Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 791296, 7 pages
http://dx.doi.org/10.1155/2013/791296
Research Article
Resilient 𝐿 2-𝐿 ∞ Filtering of Uncertain Markovian Jumping
Systems within the Finite-Time Interval
Shuping He
College of Electrical Engineering and Automation, Anhui University, Hefei 230601, China
Correspondence should be addressed to Shuping He; henyhappy@yahoo.com.cn
Received 16 November 2012; Revised 10 March 2013; Accepted 17 March 2013
Academic Editor: Gani Stamov
Copyright © 2013 Shuping He. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper studies the resilient 𝐿 2 -𝐿 ∞ filtering problem for a class of uncertain Markovian jumping systems within the finite-time
interval. The objective is to design such a resilient filter that the finite-time 𝐿 2 -𝐿 ∞ gain from the unknown input to an estimation
error is minimized or guaranteed to be less than or equal to a prescribed value. Based on the selected Lyapunov-Krasovskii
functional, sufficient conditions are obtained for the existence of the desired resilient 𝐿 2 -𝐿 ∞ filter which also guarantees the
stochastic finite-time boundedness of the filtering error dynamic systems. In terms of linear matrix inequalities (LMIs) techniques,
the sufficient condition on the existence of finite-time resilient 𝐿 2 -𝐿 ∞ filter is presented and proved. The filter matrices can be solved
directly by using the existing LMIs optimization techniques. A numerical example is given at last to illustrate the effectiveness of
the proposed approach.
1. Introduction
More recently, the finite-time stability and control problems
have received great attention in the literature; see [1–6].
Compared with the Lyapunov stable dynamical systems, a
finite-time stable dynamical system does not require the
steady-state behavior of control dynamics over an infinitetime interval and the asymptotic pattern of system trajectories. The main attention may be related to the transient
characteristics of the dynamical systems over a fixed finitetime interval, for instance, keeping the acceptable values in
a prescribed bound in the presence of saturations. However,
more details are related to the stability and control problems
of various dynamic systems, and very few reports in the
literature consider the filtering problems.
Since the Kalman filtering theory [7] has been introduced
in the early 1960s, the filtering problem has been extensively
investigated. In the filtering scheme, its objective is to estimate the unavailable state variables (or a linear combination
of the states) of a given system. During the past decades,
many filtering schemes have been developed, such as Kalman
filtering [8], 𝐻∞ filtering [9], reduced-order 𝐻∞ filtering
[10], and 𝐿 2 -𝐿 ∞ filtering [11]. Then, extension of this effort
to the problem of resilient Kalman filtering with respect to
estimator gain perturbations was considered in [12]. And the
resilient 𝐻∞ filtering [13] was also raised. Among the filtering
schemes, the resilient 𝐿 2 -𝐿 ∞ filtering was not considered. In
practical engineering applications, the peak values of filtering
error should always be considered. Compared with the 𝐻∞
filtering scheme, the external disturbances are both assumed
to be energy bounded; but 𝐿 2 -𝐿 ∞ filtering setting requires
the mapping from the external disturbances to the filtering
error is minimized or no larger than some prescribed level in
terms of the 𝐿 2 -𝐿 ∞ performance norm.
In this paper, we have studied the resilient finite-time
𝐿 2 -𝐿 ∞ filtering problem for uncertain Markouvian Jumping Systems (MJSs). Firstly, the augmented filtering error
dynamic system is constructed based on the state estimated
filter with resilient filtering parameters. Secondly, a sufficient
condition is established on the existence of the robust finitetime filter such that the filtering error dynamic MJSs are
finite-time bounded and satisfy a prescribed level of 𝐿 2 𝐿 ∞ disturbance attenuation with the finite-time interval.
And the design criterion is presented by means of LMIs
techniques. Subsequently, the robust finite-time 𝐿 2 -𝐿 ∞ filter
matrices can be solved directly by using the existing LMIs
optimization algorithms. In order to illustrate the proposed
result, a numerical example is given at last.
2
Abstract and Applied Analysis
Let us introduce some notations. The symbols R𝑛 and
stand for an 𝑛-dimensional Euclidean space and the set
R
of all 𝑛 × 𝑚 real matrices, respectively, 𝐴𝑇 and 𝐴−1 denote the
matrix transpose and matrix inverse, diag{𝐴 𝐵} represents
the block-diagonal matrix of 𝐴 and 𝐵, 𝜎max (𝐴) and 𝜎min (𝐴),
respectively, denote the maximal and minimal eigenvalues
of a real matrix 𝐴, ‖ ∗ ‖ denotes the Euclidean norm of
vectors, E{∗} denotes the mathematics statistical expectation
of the stochastic process or vector, 𝑃 > 0 stands for a
positive-definite matrix, 𝐼 is the unit matrix with appropriate
dimensions, 0 is the zero matrix with appropriate dimensions,
and ∗ means the symmetric terms in a symmetric matrix.
𝑛×𝑚
2. Problem Formulation
Given a probability space (Ω, 𝐹, 𝑃𝑟 ) where Ω is the sample
space, 𝐹 is the algebra of events, and 𝑃𝑟 is the probability
measure defined on 𝐹. Let us consider a class of linear
uncertain MJSs defined in the probability space (Ω, 𝐹, 𝑃𝑟 ) and
described by the following differential equations:
𝑥̇ (𝑡) = [𝐴 (𝑟𝑡 ) + Δ𝐴 (𝑟𝑡 )] 𝑥 (𝑡) + 𝐵 (𝑟𝑡 ) 𝑤 (𝑡) ,
(1)
𝑧 (𝑡) = 𝐸 (𝑟𝑡 ) 𝑥 (𝑡) ,
𝑟𝑡 = 𝑟0 ,
𝑡 = 0,
where 𝑥(𝑡) ∈ R𝑛 is the state, 𝑦(𝑡) ∈ R𝑙 is the measured
output, 𝑤(𝑡) ∈ 𝐿𝑚2 [0 + ∞) is the unknown input, 𝑧(𝑡) ∈ R𝑞
is the controlled output, and 𝑥0 and 𝑟0 are, respectively, the
initial states and mode. 𝐴(𝑟𝑡 ), 𝐵(𝑟𝑡 ), 𝐶(𝑟𝑡 ), 𝐷(𝑟𝑡 ), and 𝐸(𝑟𝑡 ) are
known mode-dependent constant matrices with appropriate
dimensions. The jump parameter 𝑟𝑡 represents a continuoustime discrete-state Markov stochastic process taking values
on a finite set M = {1, 2, . . . , 𝑁} with transition rate matrix
Π = {𝜋𝑖𝑗 }, 𝑖, 𝑗 ∈ M, and has the following transition
probability from mode 𝑖 at time 𝑡 to mode 𝑗 at time 𝑡 + Δ𝑡
as
𝜋𝑖𝑗 Δ𝑡 + 𝑜 (Δ𝑡) ,
𝑃𝑖𝑗 = 𝑃𝑟 {𝑟𝑡+Δ𝑡 = 𝑗 | 𝑟𝑡 = 𝑖} = {
1 + 𝜋𝑖𝑖 Δ𝑡 + 𝑜 (Δ𝑡) ,
𝑖 ≠ 𝑗,
𝑖 = 𝑗,
(2)
where Δ𝑡 > 0 and limΔ𝑡 → 0 (𝑜(Δ𝑡)/Δ𝑡) → 0.
In this relation, 𝜋𝑖𝑗 ≥ 0 is the transition probability rates
from mode 𝑖 at time 𝑡 to mode 𝑗 (𝑖 ≠ 𝑗) at time 𝑡 + Δ𝑡, and
𝑁
∑ 𝜋𝑖𝑗 = −𝜋𝑖𝑖
for 𝑖, 𝑗 ∈ M, 𝑖 ≠ 𝑗.
(3)
𝑗=1,𝑗 ≠ 𝑖
For presentation convenience, we denote 𝐴(𝑟𝑡 ), 𝐵(𝑟𝑡 ),
𝐶(𝑟𝑡 ), 𝐷(𝑟𝑡 ), 𝐸(𝑟𝑡 ), Δ𝐴(𝑟𝑡 ), and Δ𝐶(𝑟𝑡 ) as 𝐴 𝑖 , 𝐵𝑖 , 𝐶𝑖 , 𝐷𝑖 , 𝐸𝑖 ,
Δ𝐴 𝑖 , and Δ𝐶𝑖 , respectively.
And the matrices with the symbol Δ(∗) are considered as
the uncertain matrices satisfying the following conditions:
[
𝑀
Δ𝐴 𝑖
] = [ 1𝑖 ] Γ𝑖 (𝑡) 𝑁1𝑖 ,
Δ𝐶𝑖
𝑀2𝑖
Remark 1. It is always impossible to obtain the exact mathematical model of practical dynamics due to the complexity
process, the environmental noises, and the difficulties of
measuring various and uncertain parameters, and so forth;
thus, the model of practical dynamics to be controlled
almost contains some types of uncertainties. In general, the
uncertainties Δ(∗) in (1) satisfying the restraining conditions
(4) and Γ𝑖𝑇 (𝑡)Γ𝑖 (𝑡) ≤ 𝐼 are said to be admissible. The unknown
mode-dependent matrix Γ𝑖 (𝑡) can also be allowed to be state
dependent; that is, Γ𝑖 (𝑡) = Γ𝑖 (𝑡, 𝑥(𝑡)), as long as ‖Γ𝑖 (𝑡, 𝑥(𝑡))‖ ≤
1 is satisfied.
We now consider the following resilient filter:
̂̇ (𝑡) = (𝐴 𝑓𝑖 + Δ𝐴 𝑓𝑖 ) 𝑥
̂ (𝑡) + 𝐵𝑓𝑖 𝑦 (𝑡) ,
𝑥
̂ (𝑡) ,
𝑧̂ (𝑡) = (𝐶𝑓𝑖 + Δ𝐶𝑓𝑖 ) 𝑥
𝑦 (𝑡) = [𝐶 (𝑟𝑡 ) + Δ𝐶 (𝑟𝑡 )] 𝑥 (𝑡) + 𝐷 (𝑟𝑡 ) 𝑤 (𝑡) ,
𝑥 (𝑡) = 𝑥0 ,
where 𝑀1𝑖 , 𝑀2𝑖 , and 𝑁1𝑖 are known mode-dependent matrices with appropriate dimensions, and Γ𝑖 (𝑡) is the time-varying
unknown matrix function with Lebesgue norm measurable
elements satisfying Γ𝑖𝑇 (𝑡)Γ𝑖 (𝑡) ≤ 𝐼.
(4)
̂ (𝑡) = 𝑥
̂0,
𝑥
𝑟𝑡 = 𝑟0 ,
(5)
𝑡 = 0,
̂ (𝑡) ∈ R𝑛 is the filter state, 𝑧̂(𝑡) ∈ R𝑞 is the filer output,
where 𝑥
̂ 0 is the initial estimation states, and the mode-dependent
𝑥
matrices 𝐴 𝑓𝑖 , 𝐵𝑓𝑖 , and 𝐶𝑓𝑖 are unknown filter parameters to be
designed for each value 𝑖 ∈ M. Δ𝐴 𝑓𝑖 and Δ𝐶𝑓𝑖 are uncertain
filter parameter matrices satisfying the following conditions:
[
Δ𝐴 𝑓𝑖
𝑀
] = [ 3𝑖 ] Γ𝑓𝑖 (𝑡) 𝑁2𝑖 ,
Δ𝐶𝑓𝑖
𝑀4𝑖
(6)
where 𝑀3𝑖 , 𝑀4𝑖 , 𝑁2𝑖 , and Γ𝑓𝑖 (𝑡) are defined similarly as (4).
The objective of this paper is to design the resilient 𝐿 2 -𝐿 ∞
filter of uncertain MJSs in (1) and obtain an estimate ̂z(𝑡) of
the signal 𝑧(𝑡) such that the defined guaranteed performance
criteria are minimized in an 𝐿 2 -𝐿 ∞ estimation error sense.
̂ (𝑡) and 𝑟(𝑡) = 𝑧(𝑡) − 𝑧̂(𝑡), such that the
Define 𝑒(𝑡) = 𝑥(𝑡) − 𝑥
filtering error dynamic MJSs are given by
𝑒̇ (𝑡) = (𝐴 𝑓𝑖 + Δ𝐴 𝑓𝑖 ) 𝑒 (𝑡)
+ [𝐴 𝑖 + Δ𝐴 𝑖 − (𝐴 𝑓𝑖 + Δ𝐴 𝑓𝑖 ) − 𝐵𝑓𝑖 (𝐶𝑖 + Δ𝐶𝑖 )] 𝑥 (𝑡)
+ (𝐵𝑖 − 𝐵𝑓𝑖 𝐷𝑖 ) 𝑤 (𝑡) ,
𝑟 (𝑡) = (𝐶𝑓𝑖 + Δ𝐶𝑓𝑖 ) 𝑒 (𝑡) + [𝐸𝑖 − (𝐶𝑓𝑖 + Δ𝐶𝑓𝑖 )] 𝑥 (𝑡) .
(7)
Let 𝜉(𝑡) = [ 𝑥(𝑡)
𝑒(𝑡) ], and we have
𝜉̇ (𝑡) = 𝐴𝑖 𝜉 (𝑡) + 𝐵𝑖 𝑤 (𝑡) ,
𝑟 (𝑡) = 𝐶𝑖 𝜉 (𝑡) ,
(8)
Abstract and Applied Analysis
3
Lemma 5 (see [14]). Let 𝑇, 𝑀, and 𝑁 be real matrices with
appropriate dimensions. Then, for all time-varying unknown
matrix function 𝐹(𝑡) satisfying 𝐹𝑇 (𝑡)𝐹(𝑡) ≤ 𝐼, the following
relation holds:
where
0
𝐴 𝑖 + Δ𝐴 𝑖
𝐴𝑖 = [𝐴 + Δ𝐴 − (𝐴 + Δ𝐴 ) − 𝐵 (𝐶 + Δ𝐶 ) 𝐴 + Δ𝐴 ] ,
𝑖
𝑖
𝑓𝑖
𝑓𝑖
𝑓𝑖
𝑖
𝑖
𝑓𝑖
𝑓𝑖
𝐵𝑖
𝐵𝑖 = [
],
𝐵𝑖 − 𝐵𝑓𝑖 𝐷𝑖
𝐶𝑖 = [𝐸𝑖 − (𝐶𝑓𝑖 + Δ𝐶𝑓𝑖 )
𝑇 + 𝑀𝐹 (𝑡) 𝑁 + 𝑁𝑇 𝐹𝑇 (𝑡) 𝑀𝑇 > 0,
if and only if there exists a positive scalar 𝛼 > 0, such that
𝐶𝑓𝑖 + Δ𝐶𝑓𝑖 ] .
(9)
The external disturbance 𝑤(𝑡) is varying and satisfies the
constraint condition with respect to the finite-time interval
[0 𝑇] as follows:
𝑇
∫ 𝑤𝑇 (𝑡) 𝑤 (𝑡) 𝑑𝑡 ≤ 𝑊,
(10)
0
Definition 2. Given a time-constant 𝑇 > 0, the filtering
error dynamic MJSs (8) (setting 𝑤(𝑡) = 0) are said to
be stochastically finite-time stable (FTS) with respect to
̃ 𝑖 ), if
(𝑐1 𝑐2 𝑇 𝑅
̃ 𝑖 𝜉 (0)} ≤ 𝑐1 󳨐⇒ E {𝜉 (𝑡) 𝑅
̃ 𝑖 𝜉 (𝑡)} < 𝑐2 ,
E {𝜉 (0) 𝑅
𝑇
𝑇
̃ 𝑖 = diag[𝑅𝑖
where 𝑐1 > 0, 𝑐2 > 0, 𝑅
weight coefficient matrix.
Theorem 6. For given 𝑇 > 0, 𝜂 > 0, 𝑐1 > 0, and
̃ 𝑖 > 0, the filtering error dynamic MJSs (8) are stochastically
𝑅
̃ 𝑖 𝑊) and have a 𝐿 2 -𝐿 ∞
FTB with respect to (𝑐1 𝑐2 𝑇 𝑅
performance level 𝛾 > 0, if there exist positive constants, 𝛾 >
0, 𝑐2 > 0, and mode-dependent symmetric positive-definite
̃ 𝑖 , such that
matrices 𝑃
𝑒𝜂𝑇 𝑐1 𝜎𝑃̂ +
(16)
𝑊
(1 − 𝑒−𝜂𝑇 ) < 𝑐2 𝜎𝑃̂ ,
𝜂
(17)
Proof. Let the mode at time 𝑡 be 𝑖; that is, 𝑟𝑡 = 𝑟 ∈ M. Take
the stochastic Lyapunov-Krasovskii functional 𝑉(𝜉(𝑡), 𝑟𝑡 , 𝑡 >
0) : R𝑛 × M × R+ → R+ as
𝐽 = ‖𝑟 (𝑡)‖𝑇𝐸∞ − 𝛾‖𝑤 (𝑡)‖𝑇2 < 0,
sup𝑡∈[0
𝑇
̃ 𝑖,
𝐶𝑖 𝐶𝑖 < 𝛾2 𝑃
(15)
𝑅𝑖 ] > 0, and 𝑅𝑖 is the
Definition 4. For the filtering error dynamic MJSs (8), if
there exist filter parameters 𝐴 𝑓𝑖 , 𝐵𝑓𝑖 , and 𝐶𝑓𝑖 , as well as a
positive scalar 𝛾, such that the filtering error dynamic MJSs
(8) are stochastically FTB and under the zero-valued initial
condition, the system output error satisfies the following cost
function inequality for 𝑇 > 0 with attenuation 𝛾 > 0 and for
all admissible 𝑤(𝑡) with the constraint condition (10):
=
̃
̃
̃ 𝑖 𝐵𝑖 ]
̃ 𝑖 𝐴𝑖 + 𝐴𝑇 𝑃
̃ 𝑃
[𝑃
𝑖 𝑖 + ∑ 𝜋𝑖𝑗 𝑃𝑗 − 𝜂𝑃𝑖
𝑗=1
[
] < 0,
𝑇̃
−𝜂𝑇
𝐵𝑖 𝑃𝑖
−𝑒 𝐼]
[
̃ 𝑖𝑅
̃ −1/2 , 𝜎 ̂ = max𝑖∈M 𝜎max (𝑃
̃ −1/2 𝑃
̂ 𝑖 ), and 𝜎 ̂ =
̂𝑖 = 𝑅
where 𝑃
𝑃
𝑖
𝑖
𝑃
̂ 𝑖 ).
min𝑖∈M 𝜎min (𝑃
Definition 3. Given a time-constant 𝑇 > 0, the filtering error
dynamic MJSs (8) are stochastically finite-time bounded
̃ 𝑖 𝑊), wherein 𝑐1 > 0,
(FTB) with respect to (𝑐1 𝑐2 𝑇 𝑅
̃
𝑐2 > 0, 𝑅𝑖 > 0, if condition (10) holds.
𝑇
(14)
(11)
𝑡 ∈ [0 𝑇] ,
‖𝑟(𝑡)‖𝑇𝐸∞
𝑇 + 𝛼−1 𝑀𝑀𝑇 𝛼𝑁𝑇𝑁 < 0.
𝑁
where 𝑊 is a positive scalar.
where
(13)
𝑇] E[‖𝑟(𝑡)‖],
(12)
‖𝑤(𝑡)‖𝑇2
=
√∫ 𝑤𝑇 (𝑡)𝑤(𝑡) 𝑑𝑡.
0
Then, the resilient filter (5) is called the stochastic finitetime 𝐿 2 -𝐿 ∞ filter of the uncertain dynamic MJSs (1) with 𝛾disturbance attenuation.
3. Main Results
In this section, we will study the robust stochastic finite-time
resilient filtering problem for the filtering error dynamic MJSs
(8) in an 𝐿 2 -𝐿 ∞ estimation error sense. Before proceeding
with the study, the following lemma is needed.
̃ 𝑖 𝜉 (𝑡) .
𝑉 (𝜉 (𝑡) , 𝑖) = 𝜉𝑇 (𝑡) 𝑃
(18)
Then, we introduce a weak infinitesimal generator I[∗]
(see [15, 16]), acting on 𝑉(𝜉(𝑡), 𝑖), for all 𝑖 ∈ M, which is
defined as
1
I𝑉 (𝜉 (𝑡) , 𝑖) = lim
[E {𝑉 (𝜉 (𝑡 + Δ𝑡) , 𝑟𝑡+Δ𝑡 , 𝑡 +Δ𝑡) |
Δ𝑡 → 0 Δ𝑡
𝜉 (𝑡) , 𝑟𝑡 = 𝑖} − 𝑉 (𝜉 (𝑡) , 𝑖, 𝑡)] .
(19)
The time derivative of 𝑉(𝜉(𝑡), 𝑖) along the trajectories of
the filtering error dynamic MJSs (8) is given by
𝑁
̃ 𝑖 𝜉̇ (𝑡) + 𝜉𝑇 (𝑡) ∑𝜋𝑖𝑗 𝑃
̃ 𝑗 𝜉 (𝑡)
I𝑉 (𝜉 (𝑡) , 𝑖) = 2𝜉𝑇 (𝑡) 𝑃
𝑗=1
𝑁
̃ 𝑖 𝐴𝑖 + 𝐴𝑇 𝑃
̃
̃
= 𝜉𝑇 (𝑡) (𝑃
𝑖 𝑖 + ∑ 𝜋𝑖𝑗 𝑃𝑗 ) 𝜉 (𝑡)
(20)
𝑗=1
̃ 𝑖 𝐵𝑖 𝑤 (𝑡) .
+ 2𝜉𝑇 (𝑡) 𝑃
Considering the 𝐿 2 -𝐿 ∞ filtering performance for the
dynamic filtering error system (8), we introduce the following
cost function by Definition 4 with 𝑡 ≥ 0:
𝐽 (𝑡) = E [I𝑉 (𝜉 (𝑡) , 𝑖)] − 𝜂E [𝑉 (𝜉 (𝑡) , 𝑖)] − 𝑒−𝜂𝑇 𝑤𝑇 (𝑡) 𝑤 (𝑡) .
(21)
4
Abstract and Applied Analysis
It follows from relation (16) that 𝐽(𝑡) < 0; that is,
E [I𝑉 (𝜉 (𝑡) , 𝑖)] < 𝜂E [𝑉 (𝜉 (𝑡) , 𝑖)] + 𝑒−𝜂𝑇 𝑤𝑇 (𝑡) 𝑤 (𝑡) .
(22)
Then, multiplying the previous inequality by 𝑒−𝜂𝑡 , we have
−𝜂𝑡
E {I [𝑒
−𝜂(𝑇+𝑡)
𝑉 (𝜉 (𝑡) , 𝑖)]} < 𝑒
𝑇
𝑤 (𝑡) 𝑤 (𝑡) .
(23)
Therefore, the cost function inequality (10) can be guaranteed, which implies 𝐽 = ‖𝑟(𝑡)‖𝑇𝐸∞ − 𝛾‖𝑤(𝑡)‖𝑇2 < 0.
̃ 𝑖𝑅
̃ −1/2 , 𝜎 ̂ = max𝑖∈M 𝜎max (𝑃
̂𝑖 = 𝑅
̃ −1/2 𝑃
̂ 𝑖 ),
Denote that 𝑃
𝑃
𝑖
𝑖
̂ 𝑖 ). From equality (24), we have
and 𝜎 ̂ = min𝑖∈M 𝜎min (𝑃
𝑃
𝑡
̃ 𝑖 𝜉 (𝑡)} = E {𝑉 (𝜉 (𝑡) , 𝑖)} < ∫ 𝑒−𝜂𝑠 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑠
E {𝜉𝑇 (𝑡) 𝑃
0
𝜂𝑡
+ 𝑒 E {𝑉(𝜉 (0) , 𝑟0 )} < 𝑒𝜂𝑡 E {𝑉(𝜉 (0) , 𝑟0 )}
Integrating the above inequality from 0 to 𝑇, we have
𝑡
𝑒−𝜂𝑇 E {𝑉 (𝜉 (𝑇) , 𝑖)} − E {𝑉 (𝜉 (0) , 𝑟0 )}
−𝜂𝑇
<𝑒
𝑇
−𝜂𝑠
∫ 𝑒
0
𝑇
𝑤 (𝑠) 𝑤 (𝑠) 𝑑𝑠.
+ 𝑊 ∫ 𝑒−𝜂𝑠 𝑑𝑠 < 𝑒𝜂𝑡 𝑐1 𝜎𝑃̂
0
(24)
Considering 𝑉(𝜉(0), 𝑟0 ) ≥ 0, as well as the zero initial
condition; that is, 𝜉(0) = 0, for 𝑡 > 0, then it follows that
𝑇
E {𝑉 (𝜉 (𝑇) , 𝑖)} < ∫ 𝑒−𝜂𝑠 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑠.
0
̃ 𝑖 𝜉 (𝑇)} = E {𝑉 (𝜉 (𝑇) , 𝑖)} < ∫ 𝑒−𝜂𝑠 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑠.
E {𝜉 (𝑇) 𝑃
E {𝑟 (𝑇) 𝑟 (𝑇)} = E {𝜉
𝑇
𝑇
(𝑇) 𝐶𝑖 𝐶𝑖 𝜉 (𝑇)}
̃ 𝑖 𝜉 (𝑇)}
< 𝛾2 E {𝜉𝑇 (𝑇) 𝑃
= 𝛾2 E {𝑉 (𝜉 (𝑇) , 𝑖)}
(27)
𝑇
< 𝛾2 ∫ 𝑒−𝜂𝑠 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑠
0
𝑇
< 𝛾2 ∫ 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑠.
0
Since the previous inequality is always true for any 𝑇 > 0,
the following relation:
𝑇
sup E [‖𝑟 (𝑡)‖] < 𝛾√ ∫ 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑠.
(28)
0
𝑡∈[0 𝑇]
On the other hand, it results from the stochastic
Lyapunov-Krasovskii functional that
̃ 𝑖 𝜉 (𝑡)} ≥ 𝜎 ̂ E {𝜉𝑇 (𝑡) 𝑅
̃ 𝑖 𝜉 (𝑡)} .
E {𝜉𝑇 (𝑡) 𝑃
𝑃
𝑁
‖𝑟 (𝑡)‖𝑇𝐸∞ = 𝛾∑𝜋𝑖 { sup E [‖𝑟 (𝑡)‖]}
𝑡∈[0 𝑇]
𝑁
𝑇
< 𝛾∑𝜋𝑖 √ ∫ 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑠.
0
𝑖=1
𝑇
= 𝛾√ ∫ 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑠 = 𝛾‖𝑤 (𝑡)‖𝑇2 .
0
(29)
(31)
Then, we can get
̃ 𝑖 𝜉 (𝑡)} <
E {𝜉𝑇 (𝑡) 𝑅
𝑒𝜂𝑇 𝑐1 𝜎𝑃̂ + (𝑊/𝜂) (1 − 𝑒−𝜂𝑇 )
𝜎𝑃̂
.
(32)
̃ 𝑖 𝜉(𝑡)} <
It implies that for all 𝑡 ∈ [0 𝑇], we have E{𝜉𝑇 (𝑡)𝑅
𝑐2 by condition (17). This completes the proof.
Theorem 7. For given 𝑇 > 0, 𝜂 > 0, 𝑐1 > 0, and 𝑅𝑖 > 0,
the filtering error dynamic MJSs (8) are stochastically FTB with
respect to (𝑐1 𝑐2 𝑇 𝑅𝑖 𝑊) with 𝑅𝑖 ∈ R𝑛×𝑛 > 0 and have
a prescribed 𝐿 2 -𝐿 ∞ performance level 𝛾 > 0, if there exist
a set of mode-dependent symmetric positive-definite matrices
𝑃𝑖 , a set of mode-dependent matrices 𝑋𝑖 , 𝑌𝑖 , and a positive
scalar 𝜎1 and mode-dependent sequences 𝛼𝑖 , 𝛽𝑖 , 𝜆 𝑖 , satisfying
the following matrix inequalities for all 𝑖 ∈ M:
Λ 1𝑖
[Λ
[ 2𝑖
[
[∗
[
[∗
[∗
∗
𝑃𝑖 𝐵𝑖
𝑃𝑖 𝑀1𝑖
0
Λ 3𝑖 𝑃𝑖 𝐵𝑖 − 𝑌𝑖 𝐷𝑖 𝑃𝑖 𝑀1𝑖 − 𝑌𝑖 𝑀2𝑖 𝑃𝑖 𝑀3𝑖 ]
]
]
∗
−𝑒−𝜂𝑇 𝐼
0
0 ] < 0, (33)
]
∗
∗
−𝛼𝑖 𝐼
0 ]
∗
∗
∗
−𝛽𝑖 𝐼 ]
−𝑃𝑖
It is easy to get the following result:
𝑖=1
𝑊
(1 − 𝑒−𝜂𝑇 ) .
𝜂
(26)
By (15) and within the finite-time interval [0 𝑇], we can
also get
𝑇
+
(30)
𝑇
0
𝑊
(1 − 𝑒−𝜂𝑡 ) ≤ 𝑒𝜂𝑇 𝑐1 𝜎𝑃̂
𝜂
(25)
Then, it can be verified from the defined LyapunovKrasovskii functional that
𝑇
+
[
[ 0
[
[
[
[∗
[
[
[
∗
[
0
𝑇
−𝐸𝑖𝑇 + 𝐶𝑓𝑖
−𝑃𝑖
𝑇
−𝐶𝑓𝑖
∗
−𝛾2 𝐼 + 𝜆 𝑖 𝑀4𝑖 𝑀4𝑖𝑇
∗
∗
𝑁2𝑖𝑇
]
−𝑁2𝑖𝑇 ]
]
]
] < 0,
0 ]
]
]
]
−𝜆 𝑖 𝐼
]
𝑅𝑖 < 𝑃𝑖 < 𝜎1 𝑅𝑖 ,
𝑒𝜂𝑇 𝑐1 𝜎1 +
𝑊
(1 − 𝑒−𝜂𝑇 ) < 𝑐2 ,
𝜂
(34)
(35)
(36)
Abstract and Applied Analysis
5
𝑇
where Λ 1𝑖 = 𝑃𝑖 𝐴 𝑖 + 𝐴𝑇𝑖 𝑃𝑖 + ∑𝑁
𝑗=1 𝜋𝑖𝑗 𝑃𝑗 − 𝜂𝑃𝑖 + 𝛼𝑖 𝑁1𝑖 𝑁1𝑖 +
𝑇
𝑇
𝛽𝑖 𝑁2𝑖 𝑁2𝑖 , Λ 2𝑖 = 𝑃𝑖 𝐴 𝑖 − 𝑋𝑖 − 𝑌𝑖 𝐶𝑖 − 𝛽𝑖 𝑁2𝑖 𝑁2𝑖 , Λ 2𝑖 = 𝑋𝑖 +
𝑇
𝑋𝑖𝑇 + ∑𝑁
𝑗=1 𝜋𝑖𝑗 𝑃𝑗 − 𝜂𝑃𝑖 + 𝛽𝑖 𝑁2𝑖 𝑁2𝑖 .
Moreover, the suitable filter parameters can be given as
𝐴 𝑓𝑖 =
𝑃𝑖−1 𝑋𝑖 ,
𝐵𝑓𝑖 =
𝑃𝑖−1 𝑌𝑖 ,
𝐶𝑓𝑖 = 𝐶𝑓𝑖 .
̃ 𝑖 = diag{𝑃𝑖 , 𝑃𝑖 }. Then, we can
Proof. For convenience, we set 𝑃
get the following relations according to matrix inequalities
(15) and (16):
(37)
Π𝑖 + ΔΠ1𝑖 + ΔΠ2𝑖 < 0,
(38)
Σ𝑖 + ΔΣ𝑖 < 0,
(39)
where
𝑁
𝑇
∗
𝑃𝑖 𝐵𝑖
[𝑃𝑖 𝐴 𝑖 + 𝐴 𝑖 𝑃𝑖 + ∑ 𝜋𝑖𝑗 𝑃𝑗 − 𝜂𝑃𝑖
]
[
]
𝑗=1
[
]
𝑁
],
Π𝑖 = [
𝑇
[ 𝑃𝐴 − 𝑃𝐴 − 𝑃𝐵 𝐶
𝑃𝑖 𝐴 𝑓𝑖 + 𝐴 𝑓𝑖 𝑃𝑖 + ∑𝜋𝑖𝑗 𝑃𝑗 − 𝜂𝑃𝑖 𝑃𝑖 𝐵𝑖 − 𝑃𝑖 𝐵𝑓𝑖 𝐷𝑖 ]
𝑖 𝑖
𝑖 𝑓𝑖
𝑖 𝑓𝑖 𝑖
[
]
[
]
𝑗=1
−𝜂𝑇
∗
∗
−𝑒 𝐼 ]
[
𝑇
−𝑃𝑖 0 −𝐸𝑖𝑇 + 𝐶𝑓𝑖
]
[
𝑇
]
[ 0 −𝑃𝑖
−𝐶𝑓𝑖
],
[
Σ𝑖 = [
]
]
[
∗ ∗
−𝛾2 𝐼
]
[
𝑃𝑖 Δ𝐴 𝑖 + Δ𝐴𝑇𝑖 𝑃𝑖 ∗ 0
]
[
[𝑃𝑖 Δ𝐴 𝑖 − 𝑃𝑖 𝐵𝑓𝑖 Δ𝐶𝑖 0 0]
],
ΔΠ1𝑖 = [
]
[
]
[
∗
∗ 0
]
[
(40)
0
∗
0
[
]
𝑇
ΔΠ2𝑖 = [−𝑃𝑖 Δ𝐴 𝑓𝑖 𝑃𝑖 Δ𝐴 𝑓𝑖 + Δ𝐴 𝑓𝑖 𝑃𝑖 0] ,
∗
0]
[ ∗
𝑇
0 0 Δ𝐶𝑓𝑖
[
𝑇]
]
[ 0 0 −Δ𝐶𝑓𝑖
].
[
ΔΣ𝑖 = [
]
]
[
∗ ∗
0
]
[
Referring to Lemma 5, ΔΠ1𝑖 and ΔΠ2𝑖 can be presented as the
following form:
ΔΠ1𝑖 = 𝐿 1𝑖 Γ𝑖 (𝑡) 𝐿 2𝑖 + 𝐿𝑇2𝑖 Γ𝑖𝑇 (𝑡) 𝐿𝑇1𝑖 < 𝛼𝑖−1 𝐿 1𝑖 𝐿𝑇1𝑖 +𝛼𝑖 𝐿𝑇2𝑖 𝐿 2𝑖 ,
𝑇
ΔΠ2𝑖 = 𝐿 3𝑖 Γ𝑓𝑖 (𝑡) 𝐿 4𝑖 + 𝐿𝑇4𝑖 Γ𝑓𝑖
(𝑡) 𝐿𝑇3𝑖 < 𝛽𝑖−1 𝐿 3𝑖 𝐿𝑇3𝑖 +𝛽𝑖 𝐿𝑇4𝑖 𝐿 4𝑖 ,
𝑇
𝑇
ΔΣ𝑖 = 𝐿 5𝑖 Γ𝑓𝑖 (𝑡) 𝐿 6𝑖 + 𝐿𝑇6𝑖 Γ𝑓𝑖
(𝑡) 𝐿𝑇5𝑖 < 𝜆 𝑖 𝐿 5𝑖 𝐿𝑇5𝑖 +𝜆−1
𝑖 𝐿 6𝑖 𝐿 6𝑖 ,
(41)
where 𝐿 1𝑖
=
col[𝑃𝑖 𝑀1𝑖 𝑃𝑖 𝑀1𝑖 − 𝑃𝑖 𝐵𝑓𝑖 𝑀2𝑖 0],
𝐿 2𝑖 = [𝑁1𝑖 0 0], 𝐿 3𝑖 = col[0 𝑃𝑖 𝑀3𝑖 0], 𝐿 4𝑖 =
=
[−𝑁2𝑖 𝑁2𝑖 0], 𝐿 5𝑖 = col[0 0 𝑀4𝑖 ], and 𝐿 6𝑖
[𝑁2𝑖 − 𝑁2𝑖 0].
Then, inequalities (15) and (16) are equivalent to LMIs
(38) and (39) by letting 𝑋𝑖 = 𝑃𝑖 𝐴 𝑓𝑖 and 𝑌𝑖 = 𝑃𝑖 𝐵𝑓𝑖 .
̃ 𝑖 = diag[𝑅𝑖
On the other hand, we set 𝑅
(35) implies that
̂ 𝑖) ,
1 < 𝜎𝑃̂ = min 𝜎min (𝑃
𝑖∈M
𝑅𝑖 ], and LMI
̃ 𝑖 ) < 𝜎1 . (42)
𝜎𝑃̂ = max 𝜎max (𝑃
𝑖∈M
Then, recalling condition (17), we can get LMI (36). This
completes the proof.
To obtain an optimal finite-time 𝐿 2 -𝐿 ∞ filtering performance against unknown inputs, uncertainties, and model
errors, the attenuation lever 𝛾2 can be reduced to the minimum possible value such that LMIs (33)–(36) are satisfied.
The optimization problem can be described as follows:
min
𝑃𝑖 ,𝑋𝑖 ,𝑌𝑖 ,𝐶𝑓𝑖 ,𝐷𝑓𝑖 ,𝛼𝑖 ,𝛽𝑖 ,𝜆 𝑖 ,𝜌
𝜌
(43)
2
s. t. LMIs (33) – (36) with 𝜌 = 𝛾 .
6
Abstract and Applied Analysis
0.2
40
0.1
35
0
System state 𝑥2 (𝑡)
45
𝑐2
30
25
20
15
−0.2
−0.3
−0.4
10
5
−0.1
0
1
2
3
4
5
𝑐1
6
7
8
9
−0.5
10
0
0.5
1
1.5
2
2.5
3
3.5
4
3.5
4
𝑡 (s)
Real state 𝑥2 (𝑡)
Estimated state 𝑥2 (𝑡)
Figure 1: The optimal minimal upper bound 𝑐2 with different initial
𝑐1 .
0.08
0.06
Output error 𝑟(𝑡)
System state 𝑥1 (𝑡)
Figure 3: System response with state 𝑥2 (𝑡).
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
0.5
1
1.5
2
𝑡 (s)
2.5
3
3.5
4
0.04
0.02
0
−0.02
−0.04
−0.06
0
0.5
1
1.5
2
2.5
3
𝑡 (s)
Real state 𝑥1 (𝑡)
Estimated state 𝑥1 (𝑡)
Figure 4: System response with output error 𝑟(𝑡).
Figure 2: System response with state 𝑥1 (𝑡).
Mode 2:
Remark 8. In Theorems 6 and 7, if 𝑐2 is a variable to be solved,
then (17) and (36) can be always satisfied as long as 𝑐2 is
sufficiently large. For these, we can also fix 𝛾 and look for
the optimal admissible 𝑐1 or 𝑐2 guaranteeing the stochastic
finite-time boundedness of desired filtering error dynamic
properties.
Example 9. Consider a class of MJSs with two operation
modes described as follows.
Mode 1:
−2 1
],
−1 −1
𝐷1 = 0.1,
𝑀21 = −0.1,
𝐵1 = [
−0.2
],
0.1
𝐸1 = [0.1 0.2] ,
0.1
𝑀31 = [
],
−0.1
𝑁11 = [0.2 0.1] ,
0 3
],
−1 −2
𝐷2 = −0.2,
𝑀22 = 0.2,
4. Numeral Examples
𝐴1 = [
𝐴2 = [
𝐶1 = [1 0.5] ,
0
𝑀11 = [ ] ,
0.2
𝑀41 = 0.2,
𝑁21 = [0.1 −0.2] ;
(44)
𝐵2 = [
0.1
],
−0.2
𝐸2 = [0.2 −0.1] ,
𝑀32 = [
𝑁12 = [−0.1 0.1] ,
0.1
],
0.2
𝐶2 = [1 1] ,
𝑀12 = [
−0.1
],
0.1
𝑀42 = −0.2,
𝑁22 = [0.1 0.1] .
(45)
The mode switching is governed by a Markov chain that has
the following transition rate matrix:
−0.3 0.3
Π=[
].
0.5 −0.5
(46)
In this paper, we choose the initial values for 𝑊 = 2,
𝑇 = 4, 𝜂 = 0.25, and 𝑅𝑖 = 𝐼2 . Then, we fix 𝛾 = 0.8 and
look for the optimal admissible 𝑐2 with different 𝑐1 which can
guarantee the stochastic finite-time boundedness of desired
filtering error dynamic properties. Figure 1 gives the optimal
minimal admissible 𝑐2 with different initial upper bound 𝑐1 .
Abstract and Applied Analysis
7
For 𝑐1 = 1, we solve LMIs (33)–(36) by Theorem 7 and
the optimization algorithm (43) and get the following optimal
resilient finite-time 𝐿 2 -𝐿 ∞ filter:
𝐴 𝑓1 = [
−7.7193 −3.3359
],
−6.9662 −6.6546
3.0189
𝐵𝑓1 = [
],
6.7316
𝐴 𝑓2
𝐵𝑓2 = [
𝐶𝑓1 = [0.05 0.1] ;
−1.3004 0.3608
=[
],
−3.0043 −9.2166
1.1415
],
2.0586
(47)
𝐶𝑓2 = [0.1 −0.05] .
And then, we can also get the attenuation lever as 𝛾 = 0.0777
and the relevant upper bound 𝑐2 = 592.45.
To show the effectiveness of the designed optimal resilient
finite-time 𝐿 2 -𝐿 ∞ filter, we assume that the unknown inputs
are unknown white noise with noise power 0.05 over a finitetime interval 𝑡 ∈ [0 4]. For the selected initial conditions
0.5 ] and 𝑟 = 2, the simulation results of the
𝑥(0) = [ −0.5
0
jumping modes and the response of system states are shown
in Figures 2, 3, and 4. It is clear from the simulation figures
that the estimated states can track the real states smoothly.
5. Conclusion
The resilient finite-time 𝐿 2 -𝐿 ∞ filtering problems for a class
of stochastic MJSs with uncertain parameters have been studied. By using the Lyapunov-Krasovskii functional approach
and LMIs optimization techniques, a sufficient condition
is derived such that the filtering dynamic error MJSs are
finite-time bounded and satisfy a prescribed level of 𝐿 2 -𝐿 ∞
disturbance attenuation in a finite time-interval. The robust
resilient filter gains can be solved directly by using the existing
LMIs. Simulation results illustrate the effectiveness of the
proposed a pproach.
Acknowledgments
This work was supported in part by the National Natural
Science Foundation of China (Grant no. 61203051), the
Joint Specialized Research Fund for the Doctoral Program
of Higher Education (Grant no. 20123401120010), and the
Key Program of Natural Science Foundation of Education
Department of Anhui Province (Grant no. KJ2012A014).
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