Milestone about Fuzzy control

Transcription

1 UMR CNRS 8201 Milestone about Fuzzy control Thierry Marie GUERRA LAMIH, UMR CNRS 8201, Université de Valenciennes et du Hainaut-Cambrésis Le Mont Houy, Valenciennes Cedex 9, France Tel : (+33) guerra@univ-valenciennes.fr

2 Where I come from Valenciennes Paris UMR CNRS Researchers, 20 Administrative staff, 70 PhD and Postdocs Identity Human-centered Safety, Mobility and Interacting Systems

3 Fuzzy milestone As a matter of introduction Fuzzy theory is wrong, wrong, and pernicious. What we need is more logical thinking, not less. The danger of fuzzy logic is that it will encourage the sort of imprecise thinking that has brought us so much trouble. Fuzzy logic is the cocaine of science. Professor William Kahan UC Berkeley Fuzzification is a kind of scientific permissiveness. It tends to result in socially appealing slogans unaccompanied by the discipline of hard scientific work and patient observation. Professor Rudolf Kalman Univ Florida (1972 cited in [Zadeh 2011]) Fuzziness is probability in disguise. I can design a controller with probability that could do the same thing that you could do with fuzzy logic. Professor Myron Tribus, UCLA (Bayesian) on hearing of the fuzzy-logic control of the Sendai subway system IEEE Institute, may And for the 90 s control community in France it was even worse Well Fuzzy will have 50 years in

4 As a matter of summary with some fuzzyness Fuzzy control: the pioneers, the first applications Limitations, getting older How to go further? Is Fuzzy dead? Tracks 4

5 Classical approach of control + - Cont??? Controller Model based Goals: performances, robustness 5

6 Historical approach of fuzzy control + - Human??? Rules based on knowhow Highly complex systems and/or hard to model, identify Experts knowledge capture Fuzzy controller 6

7 Historical approach of fuzzy control Nonlinear mapping from control to output For example: u t f t, t 7 And by sake of comparison with linear control FC-PD

8 I Historical approach of fuzzy control Fuzzyfication Fuzzy reasoning Defuzzyfication Inputs Rules such as: Outputs Gains If x1 t is TP And x2 t is TN Then ut is N u * u * CG MM Gains u t f t, t Nonlinear mapping between control and error replaces the usual plane from linear PI 8

9 Historical approach of fuzzy control + - Human??? Rules based on knowhow Highly complex systems and/or hard to model, identify Experts knowledge capture Fuzzy controller 9 Tuning: need for optimization step or for adaptive mechanism

10 Fuzzy control history Ebrahim MAMDANI Imperial College of Science, Technology and Medicine, University of London (passed 2010) Mamdani, E.H., Advances in the linguistic synthesis of fuzzy controllers, International Journal of Man-Machine Studies, Vol. 8, pp , st known application: Homblad, P. & Ostergaard, J-J. (1982) "Control of a cement kiln by fuzzy logic", in M.M. Gupta and Elie Sanchez, eds., Fuzzy Information and Decision Processes pp , Amsterdam: North Holland. Adaptive control using mechanism to modify the rule base in real-time T.J. Procyk, E.H. Mamdani A linguistic self-organizing process controller Original Automatica, Volume 15, Issue 1, January 1979, Pages

11 Fuzzy control history: the SOC Adaptive control using mechanism to modify the rule base in real-time T.J. Procyk, E.H. Mamdani A linguistic self-organizing process controller Original Automatica, Volume 15, Issue 1, January 1979, Pages

12 Fuzzy control history: the SOC A simple model for variations: mapping Δy/Δu (Jacobian) SOC tested on numerous models in simulation (already reinforcement learning!) Linear SISO Linear MIMO + delays NL with saturations 12

13 Applications : Sendaï Metro (Japan) 1987 Seiji Yasunobu, Univ Tsukuba A Fuzzy Control for Train Automatic Stop Control Seiji YASUNOBU, Shoji MIYAMOTO (Hitachi) GOALS: security, comfort and stop precision HITACHI Engineers: comparisons with classical controllers on 300,000 simulations and 3000 real-time tests Stop precision divided by 2.5, comfort criterion x2 and consumption gain of 10% 13

14 Applications : Sendaï Metro (Japan) 1987 Seiji Yasunobu, Univ Tsukuba A Fuzzy Control for Train Automatic Stop Control Seiji YASUNOBU, Shoji MIYAMOTO (Hitachi) 14

15 Historical approach of fuzzy control Nonlinear extension of linear controllers. Output feedback approach e t y t c + - FC e t u t Model y t - FC-P, FC-PD, FC-PI, FC-PID - FC-adaptive Many works comparing linear/fuzzy like linear [Boverie et al., 1991] [Vidolov & Melin 1993] [Pigeon 1993] 15 Equivalences [Siler 1989] [Ying & al. 1993] [Foulloy & Galichet 1994]

16 Historical approach of fuzzy control 1988 Japan Laboratory for International Fuzzy Engineering (LIFE) 45 industrials (the only European was VW) Thousands of applications Washing machines, cameras (Canon), vacuum cleaner (Matsushita)... Micro-waves oven, conditioner (Mitsubishi), showers... Train, lifts, helicopter... counting > 5000 [Jamshidi 1995] Recent paper about fuzzy control applications [Precup & Hellendorn 2011] Number of fuzzy-logic-related patents issued and applied in Japan: 7,149 Number of fuzzy-logic-related patents issued in the United States: 21,878 Number of fuzzy-logic-related patents applied in the United States: 22, (Data compiled by Dr. B. Tadayon, CEO, Z Advanced Computing TM, Inc.)

17 Fuzzy control: the limits Highly complex systems and/or hard to model, identify + - Human system expertise RF Stability? Robustness? Problem: NOT «model-based» 17

18 Fuzzy control: the limits y t c et + - FC ut Sysetem yt e t Input/output methods Circle Criterion, Conicity Criterion [Ollero et al., 1993] - linear model only: reduced interest - PB : transform the nonlinear model: a nonlinear part + a linear part Anibal Ollero - Professor at the Ingeniería de Sistemas y Automática Department, University of Seville, Spain 18

19 Fuzzy control: the limits w f z Linear Model z f : q q is in S f z, z T f z 0, f 0 0 f z [Melin et al. 1998] z Input/Ouput Methods dx dt f x u x Bu Criteria based on Jacobian matrix Necessary Conditions

20 As a matter of summary with some fuzzyness Fuzzy control: the pioneers, the first applications Limitations, getting older How to go further? Is Fuzzy dead? Tracks 20

21 I Fuzzy control: going further Brunovski form x x xn f g u y x1 T n 1 x x1 x1 x 1 T * x w x x f f f f [Li-Xin Wang 1994] 1 n T x m x Takes profit from universal approximation property: u f y k e g w f x Fuzzy vector of basis functions 21 * f f x Optimal (unknown) vector of parameters Approximation error

22 I Fuzzy control: going further Brunovski form x x xn f g u y x1 T n 1 x x1 x1 x 1 1 u fˆ x v k s gˆ x u u ˆ x [Li-Xin Wang 1994] 1 n T x m x u f y k e g Indirect fuzzy adaptive control: fuzzy models used for approximating the system, i.e. and g x f x Direct fuzzy adaptive control: : fuzzy models used for approximating the control 22

23 II Fuzzy control: going further Model inversion y spk M 1 uk M yk Rule by rule [Babuska et al. 1995], using optimization tools [Park et Han 2000] A more recent tentative with Piecewise Bilinear Modeling of Nonlinear Systems [Eciolaza & Sugeno 2012]. For minimum phase systems: 1 u k yk 1 z k b 1 System y spk M ref F d F M u k y k q r 24

24 II Fuzzy control: going further Do not overestimate the solutions exact solution procedures when possible [Boukezzoula et al.2006], best approximate solution problem [Lamara et al. 2011] y k y ck y k y ck y ck R y med c k R med y k y ck y k y k 25 1 u k yk 1 z k b 1 échantillons samples échantillons

25 III Fuzzy control: going further + - System FC FC synthesis with stability and performances properties Nonlinear Fuzzy TS via identification or directly Property : Universal approximation [N guyen & al. 1993] [Castro, 1996] 26

26 Takagi-Sugeno fuzzy models [Takagi & Sugeno 1985] multi-model vision Coming from classical fuzzy models rule based: Rule i : If z t i 1 is F1 z1t and... and z p t is Then Linear model description i F z t p p 27

27 Takagi-Sugeno fuzzy models [Takagi & Sugeno 1985] multi-model vision Coming from classical fuzzy models rule based: Rule i : If z t i 1 is F1 z1t and... and z p t is Then Linear model description i F z t p p Transfert 0 y t A q y t B q u t nk i i i i State Space i i C xt x t A x t B u t y t i linear models blended together with nonlinear functions Nonlinearities in the premises. 28

28 h i TS: sector nonlinearity approach z t hi z t r 0, 1 i1 r x t 1 hi z t Ai x t Biu t i1 r y t hi z tcix t i1 Direct from NL to TS: Sector Nonlinearity Approach [Tanaka et al. 1996] u min -2-2 u max Global Local: in a compact set u min, u max

29 h i r Takagi-Sugeno fuzzy models z t 0, hi z t 1 1 i1 Polytopic point of view A1, B1 r i1 x t h z t A x t B u t A, B r r i i i Convex hull: A2, B2 Small example: k 2 x1 k sin x 1 x1 k 2 x2 k 1 2 2sin x1 k 0.5 x2 k sin x k 0,1 ; sin x k 0 1sin x k 1sin x k k sin x h1 x h x 2 2sin x1 k

30 Robust stability: the DTLTI uncertain model is said to be robustly stable if it is asymptotically stable for all Robust stability n 1, 0,, Discrete time LTI model x t A x t x x0 x t A where is an arbitrary uncertain set A Problem: the set of all stable matrices is not a convex set Simplest polytope: T T A1, A2 A1, A A1 A A A i A A 0.5, 1,2 and 0.5, 1.5 i 1 i A1 A2 ATTENTION: Stability at the vertices DO NOT imply stability of the convex set

31 Quadratic Stability of Discrete TS models Unforced discrete TS: 1 Quadratic Lyapunov function: With the convex sum property: T T, 0 r P * V x x t Px t P P T T P V x x t A PA P xt 0 h z t 0 PAi P z z i Schur PAz P complement i1 i r i1 h z t 0, h z t 1 i r i i z i1 x t h z t A x t A x t Theorem (Tanaka and Sugeno 92): the unforced DTS is GAS if there exists a matrix P such that the LMI problem holds: T P Ai P 0, i 1,, r PAi P Remark: strictly equivalent to quadratic stability of LTI uncertain model *

32 Main goal Lower the conservatism and guarantee better performance / robustness Unfeasible (necessity conditions) Feasible not reachable

33 Main goal Lower the conservatism and guarantee better performance / robustness Unfeasible (necessity conditions) Reduction of the conservatism Several ways: - Using relaxations Asymptotically closed via Polya s theorem allows reaching the needed specifications (Kim & Lee 01, Liu & Zhang 04, Teixeira et al. 03) r r r h 0, i 1,, r, h 1, h h 0 i i i j ij j1 i1 j1 (Sala & Ariño 07)

34 Main goal Lower the conservatism and guarantee better performance / robustness Several ways: Modifying the Lyapunov function: (Johansson et al. 99, Feng 03) piecewise T T, 0 V x x t Px t P P x i i i i Restriction : no improvement in the case of Sector Non Linearity approach therefore, specific TS models under consideration

35 Main goal Lower the conservatism and guarantee better performance / robustness Several ways: Modifying the Lyapunov function: Non quadratic Restriction : r T T V x x t h x Px t, P P 0 i1 i i i i (Blanco et al. 01, Tanaka et al. 01, Guerra & Vermeiren 04) In the continuous case no real improvement excepted special case Path independent Lyapunov function (Rhee & Won 06)

36 A focus on the discrete case Lower the conservatism and guarantee better performance/robustness Modifying the Lyapunov function: r T i i i i i1 1 T Non quadratic V x x t h x Px t, P P 0 r r u x hi x Fi hi x Pi x t i1 i1 (Guerra & Vermeiren 04)

37 Choice of the Lyapunov function Multiple sum extension of NQLF r r T V x x hi zt h 1 i z t P s i i x s i is Non Quadratic Lyapunov function (NQLF) r T Vx x hi zt Pix i1 gg [Ding 06] [Ding et al 10] [Kim et al 11] [Blanco et al. 01] [Tanaka et al. 01] [Guerra et al 04] [Mozelli et al. 06] Quadratic Lyapunov function (QLF) T V x x Px Piecewise Lyapunov function (PWLF) T V x x P x xx, q q [Johansson et al 99] [Feng et al. 01] 38

38 Why decreasing each step? - Modifying the stability criteria [Kruszewski & Guerra 05] V xt Classical approach x t V x t 1 V x t 0 t

39 Why decreasing each step? - Modifying the stability criteria V xt Classical approach x t V x t 1 V x t 0 V x t k-sample approach k [Kruszewski & Guerra 05] 0 x t V x t V x t t t Can be applied on any problem in the discrete case

40 b Example: stability A 1 a A The candidate Lyapunov function is: b V x T x Px V x t V x t 4 V x t 0 V x t V x t 1 V x t a Solution set for two values of k

41 Choice of the Lyapunov function Variation over k-samples k V t 0, k 1 V t Multiple sum extension of NQLF r r T V x x hi zt h 1 i z t P s i i x s i is Non Quadratic Lyapunov function (NQLF) r T Vx x hi zt Pix i1 [Kruszewski et al 05] [Guerra et al. 09] gg [Ding 06] [Ding et al 10] [Kim et al 11] [Blanco et al. 01] [Tanaka et al. 01] [Guerra et al 04] [Mozelli et al. 06] Quadratic Lyapunov function (QLF) T V x x Px Piecewise Lyapunov function (PWLF) T V x x P x xx, q q [Johansson et al 99] [Feng et al. 01] 43

42 Variation over k-samples Choice of the Lyapunov function k V t 0, k 1 V t Multiple sum extension of NQLF Multiple sum extension of Delayed NQLF r r T V x x hi zt h 1 i z t P s i i x r r T V x s x 1 i is i1 i 1 h z t h z t n P s i i x s i is Non Quadratic Lyapunov function (NQLF) r T V x x hi ztpix i1 Quadratic Lyapunov function (QLF) V x T x Px Delayed NQLF r T V x x hi zt 1 Pix i1 44

43 b New Observer design : Results Example: Consider the following TS model with a and b are free parameters. A A a B1 B2 1 0 T C b C Non Quad delayed Non Quad a

44 IC engine cylinder pressure estimation [Kerkeni 2011] ut p p yt p Periodic TS observer K p Alexis Chezières Periodic TS model Cylinder by cylinder estimation c c xt 1 A xt z t Bzt u t c t mod p c y t Czt xt 1 S K c 1 zt c t ˆt xˆ t A xˆ t B u t y t yˆ t Periodic TS observer yˆ Czt x Structure of LMI constraints m P * m 77 m S j ij T m1 m m m m m m S j Ai K j Ci S j S j P c zt c zt c zt

45 Fuzzy applications plenty Robotics and control, a very close history Trajectory planning Obstacle avoidance Cooperative robotics language for prog humanoid robot

46 As a matter of summary with some fuzzyness Fuzzy control: the pioneers, the first applications Limitations, getting older How to go further? Is Fuzzy dead? Tracks 51

47 Fuzzy control: the community Annual international conference Fuzz IEEE / WCCI (~200 attendants) International Journals IEEE Trans. On Fuzzy Systems (2012 IF 4,26) Fuzzy Sets and Systems (2012 IF 1,8) IFAC Technical Committee 3.2 (Chair Antonio Ruano) IFAC ICONS triennial But also any IFAC Journal, conf + IEEE SMC, TAC Prof. Antonio Sala (PU Valencia) Jong-Hwan Kim KAIST (Seoul) Prof. Robert Babuska (TU Delft) 52 Prof. FENG, Gang (Gary) Hong Kong H.K. Lam King s College, UK

48 Some tracks Type II Fuzzy sets - Jerry Mendel, Pr Univ South California Mixing with reinforcement learning Franck Lewis Pr Univ Texas at Arlington Problems: high sized systems output feedback with performances networked systems Any way to come back to Human? Man in the loop? Is there any place/need for fuzzy mathematical tools (uncertainty, linguistic ) We need new tools, new ideas, new gaps 53

49 Future events of TC Deadline for receipt of papers: 20 October, th IFAC Conference on Intelligent Control and Automation Science - ICONS proposition in Reims, France (K. Guelton) 54

50 Some answers Only those who attempt the absurd can achieve the impossible. Maurits Cornelis Escher If we knew what it was we were doing, it would not be called research, would it? Thank you The truth is rarely pure and never simple. Oscar Wilde 55