Logical Linguistic Controllers - Science Direct

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ScienceDirect Procedia Computer Science 103 (2017) 629 – 636

XIIth International Symposium «Intelligent Systems», INTELS’16, 5-7 October 2016, Moscow, Russia

Logical linguistic controllers S.N. Vassilyeva, I.Y. Kudinovb, Y.I. Kudinovb*, F.F. Pashchenkoa a

Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia b Lipetsk State Technical University, Lipetsk, Russia

Abstract The logical-linguistic, analytical, learning and PID fuzzy controllers are considered, based on fuzzy logics of Zadeh. An overview of the Mamdani-type controllers, controllers based on TS-model and the ANFIS architecture, using neural network structure is provided. The conditions of optimality and stability of control systems with Mamdani fuzzy controllers are analyzed. The Sugeno dynamic models and the ANFIS adaptive models and the methods of learning developed on the basis of fuzzy controllers are considered. © by Elsevier B.V. by This is an open access article under the CC BY-NC-ND license © 2017 2017Published The Authors. Published Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium «Intelligent Systems». Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” Keywords: logical-linguistic controllers, stability conditions, the Sugeno and Mamdani controllers

1. Basic concepts and determinations We will consider such basic concepts, as fuzzy sets and some operations on them, linguistic variable and fuzzy relation. We will begin with description fuzzy input X and output Y of great numbers of the generalized fuzzy controller and control system. By the fuzzy set of Х on the universal set of Х = {x: xmin d x d xmax} well-organized totality of pairs is named1 X

{x, X ( x )}, x  X

where X(x) is a membership function of x to X, representing X in an interval [0,1]. On the fuzzy sets of Х1 and Х2 the

*

Corresponding author. E-mail address: [email protected]

1877-0509 © 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” doi:10.1016/j.procs.2017.01.091

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operations of association (copulas " or", " otherwise") are certain ( X 1 ‰ X 2 )( x )

X1( x) › X 2 ( x)

(

)

(

)

max X 1 ( x ), X 2 ( x )

and crossing (copula "and") ( X 1 ˆ X 2 )( x )

X1( x) š X 2 ( x)

min X 1 ( x ), X 2 ( x ) .

A linguistic variable is determined by three  x, Tx , X , in that х - is the name a variable, Tx

^T

1 x

`

, Tx2 ,...,Txk is a

l

great therm-number of linguistic values or therms T x , l 1, k , with the corresponding a membership l

function Tx (x ) , Х. set on an universal set Х. Fuzzy relation of R on the cartesian product of sets Х u Y = {(x, y): x  X, y  Y} there is a fuzzy set in Х u Y with the membership function of R (x, y), that characterizes the degree of compatibility of pair of x, y with R. If x, y points, i.e. х  Х = {x1, xk}, y  Y = {y1,…, ys}, a that relation is a matrix with elements R ( x l , y r ), l 1, k , r 1, s 1. There is the fuzzy input set E in the fuzzy PID controllers, characterizes some generalized error of e on the universal set of E = {e: emin d e d emax} and it can be certain well-organized totality of pairs2 {e, E (e)}, e  E

E

where E(e) is a membership function of e to E, representing E in an interval [0,1]. The output fuzzy set of PID regulator of U characterizes the control of u on the universal set of U = {u: umin d u d umax} and it can be certain as {u,U (u)}, u  U .

U

By analogy with the stated it is higher possible to set the linguistic values of variables of e and u, and also fuzzy relation of R on the cartesian product of E u U = {(e, u): e  E, u  U}. 2. Structure of Logical linguistic controllers (LLC). Such fuzzy controllers contain fuzzy sets, boolean operations of association, crossing and composition with the linguistic values of variables, unclear relation form one or a few boolean operations, and inference of unclear exit rule, at the known input. First LLC3-5 rendered very strong influence on subsequent researches in area of fuzzy control system and deserve that, in the beginning to expound basic principles of their construction, and then to show, as these principles are realized in one of controllers. Principles of construction of logical-linguistic controller we will consider on the example of the simplest, generalized controller with one input х (usually adjusting error) and one output y at (regulative or control influence), bound by fuzzy rules R1 : if x is X 1 , then y is Y 1 , else R 2 : if x is X 2 , then y is Y 2 , else n

n

R : if x is X , then y is Y

(1)

n

containing the fuzzy sets ХTТх и YTТy. In the algorithm of functioning of LLC in one or another form there are procedures of transformations

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(fuzzification Fuz) of the measured value of х0 of variable х in linguistic Х c fuzzy conclusion (fuzzy inference FI) of linguistic output Y c on the known input Х c and totalities of rules of R ={R1,., Rn} and transformations (defuzzification Def) of linguistic value of output Y c in actual у0 (fig. 1).

Fig. 1. Transformation of input to the fuzzy controller.

With the value of х0 the so-called "degenerate" fuzzy set of Х corresponds the input measurable variable of х; with the membership function ­° 1, if x x 0 ® 0 °¯ 0, if x z x

X c( x )

where х0 is the point named by singleton of Х c. We will write down expression of fuzzy conclusion for LLC set by the great number of rules (1) R1 : if x is X 1 , then y is Y 1 , otherwise R 2 : if x is X 2 , then y is Y 2 , otherwise

(2) R n : if x is X n , then y is Y n , x is X c, y is Y c

Values of truth of expressions of "х is Хθ", "y is Yθ" and "х is Хc" in rule (1) and parcel of expression of conclusion (2) determined in size corresponding membership functions of Х(х), Y(y) and Х (x) for х  Х, у Y. Every rule of R it is an fuzzy implication RT

if x is X T , then y is Y T

XT oYT

LLC as procedure of conclusion of Y c uses maxmin composition Zadeh 1 Y c( y )

( X c R )( y )

if › (X c( x ) š R ( x , y )), xX

n

n

V R T ( x, y )

where R ( x, y )

(3)

V X T ( x) š Y T ( y ) .

T 1

T 1

In a point х0 expression (3) is after the substitution Х c(х0) = 1 assumes an air Y c( y )

n

V R( x0 , y)

T 1

n

V ( X ( x 0 ) š Y ( y ))

T 1

(4)

The size of output of у0 can be defined by maximization y0

max Y c( y ) yY

(5)

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For calculations of "centre of gravity" of membership functionY(у) y0

ymax

ymin

y0

³ Y c( y ) dy

³ Y c( y ) dy

(6)

Most known and often quoted LLC worked out for a control by a steam-engine18, has four input (х1 is an error of pressure, equal to deviation of current from a set value; х2 is the error of speed, determined as х1; х3 is speed of change of х1; х4 is speed of change of х2) and two weekend (у1 is a change of heat; у2 is a change of pressure of steam) of variables. Turn-downs of input Х1,., Х4 and weekend of Y1 of variables of х1,., х4, у1 is broken up on 7 intervals with next linguistic values: РВ - positive large; РМ - positive AV; PS - positive small; NO - zero; NS negative small; NM - negative AV; NB - negative large. Turn-down of Y2 the weekend of variable of у2 consists of 5 intervals with the linguistic values of NB, NS, NO, PS, PB, defined on them. The indicated linguistic values form two term-sets Т1 ={NB, NM, NS, NO, PS, PM, PB} and Т2 ={NB, NS, NO, PS, PB}. An fuzzy controller consists of two of rules, j = 1, 2, qualificatory the change of heat of у1 and pressures of у2 R1j : if x1 is X 11 j and x2 is X 21 j and R 2j : if x1 is X 12j and x2 is X 22 j and n

n

and x4 is X 41 j then y j is Y 1j , else and x4 is X 42 j then y j is Y j2 , else

n

n

(7)

n

R j j : if x1 is X 1 jj a nd x2 is X 2 jj and and x4 is X 4 jj then y j is Y j j , x1 is X 1c j and x2 is X 2c j and and x4 is X 4c j y j is Y1c j Tj

Expressions "хi is X ij " and "хi is X ijc " in the parcel of expression of conclusion (5) with the values of truth, Tj

by the set corresponding membership functions X ij (х) and X ijc (xj), j = 1, 2,, incorporated by a logical copula " and", realizing the operation of crossing. Then truth of the left part of Tj –th rule is determined as T

T

T

X 1 jj ( x1 ) š X 2 jj ( x2 ) š ... š X 4 jj ( x4 )

(8)

and truth of parcel – X 1c j ( x1 ) š X 2c j ( x 2 ) š ... š X 4c j ( x 4 ), j

(9)

1, 2

Expression of maxmin composition (3) of signs is a kind Y jc ( y j )



V

x1X 1 ......... x 4 X 4

>X c ( x ) š X c ( x

where R j x1 ,..., x4 , y j

2)

@



š ... š X 4c ( x 4 ) š R j x1 ,..., x 4 , y j



1

1



V X 1 jj ( x1 ) š X 2 jj ( x2 ) š ... š X 4 jj ( x4 ) š Y j j ( y j ) .

nj

2



T

T

T

(10)



T

Tj 1

0 0 As x1 ,..., x4 are singleton of sets X ic , that after a substitution X ic( xi ) implication of Mamdani 0

1, i

1, 4 , in (10) we will get the

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Y jc ( y j )



0 1

0 4

R j x ,..., x , y j



nj



T

T

T

T



V X 1 jj ( x10 ) š X 2 jj ( x 20 ) š ... š X 4 jj ( x 40 ) š Y j j ( y j ) , j 1,2

Tj 1

(11)

Actual output values y1 and y2 determined on the basis of the found membership function Y1c( y1 ) and 0

0

Y2c( y 2 ) by means of correlations (5) and (6).

Considered higher LLR with an implication (11) named the Mamdani controller. If in (4) to accept R(x, y) = 1, then the Mamdani controller will have static description of multiposition relay (fig. of 2а) in that the linearness and continuity of output is violated at in relation to the input of х.

Fig. 2. Static description of fuzzy controllers with the implications of Mamdany (a) and Lucasievizh (b).

Attempts to remove the indicated defects were undertaken in works6-10 and consisted in using of implication of Lukasiewicz as an fuzzy relation R(x, y) in (3)

RL ( x, y ) 1 š >1  X ( x )  Y ( y ) @

(12)

Really, if to accept RL(x, y) = 1, then implication (12) in expression (4) with one input

Y c( y )

n

V RLT ( x, y )

T 1

allows to get more perfect LLC having static description of linear function with a satiation (fig. 2b). However, much more application was found by controllers and fuzzy control systems, using an Zadeh implication. Them plenty of researches in that regulators and control system are presented by fuzzy differential x

X (t )

X (t ) D R

(13)

and difference equations

X t 1

Xt D R

(14)

In the first publications11-15 stability and controlability of the fuzzy dynamic systems of type (13) was analysed and (14). For these aims the functions of Liapunov were attracted 11, 12 and methods estimations of stability, leaning against such specific concepts of fuzzy sets, as energy of fuzzy set of Xt and fuzzy relation of R, descriptions of spades of fuzzy sets and measure of their closeness13-15.

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3. Synthesis of LLC The basic lack of offered approach consists in absence of concrete recommendations on a choice or synthesis of unclear controllers and control system, possessing certain dynamic properties (controlability, stability and quality of adjusting processes). The first attempt of synthesis of optimal in sense a minimum of error of adjusting of LLC was undertaken in close system of control (fig. 3) on the basis of set by the tables of fuzzy operators of object 16.

Fig. 3. Scheme of close system. Table 1. U

NB

NM

NS

ZE

PS

PM

x

PB

Y

Y NB NM NS ZE PS PM PB

FOU =

ZE PB NS PB NM NM NB

NS ZE NS PB ZE NS NB

NS NB ZE PB PS ZE NB

NM NB NM ZE NB PS ZE

NB NB NM PS PB ZE PB

NM NM NB PS PM NS PB

NS NB NB PS PB NB PB

and the optimal closed-loop system. Table 2. X Y NB NM NS ZE PS PM PB

F* =

NB

NM

NS

ZE

PS

PM

PB

x

Y ZE PS PS PB PB PB PB

NS ZE PS PM PM PB PB

NS NS ZE PS PS PM PB

NM NS NS ZE PS PS PM

NB NM NS NS ZE PS PS

NB NB NM NM NS ZE PS

NB NB NB NB NM NS ZE

For the compactness of exposition we will present in an analytical form tabular operators of control object (О) x

Y

FOU (Y , U )

(15)

optimal close system x

Y

F * (Y , X )

and the synthesized controller (C)

(16)

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U

FCX ( X , Y )

(17)

there are linguistic variables in that, characterizing the task of Х, output Y and his speed, control U and indignation W, take on values from терм-множества T ={NB, NM, NS, ZE, PS, PM, PB}. The operator of object (15) is built on results research of his static and dynamic descriptions. For the tabular operator of object (15) it easily to get reverse relation to a control U U

1 FOU (Y , Y )

(18)

and operator F* optimal close system, coming from the chart of linguistic dynamics (fig. 4) and next heuristic considerations.

Fig. 4. Chart of linguistic dynamics.

Points 1, 2,…, 7 on graphic arts characterize equality of linguistic values of reference Х and exit of Y, and also x

minimum speed of output Y

ZE , allowing to prevent overshoot. As far as the increase of difference between Х x

and Y, i.e. control errors, the size of speed of output Y , must increase directed toward one of these points. Direction x

and sizes of pointers Y correspond to taken on linguistic values. For example, in a point «ο» graphic arts of linguistic dynamics of optimal close system certain one set of data of tabular operator x

x

F* : X = PS, Y = PM, Y NS and the corresponding rule if X = PS and Y = PM, then Y : NS . Now let us formulate the problem of designing the optimal fuzzy controller. For all the linguistic values of the reference X and output Y, by the operator of the optimal closed-loop system (16) and the inverse operator of the object (18) to determine the control U, i.e., triples , form the operator of the controller (17). Consider the procedure of determining a control U* in the three at X* = NS, Y* = PM. The x

substitution X* = NS and Y* = PM in table F* (operator of the optimal closed-loop system) gives Y

*

NM . For

x *

Y* = PM and Y NM from the table, FOU get U* = NM, i.e., implement the inverse operator of the facility and define the desired three . 1

In general case a reverse operator FOU is not unambigiuous. The found optimal operator of controller becomes ambiguous, that considerably reduces the practical value of such going near the synthesis of LLC. Further development of methodology of synthesis of tabular operator of controller got in-process17. On the basis of static descriptions and transitional functions of aperiodic link of first-order, formative operators of object on the channels of control of FOU and indignations of FOW, and also quality description of adjusting process, it was

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succeeded to synthesize an fuzzy controller operating at the change of reference X, FPc compensator, removing influence of indignation of W on an output FK

FX ‰ FY ‰ FE and

FW ‰ FY ‰ FE .

Here ‰ - it is an operation of association of components realizing three phases of control. On the first phase of control of UX = FX(X, E) regulator at the substantial change of task of Х in the beginning the size of UX is set by maximum. As soon as the output size of Y will attain some vinicity of reference Х, by static recommendation of channel of U - Y gets out managing influence at that the set value of exit will become near to the reference. Control UW or exit the constituent of UW = FW(W) of compensator FK formed, coming from two principles of fuzzy invariance. By static recommendation of channel of W - Y is estimated possible reaction of output YW on indignation of W and the control UW, defiant change of output Y, equal on a size and reverse on a sign to the value YW, is determined. It is succeeded to provide partial indemnification of indignation or independence (invariance) of output Y the same from indignation W. The component of FY serves for the removal of overshoot, and component of FE - static error. Coming from the offered principles of forming of components of regulator and compensator, the methods of synthesis of tabular operators of FX, FW, FY, FE allowing to provide the required quality of adjusting of temperature on the output of stove of pyrolysis of acetone, were worked out17. The near going near the synthesis of tabular fuzzy regulator were offered for a control by the rectification setting 18 and by other chemical objects19. The main disadvantage LLC table type can be attributed to their limited dimensions. As for the subjectivity of the choice of intervals and the corresponding values of linguistic variables, then this circumstance in anthropocentric applications of on-board intelligence provides a natural and intellectual interface "human-machine", which promotes to combine the key strengths of the operator and the machine. This is demonstrated, for example, by technologies of case-based reasoning in on-board operatively-advising systems as "reinforcing" or " alternative-free" technologies20, 21, 22 in such difficult tasks as automation of goal setting, especially in hard real-time operating. References 1. Zadeh LA. Concept of linguistic variable and its application to making approximate decisions. Moscow:: Mir; 1976. 165 p. 2. Aliev R.A, Abdikeev NM. Shakhnazarov MM. Production system with artificial intelligence. Moscow: Radio and communication; 1990. 264p. 3. Mamdani EH, Assilian S. An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. Man-Machine Studies. 1975; 7: 1-13. 4. Kickert WJM, Van Nauta Lemke HR. Application of fuzzy controller in a warm water plant. Automatica. 1976; 12: 301-08. 5. King P.J., Mamdani E.H. Application of fuzzy control system to industrial processes // Automatica. 1977. N13, p.235-242. 6. Giles R. Lukasiewicz. Logic and fuzzy set theory. Int. J. Man-Machine Studies. 1976; 8: 313-27. 7. Braae M., Rutherford DA. Fuzzy relations in a control setting Kybernetes. 1978; 7: 185-8. 8. Kickert WJM., Mamdani EH. Analysis of a fuzzy logic controller. Fuzzy Sets and Systems. 1978; 1: 29-44. 9. Pedrycz W. On the use of fuzzy lukasiewicz logic for fuzzy control. Archiwum automatyki i telemechaniki. 1980; 25; 3: 301-14. 10. Baldwin JF, Guild NCF. Modelling controllers using fuzzy relations. Kybernetes. 1980; 9; 3: 223-29. 11. Glass M. Theory of fuzzy systems. Fuzzy Sets and Systems. 1983; 10: 65-77. 12. Glass M. Invariance and stability of fuzzy systems. J. Math. Analysis and Appl. 1984; 99; 1: 299-319. 13. Kiszka IB, Gupta MM, Nikiforuk PN. Energetistic stability of fuzzy dynamic systems. IEEE Trans. Syst. Man and Cybern. 1985 SMC-15; 5: 783-92. 14. Jain R. Outline of an approach for the analysis of fuzzy systems. Int. J. Control. 1976; 27 3: 627-40. 15. Tong RM. Analysis and control of fuzzy systems using finite discrete relations. Int. J. Control. 1978; 27; 3: 431-40. 16. Braae M, Rutherford DA. Theoretical and linguistic aspects of the fuzzy controller. Automatica. 1979; 12: 553-77. 17. Kudinov YI. The synthesis of a fuzzy control system. Izv. Theory and control systems 1999; 1: 166-172. 18. Kudinov YI., Dorokhov IN. Control of the distillation plant on the basis of fuzzy sets. Theoretical foundations of chemical engineering. 1991. 25; 4: 563-69. 19. Kudinov YI, Dorokhov IN. A new principle of construction of controllers for complex chemical-technological objects on the basis of quality information. Reports of RAS. 1994; 336; 1: 75-9. 20. Vassilyev SN, Zherlov AK, Fedosov EA, Fedunov BE. Intelligent control of dynamical systems. Moscow: Fizmatlit; 2000.. 352p. 21. Fedunov BE. On-board operatively advising expert systems of tactical level for manned aircraft. International aerospace magazine "Aviapanorama" 2016; 1: 9-20. 22. Zheltov SYu, Fedunov BE. Operational goal-setting for anthropocentric objects from the point of view of the conceptual model "Etap", I, II Izv. Russian Academy of Sciences. TiSU. 2015; 3: 57-71; 2016; 3: 55-69.