231

Adaptive Fuzzy Sliding Mode Control for Inverted Pendulum

Wu Wang Xuchang University , School of Electrical and information Engineering, Xuchang, China

e-mail: jhwlz@tom.com

Abstract—A nonlinear sliding mode control method is presented for single inverted pendulum position tracking control. Sliding mode control (SMC) is a special nonlinear control method which have quick response, insensitive to parameters variation and disturbance, online identification for plants are not needed, its very suitable for nonlinear system control, but in reality usage, the chattering reduction and elimination is key problem in SMC. By using a function-augmented sliding hyperplane, it is guaranteed that the output tracking error converges to zero in finite time which can be set arbitrarily. Fuzzy logic systems are used to approximate the unknown system functions and switch item. Robust adaptive law is proposed to reduce approximation errors between true nonlinear functions and fuzzy systems. the definition of sliding mode control was presented and on the basis of the inverted pendulum system, the sliding mode controller was designed, Stability of the proposed control scheme is proved by Lyapnouv theorem and the control scheme is applied to an linear system and inverted pendulum system respectively, simulation studies shows the methods is effective and can applied into linear or nonlinear control system. Index Terms—adaptive fuzzy control; sliding mode control; inveted pendulumr; nonlinear system; simulation

I. INTRODUCTION An inverted pendulum system is a static unstable

system, inverted pendulum has become a hot topic in control field for the similarity in control of helicopter launching of space shuttle and operation of satellite and robot walking with two legs [1]. Various control methodologies have been proposed for inverted pendulum systems and overhead trolley crane system they have things in common in the past. In the nonlinear system of inverted pendulum is linearized at zero and a popular pole placement approach is used to design a state feedback controller. Sliding mode control (SMC) is a special nonlinear control, since the publication of the pioneering paper on sliding mode control, significant results and many applications have been reported in the literature, SMC systems exhibit superb control performance which can be designed and have no relationship with controlled plant parameters and disturbance, also it have some advantages such as quick response, insensitive to parameters variation and disturbance, online identification for plants are not needed, but chattering reduction and elimination is key problem in SMC [2]. In basic SMC, big switching gain was needed to eliminate disturbance and uncertain factors, which was the main reason of chattering. Fuzzy control has many advantages such as control with mathematical models not needed, can use expert information and knowledge, and with strong

robustness, on the other hand, in practical control equipment the controller parameters are hard to get and lack of systematic analytical method, Fuzzy sliding mode control (FSMC) combine fuzzy control with sliding mode control, FSMC make control destination from trace error to sliding mode function, if make sliding function to zero, the error can to zero gradually, for high order system, FMSC can hold two dimension input. FSMC is a soft control and can make chattering reduction and elimination. [3]. O. P. Ha applied equal control switching control and fuzzy control to realize fuzzy sliding mode controller [4], K. Y. Zhuang use fuzzy control estimate uncertain of system [5], S. H. Ryu design fuzzy rules based on chattering reduction [6], S. W. Kim divide sliding mode surface with fuzzy theory [7], B. Yoo approach unknown function with fuzzy system [8], C. Y. Liang design sliding mode surface with integral sliding mode function [9], J.Y. Chen use membership function adjust switching gain [10], P.G. Grossimon [11]and Y. P. Chen used SMC into inverted pendulum control [12].

II. ADAPTIVE FUZZY SLIDING MODE CONTROL

Definition of SMC SMC is a control strategy of variable structure control,

it’s a special nonlinear control and mainly expressed as discontinuous [13].

Assume a control system can be described as: ( , , ) , ,n mx f x u t x u t= ∈ ∈ ∈R R R&

(1) Switching function can be given: ( ), ms x s∈R (2) Control function can be given:

( ) ( ) 0( ) ( ) 0

u x s xu

u x s x

+

−

⎧ >⎪= ⎨<⎪⎩

(3)

If the sliding mode exist and reachability condition satisfied, ie. Accepted for ( ) 0s x = , all movement spot can reach sliding mode surface and the sliding mode is stable, this is called sliding mode control [14].

Design of control system Consider a SISO control system as follow [15]: ( , ) ( , ) ( ) ( )f t g t u t d tθ θ θ= + +&& (4) Where ( , )f tθ and ( , )g tθ are unknown nonlinear

function and ( )d t is external disturbance [16]. The trace error can be given:

ISBN 978-952-5726-07-7 (Print), 978-952-5726-08-4 (CD-ROM)Proceedings of the Second Symposium International Computer Science and Computational Technology(ISCSCT ’09)

Huangshan, P. R. China, 26-28,Dec. 2009, pp. 231-234

© 2009 ACADEMY PUBLISHER AP-PROC-CS-09CN005

232

( ) ( ) ( )e t t r tθ= − (5) The integral sliding mode surface can be defined as:

1 20( ) ( ) [ ( ) ( ) ( )]

ts t t r t k e t k e t dtθ= − − −∫& && & (6)

In (6), 1 20, 0k k> > .

If SMC in ideal state, ( ) ( ) 0s t s t= =& So:

1 2( ) ( ) ( ) 0e t k e t k e t+ + =&& & (7)

Switching function ( )s t as input and the fuzzy rules can be given:

Rule i : IF s is isF THEN u is iα (8)

1 20, 0k k> >

With the center of gravity defuzzification method the output of controller can be given:

1

1

( )

m

i ii

fz m

ii

u sωα

ω

=

=

=∑

∑ (9)

Where iω is the weight of according Rule i .

Assume ( , )f tθ , ( , )g tθ and ( )d t known, so the control law can be given:

11 2*( ) ( , ) [ ( , ) ( ) ]u t g t f t d t r k e k e−= − − + − −&& &θ θ

(10) When ( , )f tθ , ( , )g tθ and ( )d t unknown,

*( )u t can be approximated with fuzzy system. T T( , )f zu s ξ=α α (11)

Where T T1 2 1 2[ ] , [ ]m maα α ξ ξ ξ= =L Lα ξ .

1

ii m

ii

ωξω

=

=

∑ (12)

With fuzzy approximation theory, can approach *( )u t with an optimal fuzzy system ˆ( , )fzu s α . So:

* *T*( ) ( , )fzu t u s ε ε= + = +α α (13) Where ε is approximate error and satisfied with

Eε < . Tˆ ˆ( , )fzu s =α α ξ (14)

Where α is the estimation of ∗α . Compensate the error between *u and fzu with

switching control rule vsu and the whole control law is:

( ) fz vsu t u u= + (15)

Adaptive control algorithm The adaptive fuzzy sliding mode controller as shown

in Fig. 1.

*ˆ ˆ*fzfz fz fzu u u u u ε= − = − −% (16)

Define ˆ *= −%α α α and we can get: T

fzu ε= −%% α ξ (17)

1 2( ) ( ) ( ) ( )s t e t k e t k e t= + +& && & (18) 1*( ) ( , ) [ ( , ) ( ) ( )]u t g t g t u t s t−= − &θ θ (19)

( ) ( , )[ *( )]fz vss t g t u u u t= + −& θ (20) Define Lyapunov function as:

2 T1

1

1 ( , )( )2 2

g tV t sη

= + % %θ α α (21)

T1

1

T

1

( , )( ) ( ) ( )

1( , ) ( ( )

( ) ( , )( )vs

g tV t s t s t

g t s t

s t g t u

η

ηε

= +

+

+ −

&& % %&

&% %

θ α α

= θ α ξ α)

θ

(22)

In order to realize 1 0V ≤& , the adaptive law and switching control can be given as (23), (24).

ˆ ( )s tη1= &&%α α − ξ= (23)

( )sgn( ( ))vsu E t s t= − (24) Then:

1( ) ( ) ( ) ( , ) ( ) ( , ) 0V t E t s t g t s t g tε= − − ≤& θ θ (25)

Replace ( )E t with ˆ ( )E t and then: ˆ ( )sgn( ( ))vsu E t s t= − (26)

Here define trace error: ˆ( ) ( ) ( )E t E t E t= −% (27)

21

2

2 T 2

1 2

( , )( ) ( )2

1 ( , ) ( , )( )2 2 2

g tV t V t E

g t g ts t E

η

η η

= +

= + +

%

%% %

θ

θ θα α (28)

12

( , )( ) ( ) g tV t V t EEη

= + && & % %θ (29)

Here we take adaptive Law with:

2ˆ ( ) ( )E t s tη=&

(30) And then we get:

/d dt

/d dt

θ&

e

e&

r

r

θ θvsu

fzuu

α

E

s

Figure 1. Adaptive Fuzzy Sliding Mode controller

233

ˆ( ) ( ) ( ) ( , ) ( ) ( , )ˆ( ( ) ) ( ) ( , )

( ) ( , ) ( ) ( , )

( ) ( , ) ( ) ( , )

( ) ( ) ( , ) 0

V t E t s t g t s t g t

E t E s t g t

s t g t E s t g t

s t g t E s t g t

E s t g t

ε

ε

ε

ε

= − −

+ −

= − −

≤ −

= − − ≤

& θ θ

θ

θ θ

θ θ

θ

(31)

III. APPLICATION IN INVERTED PENDULUM

Mathematical models of inverted pendulum The dynamic equation of single inverted pendulum

can be given [17]:

1 22

1 2 1 12 2

1

12

1

sin cos sin /( )(4 / 3 cos /( ))

cos /( ) ( ) ( )(4 / 3 cos /( ))

c

c

c

c

x xg x mlx x x m mx

l m x m mx m m u t d t

l m x m m

=⎧⎪ − +⎪ =⎪

− +⎨⎪ +⎪ + +

− +⎪⎩

&

&

(32) Where 1x is angular position and 2x is the velocity of

the pole respectively, 29.8m/sg = , cm is the mass of

cart and 1kgcm = , m is the mass of pole and

0.1kgm = ,here 0.5ml = is half lengthen of pole, u is control input, ( )d t is external disturbance and here are sine function ( ) 20sin(2 )d t tπ= .

Simulation and conclusions In order to prove the control ability of fuzzy sliding

mode controller, we take simulation on linear and nonlinear system respectively; the linear system can be given as [18]:

1 2

2 225 133 ( ) ( )x xx x u t d t

=⎧⎨ = − + +⎩

&

& (33)

Here we take ( ) 20sin(2 )d t tπ= , position

tracking reference input was ( ) 0.2sin( )2

r t t ππ= + ,

take the parameters 1 2150, 200k k= = , so the sliding mode surface can be given as:

0( ) ( ) [ ( ) 150 ( ) 200 ( )]

ts t t r t e t e t dtθ= − − −∫& && &

(34) The position tracking for sine function as shown in

Fig.2, the control input as shown in Fig.4, the error as shown if Fig.5 and switching gain variation as shown in Fig.6.

Also simulation with nonlinear system of inverted pendulum above mentioned, here we take the parameters

1 210, 25k k= = , position tracking reference input was

( ) 0.2sin( )2

r t t ππ= + ,so the sliding mode surface can

be given as:

0( ) ( ) [ ( ) 10 ( ) 25 ( )]

ts t t r t e t e t dtθ= − − −∫& && & (35)

The initial state of single inverted pendulum is 600

π⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

,

simulated with fuzzy sliding mode control method that proposed in this paper, the position tracking for sine function as shown in Fig.2, we can see the tracking precision is high, the control input as shown in Fig.3, the error as shown in Fig.4 and switching gain variation as shown in Fig.5.

On the basis of the mathematical models of single inverted pendulum, FSMC controller can be applied into linear or nonlinear system successfully. The limitation of uncertainty bounds was released and the time derivative of the proposed sliding manifold was continuous, the chattering phenomenon in the sliding mode control was

alleviated by fuzzy switching, the high robustness and precision performance of control performance obtained.

0 1 2 3 4 5 6 7 8 9 10-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

time(s)

Pos

ition

trac

king

FSMC for Inverted Pendulum

Figure 2. Posintion tracking for inverted pendulum

0 1 2 3 4 5 6 7 8 9 10-80

-60

-40

-20

0

20

40

time(s)

Con

trol i

nput

FSMC for Inverted Pendulum

Figure 3. Control input for inverted pendulum

234

0 1 2 3 4 5 6 7 8 9 10-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

time(s)

Pos

ition

trac

king

erro

r

FSMC for Inverted Pendulum

Figure 4. Position tracking error for inverted pendulum

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time(s)

Sw

itchi

ng G

ain

estim

atio

n E

FSMC for Inverted Pendulum

Figure 5. Switching gain for inverted pendulum

ACKNOWLEDGMENT It is a project supported by natural science research in

office of education, HeNan Province(2008A510014), Also the project was supported by XuChang University.

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