Adaptive Robust State Observers of Uncertain Dynamical
Systems with Time-Varying Delays
Hansheng Wu
Department of Information Science, Prefectural University of Hiroshima
Hiroshima City, Hiroshima 734-8558, Japan
Abstract—The problem of adaptive robust state observer design is considered for a class of uncertain dynamical systems
with Time-varying delays. A new method is presented whereby a class of memoryless adaptive robust state observers with
simpler structure is proposed. It is also shown that by employing the proposed adaptive robust state observer, the observation
error between the observer state estimate and the true state can be guaranteed to be uniformly exponentially convergent
towards a ball which can be as small as desired. Finally, a numerical example is given to demonstrate the validity of the
results.
Keywords-Adaptive control; robust control; time-delay systems; state observer; convergence
1. Introduction
It is well known that the problem of robust state observer
design for dynamical systems with significant uncertainties
has received considerable attention of many researchers, and
some approaches to designing a robust state observer has
been developed in the past decades (see, e.g. [1]-[7] for
time--delay systems). Here, it is worth pointing out that the
time delays are often assumed to be known, and such delays
are employed to construct some types of state observers, in
most of works which are concerned with state observer
design problem of time-delay dynamical systems. Moreover,
in the control literature on time-delay dynamical systems,
the terms including delayed state variables are generally
assumed either to be linear or to be linear norm-bounded in
the states.
On the other hand, for dynamical systems with
significant uncertainties, the upper bounds of the vector
norms on the uncertainties are generally supposed to be
known, and such bounds are employed to construct some
types of robust state observers (see, e.g. [1,2,3] or [6] for
time-delay systems). However, in a number of practical
control problems, such bounds may be unknown, or be
partially known. Therefore, for such a class of uncertain
dynamical systems, an adaptive scheme should be
introduced to update these unknown bounds to construct
some types of robust state observers. In general, such an
observer is called adaptive robust state observer.
In the past decades, some efforts have been made to
consider the problem of adaptive robust state observer for
uncertain dynamical systems with the unknown bounds of
uncertainties or perturbations (see, e.g. [8,9] and the
references therein). In particular, in some recent papers, the
problem of adaptive robust state observer is also considered
for uncertain time-delay systems. In [10], for example, the
problem of adaptive robust state observer design is
considered for a class of uncertain systems with delayed
state perturbations.
In this paper, similar to [10], we also consider the
problem of adaptive robust state observer design for a class
of uncertain nonlinear time-delay systems. However, being
different from [10], we assume that the time-varying delays
are any nonnegative continuous and bounded functions
which do not require that their derivatives have to be less
than one. We present a new method whereby a class of
continuous memoryless adaptive robust state observers with
a rather simpler structure is proposed. We also show that the
proposed adaptive robust state observers can guarantee that
the observation error between the observer state estimate and
the true state converges uniformly exponentially towards a
ball which can be as small as desired. In addition, because
the adaptive robust state observers proposed in the paper are
completely independent of time delays, the results obtained
here may be applicable to a class of dynamical systems with
uncertain time delays.
The paper consists of the following parts. In Section 2,
the state observation problem to be tackled is stated and
some standard assumptions are introduced. In Section 3, a
class of memoryless adaptive robust state observers is
proposed and the corresponding convergence analysis is
made. In Section 4, a numerical example is given to illustrate
the validity of the results. Finally, the paper concludes in
Section 5 with a brief discussion of the results obtained in
the paper.
2. Problem Formulation
We consider a class of uncertain dynamical systems with
time-varying delay described by the following differential-
difference equations:
))((()()(
)( thtxftButAx
dt
tdx '
(1a)
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)()( tCxty
(1b)
where Rt  is the time, n Rtx )( is the current value of
the state,
m
Rtu)(
is the control vector,
p
Rty)(
is
the output vector, A , B , C are constant matrices of
appropriate dimensions, and the nonlinear function
)('f
represents the delayed state perturbations and is
assumed to be continuous in all their arguments. Moreover,
the constant matrix C is assumed to be of full rank. In
addition, the time delay )(th is assumed to be any
continuous bounded function satisfying
hth ddd )(0 W
(2)
where W and h are any nonnegative constants, and W
is not required in general to be zero. It is worth noting that
the function h(t) is not required to be known for the system
designer.
The initial condition for system (1) is given by
),()( ttx F ],[
00
thtt 
(3)
where )(t F
is a given continuous function.
Now, the question is how to design a continuous state
observer with the output y(t) and input u(t) such that the
state estimate )(
ˆtx
can converge to the original state x(t), as
far as possible, in the presence of the delayed state
perturbations.
Before proposing our state observers, we introduce for
system (1) the following assumptions.
Assumption 2.1. The pair ),(CA given in system (1)
is completely observable. That is, there exists a matrix
K 
R^{n
u
p} such that the matrix
KCAA  :
0
is
Hurwitz.
Assumption 2.2. There exists a vector nonlinear function
pn RR o :)(
[
such that the following matching
condition can be satisfied
)))((()))((( 1thtxCPthtxf T ' [
(4)
where the positive definite matrix P is the unique solution to
the Lyapunov equation of the form
QPAPAT 00
(5)
for any given positive definite matrix nn RQ u .
Assumption 2.3. The uncertain function
)( [
is bounded
in Euclidean norm. Moreover, there exists an unknown
constant R T such that for any 0tt , we have
))(()))((( thtxthtx d  T[
(6)
Without loss of generality, we introduce the following
definition:
21
)(:

THK
\
where
K
is any given positive constant, and
H
is any
positive constant which is not required to be known for the
system designer. It is obvious from Assumption 2.3 that

\
is still an unknown positive constant.
Remark 2.1.Assumption 2.2 represents that the
uncertain time-delay dynamical systems, described by (1),
have a special structure which is generally called a matching
condition about the nonlinearity and uncertainty, and is a
rather standard assumption for the problem of robust state
observers. In fact, in number of practical control systems,
particularly mechanical systems, environmental systems,
ecological systems, industrial electronic systems, and so on,
such a condition is often satisfied (see, e.g. [1,2], [8]-[10],
and the references therein). In addition, Assumption 2.3
defines the uncertainty bands (in general state or output
dependent).
3. Main Results
In this section, we propose a class of continuous adaptive
robust state observers with a rather simple structure, which is
independent of the time delays and bounds of nonlinear
perturbations, such that the state estimate
)(
ˆtx
can
converge as far as possible to the original state )(tx . For
this, the observation error between the observer state
estimate and the true state is defined as
)(
ˆ
)()(txtxte 
(7)
Now, for the uncertain time-delay dynamical systems
described by (1) we propose the following adaptive robust
state observer without time delays:
)),(
ˆ
),(),(
ˆ
(
))(
ˆ
)(()()(
ˆ
)(
ˆ
1tttytxECP
tytyKtButxA
dt
t xd
T
\



(8a)
)(
ˆ
)(
ˆtxCty
(8b)
where
n
Rtx )(
ˆ is the state estimate vector,
p
Rty)(
ˆ is the output vector of the observer, pn
RK u 
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is constant matrix, called the gain matrix, and
)),(
ˆ
),(),(
ˆ
(tttytxE \
is an auxiliary vector function
which will be given by
)()(
ˆ
2
1
)),(
ˆ
),(),(
ˆ
(tCettttytxE \K\
(8c)
In particular, the time function

Rt)(
ˆ
\
in (8) is the
estimate of the unknown parameter

R
\
which is
updated by the following adaptation law:
2
)()(
ˆ
)(
ˆtCet
dt
td
JK\JV
\

(9)
where V and J are any given positive constants.
Thus, it is obvious from (1) and (8) that we can easily
obtain the observation error time-delay dynamical systems
of the form:
)),(
ˆ
),(),(
ˆ
(
)))((()(
)(
1
0
tttytxECP
thtxfteA
dt
tde
T
\


'
(10)
On the other hand, letting

 \\
\
)(
ˆ
)(
~tt
, we can
rewrite (9) as the following adaptation error systems:

 JV\JK\JV
\ 2
)()(
~
)(
~
tCet
dt
td
(11)
In the following, by the pair ))(
~
),(( tte \
we denote
the solutions to the time-delay error systems described by
(10) and (11). Thus, we have the following theorem which
shows that the state estimate )(
ˆtx
of adaptive robust
observer (8) with the estimate )(
ˆt
\
given in updating law
(9) can converge as far as possible to the original state )(tx .
Theorem 3.1. Consider the time-delay error systems
described by (10) and (11) satisfying Assumptions 2.1 to 2.3.
Then, the solutions ))(
~
),(,;)(
~
,( 000 ttette \\
to (10)
and (11) are uniformly bounded and the error )(te
converges uniformly exponentially towards a ball which can
be as small as desired, in the presence of the nonlinear time-
varying delayed state perturbations.
Proof: For the adaptive time-delay error systems
described by (10) and (11), we first define a quasi-Lyapunov
functional as follows.
)(
~
2
1
)()()
~
,(
21
ttPeteeV
T
\J\


(12)
where P is the solution of Lyapunov equation (5), and
J
is
any positive constant.
Let ))(
~
),(( tte \ be the solution of the time-delay error
systems described by (10) and (11) for any 0 tt t. Then, by
taking the derivative of )(V along the trajectories of (10)
and (11), from Assumption 2.2 and Assumption 2.3 it can be
obtained that for any 0 tt t,
dt
td
t
tttytxECte
thtxtcetQete
dt
edV
TT
T
)(
~
)(
~
)),(
ˆ
),(),( ˆ ()(2
))(()(2)()(
)
~
,(
1
\
\J
\
T
\




d
(13)
After some trivial manipulations, from (13) we can obtain
that for any 0 tt t,








d
N
DHO
N
O
T
1
)(
)(
1
)(
)(
11
min
)(
0
1
min
20P
etV
P
te tt
(14)
where T , H , D , and  N are any positive constants.
Then, from (14) we know that
)(te
is uniformly
bounded, and converges uniformly exponentially to a ball
)( G %
where
})1/()()(|{:)(
11
min

 d % NDHOGG Ptee
On the other hand, from the adaptation law given in (9)
the estimate value )(
ˆt
\
of the uncertain parameter

\
is
also uniformly bounded since the error )(te is bounded.
Thus, we can complete the proof of this theorem.

Remark 3.1. In the proof of Theorem 3.1, it is assumed
that the positive constant H should satisfy some conditions.
However, the adaptive robust state observers proposed in (8)
with (9) is completely independent of such a constant. Thus,
it is not necessary for the system designer to know or choose
this constant. In fact, the adaptive robust state observers
proposed in (8) with (9) can adjust automatically to counter
the destabilizing effects of the uncertainties, delayed state
perturbations, and external disturbances.
Remark 3.2. In the light of the definition of the positive
constant H , it can be observed that by decreasing
sufficiently the value of V
which is an adjustable parameter
of the adaptation law, one can obtain the radius G of the
ball as small as desired. Thus, the system designer can tune
the size of the residual set by adjusting properly the
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parameter V which is introduced in the adaptation law
described by (9).
4. Illustrative Example
To illustrate the utilization of our approach, in this
section, we consider the following numerical example.
)))((()(
1
1
)(
30
21
)( thtxftutx
dt
tdx '
»
¼
º
«
¬
ª

»
¼
º
«
¬
ª


>@ )(11)( txty
(15)
where
»
¼
º
«
¬
ª

')))((sin(
0
)))(((
1thtx
thtxf
Q
(16)
and where Q
is an uncertain parameter.
It is obvious that the pair ),(CA
of the systems
described by (15) is completely observable. Thus, we can
arbitrarily assign the eigenvalues of the matrix
KCAA 
0. Here, we will select the eigenvalues of the
matrix 0 A as >@ 32 
. Then, the corresponding gain
K
is given by >@
T
K01 , and the matrix 0 A is given by
»
¼
º
«
¬
ª


30
12
0
A
Now, we have to find a matrix Q such that Assumption
2.2 is satisfied. For this, from (5) we can have that
»
¼
º
«
¬
ª
43
38
Q
(17)
which results in
»
¼
º
«
¬
ª
11
12
P
(18)
Therefore, from (15) to (18) it can easily be verified that
Assumption 2.2 is satisfied, i.e.
)))((()))((( 1thtxCPthtxfT ' [
where
))(()))((( thtxthtx d

T[
For (8) and (9), we select the following parameters:
2.0,0.2,0.8 VJ
K
Thus, from (8) and (9), we can obtain an adaptive robust
state observer with a rather simple structure for this
numerical example. It is obvious from Theorem 3.1 that the
state estimate )(
ˆtx
of the proposed adaptive robust state
observer can converge uniformly ultimately to the original
state )(tx of the uncertain time-delay system.
For simulation, we give the uncertain parameter
5.0 Q
, and initial conditions as follows.
>@ ]0,[,)cos(0.8)cos(0.8)( httttx
T

>@
T
x0.30.2)0( ˆ
0.12)0(
ˆ
\
In addition, for simplicity, the control input will be set to
be zero, i.e. 0)( {tu
.
With the chosen parameter settings above, the results of
simulation are shown in Fig.1 and Fig.2 for this numerical
example.
It can be observed from Fig.1 that the state estimate
)(
ˆtx
of the proposed adaptive robust state observer indeed
converges uniformly ultimately to the original state )(tx of
the time-delay system described by (15) and (16). On the
other hand, it can be known from Fig.2 that similar to the
conventional adaptation law with sigma-modification, the
improved one makes indeed the estimate values of the
unknown parameters decreasing. (The figures of the
simulation will be displayed in the presentation.)
5. Concluding Remarks
The problem of adaptive robust state observers has been
considered for a class of uncertain systems with time-
varying delays. A new method has been presented whereby a
class of continuous memoryless adaptive robust state
observers with a rather simpler structure is constructed.
Since our adaptive state observers do not involve the upper
bound of uncertainties, such upper bound is not required to
be known for the system designer. It has been also shown
that the proposed adaptive robust state observers can
guarantee that the observation error between the observer
state estimate and the true state converges uniformly
exponentially towards a ball which can be as small as
desired.
In addition, because the adaptive robust state observers
proposed in the paper are completely independent of time
delays, the results obtained here may be applicable to a class
of dynamical systems with uncertain time delays.
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6. Acknowledgment
This work was supported in part by the Japan Society for
the Promotion of Science (JSPS) under Grant (C)22560405.
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