The conventional interface composition algorithm of a fuzzy controller is very time and memory consuming. As a result, it is difficult to do real time fuzzy inference, and most fuzzy controllers are realized by look-up tables. Here, researchers derive a simplified algorithm using the defuzzification mean of maximum. This algorithm takes shorter computation time and needs less memory usage, thus making it possible to compute the fuzzy inference on real time and easy to tune the control rules on line. A hardware implementation based on a simplified fuzzy inference algorithm is described

Teh, H. H.

Wang, P. Z.

Wu, Z. Q.

English

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The
Abstract
Simplification of Fuzzy Control Algorithm
and Hardware Implementation
Z. Q. Wu, P. Z. Wang and H. H. Teh
Institute of Systems Science
National University of Singapore
Heng Mui Keng Terrace,Kent Ridge
Singapore 0511
The conventional inference composition algorithm of fuzzy controller is very time and
memory consuming. As a result, it is difficult to do real time fuzzy inference and most
fuzzy controllers are realized by look-up tables. In this paper we derived a simplified
algorithm using the defuzzification mean of maximum. This algorithm takes shorter
computation time and needs less memory usage, thus making it possible to compute the
fuzzy inference on real time and easy to tune the control rules on line. The responsibility
of thisalgorithm is proved mathematically In thls paper.
Fuzzy controller has been highly developed and come to a new stage of hardware
lmplementatlon. Many fuzzy controllers(or so called fuzzy inference machines) in
hardware have been available in the market. The conventional fuzzy inference algorithm
on which most fuzzy controller based on is too complicated. Further, its hardware
implementation is very expensive and of a large volume, and the inference speed is
limited. Reducing its cost and volume and improving its inference speed are very
important to this technology. In this paper we also describe a hardware implementation
based on the above simplified fuzzy inference algorithm.
1. Fuzzy controller algorithm
Assume that the fuzzy controller has two inputs and a single output as shown
in Figure 1,
A _ I II
X __ [ Fuzzy Relation C
Input B R u _ Output
Y
Fig.1 The block graph of fuzzy controller
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https://ntrs.nasa.gov/search.jsp?R=19910012468 2020-03-19T19:00:49+00:00Z
where A and B are the linguistic variables of the inputs, with universe of
discourse X and Y respectively, and C is the linguistic variable of the output,
with universe of discourse U. We emphasize here that X and Y are not
necessarily continuous on the real line R, but arbitrary subsets of R.
Let the sets of linguistic values concerning with A, B and C respectively be as
follows
{Ai} e F(X), (i_ I) (1)
{Bj} e F(Y), (j e J) (2)
{ck) e F(U), ( k e K ) (3)
where I={1, 2, ..., m}, J={1,2, ..-, n}, K={1, 2, ..., h}, and F(X) represents the fuzzy
power set of X.
The fuzzy control rules are described in terms of a group of multi-eomplexed
fuzzy implications as follows:
If A is Ai and B is Bj then C is Ck,
(i_ I, je J,k=q_(i,j) e K)
(4)
The above fuzzy implications can be translated into a three-dimensional
_relation R according to the fuzzy Compositional Rule of Inference(CRI method).
Definition 1. R---W (AixBjx Ck)
i,j
Re F(X x Y x U),
R(x, y, u)= V (Ai(x)/k Bj(y) A Ck(u)).
i,j
(k = q_(i, j) e K )
(5)
Suppose that the inputs of the fuzzy controller at a certain instance are fuzzy
sets A* _ F(X) and B* e F(Y), according to the CRI method, the output of the
controller will be the fuzzy set denoted by C* e FCLI), i.e
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C* = (A* x B*) o R
C*(u) = sup (A*(x) A B*(y) A R(x, y, u))
x_X
yeY
= sup ((A*(x)A B*(y)) A (V (Ai(x) A Bj(y) A C._(i, j)(u))))
x_ X i,j
y_Y
= sup ( V ((A*(x) A Ai(x)) A (B*(y) A Bj(y)) A C_(i, j)(u)))
xe X i,j
ye Y
= V sup((A*(x) A Ai(x)) A sup(B*(y) A Bj(y)) A Cq,(i, j)(u)
i,j xe X ye Y
(6)
In actual applications the inputs of the controller (i.e the observed values of
the controlled process) are some definite real numbers. Suppose in a certain
instance the observed value is a pair (xo, yo), then the fuzzy sets of inputs A*
and B* are as follows,
I, x=x o " I, Y=Yo
,B*(y) = {
A*(x) = { 0, x#x o 0, Y#Yo (7)
so that sup(A*(x) A Ai(x)) = Ai(xo) (8)
x_ X
sup(B*(y) A Bj(y)) = Bj(yo) (9)
yeY
therefore
2. The
C*(u) = V (Ai(xo) A Bj(yo)) A C_(i,j)(u)
ij
(ie I,j e J, _0(i,j)e K)
responsibility of the fuzzy controller
(10)
97
The responsibility of a fuzzy controller has been defined and analyzed in depth
by P. Z. Wang and S. P. Lou[3], here we discuss the responsibility of fuzzy
controller under a weaker condition.
Definition 2 For a set of linguistic values concerning with A is {Ai}(i e D e F(X),
I={1, 2, ..-, m}, where Ai is a normally distributed fuzzy set, there
exists m+l real numbers
ro <rl <r2< ... <rm,
such that for any given x e (ri-1, ri), if j s i, Aj(x) < Ai(x) (see
Figure 2). We called Ji = (ri-1, ri) _ X, the interval of A i, i e I, and
N 1 = { ri } the net of A, where Ji and N 1 satisfy the following:
Jin Jj =,t, ,(i ,, j) (11)
m
L) Ji=X- {ri} (12)
i=l
k_t
ASffx) Ai(x) Ai+l(x)
0 ri-1 ri ri_-i x
Figure 2 Membership functions of fuzzy sets A i
For {Ai} e F(X), {Bj} e F(Y), we have
(Ai x Bj)(x, y) _- Ai(x) A Bj(y)
V i e I, V j e J, I={ 1, 2, ..., m], J={ 1,2, ..-, n}.
If there exist nets NI= {r'i}, N 2 = {r"j} and intervals {J'i}, {J"j} for A
and B respectively, then
(13)
(14)
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(X, y) _ Jst---") As(X) A Bt(y) > Ai(x) A Bj(y)
( (s, t) _ (i, j) )
(15)
where Jij 5 J'i × J"j, is called the interval of (Ai x B j).
In fact,
(x, y) _ Jsr--_ x e J's, Y e J"t-----> As(x) > Ai(x), Bt(y) > Bj(y)
(s, t) _ (i, j)
min(As(x), Bt(y)) > min(Ai(x), Bj(y))
As(x) A Bt(y) > Ai(x) A Bj(y) (16)
we define the net of A x B as
N -- {(x, y) I x = r'i, y = r"j} (17)
According to the definition of responsibility of fuzzy controller given in [3], we
state the following definition with slight changes.
Definition 3 A Fuzzy controller is said to be responsive if there exists an
interval L _ (_oo, +_ ) such that
L = {u I C*(u) = hgt C*(u)}. (18)
u_ U
where hgtC*(u) is the height of fuzzy set C*(u) and L is the
responsive interval.
If a fuzzy controller is responsive; the output of the controller, according to the
defuzzification mean of maximum, is
u o = M(L) (u oe U), (19)
where M(L) means the mid-point of L.
Theorem 1. A given fuzzy controller is responsive as long as there exists a Net
N of A x B such that the intersection of N and the universe of
discourse (X x Y) is empty, i.e
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Proof:
N n X=¢,(X.=Xx Y)
Assume that N is the Net of A x B, which satisfies formula (20), i.e
X=X-N
from formula (21), we derive that
(2O)
(21)
v j0 =X_(X ×Y)
i0j
so for any definite (x o, Yo) _ (3[_ x Y), there e_st s, t, such that
(xo, Yo) _ Jst.
From formula (16), for any s,i _ I, tj _ J, if (s, t) _, (i, j), then
(22)
As(x o) A Bt(yo) > Ai(xo) A Bj(yo) (23)
According to formula (10), the response of the fuzzy controller is as
follows
C*(u) = V (Ai(xo)) A Bj(yo) A C_(i. j)(u)
ij
By formula (23), it is obvious that
Where
M(C*) = M(O
M(C*) = {ul C*(u) = hgtC*(u)}
M(0 = {u i f(u) = hgt f(u)}
f(u) = (As(xo)) A Bt(yo) A Cp(s, 0(u)
(24)
(25)
(26)
(27)
(28)
As it is known that C_p(i, j)_ {Ck} *' • is a distributed fuzzy set whose
kernel is
obviously, there
Ker(Ck) = {u I Ck(u) = 1} _ •
exists an interval L = (_oo, +oo) such that
(29)
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L = {u I Cv(s, 0(u) -_ (As(xo)) A Bt(yo))}
= {u I f(u) = (As(xo)) A Bt(yo))}
= {u I C*(u) = hgtC*(u)}
therefore the fuzzy controller is responsive and uo = M(L).
(30)
3. The simplification of fuzzy controller algorithm
In the right-hand side of formula (10), there are I × J terms of union
operations. The ordinary algorithm does this calculations term by term and is
very time consuming. We know from Theorem 1 that when a fuzzy controller is
responsive and the defuzzification mean of maximum is used, we only need to
calculate the interval L, then the mid-point of L will be the desired output of
the fuzzy controller. Thus for all observed value (xo, yo), we only have to
calculate f(u), only one of the terms in the formula (10). This will simplify the
computation algorithm to a great extent.
Let
A__ = V Ai (31)
i¢ I
A __(x) = V Ai(x) (32)
ieI
B _ = V Bj (33)
jCJ
B_r(y) = V Bj(y) (34)
jeJ
where x _ X-{r'i}, y e Y-{r"j}. The membership functions of A_(x) and B_(y) are
shown in Figure 3-a.
101
r0 0 rl r2 r3 x, y
Figure 3-a The membership functions of A__(x) and BE(y)
1
8
P
c4,(u)
0 Ul U 0 u 2
Figure 3-b The function of pk(13)
Clearly, when (x, y) _ Jst, i.e x _ {r's-1, r's}, y _ {r"t-1, r"t}
A_(x) = As(x) > Ai(x),
B :_(y) = Bt(y) > Bj(y),
A__(x) A BE(y) = As(x) A Bt(y) > Ai(x) A Bj(y)
V s,i _ I, t,j e J, (s, t) - (i, j)
Define the separating functions t01(x), tO2(y ) respectively as follows,
¢pl(X) = i, x ¢ {r'i-1, r'i},
q32(y) = j, y _ {r"j-1, r"j}.
For C _ {Ck}, k ¢ K, we define the following function
102
(35)
(36)
(37)
(38)
(39)
where L _ (_oo, +oo),
pk([3) = M(L)= M(Ckl3), 13e [O, 1] (40)
L_ U= {ulCk__13} (41)
where Ckl3 is the 13- outed set of Ck and M(-) represents the mid-point of (-) as
shown in Figure 3-b. Since Ck(k • K) is normally distributed set, Pk(l_)is a
continuous single-valued function of 13, V 13• [0, 1]
So far as the functions Ax(x), Bx(y), q)l(X), q)2(Y) and Pk(13) are defined, we can
derive the following simplified algorithm for the responsive fuzzy controllers:
1) Given the inputs (x o, Yo) of the fuzzy controller, calculate
a = AX(xo), b = BX(y.), (42)
s = q_l(Xo), t = _02(Yo), (43)
I_ = a A b = min(a,b). (44)
2) Calculate
k=q_(s,t), k• K (45)
where the q_ is determined by the given control rules.
3) Finally, the output of the fuzzy controller can be obtained from
Uo = pk(13) (46)
Obviously, LA U= {u ICk->l_}
= {u I C_o(s, O -_ (As(xo)) A Bt(yo))}
uo = Pk(13) : M(L)
(47)
It can be seen that the result is exactly the same as that in formula (30).
The conventional fuzzy controller algorithm is very time consuming and needs
103
large memory space so that it is hardly possible to implement the fuzzy
composition inference on line in a control system. In many applications, fuzzy
controllers used look up tables instead of real time inference. Not only it is
impossible to tune the fuzzy control rules on line, it takes a great amount of
computation time to calculate the fuzzy controller look-up table. The simplified
algorithm proposed above reduces the computation greatly and its calculating
time is nearly the same as that taken by the conventional PID Control
algorithm. This makes it possible to do real time fuzzy inference in the
controller, allowing the tuning of control rules on line. If the algorithm is used
to calculate the fuzzy control look-up table, it takes less than one minute. Since
we only need to store 5 functions, namely Ax(x), BE(y), q_l(X), q_2(Y)and Ok(13)
instead of all the Ai(x), Bj(y), and Ck(u), a total of I+J+K functions.
4. Hardware Implementation
'Fuzzy controller has been highly developed and come to a new stage of
hardware implementation. Many fuzzy controllers(or so called fuzzy inference
machines) in hardware are available in the market[4][5]. The conventional
'fuzzy inference algorithm on which most fuzzy controllers are based on is too
_complicated. Further, its hardware implementation is very expensive and of a
large volume, and the inference speed is limited. Reducing its cost and volume
and improving its inference speed are very important to this technology.
As can be seen from the last section that with the proposed algorithm, the
calculation is much simpler as there is no computation of fuzzy sets and most of
the calculations involve only function operations and comparative operations.
Therefore, this fuzzy control algorithm is very easy to implement in hardware.
The main issue in a hardware design is to construct some function generators
generating Ax(x), BX(y), q)l(X), q)2(Y) and pk(t3), while the complicated fuzzy set
operation which is difficult to turn into hardware counterparts is avoided.
104
CPU
Data Bus
fb
": xo • Yo
:x) [_,(x){ {BZ(y) _2(Y)
_Iin(a,b) Rules _--
g L
II IIk
Figure 4 Block diagram of the fuzzy controller board
We have designed a fuzzy controller board for Personal Computers(PC) based
on the above algorithm. The principle of the fuzzy controller board is
illustrated in Figure 4. The board is composed of some function generators to
generate A£(x), B£(y), tpl(X), tp2(y)and Pk(l_), a comparator to do the operation
of Min(a, b) and a rule base to store the control rules. Each part is constructed
with digital IC. The detailed design of the hardware will be presented in depth
in our future papers.
The controller board is connected to the CPU through the data bus of the PC.
The generators of At(x), B£(y), tpl(X), tp2(y) and Pk(13) and the control rules can
be programmed conveniently. Using this board with its software environment
on a personal computer, it is very flexible to construct a fuzzy control system
for an industrial process in which large number of data needed to be processed.
This is the reason why we design a fuzzy controller board instead of an
independent fuzzy controller machine which is unable to process data and
information.
Due to its fuzzy inference function and ability of data processing, the fuzzy
control system can be applied not only to the control system but also to many
other areas such as expert systems, pattern recognition and decision making
where the fuzzy inference method may be employed.
105
Reference
[1] G. Q. Chen, Analysis into the fuzzy control algorithm, Information and
Control, No.5(1980)
[2] M. Mizumoto, Fuzzy reasoning with "if...then
cybern., 1, 2927-2932(1980)
else", Applied sys. and
[3] P. Z. Wang and S. P. Lou, The responsibility of a fuzzy controller, IFAC 8th
Triennial World Cong., Kyoto, Japan, August 1981.
[4] T. Yamakawa, Fuzzy Controller Hardware System, Preprints of 2nd IFSA
Congress, Tokyo, July 1987.
[5] T. Yamakawa, A Simple Fuzzy Computer Hardware System Employing Min &
Max operation--A Challenge to 6th Generation Computer, Preprints of 2nd
IFSA Congress, Tokyo, July 1987.
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