Hierarchical Fuzzy Sliding-Mode Control for Uncertain Nonlinear Under-Actuated Systems

Chiang-Cheng Chiang Department of Electrical Engineering

Tatung University Taipei, Taiwan, R.O.C.

e-mail: ccchiang@ttu.edu.tw

Yao-Wei Yeh Department of Electrical Engineering

Tatung University Taipei, Taiwan, R.O.C.

Abstract—This paper presents a hierarchical fuzzy sliding mode control scheme for a class of uncertain nonlinear under-actuated systems. First, the sliding surface of one subsystem is selected as the first layer sliding surface. Hence, we further construct a second layer sliding surface from the first layer sliding surface and the sliding surface of another subsystem till all the subsystem sliding surfaces are included. The fuzzy system and some adaptive laws are applied to approximate the unknown nonlinear functions and estimate the upper bounds of the unknown uncertainties, respectively. By means of Lyapunov stability theorem and the theory of sliding mode control, the proposed control scheme ensures the robust stability of the uncertain nonlinear under-actuated systems. Finally, simulation results show the validity of the proposed method.

Keywords—under-actuated systems; fuzzy systems; hierarchical fuzzy sliding mode control; Lyapunov stability theorem

I. INTRODUCTION In recent years, there has been growing interest in control problems of under-actuated systems [1-4]. Under-actuated systems are commonly encountered in mechanical systems which are fewer actuators than degrees-of-freedom to be controlled. The applications are used widely in practical systems such as free-flying space robots, underwater robots, manipulators with structural flexibility, overhead crane, etc.

Many papers on the control of under-actuated systems have been proposed in [5-8]. In [5], the hybrid switching control strategy was developed to cope with a class of nonlinear and under-actuated mechanical systems. The optimal control of nonholonomic, under-actuated mechanical systems was studied in [6]. The research [7] introduced a motion planning-based adaptive control method in under-actuated crane systems. In the case of the under-actuated surface ship with rudder actuator dynamics under external disturbances, the adaptive fuzzy controller was developed in [8]. Thus, the analyses and control of nonlinear under-actuated systems have been an important research area.

The sliding mode control (SMC) [9-10] has shown to be one of the effective nonlinear robust control strategies to tackle systemic parameters and external disturbances for nonlinear systems. SMC has some advantages such as insensitivity to system parameter variations, invariance to

external disturbances, good transient performance, and fast response, etc. In [11], the sliding-mode controller on the basis of the incremental hierarchical structure and aggregated hierarchical structure were used to treat the under-actuated system. The work [12] was developed a hybrid sliding-mode controller for under-actuated systems. SMC laws usually consist of two parts: switching controller design and equivalent controller design. First, the switching controller design drives the system states toward the sliding surface. Then, when the system states are on the sliding surface, the equivalent controller design guarantees the system states to keep on the sliding surface and converge to zero along the sliding surface.

The fuzzy control methods [13-14] have been adopted widely to treat the problem of under-actuated systems with unknown nonlinear functions. Especially, the decoupled fuzzy adaptive SMC method has been applied to the control of under-actuated systems with mismatched uncertainties [15]. According to the universal approximation theorem, fuzzy systems, which are constructed from a collection of fuzzy If-Then rules, are employed here as a way to approximate the unknown nonlinear functions of the system. Moreover, the adaptive laws are used to adjust the parameters of the fuzzy model. Thus, the adaptive schemes guarantee all signals to be bounded, and the tracking error of the closed-loop system will asymptotically track our desired trajectory and achieve the desired tracking performance.

The main object of this paper is on the design of the robust hierarchical fuzzy sliding mode control for uncertain nonlinear under-actuated systems. The fuzzy system and some adaptive laws are applied to approximate the unknown nonlinear functions and estimate the upper bounds of the unknown uncertainties, respectively. Furthermore, by Lyapunov stability theorem and the theory of sliding mode control, the presented hierarchical fuzzy sliding mode controller can not only guarantee the convergence to zero of each sliding surface, but also ensure the robust stability of the uncertain nonlinear under-actuated system.

This paper is organized as follows. First, the control problems of the uncertain under-actuated system and the concept of fuzzy system are introduced in Section II. The hierarchical fuzzy sliding mode control is proposed to deal

2014 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) July 6-11, 2014, Beijing, China

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with the control problem for uncertain under-actuated systems in Section III. The simulation results are illustrated to show the validity of the proposed control method in Section IV. Finally, a conclusion is given in Section V.

II. PROBLEM STATEMENT AND PRELIMINARIES

A. Problem Statement Consider a single-input-multi-output uncertain nonlinear

underactuated system expressed as follows: 1 2

2 1 1 1

3 4

4 2 2 2

2 1 2

2

1 3 2 1

( ) ( ) ( , )

( ) ( ) ( , )

( ) ( ) ( , )

( ) , , ,

n n

n n n n

Tn

x xx f b u d tx xx f b u d t

x xx f b u d t

t x x x

−

−

=⎧⎪ = + +⎪⎪ =⎪

= + +⎨⎪⎪⎪ =⎪ = + +⎩

= ⎡ ⎤⎣ ⎦

x x x

x x x

x x x

y

(1)

where [ ] 21 2 2, , , T n

nx x x R∈x = is the system state vector

which is assumed to be available for measurement, 1u R∈ is the control input, nR∈y is the system output,

1 2( ), ( ), , ( )nf f fx x x and 1 2( ), ( ), , ( )nb b bx x x are unknown real continuous nonlinear functions, and

1 2( , ), ( , ), , ( , )nd t d t d tx x x are unknown external bound uncertainties. Without loss of generality, the following assumptions are made for the controller design:

Assumption 1 : 0 ( ) , 0 ( )i i i if F b B≤ ≤ < ∞ < ≤ < ∞x x , for ∈ Γx , 1, 2, ,i n= ,

where iF and iB are known positive constants, and Γ is a set given as follows

{ },p ωΓ = ≤ Δ0x x - x . (2)

Here ω is a set of weight, and Δ is a positive constant which denotes all state variables’ boundary. 2n∈0x is a fixed point, and ,p ωx is a weighted p-norm, which is defined as

1/2

,1

ppni

pii

xω ω=

⎡ ⎤⎛ ⎞⎢ ⎥= ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∑x . (3)

If p = ∞ ,

21 2,

1 2 2max , , , n

pn

xx xω ω ω ω

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠x . (4)

If 2p = , 1ω = , ,p ωx will denote Euclidean norm x .

Assumption 2 : 0 ( , ) ( )i id t ρ≤ ≤ < ∞x x , for ∈ Γx , 1,2, ,i n= , where ( )iρ x are unknown bounded positive

smooth continuous functions.

Control Objective: Design a controller for (1) such that the system states ( )tx would converge to zero as .t → ∞

B. Description of Fuzzy Logic Systems The fuzzy logic system performs a mapping from nU R⊂

toV R⊂ . Let 1 nU U U= × × where iU R⊂ , 1,2, ,i n= . The fuzzy rule base consists of a collection of fuzzy IF-THEN rules:

( )1 1 2 2: IF is , and is , and and, is

THEN is , for 1, 2, , .

l l l ln n

l

R x F x F x F

y G l M= (5)

in which [ ]1 2, , , Tnx x x U= ∈x and y V R∈ ⊂ are the input

and output of the fuzzy logic system, liF and lG are fuzzy sets

in iU and V , respectively. The fuzzifier maps a crisp point

[ ]1 2, , , Tnx x x=x into a fuzzy set in U . The fuzzy inference

engine performs a mapping from fuzzy sets in U to fuzzy sets in V , based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The defuzzifier maps a fuzzy set in V to a crisp point in V . The fuzzy systems with center-average defuzzifier, product inference and singleton fuzzifier are of the following form:

( )( )( )

( )( )1

1

11

li

li

Mnl

iFil

M niFi

l

xy

x

θ μ

μ

==

==

=∑ ∏

∑ ∏x

, (6)

where lθ is the point at which fuzzy membership function ( )l

lG

μ θ achieves its maximum value, and we assume that

( ) 1ll

Gμ θ = . Eq. (6) can be rewritten as

( ) ( )Ty ξ=x θ x (7) where 1 2, , ,

Tlθ θ θ⎡ ⎤= ⎣ ⎦θ is a parameter vector, and

( ) ( ) ( )1 , ,TMξ ξ ξ⎡ ⎤= ⎣ ⎦x x x is a regressive vector with the regressor

( )lξ x , which is defined as fuzzy basis function

( )( )

( )( )1

11

li

li

niFil

Mn

iFil

x

x

μξ

μ

=

==

=∏

∑ ∏x

. (8)

III. CONTROLLER DESIGN AND STABILITY ANALYSIS

In view of (1), let the sliding surfaces be defined as follows: 2 1 2 , for 1,2, , . (9)i i i is c x x i n−= + = …

where ic are positive constants. Differentiating is with respect to time t, we have

2 1 2

2 ( ) ( ) ( , )i i i i

i i i i i

s c x xc x f b u d t

−= += + + +x x x

(10)

According to the equivalent control method, the equivalent control law of the subsystems can be obtained as

( )2 ( ) ( ) for 1, 2, , . (11)eqi i i i iu c x f b i n= − + =x x

Without loss of generality, the subsystem sliding surface 1s is selected as 1S .

663

First, let the unknown nonlinear functions ( ), ( ), and ( ), for 1,2, ,i i if b i nρ = …x x x , can be approximated, over a compact set Ωx , by the fuzzy systems as follows:

ˆ ( ) ( )i i

Ti f ff =x θ ξ xθ , (12)

ˆ ( ) ( )i i

Ti b bb =x ξ xθ θ , (13)

ˆ ( ) ( )i i

Ti ρ ρρ =x ξ xθ θ , (14)

where ( )ξ x is the fuzzy basis vector, ifθ ,

ibθ , and

, for 1,2, , ,i

i nρ = …θ are the corresponding adjustable parameter vectors of each fuzzy systems. It is assumed that

ifθ , ibθ , and

iρθ belong to compact sets

fiΩθ ,

biΩθ , and

,iρ

Ωθ respectively, which are defined as

{ }: ,f i i ii

Mf f fR NΩ = ∈ ≤ < ∞θ θ θ (15)

{ }: 0 ,b i i ii

Mb b bR NδΩ = ∈ < ≤ ≤ < ∞θ θ θ

( 1 6 ) { }: ,

i i ii

MR Nρ ρ ρ ρΩ = ∈ ≤ < ∞θ θ θ (17)

where if

N , ibN , and for 1,2, , ,

iN i nρ = … are the

designed parameters, and M is the number of fuzzy inference rules. We require

ibθ to be bounded from below by 0δ >

because from (28) we see that ˆ ( )ii bb x θ must be nonzero.

Let us define the optimal parameter vectors ,if

∗θ

,ib

∗θ and

, for 1, 2, , ,i

i nρ∗ = …θ as follows:

ˆarg min sup ( ) ( ) ,i i

fi fi

f i f if f∗

∈Ω ∈Ω

⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭θ x

θ xx xθ θ

(18)

ˆarg min sup ( ) ( ) ,i i

bi bi

b i b ib b∗

∈Ω ∈Ω

⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭θ x

θ xx xθ θ

( 1 9 )

ˆarg min sup ( ) ( ) ,i i

i i

i iρ ρ

ρ ρρ ρ∗

∈Ω ∈Ω

⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭θ x

θ xx xθ θ (20)

where if

∗θ , ,ib

∗θ and for 1,2, , ,

ii nρ

∗ = …θ are bounded in the

suitable closed sets ,fi

Ωθ ,

biΩθ

and iρ

Ωθ , respectively. The

parameter estimation errors can be defined as

i i if f f− *θ = θ θ , (21)

i i ib b b− *θ = θ θ , (22)

i i iρ ρ ρ− *θ = θ θ , (23) and

1 2w w w+ ≤ (24)

where

{ }11

ˆ ˆ( ) ( ) ( ) ( )r r

i ij r r f r r bj r

r

w a f f b b=

=

⎛ ⎞ ⎡ ⎤ ⎡ ⎤− + −⎜ ⎟⎜ ⎟ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠∑ ∏ * *= x x θ x x θ

(25)

21

ˆ( ) ( )r

i ij r rj r

r

w a ρρ ρ=

=

⎛ ⎞ ⎡ ⎤−⎜ ⎟⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠∑ ∏ *= x x θ

(26)

are the minimum approximation errors, which correspond to approximation errors obtained when optimal parameters are used. Then, we define

ˆw w w= − (27)

where w be as the estimate of w . Based on the fuzzy systems, the equation (11) can be

replaced as the following controller:

( )2ˆ ˆˆ ( ) ( ) , for 1, 2, , . (28)

i ieqi i i i f i bu c x f b i n= − + =x θ x θ

The ith-layer sliding surface iS and its control law iu can be defined as follows. 1 1 (29)i i i iS S sλ − −= + 1 ˆ ˆ (30)i i eqi swiu u u u−= + +

Here 1 (1,2, , )i i nλ − = … is a constant; 0 0 00 0S uλ = = =

, for (1,2, , )swiu i n= … is the switching control of the ith-layer sliding surface. From the recursive formulas (29) and (30), we have

1( )

i ii j rj r

rS a s

==

=∑ ∏

(31)

1

ˆ ˆ( )i

i swr eqrr

u u u=

= +∑

(32)

Here for a given i, ( )j ja j iλ= ≠ is a constant,

1 ( ),ja j i= = and for (1,2, , ).iu u i n= =

where

111

1

1

1

1 1

ˆ ˆ( ) ( )

ˆ ˆ

ˆ( ) ( )

ˆsgn( ) ( )ˆ( ) sgn( )

ˆ ˆ( ) ( ) ( ) ( )

r

r

r

r r

i iij r b eqlri j rr ll

swi swl i ilj r bj r

ri i

i j rj ri i i r

i ii ij r b j r bj r j r

r r

a b u

u u

a b

S ak S S w

a b a b

ρρ

=− =≠=

==

=

==

= == =

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦= − −

+− −

∑ ∑ ∏∑

∑∏

∑∏

∑ ∑∏ ∏

x θ

x θ

x θ

x θ x θ

(33) where ik is a positive constant, and the parameter update laws as follows:

( ) ( ),r r

if f i jj r

S aγ=

= ∏θ ξ x

(34)

664

( ) ( ) ,r r

ib b i jj r

S a uγ=

= ∏θ ξ x

(35)

( ),i

i jr r j rS aρ ργ

== ∏θ ξ x

(36) ˆ ,w iw Sγ= (37)

where rfγ ,

rbγ , , for 1,2, ,r

r iργ = … , and wγ are positive constants. Remark 1: Without loss of generality, the adaptive laws used in this paper are assumed that the parameter vectors are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint sets. If the parameter vectors are on the boundaries of the constraint sets but moving toward the outside of the constraint sets, we have to use the projection algorithm to modify the adaptive laws such that the parameter vectors will remain inside of the constraint sets. Readers can refer to reference [17]. The proposed adaptive laws (34)-(37) can be modified as the following form:

( )( ), if or

and ( ) 0 ,

and

( ) , if (

r r r

r r r fr

r r

r

fr

ia S Nf j i f f

j r

i TN a Sf f f j ij r

Nf fiP a S if j i Tj r a Sj i

j r

γ

γ

⎛ ⎞⎜ ⎟ <⎜ ⎟=⎝ ⎠

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟= = ≤⎜ ⎟⎜ ⎟=⎝ ⎠⎝ ⎠

=⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟⎨ ⎬ ⎛ ⎞⎜ ⎟= ⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭ ⎜ ⎟=⎝ ⎠

∏∏

∏ ∏

x θ

θ θ θ x

θ

xθ

ξ

ξ

ξξ

( ), 38) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪ ⎛ ⎞

⎜ ⎟⎪⎜ ⎟⎪⎜ ⎟⎪

>⎜ ⎟⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

x

where ( )r

if j ij r

P a Sγ=

⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭∏ xξ is defined as

( )2

( ) ( )

( ). 39

r r

r rr

r

i if j i f j ij r j r

Ti f f

f j ij rf

P a S a S

a S

γ γ

γ

= =

=

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪ =⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭

⎛ ⎞− ⎜ ⎟⎜ ⎟

⎝ ⎠

∏ ∏

∏

x x

θ θx

θ

ξ ξ

ξ

( )( ) , if or

and ( ) 0 ,

and

( ) , if

r r r

r r r br

r r

r

ia S u Nb j i b b

j r

i TN a S ub b b j ij r

Nb biP a S u ib j i

j r a jj r

γ

γ

⎛ ⎞⎜ ⎟ <⎜ ⎟=⎝ ⎠

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟= = ≤⎜ ⎟⎜ ⎟=⎝ ⎠⎝ ⎠

=⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟⎨ ⎬ ⎛ ⎞⎜ ⎟= ⎜⎪ ⎪⎝ ⎠⎩ ⎭ ⎜ =⎝

∏∏

∏ ∏

x θ

θ θ θ x

θ

x

ξ

ξ

ξ , (40)( ) 0

br

TS ui

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪ ⎛ ⎞

⎜ ⎟⎪⎜ ⎟⎪⎜ ⎟⎪ ⎟ >⎜ ⎟⎪ ⎟⎜ ⎟⎪ ⎠⎝ ⎠⎩

θ xξ

where ( )r

ib j ij r

P a S uγ=

⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭∏ xξ is defined as

2

( ) ( )

( ) . (41)

r r

r rr

r

i ib j i b j ij r j r

Ti b b

b j ij rb

P a S u a S u

a S u

γ γ

γ

= =

=

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪ =⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭

⎛ ⎞− ⎜ ⎟⎜ ⎟

⎝ ⎠

∏ ∏

∏

x x

θ θx

θ

ξ ξ

ξ

( )( ), if or

and ( ) 0 ,

and

( ) , if

r r r

r r rr

r r

r

ia S Nj i

j r

i TN a Sj ij r

Ni

P a S ij ij r a Sj i

j r

ρ

γ ρ ρ ρ

ρ ρ ρ

ρ ρ

γ ρ

⎛ ⎞⎜ ⎟ <⎜ ⎟=⎝ ⎠

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟= = ≤⎜ ⎟⎜ ⎟⎜ ⎟=⎝ ⎠⎝ ⎠

=⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟⎨ ⎬ ⎛ ⎞⎜ ⎟= ⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭ ⎜ ⎟=⎝ ⎠

∏

∏

∏ ∏

x θ

θ θ θ x

θ

xθ

ξ

ξ

ξ , (42)( ) 0

r

Tρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪ ⎛ ⎞⎪ ⎜ ⎟⎪ ⎜ ⎟⎪ ⎜ ⎟⎪ >⎜ ⎟⎪ ⎜ ⎟

⎝ ⎠⎩

xξ

where ( )r

ij ij r

P a Sργ=

⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭∏ xξ is defined as

2

( ) ( )

( ). (43)

r r

r rr

r

i ij i j ij r j r

Ti

j ij r

P a S a S

a S

ρ ρ

ρ ρρ

ρ

γ γ

γ

= =

=

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪ =⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

⎛ ⎞− ⎜ ⎟⎜ ⎟

⎝ ⎠

∏ ∏

∏

x x

θ θx

θ

ξ ξ

ξ

The main result of the robust adaptive tracking control scheme is summarized in the following theorem.

Theorem 1: Consider the single-input-multi-output uncertain under-actuated system (1). If Assumptions 1-2 are satisfied, then the proposed adaptive fuzzy sliding mode controller defined by (32) with adaptive laws (34)-(37) guarantees that all signals of the closed-loop system are bounded, and the system states ( )tx will converge to zero as t → ∞ . Proof: Consider the Lyapunov function candidate

2 2

1 1 1

1 1 1 1 1 (44)2 r r r r r r

r r r

i i iT T T

i i f f b br r rf b w

V S wρ ρργ γ γ γ= = =

⎛ ⎞= + + + +⎜ ⎟⎜ ⎟

⎝ ⎠∑ ∑ ∑θ θ θ θ θ θ

Differentiating the Lyapunov function V with respect to time, we can obtain

1 1 1

1 1 1 1 (45)r r r r r r

r r r

i i iT T T

i i i f f b br r rf b w

V S S wwρ ρργ γ γ γ= = =

= + + + +∑ ∑ ∑θ θ θ θ θ θ

From (31) and by the fact r rf f=θ θ ,

r rb b=θ θ , r rρ ρ=θ θ ,

and ˆw w= , the above equation becomes

( )( ){1 1 1 1

21

1 1 1 1 ˆ

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )

ˆ ˆ ( ) ( ) (

r r r r r r

r r r

r r r r

r

i i i ii T T Ti i j r f f b bj r

r r r rf b w

i ii j r r r r f r f r f r fj r

r

r r b r

V S a s ww

S a c x f f f f f

b u b u b

ρ ρργ γ γ γ=

= = = =

==

⎡ ⎤= + + + +⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤≤ + − + − +⎣ ⎦ ⎣ ⎦

⎡ ⎤+ − +⎣ ⎦

∑ ∑ ∑ ∑∏

∑ ∏ * *

*

θ θ θ θ θ θ

x x θ x θ x θ x θ

x x θ x }{ }

( )1

1 1 1

ˆ ˆ) ( ) ( )

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )

1 1 1 1 ˆ 46

r r r

r r r r

r r r r r r

r r r

b r b r b

i ii j r r r r rj r

ri i i

T T Tf f b b

r r rf b w

u b u b u

S a

ww

ρ ρ ρ ρ

ρ ρρ

ρ ρ ρ ρ ρ

γ γ γ γ

==

= = =

⎡ ⎤− +⎣ ⎦

⎡ ⎤⎡ ⎤+ − + − +⎣ ⎦ ⎣ ⎦

+ + + +

∑∏

∑ ∑ ∑

*

* *

θ x θ x θ

x x θ x θ x θ x θ

θ θ θ θ θ θ

According to (12)-(14), (46) becomes

665

( ){ }( ) ( )

( )

21 1

1 1

1

ˆ ˆ ˆ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

ˆ( )

r r r

r r r r

r r

i ii ii i j r r r f r b i j rj r j r

r ri ii iT T T T

i j f f i j b bj r j rr r

i i T Ti jj r

r

ii j r rj r

V S a c x f b u S a

S a S a u u

S a

S a f f

ρ

ρ ρ

ρ= =

= =

= == =

==

=

≤ + + +

⎡ ⎤ ⎡ ⎤+ − + −⎣ ⎦ ⎣ ⎦

⎡ ⎤+ −⎣ ⎦

+ −

∑ ∑∏ ∏

∑ ∑∏ ∏

∑∏

∏

* *

*

x θ x θ x θ

θ ξ x θ ξ x θ ξ x θ ξ x

θ ξ x θ ξ x

x{ }

( )

1

1

1 1 1

ˆ( ) ( ) ( )

ˆ( ) ( )

1 1 1 1 ˆ 47

r r

r

r r r r r r

r r r

i

f r r br

i ii j r rj r

ri i i

T T Tf f b b

r r rf b w

b u b u

S a

ww

ρ

ρ ρρ

ρ ρ

γ γ γ γ

=

==

= = =

⎡ ⎤ ⎡ ⎤+ −⎣ ⎦ ⎣ ⎦

⎡ ⎤+ −⎣ ⎦

+ + + +

∑

∑∏

∑ ∑ ∑

* *

*

x θ x x θ

x x θ

θ θ θ θ θ θ

According to (21)-(23) and (25)-(26), we have ( ){ }

( ) ( )2

1 1

1 1 1

1 1 1

ˆ ˆ ˆ( ) ( ) ( )

( ) ( ) ( )

1 1 1

r r r

r r r

r r r r r r

r r r

i ii ii i j r r r f r b i j rj r j r

r ri i ii i iT T T

i j f i j b i jj r j r j rr r r

i iT T T

i f f b br r rf b

V S a c x f b u S a

S a S a u S a

S w

ρ

ρ

ρ ρρ

ρ

γ γ γ

= == =

= = == = =

= = =

≤ + + +

− − −

+ + + +

∑ ∑∏ ∏

∑ ∑ ∑∏ ∏ ∏

∑ ∑

x θ x θ x θ

θ ξ x θ ξ x θ ξ x

θ θ θ θ θ θ 1 ˆ (48)i

w

wwγ

+∑

Let us define , , and r r rf bv v vρ as follows:

( )1 1

1 = ( )r r r r

r

i ii T Tf i j f f fj r

r r f

v S aγ=

= =− +∑ ∑∏ θ ξ x θ θ (49)

( )1 1

1( )r r r r

r

i ii T Tb i j b b bj r

r r b

v S a uγ=

= == − +∑ ∑∏ θ ξ x θ θ (50)

1 1

1( )r r r r

r

i ii T Ti jj r

r rv S aρ ρ ρ ρ

ργ== =

= − +∑ ∑∏ θ ξ x θ θ (51)

If the condition in the first line of (38) is true, substituting (34) into (49), we have

( ) ( )1 1

= ( ) ( ) 0 (52)r r r

i ii iT Tf i j f i j fj r j r

r rv S a S a

= == =

− + =∑ ∑∏ ∏θ ξ x θ ξ x

If the condition in the second line of (38) is true, we have

and ( ) 0.r r fr

i Tf f j ij r

N a S=

⎛ ⎞= >⎜ ⎟⎝ ⎠∏θ θ xξ

Then, substituting (39) into (49), we have

( )2 2 2* *2

1

1 1 1 ( )2 2 2

r

r r r r

r

i Ti j i fj r

fr f f f fr f

a Sv =

=

⎡ ⎤= − − + −⎢ ⎥⎣ ⎦

∏∑

θθ θ θ θ x

θξ

(53)

By the fact that *, , and ( ) 0r r r r fr

i TN N a Sf f f f j ij r

⎛ ⎞⎜ ⎟= ≤ >⎜ ⎟=⎝ ⎠∏θ θ θ xξ ,

the above equation becomes 0

rfv ≤ . (54)

Using the same method, we can prove that 0 and 0r rbv vρ≤ ≤

for all 0t ≥ . To show rb δ≥θ , we see from (40) that if

rb δ=θ , then 0rb ≥θ ; hence, we can guarantees

rb δ≥θ .

By applying (54), (37), and 0 and 0r rbv vρ≤ ≤ into (48), it

yields

( ){ }21

1

ˆ ˆ( ) ( )

ˆ ˆ ( ) (55)

r r

r

ii

i i j r r r f r bj rr

ii

i j r ij rr

V S a c x f b u

S a S wρρ

==

==

≤ + +

+ +

∑ ∏

∑∏

x θ x θ

x θ

Using the control law (28) and (32), the above equation can be rewritten

as

( )

1 11

1

ˆ ˆ ˆ ( ) ( )( )

ˆ ˆ( ) 56

r

r

i i i ili i j r b eql swllj r l rr

i ii j r ij r

r

V S a b u u

S a S wρρ

= == ≠=

==

⎧ ⎫⎡ ⎤≤ +⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

+ +

∑ ∑ ∑∏

∑∏

x θ

x θ

By applying (33), iV can be obtained as follows:

2

2 2

1 1 1

2

1 1 1

1 1 1 12 2 2 2

1 1 1 12 2 2 2

r r r r r r

r r r

r r r r r r

r r r

i i ii i i

T T Ti i f f b b

r r rf b w

i i iT T Tf f b b

r r rf b w

V k S

k S w

w

ρ ρρ

ρ ρρ

γ γ γ γ

γ γ γ γ

= = =

= = =

≤ −

= − − − − −

+ + + +

∑ ∑ ∑

∑ ∑ ∑

θ θ θ θ θ θ

θ θ θ θ θ θ

(57)

Let 2

1 1 1

1 1 1 12 2 2 2r r r r r r

r r r

i i iT T Tf f b b

r r rf b w

wρ ρρ

μγ γ γ γ= = =

= + + +∑ ∑ ∑θ θ θ θ θ θ

2 2

1 1 1

2 22 2 2 2

1 1 1

2 21

1 1 1 12 2 2 2

1 1 1 12 2 2 2

1 12 2

r r r r r r

r r r

r r r r r r

r r r

r r

r r

r r

i i iT T T

i i i f f b br r rf b w

i i iT T T

i i f f b br r rf b w

if bT

f f br f b

V k S w

k S w

ρ ρρ

ρ ρρ

μγ γ γ γ

μγ γ γ γ

γ γγ γ

= = =

= = =

=

≤ − − − − − +

= − − − − − +

− −+ +

∑ ∑ ∑

∑ ∑ ∑

∑

θ θ θ θ θ θ

θ θ θ θ θ θ

θ θ θ 22 2

1 1

1 12 2

r

r r r r

r

i iT T w

br r w

wρρ ρ

ρ

γ γγ γ= =

− −+ +∑ ∑θ θ θ

Let 22 2 2 2

1 1 1

1 1 1 12 2 2 2

r r r

r r r r r r

r r r

i i if bT T T w

f f b br r rf b w

L wρρ ρ

ρ

γ γ γ γ μγ γ γ γ= = =

− − − −= + + + +∑ ∑ ∑θ θ θ θ θ θ

2 22 2 2 2

1 1 1

2 2

1 1 1

1 1 1 12 2 2 2

1 1 1 1 12 2 2 2 2

r r r r r r

r r r

r r r r r r

r r r

i i iT T T

i i i f f b br r rf b w

i i iT T T

i f f b br r rf b w

V k S w L

c S w L

ρ ρρ

ρ ρρ

γ γ γ γ

γ γ γ γ

= = =

= = =

≤ − − − − − +

⎡ ⎤≤ − + + + + +⎢ ⎥

⎢ ⎥⎣ ⎦

∑ ∑ ∑

∑ ∑ ∑

θ θ θ θ θ θ

θ θ θ θ θ θ

where 1 1 1 1min 2 , , , , .r r r r

if b w

c kργ γ γ γ

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

Then,

.i iV cV L≤ − + (58) Based on (58), we obtain

( ) (0) .cti i

LV t e Vc

−≤ − + (59)

Then the tracking error converges to a region exponentially.

Substituting swiu into 1( )

i

i swr eqrr

u u u=

= +∑ and letting i=n,

we can obtain the hierarchical sliding mode control law

666

( )

1

1 1

1

1 1

1

1

ˆ ˆ ˆ

ˆ ˆ( ) ( ) ˆ( ) sgn( )

ˆ ˆ( ) ( ) ( ) ( )

ˆsgn( ) ( ) 60

ˆ( ) ( )

r

r r

r

r

n n

n swl swn eqll l

n nj r b eqrj r

n n nrn n

n nj r b j r bj r j r

r r

i in j rj r

ri i

j r bj rr

u u u u

a b uk S S w

a b a b

S a

a b

ρρ

−

= =

==

= == =

==

==

= + +

+= −

−

∑ ∑

∑ ∏

∑ ∑∏ ∏

∑∏

∑ ∏

x θ

x θ x θ

x θ

x θ

where .nu u= Remark 2: Note that the control law is discontinuous when the states across the sliding surface. Since the discontinuities in the control (60) give rise to chatter in the system, it has been proposed that the switching functions sgn( )iS will be replaced by a continuous approximation in an -widthε region of iS . Thus, replacing sgn( )iS with ( )isat S ε , the

( )isat S ε is described by

( )

1 if

if , 0.

1 if

i

ii i

i

S

Ssat S S

S

ε

ε ε εε

ε

>⎧⎪⎪= ≤ ∀ >⎨⎪⎪− < −⎩

(61)

IV. AN EXAMPLE AND SIMULATION RESULTS In this section, the inverted pendulum system is used to verify the performance of the proposed controller. Consider the inverted pendulum system described by

1 2

2 1 1 1

3 4

4 2 2 2

1 3

( ) ( ) ( , )

( ) ( ) ( , )

( ) , T

x xx f b u d tx xx f b u d t

t x x

=⎧⎪ = + +⎪⎨ =⎪⎪ = + +⎩

= ⎡ ⎤⎣ ⎦

x x x

x x x

y

(62) where

2

1 1 1 21

21

sin sin cos( )

4 cos3

t p

t p

m g x m L x x xf

L m m x

−=

⎛ ⎞⋅ −⎜ ⎟⎝ ⎠

x

11

21

cos( )

4 cos3 t p

xb

L m m x=

⎛ ⎞⋅ −⎜ ⎟⎝ ⎠

x

22 1 1 1

22

1

4 sin sin cos3( )

4 cos3

p p

t p

m Lx x m g x xf

m m x

− +=

−x

22

1

4( )43 cos3 t p

bm m x

=⎛ ⎞⋅ −⎜ ⎟⎝ ⎠

x

pct mmm += and 1x θ= , 2x θ= , 3x x= and 4x x= represent the angle of the pendulum with respect to the vertical axis, the angular velocity of the pendulum with respect to the vertical axis, the position of the cart and the velocity of the cart, respectively. 1 kgcm = is the mass of cart, 0.05 kgpm = is the mass of pole,

0.5 mL = is the half-length of pole, and 29.8 m sg = is the acceleration due to the gravity. u is the force input to move the cart. 1 2( , ), and ( , ),d t d tx x are the external uncertainties. The control objective is to maintain the system states ( )tx converge to zero. In the implementation, six fuzzy sets are defined over interval [-3, 3] for 1 2 3 4, , and x x x x , with labels NB, NM, NS, PS, PM, and PB, and their membership functions are

( ) ( )( )1

1 exp 5 2NB ii

xx

μ =+ +

,

( )( )( )2

1

1 exp 1.5NM i

i

xx

μ =+ − +

,

( )( )( )2

1

1 exp 0.5NS i

i

xx

μ =+ − +

,

( )( )( )2

1

1 exp 0.5PS i

i

xx

μ =+ − −

,

( )( )( )2

1

1 exp 1.5PM i

i

xx

μ =+ − −

,

( ) ( )( )1

1 exp 5 2PB ii

xx

μ =+ − −

, 1, 2,3, 4.i =

A case is simulated in this inverted pendulum system, and we apply the hierarchical fuzzy sliding mode controller in Section 3 to deal with the control problem. In this case, the sliding surfaces are selected as 1 1 1 2s c x x= + and

2 2 3 4s c x x= + , where 1 4c = and 2 1c = , the hierarchical sliding surfaces are constructed as 1 1,S s= 2 1 1 2S a s s= + , where 1 1.1a = . The initial values are chosen as

(0) [ ,0,0,0]6

Tπ= −x , 1(0) 0f =θ ,

2(0) 0f =θ ,

1(0) 5b =θ ,

2(0) 5b =θ ,

1(0) 0ρ =θ ,

2(0) 0ρ =θ , and ˆ (0) 0w = . The other

parameters are selected as 10k = , 1

0.5fγ = , 1

0.5bγ = , 2

0.5fγ = ,

20.5bγ = ,

10.5ργ = ,

20.5ργ = , and 0.2wγ = , and the boundary

layer 0.03ε = . 21 2

3( , ) ,2

d t x=x and 22 1 2( , ) ,d t x x=x are external

uncertainties. The simulation results are shown in Figs. 1-5.

667

Figs. 1-2 reveal that the state trajectories of the states 1x and

3 ,x respectively. From these simulation results, it is easily shown that the proposed controller ensures that the state trajectories converge asymptotically to zero. The performance of sliding dynamics and the control signal are shown in Figs. 3-4 and Fig. 5, respectively. The simulation results verify the effectiveness of the proposed robust hierarchical fuzzy sliding mode controller.

V. CONCLUSION In this paper, a hierarchical fuzzy sliding mode controller is

constructed to deal with the problems for a class of SIMO under-actuated systems with uncertainties. Within the scheme, the fuzzy logic systems and some adaptive laws are used to approximate the unknown nonlinear functions and the unknown upper bounds of uncertainties. Based on Lyapunov stability theorem and the theory of sliding mode control, the presented controller can not only guarantee the convergence to zero of each sliding surface, but also ensure the robust stability of the uncertain nonlinear under-actuated system. Finally, some simulation results are illustrated to confirm the effectiveness of the proposed control method.

REFERENCES [1] D. Qian, J. Yi, and D. Zhao, “Control of a class of under-actuated

systems with saturation using hierarchical sliding mode,” IEEE International Conference on Robotics and Automation, pp. 2429 - 2434, May, 2008.

[2] F. Nafa, S. Labiod, and H. Chekireb, “A structured sliding mode controller for a class of under-actuated mechanical systems,” International Workshop on Systems, Signal Processing and their Applications (WOSSPA), pp. 243 - 246, May, 2011.

[3] W. Wang, J. Yi, D. Zhao, and D. Liu, “Design of a stable sliding-mode controller for a class of second-order underactuated systems,” IEE Proc.-Control Theory Appl., Vol. 151, No. 6, pp. 683 - 690, Nov., 2004.

[4] J. X. Xu, Z. Q. Guo, and T. H. Lee, “A synthesized integral sliding mode controller for an underactuated unicycle,” International Workshop on Variable Structure Systems, pp. 352 - 357, June., 2010.

[5] M. Zhang and T. J. Tarn, “A hybrid switching control strategy for nonlinear and underactuated mechanical Systems,” IEEE Trans. Automatic Control, vol. 48, no. 10,pp. 1777 - 1782, Oct., 2003.

[6] I. Hussein and M. Bloch, “Optimal control of underactuated nonholonomic mechanical systems,” IEEE Trans. Automatic Control, vol. 53, no. 3, pp. 668 - 682, Apr., 2008.

[7] Y. Fang, B. Ma, P. Wang, and X. Zhang, “A motion planning-based adaptive control method for an underactuated crane system,” IEEE Trans. Control Systems Technology, vol. 20, no. 1, pp. 241 - 248, Jan., 2012.

[8] T. Li, B. Yu, and B. Hong,“A novel adaptive fuzzy design for path following for underactuated ships with actuator dynamics,” IEEE Conference on Industrial Electronics and Applications, pp.2796 - 2800, May, 2009.

[9] V. Sankaranarayanan and A. D. Mahindrakar, “Control of a class of underactuated mechanical systems using sliding modes,” IEEE Trans. Robotics, vol. 25, no. 2, pp. 459-467, Apr., 2009.

[10] J. M. Yang and J. H. Kim, “Sliding mode control for trajectory tracking of nonholonomic wheeled mobile robots,” IEEE Trans. Robotic and Automation, vol. 15, no. 3, pp. 578-587, June., 1999.

[11] W. Wang, X. D. Liu, and J. Q. Yi, “Structure design of two types of sliding-mode controllers for a class of underactuated mechanical systems,” IET Control Theory Appl., vol. 1, no. 1, pp. 163-172, Jan., 2007.

[12] R. Martinez, J. Alvarez, and Y. Orlov, “Hybrid sliding-mode-based

Control of underactuated systems with dry friction,” IEEE Trans. Industrial Electronics, vol. 55, no. 11, pp. 3998 - 4003, Nov., 2008.

[13] C. L. Hwang, H. M. Wu, and C. L. Shih, “ Fuzzy sliding-Mode underactuated control for autonomous dynamic balance of an electrical bicycle,” IEEE Trans. control systems technology, vol. 17, no. 3, pp. 783-795, May., 2009.

[14] C. C. Kung, T. H. Chen, and L. C. Huang, “Adaptive fuzzy sliding mode control for a class of underactuated Systems,” IEEE International Conference on Fuzzy Systems, pp. 1791 - 1796, Aug., 2009.

[15] S. Y. Shin, J. Y. Lee, M. Sugisaka, and J. J. Lee, “Decoupled fuzzy adaptive sliding mode control for under-actuated systems with mismatched Uncertainties,” IEEE International Conference on Information and Automation, vol.3, pp. 599 - 604, June., 2010.

[16] Y. Yang and J. Ren, “Adaptive fuzzy robust tracking controller design via small gain approach and its application,” IEEE Trans. Fuzzy Systems, vol. 11, no. 6, pp. 783-795, Dec., 2003.

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Fig. 1. The state trajectory of the state 1x .

Fig. 2. The state trajectory of the state 3x .

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

1.5

2

Time(sec)

posi

tion(

m)

x3

0 5 10 15 20 25 30-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Time(sec)

angl

e(ra

d)

x1

668

Fig. 3. The sliding surface 1 1 1 2s c x x= + . Fig. 5. The control input u .

Fig. 4. The sliding surface 2 2 3 4s c x x= + .

0 5 10 15 20 25 30-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Time(sec)

first

-leve

l slid

ing

surf

ace

s1

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

1.5

2

2.5

Time(sec)

seco

nd-le

vel s

lidin

g su

rfac

e

s2

0 5 10 15 20 25 30-30

-20

-10

0

10

20

30

40

50

Time(sec)

For

ce(N

)

u

669