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arX
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6v1
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Mixed H−/H∞ Fault Detection Filtering for Ito-Type Affine
Nonlinear Stochastic Systems
Tianliang Zhang 1, Feiqi Deng 1∗, Weihai Zhang 2, and Bor-Sen Chen 3
1 School of Automation Science and Engineering,
South China University of Technology, Guangzhou 510640, P. R. China
2 College of Information and Electrical Engineering,
Shandong University of Science and Technology, Qingdao 266510, P. R. China
3 Department of Electrical Engineering,
National Tsing Hua University, Hsinchu, 30013, Taiwan.
Abstract- This paper studies the mixed H−/H∞ fault detection filtering of Ito-type nonlin-
ear stochastic systems. Mixed H−/H∞ filtering combines the system robustness to the external
disturbance and the sensitivity to the fault of the residual signal. Firstly, for Ito-type affine
nonlinear stochastic systems, some sufficient criteria are obtained for the existence of H−/H∞
filter in terms of Hamilton-Jacobi inequalities (HJIs). Secondly, for a class of quasi-linear Ito
systems, a sufficient condition is given for the existence of H−/H∞ filter by means of linear ma-
trix inequalities (LMIs). Finally, a numerical example is presented to illustrate the effectiveness
of the proposed results.
K eywords: Stochastic systems, fault detection, nonlinear systems, H−/H∞ fault detection
filtering.
1 Introduction
Along with the development of modern industrial production, higher requirements for safety
and reliability have been put forward. In order to ensure safety and reliability in industrial
process, various techniques for fault detection, fault isolation and fault estimation have appeared
∗Email: t [email protected](T. Zhang), [email protected](F. Deng), w [email protected](W. Zhang),
[email protected](B. S. Chen).
1
[11, 27]. Fault detection filter (FDF) is to use the estimated value of the system state and the
measurement output to generate the residual signal to detect the system fault. According to the
real time comparison between the designed residual evaluation function and the corresponding
threshold, we are in a position to determine whether there is a fault occurring. In real world, some
unknown disturbance signals existing in dynamic systems will fluctuate the residual. Therefore,
one needs to design robust FDF to exclude these unknown external effect. H∞ control is one of
the most important robust control methods since G. Zames’ fundamental work [41] published,
which has been studied extensively [5, 12, 14, 28, 29, 42, 43, 45], and has been applied to event
triggers [13,32], fault diagnosis [18,30,48], sliding mode control method [35] and adaptive control
method [16]. H∞ FDF is to design the filter such that the L2-gain from the external interference
to the residual signal is less than the given attenuation level γ > 0, which reflects the robust
ability of the concerned systems. H− FDF is to design the filter such that the L2-gain from the
fault signal to the residual signal is larger than the given sensitivity level δ > 0, which measures
the sensitivity of the considered systems to the fault signal. Different from the sole H∞ or H−
FDF, mixed H−/H∞ FDF is a combination of H∞ FDF and H− FDF, which not only meets
the robustness requirement, but also requires the residual signal to be sensitive enough to the
fault information [4, 34]. So H−/H∞ FDF is one of the most popular robust design methods.
Stochastic Ito-type systems are ideal mathematical models in finance mathematics [37], sys-
tems biology [6, 7], benchmark mechanical systems [36], so the study for stochastic systems has
attracted many researchers’ interest, and stochastic control has become one of the most impor-
tant research fields in modern control theory [10,39,40]. In 1998, H∞ control of linear Ito systems
was first investigated in [15, 31], and from then, nonlinear H∞ control [2, 42] and filtering [43],
mixed H2/H∞ control [45] of Ito systems have been solved. H− FDF is to use the H− index to
measure the minimum influence of the fault on the residual signal [19,20,24,30,48], where [24,48]
and [19, 20] were about FDFs of linear time-invariant and time-varying deterministic systems,
respectively. Generally speaking, for linear time-invariant/time-varying deterministic systems,
the corresopnding FDF design can be turned into solving some LMIs [24, 38, 48]/differential
Riccati equations (DREs) [19, 20]. In [30], FDF of nonlinear switched stochastic systems was
discussed based on T-S fuzzy model approach. While the fault isolation problem for discrete-
time fuzzy interconnected systems with unknown interconnections was considered in [46]. The
reference [17] investigated the adaptive fuzzy output feedback fault-tolerant optimal control
problem for a class of single-input and single-output nonlinear systems in strict feedback form.
2
A robust fault detection H−/H∞ observer was constructed for a Takagi-Sugeno fuzzy model
with sensor faults and unknown bounded disturbances via an LMI formulation in [4]. The ref-
erence [34] studied H−/H∞ fault detection observer design in the finite frequency domain for a
class of linear parameter-varying descriptor systems. Recently, we analyze the H− index for a
class of linear discrete-time stochastic Markov jump systems with multiplicative noise [21]. It
can be found that, up to now, there are few works on mixed H−/H∞ FDF for nonlinear Ito-type
stochastic systems.
Motivated by the aforementioned reason, this paper studies H−/H∞ FDF of affine nonlinear
Ito systems, where a Luenberger type observer is considered in designing H−/H∞ FDF, which
can not only suppress the effect of external interference on the residual signal below a level γ > 0,
but also guarantee the residual signal to be sensitive to the fault signal. The contributions of
this paper can be summarized as follows:
• Applying Ito formula together with square completion technique, some sufficient conditions
are obtained for the existence of H−/H∞ FDF of affine stochastic Ito systems in terms of coupled
HJIs. As corollaries, the existence conditions of H−/H∞ FDF of linear stochastic Ito systems
are also presented in terms of algebraic Riccati inequalities (ARIs), which can be transformed
into solving LMIs by Matlab LMI Toolbox. How to solve the HJIs is a challenging problem,
and a potential powerful technique to solve the coupled HJIs can refer to [1] by using the neural
network approach.
• We also study H−/H∞ FDF for a class of quesi-linear stochastic Ito systems. It is shown
that for such a class of nonlinear stochastic systems, the corresponding H−/H∞ FDF can be
designed via solving some LMIs instead of HJIs, which is very convenient in practice.
The organization of this paper is as follows: In Section 2, we make some preliminaries to
introduce useful definitions and lemmas. In Section 3, some sufficient conditions for the existence
of H−/H∞ fault detection filter are presented based on HJIs. Applying LMI-based technique, in
Section 4, for a class of quasi-linear stochastic systems, the corresponding H−/H∞ FDF design
is converted into solving LMIs. Section 5 presents an example to illustrate the effectiveness of
our given results. Section 6 concludes this paper with some remarks and perspectives.
For convenience, this paper adopts the following standard notations:
R+ := [0,+∞); M ′: the transpose of the matrix M or vector M ; M > 0 (M < 0): the matrix
M is a positive definite (negative definite) real symmetric matrix; In: n × n identity matrix;
Rn: the n-dimensional real Euclidean vector space with the norm ‖x‖ =√
∑nk=0 x
2k; R
n×m: the
3
n×m real matrix space; L2F(R
+,Rnv): the space of nonanticipative stochastic process v(t) ∈ Rnv
with respect to an increasing σ-algebra Ftt≥0 satisfying ‖v(t)‖L2∞
:= E∫∞
0 ‖v(t)‖2dt < ∞; A
function f(x) is called a positive function, if f(x) > 0 for any x 6= 0, and f(0) = 0; C2(U ;X):
the class of X-valued functions V (x), which are twice continuously differential with respect to
x ∈ U , except possibly at the point x = 0; C2,1(U × R+;X): the class of X-valued functions
V (x, t), which are twice continuously differential with respect to x ∈ U , and once continuously
differential with respect to t ∈ R+, except possibly at the point x = 0.
2 Preliminaries
Consider the following affine nonlinear Ito stochastic system (the time variable t is suppressed):
dx = [f1(x) + g1(x)v + h(x)f ] dt
+ [f2(x) + g2(x)v] dw,
y = l(x) +m(x)v + n(x)f,
x(0) = x0 ∈ Rn,
(1)
where x(t) ∈ Rn is the n-dimensional state vector, y ∈ Rny is the ny-dimensional measure-
ment output, v ∈ Rnv stands for the exogenous disturbance signal with v ∈ L2F (R
+,Rnv ),
and w(t) is a 1-D standard Wiener process defined on the complete filtered probability space
(Ω,F , Ftt∈R+ ,P) with the σ-field Ft generated by w(·) up to time t. f(t) ∈ Rnf denotes
the fault information to be detected. We assume that all functions f1(x), g1(x), h(x), f(t),
f2(x), g2(x), l(x). m(x) and n(x) are continuous, which satisfy certain conditions as linear
growth condition and Lipschitz condition such that the state equation in (1) has a unique strong
solution.
In this paper, we adopt the following Luenberger-type observer as FDF of system (1):
dx(t) = f(x(t))dt+ h(x(t)) (y(t)− l(x(t))) dt,
r(t) = s(x(t))(y(t) − l(x(t))),
x(0) = 0,
(2)
where x(t) is the estimated value of x(t), r(t) is viewed as the residual signal, f , h, and s are
the filter functions to be designed.
4
Set η(t) =[
x(t)′ x(t)′]′
, then we get the following augmented system:
dη =(
f1(η) + g1(η)v + h(η)f)
dt
+(
f2(η) + g2(η)v)
dw,
r(t) = s(η, v, f),
η(0) =
x0
0
∈ Rnη ,
(3)
where
f1(η) =
f1(x)
f(x) + h(x) (l(x)− l(x))
, g1(η) =
g1(x)
h(x)m(x(t))
,
h(η) =
h(x)
h(x)n(x)
, f2(η) =
f2(x)
0
,
g2(η) =
g2(x)
0
, s(η, v, f) = s(x) [l(x) +m(x)v + n(x)f − l(x)] .
2.1 Definitions and lemmas
For our needs, we introduce some definitions as follows.
Definition 1 An Ito-type stochastic differential system
dx(t) = f(x(t)) dt+ g(x(t))dw(t),
x(0) = x0 ∈ Rn
(4)
is said to be exponentially stable in mean square sense, if there exist constants β ≥ 1 and α > 0,
such that the solution x(t) of system (4) satisfies
E‖x(t)‖2 ≤ βe−αt‖x0‖2, t ≥ 0.
The residual signal r(t) needs to be measured and calculated in real time to judge whether
there is a noticeable fault occuring. Therefore, it is worth emphasizing that the comprehensive
ability to reflect external interferences and internal fault information are important for r(t) in
the designed filter. In order to reasonably analyze and study the capacity of r(t), we introduce
H− index and H∞ index to discribe system (3).
5
Definition 2 For stochastic system (3), its sensitive operator Lf,r and H− index are respectively
defined as
Lf,r : f(t) ∈ L2F (R
+,Rnf ) 7→ r(t) ∈ L2F (R
+,Rnr)
and
‖Lf,r‖− = inf
v(t) ≡ 0, η(0) = 0,
f(t) 6≡ 0, f(t) ∈ L2F (R
+,Rnf )
‖r(t)‖L2∞
‖f(t)‖L2∞
.
Meanwhile, we define the perturbation operator Lv,r and H∞ index as
Lv,r : v(t) ∈ L2F (R
+,Rnv) 7→ r(t) ∈ L2F (R
+,Rnr )
and
‖Lv,r‖∞ = sup
f(t) ≡ 0, η(0) = 0,
v(t) 6≡ 0, v(t) ∈ L2F (R
+,Rnv)
‖r(t)‖L2∞
‖v(t)‖L2∞
,
respectively.
The purpose in this paper is to design a mixed H−/H∞ FDF for system (1). Below, we give the
definition of mixed H−/H∞ FDF.
Definition 3 A FDF (2) is called the mixed H−/H∞ FDF, if for any given scalars γ > 0 and
δ > 0, the following requirements are satisfied simultaneously.
• The augmented system (3) is internally stable, that is, when f(t) ≡ 0 and v(t) ≡ 0 in
system (3), the following system
dη(t) = f1(η(t))dt + f2(η(t)) dw(t),
η(0) =
x0
0
∈ Rnη ,
(5)
is exponentially stable in mean square sense.
6
• The augmented system (3) is externally stable, that is, when f(t) ≡ 0, η(0) = 0 in system
(3) , for any nonzero v(t) ∈ L2F (R
+,Rnv ), the L2-gain from v(t) to r(t) of the following
system
dη(t) =(
f1(η(t)) + g1(η(t))v(t))
dt
+(
f2(η(t)) + g2(η(t))v(t))
dw(t),
r(t) = s(x(t)) (l(x(t)) +m(x(t))v(t) − l(x(t)))
(6)
is less than or equal to γ > 0, i.e., the H∞ index ‖Lv,r‖∞ ≤ γ.
• The residual signal is enough sensitive to the fault, that is, when v(t) ≡ 0, η(0) = 0 in
system (3), for any nonzero f(t) ∈ L2F (R
+,Rnf ), the L2-gain from f(t) to r(t) of the
following system
dη(t) =(
f1(η(t)) + h(η(t))f(t))
dt+ f2(η(t))dw(t),
r(t) = s(x(t)) (l(x(t)) + n(x(t))f(t)− l(x(t)))
(7)
is large than or equal to δ > 0, i.e., the H− index ‖Lf,r‖− ≥ δ.
In Definition 3, γ > 0 is called as the disturbance attenuation level, and δ > 0 as the fault
sensitivity level.
The following lemmas will play important roles in this study.
Lemma 1 [45] For x, b ∈ Rn, if A is a real symmetric matrix with appropriate dimension,
A−1 exists. Then we have
x′Ax+ x′b+ b′x = (x+A−1b)′A(x+A−1b)− b′A−1b.
Lemma 2 [26,45] Suppose there exists a function V (η, t) ∈ C2,1(Rnη ×R+;R). An infinites-
imal generator LV (η, t) : Rnη ×R+ 7→ R associated with (3) is given by
LV (η, t)
=∂V ′
∂t+
∂V ′
∂η
(
f1(η) + g1(η)v + h(η)f(t))
+1
2
(
f2(η) + g2(η)v)′ ∂2V
∂η2
(
f2(η) + g2(η)v)
, (8)
and
EV (η(t), t) = EV (η(t0), t0) + E
∫ t
t0
LV (η(s), s) ds, 0 ≤ t0 < t < +∞.
7
2.2 Residual evaluation
After obtaining the gain matrices of the filter, we are in the position to discuss the residual
estimation. For the purpose of evaluating the residual signal, one tends to adopt a threshold
Jth > 0, and the Jth conforms to the following decision logic:
Jr(t) > Jth ⇒ faults ⇒ alarm,
Jr(t) ≤ Jth ⇒ fault-free,
(9)
where the residual evaluation function Jr(t) is defined by
Jr(t) :=
(
1
t
∫ t
0r′(s)r(s)ds
)
1
2
, Jr(0) = 0, (10)
and the threshold [18] is determined by
Jth := supf(t)≡0,v(t)∈L2
F(R+,Rnv )
EJr(T ), (11)
where T is the evaluation window.
3 FDF for Affine Nonlinear Stochastic Systems
In this section, we will give our main results about the mixed H−/H∞ FDF for affine nonlinear
stochastic systems.
Theorem 1 For any given disturbance attenuation level γ > 0 and fault sensitivity level δ > 0,
if there exist positive constants c1, c2, c3, ε1, ε2 and a positive Lyapunov function V (η) ∈
C2(Rnη ;R+), such that
c1‖η‖2 ≤ V (η) ≤ c2‖η‖
2, (12)
and V (η) solves the following two coupled HJIs
(1 + ε1)‖s(x) (l(x)− l(x)) ‖2 + ∂V ′
∂η f1(η)
−(
12 g2(η)
′ ∂2V∂η2
f2(η) +12 g
′1(η)
∂V∂η
)′
·(
12 g2(η)
′ ∂2V∂η2 g2(η) + (1 + ε−1
1 )‖s(x)m(x)‖2I
−γ2I)−1
(
12 g2(η)
′ ∂2V∂η2
f2(η) +12 g
′1(η)
∂V∂η
)
+12 f2(η)
′ ∂2V∂η2
f2(η) + c3‖η‖2 ≤ 0,
12 g2(η)
′ ∂2V∂η2
g2(η) + (1 + ε−11 )‖s(x)m(x)‖2I
−γ2I < 0,
(13)
8
and
(1− ε2)‖s(x) (l(x)− l(x)) ‖2 − ∂V ′
∂η f1(η)
−12 f2(η)
′ ∂2V∂η2
f2(η) −12∂V ′
∂η h(η)
·(
(1− ε−12 )‖s(x)n(x)‖2I − δ2I
)−1
·h(η)′ ∂V∂η ≥ 0,
(1− ε−12 )‖s(x)n(x)‖2I − δ2I > 0
(14)
for some filter functions f(·), h(·), s(·) with suitable dimensions. Then the desired H−/H∞
FDF is obtained by (2).
Proof : Firstly, we show that system (3) is internally stable, or equivalently, the system (5)
is exponentially stable in mean square sense. For system (5), apply Lemma 2 and consider (13),
we have
LV (η)|f≡0,v≡0 =∂V ′
∂ηf1(η) +
1
2f2(η)
′ ∂2V
∂η2f2(η) ≤ −c3‖η‖
2.
So, by Theorem 4.4 of [26] and (12), the system (5) is exponentially stable in mean square sense.
Next, we will show that the system (6) is also externally stable, i.e., ‖Lv,r‖∞ ≤ γ. From
Lemma 2, for any T > 0, v(t) ∈ L2F (R
+,Rnv), and the initial value η(0) = 0, we have
E
∫ T
0
(
‖r(s)‖2 − γ2‖v(s)‖2)
ds
= E
∫ T
0
(
‖r(s)‖2 − γ2‖v(s)‖2 + LV (η(s))|f≡0
)
ds
−EV (η(T )) + V (η(0))
= E
∫ T
0
[
‖r(s)‖2 − γ2‖v(s)‖2 +∂V ′
∂η
(
f1(η(s))
+g1(η(s))v(s)) +1
2
(
f2(η(s)) + g2(η(t))v(s))′
·∂2V
∂η2
(
f2(η(t)) + g2(η(s))v(s))
]
ds− EV (η(T )).
Note that for any ε1 > 0, we have
‖r(s)‖2 ≤ (1 + ε1)‖s(x(s) (l(x(s))− l(x(s))) ‖2
+(1 + ε−11 )‖s(x(s))m(x(s))‖2‖v(s)‖2.
9
So
E
∫ T
0
(
‖r(s)‖2 − γ2‖v(s)‖2)
ds
≤ E
∫ T
0
[
(1 + ε1)‖s(x(s) (l(x(s))− l(x(s))) ‖2)
+((1 + ε−11 )‖s(x(s))m(x(s))‖2 − γ2)‖v(s)‖2
+∂V ′
∂η
(
f1(η(s)) + g1(η(s))v(s))
+1
2(g2(η(s))v(s))
′
·∂2V
∂η2(g2(η(s))v(s)) +
1
2f2(η(s))
′ ∂2V
∂η2f2(η(s))
+f2(η(s))′ ∂
2V
∂η2g2(η(s))v(s)
]
ds −EV (η(T )). (15)
By Lemma 1,
E
∫ T
0(‖r(s)‖2 − γ2‖v(s)‖2)ds
≤ E
∫ T
0
(1 + ε1)‖s(x(s)) (l(x(s)) − l(x(s))) ‖2
+∂V ′
∂ηf1(η(s))−
(
1
2g2(η(s))
′ ∂2V
∂η2f2(η(s))
+1
2g′1(η(s))
∂V
∂η
)′
Λ−12
(
1
2g2(η(s))
′ ∂2V
∂η2f2(η(s))
+1
2g′1(η(s))
∂V
∂η
)
+ [v + Λ1]′ Λ2 [v + Λ1]
+1
2f2(η(s))
′ ∂2V
∂η2f2(η(s))
ds− EV (η(T )), (16)
where
Λ1 = Λ−12
(
1
2g2(η)
′ ∂2V
∂η2f2(η) +
1
2g′1(η)
∂V
∂η
)
,
Λ2 =1
2g2(η)
′ ∂2V
∂η2g2(η) + (1 + ε−1
1 )‖s(x)m(x)‖2I − γ2I.
10
By (13), Λ2 < 0, then inequality (16) yields that
E
∫ T
0
(
‖r(s)‖2 − γ2‖v(s)‖2)
ds
≤ E
∫ T
0
(1 + ε1)‖s(x(s)) (l(x(s)) − l(x(s))) ‖2
+∂V ′
∂ηf1(η(s))−
(
1
2g2(η(s))
′ ∂2V
∂η2f2(η(s))
+1
2g′1(η(s))
∂V
∂η
)′
Λ−12
(
1
2g2(η(s))
′ ∂2V
∂η2f2(η(s))
+1
2g′1(η(s))
∂V
∂η
)
+1
2f2(η(s))
′ ∂2V
∂η2f2(η(s))
ds− EV (η(T )), (17)
Now, the following inequality holds under the condition (13):
E
∫ T
0(‖r(s)‖2 − γ2‖v(s)‖2) ds ≤ 0.
Let T → ∞ in the above inequality, we have
E
∫ ∞
0‖r(t)‖2dt ≤ γ2E
∫ ∞
0‖v(t)‖2 dt, ∀v(t) ∈ L2
F (R+;Rnv). (18)
Thus, the external stability of the system (6) is shown.
Finally, we show the H− index to satisfy ‖Lf,r‖∞ ≥ δ. By Ito formula, for any T > 0,
t ∈ [0, T ], f(t) 6≡ 0, v(t) ≡ 0, and the initial value η(0) = 0, we have
E
∫ T
0
(
‖r(s)‖2 − δ2‖f(s)‖2)
ds
= E
∫ T
0
(
‖r(s)‖2 − δ2‖f(s)‖2 − LV (η(s))|v≡0
)
ds
+EV (η(T )) − V (η(0))
= E
∫ T
0
[
‖r(s)‖2 − δ2‖f(s)‖2 −∂V ′
∂η
(
f1(η(s))
+h(η(s))f(s))
−1
2f2(η(s))
′ ∂2V
∂η2f2(η(s))
]
ds
+EV (η(T ))
≥ E
∫ T
0
[
(1− ε2)‖s(x(s)) (l(x(s))− l(x(s))) ‖2
+(1− ε−12 )‖s(x(s))n(x(s))‖2‖f(s)‖2 − δ2‖f(s)‖2
−∂V ′
∂η
(
f1(η(s)) + h(η(s))f(s))
−1
2f2(η(s))
′ ∂2V
∂η2f2(η(t))
]
ds+ EV (η(T )). (19)
11
The above inequality makes use of the following relation (∀ε2 > 0)
‖s(x) (l(x) + n(x)f − l(x)) ‖2
≥ (1− ε2)‖s(x) (l(x)− l(x)) ‖2 + (1− ε−12 )‖s(x)n(x)‖2‖f‖2.
By Lemma 1,
E
∫ T
0
(
‖r(s)‖2 − δ2‖f(s)‖2)
ds
≥ E
∫ T
0
(1− ε2)‖s(x(s)) (l(x(s))− l(x(s))) ‖2
−c3‖η(s)‖2 −
∂V ′
∂ηf1(η(s)) −
1
2f2(η(s))
′ ∂2V
∂η2f2(η(s))
+
[
f(s)−1
2
(
I − δ2I)−1
h(η(s))′∂V
∂η
]′
·(
(1− ε−12 )‖s(x(s))n(x(s))‖2I − δ2I
)
·
[
f(s)−1
2
(
I − δ2I)−1
h(η(s))′∂V
∂η
]
−1
2
∂V ′
∂ηh(η(s))
·(
(1− ε−12 )‖s(x(s))n(x(s))‖2I − δ2I
)−1
·h(η(s))′∂V
∂η
ds. (20)
By the similar technique used in proving (18), from inequality (14), we obtain that for any
∀r(t) ∈ L2F (R
+,Rnr),
E
∫ ∞
0‖r(s)‖2 ds ≥ δ2E
∫ ∞
0‖f(s)‖2 ds.
The proof is completed.
Remark 1 In fact, when we consider H∞ index ‖Lv,r‖∞ and H− index ‖Lf,r‖−, the same
Lyapunov function is not necessary. That is, we can choose two different Lyapunov functions
V1(η) and V2(η) ∈ C2(Rnη ;R+) in calculating
‖r(s)‖2 − γ2‖v(s)‖2 + LV1(s)|f≡0 ≤ 0
and
‖r(s)‖2 − δ2‖f(s)‖2 + LV2(s)|v≡0 ≥ 0,
which would reduce the conservatism of Theorem 1.
Corollary 1 For given disturbance attenuation level γ > 0 and fault sensitivity level δ > 0,
if for any η ∈ Rnη , there exist positive constants c1, c2, c3, ε1, ε2 and two positive Lyapunov
12
functions V1 ∈ C2(Rnη ;R+) satisfying (12) and V2 ∈ C2(Rnη ;R+), which solve the following
coupled HJIs:
(1 + ε1)‖s(x) (l(x)− l(x)) ‖2 + c3‖η‖2 +
∂V ′1
∂η f1(η)
−(
12 g2(η)
′ ∂2V1
∂η2f2(η) +
12 g
′1(η)
∂V1
∂η
)′
·(
12 g2(η)
′ ∂2V1
∂η2g2(η) + (1 + ε−1
1 )‖s(x)m(x)‖2I
−γ2I)−1
(
12 g2(η)
′ ∂2V1
∂η2 f2(η) +12 g
′1(η)
∂V1
∂η
)
+12 f2(η)
′ ∂2V1
∂η2f2(η) ≤ 0,
12 g2(η)
′ ∂2V1
∂η2g2(η) + (1 + ε−1
1 )‖s(x)m(x)‖2I
−γ2I < 0,
(21)
and
(1− ε2)‖s(x) (l(x)− l(x)) ‖2 −∂V ′
2
∂η f1(η)
−12 f2(η)
′ ∂2V2
∂η2f2(η)−
12∂V ′
2
∂η h(η)(
(1− ε−12 )
·‖s(x)n(x)‖2I − δ2I)−1
h(η)′ ∂V2
∂η ≥ 0,
(1− ε−12 )‖s(x)n(x)‖2I − δ2I > 0
(22)
for some filter functions f , h, s with suitable dimensions. Then the desired H−/H∞ FDF is
given by (2).
If we consider a special case that m(x) ≡ 0 and g2(x) ≡ 0 in system (1), then, we can get
the following result.
Corollary 2 For given disturbance attenuation level γ > 0 and fault sensitivity level δ > 0, if
for any η ∈ Rnη , there exist constants c1, c2, c3, ε1, ε2 and two positive Lyapunov functions
V1 ∈ C2(Rnη ;R+) satisfying (12) and V2 ∈ C2(Rnη ;R+) solving the following two coupled HJIs:
(1 + ε1)‖s(x (l(x)− l(x)) ‖2 + c3‖η‖2 +
∂V ′1
∂ηf1(η)
+1
4γ−2
(
g′1(η)∂V1
∂η
)′ (
g′1(η)∂V1
∂η
)
+1
2f2(η)
′ ∂2V1
∂η2f2(η) ≤ 0 (23)
13
and
(1− ε2)‖s(x (l(x)− l(x)) ‖2 −∂V ′
2
∂η f1(η)
−12 f2(η)
′ ∂2V2
∂η2f2(η)−
12∂V ′
2
∂η h(η)(
(1− ε−12 )
·‖s(x)n(x)‖2I − δ2I)−1
h(η)′ ∂V2
∂η ≥ 0,
(1− ε−12 )‖s(x)m(x)‖2I − δ2I > 0
(24)
for some filter functions f , h, s with suitable dimensions. Then (2) is a desirable H−/H∞ FDF
for system (1) when m(x) ≡ 0 and g2(x) ≡ 0.
It is well known that for some practical models, not only the state, but also the external
disturbance [47] and fault signal maybe corrupted by noise. For example, the nonlinear stochas-
tic H∞ control of Ito type differential systems with all the state, control input and external
disturbance-dependent noise ( (x, u, v)-dependent noise for short) was studied in [44]. Therefore,
a mixed H−/H∞ FDF for nonlinear stochastic systems with (x, v, f)-dependent noise deserves
further study, which motivates us to consider the following system
dx(t) = (f1(x(t)) + g1(x(t))v(t) + h1(x(t))f(t)) dt
+(f2(x(t)) + g2(x(t))v(t) + h2(x(t))f(t)) dw(t),
y(t) = l(x(t)) +m(x(t))v(t) + n(x(t))f(t),
x(0) = x0 ∈ Rn.
(25)
As so, we can get the following augmented system:
dη(t) =(
f1(η(t)) + g1(η(t))v(t) + h1(η(t))f(t))
dt
+(
f2(η(t)) + g2(η(t))v(t) + h2(η(t))f(t))
dw(t),
r(t) = s(η(t)),
η(0) =
x0
0
∈ Rnη ,
(26)
where
h1(η(t)) =
h1(x)
h(x(t))n(x(t))
, h2(η(t)) =
h2(x)
0
.
14
Theorem 2 For any given disturbance attenuation level γ > 0 and fault sensitivity level δ > 0,
if for all η ∈ Rnη , the following two coupled HJIs
(1 + ε1)‖s(x) (l(x)− l(x)) ‖2 + c3‖η‖2 +
∂V ′1
∂η f1(η)
−(
12 g2(η)
′ ∂2V1
∂η2f2(η) +
12 g
′1(η)
∂V1
∂η
)′
·(
12 g2(η)
′ ∂2V1
∂η2g2(η) + (1 + ε−1
1 )‖s(x)m(x)‖2I
−γ2I)−1
·(
12 g2(η)
′ ∂2V1
∂η2 f2(η) +12 g
′1(η)
∂V1
∂η
)
+12 f2(η)
′ ∂2V1
∂η2f2(η) ≤ 0,
12 g2(η)
′ ∂2V1
∂η2g2(η) + (1 + ε−1
1 )‖s(x)m(x)‖2I
−γ2I < 0
(27)
and
(1− ε2)‖s(x) (l(x)− l(x)) ‖2 −∂V ′
2
∂η f1(η)
−12 f2(η)
′ ∂2V2
∂η2f2(η) −
(
12 h
′2(η)
∂2V2
∂η2f2(η)
+12 h
′1(η)
∂V2
∂η
)′ (
−12 h2(η)
′ ∂2V∂η2
h2(η) + (1− ε−12 )
· ‖s(x)n(x)‖2I − δ2I)−1
(
12 h
′2(η)
∂2V2
∂η2 f2(η)
+12 h
′1(η)
∂V2
∂η
)
≥ 0,
−12 h2(η)
′ ∂2V2
∂η2h2(η) + (1− ε−1
2 )‖s(x)n(x)‖2I
−δ2I > 0,
(28)
admit a set of solutions (V1 > 0, V2 > 0, f , h, s, c1 > 0, c2 > 0, c3 > 0, ε1 > 0, ε2 > 0), where
V1 ∈ C2(Rnη ;R+) satisfying (12), V2 ∈ C2(Rnη ;R+) and c1, c2, c3, ε1, ε2 are positive constants.
Then the FDF for system (25) is given by (2).
Proof : Repeating the same procedure as in Theorem 1 and Corollary 1. The proof is completed.
Below, we consider the following linear stochastic system
dx(t) = (A0x(t) +B0v(t) + C0f(t)) dt
+(A1x(t) +B1v(t) + C1f(t)) dw(t),
y(t) = A2x(t) +B2v(t) + C2f(t)
(29)
15
as well as the following linear FDF
dx(t) =(
Ax(t) + B(y(t)−A2x(t)))
dt,
r(t) = S(y(t)−A2x(t)).
(30)
Set η(t) =[
x(t)′ x(t)′]′
, then we get the following augmented system
dη(t) = (A0η(t) + B0v(t) + C0f(t))dt
+(A1η(t) + B1v(t) + C1f(t))dw(t),
r(t) = A2η(t) + B2v(t) + C2f(t),
(31)
where
A0 =
A0 0
BA2 A− BA2
, B0 =
B0
BB2
, C0 =
C0
BC2
,
A1 =
A1 0
0 0
, B1 =
B1
0
, C1 =
C1
0
,
A2 =[
SA2 −SA2
]
, B2 = SB2, C2 = SC2.
Using Theorem 2, it is easy to show the following result.
Theorem 3 For any given disturbance attenuation level γ > 0 and fault sensitivity level δ > 0,
if there exist two positive definite matrices P1 and P2 solving the following ARIs:
R1 := (1 + ε1) + P1A0 + A′
0P1 − (B′
1P1A1 + B′
0P1)′
·(B′
1P1B1 + (1 + ε−1
1)‖SB2‖
2I − γ2I)−1
·(B′
1P1A1 + B′
0P1) + A′
1P1A1 + cI ≤ 0,
(B′
1P1B1 + (1 + ε−1
1)‖SB2‖
2I − γ2I) < 0
(32)
and
R2 := (1− ε2) − P2A0 − A′
0P2 − (C′
1P2A1 + C′
0P2)
′
·(−C′
1P2C1 + (1− ε−1
2)‖SC2‖
2I − δ2I)−1
·(C′
1P2A1 + C′
0P2)− A′
1P2A1 ≥ 0,
(−C′
1P2C1 + (1− ε−1
2)‖SC2‖
2I − δ2I) > 0,
(33)
where
=
A′2S
′SA2 −A′2S
′SA2
−A′2S
′SA2 A′2S
′SA2
and c, ε1 and ε2 are positive constants. Then the FDF for system (29) is given by (30).
16
Proof : For system (31), we choose two Lyapunov functions as V1(η) = η′P1η and V2(η) = η′P2η.
By Theorem 2, we obtain
(1 + ε1)‖S(A2x−A2x)‖2 + c‖η‖2 − (B′
1P1A1η + B′
0P1η)
′
(B′
1P1B1 + (1 + ε−1
1)‖SB2‖
2I − γ2I)−1
(
B′
1P1A1η
+B′
0P1η)
+ 2η′P1A0η + η′A′
1P1A1η ≤ 0,
B′
1P1B1 + (1 + ε−1
1)‖SB2‖
2I − γ2I < 0
(34)
and
(1− ε2)‖S(A2x−A2x)‖2 − (C′
1P2A1η + C′
0P2η)
′
(−C′
1P2C1 + (1− ε−1
2)‖SC2‖
2I − δ2I)−1
(
C′
1P2A1η
+C′
0P2η
)
− 2η′P2A0η − η′A′
1P2A1η ≥ 0,
−C′
1P2C1 + (1− ε−1
2)‖SC2‖
2I − δ2I > 0.
(35)
(34) and (35) are equivalent to
η′R1η ≤ 0,
B′1P1B1 + (1 + ε−1
1 )‖SB2‖2I − γ2I < 0
(36)
and
η′R2η ≥ 0,
−C ′1P2C1 + (1− ε−1
2 )‖SC2‖2I − δ2I > 0,
(37)
respectively, where R1 and R2 are defined respectively in (32) and (33). (36) and (37) are
equivalent to (32) and (33), respectively. The proof is thus completed.
Theorem 3 is regarding H−/H∞ FDF for linear stochastic systems based on ARIs. (32) and
(33) can be converted into LMIs by the technique in the next section, hence, H−/H∞ FDF of
the system (29) can be easily designed by Matlab LMI Toolbox.
4 LMI-Based Approach for Quasi-Linear Systems
Generally speaking, for general nonlinear stochastic systems, it is not easy to design the H−/H∞
filter due to the difficulty in solving HJIs. However, for a class of special nonlinear stochastic
systems called “quasi-linear stochastic systems”, the filtering design problem can be converted
into solving LMIs as done in [43].
17
We consider the following quasi-linear stochastic system governed by Ito differential equation
dx(t) = (A0x(t) + F0(x(t)) +B0v(t) + C0f(t)) dt
+(A1x(t) + F1(x(t)) +B1v(t) + C1f(t)) dw(t),
y(t) = A2x(t) +B2v(t) + C2f(t),
(38)
where Fi(0) = 0, i = 1, 2. As a matter of fact, in (25), if one takes g1(x) = B0, h1(x) = C0,
g2(x) = B1, h2(x) = C1, l(x) = A2x, m(x) = B2, n(x) = C2, and regards A0x(t) + F0(x(t))
and A1x(t)+F1(x(t)) as the Taylor’s series expansions of f1(x) and f2(x), respectively, then the
state equation of (25) comes down to the first equation of (38).
For the quasi-linear stochastic system (38), we consider linear FDF (30). Set η(t) =[
x(t)′ x(t)′]′
, then we get the following augmented system:
dη(t) = (A0η(t) + F0(η(t)) + B0v(t) + C0f(t))dt
+(A1η(t) + B1v(t) + F1(η(t)) + C1f(t))dw(t),
r(t) = A2η(t) + B2v(t) + C2f(t),
(39)
where
F0(η(t)) =
F0(x(t))
0
, F1(η(t)) =
F1(x(t))
0
.
For any given disturbance attenuation level γ > 0 and fault sensitivity level δ > 0, we want
to seek the filtering parameters A, B and S, such that ‖Lv,r‖∞ < γ and ‖Lf,r‖− > δ.
Assumption 1 Suppose there exists a scalar α > 0, such that
‖Fi(η)‖ ≤ α‖η‖, i = 1, 2. (40)
Lemma 3 Given scalars γ > 0, δ > 0, if the following three matrix inequalities
0 < P ≤ βI, (41)
A′2A2 + A′
0P + PA0 + P + 4α2βI PB0 A′1P A′
1P 0
∗ −γ2I + B′2B2 0 B′
1P B′1P
∗ ∗ −P 0 0
∗ ∗ ∗ −P 0
∗ ∗ ∗ ∗ −P
< 0 (42)
18
and
A′0P + PA0 + P + 4α2βI − A′
2A2 PC0 A′1P A′
1P 0
∗ δ2I − C ′2C2 0 C ′
1P C ′1P
∗ ∗ −P 0 0
∗ ∗ ∗ −P 0
∗ ∗ ∗ ∗ −P
< 0 (43)
admit a pair of positive solutions (P > 0, β > 0), then the augmented system (39) is internally
stable. Moreover, ‖Lv,r‖∞ < γ and ‖Lf,r‖− > δ.
Proof : Firstly, we prove that system (39) is internally stable, i.e., the following system
dη(t) = (A0η(t) + F0(η(t))) dt + (A1η(t) + F1(η(t))) dw(t) (44)
is exponentially stable in mean square sense. For convenience, in this section, we denote L1 as
the infinitesimal generator of system (39). To prove the internal stability of (39), we take the
Lyapunov function as V (η) = η′Pη, where P > 0 is the solution of (41)-(43). By Ito formula,
for system (44),
L1V (η)|v≡0,f≡0
= η′(A′0P + PA0 + A′
1PA1)η + 2η′PF0 + F ′1PF1 + 2η′A′
1PF1. (45)
By (40), it follows that
2η′PF0 + F1PF1 + 2η′A′1PF1 ≤ η′Pη + F ′
0PF0
+2F1PF1 + η′A′1PA1η ≤ η′(P + A′
1PA1 + 3α2βI)η.
Substituting the above inequality into (45), it yields that
L1V (η)|v≡0,f≡0 ≤ η′(A′0P + PA0 + 2A′
1PA1 + P + 3α2βI)η. (46)
Obviously, under the condition (42), we have
−θ := λmax(A′0P + PA0 + 2A′
1PA1 + P + 3α2βI) < 0.
So
L1V (η)|v≡0,f≡0 ≤ −θ‖η‖2.
According to [26], system (39) is exponentially stable in mean square sense.
19
Secondly, we prove ‖Lv,r‖∞ < γ with η(0) = 0 and f(t) ≡ 0 in (39). Note that when f(t) ≡ 0
and η(0) = 0, (39) becomes
dη(t) = (A0η(t) + F0(η(t)) + B0v(t)) dt
+(A1η(t) + B1v(t) + F1(η(t))) dw(t),
η(0) = 0,
r(t) = A2η(t) + B2v(t).
(47)
For system (47), using Ito formula to V (η) = η′Pη, we have
E
∫ T
0(‖r(t)‖2 − γ2‖v(t)‖2) dt
= E
∫ T
0[L1V (η(t))|f≡0 + ‖r(t)‖2 − γ2‖v(t)‖2]dt
−EV (η(T ))
≤ E
∫ T
0
[
η′(t)(
A′2A2 + A′
0P + PA0 + 2A′1PA1 + P
+3α2βI)
η(t) + 2η(t)′PB0v(t) + v(t)′B′1PB1v(t)
+2η(t)′A′1PB1v(t) + 2F ′
1PB1v(t) + v(t)′
·(−γ2I + B′2B2)v(t)
]
dt− EV (η(T ))
≤ E
∫ T
0
[
η′(
A′2A2 + A′
0P + PA0 + 2A′1PA1 + P
+4α2βI)
η + 2η(t)′PB0v(t) + 2v(t)′B′1PB1v(t)
+2η(t)′A′1PB1v(t) + v(t)′(−γ2I + B′
2B2)v(t)]
dt
−EV (η(T ))
= E
∫ T
0
[
η(t)′ v(t)′]
∏
η(t)
v(t)
dt−EV (η(T ))
≤ E
∫ T
0
[
η(t)′ v(t)′]
∏
η(t)
v(t)
dt,
where
∏
:=
Π11 PB0 + A′1PB1
∗ −γ2I + B′2B2 + 2B′
1PB1
with
Π11 = A′2A2 + A′
0P + PA0 + 2A′1PA1 + P + 4α2βI.
20
By Schur’s complement and inequality (42), we get∏
< 0. Thus, for any T > 0,
E
∫ T
0‖r(t)‖2 dt
≤ γ2E
∫ T
0‖v(t)‖2 dt+ λmax(
∏
)E
∫ T
0‖Z(t)‖2 dt
≤ (γ2 + λmax(∏
))E
∫ T
0‖v(t)‖2 dt,
where Z(t) =[
η(t)′ v(t)′]′
. Let T → ∞ in above, then for any nonzero v ∈ L2F (R
+,Rnv ),
‖Lv,r‖∞ ≤ (γ2 + λmin(∏
))1/2 < γ, so the external stability of (39) is proved.
Finally, we show ‖Lf,r‖− > δ when v ≡ 0 and η(0) = 0 in (39). Consider system
dη(t) = (A0η(t) + F0(η(t)) + C0f(t))dt
+(A1η(t) + F1(η(t)) + C1f(t))dw,
η(0) = 0,
r(t) = A2η(t) + C2f(t).
(48)
According to Lemma 2, for system (48), we are in a position to show that
E
∫ T
0(‖r(t)‖2 − δ2‖f(t)‖2) dt
= E
∫ T
0[‖r(t)‖2 − δ2‖f(t)‖2 − L1V (η(t))|v≡0] dt
+EV (η(t)) − EV (η(0))
≥ E
∫ T
0
[
−η′(t)(
−A′2A2 + A′
0P + PA0 + 2A′1PA1
+P + 4α2βI)
η(t)− 2η(t)′PC0f(t)
−2f(t)′C ′1PC1f(t)− 2η(t)′A′
1PC1f(t)
+f(t)′(−δ2I + C ′2C2)f(t)
]
dt+ EV (η(T ))
= E
∫ T
0
[
η(t)′ f(t)′]
⊔
η(t)
f(t)
dt+ EV (η(T )), (49)
where, by (43),
⊔
:=
Φ11 −PC0 − A′1PC1
∗ −δ2I + C ′2C2 − 2C ′
1PC1
> 0
with
Φ11 = A′2A2 − A′
0P − PA0 − 2A′1PA1 − P − 4α2βI.
21
So
E
∫
T
0
‖r(t)‖2 dt
≥ δ2E
∫
T
0
‖f(t)‖2 dt+ λmin(⊔
)E
∫
T
0
‖M(t)‖2 dt, (50)
where M(t) =[
η(t)′ f(t)′]′
. For f(t) 6≡ 0, f ∈ L2F (R
+,Rnf ), let T → ∞ in (50), we have
E
∫ ∞
0‖r(t)‖2 dt
≥ δ2E
∫ ∞
0‖f(t)‖2 dt+ λmin(
⊔
)E
∫ ∞
0‖M(t)‖2 dt
≥ δ2E
∫ ∞
0‖f(t)‖2 dt+ λmin(
⊔
)E
∫ ∞
0‖f(t)‖2 dt
which yields that ‖Lf,r‖− ≥ (δ2 +λmin(⊔
))1/2 > δ due to λmin(⊔
) > 0. The proof is completed.
Based on Lemma 4, an H−/H∞ FDF can be designed in terms of LMIs for system (38). For
convenience, we set P a real symmetric diagonal matrix such as P =
P1 0
0 P2
> 0, then
A0 := PA0 =
P1A0 0
BA2 A− BA2
,
B0 := PB0 =
P1B0
BB2
,
C0 := PC0 =
P1C0
BC2
, A1 := PA1 =
P1A1 0
0 0
,
B1 := PB1 =
P1B1
0
, C1 := PC1 =
P1C1
0
,
A2 := A′2A2 =
A′2SA2 −A′
2SA2
−A′2SA2 A′
2SA2
,
B2 := B′2B2 = B′
2SB2, C2 := C ′2C2 = C ′
2SC2.
where[
A B]
= P2
[
A B]
, S = S′S.
Theorem 4 If there exists the solution (P1 > 0, P2 > 0, β > 0, A, B, S) solving the following
LMIs:
0 <
P1 0
P2
≤ βI, (51)
22
~11 B0 A′1 A′
1 0
∗ −γ2I + B2 0 B′1 B′
1
∗ ∗ −P 0 0
∗ ∗ ∗ −P 0
∗ ∗ ∗ ∗ −P
< 0 (52)
and
ℓ11 C0 A′1 A′
1 0
∗ δ2I − C2 0 C′1 C′
1
∗ ∗ −P 0 0
∗ ∗ ∗ −P 0
∗ ∗ ∗ ∗ −P
< 0, (53)
with
~11 = A2 +A′0 +A0 + P + 4α2βI
and
ℓ11 = −A2 +A′0 +A0 + P + 4α2βI,
then (30) is a mixed H−/H∞ FDF of the system (38). In this case, the admissible filter matrices
can be given by
S = S1
2 , A = P−12 A, B = P−1
2 B.
5 Numerical Example
In this section, one numerical example is provided to illustrate the effectiveness of our main
results.
23
Example 1 Consider the nonlinear stochastic system (38) with the following parameters:
A0 =
−6.01 −2.94
−2.94 −6.17
, A1 =
0.91 −0.44
−1.31 −0.39
,
A2 =
0.43 0.15
−0.09 0.07
, B0 =
1.37
−0.41
,
B2 =
0.35
−0.6
, C0 =
1.21
−0.11
, C2 =
−3.67
0.51
,
F1(x(t)) = B1 = C1 =
0
0
,
F0(x(t)) = 0.5
sin(x1(t))
sin(x2(t))
.
In addition, we choose the H∞ performance level γ = 1 and H− performance level δ = 0.5. For
the above parameters, by using Matlab LMI Toolbox, the solutions of LMIs in Theorem 4 for
P1 > 0, P2 > 0, β > 0 , A, B, S
are obtained as
P1 =
0.5248 −0.0397
−0.0397 0.3805
, P2 =
0.4799 0
0 0.4799
,
A =
−2.0747 −0.8242
0.8209 −2.0835
, B =
0.0032 −0.0529
−0.0054 −0.0441
,
S =
0.4052 0.2562
0.2562 0.2745
, β = 6.
Thus, the desired filter matrices of the H−/H∞ FDF are as follows:
A =
−4.3228 −1.7172
1.7103 −4.3411
, B =
0.0067 −0.1102
−0.0113 −0.0919
,
S =
0.0067 −0.1102
−0.0113 −0.0919
.
We use Matlab to simulate the state trajectories x(t) and the filter trajectories x(t) of system
(39) under v(t) ≡ 0, f(t) ≡ 0 and x0 =
1
−1
; see Figures 1 and 2. From Figures 1-2, it is
easy to see that system (39) is internally stable.
24
0 5 10 15 20 25 30 35 40 45 500
0.5
1
1.5
t(sec)
x 1(t)
0 5 10 15 20 25 30 35 40 45 50−1
−0.8
−0.6
−0.4
−0.2
0
t(sec)
x 2(t)
1 2 3 4 5 6
0.25
0.3
0.35
0.4
0.45
0 2 4 6 8 10 12
−0.35
−0.3
−0.25
−0.2
−0.15
Figure 1: State trajectories x(t) of the system (38) with f(t) ≡ 0, v(t) ≡ 0.
0 5 10 15 20 25 30 35 40 45 500
0.005
0.01
0.015
t(sec)
x1(t)
0 5 10 15 20 25 30 35 40 45 500
0.005
0.01
0.015
0.02
t(sec)
x2(t)
0.5 1 1.5
0.00980.01
0.01020.01040.0106
0 0.5 1 1.5 2 2.5 30.01520.01540.01560.01580.016
0.0162
Figure 2: State trajectories x(t) of the system (30).
To show the effectiveness of the designed H−/H∞ filter, we assume v(t) = 0.9t ∈ L2F (R
+,R)
and f(t) = f0.4(t) =
0.4, t ∈ [10, 20]
0, else
∈ L2F (R
+,R). The residual evaluation function is
Jr(t) =
(
E
1
t
∫ t
0r′(s)r(s)ds
)
1
2
.
After 100 times Monte Carlo simulations without fault influence, Jth = 0.4 with evaluation
window T = 5. The residual evaluation function Jr(t) and fault signal f(t) are depicted in
Figure 3. However, if the fault signal is weaker, the FDF may fail to alarm, which can be seen
from Figure 4 when we set f(t) = f0.1(t) =
0.1, t ∈ [10, 20]
0, else
∈ L2F (R
+,R). In practice, one
25
0 20 40 60 80 1000
0.1
0.2
0.3
0.4
t(sec)
J r(t)
and
J th=
0.03
3
0 20 40 60 80 100−1
−0.5
0
0.5
1
t(sec)
f(t)
5 100.02
0.03
0.04J
r(t)
Jth
=0.033
Figure 3: Jr(t), Jth and f(t) = f0.4(t).
0 20 40 60 80 1000
0.02
0.04
0.06
0.08
t(sec)
J r(t)
and
J th=
0.03
3
0 20 40 60 80 100−1
−0.5
0
0.5
1
t(sec)
f(t)
19 20 21
0.031
0.032
0.033
t(sec)J
r(t)
Jth
=0.033
Figure 4: Jr(t), Jth and f(t) = f0.1(t).
needs to select a suitable combination of γ and δ according to practical engineering requirements.
In this example, if we set γ = 0.78, δ = 1.41, f(t) = f0.1(t), the H−/H∞ FDF parameters can
be computed as
A =
−4.8359 −3.3771
3.3761 −4.509
, B =
−0.0177 −0.1615
−0.0124 −0.1198
,
S =
0.6030 0.2787
0.2787 0.4084
as well as Jth = 0.0205. From Figure 5, we can see that the fault sensitivity is improved.
26
0 20 40 60 80 1000
0.02
0.04
0.06
t(sec)
J r(t)
and
J th=
0.02
05
0 20 40 60 80 100−1
−0.5
0
0.5
1
t(sec)
f(t)
5 10 150.015
0.02
t(sec)
Jth
=0.0205
Jr(t)
Figure 5: Jr(t), Jth and f(t) = f0.1(t) for γ = 0.78 and δ = 1.41.
6 Conclusion
In this paper, theH−/H∞ FDF design for Ito-type affine nonlinear stochastic systems and quasi-
linear systems have been discussed, and sufficient conditions for the existence of the desired FDF
have been given via HJIs and LMIs, respectively. The key to the FDF design of affine nonlinear
stochastic systems is how to solve the coupled HJIs, this is a very challenging problem, and
some potential effective approaches to overcome this difficulty may refer to [8, 25] for global
linearization method, [1] for neural network method, and [9] for fuzzy interpolation method.
In addition, from our simulation example, in order to select a more suitable combination of
(γ, δ), a co-design algorithm for H− index and H∞ index is necessary. Wherever possible, the
smaller γ > 0 and the larger δ > 0, the better the performance of FDF. However, from the
second inequalities of HJIs (13) and (14), γ > 0 cannot be arbitrarily small and δ > 0 cannot
be arbitrarily large. In order to obtain the optimal selection (γ∗, δ∗), we may turn to Pareto
optimization method [22, 23] together with convex optimization [3]. Pareto optimization is a
co-operative game, its application to H−/H∞ FDF design will be our future work.
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