Fuzzy Control. PI vs. Fuzzy PI-Control. Olaf Wolkenhauer. Control Systems Centre UMIST.

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1 Fuzzy Control PI vs. Fuzzy PI-Control Olaf Wolkenhauer Control Systems Centre UMIST

2 2 Contents Learning Objectives 4 2 Feedback Control 5 3Fuzzy PI-Controller 6 4 Example: First-Order System with Dead-Time 5 4. Fuzzy Rule-Base Centre Average Defuzzification Control Surface: Zadeh Logic Analysis Summary First-OrderDelayedProcess Example: Coupled Tanks Design of a fluid level proportional controller Design of a proportional plus integral controller... 35

3 3 5.3 Design of a fuzzy-pi-controller

4 Section : Learning Objectives 4. Learning Objectives Fuzzy rule-based systems can also be used to devise control laws. Fuzzy control can be particular useful if no linear parametric model of the process under control is available. Fuzzy control is not model-free as a good understanding of the process dynamics may be required. Fuzzy control lacks of design methodologies. Fuzzy controllers are easy to understand and simple to implement.

5 Section 2: Feedback Control 5 2. Feedback Control Feedback: When we desire a system to follow a given pattern the difference between this pattern and the actual behaviour is used as a new input to cause the part regulated to change in such a way as to bring its behaviour closer to that given by the pattern. reference error Controller control action Process output (feedback loop)

6 Section 3: Fuzzy PI-Controller 6 3. Fuzzy PI-Controller setpoint error G e fuzzification rule-base G r defuzzification G u rate of change process

7 Section 3: Fuzzy PI-Controller 7 Conventional PI-Control: du(t) dt where e(t) = s(t) y(t). = K p de(t) dt + K i e(t) () To obtain a control action the term du(t)/dt is integrated. A fuzzy-pi-controller is developed analogously : deriv (k) =K p rate (k)+k i error (k) (2) error, rate, deriv are fuzzy (or linguistic) variables partioning the underlying spaces by piecewise linear (triangular) fuzzy sets as shown in figure.

8 Section 3: Fuzzy PI-Controller 8 µ µ negative zero positive negative.large negative.small zero positive.small positive.large error, rate deriv Figure : Fuzzy sets for the variables error, rate and the output.

9 Section 3: Fuzzy PI-Controller 9 Scale (2) toarangefrom to+. Use scaling factors G u, G r and G e, where deriv = G u deriv, G r rate = rate, G e error = error Substituting these into (2) deriv(k) = K p rate(k)+ K i error(k) (3) G u G r G u G e The constants K p (G u G r )andk i /(G u G e ) are assumed equal to.5 to make deriv fall into the interval [, ]. The fuzzy controller is then equivalent to a conventional PIcontroller with proportional gain K p =.5 G u G r and integral gain K i =.5 G u G e. Note: there are infinitely many combinations of G e, G r,andg u to hold true for these expressions.

10 Section 3: Fuzzy PI-Controller The complete rule-base : R : R 2 : R 3 : R 4 : R 5 : IF error is negative AND rate is negative, THEN deriv is negative large IF error is negative AND rate is zero, OR error is zero AND rate is negative, THEN deriv is negative small IF error is negative AND rate is positive, OR error is zero AND rate is zero, OR error is positive AND rate is negative, THEN deriv is zero IF error is zero AND rate is positive, OR error is positive AND rate is zero, THEN deriv is positive small IF error is positive AND rate is positive, THEN deriv is positive large

11 Section 3: Fuzzy PI-Controller negative, zero, positive, etc. are fuzzy sets. The logical connectives AND and OR are are t-andt-conorms. firing level of the i th rule, denoted µ DERIVi (deriv). Assuming n s fuzzy sets for error, error rate of change we require 2(n s ) fuzzy sets (and rules) for the output deriv. error N Z P P Z PS PL rate Z NSZ PS N NL NSZ

12 Section 3: Fuzzy PI-Controller 2 The principal values for which µ DERIVi (deriv) =, are equally spaced, but at half the interval of the antecedent fuzzy sets. With three fuzzy sets on the input spaces. The principal values of the i th member of the fuzzy partition DERIV i are given by +(i )/(n s ) Linear defuzzification strategy : deriv(k) = 2n s i= µ DERIVi (deriv) ( ) (i ) + n s. (4) This value is integrated and scaled to obtain the control action required to drive the plant.

13 Section 3: Fuzzy PI-Controller 3 Zadeh-logic : Conjunction: T ( µ A ( ),µ B ( ) ) = min ( µ A ( ),µ B ( ) ) Disjunction: S ( µ A ( ),µ B ( ) ) = max ( µ A ( ),µ B ( ) ) deriv error error.5.5 rate rate

14 Section 3: Fuzzy PI-Controller 4 Mixed-logic : Rule and 5: Zadeh-logic Rule 2 and 4: Lukasiewicz-logic Conjunction: T ( µ A ( ),µ B ( ) ) = max (, (µ A ( )+µ B ( )) ) Disjunction: S ( µ A ( ),µ B ( ) ) = min (,µ A ( )+µ B ( ) ) deriv error error.5.5 rate rate

15 Section 4: Example: First-Order System with Dead-Time 5 4. Example: First-Order System with Dead-Time Replace linear defuzzification by non-linear strategy [5]. Notation : (sampling period equals one) error (k) = s(k) y(k) error(k) = G e error (k) rate (k) = error (k) error (k ) rate(k) = G r rate (k) deriv (k) = G u deriv(k) u(k) = u(k ) + deriv (k). Input and output fuzzy sets.

16 Section 4: Example: First-Order System with Dead-Time 6 µ µ Input (error,rate) and output fuzzy sets: negative positive negative zero positive error, rate deriv

17 Section 4: Example: First-Order System with Dead-Time Fuzzy Rule-Base There are three fuzzy control rules composed out of four : R : R 2 : R 3 : R 4 : IF error is negative AND rate is negative, THEN deriv is negative IF error is negative AND rate is positive, THEN deriv is zero IF error is positive AND rate is negative, THEN deriv is zero IF error is positive AND rate is positive, THEN deriv is positive

18 Section 4: Example: First-Order System with Dead-Time Centre Average Defuzzification Instead of deriv(k) = 2n s i= ( µ DERIVi (deriv) + normalise the membership degrees to one : ) (i ) n s. (4) deriv(k) = 2n s i= ( ) µ DERIVi (deriv) + i n s 2n s i=... called Center Average Defuzzification. µ DERIVi (deriv) (5)

19 Section 4: Example: First-Order System with Dead-Time Control Surface: Zadeh Logic deriv error error.5.5 rate rate Compare with linear defuzzification!

20 Section 4: Example: First-Order System with Dead-Time Analysis The membership functions associated with error and rate of change of the error : µ error is pos ( e(k) ) = error(k)+ 2 µ error is neg ( e(k) ) = error(k)+ 2 µ rate is pos ( r(k) ) = rate(k)+ 2 µ rate is neg ( r(k) ) = rate(k)+ 2 = G e error (k)+ 2 = G e error (k)+ 2 = G r rate (k)+ 2 = G r rate (k)+ 2 (6) (7) (8). (9)

21 Section 4: Example: First-Order System with Dead-Time 2 Partition of the phase-plane into sectors : Sector R R 2 R 3 R 4 error is neg. rate is neg. error is neg. rate is pos. 2 error is neg. rate is neg. error is neg. rate is pos. 3 rate is neg. rate is neg. error is neg. error is pos. 4 rate is neg. rate is neg. error is neg. error is pos. 5 rate is neg. error is pos. rate is pos. error is pos. 6 rate is neg. error is pos. rate is pos. error is pos. 7 error is neg. error is pos. rate is pos. rate is pos. 8 error is neg. error is pos. rate is pos. rate is pos.

22 Section 4: Example: First-Order System with Dead-Time 22 Partioning of the phase-plane for a fuzzy PI-controller : G e error G r rate 6 5

23 Section 4: Example: First-Order System with Dead-Time 23 From equations (6)-(9) and (5) we obtain the following equations for the output : Sector and 2 : ( ) ( ) µ error is neg e(k) + µrate is pos r(k) deriv(k) = ( ) ( ) ( ) µ er. is neg e(k) + µrate is neg r(k) + µrate is pos r(k) = G e error (k)+g r rate (k) 3 G e error () (k) Sector 3and 4 : ( ) ( ) µ rate is neg r(k) + µerr is pos e(k) deriv(k) = ( ) ( ) ( ) µ rate is neg r(k) + µer. is neg e(k) + µer. is pos e(k) = G r rate (k)+g e error (k) 3 G r rate () (k)

24 Section 4: Example: First-Order System with Dead-Time 24 Sector 5 and 6 : ( ) ( ) µ rate is neg r(k) + µerr is pos e(k) deriv(k) = ( ) ( ) ( ) µ rate is neg r(k) + µrate is pos r(k) + µer. is pos e(k) = G r rate (k)+g e error (k) 3+G r error (2) (k) Sector 7 and 8 : ( ) ( ) µ error is neg e(k) + µrate is pos r(k) deriv(k) = ( ) ( ) ( ) µ er. is neg e(k) + µer. is pos e(k) + µrate is pos r(k) = G e error (k)+g r rate (k) 3+G r rate (3) (k)

25 Section 4: Example: First-Order System with Dead-Time 25 If G r rate (k) G e error (k), we then have deriv(k) = G e error (k)+g r rate (k) 3 G e error (4) (k) and if G e error (k) G r rate (k), deriv(k) = G e error (k)+g r rate (k) 3 G r rate. (5) (k)

26 Section 4: Example: First-Order System with Dead-Time 26 As we have seen in the previous section, the fuzzy PI-controller with linear defuzzification and mixed logic is equivalent to a nonfuzzy PI-controller with proportional gain K p =.5 G u G r and integral gain K i =.5 G u G e : deriv (k) =K p rate (k)+k i error (k). (6) Comparing (6) with equations (4) and (5), we notice that the fuzzy PI-controller with nonlinear defuzzification and Zadehlogic for rule evaluation is equivalent to a linear PI-controller with changing gains K p and K i : G r G u K p = 3 G e error (k) G e G u K i = 3 G e error (k) (7) (8)

27 Section 4: Example: First-Order System with Dead-Time 27 when G r rate (k) G e error (k), and G r G u K p = 3 G r rate (k) G e G u K i = 3 G r rate (k) when G e error (k) G r rate (k). (9) (2) If we define the static gains K ps and K is as the proportional and integral gains when both error and rate are equal to zero, we have : K ps K is = G r G u 3 = G e G u 3 (2) (22)

28 Section 4: Example: First-Order System with Dead-Time 28 and find for the conventional PI-controller deriv(k) = K p s G u rate (k)+ K i s G u error (k) = G r rate (k)+g e error (k). (23) 3 Comparing equality (23) with equations (4) and (5), the following inequalities are obtained : 3 G e error (k) 3 when G r rate (k) G e error (k), and 3 G r rate (k) 3 when G r error (k) G e rate (k).

29 Section 4: Example: First-Order System with Dead-Time Summary The (absolute value of the) incremental control action of the fuzzy PI-controller is equal or greater the (absolute value of the) incremental control action of the nonfuzzy PI-controller when G e error (k) andg r rate (k). We can conclude that the larger (absolute values of) error (rate) values, the larger is the difference between the outputs of the two controllers. The nonlinearity of the fuzzy PI-controller can therefore be used to improve the control performance in comparison to a nonfuzzy and linear PI-controller.

30 Section 4: Example: First-Order System with Dead-Time First-Order Delayed Process Comparison: Fuzzy PI-controller with Zadeh logic and nonlinear defuzzification vs linear PI-controller. Static proportional gain K ps and the static integral gain K is of the fuzzy controller were set equal the proportional and integral gains K p =2.38 and K i =4.43 of the conventional PI-controller. The process plant is taken to be a first order system with time delay and transfer function ; Y (s) U(s) = s + e.2s

31 Section 4: Example: First-Order System with Dead-Time 3 Step responses PI 3 Output Fuzzy PI time (sec)

32 Section 4: Example: First-Order System with Dead-Time 32 Phase plane and trajectory error(nt) rate(nt)

33 Section 5: Example: Coupled Tanks Example: Coupled Tanks The control input is the pump drive voltage. The sensed output is the water depth in tank 2. Q i Drain tap Cross-sectional area = a o Discharge coefficient = C do H Q 2 H 2 Q o Tank Volume of fluid = V Cross-sectional area = A Inter-tank hole Tank 2 Cross-sectional area = a 2 Volume of fluid = V 2 Discharge coefficient = C Cross-sectional area = A d 2

34 Section 5: Example: Coupled Tanks Design of a fluid level proportional controller Plot the root locus of the open loop transfer function G v (z), select the gain K p that gives a closed loop damping factor of ϑ =.7. Read off the c.l. natural frequency ω n. The closed loop transfer function is : H v (z) = v d 2 (z) v r (z) = K p G v (z) (24) +K p G v (z) where : G v (z) = z ( ) Gv (s) Z (25) z s The steady state error can be calculated using : e ss =[v r (z) v d2 (z)] z = v r () [ H v (z)] z (26) Where v r () is the steady state reference input.

35 Section 5: Example: Coupled Tanks Design of a proportional plus integral controller Set the integral action time constant to a reasonable value (in this case T i = 5s). Plot the root locus of the open loop system in cascade with the compensator + z T i z, and select the gain k that gives a closed loop damping factor of ϑ =.7. The proportional and integral gains K p,andk i can be computed by comparing coefficients of the compensator transfer functions: ( K + z = K p + K iz (z ) T i (z ) The value of ω n can be read off the root locus plot. ) (27)

36 Section 5: Example: Coupled Tanks 36 Writing the open loop system in series with a proportional plus integral action compensator as G c (s), where : G c (s) = K (st g pg d2 i +) K 2 st i T T 2 s 2 +(T + T 2 ) s + (28) The closed loop transfer function is : H v (s) = Kg pg d2 k 2 st i + ) T i T T 2 s 3 + T i (T + T 2 ) s 2 + T i (+ Kgpg d 2 K 2 s + Kgpg d 2 K 2 (29) Assuming : T > ; T 2 > ; g p > ; g d2 > ; k 2 > ; T i > ; K> (3)

37 Section 5: Example: Coupled Tanks 37 The closed loop system is stable for : ( T i (T + T 2 ) + Kg ) pg d2 K 2 >T T 2 Kg p g d2 K 2 (3) Hence the closed loop system can become unstable for sufficiently large gain if : T i < T T 2 (32) T + T 2 and the gain required to make the closed loop system unstable is : K = K 2 T i (T + T 2 ) (33) g p g d2 T T 2 T i (T + T 2 ) The closed loop system under proportional plus integral control has three poles, and one zero. Using the design procedure outlined above, two of the poles will be complex conjugate (at ϑ =.7), the remaining pole and zero are negative real.

38 Section 5: Example: Coupled Tanks Design of a fuzzy-pi-controller. The constants KpGr G u and KiGe G u are set to make the output to fall within the interval [, +]. Choose fuzzy sets for the input and output as in figure. For example, G e =.2, G r =,andg u = 5. Fix the number of (triangular, fully overlapping) fuzzy sets partitioning the input spaces. On the basis of the phase-plane characteristic and/or trajectory decide upon the fuzzy logic employed. This will decide the overall gain structure of the phase-plane. Adjust input-output gains to have trajectories of the system to fall within the [, ] range. Change positions of principal values for input fuzzy sets to finetune gain structure in the quadrants of the phase-plane.

39 Section 5: Example: Coupled Tanks 39 Simulink block diagram : uod_rate z Discrete Filter Gr= Kp/(.5*Gu) uod Rate To Workspace3 Step Sum Mux Mux fuzzy_controller S Function Tz z Discrete Time Integrator 5 Gu Zero Order Hold / s s+ Coupled tanks.2 FC_output Ge= Ki/(.5*Gu) uod Error uod_error To Workspace2 To Workspace4 FC output 7.5 /2.37 Mux Output_H2 Sum Kp Sum2 Zero Order Hold s s+ Coupled tanks Mux To Workspace.5 T(z+) 2(z ) Output H2 Ki Discrete Time Integrator

40 Section 5: Example: Coupled Tanks 4 Step responses: non-fuzzy PI-controller, K i =.5, K p = 7.5, (dashed line) and its fuzzy equivalent : Output time (sec)

41 Section 5: Example: Coupled Tanks 4 References [] Babuska, R. : Fuzzy Modelling for Control. Kluwer, 998. See [2] Passino, K.M. and Yurkovich, S. : Fuzzy Control. Addisson Wesley, 997. [3] Siler, W. and Ying, H. : Fuzzy Control Theory: The Linear Case. Fuzzy Sets and Systems, 33: , 989. [4] Wolkenhauer, O. : Data Engineering. [5] Ying, H. and Siler, W. and Buckley, J.J. : Fuzzy Control Theory: A Nonlinear Case. Automatica, 26 (3): 53 52, 99. 5

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