Distributed Fuzzy Control of Multivariable Processes

Copyright © IFAC Algorithms and Architectures for Real-Time Control. Ostend. Belgium. 1995

DISTRIBUTED FUZZY CONTROL OF MULTIVARIABLE PROCESSES Alexander Gel:0V, Hans-Joachim Nern

University of Wuppertal Department of Automatic Control and Technical Cybernetics Fuhlrottstreet 10. 42097 Wuppertal. Germany "

Abstract: The paper describes briefly some recent results in distributed fuzzy control of multivariable processes. DefInitions and theorems with regard to this type of control are formulated. Methods of decentralized and multilayer fuzzy control are presented. Numerical examples are shown for illustration of the theoretical results. The number of fuzzy relations and the respective on-line computations is reduced . In this way. the real time control implementation is facilitated. Keywords : Fuzzy control, hierarchically intelligent control, multi variable systems, multilevel control. decentralized control. knowledge-based systems .

1. INTRODUCfION

corresponding control problem becomes non-trivial and does not seem to have an easy solution.

It is well known that many real control processes are characterized by both quantitative and qualitative complexity. The quantitative complexity is expressed in a large number of state variables. respectively high dimensional mathematical model. The qualitative complexity is expressed in uncertain behaviour, respectively approximately known mathematical model.

Modern control theory and practice have reacted accordingly to the above mentioned new challenges of the day by utilizing the latest achievements in computer technology and artificial intelligence distributed computation and intelligent operation. In this respect, a new fIeld has emerged in the last decade, called WDistributed intelligent control systems W (Saridis, 1988). However, most of the familiar works in this fIeld are still either on empirical or conceptual level and this is a signifIcant drawback.

If the above two aspects of complexity are considered separately. the corresponding control problem seems to be easy to tackle with. On one

hand, large scale systems theory has existed for more than 20 years since the pioneering work (Mesarovic et al. 1973) and has proved its capabilities in solving high dimensional control problems on the basis decomposition, of hierarchy. decentralization and multilayers (Jamshidi. 1983). On the other hand, the fuzzy linguistic (Zadeh. approach, starting from 1973), is almost at the same age and has shown its advantages in solving approximately formulated control problems on the basis of linguistic reasoning and logical inference (Pedrycz, 1993). However, if both aspects of complexity are considered together, the

One of the most important problems in fuzzy control is the computational complexity of control algorithms which has a significant impact on their real time implementation. Many investigations have been made recently with the aim to reduce this complexity by spatial or temporal decomposition of the rule basis into subsystems or layers (Xu, 1991; Raju et aI, 1992; De Silva. 1993; Sustal, 1993). However, these and most other similar works have limited application which is due to the fact that the fuzzy linguistic approach has been developed

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mainly for simple and low-order systems . Similar conclusions are presented in (Titli, 1992; Gegov, 1996) wbere tbe necessity of extending tbis approacb for multi variable and large scale systems is pointed out.

problem has additional computational complexity as tbe power of the universal sets f, defining the dimension of the fuzzy relations, could be also substantial. Therefore, it would be reasonable to simplify the control law (2) by applying spatial or temporal decomposition, i.e. to replace it by suitable decentralized or multilayer control laws. These two types of decomposition are mutually complementing to each other alternatives of the distributed control approach.

This paper describes briefly some recent results in distributed fuzzy control of multivariable systems. Tbese results bave been obtained witbin a systematic investigation, extending over tbe empirical and conceptual level of tbe majority of works in tbis field till now. Tbe paper is structured as follows : section 2 gives problem statement, section 3 is concerned witb decentralized fuzzy control of multi variable systems, section 4 considers tbe problem of multilayer fuzzy control of sucb systems and section 5 gives analysis of results.

3. DECENTRALIZED FUZZY CONTROL One way to reduce tbe number of fuzzy relations is to find the conditions under which the control law (2) may be spatially decomposed (decentralized) into subsystems (Gegov and Frank, 1995). For this purpose, the system (1) should be decomposed into single-in put-single-output (SISO) subsystems as similar considerations apply to non-SISO (NSISO) subsystems. The number of subsystems is denoted by N wbere N=n=m in the case of SISO subsystems

2. PROBLEM STATEMENT A multi variable process (system) described by tbe linguistic control rules

can

be

If x 1(1) .and. xn (1)' tben u 1(1) .and. um (I)' - - - - - - - - - - - - - - - - - -

(1)

Tbe control law (2) can be represented by

if x 1(b) .and. xn(b) , then ul(b) .and. um(h) where xj(s)' j=l,n and ui(s)' i=l,m are respectively the j-th input (state) and the i-th output (control) fuzzy variables in tbe s-th rule, s=l,h (Gupta et aI, 1986). Both variables x and u are defined in universal sets X and U of equal power f, i.e. XEE n , UEE m , X,UEE f where E is a vector space. For simplicity of notations, tbe above rules contain only the fuzzy variables and not tbeir linguistic values.

[ ~I]T [~I]T. [~11'" ~IN] uN

xN RNl ... RNN where • is the «(), 0) operator.

The detailed presentation of (4) is given by

~~]T _- N . n [~J]T .

[

~~

1

o

J

. /~ J1

(5)

. ~~~ J1

Tbe development of (5) for a given element of the universal set u~, i=I,N, t=l,f leads to the following expression:

(2)

u~

The symbol .. 0" in (2) denotes the max-min composition and Rji E Ef.f, j = l,n, i = I,m are two-dimensional fuzzy relations which can be calculated as follows: h S~I(Xj(S) () ui(s» ' j=l,n, i=l,m.

~~

j=1

[~H . . ~H]

where the upper index t = l,f stands for the respective element of the universal set and rn, j,i=I,N, s,t=l,f are elements of the fuzzy relation Rji"

The system (1) can be also presented by m single-output systems (i~e, 1990). In this case, the wif w parts are repeated for each output variable up i = I,m and the following control law is obtained: u·1

(4)

1

N

s j=1 s=l J

n

f

[U (x . () rst)l

(6)

ji'

If the system (1) is decentralized into N SISO

subsystems, be

(3)

tbe

corresponding

control

law

ui = xi 0 RH' iE[l,N]

where the notation i E [1 ,N] is equivalent i = 1,N in the case of full decentralizability.

It is evident from the control law (2) that the number of fuzzy relations is n.m. However, this could be quite a big number for multi variable systems and time constraints in on-line control computations may be violated. Moreover, the

will

(7)

to

The considered centralized and decentralized control laws (2) and (7) are shown schematically on Figures 1-2 .

456

of the interactional fuzzy relations Rji' j = 1,N, j*i. 1

A numerical example is presented below for illustration of the theoretical results. The linguistic control rules in this example can be observed in multi-tank systems where the state variables are the liquid levels in each tank and the control variables are the inflow rates of the liquid in the tanks. The task of the control system in this case is to maintain the liquid at a desired level despite of the presence of disturbing leaks .

~I

Fig.l. Multivariable system by centralized fuzzy control

xl~

--+1

0

RI 1

1--+ --+

A two-tank system is considered which is illustrated on Figure 3. Each tank can be filled with liquid through a separate inflow channel and besides this the tanks are connected. Therefore, the respective fuzzy control system has two inputs and two outputs. The state and control variables can take the following linguistic values : S - small, M - medium, B big. These values are presented by :

u1

Fig.2. Multivariable system by decentralized fuzzy control

S

[1.0, 0.5, 0.0), M = [0.5, 1.0, 0.5],

B

[0 .0, 0.5, 1.0].

(9)

Definition 1. The relation R .., j=l,n, i=l,m is )1

relation in a a local (interactional) fuzzy multivariable decentralized presentation of a system if the corresponding state and control variables x . and u . belong (do not belong) to ) 1 one and the same subsystem.

_I

1_

"1-->-- __ '!

_I~I

-I 1

1

2. The multivariable fuzzy system described by the control law (2), is (fully) decentralizable in the form partially (7) if the linguistic values of at least one control variable (all control variables) in both Definition

__', _--<--., 1 1-

1

Fig.3 A two-tank fuzzy control system: decentralized case

(1),

The system is linguistic rules:

cases coincide.

described

by

the

following

If x 1(1)=M, ~(1)=M, then u 1(1)=B, u2 (1)=B, (10)

Theorem 1. The multivariable fuzzy system (1), described by the control law (2), is partially (fully) decentralizable in the form (7) if all state variables are represented by normal fuzzy sets and the following condition holds for all elements of the universal set u~, t=l/ of at least one control variable (all control variables) u ' i=l,N: i rst s-l/:s It -1" N )' *i , s=lJ ii' ji' )' -

if x 1(2) =B, ~(2) =B, then u 1(2) =M, u2 (2) =M . The respective fuzzy relations are calculated by (3) as follows: R11 = Rzl = Rl2 = R22 = (M 0.0 0.5 0.5 ] 0.5 0.5 1.0 . [ 0.5 1.0 0.5

n

B) U (B

n

M) = (11)

(8)

It is evident from (11) that condition (8) holds in the form of an equality and therefore the system (10) can be fully decentralized in two SISO subsystems .

The proof of the above theorem is given in (Gegov and Frank, 1995). The verbal interpretation of (8) is the following: for each element of the universal set u~, iE[I,N), t=l,f, each element in the t-th column of the local fuzzy relation R ii , i E (1 ,N] should be smaller than or equal to any element in the t-th columns

To verify the above result, it is supposed that the linguistic values of state variables are x("'B and x2 =B. The calculation of the linguistic

457

values of the control variables u l and u2 by the centralized and the decentralized control laws (2) and (7) leads to the following coinciding results:

n (B

u 1 = (B 0 R 11)

0

R2l )

on

.. [0.5, 1.0, 0.5] = M,

u2 = (B

0

R 12)

n (B

The relations in the first N-l equations in (17) are calculated by h (18) 0j_I,j = k~I(Xj-I(k) () Xj(k» ' i=I,m the basis of the linguistic control rules The relation in the last equation of (17) is calculated by (3). (1).

(12) 0

R22 )

= [0.5, 1.0, 0.5] = M,

uI = B

0

u2

0

B

R11 = [0.5, 1.0, 0.5] = M,

The considered original and multilayer control laws (2) and (17) are shown schematically on Figures 4-5.

(13)

R22 = [0 .5 , 1.0, 0.5] = M.

4. MULT1LAYER FUZZY CONTROL

o RIl

Another way to reduce the number of fuzzy relations is to find the conditions under which the control law (2) may be temporally decomposed into layers (Gegov and Frank, 1994) . For this purpose, the system (I) should be decomposed into 5150 layers as similar considerations apply to N5150 layers . The number of layers is denoted by N where N=n in the case of 5150 layers. The input of each layer is output of the next layer. The only exceptions in this respect are the input of the first layer and the output of the last layer which are not an output and an input of other layers . The control law (2) can be represented by T [ ~ll ~Im .

. ..

~I

o Rn 1

o RIm o Rnm

h~I -;;-I ~

I - I ~-

h~I -;;-I~

I-I~

~

uI

~

um

Fig.4. Multivariable system by original fuzzy control

(14)

RNI ... RNm

Fig.S. Multivariable system by multilayer fuzzy control

The detailed presentation of (14) is given by

~~.. f T= j =n 1 [~]lT ~Hl · . o [~H . . .f . ff

[

Definition 3. The fuzzy relation Rki • i=l.m is dominant in a multi/ayer presentation of a multi variable system if the other fuzzy relations R • j *- k do not affect the linguistic ji values of the control variable ui ' i=l .m.

(15)

N

ui

fl

Xj

.

r ji ... r ji

The development of (IS) for a given element of the universal set u~ , i= I,m, t= I,f leads to the following expression : t

u· 1

N

f

n

s

st J

[U (x . () r )] .

j=I s=I J

Definition 4. The multivariable system (1). described by the control law (2), is partially (fully) decomposable into layers in the form (17) if the linguistic values of at least one control variable (all control variables) in both

(16)

Let the system (1) be decomposed into N 5150 layers with regard to each control variable up i = 1,m as follows

cases coincide.

The order of state variables in the respective layers in (17) is conditional, i.e . the lower indices of these variables correspond to the indices of the respective layers and not to the original indices in the linguistic control rules (1). Therefore, there may exist different permutations of these variables .

(17) XN = x N_I

0

ui = x N

0N,i .

0

0N-l.W

ff where 0 . 1 .eEf . f , j=2,N and ON l· eE . . ' J- ,J

458

Let a given permutation of the state variables v in (17) be denoted by Px ' vE[l,N!] where number of the N! = 1.2 ... N-1.N is all permutations. The original index of the input state variable of the first layer in the v-th permutation is denoted by vI.

linguistic control rules in this example can be also observed in multi-tank systems , as it was already shown in the previuos section. A two-tank system is considered which is illustrated on Figure 7. One of the tanks can be filled with liquid through a separate inflow channel while the other tank is only interconnected . Therefore, the respective fuzzy control system has two inputs and one output. The state and control variables can take the following linguistic values : S small, M medium, B - big. These values are presented by:

(i), Theorem 2. The multivariable system described by the control law (2), is partially (fully) decomposable into layers in the form (17) if all state variables are represented by normal fuzzy sets and the following conditions hold for all elements of the universal set u~, t = i J of at least one control variable (all control variables) ui' i=i,m and for at least vE[i,N!j: one permutation

P;,

r~> s=iJ

S

rj~, ViE[i,NJ, j=i,N, j.;:v i ,

S

[1.0, 0.5, 0.5] , M = [0.5, 1.0, 0.5].

B

[0.5, 0.5, 1.0].

-,

(19)

N R

. = ( 0 D . i .) 0 DN ., ViE[l,NJ. vii j=2)- J ,I

(20)

uC--+_

ui = xl = xl

0

0

then

-,

x

2

"1

Fig.7. A two-tank fuzzy control system: multilayer case The system is linguistic rules :

j

substituted

,-

- ,1

If all intermediate state variables x ' j = I,N in

successively

XI

- -- - -------,

The proof of the above theorem is given in (Gegov and Frank, 1994) . Condition (19) has similar interpretation as condition (8) while condition (20) reflects additional requirements with regard to the layers.

(17) are expression

(22)

the

described

by

the

If X 1(1)=M, x2 (1)=M, then u 1(1)=B.

D 1,2

0

D 1 ,i = xl

...

*

0

D N- 1,N

0

D 1 , i' i=l,m

following (23)

if x I (2) =B, ~(2) =B, then u 1(2) =M.

DN,i (21)

The respective fuzzy relations in the law (2) are calculated by (3) as follows :

is obtained where the operators "0" and "*" are equivalent due the block-wise influence of the latter. This expression is an equivalent unilayer form of (17), as shown on Figure 6.

Rn ~

= ~1 = (M

control

() B) U (B () M)

0.5 0.5 0.5 ] 0.5 0.5 1.0 . [ 0.5 1.0 0.5

(24)

The system should be decomposed in two SISO layers, each of them with one input state variable. In this case, the respective relations are calculated as follows : 0.5 0.5 0.5 ] D1 2 = (M () M) U (B () B) = 0.5 1.00.5 ,(25) [ , 0.50.5 1.0

Fig.6. Equivalent unilayer presentation Expression (21) shows the relation between xl and u ' i= I ,m as all other state variables are i taken into account implicitly by Dj _1,j' j=2,N and D ·. The index 1 of the state variable is N ,I conditional and denotes the first element in the considered permutation P~, vE[I,N!] of xj , j=l ,N.

(M () B) U (B () M) =

0.5 0.5 0.5 ] 0.5 0.5 1.0 , [ 0.5 1.0 0.5

0.5 0.5 0.5 ] 0.5 0.5 1.0 . [ 0.5 1.00.5

(26)

It is evident from (25) and (26) that conditions (19)-(20) hold and therefore the system (23) may be fully decomposed in two SISO subsystems.

A numerical example is presented below for theoretical results. The of the illustration

459

To verify the above result, it is supposed that the linguistic values of state variables are xI =M and x2 =M . The calculation of the linguistic values of the control variable u 1 by the original and the multilayer control laws (2) and (21) leads to the following coinciding results : u 1 = (M

0

Rn)

[0.5, 0.5, 1.0]

n

(M

0

Fuzzy Control of Multivariable Systems. Kluwer Academic Publishers, Dordrecht. Gegov, A. and P. Frank (1994, submitted for possible publication). Multilayer fuzzy control of multi variable systems by passive decomposition. IEEE Transactions on Systems Man and Cybernetics. Gegov, A. and P. Frank (1995, to appear). Decentralized fuzzy control of multi variable systems by passive decomposition. Intelligent Systems Engineering. Gupta, M., 1. Kiszka and G. Trojan (1986) . of fuzzy control Multivariable structure systems, IEEE Transactions on Systems, Man and Cybernetics, 1615, 638-655. 1amshidi, M. (1983) . Large Scale Systems: Modelling and Control. North Holland , Amsterdam. Lee, C. (1990). Fuzzy logic in control systems: fuzzy logic controller , part 11. IEEE Transactions on Systems, Man and Cybernetics, 20/2, 419-435 . Mesarovic , M., D. Macko and Y. Takahara (1970) . Theory of Hierarchical Multilevel Systems. Academic Press , New York. Pedrycz, W. (1993) Fuzzy Control and Fuzzy Systems. 10hn Wiley & Sons, New York.

~l) =

B,

(27)

[0.5 , 0.5, 1.0] = B.

(28)

5. ANALYSIS OF RESULTS The main advantage of the presented results is the reduction of the number of fuzzy relations which is due to the distributed (decentralized or multilayer) computation of control actions . In this case , the number of on-line computations is also reduced and the real-time control implementation is facilitated. For instance, in the case of full decomposability, the number of fuzzy relations in the decentralized control law (7) is N and in the multilayer control law (21) this number is 1. It is evident 'that there is a significant reduction in both these cases in comparison to the respective number in the control law (2) which is n.m.

Raju, G. , 1. Zhou and R. Kisner (1992) . International Hierarchical fuzzy control. Journal of Control, 5415, 1201-1216 . Saridis , G. (1988). Intelligent machines : distributed vs. hierarchical intelligence. IFACIIMACS Symposium on Distributed Intelligence Systems: Methods and Applications, Varna, Bulgaria, 1, 29-34 . Sustal, 1. (1993). On the equivalence of some fuzzy control systems. European Congress on Fuzzy and Intelligent Technologies, Aachen, Germany, 3, 1474-1477. Titli, A. (1992). Facing up to complex problems by introducing fuzzy logic in control. IFACIIFORSIIMACS Symposium on Large Scale Systems: Theory and Applications, Beijing, China, 1, 202-206. Xu, C. (1991) . Linguistic decoupling control of fuzzy multi variable processes. Fuzzy Sets and Systems , 49, 209-217 . Zadeh, L. (1973). The Concept of a Linguistic Applications Variable and its to Approximate Reasoning. American Elsevier, New York.

The results can be applied if certain conditions are fulfilled . These conditions are checked entirely off-line and therefore the respective computational time is not of great importance. However, it would be reasonable to fmd possibilities for relaxing the conditions which would expand the application scope of the results. This relaxation could be achieved for instance by suitable choice of the fuzzy membership values of variables in the fuzzification step. The relaxation would give a more approximative character of the decomposition algorithms which are too accurate for the considered fuzzy approach at this stage. 6. ACKNOWLEDGEMENT The first author would like to thank the Alexander von Humboldt Foundation from Germany for the financial support with regard to this work. 7. REFERENCES De Silva , decoupling in American Control 1, 760-764. Gegov, A.

C. (1993) . Knowledge base fuzzy logic control systems. Conference, San Francisco, USA, (1996,

to

appear) .

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