Direct digital control, auto-tuning and supervision using fuzzy logic

Fuzzy Sets and Systems 30 (1989) 135-153 North-Holland

135

D I R E C T D | G H ' A L C O N T R O L , AUTO~TUI~NG AND SUPERVISION USING F U Z Z Y LOGIC A. OLLERO and A.J. GARCfA-CEREZO Departamento de lngenieria El~ctrica y de Computadores y Sistemos, Universidad de Santiago, .F_.scuelaTcfcnica Superior de Ingenieros industriales, Apartado 62, Vigo, Spain Revised December 1986 R~vised August 1987

Abstract: We describe algorithms for the design and implementation of fuzzy logic controllers and present results of their application to temperature control of a heated air-stream. Auto-tuning methods are studied for adjustment of conventional parameters to compensate for actual values of control variables, or of external variables; and for improvement of performance from observation of results. Conditions for verifying the consistency of the linguistic protocol with its implementation are presented in an appendix

Keywords: Fuzzy logic control; auto-tuning of controller parameters; supervision of control systems.

1. Introduction

It is well known that modern control theory has failed to cope w~h the practicalities of many industrial processes despite the development of an important body of mathematical knowledge. Indeed, the position of many control practitioners is essentially the following: non-complex systems can be controlled with classical (analog or digital) PID controllers which can be tuned according to heuristic rules without first modelling the process to be controlled; on the other hand there are no appropriate mathematical models of many complex systems which could be easily handled by means of control theory to obtain reliable control laws. Indeed, the control of a great number of industrial processes requires the attention of p~oces~ operators of industrial processes ~equires the attention of process operators to generate control actions directly or to supervise the performance of simple automatic controllers. Today, most analog controllers are being replaced by digital ones, particularly by using microprocessors. Moreover, computer facilities are greatly appreciated for data logging and supervision. However, despite some exceptions (self-tuning controllers, for example), direct control algorithms of most industrial digital controllers of continuous processes perform basically discretized versions of classical analog functions. It must be noticed that direct regulation procedures are only a reduced part of the programs of a typical industrial digital contlollei'. 0165-0114/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

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A. Ollero, A.]. Garda-Cerezo

In recent decades several researchers have been exploring alternative ways of automatic control without using a mathematical model of the process. Artificial Intelligence methods provide interesting ideas to perform automatically the actions of human operators [1]. Particularly, fuzzy control techniques [13, 14, 15, 18], have been applied successfully [9, 10, 12, 17]. We notice that most applications of fuzzy control are concerned with direct control and do not exploit other possibilities in auto-tuning and supervision of controllers. However approximate reasoning [23], as used in fuzzy control, seems to be an interesting tool to perform these tasks automatically. Indeed, artificial intelligence and expert systems have been recognized as vehicles to incorporate experience from control system designers [3], on a supervisory level of automatic control, and the same ideas apply when considering the information from process operators. In the following section we present results in the application of fuzzy control methods both in direct control and s,pervision. 2. Design and bnp|ementadonof ~rect fuz~ logic com~geu.s Fuzzy logic controllers can be considered as non-linear digital controllers in which the control signal to be applied on the process is generated by means of a multidimensional look-up table inference matrix obtained from a linguistic protocol provided by plant engineers and process operators. Thus, fuzzy controllers can be viewed as models of the human operators determining the appropriate values of the control signal u, or its increment u, from the observation of process variables, as for example the error signal e of a control loop, its increments Ae, and the sum of errors s. Taking into account the input variables (e, Ae, s) and output variables (u, Au) of this model, it is possible to define [19] proportional controllers: (la)

u ( k ) ffi~(e(k));

integral controllers: Au(k) = ¢(e(k))

or . ( k ) = ¢~(s(k));

(lb)

proportional-derivative controllers (PD): (lc)

u(k) = ~(e(k), Ae(k));

proportional-integral controllers (PI): u ( k ) ffi ~ ( e ( k ) , s ( k ) )

or A u ( k ) = ~ ( e ( k ) , A e ( k ) )

(ld)

where A e ( k ) = e(k) - e(k - 1)~

A u ( k ) = u ( k ) - u(k - 1),

k

s ( k ) = ~ e(i) - s ( k - I) + e(k), iffiO

(2)

Fuzzy logic control

-<

137

L-I

r--

Inference PLANT

matrix

Fig. 1. Pl fuzzy control system.

and ~(.) is the inference matrix which provides discrete values of u from discrete values of the controllers inputs. Figure 1 shows a PI fuzzy logic controller. The design of a fuzzy logic controller includes the specification of a protocol R,

R={r,},

i ffil,...,,~,

(3)

consisting of ns linguistic statements relating the input of the controller and the appropriate outputs. Consider a controller with two inputs variables x~ and x2 and one output variable (control signal) u. The rules have the form: "if xl is vxl and x2 is vx2 then u is v u "

(4)

where vxl, vx2 and vu are angmst~.~ vah~es of x~, x2 and u respectively. For example, in the PD fuzzy logic controller: "if e is POSITIVE-SMALLand e is LARGE-INCREASEthen u is posmve". Notice that rules reflect 'situation-action' pairs to cope with the process operator experience. The situation side of the pair is often a compound statement of the input variables. A set of basic linguistic terms can be used as linguistic values of each ~ariable: VXi ffi { v x s l , . . . , vx~t} for the variable xl.

(5)

The semantics of basic linguistic terms is provided from its definition as fuzzy subsets of the interval in which the corresponding variables take values. The membership function values are assigned according to the process operators. Moreover, linguistic hedges [22] sucE as v~RY, MORE-OR-LeSS,RA~eR can be used to obtain linguistic values by modifying basic linguistic terms. The effect of these hedges can be interpreted by using non-linear operators over the basic linguistic terms. The protocol can be considered as a composed fuzzy relation with the membership values ~R(x~, • • • , z . , . ) = m a x ( ~ . j ( x l , . . . ,

x., .))

(6)

where # , j ( x , , . . . , x., u) = n~in(gj,(x~),..., #j.(x,); #/.(u)).

(7)

138

A. Ollero, A.J. Garcfa-Cerezo

~# and ~ju are respectively the membership functions of the linguistic values assigned to x~ and u in the statement j. From a set ( x l , . . . , x,) of n measures (numerical values) of the process, the result of the inference is a fuzzy subset g(u) of the interval in which take values the controller output u. This fuzzy subset must be reduced to the most significative real value. One method is to choose the value which corresponds to the maximum of the m e m b e ~ O function, or to obtain the average when there are several maxima. ,""~*~~'-~ ~:~ ~or~ conser~,ative method is to form an average based on the shape o~ ~;~ membership function, center-of-area for example. The result of the above mentioned designing procedure is the multidimensional look-up table [4, 8, 20], or inference matrix ~(.): ,

=

. . . , x.),

(8)

relating the discretized measured values of input variables x~ with the appropriate control actions u. The performance of the fuzzy logic controller depends on the inference matrix and its consistency with the protocol. In the Appendix we analyze this consistency. Definition of a fuzzy subset describing linguistic terms involves the choice of the number of quantization levels for each variable. This number varies between 5 and 30 levels in different implementations [18]. It must be noticed that quantization levels define the dimensions of ~(.). For a fuzzy controller with two inputs xt, x2 and one output u, dimensions of ~ are (n~, x n~z) where nx, and n~2 are respectively the number of quantization levels of xt and x2. Let i~, i2, i,, be indices denoting the quantization levels of x~, x2, and u respectively. The algorithm to determine O(xt, x2) from a linguistic protocol and the membership function of fuzzy subsets describing linguistic terms, can be represented as shown in Figure 2. begin for il -- 1 , . . . , nx, do for ia-- 1 , . . . , n , , do begin for i, = 1 , . . . , n, do begin a::O; for each linguistic statement j = 1 , . . . , n~ do begin b := min(/Aj~(il) , t41a(i2),/41.(i.));

a:=max(a,b); g(i.):=a

end end; reduce vector g(iu) to a scalar value; {z := redt,ct[g]}; ~(il, i2): = z end

end;

Fig. 2. Determinationof ~(x, x2) and membershipfunctions.

FuzZy logic control

139

begin controller initialization; for each sampling time kT do begin sample x~(kT), x2(kT); approximate x~(kT), x2(kT} to discretization points il, i2,

u:=¢(il, i2); apply u end end; Controller initialization procedure: begin load matrix 7('); select sampling time T; real time clock initialization end;

Fig. 3. Implementation algorithm (rectangular).

Implementation of the controller can be performed by approximating the sampled values of the controller inputs to the nearest discret~ation points of ~(.) as is shown in Figure 1 for a type PI fuzzy logic controller. In Figure 3 we show the corresponding implementation algorithm.

2.1.

Trapezoidalimplementation

The performance of the above controller depends on the number of quantization levels used to form matrix ¢(.). An alternative to reduce this effect is to make ase of trapezoidal discretization instead of the rectangular rule. Let (i~, iz) and (jl, j2) be respectively the immediately inferior and superiol" discretizing points for the actual values of (xl, x2). The algorithm for the fuzzy logic controller implementation with the trapezoidal rule can be given as in Figure 4. This algorithm can be easily extended for multiple input systems. The main advantages of this approach are: - The effects of quantization (limit cycles, steady state errors, oscillations) are eliminated or greatly reduced. - The number of quantization levels in the definition of linguistic terms can be reduced. Thus, we save memory when storing the inference matrix ¢(.). begin controller initialization sample xl(kT), x2(kT); determine il,/2 and jl, J2; u:=U2-x2

.,[~(i,,i2) ~(j,,i2l][J,-x,]. x2-12J[~(il,j2 ) ~(j,,j2)JLxl-il '

apply u end;

Fig. 4. Implementation algorithm (trapezoidal).

A. Ollero, A J . Garcfa-Cerezo

140

Sensor

I A/D

]

D/A

8 bits microcomputer

Fig. 5. Heater air-stream prototype.

The execution time of this implementation is greater but even as small as for control of systems with time constants on the order of 1/100 seconds by using 8-bit microcomputers. From t~e point of view of memory requirements, it can be noticed that fuzzy logic controllers with two input variables and one output, with 16 discretizing points (256 memory locations for ~), can be implemented on IK EPROM including program and data. Example. Consider the temperature control oi' a heated air stream. This process is found in many industrial systems such as furnaces, boilers, air conditioning, etc. The following experiences have been gained from the scale prototype shown in Figure 5. The digital controller receives from an analog-digital converter the sampled signal of the air-stream temperature sensor and the control signal is applied with a digital-analog converter to the thyristor power amplifier. A fuzzy logic controller has been designed, using as inputs the temperature error with respect to the set-point, and the sum of errors. The protocol used is shown in Table 1. Fuzzy subsets describing linguistic terms are shown in Figure 6. The inference matrix obtained with the algorithm above described is illustrated in Figure 7. Table I If If If If If If If If If

e e • • e e • e •

is is is is is is is is is

POSmvE.BIG a n d s is POSITIVE-BIG then a is POSITIVE-BIG. POsrnvE.BtG and s is POSITIVE-MEDIUM then a is POSITIVE-MEDIUM. POSITIVE.BIG and s is NEGATIVE t h e n a is POSITIVE-SMALL. POSITIVE-MEDIUM and s is POSITIVE.BIG then a is POSITIVE-MEDIUM. POSITIVE-MEDIUM and s is POSITIVE.MEDIUM then a is POSITIVE-MEDIUM. POSITIVE-MEDIUM and s is NEGATIVE t h e n a is ZERO. NEGATIVE a n d s is POSITIVE.BIG t h e n a is ZERO. NEGATIVE a n d s is POSITIVE-MEDIUM t h e n a is ZERO. NEGATIVE agld s is NEGATIVE then a is ZERO.

Fuzzy logic control

141

1

,8 .G ,4 ,2 0

. . . . . . . . . . . .

-3

e

0

7

.8 ,G ,A

I

! NEGATIVE I1 POSITIVE MEDIUM III POSITIVE BIG

$ -1

22

IV ZERO V POSITIVE SMALL I ,8

,G

,4 ,2 O

I

I

I

0

I

!

I

I

t

I

I

a

7

Fig. 6. Definition of basic linguistic terms.

Figure 8 shows the results of the application of the fuzzy logic controller with both rectangular and trapezoidal implementation algorithms. Notice that rectangular discretization deadband produces a steady-state permanent error. Moreover, the round-off by using rectangular discretization may lead to limitcycle oscillations.

a

0 22

/

Actuation

e

Error

s

Sum

-3 1

Fig. 7. 3D representation of inference matrix.

of e r r o r

A. Ollero, A.7. Garcfa-Cerezo

142

v°,I 3

"

/

_

/ .

.

.

.

Rectangular Trapezoidal

.

aproximation aprox,mation

I

'~

I

0

I

|

I

25

I

'

50

,

!

,

75

o'

,

SeC.

100

Fig. 8. Results of implementation.

3. Sepen~sion n d on-line ~tm~mgof linear controller parameters In recent years the supervision ~:nd tuning of controller parameters has received increased attention. In PID process control there are classical tuning methods [24] which have been applied extensively for tuning parameters to fixed values. Moreover, self-tuning controllers based on process parameter estimation [2] and other control adaptive schemes [11, 7] have been proposed. These automatic tuning systems are based on the knowledge of the process model and assumptions such as distribution of the noise, character of the reference signal, and

,e

,6 ,2 0

e

-3

0

7

t~

22

PM

Z

PB 16,1

PM

Z

PM 7, 72

Z

Z

PS -1

-3

0

4

Fig. 9. Partition of Table 1 protocol.

Fuzzy logic control

143

uamodelled high-frequency process dynamics. The adaptive control systems contain many parameters, and the designer and process operator must decide on values when it is difficult to understand the relationship between parameter settings and control system actions. However, it must be pointed out that violations of the assumptions are very c()mmon in practical implementations [21]. When the process model it not well known and the underlying assumptions of the adaptive control design methods are violated, the only way for automatic tuning seems to be the consideration, in a supervisory level, of the heuristic knowledge of the designers and process operators. In the following we propose three techniques based on approximate reasoning as used in fuzzy controllers.

3.1. Fuzzy auto-tuning of controller parameters Consider a non-linear system with approximate linear behaviour for different working areas. In particular consider the two-dimensional position control system shown in Figure 10 with linear state feedback

u(k + 1)=f(x(k), u(k)), u(k) =

(9)

-h,(x,(k) - ref) - h2x2(k),

where x1(k) is the position and xa(k) the velocity. Assume that this system is approximately linear in fuzzy interval values of the control variab|e u. It is intended to obtain state variable trajectories close to the target trajectories in Figure 11, these trajectories corresponding to a linear quadratic control system assuming a linearized model in all the state space. Assume that process operator knows that the following feedback gains have acceptable performance according with target trajectories: h = [5.2, 3.7]

f o r t~ POSITIVE-SMALL,

h = [7.3, 5.2l

f o r u NEGATIVE-BIG o r POSITIVE.BIG,

h = [12.2, 8.5]

f o r U NEGATIVE-SMALL.

Definition of linguistic terms concerning K is performed taking into account the

x(Kot }- F(x{K},u(K })~---X2

4-

velocity _

sition Fig. iO. Linear state feedback system.

set

point

A. Ollero, A.J. G~c[a-Cere~:,~

144

I

Fig. 11. Target state variable trajectories.

above mentioned experiences. Linguistic terms for the control variable and K are shown in Figure 12. Consider a state variable feedback defined as - ref) + h 2 x 2 ( k )).

u(k) = - K(ht(xt(k)

(11)

The protocol to infer changes in the gain K can be formulated as follows: If u is NEGATIVE-BIG or POSITIVE.BIG then K is LARGE, If u is NEGATIVE-SMALLthen K is MEDIUM,

(12)

If u is VOSI'HVI£-SMALLthen K is SMALl[..

[ 10. 0,8 O~ 0,~ 0,2 0

II

Ill

IV I Negative big il Negotiv, e small !|! Positive stool! IV Positive big

I

I

l

l

I

!

I

I

w

.

I

i

2 3 /, S 6 7 8 9 10 I1

I 1,0

II

Ill

:

0,8 0,6 0,4 0,2 0

| Sinai! il Medium II! Lorge !

I

2

3

¢

5 6

1

7 6

!

. . . .

I

9 10 II

Fig. 12. Definition of basic linguistic terms.

Fuzzy logic control

145

K 12 10 8 G

4 2 0

,u

+2

0

-2

Fig. 13. Inference vector.

Notice that, in this one-input one-output protocol, the inference matrix is a vector O(u). By using the procedure presented in Section 2, the vector shown in Figure 13 is obtained. The control system with fuzzy auto-tuning of K is shown in Figure la,. In the Figure 15, we present the result of the implementation on the position control system.

xIk+l ) = f(x(kI, u(k))

.t set point

ition Fig. 14. Control system with automatic tuning of a gain. 2

1

~ L~2

o

~,2

o--~ Ii velo,c i ty 2,4

3,6

Fig. 15. Result of implementation.

s~r.

146

A. OIlero, A.J. Garcfa-Cerezo

We must notice that more complex fuzzy auto-tuning systems can be developed. For example, modifications on K can be inferred from O(u, xl, x2) if designers and process operators can provide the appropriate protocol and definitions of linguistic terms.

3.2. Fuzzy auto-tuning from external variables The operator tuning of controller parameters is also performed taking into account external variables without consideration in typical process models. For example, the dynamics of the heater temperature control system in Section 2 depends on the external temperature T~, and the tuning of a conventional controller, by considering appropriately this variable, improves the performance. Consider a proportional-integral controller (PI) for the heater temperature control system: (13)

u(k) = ht " e(k) + hz "s(k),

where e(k) is the temperature error with respect to the set point and s(k) is the sum of errors. When the external tempcrature Te changes significant, the control parameters can be updated on-line by means of factors of change cht and oh2: ht - ht * cht,

hz = h2 * oh2.

(14)

1,0 0,0 0.6

'k___~_

0,/,

I LOW II M E D I U M I[IHIGH

0,2 0,0

--,

/

.

1~

,

/

.

.~-,

16

,

,,Te

20

1,0 0,8

] SMALL-DECREASE 11 NIL III L A R G E - I N C R E A S E

0,6 0,~. 0,2 0,0

1

!

i

,0 7

1

,

,oh 2

1,07

1,0 0.8 0,6

I MEDIUM-DECREASE 1I NIL Ill M E D I U M - I N C R E A S E

O.t, 0.2 0,0

chl .053

1

1,053

Fig. 16. Definition of basic terms of lingustic variables.

Fuzzy logic control

147

ch2

ch 1

1

11"

1 _

9 8

7 •

12

i

l

i

1"6

I

i

~

20

li

i

i

1'6

w

i

e

]'e

2~}

Fig. 17. cht and ch 2 inference vector.

The process operator experience can be expressed by means of the protocol If T¢ is LOW then chs MEDIUM-DECREASEand If Ire is MEDIUMthen chs NliLand

ch 2 SMALL-DECREASE,

ch2 NIL,

If T~ is HIGH then ch~ MEDIUM-INCREASEand ch2 LARGE-INCREASE.

Definition of linguistic terms is shown in Figure 16. In this way, it is possible to obtain ch, ---

oh,-

(15)

where qb---(qbl, ¢2) is an inference 1 × 2 matrix. In Figure 17 we illustrate the values of oh1 and ch2 for different values of T~. Figure 18 shows the control strategy. The result of the implementation of this control system for two values of the external temperature T~ are shown in Figure 19.

Te

._~

Heater

!c

Fig. 18. Fuzzy auto-tuning from externa| variables.

T

A. Oflero, A.J. Garcfa-Cerezo

148

i

\

,,

Volt "f

"~. "..

1.7

•

\".. ",,

#

"~:>. ',,,

,.

,,-.'~.. ....

,.-

,/,

r.

:

Target

:

Response for Te-16oC Auto tuning response { Te:16oC |

........................

response.

!

/ 0

I

0

I

2S

50

~

!

I

75

100

~

sec.

Fig. 19. Results of implementation.

3.3. On line improving of control systems designs Let a control ,Jystem evolution to the steady state be characterized by a set of typical response parameters P - ( P l , P 2 , . . . ,Pro) such as overshoot, settling time, rise time, steady state errors etc. Moreover, let h - (h~, h 2 , . . . , hn) be a vector of constant controller parameters. These parameters must be chosen according with control theory, simulations, and both designers and plant engineers experience. AssL~me that, in current operation with h = k °, the resulting response parameters are p _pO. Obviously, if the process model is a perfect one, the results pO would be the same than obtained in simulation daring the system design phase. However, this rarely occurs in practice. Indeed, inaccuracies in the model formulation and the effect of variables without consideration in the model, normally produce a different value for p. The implementation of an industrial control system involves several experirnental operations in which the engineers and process operators acquire knowledge for appropriate tuning of controller parameters. Furthermore, after the implementation, changes in working conditions from one operation to another may produce changes in next operation response parameters with the same controller parameters. Thus~ an experienced process operator must compensate an eventual response de~;~adation by modifying h appropriately. Although control theory provides some precise relations between h and p, the actual relation for a particular application is rarely explicit. However, plant engineers and process operations normally may provide, from their experience, a protocol in qualitative linguistic terms. This protocol can be used to improve automatically the next operation parameters p from previous operations.

Fuzzy logic control

P 2error PB

NB

P2

NB

Z

PB

PM

PM

PM

Z

Z

NB

NB

Z

NB_

149

, HB

Z

......PI3

PB

NB

NM

PS

Z

NM

Z

PM

NB

NS

PM

PB

ch I

Pl

error

PB POSITIVE-BIG PM POSITIVE - MEDIUM PS POSITIVE- SMALL

Z

ZERO

error

ch z

Pl • rror

NS NEGATIVE-SMALL NM NEGATIVE- MEDIUM NB NEGATIVE-BIG

FiB. 20. ch~ and ch2 protocols.

For example, consider the position control system and two response parameters: p~ for the overshoot and P2 for the settling time. It is intended to obtain Pl = 30%,

P2 = 16" T

(16)

where T is the sampled time. The control parameters h~ and h2 of the state feedback control system are updated for the next operation according to (14)o In Figure 20 the protocols to infer ch~ and ch2 from p~ and P2 are shown in tabular form. The obtained inference matrices are illustrated in Figure 21.

I

e rror P t e rror

error P1 error

Fig. 21. ch~ and ¢h2 inference matrix.

A. Ollero, A.J. Garda-Cerezo

150

Or, 03 ~ 012

0,1

4

-

-

0,0

......

'

18

r,

,

,

20

22

. . . . 24

26

26

30

Sett, time "1"

Fig. 22. Overshoot and settling time evolution.

Figure 22 presents the overshoot and settling time obtained in several transitions to the steady state. Notice that these operations evolve from (P~, Pc) = (9%, 29. T) to (Pt, P2) = (28%, 18. T) which is a better compromise in accordance with the specifications (16). Notice that many control systems, as for example position control systems, must operate many times in an industrial process. Thus, automatic improvements in successive operations are greatly valuable. To conclude we notice that the above mentioned functions can be included on a supervisory level with other typical functions such as operating modes transitions (automatic-manual, regulation-tracking, etc.), initialization, monitoring and information display. 4. Conclusions Fuzzy logic controllers can be considered as non-linear controllers in which the control signals are generated by means of a m~l~idimensional table or inference ~atrix. We present an algorithm to obtain this ma~,,x from a linguistic protocol and definitions of fuzzy subsets describing linguistic terms. Moreover we present two algorithms to implement the fuzzy controller. The first one uses a rectangular discretization of proces~ signals by round-off to the nearest discretization points of the inference matrix. In the second one a trapezoidal discretization rule is used. This implementation increases the execution time but decreases the effe~ of discretization improving the response, p,trticularly when considering the steady state errors and limit cycle oscillations produced in the classical rectangular implementation. Moreover, the trapezoidal implementation can be used to minimize storage requirements by decreasing the number of quantization levels. Moreover, applications of fuzzy logic contrgllers fo~~ ~.ato-tuning and on-line improvements of conventional linear regulators are proposed. It is intended to substitute the task of a process operator when supervising regulators. Thus,

15~

Fuzzy logic control

taking into account experience of plant engineers and process operators M a very simple protocol, it is possible to automate tuning of controller parameters, depending on the actual values of pro~ess variables and the influence of external variables. Furthermore, the parameters of a controller can be updated from the values obtained in a set of response parameters such as overshoot, settling time, steady state errors, etc. to improve next operation response. The proposed methods can be included on supervision level with other logical functions. The presented methods have been illustrated by means of practical temperature control experiences with a heated air-stream prototype, and simulations of a position control system. To conclude we notice that the presented algorithms and procedures have been Frogramed in Pascal and implemented, under both CPM and DOS operating systems, on commercial microcomputers.

AppendS: Ma|rix |nference na|ys|s Consider a two-input (xt, x2) one-output (u) fuzzy logic controller, and let VXt, VX2, VU be the set of linguistic terms used in the protocol: card(VXt) = tl,

card(VX2) = t2, card(VU) = t,.

(17)

Now let X~ and X2 be respectively the universe of discourse of xt and x2. Then, it is possible to obtain a partition of Xt" X2 into ns regions corresponding to the ns statements of the protocol. This partition can be performed by associating the region Ai to the statement r~ with a greater influence on A. That is, the statemem in which the condition

l~,k(xt, x2)>l~,i(xt,x2),

i=l,...,n~,

i~k,

(18)

is satisfied for every (xt, x2) included in Ai. In practice, only discretized values (it, i2) within the limits of the region, are considered. Figure 9 showr the partition in the heated air-stream example. Now consider the elements of VU ordered according to its reduction to scalar values: VU = { r u t , . . . ,

vu,},

(19) reduct[vu~]-< reduct[vul],

i, j = 1 , . . . , t,,

j < i.

Let t(it, i2) be the ordindl 1 <-t(it, i2)<-t, corresponding to the element of VU in the statement associated with the region in which the point (/t,/2) lies. Thus, the inference nx, "nx2 matrix ~ is said to be consistent with the protocol if the following condition is satisfied: ~>0

(20)

for it--1,...,nxt, i2--1,...,nx2, kl--1,...,nxt, k2--l,...,nx~ This condition assures that a qualitatiw ~. change of the action described in the

A. Ollero, AJ. Garcfa-Cerezo

152

protocol (e.g. when passing from a statement with vosrnw-sMA~ to other with posmw.mG) produces a change in the same sense on the control signal generated with the inference matrix ~(.). The consistency can also be verified in only one direction of the state space x t - x2 by means of the condition t(il, i2) - t ( k l , i2)

~>0, il = 1 , . . . , n~; kl = il + 1.

(21)

~ ( i l , i2) "- ~ ( k l , i2)

Notice that points in the same region give t(i~, i2)=t(k~, k2). To verify the consistency when moving into a region it is necessary to use the ordinals corresponding to linguistic terms of adjacent regions in the direction of the movement. In practice, we can observe that inconsistencies appear in the proximities of the protocol partition. The above described results have also been used to design a self-organizing fuzzy logic controller [5].

Aeknow|edlgement The authors thank the anonymous referees for their careful reading of the manuscript, and their useful suggestions. This research was performed under Comisi6n Asesora de lnvestigaci6n Cientifica y T~cnica, Project CAICYT..II02(3).

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