Stability Analysis of Fuzzy Controller and Implications for Design

Copyright e IFAC Intelligent Components and Instruments for Control Applications. Budapest, Hungary. 1994

STABILITY ANALYSIS OF FUZZY CONTROLLER AND IMPLICATIONS FOR DESIGN L.JURISICA and M.SEDLAcEK Slovak Technical University. Faculty ofElectrical Engineering. Department of Automation and Control. 3 Ilkovicova St.• Bratislava 81219. SlovakRepublic Abstr.cL Some techniques for the stability analysis of the control system with fuzzy logic controller are presented. The techniques are based on classical methods and they require the fuzzy controller. satisfying the sector condition. The structure of fuzzy controller. satisfying this requirement is proposed. For this controller. the analytical dependence between the stability margin and controller parameters is developed and the technique for tuning the controller parameters is also developed. The analogy with

switching curve is used to tune the

remaining degrees offreedom of the controller. The method is documented on a d.c. velocity servosystem.

Key Words. Fuzzy control; stability criterion; parameter tuning; d.c.motor; uncertain system.

1. INTRODUCTION

stability analysis of general nonlinear plant and non-memoryless fuzzy logic controller.

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 nature 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).

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.

3. FORMULATION OF THE PROBLEM 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 (R.'\y, 1984), Popov criterion (Yamashita, 1991) , and others. Most of these solutions suffer either from their enormous complexity or from their intuitive nature.

The paper is dealing with the feedback structure of a linear time invariant (LT!) plant (operator G) and a fuzzy logic feedback controller (cI». 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: dx/dt=Ax+bu (1) y c: eT x u = <1>(x) x=(x1,xJ

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 Small Gain Theorem) in a system with fuzzy logic controller. Then. in Definitions 1, 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 1. Finally, extension of method of Lim (1992) is proposed in Theorem 3, allowing the

3,1 Notation of the Fuzzy Controller

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. Let x E X" are the inputs of the fuzzy controller: X"= [XIDIiD' xl....J x ... x [x....." x....J (2)

73

Then the fuzzy controller is described by the operator, in general nonlinear: u = <1>(x ,t) (3) where u is an output of the controller. For the internal structure of the fuzzy controller holds the following notation: Let the j-th fuzzy set of the i-th input in X" has assigned the index z/. For the index holds: z.;i E [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 J.l.i (xJ . Following the same reasoning, let the j-th fuzzy set of the output is denoted by the index z'tJFor the index holds: : z..,j E [Pn+. , q"... ] (5) where Pn+. and qn+. are the limits. Then the j-h fuzzy set of the output has associated the membership function ~L"...i (x).

and the limits are: P.= P2= P3= -2 (7) q.=qz=%=+2 (8) From the Table 1 follows, that the rule base can be expressed as: ~ (z., zJ = sat(z. + zJ (9) where:

(-1)

BM

(+1) PM

{D

z

%

for for

P3::;; i ::;; % %< i

4 .1 Stability Theorems for the

(10)

Fuzzy Controller

Proposition 1 (Stability of 2nd order process): If x2 [<1>(x., xJ - <1>(x.,O)] < 0 , then the second-order system (1) is stable. }EQt The proof is based on the Lyapunov direct method with quadratic-integral energy function.

Vex, t) = a(x.z +

x.

~ x/)+y

J<1>(o,O)do

(11)

o a,

~,

'Y > 0 From the conditions of Lyapunov stability follows:

0::;; <1>(x., x2)/x. ::;; k x 2 (<1>(x.,x 2)-<1>(x.,0» ~ 0 Q.E.D.

(12)

(13)

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

(+2)

PB

Ray (1984) and others suggested to use the circle criterion to analyse the stability of the system with the fuzzy controller. This conjecture has only intuitive nature, since the original version of the 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 circle criterion for the system with fuzzy controller is a subject of Proposition 2.

Fig.I . Index notation of membership functions Table 1 Example of the rule base

NB

i
4. CONTRIBUTIONS

Example of the used notation: Consider the fuzzy controller with two inputs x. and x2 and the output x3=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.I. The rule base is given in Table 1.

(-2)

for

sat(i) =

The rule base of the fuzzy controller is given by the hypercube of rules e, 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 e as super- and subscripts it will be used the symbol 0 instead ofe.

liB

P3

NM Z

PM

NB NB

NB NB

NM Z

NM NB

NB NM Z

PM

~

NMZ

PM

PB

PM PB

PB

Proposition 2 (Circle criterion): If

PB

PB

(i) k. ~ (x) / x. ~ k z (14) (ii) Nyquist curve ofG avoids circle (-Ilk., -IIkJ (iii) c-shifted loop ofG (denoted GJ is stable then the system (1) is stable.

NB

PM NM Z PB

Z

PM PB

PB

J, x2 Using the proposed notation: z."'" ~= ~= (-2, -1,0, +1, +2)

(6)

74

ftQQt The proof is based on the Small Gain

- o(i,j) = 0 for i ~O j~-i - o(i, j) = 0 for i ~O j~-i -o(i,j)=o(i,j+I)-1 fori>O j~-i - o(i, j) = o(i, j-l) + I for i < 0 j~-i where o(ij) is an index method for the notation of the rule base.

Theorem and the coni city concept (Zames, 1966). The feedback system is considered: y=G(r-<1>(y)) (IS) where G is a linear plant operator, <1> is a nonlinear c0o/l'0~er. For the norm limits holds: G IIYa (16) 11<1>(,)11 ~ Y4> IIYII (17) where 11.11 is norm, Ya and Y4> are induced norms. Then the transformation to the equivalent c-shifted loop is done. Then y=G(r-<1>(y)-cy+cy)= G*(r-a) (18) G*=G/(l+cG) Consideration that the gain is to be limited 11n(.)1I / IIYII ~ 0 (19) is leading to the condition i) of Proposition 2. From the Small gain theorem follows the requirement IIG*II ~ 1/0 (20) Considering G(jw)=x+jy follows (21) [x+ I12(I1a+ I1f3W + .; ~ [112 (I1a-I1f3W which is a circle criterion. Q.E.D.

II l eI)

lied

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

"11

.. "1\+1

-2

-1

0 +1 +2

al"1

-2m 2.tl

~tl

r-t1

0

0

0

0

0

~tl

11

I

D

D

:

0

.1

0

•

:

1n"3 1It"2

..+1 z..t 2.t

..

'2

-1

·.

..

...2 ...1

:

.

, ,

'

~

-J

-3

D

..

11

...

-2

D

0

tl

..2 ....1

o

..tl

-2

-1

D

tl

t2

..1

11

+1 +2

"1IItl ..t2

-1

0

D

t2

t3

•

.tl

"1IIt2 ..tJ

D

0

D

tJ

t~

11

..1

2a-2 2a-l

..1 •• 2

2...1 2nI

111"1

.. · .

.'1

-1

+m

••

D

, ,

• •

0 0

• •

.. . .

••1 .*2

~

4.2 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). I 11, -Ill

-18+1

.- 1 '

1

.. 1

~finition

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

III

Lemma 1 :(Sector condition) Let the u = <1>(x) is a specialised FLC. Then: i) 0 ~ <1>(x., xJ/XI ~ k for XEX2, k>O ii) <1>(0, xJ = 0 I

I .,

.,

.,

cl - ' , cl

I

.,

I ,

,

+', c.-~

,

I

c.+~

cj

(24)

(25)

ftQQt The proof is based on Definitions I, 2, 3 and systems of inequalities. The proof is restricted to the

I .., ..I cl - ' ,

interval: 0 ~ XI ~ XI...,. because of sgn((XI' xJ) = sgn(x l ) First, it can be shown, that <1>(0, xJ = 0 Further, it can be proved that <1>(XI' xJ ~ 0 And finally, it can be proved that <1>(xl' ~") ~ <1>(XI' xJ for ~.= ~ It all implies the condition: ~ <1>(XI' xJ/XI ~ k Q.E.D.

Fig. 2. Notation of the input membership functions First, regular membership functions are introduced.

Definition 1: (Regular membership functions) Regular membership functions are triangular membership functions and for the FLC holds: i) x = (e, de / dt) ii) ql = Ch = -PI = -P2 = m; P3 = 2p,= -~; ill) c)i ~ C).i+I_ b.i+I< c.) i + bi~ Ci+1 (22) ) ) ) f~ j = 1,2; 'liE [Pj'~] iv) c j J = X;mu.; c j J= X;.-; for j = 1,2 v) c.) i = _C.i~O (23) ) b)i "", _b.i~O for j = 1,2; iE [0, <\] )

o

Fact 1 (Upper stability margin): The bounds for SFLC are k,= 0; ~ = c/,m.ml/Cll

fmQt The proof is based on Definitions 1, 2, 3 and systems of inequalities. From the Lemma 1 follows: <1>(x l, x->~ <1>(x l , ~) It is searched for ~O such that:

D~finition

2' (Specialised knowledge base) Specialised knowledge base for the fuzzy controller with regular membership functions is defined as: 75

o ~ (x., xJ/ x. ~ (x., ~/ x. ~ k

frQQf: Proof is based on Lyapunov direct method

Analysing two possible situations: i) 0 ~ x.~ c.. ii) c.·< x.~ c.M and considering regular membership functions, the worst case is approached. It is leading to the result: o~ (Xj, xJ/x. ~ (x., x-m.J/x.
with energy function:

y

f

Vex) = xTpx + ~ (cr,-a. sgn(cr» dcr

(33)

o

Then: dV/dt = - xTRx - XTQX + 2uxT(pg-Hl) n'l' 8/8x[Hx](Fx+gu) + u(2y+uIk) - u 21k (34) where 'I' = -(y,-a. sgn(y» (35) Defining n = [XT, U, '¥] E> [XT, U, '¥]T (36) dV/dt=- xTRx - [XT, U, '¥] E> [XT, U, '¥]T + u(2y+uIk) + '1'2Ik (37) which gives the conditions of the theorem. Q.E.D.

In Theorem 1 it will be shown the suitability of the SFLC for stability analysis using the Proposition 1. Theorem 1 .. If G is a second order system and u= (x) is the SFLC, then the system (1) is stable.

frQQt 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 x.~O and X2~O is (x.,xJ~ (x.,O) ii) for x.~O and ~~O is (x.,xJ~ x.,D) iii) for x.~O and ~~O is (x.,xJ~ (x.,D) iv) for x.~O and X2~O is (X.,X2)~ (x.,D) follows x2 ((x.,xJ-(x.,O» ~ 0 Q.E.D.

5. DESIGN PROCEDURE AND APPLICATIONS There is no general method for the design of FLC today. It is necessary to answer following questions: 1) What are the inputs of FLC 2) What is the output of FLC 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

In Theorem 2 it will be shown the suitability of the SFLC for stability analysis using the Proposition 2. Theorem 2 .. If G satisfies conditions of Proposition 2 and u= (x) is the SFLC, then system (1) is stable.

frQQt In Lemma 1 it is shown that the SFLC satisfies the condition (i) of Proposition 2 and thus the overall system is stable.

In the proposed design technique the preliminary answers are following. The details are afterwards. ad 1) It is considered the most widely used choice of e and de/dt 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 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. 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.

The stability analysis of the system with nonlinear process and the FLC is rare in literature. As an extension of method of Lirn (1992) it is developed and presented in Theorem 3 the suitable criterion.

Theorem 3 (Nonlinear plant and FLC): Denote Plant: d xldt = F(x) x + g(x) u (27) y= H(x) x (28) Controller: u = -(y, dy/dt) 0~(y, dy/dt) / y~k -a. ~ dy/dt ~ a. (29) (y, dy/dt) ~ (y, -a.) for y~O (30) (y, dy/dt) ~ (y, a.) for yO such that (31) (H-gTpl)Q.• (H_gTpl)T < Ilk and _~2/(4k) dldx [Hx] FFTdldx [HxF - Ilk ~ + 21k HPg -Ilk (Pg)TPg>O (32) Then the system is globally asymptotically stable.

76

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 the SFLC'holds the sector condition 0::;; (x l, X;)/XI ::;; k and further from the Fact 1 follows the slope o: ; (x l, xJ/ XI ::;; C3o
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. 3. Selection of the switching curve in a phase plane. 4 . Selection of number of membership functions, according to desired accuracy of approximation of the switching curve and real-time requirements. 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. 6. Choosing the parameters of output membership functions, satisfying stability requirements.

6. EXAMPLE The technique is documented step by step in an example of the d.c. velocity servosystem. The d.c. motor Servalco 300 W is considered with the current loop as in Jurisica (1992).The overall process has additive signal disturbance-load torque Mz 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 4. Theorem 4: (Switching curve for the SFLC) The area ~ = {e, de/dt ; u=(x)=O}at the SFLC is bounded by a staircase function: (41) 0::;; e ::;; Cli +1 - bt l For ~O: and (de/dt)..... ::;;de/dt::;; ctl - bt l

'1

Fig.4. Control system

fmQf.

We are looking for u=Cl>(x)=O. Since c30 = 0, we search z/ and ~j that have corresponding ~t.:0. From the Definition 2 follows that it holds only for i=j. For the rest of them is ~k :;t:O, i.e. u=(x):;t:O. This is leading to the inequalities: for ~O·. c Ii·1 + b Ii· 1 ::;; e ::;; c Ii+1 ~ b Ii+1 and ctl + bt l ::;; de/dt ::;; ctl - bt l

Cu

where Go(s)= lOJKTs(1 + 2Tes

(42)

is a nominal process and AG=[O,9] is the uncertainty. Further: GI(s)=Go(s) (HAG) -I G2(s)= co(s)!Mz(s)= IOJs

Further holds pro i ~ 0; j ::;; -i - o(i, j) = 0 pro i ::;; 0; j ~ -i - o(i, j) = 0 From this follows, that ~t.:0 also for i:;t:j, i.e. for i>O in interval ie [OJ] or je [-m, i] . This is leading to the restrictions: 0::;; e::;; ctl - bt l for ~O : and (de/dt)..... ::;; de/dt ::;;~i+1 - bt l

The design methodology .

is

done

(43) using

the

proposed

1) Finding the stability limit: According to the circle criterion the value A. = inf{Re [GICs)]} is searched. It gives A. = -0.0367; k,..,. = Ill. =27.22 0::;; (x l, X2)/X I ::;; 27.22 (44) 2) The SFLC is chosen. 3) Selection of the switching curve: The selection is done according to the general methodology for the time-optimal control from the literature.

Q.E.D. The final design algorithm is as follows:

77

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

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. 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.

7. SUMMARY Considering two classical approaches - the Lyapunov method and the Small Gain Theorem two propositions are derived, resulting in the structure of the so-called specialised fuzzy logic controller. This is a fuzzy controller which stabilises given LTI plant and generates bounds for tuning of the controller parameters. Using the proposed de~ign procedure, one can guarantee stability of the system and enormously decrease tuning effort and time. Method for the nonlinear plant with controller is finally nonmemoryless fuzzy presented. Results are documented by a design of velocity d.c. servosystem Acknowledeement: The research of M.Sedla.cek was supported by the Daimler-Benz Foundation, Germany. The support is greatly acknowledged.

Table 2 Input membership function parameters c Ii

i

b Ii

0

0

1

b 2i

Ci 2

0.5

0

500

1.5

1.5 1500

1500

2

3.5

1.5 3000

1000

3

6

1.5 4000

500

4

7.5

1 5000

750

5

10

2 6000

500

8. REFERENCES

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

.

Albertos, P. (1992): Fuzzy neural control. In: IFAC Symp. Low Cost Automation, Wien, p.143-156. Braae, M. and D.A.Rutherford (1979): Theoretical and linguistic aspects of the fuzzy controllers. Automatica, IS, pp.553-577. Jurisica L. and M.Sedlacek (1992): Implementation of fuzzy controlled d.c. servosystem. In: IFAC Workshop on Intelligent Motion Control, Perugia, pp. 11-107-112. Lee Ch.Ch. (1990): Fuzzy logic in control systems-

I

I I

I I

1,11. IEEE Transactions on Systems, Man and Cybernetics, 20, No.2, pp.404-435. Lim J.T. (1992): Absolute stability of class of nonlinear plants with fuzzy logic controllers. Electronic Letters, 28, No.21, pp. 1968-1970. Kiszka J.B, Gupta M.M. and P.N.Nikiforuk (1985): Energetistic stability of fuzzy dynamic systems. 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 SI SO linear system with fuzzy controller. IEEE Transactions on Systems, Man and Cybernetics, 14, No.6, pp.932-939. Tang K.L. and R.J.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.Rori (1991) : Stability analysis of fuzzy control system. Proceedings IECON'91 , Kobe . October, Vo1.2, pp. 1579-1584. Zames G. (1966): On the input-output stability of time-varying nonlinear feedback system. Part I and 11. IEEE Transactions on Automatic Control, 11. pp.228-238 and pp. 465-476.

I I I

- la

.... I

-3

I

-2

I

-I

0.1

":I."

.. 2

~. Fig.5 Influence of the fuzzy set parameters 6) Selection of output membership parameters: According to point 1 0:$ cI>(x., ~)/xl :$ 27.22 and the Fact 1 gives: o:$ cI>(XI' xJI XI :$ c)o(m.JS1)/c ll Then for the Cll follows 40.83 ~ c)o(Dl,Dl)

function

and the output membership function parameters are in Table 3. Table 3: Output membership function parameters 1

2

3

4

5

6

7

8

9 10

i

0

c) i

0 15 17 19 20 22 24 28 30 35 40

78