Abstract—This paper designs a time-varying nonlinearswitching manifold in an optimal fashion for a class of nonlinear systems by developing the successive approximation approach of differential equation into infinite-time horizon. Based on the reaching law approach, we obtain a control input which drivesthe system’s state trajectories to reach the nonlinear sliding surface. The stability of the nonlinear sliding mode is analyzed.The convergence velocity of every state trajectory on the idealsliding surface can be adjusted through choosing quadratic performance index. Simulation example is employed to test the validity of the proposed design algorithm..

I. INTRODUCTIONHE control with sliding mode (SMC), which is commonly known as sliding mode control, is a nonlinear control strategy that is well known for its fast response, good

transient performance, easy realization and insensitiveness to the matching parameter variations and external disturbances. Many approaches had been proposed for the design of sliding surface. Normally, it was designed as linear sliding surface[2, 3]. A necessary and sufficient condition for the existence of a linear sliding surface depending on outputs and compensator states was derived [2]. Based on a singular system approach and linear matrix inequality, a sufficient condition which guarantees the existence of linear switching surface and the stochastic stability of sliding mode dynamics was given in [3]. Pole assignment and optimal quadratic methods were presented in many researches (see, for example, [4] and [5]). To remove the reaching phase problems and eliminate chatting, a special integral sliding surface was introduced in [6] and [7]. [8] proposed a coupled sliding-mode control method for the periodic orbit generation and the robust exponential orbital stabilization of inverted-pendulum systems. More recently, combinations of fuzzy control and VSC approaches had achieved superior performance [9], [10]. Differing from the classical first-order sliding mode, a new second-order sliding mode control had been proposed [11], [12].

For a class of nonlinear systems that could be recursively approximated as linear time-varying systems, a sliding modecontrol and corresponding time-varying sliding surfaces had been designed for each approximated system so that a given optimization criterion was minimized [13]. For a general

R. Dong is with the Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China. (Tel: +86 373 3040767;E-mail: drui@163.com.)M.-Q. Fan is with Department of Electrical Engineering and Information

Technology, Shandong University of Science and Technology, Jinan, 250031, China. (E-mail: fmq@sdust.edu.cn.)

Y.-R. Guo is with the Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China. (yr640323@163.com)

nonlinear system, Tang [14] proposed the successive approximation approach (SAA) of optimal control in finite horizon. By using the finite-step iterations, a suboptimal control law was obtained.

In this paper, the SAA is developed into infinite-time horizon. A new sliding mode is designed based on the optimal control for nonlinear systems. We first treat some statevariables as virtual “control input” and give a quadratic performance index in infinite-time horizon. Using the SAA of differential equation theory, we obtain an optimal nonlinear switching manifold. The stability of the sliding mode is assured by the optimal switching surface chosen. Convergence velocity of the state trajectories of the closed-loop system in idea sliding surface is assured by minimizing the quadratic performance index. Finally, a simulation example is employed to test the validity of this approach in control design.

II. PROBLEM STATEMENT

Consider the following nonlinear system

1 1 1 1 2 2

2 3 1 2 1 2

1 10 2 20

( ) ( ) ( ) ( ),( ) ( , ) ( , ) ( ),(0) , (0) ,

x t Ax t f x Cf xx t f x x B x x u tx x x x

= + += +

= =

�� (1)

where ( ) inix t R∈ , 1 2( ) [ ( ) ( )]T T Tx t x t x t= is the state vector,

2( ) nu t R∈ is the control vector, 1 1n nA R ×∈ and 1 2n nC R ×∈

are real constant matrices, 0ix are the known initial vectors

( 1,2)i = � 1 2 3 1 2(0) 0, (0) 0, (0,0) 0,det ( , ) 0.f f f B x x= = = ≠

Assumption 1: The pair ),( CA is completely

controllable.

Assumption 2: There exist some positive constant α such

that

11 1 1 1 1 1 1 1 1( ) ( ) || ||, , nf x f y x y x y Rα− ≤ − ∈ �2�

where ⋅ denotes some appropriate vector norm.

Assumption 3: The matrix 2 2 2( )df x dx is nonsingular.

Assumption 4: The function 2 2( )f x has unique

equilibrium point; that is 2 2( ) 0f x = if and only if 2 ( ) 0x t = .

Optimal Sliding Mode Design for Nonlinear SystemsRui Dong, Mingqu Fan, Yunrui Guo

T

414

Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

978-1-4244-6337-4/10/$26.00 @2010 IEEE

Considering system (1), choose the following quadratic

performance index:

1 1 2 2 2 20

1 [ ( ) ( ) ( ) ( )]2

T TJ x t Qx t f x Rf x dt∞

= +∫ (3)

where 1 1 2 2Q ,n n n nR R R× ×∈ ∈ are positive-definite matrices.

The optimal nonlinear sliding mode design problem is to

find a virtual “optimal control law” 2 2( )f x which minimizes

the quadratic performance index (3) subject to the dynamic

equality constraint (1). According to the necessary conditions

for the optimality, we can obtain the following nonlinear

two-point boundary value������� problem

11 1 1

1 1 1 1 2 2

1 10

( ) ( ) ( ) ( (x )) ( ),

( ) ( ) ( ) ( ), 0,( ) 0, (0) ,

Txt Qx t A t f t

x t Ax t f x Cf x tx x

λ λ λ

λ

− = + +

= + + >∞ = =

�

� (4)

where 11 1 1 1 1(x ) ( )Txf df x dx= . The virtual “optimal control

law” can be expressed as

2 2( ) ( ) 0TRf x C tλ+ = (5)

and the controller can be obtained based on the sliding

reaching law as following

( , ) ( )s s t Ks Esign sγ= = − −� (6)

where 2 21 2 1 2{ , , , }, { , , , }n nK diag k k k E diag ε ε ε= =� � ,

with 2, 0, ( 1, 2, , )i ik i nε > = � ,

21 2( ) [ ( ), ( ), , ( )] ,Tnsign s sign s sign s sign s= �

1 0( ) 0 0

1 0

i

i i

i

ssign s s

s

>= =− <

Unfortunately, for the nonlinear TPBV problem in (4), with the exception of the simplest case, there is no analytic solution. Therefore, it is necessary to find an approximate approach to solve nonlinear TPBV problem (4) for the optimal nonlinear switching manifold design. In this paper, we will develop the SAA into infinite-time horizon.

III. PRELIMINARIES AND ASSUMPTIONS

Consider the unforced nonlinear system described by0

0 0

( ) ( ) ( , ), ,( ) ,z t Gz t h z t t tz t z

= + >

=

� (7)

where nz R∈ is the state vector, 0z is the initial state vector,

0

1: ( )n nth C R R R× → which satisfies the Lipschitz conditions

on 0

ntR R× ,

0 0( , )tR t= ∞ , n nG R ×∈ and all eigenvalues of G satisfy

Re( ( )) 0, 1,2, ,i G i nλ < = � (8)Lemma 1: The vector function sequence { }( ) ( )kz t is

represented by the expression

0

(0)0

( ) ( 1)0 0 0

( )0 0

( ) 0,

( ) ( ) ( ) ( ( ), ) ,

( ) , 1,2,

tk k

t

k

z t t t

z t t t x t r h z r r dr t t

z t z k

−

= ≥

=Φ − + Φ − >

= =

∫�

(9)

where ( ) exp( )t GtΦ = is state-transition matrix with respect

to the matrix G . Then the sequence { }( ) ( )kz t uniformly

converges to the solution of system (7) for any 0( , )t t∈ ∞ .

Proof. Because h is Lipschitz in z on0

ntR R× , there

exist some positive constants α and β such that

1 2 1 2

( , )

( , ) ( , )

h z t z

h z t h z t z z

α

β

≤

− ≤ −(10)

And based on (8), one can obtain 0( ) ,t M t tΦ ≤ ≥ (11)

where M is a positive constant, ⋅ denotes any appropriate

vector or matrix norm. We first consider the situation of 0[ , ]t t T∈ , where T is a constant. For any j and k ,

considering { }( ) ( )kz t as a sequence and using the arguments

as in the proof of Lemma 1 of [14], we obtain( ) ( ) 1 1 0

01

2 10 0

0

0

( )( ) ( )

!( )

exp( ( )),( 1)!

[ , ], 0,1, 2, .

ik jk j k i i

i kk k k

T tz t z t z M

iz M T t

M T tk

t t T k

αβ

α ββ

++ − +

= +

+ +

−− ≤

−≤ −

+∈ =

∑

�

(12)

Inequality (12) implies( ) ( )

0lim ( ) ( ) 0, 0, [ , ]k j k

kz t z t j t t T+

→∞− = ∀ > ∈ (13)

Now we consider the case of T → ∞ . Choosing 2k T= , from inequality (12) we get

2 2 2

( ) ( )

( 2) 10 0

0

( 2)0

2

20

( ) ( )

( )exp( ( ))

( 1)!

exp( )( )!

( ) exp( )( 2)!

k j k

k k k

T T T

TT

z t z t

z M T tM T t

k

z M TMT

T

Mz MT

α ββ

α ββ

βα β β

+

+ +

+

− ≤

−− ≤

+

≤

−

(14)

From (13)��14� and the following fact

415

1( )lim 0( 2)!

T

T

MTβ

→∞=

− (15)

we can see ( ) ( )lim ( ) ( ) 0,k j k

kz t z t+

→∞− = 00, [ , )j t t∀ > ∈ ∞ .

Thus sequence { }( ) ( )kz t is uniformly convergent.

Now we consider sequence ( ){ ( )}kz t� . For any j and k ,

from (10) we get( ) ( ) ( ) ( )

min( ) ( ) ( ( ) ) ( ) ( )k j k k j kz t z t G z t z tλ β+ +− ≤ + −� � (16)

Obviously, sequence { }( ) ( )kz t� is uniformly convergent.

Thus the limit of the sequence { }( ) ( )kz t is the solution of

system (7).

Remark 1: In practical control process, any system can’t run

endlessly. If T is large enough, we may consider T → ∞ .

We can adjust the iteration times according to the control time

T and the demand for precision.

IV. OPTIMAL SMC DESIGNING PROCESS

Let1( ) ( ) ( ), (0, )t Px t g t tλ = + ∈ ∞ (17)

where 1 1n nP R ×∈ is a positive-definite matrix to be found. 1(t) ng R∈ is the thk adjoint vector. Calculating the

derivative to the both sides of (17), we get 1

11

1 1

( ) ( ) ( )

(t) ( ( )) ( )

T

T

t PA PCR C P x tPCR C g Pf x t g t

λ −

−

= − −

+ +

�

� (18)

Substituting (17) into the first equation of (4) and comparing with (18), one can obtain

1

1 11

1 1 1 1 1

( ) ( ) ( ) ( )( ( )) ( ) ( ( ( )))( ( ) ( )) 0

T T T T

x

PA A P PCR C P Q x t PCR C g t A g tPf x t g t f x t Px t g t

− −+ − + − +

+ + + + ≡� (19)

From equation (19) we obtain the following Riccati matrix equation:

1 0T TPA A P PCR C P Q−+ − + = (20)According Assumption 1, the unique positive-definite matrix solution P exists. Comparing the third equation of (4) with (19) and (20), we can obtain the following adjoint vector differential equation

1

11 1

1 1 1

( ) ( ) ( ) ( ( ))( ( ( )))( ( ) ( ))

( ) 0.

T T

x

g t A CR C P g t Pf x tf x t Px t g t

g

−= − − − −+

∞ =

� (21)

Construct the following sequence(1,0)1(1, ) 1 (1, ) (1, 1)1 1 1 1(1, )1 10

( ) 0

( ) ( ) ( ) ( ( ))

(0) , 1,2,

k T k k

k

x tx t A CR C P x t f x tx x k

− −

=

= − +

= = ⋅⋅⋅

� (22)

For the thk iteration, (1, 1)1 ( )kx t− is known and (22) is a

nonhomogeneous linear vector differential equation, so we can obtain its solution (1, )

1 ( )kx t . According to the optimal control theory, one can get all eigenvalues of the matrix

1( )T TA CR C P−− satisfy 1Re[ ( )] 0Ti A CR C Pλ −− < ,

11, 2, ,i n= � . From Assumption 2 we know 1 1( )f x satisfies the Lipschitz conditions. Based on Lemma 1, the solution sequence { }(1, )

1 ( )kx t of system (22) uniformly converges to

the solution of the following system1

1 1 1 1

1 10

(t) ( ) ( ) ( ), 0(0)

Tx A CR C P x t f x tx x

−= − + >

=

� (23)

Let 1(* )(1, )1 1lim ( ) ( )k

kx t x t

→∞= and construct a sequence as

1 1 1*11

(1,0)

(1, ) 1 (1, )

* * * ( 1)1 1 1 11

(1, )

( ) 0( ) ( ) ( )

( ) ( ( ))( ( ) ( ))

( ) 0

k T T k

kx

k

g tg t A CR C P g tPf x f x Px t g t

g

−

−

=

= − − −

− +

∞ =

�(24)

For the kth iteration,1 1 1

*11

* * * ( 1)1 1 1 11( ) ( ( ))( ( ) ( ))k

xPf x f x Px t g t−− − + is known, so we

can obtain its solution sequence (1, ) ( )kg t . Let 1( ) exp[( ) ]T Tt CR C P A t−Φ = − . There exists a positive

constant 1M such that 1( ) , [ , )t r M r tΦ − ≤ ∈ +∞ . For

same reason, the sequence { }(1, ) ( )kg t uniformly converges to the solution of the following system

1

1 1*11

*11 1

* *1 11

( ) ( ) ( ) ( )

( ( ))( ( ) ( ))

( ) 0

T T

x

g t A CR C P g t Pf xf x Px t g t

g

−= − − − −

+

∞ =

�

(25)

Letting 1(* )(1, )lim ( ) ( )k

kg t g t

→∞= and solving the following

sequence

1

(2,0)1(2, ) (2, ) (2, 1)1 1 1 1

*1 (2, )1

(2, )1 10

( ) 0

(t) ( ) ( )

( ( ) ( )), 0

(0) , 1,2,

k k k

T k

k

x tx Ax t f xCR C Px t g t tx x k

−

−

=

= + −

+ >

= = ⋅⋅ ⋅

� (26)

we get a solution sequence (2, )1{ ( )}kx t . Define 2(* )

1 ( )x t as 2(* ) (2, )

1 1( ) lim ( )k

kx t x t

→∞= . Construct the following sequences:

0

1

*( ,0)1( , ) ( , ) ( , 1)1 1 1 1

*1 ( , )1

( , )1 10

( ) 0, ( ) 0

(t) ( ) ( )

( ( ) ( )), 0

(0) , , 1,2,

h

h

h k h k h k

T h k

h k

x t g tx Ax t f x

CR C Px t g t tx x k h

−

−

−

= =

= + −

+ >

= = ⋅⋅⋅

� (27)

*1

( ,0)

*( , ) 1 ( , )1 1

* * ( , 1)1 11

( , )

( ) 0

( ) ( ) ( ) ( )

( ( ))( ( ) ( ))

( ) 0, , 1,2,

h

h hp

h

h k T T h k

h k

x

h k

g tg t A CR C P g t Pf x

f x Px t g t

g k h

−

−

=

= − − − −

+

∞ = =

�

�

(28)

416

Repeating the foregoing iterations we can obtain sequences ( , )1{ ( )}h kx t and ( , ){ ( )}h kg t from (27) and (28) respectively.According the foregoing discussion, we have the

following:Theorem 1: Define (* )(*)

1 1( ) lim ( )h

hx t x t

→∞= and

(* )(*) ( ) lim ( )h

hg t g t

→∞= . Then (*)

1 ( )x t and (*) ( )g t are the

solutions of the nonlinear TPBV problem (4). Thus the corresponding ( , )H K th nonlinear switching manifold sequence satisfies

( ) ( , ) ( , )2 2 1( ) ( ( ) ( )) 0H T H K H KRf x C Px t g t+ + = (29)

and the optimal sliding surface is ( , )

2 2 1( ) ( ( ) lim lim ( )) 0T H K

H Kf x C Px t g t

→∞ →∞+ + = (30)

In fact, we cannot obtain optimal nonlinear switching manifold (30). We may find a “suboptimal nonlinear switching manifold” in practical applications by replacing

( , )lim lim ( )H K

H Kg t

→∞ →∞with ( , ) ( )H Kg t in (30). So, the ( , )H K th

order suboptimal nonlinear switching manifold can be rewritten as:

( , )2 2 1( ) ( ) ( ) 0T T H KRf x C Px t C g t+ + = (31)

Algorithm 1. (Find a suboptimal nonlinear switching manifold)Step 1: Solve the positive-definite matrix P from Riccati

matrix equation (20). Let (1,0)1 ( ) 0x t = and 1k h= = . Give

two positive real constants δ and ε .Step 2: Obtain the kth state vector (1, )

1 ( )kx t from (27).

Step 3: If (1, ) (1, 1)1 1( ) ( )k kx t x t δ−− < , let

(* ) (1, )1 1( ) , 1h kx t x k= = ; else replace k by 1+k then go to

step 2.Step 4: Let ( ,0) ( ) 0,hg t = and then obtain the kth adjoint

vector ( , ) ( )h kg t from (28).

Step 5: If ( , ) ( , 1)( ) ( )h k h kg t g t δ−− < , let (* ) ( , )( ) , 1h h kg t g k= = ; else replace k by 1+k then go to

step 4. Step 6: Letting H h= , calculate ( )

2 2( )Hf x from (29).Calculate ( )HJ from

( ) ( ) ( )1 1 2 2 2 20

1 [ ( ) ( ) ( ( )) ( )]2

H T H T HJ x t Qx t f x Rf x dt∞

= +∫ (32)

Step 7: If ( ) ( 1) ( )( ) ,H H HJ J J ε−− < then stop and output

the “suboptimal control law” ( )2 2( )Hf x .

Step 8: Find the kth approximation ( , )1 ( )h kx t of the state

vector 1( )x t from the state equation�27�.

Step 9: If ( , ) ( , 1)1 1( ) ( )h k h kx t x t δ−− < , let

(* ) ( , )1 1( ) ( ), 1h h kx t x t k= = ; else replace k by 1+k then go to

step 8. Step 10: Replace h by 1h + then go to step 4.

V. CONTROL LAW DESIGN AND STABILITY ANALYSIS

Choose the following switching function:( , )

2 2 1( ) ( ( ) ( ))T H KS Rf x C Px t g t= + + (33)The following control is employed:

{

( )

}

1

2 21 2 1

2

( , )1 1 2 2 1

2 23 1 2 2 2

2( , ) ( , )

( )( ) ( , ) ( ) ( )

( ) ( ) ( ( ) ( ))

( ), ( ) ( )

( ) ( )

TT T

T T H K

TT

T H K T H K

df xu t R B x x C PA KC P x tdx

C Pf x Esign Rf x C Px t g t

df xR f x x C PC KR f xdx

KC g t C g t

−

= − + +

+ + + +

+ + +

+ �(34)

Theorem 2: Consider system (1) with control (34). Assume that the switching manifold is chosen as (33). Then the state trajectories of system (1) can reach switching surface 0s =in finite time.Theorem 3: The closed-loop system (1) with the control

(34) is asymptotically stable.Proof: In quadratic performance index (3), Q and R are

positive-definite matrices, so 11 2( ) nx t L∈ ; that is,

1 10( ) ( )Tx t x t dt

∞< ∞∫ (35)

2 2 2 20( ) ( )Tf x f x dt

∞< ∞∫ . (36)

According to Assumption 2, 11 1 2( ) nf x L∈ , so we can get

11 2( ) nx t L∈� from the first equation of system (1). Based on

Schwarz inequality, we obtain 1 1 1( ) ( )Tx t x t L∈� ; that is,1 12 2

1 1 1 1 1 10 0 0( ) ( ) ( ) ( ) ( ) ( )T T Tx t x t dt x t x t dt x t x t dt

∞ ∞ ∞ ≤ < ∞ ∫ ∫ ∫� � �

(37)It is obvious that

1 1 1 11( ) ( ) ( ( ) ( ))2

T Tdx t x t x t x tdt

=� . (38)

Integrating (38) from 0 to ∞ , we obtain

1 1 1 1 10 100

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

tT T Tx t x t x t x t dt x x= +∫ � (39)

So, there exists a constant c such that 1 1lim ( ) ( )T

tx t x t c

→∞= .

Because 11 2( ) nx t L∈ , we get 0c = �that is� 1lim ( ) 0

tx t

→∞= .

From (5) and the third equation of (4) we get 2 2lim ( ) 0t

f x→∞

= .

Based on Assumption 4, we obtain 2lim ( ) 0t

x t→∞

= .

VI. SIMULATION

Consider nonlinear system�1� with

417

11 21 11 121 2 1 1

12 22 12 11

11 213 1 2 2 2

22 12 22

11

12 2 2

2

sin( )0 1, , , ( ) ,

sin( )-1 1

0 1 1 0( ) , , ( , ) ,

1 0 1

2(0) -1 1(0) 0 , ( )(0) 0

x x x xA x x f x

x x x x

x xB x C f x x

x x x

xx f xx

+ = = = = +

= = = +

− = =

21

22

121

1

x

x

e

e

+ − +

The matrices Q , R , K and E in “quadratic cost function” (3) and in sliding reachability condition (6) arechosen as

3 0 0.01 0 1 0, , .

0 6 0 0.01 0 1K E Q R

= = = =

Choosing 0.01,δ = we can get (1) 3.583,J =

(2) 2.8273J = , (3) 2.3033,J = (4) 2.2004J = . Choosing

0.05ε = , we obtain the relative error of the “performance

index” values satisfies (4) (3) (4)( )J J J ε− < . It indicates

that the 4th nonlinear switching surface(4,3)

2 2 1( ) ( ) ( ) 0T TRf x C Px t C g t+ + = is close enough to the

“optimal solution” 2 2 1( ) ( ) ( ) 0T TRf x C Px t C g t+ + = . So we

obtain the switching function defined as equation (33). Based

on reaching law approach (6), a control law that forces the

system state to reach the sliding surface from initial system

state within finite time is obtained. Simulation results are

shown as following

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

t

x11

k=1k=2k=3k=4

Fig. 1. State variable 11x of the closed-loop in the thk iteration

0 1 2 3 4 5 6 7-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t

x12

k=1k=2k=3k=4

Fig. 2. State variable 12x of the closed-loop in the thk iteration

0 1 2 3 4 5 6 7-2

-1.5

-1

-0.5

0

0.5

t

x21

k=1k=2k=3k=4

Fig. 3. State variable 21x of the closed-loop in the thk iteration

0 1 2 3 4 5 6 7-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

x22

k=1k=2k=3k=4

Fig. 4. State variable 22x of the closed-loop in the thk iteration

Fig. 1-Fig. 4 clearly show that the state variables of the

closed-loop system are asymptotically stable. Furthermore, it

is obvious that the more iterative steps, the higher control

precision.

418

When 2.08t ≥ , the state variables 11 12 21( ), ( ), ( )x t x t x t and

22 ( )x t of the closed-loop system reach the sliding surface

from their initial states and remain on it. Choose the

following quadratic performance index:

( ) ( ) ( )1 1 2 2 2 22.08

1 [ ( ) ( ) ( ( )) ( )] .2

N T N T NJ x t Qx t f x Rf x dt∞

= +∫

We get (1) (2)1.1520, 0.9026,J J= =(3) (4)0.9014, 0.9006J J= = . It means that the sliding mode

designed can ensure the state variables of the closed-loop system converge to zero fast on idea sliding surface.

ACKNOWLEDGMENT

This work is supported by National Nature Science Foundation under Grant 60574023 and Key Natural Science Foundation of Shandong Province under Grant Z2005G01.

REFERENCES

[1] V. I. Utkin, “Variable structure systems with sliding mode,” IEEE Transactions on Automatic Control , vol. 22, no.2, pp. 212�222, 1977.

[2] H. H. Choi, “Sliding-mode output feedback control design,” IEEE Transactions on Industrial Electronics, vol. 55, no. 11, pp. 4047-4054, 2008.

[3] S. P. Ma, E. K. Boukas, “A singular system approach to robust sliding mode control for uncertain Markov jump systems,” Automatica, vol. 45, no. 11, pp. 2707-2713, 2009.

[4] K. S. Kim, Y. Q. Park, “Sliding mode design via quadratic performance optimization with pole-clustering constraint,” SIAM Journal on Control and Optimization, vol. 43, no.2, pp. 670-684, 2005.

[5] H. Q. Ha, H. Trinh H, H. T. Nquyen, H. D. Tuan, “Dynamic output feedback sliding-mode control using pole placement and linear functional observers,” IEEE Transactions on Industrial Electronics , vol. 50, no. 5, pp. 1030-1037, 2003.

[6] L. G. Wu, W. Q. Zheng, “Passivity-based sliding mode control of uncertain singular time-delay systems,” Automatica, vol. 45, no. 9, pp. 2120-2127, 2009.

[7] A. Poznyak, L. Fridman, F. J. Bejarano, “Mini-max integral sliding-mode control for multimodel linear uncertain systems,” IEEE Transactions on Automatic Control, vol. 49, no. 1, pp. 97-102, 2004.

[8] M. S. Park, D. Y. Chwa, “Orbital stabilization of inverted-pendulum systems via coupled sliding-mode control,” IEEE Transactions on Industrial Electronics, vol. 56, no. 9, pp. 3556-3570, 2009.

[9] Y. C. Hsu, G. R. Chen, H. X. Li, “A Fuzzy Adaptive Variable Structure Controller with Applications to Robot Manipulators,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 31, no. 3, pp. 331-340, 2001.

[10] M. M. Abdelhameed, “Enhancement of sliding mode controller by fuzzy logic with application to robotic manipulators,” Mechatronicsvol. 15, no. 4, pp. 439-458, 2005.

[11] G. Bartolini, A. Pisano, E. Usai, “An Improved Second-Order Sliding-Mode Control Scheme Robust Against the Measurement Noise,” IEEE Transactions on Automatic Control, vol. 49, no. 10, pp. 1731-1736, 2004.

[12] G. Bartolini G, Punta E, Zolezzi T. Simplex Methods for Nonlinear Uncertain Sliding-Mode Control. IEEE Transactions on Automatic Control 2004; 49(6):922-933.

[13] M. U. Salamci, M. K. Ozgoren, S. P. Banks, “Sliding mode control with optimal sliding surfaces for missile autopilot design,” Journal of Guidance, Control, and Dynamics , vol. 23, no. 4, pp. 719-727, 2000.

[14] G. Y. Tang, “Suboptimal control for nonlinear systems: a successive approximation approach,” Systems and Control Letters, vol. 54, no.5, pp. 429-434, 2005.

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