Application of fuzzy logic controllers in single-loop tuning of multivariable sytem design

Computers in Industry 17 (1991) 33-41 Elsevier

33

Forum on Fuzzines

Application of fuzzy logic controllers in single-loop tuning of multivariable sytem design Chieh-Li Chen Institute of Astronautics & Aeronautics, National Cheng-Kung University, Tainan, Taiwan, ROC 70101

Pey-Chung Chen Department of Mechanical Engineering, National Cheng-Kung University, Tainan, Taiwan, ROC 70101

The Rosenbrock-Nyquist Array design method is a very valuable design technique for a wide range of practical multivariable feedback control systems. The system designed using this design approach has many nice properties, such as the integrity property. In this design approach, the dominance condition plays an important role. However, the dominance condition achieved using a pre-compensator may be altered and instability or unacceptable performance may occur when facing model uncertainties. In this paper, a fuzzy-PID controller structure is proposed. Using this controller structure, one can design the fuzzy controller more easily and the resulting system performance can also be improved. A hybrid design scheme of using a fuzzy controller as well as the Rosenbrock design methodology is also presented for the multivariable control system design. A numerical example is used to illustrate the robustness enhancement and the effectiveness of the proposed hybrid design scheme.

Keywords: Multivariable control, Robustness, Fuzzy logic controller.

I. Introduction

The multivariable feedback design problem is reduced to a set of single-loop design subproblems using the Rosenbrock-Nyquist Array (RNA) design methodology. The RNA design methods, i.e.

the Inverse Nyquist Array (INA) method and Direct Nyquist Array (DNA) method, consist essentially of determining a multivariable pre-compensator matrix K p ( s ) t o be as simple as possible and such that the resulting system Q(s)= G(s) )< K p ( s ) is diagonally dominant. When this condition has been achieved, a diagonal compensator matrix Kd(s ) can be used to implement single-loop compensators as required to meet the overall design specifications. This is why the RNA design method is so attractive and has been widely used in multivariable feedback design [1-7]. Various attempts have been made to determine a pre-compensator such that the resulting system is diagonally dominant, such as the pseudo-diagonalisation technique [8], the Align algorithm [9], diagonal compensator design procedures [1012], a constant compensator design algorithm [13], and a P I / P I D controller design procedure [14,15]. Using these diagonal dominance procedures, the dominance condition can be achieved and the multivariable system design is completed by a set of single-input single-output (SISO) designs. However, it is argued that for some multivariable systems the RNA design technique can lead to controller designs whose performances are very sensi-

0166-3615/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved

34

Forum on Fuzziness

tive to modelling uncertainties [16]. This lack of robustness, which is not detected by RNA, can be attributed to the fact that the method does not account for the model uncertainty which can adversely affect the diagonal dominance and drastically alter the stability and performance properties of multivariable control systems. The fuzzy logic control algorithm based on fuzzy set theory [17] can be regarded as a set of heuristic decision rules such that non-mathematical control algorithms can be inplemented easily in a computer. They are straightforward and should not involve any computational problems. Further, they are implicitly assumed robust because they are based on human experience. With these advantages, the fuzzy logic controller has made itself more popular in both theoretical researches and industrial applications [18]. For multivariable systems, the interactions between the input-output pairs are quite difficult to be modeled using single linguistic formulations. Therefore, applications of fuzzy controllers to multi-input multi-output system design are quite rare compared with those to single-input (or multi-input) single-output system cases. In this paper, a hybrid controller structure is proposed to improve robustness of a multivariable control system design using Rosenbrock's method-

Chieh-Li Chen received the Bachelor's degree in Mechanical Engineering from Taiwan University, Taiwan, in 1983, and the MS and PhD degrees from the Control Systems Centre, UMIST, Manchester, in 1987 and 1989, respectively. Currently he is an associate Professor at the Institute of Aeronautics and Astronautics, National Cheng-Kung University, Taiwan. His main research interests include fuzzy engineering, robust control and computer-aided design.

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ology. This hybrid design seems to be a simple way to cope with multivariable system design when the RNA approach is used. A numerical example is also given to illustrate the effectiveness of this proposed control scheme.

2. Fuzzy control algorithm and implementation The fuzzy controller mechanism includes membership functions, inference rules (fuzzy relations or fuzzy look-up table), and a defuzzification process. Although, a variety of approaches to implement the fuzzy controller have been presented, they are based on the look-up table approach. The fuzzy inference rules and membership functions are obtained by a trial-and-observation procedure. This makes the fuzzy controller design a time-consuming task. In this section, an alternative controller structure called F u z z y - P I D controller, which can be considered as a conventional PID controller with condition-dependent parameters, is presented. The proposed controller has the advantages of both PID and fuzzy logic controllers. This will make the selection of membership functions easier and the resulting control system is robust in the sense of the fuzzy control scheme. Another attractiveness of this controller is that its synthesis does not require the existence of a mathematical model. Hence, the discussion on stability seems somewhat irrelevant for the fuzzy type controllers. They are implicitly assumed robust because they are based on human experience [19]. 2.1. F u z z y algorithm a n d implementation

Consider the following fuzzy inference rules: R:

Pey-Chung Chen received the Bachelor's degree in Mechanical Engineering from T a m k a n g University, Taiwan, in 1984, and the MS degree from the Department of Mechanical Engineering, National Cheng-Kung University, Taiwan, in 1988. He is now a PhD student at the Department of Mechanical Engineering, National ChengKung University, Taiwan. His main research interests include variable structure control system design, fuzzy set theory and its application to automation.

If F ( x I is A 1. . . . . x k is Ak),

(])

then y = G ( x 1. . . . . x k ) , where y denotes the variable of the consequence whose value is inferred, x~ . . . . . x k denote the variables of the premise which also appear in the part of the consequence, A 1. . . . . A k denote the fuzzy sets representing a fuzzy subset in which the fuzzy inference rule R can be applied for reasoning, F ( . ) denotes the logical function connecting the proposition and the premise, and G(-) denotes the

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C.-L. Chen, P.-C. Chen / Fuzzy logic controllers

function that implies the value of y when x~ . . . . . x k satisfy the premise. Given inference rule R i (i = 1 . . . . . n) and given (x 1 = x~ ..... x k = x~) then the value of y is inferred as shown in the following steps: (i) For each inference rule R g, y~ is calculated by the G' in the consequence: yi

i

o

X

(2)

o

= G (x I ..... X k ) .

(ii) The truth value of the proposition y = y~ is calculated by the equation:

35

0,0

Ixl Fig. 1. Description of fuzzy sets for B, M and S.

~ ( y = y i ) = (x? is A~ a n d . . . a n d x~ is A~)

= [~A~(x~)~ . . . / , ~A~(x~,)],

(3)

where f f ( y = J ) means the truth value of the proposition y =y~, A stands for the minimum operator, (x ° is A t) = #A,(X °) represents the grade of the membership of x °. (iii) The final output y inferred from n implications is given as the average of all y~ with the weights iz(y = yi), i.e.,

y'p,(y=yi)Xy' Y=

y'Cy

(4)

=yi)

2.2. The fuzzy-PID control algorithm In this section, the idea of combining the P I D controller concept with fuzzy set theory will be introduced. First, the membership functions of three fuzzy sets B (big), M (medium) and S (small) are defined as follows:

~s(X)=lJ,m(X)=~.tb(X)=O ~,~(x) =

Ixl=O,

I x l - Ms u~ °s

(6)

I'[( )1 mx,=exp[(x °m t,~(x) =

exp --

{'[ exp

- ( Ix]-- Mb °b

Nb

)

(5)

Ix[ > x~,

These membership functions are shown in Fig. 1. (Note: Alternative membership functions can be used.) For a conventional P I D controller, the control law is given as: u = Kpe +

Kife

dt +

(9)

KdO,

where e = y - r , y is the system output, r is the reference input, e is the deviation of y from r and U is the control output of the PID controller. Further, let the membership function variable x denote any of the variables e, f e d t and ~. By substituting these real parameters for x into (5)(8), the corresponding membership functions are obtained as: ~ ( z , ) = ~,m(Z,) = ~b(Z,) = 0

{1[( )]

~(z,)=

exp-

Iz']-Z's O'zl s

.m(Z~)=exp

-

[zs!--

N:'s

Zi] = 0,

(10) 0<[z,]~zis (11) Izil>z,s,

[zi[ > 0,(12)

OZim

]z,] > Z,b

Ixl>O, (7)

]

~tb(Zi) =

exp-

]Ziv~,hZib

0 <[z,[~z~b

[XI>Xb (13)

O
where z i (i = 1, 2, 3) represents e, f e d t and ~. The three variables e, f e d t and ~ each have three fuzzy sets B, M and S. Therefore, 3 3 in-

36

Forumon Fuzziness

Computers in lndusto'

Referencesignal r ._.

"L~!

+ I,

~-1

~

~

Kd(+')/

"l

l

i

I

~

--

Kp(s) /

I QCs) ~

N_

Output y

6(s~ .....

J

[

,.

/

Fig. 2. Blockdiagram of multivariablesystem with conventionalcontrol scheme.

ference rules are obtained; for example, the ith inference rule Ri:

If (e is small and

fedt is medium

and O is big) then

U,=Kp#~(e)e+Ki#m(ie)fedt

(14)

+Kjt~h(0)0 and p,(U i ) = p.s(e) A ,ttm(Ie ) A /.tb(~; ). For each stage, the defuzzified control output U is defined as: 33 / 33 U = E V' E / ~ ( U ' ) , (15) i=l i=l where # ( U ' ) is the membership function of U'.

3. The Rosenbrock methodology and the hybrid control scheme

As mentioned in Section 1, the RosenbrockNyquist Array design methodology can be stated

References i g n a l . ~ . r +~)~[

I

as follows: For a given functionally controllable multivariable system, first a pre-compensator matrix Kp(s) is designed to make the resulting system Q(s) = G(S)Kp(s) diagonally dominant. Then, a set of single-loop controllers (a diagonal compensator matrix Ka(s ) as shown in Fig. 2) is designed for each loop to meet design requirements. The block diagram of this multivariable control scheme is shown in Fig. 2. Though the RNA method provides valuable insight into the stability and performance analysis of feedback control systems, it is argued that when facing model uncertainties the performance of a multivariable control system designed by the RNA method could be drastically altered. In order to solve this problem, a hybrid control scheme which combines the classical design methodology and the fuzzy controller structure is proposed. For a given functionally controllable multivariable system, first a pre-compensator matrix Kp(S) is designed to make the resulting system Q ( s ) = G ( s ) K p ( s ) diagonally dominant. Subsequently, a set of fuzzy logic controllers (as described in Section 2) is used instead of the conventional diagonal compensator matrix Kd(S). The block diagram of this hybrid control scheme is shown in Fig. 3. Using this control scheme, the performance of this multivariable system is insensitive to rood-

L Fuzzylogic

co+o+

Kp(s)

.....

i

[ ~L G(s)

Output y

Q(s)

Fig. 3. Blockdiagram of multivariablesystemwith hybrid control scheme.

Computers in Industry

C.-L. Chen, P.-C. Chen / Fuzzy logic controllers

elling errors. The robust fuzzy logic controllers, therefore, improve the robustness of the multivariable control system. For illustrative purpose, a numerical example is considered in the following section.

37

with nominal values PI°I = 1.15 and P~°2 = 0.264, - 15s -

1.42

gl2(S) = s3 + 12.8s 2 + 13.6s + 2.36 ' 1.95s 2 + 2.12s + P22 g21(s) = s 3 + 9.15s 2 + P21 s + 1.616 ' with nominal values P~ = 9.39 and /'202= 0.49,

4. N u m e r i c a l

example g22(s)

Consider the automotive gas-turbine shown in Fig. 4 and described as follows [5]:

where

gll(s)

=

7.14s 2 + 25.8s + 9.35 116.4s 2 + l l l . 6 s + 18.8

s 4 + 20.8S 3 +

where Yl is the gas generator speed, Y2 is the inter-turbine temperature, u 1 is the fuel p u m p excitation signal, and u2 is the nozzle actuator excitation signal. It is obvious that the u n c o m p e n s a t e d system is not diagonally dominant. Choose 1 [ 0.0232s + 0.793 Kp ( s ) = s [ - 0.5305s - 1

0.806s + P12 = s 2 + P l l s + 0.202 '

Fuel actuator

Fuel pump

signal

4

RGEf ,J

(

,/

J

CT Speed meter I compressor CT compressor-turbine PT power-turbine RG regenerator

C

Air

l e= o*el Nozzle actuator signal Gas

enerator

speed

'

Inlet temperature

Fig. 4. S c h e m a t i c o f a u t o m a t i c g a s - t u r b i n e .

s + 0.1074 ] 0.6178s + 0.21261 (17)

38

Forum on Fuzziness

Computers in Industry

and plot the Nyquist Array with the circles of the resulting system Q(s) = G(s)Kp(s). As shown in Fig. 5, the system Q(s) is now diagonally dominant. If a diagonal compensator Kd(S ),

Ka(s ) =

10

0.3833s + 1 0.3833s + 5.83

0 0.1587s + 1 30 3 0303s + 1 (18)

is further introduced, the time responses of the closed-loop system are as shown in Fig. 6. Using the proposed control scheme, with the s a m e K p ( s ) , the time responses of the closed-loop system are as shown in Fig. 7. From Figs. 6 and 7, it is shown

[1.1

]

that good system performance can be obtained using both control schemes. Suppose the first-order coefficients of the denominators of element (1,1), Pll, and element (2,1), P21 (whose nominal values are 1.15 and 9.39, respectively) in the transfer-function matrix G(s) have been perturbed to 3.45 and 18.78, respectively. Then, the time responses of this perturbed closed-loop system using the conventional controller and the proposed hybrid control scheme are as shown in Figs. 8 and 9, respectively. The results show that the proposed hybrid control scheme is insensitive to model uncertainties or plant parameters variations. The stability region in the space of these two perturbed parameters (P~ and P21) using both control schemes is shown in Fig. 10. Figure 11 shows the stability region in the

elemeqi;.

[1.2]

element

0.4

.............................................

0.2

.............................................

-5 (_q

-10

< ~E

0

-0.2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-15 -0.4

. . . . . . . . . . . . . . . . . . . . . . ~. . . . . . . . . . . . . . . . . . . . . .

i

I

-20

-5

0

5

-0.5

0 REAL

REAL 2.1]

3

element

[2.2]

0

0.5

e]ement

2.5 -0.5 2 1.5

1 0,5 0 -2

iii iiiiii iiiiiiii iii

-1

-1 ,5

i

-2 -1

0 REAL

2

0

0.1

i 0.2 REAL

0.3

0.4

Fig. 5. Direct Nyquist array of system Q(s ) = G ( s ) K p ( s ) with corresponding column Gershgorin circles•

C.-L. Chen, P.-C. Chen / Fuzzy logic controllers

Computers in Industry

39

1.0

1.0 ¸

.... ~"

1-

I1 ~

RflL L

0.0 R I is the refcte.ac¢ s~gnal o f loop 1.

R2 is the m f e ~ -~

i

i

n i i i 2.00

OJ~

RI

signalof Ioo~ 2+ 3 i

~ i i 4,.00

i

thc rgfgtcace signal o f loop 1

R 2 is the reference signal o f lOOp 2.

i

l j l 6.00

~ i

i i i 8.00

i

i

-0,~

i 10.00

i

i

i

~ i

~ i

i

i

i

i

i

i

i

j

~ i

i

i

i

i

i

i

i

<'m3

1/2

1/2 1.0-

g :/

1 J

I

R2~

I I RfL

Rf L

o.o R I is the t ¢ ~

s i g n a l o f loop 1.

R 2 is iI1+ m f ~

sigmll o f loop 2.

i

t

i

,

0.00

i

i

i

i

i

2.OO

]

i

i

,

R I is II~ mf~renc~ signal o f loop 1. R2 is ~¢ reference signal o f loop 2. ,

4,OO

i

i

i

i

6.OO

i

i

i

i

i

~

i

l

10.00

Fig. 6. Time responses for conventional design scheme.

I l I L I I I I I I I I I I L I I £ 1 1

0.00

2.OO

S~I

i l l

6.OO

I 0+00

&O0

Fig. 8. Time responses for conventional design scheme with the first-order coefficients of the denominators of element (1,1) and element (2,1) perturbed to 3.45 and 18.78 respectively.

Vf

1/f 1.0.

1.0

I

}°

1 J

Rf

0.0

RI

0.0-

R I i s the mf~t~nc~ s i s nltl o f loop 1.

R l is m e ~ f ~ a c e signal o f ] ~ p I

R2 is II~ t e f m n ¢ ~ sigmflo f loop 2. -~

l O.OO

l

l

l

l l 2.OO

l

l

l

] l 4,OO

l

l

R2 is the ~fe~ence signal o f loop 2 l

l l 6,OO

l

l

--~

i 1 1 1 1 1 S.O0 IO.O0

i

i

i

0.00

i

i

i

i

L

200

t

i

t

i

L

4.00

i

]

i

i

i

~

i

[

L

i

i

~.00

i

10.00

(.o)

1/2 1.0

g

}°

}° RIL

o.o

RIL

o.0 R I is the ~ f ~ e n o ~ signal o f loop I R2 is the r~ f c t ~ ¢ signal o f loop 2

R I is the r~fercn~ mgmd of Loop I. R2 is the tcfereac¢ signal o f loop 2.

0.00

i i l l ] l l l 2.OO

l l l 4+0¢

l

l

t

R8

l

l l 4LO0

h

l

l

l L ~+00

l

h

i

l

10.00

Fig. 7. Time responses for hybrid design scheme.

0.00

+ i

I i i ~.00

i

i

i i i +.OO

i

i

i [ i e.O0

(.o)

i

i

i i i 8.00

i

i

i IO.O0

Fig. 9. Time responses for hybrid design scheme with the first-order coefficients of the denominators of element (1,1) and element (2,1) perturbed to 3.45 and 18.78 respectively.

40

Forum on Fuzziness

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References 21@~.0

[]

stable rag/on It,r convenuon~l design s~h¢I,,¢

~x~,~ : stable region [or hybrid d,~sign~hcm¢ -22.~0

-~.0

17'&0

37~.0

5"7~0

P1f

Fig. 10. Comparison of stable region between the conventional and the hybrid design schemes in the space of system parameters Pll and P2t.

--55.O []

stable region for conventionzfldestgn scheme

~:

stable region for hybrid design ~hcme

-l~k~-o 50. ~ . . . . . . . .

O"/ . . . . . . . .

~Oi.° . . . . . . . .

IO0"O

P12

Fig. 11. Comparison of stable region between the conventional and the hybrid design schemes in the space of system parameters P~2 and P22.

space of parameters P12 and P22 (with nominal values 0.264 and 0.49, respectively) for both control schemes. These figures indicate that the proposed control scheme is robust to large parameters variations. Therefore, the robustness of the Rosenbrock design methodology is improved using the proposed hybrid control scheme.

5. Conclusions In this paper, a fuzzy-PID controller structure is proposed. A hybrid control scheme has also been presented to improve the robustness of a control system design using the Rosenbrock methodology. With this hybrid control scheme, Rosenbrock's approach could be applied to practical systems, which suffer from the problems of plant parameter variations or nonlinearity.

[1] H.H. Rosenbrock, "'Design of multivariable control systems using the inverse Nyquist array", Proc. lEE, Vol. 116, No. 11, 1969, pp. 1929-1936. [2] N. Munro, "Multivariable design using the inverse Nyquist array", Comput. Aided Des., Vol. 4, No. 4, 1972. [3] S.I. Ahson and H. Nicholson, "Improvement of turbo-alternator response using the inverse Nyquist array method". Int. J. Control, Vol. 23, 1973, pp. 657-672. [4] J.C. Kantor and R.P. Andres, "'A note on the extension of Rosenbrock's Nyquist array techniques to a larger class of transfer function matrices", Int. J. Control, Vol. 30, No. 3, 1979, pp. 387-393. [5] R.V. Patel and N. Munro, Multwariable System Theory and Design, Pergamon, Oxford, 1982. [6] V. Neboyan and H.N. Koivo, "Design of a return flux control system by the inverse Nyquist array method", Int. J. Control, Vol. 37, No. 3, 1983, pp. 535-552. [7] P.T. Kidd, "Design of a controller for a multivariable system with varying parameters using the extended direcl Nyquist array", Int. J. Control, Vol. 43, No. 3, 1986, pp. 901-920. [8] D.J. Hawkins, "Pseudodiagonalisation and the inverse Nyquist array mehtod", Proc. lEE, Vol. 119, No. 3, 1972, pp. 337-342. [9] B. Kouvaritakis, Characteristic locus methods for multivariable system design, Ph. D. Thesis, UMIST, Manchester, UK, 1974. [10] A.I. Mees, "Achieving diagonal dominance", Svst. Control Lett., Vol. 1, No. 3, 1981, pp. 155-158. [11] D.J.N. Limebeer, "The application of generalized diagonal dominance to linear system stability theory", Int. J. Control, Vol. 36, No. 1, 1982. [12] N. Munro, "Recent extensions to the Inverse Nyquist Array design method", 24th I E E E Conf. on Design and Control, Miami, FL USA, 1985, pp. 1852-1857. [13] S.L. Wang and P.A. Kai, "Design of diagonal dominance by compensator", Int. J. Control, Vol. 38, No. 1, 1983, pp. 221-227. [14] C.L. Chen and N. Munro, "Procedure to achieve diagonal dominance by using a P1/PID controller structure", Int. J. Control, Vol. 50, No. 5, 1989, pp. 1771-1792. [15] C.L. Chert, Robustness analysis and frequency domain design of multivariable control system, Ph.D. Thesis, UMIST, Manchester, UK, 1989. [16] J.C. Doyle and G. Stein, "Multivariable feedback design: concept for a classical/modern synthesis", I E E E Trans. Automat. Control, Vol. AC-26, No. 1, 1981, pp. 4-17. [17] L.A. Zadeh, "Fuzzy sets", l n f Control, Vol. 8, 1965, pp. 338-353. [18] Michio Sugeno (ed.), Industrial Applications of Fuzz)' Control, Elsevier, Amsterdam, 1985. [19] D. Dubois and H. Prade, Fuzzy Sets and Systems, Theo O' and Applications, Academic Press, New York, 1980, pp. 297-306.

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C.-L. Chen, P.-C. Chen / Fuzzy logic controllers

Appendix A P a r a m e t e r s o f f u z z y l o g i c c o n t r o l l e r a r e as follows: Kpl 1 = 1.25,

Kp22 = 1.0,

Kil 1 = 0,

Ki2 2 =

Kin1 = 0 . 2 2 5 ,

Kd22 : 0 . 0 4 .

e~ = I e S = e;s = 0 . 1 ,

= em = 0.8,

eb = Ieb

= eb =

1.6,

Oe~ = %m = Oeh = 0 . 3 6 , O'le s = O l e m = Olet, =

0.01,

Parameters of membership a n d B a r e as f o l l o w s .

em = Iem

0.36,

°e~ = %m = O0~ = 0 . 3 6 , functions

f o r S,

M

IVe = N e m = N e b ~ - 4 , Nl~ ~ = Nle m = NIe ~ = 4,

No = U~m = U~. = 4 .

41