1. UNIVERSITAT POLITÈCNICA DE CATALUNYA Programa 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 HuberFrançois E. Cellier (Univ. Politècnica de Catalunya) (Univ. of Arizona) Barcelona, Novembre 2001

2. 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

3. 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

4. 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

5. 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

6. About the FIR methodology Fuzzification Module (recoding) Qualitative Modelling Engine Defuzzification Module (regeneration) Qualitative data Qualitative predictions Qualitative Simulation Engine Model (mask + behaviour) FIR Quantitative predictions Prediction error real-valued trajectories from the system variables

7. About the FIR methodology The Qualitative modelling engine

8. 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

9. 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

10. 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

11. Developed methods New 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

12. Developed methods New 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

13. Developed methods New sub-optimal search algorithm Application to a garbage incinerator plant y(t) = f{y(t-1),y(t-4),x 2 (t-8)} Q opt =0.6548 Q=0.6312 Q loss = 3.60%

14. 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

15. Developed methods Method 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

16. Developed methods Method based on spectral coherence functions Start Compute the cross- coherence function, Cxy and significant peaks for the pair Obtain the significant delays for which each x i 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 variable i = n ? no yes

17. Developed methods Method based on spectral coherence functions Application to a garbage incinerator plant y(t) = f{y(t-1),y(t-8),x 2 (t-9), x 7 (t)} Q opt =0.6548 Q=0.6274 Q loss = 4.18%

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

19. Developed methods Subsystem decomposition methods Reconstruction analysis based method Using FIR to find the structure of a system Statistical method combined with FIR

20. Developed methods Statistical 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

21. Developed methods Statistical method combined with FIR Inclusion of time

22. Developed methods Statistical 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

23. Developed methods Statistical method combined with FIR Linear relationship search. Application to a garbage incinerator plant

24. Developed methods Statistical method combined with FIR Non-linear relationship search Complete subsets of linearly related variables with possible non-linear relations between them - The correlation among (X i *,ξ m ) is calculated, where: X i * =spline(X i ), is a non-linear transformation of variable X i ξ m = linear(X m1 … X mj ) is a linear combination of the j variables from m-th subset.

25. Developed methods Statistical 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 Yes

26. Developed methods Statistical method combined with FIR Application to a garbage incinerator plant y(t) = f{y(t-1), x 1 (t-9), x 2 (t),x 7 (t-14) } S2 --> 30 Models S4 --> 561 Models Classical FIR --> 428.812.560 Models

27. 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

28. Application Gas turbine for electric power generation Gas Fuel system Compressor Combustion Chamber Turbine section Gearbox Generator IGV Exhaust to atmosphere air filter Liquid Fuel system P0P0 T0T0 Q0Q0 QgQg QlQl P1P1 P2P2 T1T1 T2T2 T3T3 P3P3 Electric power to the grid 215 variables reduced to 64 using prior knowledge of the system

29. Application Gas turbine for electric power generation

30. 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.

31. Main contributions Reducing 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

32. Other results FIR 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

33. 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

34. UNIVERSITAT POLITÈCNICA DE CATALUNYA Programa 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 HuberFrançois E. Cellier (Univ. Politècnica de Catalunya) (Univ. of Arizona) Barcelona, Novembre 2001

36. About the FIR methodology Qualitative Simulation Engine Distance Computation Forecasted value Qualitative data Output Forecast Computation 5-nearest neighbours Input patterns matching

37. Developed methods Using FIR to find the structure of a system Perform a cheap variable selection Calculate optimal FIR model of complexity 5 X 20 = f 1 (X 4, X 5, X 6, X 12, X 19 ) 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 X 20 = f 2 (X 4, X 5, X 6, X 19 )  Q = 0.1752 X 20 = f 3 (X 4, X 5, X 12, X 19 )  Q = 0.1735 X 20 = f 4 (X 5, X 6, X 12, X 19 )  Q = 0.1521 X 20 = f 5 (X 4, X 6, X 12, X 19 )  Q = 0.1339 X 20 = f 6 (X 4, X 5, X 6, X 12 )  Q = 0.1165 f 11 f 10 x 7, x 8,x 11,x 14 x 8,x 10,x 11,x 14,x 18 x 19 x5x5 f9f9 x 2,x 9,x 13,x 14 x4x4 f8f8 x 7, x 8, x 13 x6x6 f7f7 x 7,x 9,x 13 f1f1 x 12 x 20 x2x2 x7x7 x8x8 x9x9 x 10 x 11 x 13 x 14 x 18