Design of fuzzy sliding-mode control systems

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 95 (1998) 295-306

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, B a o 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 mathematically. On the other hand, the mathematical treatment *Corresponding author. Fax: +61 79 309729, E-mail: x.yu@ cqu.edu.au. 0165-0114/98/$19.00 (~ 1998 Elsevier Science B.V. All rights reserved PHS0165-0114(96)00278-3

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 question. One of the effective approaches is the fuzzy logic approach. By employing the fuzzy logic, the set oflinearized 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

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

296

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. Simulation results are presented to show the effectiveness of the control.

2. Sliding-mode control Consider the linear controllable system (1)

~c = A x + Bu,

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) u =

fors(x) > 0 , fors(x) < 0 ,

u-(x)

~(x) = 0

(3)

(4)

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

= 0,

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

Px = 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

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~

THEN 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 indicating influence from u(t) to yi(t), z = (Zl . . . . . zn) T represents some measurable system variables. Denote 12i(z(t)) as the normalized fuzzy membership function of the inferred fuzzy set F i where

(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 ( C B ) is nonsingular) Ueq = - ( C B ) - 1 C A x

(7)

(2)

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,

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

plane and remaining q eigenvalues remain zero [14].

where s ( x ) is the set of switching hyperplanes and defined as s ( x ) = C x = 0,

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

(6)

F i = f i Fj

(10)

j=l

and

£ i=1

t~e = 1.

(11)

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

Using the weighted average fuzzy inferences approach, we obtain the following global fuzzy statespace model: it(t) = Ax(t) + Bu(t),

(12)

297

where the superscript T stands for transpose. Differentiating V(x) leads to (16)

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

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

y = Dx(t) + Eu, = sT(x)C(Aix + Biu)

where

(17)

= -KiUsllUxU. A:

L

I~,Ai,

B= L,uiBi,

i=1

D : ~

i=1

l~iDi,

(13)

E : ~-~ IziEi.

i=1

Assumption 3. It is assumed that for i = 1,..., m, CBi is nonsingular: thus the existence of the S M C , u i, ui = -(CBi)

i=l

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 f u z z y model is controllable, i.e. the matrices Mi = [Bi, AiBi, A2Bi . . . . . An-lBi] for i = 1. . . . . m have full ranks, i.e. rank(Mi) : n.

-1 CAix -

(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= ~

(18)

Iziui(x).

i=l

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

Assumption 2. The global f u z z y model (9) is controllable in the state space, L e. the matrix M = [B, AB, A 2B . . . . . An- l B ] has full 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)

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.

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,

Theorem 1. For the fuzzy system (12), if the S M C

V(x) = ½J(x)s(x),

then the system is asymptotically stable.

(15)

u i for ith subsystem is ui = -(CB,)-'CA,x

-

(CB,)

1KisIIxll/llsll

and

CBi = CBj = CB for i C j

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

298

Proof. Let the Lyapunov function candidate be

V(x) -- ½sT(x)s(x).

where (19)

k = {j:max[pl,...,pj,...,pm]}. Under certain conditions, the influence on the asymptotical stability of other subsystems cannot be overcome.

Therefore,

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

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

= sT(x)(CAx + CBu)

2min(CBi(CBk)-I + (CBi(CBk)-I)T) > 0, Kk > K° = [2iAix + CB Z

C

= sT(x)

[2iui

IICAi - CBi( CBk )- 1C Ak II /.min(CBi(CBk)-1 q_ (CBi (CBk) -1 )T)' (23)

i=l m

then the system is asymptotically stable.

#i(CAix + CBiui)

= sT(x) ~

(22)

i=l

Proof. Let the Lyapunov function be

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

v = lsT(x)s(x).

i=1

= -~__, ~Killsllllxll < o

(20)

i=l

DifferentiatingV along the fuzzy globaldynamics (9) yields ~Z ~__sT(x)g(X)

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

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

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) uk, i.e.

Ri: IF

--( CBk )- ~K~ slIxlI/ [ISII)] =sT(x) [ ~

pi(CAix +

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

1

=~

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

i=1

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

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

< ~ ~illCAi -

THEN ui = - (CBk)- 1CAkx

- - "~min (

- (CBk)-~Kksllxll/llsll,

cgi(CBk)-lCAkll

IlsllIlxll

i=l

(21)

CBi ( CBk )- i

+(CBi(CB~ )-l )T)K~ IIsIIIlxll.

(24)

299

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

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

B~ =

V O.

A2 =

,

D1 = [ 0 . 5 1 0.5],

FIy] 0

,

-1

B2=

,

D1 = [ 1 0.8 0.1].

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

1 re(y)

=

1 -

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

1 ~2(Y) =

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

respectively. The switching plane is chosen as s(x) = xj + x2 + x3 = 0

5. Computer simulations We studythe 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.

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 slidingmode controls for the two subsystems are u 1 = -(CB1 )-lCAlx

-- ( C B 1 ) - l K 1 sgn(s)l]x H,

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

Example 1. Consider the fuzzy system

Apparently, CB~ = CB2 = 1. The fuzzy control is y ( t ) is S M A L L

R 1 " IF

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

- ~2(CB2) -1CA2x

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

2(t) = A l x ( t ) + B l u ( t ) , Yl (t) : DlX(t),

(25)

y ( t ) is L A R G E

R 2" I F THEN

~(t) = A 2 x ( t ) ÷ B2u(t),

y2(t) = D 2 x ( t ) , where AI=

[01 0 -0.125

Fig. 5 shows the performance of the fuzzy slidingmode 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 (26)

cart is considered [3]. The dynamics of the pendulum is given as follows: Xl

0 0.311

, -

=

X2,

g sin(x1 ) - a m l x 2 sin(2xl )/2 - a c o s x l u 4l/3 - a m l

COS2(Xl)

(27)

300

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

0,8 xl 0.6 0.4

°: .o.ol/ .... -

0 -1

0

. 2

8 4

~ 6 Time

(a )

8

10

12

Fig. l(a). System state responses.

1

0.8

0.6

0.4 >, 0,2

11

-0.~

-0.4

i

i

i

i

i

2

4

6

8

10

12

Time

(b)

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

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

-

0

1

2

5

4

;

(C)

~ 6 Time

8

12

10

Fig. l(c), Control.

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

R 1. I F

xl is about 0, x2 is about 0

THEN £c(t) = A l x ( t ) + BlU(t),

R2:

IF

xl is about 0, x 2 is about ± 4

THEN ~(t) = A z x ( t ) + B2u(t), R 3" I F

xl is about + n/3,

X2

is about 0

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

R4:

IF

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

THEN £c(t) = A4x(t) + B4u(t),

Rs:

IF

with

[0 Io A3E° io [0 AI =

17.2941

A2 =

14.4706

11 1] 11 1j 11

E0] E ol E°I [o 1 E0]

0 '

B1 =

-0.1765

'

0 '

B2 ---- - 0 . 1 7 6 5

'

B3 --

'

5.8512

0 '

A4 =

7.2437

0.5399

'

B4 =

A5 =

7.2437

0.5399

'

B5 =

-0.0779

-0.0779

'

-0.0779

"

The switching function s is chosen to be s = 10Xl + X2 = 0. The fuzzy m e m b e r s h i p functions are chosen to be as in Fig. 2. It is easy to check that the condition ,~min(CBi(CBk)-I ÷ (CBi(CBk)-I)T) > 0

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

or Xl is about - n/3, x2 is about + 4 THEN £c(t) = Asx(t) + Bsu(t),

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 o f these values, i.e.

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

302

0.! O.E

0,7 Q.O,E

~o.s E

0.4 0.3 0.2 0.1 0

(a)

-4

-6

-2

0

2

4

xl

Fig. 2(a). F u z z y sets o f state x 1.

1

0.9 0.8 0.7

.~0.5 E

(1.4

0.~ 0"2t 0.1 0

-4

-3

-2

-1

(b)

0 x2

2

Fig. 2(b). F u z z y sets o f s t a t e x 2 .

K1° = 6.7543. Similarly, I,-o " ~ 2 , 'i,-0 ~ 3 , ' ~Ic0 4 , '~-0 ~ 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

"

X2

"

-1.5 "'.. -2

(a)

I

I

I

I

I

1

i

i

i

0.2

0.4

0.6

0.8

1 Time

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

I

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

(b )

Time

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

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).

304

X. Yu et al./Fuzzy Sets and Systems 95 (1998) 295-306 250 200 150 100 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

.°o

C

u~ -0.5 '

x2

"% -1.E

-2 (a)

I

I

l

I

I

I

I

I

I

0.2

0,4

0.6

0.8

I

1.2

1.4

1.6

1.8

Time Fig. 4(a). System state responses.

X. Yu et aL /Fuzzy Sets and Systems 95 (1998) 295-306 12

305

i

lO

6

(n 4

-2

I

I

I

I

I

I

I

I

I

0.2

0.4

0.6

0.8

1 Time

1.2

1.4

1.6

1.8

(b)

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

250 200 150 100 50 0 -50 -100 -150 -200 -250 0

(c)

I

I

I

I

I

I

I

I

I

0.2

0,4

0.6

0.8

1 Time

1.2

1.4

1.6

1.8

Fig. 4(c). Control.

306

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

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 slidingmode 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 [1] R.A. DeCarlo, S.H. Zak and G.P. Matthews, Variable structure control of nonlinear multivariable systems: a tutorial, Proc. IEEE 76 (1988) 212-232. [2] O.M.E. E1-Ghezawi, A.S.I. Zinober and S.A. Billings, Analysis and design of variable structure systems using a geometric approach, lnternat. J. Control (1983) 657-671. [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). [6] J.-J.E. Slotine and W. Li, Applied Nonlinear Control (Prentice-Hall, Englewood Cliffs, NJ, 1991). [7] M. Sugeno and G.T. Kang, Fuzzy modeling and control of multilayer incinerator, Fuzzy Sets and Systems 18 (1986) 329-346. [8] M. Sugeno and G.T. Kang, Fuzzy identification of fuzzy model, Fuzzy Sets and Systems 28 (1988) 15-33. [9] T. Tagagi and M. Sugeno, Fuzzy identification of systems and its application to modeling and control, IEEE Trans. System Man Cybernet. SMC-15 (1985) 110-132. [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).