Abstract— This paper presents a sliding mode observer based

controller for a class of uncertain systems. Exponential stabilizability for the uncertain systems is considered and the rate of convergence is estimated. The proposed observer based controller is designed using linear matrix inequality (LMI) approach. The observer and controller gains are obtained from the feasible solution of LMI. A numerical example is given to illustrate the performance of observer based controller with simulation results.

Index Terms-- Linear matrix inequality, Lyapunov theory, observer, sliding mode control, state perturbation.

I. INTRODUCTION HE sliding mode control (SMC) technique for a class of uncertain systems has been studied extensively in

literature [1]-[6]. SMC is an established method of controlling uncertain systems [7-8]. The invariance properties of SMC with respect to so-called matched uncertainty have encouraged many researchers to apply sliding mode techniques to several applications [7-10]. The most outstanding features of SMC are its insensitivity to parameter variations, external disturbance rejection ability and fast dynamic response [10]. Other remarkable advantages of this control approach are the simplicity of its implementation and the order reduction of the closed-loop systems.

In the sliding mode observer (SMO), the error between the observer output and the system output is fed back via a discontinuous switched signal instead of feeding it back linearly. SMO has a unique feature of generating sliding mode on the error between the measured plant output and the observed output. Spurgeon describes an overview of linear and nonlinear SMOs in her survey paper [11]. The effectiveness of the methodology through simulation studies for the observer design for nonlinear systems was considered in [12]-[17]. A comparative study of an SMO and Kalman filters for induction machine is given in [18].

Furthermore, output feedback controls via linear matrix inequality (LMI) have been used efficiently in the recent years [19]. LMI approach is a powerful tool in the control theory and applications; such as the sliding-mode design, static output feedback control and stability analysis of time-delay systems. Sliding mode observer based controllers using LMI

Vikas Sharma and Bhanu Pratap are with the Department of Electrical Engineering, M. N. National Institute of Technology, Allahabad, India. (e-mail: vikassharma.ecb@gmail.com, bhanumnnit@gmail.com).

are proposed in [20]-[23].

In this paper an SMO based SMC for a class of uncertain system using LMI is proposed. The aim is accomplished by utilizing techniques prevalent in variable structure systems (VSS) theory. A numerical example is given to illustrate the performance of proposed observer based control.

The remainder of the paper is arranged as follows. The problem statement is introduced in Section II. Robust observer based controller is designed using LMI in Section III. In Section IV, the performance of observer based controller is shown by simulation results. Finally conclusions are given in the last section.

II. PROBLEM STATEMENT Consider the following uncertain system:

( )x A A x Bu= + Δ + (1a)

( )y C C x Du= + Δ + (1b)

where nx ∈ℜ is state vector, mu ∈ℜ is input vector, py ∈ℜ

is output vector, n nA ×∈ℜ , n mB ×∈ℜ , p nC ×∈ℜ , and p mD ×∈ ℜ are known matrices. AΔ and CΔ are two

perturbed matrices and satisfy 1 1 1A M F NΔ = , 2 2 2C M F NΔ =

with Ti iF F I≤ , 1, 2i = , where the matrices iM and iN ;

1, 2i = , are known with appropriate dimension.

The parameter uncertainties AΔ and CΔ will represent the impossibility for exact mathematical model of a dynamic system due to the system complexity. The uncertainty has been widely used in many practical systems which can be either exactly modeled or over bounded by the condition

Ti iF F I≤ . The matrices 1F and 2F contain the uncertain

parameters, and constant matrices iM and iN ; 1, 2i = , specify how the uncertain parameter iF affect the nominal matrix of (1) [19].

The objective of this paper is to design a robust observer based controller for an uncertain system (1) that forces the plant output y and state estimation error ˆx x− tends to zero.

Sliding Mode Observer Based Controller of Uncertain Systems: An LMI Based Approach

Vikas Sharma, Student Member, IEEE, and Bhanu Pratap, Student Member, IEEE

T

2012 2nd International Conference on Power, Control and Embedded Systems

978-1-4673-1049-9/12/$31.00 ©2012 IEEE

2

III. ROBUST OBSERVER BASED CONTROLLER DESIGN USING LMI APPROACH

A. Sliding Mode Controller:

The switching function is defined as [5]

0TcS K x= = (2)

where T m nK ×∈ℜ is controller gain. Differentiating (2) and using (1) we get,

( ) 0TcS K A A x Bu= + Δ + =⎡ ⎤⎣ ⎦ (3)

The controller u is obtained as

( ) ( )1T Tsu K B K A A x u

−⎡ ⎤= − + Δ +⎣ ⎦ (4)

where ( )sgns c cu K S= − is the sliding control and m m

cK ×∈ℜ .

Theorem 1: The controller given in (4) brings the system trajectory to the surface given in (2) in finite time, and thereafter, the trajectory slides along the surface to the origin asymptotically [5].

Proof: Consider a Lyapunov function

( ) 12

Tc c cV S S S= (5)

Differentiating (5) and using (3)

( ) ( )sgnTc c c cV S S K S= − (6)

( )c c cV S K S= − (7)

B. Sliding Mode Observer:

The sliding mode observer is given by [14]

( ) ( )1

ˆ ˆ ˆ

ˆ ˆ

To

o

R C CSx Ax Bu L y y

CSy Cx Du

δ−

= + + − −

= +

(8)

where ˆ nx ∈ℜ is estimate of x , ˆ py ∈ℜ is estimate of y , n pL ×∈ℜ is the observer gain, R is symmetric positive

definite matrix and δ is the bound of uncertainty.

Now the switching function is defined as

ˆoS x x x= − ≡ (9)

Differentiating (9) and using (1) and (8) we get,

( ) ( ) ( )1

ˆ ˆT

oo

o

R C CSS A A x Ax L y y

CSδ

−

= + Δ − − − + (10)

( ) ( ) ( )1 To

o oo

R C CSS A LC S A L C x

CSδ

−

= − + Δ − Δ + (11)

C. Observer Based Controller:

Since the controller in (4) is not free from uncertainties and hence cannot be implemented. A sliding surface matrix is proposed to ensure the uncertainties to belong to the null space of TK and TK B to be invertible. Without loss of generality,

TK B can be assumed to be an identity matrix of appropriate dimension [5]. Thus, the equivalent control using observer is given as

ˆTequ K Ax⎡ ⎤= −⎣ ⎦ (12)

Now using (12), (1a) becomes

( ) ˆTx A A x B K Ax⎡ ⎤= + Δ + −⎣ ⎦ (13)

Using (9) and (13) gives

( )T Tox A BK A A x BK AS= − + Δ + (14)

D. Stability Analysis:

Consider the Lyapunov function,

T To oV x Px S RS= + (15)

Differentiating (15)

T T T To o o oV x Px x Px S RS S RS= + + + (16)

Using (11) and (14), (16) becomes

( )( )

( ) ( ) ( )

( ) ( ) ( )

1

1

T T To

TT To

ToT

o oo

TTo

o oo

V x P A BK A A x BK AS

A BK A A x BK AS Px

R C CSS R A LC S A L C x

CS

R C CSA LC S A L C x RS

CS

δ

δ

−

−

⎡ ⎤= − + Δ +⎣ ⎦

⎡ ⎤+ − + Δ +⎣ ⎦⎡ ⎤

+ − + Δ − Δ +⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤+ − + Δ − Δ +⎢ ⎥⎢ ⎥⎣ ⎦

( )( )

( ) ( ){ } ( )

( )

1 1 1 1 1 1

2

T T T To

T T T T T T To

T T T T T

TT To o o

TT T T T T o

o oo

V x PA PBK A x x PBK AS

x A P A KB P x S A KB Px

x PM F N x x N F M Px

S A LC R R A LC S S R A L C x

C CSx A C L S S

CSδ

= − +

+ − +

+ +

+ − + − + Δ − Δ

+ Δ − Δ +

( )( )

1 1

1 1 1 1 1 1 1 1 1 1

11 1 1 1 1 1 1 1

T T T To

T T T T T T To

TT T

T T T T T

V x PA PBK A x x PBK AS

x A P A KB P x S A KB Px

M Px F N x M Px F N x

x N F F N x x PM M Px

ε ε ε ε

ε ε

− −

−

= − +

+ − +

⎡ ⎤ ⎡ ⎤− − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦+ +

3

( ) ( ){ }1

2 1 2 1 1

1

2 1 2 1 1

12 1 1 1 1 2 1 1

1

3 2 3 2 2

1

3 2 3 2 2

13 2 2 2 2 3 2 2

2TTo o o

TT

o

To

T T T T To o

TT T

o

T To

T T T T T To o

S A LC R R A LC CS S

M RS F N x

M RS F N x

x N F F N x S RM M RS

M L RS F N x

M L RS F N x

x N F F N x S RLM M L RS

δ

ε ε

ε ε

ε ε

ε ε

ε ε

ε ε

−

−

−

−

−

−

+ − + − −

⎡ ⎤− − ×⎢ ⎥⎣ ⎦⎡ ⎤−⎢ ⎥⎣ ⎦+ +

⎡ ⎤− + ×⎢ ⎥⎣ ⎦⎡ ⎤+⎢ ⎥⎣ ⎦+ +

Now using inequality [24]

( )2 T To o oCS S C CSδ δ≤ (17)

Thus V is given as

( )

{ }1

1 1 1 1 1 1

12 1 1 2 1 1

13 2 2 3 2 2

T T T T T

T T T T To o

T T T T

T T T T To o

T T T To o

T T T T To o

V x A P A KB P PA PBK A x

x PBK AS S A KB Px

x N N x x PM M Px

S A R RA C L R RLC C C S

x N N x S RM M RS

x N N x S RLM M L RS

ε ε

δ

ε εε ε

−

−

−

≤ − + −

+ +

+ +

+ + − − +

+ +

+ +

(18)

V can be rewritten as

11 12

12 22

T T T To o T

o o

x xx S x S

S SΓ Γ⎡ ⎤ ⎡ ⎤⎡ ⎤

⎡ ⎤ ⎡ ⎤Γ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ Γ Γ⎣ ⎦⎣ ⎦ ⎣ ⎦ (19)

where

( )11

11 2 1 1 3 2 2 1 1 1

T T T T

T T T

A P A KB P PA PBK A

N N N N PM M Pε ε ε ε −

Γ = − + −

+ + + +

12TPBK AΓ =

12T T TA KB PΓ =

22

1 12 1 1 3 2 2

T T T T

T T T T To o o o

A R RA C L R RLC C C

S RM M RS S RLM M L RS

δε ε− −

Γ = + − − +

+ +

Lemma 1: The LMI [19]

11 12

12 22

0T

Z I ZZ Z

ρ+⎡ ⎤<⎢ ⎥

⎣ ⎦ (20)

is equivalent to

22 0Z < , 111 12 22 12

TZ Z Z Z Iρ−− < − (21)

where 11 11TZ Z= , 22 22

TZ Z= . Now, we present some LMI results for the exponential

stability of system (5) and (11). The control gain K and

observer gain L could be solved by the following results.

Theorem 2: System (1) is exponentially stabilizable by (2) and (9) provided that there exist some positive constants 1ε ,

2ε , 3ε , ρ , two positive definite matrices , n nP R ×∈ℜ , and ˆ n mK ×∈ℜ , n pX ×∈ℜ , ˆ m mP ×∈ℜ , such that

11 1

22 1 2

1 1

1 2

2 3

ˆ 0 0ˆ 0

00 0 00 0 00 0 0

T T T

T

T

T T

X BKA PM

A K B X RM XMM P I

M R IM X I

εε

ε

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ <−⎢ ⎥⎢ ⎥−⎢ ⎥−⎣ ⎦

(22)

ˆPB BP=

where

11 11ˆ ,X X Iρ= +

( )11

1 2 1 1 3 2 2

ˆ T T T T

T T

X A P A KB P PA PBK A

N N N Nε ε ε= − + −

+ + +

22 22ˆX X Iρ= +

22ˆ T T T TX A R RA C L R RLC C C Iδ ρ= + − − + +

The stabilizing observer based controller gains are given by 1ˆ ˆK P K−= and 1L R X−= , and the systems convergence rate

is ( ) ( )( )max max/ 2 max ,P Rρ λ λ⋅ ⎡ ⎤⎣ ⎦ .

Proof: By the Lemma 1, the LMI condition (22) implies

( )( )

11

22

11

12

1 23

1

1

1

ˆ ˆ ˆ

ˆ ˆ ˆ

0 00 0

0 00

0 0

000

T

T T T T

T

T

T T

X B K A XC

A K C X B X

IPM

IRM XM

I

M PM R

M X

εε

ε

−

⎡ ⎤−⎢ ⎥Γ =⎢ ⎥−⎣ ⎦

−⎡ ⎤⎡ ⎤ ⎢ ⎥− × − ×⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥−⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(23)

The matrix Γ is equal to the matrix Γ with 1ˆ ˆK P K−= , and 1L R X−= . By (15), (19) and (23), we have

( ) [ ] [ ]( )

( ) [ ]

2

min

2

max

min 1,

max 1,

T To o

To

R x S V x S

R x S

λ

λ

⋅ ≤⎡ ⎤⎣ ⎦

≤ ⋅⎡ ⎤⎣ ⎦

and [ ]( ) [ ]2T T

o oV x S x Sρ≤ − ⋅

We conclude that (11) and (14) is exponentially stabilizable

4

by (8) and (12) with the convergence rate

( ) ( )( )max max/ 2 max ,P Rρ λ λ⋅ ⎡ ⎤⎣ ⎦ [19].

IV. SIMULATION RESULTS Consider (1) with the following parameters:

1 1 10 2 11 2 5

A⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥− −⎣ ⎦

; 1 00 10 0

B⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

;

[ ]1 0 1C = ; [ ]0 0D = ;

( )

( )( )

0 00 0

0 0

a tA b t

c t

⎡ ⎤⎢ ⎥Δ = ⎢ ⎥⎢ ⎥⎣ ⎦

; ( )0 0C d tΔ = ⎡ ⎤⎣ ⎦ ;

where ( ) ( )cos 3a t t= ; ( ) ( )1 0.3sin 2b t t= − ;

( ) ( )1.3sin 4c t t= ; ( ) ( )1.2sin 3d t t− ;

and ( )a t α≤ , ( )b t β≤ , ( )c t γ≤ , ( )d t ζ≤ .

1 2M M I= =

1

/ 0 00 / 00 0 /

aF b

c

αβ

γ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

; 2 /F d ζ=

1

0 00 0

0 0N

αβ

γ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

; [ ]2 0 0N ζ= ; 1α β γ ζ= = = =

By the Theorem 2, (22) is satisfied and (1) would be exponentially stabilized by (8) and (12). The controller and observer gain are obtained as

22.7207 5.8503 3.932811.3944 2.9359 1.9723

TK⎡ ⎤

= ⎢ ⎥⎣ ⎦

;

92.97921.687340.8147

L⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦

;

and other design parameters are obtained by LMI (22) are given as

9.6724 7.5102 5.37367.5102 114.362 10.7149 ;5.3736 10.7149 32.4956

P− −⎡ ⎤

⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦

10.2925 1.015 5.22411.015 72.5821 9.2891 ;5.2241 9.2891 38.7712

R− −⎡ ⎤

⎢ ⎥= − −⎢ ⎥⎢ ⎥− −⎣ ⎦

1 79.8318ε = ; 2 61.9101ε = ; 3 275.6177ε = ; 10.34ρ = ;

30.588δ = . { }2cK diag= .

0 1 2 3 4 5

-5

0

5

10x 10

-3

Time (sec)

Reg

ulat

ion

of S

tate

s

x1

x2

x3

Fig. 1 Regulation of states 1x , 2x , 3x

0 1 2 3 4 5

-0.1

-0.05

0

0.05

Time (sec)

Con

trol

Eff

ort

(u1)

u1

Fig. 3 Control Effort 1u

0 1 2 3 4 5

-0.1

-0.05

0

0.05

Time (sec)

Con

trol

Eff

ort

(u2)

u2

Fig. 4 Control Effort 2u

The initial conditions of the plant and observer are ( )1 0 0.1x = , ( )2 0 0x = , ( )3 0 0.1x = , and ( )1̂ 0 0x = ,

( )2ˆ 0 0x = , ( )3ˆ 0 0x = . Fig. 1 and 2 shows the regulation of

outputs 1x and 3x . Control efforts 1u and 2u are shown in Fig. 3 and 4 respectively.

5

V. CONCLUSION In this paper a sliding mode observer based controller for a

class of uncertain system is presented. A full-order observer-based control is designed using LMI approach, which guarantees the exponential stabilization of feedback system. Finally, simulation results are given to show the effectiveness of proposed observer based control.

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