IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 2, FEBRUARY 2005
Robust
589
H
Filtering for Nonlinear
Stochastic Systems
Weihai Zhang, Bor-Sen Chen, Fellow, IEEE, and Chung-Shi Tseng, Member, IEEE
Abstract—This paper describes the robust
filtering analysis and synthesis of nonlinear stochastic systems with state and
exogenous disturbance-dependent noise. We assume that the state
and measurement are corrupted by stochastic uncertain exogenous
disturbance and that the system dynamic is modeled by Itô-type
stochastic differential equations. For general nonlinear stochastic
systems, the
filter can be obtained by solving second-order
nonlinear Hamilton–Jacobi inequalities. When the worst-case disturbance is considered in the design procedure, a mixed 2
filtering problem is also solved by minimizing the total estimation
error energy. It is found that for a class of special nonlinear stofiltering design can be given via solving
chastic systems, the
several linear matrix inequalities instead of Hamilton–Jacobi inequalities. A few examples show that the proposed methods are effective.
Index Terms—Hamilton–Jacobi inequality,
matrix inequality, nonlinear stochastic systems.
filtering, linear
I. INTRODUCTION
I
N the past few decades, the
control, since it was first
formulated by [1], has been extensively developed (see the
celebrated paper [2] for its state-space treatment), and the discontrol can be found in [3]
cussion on output feedback
for nonlinear or [4] and [5] for linear uncertain systems. The
filtering problem is to design an estimator to esso-called
timate the unknown state combination via output measurement,
gain (from the external disturbance to
which guarantees the
;
the estimation error) to be less than a prescribed level
see [6] and [7] for a discrete-time investigation and [8]–[10] for
filtering investigation. In contrast with
a general nonlinear
the well-known Kalman filter, one of the main advantages of
filtering is that it is not necessary to know exactly the statistical properties of the external disturbance but only assumes
the external disturbance to have bounded energy. See [11]–[13]
filtering in signal processing.
for practical applications of
filtering and control problems
In recent years, stochastic
with system models expressed by Itô-type stochastic differential
Manuscript received June 16, 2003; revised January 20, 2004. This work
was supported by National Science Council under Contract NSC 91-2213-E007-014 and by the Chinese Natural Science Foundation under Grant 60474013.
The associate editor coordinating the review of this paper and approving it for
publication was Dr. Yuan-Pei Lin.
W. Zhang is with the Shenzhen Graduate School, Harbin Insitute of Technology, HIT Campus, Shenzhen 518055 China (e-mail: [email protected]).
B.-S. Chen is with the Department of Electrical Engineering, National Tsing Hua University, 30043 Hsin-Chu, Taiwan, R.O.C. (e-mail:
[email protected]).
C.-S. Tseng is with the Department of Electrical Engineering, Ming Hsin University of Science and Technology, 30401 Hsin Feng, Taiwan, R.O.C. (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TSP.2004.840724
equations have become a popular research topic and has gained
extensive attention; see [14]–[19] and the references therein. We
filsummarize below the recent development on stochastic
tering problem.
In [15], a bounded real lemma was presented for linear continuous-time stochastic systems, according to which full- and
estimation problems for stationary
reduced-order robust
continuous-time linear stochastic uncertain systems were discussed by [14] and [17], respectively. All the above works are
limited to the linear stationary stochastic systems, whereas [18]
investigated the same problem for a class of special nonlinear
stochastic systems. In [16] and [19], the linear and nonlinear
control problems have been discussed.
stochastic
control
There have been a lot of studies on nonlinear
or state estimation in deterministic systems; see, e.g., [3], [6],
[8], and [9], and the references therein. However, it should be
noted that up to now, there is little corresponding work on the
general nonlinear stochastic systems governed by Itô-type stochastic differential equations, which has many applications in
practice [20]. Unlike the deterministic case, the Hamilton–Jacobi inequality (HJI) associated with the nonlinear stochastic
filter is a second-order (not first-order) nonlinear partial differential inequality due to the effect of the diffusion term, which
filtering problem more complex.
makes the stochastic
In the present paper, we discuss the infinite horizon robust
state estimation for nonlinear stochastic uncertain systems,
assuming that the system state is corrupted not only by white
noise, but also by exogenous disturbance signal, and the measurement output is also corrupted by exogenous disturbance.
Our goal in this paper is to construct an asymptotically stable
(in some sense) observer that leads to a stable estimation error
-gain with respect to uncertain disturbance
process whose
signal is less than a prescribed level.
In Section II, for nonlinear perturbed stochastic systems with
state and external disturbance-dependent noise, the robust
estimation problem is investigated. As in deterministic case,
filtering problem, stawhen we study the infinite horizon
bility is an essential requirement. That is, we should search for
filter, which makes the augmented
a stable (in some sense)
system to be asymptotically stable in probability. Unlike most
previous work on nonlinear filtering problem (e.g., [9] and [10]),
a very general form of the filter is considered, which needs us
, and via solving
to determine three parameter functions
a second-order nonlinear HJI. An exact form of HJI associated
filtering is derived. Meanwhile, a subopwith nonlinear
filter is also studied, that is, of all the
timal mixed
filters, we seek one to minimize an upper bound of the total es-
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timation error energy when the worst-case disturbance (from
to ) is considered in the design procedure.
For a class of special nonlinear stochastic systems, Section III
presents a linear matrix inequality (LMI)-based algorithm for
filtering design, and solving the LMIs is much
its robust
easier than solving the HJI. Section IV presents some examples
to illustrate our proposed methods. Section V concludes this
paper.
For convenience, we adopt the following notations:
Trace (transpose) of matrix .
Tr
Positive semidefinite (positive
definite) matrix .
Identity matrix.
Euclidean 2-norm of the -dimensional real vector .
space of nonanticipative stowith rechastic processes
spect to filtration
satisfying
.
Class of functions
twice
continuously differential with re, except possibly
spect to
at the origin.
II. NONLINEAR
filtering problem, it is inevitably related to stostochastic
chastic stability, see [21] for the definitions of “globally asymptotic stability in probability” and “exponentially mean square
stability.”
The following proposition is a special case of [16], which
plays an important role in this paper.
Proposition 1: For system (1), if the state information
is completely available and there exists a positive function
solving the following HJI:
(3)
then
(4)
with initial state
.
holds for some
For convenience, we give its proof as follows.
be the infinitesimal generator of (1), which
Proof: Let
is defined as
FILTERING
Consider the following nonlinear stochastic system (the time
variable is suppressed):
Applying “completing the square,” we have
(1)
is called the system state,
In the above,
is the measurement,
is the state combination to
stands for the exogenous
be estimated, and
disturbance signal.
, and
are smooth functions
.
is a standard onewith
dimensional (1-D) Wiener process defined on the probability
relative to an increasing family
of
space
-algebras
. In (1), the state equation, in engineering
terminology, can be written as [20]
(2)
which means that the state and external disturbance are dependent on the same noise, where is a stationary white noise. In
particular, along the lines of this paper, it is easy to treat with
filtering problem for the system that the state
the nonlinear
and external disturbance are dependent on uncorrelated noise.
When the system state is not completely available, to estimate
from the observable information
with an -gain (from to the estimation error) less than a pre, a nonlinear stochastic
filter should
scribed level
be constructed. Since this paper deals with the infinite horizon
ZHANG et al.: ROBUST
FILTERING FOR NONLINEAR STOCHASTIC SYSTEMS
Considering (3), we immediately have
591
one can see that, for any
with
, where
That is, achieves the maximal possible energy gain from the
disturbance input to the controlled output . Thereefore, in
can be viewed as the worst case disturbance.
this case,
In what follows, we construct the following estimator equation for the estimation of
(5)
By integrating and taking expectation from 0 to , we have
, and
, which are to be determined, are
where
matrices of appropriate dimensions with sufficient smoothness.
; then, we get the following augmented system:
Set
(6)
where
Because
, therefore
In addition, let
Letting
in the above, (4) is followed, accordingly, by
Proposition 1.
Remark 1: From the Proof of Proposition 1, it is easily seen
and
can be weakened to become
that in (3),
and
, respectively. Here, we take a stronger
and
) only for the purpose of
condition (i.e.,
simplicity in applications.
Remark 2: Proposition 1 can be called a bounded real lemma
(BRL) of nonlinear stochastic systems; a linear stochastic BRL
can be found in [15]. In addition, one anonymous reviewer
pointed out that a more general BRL for nonlinear stochastic
systems was derived in [22]. Within the frame of stochastic
game theory, [23] studied a certain type of minimax dynamic
game for stochastic nonlinear systems, and an Hamilton-Jacobi
equation, which contains an Hamiltonian function, was derived.
Remark 3: If we let
denote the estimator error; then, the nonlinear stochastic
filtering can be stated as follows: Find filter gain matrices
, and
in (5) such that we have the following.
of the augmented system (6)
1) The equilibrium point
is globally asymptotically stable in probability in the case
.
, the fol2) For a given disturbance attenuation level
lowing relation holds:
(7)
Some main results of this section are listed as follows.
Theorem 1: For given disturbance attenuation level
if there exists a positive Lyapunov function
solving the following HJI:
,
(8)
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 2, FEBRUARY 2005
for some matrices
, and of suitable dimensions, then the
filtering problem is solved by (5), where
stochastic
Corollary 2: If there exists a positive constant
and
such that the
a positive Lyapunov function
conditions where filter gain matrices
, and
of suitable
, then
dimensions hold for some disturbance attenuation
filtering problem is solved by (5).
the stochastic
)
To prove Theorem 1, the following lemma on the globally
of
asymptotic stability of the equilibrium point
(12)
)
(9)
is needed.
Lemma 1 [21]: Assume there exists a positive Lyapunov
function
satisfying
for all
nonzero
; then, the equilibrium point
of (9) is
globally asymptotically stable in probability.
Proof of Theorem 1: We first show that (7) holds. Applying Proposition 1 to the system (6), we immediately have
that (7) holds if
(13)
)
(10)
(14)
for
admits solutions
and
. By a series of computations, (10) is equivalent
to (8).
Second, we show that
of the augmented system (6)
to be globally asymptotically stable in probability in the case
. By Lemma 1, we only need to prove
for
some
, i.e.,
Proof of Corollary 2: Applying the following well-known
fact:
(15)
it follows that
(11)
is defined as the infinitesimal generator of system (6).
where
While (11) is obvious because of (8), the Proof of Theorem 1 is
complete.
More specifically, if
, i.e., only the state-dependent
noise, Theorem 1 yields the following corollary.
in (1), if the following HJI
Corollary 1: For
By condition
(16)
admits solutions
for
and for some
, then the stochastic
problem is solved by (5), where in this case
filtering
by
Since it is easy to test that
and
, this corollary is shown.
In general, (13) and (14) are a pair of coupled HJIs, but if
as the form of
, then (13)
we take
and (14) become decoupled and can be solved independently.
, Corollary 2 yields the following.
Especially, for
ZHANG et al.: ROBUST
FILTERING FOR NONLINEAR STOCHASTIC SYSTEMS
Corollary 3: The consequence of Corollary 2 still
there exists a positive Lyapunov function
holds if
solving the HJI
(17)
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Generally speaking, the design of robust
filter is not
filters, we seek one that minimizes the
unique. Of all the
when the worst-case disturtotal error energy
bance (from to ) is considered in the design procedure with
. This is the
initial state
filtering design problem. In the sense
so-called mixed
of Remark 3, it can be seen that the worst-case disturbance from
to is of the following form:
also solves HJI
(18)
with some matrices
, and of suitable dimensions.
. AddiProof: Note that in this case, we can take
tionally
(22)
where
the above
is an admissible solution of (10) or (8). Substituting
into (6) yields
(23)
Theorem 2: For any prescribed disturbance attenuation level
, if there exists a positive Lyapunov function
solving the following HJI:
The rest is omitted.
Under a standard assumption [24]
, repeating the
same procedure as in Corollary 2, we have the following result.
–
of Corollary 2 are replaced by
Corollary 4: If
, and
, as shown below, respectively, then for
, the consequence of Corollary 2 still holds.
any
)
(24)
for some filter gain matrices
, and
of suitfilter can
able dimensions, then a suboptimal mixed
be synthesized by solving the following constraint optimization
problem:
(19)
)
s.t.
(20)
)
(25)
from (24); therefore, the
Proof: First, we have
filtering problem is solved by (5).
when of (22)
Second, we assert
. To prove
is implemented in (23), i.e.,
, by Itô’s formula,
this assertion, we first note that for any
we have
(21)
Proof: We only need to note that under the condition of
because of the assumption
.
Remark 4: In most literature (e.g., [3]) on deterministic nonlinear
control or filtering, one often assumes
for simplicity, which implies that the disturbances
and
to state
and measurement
, respectively, are independent. Under the above assumption, (19) comes down to
(26)
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 2, FEBRUARY 2005
In addition, still using Itô’s formula, for
that
, it follows
or mixed
(24). From Theorem 2, to synthesize nonlinear
filter, one should solve HJI (8) or constraint optimization problem (25); neither is easy. Maybe, only in some special
cases, such as for linear time-invariant systems, one can give an
applicable design algorithm. Moreover, from the Proof of Theis a tighter upper bound of
due
orem 2,
. Especially when
, then
to
from (26), it follows that
(27)
Letting
for
Therefore,
with respect to
, (26) follows
We find that for general nonlinear stochastic system (1), to
filter, one needs to solve HJI (8), which is
design its robust
not an easy thing. However, for a class of special nonlinear stochastic systems, the above-mentioned problem can be converted
into solving LMIs, as done in [18]; therefore, a numerical algorithm is admissible. It is well known that using the LMI-based
control for both deterministic and stotechnique to study
chastic systems has become a popular approach in recent years;
see [15], [22], [27], and the references therein. We consider
below the following special nonlinear stochastic system governed by Itô differential equation:
(28)
(30)
. Moreover
a.s.
III. LMI-BASED APPROACH FOR A CLASS OF
FILTERING
NONLINEAR
is a non-negative supermartingale
. Additionally, from (24), we have
where
with measurement output
(31)
(29)
Hence, (23) is globally asymptotically stable in probability. By Doob’s convergence theorem [25],
. Moreover,
.
in (26) and applying the above asserFinally, taking
tion yields
and
are 1-D standard Wiener processes.
where
and
to
Without loss of generality, we also assume
, , and are conbe mutually uncorrelated.
stant matrices of suitable dimensions, and
still represents the exogenous disturbance signal. Equation (30)
is a special case of the state equation of (1) with only state-dependent noise, for the purpose of simplicity. As a matter of
, and regards
fact, in (1), if one takes
and
as the Taylor’s series expansion of
and
, respectively, then the state equation of (1) comes
down to (30). In (31), we assume that the measurement output,
as in [14], is expressed by a stochastic differential equation.
For the special nonlinear systems (30) and (31), we take the
(see [26] for the
following linear filter for the estimation of
treatment of discrete-time nonlinear systems):
(32)
Theorem 2 is concluded, i.e., by solving (25), a suboptimal
filter is obtained.
mixed
It should be mentioned that HJI (8) for nonlinear
filtering
is implied by (24); therefore, for a suboptimal mixed
filtering problem, we only need to minimize
subject to
where
. Still, letting
, then
(33)
ZHANG et al.: ROBUST
FILTERING FOR NONLINEAR STOCHASTIC SYSTEMS
where
595
Applying (15) together with (37) and (39), we have the estimafor simplicity)
tions as (taking
(41)
(34)
For a prescribed disturbance attenuation level
and
such that
to find constant matrices
(42)
(43)
, we want
Substituting (41)–(43) into (40), we have
(35)
. Define the
holds for any
index as
(44)
performance
Obviously, if (38) holds, then there exists
such that
(36)
Obviously, the
filtering performance (35) holds iff
.
-based robust state estimation problems
As in [14], the
,
are formulated as follows: Given a prescribed value
find an estimator in (32) leading (33) to be exponentially mean
; Moreover,
for all
square stable in the case of
with
.
nonzero
In this section, a sufficient condition is given for stochastic
filtering design of systems (30) and (31). Now, we first give
a lemma as follows.
Lemma 2: Suppose there exists a scalar
such that
Therefore,
, which yields (33) being exby Lemma 2.
ponentially mean square stable for
Second, we prove
for all nonzero
with
. Note that for any
(37)
If the following matrix inequalities
(38)
(39)
have solutions
square stable when
where
, then (33) is exponentially mean
, and the
performance
,
.
Remark 5: Equation (37) is, in fact, a very loose condition,
satisfies the globally Lipschitz
which means that
filcondition at the origin. If, in the definition of stochastic
tering, we demand (33) to be locally exponentially mean square
, then it is only necessary that (37) holds in the
stable
neighborhood of the origin.
Proof: We first prove (33) to be exponentially mean
. Take the Lyapunov candidate as
square stable when
, where
is a solution to (38) and (39), and
be the infinitesimal operator of (33); then
let
Therefore, if
(45)
(40)
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then there exists
for any nonzero
, which yields
. As (45) is equivalent to (38) by Schur’s
complement [27], the Proof of Lemma 2 is completed.
Lemma 2 is only of theoretical value because it is inconvefilter. The following result is more
nient for designing the
suitable to use in practice.
Theorem 3: Under the conditions of Lemma 2, if the following LMIs
diag
, then (39) is equivalent to (46).
If we take
Substituting (34) into (49), we have
(50)
where we have the the equation at the bottom of the page, which
is equivalent to
(46)
(47)
(51)
have solutions
, then (33) is exponentially mean square stable for
filtering performance
holds, and
the
,
(48)
filter.
is the corresponding
Proof: By Schur’s complement, (38) is equivalent to
(49)
Letting
, (51) becomes (47). From
, and therefore, an
our assumption,
filter is constructed as in the form of (48), and the proof of
Theorem 3 is completed.
Based on the above discussion, we summarize the following
design algorithm.
Design Algorithm:
and by solving
Step i) Obtain solutions
, and
.
LMIs (46) and (47),
, and substitute
Step ii) Set
the just obtained
into (32); then, (32) is the
filter.
desired
Remark 6: If, in (30) and (31), the external disturbance signal
is regarded as a white noise, following the line of [14], one
can further discuss the mixed
filtering problem for this
filkind of special nonlinear system. That is, of all the
ters, we select one that minimizes the estimation error variance
.
ZHANG et al.: ROBUST
FILTERING FOR NONLINEAR STOCHASTIC SYSTEMS
IV. SOME ILLUSTRATIVE EXAMPLES
Below, we give some examples to illustrate our developed
theory in the above sections.
Example 1 (1-D Nonlinear
Filtering Design): Suppose
a stochastic signal is generated by the following nonlinear
stochastic system driven by a Wiener process and corrupted by
a stochastic external disturbance ; we therefore construct an
filter to estimate from the measurement signal .
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and solving these two LMIs together with constraints
and
, we have
Then
Assuming that the disturbance attenuation is prescribed as
, and the state estimator is as the form of (5), the augmented
system takes the form of (6) with
(52)
filter.
is the corresponding
V. CONCLUSION
Setting
, it is then easy to test that HJI
if we take
.
filter is obtained as the form of
Therefore, the robust
Clearly, there may be more than one solution to HJI (8). In genstate estimator is not unique.
eral, the robust
Filtering Design for Special Nonlinear
Example 2 (
System): Consider the
filtering design of nonlinear stochastic signal processes (30) and (31) with
filtering
In this study, we have discussed the robust
problem for affine nonlinear stochastic systems with state
and external disturbance-dependent noise, and a bounded real
lemma for stochastic nonlinear systems is derived. To obtain
filter of nonlinear stochastic systems, one should solve
the
an HJI, which is a second-order nonlinear partial differential
filtering analysis
inequalities. Meanwhile, the mixed
is also discussed, where the
performance is minimized
when the effect of the worst-case disturbance is considered.
filtering design for a class of special nonlinear
Finally, the
systems is also solved via the LMI Toolbox technique [28]
instead of solving the HJI, giving a more convenient algorithm
for practical applications [27].
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H
Weihai Zhang was born in 1965 in Shandong
Province, China. He received the M.Sc. degree from
Hangzhou University, Hangzhou, China, and the
Ph.D. degree from Zhejiang University, Hangzhou,
in 1994 and 1998, respectively.
From May 2001 to July 2003, he was a Postdoctoral Researcher at National Tsing Hua University,
Hsin-Chu, Taiwan, R.O.C. He is now a Research
Fellow with Shenzhen Graduate School, Harbin Insitute of Technology, HIT Campus, Shenzhen, China.
His research interests are in linear and nonlinear
stochastic control, robust filtering, and stochastic stability.
Bor-Sen Chen (M’82–SM’89–F’01) received the
B.S. degree from Tatung Institute of Technology,
Taipei, Taiwan, R.O.C., the M.S. degree from
National Central University, Chungli, Taiwan, and
the Ph.D degree from the University of Southern
California, Los Angeles, in 1970, 1973 and 1982,
respectively.
He was a Lecturer, Associate Professor, and
Professor at the Tatung Institute of Technology from
1973 to 1987. He is currently a Tsing Hua Professor
of electrical engineering and computer science at
National Tsing Hua University, Hsinchu, Taiwan. His current research interests
are in control engineering, signal processing and system biology. He is a
member of the Editorial Advisory Board of two journals (the International
Journal of Fuzzy Systems and the International Journal of Control, Automation
and Systems) and editor of the Asian Journal of Control.
Dr. Chen has received the Distinguished Research Award from National Science Council of Taiwan four times. He is a Research Fellow of the National
Science Council of Taiwan and holds the excellent scholar Chair in engineering.
He has also received the Automatic Control medal from the Automatic Control
Society of Taiwan in 2001. He is an associate editor of IEEE TRANSACTIONS ON
FUZZY SYSTEMS He is a Fuzzy Systems Technical Committee member of the
IEEE Neural Network Council.
Chung-Shi Tseng (M’01) received the B.S. degree
from the Department of Electrical Engineering,
National Cheng Kung University, Tainan, Taiwan,
R.O.C., the M.S. degree from the Department of
Electrical Engineering and Computer Engineering,
University of New Mexico, Albuquerque, NM, and
the Ph.D. degree in the electrical engineering from
National Tsing-Hua University, Hsin-Chu, Taiwan.
He is now an Associate Professor at Ming Hsin
University of Science and Technology, Hsin-Feng,
Taiwan. His research interests are in signal processing, nonlinear robust control, fuzzy control, and robotics.