# Stability Based Design of Fuzzy Logic Controller

## Stability Based Design of Fuzzy Logic Controller

Copyright e IFAC New Trends in Design of Control Systems. Smoleoice. Slovak Republic. 1994 STABILITY BASED DESIGN OF FUZZY LOGIC CONTROLLER M.SEDLACE...

Copyright e IFAC New Trends in Design of Control Systems. Smoleoice. Slovak Republic. 1994

STABILITY BASED DESIGN OF FUZZY LOGIC CONTROLLER M.SEDLACEK Slovak Technical University. Faculty of Electrical Engineering. Depanmenr of Automation and Control. 3 llkovicova St.. Bratislava 812 19. Slovak Republic Abrtnct. There are presented some methods for the stability analysis of the control system with fuzzy logic controller. The methods are based on classical methods and they require fuzzy logic controller. satisfying the sector condition. The necessary structure of fuzzy controller. satisfying this requirement is proposed. For this controller. tbe method for tuning the controller parameters is developed. The method is based on stability analysis and guarantees the stability of the system. The analogy with

switching curve is used to tune tbe

remaining degrees of freedom oftbe controller. The method is documented on a d.c . velocity servosystem. Key Word5. Fuzzy control: Stability: Tuning: d.c . motor; Uncertain system.

l. INTRODUcrION

2. MAIN TOPICS It is first, in Propositions 1 and 2, shown the possibility of the application of two classical stability analysis methods (the Lyapunov method and the Circle criterion) in a system with fuzzy logic controller. Then, in Definitions I, 2 and 3, the structure of the fuzzy logic controller is generated, satisfying stability requirements. Finally, in Theorems 1 and 2 it is shown that the system with this controller is stable and the bounds for parameters of the controller are derived in Theorem 4 and Fact l.

One of the major problems in the field of fuzzy logic controllers is the lack of general methodology for their design and stability analysis. This is due to their non-parametric feature as linguistic knowledge based controllers. Much effort has been spent on the problem of synthesis of the fuzzy controller. The basic methodologies for their design can be found in (Lee, 1990)~ in (Albertos, 1992). Stability analysis of the fuzzy logic controller is a problem, not solved in general until now. Particular solutions include linguistic phase plane (Braae, 1979), energetic criterion (Kiszka 1985), Lyapunov method, circle criterion (Ray, 1984), Popov criterion (Yarnashita, 1991) , and others. Most of these solutions suffer either from their enormous complexity or from their intuitive nature.

The parallel with a switching curve in a phase plane is outlined and used to synthesize the remaining degrees of freedom . The stability-based design is documented in an example of a velocity d.c. servosystem with changing moment of inertia. Thus it is possible to make the tuning of the parameters more effective.

There is no general methodology for the design of the fuzzy controller. There is a lack of some practical methods, known from the linear control theory. In a design of fuzzy logic controller one has to solve, among others the following problems: - Number and shape of membership functions - Parameters of membership functions - Rules for the knowledge base and others. The great number of degrees of freedom makes the design too time-consuming and heuristic. The goal of this paper is to use some classical methods for stability analysis for the fuzzy logic controller, analyze the possibility of their use and develop the corresponding easy-to-use method for tuning of the controller parameters.

3. FORMULATION OF THE PROBLEM The paper is dealing with the feedback structure of a linear time invariant (LTI) plant (operator G) and a fuzzy logic feedback controller (). For clarity, it will be further restricted G to a second order proper system, and fuzzy controller to be with input error and derivative of error. Then: ch / dt = AI + bu (1) y=ex u = (x) x = (x.' x 2)

233

Table 1 Example of the rule base 3.1 Notation of the Fuzzy Controller NB

There is no widely used notation for the description of functioning of the fuzzy controller. In this paper it is introduced the method, which is a combination of the index method and the parametric function method. These methods have the advantage to be more rigorous, but less close to human reasoning.

liB

~

I

Y/\( o

NB

NM Z

~

Z

NM Z

NM Z

PM

PM

PB

PM PB

PB

PB

PB

PM PB

P3

for

i<~

~

CL

for for

sat(i)=

(6)

(n

\$ i \$ CL

(10)

CL < i

Thus the knowledge base of the fuzzy logic controller can be expressed in a compact form.

4. ST ABILITY ANALYSIS BY CLASSICAL

METHODS Proposition J (Stability of 2nd order process): If X2 [Cl>(XI' x 2) - Cl>(xl,O)] < 0 • then the second-order system (1) is stable.

fmQt The proof is based on the Lyapunov direct method with quadratic-integral energy function . XI Vex, t) = a(X 12 + P X22)+y Cl,

f Cl>(0", 0)d0"

(11)

o

p, 'Y > 0

From the conditions of Lyapunov stability follows: 0\$ Cl>(x l ,

X2 )/X I

\$ k

x 2 (Cl>(X I,X 2}-Cl>(X I,0» ~ 0

(12) (13)

Q.E.D. Requirements of the Lyapunov method (Cl class) have been now omitted, but are of significance for the fuzzy controller and are satisfied using the further proposed specialised fuzzy controller.

( X, )

\jjYlIM:ZPM

NB

Q,=CU=CL=+2 (8) From the Table 1 follows, that the rule base can be expressed as: ~ (ZI' ~) = sat(ZI + ~) (9) where:

Example of the used notation: Consider the fuzzy controller with two inputs XI and X2 and the output x 3=u. Both inputs and output have the fuzzy sets: NB (-2) - negative big, NM (-1) - negative medium, Z (0) - zero, PM (+1) - positive medium. PB (+2)positive big. The membership functions are in Fig. 1. The rule base is given in Table 1.

(+1)

NB

~=~=~=~

The rule base of the fuzzy controller is given by the hypercube of rules 6, which is assigning to every combination of the indexes of input fuzzy sets the corresponding index of the output fuzzy set. Because of poor accuracy in using 6 as super- and subscripts it will be used the symbol 0 instead of 6 .

(O ~

~ ~ PM ,!,X2

Following the same reasoning, let the j-th fuzzy set of the output is denoted by the index z"./For the index holds: : z,.../ e [Pn+I' q"..1] (5) where Pn+1 and q"..1 are the limits. Then the j-h fuzzy set of the output has associated the membership function ~n+lj (x).

(-1)

PB

Using the proposed notation: ZI = ~= ~= (-2, -1, 0, +1, +2) and the limits are:

Let the j-th fuzzy set of the i-th input in X" has assigned the index z( For the index holds: Zie[Pi,q] (4) where Pi and q are the limits. Then the j-th fuzzy set of the i-th input has associated the membership function ~i (X;) •

(-2)

PM

~ NB NB NB NM Z

Let x e X" are the inputs of the fuzzy controller: X"= [Xl ..... ' XI...,.] X .. . X [x"".,., x,....,.] (2) Then the fuzzy controller is described by the operator, in general nonlinear: u = Cl>(x ,t) (3) where u is an output of the controller. For the internal structure of the fuzzy controller holds the following notation:

.I '"

NM Z

r PS

(+2)

Ray (1984) and others suggested to use the circle criterion to analyze the stability of the system with the fuzzy controller. This conjecture has only intuitive nature, since the original version of the

Xl

Fig. 1. Index notation of membership functions 234

circle criterion was derived for the static nonlinearity and the extension to the dynamic nonlinearity, which the fuzzy controller is, is not straightforward. The circle criterion can be proved using the small gain theorem (Zames, 1966), omitting the requirement to use static nonlinearity. Thus the circle criterion is proved to be suitable for the stability analysis of the system with a fuzzy controller. The simplified version of the circle criterion for the system with fuzzy controller is a subject of Proposition 2.

forj= 1,2: iE[O,~]

[kfinition 2' (Specialised knowledge base) Specialised knowledge base for the fuzzy controller with regular membership functions is defined as: -o(i,j)=O fori~O j\$-i -o(i,j)=O fori\$O j~-i - o(i,j) = o(i,j+1) - 1 for i > 0 j~-i - o(i, j) = o(i, j-l) + 1 for i < 0 j\$-i where o(ij) is an index method for the notation of the rule base.

Proposition 2 (Circle criterion): H (i) k, \$ (x) / x, \$ ~ (14) (ii) Nyquist curve ofG avoids circle (-lIk" -1~) (iii) c-shifted loop ofG (denoted Gc) is stable then the system (1) is stable.

An example of the Specialised knowledge base is the structure depicted in Fig. 3.

.

.. ..+1 oZ•

Z.+1

.. +1 ZIIt+ Z.+l

Circle criterion says, that if the fuzzy logic controller satisfies the conditions of the criterion, it is possible to analyze the stability. The necessary condition is a sector condition. In the next it is introduced the concept of specialised fuzzy logic controller - SFLC. It is a fuzzy controller, satisfying the sector condition. It is based on modified McVicar-Whelan matrix and the aim is to keep it as general as possible._

.. . . -2

.z

.1

-1

...1

o +1 +2

-2

-1

0

...2 rflI·l

D

D

I

... 1

0

D

8

...

.. ..

m-I •

+1 +2

I

I

0

0

0 1·1

..

!

I'

I -~

·3

I

0

0

0 I

..31 ...Z

-ill

·3

·2

I 0

0

+1

... Z ! nt-l

..+1

·z

·1

0

+1

+2

...1 ... 2

·1

0

0

+2

+3

" 1...1

".z

+~

.'I I ..·z

...

..

..

.•.

• -1

-1

I

D

D

+m

8

0

0

0

... 3

0

• .. • .

+3

11+1

I

..

...1 1

11

•• 1

2..2 2..1

1I.z

Z...1

z.

~

5. SPECIALISED FUZZY CONTROLLER

Fig. 3. Specialised knowledge base Let us consider the notation of membership functions as in Fig. 2 . Following a similar reasoning to Tang (1989) we have defined the so called specialised fuzzy controller (SFLC).

De.finition 3 (SFLC): The specialised fuzzy logic controller (SFLC) is with: (i) Specialised knowledge base (ii) t-conorm is a product operation

I

-1

0

J.L I

... 1 •

Lemma J :(Sector condition) Let the u = (x) is a specialised FLC. Then: i ) 0 \$ (x" x 2)/x, \$ k forxE~, k>O ii) (O, x 2) = 0

, .1

.,

Cl-Ill

.1

.1

I

I

ct"l ~-1I1

I

I

~+1I1

systems of inequalities. The proof is restricted to the interval: 0 \$ x, \$ x, .... because of sgn(cl>(x" x 2 = sgn(x,) First, it can be shown, that (O, x 2) = 0 Further, it can be proved that (x" x 2) ~ 0 And finally, it can be proved that (x" X 2) ~ (x" x 2) for x 2 = Xz...,. It all implies the condition: 0\$ (x" x 2)/x, \$ k Q.E.D .

Cl-Ill

»

First, regular membership functions are introduced.

Definition J . (Regular membership functions) Regular membership functions are triangular membership functions and for the FLC holds: i) x = (e, de / eft) ii) q, = Cb = -PI = -P2 = m; PJ = 2p,= -<13; iii) c'J -< cJi+'_ bJi+' < c'J + bJi < CJi+' (15) -

v)

for j = 1,2; iE [pj'~] /i} = X-'1IWl'. . c)q, = x-...,mu.'. for c/=-c/~O

(18)

ftQQt The proof is based on Definitions I, 2, 3 and .. 1 .. 1

Fig. 2. Notation of the input membership functions

iv)

(17)

J' = 1,2

&ru..(Upper stability margin): TheboundsforSFLC arek,=O;

(16)

235

~=C3o(lD,III)/C,'

frQQt The proof is based on Definitions 1, 2, 3 and systems of inequalities. From the Lemma 1 follows : (x l, ~)~ (x l, "-2) It is searched for ~o such that: 0::; (x l, x 2)/ XI ::; (x l, x'lma.Y XI ::; k Analysing two possible situations: i) 0::; XI::; Cl l ii)cll(XI ' X2)!X I ::; (x l, X~)!XI <~ '*'JJI1)/ c/ (19) Q.E.D.

ad 1) It is considered the most widely used choice of

e and deldt as the inputs of the FLC. The reason comes from the intuitive design of the rule base in the phase plane. ad 2) The choice of output influences the type of

FLC-either PI or PD. With no loose of generality the

case (x)=u is considered.. ad 3) The choice of rule base is a crucial element of the design. In the specialised FLC the rule base is strictly limited to the structure of Definition 2. ad 4) In the following it will be shown, that the

In Theorem 1 it will be shown the suitability of the SFLC for stability analysis using the Proposition 1.

input membership functions form the area u=O in the phase plane. According to fuzzy VSS this area will be called switching curve. Then the number of input membership functions determines the accuracy of approximation of the switching curve.

Theorem l ' If G is a second order system and u= (x) is the SFLC, then the system (1) is stable. ~ It is shown that G and satisfy conditions of Proposition 1. The sector condition was proved in Lemma 1. Remains to prove the second condition of Proposition 1. From Definition 3 follows: i) for XI~O and X2~0 is (XI,X2~ (xl,O) ii) for XI~O and x 2::;0 is (X I,X 2)::; (xl,O) iii) for xl::;O and X2~0 is (XI,X2~ (xl,O) iv) for xl::;O and x2::;0 is (x l,x2)::; (xl,O) follows x 2 ((x l,X 2)-(X I,0» ~ 0 Q.E.D.

ad 5) According to the previous point the parameters of input membership functions determine the shape and slope of the switching curve. The analytical dependence between parameters and the curve will be shown. Thus from the desired closed loop response the parameters of input membership functions can be directly found. ad 6) The number of output membership functions is determined by the Definition 2 of the specialised fuzzy controller. In Lemma 1 it was shown that for

In Theorem 2 it will be shown the suitability of the SFLC for stability analysis using the Proposition 2.

the SFLC holds the sector condition o::; (XI' XJ/XI ::; k and further from the Fact 1 follows the slope o ::; (x l, x2)! XI ::; ~ o(m.m)/C ll

Theorem 2 ' If G satisfies conditions of Proposition 2 and u= (x) is the SFLC, then system (1) is stable.

The biggest slope of the sector, satisfying the circle criterion, is a stability margin. For the SFLC the one parameter goes to minus infinity and the circle degenerates to the line, perpendicularly crossing the real axis. The biggest slope is: k".. < -l/inf[Re{G(jw)}] (20) This value can be directly used for tuning the output membership functions . From the known c I I and the Fact one has c 3o(m.m) : c3o(m.m) ~k.... C l l (21) and the rest of parameters comes from Definition 1: C3'=c3-i (22) c3' < ~ o(m.m)

IEQ[ In Lemma 1 it is shown that the SFLC

satisfies the condition (i) of Proposition 2 and thus the overall system is stable.

6. DESIGN METHOD

There is no general method for the design ofFLC today. It is necessary to answer following questions: I) What are the inputs of FLC 2) What is the output ofFLC 3) What rules are in the knowledge base 4) How many input membership functions 5) What are the parameters of them 6) How many output membership functions and what parameters do they have 7) The shape of membership functions 8) The type of fuzzy operators 9) The choice of defuzzyfication strategy

ad 7) Regular membership function are considered. ad 8) The operator of logical AND has significant

influence on the control performance. Definition 3 an algebraic product is used.

In the proposed design technique the preliminary answers are following. The details are afterwards.

236

From

ad 9) The method COG (Centre of Gravity) is considered. because of simplicity in the case of fuzzy singletons as output membership functions.

7. EXAMPLE

The technique is documented step by step in an example of the d.c. velocity servosystem. The d.e. motor Servalco 300 W is considered with the current loop as in Jurisica (I992).The overall process has additive signal disturbance-load torque M z and the parameter uncertainty-changing moment of inertia J in a range 1: 10. Considering the multiplicative model of uncertainty, the control system is depicted in Fig.4.

What remains is the analytical dependence between the switching curve and the parameters of input membership functions . It is a subject of Theorem 3. Theorem 3: (Switching curve for the SFLC) The area S = {e, de/dt ; \FCl)(x)=O}at the SFLC is bounded by a staircase function: (23) ~ e ~ ctl - bt' For ~o : and (deldt)..... ~deldt ~ ct l - bt '

..

°

£mQL We are looking for u=4>(x)=O. Since ~ 0 = 0, we search zl i and ~j that have corresponding z./=O. From the Definition 2 follows that it holds only for i=j . For the rest of them is ~k *0, i.e. u=4>(x>*O. This is leading to the inequalities: 1 1 i 1 i 1 CI •• + b 1i . ~ e ~ C l + _ b 1+ for i~O : l and ct + bt ' ~ deldt ~ ctl - bt ' Further holds - o(i, j) = - o(i,j) =

°°

y

Fig.4. Control system pro i ~ 0; pro i ~ 0;

j j

s ~

-i -i

ell

where Go(s)= 101K/s(l + 2 T e s )

From this follows, that ~k=(J also for i*j, i.e. for i>O in interval ie [OJ] or je [-m, i]. This is leading to the restrictions: for i~O: ~ e ~ ctl - bt ' and (deldt) ..... ~ deldt ~ctl - bt' Q.E .D.

(24)

is a nominal process and 9] is the uncertainty. Further:

~G=[O,

°

(25)

The final design algorithm is as follows: The design is done using the proposed methodology.

1. Finding the maximum slope for the sector condition, given by the circle criterion. 2. Choosing FLC, satisfying the sector condition. According to Lemma 1 the SFLC is suitable.

1) Finding the stability limit: According to the circle criterion the value A = inf{Re [G1(s)]} is searched. It gives 1.= -0.0367 ; k".,. = 111. =27.22 ~ Cl>(x l, X2 )/X I ~ 27.22 (26)

3. Selection of the switching curve in a phase plane.

2) The SFLC is chosen.

4. Selection of number of membership functions, according to desired accuracy of approximation of the switching curve and real-time requirements.

3) Selection of the switching curve: The selection is done according to the general methodology for the time-optimal control from the literature.

°

5. Selection of the parameters of the input membership functions, which approximate the given switching curve. The selection is based on the dependence, given analytically in Theorem 4.

4) Selection of number of membership functions : As

a compromise between the demand for accurate approximation of the switching curve and the real-time requirements, the number m=5 was chosen.

6. Choosing the parameters of output membership functions, satisfying stability requirements.

5) Selection of input membership function parameters: The parameters of input membership functions are in Table 2, chosen according to Theorem 4 to approximate the chosen switching curve.

237

and generates bounds for tuning of the controller parameters. Using the proposed design procedure, one can guarantee stability of the system and enormously decrease tuning effort and time. Results are documented by a design of velocity d.e. servosystem.

Table 2 Input membership function parameters i

bI'

C'I

b 2'

C'2

0

500

1.5

1.5 1500

1500

2

3.5

1.5 3000

1000

3

6

1.5 4000

500

4

7.5

I 5000

750

5

10

2 6000

500

0

0

1

0.5

AcknQwled2ement: The research of M.Sedlatek was supported by the DaimJer-Benz Foundation, Germany. The support is greatly acknowledged. 9. REFERENCES

The area ~: {e, de; <1>( x)=O} for the given parameters is depicted in Fig.5.

Albertos, P. (1992): Fuzzy neural control. In: IF AC Symp. Low Cost Automation, Wien, p.143-156. Braae, M. and D.A.Rutherford (1979): Theoretical and linguistic aspects of the fuzzy controllers. Automatica, 15, pp.553-577. Jurisica L. and M.Sedlacek (1992): Implementation of fuzzy controlled d.e. servosystem. In: IF AC Workshop on Intelligent Motion Control, Perugia, pp. 11-107-112. Lee Ch,Ch. (1990): Fuzzy logic in control systemsI,ll. IEEE Transactions on Systems, Man and Cybernetics, 20, No,2, pp.404-435 . Lim 1.T. (1992): Absolute stability of class of nonIinear plants with fuzzy logic controllers. Electronic Letters, 28, No.21, pp. 1968-1970. Kiszka 1.B, Gupta M.M. and P.N.Nikiforuk (1985): Energetistic stability of fuzzy dynamic systems.

·1

I

I I

I

I I I I

.~

-I.

I -)

.~

~. Fig.5 Influence of the fuzzy set parameters 6) Selection of output membership parameters: According to point I 0::; <1>(x l , x 2 )Ix I ::; 27.22 and the Fact 1 gives: 0::; <1>(x l , x 2 )/ XI ::; S o(=>/C l l Then for the Cl l follows 40.83 ~ c 30(=>

IEEE Transactions on Systems. Man and Cybernetics,S, No.6, pp.783-792. Ray K.S . (1984): L 2- stability and the related design concept for SISO linear system with fuzzy controller. IEEE Transactions on Systems, Man and Cybernetics, 14, No.6, pp.932-939. Tang K.L. and R1.Mulholland (1987): Comparing fuzzy logic with classical controller design. IEEE Transactions on Systems. Man and Cybernetics, 17., NO.6, pp. 1085-1087. Yamashita Y. and T.Hori (1991): Stability analysis offuzzy control system. Proceedings IECON'91, Kobe, October, Vo1.2, pp.1579-1584. Zames G. (1966): On the input-output stability of time-varying nonIinear feedback system. Part I and 11. IEEE Transactions on Automatic Control, 11, pp.228-238 and pp. 465-476.

function

and the output membership function parameters are in Table 3. Table 3; Output membership function parameters I

0

c'3

0 15 17 19 20 22 24 28 30 35 40

I

2

3

4

5

6

7

8

9 10

Thus the design is completed. Results in simulations shown very good performance of the system. 8. SUMMARY

Considering two classical approaches - the Lyapunov method and the Circle criterion - two propositions are derived, resulting in the structure of the ~called specialised fuzzy logic controller. This is a fuzzy controller which stabilises given LTI plant

238