Compute the corresponding FIR

models

End

Last input variablei = n ?

no

yes

Developed methodsMethod based on spectral coherence functions

Application to a garbage incinerator plantNumber of visited models

10

1depthdepthmasksvisited

Number of models tovisit using a depth 10full candidate mask

Percent ofcomputationalleviation

Energy method 4.785.690 64.704.850 92.58%Energy method combined withthe new sub-optimal search

algorithm

2.229.636 64.704.850 96.54%

y(t) = f{y(t-1),y(t-8),x2(t-9), x7(t)}

Qopt=0.6548

Q=0.6274

Qloss = 4.18%

Index of subjects

• About the Fuzzy Inductive Reasoning methodology

• Main lines of research in the dissertation

• Developed methods– Sub-optimal mask search algorithm– Method based on spectral coherence functions– Subsystem decomposition methods

• Applications

• Conclusions

Developed methodsSubsystem decomposition methods

• Reconstruction analysis based method

• Using FIR to find the structure of a system

• Statistical method combined with FIR

Developed methodsStatistical method combined with FIR

• Inclusion of time in the analysis

• Variable selection to eliminate redundancy

• Linear relationship search

• Non-linear relationship search

• FIR modelling from formed subsystems

Developed methodsStatistical method combined with FIR

Inclusion of time

Developed methodsStatistical method combined with FIR

Variable selection. Linear relationship search

• Forming subsets of linearly related variables- Singular value decomposition of the remaining correlation matrix

- Projection of the eigen-vectors onto the principal axes

-Groups of variables are formed

•A cheap variable selection by means of a correlation analysis is performed to eliminate information redundancy

Developed methodsStatistical method combined with FIR

Linear relationship search. Application to a garbage incinerator plant

X9

X14

X8

X1 X9

X3 X14

X8 X15 X16 X17

Groups of linearly correlated variables

S1 X5 X8 X14

S2 X4 X6 X7 X12 X13

S3 X2 X11 X12

S4 X6 X7 X13

S5 X8 X14

Resulting subsets of linearly

related variables

Developed methodsStatistical method combined with FIR

Non-linear relationship search

• Complete subsets of linearly related variables with possible non-linear relations between them

- The correlation among (Xi*,ξm) is calculated, where:

Xi*=spline(Xi), is a non-linear transformation of variable Xi

ξm = linear(Xm1… Xmj) is a linear combination of the j variables from m-th subset.

Developed methodsStatistical method combined with FIR

FIR modelling

Use cluster variables as mask candidate matrix

Number of variables >5 ?

Calculate optimal FIR model of complexity 5

Use cluster variables as FIR model

Add to list of good FIR models

Last cluster ?

Use FIR simulation to determine best model

Yes

No

NoYes

Developed methodsStatistical method combined with FIR

Application to a garbage incinerator plant

y(t) = f{y(t-1), x1(t-9), x2(t),x7(t-14) }

Final obtained subsystems

S1 x10(t-14) x18(t-5) x18(t-7) x18(t-8) x18(t-10)x18(t-14)x19(t-14) y(t-10)

S2 x4(t-1) x6(t-14) x18(t-13) y(t) y(t-1) y(t-4)

S3 x5(t-14) x9(t-14) x10(t-14) x18(t-1) x18(t-2) x18(t-3) x18(t-4) x18(t-14) x19(t-14)

S4 x1 x1(t-9) x2 x6(t-14) x7(t-14) x8 x11(t-2) x12 x14 y(t) y(t-1) y(t-4)

S5 x5(t-14) x18 x18(t-1) x18(t-11) y(t-10)

S2 --> 30 ModelsS4 --> 561 Models

Classical FIR --> 428.812.560 Models

Index of subjects

• About the Fuzzy Inductive Reasoning methodology

• Main lines of research in the dissertation

• Developed methods– Sub-optimal mask search algorithm– Method based on spectral coherence functions– Subsystem decomposition methods

• Applications

• Conclusions

ApplicationGas turbine for electric power generation

Gas Fuel

system

Compressor

CombustionChamber

Turbinesection

Gearbox GeneratorIGV

Exhaust to atmosphere

airfilter

Liquid Fuel

system

P0

T0

Q0

QgQl

P1P2

T1T2

T3 P3

Electric power to the grid

215 variables reduced to 64 using prior knowledge of the system

ApplicationGas turbine for electric power generation

Classical FIR Energy method Energy method + Newsearch algorithm

Subsystemdecomposition method

Number ofmodels tocompute

644.938.981.680 1.218.190.170 3.548.379 255 + 516 = 816

Computationtime required

711 years(estimation)

1 year and 125days (estimation)

About 35 hours About 30 hours(29.9 + 0.1)

Achievedreduction

- 99.98% 99.99% 99.99%

Conclusions•Since the FIR modelling engine is of exponential complexity, new techniques had to be devised that would reduce the number of masks to be visited•This can be accomplished either by reducing the number of ‘-1’ elements in the mask candidate matrix, or by decomposing the system into subsystems.•The so enhanced FIR toolbox can now easily cope with large-scale systems comprising of dozens if not hundreds of variables.•The new tools were built in a modular fashion so that they can be combined to form a variety of search-space reduction algorithms.

Main contributionsReducing FIR model search space

• New sub-optimal mask search algorithm

• Spectral coherence functions based method

• Subsystem decomposition:• Using Fuzzy Reconstruction Analysis

– Re-implementation of the FRA module

• Using FIR to find the structure of a system

• Using statistical techniques

Other resultsFIR methodology

• Improvement to the FIR simulation engine - Corrected five-neighbours prediction formula

• New use of the unreconstructed variance methodology

• The concept of variable acceptability, ‘envelopes’• A variable similarity measure based on a modified

Hr value

Future research

•More thorough validation of the search-space reduction algorithms

•Investigate alternative algorithms to include time in the analysis

•Parameters intrinsic to the sub-optimal search algorithm and the energy method

•Is a subsystem decomposition preferable to a whole model?

•Parameters intrinsic to FIR

UNIVERSITAT POLITÈCNICA DE CATALUNYAPrograma de doctorat: Tecnologies avançades de la producció

Qualitative Modelling of Complex Systems by Means of Fuzzy Inductive Reasoning. Variable Selection and

Search Space Reduction.

Josep Maria Mirats Tur

Directors:

Rafael Huber François E. Cellier

(Univ. Politècnica de Catalunya) (Univ. of Arizona)

Barcelona, Novembre 2001

About the FIR methodologyQualitative Simulation Engine

Distance Computation

Forecasted value

Qualitative data

Output Forecast Computation

5-nearest neighbours

Input patterns matching

x1 x2 x3 y1 1 2 3 11 2 1 32 2 1 3 i1 i2 i3 i4 o1

1 1 3 ? 1 1 2 1 11 1 2 ? 1 2 3 1 22 2 3 ? : : : : :2 1 2 ? 1 2 3 1 2: : : : 3 2 1 1 3

Input state vector

Developed methodsUsing FIR to find the structure of a system

Perform a cheap variable selection

Calculate optimal FIR model of complexity 5

X20 = f1 (X4, X5, X6, X12, X19) Q = 0.2089

For each input, starting by the less important

one, obtain a FIR model

Determine the relative importance of the inputs used by the model

X20 = f2 (X4, X5, X6, X19)Q = 0.1752X20 = f3 (X4, X5, X12, X19)Q = 0.1735X20 = f4 (X5, X6, X12, X19)Q = 0.1521X20 = f5 (X4, X6, X12, X19)Q = 0.1339X20 = f6 (X4, X5, X6, X12)Q = 0.1165

f11

f10

x7, x8,x11,x14

x8,x10,x11,x14,x18

x19

x5

f9x2,x9,x13,x14

x4

f8x7, x8, x13

x6

f7x7,x9,x13 f1

x12x20

x2

x7

x8

x9

x10

x11

x13

x14

x18