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# [Chen G.R.] Fuzzy PID controllers

publicité ```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
Outline of the Presentation
• Overview of the Fuzzy Logic Technology
• Overview of Conventional PID Controllers
• Introduction to Fuzzy PID Controllers
• Some Successful Examples of Applications
• Concluding Remarks
Overview of the Fuzzy Logic Technology
Closed-Loop Set-Point Tracking System
Consider the typical set-point tracking system:
r
+
Σ
(reference
signal)
e
−
u
Controller
(error
signal)
y
Plant
(control
signal)
(output
signal)
Figure 1 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
Fuzzification
Fuzzy Rule Base
Defuzzification
controller
input
u
controller
output
Fuzzy Logic Controller (FLC)
Figure 2 General structure of a fuzzy logic controller
temperature
b
r = 45 o
c
y (t )
d
e (t )
a
0
t
Figure 3 Temperature set-point tracking example
(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) Therefore, the output y is at position a
Fuzzy Logic Rule Base:
R1:
IF e > 0 AND e& < 0 THEN u(t+) = u(t)
R2:
IF e < 0 AND e& < 0 THEN u(t+) = − u(t)
R3:
IF e < 0 AND e& > 0 THEN u(t+) = u(t)
R4:
IF e > 0 AND e& > 0 THEN u(t+) = − u(t)
Fuzzy Controller Design
A. Fuzzification
Purpose: Enable the input physical signal to use the rule base
Approach: Use membership functions
1
0
µ
µ
PL
µPS
NL
µNS 1
.
e, e
H
(a)
−H
0
(b)
Figure 4 Four membership functions for signals e and e&
B. Programmable Rule Base
R1:
IF e = PL AND e& < 0 THEN u(t+) = µP L(e) . u(t)
R2:
IF e = PS AND e& < 0 THEN u(t+) = (1−µPS(e)) . u(t)
R3:
IF e = NL AND e& < 0 THEN u(t+) = −µNL(e) . u(t)
R4:
IF e = NS AND e& < 0 THEN u(t+) = −(1−µNS(e)) . u(t)
R5:
IF e = NL AND e& > 0 THEN u(t+) = µNL(e) . u(t)
R6:
IF e = NS AND e& > 0 THEN u(t+) = (1−µNS(e)) . u(t)
R7:
IF e = PL AND e& > 0 THEN u(t+) = −µPL(e) . u(t)
R8:
IF e = PS AND e& > 0 THEN u(t+1) = −(1−µPS(e)) . u(t)
.
e, e
To implement the FLC on a digital computer:
u(t) = u(kT)
and
u(t+) = u((k+1)T)
where T is the sampling time.
R1:
IF e(kT) = PL AND e& (kT) < 0
THEN u((k+1)T ) = µPL(e(kT)) . u(kT)
R2:
IF e(kT) = PS AND e& (kT) < 0
THEN u((k+1)T) = (1−µPS(e(kT))) . u(kT)
R3:
IF e(kT) = NL AND e& (kT) < 0
THEN u((k+1)T ) = −µNL(e(kT)) . u(kT)
R4:
IF e(kT) = NS AND e& (kT) < 0
THEN u((k+1)T ) = −(1−µNS(e(kT))) . u(kT)
IF e(kT) = NL AND e& (kT) > 0
THEN u((k+1)T ) = µNL(e(kT)) . u(kT)
R5:
R6:
IF e(kT) = NS AND e& (kT) > 0
THEN u((k+1)T ) = (1−µNS(e(kT))) . u(kT)
R7:
IF e(kT) = PL AND e& (kT) > 0
THEN u((k+1)T ) = −µP L(e(kT)) . u(kT)
R8:
IF e(kT) = PS AND e& (kT) > 0
THEN u((k+1)T) = −(1−µPS(e(kT))) . u(kT)
where e& (kT) ≈
1
T
[e(kT) − e((k−1)T)], with initial conditions
y(0) = 0, e(−T) = e(0) = r − y(0), e& (0) =
1
T
[e(0) − e(−T)] = 0
C. Defuzzification
Select membership functions for the different control outputs from
the rule base
µN
1 µZO
–H
0
µP
u
H
Figure 5 Typical membership functions for u
Then, the overall control signal, u, is generated by a weighted
average formula:
N
∑ µ u (kT )
i =1
u((k+1)T) =
i i
N
∑µ
N
,
i
(µi ≥ 0,
∑µ
i =1
i
> 0)
i =1
where control outputs ui(kT), i = 1, ..., N=8 are from the rule base.
Overview of Conventional PID Controllers
In the time domain:
(i) P-controller
u(t) = KP e(t)
t
(ii) I-controller
u(t) = KI
∫ e(τ) dτ
0
d
(iii) D-controller u(t) = KD dt e(t)
Control gains, KP, KI, and KD, are constants to be determined in the
design for set-point tracking and stability consideration.
r
+
e
–
KP
u
system
y
(a) Proportional controller
r
+
e
–
t
KI∫
u
system
y
0
(b) Integral controller
r
+
e
–
d
KD
dt
u
system
(c) Derivative controller
Figure 6. Conventional PID controllers
y
In the frequency domain:
(i) P-controller
U(s) = KP E(s)
K
U(s) = sI E(s)
(iii) D-controller U(s) = KD s E(s)
(ii) I-controller
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.
r
+
e
−
KI∫ t
0
+
u
system
y
+
KP
(a) PI controller
d
KDdt
r
+
e
+
−
u
system
y
+
KP
(b) PD controller
KP
r
+
e
−
t
K∫I 0
+
+
u
system
y
+
d
KD dt
(c) PID controller
t
r
+
e
−
K∫I 0
KP
+
+
u
system
−
+
d
KDdt
(d) PI+D controller
Figure 7 Some typical combination of P, I, D controllers.
y
Introduction to Fuzzy PID Controllers
A. Discretization of Convetional PID Controllers
First, digitize the conventional analog PID controllers by
2 z −1
S = T z +1
where T > 0 is the sampling time.
For the PI controller:
u(nT) = u(nT–T) + T∆u(nT)
~
KI
~
∆u(nT) = K P v(nT) +
e(nT)
T
1
T
e(nT)
+
~
KI
+
z−1
−
∆u(nT)
T
+
+
1
T
v(nT)
~
KP
Figure 8. The digital PI controller
u(nT)
+
z−1
u(nT−T)
Similarly, for the PD controller:
~
~
∆u(nT) = K P d(nT) + K D v(nT)
where
e(nT ) + e(nT − T )
T
e(nT ) − e(nT − T )
v(nT) =
T
d(nT) =
1
T
+
d(nT)
~
KP
+
z− 1
e(nT)
+
∆u(nT)
T
+
+
z− 1
u(nT)
−
z−1
+
−
1
T
u(nT−T)
v(nT)
~
KD
Figure 9. The digital PD controller
B. Example: Designing the Fuzzy PI Controller
r(nT) e(nT)
+
1
T
-
d(nT)
Ki
Fuzzy PI
Controller
z-1
+
+
KuPI
u(nT)
-
1
T
z-1
Kp
ev (nT
Figure 10. The Fuzzy PI control system
Fuzzification and Defuzzification fuzzy membership functions
negative
positive
1
0.5
-L
0
L
∆ u PI (nT )
Figure 11. Output membership functions
negative
positive
1
0.5
-L
0
L
ep , ev
Figure 12. Input membership functions
Process
y(nT)
K p ev (nT )
IC18
IC12
L IC11
IC4
IC3
IC17
IC13
IC10
IC5
IC2
-L
K i d (nT )
L
IC6
IC1
IC14
IC9
IC7
IC8
-L
IC15
IC19
IC16
IC20
Figure 13. IC regions for the fuzzy PI controller
∆u PI (nT )
=
=
=
=
=
=
=
=
=
[
L K i d ( nT ) + K p ev ( nT )
[
2(2 L − K i d ( nT ) )
L K i d ( nT ) + K p ev ( nT )
2(2L − K p e v ( nT ) )
[
],
in IC1, IC2, IC5, IC6
],
in IC3, IC4, IC7, IC8
]
1
L + K p ev ( nT ) ,
2
1
[L + K i d (nT )] ,
2
1
− L + K p ev ( nT ) ,
2
1
[− L + K i d ( nT )] ,
2
[
]
0,
−L,
L,
Note: Rigorous stability can be guaranteed
in IC9, IC10
in IC11, IC12
in IC13, IC14
in IC15, IC16
in IC18, IC20
in IC17
in IC19
Some Successful Examples of Applications
There are many successful examples of fuzzy PID controllers.
1. Robotics
* Six-legged Insect Robot
(Dr. P. Sooraksa, G. R. Chen, and students)
King Mongkut’s Institute of Technology, Bangkok, Thailand
* Multi-purpose Autonomous Robust Carrier for Hospitals
(MARCH)
(Dr. P. Sooraska, Prof. S. K. Tso, Dr. B. L. Luk, G. R. Chen, and students)
Centre for Intelligent Design, Automation and Manufacturing
City University of Hong Kong
* Uncertain Robot-Arm Control
(next page)
(G. R. Chen and students)
University of Houston, Texas, USA
* Uncertain Double-Pendulum Control
(DEMO)
(Prof. Edgar Sanchez, G. R. Chen, and students)
CINVESTAV, Mexico
2. Temperature Control
(next page)
(G. R. Chen and students)
University of Houston, Texas, 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 … …
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 3, MAY 1997
Fig. 6. The output membership functions for the I component.
(R4) IF
AND
Then PD-output
.
In these rules,
is the error, where
is the
is the rate of the change of the error,
set-point,
“PD-output” is the fuzzy PD control output
,“
”
means “error positive” and “
” means “output positive,”
etc. Finally, “AND” is the Zadeh’s logical “AND” defined by
.
Similarly, from the membership functions of the fuzzy
I controller, the following control rules are used for the I
component, where
is the delayed error signal.
(R5) IF
AND
THEN I-output
.
(R6) IF
AND
THEN I-output
.
(R7) IF
AND
THEN I-output
.
(R8) IF
AND
THEN I-output
.
In the above rules, “I-output” is the fuzzy I control output
, and the other terms are defined similarly to the PD
component.
These eight rules altogether yield the control actions for the
fuzzy PD I control law.
3) Defuzzification: In the defuzzification step, for both
fuzzy PD and I controllers, the commonly used “center of
mass” formula is employed to defuzzify the incremental
control of the fuzzy control law, (15) as shown in (17)
Fig. 7. Regions of the fuzzy PD and I controllers’ input-combination values.
Fig. 8. Block diagram and physical setup for the robot arm experiment.
geometry of Fig. 7:
we obtain the following nine formulas for the ten IC regions:
input membership value membership value of output
membership value of output
(17)
For the fuzzy PD controller, the value-ranges of the two
inputs, the error and the rate of change of the error, are
actually decomposed into ten adjacent input-combination (IC)
regions, as shown in Fig. 7. The control rules for the fuzzy
PD controller (R1)–(R4), with membership functions and IC
regions together, are used to evaluate appropriate fuzzy control
law’s for each region.
Now, by applying the values
,
,
,
and the following straight line formulas obtained from the
in IC A
(18)
in IC B
(19)
in IC C
(20)
in IC D
(21)
in IC E
(22)
in IC F
(23)
in IC G, J
(24)
in IC H
(25)
in IC I
(26)
Similarly, defuzzification of the fuzzy I controller follows
the same procedure as described above for the PD component,
except that the input signals in this case are different. The
376
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 3, MAY 1997
Fig. 10.
Tracking performance of the fuzzy PID-controlled system (I).
Fig. 11.
Tracking performance of the conventional PID-controlled system (II).
the mass-center of the axis of rotation, and
the gravity
constant.
The actual data that we used in the experiment were:
(kgm ),
(kgm ),
(Nm),
IV. EXPERIMENT
AND
RESULTS
In the experiment, the robot arm system was run using two
control methods: using the conventional digital PID controller
and the fuzzy PID controller, respectively, both based on
the same configuration and under the same conditions. The
conventional digital PID controller was unable to satisfactorily
run the robot arm system (with a nonlinear load) no matter
what combinations of control gains were used. Plots of the
system output with the tuned PID gains that achieved the best
results by trial-and-error are shown in Fig. 9, where and below
s. The fuzzy PID controller, in contrast, controlled
the robot arm system with reasonably good results, obtained
also by trial-and-error, as shown in Fig. 10. In this and the
,
,
other fuzzily controlled simulation figures,
, and
and
are fuzzy gains defined in
Section II-B (see Fig. 2). In Figs. 9 and 10, the robot arm was
not connected to the spring, where the units used are oltage
400 r/min) versus second.
(1 V
Another comparison, at a different speed and with the spring
added to the arm, is shown in Figs. 11 and 12, where the units
used are voltage (1 V 400 r/min) versus s in Fig. 11 while
versus second in Fig. 12. In these experiments, parameters
for the fuzzy PID control gains were found such that there
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
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