Fuzzy Membership Function Optimization for System Identification Using an Extended Kalman Filter

Transcription

1 Fuzzy Membership Fuctio Optimizatio for System Idetificatio Usig a Eteded Kalma Filter Srikira Kosaam ad Da Simo Clevelad State Uiversity NAFIPS Coferece Jue 4, 2006 Embedded Cotrol Systems Research Lab

2 Overview iputs Real system Fuzzy logic system Adaptatio algorithm Embedded Cotrol Systems Research Lab

3 Overview 1. Itroductio 2. Membership optimizatio 3. Eteded Kalma filter 4. Fuzzy model idetificatio 5. Eperimetal results 6. Coclusio ad future work Embedded Cotrol Systems Research Lab

4 Itroductio Fuzzy system performace depeds o Rule base Membership fuctios Give a rule base, is membership fuctio idetificatio trivial? I most real world problems it is ot Membership fuctio tuig improves performace Embedded Cotrol Systems Research Lab

5 Itroductio Fuzzy membership fuctio idetificatio Derivative-based methods Simple Least squares Backpropagatio Optimal filterig Derivative-free methods Geetic algorithms Swarm optimizatio Other methods of computer itelligece Embedded Cotrol Systems Research Lab

6 Itroductio Advatage of Kalma filterig Use of optimal state estimatio theory Ituitively attractive i that we are idetifyig ukow parameters usig available theory Should work better tha first-order methods (e.g., gradiet descet) Opes up ew research i the area of membership fuctio optimizatio Embedded Cotrol Systems Research Lab

7 Overview 1. Itroductio 2. Membership optimizatio 3. Eteded Kalma filter 4. Fuzzy model idetificatio 5. Eperimetal results 6. Coclusio ad future work Embedded Cotrol Systems Research Lab

8 Membership Optimizatio Cosider triagular membership fuctios for simplicity ad proof of cocept Each membership fuctio is characterized by its cetroid, lower half-width ad upper halfwidth (c, a, ad b, respectively) c a c cb Embedded Cotrol Systems Research Lab

9 Membership Optimizatio Degree of membership for the i th membership fuctio of the j th iput is 1 ( cij)/ aij if aij ( cij) 0 fij( ) 1 ( cij)/ bij if 0 ( cij) bij 0 otherwise Similar equatio for output membership fuctios Embedded Cotrol Systems Research Lab

10 Membership Optimizatio The mappig from the fuzzy output to a crisp value is give by crisp output m( γ ) γ J j 1 j j m( γ j 1 j) where m( ) is the output membership fuctio, γ j ad J j are the cetroid ad area of the j th output fuzzy membership fuctio, ad is the umber of fuzzy output sets J j j Embedded Cotrol Systems Research Lab

11 Membership Optimizatio Derivative-based traiig Cosider E 1 2N N q 1 g q ( y q yˆ q 2 ) g is the user defied weightig fuctio y is the target value ad y-hat is the output of the fuzzy system E ca be optimized w/r to cetroids ad half widths Embedded Cotrol Systems Research Lab

12 Overview 1. Itroductio 2. Membership optimizatio 3. Eteded Kalma filter 4. Fuzzy model idetificatio 5. Eperimetal results 6. Coclusio ad future work Embedded Cotrol Systems Research Lab

13 Eteded Kalma filter Proposed by Rudolf Kalma Highly celebrated for the purpose of optimal state estimatio Developed i late 1950 s ad early 1960 s Embedded Cotrol Systems Research Lab

14 Embedded Cotrol Systems Research Lab Eteded Kalma filter Cosider where is the state, w is the process oise, d is the measuremet, ad v is the measuremet oise v h d w f ) ( ) ( 1

15 Eteded Kalma filter Cosider Ε[( )( ) ] P T Ε ( w w ) T l T l Ε ( v v ) Qδ Rδ l l Embedded Cotrol Systems Research Lab

16 Embedded Cotrol Systems Research Lab Eteded Kalma filter The problem addressed by the KF is to If the oliearities are smooth, the system ca be simplified to ) 0 ( give of ˆ fid 1 1,..., j d j H h v H d F f w F ˆ ) ( ˆ ˆ ) ( ˆ 1

17 Embedded Cotrol Systems Research Lab Eteded Kalma filter Kowig the system dyamics, the desired estimate ca be computed as T T T Q F P H K P F P h d K f P H H R P H K ) ( )] ( ˆ [ ) ( ˆ ˆ ) (

18 Eteded Kalma filter If the computatioal effort is too high, ad the derivative of h( ) w/r to does ot chage too much durig optimizatio, the followig assumptio ca be made: H H 0 h( ) ˆ0 Embedded Cotrol Systems Research Lab

19 Overview 1. Itroductio 2. Membership optimizatio 3. Eteded Kalma filter 4. Fuzzy model idetificatio 5. Eperimetal results 6. Coclusio ad future work Embedded Cotrol Systems Research Lab

20 Fuzzy model idetificatio Cosider a permaet maget sychroous motor (a oliear system) The fuzzy system model ca be used to predict the motor output, which ca the be used for closed-loop cotrol Embedded Cotrol Systems Research Lab

21 Fuzzy model idetificatio Cosider a two-iput, oe-output fuzzy logic system Cosider a fuzzy system which has µ fuzzy sets for the first iput, ν fuzzy sets for secod iput, ad κ fuzzy sets for the output Embedded Cotrol Systems Research Lab

22 Fuzzy model idetificatio Embedded Cotrol Systems Research Lab

23 Fuzzy model idetificatio If both the half-widths ad cetroids for the iputs ad outputs are cosidered as parameters defiig the fuzzy system, the state vector of the dyamic system for the KF ca be give as [ a a p b b q r 1 c c a... a p κ µ 1 ν 2 q κ b b ν 2 r µ 1 κ ] c c T µ 1 ν elemets Embedded Cotrol Systems Research Lab

24 Embedded Cotrol Systems Research Lab Fuzzy model idetificatio System descriptio For implemetig the Kalma filter, we add artificial oise to the model ) ( 1 h d v h d w ) ( 1

25 Overview 1. Itroductio 2. Membership optimizatio 3. Eteded Kalma filter 4. Fuzzy model idetificatio 5. Eperimetal results 6. Coclusio ad future work Embedded Cotrol Systems Research Lab

26 Eperimetal results 0.3 Actual measuremet Actual Measuremet 0.25 Widig Curret Time Embedded Cotrol Systems Research Lab

27 Eperimetal results 0.3 Referece measuremet Referece Measuremet 0.25 Widig Curret Time Embedded Cotrol Systems Research Lab

28 Eperimetal results KF traiig progress Error Fuctio Value Iteratio Embedded Cotrol Systems Research Lab

29 Eperimetal results Fuzzy output 0.3 Fuzzy Estimate 0.25 Widig Curret Time Embedded Cotrol Systems Research Lab

30 Eperimetal results Output membership fuctios before ad after traiig 1 Output membership fuctios before traiig 1 Output membership fuctios after traiig Level of Membership membership level Embedded Cotrol Systems Research Lab

31 Coclusio ad future work Optimizatio of fuzzy parameters ca be posed as a optimal filterig problem Fuzzy systems ca be used as black-bo models for geeral dyamic systems Kalma filter parameters ca be further tued Other filterig algorithms could be used (e.g., H-ifiity ad usceted Kalma filters) Establish covergece/stability results of the Kalma filter for this applicatio Embedded Cotrol Systems Research Lab