[lecture notes in control and information sciences] uncertain logics, variables and systems volume...

Lecture Notesin Control and Information Sciences 276

Editors: M. Thoma · M. Morari

SpringerBerlinHeidelbergNewYorkBarcelonaHong KongLondonMilanParisTokyo

Z. Bubnicki

Uncertain Logics,Variables and SystemsWith 36 Figures

1 3

Series Advisory BoardA. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic ·A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

AuthorProf. Zdzislaw BubnickiWroclaw University of TechnologyInstitute of Control and Systems EngineeringWyb. Wyspianskiego 2750-370 Wroclaw, Poland

Cataloging-in-Publication Data applied forDie Deutsche Bibliothek – CIP-EinheitsaufnahmeBubnicki, Zdzislaw:Uncertain logics, variables and systems / Z. Bubnicki. - Berlin ; Heidelberg ;NewYork ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo :Springer, 2002

(Lecture notes in control and information sciences ; 276)(Engineering online library)ISBN 3-540-43235-3

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Foreword

The ideas of uncertain variables based on uncertain logics have been introducedand developed for a wide class of uncertain systems. The purpose of this mono-graph is to present basic concepts, definitions and results concerning the uncertainvariables and their applications to analysis and decision problems in uncertainsystems described by traditional mathematical models and by knowledge repre-sentations.

I hope that the book can be useful for graduate students, researchers and allreaders working in the field of control and information science. Especially forthose interested in the problems of uncertain decision support systems and uncer-tain control systems.

I wish to express my gratitude to my co-workers from the Institute of Controland Systems Engineering of Wroclaw University of Technology, who assisted inthe preparation of the manuscript. My special thanks go to Dr L. Siwek for thevaluable remarks and for his work concerning the formatting of the text.

This work was supported in part by the Polish Committee for Scientific Re-search under the grants no. 8 T11A 022 14 and 8 T11C 012 16.

Preface

Uncertainty is one of the main features of complex and intelligent decision makingsystems. There exists a great variety of definitions and descriptions of uncertain-ties and uncertain systems. The most popular non-probabilistic approaches arebased on fuzzy sets theory and related formalisms such as evidence and possibilitytheory (e.g. [1, 2, 37-42, 51-55]). The different formulations of decision makingproblems and various proposals for reasoning under uncertainty are adequate tothe different formal models of uncertainty. Special approaches have been pre-sented for uncertainty in expert systems [47] and for uncertain control systems(e.g. [21, 44, 50]). This work concerns a class of uncertain systems containing un-known parameters in their mathematical descriptions: in traditional mathematicalmodels or in knowledge representations in the knowledge-based systems. For suchsystems a concept of so called uncertain variables and its application to the analy-sis and decision making problems have been developed [15, 16, 17, 20, 26, 28-36]. The purpose of this work is to present a basic theory of the uncertain variablesand a unified description of their applications in the different cases of the uncer-tain systems.

In the traditional case, for a static system described by a function ),( xuy Φ=where xyu ,, are input, output and parameter vectors, respectively, the decision

problem may be formulated as follows: to find the decision *u such that *yy =

(the desirable output value). The decision *u may be obtained for the known Φand x . Let us now assume that x is unknown. In the probabilistic approach xis assumed to be a value of a random variable x~ described by the probabilitydistribution. In our approach the unknown parameter x is a value of an uncertainvariable x for which an expert gives the certainty distribution )~()( xxvxh ==where v denotes a certainty index of the soft property: " x is approximatelyequal to x " or " x is the approximate value of x ". The uncertain variables, re-lated to random variables and fuzzy numbers, are described by the set of valuesX and their certainty distributions which correspond to probability distributions

for the random variables and to membership functions for the fuzzy numbers. Todefine the uncertain variable, it is necessary to give )(xh and to determine the

certainty indexes of the following soft properties: " xDx ∈~ " for XDx ⊂ which

means: "the approximate value of x belongs to xD " or " x belongs approxi-

mately to xD ", and " xDx ∉~ " = " )~( xDx ∈¬ " which means " x does not belong

approximately to xD ". To determine the certainty indexes for the properties:

VIII Preface

)~( xDx ∈¬ , )~()~( 21 DxDx ∈∨∈ and )~()~( 21 DxDx ∈∧∈ where XDD ⊆21, ,

it is necessary to introduce an uncertain logic which deals with the soft predicatesof the type " xDx ∈~ ". Four versions of the uncertain logic have been introduced

(Sects. 2.1 and 2.2) and then two of them have been used for the formulation oftwo versions of the uncertain variable (Sects. 2.3 and 2.4).

For the proper interpretation (semantics) of these formalisms it is convenient toconsider )(ωgx = as a value assigned to an element Ωω ∈ (a universal set). For

fixed ω its value x is determined and xDx ∈ is a crisp property. The property

xx DxDx ∈=∈~ = "the approximate value of x belongs to xD " is a soft prop-

erty because x is unknown and the evaluation of " xDx ∈~ " is based on the

evaluation of xx =~ for the different Xx ∈ given by an expert. In the first ver-

sion of the uncertain variable )~()~( xx DxvDxv ∉≠∈ where xx DXD −= is the

complement of xD . In the second version called C-uncertain variable

)~()~( xcxc DxvDxv ∈=∉ where cv is the certainty index in this version:

)]~()~([21

)~( xxxc DxvDxvDxv ∉+∈=∈ . The uncertain variable in the first ver-

sion may be considered as a special case of the possibilistic number with a specificinterpretation of )(xh described above. In our approach we use soft properties of

the type " P is approximately satisfied" where P is a crisp property, in particu-lar "" xDxP ∈= . It allows us to accept the difference between xDx ∈~ and

xDx ∉~ in the first version. More details concerning the relations to random vari-

ables and fuzzy numbers are given in Chap. 6. Now let us pay attention to the fol-lowing aspects which will be more clear after the presentation of the formalismsand semantics in Chap. 1:1. To compare the meanings and practical utilities of different formalisms, it is

necessary to take into account their semantics. It is specially important in ourapproach. The definitions of the uncertain logics and consequently the uncer-tain variables contain not only the formal description but also their interpreta-tion. In particular, the uncertain logic may be considered as special cases ofmulti-valued predicate logic with a specific semantics of the predicates. It isworth noting that from the formal point of view the probabilistic measure is aspecial case of the fuzzy measure and the probability distribution is a specialcase of the membership function in the formal definition of the fuzzy numberwhen the meaning of the membership function is not described.

2. Even if the uncertain variable in the first version may be formally consideredas a very special case of the fuzzy number, for the simplicity and the unifica-tion it is better to introduce it independently (as has been done in the work) andnot as a special case of the much more complicated formalism with differentsemantics and applications.

3. The uncertainty is understood here in a narrow sense of the word and concernsan incomplete or imperfect knowledge of something which is necessary to

Preface IX

solve the problem. In our considerations it is the knowledge on the parametersin the mathematical model of the decision making problem, and is related to afixed expert who gives the description of the uncertainty.

4. In the majority of interpretations the value of the membership function meansa degree of truth of a soft property determining the fuzzy set. In our approach," xDx ∈ " and " xDx ∈ " are crisp properties, the soft property " x

Dx ∈~ " is in-

troduced because the value of x is unknown and )(xh is a degree of cer-

tainty (or )(1 xh− is a degree of uncertainty).

In Chaps. 2–5 the application of the uncertain variables to basic analysis anddecision making problems is presented for the systems with the different forms ofthe mathematical descriptions. Additional considerations concerning special andrelated problems are presented in Chap. 7.

Contents

1 Uncertain Logics and Variables........................................................................ 11.1 Uncertain Logic........................................................................................... 11.2 Other Versions of Uncertain Logic ............................................................. 51.3 Uncertain Variables..................................................................................... 91.4 Additional Description of Uncertain Variables ......................................... 12

2 Analysis and Decision Making for Static Plants............................................ 172.1 Analysis Problem for a Functional Plant ................................................... 172.2 Decision Making Problem for a Functional Plant ..................................... 182.3 External Disturbances................................................................................ 202.4 Nonparametric Uncertainty ....................................................................... 25

3 Relational Systems ........................................................................................... 313.1 Relational Knowledge Representation ...................................................... 313.2 Analysis and Decision Making for Relational Plants ................................ 333.3 Determinization......................................................................................... 393.4 Analysis for Relational Plants with Uncertain Parameters ........................ 413.5 Decision Making for Relational Plants with Uncertain Parameters.......... 463.6 Computational Aspects.............................................................................. 50

4 Systems with Logical Knowledge Representation ......................................... 574.1 Logical Knowledge Representation .......................................................... 574.2 Analysis and Decision Making Problems.................................................. 594.3 Logic-Algebraic Method ........................................................................... 614.4 Analysis and Decision Making for the Plant

with Uncertain Parameters ........................................................................ 644.5 Uncertain Logical Decision Algorithm ..................................................... 66

5 Dynamical Systems .......................................................................................... 695.1 Relational Knowledge Representation ..................................................... 695.2 Analysis and Decision Making for the Dynamical Plants

with Uncertain Parameters ........................................................................ 745.3 Closed-Loop Control System. Uncertain Controller ................................. 805.4 Examples ................................................................................................... 815.5 Stability of Dynamical Systems with Uncertain Parameters ..................... 84

XII Contents

6 Comparison, Analogies and Generalisation .................................................. 916.1 Comparison with Random Variables and Fuzzy Numbers........................ 916.2 Application of Random Variables ............................................................. 956.3 Application of Fuzzy Numbers ................................................................. 966.4 Generalisation. Soft Variables................................................................. 103

7 Special and Related Problems....................................................................... 1097.1 Pattern Recognition ................................................................................. 1097.2 Control of the Complex of Operations .................................................... 1167.3 Descriptive and Prescriptive Approaches................................................ 1197.4 Complex Uncertain System..................................................................... 1217.5 Learning System...................................................................................... 124

Index................................................................................................................... 129

References.......................................................................................................... 131

1 Uncertain Logics and Variables

1.1 Uncertain Logic

Our considerations are based on multi-valued logic. To introduce terminology andnotation employed in our presentation of uncertain logic and uncertain variables,let us remind that multi-valued (exactly speaking – infinite-valued) propositionallogic deals with propositions ...),,( 21 αα whose logic values ]1,0[)( ∈αw and

)(1)( αα ww −=¬ ,

)(,)(max )( 2121 αααα www =∨ , (1.1)

)(,)(min)( 2121 αααα www =∧ .

Multi-valued predicate logic deals with predicates )(xP defined on a set X , i.e.

properties concerning x, which for the fixed value of x form propositions inmulti-valued propositional logic, i.e.

)]([ xPw

= Xxxp ∈∈ each for ]1,0[ )(µ . (1.2)

For the fixed x, )(xpµ denotes degree of truth, i.e. the value )(xpµ shows to

what degree P is satisfied. If for each Xx ∈ 1,0)( ∈xpµ then )(xP will be

called here a crisp or a well-defined property, and )(xP which is not well-

defined will be called a soft property. The crisp property defines a set

1)]([: =∈= xPwXxDx

= )(: xPXx ∈ . (1.3)

Consider now a universal set Ω , Ωω ∈ , a set X which is assumed to be a metric

space, a function Xg →Ω: , and a crisp property )(xP in the set X. The property

P and the function g generate the crisp property ),( PωΨ in Ω : "For the value

)(ωgx =

= )(ωx assigned to ω the property P is satisfied", i.e.

)]([),( ωωΨ xPP = .

1

Z. Bubnicki (Ed.): Uncertain Logics, Variables and Systems, LNCIS 276, pp. 1−16, 2002. Springer-Verlag Berlin Heidelberg 2002

2 1 Uncertain Logics and Variables

Let us introduce now the property "~"),( xxxxG == for Xxx ∈, , which

means: " x is approximately equal to x ". The equivalent formulations are: " x is

the approximate value of x " or " x belongs to a small neighbourhood of x " or

" the value of the metric ),( xxd is small ". Note that ),( xxG is a reflexive, sym-

metric and transitive relation in XX × . For the fixed ω , ]),([ xxG ω

= )(xGω

is a soft property in X. The properties )(xP and )(xGω generate the soft property

),( PωΨ in Ω : " the approximate value of )(ωx satisfies P " or " )(ωx ap-

proximately satisfies P ", i.e.

)(]~)([)()(),( xPxxxPxGP ∧==∧= ωωΨ ω (1.4)

where x is a free variable. The property Ψ may be denoted by

"~)("),( xDxP ∈= ωωΨ (1.5)

where xD is defined by (1.3) and " xDx ∈~ " means: " the approximate value of

x belongs to xD " or " x approximately belongs to xD ". Denote by )(xhω the

logic value of )(xGω :

)]([ xGw ω

= )(xhω , )0)(( ≥∈

xhXx

ω , (1.6)

1)(max =∈

xhXx

ω . (1.7)

Definition 1.1 (uncertain logic): The uncertain logic is defined by a universalset Ω , a metric space X, crisp properties (predicates) )(xP , the properties )(xGω

and the corresponding functions (1.6) for Ωω ∈ . In this logic we consider soft

properties (1.4) generated by P and ωG . The logic value of Ψ is defined in the

following way

)],([ Pw ωΨ

=∅=

∅≠

= ∈x

xDxD

DxhPv x

for

for

0

)(max)],([

ωωΨ (1.8)

and is called a degree of certainty or certainty index. The operations for the cer-tainty indexes are defined as follows:

)],([1)],([ PvPv ωΨωΨ −=¬ , (1.9)

)],([, )],([max)],(),([ 22112211 PvPvPPv ωΨωΨωΨωΨ =∨ , (1.10)

2

1.1 Uncertain Logic 3

=∧

=∧otherwise)],([, )],([min

0)( eachfor if0)],(),([

2211

212211 PvPv

PPwxPPv

ωΨωΨωΨωΨ

(1.11)

where 1Ψ is Ψ or Ψ¬ , and 2Ψ is Ψ or Ψ¬ .

Using the notation (1.5) we have

]~)([1]~)([ xx DxvDxv ∈−=∉ ωω , (1.12)

]~)([,]~)([max]~)(~)([ 2121 DxvDxvDxDxv ∈∈=∈∨∈ ωωωω , (1.13)

]~)([],~)([min]~)(~)([ 2121 DxvDxvDxDxv ∈∈=∈∧∈ ωωωω (1.14)

for ∅≠∩ 21 DD and 0 for ∅=∩ 21 DD – where 1~ D∈ and 2

~ D∈ may be

replaced by 1~ D∉ and 2

~ D∉ , respectively.

From (1.7) and (1.8) 1]~[ =∈ Xxv . Let txD , for Tt ∈ be a family of sets

xD . Then, according to (1.13) and (1.14)

]~)([ max ]~)( [ ,

, tx

Tttx

TtDxvDxv ∈=∈

∈∈ωω , (1.15)

]~)([ min ]~)( [ ,, tx

Tttx

TtDxvDxv ∈=∈

∈∈ωω . (1.16)

One can note that "~)(")( xxxG == ωω is a special case of Ψ for xDx =(a singleton) and

)(]~)([ xhxxv ωω == , )(1])([ xhxxv ωω −=≅/ . (1.17)

According to (1.4), (1.5), (1.17), (1.15), (1.16)

)( max ]~)( []~)([

xhxxvDxvxx

DxDxx ωωω

∈∈===∈

which coincides with (1.8), and

)( max 1)](1[ min ])( []~)([

xhxhxxvDxvxxx

DxDxDxx ωωωω

∈∈∈−=−=≅/=∉

which coincides with (1.8) and (1.12). From (1.8) one can immediately deliverthe following property: if 21 PP → for each x (i.e. 21 DD ⊆ ) then

]~)([]~)([or)],([)],([ 2121 DxvDxvPvPv ∈≤∈≤ ωωωΨωΨ . (1.18)

3


Theorem 1.1:

)],( ),([)],([ 2121 PPvPPv ωΨωΨωΨ ∨=∨ , (1.19)

)],([ , )],([min)],([ 2121 PvPvPPv ωΨωΨωΨ ≤∧ . (1.20)

Proof: From (1.8) and (1.10)

.)],([)(max

)(max,)(maxmax)],(),([

21

21

21

21

PPvxh

xhxhPPv

DDx

DxDx

∨==

=∨

∪∈

∈∈

ωΨ

ωΨωΨ

ω

ωω

Inequality (1.20) follows immediately from 121 DDD ⊆∩ , 221 DDD ⊆∩and (1.18).

Theorem 1.2:)],([)],([ PvPv ωΨωΨ ¬≥¬ . (1.21)

Proof: Let PP =1 and PP ¬=2 in (1.19). Since 1)( =¬∨ PPw for each x

( XDx = in this case),

)],([ , )],([max)],(),([1 PvPvPPv ¬=¬∨= ωΨωΨωΨωΨ

and

)],([)],([1 )],([ PvPvPv ωΨωΨωΨ ¬=−≥¬ .

Inequality (1.21) may be written in the form

]~)([1]~)([ ]~)([ xxx DxvDxvDxv ∈−=∉≥∈ ωωω (1.22)

where xx DXD −= .

As was said in Preface, the definition of uncertain logic should contain two parts:a mathematical model (which is described above) and its interpretation (seman-tics). The semantics is here the following: the uncertain logic operates with crisppredicates )]([ ωxP , but for the given ω it is not possible to state whether )(xP is

true or false because the function )(ωgx = and consequently the value x corre-

sponding to ω is unknown. The exact information, i.e. the knowledge of g is re-

placed by )(xhω which for the given ω characterizes the different possible ap-

proximate values of )(ωx . If we use the terms: knowledge, information, data etc.,

it is necessary to determine the subject (who knows?, who gives the information?).

4

1.2 Other Versions of Uncertain Logic 5

In our considerations this subject is called an expert. So the expert does not knowexactly the value )(ωx , but "looking at" ω he obtains some information

concerning x , which he does not express in an explicit form but uses it to formu-late )(xhω . Hence, the expert is the source of )(xhω which for particular x evalu-

ates his opinion that xx =~ . That is why )(xhω and consequently )],([ Pv ωΨare called degrees of certainty. For example Ω is a set of persons, )(ωx denotes

the age of ω and the expert looking at the person ω gives the function )(xhω

whose value for the particular x is his degree of certainty that the age of this per-son is approximately equal to x. The predicates ),( PωΨ are soft because of the

uncertainty of the expert. The result of including )(xhω into the definition of un-

certain logic is that for the same ),( XΩ we may have the different logics speci-

fied by different experts.The logic introduced by Definition 1.1 will be denoted by L-logic. In the nextpart we shall consider other versions of uncertain logic which will be denoted by

cnp LLL and, .

1.2 Other Versions of Uncertain Logic

Definition 1.2 (Lp-logic): The first part is the same as in Definition 1.1. Thecertainty index

)(max)],([)],([ xhPvPvxDx

p ωωΨωΨ∈

== . (1.23)

The operations are defined in the following way

),(),( PP ¬=¬ ωΨωΨ , (1.24)

),(),(),( 2121 PPPP ∨=∨ ωΨωΨωΨ , (1.25)

),(),(),( 2121 PPPP ∧=∧ ωΨωΨωΨ . (1.26)

Consequently, we have the same equalities for pv , i.e.

)],([)],([ PvPv pp ¬=¬ ωΨωΨ , (1.27)

)],([)],(),([ 2121 PPvPPv pp ∨=∨ ωΨωΨωΨ , (1.28)

)],([)],(),([ 2121 PPvPPv pp ∧=∧ ωΨωΨωΨ . (1.29)

In a similar way as for L-logic it is easy to prove that:

5


If 21 PP → then )],([)],([ 21 PvPv pp ωΨωΨ ≤ , (1.30)

)],([)],,([max)],([ 2121 PvPvPPv ppp ωΨωΨωΨ =∨ , (1.31)

)],([)],,([min)],([ 2121 PvPvPPv ppp ωΨωΨωΨ ≤∧ , (1.32)

)],([1)],([ PvPv pp ωΨωΨ −≥¬ . (1.33)

Definition 1.3 (Ln-logic): The certainty index of Ψ is defined as follows

)( max1)],([1)],([

xhPvPvxDx

pn ωωΨωΨ∈

−=¬−= . (1.34)

The operations are the same as for pv in Lp-logic, i.e. (1.24), (1.25), (1.26) and

(1.27), (1.28), (1.29) with nv in the place of pv .

It may be proved that:

If 21 PP → then )],([)],([ 21 PvPv nn ωΨωΨ ≤ , (1.35)

)],([)],,([max)],([ 2121 PvPvPPv nnn ωΨωΨωΨ ≥∨ , (1.36)

)],([)],,([min)],([ 2121 PvPvPPv nnn ωΨωΨωΨ =∧ for 0)( 21 >∧ PPw ,

(1.37)

)],([1)],([ PvPv nn ωΨωΨ −≤¬ . (1.38)

The statement (1.35) follows immediately from (1.30) and (1.34). Property(1.36) follows from 121 DDD ⊇∪ , 221 DDD ⊇∪ and (1.35). From (1.34)

we have

)],([1 )],,([1max1

)( max ,)( maxmax1)(max1)],([

21

212121

PvPv

xhxhxhPPv

nn

DxDxDDxn

ωΨωΨ

ωΨ ωωω

−−−=

−=−=∧∈∈∪∈

which proves (1.37). Substituting (1.34) into (1.33) we obtain (1.38).In Definition 1.2 the certainty index is defined in "a positive way", so we mayuse the term: "positive" logic ( pL ). In Definition 1.3 the certainty index is de-

fined in "a negative way" and consequently we may use the term: "negative" logic

( nL ). In (1.23) the shape of the function )(xhω in xD is not taken into account,

and in (1.34) the function )(xhω in xD is neglected. They are the known disad-

vantages of these definitions. To avoid them consider the combined logic denotedby cL .

6

1.2 Other Versions of Uncertain Logic 7

Definition 1.4 (Lc-logic): The certainty index of Ψ and the negation Ψ¬ are

defined as follows:

)]( max1)( max[2

1

2

)],([)],([)],([

xhxh

PvPvPv

xx DxDx

npc ωω

ωΨωΨωΨ

∈∈−+=

+= ,

(1.39)

),(),( PP ¬=¬ ωΨωΨ . (1.40)

The operations for cv are determined by the operations for pv and nv .

Since Lc-logic will be used in the next section for the formulation of the uncertainvariable, it will be described in more detail than pL and nL . According to (1.40)

)],([)],([ PvPv cc ¬=¬ ωΨωΨ .

Using (1.39) and (1.28), (1.29) for pv and nv , it is easy to show that

)],([)],(),([ 2121 PPvPPv cc ∨=∨ ωΨωΨωΨ , (1.41)

)],([)],(),([ 2121 PPvPPv cc ∧=∧ ωΨωΨωΨ . (1.42)

Lc-logic may be defined independently of pv and nv , with the right hand side of

(1.39) and the definitions of operations (1.40), (1.41), (1.42). The operationsmay be rewritten in the following form

xx DxDx ∈=∉ ~~ , (1.43)

]~)([]~)(~)([ 2121 DDxvDxDxv cc ∪∈=∈∨∈ ωωω , (1.44)

]~)([]~)(~)([ 2121 DDxvDxDxv cc ∩∈=∈∧∈ ωωω . (1.45)

From (1.8), (1.34) and (1.39)

1]~)([ =∈ Xxvc ω , 0]~)([ =∅∈ωxvc . (1.46)

One can note that "~")( xxxG ==ω is a special case of Ψ for xDx = and

according to (1.39)

)](max1)([2

1]~)([

xhxhxxv

xXxc

ωωω−∈

−+== , (1.47)

)](1)(max[2

1])([

xhxhxxv

xXxc ωωω −+=≅/

−∈

. (1.48)

7


It is worth noting that if )(xhω is a continuous function then

)(2

1]~)([ xhxxvc ωω == .

Using (1.30) and (1.35), we obtain the following property: If for each x

21 PP → (i.e. 21 DD ⊆ ) then

)],([)],([ 21 PvPv cc ωΨωΨ ≤ or ]~)([]~)([ 21 DxvDxv cc ∈≤∈ ωω .(1.49)

Theorem 1.3:

)],([,)],([max)],([ 2121 PvPvPPv ccc ωΨωΨωΨ ≥∨ , (1.50)

)],([,)],([min)],([ 2121 PvPvPPv ccc ωΨωΨωΨ ≤∧ . (1.51)

Proof: Inequality (1.50) may be obtained from 121 DDD ⊇∪ , 221 DDD ⊇∪and (1.49). Inequality (1.51) follows from 121 DDD ⊆∩ , 221 DDD ⊆∩ and

(1.49). The property (1.50) can also be delivered from (1.39), (1.31), (1.36),and the property (1.51) – from (1.39), (1.32), (1.37).

Theorem 1.4:

)],([1)],([ PvPv cc ωΨωΨ −=¬ . (1.52)

Proof: From (1.34) and (1.39)

)],([1)],([2

1)],([ PvPvPv ppc ¬−+= ωΨωΨωΨ .

Then

)],([1)],([1)],([21)],([ PvPvPvPv cppc ωΨωΨωΨωΨ −=−+¬=¬ .

Till now it has been assumed that Xxx ∈),(ω . The considerations can be ex-tended for the case Xx ∈)(ω and XXx ⊂∈ . It means that the set of approxi-mate values X evaluated by an expert may be a subset of the set of the possiblevalues of )(ωx . In a typical case ...,,, 21 mxxxX = (a finite set), Xxi ∈ for

mi ,1∈ . In our example with persons and age an expert may give the values)(xhω for natural numbers, e.g. 22,21,20,19,18=X .

8

1.3 Uncertain Variables 9

1.3 Uncertain Variables

The variable x for a fixed ω will be called an uncertain variable. Two versionsof uncertain variables will be defined. The precise definition will contain: )(xh

given by an expert, the definition of the certainty index )~( xDxw ∈ and the defi-

nitions of )~( xDxw ∉ , )~ ~( 21 DxDxw ∈∨∈ , )~ ~( 21 DxDxw ∈∧∈ .

Definition 1.5 (uncertain variable): The uncertain variable x is defined by theset of values X, the function )()( xxvxh ≅= (i.e. the certainty index that xx ≅ ,

given by an expert) and the following definitions:

∅=

∅≠=∈ ∈

, for 0

for )( max )~(

x

xDx

xD

DxhDxv x (1.53)

)~(1 )~( xx DxvDxv ∈−=∉ , (1.54)

)~(, )~(max )~ ~( 2121 DxvDxvDxDxv ∈∈=∈∨∈ , (1.55)

∅=∩∅≠∩∈∈

=∈∧∈ for 0

for )~(, )~( min )~ ~(

21

212121 DD

DDDxvDxvDxDxv

(1.56)

The function )(xh will be called a certainty distribution.

The definition of the uncertain variable is based on the uncertain logic, i.e.L-logic (see Definition 1.1). Then the properties (1.17), (1.18), (1.19), (1.20),(1.22) are satisfied. The properties (1.19) and (1.20) may be presented in the fol-lowing form

)~(, )~(max ) ~( 2121 DxvDxvDDxv ∈∈=∪∈ , (1.57)

)~(, )~(min ) ~( 2121 DxvDxvDDxv ∈∈≤∩∈ . (1.58)

Example 1.1: 7,6,5,4,3,2,1 =X and the corresponding values of )(xh are

)2.0,4.0,5.0,6.0,1,8.0,5.0( , i.e. 5.0)1( =h , 8.0)2( =h etc. Let

6,5,4,2,1 1 =D , 5,4,3 2 =D . Then 6,5,4,3,2,1 21 =∪ DD ,

5,4 21 =∩ DD , =∈ )~( 1Dxv 8.04.0,5.0,6.0,8.0,5.0max = ,

=∈ )~( 2Dxv 15.0,6.0,1max = , =∪∈ ) ~( 21 DDxv ,6.0,1,8.0,5.0max

14.0,5.0 = =∈∨∈ )~ ~( 21 DxDxv 11,8.0max = , =∩∈ ) ~( 21 DDxv

6.05.0,6.0max = , =∈∧∈ )~ ~( 21 DxDxv 8.01,8.0min = .

9


Example 1.2: The certainty distribution is shown in Fig. 1.1. Let Dx = [0, 4].Then

)~( xDxv ∈ = 8.0])4,0[~( =∈xv ,

)~( xDxv ∈ = ( ) 1]16,4[~ =∈xv ,

)~( xDxv ∉ = 011 ])16,4[~( =−=∉xv < )~( xDxv ∈ ,

)~( xDxv ∉ = 0.28.01 ])4,0[~( =−=∉xv < )~( xDxv ∈ .

Fig. 1.1. Example of certainty distribution

Definition 1.6 (C-uncertain variable): C-uncertain variable x is defined by theset of values X, the function )()( xxvxh ≅= given by an expert, and the follow-

ing definitions:

)]( max1+ )( max[2

1 )~( xhxhDxv

xx DxDxxc

∈∈−=∈ , (1.59)

)~(1 )~( xcxc DxvDxv ∈−=∉ , (1.60)

) ~()~ ~( 2121 DDxvDxDxv cc ∪∈=∈∨∈ , (1.61)

) ~()~ ~( 2121 DDxvDxDxv cc ∩∈=∈∧∈ . (1.62)

The definition of C-uncertain variable is based on Lc-logic (see Definition 1.4).Then the properties (1.46), (1.47), (1.48) are satisfied. According to (1.40) and(1.52)

)~( )~( xcxc DxvDxv ∈=∉ . (1.63)

Inequalities (1.50) and (1.51) may be presented in the following form

)~(, )~(max ) ~( 2121 DxvDxvDDxv ccc ∈∈≥∪∈ , (1.64)

)~(, )~(min ) ~( 2121 DxvDxvDDxv ccc ∈∈≤∩∈ . (1.65)

10

1.3 Uncertain Variables 11

The function )( xxvc ≅ = )(xhc expressed by (1.47) may be called a

C-certainty distribution. Note that the certainty distribution )(xh is given by an

expert and C-certainty distribution may be determined according to (1.47), using)(xh . The C-certainty distribution does not determine the certainty index

)~( xc Dxv ∈ . To determine cv , it is necessary to know )(xh and to use (1.59).

According to (1.64)

)~( )( max xccDx

Dxvxhx

∈≤∈

.

The formula (1.59) may be presented in the following way

∈−∈=−

∈=∈

∈

∈∈

. otherwise )~(21)~()( max

211

1=)( max if )~(21)( max

21

= )~(

xxDx

Dxx

Dxxc

DxvDxvxh

xhDxvxh

Dxv

x

xx

(1.66)

The formula (1.66) shows the relation between the certainty indexes v and cv

for the same xD : if Dx ≠ X and Dx ≠ ∅ then vc < v. In particular, (1.47) becomes

−

==

−∈

−∈

. otherwise )(max211

1)(max if )(21

)(

xh

xhxhxh

xXx

xXxc

(1.67)

In the continuous case

)(xhc = )(21 xh

and in the discrete case

−

==

≠

≠

. otherwise )(max211

1)(max if )(21

)(xh

xhxh

xh

i

i

xx

xxi

ic

Example 1.3: The set X and )(xh are the same as in Example 1.1. Using

(1.67) we obtain 25.0)1( =ch , 4.0)2( =ch , 6.028.01)3( =−=ch , 3.0)4( =ch ,

25.0)5( =ch , 2.0)6( =ch , 1.0)7( =ch . Let 1D and 2D be the same as in Ex-

ample 1.1. Using (1.66) and the values v obtained in Example 1.1 we have:

4.021)~( 1 ==∈ vDxvc , 6.0

28.01)~( 2 =−=∈ Dxvc , =∈∨∈ )~ ~( 21 DxDxvc

11


=) ~( 21 DDxvc ∪∈ 0.9=22.01 − ,

=∈∧∈ )~ ~( 21 DxDxvc = ) ~( 21 DDxvc ∩∈ 3.02

0.6 = . In this case, for both

1D and 2D , )( max)~( xhDxv cc =∈ for Dx∈ . Let 4,3,2 =D . Now

0.75=25.01)~( −=∈ Dxvc and = )( max xhc 6.00.30.6,0.4,max = < cv .

Example 1.4: The certainty distribution and Dx are the same as in Example 1.2.

)~( xc Dxv ∈ = 4.0]08.0[21)]~()~([

21 =+=∉+∈ xx DxvDxv ,

)~( xc Dxv ∈ = 6.0)~(1)~( =∈−=∉ xcxc DxvDxv ,

)~( xc Dxv ∉ = )~( xc Dxv ∈ = 0.4,

)~( xDxv ∉ = )~( xc Dxv ∈ = 0.6.

The uncertain logic and Lc-logic have been chosen as the bases for the uncertainvariable and C-uncertain variable, respectively, because of the advantages ofthese approaches. In both cases the logic value of the negation is

)~(1)~( xx DxwDxw ∈−=∉ (see (1.54) and (1.60)). In the first case it is easy

to determine the certainty indexes for 21~ ~ DxDx ∈∨∈ and 21

~ ~ DxDx ∈∧∈ and

all operations are the same as in (1.1) for multi-valued logic. In the second casein the definition of the certainty index )~( xc Dxv ∈ the values of )(xh for xD

are also taken into account and the logic operations (negation, disjunction andconjunction) correspond to the operations in the family of subsets xD (comple-

ment, union and intersection). On the other hand, the calculations of the certaintyindexes for disjunction and conjunction are more complicated than in the first caseand are not determined by the certainty indexes for 1

~ Dx ∈ , 2~ Dx ∈ , i.e. they can-

not be reduced to operations in the set of certainty indexes )~( Dxvc ∈ . These

features should be taken into account when making a choice between the applica-tion of the uncertain variable or C-uncertain variable in particular cases.

1.4 Additional Description of Uncertain Variables

For the further considerations we assume kRX ⊆ (k-dimensional real numbervector space) and we shall consider two cases: the discrete case with

,...,, 21 mxxxX = and the continuous case in which )(xh is a continuous func-

tion.

12

1.4 Additional Description of Uncertain Variables 13

Definition 1.7: In the discrete case

∑=

=m

jj

ii

xh

xhxh

1)(

)()( , mi ,1∈ (1.68)

will be called a normalized certainty distribution. The value

∑=

=m

iii xhxxM

1)( )( (1.69)

will be called a mean value of the uncertain variable x . In the continuous case thenormalized certainty distribution and the mean value are defined as follows:

∫=

X

dxxh

xhxh

)(

)()( , ∫=

X

dxxhxxM )( )( . (1.70)

For C-uncertain variable the normalized C-certainty distribution )(xhc and the

mean value )(xM c are defined in the same way, with ch in the place of h in

(1.68), (1.69) and (1.70).

In the continuous case hc(x) = )(21 xh , then )(xhc = )(xh and Mc = M. In the

discrete case, if x* is a unique value for which h(x

*) = 1 and

1)(max*

≈≠

xhxx

then Mc ≈ M. As a value characterizing )(xh or )(xhc one can also use

)( max arg * xhxXx∈

= or )( max arg * xhx cXx

c∈

= .

Replacing the uncertain variable x by its deterministic representation )(xM or*x may be called a determinization (analogous to defuzzification for fuzzy num-

bers).

Consider now a function YX → : Φ , kRY ⊆ , i.e. )(xy Φ= . We say that the

uncertain variable ><= )(, yhYy y is a function of the uncertain variable

><= )(, xhXx x , i.e. )(xy Φ= where the certainty distribution )(yhy is de-

termined by )(xhx and Φ :

)( max)()()(

xhyyvyh xyDx

yx∈

=≅= (1.71)

13


where

)( : )( yxXxyDx =∈= Φ .

If y = Φ (x) is one-to-one mapping and x = )(1 y−Φ then

Dx( y) = )(1 y−Φ

and

hy( y) = hx[ )(1 y−Φ ].

In this case, according to (1.68) and (1.69)

1

11])([)()()( −

==∑∑=m

jjx

m

iixiy xhxhxyM Φ .

For C-uncertain variables C-certainty distribution )()( yyvyh ccy ≅= may be

determined in two ways:1. According to (1.67)

−

==

−∈

−∈

. otherwise )( max211

1)( max if )(21

)(

I

yh

yhyhyh

yyYy

yyYy

y

cy

(1.72)

where )( yhy is determined by (1.71).

2. According to (1.66)

−

==

∈

∈∈

. otherwise )( max211

1)( max if )( max21

)(

)(

)(II

xh

xhxh

yh

xyDx

xDx

xyDx

cy

x

xx (1.73)

Theorem 1.5:

)()( III yhyh cycy = .

Proof: It is sufficient to prove that

=−∈

)( max

yhyyYy

)( max xhxDx x∈

.

From (1.71)=

−∈)( max

yhy

yYy

])( max[ max)(

xhxyDxyYy x

∈−∈

.

14

1.4 Additional Description of Uncertain Variables 15

Note that if 21 yy ≠ then ∅=∩ )( )( 21 yDyD xx . Consequently,

∅=∩ )( )( yDyD xx

and

)( )(

yDyD x

yYy

x =−∈

.

Therefore

)( max ])( max[ max)()(

xhxh xyDx

xyDxyYy xx ∈∈−∈

=

.

It is important to note that )( yhcy is not determined by )(xhcx . To determine

)( yhcy it is necessary to know )(xhx and to use (1.73), or (1.71) and (1.72).

Let us now consider a pair of uncertain variables >×<= ),( ,),( yxhYXyx

where )],(),[(),( yxyxvyxh ≅= is given by an expert and is called a joint

certainty distribution. Then, using (1.1) for the disjunction in multi-valued logic,we have the following marginal certainty distributions

),( max )()(

yxhxxvxhYy

x∈

=≅= , (1.74)

),( max )()(

yxhyyvyhXx

y∈

=≅= . (1.75)

If the certainty index ])([ xxv ≅ω given by an expert depends on the value of y

for the same ω (i.e. if the expert changes the value )(xhx when he obtains the

value y for the element ω "under observation") then )( yxhx may be called a

conditional certainty distribution. The variables x , y are called independent

when

)()( xhyxh xx = , )()( yhxyh yy = .

Using (1.1) for the conjunction in multi-valued logic we obtain

)( ),(min)( ),(min) (),( yxhyhxyhxhyyxxvyxh xyyx ==≅∧≅= . (1.76)

Taking into account the relationships between the certainty distributions one cansee that they cannot be given independently by an expert. If the expert gives

)(xhx and )( xyhy or )( yhy and )( yxhx then ),( yxh is already determined

by (1.76). The joint distribution ),( yxh given by an expert determines )(xhx

(1.74) and )( yhy (1.75) but does not determine )( yxhx and )( xyhy . In such

a case only sets of functions )( yxhx and )( xyhy satisfying (1.76) are deter-

15


mined. For the function )(xy Φ= where x is a pair of variables ),( 21 xx ,

Xx ∈2,1 , according to (1.71)

),(max)( 21)(),( 21

xxhyhyDxx

y∈

= ,

),( 21 xxh is determined by (1.76) for 1xx = , 2xy = , and

),( : ),()( 2121 yxxXXxxyD =×∈= Φ .

16

2 Analysis and Decision Making for Static Plants

2.1 Analysis Problem for a Functional Plant

Let us consider a static plant with input vector Uu ∈ and output vector Yy ∈ ,

where U and Y are real number vector spaces. When the plant is described bya function )(uy Φ= , the analysis problem consists in finding the value y for the

given value u. Consider now the plant described by ),( xuy Φ= where Xx ∈ is

an unknown vector parameter which is assumed to be a value of an uncertain vari-able x with the certainty distribution )(xhx given by an expert. Then y is a value

of an uncertain variable y and for the fixed u, y is the function of x :

),( xuy Φ= .

Analysis problem may be formulated as follows: For the given Φ, )(xhx and u

find the certainty distribution )(yhy of the uncertain variable y . Having )(yhy

one can determine My and

)(maxarg* yhy yYy∈

= , i.e. 1)( * =yhy .

According to (1.71)

)(max)();();(

xhyyvuyh xuyDx

yx∈

=≅= (2.1)

where ),(:);( yxuXxuyDx =∈= Φ . If Φ as a function of x is one-to-one

mapping and ),(1 yux −= Φ then

)],([);( 1 yuhuyh xy−= Φ

and ),( ** xuy Φ= where )(maxarg* xhx x= . From the definition of the certainty

distributions h and hc it is easy to note that in both continuous and discrete cases

17


18 2 Analysis and Decision Making for Static Plants

**cyy = where )(maxarg* yhy cyc = and )( yhcy is a certainty distribution of y

considered as C-uncertain variable.

Example 2.1: Let 2, Rxu ∈ , u = (u(1), u

(2)), x = (x(1), x

(2)), 1Ry ∈ ,

)2()2()1()1( uxuxy += ,

x(1)∈ 3, 4, 5, 6 , x

(2)∈ 5, 6, 7 and the corresponding values of 1xh , 2xh given

by an expert are (0.3, 0.5, 1, 0.6) for )1(x and (0.8, 1, 0.4) for )2(x . Assume that)1(x and )2(x are independent, i.e. )(),(min),( )2(

2)1(

1)2()1(

jxixjix xhxhxxh = . Then

for ∈= ),( )2()1( xxx (3,5), (3,6), (3,7), (4,5), (4,6), (4,7), (5,5), (5,6), (5,7), (6,5),

(6,6), (6,7) the corresponding values of hx are (0.3, 0.3, 0.3, 0.5, 0.5, 0.4, 0.8, 1,

0.4, 0.6, 0.6, 0.4). Let 2)1( =u , 1)2( =u . The values of )2()1(2 xxy += corre-

sponding to the set of pairs ),( )2()1( xx are the following: 11, 12, 13, 13, 14, 15,

15, 16, 17, 17, 18, 19. Then hy(11) = hx(3,5) = 0.3, hy(12) = hx(3,6) = 0.3,hy(13) = max hx(3,7), hx(4,5) = 0.5, hy(14) = hx(4,6) = 0.5, hy(15) =max hx(4,7), hx(5,5) = 0.8, hy(16) = hx(5,6) = 1, hy(17) = max hx(5,7),hx(6,5) = 0.6, hy(18) = hx(6,6) = 0.6, hy(19) = hx(6,7) = 0.4.

For )(yhy we have y*=16. Using (1.68) and (1.69) for y we obtain 5=yh ,

40.155

77 ==yM .

Using (1.72) we obtain the corresponding values of )(yhyc : (0.15, 0.15, 0.25,

0.25, 0.4, 6.028.01 =− , 0.3, 0.3, 0.2). Then 16** == yyc , 6.0)16( =≅yvc ,

6.2=ych , yyc MM ≈= 43.15 .

2.2 Decision Making Problem for a Functional Plant

For the functional system )(uy Φ= the basic decision problem consists in finding

the decision u for the given desirable value y . Consider now the system with the

unknown parameter x, described in 2.1.

Decision problem may be formulated as follows:

Version I: To find the decision u maximizing )ˆ( yyv ≅ .

18

2.2 Decision Making Problem for a Functional Plant 19

Version II: To find u such that yuyM y ˆ);( = where yM is the mean value of

( )xuy ,Φ= for the fixed u.

Version III: To find u minimizing );( usM s where )ˆ,( yys ϕ= is a quality in-

dex, e.g. )ˆ()ˆ( T yyyys −−= where y and y are column vectors.

When x is assumed to be a C-uncertain variable, the formulations are the samewith cv , cyM , csM instead of v, yM , sM . It is worth noting that the decision

problem statements are analogous to those in the probabilistic approach where xis assumed to be a value of a random variable with the known probability distri-bution. In each version );( uyhy should be determined according to (2.1). Then,

in version I u is the value of u maximizing );ˆ( uyhy , i.e. u is the solution of

the equation yu ˆ)( =ε where )(* uy ε= is a value of y maximizing );( uyhy . In

version II u is obtained as a solution of the equation yuyM y ˆ);( = .

In version III for the determination of );( usM s one should find

);( max);()(

uyhush ysDy

sy∈

= (2.2)

where )ˆ,( : )( syyYysD y =∈= ϕ .

When x is considered as C-uncertain variable, it is necessary to determine cyh

using (1.72) or (1.73) and in version II to find );( uyMcy . In version III it is nec-

essary to find csh according to (1.72) or (1.73) with ),( xs instead of ),( xy and

then to determine ( )usM cs ; . Using (1.72) it is easy to see that if *y is the only

value for which 1=yh then ** yyc = where *cy is the value maximizing cyh , and

ycy MM ≈ . Consequently, in this case the results u in version I are the same and

in version II are approximately the same for the uncertain and C-uncertain vari-able, and for uu ˆ= in version I 1)ˆ( =≅ yyv , 1)ˆ( <≅ yyvc .

Example 2.2: Let u, y, x ∈ 1R , xuy = , 7,6,5,4,3 =X and the corre-

sponding values of )(xhx are )1.0,1.0,3.0,1,5.0( . Using (1.67) or (1.72) for x

we obtain the corresponding values of cxh : )05.0,05.0,15.0,75.0,25.0( .

In version I uy 4* = , i.e. yu ˆ25.0ˆ = , 1)ˆ( =≅ yyv , 75.0)ˆ( =≅ yyvc .

19


In our example 7,6,5,4,3 uuuuuY = , the values of );( uyhy are the same

as xh , the values of yh are )05.0,05.0,15.0,5.0,25.0( and the values

of cycy hh 25.1= are the same as cxh . In version II, using (1.69) for y we obtain

uM y 15.4= and uM cy 12.4= . Then in both cases the result is approximately

the same: yu ˆ24.0ˆ ≈ .

Let in version III 2)ˆ( yys −= . Then );( usM s is equal to

+− 2)ˆ3(25.0 yu +− 2)ˆ4(5.0 yu +− 2)ˆ5(15.0 yu +− 2)ˆ6(05.0 yu 2)ˆ7(05.0 yu −

= )(uM s except for

ji xx

yu

+=

ˆ2

= du , ji xx ≠ , 7,6,5,4,3 , ∈ji xx . (2.3)

For example, for 31 =x , 42 =x and duu = we have 22 )ˆ4()ˆ3( yuyu −=− .

Consequently, 2)ˆ4( yus −∈ , 2)ˆ5( yu − , 2)ˆ6( yu − , )ˆ7( 2yu − , according to

(2.2) the values of sh are ]1.0,1.0,3.0),1,5.0max([ and

])ˆ7(1.0)ˆ6(1.0)ˆ5(3.0)ˆ4[(5.1

1);( 2222 yuyuyuyuusM dddds −+−+−+−=

.

Then );( usM s is a discontinuous function of u. The value of u minimizing

)(uM s is

yyu ˆ26.0ˆ8.15

15.4min ≈= .

From the sensitivity point of view it is reasonable not to take into account thepoints of discontinuity du (2.3) and to accept minˆ uu = .

2.3 External Disturbances

The considerations may be easily extended for a plant with external disturbances,described by a function

),,( xzuy Φ= (2.4)

where Zz ∈ is a vector of the disturbances which can be measured.

20

2.3 External Disturbances 21

Analysis problem: For the given Φ, )(xhx , u and z find the certainty distribution

)( yhy .

According to (1.71)

)(max)],;(~[)~(),;(),;(

xhzuyDxvyyvzuyh xzuyDx

xyx∈

=∈=== (2.5)

where

),,(:),;( yxzuXxzuyDx =∈= Φ .

Having ),;( zuyhy one can determine the mean value

∫∫ −⋅=Y

yY

yy dyzuyhdyzuyyhzuyM 1]),;([),;(),;(

= ),( zubΦ (2.6)

(for the continuous case) and

),;(maxarg* zuyhy yYy∈

= ,

i.e. such a value *y that 1),;( * =zuyhy . If Φ as a function of x is one-to-one

mapping and ),,(1 yzux −= Φ then

)],,([),;( 1 yzuhzuyh xy−= Φ (2.7)

and ),,( ** xzuy Φ= where *x satisfies the equation 1)( =xhx . It is easy to note

that **cyy = where

),;(maxarg* zuyhy cyYy

c ∈=

and cyh is a certainty distribution for C-uncertain variable.

Decision problem: For the given Φ, )(xhx , z and y

I. One should find u

= au maximizing )ˆ( yyv ≅ .

II. One should find u

= bu such that yyM y ˆ)( = .

The versions I, II correspond to the versions I, II of the decision problem pre-sented in 2.2. The third version presented in 2.2 is much more complicated andwill not be considered. In version I

21


),(maxarg zuu aUu

a Φ∈

=

= )(zaΨ (2.8)

where ),;ˆ(),( zuyhzu ya =Φ and yh is determined according to (2.5). The result

au is a function of z if au is a unique value maximizing aΦ for the given z.

In version II one should solve the equation

yzub ˆ),( =Φ (2.9)

where the function bΦ is determined by (2.6). If the equation (2.9) has a unique

solution with respect to u for a given z then as a result one obtains )(zu bb Ψ= .

The functions aΨ and bΨ are two versions of the decision algorithm )(zu Ψ=in an open-loop decision system (Fig. 2.1). It is worth noting that au is a decision

for which 1)ˆ( =≅ yyv .

Ψ ( ) Φ ( )

Fig. 2.1. Open-loop decision system

The functions aΦ , bΦ are results of two different ways of determinization of

the uncertain plant, and the functions aΨ , bΨ are the respective decision algo-

rithms based on the knowledge of the plant (KP):

><= xh,KP Φ . (2.10)

Assume that the equation

yxzu ˆ),,( =Φ

has a unique solution with respect to u:

u= ),( xzdΦ . (2.11)

The relationship (2.11) together with the certainty distribution )(xhx may be con-

sidered as a knowledge of the decision making (KD):

><= xd h,KD Φ , (2.12)

22

2.3 External Disturbances 23

obtained by using KP and y . The equation (2.11) together with xh may also be

called an uncertain decision algorithm in the open-loop decision system. The de-terminization of this algorithm leads to two versions of the deterministic decisionalgorithm dΨ , corresponding to versions I and II of the decision problem:

I.

);(maxarg zuhu uUu

ad∈

=

= )(zadΨ (2.13)

where

)(max);();(

xhzuh xzuDx

ux∈

= (2.14)

and

),(:);( xzuXxzuD dx Φ=∈= .

II.

);( zuMu ubd =

= )(zbdΨ . (2.15)

The decision algorithms adΨ and bdΨ are based directly on the knowledge of

the decision making. Two concepts of the determination of deterministic decisionalgorithms are illustrated in Figs. 2.2 and 2.3. In the first case (Fig. 2.2) the deci-sion algorithms )(zaΨ and )(zbΨ are obtained via the determinization of the

knowledge of the plant KP. In the second case (Fig. 2.3) the decision algorithms)(zadΨ and )(zbdΨ are based on the determinization of the knowledge of the

decision making KD obtained from KP for the given y . The results of these two

approaches may be different.

Theorem 2.1: For the plant described by KP in the form (2.10) and for KD in the

form (2.12), if there exists an inverse function ),,(1 yzux −= Φ then

)()( zz ada ΨΨ = .

Proof: According to (2.7) and (2.13)

)]ˆ,,([),;ˆ( 1 yzuhzuyh xy−= Φ ,

)]ˆ,,([);( 1 yzuhzuh xu−= Φ .

Then, using (2.8) and (2.13) we obtain )()( zz ada ΨΨ = .

23


ba ΨΨ ,z u

z

yba ΦΦ ,

>< xh,Φ

y

Fig. 2.2. Decision system with determinization – the first case

b da d ΨΨ ,z du

z

>< xd h,Φ

y

Φ

y

Fig. 2.3. Decision system with determinization – the second case

Example 2.3: Let u, y, x, z 1R∈ and

zuxy += .

Then

zxMuyM xy += )()(

and from the equation yyM y ˆ)( = we obtain

24

2.4 Nonparametric Uncertainty 25

)(

ˆ)(

xM

zyzu

xbb

−==Ψ .

The uncertain decision algorithm is

x

zyxzu d

−==ˆ

),(Φ

and after the determinization

)()(

ˆ)(

1z

xM

zyzu b

xbdbd ΨΨ ≠−== − .

This very simple example shows that the deterministic decision algorithm )(zbΨobtained via the determinization of the uncertain plant may differ from the deter-ministic decision algorithm )(zbdΨ obtained as a result of the determinization of

the uncertain decision algorithm.

2.4 Nonparametric Uncertainty

The certainty distribution ),;( zuyhy may be given directly by an expert as a

nonparametric description of the uncertain plant. If u and z are considered as val-ues of uncertain variables u and z , respectively, then

),|(),;( zuyhzuyh yy = ,

i.e. ),|( zuyhy is a conditional certainty distribution. If the certainty distribution

)(zhz for z is also given by an expert then it is possible to find the uncertain de-

cision algorithm in the form of a conditional certainty distribution )|( zuhu , for

the given desirable certainty distribution )(yhy required by a user.

Decision problem: For the given ),|( zuyhy , )(zhz and )(yhy one should de-

termine )|( zuhu .

According to the relationships (1.74), (1.75) and (1.76)

),,(max)(,

zuyhyh yZzUu

y∈∈

=

where ),,( zuyhy is the joint certainty distribution for ),,( zuy , i.e.

),|(),,(minmax),,(,

zuyhzuhzuyh yuzZzUu

y∈∈

=

25


and the joint certainty distribution

)|(),(min),( zuhzhzuh uzuz = . (2.16)

Finally

),|(),|(),(minmax)(,

uzyhzuhzhyh yuzZzUu

y∈∈

= . (2.17)

Any distribution )|( zuhu satisfying the equation (2.17) is a solution of our deci-

sion problem. It is easy to note that the solution of the equation (2.17) with respectto ),( zuhu is not unique, i.e. the equation (2.17) may be satisfied by different

conditional certainty distributions )|( zuhu . Having )|( zuhu one can obtain the

deterministic decision algorithm after the determinization of the uncertain decisionalgorithm described by )|( zuhu , according to (2.13) or (2.15) with )|( zuhu in-

stead of );( zuhu . The decision algorithms )(zadΨ or )(zbdΨ are then obtained

as a result of the determinization of the knowledge of the decision makingKD = >< )|( zuhu , which is determined from the knowledge of the plant

KP = >< )(),,|( zhzuyh zy

for the given )(yhy (Fig. 2.4). It is worth noting that the deterministic decision

algorithm obtained in this way has no clear practical interpretation. It is introducedhere mainly for the comparison with a fuzzy approach presented in Chap. 6.

b da d ΨΨ ,

)|( zuhu

>< )(),,|( zhzuyh zy

)( yh y

z du

z

y

Fig. 2.4. Open-loop decision system under consideration

26


The determination of )|( zuhu may be decomposed into two steps. In the first

step we determine the sets of the joint certainty distributions ),( zuhuz satisfying

the equation

),|(),,(minmax)(,

zuyhzuhyh yuzZzUu

y∈∈

= (2.18)

and in the second step we determine )|( zuhu from equation (2.16). It is easy to

see that if the functions )(yhy and ),|( zuyhy have one local maximum equal to

1 then the point (u, z) maximizing the right hand side of equation (2.18) satisfiesthe equation

),|(),( zuyhzuh yuz = .

Hence, for this point we have

),|()( zuyhyh yy = . (2.19)

Theorem 2.2: The set of functions )|( zuhu satisfying the equation (2.16) is

determined as follows:

∈≥

∉=

),(),(for),(

),(),(for),()|(

zuDzuzuh

zuDzuzuhzuh

uz

uzu

where

),()(:),(),( zuhzhZUzuzuD uzz =×∈= .

Proof: From (2.16) it follows that

)],()([ zuhzh uzzZzUu

≥∈∈

.

If ),()( zuhzh uzz > then, according to (2.16), )|(),( zuhzuh uuz = . If

),()( zuhzh uzz = , i.e. ),(),( zuDzu ∈ then ),()|( zuhzuh uzu ≥ .

In particular, as one of the solutions of the equation (2.16) we may accept

),()|( zuhzuh uzu = . (2.20)

Consequently, we may apply the following procedure for the determination of theuncertain decision algorithm:

1. To solve the equation (2.19) with respect to y and to obtain ),(* zuy .

2. To put ),(* zuy into )(yhy in the place of y and to obtain

27


)],([),( * zuyhzuh yuz = .

3. To assume ),()|( zuhzuh uzu = .

Let us note that under the assumption (2.20) the knowledge of )(zhz is not

necessary for the determination of the uncertain decision algorithm.

Example 2.4: Consider a plant with 1,, Rzyu ∈ , described by the conditional

certainty distribution given by an expert:

)(1)(),|( 2 zbudyzuyhy −−−+−−=

for

2

10 ≤≤ u , bzb ≤≤−

2

1,

dzbxydzbx +−−−≤≤+−−−− )(1)(1 ,

and 0),|( =zuyhy otherwise.

For the certainty distribution required by a user (Fig. 2.5):

+−−

=0

1)()(

2cyyhy

,otherwise

11for +≤≤− cyc

one should determine the uncertain decision algorithm in the form

),()|( zuhzuh uzu = .

−


Let us assume that

21 +≤≤+ cdc . (2.21)

28


Then the equation (2.19) has a unique solution which is reduced to the solution ofthe equation

)(1)(1)( 22 zbudycy −−−+−−=+−−

and

)(2

1

)(2

22*

cd

zbucd

cd

zbucdy

−−+++=

−−++−= . (2.22)

Using (2.22) and (2.21) we obtain

≤≤−≤≤

−+−−≤−+−

−++−−=

==

.otherwise 021,

210

,)]1([1for1])(2

)([

)(),()|(

222

*

bzbu

bcdzucd

zbucd

yhzuhzuh yuzuz

29

3 Relational Systems

3.1 Relational Knowledge Representation

The considerations presented in the previous chapter may be extended for staticrelational systems, i.e. the systems described by relations which are not reduced tofunctions. Let us consider a static plant with input vector Uu ∈ and output vector

Yy ∈ , where U and Y are real number vector spaces. The plant is described by a

relation

yu ρ

= YUyuR ×⊂),( (3.1)

which may be called a relational knowledge representation of the plant. It is anextension of the traditional functional model )(uy Φ= considered in the previous

chapter. The description (3.1) given by an expert may have two practical interpre-tations:1. The plant is deterministic, i.e. at every moment n

)( nn uy Φ= ,

but the expert has no full knowledge of the plant and for the given u he candetermine only the set of possible outputs:

),(),(::)( yuRyuYyYuDy ∈∈⊂ .

For example, in one-dimensional case ucy = , the expert knows that

0,; 2121 >≤≤ ccccc . Then as the description of the plant he gives a rela-

tion presented in the following form

≤≤≤

≥≤≤

0for

0for

12

21

uucyuc

uucyuc. (3.2)

The situation is illustrated in Fig. 3.1, in which the set of points ),( nn yu is

denoted.2. The plant is not deterministic, which means that at different n we may observe

31


32 3 Relational Systems

different values ny for the same values nu . Then ),( yuR is a set of all possi-

ble points ),( nn yu , denoted for the example (3.2) in Fig. 3.2.

uc 2

uc1

cu

u

y

Fig. 3.1. Illustration of a relation – the first case

uc 2

uc1

u

y

Fig. 3.2. Illustration of a relation – the second case

In the first case the relation (which is not a function) is a result of the expert’suncertainty and in the second case – a result of the uncertainty in the plant. Forsimplicity, in both cases we shall say about an uncertain plant, and the plant de-

32

3.2 Analysis and Decision Making for Relational Plants 33

scribed by a relational knowledge representation will be shortly called a relationalplant. In more complicated cases the relational knowledge representation given by anexpert may have a form of a set of relations:

YWUywuRi ××⊂),,( , ki ...,,2,1= (3.3)

where Ww ∈ is a vector of additional auxiliary variables used in the description ofthe knowledge. The set of relations (3.3) may be called a based knowledge repre-sentation. It may be reduced to a resulting knowledge representation ),( yuR :

),,(),,(:),(),( ywuRywuYUyuyuRWw

⊂×∈=∈

where

k

ii ywuRywuR

1

),,(),,(=

= .

The relations ),,( ywuRi may have a form of a set inequalities and/or equalities

concerning the components of the vectors u, w, y.

3.2 Analysis and Decision Making for Relational Plants

The formulations of the analysis and decision making problems for a relationalplant analogous to those for a functional plant described by a function )(uy Φ=are adequate to the knowledge of the plant [14].

Analysis problem may be formulated as follows: For the given ),( yuR and

UDu ⊂ find the smallest set YDy ⊂ such that the implication

yu DyDu ∈→∈ (3.4)

is satisfied.The information that uDu ∈ may be considered as a result of observation. For

the given uD one should determine the best estimation of y in the form of the set

of possible outputs yD . It is easy to note that

),(),(: yuRyuYyDuDu

y ∈∈=∈

. (3.5)

This is then a set of all such values of y for which there exists uDu ∈ such that

),( yu belongs to R. In particular, if the value u is known, i.e. uDu =

33


(a singleton) then

),(),(:)( yuRyuYyuDy ∈∈= (3.6)

where )(uDy is a set of all possible y for the given value u. The analysis problem

is illustrated in Fig. 3.3 where the shaded area illustrates the relation ),( yuR and

the interval yD denotes the solution for the given interval uD .

u

y

yD

uD

Fig. 3.3. Illustration of analysis problem

Example 3.1: Let us consider the plant with two inputs )1(u and )2(u , describedby inequality

)2(2

)1(2

)2(1

)1(1 uducyuduc +≤≤+ ,

and the set uD is determined by inequalities

α≤+ )2()1( buau (3.7)

)1(min

)1( uu ≥ , )2(min

)2( uu ≥ . (3.8)

For example, y may denote the amount of a product in a production process, )1(u

and )2(u – amounts of two kinds of a raw material, and the value )2()1( buau + –a cost of the raw material. Assume that 1c , 2c , 1d , 2d , a, b, α are positive num-

bers and 21 cc < , 21 dd < . It is easy to see that the set (3.5) is described by in-

34


equality

max)2(

min1)1(

min1 yyuduc ≤≤+ (3.9)

where

)(max )2(2

)1(2

,max

)2()1(uducy

uu

+= (3.10)

subject to constraints (3.7) and (3.8).The maximization in (3.10) leads to the following results:

If

b

a

d

c≤

2

2

then

)( )1(min

2)1(min2max au

b

ducy −+= α . (3.11)

If

b

a

d

c ≥2

2

then

)2(min2

)2(min

2max )( udbu

a

cy +−= α . (3.12)

For the numerical data 11 =c , 22 =c , 21 =d , 42 =d , 1=a , 4=b , 3=α ,

1)1(min =u , 5.0)2(

min =u

2

1

2

2 =d

c,

2

2

4

1

d

c

b

a <= .

From (3.12) we obtain 4max =y and according to (3.9)

2)2(min1

)1(min1min =+= uducy . The set yD is then determined by inequality

42 ≤≤ y .

Decision problem: For the given ),( yuR and YDy ⊂ find the largest set

UDu ⊂ such that the implication (3.4) is satisfied.

The set yD is given by a user, the property yDy ∈ denotes the user’s require-

ment and uD denotes the set of all possible decisions for which the requirement

35


concerning the output y is satisfied. It is easy to note that

)(: yyu DuDUuD ⊆∈= (3.13)

where )(uDy is the set of all possible y for the fixed value u, determined by

(3.6). The solution may not exist, i.e. ∅=uD (empty set). In the example illus-

trated in Fig. 3.2, if ],[ maxmin yyDy = and 0, 21 >cc then

],[2

max

1

minc

yc

yDu =

and the solution exists under the condition

2

max

1

min

c

y

c

y ≤ .

The analysis and decision problems for the relational plant are the extensions ofthe respective problems for the functional plant, presented in Sect. 2.1. The prop-erties "" uDu ∈ and "" yDy ∈ may be called the input and output properties, re-

spectively. For the functional plant we considered the input and output properties

in the form: "" *uu = and "" *yy = where *u , *y denote fixed variables. For

the relational plant the analysis problem consists in finding the best output prop-erty (the smallest set yD ) for the given input property, and the decision problem

consists in finding the best input property (the largest set uD ) for the given re-

quired output property. The procedure for determining the effective solution uD

and yD based on the general formulas (3.5) and (3.13) depends on the form of

),( yuR and may be very complicated. If ),( yuR and the given property (i.e. the

given set uD or yD ) are described by a set of equalities and/or inequalities con-

cerning the components of the vector u and y then the procedure is reduced to"solving" this set of equalities and/or inequalities.

Example 3.2: Consider a plant described by a relation

)()( 21 uGyuG ≤≤ (3.14)

where 1G and 2G are the functions

+→ R:1 UG , +→ R:2 UG ; ),0[R ∞=+ ,

and

)]()([ 21 uGuGUu

≤∈

.

36


For example, y is the amount of a product as in Example 3.1 and the componentsof the vector u are features of the raw materials. For a user’s requirement

maxmin yyy ≤≤ ,

i.e. ],[ maxmin yyDy = , we obtain

])([])([: max2min1 yuGyuGUuDu ≤∧≥∈= .

In particular, if the relation (3.14) has a form

uucyuuc T2

T1 ≤≤ , 01 >c , 12 cc > (3.15)

where ku R∈ and

2)(2)2(2)1(T )(...)()( kuuuuu +++=

then uD is described by inequality

2

maxT

1

minc

yuu

cy

≤≤

and the decision u satisfying the requirement (3.15) exists iff

1

min

2

maxc

yc

y≥ .

The considerations may by extended for a plant with external disturbances, de-scribed by a relation ZYUzyuR ××⊂),,( where Zz ∈ is a vector of the distur-

bances which may be observed. The property zDz ∈ for the given ZDz ⊂ may

be considered as a result of the observations. Our plant has two inputs ),( zu and

analysis problem is formulated in the same way as for the relation ),( yuR , with

zu DDzu ×∈),( in the place of uDu ∈ . The result analogous to (3.5) is

),,(),,(: zyuRzyuYyDzu DzDu

y ∈∈=∈∈

.

Decision problem: For the given ),,( zyuR , yD (the requirement) and zD (the

result of observations), find the largest set uD such that the implication

yzu DyDzDu ∈→∈∧∈ )()(

is satisfied. The general form of the solution is as follows

37


]),([: yyDz

u DzuDUuD

z

⊆∈=∈

(3.16)

where

),,(),,(:),( zyuRzyuYyzuDy ∈∈= . (3.17)

It is then the set of all such decisions u that for every zDz ∈ the set of possible

outputs y belongs to yD . For the fixed z (the result of measurement) the set uD

is determined by (3.16) with the relation

),,( zyuR

= YUzyuR ×⊂);,( .

In this notation z is the parameter in the relation );,( zyuR . Then

),(:)( yyu DzuDUuzD ⊆∈=

= ),( uzR (3.18)

where ),( zuDy is defined by (3.17). The formula (3.18) defines a relation be-

tween z and u denoted by ),( uzR . The relation ),( uzR may be called a knowl-

edge representation for the decision making (the description of the knowledge ofthe decision making) or a relational decision algorithm. The block scheme of theopen-loop decision system (Fig. 3.4) is analogous to that in Fig. 2.1 for a func-tional plant. The knowledge of the decision making

>< ),( uzR

= KD

has been obtained for the given knowledge of the plant

>< ),,( zyuR

= KP

and the given requirement yDy ∈ .

),( uzR ),,( zyuR

z

yuz

Fig. 3.4. Open-loop decision system

38

3.3 Determinization 39

3.3 Determinization

The deterministic decision algorithm based on the knowledge KD may be ob-tained as a result of determinization of the relational decision algorithm ),( uzR

by using the mean value

1

)()(

][)(~ −∫∫ ⋅=zDzD uu

duduuzu

= )(~

zΨ .

For the given desirable value y we can consider two cases analogous to the con-

cepts described in Sect. 2.3 and illustrated in Fig. 2.2 and Fig. 2.3. In the first casethe deterministic decision algorithm )(zΨ is obtained via the determinization of

the knowledge of the plant KP and in the second case the deterministic decisionalgorithm )(zdΨ is based on the determinization of the knowledge of the deci-

sion making KD obtained from KP for the given y . In the first case we determine

the mean value

1

),(),(

][)(~ −∫∫ ⋅=zuDzuD yy

dydyyzy

= ),( zuΦ (3.19)

where ),( zuDy is described by formula (3.17). Then, solving the equation

yzu ˆ),( =Φ (3.20)

with respect to u, we obtain the deterministic decision algorithm )(zu Ψ= , under

the assumption that the equation (3.20) has a unique solution.In the second case we use

),ˆ,( zyuR

= ),( uzRd , (3.21)

i.e. the set of all pairs ),( zu for which it is possible that yy ˆ= . The relation

UZuzRd ×⊂),( may be considered as the knowledge of the decision making

KD, i.e. the relational decision algorithm obtained for the given KP and the valuey . The determinization of the relational decision algorithm dR gives the deter-

ministic decision algorithm:

1

)()(

][)( −∫∫ ⋅=zDzD

d

udud

duduuzu

= )(zdΨ (3.22)

39


where

),(),(:)( uzRzuUuzD dud ∈∈= .

The equations (3.19), (3.20), (3.22) are analogous to the equations (2.6), (2.9),(2.15) presented in Sect. 2.3. Two cases of the determination of the deterministicdecision algorithm are illustrated in Figs. 3.5 and 3.6, analogous to Figs. 2.2 and2.3. The results of these two approaches may be different, i.e. in general

)()( zz dΨΨ ≠ (see Example 3.3).

Ψu

y

),,( zyuR

Φ

z

z y

Fig. 3.5. Decision system with determinization – the first case

du

y

),,( zyuR

dΨ

),( uzR d

z

z y

Fig. 3.6. Decision system with determinization – the second case

40

3.4 Analysis for Relational Plants with Uncertain Parameters 41

Example 3.3: Consider the plant with u, z, y 1R∈ (one-dimensional variables),described by the inequality

zcuyzcu +≤≤+ 2 , 0>c . (3.23)

For ],[ maxmin yyDy = and the given z the set (3.18) is determined by the ine-

quality

czy

uc

zy2

maxmin −≤≤

−.

The determinization of the knowledge KP according to (3.19) gives

),(23~ zuzcuy Φ=+= .

From the equation yzu ˆ),( =Φ we obtain the decision algorithm

zcy

zu3

)ˆ(2)(

−==Ψ .

Substituting y into (3.23) we obtain the relational decision algorithm ),( uzRd in

the form

czy

uc

zy −≤≤− ˆ2

ˆ

and after the determinization

)(4

)ˆ(3)( z

czy

zu dd ΨΨ ≠−== .

3.4 Analysis for Relational Plants with Uncertain Parameters [26, 35]

Let us consider the plant described by a relation YUxyuR ×⊆);,( where

Xx ∈ is an unknown vector parameter which is assumed to be a value of an un-certain variable x with the certainty distribution )(xhx given by an expert. Now

the sets of all possible values y in (3.5) and (3.6) depend on x. For the given setof inputs uD we have

);,(),(:)( xyuRyuYyxDuDu

y ∈∈=∈

and for the given value u

41


);,(),(:);( xyuRyuYyxuDy ∈∈= .

The analysis may consist in evaluating the input with respect to a set YDy ⊂given by a user. We can consider two formulations with the different practical in-terpretation: the determination of )](~[ xDDv yy ⊆ (version I) or the determina-

tion of ])([ yy DxDv ⊆ (version II). The analogous formulations may be consid-

ered for the given u, with );( xuDy in the place of )(xDy .

Analysis problem – version I: For the given );,( xyuR , )(xhx , u and YDy ⊂one should determine

)];(~[ xuDDv yy ⊆

= ),( uDg y . (3.24)

The value (3.24) denotes the certainty index of the soft property: “the set of allpossible outputs approximately contains the set yD given by a user” or “the ap-

proximate value of x is such that );( xuDD yy ⊆ ” or “the approximate set of

the possible outputs contains all the values from the set yD ”. Let us note that

)],(~[)];(~[ uDDxvxuDDv yxyy ∈=⊆ (3.25)

where

);(:),( xuDDXxuDD yyyx ⊆∈= . (3.26)

Then

)(max),(),(

xhuDg xuDDx

yyx∈

= . (3.27)

In particular, for yDy = (a singleton), the certainty index that the given value

y may occur at the output of the plant is

)(max),(),(

xhuyg xuyDx x∈

= . (3.28)

where

);(:),( xuDyXxuyD yx ∈∈= . (3.29)

When x is considered as C-uncertain variable, it is necessary to determine

)(max)],(~[),(

xhuDDxv xuDDx

yxyx∈

=∈

42


where ),(),( uDDXuDD yxyx −= . Then, according to (1.59)

)],(~[1)],(~[21)];(~[ uDDxvuDDxvxuDDv yxyxyyc ∈−+∈=⊆ .

The considerations may be extended for a plant described by a relation);,,( xzyuR where Zz ∈ is the vector of disturbances which may be measured.

For the given z

);,,(),,(:);,( xzyuRzyuYyxzuDy ∈∈=

and

)(max)];,(~[),,(

xhxzuDDv xzuDDx

yyyx∈

=⊆

where

);,(:),,( xzuDDXxzuDD yyyx ⊆∈= .

Consequently, the certainty index that the approximate set of the possible outputscontains all the values from the set yD depends on z.

For the given set uD , the formulas analogous to (3.24) – (3.29) have the fol-

lowing form:

)](~[ xDDv yy ⊆

= ),( uy DDg ,

)],(~[)](~[ uyxyy DDDxvxDDv ∈=⊆ ,

)(:),( xDDXxDDD yyuyx ⊆∈= ,

)(max),(),(

xhDDg xDDDx

uyuyx∈

= ,

)(max),(),(

xhDyg xDyDx

uux∈

= ,

)(:),( xDyXxDyD yux ∈∈= .

Analysis problem – version II: For the given );,( xyuR , )(xhx , u and YDy ⊂one should determine

]~);([ yy DxuDv ⊆

= ),( uDg y . (3.30)

The value (3.30) denotes the certainty index of the soft property: “the set yD

43


given by a user contains the approximate set of all possible outputs”. The formulascorresponding to (3.25), (3.26) and (3.27) are as follows:

)],([]~);([ uDDxvDxuDv yxyy ∈=⊆

where

);(:),( yyyx DxuDXxuDD ⊆∈= , (3.31)

)(max),(),(

xhuDg xuDDx

yyx∈

= . (3.32)

For the given set uD one should determine

)(max)],(~[]~)([),(

xhDDDxvDxDv xDDDx

uyxyyuyx∈

=∈=⊆ (3.33)

where

)(:),( yyuyx DxDXxDDD ⊆∈= . (3.34)

In the case where x is considered as C-uncertain variable it is necessary to find v(3.33) and

)(max)],(~[),(

xhDDDxv xDDDx

uyxuyx∈

=∈ (3.35)

where ),(),( uyxuyx DDDXDDD −= . Then, according to (1.59)

)],(~[1)],(~[21]~)([ uyxuyxyyc DDDxvDDDxvDxDv ∈−+∈=⊆ . (3.36)

The considerations for the plant described by );,,( xzyuR are analogous to those

in version I.

Example 3.4: Let 1R,, ∈xyu , the relation R is given by inequality

xuyxu 2≤≤ , ],[ 21 uuDu = , 01 >u , ],[ 21 yyDy = , 01 >y . For these data

( ) ]2,[ 21 xuxuxDy = and (3.34) becomes ]2

,[),(2

2

1

1uy

uy

DDD uyx = . Assume

that x is a value of an uncertain variable x with triangular certainty distribu-

tion: xhx 2= for 21

0 ≤≤ x , 22 +−= xhx for 121 ≤≤ x , 0=xh otherwise

(Fig. 3.7). Using (3.33) we obtain for 1221 2 yuyu ≥

44


≥

≤≥−

≥≤

≤

=

∈=⊆

.when0

and2when)1(22

and2when1

when

)],(~[]~)([

11

11111

1

2211

222

2

uy

uyuyuy

uyuy

uyuy

DDDxvDxDv uyxyy

)(xhx

1

21 x

uy2

uy2

2


For 1221 2 yuyu < ∅=),( uyx DDD and 0]~)([ =⊆ yy DxDv . For example,

for 51 =u , 62 =u , 41 =y , 122 =y we have 112 uy ≥ , 11 uy ≤ and

[ ] 4.0)12,4~)(( =⊆xDv y . When 21 =y we have 112 uy ≤ , 22 uy ≥ and

1=v . To apply the description for C-uncertain variable one should determine

)],([ uyx DDDxv ∈ according to (3.35):

.2

and2

when

when

when

1

)2,2

(max

1

)],([

11

2211

22

2

2

1

1

uy

uyuy

uy

uy

uy

DDDxv uyx

≥

≥≤

≤

+−

=∈

Then, using (3.36) we obtain ]~)([ yyc DxDv ⊆ . For the numerical data in the first

case ( 41 =y ) 1)],([ =∈ uyx DDDxv , 2.0=cv . In the second case ( 21 =y )

8.0)],([ =∈ uyx DDDxv , 6.0=cv .

45


3.5 Decision Making for Relational Plants with Uncertain Parameters [35, 36]

We can formulate the different versions of the decision problem with differentpractical interpretations – corresponding to the formulations of the analysis prob-lem presented in Sect. 3.4.Decision problem – version I: For the given );,( xyuR , )(xhx and YDy ⊂find

)];(~[maxargˆ xuDDvu yyUu

⊆=∈

. (3.37)

In this formulation u is a decision maximizing the certainty index that the ap-proximate set of the possible outputs contains the set yD given by a user. To ob-

tain the optimal decision one should determine the function g in (3.27) and tomaximize it with respect to u, i.e.

)(maxmaxargˆ),(

xhu xuDDxUu yx∈∈

= (3.38)

where ),( uDD yx is defined by (3.26).

Decision problem – version II: For the given );,( xyuR , )(xhx and YDy ⊂find

]~);([maxargˆ yyUu

DxuDvu ⊆=∈

. (3.39)

Now u is a decision maximizing the certainty index that the approximate set ofall possible outputs (i.e. the set of all possible outputs for the approximate value ofc ) belongs to the set yD given by a user. To obtain the optimal decision one

should determine the function g in (3.32) and to maximize it with respect to u,i.e. u is determined by (3.38) where ),( uDD yx is defined by (3.31). It is worth

noting that in both versions the solution may not be unique, i.e. we may obtain the

set uD of the decisions (3.37). Denote by *x the value maximizing )(xhx , i.e.

such that 1)( * =xhx . Then

),(: * uDDxUuD yxu ∈∈=

and for every uDu ∈ maximal value of the certainty index in (3.37) and (3.39) is

equal to 1. To determine the set of optimal decisions uD it is not necessary to

know the form of the function )(xf x . It is sufficient to know the value *x .

In the case where x is considered as C-uncertain variable one should determine

46

3.5 Decision Making for Relational Plants with Uncertain Parameters 47

)],(~[1)],(~[21

)],(~[ uDDxvuDDxvuDDxv yxyxyxc ∈−+∈=∈ (3.40)

where

)(max)],(~[),(

xhuDDxv xuDDx

yxyx∈

=∈ . (3.41)

Then the optimal decision cu is obtained by maximization of cv :

)],(~[maxˆ uDDxvu yxcUu

c ∈=∈

where ),( uDD yx is defined by (3.26) in version I or by (3.31) in version II.

In the similar way as in Sect. 3.4, the considerations may be extended for the plantwith the vector of external disturbances z, described by );,,( xzyuR . Now the set

)(zDu of the optimal decisions depends on z. In the case of the unique solution

u for every z, we obtain the deterministic decision algorithm )(ˆ zu Ψ= in an

open-loop decision system. It is the decision algorithm based on the knowledge ofthe plant ><= xhR,KP . For the fixed x and z we may solve the decision

problem such as in Sect. 3.2, i.e. determine the largest set );( xzDu such that the

implication

yu DyxzDu ∈→∈ );(

is satisfied. According to (3.18)

);,(:);( yyu DxzuDUuxzD ⊆∈=

= );,( xuzR

where

);,,(),,(:);,( xzyuRzyuYyxzuDy ∈∈= .

Then we can determine the optimal decision

)];(~[maxarg xzDuvu uUu

d ∈=∈

= )(zdΨ (3.42)

where

)],,(~[)];(~[ zuDDxvxzDuv yxdu ∈=∈

and

);(:),,( xzDuXxzuDD uyxd ∈∈= .

47


Hence

)(max)];(~[),,(

xhxzDuv xzuDDx

uyxd∈

=∈ . (3.43)

In general, we may obtain the set udD of decisions du maximizing the certainty

index (3.43). Let us note that the decision algorithm )(zdΨ is based on the

knowledge of the decision making ><= xhR ,KD . The relation R or the set

);( xzDu may be called an uncertain decision algorithm in the case under consid-

eration. It is easy to see that in this case uud ˆ= for every z, i.e. )()( zzd ΨΨ =where )(ˆ zu Ψ= is the optimal decision in version II. This follows from the fact

that

yyu DxzuDxzDu ⊆↔∈ );,();( ,

i.e. the properties );( xzDu u∈ and yy DxzuD ⊆);,( are equivalent. The opti-

mal decision in version II duu =ˆ is then the decision which with the greatest cer-

tainty index belongs to the set of decisions );( xzDu for which the requirement

yDy ∈ is satisfied. The determination of duu =ˆ from (3.42) and (3.43) may be

easier than from (3.39) with );,( xzuDy in the place of );( xuDy . In the case

without z the optimal decision (3.39) may be obtained in the following way:

)](~[maxargˆ xDuvu uu

∈=

where

)(max)],(~[)](~[),(

xhuDDxvxDuv xuDDx

yxduyxd∈

=∈=∈ , (3.44)

and

∈∈=

⊆∈=

∈∈=

).;,(),(:);(

,);(:)(

),(:),(

xyuRyuYyxuD

DxuDUuxD

xDuXxuDD

y

yyu

uyxd

(3.45)

Example 3.5 (decision problem – version II): Let u, y, x 1R∈ and ( )xyuR ,,

be given by the inequality

1 3 22 ++≤≤− xuyux .

48

3.5 Decision Making for Relational Plants with Uncertain Parameters 49

For [ ]2 ,0=yD the set )(xDu (3.45) is determined by

3 xu ≤ and 1 22 ≤+ xu . (3.46)

Assume that x is a value of an uncertain variable x with triangular certainty dis-

tribution: xhx 2= for 21

0 ≤≤ x , 22 +−= xhx for 121 ≤≤ x , 0=xh other-

wise. From (3.46) we have ]1,3

[)( 2uuuDx −= and the set of all possible

u : ]103,1[

−=u (the value 10

3 is obtained from the equations xu 3= ,

122 =xu + ). It is easy to see that ( )uDx∈21

iff 21

1 2 ≥− u . Then, according

to (3.44)

)](~[ xDuv u∈=

−

≤≤−=

.

in otherwise 12 23

23for 1

)(2

uu

uuv (3.47)

For example ( ) 15.0 =v , ( ) 88.09.0 ≈v . As the decision u we can choose any

value from ]23,

23[− and the property yy DxuD ⊆~);( is satisfied with

certainty index equal to 1. To apply the description for C-uncertain variable it isnecessary to determine )],(~[ uDDxv yx∈ . Using (3.41) let us note that for

23<u 122,

32max)],(~[ 2uuuDDxv yx −−=∈ . Then

≤≤−−−=∈

.

in otherwise 1 23

23for 122 ,

32 max )],(~[

2

u

yxuuu

uDDxv

(3.48)

Substituting (3.47) and (3.48) into (3.40) we obtain ( )uvc . For example

( )65

5.0 =cv , ( ) 44.09.0 ≈cv . It is easy to note that in this case 0ˆ =cu and

1 )ˆ( =cc uv .

Example 3.6 (decision problem – version II): R and )(xhx are the same as in

Example 3.4, ],[ 21 yyDy = , 01 >y , 12 2 yy > . Then ]2

,[)( 21x

yxy

xDu = ,

]2

,[)( 21u

yuy

uDx = and )(uv in (3.44) is the same as ]~)([ yy DxDv ⊆ in Exam-

49


ple 3.4, with uuu == 21 . Thus, u is any value from ],2[ 21 yy and 1)ˆ( =uv . In

the case of C-uncertain variable )],(~[ uDDxv yx∈ is the same as in Example 3.4,

with uuu == 21 . Using (3.40) we obtain

( )

.

5.0

5.0

when

when

when

0

1

2

1

211

21

1

2

yu

yyuy

yyu

uyu

y

uvc

≤

+≤≤

+≥

−=

It is easy to see that 21 5.0ˆ yyuc += and ( )21

22

ˆyy

yuv cc += . For example, for

21 =y , 122 =y we obtain ]12,4[ˆ ∈u and 1=v , 8ˆ =cu and 75.0=cv . The

function )(uvc is illustrated in Fig. 3.8.

8ˆ =cu

Fig. 3.8. Example of the relationship between vc and u

3.6 Computational Aspects

The application of C-uncertain variables with the certainty index cv instead of v

means better using the expert’s knowledge, but may be connected with muchgreater computational difficulties. In the discrete case, when the number of possi-

50

3.6 Computational Aspects 51

ble values x is small, it may be acceptable to determine all possible values of cv

and then to choose the value cu for which cv is the greatest. Let us explain it for

the decision problem in version II. Assume that X and U are finite discrete sets:

,...,, 21 mxxxX = , ,...,, 21 puuuU = .

Now the relation );,( xyuR is reduced to the family of sets

YxuD jiy ⊂);( , pi ,1∈ , mj ,1∈ ,

i.e. the sets of possible outputs for all the pairs ),( ji xu .

The algorithm for the determination of u is as follows:1. For iu )...,,2,1( pi = prove if

yjiy DxuD ⊆);( , mj ...,,2,1= (3.49)

2. Determine

)(max),(

xhv xuDDxiiyx∈

=

where ),( iyx uDD is the set of all jx satisfying the property (3.49)

3. Choose iuu =ˆ for *ii = where *i is an index for which iv is the greatest.

The algorithm for the determination of cu is then the following:

1) For iu )...,,2,1( pi = prove if

yjiy DxuD ⊆);( , mj ...,,2,1= .

If yes then ),( iyxj uDDx ∈ . In this way, for mj = we obtain the set

),( iyx uDD as a set of all jx satisfying the property (3.49).

2) Determine civ according to (1.66) and (3.40):

−

∈=

∈

∈

otherwise)(max211

),( if )(max21

),(

*

),(

xh

uDDxxh

v

xuDDx

iyxxuDDx

ci

iyx

iyx

where Xx ∈* is such that 1)( * =xhx and ),(),( iyxiyx uDDXuDD −= .

3) Choose *ii = such that civ is the maximum value in the set of civ determined

in the former steps. Then ic uu =ˆ for *ii = .

Let us consider the relational plant with one-dimensional output, described by the

51


inequality

);();( 21 duyeu ΦΦ ≤≤

where 11 R: →UΦ , 1

2 R: →UΦ , e and d are the subvectors of the parameter

vector ),( dex = ,

...,,, 21 seeeEe =∈ , ...,,, 21 ldddDd =∈ .

Now lsm ⋅= where m is a number of the pairs ),( δγ de ; s,1∈γ , l,1∈δ . If

],[ maxmin yyDy = then the set ),;( δγ deuD iy is described by the inequalities

min1 );( yeui ≥γΦ and max2 );( ydui ≤δΦ .

Assume that e and d are independent uncertain variables. Then, according to(1.76)

)(),(min),()( dhehdehxh dex == .

Let νee =* and µdd =* , i.e. 1)( =νehe and 1)( =µdhd .

The algorithm for the determination of the optimal decision cu in this case is

as follows:1) For iu prove if

min1 );( yeui ≥νΦ and max2 );( ydui ≤µΦ .

If yes, go to 2). If not got to 4).2) Prove if

min1 );( yeui ≤γΦ or max2 );( ydui ≥δΦ (3.50)

for

s...,,1,1...,,2,1 +−= ννγ ,

l...,,1,1...,,2,1 +−= µµδ .

3) Determine

)(),(minmax21

1),(

δγ dhehv deDde

cix∈

−=

where xD is the set of all pairs ),( δγ de satisfying the property (3.50).

4) Prove if

52


min1 );( yeui ≥γΦ and max2 );( ydui ≤δΦ (3.51)

for

s...,,1,1...,,2,1 +−= ννγ ,

l...,,1,1...,,2,1 +−= µµδ .

5) Determine

)(),(minmax21

),(δγ dhehv de

Ddcci

x∈=

where xD is the set of all pairs ),( δγ de satisfying the property (3.51).

6) Execute the points 1 – 4 for pi ...,,2,1= .

7) Choose

cipi

vi,1

* maxarg∈

= .

The result (the optimal decision) is: iuu =ˆ for *ii = .

The algorithm is illustrated in Fig. 3.9. For the great size of the problem (the great

value p) the method of integer programming may be used to determine *i .

Example 3.7: One-dimensional plant is described by the inequality

xuyxu 2≤≤ ,

3,2,1∈u , 6,5,4,3∈x

and the corresponding values of )(xhx are )4.0,1,6.0,5.0( . The requirement is

]10,5[=∈ yDy . Then ),( uDD yx is determined by

ux

u105 ≤≤ .

For 1=u

1)6(),5(max]10,5[~1 ==∈= xx hhxvv .

For 2=u

1)5(),4(),3(max]5,5.2[~2 ==∈= xxx hhhxvv .

For 3=u

53


min1 );( yeui ≥νΦ

max2 );( ydui ≤µΦ

" # % & '

( ) * )

" # % & '

, ) - )

/ & 1 1 3civ

pi <

4 1 &

cipi

vi,1

* maxarg∈

=

6 /

/ & % & 1 7 & ' &

*ˆ iiuu i == : ; <

= 1 3

1+→ ii

seui ,1),;(1 ∈γΦ γ

ldui ,1),;(2 ∈δΦ δ

pi ,1∈

sehe ,1),( ∈γγ

ldhd ,1),( ∈δδ

maxmin , yy

? @ A CD E G H I C K L C

N ' O

O N '

u

Fig. 3.9. Block scheme of decision algorithm

54


5.0)3(]3

10,35[~3 ==∈= xhxvv .

Then 2or 1ˆ =u , and 1)ˆ( =uv .

Now let us assume that x is C-uncertain variable.For 1=u

7.0)]4(),3(max11[21]10,5[~1 =−+=∈= xxcc hhxvv .

For 2=u

8.0)]6(11[21]5,5.2[~2 =−+=∈= xcc hxvv .

For 3=u

25.0)]6(),5(),4(max15.0[21]

310,

35[~3 =−+=∈= xxxcc hhhxvv .

Then 2ˆ =cu and 8.0)ˆ( =cc uv , i.e. for 2=u the certainty index that the set of

possible outputs belongs to the set ]10,5[ is equal to 0.8.

55

4 Systems with Logical Knowledge Representation

4.1 Logical Knowledge Representation

Now we shall consider the knowledge representation in which the relations iR

(3.3) have the form of logic formulas concerning u, y, w. Let us introduce thefollowing notation:1. )(uujα – simple formula (i.e. simple property) concerning u, 1...,,2,1 nj = ,

e.g. "2")( T1 ≤= uuuuα .

2. ),,( ywuwrα – simple formula concerning u, w and y, 2...,,2,1 nr = .

3. )( yysα – simple formula concerning y, 3...,,2,1 ns = .

4. ),...,,(121 unuuu αααα = – subsequence of simple formulas concerning u.

5. ),...,,(221 wnwww αααα = – subsequence of simple formulas concerning u, w

and y.6. ),...,,(

321 ynyyy αααα = – subsequence of simple formulas concerning y.

7. ),,( ywuα

= ),,(),...,,( 21 ywun αααααα = – sequence of all simple formu-

las in the knowledge representation, 321 nnnn ++= .

8. )(αiF – the i-th fact given by an expert. It is a logic formula composed of the

subsequence of α and the logic operations: ∨ – or, ∧ – and, ¬ – not, → – if ...then, ki ...,,2,1= .

For example 4211 ααα →∧=F , 232 αα ∨=F where "2" T1 ≤= uuα ,

"3or small is re temperatuthe" T2 ≤= yyα , "" TT

3 wwyy >=α ,

"4" T4 == yyα .

9. )(...)()()( 21 αααα kFFFF ∧∧∧= .

10. )( uuF α – input property, i.e. the logic formula using uα .

11. )( yyF α – output property.

57


58 4 Systems with Logical Knowledge Representation

12. 1,0∈ma – logic value of the simple property mα , nm ...,,2,1= .

13. ),...,,( 21 naaaa = – zero-one sequence of the logic values.

14. )(uau , ),,( ywuaw , )( ya y – zero-one subsequences of the logic values cor-

responding to )(uuα , ),,( ywuwα , )( yyα .

15. )(aF – the logic value of )(αF .

All facts given by an expert are assumed to be true, i.e. 1)( =aF .

The description

>< )(, αα F

= KP

may be called a logical knowledge representation of the plant. For illustrationpurposes let us consider a very simple example:

),( )2()1( uuu = , ),( )2()1( yyy = , 1R∈w ,

"0" )2()1(1 >+= uuuα , "2" )2(

2 >= uuα , "" )1()2(1 yyy <=α ,

"4" )2()1(2 =+= yyyα , "02" )2()1(

1 <+−= ywuwα ,

"" )1()2(2 yuw >=α ,

21111 wywuF αααα ¬∨→∧= , )()( 12222 uywuF αααα ¬∧∨∧= ,

21 uuuF αα ∨= , 2yyF α¬= .

The expressions )(aF have the same form as the formulas )(αF , e.g.

211112111 ),,,( wywuywwu aaaaaaaaF ¬∨→∧= .

The logic formulas )(αjF , )( uuF α and )( yyF α are special forms of the rela-

tions introduced in Sects. 3.1 and 3.2. Now the relation (3.3) has the form

1)],,([:),,(),,( =××∈= ywuaFYWUywuywuR ii , ki ,1∈ . (4.1)

The input and output properties may be expressed as follows:

uDu ∈ , yDy ∈

where1)]([: =∈= uaFUuD uuu , (4.2)

1)]([: =∈= yaFYyD yyy . (4.3)

The description with )(aF , )( uu aF , )( yy aF may be called the description on

the logical level. The expressions )(aF , )( uu aF and )( yy aF describe logical

structures of the plant, the input property and the output property, respectively.The description on the logical level is independent of the particular meaning of the

58

4.2 Analysis and Decision Making Problems 59

simple formulas. In other words, it is common for the different plants with differ-ent practical descriptions but the same logical structures. On the logical level ourplant may be considered as a relational plant with the input ua (a vector with 1n

zero-one components) and the output ya (a vector with 3n zero-one compo-

nents), described by the relation

1),,( =ywu aaaF (4.4)

(Fig. 4.1). The input and output properties for this plant corresponding to theproperties uDu ∈ and yDy ∈ for the plant with input u and output y are as

follows

uuu SSa ⊂∈ , yyy SSa ⊂∈

where uS , yS are the sets of all zero-one sequences ua , ya , respectively, and

1)(: =∈= uuuuu aFSaS , 1)(: =∈= yyyyy aFSaS . (4.5)

ua ya

1),,( =ywu aaaF

Fig. 4.1. Plant on logical level

4.2 Analysis and Decision Making Problems

The analysis and decision making problems for the relational plant described bythe logical knowledge representation are analogous to those for the relational plantin Sect. 3.2. The analysis problem consists in finding the output property for thegiven input property and the decision problem is an inverse problem consisting infinding the input property (the decision) for the required output property.Analysis problem: For the given )(αF and )( uuF α find the best property

)( yyF α such that the implication

)()( yyuu FF αα → (4.6)

is satisfied.If it is satisfied for 1yF and 2yF , and 21 yy FF → , then 1yF is better than

2yF . The property yF is then the best if it implies any other property for which

59


the implication (4.6) is satisfied. The best property yF corresponds to the smallest

set yD in the formulation presented in Sect. 3.2.

Decision problem: For the given )(αF and )( yyF α (the property required by a

user) find the best property )( uuF α such that the implication (4.6) is satisfied.

If it is satisfied for 1uF and 2uF , and 12 uu FF → , then 1uF is better than 2uF .

The property uF is then the best if it is implied by any other property for which

the implication (4.6) is satisfied. The best property uF corresponds to the largest

set uD in the formulation presented in Sect. 3.2.

Remark 4.1: The solution of our problem may not exist. In the case of theanalysis it means that there is a contradiction between the property )( uuF α and

the facts ),,( ywuF ααα , i.e. the sequence ua such that

1),,()( =∧ ywuuu aaaFaF does not exist. In the case of the decision making it

means that the requirement yF is too strong. The existence of the solution will be

explained in the next section.

Remark 4.2: Our problems are formulated and will be solved on the logiclevel. Consequently they depend on the logical structures (the form of F and yF

or uF ) but do not depend on the meaning of the simple formulas. The knowledge

representation KP and the problem formulations may be extended for differentvariables, objects and sets (not particularly the sets of real number vectors) used inthe description of the knowledge. For example, in the example in the previoussection we may have the following simple formulas in the text given by an expert:

=1uα ”operation 1O is executed after operation 2O ”,

=2uα ”temperature is small”,

=1wα ”pressure is high”,

=2wα ”humidity is small”,

=1yα ”state S occurs”,

=2yα ”quality of product is sufficient”.

Then the facts 1F and 2F in this example mean:

=1F ”If operation 1O is executed after operation 2O and pressure is high then

state S occurs or humidity is not small”,=2F ”Temperature is small and humidity is small or quality is sufficient and op-

eration 1O is not executed after operation 2O ”.

Remark 4.3: The possibilities of forming the input and output properties arerestricted. Now the sets uD and yD may be determined by the logic formulas

60

4.3 Logic-Algebraic Method 61

)( uuF α and )( yyF α using the simple formulas uα and yα from the sequence

of the simple formulas α used in the knowledge representation.

4.3 Logic-Algebraic Method

The solutions of the analysis and decision problems formulated in Sect. 4.2 maybe obtained by using so called logic-algebraic method [9, 13, 14, 19]. It is easy toshow that the analysis problem is reduced to solving the following algebraic equa-tion

1),,(~ =ywu aaaF (4.7)

with respect to ya , where

),,()(),,(~

ywuuuywu aaaFaFaaaF ∧= .

Now ),,( ywu aaaF , )( uu aF and )( yy aF are algebraic expressions in two-

value logic algebra. If yS is the set of all solutions then yF is determined by yS ,

i.e. 1)( =↔∈ yyyy aFSa . For example, if ),,( 321 yyyy aaaa = and

)0,1,0(),0,1,1(=yS then ∨¬∧∧= )()( 321 yyyyyF αααα

)( 321 yyy ααα ¬∧∧¬ .

In the decision making problem two sets of the algebraic equations should besolved with respect to ua :

=

=

1)(

1),,(

yy

ywu

aF

aaaF,

=

=

0)(

1),,(

yy

ywu

aF

aaaF(4.8)

If 1uS , 2uS are the sets of the solutions of the first and the second equation, re-

spectively – then )( uuF α is determined by 21 uuu SSS −= [13] in the same

way as yF by yS in the former problem.

The generation of the set yS requires the testing of all sequences

),,( ywu aaaa = and the execution time may be very long for the large size of

the problem. The similar computational difficulties may be connected with thesolution of the decision problem. The generation of yS (and consequently, the

solution yF ) may be much easier when the following decomposition is applied:

61


),(...),(),( 1212101 NNNu aaFaaFaaFFF −∧∧∧=∧ (4.9)

where yaa =0 , 1F is the conjunction of all facts from F~

containing the vari-

ables from 0a , 1a is the sequence of all other variables in 1F , 2F is the conjunc-

tion of all facts containing the variables from 1a , 2a is the sequence of all other

variables in 2F etc. As a result of the decomposition the following recursive pro-

cedure may be applied to obtain ySS =0 :

]1),([: 1111 =∈= −∈

−−− mmmSa

mmm aaFSaSmm

, (4.10)

where mS is the set of all ma , 1...,,1, −= NNm , NN SS = .

The recursive procedure (4.10) has two interesting interpretations:A. System analysis interpretation.

Let us consider the cascade of relation elements (Fig. 4.2) with input ma , out-

put 1−ma (zero-one sequences), described by the relations 1),( 1 =− mmm aaF

( 1...,,1, −= NNm ). Then 1−mS is the set of all possible outputs from the ele-

ment mF and 0S is the set of all possible outputs from the whole cascade.

1−NaNF 1−NF 1F

Na 2−Na 1a 0a

Fig. 4.2. Relational system

B. Deductive reasoning interpretation.The set 1−mS may be considered as the set of all elementary conclusions from

mN FF ∧∧ ... , and 0S is the set of all elementary conclusions from the facts

FFu ∧ .

A similar approach may be applied to the decision problem. To determine 1yS

and 2yS we may use the recursive procedure (4.10) with F in (4.9) instead of

FFu ∧ and with ),(0 yu aaa = . After the generation of 0S from (4.10) one

can determine 1uS and 2uS in the following way:

]),[(: 01 SaaaS yuSa

uuyy

∈=∈

,

62

4.3 Logic-Algebraic Method 63

]),[(: 0ˆ

2 SaaaS yuSSa

uuyyy

∈=−∈

where 1)(: == yyyy aFaS and yS is the set of all ya .

The different versions of the presented procedures have been elaborated andapplied in the general purpose expert systems CONTROL-LOG and CLASS-LOG, specially oriented for the applications to a class of knowledge-based controlsystems and to classification problems.

The main idea of the logic-algebraic method presented here for the generationof the solutions consists in replacing the individual reasoning concepts based oninference rules by unified algebraic procedures based on the rules in two-valuelogic algebra. The results may be considered as a unification and generalization ofthe different particular reasoning algorithms (see e.g. [3] ) for a class of the sys-tems with the logical knowledge representation for which the logic-algebraicmethod has been developed. The logic-algebraic method can be applied to the de-sign of complex knowledge-based computer systems [43, 45, 46, 48].

Example 4.1 (analysis): The facts F~

are the following:

4131 )( ααα →¬∨=F , 3712 )( ααα ¬∨∧¬=F , 2193 )( ααα →∧=F ,

5744 )( ααα ∨¬∧=F , )( 8465 ααα ∧→=F , )( 6426 ααα ∧¬→=F ,

10237 )( ααα ∨∧=F , ),( 109 ααα =y .

It is not important which simple formulas from 1α ÷ 8α are uα and which fact

from the set 1F , 2F , 4F , 5F , 6F (not containing yα ) is the input property. It is

easy to see that),,(),,(),( 103279213101 aaaFaaaFaaF ∧= , ),,( 3211 aaaa = ,

),,(),,(),,(),( 642673124311212 aaaFaaaFaaaFaaF ∧∧= ,

),,( 7642 aaaa = ,

),,(),,(),( 86457544323 aaaFaaaFaaF ∧= , ),( 853 aaa = .

In our case 3=N , )0,0(),1,0(),0,1(),1,1(=NS . According to (4.10) one

should put successively the elements of NS into 3F and determine all 0-1 se-

quences ),,( 764 aaa such that 13 =F . These are the elements of 2S . In a similar

way one determines 1S and finally )1,1(),1,0(0 =S . Then

10109109 )()( ααααα =∧∨∧¬=yF .

Example 4.2 (decision making): The facts F in the knowledge representationKP are the following:

63


)( 6411 ααα ¬∨∧=F , 6422 )( ααα →∧=F , 5343 ααα ∨¬∨¬=F ,

)( 5344 ααα ¬∨∧=F , 7245 )( ααα →¬∧=F , ),( 21 ααα =u ,

),( 76 ααα =y .

Now ),,,(),( 76210 aaaaaaa yu == , 5211 FFFF ∧∧= , 432 FFF ∧= ,

41 aa = , ),( 532 aaa = .

Using (4.10) (two steps for 1,2=m ) we obtain ),0,1,1,1(),1,1,1,1(0 =S

)1,0,0,1(),1,1,0,1( . We can consider the different cases of ),( 76 ααyF . It is

easy to see that for 76 αα ∨=yF we have )1,0(),0,1(),1,1(=yS ,

)0,1(),1,1(1 =uS , 2uS is an empty set, 1uu SS = and

12121 )()( ααααα =¬∧∨∧=uF . If 6α=yF then 21 αα ∧=uF , if

7α=yF then 21 αα ¬∧=uF , if 76 αα ∧=yF then 21 uu SS = , uS is an

empty set and the solution uF does not exist.

The formulas α and the facts may have a different practical sense. For exam-

ple, in the second example 1,, Rcyu ∈ and: "3"1 cu ≤=α ,

"1" 222 ≤+= cuα , "3 =α pressure is high " , "4 =α humidity is small " ,

"5 =α temperature is less than cyu ++ " , "25.0)5.0(" 226 ≤−+= cyα ,

""7 cyc ≤≤−=α for a given parameter c. For example, the fact 2F means

that: 1if" 22 ≤+ cu and humidity is small then "25.0)5.0( 22 ≤−+ cy , the

fact 3F means that: " humidity is not small or pressure is not high or temperature

is less than "cyu ++ . The required output property 6α=yF is obtained if

21 αα ∧=uF , i.e. if cu 3≤ and 122 ≤+ cu .

4.4 Analysis and Decision Making for the Plant with Uncertain Parameters

Now let us consider the plant described by a logical knowledge representationwith uncertain parameters in the simple formulas and consequently in the proper-ties yu FFF ,, [11, 26]. In general, we may have the simple formulas );( xuuα ,

);,,( xwyuwα and );( xyyα where Xx ∈ is an unknown vector parameter

which is assumed to be a value of an uncertain variable x with the certainty dis-tribution )(xhx given by an expert. For example,

"2" TT1 xxuuu ≤=α , "" TT

1 xxyyw ≤=α , "4" TT1 <+= xxyyyα .

64

4.4 Analysis and Decision Making for the Plant with Uncertain Parameters 65

In particular, only some simple formulas may depend on some components of thevector x.

In the analysis problem the formula )];([ xuF uu α depending on x means that

the observed (given) input property is formulated with the help of the unknownparameter (e.g. we may know that u is less than the temperature of a raw materialx, but we do not know the exact value of x). Solving the analysis problem de-scribed in Sects. 4.2 and 4.3 we obtain )];([ xyF yy α and consequently

1)];([:)( =∈= xyaFYyxD yyy .

Further considerations are the same as in Sect. 3.4 for the given set uD . In ver-

sion II (see (3.33) and (3.34)) we have

)(max])([)(

xhDxDv xDDx

yyyx∈

=⊆

where yD is given by a user and

)(:)( yyyx DxDXxDD ⊆∈= .

In the decision problem the formula )];([ xyF yy α depending on x means that the

user formulates the required output property with the help of the unknown pa-rameter (e.g. he wants to obtain y less than the temperature of a product x). Solv-ing the decision problem described in Sects. 4.2 and 4.3 we obtain )];[( xuFu and

consequently

1)];([:)( =∈= xuaFUuxD uuu . (4.11)

Further considerations are the same as in Sect. 3.5 for version II (see (3.44)). Theoptimal decision, maximizing the certainty index that the requirement

)];([ xyF yy α is satisfied, may be obtained in the following way:

)(maxmaxargˆ)(

xhu xuDxu xd∈

=

where

)(:)( xDuXxuD uxd ∈∈= .

Example 4.3: The facts are the same as in example 4.2 where c

= x. In the ex-

ample 4.2 for the required output property 6α=yF the following result has been

obtained:

65


If

xu 3≤ and 122 ≤+ xu (4.12)

then

25.0)5.0( 22 ≤−+ xy .

The inequalities (4.12) determine the set (4.11) in our case. Assume that x is avalue of an uncertain variable with triangular certainty distribution: xhx 2= for

210 ≤≤ x , 22 +−= xhx for 2

21 ≤≤ x , 0=xh otherwise. Then we can use

the result in Example 3.5. As the decision u we can choose any value from

]23,

23[− and the requirement will be satisfied with the certainty index equal to

1. The result for C-uncertain variable is 0ˆ =cu and 1)ˆ( =cc uv .

4.5 Uncertain Logical Decision Algorithm

Consider the plant with external disturbances Zz ∈ . Then in the logical knowl-edge representation we have the simple formulas );,( xzuuα , );,,,( xzywuwα ,

);,( xzyyα and );( xzzα to form the property )( zzF α concerning z.

The analysis problem analogous to that described in Sects. 4.2 and 4.3 for thefixed x is as follows: For the given ),,,( zywuF αααα , )( zzF α and )( uuF α

find the best property )( yyF α such that the implication

)()()( yyuuzz FFF ααα →∧ (4.13)

is satisfied. In this formulation )( zzF α denotes an observed property concerning

z. The problem solution is the same as in Sect. 4.3 with uz FF ∧ in the place of

uF . As a result one obtains )];,([ xzyF yy α and consequently

1)];,([:);( =∈= xzyaFYyxzD yyy .

Further considerations are the same as in Sect. 3.4.The decision problem analogous to that described in Sects. 4.2 and 4.3 for the

fixed x is as follows: For the given ),,,( zywuF αααα , )( zzF α and )( yyF α

find the best property )( uuF α such that the implication (4.13) is satisfied.

66

4.5 Uncertain Logical Decision Algorithm 67

The problem solution is the same as in Sect. 4.3 with FFz ∧ in the place of F.

As a result one obtains

1)];,([:);( =∈= xzuaFUuxzD yuu . (4.14)

Further considerations are the same as in Sect. 3.5 for version II.To obtain the solution of the decision problem another approach may be ap-

plied. For the given F and yF we may state the problem of finding the best input

property ),( zudF αα such that the implication

)(),( yyzud FF ααα →

is satisfied. The solution may be obtained in the same way as in Sect. 4.3 with),( zu αα and dF in the place of uα and uF , respectively. The formula

),( zudF αα may be called a logical knowledge representation for the decision

making (i.e. the logical form of KD) or a logical uncertain decision algorithm cor-responding to the relation R or the set );( xzDu in Sect. 3.5. For the given

)( zzF α , the input property may be obtained in the following way: Denote by S

the set of all ),( zu aa for which 1=dF and by zS the set of all za for which

1=zF , i.e.

1),(:),( == zudzud aaFaaS .

1)(: == zzzz aFaS .

Then )( uuF α is determined by the set

),(: dzuSa

uuu SaaSaSzz

∈∈=∈

. (4.15)

The formula (4.14) is analogous to the formula (3.16) for the relational plant. Itfollows from the fact that on the logical level our plant may be considered as a re-lational plant with the input ua , the disturbance za and the output ya (see

Fig. 4.1).

67

5 Dynamical Systems

5.1 Relational Knowledge Representation [12]

The relational knowledge representation for the dynamical plant may have theform analogous to that for the static plant presented in Sect. 3.1. The deterministicdynamical plant is described by the equations

==+

)(

),,(1

nn

nnn

sy

usfs

η (5.1)

where n denotes the discrete time and Ssn ∈ , Uun ∈ , Yyn ∈ are the state, the

input and the output vectors, respectively. In the relational dynamical plants thefunctions f and η are replaced by relations

×⊆××⊆+

.),(

,),,( 1

YSysR

SSUssuR

nnII

nnnI (5.2)

The relations IR and IIR form a relational knowledge representation of the dy-

namical plant. For a nonstationary plant the relations IR and IIR depend on n .

The relations IR and IIR may have the form of equalities and/or inequalities

concerning the components of the respective vectors. In particular the relations aredescribed by inequalities

),()(

),,(),(

21

211

nnn

nnnnn

sys

usfssuf

ηη ≤≤≤≤ +

i.e. by a set of inequalities for the respective components of the vectors. The for-mulations of the analysis and decision problems may be similar to those in Sect.3.2. Let us assume that .00 SDs s ⊂∈Analysis problem: For the given relations (5.2), the set 0sD and the given se-

quence of sets UDun ⊂ ...),1,0( =n one should find a sequence of the smallest

sets YDyn ⊂ ...),2,1( =n for which the implication


70 5 Dynamical Systems

ynnnunuu DyDuDuDu ∈→∈∧∧∈∧∈ −− )(...)()( 1,11100

is satisfied.It is an extension of the analysis problem for the deterministic plant (5.1), con-

sisting in finding the sequence ny for the given sequence nu and the initial state

0s , and for the known functions f ,η . For the fixed moment n our plant may be

considered as a connection of two static relational plants (Fig. 5.1). The analysisproblem is then reduced to the analysis for the relational plants IR and IIR , de-

scribed in Sect. 3.2. Consequently, according to the formula (3.5) applying to IR

and IIR , we obtain the following recursive procedure for ...,2,1=n :

1. For the given unD and snD obtained in the former step, determine the set

1, +nsD using ),,( 1+nnnI ssuR :

)].,,(),,[(: 1111, ++∈∈

++ ∈∈= nnnInnnDsDu

nns ssuRssuSsDsnnunn

(5.3)

2. Using 1, +nsD and ),( 11 ++ nnII ysR , determine 1, +nyD :

)].,(),[(: 111111,1,1

++++∈

++ ∈∈=++

nnIInnDs

nny ysRysYyDnsn

(5.4)

For 0=n in the formula (5.3) we use the given set 0sD .

),,( 1+nnnI ssuR ),( 11 ++ nnII ysR1+ns 1+nynu

ns

Fig. 5.1. Dynamical relational plant

Decision problem: For the given relations (5.2), the set 0sD and the sequence of

sets YDyn ⊂ )...,,2,1( Nn = one should determine the sequence unD

)1...,,1,0( −= Nn such that the implication

)(...)()(

)(...)()(

,2211

1,11100

NyNyy

NuNuu

DyDyDy

DuDuDu

∈∧∧∈∧∈→

∈∧∧∈∧∈ −−

is satisfied.

5.1 Relational Knowledge Representation 71

The set ynD is given by a user and the property ynn Dy ∈ )...,,2,1( Nn =denotes the user’s requirement. To obtain the solution one can apply the followingrecursive procedure starting from 0=n :1. For the given 1, +nyD , using ),( 11 ++ nnII ysR determine the largest set 1, +nsD

for which the implication

1,11,1 ++++ ∈→∈ nynnsn DyDs

is satisfied. This is a decision problem for the part of the plant described by

IIR (see Fig. 5.1). According to (3.13) with 1+ns , 1+ny in the place of ),( yu

we obtain

)(: 1,11,11, +++++ ⊆∈= nynnynns DsDSsD (5.3a)

where

).,(),(:)( 1111111, +++++++ ∈∈= nnIInnnnny ysRysYysD

2. For 1, +nsD obtained at the point 1 and snD obtained in the former step, using

),,( 1+nnnI ssuR determine the largest set unD for which the implication

1,1)()( ++ ∈→∈∧∈ nsnsnnunn DsDsDu

is satisfied. This is a decision problem for the part of the plant described by

IR . According to (3.16) with ),,( 1 nnn ssu + in the place of ),,( zyu we obtain

]),([: 1,1, ++∈

⊆∈= nsnnnsDs

nun DsuDUuDsnn

(5.4a)

where

).,,(),,(:),( 1111, ++++ ∈∈= nnnInnnnnnns ssuRssuSssuD

Remark 5.1: In the formulation of the decision problem we did not use thestatement “the largest set unD ”. Now the set of all possible decisions means the

set of all sequences 110 ...,,, −Nuuu for which the requirements are satisfied.

Using the recursive procedure described above we do not obtain the set of all pos-sible decisions. In other words, we determine the set of sequences

110 ...,,, −Nuuu belonging to the set of all input sequences for which the re-

quirements concerning ny are satisfied.

Remark 5.2: The relations IR and IIR may be given by the sets of facts in a

similar way as described in Sect. 4.1. The formulation and solution of the analysisand decision problems for the plant described by dynamical logical knowledge


representation are analogous to those presented above and for the fixed n are re-duced to the analysis and decision problems considered for the static plant in Sect.4.2.

The considerations presented in this section will be used for the plants with un-certain parameters in the knowledge representation, described in the next section.

Example 5.1: As a very simple example let us consider first order one-dimensional plant described by inequalities

nnnnn ubsasubsa 22111 +≤≤+ + ,

12111 +++ ≤≤ nnn scysc .

It is known that 02001 sss ≤≤ ; 1b , 2b , 1c , 02 >c . The requirement concern-

ing ny is as follows

),( maxmin1

yyy nn

≤≤≥

i.e. ],[ maxmin yyDyn = for every n . For the given maxmin0201 ,,, yyss and the

coefficients 212121 ,,,,, ccbbaa one should determine the sequence unD such

that if unn Du ∈ for every n then the requirement is satisfied. For 0=n the set

1sD according to (5.3a) is determined by inequalities

max12 ysc ≤ , min11 ysc ≥ .

Then

].,[2

max

1

min1 c

y

c

yDs =

Using (5.4a) for 0u we obtain the following inequalities

,2

max02022 c

yubsa ≤+

1

min01011 c

yubsa ≥+

and

].,[2

022

22

max

1

011

11

min0 b

sa

cb

y

b

sa

cb

yDu −−=

For 1≥n 11, sns DD =+ and according to (5.4a) unD is determined by inequali-

ties

5.1 Relational Knowledge Representation 73

,2

max2

2

max2 c

yub

c

ya n ≤+

.1

min1

1

min1 c

yub

cy

a n ≥+

Consequently

].)1(

,)1(

[22

2max

11

1mincb

aycb

ayDun

−−=

The final result is then as follows: If

2

022

22

max0

1

011

11

minb

sa

cb

yu

b

sa

cb

y−≤≤− (5.5)

and for every 0>n

22

2max

11

1min )1()1(cb

ayu

cb

ayn

−≤≤

− (5.6)

then the requirement concerning ny will be satisfied. The conditions for the exis-

tence of the solution are the following:

,2

022

22

max

1

011

11

minb

sa

cb

y

b

sa

cb

y−≤− (5.7)

,2

max

1

minc

y

c

y≤ (5.8)

.)1()1(

22

2max

11

1mincb

ay

cb

ay −≤

− (5.9)

If 0min >y and 12 <a then these conditions are reduced to the inequality

),max(min

max βα≥y

y

where

.,1

1

1

2

2

1

11

22c

c

a

a

cb

cb=

−−

⋅= βα

Then unD are not empty sets if the requirement concerning ny is not too strong,

i.e. the ratio 1minmax

−⋅ yy is respectively high. One should note that the inequali-


ties (5.7), (5.8), (5.9) form the sufficient condition for the existence of the se-quence nu satisfying the requirement.

5.2 Analysis and Decision Making for the Dynamical Plants with Uncertain Parameters

The analysis and decision problems for dynamical plants described by a relationalknowledge representation with uncertain parameters may be formulated andsolved in a similar way as for the static plants in Sects. 3.4 and 3.5. Let us con-sider the plant described by relations

×⊆××⊆+

,);,(

,);,,( 1

YSwysR

SSUxssuR

nnII

nnnI (5.10)

where Xx ∈ and Ww ∈ are unknown vector parameters which are assumed tobe values of uncertain variables ( )wx, with the joint certainty distributions

),( wxh . We shall consider the analysis and decision problems in version II (see

Sects 3.4 and 3.5) which has better practical interpretation. The considerations inversion I are analogous.Analysis problem: For the given relations (5.10), ),( wxh , 0sD and the se-

quences unD , ynD one should determine

nynyn vDwxDvÿ

]~),([ =⊆

where ),( wxDyn is the result of the analysis problem formulated in the previ-

ous section, i.e. the set of all possible outputs ny for the fixed x and w .

In a similar way as for the static plant considered in Sect. 3.4 (see the formulas(3.33) and (3.34)) we obtain

),(max)],(~),[(),(),(

1,1,

wxhDDDwxvvnuyn DDDwx

nuynn−∈− =∈= (5.11)

where

.),(:),(),( 1, ynynnuyn DwxDWXwxDDD ⊆×∈=−

In the case where ( )wx, are considered as C-uncertain variables it is necessary to

find nv (5.11) and

),(max)],(~),[(),(),(

1,1,

wxhDDDvxv

nuyn DDDwxnuyn

−∈− =∈

5.2 Analysis and Decision Making for the Dynamical Plants with Uncertain Parameters 75

where ).,(),( 1,1, −− −×= nuynnuyn DDDWXDDD Then

)],(~),[(1

)],(~),[(21]~),([

1,

1,

−

−

∈−+

∈=⊆

nuyn

nuynynync

DDDwxv

DDDwxvDwxDv

(see (3.35) and (3.36)).For the given value nu , using (5.3) and (5.4) for the fixed ),( wx we obtain

);,,(),,(:);( 11)(

11, xssuRssuSsxuD nnnInnnxDs

nnnssnn

++∈

++ ∈∈= ,

)];,(),[(:),;( 1111)(

11,1,1

wysRysYywxuD nnIInnxDs

nnnynsn

++++∈

++ ∈∈=++

.

(5.12)

The formulation and solution of the analysis problem are the same as describedabove with nu , ),;( 1 wxuD nyn − and ),( 1−nyn uDD instead of ),(, wxDD ynun

and ),( 1, −nuyn DDD , respectively.

Decision problem: For the given relations (5.10), ),( wxh , 0sD and the sequence

ynD )...,,2,1( Nn = find the sequence of the optimal decisions

]~),;([maxargˆ 1,1, ++∈

⊆= nynnyUu

n DwxuDvun

for 1...,,1,0 −= Nn , where ),;(1, wxuD nny + is the result of the analysis prob-

lem (5.12). Then

),(maxmaxargˆ),(),( 1

wxhunynn uDDwxUu

n−∈∈

=

where

),;(:),(),( 11 ynnynnyn DwxuDWXwxuDD ⊆×∈= −− .

In the similar way as in Sect. 3.5 the determination of nu may be replaced by the

determination of ndn uu ˆˆ = where

)],(~[maxargˆ wxDuvu unnUu

ndn

∈=∈

where ),( wxDun is the result of the decision problem considered in the previous

section for the fixed x and w . Then


),(maxmaxargˆˆ)(),(

wxhuundn uDwxUu

nnd ∈∈==

where

),(:),()( wxDuwxuD unnnd ∈= .

Example 5.2: Let us assume that in the plant considered in Example 5.1 the pa-

rameters 1

ÿ

1 xc = and 2

ÿ

2 xc = are unknown and are the values of independent un-

certain variables 1x and 2x , respectively. The certainty distributions )( 11 xhx and

)( 22 xhx have the triangular form with the parameters 1d , 1γ for 1x (Fig. 5.2)

and 2d , 2γ for 2x ; 11 d<γ , 22 d<γ . Using the results (5.5) and (5.6) one may

determine the optimal decisions nu , maximizing the certainty index

)()],(~[ 21 nunn uvxxDuv =∈ . From (5.6) we have

.)](~[)],(~[min

)](~[)](~[)(

2211

2211

nn

nnn

uDxvuDxv

uDxuDxvuv

∈∈=∈∧∈=

(5.13)

)( 11 xhx

1

1d11 γ−d 11 γ+d 1x


Under assumption 12,1 <a , the sets )(1 nuD and )(2 nuD for 0>n are deter-

mined by the inequalities

nux α≥1 ,

nux

β≤2 , (5.14)

respectively, where

1

1min )1(b

ay −=α ,

2

2max )1(

b

ay −=β .

The certainty indexes

)()(max)](~[ 1

ÿ

11)(

1111

nxuDx

n uvxhuDxvn

==∈∈

and

)()(max)](~[ 2

ÿ

22)(

2222

nvxhuDxv xuDx

nn

==∈∈

may be obtained by using (5.14) and 1xh , 2xh :

,for

for

for

0

1

1

)(

1

111

1

2

1

11

γα

αγ

α

α

γγα

+≤

≤≤+

≥

++−=

du

du

d

du

d

uuv

n

n

n

nn

.for

for

for

0

1

1

)(

22

222

2

2

2

22

γβ

γββ

β

γγβ

−≥

−≤≤

≤

−+=

du

du

d

du

d

uuv

n

n

n

nn

Now we can consider three cases illustrated in Figs. 5.3, 5.4 and 5.5:

)( nuv

1

11 γα+d 22 γ

β−d1d

α2d

β nu

)(1 nuv

)(2 nuv

Fig. 5.3. Relationship between v and u – the first case



)( nuv

1

11 γα+d 22 γ

β−d1d

α2d

β nunu

)(1 nuv

)(2 nuv

Fig. 5.4. Relationship between v and u – the second case

)( nuv

1

11 γα+d22 γ

β−d 1d

α2d

β nu

)(1 nuv

)(2 nuv

Fig. 5.5. Relationship between v and u – the third case

1.

21 ddβα ≤ .

Then)(),(minmaxargˆ 21 nn

un uvuvu

n

=

is any value satisfying the inequality

21 du

d nβα ≤≤

and 1)ˆ( =nuv .

2.

21 ddβα > ,

2211 γβ

γα

−<

+ dd.

Then

1221

2121 )(),(minmaxargˆ

dduvuvu nn

un

nγγ

αγβγ++

==

and

1)ˆ(21

21 ++−

=γαγβ

αβ dduv n .

3.

221 γβ

γα

−≥

+ dd

Then for every nu

0)(),(min)( 21 == nnn uvuvuv

which means that the decision for which the requirement is satisfied with the certainty index greater than 0 does not exist.

The results for 0u based on the inequality (5.5) have a similar form. It is im-

portant to note that the results are correct under the assumption

22

max

11

minγγ +

≤− d

y

dy

which means that the condition (5.8) is satisfied for every 1x and 2x . Otherwise

)( nuv may be smaller:

),(),(min)( 321 vuvuvuv nnn =

where 3v is the certainty index that the condition

2

max

1

minx

y

x

y≤

is approximately satisfied, i.e.

)~(1

23 D

x

xvv ∈=

where 11 RRD ×⊂ is determined by the inequality

min

max

1

2y

y

x

x≤ .



5.3 Closed-Loop Control System. Uncertain Controller[17, 34]

The approach based on uncertain variables may be applied to closed-loop controlsystems containing continuous dynamic plant with unknown parameters which areassumed to be values of uncertain variables. The plant may be described by a clas-sical model or by a relational knowledge representation. Now let us consider twocontrol algorithms for the classical model of the plant, analogous to the algorithmsΨ and dΨ presented in Sect. 2.3: the control algorithm based on KP and the

control algorithm based on KD which may be obtained from KP or may be givendirectly by an expert. The plant is described by the equations

]);(),([)( xtutsfts =ÿ

,

)]([)( tsty η=

where s is a state vector, or by the transfer function );( xpKP in the linear case.

The controller with the input y (or the control error ε ) is described by the analo-gous model with a vector of parameters b which is to be determined. Conse-quently, the performance index

∫ ==T

xbdtuyQ0

þ

),(),( Φϕ

for the given T and ϕ is a function of b and x. In particular, for one-dimensional

plant

∫∞

==0

2 ),()( xbdttQ Φε .

The closed-loop control system is then considered as a static plant with the inputb, the output Q and the unknown parameter x, for which we can formulate andsolve the decision problem described in Sect. 2.2. The control problem consistingin the determination of b in the known form of the control algorithm may be for-mulated as follows.Control problem: For the given models of the plant and the controller find the

value b minimizing )(QM , i.e. the mean value of the performance index.

The procedure for solving the problem is then the following:1. To determine the function ),( xbQ Φ= .

2. To determine the certainty distribution );( bqhq for Q using the function Φ

and the distribution )(xhx in the same way as in the formula (2.1) for y .

3. To determine the mean value );( bQM .

5.4 Examples 81

4. To find b minimizing );( bQM .

In the second approach corresponding to the determination of dΨ for the static

plant, it is necessary to find the value b(x) minimizing ),( xbQ Φ= for the fixed

x. The control algorithm with the uncertain parameter b(x) may be considered as aknowledge of the control in our case, and the controller with this parameter maybe called an uncertain controller. The deterministic control algorithm may be ob-tained in two ways, giving the different results. The first way consists in substi-tuting )(bM in the place of b(x) in the uncertain control algorithm, where )(bM

should be determined using the function b(x) and the certainty distribution )(xhx .

The second way consists in determination of the relationship between)(uMud = and the input of the controller, using the form of the uncertain con-

trol algorithm and the certainty distribution )(xhx . This may be very difficult for

the dynamic controller.The problem may be easier if the state of the plant s(t) is put at the input of the

controller. Then the uncertain controller has the form

),( xsu Ψ=

which may be obtained as a result of nonparametric optimization, i.e. Ψ is the

optimal control algorithm for the given model of the plant with the fixed x and forthe given form of a performance index. Then

)();(ÿ

ssuMu dd Ψ==

where );( suM is determined using the distribution

)(max)];(~[);();(

xhsuDxvsuh xsuDx

xux∈

=∈=

and

),(:);( xsuXxsuDx Ψ=∈= .

5.4 Examples

Example 5.3: The data for the linear control system under consideration(Fig. 5.6) are the following:


zyzu + 0ˆ =y

y−=ε

u

);( xpKP

);( bpKC

Fig. 5.6. Closed-loop control system

)1)(1();(

21 ++=pTpT

xxpKP , pbbpKC =);(

z(t) = 0 for t < 0, z(t) = 1 for t ≥ 0, )(xhx has a triangular form presented in

Fig. 5.2 with adÿ

1 = and dÿ

1 =γ .

It is easy to determine

∫∞

=−++

==0 2121

212

2 ),()(2

)()( xb

TxbTTTxbTTx

dttQ Φε . (5.15)

The minimization of Q with respect to b gives

xxb α=)( ,

21

21TT

TT +=α ,

i.e. the uncertain controller is described by

xppxb

pKCα== )(

)( .

The certainty distribution )(bhb is as follows:

∞<≤−

−≤≤++−

≤≤++−+≤<

=

.for0

for1

for1

0for0

)(

bda

dab

adbab

ab

dadbab

dab

bhb

α

ααα

ααα

α

From the definition of a mean value we obtain

5.4 Examples 83

22

22

22

ln2

)2()(

daa

a

addbM

−

+= α . (5.16)

Finally, the deterministic controller is described by

pbM

pK dC)(

)(, = .

To apply the first approach described in the previous section, it is necessary tofind the certainty distribution for Q using formula (5.15) and the distribution

)(xhx , then to determine );( bQM and to find the value b minimizing );( bQM .

It may be shown that )(ˆ bMb ≠ given by the formula (5.16).

Example 5.4: Let us consider the time-optimal control of the plant with2);( −= xpxpKP (Fig. 5.7), subject to constraint Mtu ≤|)(| .

u y 0

εε

ÿ

);( xpK P

dtd

Fig. 5.7. Example of control system

It is well known that the optimal control algorithm is the following

))2(sgn()( 1−+= xMMtu εεεþþ

.

For the given )(xhx we can determine ),;( εεÿ

uhu which is reduced to three val-

ues )~(1 Muvv == , )~(2 Muvv −== , )0~(3 == uvv . Then

132121 ))(()()( −++−== vvvvvMuMtud .

It is easy to see that

)(max1

1 xhv xDx x∈

= , )(max2

2 xhv xDx x∈

=


where

)2(sgn: 11

−−>= εεεε MxxDxÿÿ

,

)2(sgn: 12

−−<= εεεε MxxDxÿÿ

and ))2(( 13

−−= εεε Mhv xÿÿ

.

Assume that the certainty distribution of x is the same as in Example 5.3. For0>ε , 0<ε

ÿ

and axg < it is easy to obtain the following control algorithm

−≥−−

−−≤

==g

g

g

g

d xadxad

xaM

xadM

uMufor

)(23

for

)(

where 1)2( −−= εεε Mxgÿÿ

. For example for 5.0=M , 3−=εÿ

, 1=ε , 16=a

and 10=d we obtain 2.0=du .

5.5 Stability of Dynamical Systems with Uncertain Parameters

The uncertain variables and certainty distributions may be used in a qualitativeanalysis which consists in proving if the system with uncertain parameter x satis-fies a determined property )(xP . In such a case, knowing the certainty distribu-

tion )(xhx we can calculate the certainty index that the property under considera-

tion is approximately satisfied:

)(max)~( xhDxv xDx

xx∈

=∈

where

)(: xPXxDx ∈=(see (1.3)). Let us explain it for the stability of the discrete system

),(1 xsfs nn =+ (5.17)

where kn Ss R=∈ is a state vector and Xx ∈ is an unknown vector parameter

which is assumed to be a value of an uncertain variable described by the knowncertainty distribution )(xhx . The considerations are analogous to those for the

system with random parameters presented in [21, 27]. Let us assume that the sys-

5.5 Stability of Dynamical Systems with Uncertain Parameters 85

tem (5.17) has one equilibrium state equal to 0 (a vector with zero components). For the linear time-invariant system

nn sxAs )(1 =+

the necessary and sufficient condition of stability is as follows

1|)]([|,1

<∈

xAiki

λ (5.18)

where iλ denotes an eigenvalue of the matrix )(xA . The inequality (5.18) is a

property )(xP in this case and the certainty index sv that the system is stable

may be obtained in the following way

)(max xhv xDx

sx∈

=

where

1|)]([|:,1

<∈=∈

xAXxD iki

x λ .

Consider the nonlinear and/or time-varying system described by

nnnn sxcsAs ),,(1 =+ (5.19)

where Ccn ∈ is a vector of time-varying parameters and the uncertainty con-

cerning nc is formulated as follows

cnn

Dc ∈≥ 0

(5.20)

where cD is a given set in C. The system (5.19) is globally asymptotically stable

(GAS) iff ns converges to 0 for any 0s . For the fixed x, the uncertain system

(5.19), (5.20) is GAS iff the system (5.19) is GAS for every sequence nc satisfy-

ing (5.20). Let )(xW and )(xP denote properties concerning x such that )(xW is

a sufficient condition and )(xP is a necessary condition of the global asymptotic

stability for the system (5.19), (5.20), i.e.

→)(xW the system (5.19), (5.20) is GAS,

the system (5.19), (5.20) is GAS )(xP→ .


Then the certainty index sv that the system (5.19), (5.20) is GAS may be esti-

mated by the inequality

psw vvv ≤≤

where

)(max xhv xDx

wxw∈

= , )(max xhv xDx

pxp∈

= ,

)(: xWXxDxw ∈= , )(: xPXxDxp ∈= ,

wv is the certainty index that the sufficient condition is approximately satisfied

and pv is the certainty index that the necessary condition is approximately satis-

fied. In general, xpxw DD ⊆ and xwxp DD − may be called "a grey zone" which

is a result of an additional uncertainty caused by the fact that )()( xPxW ≠ . In

particular, if it is possible to determine a sufficient and necessary condition)()( xPxW = then pw vv = and the value sv may be determined exactly. The

condition )(xP may be determined as a negation of a sufficient condition that the

system is not GAS, i.e. such a property )(xPneg that

→)(xPneg there exists nc satisfying (5.20) such that (5.19) is not GAS.

For the nonlinear and time-varying system we may use the stability conditions inthe form of the following theorems presented in [4, 5, 6, 21]:

Theorem 5.1: If there exists a norm |||| ⋅ such that

1||),,(|| <∈∈

xcsASsDc c

then the system (5.19), (5.20) is GAS. ÿ

The final form of the set

1||),,(||: <∈=∈∈

xcsAXxDSsCc

xw

depends on the form of the norm. In particular the norm |||| A may have the form

)(|||| Tmax2 AAA λ= (5.21)

where maxλ is the maximum eigenvalue of the matrix AAT ,


∑=≤≤

=k

jij

kiaA

111 ||max|||| , ∑

=≤≤∞ =

k

iij

kjaA

11||max|||| . (5.22)

Theorem 5.2: Consider a linear, time-varying system

nnn sxcAs ),(1 =+ . (5.23)

If the system (5.23), (5.20) is GAS then

1|)],([|max <∈

xcAiiDc c

λ (5.24)

where )(Aiλ are the eigenvalues of the matrix A ( ki ...,,2,1= ). ÿ

Theorem 5.3: The system (5.19), (5.20) where

)](),,()([: xAxcsAxACcDSs

c ≤≤∈=∈

(5.25)

is GAS if all entries of the matrices )(xA and )(xA are nonnegative and

1||)(|| <xA . (5.26)

ÿ

The inequality in (5.25) denotes the inequalities for the entries:

)(),,()( xaxcsaxa ijijij ≤≤ .

Theorem 5.4: Assume that all entries of the matrix )(xA are nonnegative. If the

system (5.19), (5.25) is GAS then

∑=

<k

iij

j

xa1

1)( (5.27)

and

∑=

<k

jij

i

xa

1

1)( . (5.28)

ÿChoosing different sufficient and necessary conditions we may obtain the differentestimations of the certainty index þ ý . For example, if we choose the condition

(5.26) with the norm ∞⋅ |||| in (5.22) and the condition (5.27) then


∑=

<∈=k

iij

jxw xaXxD

1

1)(: , (5.29)

neg,xxp DXD −=

where

∑=

≥∈=k

iij

jx xaXxD

1neg, 1)(: . (5.30)

Example 5.5: Consider an uncertain system (5.19) where 2=k and

+

+=

xcsacsa

csaxcsaxcsA

nnnn

nnnnnn ),(),(

),(),(),,(

2221

1211

with the uncertainty (5.25), i.e. nonlinearities and the sequence nc are such that

cDc∈ sDs∈ijijij acsaa ≤≤ ),( , 2,1;2,1 == ji .

Assume that 0≥x and 0≥ija . Applying the condition (5.26) with the norm

∞⋅ |||| in (5.22) yields

12111 <++ axa , 12212 <++ xaa

and xwD in (5.29) is defined by

),max(1 22122111 aaaax ++−< .

Applying the negation of the condition (5.27) yields

12111 ≥++ axa , 12212 ≥++ xaa .

Then neg,xD in (5.30) is determined by

),min(1 22122111 aaaax ++−≥

and the necessary condition (5.27) defining the set neg,xxp DXD −= is as fol-

lows

),min(1 22122111 aaaax ++−< .


For the given certainty distribution )(xhx we can determine

)(max0

xhv xxx

ww≤≤

= , )(max0

xhv xxx

pp≤≤

= (5.31)

where

),(max1 22122111 aaaaxw ++−= ,

),(min1 22122111 aaaax p ++−= .

Assume that )(xhx has triangular form presented in Fig. 5.8. The results obtained

from (5.31) for the different cases are as follows:1. For γ+≥ dxw

1== pw vv .

2. For γ+≤≤ dxd w

γγdx

v ww ++−= 1

ÿ

= 1v ,

+≥

++−=.otherwise

for

1

1 γ

γγ

dx

dxvp

pp

3. For dxd w ≤≤− γ

γγdx

v ww −+= 1 ,

−+

+≤≤++−

+≥

=

.otherwise1

for 1

for 1

γγ

γγγ

γ

dx

dxddx

dx

v

p

pp

p

p

4. For γ−≤ dxw

0=wv ,


≤≤−−+

+≤≤++−

+≥

=

.otherwise0

for 1

for 1

for 1

dx ddx

dxddx

dx

v

pp

pp

p

p

γγγ

γγγ

γ

)(xhx

1

dγ−d γ+d x


For example, if dxd w ≤≤− γ and dx p ≤ then the certainty index sv that

the system is globally asymptotically stable satisfies the following inequality

γγγγdx

vdx ps

w −+≤≤−+ 11 .

6 Comparison, Analogies and Generalisation

6.1 Comparison with Random Variables and Fuzzy Numbers

The formal part of the definitions of a random variable, a fuzzy number and anuncertain variable is the same: >< )(, xX µ , that is a set X and a function

1: RX →µ where )(0 xµ≤ for every Xx ∈ . For the fuzzy number, the un-

certain variable and for the random variable in the discrete case, 1)( ≤xµ . For the

random variable the property of additivity is required, which in the discrete case,...,, 21 mxxxX = is reduced to the equality

1)(...)()( 21 =+++ mxxx µµµ . Without any additional description, one can

say that each variable is defined by a fuzzy set >< )(, xX µ . In fact, each defini-

tion contains an additional description of semantics which discriminates the re-spective variables. To compare the uncertain variables with probabilistic and

fuzzy approaches, take into account the definitions for 1RX ⊆ , using ωΩ , and

)()( ωω xg = introduced in Sect. 1.1. The random variable x~ is defined by X

and probability distribution )()( xFx =µ (or probability density )(')( xFxf =if this exists) where )(xF is the probability that xx ≤~ . In discrete case

)~()()( iii xxPxpx ===µ (probability that ixx =~ ). For example, if Ω is a

set of 100 persons and 20 of them have the age 30)( =ωx , then the probability

that a person chosen randomly from Ω has 30=x is equal to 2.0 . In general,the function )(xp (or )(xf in a continuous case) is an objective characteristic of

Ω as a whole and )(xhω is a subjective characteristic given by an expert and de-

scribing his or her individual opinion of the fixed particular ω .To compare uncertain variables with fuzzy numbers, let us recall three basic

definitions of the fuzzy number in a wide sense of the word, that is the definitions

of the fuzzy set based on the number set 1RX = .

91


92 6 Comparison, Analogies and Generalisation

1. The fuzzy number )(ˆ dx for the given fixed value Xd ∈ is defined by X

and the membership function ),( dxµ which may be considered as a logic

value (degree of truth) of the soft property “if xx =ˆ then dx =~ˆ ”.2. The linguistic fuzzy variable x is defined by X and a set of membership

functions )(xiµ corresponding to different descriptions of the size of x

(small, medium, large, etc. ). For example, )(1 xµ is a logic value of the soft

property “if xx =ˆ then x is small”.3. The fuzzy number )(ˆ ωx (where Ωω ∈ was introduced at the beginning of

Sect. 1.1) is defined by X and the membership function )(xωµ which is a

logic value (degree of possibility) of the soft property “it is possible that thevalue x is assigned to ω ”.

In the first two definitions the membership function does not depend on ω ; inthe third case there is a family of membership functions (a family of fuzzy sets)for Ωω ∈ . The difference between )(ˆ dx or the linguistic fuzzy variable x and

the uncertain variable )(ωx is quite evident. The variables )(ˆ ωx and )(ωx are

formally defined in the same way by the fuzzy sets >< )(, xX ωµ and

>< )(, xhX ω respectively, but the interpretations of )(xωµ and )(xhω are dif-

ferent. In the case of the uncertain variable there exists a function )(ωgx = , the

value x is determined for the fixed ω but is unknown to an expert who formu-lates the degree of certainty that xx =~)(ω for the different values Xx ∈ . In the

case of )(ˆ ωx the function g may not exist. Instead we have a property of the type

“it is possible that ),( xP ω ” (or, briefly, “it is possible that the value x is as-

signed to ω ”) where ),( xP ω is such a property concerning ω and x for which it

makes sense to use the words “it is possible”. Then )(xωµ for fixed ω means the

degree of possibility for the different values Xx ∈ given by an expert. The ex-ample with persons and age is not adequate for this interpretation. In the popularexample of the possibilistic approach =),( xP ω “John )(ω ate x eggs at his

breakfast”.

From the point of view presented above, )(ωx may be considered as a special

case of )(ˆ ωx (when the relation ),( xP ω is reduced to the function g ), with a

specific interpretation of )()( xhx ωωµ = . A further difference is connected with

the definitions of )~( xDxw ∈ , )~( xDxw ∉ , )~~( 21 DxDxw ∈∨∈and )~~( 21 DxDxw ∈∧∈ . The function )()~(

xx DmDxw =∈ may be consid-

ered as a measure defined for the family of sets XDx ⊆ . Two measures have

been defined in the definitions of the uncertain variables: )()~(

xx DmDxv =∈

92

6.1 Comparison with Random Variables and Fuzzy Numbers 93

and )()~(

xcxc DmDxv =∈ . Let us recall the following special cases of fuzzy

measures (see for example [41]) and their properties for every 1D , 2D .1. If )( xDm is a belief measure, then

)()()()( 212121 DDmDmDmDDm ∩−+≥∪ .

2. If )( xDm is a plausibility measure, then

)()()()( 212121 DDmDmDmDDm ∪−+≤∩ .

3. A necessity measure is a belief measure for which).(),(min)( 2121 DmDmDDm =∩

4. A possibility measure is a plausibility measure for which).(),(max)( 2121 DmDmDDm =∪

Taking into account the properties of m and cm presented in Definitions 1.5

and 1.6 and in Theorems 1.1, 1.2 and 1.3, 1.4, it is easy to see that m is a possi-

bility measure, that )~(1

xn Dxvm ∈−= is a necessity measure and that cm is

neither a belief nor a plausibility measure. To prove this for the plausibility meas-

ure, it is enough to take Example 1.3 as a counter-example:

.9.06.04.0)()()(3.0)( 212121 −+=∪−+>=∩ DDmDmDmDDm cccc

For the belief measure, it follows from (1.66) when 1D and 2D correspond to

the upper case, and from the inequality <=∪ )(),(max)( 2121 DmDmDDm

)()( 21 DmDm + for ∅=∩ 21 DD .

The interpretation of the membership function )(xµ as a logic value w of a

given soft property )(xP , that is )]([)( xPwx =µ , is especially important and

necessary if we consider two fuzzy numbers ),( yx and a relation ),( yxR or a

function )(xfy = . Consequently, it is necessary if we formulate analysis and de-

cision problems. The formal relationships (see for example [39, 40])

])(:)([max)( yxfxy xx

y == µµ

for the function and

]),(:)([max)( Ryxxy xx

y ∈= µµ

for the relation do not determine evidently )(yPy for the given )(xPx . If

)]([)( xPwx xx =µ where =)(xPx “if xx =ˆ then dx =~ˆ ”, then we can accept

that )]([)( yPwy yy =µ where =)(yPy “if yy =ˆ then )ˆ(~ˆ xfy = “ in the case

of the function, but in the case of the relation )(yPy is not determined. If

93


=)(xPx ”if xx =ˆ then x is small” , then )(yPy may not be evident even in the

case of the function, for example .sin xy = For the uncertain variable

)~()()( xxvxhx xx ===µ with the definitions (1.53) – (1.56), the property

)(yPy such that )]([)( yPvy yy =µ is determined precisely: in the case of the

function, )~()()( yyvyhy yy ===µ and, in the case of the relation, )(yyµ is

the certainty index of the property =)(yPy ” there exist x such that

),(~),( yxRyx ∈ ”.

Consequently, using uncertain variables it is possible not only to formulate theanalysis and decision problems in the form considered in Chaps. 2 and 3 but alsoto define precisely the meaning of these formulations and solutions. This corre-sponds to the two parts of the definition of the uncertain logic mentioned in Sect.1.1 after Theorem 1.2: a formal description and its interpretation. The remark con-cerning ω in this definition is also very important because it makes it possible tointerpret precisely the source of the information about the unknown parameter xand the term “certainty index”.

In the theory of fuzzy sets and systems there exist other formulations of analy-sis and decision problems (see for example [39]), different from those presented inthis paper. The decision problem with a fuzzy goal is usually based on the given

)(yyµ as the logic value of the property “ y is satisfactory” or related properties.

The statements of analysis and decision problems in Chap. 3 for the system withthe known relation R and unknown parameter x considered as an uncertain vari-able are similar to analogous approaches for the probabilistic model and togetherwith the deterministic case form a unified set of problems. For ),( xuy Φ= and

given y the decision problem is as follows.

1. If x is known (the deterministic case), find u such that yxu =),(Φ .

2. If x is a value of random variable x~ with given certainty distribution, find u ,maximizing the probability that yy =~ (for the discrete variable), or find u

such that yuy =),~(E where E denotes the expected value of y~ .

3. If x is a value of uncertain variable x with given certainty distribution, findu , maximizing the certainty index of the property yy =~ , or find u such that

yuM y =)( where M denotes the mean value of y .

The definition of the uncertain variable has been used to introduce an C-uncertain variable, especially recommended for analysis and decision problemswith unknown parameters, because of its advantages mentioned in Sect. 1.3. Notonly the interpretation but also a formal description of the C-uncertain variablediffer in an obvious way from the known definitions of fuzzy numbers (see Defi-nition 1.6 and the remark concerning the measure cm in this section).

94

6.2 Application of Random Variables 95

To indicate the analogies with the probabilistic approach and the approachbased on the fuzzy description, in the next two sections we shall consider the non-parametric decision problems analogous to those presented in Sect. 2.4.

6.2 Application of Random Variables

Let us consider the static plant with the input Uu ∈ , the output Yy ∈ and the

vector of external disturbances Zz ∈ and let us assume that ),,( zyu are values

of random variables )~,~,~( zyu . The knowledge of the plant given by an expert

contains a conditional probability density ),|( zuyf y and the probability density

)(zf z for z~ , i.e.

><= )(),,|(KP zfzuyf zy .

Then it is possible to determine a random decision algorithm in the form of a con-ditional probability density )|( zufu , for the given desirable probability density

)(yf y required by a user.

Decision problem: For the given ),|( zuyf y , )(zf z and )(yf y one should de-

termine )|( zufu .

The relationship between the probability densities )(yf y and ),( zufuz is as

follows

∫ ∫=U Z

yuzy dudzzuyfzufyf ),|(),()(

where)|()(),( zufzfzuf uzuz = (6.1)

is the joint probability density for )~,~( zu . Then

∫ ∫=U Z

yuzy dudzzuyfzufzfyf ),|()|()()( . (6.2)

Any probability density )|( zufu satisfying the equation (6.2) is a solution of

our decision problem. It is easy to note that the solution of the equation (6.2) maybe not unique. Having )|( zufu one can obtain the deterministic decision algo-

rithm )(zΨ as a result of the determinization of the uncertain decision algorithm

described by )|( zufu . Two versions corresponding to the formulation in (2.13)

and (2.15) are the following:

95


I.

)()|(maxarg

zzufu auUu

a Ψ==∈

.

II.

∫ ===U

bub zduzuufzuu )()|()|~(E

Ψ

where E denotes the conditional expected value. The decision algorithms )(zaΨor )(zbΨ are based on the knowledge of the decision making

><= )|(KD zufu (or the random decision algorithm) which is determined from

the knowledge of the plant KP for the given )(yf y (Fig. 6.1). The relationships

(6.1) and (6.2) are analogous to (2.16) and (2.17) for the description using un-certain variables.

>< )(),,|(

KP

zfzuyf zy)|(

KD

zufu

ba ΨΨ ,

)(yf y

z u

z

y

Fig. 6.1. Decision system with probabilistic description

6.3 Application of Fuzzy Numbers

Let us consider two fuzzy numbers defined by set of values 1RX ⊆ , 1RY ⊆and membership functions )(xxµ , )(yyµ , respectively. The membership func-

tion )(xxµ is the logic value of the soft property )(xxϕ = ”if xx =ˆ then x is

1d ” or shortly “ x is 1d ”, and )(yyµ is the logic value of the soft property

)(yyϕ = “ y is 2d ”, i.e.

96

6.3 Application of Fuzzy Numbers 97

)()]([ xxw xx µϕ = , )()]([ yyw yy µϕ =

where 1d and 2d denote the size of the number, e.g. )(xxϕ = ” x is small”,

)(yyϕ = ” y is large”. Using the properties xϕ and yϕ we can introduce the

property yx ϕϕ → (e.g. “if x is small then y is large”) with the respective

membership function

)|(][

xyw yyx µϕϕ =→

and the properties

yx ϕϕ ∨ and ][ yxxyx ϕϕϕϕϕ →∧=∧

for which the membership functions are defined as follows

)(),(max][ yxw yxyx µµϕϕ =∨ ,

),()|(),(min][

yxxyxw xyyxyx µµµϕϕ ==∧ . (6.3)

If we assume that

][][ xyyyxx ϕϕϕϕϕϕ →∧=→∧

then

)|(),(min)|(),(min),( yxyxyxyx xyyxxy µµµµµ == . (6.4)

The properties xϕ , yϕ and the corresponding fuzzy numbers x , y are called in-

dependent if

)(),(min),(][ yxyxw yxxyyx µµµϕϕ ==∧ .

Using (6.4) it is easy to show that

),(max)( yxx xyYy

x µµ∈

= , (6.5)

),(max)( yxy xyXx

y µµ∈

= . (6.6)

The equations (6.4) and (6.5) describe the relationships between xµ , yµ , xyµ ,

)|( yxxµ analogous to the relationships (1.76), (1.74), (1.75) for uncertain vari-

ables, in general defined in multidimensional sets X and Y . For the given),( yxxyµ the function )|( xyyµ is determined by the equation (6.3) in which

97


),(max)( yxx xyYy

x µµ∈

= .

Theorem 6.1: The set of functions )|( xyyµ satisfying the equation (6.3) is

determined as follows:

∈≥

∉=

),(),(for),(

),(),(for),()|(

yxDyxyx

yxDyxyxxy

xy

xyy µ

µµ

where),()(:),(),( yxxYXyxyxD xyx µµ =×∈= .

Proof: From (6.3) it follows that

)],()([ yxx xyxYyXx

µµ ≥∈∈

.

If ),()( yxx xyx µµ > then, according to (6.3), )|(),( xyyx yxy µµ = . If

),()( yxx xyx µµ = , i.e. ),(),( yxDyx ∈ then ),()|( yxxy xyy µµ ≥ .

In particular, as one of the solutions of the equation (6.3), i.e. one of the pos-sible definition of the membership function for an implication we may accept

),()|( yxxy xyy µµ = . (6.7)

If )(),(min),( yxyx yxxy µµµ = then according to (6.7)

)(),(min)|( yxxy yxy µµµ =

and according to (6.3)

)()|( yxy yy µµ = .

The description concerning the pair of fuzzy numbers may be directly applied toone-dimensional static plant with one input Uu ∈ , one disturbance Zz ∈ and

one output Yy ∈ ),,( 1RYZU ⊆ . The nonparametric description of uncertainty

using fuzzy numbers may be formulated by introducing three soft properties)(uuϕ , )(zzϕ and )(yyϕ . This description is given by an expert and contains the

membership function

),|(][ zuyw yyzu µϕϕϕ =→∧

and the membership function )()]([ zzw zz µϕ = , i.e. the knowledge of the plant

98


><= )(),,|(KP zzuy zy µµ .

For examples, the expert says that “if u is large and z is medium then y is

small” and gives the membership function ),|( zuyyµ for this property and the

membership function )(zzµ for the property “ z is medium”. In this case the de-

cision problem may consists in determination of such a membership function)|( zuuµ for which the membership function for the output property

)()]([ yyw yy µϕ = will have a desirable form. In some sense, this is a problem

analogous to that in the previous section for random variables and to that in Sect.2.4 for uncertain variables. The essential difference consists in the fact that the re-quirements in the form of )(yhy or )(yf y have been concerned directly with a

value of the input and now the requirement )(yyµ concerns the output property

)(yyϕ .

Decision problem: For the given ),|( zuyyµ , )(zzµ and )(yyµ one should

determine )|( zuuµ .

Let us introduce ][),,(,, yzuyzu wyzu ϕϕϕµ ∧∧= . According to the general

relationships (6.5) and (6.4)

),|(),,(minmax),,(max)(,

,,,

zuyzuyzuy yuzzu

yzuZzUu

y µµµµ ==∈∈

(6.8)

or

),|(),|(),(minmax)(,

zuyzuzy yuzZzUu

y µµµµ∈∈

= . (6.9)

As a solution we may accept any function )|( zuuµ satisfying the equation (6.9).

The solution may be obtained in two steps. In the first step we determine the set offunctions ),( zuuzµ satisfying the equation (6.8) and in the second step we deter-

mine )|( zuuµ from the equation

)|(),(min),( zuzzu uzuz µµµ = . (6.10)

If the definition (6.7) is accepted then it is sufficient to determine)|(),( zuzu uuz µµ = in the first step. It is easy to see that if the functions )(yyµ

and ),|( zuyyµ have one local maximum equal to 1 then the point ),( zu maxi-

mizing the right hand side of the equation (6.8) satisfies the equation

),|(),( zuyzu yuz µµ = .

Hence, for this point we have

99


),|()( zuyy yy µµ = . (6.11)

Consequently, the procedure for the determination of uzµ analogous to that for

the determination of uzh presented in Sect. 2.4, is as follows:

1. To solve the equation (6.11) with respect to y and to obtain ),(* zuy .

2. To put ),(* zuy into )(yyµ in the place of y and to obtain

)],([),( * zuyzu yuz µµ = .

3. To assume ),()|( zuzu uzu µµ = as one of the solutions of the equation

(6.10).The function )|( zuuµ may be considered as the knowledge of the decision

making ><= )|(KD zuuµ or the fuzzy decision algorithm (fuzzy controller in

open-loop control system). According to another version, the knowledge of thedecision making is the function ),( zuuzµ in (6.10), i.e.

><= )(),|(KD zzu zu µµ . The determinization, i.e. the determination of the

mean value, gives the deterministic decision algorithm

)()ˆ( zuMud Ψ==

where the definition of the mean value )ˆ(uM for a fuzzy number is the same as

for an uncertain variable (see Sect. 1.4) with the membership function µ in the

place of the certainty distribution h . Using )|( zuuµ or ),( zuuzµ with the fixed

z in the determination of )ˆ(uM one obtains two versions of )(zΨ . The both

versions are the same if we assume that ),()|( zuzu uzu µµ = . Let us note that in

the analogous problems for uncertain variables (Sect. 2.4) and for random vari-ables (Sect. 6.2) it is not possible to introduce two versions of KD consideredhere for fuzzy numbers. It is caused by the fact that ),( zuuzµ and )|( zuuµ does

not concern directly the values of the variables (as probability distributions orcertainty distributions) but are concerned with the properties uϕ , zϕ and

][),( zuuz wzu ϕϕµ ∧= , ]|[][)|( zuuzu wwzu ϕϕϕϕµ =→= .

The deterministic decision algorithm is based on the knowledge of the decisionmaking KD which is determined from the knowledge of the plant KP for the given

)(uuµ (Fig. 6.2).

The considerations may be extended to multidimensional case with vectorszyu ,, . To formulate the knowledge of the plant one introduces soft properties of

the following form:

100


)( juiϕ = ” )(iu is jd ”, )( jziϕ = ” )(iz is jd ”, )( jyiϕ = ” )(iy is jd ”

where )()()( ,, iii yzu denote i-th components of yzu ,, , respectively and

...,,, 21 mj dddd ∈ denotes the size of the number (e.g. small, medium, large,

etc.). Each property is described by a membership function. Consequently, in the

>< )(),|(

or)|(

KD

zzu

zu

zu

u

µµµ

Ψ

)(uuµ

z u

z

y

>< )(),,|(

KP

zzuy zy µµ

Fig. 6.2. Decision system with fuzzy description

place of one implication yzu ϕϕϕ →∧ now we have a set of implications for

)()()( spl ynzmuk ϕϕϕ →∧ the different components and properties, e.g. “if

)2(x is small and )4(z is large then )1(y is medium”. The formulation of the deci-

sion problem and the corresponding considerations are the same as for one-dimensional case with

)](...)()([

)](...)2()1([)(

2211,1,...

,1

1

susuumjj

uiuiuisi

u

jjj

mu

s

ϕϕϕ

ϕϕϕϕ

∧∧∧=

∨∨∨=

∈

∈

where s is a number of components in the vector x , and with )(zzϕ , )(yyϕ in

the analogous form. The formulas (6.8) – (6.11) have the identical form for themultidimensional case where uµ , zµ , yµ , uyzµ , uzµ and ),|( zuyyµ are the

membership functions of uϕ , zϕ , yϕ , yzu ϕϕϕ ∧∧ and yzu ϕϕϕ →∧ ,

respectively. The determinization of the fuzzy decision algorithm consists in the

101


determination of )ˆ( )(iuM for the fixed z and each component of the vector u ,

using the membership functions ),( )( zu iuiµ or )|( )( zu i

uiµ where

),(),...,,2(),,1(max),( )( zmzzzu uiuiuii

ui µµµµ = .

Example 6.1: Consider a plant with 1,, Rzyu ∈ described by the following

KP:“ If u is small nonnegative and z is large but not greater than b (i.e. zb − is smallnonnegative) then y is medium”. Then

)(uuϕ = “ u is small nonnegative”,

)(zzϕ = “ z is large, not greater than b”,

)(yyϕ = “ y is medium”.

The membership function ][ yzuw ϕϕϕ →∧ is as follows:

)(1)(),|( 2 zbudyzuyy −−−+−−=µfor

210 ≤≤ u , bzb ≤≤−

21 ,

dzbxydzbx +−−−≤≤+−−−− )(1)(1

and 0),|( =zuyyµ otherwise.

For the membership function required by a user

+≤≤−+−−

=,otherwise

11for

0

1)()(

2 cyccyyyµ

one should determine the fuzzy decision algorithm in the form),()|( zuzu uzu µµ = .

Let us assume that

21 +≤≤+ cdc .

Then the equation (6.11) has a unique solution which is reduced to the solutionof the equation

)(1)(1)( 22 zbudycy −−−+−−=+−− .

Further considerations are the same as in Example 2.4 which is identical from theformal point of view. Consequently, we obtain the following result:

102

6.4 Generalisation. Soft Variables 103

≤≤−≤≤

−+−−≤−+−

−++−−=

.otherwise021,

210

,)]1([1for1])(2

)([

),(

222

bzbu

bcdzucd

zbucd

zuuzµ

Applying the determinization (defuzzification) we can determine)();ˆ( zzuMu Ψ== , i.e. the deterministic decision algorithm in an open-loop

system.

6.4 Generalisation. Soft Variables

It is worth noting the analogies between the relationships (2.17), (6.2) and (6.9),for uncertain variables, random variables and fuzzy numbers, respectively. Theuncertain variables, the random variables and the fuzzy numbers may be consid-ered as special cases of more general description of the uncertainty in the form ofsoft variables and evaluating functions [28, 29] which may be introduced as a toolfor a unification and generalisation of decision making problems based on the un-certain knowledge representation.

Definition 6.1: A soft variable ><=∨ )(, xgXx is defined by the set of values

X and an evaluating function +→ RXg : . The evaluating function satisfies the

following conditions:

∫ ∞<X

xxg )(

for the continuous case and

∑∞

=∞<

1

)(i

ii xgx

for the discrete case, i.e. for ...,,, 21 ∞= xxxX .

For two soft variables ),( ∨∨ yx we can introduce the joint evaluating function

),( yxgxy and the conditional evaluating functions )|( yxgx , )|( xyg y . For ex-

ample )|( yxgx denotes the evaluating function of ∨x for the given value y. The

evaluating function may have different practical interpretation (semantics). Inthe random case the soft variable is the random variable described by the prob-ability density )()( xfxg = , in the case of the uncertain variable )()( xhxg = is

the certainty distribution and in the case of the fuzzy description the soft variable

103


is the fuzzy number described by the membership function)]([)()( xwxxg ϕµ == where w denotes a logic value of a given soft property

)(xϕ .

Let us consider the plant with the input vector Uu ∈ , the output vector Yy ∈and the vector of disturbances Zz ∈ , and assume that ),,( zyu are values of soft

variables ),,( ∨∨∨ zyu . Denote by guD , gyD and gzD the sets of the evaluating

functions )(ugu , )(yg y and )(zgz , respectively. The relation

gzgyguzyug DDDgggR ××⊂),,( , (6.12)

i.e. the relationship between the evaluating functions may be considered as theknowledge representation of the plant (KP). It is easy to note that gR , ug , yg ,

zg are generalisations of the statements: “ R is a set of all possible values

),,( zyu ”, “ uD is a set of all possible values u” etc., introduced in Sect. 3.1. For

example, if uD is a set of all possible values u then .)( constugu = for uDu ∈and 0)( =ugu for uDu ∉ . If z is fixed then the relation gR (6.12) is reduced

to );,( zggR yug . For this case let us formulate the decision problem for the re-

quired property concerning y in the form gyy Dg ∈ where gygy DD ⊂ is given

by a user.Decision problem: For the given );,( zggR yug , gyD and z find the largest set

)(zDgu such that the implication

gyyguu DgzDg ∈→∈ )(

is satisfied.Our problem is analogous to that presented in Sect. 3.2 (see 3.18) and

);();(:)(

zgRDzgDDgzD uggyugyguugu =⊆∈= (6.13)

where

);,(),(:);( zggRggDgzgD yugyugyyugy ∈∈=

and );()( zgRzD ugu = denotes the set of the evaluating functions );( zugu

which may be considered as the knowledge of the decision making KD deter-mined from the given knowledge of the plant KP. For the set )(zDgu we can de-

termine the set MS of the mean values )(zMu for all )(zDg guu ∈ and use the

104


mean value as a final decision. Consequently, as a result based on KP we obtainthe set ΨS of the decision algorithms )(zΨ :

everyfor)(: zSzS M∈= ΨΨΨ .

Denote by )(ugu the evaluating function in the case when the set of possible val-

ues of ∨u is reduced to one value u. In this case uMu = and gR is reduced to the

set ),( zuDgy of the evaluating functions ),;( zuyg y . Now we can propose the

determinization of KP and KD using the mean values.Decision problem with the determinization: For the given ),( zuDgy , the re-

quired output value *y and z, find the decision u such that *),( yzuM y = .

As a result we obtain the set ΨS of the decision algorithms )(zu Ψ= corre-

sponding to all ),;( zuyg y in the set ),( zuDgy . As KD we can accept

),,( zggR yug with yy gg = for *yy = , (i.e. ∨y has only one possible value

equal to *y ) which is reduced to the set )(zDgu of the evaluating functions

);( zugu . The determinization of KD gives the set dSΨ of the decision algo-

rithms based on KD:

)()(

zzMu dud Ψ==

where )(zMu is the mean value for );( zugu belonging to the set )(zDgu . The

solutions based on KP and KD may be not equivalent, i.e. in general dSS ΨΨ ≠ .

This was shown for the uncertain variables, that is in the case where the evaluatingfunctions are the certainty distributions (see Example 2.3 in Sect. 2.3 for theparametric case).

The relation gR may have the form of a function (a one-to-one mapping):

);( zgTg yuu = or );( zgTg uyy = (6.14)

i.e.

gugyu DDT →: or gyguy DDT →: .

Then (6.13) is reduced to

);()( zgTgzD yuugu ==

or);(:)( yuyugu gzgTgzD == . (6.15)

105


In particular, if )(ygg yy = or )(ugg uu = then according to (6.14)

),;( zyugg uu = or ),;( zuygg yy = ,

respectively.The relationships between the general formulation using soft variables and the

respective formulations with uncertain, random and fuzzy variables may be showndirectly for the fixed z. Then in the place of (2.17), (6.2) and (6.9) we have

);|(),;(minmax)(,

zuyhzuhyh yuZzUu

y∈∈

= , (6.16)

∫ ∫=U Z

yuy dudzzuyfzufyf );|();()( , (6.17)

);|(),;(minmax)(,

zuyzuy yuZzUu

y µµµ∈∈

= , (6.18)

respectively. In this formulations );( zuhu and );|( zuyhy denote the certainty

distributions for the fixed value z ; );( zufu , );|( zuyf y are the probability den-

sities and );( zuuµ , );|( zuyyµ are the membership functions for the fixed

value z. In the first case we assume that the soft variables ),( ∨∨ yu are the uncertain

variables, the evaluating functions take the form of the certainty distributions

)()( yhyg yy = , );|(),;( zuyhzuyg yy = (6.19)

and the function yT is determined by (6.16).

The knowledge of the plant KP is then reduced to );|(),;( zuyhzuyg yy = .

For the required distribution )(yhy , according to (6.15), the result of the decision

problem based on KP is the set )(zDgu of the distributions );( zuhu satisfying

the equation (6.16).In the second case

)()( yfyg yy = , );|(),;( zuyfzuyg yy = ,

the function yT is determined by (6.17) and the result of the decision problem

based on ><= );|(KP zuyf y is the set )(zDgu of the probability densities

);( zufu satisfying the equation (6.17) for the required density )(yf y .

In the third case

106


)()( yyg yy µ= , );|(),;( zuyzuyg yy µ= ,

yT is determined by (6.18) and as the result of the decision problem based on

><= );|(KP zuyyµ we obtain the set )(zDgu of the membership functions

);( zuuµ .

An evaluating function );( zugu chosen from the set )(zDgu may be called a

soft decision algorithm in an open-loop decision system. The uncertain decision

algorithm );( zuhu , the random decision algorithm );( zufu and the fuzzy deci-

sion algorithm );( zuuµ may be considered as special cases of the soft decision

algorithm. If z is assumed to be a value of a soft variable ∨z (in particular, a value

of an uncertain variable z , a random variable z~ or a fuzzy number z ) then we

can introduce the evaluating function )(zgz (in particular, )(zhz , )(zf z or

)(zzµ ). Instead of (6.16), (6.17), (6.18) we have the formulations (2.17), (6.2) and

(6.9) which may be generalised by introducing soft variables and the conditional

evaluating functions.

107

7 Special and Related Problems

7.1 Pattern Recognition

Let an object to be recognized or classified be characterized by a vector of features

Uu ∈ which may be observed, and the index of a class j to which the object

belongs; JMj

...,,2,1 =∈ , M is a number of the classes. The set of the ob-

jects may be described by a relational knowledge representation JUjuR ×∈),(

which is reduced to the sequence of sets

MjUjDu ...,,2,1,)( =⊂ ,

i.e.),(),(:)( juRjuUujDu ∈∈= .

Assume that as a result of the observation it is known that UDu u ⊂∈ . The rec-

ognition problem may consist in finding the set of all possible indexes j , i.e. the

set of all possible classes to which the object may belong [32, 49].

Recognition problem: For the given sequence MjjDu ,1),( ∈ and the result of

observation uD find the smallest set JD j ⊂ for which the implication

ju DjDu ∈→∈

is satisfied.This is the specific analysis problem for the relational plant (see Sect. 3.2) and

)(: ∅≠∩∈= jDDJjD uuj

where ∅ denotes an empty set. In particular, if uDu = , i.e. we obtain the ex-

act result of the measurement then

)(: jDuJjD uj ∈∈= .

109


110 7 Special and Related Problems

Now let us assume that the knowledge representation contains a vector of un-known parameters Xx ∈ and x is assumed to be a value of an uncertain vari-able x described by a certainty distribution )(xhx given by an expert. The recog-

nition problem is now formulated as a specific analysis problem (version I)considered in Sect. 3.4.Recognition problem for uncertain parameters: For the given sequence

)(),;( xhxjD xu , uD and the set JD j ⊂ˆ given by a user one should find the

certainty index that the set jD belongs to the set of all possible classes

);(:)( ∅≠∩∈= xjDDJjxD uuj . (7.1)

It is easy to see that

)]ˆ(~[)](~ˆ[ jxjj DDxvxDDv ∈=⊆ (7.2)

where

)(ˆ:)ˆ( xDDXxDD jjjx ⊆∈= . (7.3)

Then

)(max)](~ˆ[)ˆ(

xhxDDv xDDx

jjjx∈

=⊆ . (7.4)

In particular, for ˆ jD j = one can formulate the optimization problem consisting

in the determination of a class j maximizing the certainty index that j belongs

to the set of all possible classes.Optimal recognition problem: For the given sequence );( xjDu , )(xhx and uD

one should find *j maximizing

)()](~[

jvxDjv j =∈ .

Using (7.2), (7.3) and (7.4) for ˆ jD j = we obtain

)(max)](~[)()(

xhjDxvjv xjDx

xx∈

=∈= (7.5)

where)(:)( xDjXxjD jx ∈∈= (7.6)

and )(xD j is determined by (7.1). Then

)(maxmaxarg)(maxarg)(

* xhjvj xjDxjj x∈

== . (7.7)

110

7.1 Pattern Recognition 111

Assume that the different unknown parameters are separated in the different sets,i.e. the knowledge representation is described by the sets );( ju xjD where

jj Xx ∈ are subvectors of x , different for the different j . Assume also that jx

and ix are independent uncertain variables for ji ≠ and jx is described by the

certainty distribution )( jxj xh . In this case, according to (7.1)

∅≠∩⇔∈ );()( juuj xjDDxDj .

Then

)](~[)];(~[)( jDxvxjDuvjv xjjjuDu u

∈=∈=∈

(7.8)

where

);(:)( juDu

jjxj xjDuXxjDu

∈∈=∈

. (7.9)

Finally

)(maxmaxarg)(

*jxj

jDxjxhj

xjj ∈= . (7.10)

In particular, for uDu = (7.1), (7.8) and (7.9) become

);(:)( xjDuJjxD uj ∈∈=

)(max)](~[)];(~[)()(

jxjjDx

xjjju xhjDxvxjDuvjvxjj ∈

=∈=∈= , (7.11)

);(:)( jujjxj xjDuXxjD ∈∈= . (7.12)

The procedure of finding *j based on the knowledge representation

>∈< )(;,1),;( xhMjxjD xu or the block scheme of the corresponding rec-

ognition system is illustrated in Fig. 7.1. The solution may be not unique, i.e.

)( jv may take the maximum value for the different *j . The result 0)( =jv for

each Jj ∈ means that the result of the observation uDu ∈ is not possible or

there is a contradiction between the result of the observation and the knowledgerepresentation given by an expert.

If x is considered as C-uncertain variable then

)(maxarg* jvj cj

c =

111


" # % & )

>∈< )(;,1),;( xhMjxjD xu

j u uD

" # % & + % &

*j)( jv

Mj ,1∈

Fig. 7.1. Block scheme of recognition system

where

)](~[1)](~[21)( jDxvjDxvjv xxc ∈−+∈= ,

)()( jDXjD xx −= . Finally

)](max1)(max[21)(

)()(xhxhjv x

jDxx

jDxc

xx ∈∈−+= . (7.13)

The certainty indexes )( jvc corresponding to (7.8) and (7.11) have the analogous

form.

Example 7.1: Let 1, Rxu j ∈ , the sets );( ju xjD be described by the inequali-

ties

Mjxux jj ...,,2,1,2 =≤≤

and the certainty distributions )( jxj xh have a parabolic form for each j (Fig.

7.2):

otherwise

11for

0

1)()(

2 +≤≤−

+−−

= jjjjjjxj

dxddxxh

where 1>jd .

In this case the sets (7.12) for the given u are described by the inequality

uxuj ≤≤

2.

112


)( jxj xh

jd1−jd 1+jd jx

1

Fig. 7.2. Parabolic certainty distribution

Applying (7.11) one obtains )( jv as a function of jd illustrated on Fig. 7.3:

+≥

+≤≤

≤≤

≤≤−

−≤

+−−

+−−

=

.1

12

21

2

12

for

for

for

for

for

0

1)(

1

1)2

(

0

)(2

2

ud

udu

udu

udu

ud

du

du

jv

j

j

j

j

j

j

j

For example, for 3=M , 5=u , 21 =d , 2.52 =d , 63 =d we obtain

75.0)1( =v , 96.0)2( =v and 0)3( =v . Then 2* =j , which means that for

5=u the certainty index that 2=j belongs to the set of the possible classes has

)( jv

2u1

2−u u 1+u

jd

1

Fig. 7.3. Relationship between v and the parameter of certainty distribution

113


the maximum value equal to 0.96. For ],2

[,, 321 uuddd ∈ one obtains 1* =j

or 2 or 3 and 1)( * =jv .

Let us consider x as a C-uncertain variable for the same numerical data. Toobtain )( jvc according to (7.13) it is necessary to determine

)()(max)](~[

)(jvxhjDxv njxj

jDxxjj

xjj

==∈∈

. (7.14)

In our case the set )()( jDXjD xjjxj −= is determined by the inequalities

2ux j < or ux j > .

Using (7.14) we obtain 1)3()2()1( === nnn vvv . Then

)(21)](1)([

21)( jvjvjvjv nc =−+= , (7.15)

i.e. 375.0)1( =cv , 48.0)2( =cv , 0)3( =cv and 2* =cj with the certainty index

48.0)( * =jvc .

For 31 =d , 2.32 =d , 43 =d we obtain 1)3()2()1( === vvv and

75.01)35.2()1( 2 =+−−=nv ,

51.01)2.35.2()2( 2 =+−−=nv ,

0)3( =nv .

Then

625.0)75.011(21)1( =−+=cv ,

745.0)51.011(21)2( =−+=cv ,

1)3( =cv

and 3* =cj with the certainty index 1)( * =cc jv .

Example 7.2: Assumed that in the Example 7.1 the certainty distributions havean exponential form:

2)()( jj dxjxj exh

−−= .

114


Applying (7.11) one obtains )( jv as a function of jd :

≥

≤≤

≤

=−−

−−

.2

2

for

for

for

1)(2

2

)(

)2

(

ud

udu

ud

e

e

jv

j

j

j

du

du

j

j

For 3=M , 5=u , 21 =d , 2.52 =d , 63 =d we obtain

25.0)1( −= ev , 4.0)2( −= ev , 1)3( −= ev .

Then 2* =j with the certainty index 67.0)( 4.0* == −ejv . For

1d , 2d , ],2

[3 uud ∈ one obtains 1* =j or 2 or 3 and 1)( * =jv .

Now let us consider x as an C-uncertain variable. Using (7.14) we obtain

≥

≤≤

≤≤

≤

=−−

−−

.for143for

43

2for

2for1

)(2

2

)(

)2

(

ud

udue

udue

ud

jv

j

jdu

jdu

j

nj

j

Then the formula

)](1)([21)( jvjvjv nc −+=

gives the following results:

≥

≤≤−

≤≤−

≤

=

−−

−−

−−

−−

.for21

43for

211

43

2for

211

2for

21

)(

2

2

2

2

)(

)(

)2

(

)2

(

ude

udue

udue

ude

jv

jdu

jdu

jdu

jdu

c

j

j

j

j

Substituting the numerical data 5=u , 21 =d , 2.52 =d , 63 =d one obtains

25.021)1( −= evc , 04.0

21)2( −= evc , 1

21)3( −= evc .

115


Then 2** == jjc with the certainty index 335.021)( 4.0* == −ejv cc . The re-

sults for 31 =d , 2.32 =d and 43 =d are as follows:

25.0211)1( −−= evc , 49.0

211)2( −−= evc , 1

211)3( −−= evc

and 3* =cj with the certainty index 0.816211)( 1* =−= −ejv cc .

In this particular case the results *j and *cj are the same for the different

forms of the certainty distribution (see Example 7.1).

7.2 Control of the Complex of Operations

The uncertain variables may be applied to a special case of the control of the com-plex of parallel operations containing unknown parameters in the relationalknowledge representation. The control consists in a proper distribution of a givensize of a task taking into account the execution time of the whole complex. It maymean the distribution of a raw material in the case of a manufacturing process or aload distribution in a group of parallel computers. In the deterministic case wherethe operations are described by functions determining the relationship between theexecution time and the size of the task, the optimization problem consisting in thedetermination of the distribution minimizing the execution time of the complexmay be formulated and solved (see e.g. [7]). In the case of the relational knowl-edge representation with uncertain parameters the problem consists in the deter-mination of the distribution maximizing the certainty index that the requirementgiven by a user is satisfied [24, 30, 31]. This is a specific form of the decisionproblem described in Sect. 3.5.

Let us consider the operations described by the relations

++ ×⊂ RRxTuR iiii );,( , ki ...,,2,1= (7.16)

where iu is the size of the task, iT denotes the execution time and ix is an un-

known parameter which is assumed to be a value of an uncertain variable ix with

the certainty distribution )( ixi xh given by an expert. From (7.16) we obtain the

set of the possible values iT for the given value iu :

);,(),(:);(, iiiiiiiiiiT xTuRTuTxuD ∈= .

The complex of operations is considered as a plant with the input)...,,,( 21 kuuuu = , the output Ty = where T is the execution time of the whole

complex:

116

7.2 Control of the Complex of Operations 117

...,,,max 21 kTTTT = , (7.17)

and the requirement ],0[ α∈T , i.e. α≤T where α is a number given by a user.

Decision problem: For the given iR , )( ixi xh )...,,2,1( ki = and α find the dis-

tribution )ˆ...,,ˆ,ˆ(ˆ 21 kuuuu = maximizing the certainty index )(uv that the ap-

proximate set of the possible outputs Ty ∈ belongs to the interval ],0[ α , subject

to constraints

Uuuu k =+++ ...21 , 0,1

≥∈

iki

u (7.18)

where U is the global size of the task to be distributed.From (7.17) it is easy to note that the requirement α≤T is equivalent to the

requirement

]),0[(...]),0[(]),0[( 21 ααα ∈∧∧∈∧∈ kTTT .

Then)(minmaxarg)(maxargˆ ii

iUuUuuvuvu

∈∈==

where U is determined by the constraints (7.18) and

],0[~);()( , α⊆= iiiTii xuDvuv . (7.19)

Consider as a special case the relations (7.16) described by the inequalities

iii uxT ≤ , 0>ix , ki ...,,2,1= . (7.20)

The inequality (7.20) determines the set of possible values of the execution time ini-th operation for the fixed value of the size of the task iu , e.g. the set of possible

values of the processing time for the amount of a raw material equal to iu , in the

case of a manufacturing process. Then the certainty index (7.19) is reduced to thefollowing form

)(max)](~[)~()()(

ixiuDx

ixiiiii xhuDxvTvuvixii ∈

=∈=≤= α (7.21)

where )( ixi uD is described by the inequality

1−≤ ii ux α .

Example 7.3: Let )( ixi xh have a triangular form (Fig. 7.4). Then, using (7.21) it

is easy to obtain

117


xih

zia ibix

2ii ba

z+

=1

Fig. 7.4. Triangular certainty distribution

≥

+≤

+−

= −

−

otherwise

)(2

for

for

0

1

)( 1

1

ii

iii

iii

ii au

bau

BuA

uv α

α

where

)()(

ii

iiii ab

baaA −

+= α ,

ii

iii ab

baB

−+

= .

For 2=k the decision 1u may be found by solving the equation

)()( 1211 uUvuv −= .

The result is as follows:

1. For )11(21 aa

U +≥ α

0)( =uv for any 1u which means that α is too small to satisfy the

requirement.

2. For ]11[22211 baba

U +++≤ α

the optimal decision 1u is any value from the interval

]2,2[1122 baba

U ++− αα

and

1)(max *

== vuvu

.

3. Otherwise,

1212211 ))((ˆ −++−= AAUABBu (7.22)

118

7.3 Descriptive and Prescriptive Approaches 119

and 111* uABv −= .

For example, if 2=U , 2=α , 11 =a , 31 =b , 22 =a and 42 =b then 1u is

determined from (7.22) and 25.1ˆ1 =u , 75.0ˆ2 =u . For this distribution the re-

quirement α≤T is approximately satisfied with the certainty index 75.0* =v .

7.3 Descriptive and Prescriptive Approaches

In the analysis and design of knowledge-based uncertain systems it may be im-portant to investigate a relation between two concepts concerning two differentsubjects of the knowledge given by an expert [29]. In the descriptive approach anexpert gives the knowledge of the plant KP, and the knowledge of the decisionmaking KD is obtained from KP for the given requirement. This approach iswidely used in the traditional decision and control theory. The deterministic deci-sion algorithm may be obtained via the determinization of KP or the determiniza-tion of KD based on KP. Such a situation for the formulation using uncertain vari-ables is illustrated in Figs. 2.2, 2.3, 2.4, 3.5 and 3.6. In the prescriptive approach

the knowledge of the decision making KD is given directly by an expert. This ap-proach is used in the design of fuzzy controllers where the deterministic controlalgorithm is obtained via the defuzzification of the knowledge of the control givenby an expert. The descriptive approach to the decision making based on the fuzzydescription may be found in [39].

Generally speaking, the descriptive and prescriptive approaches may be called

equivalent if the deterministic decision algorithms based on KP and KD are thesame. Different particular cases considered in the previous chapters may be illus-trated in Figs. 7.5 and 7.6 for two different concepts of the determinization. Fig.7.7 illustrates the prescriptive approach. In the first version (Fig. 7.5) the ap-proaches are equivalent if )()( zz dΨΨ = for every z. In the second version (Fig.

7.6) the approaches are equivalent if KDKD = . Then )()( zz dΨΨ = for every

z.Let us consider more precisely version I of the decision problem described in

Sect. 2.3. An expert formulates ><= xh,KP Φ (the descriptive approach) or

><= xd h,KD Φ (the prescriptive approach). In the first version of the deter-

minization illustrated in Fig. 2.2 the approaches are equivalent if)()( zz ada ΨΨ = for every z, where )(zaΨ is determined by (2.8) and )(zadΨ is

determined by (2.13) with ),( xzdΦ instead of ),( xzdΦ obtained as a solution of

the equation

yxzu ˆ),,( =Φ . (7.23)

119


Ψ

z

uz

# $

Fig. 7.5. Illustration of descriptive approach – the first version

dΨ

z

duz

# $

Fig. 7.6. Illustration of descriptive approach – the second version

dΨ

z

duz

# $

KD

Fig. 7.7. Illustration of prescriptive approach

120

7.4 Complex Uncertain System 121

In the second version of the determinization illustrated in Fig. 2.3 the approachesare equivalent if the solution of the equation (7.23) with respect to u has the form

),( xzdΦ , i.e.

yxzxzd ˆ],),,([ =ΦΦ .

For the nonparametric problem described in Sect. 2.4 only the second version ofthe determinization illustrated in Fig. 2.4 may be applied. If we accept (2.20) as asolution of the equation (2.16) then ><= ),|(KP zuyhy and

><= )|(KD zuhu are equivalent for the given required distribution )(yhy if

),()|( zuhzuh uzu = satisfies the equation (2.18). The similar formulation of the

equivalency may be given for the random and the fuzzy descriptions presented inSects. 6.2 and 6.3, respectively.

The generalisation for the soft variables and evaluating functions described inSect. 6.4 may be formulated as a principle of equivalency.

Principle of equivalency: If the knowledge of the decision making KD givenby an expert has a form of the set of evaluating functions )(zDgu and

)()( zDzD gugu ⊆ where )(zDgu is determined by (6.13), then ΨΨ SS ⊆ where

ΨS is the set of the decision algorithms corresponding to )(zDgu . In particular, if

an expert gives one evaluating function );( zuggu , i.e. );()( zugzD gugu =

and )();( zDzug gugu ∈ then the decision algorithm based on the knowledge of

the decision making given by an expert is equivalent to one of the decision algo-rithms based on the knowledge of the plant.

7.4 Complex Uncertain System

As an example of a complex uncertain system let us consider two-level systempresented in Fig. 7.8, described by a relational knowledge representation with un-certain parameters where ii Uu ∈ , ii Yy ∈ , ii Zz ∈ , Yy ∈ [22, 30]. For exam-

ple, it may be a production system containing k parallel operations (productionunits) in which iy is a vector of variables characterizing the product (e.g. the

amounts of some components), iu is a vector of variables characterizing the raw

material which are accepted as the control variables and iz is a vector of distur-

bances which are measured. The block P may denote an additional production unitor an evaluation of a vector y of global variables characterizing the system as awhole.

121


P

1P 2P kP

ku2u1u

1z 2z kz1y 2y ky

y

. . .

Fig. 7.8. Example of complex system

Assume that the system is described by a relational knowledge representationwhich has a form of a set of relations:

×⊂

∈××⊂

+ YYxyyR

kiZYUxzyuR

ky

iiiiiiii

);,(

,1,);,,(

1

(7.24)

where ii Xx ∈ ( 1...,,2,1 += ki ) are vectors of parameters,

Yyyyy k ∈= )...,,,( 21 .

Each relation may be presented as a set of inequalities and/or equalities concern-ing the components of the respective vectors. The unknown parameters ix are as-

sumed to be values of uncertain variables described by certainty distributions)( ixi xh given by an expert. The relations (7.24) may be reduced to one relation

ZYUxzyuR ××∈);,,(

where

Uuuuu k ⊂= )...,,,( 21 , Zzzzz k ⊂= )...,,,( 21 ,

Xxxxx k ⊂= + )...,,,( 121 .

Now the decision problem may be formulated directly for the system as a whole,i.e. for the plant with the input u, the output y, the disturbance z and the uncertainvector parameter x. The formulation of the solution of the decision problem withthe requirement yDy ∈ given by a user have been described in Sect. 3.5. If the

uncertain variables 121 ...,,, +kxxx are independent then

)(...,),(),(min)( 11,2211 ++= kkxxxx xhxhxhxh .

122

7.4 Complex Uncertain System 123

The direct solution of the decision problem for the system as a whole may be verycomplicated and it may be reasonable to apply a decomposition, i.e. to decomposeour decision problem into separate subproblems for the block P and the blocks iP .

1. The decision problem for the block P: For the given );,( 1+ky xyyR ,

)( 11, ++ kkx xh and yD find y maximizing the certainty index

]~);([ 1 yky DxyDv ⊆+

where);,(),(:);( 11 ++ ∈∈= kyky xyyRyyYyxyD .

2. The decision problem for the blocks iP ),1( ki ∈ : For the given

);,,( iiiii xzyuR , )( ixi xh and yiD find ix maximizing the certainty index

]~);,([ yiiiiyi DxzuDv ⊆where

);,,(),,(:);,( iiiiiiiiiiiiiyi xzyuRzyuYyxzuD ∈∈= .

The decision problem for the block iP with the given yiD is then the same as

the problem for the system as a whole with the given yD . The sets yiD are

such that

yykyy DDDD ⊆××× ...21 (7.25)

where yD is the set of the solutions y of the decision problem for the block

P.In general, the results of the decomposition are not unique (the condition (7.25)may be satisfied by different sets yiD ) and differ from the results obtained from

the direct approach for the system as a whole.If there is no unknown parameter in the block P then the decomposition may

have the following form:1. The decision problem for the block P: For the given ),( yyRy and yD find the

largest set yD such that the implication

yy DyDy ∈→∈ ˆ

is satisfied. This is a decision problem for the relational plant described inSect. 3.2.

123


2. The decision problem for the block iP is the same as in the previous formula-

tion, with yD instead of yD .

7.5 Learning System

When the information on the unknown parameters x in the form of the certaintydistribution )(xhx is not given, a learning process consisting in step by step

knowledge validation and updating based on results of current observations maybe applied [10, 18, 19, 23, 24, 25]. The results of the successive estimation of theunknown parameters may be used in the current determination of the decisions inan open-loop or closed-loop learning decision making system. This approach maybe considered as an extension of the known idea of adaptation via identificationfor the plants described by traditional mathematical models (see e.g. [8]). Let usexplain this idea for the plant described by the relation );,( xyuR , considered in

Chap. 3. For the given yD one may determine the largest set of the decisions

UxDu ⊂)( such that the implication

yu DyxDu ∈→∈ )(

is satisfied. The learning process may concern the knowledge of the plant);,( xyuR or directly the knowledge of the decision making )(xDu . Let us con-

sider the second case and assume that )(xDu is a continuous and closed domain

in U, the parameter x has the value xx = and x is unknown. In each step of thelearning process one should prove if the current observation “belongs” to theknowledge representation determined to this step (knowledge validation) and ifnot – one should modify the estimation of the parameters in the knowledge repre-sentation (knowledge updating). The successive estimations will be used in thedetermination of the decision based on the current knowledge in the learning sys-tem.When the parameter x is unknown then for the fixed value u it is not known if uis a correct decision, i.e. if )(xDu u∈ and consequently yDy ∈ . Our problem

may be considered as a classification problem with two classes. The point ushould be classified to class 1=j if )(xDu u∈ and to class 2=j if

)(xDu u∉ . Assume that we can use the learning sequence

nnn Sjujuju

2211 ),(...,),,(),,( =

where 2,1∈ij are the results of the correct classification given by an external

trainer or obtained by testing the property yi Dy ∈ at the output of the plant. Let

124

7.5 Learning System 125

us denote by iu the subsequence for which 1=ij , i.e. )(xDu ui ∈ and by iu the

subsequence for which 2=ij , and introduce the following sets in X:

)(:)( xDuXxnD uix ∈∈= for every iu in nS ,

)(ˆ:)(ˆ xDUuXxnD uix −∈∈= for every iu in nS .

It is easy to see that xD and xD are closed sets in X. The set

)()(ˆ)(

nnDnD xxx ∆=∩

is proposed here as the estimation of x . For example, if 1Rx ∈ and )(xDu is

described by the inequality 2T xuu ≤ then

),[)( min, ∞= nx xnD , ),0[)(ˆmax,nx xnD =

),[)( max,min, nnx xxn =∆

where

iii

n uux T2min, max= , ii

in uux ˆˆmin T2

max, = .

The determination of )(nx∆ may be presented in the form of the following recur-

sive algorithm:If 1=nj ( nn uu = )

1. Knowledge validation for nu . Prove if

)]([)1(

xDu unnDx x

∈−∈

.

If yes then )1()( −= nDnD xx . If not then one should determine the new )(nDx ,

i.e. update the knowledge.2. Knowledge updating for nu

)(:)1()( xDunDxnD unxx ∈−∈= .

Put )1(ˆ)(ˆ −= nDnD xx .

If 2=nj ( nn uu ˆ= )

3. Knowledge validation for nu . Prove if

)]([)1(ˆ

xDUu unnDx x

−∈−∈

.

125


If yes then )1(ˆ)(ˆ −= nDnD xx . If not then one should determine the new )(ˆ nDx ,

i.e. update the knowledge.4. Knowledge updating for nu

)(:)1(ˆ)(ˆ xDUunDxnD unxx −∈−∈= .

Put )1()( −= nDnD xx and )(ˆ)()( nDnDn xxx ∩=∆ .

For 1=n , if 11 uu = determine

)(:)1( 1 xDuXxD ux ∈∈= ,

if 11 uu = determine

)(:)1(ˆ1 xDUuXxD ux −∈∈= .

If for all pi ≤ ii uu = (or ii uu ˆ= ), put XpDx =)(ˆ (or XpDx =)( ).

The successive estimation of x may be performed in a closed-loop learning sys-tem where iu is the sequence of the decisions. The decision making algorithm is

as follows:1. Put nu at the input of the plant and measure ny .

2. Test the property yn Dy ∈ , i.e. determine nj .

3. Determine )(nx∆ using the estimation algorithm with the knowledge valida-

tion and updating.4. Choose randomly nx from )(nx∆ , put nx into );,( xyuR and determine

)(xDu , or put nx directly into )(xDu if the set )(xDu may be determined

from R in an analytical form.5. Choose randomly 1+nu from )( nu xD .

At the beginning of the learning process iu should be chosen randomly from U.

The block scheme of the learning system is presented in Fig. 7.9 where G1 and G2

are the generators of random variables for the random choosing of nx from

)(nx∆ and 1+nu from )( nu xD , respectively. Under some assumptions concern-

ing the probability distributions describing the generators G1 and G2 it may beproved that )(nx∆ converges to x with probability 1 [25].

The a priori information on the unknown parameter x in the form of the cer-tainty distribution )(xhx given by an expert may be used in the learning system in

two ways:

126

7.5 Learning System 127

" # $

& & '

* + , - . / 0 1

2 3 '

4 ' 5 8 )( 1−nu xDyD

1−nx

nx )(nx∆

ny

nj

nu

nu

?yn Dy ∈

Fig. 7.9. Block scheme of learning system

1. The parameters of the distribution )(xhx may be successively adjusted in a

closed-loop adaptive system.2. The expert may change successively the form or the parameters of the distri-

bution )(xhx using the results of current observations.

Problems concerning the application of uncertain variables in complex uncer-tain systems with a distributed knowledge [22] and the application of uncertainvariables in learning systems with the successive knowledge updating form twomain directions of further researches in the area considered in this work.

127

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Index

analysis problem 17, 21, 33, 42, 59,64, 69, 74

C-certainty distribution 11certainty distribution 9

conditional 15joint 15, 25marginal 15normalized 13

certainty index 2classification problem 124CLASS-LOG 53combined logic 6complex of operations 116complex uncertain system 121continuous case 12control algorithm 80

deterministic 81control system 80controller

uncertain 89fuzzy 100

CONTROL-LOG 53crisp property 1C-uncertain variable 10

decision algorithmdeterministic 21, 22, 23, 25fuzzy 100random 95, 96relational 38, 39soft 107uncertain 23, 48uncertain logical 67

decision problem 18, 21, 35, 37, 46, 60,65, 66, 70, 75, 95, 99, 104, 117

decomposition 61, 123

deductive reasoning 62degree of certainty 2, 5degree of possibility 92degree of truth 92descriptive approach 119deterministic control algorithm 81determinization 22, 23, 25, 39discrete case 12

evaluating function 103execution time 116expert 5external disturbances 20, 37, 47

fact 57fuzzy controller 100fuzzy number 91, 96

input property 36, 57, 59

knowledge of the decision making 22,38, 39

knowledge of the plant 22, 38knowledge representation

based 33dynamical logical 71logical 57, 58of the dynamical plant 69relational 31, 109, 122resulting 33

knowledge updating 124, 125knowledge validation 124, 125knowledge-based recognition 112

L-logic 5Lc-logic 7

130 Index

Ln-logic 6Lp-logic 5learning process 124learning system 126linguistic fuzzy variable 92load distribution 116logic formula 57logic value 1, 2, 3logical level 58logical structures 58logic-algebraic method 61, 63

mean value 13measure 93

belief 93fuzzy 93necessity 93plausibility 93possibility 93

membership function 92, 96

operation 116optimal decision 46, 48, 52, 65optimal recognition 110output property 36, 57, 59, 99

pattern recognition 109plant

dynamical 69, 74functional 17, 18knowledge 22, 38relational 33, 69static 17uncertain 32

predicate 1prescriptive approach 119principle of equivalency 121probability density 91, 95

random variable 91, 95recognition 109recognition problem 109recursive algorithm 125recursive procedure 62, 71relation 31

relational plant 33, 69relational system 31

simple formula 57soft decision algorithm 107soft property 1soft variable 103stability 84stability condition 85system

dynamical 69functional 17, 18nonlinear 85relational 31time-varying 85, 86uncertain complex 121

task distribution 116two-level system 121

uncertain controller 81uncertain decision algorithm 23, 48uncertain logic 1, 2uncertain plant 32uncertain complex system 121uncertain variable 9uncertainty 86

[lecture notes in control and information sciences] uncertain logics, variables and systems volume...

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