Automated rule-based model generation for uncertain complex dynamic systems

EngngApplic.Artif. Intell.Vol.4, No. 5, pp. 359-366,1991 Printed in Great Britain. All rightsreserved

0952-1976/91$3.00+0.00 Copyright© 1991PergamonPresspie

Contributed Paper

Automated Rule-based Model Generation for Uncertain Complex Dynamic Systems CELAL BATUR University of Akron ARVIND SRINIVASAN University of Akron CHIEN-CHUNG CHAN University of Akron This paper is an attempt to model a dynamic system by a set of production rules. These rules are automatically induced from a set of training data by the ID3 algorithm of Quinlan. The effects of quantization and model order on the model performance are discussed. The algorithm has been applied to a simulated linear system and a real gas furnace data.

Keywords: System modeling, ID3, production rules, decision trees. INTRODUCTION

generation of a rule-based model of these complex systems. 2. The final product of the linguistic model is in terms of production rules. This facilitates easy interpretation and maintenance by the plant personnel. This is in contrast with the three other approaches shown in Fig. 1. 3. Creation of the linguistic model proposed in this paper does not involve interviewing a human operator to build up the knowledge base. The knowledge acquisition is an automatic process, acquired by observing the input and output behavior of the system. Therefore, classical problems associated with interviewing plant operators with regard to their expertise can be greatly alleviated.

Modeling of non-deterministic dynamic systems is generally performed under three methodologies. Depending on the nature of assumptions involving uncertainties in the systems, these methodologies can be classified as: (a) parametric model building, (b) fuzzy model building, and (c) neural-network-based model building. Figure 1 shows the main building blocks of each class with the resulting transfer function. The major contribution of this paper is to introduce a linguistic model-building approach for uncertain dynamic systems. The final product of this technique is a set of production rules obtained by an automated knowledge-acquisition methodology. Figure 1 also shows the main building blocks of this approach. The motivation behind the linguistic model building process is threefold.

The paper is organized as follows. Previous pertinent works related to modeling uncertain dynamic systems are summarized in the following section. The problem statement is given, and automated rule-based modelgeneration techniques, and some anticipated problems, are introduced and discussed. Two examples, one involving a simulated system, the other a real life gas furnace, are presented. Finally, the overall conclusion is outlined.

1. Complex dynamic systems, such as human operators acting as process controllers, do not lend themselves to being expressed in terms of exact mathematical relations. Examples of process controllers involving experienced human operators can be found in several petro-chemical and kiln industries. The major benefit of the proposed technique will be the

PREVIOUS WORK

Correspondence should be sent to: C. Batur, Department of Mechanical Engineering, University of Akron, Akron, OH 44325-4002, U.S.A.

Parametric experimental model building for linear dynamic systems has been extensively applied within

359

CELAL BATUR et al.: AUTOMATED RULE-BASED MODEL GENERATION

360

Experimental modeling of uncertain dynamic systems

I Ooto acquisitioo I I model building

properties of

uncertainties

I

model building

I

parametric modelstructure

I

Assumed example structure for training instances

fuzzy model structure

Transfer J .~ function model for dynamics

Linguistic model building

distribution of uncertainties

I _~

1

Neural network based model building

1

Assumed networkstructure

Assumed example [ structure for training instances

[

1

_~ Neural network transfer function model

Fuzzytransfer function

DecisiOnortree

I

production rules

Fig. 1. Modeling dynamic systems.

the last two decades) Self-tuning and adaptive control strategies are mostly based on parametric dynamic models. 2 Fuzzy models exploit the vagueness associated with linguistic qualifiers such as small, large, etc. Several fuzzy controllers are reported. 3 There are however a few fuzzy modeling applications:' 5 Neural-network-based dynamic model building is a non-linear least squares fit to system dynamics. Mostly inverse process dynamics is approximated by neural networks as process controllers. 6 More recently Levin e t al. 7 employed neural networks for process models. Similarities between parametric and neural network modeling are shown in Refs 8 and 9. Inductive production rules learning from examples has been applied to the synthesis of a clinker kiln control algorithm in a cement plant. 1° Some results of Table 1. A training set consists of 12 examples

xl x2 x3 x4 x5 x6 x7 xs x9 Xl0 Xn x12

y(t- 1)

y(t- 2)

u(t- 1)

u(t- 2)

1 0 0 0 0 1 1 0 0 0 0 2

0 3 0 1 0 0 0 0 2 0 0 2

0 0 0 0 2 0 0 1 0 1 1 0

0 0 0 0 0 0 0 1 2 2 2 2

Fig. 2. Decision tree induced from Table 1 by ID3.

combining decision tree and reinforcement learning to learn to control dynamic systems were reported in Ref. 11.

y(t) 0 1 1

0

-

=

1 1

0 0 0 0 0

Fig. 3. Partial decision tree after u(t-2) is used to split the training set.

CELAL B A T U R et al.: A U T O M A T E D RULE-BASED MODEL GENERATION

System input/°utput~ Continuous M Quantized ~ training set training set Structural Ouontizotion parameters level (mtn, k)

361

Decision ~_~ Production tree rules

Fig. 4. Requirements and steps for model generation.

PROBLEM STATEMENT

A model for a given dynamic system is assumed to be in the following form

y(t) =f(Yt-1, Ut-1, O) + n(O where the vectors Yt-1 and Ut-~ indicate the past values of outputs y(t) and inputs u(t) as Y,-1 = ( y ( t - 1), y ( t - 2) . . . . . y ( t - n)) r

Ut-, = ( u ( t - k - 1), u ( t - k - 2 ) , . . . , u ( t - k - m)) The process order is effectively defined by the (m, n) pair and k indicates the dead time. The term n(t) is the process noise and 0 is a set of parameters describing the dynamics through function f(.). Under this configuration the identification problem can be stated as follows. Given a set of observation as @(t) = {Yt, Ut; t = 1, 2 . . . . , N}, determine (m, n, k) and a set of production rules that would describe the dynamics of the system. To compare this new methodology with the existing modeling techniques, a performance measure will be defined as N

V= E (Y(J)_ j=l

p(j))2

where )~(j) is the prediction of the process output obtained by sending O ( j - 1) to the derived production

rules. Notice that the performance index can also be interpreted as the variance of one-step-ahead prediction. AUTOMATED MODEL GENERATION In the proposed linguistic model building approach, a model is represented by a set of production rules induced from data. Production rules are the rules of the form if condition then decision, which means that if the condition is true then the decision is taken. The data used to generate rules are called training examples or a training set; they may be provided by an expert or taken from databases. Training examples are characterized by a finite collection of attributes with numeric or symbolic values. Among the attributes, two kinds of attributes can be distinguished, called condition and decision attributes. For instance, in a dynamic system, past inputs and outputs are described by condition attributes and the current output by a decision attribute. More specifically, given a set of N observations of a system of the form described above, the training set is of size N and each training example is described by m + n condition attributes y ( t - 1 ) , y(t-2) .... ,y(t-n), u(t-k-1),..., u(t-k-m) and one decision attribute y(t). Many machine learning programs have been designed to learn production rules from examples. 12-15 In this paper, one of the well-known programs called ID316.17 is used for model generation. The basic idea of ID3 is introduced below, followed by identification and discussion of some problems of applying ID3 for model generation. ID3 program

Median

o E

/I

Yi

',-[

Decisionvalue

Fig. 5. Suggested solution for data inconsistency. EAAI4:5-8

The goal of ID3 is to learn classification rules from a set of examples. The classification knowledge learned by ID3 is represented as a decision tree in which nonleaf nodes are labeled with condition attributes, branches are labeled with values of condition attributes, and leaves are labeled with values of the decision attributes. To illustrate the method of ID3, consider a training set of 12 examples shown in Table 1, where each example xl is described by four condition attributes y ( t - 1), y ( t - 2), u ( t - 1), and u ( t - 2) and one decision attribute y(t). A decision tree induced from the examples by ID3 is shown in Fig. 2, where circles denote non-leaf nodes of the tree, ovals denote leaves, and

362

CELAL BATUR et al.: AUTOMATED RULE-BASED MODEL GENERATION Simulated

system

-0.5 -1.0

2 1 0 -1 -2

I 50

0

I 100

I 150

I 200

250

Time

Fig. 6. The response of the system. branches of the tree are labeled with attribute-value pairs. The underlying algorithm of ID3 is a recursive treegrowing procedure. The procedure works as follows: initially, if all examples in the training set have the same decision value, the program returns an induced decision tree consisting of one node labeled with the decision value and the tree-growing process stops. Otherwise, one of the most informative condition attributes is selected to split the training set. In ID3, an information-theoretic measure called maximum expected information gain is used for attribute selec-

, \

'i 30 25

H°=-

E PY(o=il°g2Py(O=i' i~ ly(01

= - ( 5 / 1 2 log2 5/12 + 7/12 log2 7/12) = 1.34 bits, where ly(t)l denotes the range of y(t) and Py(o=;is the probability of occurrence of objects having the decision value y(t) = i. In the example, the value of Pro)=1 is 5/12 and Pr(0=0 is 7/12. If attribute y ( t - 1) is selected to partition the training set, the resulting entropy is given by:

Condition attributes: U (t-I),y (t-i) Decision attribute: y (t) 35

tion. An attribute is said to provide maximum expected information gain, if by testing that attribute, the entropy of the training set can be reduced most. For example, the initial entropy of the training set is given by:

------: ::',.2

XXx\~

20 no t5

n(y(t)ly(t-

1)) = -

E

PiHi

jEly(t-1)l

= -[8/12(2/8 log2 2/8 + 6/8 log2 6/8)

Ol 2

I 4

\

\

6 8 10 12 Number of quantization levels

I 14

I

16

Fig. 7. Performance of the linguistic model for the simulated system. y (t-I) i L(t_~ u(t-!) I utt-2~l -I

] yqtt-t) ~[ [ y,(t-2) ~l Quontizer

Decision tree l uq(t-1) ~] or I uQtt-2) _-7 production rules I -[

Fig. 8. Generation of model output.

it)

+ 3/12.0 + 1/12-0] = 0.54 bits, where Pj is the probability of occurrence of objects having the condition attribute value y ( t - 1) = j and Hj is the entropy of the jth subset. The expected information gain by testing attribute y ( t - 1) is defined as

H o- H(y(t) iY(t- 1)) = 1.34 - 0.54 = 0.8 bits.

CELAL B A T U R et al.: A U T O M A T E D RULE-BASED MODEL GENERATION

363

Fig. 9. The decision tree for the simulated system. The condition attributes are u ( t - 1) and y ( t - 1). Quantization level is lq = 3.

4 2 O -2 -4

65

60 55 0

50 45

0

,

,

,

20

40

60

o

80

°

.

100

120

140

160

180

Time

Fig. 10. Gas furnace data.

900 80O ~"

~

Condition ottributes: u [ t - i ) , y ( t - i ) Decision ottribute : y (t)

~,~

7O0

5O0 40O a. 3(30 20O 100

.

--

~

..°°°

"'°°°

---2.2."- . . . . . . . . . . . . . . .

0 2

I

I

4

6

I

I

I

8 I0 12 Number of quontizotion levels

Fig. 11. Performance index for the gas furnace data.

I

14

t 16

200

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CELAL BATUR et al.: AUTOMATED RULE-BASED MODEL GENERATION

(y(t-l) = 3) & (u(t-2) = 3) & (u(t-l) = 3) -4 (y(t) = 3) (.y(t-1) = 3) & (u(t-2) = 3) & (u(t-l) = 2) -4 (y(t) = 3) (y(t-l) = 3) & (u(t-2) = 3) & (u(t.l) = 4) & (y(t.2) = 2) --* (y(t) = 3) (y(t-1) = 3) & (u(t-2) = 3) & (u(t-l) = 4) & (y(t-2) = 3) .--* (y(t) = 4) (2,,(t-l) ffi 3) & (u(t-2) = 2) & (u(t-1) = 3) -4 (y(t) = 4)

(y(t-l) = 3) & (u(t-2) = 2) & (u(t4) = 2) -4 (y(t) = 3) (y(t-Â) = 3) &.(u(t-2) = 2) & (u(t-l) = 1) -4 (y(t) = 4) (y(t-l) ffi 3) & (u(t-2) = 5) --* (y(t) = 3) (.y(t-1)= 3) & (u(t-2) = I) -4 (y(t) = 4) (y(t-l) = 3) & (u(t-2) = 4) -4 (y(t) = 3) (y(t-Â) = 3) & (u(t.2) = 6) & (y(t-2) = 3) -4 (y(t) = 2) (.V(t-D = 3) R (u(t-2) = 6) & (y(t-2) =,4) --*-(.v(0= 3) (y(t-l) ffi4) & (u(t-2) = 4) & (u(t-l) = 3) -4 (y(t) = 3)

(y(t-1) = 4) & (u(t.2) = 4) & (u(t-1) = 2) -4 (y(t-1) = 4) & (u(t-2) = 4) & (u(td) = 5) -4 (y(t-l) = 4) & (u(t-2) = 4) & (u(t-1) = 6) -4 (,v(td) = 4) a, (u(t-2) = 4) & (u(t-D = 4) ~ (y(t-D = 4) & (u(t.2) = 3) -4 (y(t) = 4) (.v(t-l)= 4) & (u(t-2) = 2) & (u0-1) = 3) -4 (y(t-D = 4) & (u(t.2) = 2) & (u(t-1) = 2) -4

(y(t) = 4) (y(t) = 4) (y(t) = 3)

(y(t) = 4) (.v(t)= 5) (y(t) =

4)

fy(t-l) = 4) & (u(t-2) = 2) & (u(t-1) = 1)--¢ (y(t)= 4) (y(t-D = 4) & (u(t.2) = 5) -4 (y(t) = 3)

this p o i n t is a t r e e s h o w n in Fig. 3, w h e r e e x a m p l e s in t h e t r a i n i n g set a r e d e n o t e d b y i n t e g e r s f r o m 1 to 12, a n d t h e r o o t n o d e is l a b e l e d with a t t r i b u t e u ( t - 2 ) . I n Fig. 3, t h e t r a i n i n g set is p a r t i o n e d i n t o t h r e e subsets b y v a l u e s o f t h e a t t r i b u t e u ( t - 2). E x a m p l e s in each subset have the same value of attribute u ( t - 2 ) , but their decision values may not be the same. A t t h e n e x t level, the s a m e p r o c e d u r e is a p p l i e d r e c u r s i v e l y to split all t h e s u b s e t s c o n t a i n i n g e x a m p l e s with d i f f e r e n t d e c i s i o n values. I n Fig. 3, o n l y t h e l e f t m o s t n o d e o f t h e r o o t n o d e n e e d s to b e split a g a i n , b e c a u s e the d e c i s i o n v a l u e o f e x a m p l e s 1, 3, 4, 6 a n d 7 is 1 a n d t h e d e c i s i o n v a l u e o f e x a m p l e s 2 a n d 5 is 0. T h e t r e e - g r o w i n g p r o c e s s s t o p s w h e n all t h e s u b s e t s c o n t a i n e x a m p l e s with t h e s a m e d e c i s i o n v a l u e . A f t e r using I D 3 to g e n e r a t e a d e c i s i o n t r e e , a set o f p r o d u c t i o n rules can b e o b t a i n e d b y c o n v e r t i n g e a c h p a t h , s t a r t i n g at t h e r o o t n o d e a n d e n d i n g at a l e a f n o d e , in t h e t r e e into o n e rule. T h e c o r r e s p o n d i n g p r o d u c t i o n rules o f t h e d e c i s i o n t r e e in Fig. 2 is given in t h e following: if u ( t - 2) = 1 t h e n y(t) = O,

(y(td) = 4) & (uCt-2)= o) •-* (.v(t) = 5) (y(t-1) =4) & (u(t-2) = 1) & CuCtd) = o) -4 (,v(t)= 4) (y(td) = 4) & (uCt.2) = I) & (u(t-D = 1) ~ (y(t) =4) ~ t d ) = 4) R (u(t.2) = 1) & (u(t-l) = 2) -4 (y(t) = 5) (.v(t-1) = 5) & (u(t-2) = 3) & (u(t-1) = 3) -4 (y(0 = 5) (y(t-1) = 5) & (u(t-2) = 3) & (u(t.1) = 4) & (y(t-2) = 4) --4 (y(t) = 5) (.y(t-1) = 5) & (u(t-2) = 3) & (u(t-1) = 4) & (y(t.2) = 5) --¢ (y(t) ffi4) (y(t-1) = 5) & (u(t-2) = 3) & (u(t-l) = 2) -4 (y(t) = 5) (y(t-l) = 5) & (u(t-2) = 4) & (u(t-1) = 3) -4 (y(t) =

if u ( t - 2) = 2 t h e n y(t) = O, if u ( t - 2) = 0 a n d y ( t - 1) = 1 t h e n y(t) = 1, if u ( t - 2) = 0 a n d y ( t - 1) = 0 a n d y ( t - 2) = 3 t h e n y(t) = O, if u ( t - 2) = 0 a n d y ( t - 1) = 0 a n d y ( t - 2) = 1

4)

(y(t-1) = 5) & (u(t-2) = 4) & (u(t-D = 4) -4 (.v(t)= 5) (y(t-l) = 5) & (uCt.2) = 2) -4 (y(t) = 5) (y(t-l) = 5) & (u(t-2) = 0) & (yCt-2)= 5) -4 (y(t) = 6) (y(t-D = 5) & (u(t-2) = 0) & (y(t-2) = 4) -4 0,(t) = 5) (y(l-l) = 5) & (u(t-2) = 1) -4 (y(t) = 5) (y(t-D = 6) --* (Y(0 = 6) (y(td) = 2) & (u(t-2) = 4) -4 (y(t) = 2) (y(t.l) = 2) & (u(t-2) = 5) -4 (y(t) = 2) (y(t-l) = 2) & (u(t-2) = 6) -4 (y(t) = 2) (y(t-D = 2) & (u(t-2) = 3) -4 (y(t) = 3) Fig. 12. Production rules for the gas furnace data. The condition

attributes are y(t-1), y ( t - 2 ) , u(t-1), and u(/-2). Ouantization level is lq=6.

t h e n y(t) = 1, if u ( t - 2) = 0 a n d y ( t - 1) = 0 a n d y ( t - 2) = 0 a n d u ( t - 1) = 0 t h e n y(t) = 1, if u ( t - 2) = 0 a n d y ( t - 1) = 0 a n d y ( t - 2) = 0 a n d u ( t - 1) = 2 t h e n y(t) = O. P r o d u c t i o n rules o b t a i n e d b y this s t r a i g h t f o r w a r d conversion may contain redundant attribute-value pairs. S o m e t e c h n i q u e s for solving this p r o b l e m w e r e i n t r o d u c e d in R e f s 13 a n d 15. Some

S i m i l a r l y , t h e e x p e c t e d i n f o r m a t i o n gain b y testing e a c h o f t h e r e m a i n i n g a t t r i b u t e s a r e c o m p u t e d as

Ho - H(y(t)ly(t- 2)) = 1.34 - 0.75 = 0.59 bits, Ho - H(y(Olu(t- 1)) = 1.34 - 0.64 = 0.7 bits, Ho - H(y(t) lu(t- 2)) = 1.34 - 0.50 = 0.84 bits. Since [H0 - H(y(t) lu(t- 2))] is m a x i m u m , t h e m a x i m u m e x p e c t e d i n f o r m a t i o n gain c r i t e r i o n suggests t h a t a t t r i b u t e u ( t - 2) b e s e l e c t e d as t h e first a t t r i b u t e to split t h e t r a i n i n g set. T h e p a r t i a l d e c i s i o n t r e e o b t a i n e d at

anticipated

problems

T h e r e q u i r e m e n t s a n d s t e p s f o r g e n e r a t i n g a ruleb a s e d m o d e l is s u m m a r i z e d in Fig. 4. T h i s s h o w s t h a t t h e u s e r m u s t p r o v i d e t h e (m, n, k) t r i p l e a n d t h e q u a n t i z a t i o n level to b e u s e d for g e n e r a t i n g a q u a n t i z e d t r a i n i n g set. T h e set is u s e d b y I D 3 to g e n e r a t e a d e c i s i o n t r e e , which is t h e n c o n v e r t e d i n t o a set o f p r o d u c t i o n rules. F r o m Fig. 4, s o m e p r o b l e m s o f a p p l y ing ID3-1ike p r o g r a m s for g e n e r a t i n g r u l e - b a s e d m o d e l s can also b e identified. A t t r i b u t e s with c o n t i n u o u s v a l u e s n e e d to b e q u a n tized. This will limit t h e size o f t h e d e c i s i o n t r e e b u t it m a y also i n t r o d u c e i n c o n s i s t e n c y into t h e q u a n t i z e d

CELAL

BATUR

et al.: A U T O M A T E D

training set. For example, identical sets of condition attribute values may lead to different decision values. To further illustrate this case, consider a situation where identical sets of condition attribute values yield Yi as a decision value 7y, times where i = 1, 2 , . . . , ore. Under this condition two different solutions can be conjectured:

RULE-BASED

MODEL

GENERATION

365

where (n, m) define the effective model order and (k) is the process dead time. The continuous input/output data should further be quantized into a finite number of classes. This limits the size of the decision tree and correspondingly the number of production rules representing the dynamics. The quantization process is described by:

A1. The maximum number of occurrences in yi can be considered as the decision value. A2. The mean or median of the histogram can be used as the decision value.

uq(0=L(u(0) yq(t) =fq(y(t))

These solutions are illustrated in Fig. 5. In practice, the training set may not completely characterize the system dynamics. This happens, for example, if the operator input does not persistently excite all the modes of the system. This in turn means that the resulting set of production rules may not completely describe the behavior of the system. This is a common problem shared by other modeling methodologies and it can be avoided if the identification experiment can be planned. Finally, the training set may be noisy. Some statistical-based methods for pruning decision trees were introduced to deal with noisy training data. 17-21 It has been reported that pruning of decision trees is more effective on independent attributes. 22 However, in dynamic systems, input/output attributes of a system are usually interdependent.

where ttq(t) and yq(t) a r e the quantized input and output respectively. If the number of quantization levels is lq then the quantization function fq(.) can be described as:

BENCHMARKING EXAMPLES

Under this quantized operation, the training instance set takes the following modified form

Two systems are considered to test the proposed technique.

if ui <~u(t) < Ui+l then u(t) = ui

i=1,2,...

if yi <-Y(t)
i = 1 , 2 . . . . . lq

where u i and Yi indicate the magnitudes of input and output signal at quantized levels respectively, i.e. Ureax - - Umin

Ui+ 1 = Ui -~

y(t) = 0.5 y ( t - 1) + u ( t - 1).

1 - e -"

2

s

1.433 s + 1"

U(s) -

Figure 6 shows the response of this system to a pseudo random binary input. As in the other modeling techniques, one has to decide on the number of past inputs, past outputs and the process delay time which effectively determine the model output. These unknowns can be conveniently expressed in a training instance set as:

• e(t) = ( y ( t - 1 ) , . . . , y ( t - n), u ( t - k - 1) . . . . , u ( t - k - m)),

lq

dpqe(t) = ( y q ( t - 1 ) , . . . , y q ( t - n), Uq(t-k- 1),..., Uq(t-k-m)). Figure 7 shows the performance of the linguistic model measured by the index 1

This corresponds to a continuous first-order system proceeding a zero order hold and the input/output data are sampled with 1 s time interval, i.e., Y(s)

tq Ymax - - Ymin

Yi+ 1 = Yi -1

Simulated system

The system 1 is a linear process presented in its discrete form as

,lq

N

V = 3 1 Z (y(t) - y~(t))z t=l

where )~(t) is the response of the linguistic model receiving the same process input. This index can be interpreted as the variance of one-step-ahead prediction error. Figure 8 shows explicitly how the model output p(t) is generated. It is obvious that for lq i> 9, the change in the performance index is not significant. The decision tree for the simulated system is shown in Fig. 9. Gas furnace

The system 2 of our second example involves real life gas furnace data reported in Ref. 23. The plot of the system response is shown in Fig. 10. The values of the performance index, based on five

366

CELAL BATUR et al.: AUTOMATED RULE-BASED MODEL GENERATION

d i f f e r e n t q u a n t i z a t i o n levels, a r e s h o w n in Fig. 11. T h e t h e o r e t i c a l m i n i m u m v a l u e o f t h e p e r f o r m a n c e i n d e x is n o t k n o w n a p r i o r i in this e x a m p l e . H o w e v e r f o l l o w i n g l q ~ 6 t h e d e c r e a s e in t h e p e r f o r m a n c e i n d e x is n o t significant. T h e p r o d u c t i o n rules for t h e gas f u r n a c e d a t a a r e s h o w n in Fig. 12.

C O N C L U D I N G REMARKS This p a p e r a t t e m p t s to m o d e l a d y n a m i c s y s t e m b y a set o f p r o d u c t i o n rules. In c o n t r a s t to o t h e r m o d e l i n g t e c h n i q u e s , n o fixed s t r u c t u r e h a s to b e a s s u m e d for t h e m o d e l . T h e p r o d u c t i o n rules a r e i n d u c e d f r o m a set o f training data by the ID3 algorithm of Quinlan. The p e r f o r m a n c e o f t h e d y n a m i c m o d e l s is j u d g e d b y t h e error variance between the process and the model o u t p u t . T h e effect o f q u a n t i z a t i o n levels o n t h e p e r f o r m a n c e is d i s c u s s e d a n d d e m o n s t r a t e d . A s is e x p e c t e d , i n c r e a s i n g q u a n t i z a t i o n level r e d u c e s p r o c e s s - m o d e l error measured by the error variance. T h e t e c h n i q u e d o e s n o t r e q u i r e t h e specification o f an explicit m o d e l s t r u c t u r e , b u t a n u m b e r o f p a s t p r o c e s s i n p u t s a n d o u t p u t s has to b e specified as a set o f c o n d i t i o n a t t r i b u t e s . This c h o i c e c o r r e s p o n d s to t h e m o d e l o r d e r d e t e r m i n a t i o n in o t h e r p r o c e s s m o d e l i n g t e c h n i q u e s . T h e effect o f t h e n u m b e r o f c o n d i t i o n a t t r i b u t e s o n t h e e r r o r v a r i a n c e is s h o w n . It is d e m o n strated by examples that by increasing the number of c o n d i t i o n a t t r i b u t e s o n e can r e d u c e t h e e r r o r v a r i a n c e . T h e a d v a n t a g e o f this t e c h n i q u e is t h a t it is c o m p l e t e l y d a t a d r i v e n a n d r e q u i r e s a m i n i m u m n u m b e r o f assumptions for the model structure.

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