Author's Accepted Manuscript

Non-fragile Observer-based Sliding Mode Controlfor A Class of Uncertain Switched Systems

Yonghui Liu, Yugang Niu, Yuanyuan Zou

PII: S0016-0032(13)00357-8DOI: http://dx.doi.org/10.1016/j.jfranklin.2013.09.020Reference: FI1886

To appear in: Journal of the Franklin Institute

Received date: 25 May 2013Revised date: 26 August 2013Accepted date: 19 September 2013

Cite this article as: Yonghui Liu, Yugang Niu, Yuanyuan Zou, Non-fragile Observer-based Sliding Mode Control for A Class of Uncertain Switched Systems, Journal of theFranklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2013.09.020

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Non-fragile Observer-based Sliding Mode Control

for A Class of Uncertain Switched Systems

Yonghui Liu∗ Yugang Niu ∗† Yuanyuan Zou∗

Abstract

In this work, the problem of non-fragile sliding mode control is investigated for a class of

uncertain switched systems with state unavailable. First, a non-fragile sliding mode observer is

constructed to estimate the unmeasured state. And then, a state-estimate-based sliding mode

controller is designed, in which a weighted sum approach of the input matrices is utilized to

obtain a common sliding surface. It is shown that the reachability of the specified sliding

surface can be ensured by the present sliding mode controller. Moreover, the exponential

stability of the sliding mode dynamics is analyzed by adopting the average dwell time method.

Finally, a numerical simulation is given to demonstrate the effectiveness of the results.

Keywords: switched systems, unmeasured state, sliding mode control, non-fragility

1 Introduction

In the past decades, switched systems have played an important role in physical world, such as

power systems, aircraft control systems, chemical control processes and communication network

systems [1, 2]. Due to its success in application and importance in theory, more and more attention

has been drawn onto this field and many results on stability and stabilization have been obtained,

see [3, 4, 5, 6, 7] and the reference therein.

As well known, sliding mode control (SMC) has been recognized as an effective robust control

approach due to its excellent advantages of strong robustness against model uncertainties, parameter

variations, and external disturbances. Therefore, it has been used widely in control systems. More

recently, the application of SMC has been extended to switched systems in [8, 9, 10]. Among them,

Wu and Lam [8] considered SMC for a class of switched systems with state delay, whose method

∗Key Laboratory of Advanced Control and Optimization for Chemical Process (East China University of Science

and Technology), Ministry of Education, Shanghai 200237, China†Author for Correspondence: E-mail: acniuyg@ecust.edu.cn

1

was further extended to stochastic switched systems in [9]. Besides, Lian et al. [10] discussed the

robust H∞ SMC for a class of uncertain switched systems. In the previous works [8, 9, 10], the

control systems were assumed to have the same input matrices. To remove this restriction, Liu et

al. [11] proposed a weighted sum approach, and then, a common sliding function was constructed

for the case that the input matrix Bi for each subsystem may be different.

On the other hand, in the ideal case, the controller/observer is always assumed to be implemented

exactly. However, in practice, the variations are usually inevitable, which may deteriorate the

performance of the control systems and even lead to instability [12, 13, 14]. Hence, non-fragile

control has been a hot issue and many constructive results have been proposed in, e.g., [15, 16, 17,

18, 19, 20]. In [17], the robust H∞ SMC for a class of uncertain time-delay systems is considered

based on the non-fragile observer, which was further extended to stochastic systems in [19]. Besides,

Wang and Zhao [20] investigated the non-fragile guaranteed cost control for a class of switched linear

systems.

However, to the authors’ best knowledge, the research on non-fragile SMC of switched systems

is still open. Especially, when the input matrix for each subsystem may be different, some existing

results [8, 9, 10] cannot be simply extended to such class of systems. This motivates the present

study.

In this work, the problem of non-fragile SMC is investigated for a class of uncertain switched

systems, in which the system states are unavailable. Thus, a non-fragile observer is first designed.

Since the input matrix of each subsystem may be different, which also makes some existing results

not be utilized directly. To overcome this difficulty, a weighted sum approach of the input matrices

is utilized to construct a common sliding surface. And then, a state-estimation-based sliding mode

controller is designed. It is shown that the reachablbity of the specified sliding surface can be ensured

by the present sliding mode controller. Furthermore, by adopting the average dwell time strategy,

the sufficient condition on the exponential stability of the sliding mode dynamics is obtained.

This paper is organized as follows. In Section 2, the uncertain switched systems is described

and the problem is formulated. By proposing a non-fragile observer, a switching signal and a SMC

law are designed in Section 3. A simulation example is given in Section 4, which is followed by the

conclusion in Section 5.

The following notations are used throughout this paper: Rn denotes the real n-dimensional

space; Rm×n denotes the real m × n matrix space; ∥ · ∥ denotes the Euclidean norm of a vector

or its induced matrix norm. For any vector x = [x1 x2 · · · xn]T , xT is its transpose and sgn(x) =

[sgn(x1) sgn(x1) · · · sgn(xn)]T . For a real symmetric matrix,M > 0 (< 0) means positive (negative)

definite. I is used to represent an identity matrix of appropriate dimensions, diag(·) denotes a

diagonal matrix, the vector 1n ∈ Rn is consisted of ones, and ei ∈ Rn is the i-th standard base

2

vector. λmax(·) and λmin(·) represent the maximum and minimum eigenvalue of a real symmetric

matrix, respectively, rank (·) denotes the rank of a matrix, the part in a matrix induced by symmetry

is denoted by ∗, and ⊗ stands for the Kronecker product. Matrices, if their dimensions are not

explicitly stated, are assumed to have compatible dimensions.

2 Problem Statement

Consider the following uncertain switched systems:

x(t) = (Aσ +∆Aσ)x(t) +Bσ(u(t) + fσ(x(t))), (1)

y(t) = Cσx(t), (2)

where x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input, y(t) ∈ Rp is the system output,

fσ(x(t)) is the external disturbance, ∆Aσ is the parameter uncertainty, {Aσ, Bσ, Cσ : σ ∈ Γ}

is a family of known matrices depending on an index set Γ = {1, 2, . . . , s}, and σ(t) : R → Γ

is a piecewise constant function of time t, called as switching signal. The switching sequence

{(i0, t0), (i1, t1), . . . , (iN , tN) | ik ∈ Γ}, corresponding to the switching signal σ(t) = ik, means that

the ik-th subsystem is activated as t ∈ [tk, tk+1).

For each possible value σ(t) = i, i ∈ Γ, the parameters associated with the i-th subsystem are

denoted as

Aσ , Ai, Bσ , Bi, ∆Aσ , ∆Ai, fσ(x(t)) , fi(x(t)), Cσ , ∆Ci.

In this work, it is assumed that the admissible uncertainty ∆Ai satisfies ∆Ai = Ei1Fi1(t)Hi1,

where Ei1 and Hi1 are known constant matrices, and Fi1(t) is an unknown matrix function satis-

fying F Ti1(t)Fi1(t) ≤ I. Besides, the disturbance fi(x(t)) is assumed to be norm bounded, that is,

∥fi(x(t))∥ ≤ di, with di a positive scalar.

It is noted that, in practice, the system states are not always available due to the limit of

physical condition or expensive to measure. Hence, this work investigates the problem of SMC

for the switched systems with unmeasured states, and a state-observer-based SMC method will be

proposed to guarantee the exponential stability of the switched systems (1)–(2). To this end, some

preliminaries are first given as follows.

Assumption 1 The matrix Bi is full column rank, that is, rank(Bi) = m.

Lemma 1 [21] The real matrices D, H, and F (t) are of appropriate dimensions with F (t) satisfying

F T (t)F (t) ≤ I. Then, for any ε > 0, we have

DF (t)H +HTF T (t)DT ≤ ε−1DDT + εHTH.

3

Definition 1 [2] For any T2 > T1 > 0, let Nσ denote the number of switchings of σ(t) over (T1, T2).

If

Nσ ≤ N0 +(T2 − T1)

Tσ

holds for Tσ > 0 and N0 ≥ 0, then Tσ is called the average dwell time.

In this work, let N0 = 0, as is usually used in the previous works.

Definition 2 The equilibrium x∗ = 0 of system (1) is said to be exponentially stable if the solution

x(t) satisfies

∥x(t)∥ ≤ ρ∥x(t0)∥e−λ(t−t0), ∀ t ≥ t0,

for scalars ρ ≥ 1 and λ > 0.

3 Non-fragile observer-based SMC

In this section, a non-fragile observer will be introduced and a state-estimate-based SMC law will

be proposed. Moreover, by utilizing the average dwell time method, the exponential stability of the

sliding motion is analyzed.

3.1 Non-fragile observer

In this work, the following non-fragile observer is proposed:

˙x(t) = Aix(t) + Bi(u(t) +ϖi(t)) + Bi(Li +∆Li)(y(t)− Cix(t)), (3)

where x(t) is the estimation of the state x(t), the robust term ϖi(t) will be designed later to

counteract the disturbance fi(x(t)), the observer gain Li will be given later, and ∆Li is a perturbed

matrix, which is assumed to satisfy

∆Li = Ei2Fi2(t)Hi2, (4)

where Ei2 and Hi2 are constant matrices, and Fi2(t) is an unknown time-varying matrix satisfying

F Ti2(t)F

Ti2(t) ≤ I.

Denote e(t) = x(t)− x(t), then the following estimation error system is obtained:

e(t) = (Ai +∆Ai)e(t) + ∆Aix(t) +Bi(fi(x(t))−ϖi(t))−Bi(Li +∆Li)Cie(t). (5)

In the sequel, a sliding surface will be designed and the corresponding sliding mode dynamics

will be obtained.

4

3.2 Sliding surface design

It should be pointed out that in systems (1)–(2), the input matrix Bi for each subsystem is not

necessarily the same. Hence, the proposed method in this work has a much wider application in

practice than some existing ones as in [8, 9, 10]. However, it also brings challenge in designing

a common sliding surface. To overcome this difficulty, the following weighted sum of the input

matrices, as in [11], is introduced:

B ,s∑

i=1

αiBi,

where αi is a parameter satisfying α ≤ αi ≤ α, i = 1, 2, . . . , s, with α and α known scalars.

Then, define

M , 1

2[B − sα1B1 B − sα2B2 · · · B − sαnBs] ,

V (i) , (Is − 2eieTi )⊗ Is, N , 1s ⊗ Im,

β , max{|sα− 1|, |sα− 1|} · max1≤i≤s

{∥Bi∥},

V(i) ,

V (i) 0

0 1β(1− sαi)Bi

,

M ,[M βIn

], N ,

N

Im

.

It can be shown that Bi = B +MV(i)N , and ∥V(i)∥ ≤ 1.

Remark 1 The upper bound of V(i) is achieved if and only if there exists an index i such that

αi = α or αi = α and ∥Bi∥ = max1≤j≤s

{∥Bj∥}. Moreover, for αi = 1/s, i = 1, . . . , s, it can be verified

that Bi = B +MV (i)N with ∥V (i)∥ ≤ 1.

Remark 2 It is worth noting that, under Assumption 1, B is full column rank. That is, for general

choice of scalars αi, i = 1, . . . , s, it can be obtained that B is full column rank.

By taking the above transformation into account, a common sliding function in the state estimation

space is designed as

S(t) = Dx(t) +K

∫ t

t0

x(τ)dτ, (6)

where D = (BTB)−1BT , and the matrix K will be given later.

According to (3), we have

S(t) = (DAi +K)x(t) +DBi(u(t) +ϖi(t)) +DBi(Li +∆Li)Cie(t). (7)

5

In view of sliding mode theory, we have S(t) = 0 and S(t) = 0 when the state trajectory enters the

sliding surface. Hence, the equivalent control is obtained as

ueq(t) = −ϖi(t)− (DBi)−1(DAi +K)x(t)− (Li +∆Li)Cie(t), (8)

which, substituted into (3), yields the following sliding mode dynamics:

˙x(t) = (Ai −Bi(DBi)−1(DAi +K))x(t). (9)

Remark 3 It is noted from (8) that the matrix DBi is required to be non-singular. Hence, the

parameters αi, i = 1, 2, . . . , s, should be selected such that the non-singularity condition can be

satisfied.

3.3 Reachability analysis

In the following part, by means of the state estimate in (3), a sliding mode controller will be designed

to ensure the reachability of the sliding surface S(t) = 0.

To this end, the following sliding mode controller is designed:

u(t) = −(DBi)−1(DAi +K)x(t)− Li(y(t)− Cix(t))

− (∥Ei2∥∥Hi2Cie(t)∥+ ξi + di + µi) sgn((DBi)TS(t)), (10)

and the robust term ϖi(t) in (3) is designed as

ϖi(t) = (ξi + di)sgn(Xi(y(t)− Cix(t))), (11)

where ξi and µi are positive scalars and the matrix Xi will be given in Theorem 2 later.

In the following theorem, we will analyze the reachability of the sliding surface S(t) = 0.

Theorem 1 Consider the switched systems (1)–(2) satisfying Assumption 1. For the designed

sliding function (6), if the SMC law is designed as (10)–(11), then the state trajectory can be driven

onto the specified sliding surface S(t) = 0 in finite time.

Proof: Choose the Lyapunov function as

V (t) =1

2ST (t)S(t). (12)

Thus, it follows from (7) that

V (t) = ST (t)[(DAi +K)x(t) +DBi(u(t) +ϖi(t)) +DBi(Li +∆Li)Cie(t)

]. (13)

6

Substituting (10) and (11) into (13), we have

V (t) = ST (t)DBi

[− (∥Ei2∥∥Hi2Cie(t)∥+ ξi + di + µi)sgn((DBi)

TS(t))

+ϖi(t) + ∆LiCie(t)]

≤ −µi∥DBi∥∥S(t)∥,

which means that the state trajectory of the systems (1)–(2) will be driven onto the specified sliding

surface S(t) = 0 in finite time and remain there in the subsequent time. Hence, the reachability of

the sliding surface can be guaranteed.

Remark 4 It should be mentioned that the designed SMC law (10), including the term (DBi)TS(t),

is different from the traditional sliding mode controller. Actually, the sliding mode manifold S(t) = 0

is not changed, since DBi is non-singular.

3.4 Stability

Next, the stability of the closed-loop system composed of estimation error systems (5) and sliding

mode dynamics (9) will be analyzed .

Theorem 2 Consider the switched systems (1)–(2) satisfying Assumption 1. Given scalar γ > 0

and matrix K such that Ai −Bi(DBi)−1(DAi +K) is stable, if there exist matrices Pi > 0, Xi, Li,

and scalars εi1 > 0, εi2 > 0 and εi3 > 0, i ∈ Γ, satisfying the following linear matrix inequalities

(LMIs):

Θ1i 0 0 0 0

∗ Θ2i PiEi1 PiEi1 PiBiEi2

∗ ∗ −εi1I 0 0

∗ ∗ ∗ −εi2I 0

∗ ∗ ∗ ∗ −εi3I

< 0, (14)

BTi Pi = XiCi, i ∈ Γ, (15)

where

Θ1i = Pi(Ai −Bi(DBi)−1(DAi +K)) + (Ai −Bi(DBi)

−1(DAi +K))TPi + γPi + εi2HTi1Hi1,

Θ2i = PiAi + ATi Pi − LiCi − CT

i LTi + γPi + εi1H

Ti1Hi1 + εi3(Hi2Ci)

THi2Ci,

then with the parameter

µ = maxi,j∈Γ,i=j

λmax(Pi)

λmin(Pj), (16)

and the average dwell time

Tσ >lnµ

γ, (17)

7

the closed-loop system is exponentially stable for arbitrary switching signal σ(t). Furthermore, the

norm of the state ζ(t) = [xT (t) eT (t)]T obeys

∥ζ(t)∥ ≤ ηe−κt∥ζ(t0)∥, (18)

where

κ =1

2(λ− lnµ

Tσ

), η =

√b

a≥ 1,

a = mini ∈ Γ

{λmin(Pi)}, b = maxi ∈ Γ

{λmax(Pi)}. (19)

Moreover, the observer gain in (3) is given by Li = B+i P

−1i Li, where B+

i is the Moore-Penrose

inverse of matrix Bi.

Proof: For the closed-loop system composed of (5) and (9), choose the Lyapunov function for the

i-th subsystem as

Vi(t) = xT (t)Pix(t) + eT (t)Pie(t). (20)

It can be derived from (5) and (9) that

Vi(t) = 2xT (t)Pi˙x(t) + 2eT (t)Pie(t)

= xT (t)[Pi(Ai −Bi(DBi)

−1(DAi +K)) + (Ai −Bi(DBi)−1(DAi +K))TPi

]x(t)

+2eT (t)Pi[(Ai +∆Ai)e(t) + ∆Aix(t) +Bi(fi(x(t))−ϖi(t))

−Bi(Li +∆Li)Cie(t)]. (21)

By means of (15), we have

eT (t)PiBi(fi(x(t))−ϖi(t)) = eT (t)CTi Xi(fi(x(t))−ϖi(t))

≤ −ξi∥Xi(y(t)− Cix(t))∥

< 0. (22)

In view of Lemma 1, one further has for εi1 > 0, εi2 > 0, and εi3 > 0

2eT (t)Pi∆Aie(t) ≤ ε−1i1 e

T (t)PiEi1(PiEi1)T e(t) + εi1e

T (t)HTi1Hi1e(t), (23)

2eT (t)Pi∆Aix(t) ≤ ε−1i2 e

T (t)PiEi1(PiEi1)T e(t) + εi2x

T (t)HTi1Hi1x(t), (24)

−2eT (t)PiBi∆LiCie(t) ≤ ε−1i3 e

T (t)PiBiEi2(PiBiEi2)T e(t) + εi3e

T (t)(Hi2Ci)THi2Cie(t). (25)

Combining (21)–(25), it yields

Vi(t) + γVi(t) ≤ ζT (t)Σiζ(t), (26)

8

where Σi = diag(Σ1i, Σ2i), with

Σ1i = Pi(Ai −Bi(DBi)−1(DAi +K)) + (Ai −Bi(DBi)

−1(DAi +K))TPi + γPi + εi2HTi1Hi1,

Σ2i = PiAi + ATi Pi − PiBiLiCi − CT

i (PiBiLi)T + γPi + ε−1

i1 PiEi1(PiEi1)T + εi1H

Ti1Hi1

+ε−1i2 PiEi1(PiEi1)

T + ε−1i3 PiBiEi2(PiBiEi2)

T + εi3(Hi2Ci)THi2Ci.

Let PiBiLi = Li. Then employing Schur’s complement, it can be seen that Σi < 0 is implied by

(14), which together with (26) gives

Vi(t) ≤ −γVi(t).

Thus, there holds

Vi(t) ≤ e−γ(t−t0)Vi(t0), (27)

which means that each subsystem of the closed-loop system is exponentially stable.

Suppose that tk, k ∈ {1, 2, . . . , Nσ}, is the switching instant and the closed-loop system switches

from the j-th subsystem to the i-th one. Hence, one has σ(t−k ) = j and σ(t+k ) = i. In view of (16)

and (27), it follows that

Vi(t) ≤ e−γ(t−tk)Vi(tk), and Vi(tk) ≤ µVj(t−k ). (28)

Let Nσ(t0, t) ≤ N0 + (t− t0)/Tσ. According to (28), we have

Vi(t) ≤ e−γ(t−tk)µVj(t−k )

...

≤ e−γ(t−t0)µNσ(t0,t)Vσ(t0)(t0)

≤ e−(γ−lnµ/Tσ)(t−t0)Vσ(t0)(t0). (29)

Considering (19), one has

a∥ζ(t)∥2 ≤ Vi(t), and Vσ(t0)(t0) ≤ b∥ζ(t0)∥2, (30)

which together with (29) and (30) yields

∥ζ(t)∥2 ≤ 1

aV (i, t) ≤ b

ae−(γ−lnµ/Tσ)(t−t0)∥ζ(t0)∥2. (31)

In view of (17) and (31), it can be shown that the closed-loop system is exponentially stable, which

completes the proof.

3.5 Algorithm

In the sequel, a algorithm is given to solve LMIs with equality constrain as (14)–(15).

Consider the equality condition

BTi Pi = XiCi, i ∈ Γ.

9

where Pi > 0 and Xi satisfies LMI (14), which can be equivalently converted to

trace((BT

i Pi −XiCi)T (BT

i Pi −XiCi))= 0.

For δi > 0, consider the following matrix inequality:

(BTi Pi −XiCi)

T (BTi Pi −XiCi) ≤ δiI. (32)

By schur’s complement, (32) is equivalent to−δiI CTi Xi − PiBi

∗ −I

< 0. (33)

Now, the problem of non-fragile observer-based SMC is changed to the following minimization

problem:

min δi subject to (14) and (33), (34)

which is a minimization problem involving linear objective and LMI constrains and can be solved

by using LMI toolbox in Matlab.

4 Numerical simulation

Consider the switched systems (1)–(2) with the following parameters:

Subsystem 1:

A1 =

1.5 2.5 −12.5

−5.5 −7.5 −1.5

4.5 −2.5 −6.5

, B1 =

1

2

−1

, C1 =

1

−0.5

−1

T

,

E11 = [0.5 − 0.5 − 0.5]T , F11(t) = sin(t), H11 =[− 0.2 0.2 0.4

],

E12 = 0.5, F12(t) = sin(t), H12 = −0.2, f1(t) =1

1 + t2.

Subsystem 2:

A2 =

2.5 −3.5 5.5

4.5 −7.5 −5.5

5 −5.5 −12.5

, B2 =

3

1

1

, C2 =

0

2

1

T

,

E21 = [0.5 0.5 − 0.5]T , F21(t) = cos(t), H21 =[− 0.2 0.2 − 0.2

],

E22 = 0.5, F22(t) = cos(t), H22 = −0.2, f2(t) =1

1 + t2.

10

By choosing α1 = α2 = 12, it is easily shown that DBi is non-singular. Moreover, for γ = 1.5

and K =[2.5 3 − 5], solving the minimization problem (34) yields:

ε11 = 0.9487, ε12 = 0.9449, ε13 = 0.9403,

ε21 = 0.8603, ε22 = 0.8783, ε23 = 0.9657,

X1 = 0.0756, X2 = 0.0083,

L1 =

0.8319

0.8783

−0.6463

, L2 =

0.3901

−0.2803

−0.1298

,

P1 =

0.1526 −0.0289 −0.0597

−0.0289 0.0751 0.0584

−0.0597 0.0584 0.1041

,

P2 =

0.0298 −0.0637 0.0176

−0.0637 0.3130 −0.1292

0.0176 −0.1292 0.1329

,

δ1 = 2.6958e × 10−11 and δ2 = 6.8394e × 10−17 (that is, the constrains BTi Pi = XiCi, i ∈ Γ are

satisfied).

According to Theorem 1, the parameters µ and Tσ are designed, respectively, as follows:

µ = maxi,j∈Γ,i =j

λmax(Pi)

λmin(Pj)= 14.8288, and Tσ >

lnµ

γ= 1.7977.

Thus, the average dwell time can be selected as

Tσ = 2.

In view of (10) and (11), the sliding mode controller is designed as

u(t) =

−2.0750x1(t)− 2.5000x2(t) + 11.7000x3(t)− 13.667(y(t)− x1(t) + 0.5x2(t) + x3(t))

−(0.1∥y(t)− x1(t) + 0.5x2(t) + x3(t)∥+ ξ1 + d1 + µ1)sgn((DB1)TS(t)), if i = 1

−3.6500x1(t)− 0.0667x2(t) + 3.8000x3(t)− 5.4202(y(t)− 2x2(t)− x3(t)))

−(0.1∥y(t)− 2x2(t)− x3(t))∥+ ξ1 + d1 + µ2)sgn((DB2)TS(t)), if i = 2.

Suppose that the initial state x(t0) =[0.4 − 0.3 − 0.4

]Tand x(t0) =

[0.5 − 0.5 − 0.2

]T.

For the parameters d1 = d2 = 1, ϵ1 = ϵ2 = 0.01 and µ1 = µ2 = 0.001, the simulation results

with the proposed SMC law are illustrated in Figures 1–5. The switching signal is given in Fig. 1.

The control signal is depicted in Fig. 3. It has been shown from Figures 2 and 4 that the state

trajectories are first driven onto the specified sliding surface S(t) = 0 and then asymptotically tend

to zero along this sliding surface. Moreover, the state estimation errors are also asymptotically

11

stable. These have shown that the proposed method in this work can effectively cope with the

effects of non-fragile observer, parameter uncertainties, and external disturbances.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(Sec)

σ(t)

Fig.1. Switching signal σ(t).

0 1 2 3 4 5 6 7 8 9 10−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time(Sec)

x

1(t)

x2(t)

x3(t)

Fig.2. State trajectories x(t).

12

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

0.4

Time(Sec)

e

1(t)

e2(t)

e3(t)

Fig.3. Error estimation e(t).

0 1 2 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time(Sec)

S(t)

Fig.4. Sliding variable S(t).

13

0 1 2 3 4 5 6 7 8 9 10−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Time(Sec)

u(t)

Fig.5. Control signal u(t).

5 Conclusion

In this paper, we have concerned with non-fragile SMC for a class of uncertain switched systems

with state unavailable. Since the input matrix for each subsystem may be different, a weighted sum

of the input matrices was proposed, by which a common sliding surface based on the non-fragile

observer has been designed. Furthermore, a switching signal and a SMC law were designed such

that the exponential stability of the closed-loop system can be guaranteed despite the presence of

non-fragile observer, parameter uncertainties, and external disturbances.

Acknowledgements

This work was supported by NNSF from China (61074041, 61273073, 61374107), and the Funda-

mental Research Funds for the Central Universities.

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