Design of fuzzy sliding-mode control systems

E L S E V I E R Fuzzy Sets and Systems 95 (1998) 295-306

FUZZY sets and systems

Design of fuzzy sliding-mode control systems X i n g h u o Y u a'*, Z h i h o n g M a n b, Bao l i n W u a

a Department of" Mathematics and Computing, Central Queensland University, Rockhampton, QLD 4702, Australia b Department of Computer and Communication Engineering, Edith Cowan University, Perth, WA 6027, Australia

Received July 1995; revised August 1996

Abstract

In this paper the design of fuzzy sliding-mode control is discussed. For a complex physical system represented by an aggregated fuzzy global model which compromises a set of linear models, conditions for the fuzzy sliding mode control to stabilize the global fuzzy model are given. Simulations are presented to show the effectiveness of the control strategy. @ 1998 Elsevier Science B.V. All rights reserved.

Keywords: fctrl; clus; proc; model

I. Introduction

Sliding-mode control (SMC) systems have been studied extensively and received many applications [12, 14]. The sliding mode is attained by designing the control laws which drive the system state to reach and remain on the intersection of a set of prescribed switching surfaces. When in the sliding mode, the system exhibits invariance properties, such as robustness to certain internal parameter variations and external disturbances. The dynamic performance ofa SMC system is determined by the prescribed switching surfaces upon which the control structure is switched. Most commonly used switching surfaces are linear hyperplanes, and the SMC of linear systems has been well studied.

However, most physical systems are nonlinear and complex that may not be easily modeled mathemati- cally. On the other hand, the mathematical treatment

*Corresponding author. Fax: +61 79 309729, E-mail: x.yu@ cqu.edu.au.

of nonlinear systems is still a problem in modern control theory.

For complex nonlinear systems, it is possible that a complex nonlinear system is linearized around given operating points such that the well-developed linear control theory can be applied in the local region with apparent ease. Such a treatment is quite common in practice.

There may exist a number of operating points in the complex nonlinear systems that should be considered during controlling a nonlinear system. How to aggregate each locally linearized model into a global model representing the nonlinear system is a ques- tion. One of the effective approaches is the fuzzy logic approach. By employing the fuzzy logic, the set oflin- earized mathematical models can be integrated into a global model that is equivalent to the nonlinear system. Various fuzzy models and their control have been discussed, for example, [7-10, 3].

In this paper we shall discuss how to design a fuzzy SMC control such that the global fuzzy model presents desired dynamic characteristics. We first design SMC

0165-0114/98/$19.00 (~ 1998 Elsevier Science B.V. All rights reserved PHS0165-0114 (96 )00278 -3

296 X. Yu et al./Fuzzy Sets and Systems 95 (1998) 295-306

control for each linear subsystem of the global fuzzy model. We then discuss the conditions for "fuzzily" amalgamated SMC control to stabilize the global fuzzy model. Robustness issue are examined as well. Simu- lation results are presented to show the effectiveness of the control.

2. Sliding-mode control

Consider the linear controllable system

~c = A x + Bu, (1)

where A is an n x n matrix and B an n x q matrix. The SMC u E ~q is characterized by the control structure defined by

u+(x) fors(x) > 0 , u = u - ( x ) fors(x) < 0 , (2)

where s ( x ) is the set of switching hyperplanes and defined as

s ( x ) = C x = 0, ( 3 )

where C is a constant q x n matrix to be determined. The design of SMC involves two phases. The first

phase is to select the switching hyperplanes s ( x ) to prescribe the desired dynamic characteristics of the controlled system. The second phase is to design the discontinuous control such that the system enters the sliding mode s ( x ) = 0 and remains in it forever. The well-known condition f f s < 0 is usually used for the design [12] where the superscript T stands for transpose.

When in sliding, the system satisfies

s ( x ) = 0, ~(x) = 0 (4)

and the system exhibits invariance properties, yield- ing motion independent of certain parameter variations and disturbances [12]. From the equations in (4), one can see that

= c i = 0, ( 5 )

and the equations governing the system dynamics may be obtained by substituting a so-called equivalent control [12], denoted by Ueq, for the original control u (assume the matrix (CB) is nonsingular)

Ueq = - ( C B ) - 1 C A x (6)

such that under the control the dynamics in the sliding mode becomes

k = [I - B ( C B ) - 1 C ] A x = PAx , (7)

where P = [I - B ( C B ) -1C] . Notice that during sliding, the system dynamics becomes n - q dimensions due to the constraint C x = 0 and confined to s ( x ) = 0 and ~(x) = 0. The matrix P is actually a projec t ion

operator along the range space of B onto the null space of C [2], i.e.

P B = O, P x = x Vx E ~ subject to C x = O.

(8)

C can be designed such that the n - q eigenvalues of Aeq are allocated on the left-hand side of the complex plane and remaining q eigenvalues remain zero [14].

3. Fuzzy system modeling

In this paper, we consider the following fuzzy system to model a complex system [3]

Ri: I F Zl is F I A N D . • • zn is F~

T H E N

Yc(t) =- A i x ( t ) + Biu( t ) ,

yi( t ) = Dix( t ) + Eiu

for i = 1 ,2 , . . . ,m (9)

where R i represents the ith fuzzy inference rule, m the number of inference rules, Fj ( j = 1 . . . . ,n) the fuzzy sets, x ( t ) the system state, u the system input, and Yi the system output. The matrices Ai, Bi, Di are n x n, n x q, p x n, respectively, and Ei a constant vector in- dicating influence from u( t ) to y i ( t ) , z = (Zl . . . . . zn) T represents some measurable system variables.

Denote 12i(z(t)) as the normalized fuzzy member- ship function of the inferred fuzzy set F i where

F i = f i Fj (10) j=l

and

£ t~e = 1. ( 1 1 )

i=1

X. Yu et al./Fuzzy Sets and Systems 95 (1998) 295-306 297

Using the weighted average fuzzy inferences approach, we obtain the following global fuzzy state- space model:

it(t) = Ax( t ) + Bu(t), (12)

y = Dx( t ) + Eu,

where

A : L I~,Ai, B = L , u i B i , i=1 i=1

D : ~ l~iDi, E : ~-~ IziEi. i=1 i=l

(13)

In the following we use the symbols A, B, D, E as the fuzzy model matrices. Before we proceed, we assume that

Assumption 1. Each linear subsystem o f the global fuzzy model is controllable, i.e. the matrices Mi = [Bi, AiBi, A2Bi . . . . . An-lBi] for i = 1 . . . . . m have

ful l ranks, i.e. rank(Mi) : n.

Assumption 2. The global fuzzy model (9) is controllable in the state space, L e. the matrix M = [ B, AB, A 2 B . . . . . An- l B ] has fu l l rank, i.e. r a n k ( M ) : n, in the state space [5].

4. Design of fuzzy SMC control

For each subsystem of the fuzzy model (9), from Section 2, we can always design the SMC U i such that

s(x) = O, ~(x) = O.

Indeed, one common SMC candidate is

u' = - ( C B i ) - ' C A i x - (CBi)-~K~sllxll/llsll, (14)

where the scalar Ki > 0 and II II represents the Euclidean norm. Here the norm Ilxll is considered to limit the chattering when x --+ 0.

Such control ensures the subsystem to reach and remain in the sliding mode s ( x ) = 0; thus local asymptotical stability is guaranteed. Indeed, with the Lyapunov function,

V(x) = ½J(x)s(x), (15)

where the superscript T stands for transpose. Differ- entiating V(x) leads to

1 ) = sT(x)g(x). (16)

Substituting (14) and (9) into (16) yields

= sT(x)C(Aix + Biu)

= -KiUsllUxU. (17)

Assumption 3. It is assumed that for i = 1,.. . , m, CBi is nonsingular: thus the existence o f the SMC, u i,

u i = - ( C B i ) -1 C A i x - (CB~)-~K~sllxll/llsll for ith subsystem under R i is guaranteed.

One natural candidate of the global control for the fuzzy model may be

u = ~ Iziui(x). (18) i=l

The problem of interest is that whether the control (18) is able to ensure the globally asymptotical stability.

Remark 1. In the global fuzzy model (12), even in each subregion, the subsystem is asymptotically stable, the global asymptotical stability is not guaranteed in the overlapping regions of fuzzy sets where several subsystems are activated at the same time to certain degrees. The asymptotical stability of the subsystem in its activating region is at least weakened by those subsystems whose activating regions overlap with it.

Generally speaking, it is impossible to stabilize the global fuzzy system by means of amalgamating the sliding-mode control for each linearized subsystem. However, under certain conditions, this can be done. The following theorem provides the results.

Theorem 1. For the fuzzy system (12), i f the S M C u i for ith subsystem is

u i = - ( C B , ) - ' C A , x - ( C B , ) 1KisIIxll/llsll a n d

CBi = CBj = CB for i C j

then the system is asymptotically stable.

2 9 8 X. Yu et al./Fuzzy Sets and Systems 95 (1998) 295-306

Proof. Let the Lyapunov function candidate be

V(x) -- ½sT(x)s(x). (19)

Therefore,

~'(X) ~-" sT(x),~(X)

= sT(x)(CAx + CBu)

= s T ( x ) C [2iAix + CB Z [2iui i = l

m

= sT(x) ~ #i(CAix + CBiu i) i = l

= --sT(x) ~ ~gisllxll/llsll i=1

= -~__, ~Killsllllxll < o i = l

(20)

since for s j ¢ 0, x ¢ 0, then (20) indicates that the fuzzy system is globally asymptotically stable. []

Remark 2. Theorem 1 presents the conditions to guarantee the asymptotical stability. However the conditions CBi = CBj for i # j are stringent to apply. In the following, we shall provide a robust SMC strategy.

Before we proceed further, denote 2(V) as an eigenvalue of the matrix V, and Amax(V) and 2min(V) as the largest eigenvalue(s) and the smallest eigenvalue(s) of the matrix V, respectively.

As we see from the global fuzzy model (12), for the rule R ~, the system may be dominated by the control for another rule Rk ( i # k) u k, i.e.

Ri: IF

THEN

z~ is F I AND...Zn is F /

u i = - (CBk)- 1CAkx

- (CBk)-~Kksllxll/llsll, (21)

where

k = { j :max[p l , . . . , p j , . . . , pm]} .

Under certain conditions, the influence on the asymptotical stability of other subsystems cannot be over- come.

Theorem 2. For the global fuzzy model (12), tf

2min(CBi(CBk)-I + (CBi(CBk)-I)T) > 0, (22)

Kk > K° = II CAi - CBi( CBk )- 1 C Ak II

/.min(CBi(CBk )-1 q_ (CBi (CBk) -1 )T)' (23)

i=1

-- sT (x )CB~( CBk )- ~ Kksllxll/llsll]

< ~ ~illCAi - c g i ( C B k ) - l C A k l l Ilsll Ilxll i = l

- - "~min ( CBi ( CBk )- i

+(CBi(CB~ )- l )T)K~ IIs II Ilxll. (24)

= ~ ]2i[sT(x)(CAi - CBi(CBk)-l CAk )x

then the system is asymptotically stable.

Proof. Let the Lyapunov function be

v = lsT(x)s(x).

Differentiating V along the fuzzy global dynamics (9) yields

~Z ~__ sT(x)g(X)

= sT(x)C(Ax + Bu)

=sT(x)C[ ~--~ltiAixq-~-'~#iBi(-(CBk)-lCAkxi=l i=1

--( CBk )- ~ K~ slIxlI/ [ISII ) ]

=sT(x) [ ~ pi(CAix +

- ( CB~ ) - ' X~sllxll/llsll ) ) l 1

X. Yu et al./Fuzzy Sets and Systems 95 (1998) 295-306 299

Substituting the conditions (22) and (23) into (24) yields

V < O

The asymptotical stability of the global fuzzy system is then guaranteed. []

When in sliding, the system satisfies s(x) = 0 and ~(x) = 0. The next problem is the design of the switching hyperplanes s (x ) = 0. Many approaches have been proposed to design C of s(x) , for example, the pole placement approach [13] and the hyperplane design approach [15]. All the methods can be used for the design of C for the fuzzy system. However, the following procedure should be noted. Design C for each linearized subsystem such that it is asymptotically stable, and also it satisfies the condition

).min( CBi( CBk ) - I + ( CBi( CBk ) - I )T ) > O.

5. Computer simulations

We study the following two cases to show the effectiveness of the fuzzy sliding-mode control proposed. The first example is to confirm Theorem 1 and the second example to Theorem 2.

Example 1. Consider the fuzzy system

R 1 " IF y ( t ) is SMALL

T H E N

2( t ) = A l x ( t ) + B lu ( t ) ,

Yl ( t ) : DlX(t ) , (25)

R 2" I F y ( t ) is L A R G E

T H E N

where

A I =

~(t ) = A2x( t ) ÷ B2u(t) ,

y2(t) = D2x( t ) , (26)

[01 0 0 ,

-0.125 0.311 -

B~ = , D1 =[0 .5 1 0.5],

F Iy] A2 = 0 ,

- 1

B 2 = , D1 = [ 1 0.8 0.1].

The fuzzy sets of SMALL and L A R G E are represented as

1 r e ( y ) = 1 -

1 + e x p ( - 2 ( y - 0.5)) '

1 ~2(Y) = 1 + exp ( -Z (y - 0.5)) '

respectively. The switching plane is chosen as

s(x) = xj + x2 + x3 = 0

which can be easily verified to be asymptotically stable. Indeed, it can be rewritten as £j + )71 + xl = 0 which is an asymptotical stable dynamics. The sliding- mode controls for the two subsystems are

u 1 = - ( C B 1 ) - l C A l x -- (CB1) - lK1 sgn(s)l]x H,

u 2 = -- ( C B 2 ) - I C A 2 x - (CB2)-~K2 sgn(s)tlxll.

Apparently, CB~ = CB2 = 1. The fuzzy control is

u =- - / 2 1 ( C B 1 ) - I C A I x - ~2(CB2) - 1 C A 2 x

- ( C B 1 ) - l ( # l K l + #2K2) sgn(s)Hxl].

Fig. 5 shows the performance of the fuzzy sliding- mode control. The initial state is x(0) = (1,0,0) T. KI = K2 = 10. The sampling period for the simulation is chosen to be h = 0.002. The trajectory fast reaches the sliding mode s = 0 and converges to the system origin within the sliding mode.

Example 2. Balancing of an inverted pendulum on a cart is considered [3]. The dynamics of the pendulum is given as follows:

Xl = X2,

g sin(x1 ) - amlx 2 sin(2xl )/2 - a c o s x l u (27)

4l/3 - aml COS2(Xl )

300 X. Yu et aL /Fuzzy Sets and Systems 95 (1998) 295-306

0,8

xl 0.6

0.4

°:

.o.ol/ ....

- 0 . 8 ~

-1 0 2 4 6 8 10 12

( a ) Time

Fig. l(a). System state responses.

1

0.8

0.6

0.4

>,

0,2

-0.~

-0.4

(b)

11 i i i i

2 4 6 8 Time

Fig. l(b). Switching function and output response.

i

10 12

where xi represents the angle o f the pendulum from the vertical axis, and x2 the angular velocity. 9 = 9.8 m/s 2 is the gravity constant, m is the mass o f the pendulum, 2l is the length o f the pendulum, a = 1/(m +M) where M is the mass of the cart, and u is the force

applied to the cart. In this study we choose m = 2 kg, M = 8kg, 21 = l m .

The fuzzy model o f this pendulum is obtained by linearizing the nonlinear equations over a number o f

X. Yu et al./Fuzzy Sets and Systems 95 (1998) 295 306 301

15

10

o

-1G

- 1 5 ; ~ 0 2 4 6 8 10

(C) Time

Fig. l(c), Control.

12

operat ion points in the phase plane (x l, x2) [3]:

R 1. IF

THEN xl is about 0, x2 is about 0

£c(t) = A lx ( t ) + BlU(t),

R2: IF

THEN xl is about 0, x 2 is about ± 4

~(t) = Azx ( t ) + B2u(t),

R 3" IF

THEN xl is about + n/3, X 2 is about 0

A(t) = A3x( t ) + B3u(t),

R4: IF

THEN

xl is about + n/3, X 2 is about + 4 or xl is about - n/3, x2 is about - 4

£c(t) = A4x( t ) + B4u(t),

Rs: IF

THEN

Xl 1S about + n/3, x2 is about + 4 or Xl is about - n/3, x2 is about + 4

£c(t) = Asx( t ) + Bsu(t),

with

[0 11 E0] AI = 17.2941 0 ' B1 = - 0 . 1 7 6 5 '

Io 1] E ol A2 = 14.4706 0 ' B2 ---- - 0 . 1 7 6 5 '

A3E° 11 E°I 5.8512 0 ' B3 -- - 0 . 0 7 7 9 '

io 1j [o 1 A4 = 7.2437 0.5399 ' B4 = - 0 . 0 7 7 9 '

[0 11 E0] A5 = 7.2437 0.5399 ' B5 = - 0 . 0 7 7 9 "

The switching funct ion s is chosen to be s = 10Xl +

X2 = 0. The fuzzy membersh ip funct ions are chosen to be as in Fig. 2.

It is easy to check that the condi t ion

,~min(CBi(CBk)-I ÷ (CBi (CBk) - I )T) > 0

is satisfied for all i and k. Also K ° is 1.4118, 6.6427, 6.9543, 6.9543 for i = 2, 3, 4, 5, respectively; hence we choose the largest value of these values, i.e.


-4 -2 0 2 4 xl

0.!

O.E

0,7

Q.O,E

~o.s E

0.4

0.3

0.2

0.1

0 -6

(a)

Fig. 2(a). Fuzzy sets o f state x 1 .

1

0.9

0.8

0.7

.~0.5 E

(1.4

0.~

0"2 t

0.1

0 -4

(b) -3 -2 -1 0 2

x2

Fig. 2(b). Fuzzy sets o f s t a t ex2 .

K1 ° = 6.7543. Similarly, I,-o i,-0 Ic0 ~-0 " ~ 2 , ' ~ 3 , ' ~ 4 , ' ~ 5 a r e found t o

be 7.0078, 2.9318, 3.0930, 3.0930, respectively. So to obtain an effective fuzzy sliding control, the Kk is chosen to be kk = 10 which is greater than all/£9.

The performance of the fuzzy sliding-mode controller is shown in Figs. 3 and 4. In Fig. 3, the initial state is chosen to be x(0) = (65 °, 0), the switching line is reached fast and the system converges to the

X. Yu et al./Fuzz), Sets and Systems 95 (1998) 295-306 303

0

-0.5

-1.5

-2

(a)

" X2 "

" ' . .

I I I I I

0.2 0.4 0.6 0.8 1 Time

1 i i i

1.2 1.4 1.6 1.8 2

Fig. 3(a). System state responses.

12

10

6

o9 4

.~ i I I I I I i I

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ( b ) Time

Fig. 3(b). Switching function and output response.

I

1.8

system equilibrium (0,0). In [3] it is reported that using the fuzzy linear feedback control, the pendulum can only be stabilized from the initial states of xl(0) C ( - 8 4 °, 84 °) and x2(O) = 0. We tested a

number of different initial states and find the fuzzy sliding-mode controller works for all of them. Fig. 4 shows the simulation results for the initial state x(0) = (90°,0.5).


250

200

150

100

5O

-5G

-10C

-15C

-20C

-250'

(c)

I I I I I I I I I

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time

Fig. 3(c). Control.

1.5

1

0.5

C

u~ -0.5

-1.E

-2

(a)

' x2

"%

I I l

0.2 0,4 0.6

. ° o

I I I I I I

0.8 I 1.2 1.4 1.6 1.8 Time

Fig. 4(a). System state responses.

12

lO

6

(n 4

-2

(b)

X. Yu et aL /Fuzzy Sets and Systems 95 (1998) 295-306

i

I I I I I I I I I

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time

Fig. 4(b). Switching function and output response.

305

250

200

150

100

50

0

-50

-100

-150

-200

-250 0

(c)

I I I I I I I I

0.2 0,4 0.6 0.8 1 1.2 1.4 1.6 Time

I

1.8

Fig. 4(c). Control.


6. Conclusion

A fuzzy sliding-mode control strategy has been developed in this paper. This control consists of "fuzzily" amalgamated sliding-mode controls of the system linearized around a set of operating points. The sliding-mode control for each individually linearized model is well known. It has been shown that under certain conditions the amalgamated sliding- mode control can stabilize the general fuzzy model. The sufficient condition for a robust fuzzy control has also been given.

Acknowledgements

The first author wishes to thank the Australian Research Council for a grant. The authors are grateful to Dr Gang Feng for providing their results on which this work is based.

References

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[3] G. Feng, S.G. Cao, N.W. Rees and C.K. Chak, Design of fuzzy control systems based on state feedback, J. Intelligent Fuzzy Systems, accepted for publication.

[4] A. Ishigame, T. Furukawa, S. Kawamoto and T. Taniguchi, Sliding mode controller design based on fuzzy inference for nonlinear systems, IEEE Trans. Ind. Electron. 40 (1993) 64-70.

[5] A. Isidoti, Nonlinear Control Systems (Springer, Berlin, 1989).

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[10] K. Tanaka and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems 45 (1992) 135-156.

[11] R.M. Tong, A control engineering review of fuzzy systems, Automatica 13 (1977) 559-568.

[12] V.I. Utkin, Sliding Modes in Control Optimization (Springer, Berlin, 1992).

[13] V.I. Utkin and K.D. Yang, Methods for constructing discontinuity planes in multidimensional variable structure systems, Automat. Remote Control 39 (1978) 1466-1470.

[14] A.S.I. Zinober, Deterministic Control of Uncertain Systems (Peter Peregrinus Ltd., London, 1990).

[15] A.S.I. Zinober and C.M. Doting, Hyperplane design and CAD of variable structure control systems, in: A.S.I. Zinober, Ed., Deterministic Control of Uncertain Systems (Peter Peregrinus Ltd., Londin, 1990).

Design of fuzzy sliding-mode control systems

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