Intelligent Control Methods Lecture 10: Fuzzy Control (1) Slovak University of Technology Faculty of Material Science and Technology in Trnava

2 Introduction Classical control theory:  mathematical description of processes (differential equations) Fuzzy control:  normally used by people, based on experiences and expert-knowledge, (which are described by linguistic tools, not by equations, mathematical tools are replaced by fuzzy logic).  Examples: reverted pendulum (described by 4 non-linear differential equations!) car parking: turn the driving wheel just a bit to the left and turn back (not: turn the wheel 16 o 33“ and drive back 2,675 m)

3 h(t) e(t)u(t) y(t) - R(s)S(s) Klassical controller states (calculates) the control action u(t) according to e(t): For example PI-controller: Fuzzy controller states the actuating signal according to control strategy based on rules IF – THEN. F.E: IF difference is big and difference of difference is small THEN difference of control action is big. Usual by people decision and performance: (If a car rides faster than we want (e) and it reduced gently (Δe), we brake stronger (Δu)).

4 Control strategy: Rules IF – THEN (in a form similar to normal speech). Derived according to some type of classical controller.

5 P-controller: u(t) = K P e(t) u(t) – control action e(t) – control difference Fuzzy P-controller: IF e is A e THEN u is B u A e, B u –linquistic expressions giving the value of control difference and control action.

6 PD-controller: u(t) = K P e(t) + K D Δ e(t) Fuzzy PD-controller: IF e is A e AND Δ e is A Δ e THEN u is B u

7 PI-controller: Δ u(t) = K I e(t) + K P Δ e(t) Fuzzy PI-controller: IF e is A e AND Δ e is A Δ e THEN Δ u is B Δ u Often case, Δu is more native for people (valve or gas pedal opening or closing) than the absolute value u (valve open 62 %, pedal pressed 16 o ).

8 PID-controller: Δ u(t) = K I e(t) + K P Δ e(t) + K D Δ 2 e (t) Fuzzy PID-controller: IF e is A e AND Δ e is A Δ e AND Δ e 2 is A Δ 2 e THEN Δ u is B Δ u Assigned for non-linear and unstabil processes. Problem with great number of antecedents combinations.

9 Matematical background of fuzzy control (1): Clasical (crisp) sets: A 1 = {ball, cylinder, cube} set of figures given by elements listing A 2 = {x  Z / 6 < x < 10} set of numbers given by property T 1for x  A charakteristic function of a x  A  A (x) = set A F 0for x  A (gives membership of elements to the set A)

10 Matematical background of fuzzy control (2): U A negation U AB intersection U AB union

11 Matematical background of fuzzy control (3): Pojem relácie (v prípade ostrých množín): Let X and Y are definition scopes and let their cartesian product is U = X x Y. Then a binary relation R is each subset R  U. (the same definition is valid for n-dimensional relations) Example: X = {Jana, Iveta, Eva} and Y = {Peter, Ján, Milan, Igor} are definition scopes (universes) R = {(Jana, Igor), (Iveta, Peter), (Eva, Ján)} is relation „married couples“ defined on X x Y.

12 Matematical background of fuzzy control (4): Fuzzy set definition: Fuzzy set is the set of elements, which can belong into the set partially. The membership of element into the set is given by membership function (what is generalised characteristic function of the set).  F :U  F = {(u,  F (u)/u  U} =  F (u 1 )/u 1 +  F (u 2 )/u  F (u n )/u n

13 Matematical background of fuzzy control (5): Fuzzy set example: Let the temperature in a room is ( o C), i.e. U = Membership functions into sets Cald, Good, Hot are: 1   c (25) = 0,0  g (25) = 0,3  H (25) = 0,7

14 Matematical background of fuzzy control (6): Typical membership functions (linear, therefore simple):  (u) 1 0 for u   (u, ,  ) = (u-  )/(  -  ) for  u  1 for u    u  (u) 1 1 for u  L(u, ,  ) = (  - u)/(  -  ) for  u  0 for u    u

15 Matematical background of fuzzy control (7): Typical membership functions:  (u) 1 0 for u   (u, , ,  ) = (u-  )/(  -  ) for  u  (  -u)/(  -  ) for  u     u 1 for u   (u) 0 for u  1 (u-  )/(  -  ) for  u   (u, , , ,  ) = 1 for  u  (  -u)/(  -  ) for  u     u 0 for u 

16 Matematical background of fuzzy control (8): Often (general) case of description of definition scope by fuzzy sets without considering the physical parameters:  1 NB NM NS Z PS PM PB u NB (Negative Big): L(u,-6,-4) NM (Negative Medium):  (u,-6,-4,-2) NS (Negative Small):  (u,-4,-2,0) Z (Zero):  (u,-2,0,2) PS (Positive Small):  (u,0,2,4) PM (Positive Medium):  (u,2,0,4) PB (Positive Big):  (u,4,6)

17 Operations with fuzzy sets:  A B x  A’ x Complement (negation):  A’ (x) = 1 -  A (x)

18 Operations with fuzzy sets:  A B x x Intersection:  A  B (x) = min (  A (x),  B (x)) Union:  A  B (x) = max (  A (x),  B (x))

19 Fuzzy relation: Let U and V are definition scopes and let it is given the function  R : UxV   0,1 . Binary fuzzy relation R is fuzzy set of ordered couples If the definition scopes are continuous, then:

20 Fuzzy relation (example): X = {Jana, Iveta, Eva} and Y = {Peter, Ján, Milan, Igor} are definition scopes. Relation „Friends“ defined on X x Y: PeterJánMilanIgor Jana 0,80,90,10,3 Iveta 0,50,60,30,7 Eva 0,20,10,80,4

21 Operations with fuzzy relations: (intersection and union) Let R and S are binary relations defined on X x Y. Then membership functions for intersection and union of relations R and S are defined for all x,y as follow: Intersection:  R  S (x,y) = min (  R (x,y),  S (x,y)) Union:  R  S (x,y) = max (  R (x,y),  S (x,y))

22 Operations with fuzzy relations (example for intersection and union): PeterJánMilanIgor Jana0001 Iveta1000 Eva0100 X = {Jana, Iveta, Eva} and Y = {Peter, Ján, Milan, Igor} are definition scopes. Relations „Married couples“ and „Friends“ defined on X x Y: Married couples (M): Friends (F): PeterJánMilanIgor Jana0,80,90,10,3 Iveta0,50,60,30,7 Eva0,20,10,80,4 PeterJánMilanIgor Jana0000,3 Iveta0,5000 Eva00,100 PeterJánMilanIgor Jana0,80,90,11 Iveta10,60,30,7 Eva0,210,80,4 Married c. and friends (  M  F (x,y)) M.c. or friends (  M  F (x,y))

23 Operations with fuzzy relations (2): Projection Let R is binary relation defined on X x Y. Then projection R into Y is fuzzy set I.e.: Projection R into Y means the finding of maximal value  R in each column y 1, y 2,... y n in the table and assignment of this value to element y j. PeterJánMilanIgor Jana0,80,90,10,3 Iveta0,50,60,30,7 Eva0,20,10,80,4 Proj R in Y = 0,8/Pe + 0,9/Já + 0,8/Mi + 0,7/Ig Proj R in X = 0,9/Ja + 0,7/Iv + 0,8/Ev

24 Operations with fuzzy relations (3): Extension Opposit operation for projection: Let F is a fuzzy set defined on Y. Then cylindric extension F to X x Y is the set of all couples (x,y)  X x Y with membership function  CE(F) (x,y), i.e.: I.e.: Cylindric extension means the building of a table from the function. PeterJánMilanIgor Jana0,80,70,30,6 Iveta0,80,70,30,6 Eva0,80,70,30,6 F = 0,8/Pe + 0,7/Já + 0,3/Mi + 0,6/Ig