Abstract— This paper proposes an approach to design sliding mode control for a class of uncertain nonlinear systems, where the uncertainty is a norm bounded type. Firstly, we choose the sliding surface which gives a good behaviour during sliding mode. It is formulated as an assignment of the poles of uncertain nonlinear system in a convex optimization area. Secondly, we design a nonlinear control law leading the state trajectory on the sliding surface in a finite time. A numerical application of inverted pendulum is given to validate the theoretical results of our approach. Keywords- Variable Structure Control, Sliding Mode Control, Norm Bounded Uncertainty, Robustness.

I. INTRODUCTION The sliding mode control of uncertain systems is a main challenge of modern automatic [1]-[2] - [3]. This command belongs to class of variable structure control (VSC) [10]. This latter, is characterized by its robustness and its simplicity of implementation. Many approaches based on sliding mode control have been proposed to treat several control problems such as the uncertain systems with uncertainties type norm bounded [4], and [5] - [6] have been proposed a new approach for linear systems with the same type of uncertainty. However, little number of works has been proposed on the sliding mode control of nonlinear systems based on Takagi-Sugeno (T-S) fuzzy model [7] - [8]. In this paper, new sliding mode control for nonlinear systems with uncertainties parametric type norm bounded using T-S fuzzy models have been proposed. This paper is organized as follows. In section II, the problem formulation is given. The proposed approach is detailed in section III and IV. A numerical example will be treated to validate the theoretical concepts is presented in section V. Finally, section VI summarizes the important features of proposed approach.

II. PROBLEM FORMULATION The global operation of a nonlinear dynamic system can be divided into several local linear operating regions. A fuzzy dynamic model has been proposed by Takagi and Sugeno to interpolate locally linear input/output relations via membership functions for nonlinear systems [9]. This fuzzy dynamic model is

described by fuzzy If-Then rules and will be employed here to approximate the dynamics of a nonlinear system. Consider the following fuzzy system to model a complex system:

Ri: If 1( )z t is 1iF and … ( )nz t is i

nF

Then ( ) ( ) ( )( ) ( )

, 1, 2,...( )

i i

i i

x t A x t B u ti r

y t C x t D u t

= +⎧⎪ =⎨= +⎪⎩

(1)

where iR represents the ith fuzzy inference rule, r the number of

inference rules, ( )12ijF j , ,..,n= the fuzzy sets, ( ) nx t ∈ℜ is the state

vector, ( ) mu t ∈ℜ is the input vector, ( ) py t ∈ℜ represents the

output vector. n niA ×∈ℜ , n m

iB ×∈ℜ , p niC ×∈ℜ and p m

iD ×∈ℜ are

known matrices. ( )1T

nz z ,..., z= represents some measurable system variables. Denote ( ( ))i z tμ as the normalized fuzzy membership function of

the inferred fuzzy set iF , where

1

ni i

jj

F F=

=∩ (2)

and

{ }1

( ( )) 1

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

ri

i

i

z t

z t i r

μ

μ=

⎧∑ =⎪

⎨⎪ ≤ ≤ ∀ ∈⎩

(3)

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

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

x t Ax t Bu ty t Cx t Du t

= +⎧⎨ = +⎩

(4)

where

1 1

1 1

r r

i i i ii ir r

i i i ii i

A ( z( t ))A , B ( z( t ))B

C ( z( t ))C , D ( z( t ))D

μ μ

μ μ

= =

= =

= =

= =

∑ ∑

∑ ∑ (5)

Let us consider a continuous linear uncertain system described by:

( )( ) ( ) ( )x t A A x t Bu t= + Δ + (6) here AΔ is the uncertainty of norm-bounded type written as

A G EΔ = ∇ (7)

Sliding Mode Control of Nonlinear Uncertain System Based on Takagi-Sugeno Fuzzy Model

Lotfi Chaouech High school of science and

techniques of Tunis Avenue Taha Hussein, B.P.56,

Bab Menara 1008 Tunis, Tunisia Email: chaouech.lotfi@yahoo.fr

Oussama Saadaoui High school of science and

techniques of Tunis Avenue Taha Hussein, B.P.56,

Bab Menara 1008 Tunis, Tunisia Email: oussa2302@hotmail.com

Abdelkader Chaari High school of science and

techniques of Tunis Email: assil.chaari@esstt.rnu.tn

Telephone: (216) 71–496066 Fax: (216) 71–391166

U.S. Government work not protected by U.S. copyright

where { }/d e Te eD R I×

∇ ×∇∈ = ∇∈ ∇ ∇ ≤ , n dG R ×∈ and e nE R ×∈

define the structure of the uncertainty. Assumption 1: the matrix iB has full rank. Assumption 2 : each linear sub-system of the global model is controllable if ( )irank V n= .

with -1, , ..., ni i i i i iV B A B A B⎡ ⎤= ⎣ ⎦

for 1,...,i r=

We consider the matrix iC such as the switching surfaces

{ }1

: 0li i n

j ij

S S x R C x=

= ∩ = ∈ = are usually intersecting hyper-

planes passing through the state space origin. The sliding mode occurs when the state reaches and remains in the intersection iS of the l hyper-planes. Geometrically, the subspace iS is the null space (or Kernel) of iC .

Differentiating iS with respect the time 0i

iS C x= = (8) From (6) and (8), we get:

( ) ( ) ( ) 0ii i i i iS C A A x t C B u t= + Δ + = (9)

if -1( )i iC B exists, then

-1-( ) ( ) -eq i i i i i i iu C B C A G E x k x= + ∇ =

(10)

with

-1( ) ( )i i i i i i ik C B C A G E= + ∇ (11)

As a result, the dynamics ( )( )-1- ( )i i i i i i ix I B C B C A G E x= + ∇

describes the motion on the sliding surface which is independent of the actual value of the control and depends only on the choice of the matrix iC .

III. DESIGN OF THE SLIDING SURFACE The canonical form used in [13] for VSC design can be extended to uncertain systems in order to select the gain matrix iC . Assumption3: there exists an ( )n n× orthogonal transformation

matrix iT such that iY T x= and 2,

0i i

iT B

B⎡ ⎤

= ⎢ ⎥⎣ ⎦

where iB has

full rank m and 2,iB is non-singular. The transformed state variable vector is defined as

iY T x= (12) The state equation becomes

( ) Ti i i i i i iY T A G E T Y T B u= + ∇ + (13)

The sliding condition is

0Ti i iC x C T Y= = with 1, 2,

Ti i i iC T C C⎡ ⎤= ⎣ ⎦

(14)

If the transformed state is partitioned as 1, 2,T T T

i iY Y Y⎡ ⎤= ⎣ ⎦ ,

with -1,

n miY R∈ and 2,

miY R∈ such as

11, 12,

21, 22,

i iTi i i

i i

A AT AT

A A⎡ ⎤

= ⎢ ⎥⎣ ⎦

(15)

and 1, 1, 1, 2,

2, 1, 2, 2,

i i i iTi i i

i i i i

G E G ET AT

G E G E∇ ∇⎡ ⎤

Δ = ⎢ ⎥∇ ∇⎣ ⎦ (16)

Then the system is given by

1 11, 1 12, 2 1 2

2 21, 1 22, 2 2, 1 2

( , ) ( , )

i i i

i i i i

Y A Y A Y f Y YY A Y A Y B u g Y Y⎧ = + +⎪⎨ = + + +⎪⎩

(17)

with 1 2 1, 1, 1 1, 2, 2

1 2 2, 1, 1 2, 2, 2

( , )( , )

i i i i i

i i i i i

f Y Y G E Y G E Yg Y Y G E Y G E Y

= ∇ + ∇⎧⎪⎨ = ∇ + ∇⎪⎩

(18)

where ( - ) ( - )

1, 2, 1, 2,, , e n m e m n m d m di i i iE R E R G R and G R× × × ×∈ ∈ ∈ ∈

The functions f and g represent the uncertainties in the system. Well they satisfy the following assumptions conditions: Condition1:

2 21 2 , 1 2

, 1, 1, 2, 1, 1, 2,

( ,

( ) ( )

i f i

f i i i i i i i

f Y Y k Y Y

K G E E G E Eσ σ

⎧ ≤ +⎪⎨

⎡ ⎤ ⎡ ⎤= =⎪ ⎣ ⎦ ⎣ ⎦⎩

(19)

Condition 2: 2 2

1, 2, , 1 2

, 2, 1, 2, 2, 1, 2,

( ,

( ) ( 0i i i i

i i i i i i i

g Y Y k Y Y

K G E E G E Eα

α σ σ

⎧ ≤ +⎪⎨

⎡ ⎤ ⎡ ⎤= = ≥⎪ ⎣ ⎦ ⎣ ⎦⎩(20)

⋅ and ( )σ ⋅ denote the 2-norm and the largest singular value, respectively. Assumption 4: i iC B non singular implies that iC must also be non singular and the condition defining the sliding mode is:

-12 2, 1, 1 1

-12, 1,

- -

i i i

i i i

Y C C Y F Y

F C C

⎧ = =⎪⎨

=⎪⎩ (21)

with iF is matrix of dimension ( - )m n m× and the order of the uncertain system is ( - )n m . The sliding mode is governed by the following system equations

1 11, 1, 1, 1 12, 1, 2, 2

2 1

( ) ( )

- i i i i i i

i

Y A G E Y A G E Y

Y FY

⎧ = + ∇ + + ∇⎪⎨

=⎪⎩ (22)

where 2y is a feedback control. The closed loop system is

( )1 11, 12, 1, 1, 2, 1- ( - )i i i i i i iY A A F G E E F Y= + ∇

(23) This indicates that the design of a stable sliding mode requires the selection of gain matrix iF such that

11, 12, 1, 1, 2,- ( - )i i i i i i iA A F G E E F+ ∇ has ( - )n m left-half-plane eigen-values. The performance can be easily taken into account via root clustering with LMI concept. The following theorem gives a sufficient condition in order to determine the gain matrix iF such that the closed-loop uncertain system poles are in the sector ( , )σ θΩ .

Theorem 1 [14] If there exists a positive definite symmetric matrix

( ) ( )2 , n m n mw I W W − × −= ⊗ ∈ℜ and ( )

2 , m n mr I R R × −= ⊗ ∈ℜ such that:

11 11 12 12 1 1 2

1

1 2

( )( , ) 0 0

0

T T T T

T

A w wA A r r A G E w E rH w r G I

E w E r I

θ θ θ θ θ θ θ

θ

θ θ

⎡ ⎤+ + + +⎢ ⎥= − <⎢ ⎥⎢ ⎥+ −⎣ ⎦

(24)

then the gain matrix 1F RW −= − places the closed-loop system poles of the uncertain system (22) in the sector ( , )σ θΩ . To determine a stabilizing gain, we can define a parametric convex optimization problem, of the type:

( , ) 0i

i

Ti i

i

H w r

w I

w rr Q

σ

⎧ <⎪⎪⎪ >⎪⎨⎪⎪ ⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

(25)

where the matrix Q is a new variable and σ represent design parameters. Once the stabilizing matrix iF is determined, the matrix iC can be obtained by

[ ]i i m iC F I T= (26)

IV. CONTROL LAW DESIGN To research the sliding surface " iC " and ensure that the trajectories are directed towards the switching surface from any point in the state space. It can be summarized by the construction of the control law nonlinear leading the state trajectory on the sliding surface in a finite time gT and the binding to remain on the surface. The proposed control law consists of the sum of a linear control law ,L iu and a nonlinear part ,N iu , which has the following form:

, ,( ) ii L i N i

i

N xu x L x u u

M xρ

δ= + = +

+ (27)

where , i i iL N and M are an appropriate matrix. ρ is a design parameter and δ is a smoothing parameter.

Starting from the transformed state 1, 2,TT T

i iY Y⎡ ⎤⎣ ⎦ , we form a

second transformation 2, : n niT ℜ → ℜ such that:

2, 1 2TT T

iz T Y z z⎡ ⎤= = ⎣ ⎦ (28)

with 1 n mz −∈ ℜ and 2 mz ∈ ℜ where

-2,

0n mi

i m

IT

F I⎡ ⎤

= ⎢ ⎥⎣ ⎦

(29)

and

--12,

0-n m

ii m

IT

F I⎡ ⎤

= ⎢ ⎥⎣ ⎦

(30)

Then the state variables 1z and 2z are

1 1

2 1 2

i

z Yz F Y Y

=⎧⎨ = +⎩

(31)

The transformed system is given by

1 1, 1 2, 2

2 3, 1 4, 2 2,

i i i

i i i i

z z z fz z z B u g

⎧ = ∑ +∑ +⎪⎨ = ∑ +∑ + +⎪⎩

(32)

with 1, 11, 12,

2, 12,

3, 1, 22, 21,

4, 22, 12,

-

-

i i i i

i i

i i i i i i

i i i i

A A F

A

F A F A

A A F

∑ =

∑ =

∑ = ∑ +

∑ = +

1, 1, 2, 1 1, 2, 2

2, 1, 2, 1 2, 2, 2

( - )

( - ) i i i i i i i

i i i i i i i i i

f G E E F z G E z

g G E E F z G E z F f

⎧⎪⎪⎪⎪⎨⎪⎪ = ∇ + ∇⎪⎪ = ∇ + ∇ +⎩

(33)

From conditions (1) and (2), the transformed uncertainty functions if and ig have to satisfy the following conditions: Condition 3:

2 21 2 1 2 1( , ) -i f if z z K z z F z≤ + (34)

Condition 4:

( ) 2 21 2 1( ) -i f i ig K K F z z F zα σ≤ + + (35)

A. Design of the linear control law ,L iu

The linear control law ,L iu is obtained by taking 2 2 0z z= = , which is defined as

-1 *, 2, 3, 4, 4, 2,( ) - ( - )L i i i i i i i iu x B T T x L x⎡ ⎤= ∑ ∑ ∑ =⎣ ⎦

(36)

with -1 *2, 3, 4, 4, 2,- ( - )i i i i i i iL B T T⎡ ⎤= ∑ ∑ ∑⎣ ⎦

(37)

where *4,

m mi R ×∑ ∈ is a matrix such that its eigen-values are

in the left half complex plane.

B. Design of the non-linear control law ,N iu

Let us define a Lyapunov function

2, 2 2 2, 21( )2

Ti iV z z P z=

(38)

where 2,iP is a positive definite matrix solution of

* *2, 4, 4, 2, 0

T

i i i i mP P I∑ +∑ + = (39) with 2, 2 0iP z =

if and only if

2 0z = . By differentiating Lyapunov equation (38)

2, 2 2 2, 2 2 2, 21 1( )2 2

T Ti i iV z z P z z P z= +

(40)

and taking *2, 4 -

2m

iI

P ⎛ ⎞∑ = ⎜ ⎟⎝ ⎠

, we obtain

22, 2 2 2 2, 2, ,

1( ) -2i i i N iV z z z P B u= +

(41)

Using equation (27), the following equation can be obtained: 2, 2 -1

2 2, , 2,2, 2

- ii N i i

i

P zz P u B

P zρ

δ=

+ (42)

and then, the existence of 2, 2( ) 0iV z < is provided. Finally, using iT et 2,iT , we find

( )1 2( , ) it x T xρ γ γ γ= + (43)

1

2

( )

1 0

f iK K Fαγ σγγ

= +⎧⎪

>⎨⎪ >⎩

(44)

-12, 2, 2,- 0i i i i iN B P T T⎡ ⎤= ⎣ ⎦

(45)

2, 2,0i i i iM P T T⎡ ⎤= ⎣ ⎦ (46)

V. NUMERICAL APPLICATION Let us consider the problem of the tracking on an inverted pendulum with a cart. The nonlinear system is given by the following equations [11]:

1 22

1 2 1 12 2

1

sin( ) - sin(2 ) / 2 - cos( )4 / 3- cos ( )

x x

g x amlx x a x ux

l aml x

=⎧⎪⎨ =⎪⎩

(47)

where 1( )x t denotes the angle θ of the pendulum (in radians) and 2 ( )x t is the angular velocity. This system is comprised of a cart of mass 1M Kg= which move horizontally under the action of a force F . The cart is surmounted by a pendulum length 0.5l m= , 0.1m Kg= ,

1( )

aM m

=+

and -29.8g ms= is the constant gravity.

The Takagi–Sugeno fuzzy model of inverted pendulum with a cart is described as follows [12]:

Rule 1 : If 1( )x t is about 0 then

1 1 1( ) ( ) ( )x A A x t B u t= + Δ +

Rule 2 : If 1( )x t is about 2π±

then

2 2 2( ) ( ) ( )x A A x t B u t= + Δ + where

1 1

0 1 0 , B3 30

4 4A g aaml aml

l l

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = −⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

2 22 2

0 1 0 , B2 4 30

3 4A g l aaml aml

lββ β

π

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = −⎛ ⎞⎢ ⎥ ⎢ ⎥− −⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦

with

cos(88 )β = °

1 1 1 2 2 2, A G E A G EΔ = ∇ Δ = ∇

[ ] [ ]1 2 1 20

, 1 0 , 1 01

G G E E⎡ ⎤= = = =⎢ ⎥⎣ ⎦

The figure 1 shows the triangular membership functions used in this example.

Fig .1. Membership functions of the proposed model

The nonlinear system is modeled as follows:

( )1

( ) ( ( )) ( ) ( ) ( ) , 2r

i i i i ii

x t z t A G E x t B u t rμ=

= + ∇ + =∑

(48)

We simulated 3 experimental cases: case 1, case 2 and case 3. In case 1, we present the result obtained without uncertainty. The parameters defining the regions LMI to determine the gain iF are:

( )-4, 8

radπσ θ= =

(49)

Using LMI feasibility problem (24) to (25), the MATLAB LMI Toolbox solver after 26 iterations, gives

1 2 4.6900 F F= =

[ ]1 2 4.6900 1 C C= =

We get for { }*4, 1 , =2 and =0.005 i i iρ δΣ = −

the following

results: [ ]1 13.7284 4.0380L = and [ ]2 185.7226 119.4225L =

[ ]1 2 2.3450 0.5 M M= =

[ ]1 1.6642 0.3548 N = and [ ]2 49.2171 10.4942 N = (50)

0 2 4 6 8 10 12-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

time(s)

X1

Angular position

ideal system reference signal

0 2 4 6 8 10 12-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

time(s)

X2

Angular velocity

Fig.2. Evolution of 1x (upper) and 2x (down) of the ideal system

(case1)

2π

2π− 1x

0 . 2

0 . 4

0 . 6

0 . 8

1

11−1 . 5− 1 . 50 . 50 . 5− 0

1R u l e

2R u l e

0 2 4 6 8 10 12-10

-8

-6

-4

-2

0

2

4

6

8

10

time(s)

U

Control law

0 2 4 6 8 10 12-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time(s)

S

Sliding surface

Fig.3. Evolution of the control u (upper) and the sliding surface S

(down) of the ideal system (case 1)

-7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Re

Im

Fig.4. Closed-loop modes (case 1)

Figs. 2, 3, and 4 show the robustness of the proposed sliding mode control law without uncertainty. In cases 2 and 3, we analyze the robustness of our approach taking into account the parametric uncertainty being norm bounded. We take the same simulation conditions of the case 1 (ideal case). We introduced variations in the mass of cart of +40% and +90% for cases 2 and 3, respectively. Faced with a variation of 40% of the mass of the cart, we

obtain to { }*4, 1 , =2 and =0.005 i i iρ δΣ = −

the following

results: [ ]1 14.0122 4.0380L = and [ ]2 194.1179 119.4226L = (51)

[ ]1 2 2.3450 0.5 M M= = (52)

[ ]1 1.6642 0.3548 N = and [ ]2 49.2171 10.4942 N = (53)

Faced with a variation of 90% of the mass of the cart, we

obtain to { }*4, 1 , =2 and =0.005 i i iρ δΣ = −

the following

results: [ ]1 14.3671 4.0380L = and [ ]2 204.6121 119.4226L = (54)

[ ]1 2 2.3450 0.5 M M= = (55)

[ ]1 1.6642 0.3548 N = and [ ]2 49.2171 10.4942 N = (56)

As shown in Figs. 5 and 6 our proposed sliding mode control exhibit better performance for the +40% of the variations in the mass of cart. A similar analysis can be seen also in Figs. 7 and 8 for the variations in the mass of cart of +90%. On the whole, we note that, whatever the variation level is, our proposed sliding mode control always keep the best robustness.

0 2 4 6 8 10 12-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

time(s)

X1

Angular position

ideal system

reference signaluncertain system (+40% of M)

0 2 4 6 8 10 12-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

time(s)

X2

Angular velocity

ideal system

uncertain system

Fig.5. Evolution of 1x (upper) and 2x (down) of the uncertain system

(case 2)

0 2 4 6 8 10 12-10

-8

-6

-4

-2

0

2

4

6

8

10

time(s)

U

Control law

ideal systemuncertain system

0 2 4 6 8 10 12-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time(s)

S

Sliding surface

ideal system

uncertain system

Fig. 6. Evolution of the control u (upper) and the sliding surface S (down) of the uncertain system (case 2).

0 2 4 6 8 10 12-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

time(s)

X1

Angular position

ideal system reference signaluncertain system (+90% of M)

0 2 4 6 8 10 12-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time(s)

X2

Angular velocity

ideal system

uncertain system

Fig.7 Evolution of 1x (upper) and 2x (down) of the uncertain system

(case 3)

0 2 4 6 8 10 12-15

-10

-5

0

5

10

15

time(s)

U

Control law

ideal system

uncertain system

0 2 4 6 8 10 12-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time(s)

S

Sliding surface

ideal system

uncertain system

Fig.8 Evolution of the control u (upper) and the sliding surface S

(down) of the uncertain system (case 3). According to the paces we notice that the behavior of the system in both cases 2 and 3 (response of ideal system or with parametric variations) is almost similar which compiles robustness of the sliding mode control towards the parametric variations.

VI. CONCLUSION A sliding mode control design approach for nonlinear-time invariant systems affected by uncertain parametric norm bounded type has been proposed in this paper. The first step consists in the synthesis of the sliding surface which gives a good behavior during the sliding mode. It corresponds to the study of the problem of existence. In the second step, a

nonlinear control scheme is introduced. It provides a bounded motion about the ideal sliding mode. Numerical simulations have been presented showing the efficiency and the robustness of the proposed method.

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