Tuning of a Decentralized Multivariable Fuzzy Controller Alberto SORIA LOPEZ, Jean-Claude LAFONT Institut National des Télécommunications - Evry e-mail: [email protected] Claude BARRET Centre d’Études Mécaniques d’Ile-de-France - Université d’Evry Val d’Essonne e-mail: [email protected]

ABSTRACT: Multivariable centralized and decentralized fuzzy relation matrixes are calculated in order to define a passive decentralization index. This index is then used as objective function information in the tuning of the decentralized fuzzy controller’s parameters. Results on a MIMO system are shown. Introduction Research on the application of fuzzy set theory to the design of control systems has led to interest in decomposition of multivariable fuzzy systems. Decomposition of multivariable control rules is preferable since it alleviates the complexity of the problem. Gupta et al. [4] proposed a structure and analysis of multivariable fuzzy systems; this method permits a simplification from multidimensional fuzzy relations to a two dimension relation. PYEONG et al. [5] proposed a simplification index that measures the inference error when using the two dimension relations rather than the multidimensional fuzzy relation. Gegov [2, 3] has developed further the theoretical analysis of multivariable systems taking up subjects as decentralization, large scale systems theory, etc. Here we are interested in the synthesis of fuzzy logic controllers (FLC) for multivariable systems. In a former communication [8], we have proposed a tuning structure for a FLC. However, the proposed decentralized structure (two independent FLC’s) does not take in account the validity of such a structure. Our objective here is to work on this point integrating an analysis on the structure used. Given an optimization technique for the tuning of the FLC, we propose to incorporate a decentralization index in the objective function in order to take in account coupling effects during the optimization process .

First of all, we present the definition of the two-dimension fuzzy relations. Next, an analysis of these relations will permit to establish the conditions to decentralize them. Based on this analysis we define a decentralization index CDP . Afterwards, we detail tuning of the controller and we present some results. Fuzzy Relations The definition and analysis is taken from the work done by Gegov [2, 3]. A multivariable system can be controlled by the following linguistic rules:

if x1(1) andKand x j (1) then x1(1) andKand x i (1) 1)

M if x1( k ) andKand x j ( k ) then x1( k ) andKand x i ( k )

where x j ( s ) is the j th input (state) and the i th output (control) fuzzy variables in the k th rule. Both

variables

are

defined

in

universe

of

discourse

of

equal

power

x ∈ E N , u ∈ E M ; XU ∈ E f where E is a vector space. In this case we have N inputs (index j = 1K N ) and M outputs (index i = 1K M ) and S rules (index k = 1... S ). The system described by ( 1 ) can be represented approximately by M single output systems and the following control law is obtained: N

ui =

U (x j =1

j

o R ji

)

; j = 1.. M

2)

The o symbol denotes the max-min composition and R ji is a two dimension fuzzy relation calculated by: S

(

R ji = U x j ( k ) I ui( k ) k =1

)

; j = 1.. N ; i = 1.. M ; R ji ∈ E f

The control law in ( 2 ) can be presented as:

3)

T

T

 u1   x1   R11       M  =  M  ∗ M uN   x N   R1N

L R1 M   M  → u = x∗ m L RNM 

4)

where ∗ is the o, ∩ operator. The detailed presentation of the elements of relation R ji is:

r

ab ji

S

(

= U x aj ( k ) I uib( k ) k =1

)

; j = 1.. N ; i = 1.. M ; a, b = 1K f

5)

The upper index stands for the respective element in the universe of discourse and r jiab are elements of the fuzzy relation R ji .

Fuzzy Relation Analysis The objective of this analysis is to establish the conditions allowing the application of the control law:

ui = x i o Rii

; i = 1K N

6) As shown in Figure 1, the “passive decomposition” of fuzzy relations Rij is the elimination of the relations Rij , i ≠ j letting the original control law unchanged during decomposition. x1 x2

R 11 ^

. . . .

R N1

. xN

x1

u

x2

u

.

2

2 .

R 1M

.

u1

^

==>

P a s s iv e D e c o m p o s itio n

==>

.

.

.

.

.

xN

R NM

u1

R 11

.

R NM

Figure 1. Passive decomposition principle The multivariable fuzzy system ( 1 )

described by the control law ( 4 ) is totally

decentralizable if the input and output variables are normal fuzzy sets and the following condition holds for all elements of the universe of discourse uit , t = 1K f of all variables ui , i ∈[1K N ] :

riiab ≤ r jiab

; a, b = 1K f ; j ≠ i 7)

The details of the proof of this theorem can be found in Gegov [2, 3].

Passive Decentralization Index We can define the distance that exists between riiab and r jiab ;the obtained value can represent the decentralization difference, based on equation, given by : M N

CDP = ∑ riiab − r jiab

; j ≠ i; a , b = 1K f

i =1 j =1

8)

Tuning of the Fuzzy Logic Controller (FLC) We use an optimization techniques to tune the parameters of the FLC. The structure of such an approach is presented in Figure 2. We take a Mamdani’s type controller since we have to calculate the two dimension fuzzy relations given by equation ( 4 ). Objective Function Optimizer Set Point

e

Fuzzy Logic u Controller

System

y

Figure 2. Structure of the optimization technique. It includes two parts: first, the evaluation of an objective function, and, secondly, the use of an optimization algorithm for the tuning of the parameters. This algorithm has to improve or minimize, by changing parameters of the FLC, the objective function. This iterative process is repeated until a certain performance criterion is met: a sufficient number of loops are performed and an acceptable value of the objective function is obtained.

Structure of the FLC Figure 3 shows a PD type controller structure. Three fuzzy sets are used for the Error and dError/dt: Negative, Zero, Positive. For the output, five fuzzy sets are used : Negative Medium, Negative Small, Zero, Positive Small and Positive Medium. Gaussian membership functions are applied. The universe of discourse for the two input variables are in the range [11 , ].

*

K1

+ +

K2

+

*

1/s

+

3 Fobj

K3

K4

sin(2*pi*u[1]/25) f(u)

Fuzzy Controller 1 +

Mux

du/dt

1/s Integrator 1

Yp1 1

12:34 Digital Clock

Fuzzy Controller 2 f(u) cos(2*pi*u[1]/25)

2

+ -

Mux

du/dt

1/s Integrator 2

MIMO Narr. & Parth

Yp2

Figure 3. Structure of the Fuzzy Control Scheme.

The rules employed are given in Figure 4.

Error/dError

Negative

Zero

Positive

Negative

Negative

Negative

Zero

Zero

Positive

Medium Zero

Small

Negative Small

Positive

Small Zero

Positive Small

Positive Medium

Figure 4. Control Rules.

Parameters The parameters for each FLC are the widths and centers of the fuzzy sets of the input (Error and dError/dt) and output. However the center of the fuzzy set Zero are imposed to be zero. This

makes a total of 38 parameters that we must optimize. The initial parameters for the fuzzy sets are initialized randomly the two cases.

Objective Function (OF) During each iteration of the optimization process, it is necessary to evaluate the OF. It includes the passive decentralization index CPD and the Integral of the Squared Error, in a fixed horizon time, for each output value, as shown in Figure 3. To obtain the elements of the fuzzy sets ( r jiab ) in equations ( 3 ) and ( 5 ), Gaussian membership functions are sampled, with 25 points (i.e. the number of element of the fuzzy sets, f = 25 ) for each membership function. Normalizing coefficients k1 , k2 , k3 , k4 and k5 are weighting values for each component of the OF, given by: tf

f obj =

∫ (k e

2 1 1

t0

tf

(

)

+ k2e2 ) dt+ ∫ k3 u1 + k4 u2 dt + k5CPD 2

t0

( 9)

Optimization Algorithm The purpose of this algorithm is to change the parameters of the FLC in such a way that the OF has a minimal value. Among the methods of optimization we can distinguish those that use the calculation or an estimation of the gradient and those that do not. The optimization method used is a genetic algorithm (see [1]), that belongs to the second category mentioned above. The results obtained with this simple algorithm (with standard mutation and one-point crossover operators and tournament selection), are better in our experience than with other type of optimization algorithms such as the Simplex method [7]. The simple version used here can however be modified with ad hoc genetic operators (cross over and mutation) to eventually improve the FLC tuning process.

Simulation System The system used for this study was proposed by Narendra & defined by

Parthasarathy [6]. The MIMO system is

y p1 ( k )      y p 1 ( k + 1)   1 + y 2p 2 ( k )   u1 ( k )  =  y ( k + 1)  y ( k ) y ( k )  +   p1 p2 u2 ( k )  p2    2  1 + y p 2 ( k ) 

10 )

The system has two inputs and two outputs with non-linear coupling between y p1 ( k ) and y p 2 ( k ) .The input signals used for the simulation are

 2πk  and  2πk  .The u1 ( k ) = sin  u2 ( k ) = cos    25   25 

discrete time step used is k = 0.25 .

Results Figure 5 presents results after optimization (system’s outputs and set point) obtained with an OF without the decomposition index CDP . In this case k5 = 0 .

∫e

2

dt = 1.6132

3.6591

yp1

1.5

yp2

1.5 yp1 SP1

1 0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

0

10

20

30

40

yp2 SP2

1

50

-1.5

0

10

20

30

40

50

Figure 5. Results without decentralization index Figure 6 presents results after optimization obtained with and OF with the decomposition index CDP defined by equation ( 9 ).

∫e

2

dt =

2.1331

∫

d

3 4273

yp1

1.5

yp2

1.5

yp2 SP2

yp1 SP1

1

1

0.5

0.5

0 0

-0.5

-0.5

-1 -1.5

0

10

20

30

40

50

-1

0

10

20

30

40

50

Figure 6. Results with decentralization index From Figure 5 and Figure 6, we can observe that values and performance obtained in the two cases are comparable. These results suggest that the information given by the decentralization index CDP to the optimization process is incomplete and therefore a clear discrimination between the two cases is difficult. Nevertheless, the results obtained with the proposed decentralized structure shown in Figure 3, are satisfactory in the two cases.

Conclusion A decentralization index based on the analysis of fuzzy relations has been proposed to examine the decentralized structure of the FLC. We have introduced this index into the objective function and used it for FLC optimization. This information was then employed for the tuning of a FLC, applying an optimization approach. We have used existing theoretical work to passively analyze multivariable fuzzy systems. Within our optimization structure, practical application rests still difficult to implement. Indeed, passive decomposition analysis imposes certain restrictions on the values of the fuzzy sets in the optimization process. The proposed decentralization is a measure of such restriction. The analysis used here is the most simple computationally talking, yet with the obtained results it is necessary to assume a more complex type of analysis. A first direction is to take in account the inference error (simplification of the multidimensional fuzzy relations to two dimension relations) when tuning the FLC. A second direction is to use an active decomposition analysis on the FLC’s structure. This type of decomposition would introduce decoupling fuzzy relations; in turn, it will demand the introduction of more tuning parameters.

References

[1] FLEXIBLE INTELLIGENT GROUP ; Flex Tool(GA)M 2.1. Release Notes. Tuscaloosa, AL. USA. 1995. [2] GEGOV, Alexander ; Distributed Fuzzy Control of Multivariable Systems. Kluwer Academic Publishers. Netherlands,1996. p. 4-18, 50-66. [3] GEGOV, A. and Frank, P. ;“Decentralized Fuzzy Control for Multivariable Systems by Passive Decomposition”. Intelligent Systems Engineering. Winter 1994. p. 194-200. [4] GUPTA, Madan; KISZKA, Jerzy B. and TROJAN, G. M. ;“Multivariable Structure of Fuzzy Control Systems”. 1986. IEEE Transactions on Systems, Man, and Cybernetics. Vol. SMC-16, No 5. September/October 1986. p. 638-656. [5] JEON, Gi J.; LEE, Kyun K. and LEE, Pyeong G. ; “An Index of Applicability for the Decomposition Method of Multivariable Fuzzy Systems”.1995. IEEE Transactions on Fuzzy Systems. Vol. 3, No 3. August 1995. p. 364-369. [6] NARENDRA, K S. and PARTHASARATHY, K.; “Identification and Control of Dynamical Systems Using Neural Networks”. IEEE Transactions on Neural Networks. Vol.1, No 1. March 1990. [7] NELDER, J. and MEAD, R. “ A Simplex Method for Function Minimization”, Computer Journal, Vol.7, p. 308-313. [8] SORIA LOPEZ, Alberto ; “Tuning of a Multivariable Fuzzy Logic Controller”, Procs. EUFIT 96. Vol. 2. 1996. p. 965-969.