Fuzzy PID Controllers for Industrial Applications

Transcription

1 Fuzzy PID Controllers for Industrial Applications G. Ron Chen Lecture for EE 6452 City University of Hong Kong Summary Proportional-Integral-Derivative (PID) controllers are the most widely used controllers in industries today Statistics: > 90% controllers in industries are PID or PIDtype of controllers (the rest are programmable logical controllers (PLC)) Merits of PID controllers: simple, cheap, reliable, and effective For lower-order linear time-invariant systems and processes, PID controllers have good set-point tracking performance with guaranteed stability Fuzzy logic provides a certain level of artificial intelligence to the conventional PID controllers Fuzzy PID controllers have self-tuning ability and on-line adaptation to nonlinear, time-varying, and uncertain systems Fuzzy PID controllers provide a promising option for industrial applications with many desirable features

2 Outline of the Presentation Overview of the Fuzzy Logic echnology Overview of Conventional PID Controllers Introduction to Fuzzy PID Controllers Some Successful Examples of Applications Concluding Remarks

3 Overview of the Fuzzy Logic echnology Closed-Loop Set-Point racking System Consider the typical set-point tracking system: r (reference signal) S e (error signal) Controller u (control signal) Plant y (output signal) Figure A typical closed-loop set-point tracking system Objective: e(t) := r(t) y(t) 0 (t ) Approach: Design a fuzzy logic controller (FLC) e controller input Fuzzification Fuzzy Rule Base Defuzzification Fuzzy Logic Controller (FLC) u controller output Figure 2 General structure of a fuzzy logic controller

4 temperature r = 45 o b c d y ( t ) e ( t ) a 0 Figure 3 emperature set-point tracking example t (i) If e > 0 then e = r y > 0 or r > y the output y is at position a or d (ii) Furthermore, if &e < 0 then &e = r & y& = 0 y& or y& > 0 (iii) herefore, the output y is at position a Fuzzy Logic Rule Base: R : IF e > 0 AND &e < 0 HEN u(t) = u(t) R 2 : IF e < 0 AND &e < 0 HEN u(t) = u(t) R 3 : IF e < 0 AND &e > 0 HEN u(t) = u(t) R 4 : IF e > 0 AND &e > 0 HEN u(t) = u(t)

5 Fuzzy Controller Design A. Fuzzification Purpose: Enable the input physical signal to use the rule base Approach: Use membership functions µ PS µ PL µ NL µ NS 0 H (a) e, ė (b) H Figure 4 Four membership functions for signals e and &e 0 e, ė B. Programmable Rule Base R : IF e = PL AND &e < 0 HEN u(t ) = µ PL (e). u(t) R 2 : IF e = PS AND &e < 0 HEN u(t ) = ( µ PS (e)). u(t) R 3 : IF e = NL AND &e < 0 HEN u(t ) = µ NL (e). u(t) R 4 : IF e = NS AND &e < 0 HEN u(t ) = ( µ NS (e)). u(t) R 5 : IF e = NL AND &e > 0 HEN u(t ) = µ NL (e). u(t) R 6 : IF e = NS AND &e > 0 HEN u(t ) = ( µ NS (e)). u(t) R 7 : IF e = PL AND &e > 0 HEN u(t ) = µ PL (e). u(t) R 8 : IF e = PS AND &e > 0 HEN u(t) = ( µ PS (e)). u(t)

6 o implement the FLC on a digital computer: u(t) = u(k) and u(t) = u((k)) where is the sampling time. R : IF e(k) = PL AND &e (k) < 0 HEN u((k) ) = µ PL (e(k)). u(k) R 2 : IF e(k) = PS AND &e (k) < 0 HEN u((k)) = ( µ PS (e(k))). u(k) R 3 : IF e(k) = NL AND &e (k) < 0 HEN u((k) ) = µ NL (e(k)). u(k) R 4 : IF e(k) = NS AND &e (k) < 0 HEN u((k) ) = ( µ NS (e(k))). u(k) R 5 : IF e(k) = NL AND &e (k) > 0 HEN u((k) ) = µ NL (e(k)). u(k) R 6 : IF e(k) = NS AND &e (k) > 0 HEN u((k) ) = ( µ NS (e(k))). u(k) R 7 : IF e(k) = PL AND &e (k) > 0 HEN u((k) ) = µ PL (e(k)). u(k) R 8 : IF e(k) = PS AND &e (k) > 0 HEN u((k)) = ( µ PS (e(k))). u(k) where &e (k) [e(k) e((k ))], with initial conditions y(0) = 0, e( ) = e(0) = r y(0), &e (0) = [e(0) e( )] = 0

7 C. Defuzzification Select membership functions for the different control outputs from the rule base µ N µ ZO µ P H 0 H u Figure 5 ypical membership functions for u hen, the overall control signal, u, is generated by a weighted average formula: u((k)) = N i= µ u ( k) i N i= i µ i, (µ i 0, µ i > 0) N i = where control outputs u i (k), i =,..., N=8 are from the rule base.

8 Overview of Conventional PID Controllers In the time domain: (i) P-controller (ii) I-controller u(t) = K P e(t) u(t) = K I e( τ) dτ t 0 d (iii) D-controller u(t) = K D dt e(t) Control gains, K P, K I, and K D, are constants to be determined in the design for set-point tracking and stability consideration. r e K u P system y (a) Proportional controller t r e K u I system y 0 (b) Integral controller r e d K u D system y dt (c) Derivative controller Figure 6. Conventional PID controllers

9 In the frequency domain: (i) P-controller U(s) = K P E(s) (ii) I-controller K U(s) = I E(s) (iii) D-controller U(s) = K D s E(s) Use Laplace transform L{ } for continuous-time signals: U(s) = L{ u(t) } and E(s) = L{ e(t) } Use z-transform Z{ } for discrete-time signals. s

10 K I t 0 r e u y K P system (a) PI controller r e u y d K Ddt K P system (b) PD controller K P r e u y K I t 0 system d K D dt (c) PID controller K I t 0 r e u y K P d K D dt system (d) PID controller Figure 7 Some typical combination of P, I, D controllers.

11 Introduction to Fuzzy PID Controllers A. Discretization of Convetional PID Controllers First, digitize the conventional analog PID controllers by S = 2 z z where > 0 is the sampling time. For the PI controller: u(n) = u(n ) u(n) u(n) = ~ KP v(n) ~ K I e(n) e(n) z v(n) K ~ P K ~ I u(n) u(n) u(n ) z Figure 8. he digital PI controller

12 Similarly, for the PD controller: u(n) = ~ KP d(n) ~ K D v(n) where d(n) = v(n) = e ( n ) e( n ) e ( n ) e( n ) d(n) K ~ P e(n) z z v(n) u(n) K ~ D u(n ) z u(n) Figure 9. he digital PD controller

13 B. Example: Designing the Fuzzy PI Controller r(n) e(n) d(n) - K i - z - Fuzzy PI Controller K upi - u(n) Process y(n) K p z - e v (n Figure 0. he Fuzzy PI control system Fuzzification and Defuzzification fuzzy membership functions negative positive 0.5 -L 0 L ( ) u PI n Figure. Output membership functions negative positive 0.5 -L 0 L e p, e v Figure 2. Input membership functions

14 IC8 IC2 L IC IC7 IC3 IC4 IC3 IC0 IC5 IC2 d( n ) -L L IC4 IC6 IC IC9 IC7 IC8 -L IC9 IC5 IC6 IC20 Figure 3. IC regions for the fuzzy PI controller K p e v ( n ) K i u PI ( n ) = L[ K id( n) K pev ( n) ] 2( 2L K i d( n) ) L[ K id( n) K pev ( n) ] = 2( 2L K p ev ( n) ) = [ L K e ( n) ] = [ L K d ( n )] p v, in IC, IC2, IC5, IC6, in IC3, IC4, IC7, IC8, in IC9, IC0, in IC, IC2 = [ L K e ( n) ] i, in IC3, IC4 p v = [ L K d( n )], in IC5, IC6 2 i = 0, in IC8, IC20 = L, in IC7 = L, in IC9 Note: Rigorous stability can be guaranteed

15 Some Successful Examples of Applications here are many successful examples of fuzzy PID controllers.. Robotics * Six-legged Insect Robot (Dr. P. Sooraksa, G. R. Chen, and students) King Mongkut s Institute of echnology, Bangkok, hailand

16 * Multi-purpose Autonomous Robust Carrier for Hospitals (MARCH) (Dr. P. Sooraska, Prof. S. K. so, Dr. B. L. Luk, G. R. Chen, and students) Centre for Intelligent Design, Automation and Manufacturing City University of Hong Kong

17 * Uncertain Robot-Arm Control (next page) (G. R. Chen and students) University of Houston, exas, USA * Uncertain Double-Pendulum Control (DEMO) (Prof. Edgar Sanchez, G. R. Chen, and students) CINVESAV, Mexico 2. emperature Control (next page) (G. R. Chen and students) University of Houston, exas, USA Other Application Examples A. Process Control (temperature, flow rate, injection) B. Vehicles Control (parking, docking, backing truck-trailers) C. Electronic Appliances (washer, dryer, camcorder-camera) D. Life-Support Systems (space shuttle, submarine) E. many more

18 Fig. 8. Block diagram and physical setup for the robot arm experiment.

19 376 IEEE RANSACIONS ON CONROL SYSEMS ECHNOLOGY, VOL. 5, NO. 3, MAY 997 Fig. 0. racking performance of the fuzzy PID-controlled system (I). Fig.. racking performance of the conventional PID-controlled system (II).

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23 Concluding Remarks Fuzzy logic provides a certain level of artificial intelligence to the conventional PID controllers, leading to the effective fuzzy PID controllers Fuzzy PID controllers are easy to use (plug-and-play) Fuzzy PID controllers can be implemented by both o software (matured) o hardware (pre-matured) Fuzzy PID controllers have strong self-tuning ability and online adaptation to nonlinear, time-varying, and uncertain systems. Fuzzy PID controllers have a promising future for various industrial applications