INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 9(4):482-496, August, 2014.

Observer-Based Non-Fragile Passive Control for Uncertain
Nonlinear Sampled-Data System with Time-Delay

S.G. Wang, Y.S. Li, Y. Wu

Shigang Wang
School of Mechanical & Electric Engineering
Heilongjiang University
No.74 XueFu Road, Harbin, 150080, P. R. China
wsg6363@gmail.com

Yingsong Li*
College of Information and Communications Engineering
Harbin Engineering University
No.145 NanTong Street, Harbin, 150001, P. R. China
*Corresponding author: liyingsong82@gmail.com

Yi Wu
School of Information Science and Technology
Heilongjiang University
No.74 XueFu Road, Harbin, 150080, P. R. China
wy51cn@yahoo.com.cn

Abstract: The problem of observer-based passive control for uncertain nonlinear
sampled-data systems with time delay is investigated by using non-fragile passive
control. Attention is focused on the design of a non-fragile passive observer and a
controller which guarantees the passivity of the closed-loop system for all admissible
uncertainties. A sufficient condition for passivity and asymptotic stability of the
combined system is derived via linear matrix inequality (LMI). Finally, a simulation
example is presented to show the validity and advantages of the proposed method.
Keywords: Uncertain sampled-data system, Time-delay systems, State observer,
Non-fragile passive control, Linear matrix inequality

1 Introduction

In the past few years, sampled-data systems are widely encountered in the area of control
theory and control engineering, such as welding process, aerospace, signal processing, earthquake
prediction, due to its character as continuous control plant and discrete controller [1]. Since time
delay and inherent nonlinearity often occurs and causes serious deterioration of the stability of
various engineering system, considerable research has been done and numerous results have been
obtained relating to the analysis and synthesis of uncertain nonlinear sampled-data systems with
time-delay, see e.g. [2–7]. Among these results, non-fragile passive control problem have attracted
particular attention. Passivity is part of a broader and more general theory of dissipativity
and non-fragility is a scheme of solving robustness of controller and observer themselves [8–11],
they maintain the system’s internal stability. Because non-fragile passive control has attractive
features such as fast response, good transient response and insensitivity to variations in system
parameters and external disturbance [12–16], which is likely to be an effective technique of control
for uncertain nonlinear sampled-data system with time-delay.

On the other hand, all above works are based on an implicit assumption that the states are all
known. However, this unrealistic assumption is not always verified. and hence, the construction
of the unmeasured states through the knowledge of the system’s inputs and outputs still an
unavoidable task to solve any desired control problem [17–21]. However, to our knowledge, there

Copyright © 2006-2014 by CCC Publications



Observer-Based Non-Fragile Passive Control for Uncertain Nonlinear Sampled-Data System
with Time-Delay 483

have been few results in literature of any investigation for observer-based non-fragile passivity
uncertain nonlinear sampled-data system with time-delay.

The above situation is exactly what concerns and interests us. a novel approach of non-fragile
control combined with passive control is proposed for stabilizing a class of uncertain nonlinear
systems with time-delay. By utilizing a non-fragile state observer, a novel control law is estab-
lished such that the resulting closed-loop system is strictly passive. A sufficient condition for
the passivity and asymptotic stability of the augmented system is derived via LMI. Finally, an
example is simulated to illustrated the advantage of the proposed method.

2 Problem Statement and Preliminaries

Consider the plant of uncertain nonlinear sampled-data system with time-delay described by




ẋ(t) = (A0 + ∆A0)x(t) + (A1 + ∆A1)x(t − τ) + B0u(t) + f(x, u, t) + B1ω(t)
y(t) = C1x(t) + H2u(t)

z(t) = C2x(t) + H3ω(t)

x(t) = x0 , t ∈ [−τ, 0]

(1)

where x(t) ∈ Rn is the state, and u(t) ∈ Rm is the control input, y(t) ∈ Rp is regulated
output, z(t) ∈ Rq is measured output, ω(t) ∈ Rr is the external disturbance input that be-
longs to L2[0, ∞], A0, A1, B0, B1, C1, C2, H2, H3 are known real constant matrices of appropriate
dimension, ∆A0, ∆A1, are uncertain matrices. f is the uncertain nonlinear function vector,
f(0, 0, t0) = 0, and f satisfies the Lipschitz condition.

Assumption 2.1. The continuous plant is time-driven with a constant sampling period
h(h > 0).

Discretizing system (1) in one period, we can obtain the discrete state equation of the
plant of sampled-data system




x(k + 1) = (G0 + ∆G0)x(k) + (G1 + ∆G1)x(k − d) + H0u(k) + f̄(xk, xk−d, k) + H1ω(k)
y(k) = C1x(k) + H2u(k)

z(k) = C2x(k) + H3ω(k)

x(k) = x0 , k ≤ 0
(2)

where
G0 = e

A0h , G1 =
∫ h
0
eA0(h−w)dwA1

H0 =
∫ h
0
eA0(h−w)dwB0 , H1 =

∫ h
0
eA0(h−w)dwB1

f̄(x, u, k) =
∫ h
0
eA0wdwf(x, u, t)

f̄(x̂, u, k) =
∫ h
0
eA0wdwf(x̂, u, t)

Consider non-fragile observer described by{
x̂(k + 1) = G0x̂(k) + G1x̂(k − d) + H0u(k) + (L + ∆L)(y(k) − ŷ(k)) + f̄(x̂k, xk−d, k)
ŷ(k) = C1x̂(k) + H2u(k)

(3)



484 S.G. Wang, Y.S. Li, Y. Wu

where x̂(k) ∈ Rn is state of observer, and L is observer gain, ∆G0, ∆G1, are uncertain matrices
and ∆L are observer gain perturbation which are assumed to be of the following form:

[∆G0 ∆G1 ∆L] = M0F(k)[E0 E1 E2] (4)

On the other hand non-fragile controller described by

u(k) = (K + ∆K)x̂(k) (5)

where K is controller gain, ∆K represents corresponding gain perturbation, and generally,
there exist the following two classes of perturbation in ∆K:

Type1: ∆K is of the additive form:

∆K = M0F(k)E3 (6)

Type2: ∆K is of the multiplicative form:

∆K = M0F(k)E4K (7)

where M0, E0, E1, E2, E3, and E4 are real matrices with appropriate dimension and F(k) ∈
Rk×l is an unknown time-varying matrix function satisfying

F T(k)F(k) ≤ I

Assumption 2.2. f̄(xk, xk−d, k), f̄(x̂k, x̂k−d, k) satisfies the quadratic inequality in the do-
mains of continuity, that is

f̄T(xk, xk−d, k)f̄(xk, xk−d, k) ≤ δ21xT(k)MT1 M1x(k) + δ
2
d1x

T(k − d)MTd1Md1x(k − d) (8)

Let ξT1 =
[
eT(k) x̂T(k) eT(k − d) x̂T(k − d) f̄T(xk, xk−d, k)

]
, then (8) can be conve-

niently written as

ξT1 (k)




−δ21MT1 M1 ⋆ ⋆ ⋆ ⋆
−δ21MT1 M1 −δ

2
1M

T
1 M1 ⋆ ⋆ ⋆

0 0 −δ2d1M
T
d1Md1 ⋆ ⋆

0 0 0 −δ2d1M
T
d1M ⋆

0 0 0 −δ2d1M
T
d1M I


 ξ1(k) ≤ 0 (9)

In addition,

f̄T(x̂k, x̂k−d, k)f̄(x̂k, x̂k−d, k) ≤ δ22x̂T(k)MT2 M2x̂(k) + δ
2
d2x̂

T(k − d)MTd2Md2x̂(k − d) (10)

Let ξT2 =
[
eT(k) x̂T(k) eT(k − d) x̂T(k − d) f̄T(x̂k, x̂k−d, k)

]
, then (10) can be con-

veniently written as

ξT2 (k)




0 ⋆ ⋆ ⋆ ⋆

0 −δ22MT2 M2 ⋆ ⋆ ⋆
0 0 0

0 0 0 −δ2d2M
T
d2Md2 ⋆

0 0 0 0 I


 ξ2(k) ≤ 0 (11)



Observer-Based Non-Fragile Passive Control for Uncertain Nonlinear Sampled-Data System
with Time-Delay 485

where δ1, δ2, δd1, δd2 are the bounding parameter, and M1, M2, Md1, Md2 are constant matrices
such that f(0, 0, k) = 0 and x = 0 is an equilibrium of system (2) for dk = 0.

The objective of this paper is to design observer-based non-fragile passive controller, substi-
tute non-fragile observer (4) and controller (5) into system (2), and let
e(k) = x(k) − x̂(k), then resulting error closed-loop system is obtain by




e(k + 1) = (G0 − LC1 + ∆G0 − ∆LC1)e(k) + ∆G0x̂(k) + (G1 + ∆G1)e(k − d)+
∆G1x̂(k − d) + H1ω(k) + f̄(x) − f̄(x̂)

x̂(k + 1) = (G0 + H0K + H0∆K)x̂(k) + G1x̂(k − d) + (L + ∆L)C1e(k) + f̄(x̂(k))
(12)

Before proceeding to this main results, the following useful assumption and lemmas are need.

Assumption 2.3. Suppose that the matrix C1 has full row rank (i.e. rank(C1)=p).
for convenience of discussion, the singular value decomposition of C1 as follows:

C1 = U
[
S 0

]
V T

where is S ∈ Rp×p a diagonal matrix with positive diagonal elements in decreasing order, 0 ∈
Rp×(n−p) is a zero matrix, and U ∈ Rp×p and V ∈ Rn×n are unitary matrices.

Lemma 1. [22] For a given C1 ∈ Rp×n with rank(C1)=p,assume that X ∈ Rn×n is a symmetric
matrix, then there exists a matrix X̂ ∈ Rp×p such that C1X = X̂C1 if and only if

X = V

[
X̂11 0

0 X̂22

]
V T

where x̂11 ∈ Rp×p and x̂22 ∈ R(n−p)×(n−p)

Lemma 2. [23] (Schur complement) For a given symmetric matrix S = ST =

[
S11 S12

ST12 S22

]
with S11 ∈ Rr×r, the following conditions are equivalent:

(1) S < 0
(2) S11 < 0, S22 − ST12S

−1
11 S12 < 0

(3) S22 < 0, S11 − S12S−122 S
T
12 < 0

Lemma 3. [24] For given matrices Q = QT, H, and E, with appropriate dimensions

Q + HF(k)E + ETF T(k)HT < 0

holds for all F(k) satisfying F T(k)F(k) ≤ I if and only if there exists ε > 0

Q + εHHT + ε−1ETE < 0

Definition 4. The systems (2) is called passive if there exists a scalar β ≥ 0 such that

∞∑
k=0

ω(k)z(k) ≥ β , ∀ω ∈ l2[0, ∞]

where β is some constant which depends on the initial condition of the system.



486 S.G. Wang, Y.S. Li, Y. Wu

3 Main Results

Theorem 5. For system(2) and observer (3), if there exist two symmetric and positive matrices
R̄ ∈ Rn×n, P̄ ∈ Rn×n, two real matrices Y1 ∈ Rm×n, Y2 ∈ Rn×p and three positive constants ε1,
ε2, and ε3 such that the following holds:

Ξ =

[
Ξ11 Ξ

T
21

Ξ21 Ξ22

]
< 0 (13)

then there exist two gains K = Y1P̄ −1, and L = Y2USX̂
−1
11 S

−1UT, such that system is asymp-
totically passive stable.
where

Ξ11 =




W1 − R̄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 W2 − P̄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −Q1 ⋆ ⋆ ⋆ ⋆ ⋆
0 0 0 −Q2 ⋆ ⋆ ⋆ ⋆

−C2R̄ −C2P̄ 0 0 −H2 − HT2 ⋆ ⋆ ⋆
0 0 0 0 0 0 ⋆ ⋆

0 0 0 0 0 0 0 ⋆

G0R̄ − Y2C1 0 G0 0 H1 I −I −R̄




Ξ21 =




Y2C1 G0P̄ + H0Y1 0 G1 0 0 I 0

0 0 0 0 0 0 0 ε1M
T
0

E0R̄ − E2C1R̄ E0P̄ E1 E1 0 0 0 0
0 0 0 0 0 0 0 0

E2CR̄ 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 E3P̄ 0 0 0 0 0 0

δ1M1R̄ 0 δd1Md1
δd1Md1+

δd2Md2
0 0 0 0




Ξ22 =




−P̄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 −ε1I ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −ε1I ⋆ ⋆ ⋆ ⋆ ⋆

ε1M
T
0 0 0 −ε1I ⋆ ⋆ ⋆ ⋆

0 0 0 0 −ε1I ⋆ ⋆ ⋆
ε1(H0M0)

T 0 0 0 0 −ε1I ⋆ ⋆
0 0 0 0 0 0 −ε1I ⋆
0 0 0 0 0 0 0 −ε1I




Proof: Choose a Lyapunov function candidate for the system (12) as follow:



Observer-Based Non-Fragile Passive Control for Uncertain Nonlinear Sampled-Data System
with Time-Delay 487

V = V1 + V2 + V3 + V4

V1 = e
T(k)Re(k)

V2 =

k−1∑
i=k−h

eT(i)Q1e(i)

V3 = x̂
T(k)Px̂(k)

V4 =
k−1∑

i=k−h
xT(i)Q2x(i)

where R = RT > 0, Q1 = QT1 > 0, Q2 = Q
T
2 > 0, and P = P

T > 0.
Define vector

ξ(k) =




e(k)

x̂(k)

e(k − d)
x̂(k − d)
ω(k)

f̄(xk, xk−d, k)

f̄(x̂k, x̂k−d, k)




, θ1 =




(G0 − LC1 + ∆G0 − ∆LC1)T

∆GT0
(G1 + ∆G1)T

∆GT1
HT1
I

−I




T

θ2 =
[
H(L + ∆L)C G0 − H0K 0 G1 0 0 I

]
Therefore,

∆V = ∆V1 + ∆V2 + ∆V3 + ∆V4

= ξT(k)θT1 Rθ1ξ(k) + e
T(k)Q1e(k) − eT(k − d)Q1e(k − d)+

ξT(k)θT2 Pθ2ξ(k) + x̂
T(k)Q2x̂(k) − x̂T(k − d)Q2x̂(k − d)

= ξT(k)Π1ξ(k)

(14)

On one hand, the sufficient condition of stability ∆V < 0, implies that Π1 < 0, that is

Π1 :=




Q1 − R ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 Q2 − P ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −Q1 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 0 −Q2 ⋆ ⋆ ⋆ ⋆ ⋆
0 0 0 0 0 ⋆ ⋆ ⋆ ⋆

0 0 0 0 0 0 ⋆ ⋆ ⋆

0 0 0 0 0 0 0 ⋆ ⋆

G0 − LC1+
∆G0 − ∆LC1

∆G0 G1 + ∆G1 ∆G1 H1 I −I −R−1 ⋆

(L + ∆L)C G0 − H0K 0 G1 0 0 I 0 −P −1




< 0

(15)



488 S.G. Wang, Y.S. Li, Y. Wu

On the other hand, utilizing (14) with ω(k) ∈ l2[0, +∞] ̸= 0, one is obtained by

∆V − 2zT (k)ω(k) ≤

ξT(k)







Q1 − R ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 Q2 − P ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −Q1 ⋆ ⋆ ⋆ ⋆
0 0 0 −Q2 ⋆ ⋆ ⋆
0 0 0 0 0 ⋆ ⋆

0 0 0 0 0 0 ⋆

0 0 0 0 0 0 0



+

θT1 Rθ1 + θ
T
2 Pθ2 − 2θ

T
3 θ4

}
ξ(k) = ξT(k)Π2ξ(k)

(16)

If Π2 < 0, then ∆V (k) − 2zT(k)ω(k) < 0 and from which it follows that

∞∑
k=0

ω(k)z(k) > 1/2

∞∑
k=0

∆V = 1/2[V (0) − V (∞)] (17)

Due to V (k) > 0 for x ̸= 0 and V (k) = 0 for x = 0, if follows as k → ∞ that system (12) is
strictly passive. In virtue of Definition 4, the strictly passive condition is guaranteed if Π2 < 0
and it can be expressed conveniently as




Q1 − R ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 Q2 − P ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −Q1 ⋆ ⋆ ⋆ ⋆
0 0 0 −Q2 ⋆ ⋆ ⋆
0 0 0 0 0 ⋆ ⋆

0 0 0 0 0 0 ⋆

0 0 0 0 0 0 0



+ θT1 Rθ1 + θ

T
2 Pθ2 − 2θ

T
3 θ4 < 0 (18)

Application of the Lemma 2 to (18) puts it into the form:




Q1 − R ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 Q2 − P ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −Q1 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 0 −Q2 ⋆ ⋆ ⋆ ⋆ ⋆

−C2 −C2 0 0 −H2 − HT2 ⋆ ⋆ ⋆ ⋆
0 0 0 0 0 0 ⋆ ⋆ ⋆

0 0 0 0 0 0 0 ⋆ ⋆

G0 − LC1+
∆G0 − ∆LC1

∆G0 G1 + ∆G1 ∆G1 H1 I −I −R−1 ⋆

(L + ∆L)C1 G0 − H0(K + ∆K) 0 G1 0 0 I 0 −P −1




< 0

(19)
Substituting the uncertainty structure into (19) and rearranging, we get



Observer-Based Non-Fragile Passive Control for Uncertain Nonlinear Sampled-Data System
with Time-Delay 489




Q1 − R ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 Q2 − P ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −Q1 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 0 −Q2 ⋆ ⋆ ⋆ ⋆ ⋆

−C2 −C2 0 0 −H2 ⋆ ⋆ ⋆ ⋆
0 0 0 0 0 0 ⋆ ⋆ ⋆

0 0 0 0 0 0 0 ⋆ ⋆

G0 − LC1 0 G0 0 H1 I −I −R−1 ⋆
LC G0 − H0K 0 G1 0 0 I 0 −P −1




+θ5F(k)θ6 + θ
T
6 F

T(k)θT5 + θ7F(k)θ8 + θ
T
8 F

T(k)θT7 + θ9F(k)θ10+

θT10F
T(k)θT9 < 0

(20)

where

θ5 =
[
0 0 0 0 0 0 0 MT0 0

]T
θ6 =

[
E0 − E2C1 E0 E1 E1 0 0 0 0 0

]
θ7 =

[
0 0 0 0 0 0 0 0 MT0

]T
θ8 =

[
E2C1 0 0 0 0 0 0 0 0

]
θ9 =

[
0 0 0 0 0 0 0 0 (H0M0)T

]T
θ10 =

[
0 E3 0 0 0 0 0 0 0

]

Then by Lemma 3, the inequality (20) holds if and only if for some εi > 0 (i=1,· · · , 3)

Π3 + ε1θ5θ
T
5 + ε

−1
1 θ

T
6 θ6 + ε2θ7θ

T
7 + ε

−1
2 θ

T
8 θ8 + ε3θ9θ

T
9 + ε

−1
3 θ

T
10θ10 < 0 (21)

where

Π3 :=




Q1 − R ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 Q2 − P ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −Q1 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 0 −Q2 ⋆ ⋆ ⋆ ⋆ ⋆

−C2 −C2 0 0 −H2 ⋆ ⋆ ⋆ ⋆
0 0 0 0 0 0 ⋆ ⋆ ⋆

0 0 0 0 0 0 0 ⋆ ⋆

G0 − LC1 0 G0 0 H1 I −I −R−1 ⋆
LC1 G0 − H0K 0 G1 0 0 I 0 −P −1




On using the Lemma 2, it becomes that



490 S.G. Wang, Y.S. Li, Y. Wu




Π3 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

ε1θ
T
5 −ε1I ⋆ ⋆ ⋆ ⋆ ⋆

θ6 0 −ε1I ⋆ ⋆ ⋆ ⋆
ε2θ

T
7 0 0 −ε2I ⋆ ⋆ ⋆

θ8 0 0 0 −ε2I ⋆ ⋆
ε3θ

T
9 0 0 0 0 −ε3I ⋆

θ10 0 0 0 0 0 −ε3I



< 0 (22)

Thirdly, introduce nonlinearities (8) and (10) into (22), then




Π4 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

ε1θ
T
5 −ε1I ⋆ ⋆ ⋆ ⋆ ⋆

θ6 0 −ε1I ⋆ ⋆ ⋆ ⋆
ε2θ

T
7 0 0 −ε2I ⋆ ⋆ ⋆

θ8 0 0 0 −ε2I ⋆ ⋆
ε3θ

T
9 0 0 0 0 −ε3I ⋆

θ10 0 0 0 0 0 −ε3I



< 0 (23)

where

Π4 :=




φ1 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

−δ21MT1 M1 φ2 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 φ3 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

0 0 −δ2d1M
T
d1Md1 φ4 ⋆ ⋆ ⋆ ⋆ ⋆

−C2 −C2 0 0 −H2 ⋆ ⋆ ⋆ ⋆
0 0 0 0 0 0 ⋆ ⋆ ⋆

0 0 0 0 0 0 0 ⋆ ⋆

G0 − LC1 ∆G0 G0 0 H1 I −I −R−1 ⋆
LC1 G0 − H0K 0 G1 0 0 I 0 −P −1



< 0

φ1 = Q1 − R − δ21MT1 M1
φ2 = Q2 − P − δ21MT1 M1 − δ

2
2M

T
2 M2

φ3 = −Q1 − δ2d1M
T
d1Md1

φ4 = −Q2 − δ2d1M
T
d1Md1 − δ

2
d2M

T
d2Md2

Pre-multiplying and post-multiplying (23) by diag

{
R−1 P −1

13 columns︷ ︸︸ ︷
I · · · I

}
, and from

Lemma 1, the condition C1R̄ = ˆ̄RC1 holds, moreover, setting Y2 = L ˆ̄R. In the meanwhile, define
P̄ = P −1, R̄ = R−1, Y1 = KP̄, W1 = R

−1Q1R
−1, W2 = P

−1Q2P
−1, it is seen that (23)<0 is

equivalent to (13), the means that the system (12) is asymptotically passive stable.

Theorem 6. For system(2) and observer (3), if there exist two symmetric and positive matrices
R̄ ∈ Rn×n, P̄ ∈ Rn×n, two real matrices Y3 ∈ Rm×n, Y4 ∈ Rn×p and three positive constants ε4,



Observer-Based Non-Fragile Passive Control for Uncertain Nonlinear Sampled-Data System
with Time-Delay 491

ε5, and ε6 such that the following holds:

Ω =

[
Ω11 Ω

T
21

Ω21 Ω22

]
< 0 (24)

then there exist two gains K = Y3P̄ −1, and L = Y4USX̂
−1
11 S

−1UT, such that system is asymp-
totically passive stable.
where

Ω11 =




W3 − R̄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 W4 − P̄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −Q1 ⋆ ⋆ ⋆ ⋆ ⋆
0 0 0 −Q2 ⋆ ⋆ ⋆ ⋆

−C2R̄ −C2P̄ 0 0 −H2 − HT2 ⋆ ⋆ ⋆
0 0 0 0 0 0 ⋆ ⋆

0 0 0 0 0 0 0 ⋆

G0R̄ − Y4C1 0 G0 0 H1 I −I −R̄




Ω21 =




Y4C1 G0P̄ + H0Y3 0 G1 0 0 I 0

0 0 0 0 0 0 0 ε4M
T
0

E0R̄ − E2C1R̄ E0P̄ E1 E1 0 0 0 0
0 0 0 0 0 0 0 0

E2CR̄ 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 E4Y3 0 0 0 0 0 0

δ1M1R̄ 0 δd1Md1
δd1Md1+

δd2Md2
0 0 0 0




Ω22 =




−P̄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 −ε4I ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
0 0 −ε4I ⋆ ⋆ ⋆ ⋆ ⋆

ε4M
T
0 0 0 −ε5I ⋆ ⋆ ⋆ ⋆

0 0 0 0 −ε5I ⋆ ⋆ ⋆
ε4(H0M0)

T 0 0 0 0 −ε6I ⋆ ⋆
0 0 0 0 0 0 −ε6I ⋆
0 0 0 0 0 0 0 I




Proof: Theorem 6 ’s proof is same to Theorem 5, so is omitted.

4 Numerical Example

Consider the plant of sampled-data system (1) with the parameters:



492 S.G. Wang, Y.S. Li, Y. Wu

A0 =




0 1 0

0 0 1

1 −3 1


 , A1 =




0 1 0

1 0 0

1 −3 1


 , B0 =




0.1 0.1

0 1.2

0.5 0.1


 , B1 =




0.1 0.2

0 1.5

0 0.1




C1 =

[
1.5 0.6 1.0

0.5 0.6 0

]
, C2 =

[
3 4 1

]
, H2 =

[
0.1 0.1

0.1 0.1

]
, H3 = 0.5

By Theorem 1 has a solution:

P̄ =




3.8697 −2.7891 −0.4465
−2.7891 2.0973 −0.0138
−0.4465 −0.0138 1.3967


 , Y1 =

[
−13.8330 9.8716 2.0314
1.4006 −0.8149 −0.9412

]

R̄ =




0.3864 −0.3364 −0.3658
−0.3364 0.3753 0.2113
−0.3658 0.2113 0.4907


 , X̂11 =

[
0.0059 0.0333

0.0333 0.2139

]
, X̂22 = 1.0326

The non-fragile passive observer-based control are given by

L = Y2USX̂
−1
11 S

−1UT =




−6.5298 −9.0418
2.7828 3.9663

8.6606 11.5095


 , K = Y1P̄ −1 =

[
−1.3147 2.9654 1.0635
0.3124 0.0231 −0.5738

]

We present design method of observer-based non-fragile passive controller in this paper, the
simulation results are given in Figure 1. From Figure 1, it can be seen the state estimation x̂(t)
has a good trace performance with the external disturbance and input nonlinearity.

5 Application to Stabilization of An Inverted Pendulum on A
Cart

An inverted pendulum on a cart [25] is depicted in Figure 2
In this model, a pendulum is conjuncted to the topside of a cart by a pivot, which is allowed

to swing in the xy−plane. A force u acts on the cart in the x direction, in order to keep the
pendulum balance upright. x(t) is the displacement between central mass of cart and the origin
0; θ is the angle of the pendulum from the top vertical.

Which is described by the following dynamics by applying Newtons Second Law

(M + m) ẍ + mlθ̈ cos θ − mlθ̇2 sin θ = u

mlẍ cos θ +
4

3
ml2θ̈ − mgl sin θ = 0



Observer-Based Non-Fragile Passive Control for Uncertain Nonlinear Sampled-Data System
with Time-Delay 493

1
x

1
x̂ with oberver L

1
x̂ with observer L

2
x

2
x̂ with oberver L

2
x̂ with observer L

3
x

3
x̂ with oberver L

3
x̂ with observer L

Figure 1: The simulation of non-fragile observer with additional perturbation

Now, by selecting state variables z =
[
z1 z2

]T
=

[
θ θ̇

]T
and by linearizing the above

model at the equilibrium point z = 0, we obtain the following state-space model:

ż(t) =

[
0 1

3(M+m)g
l(4M+m)

0

]
z(t) +

[
0

− 3
l(4M+m)

]
u(t) (25)

Here the parameters are selected in Table 1, by assuming the sampling time to be
Ts = 0.1 s, the discretized model for the above pendulum system in (21) is given by

x(k + 1) =

[
1.0877 0.1029

1.7794 1.0877

]
x(k) +

[
−0.0000
−0.0182

]
u(k) (26)

The poles of the system are 0.6598 and 1.5156, thus this discretized system is unstable. It is
assumed that a non-fragile control law with additive form is given by

u(k) =
[
219.023 49.786

]
x(k) (27)

The other non-fragile control law with multiplicative form is given by

u(k) =
[
199.755 52.011

]
x(k) (28)

Table 1. An inverted pendulum parameters



494 S.G. Wang, Y.S. Li, Y. Wu

m

y

( )u t
xM

( )x t

Figure 2: A pendulum system perturbation

System parameter Values

Mass of the cart M (kg) 8.0
Mass of the pendulum (kg) 2.0
Half length of the pendulum (m) 0.5
Acceleration of gravity (m/s2) 9.8

1
x

2
x

(a) regular controller

1
x

2
x

(b) non-fragile controller

1
x

2
x

(c) non-fragile controller with non-
liearity

(d) control input

Figure 3: State response of the pendulum system and control input

The simulation results are given in Figure 3. In fact, for sampling period Ts = 0.1s, LMI
(13) remain solvable. State response curve of regular controller is divergent in Figure 3(a),
however, the curve is convergent for non-fragile controller in Figure 3(b). Furthermore, it is still
convergent, when there exists a nonlinear perturbation in Figure 3(c). A corresponding control
input is shown in Figure 3(d).



Observer-Based Non-Fragile Passive Control for Uncertain Nonlinear Sampled-Data System
with Time-Delay 495

6 Conclusions

The problem of observer-based non-fragile passive control of uncertain nonlinear sampled-
data system with time-delay has been studied. A LMI based approach to designing state observer
and non-fragile controller, which ensure the passivity of the resulting error closed-loop system
has been developed. A numerical example has been provided to demonstrate the effectiveness
and applicability of the proposed approach.

Acknowledgment

This work is partially supported by National Defense 973 Basic Research Development Pro-
gram of China (6131380101), National Natural Science Foundation of China (61203121,51307045),and
the authors are indebted to the editor and reviewers for their valuable comments and suggestions.

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