INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
Int. J. Robust Nonlinear Control
2010;
20
:504–514Published online 14 April 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1443
Observer design for nonlinear discretetime systems: Immersion and dynamicobserver error linearization techniques
Jian Zhang
1
,
∗
,
†
Gang Feng
2
and Hongbing Xu
1
1
School of Automation Engineering
,
University of Electronic Science and Technology of China
,
Chengdu
,
Sichuan 610054
,
People’s Republic of China
2
Department of Manufacturing Engineering and Engineering Management
,
The City University of Hong Kong
,
Hong Kong
SUMMARYThis paper focuses on the observer design for nonlinear discretetime systems by means of nonlinear observer canonicalform. At ﬁrst, sufﬁcient and necessary conditions are obtained for a class of autonomous nonlinear discretetime systems tobe immersible into higher dimensional observer canonical form. Then a method called dynamic observer error linearizationis developed. By introducing a dynamic auxiliary system, the augmented system is shown to be locally equivalent to thegeneralized observer form, whose nonlinear terms contain auxiliary states and output of the system. A constructive algorithmis also provided to obtain the state coordinate transformation. These results are an extension of their counterparts of nonlinearcontinuoustime systems to nonlinear discretetime systems (
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q
2009 John Wiley & Sons, Ltd.
Received 31 July 2008; Revised 11 December 2008; Accepted 5 January 2009KEY WORDS
: nonlinear; discretetime; observer
1. INTRODUCTIONObserver design for nonlinear systems has been anactive research topic for decades. In general, thereare two main approaches. In the ﬁrst approach
[
1–3
]
,
∗
Correspondence to: Jian Zhang, School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, People’s Republic of China.
†
Email: zhangj@uestc.edu.cnContract
/
grant sponsor: Research Grants Council of the HongKong Special Administrative Region of China; contract
/
grantnumber: CityU112907
nonlinear systems are brought into a socalled observerform by means of a nonlinear state transformation.This special form yields a completely linear errordynamics of state such that linear observer theorycan be applied. However, for this approach, theconditions for achieving the desired state coordinate transformation are in general difﬁcult to satisfy,which requires the solution of a set of partial differential equations. More recently, signiﬁcant researchefforts have been directed to alleviate the difﬁcultyby developing more generalized state transformationprocedures and more generalized canonical forms sothat the larger class of nonlinear systems can be dealtwith
[
4,5
]
.
Copyright
q
2009 John Wiley & Sons, Ltd.
OBSERVER DESIGN FOR NONLINEAR DISCRETETIME SYSTEMS
505On the other hand, the second approach is concernedwith designing observers for nonlinear systems withoutthe need of state transformation. The highgainobservers in
[
6–8
]
are attractive because of theirrobustness. Available design techniques include poleplacement approach
[
6
]
, Riccati equation approach
[
7
]
and Lyapunov equation approach
[
8
]
. However, theirapplication is quite limited due to the strict conditionsthat the system must be in semitriangular structure.An observer for Lipschitz nonlinear systems has beenstudied in
[
9,10
]
, where the system was split into alocally or globally Lipschitz part, and a linear partwas assumed to be observable from the output. Theobserver can be designed by solving a Lyapunov equation or a Riccati equation. Unfortunately, the approachsuffers from a limitation that the Lipschitz constanthas to be small.The above results have been obtained for the continuous case. Later on, the observer design for nonlineardiscretetime systems has also attracted much interestof many researchers particularly since digital technology is increasingly used at the implementationstage of modelbased control law. Based on geometriccontrol theory, sufﬁcient and necessary conditions weregiven in
[
11,12
]
under which a nonlinear discretetime systems may be transformed into a system inobserver form via a state transformation. In contrast tothese approaches, some of the restrictive assumptionswere relaxed by using past values of the output in
[
13
]
. Another nonlinear observer design method wasproposed in
[
14
]
that can be viewed as the discretetimeversion of the continuoustime observer presented in
[
15
]
. Unfortunately, the restrictive global Lipschitzconditions inevitably arise in the discretetime versionas well. The paper
[
16
]
presented a robust observerfor nonlinear discretetime systems, which is basedon inverting the map from a past system state to theoutput sequence. An actual estimation is calculatedby repeatedly applying the system function to theestimation of the past system state. In
[
17
]
, the authorsproposed a nonlinear discretetime observer based onthe Newton algorithm for the simultaneous solution of a set of nonlinear equations, which yields an asymptotic observer for a broad class of systems. However,this approach requires timewise iterative solution of a nonlinear algebraic equation, and the convergenceconditions derived can not be easily checked in practice. In addition, a simple and useful reducedorderobserver for a large class of nonlinear discretetimesystems can be found in
[
18
]
, where a set of asymptotic convergence conditions for observer errors wasestablished, which appeared to be nonrestrictive in thepresence of mild nonlinearities. The observer designproblem was translated into the problem of solving asystem of ﬁrst order linear nonhomogeneous functionalequations in
[
19
]
, and a set of necessary and sufﬁcientconditions for the solvability of these equations wasderived based on functional equations theory.More recently, immersions rather than diffeomorphism techniques have been investigated in theobserver design for nonlinear continuoustime systems,which is motivated by the fact that there exists aclass of nonlinear systems that cannot be transformedinto observer form via diffeomorphism, but can beimmersed into a higher dimensional observer canonicalform. Several constructive algorithms for immersionare given in
[
20–24
]
. Similar to the idea of dynamicfeedback, a new framework called dynamic observererror linearization has been introduced for observerdesign recently
[
25,26
]
. They provide sufﬁcient andnecessary conditions under which nonlinear systemscan be transformed into higherdimensional observerforms by adding auxiliary dynamic systems such as achain of integrators or a stable linear system. Theseapproaches are recognized as efﬁcient methods torelax the structural restriction in the observer errorlinearization.In this paper, we extend the immersion and dynamicobserver error linearization techniques to nonlineardiscretetime systems. A number of novel results forobserver design of nonlinear discretetime system arepresented. Motivated by
[
27,28
]
, we also provide aconstructive algorithm for dynamic observer errorlinearizationbasedoninput–outputdifferenceequation.The paper is organized as follows. Some preliminaries are provided in Section 2. In Section 3, a numberof novel results of observer design for nonlineardiscretetime systems as well as a constructive algorithm for dynamic observer error linearization arepresented. Some illustrative examples are also includedto demonstrate the proposed methods. Conclusions aregiven in Section 4.
Copyright
q
2009 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control
2010;
20
:504–514DOI: 10.1002/rnc
506
J. ZHANG, G. FENG AND H. XU
2. PRELIMINARIESConsider a class of nonlinear discretetime systemsdescribed by the following form
x
(
k
+
1
)
=
f
(
x
(
k
))
y
(
k
)
=
h
(
x
(
k
))
(1)where
x
(
k
)
∈
R
n
,
f
:
R
n
→
R
n
and
h
:
R
n
→
R
, aresmooth functions, with
f
(
0
)
=
0 and
h
(
0
)
=
0.
Deﬁnition 1
Deﬁne the observability map
:
R
n
→
R
n
by
(
x
)
=
dhd
(
h
◦
f
)...
d
(
h
◦
f
n
−
1
)
(2)where
f
k
denotes the
k
times composite function
f
◦···◦
f
(
k
times),andassumetheorigintobeequilibriumpoint of system (1). Then we call system (1) locallyobservable on an open subset
U
∈
R
n
containing thesrcin if rank
(
(
x
))
=
n
,
∀
x
∈
U
.
Deﬁnition 2
The following canonical structure is called nonlineardiscretetime observer canonical form, or in shortobserver form:
z
(
k
+
1
)
=
Az
(
k
)
+
(
y
(
k
))
y
(
k
)
=
Cz
(
k
)
(3)where
A
=
0 1
···
0
...... ............
10
··· ···
0
C
= [
1 0
···
0
]
where
T
=[
1
2
···
n
]
is a vector of scalar functions of
y
(
k
)
.If the system (1) can be transformed to the observerform (3) by means of state and output transformations,then an observer can be constructed easily so that theobserver error satisﬁes a linear dynamical system equation. A necessary and sufﬁcient condition on the existence of state transformations is given in the followinglemma.
Lemma 1
(
Lin and Byrnes
[
12
]
)Suppose that the nonlinear discretetime autonomoussystem of the form (1) is observable on an open subset
U
∈
R
n
containing the srcin. Then system (1) is locallyequivalent to a system in the nonlinear discretetimeobserver form (3) via a state transformation if and onlyif the Hessian matrix of the function
h
◦
f
n
◦
−
1
(
s
)
isdiagonal, where
x
=
−
1
(
s
)
is the inverse map of
s
=[
h
(
x
),
h
◦
f
(
x
),
···
h
◦
f
n
−
1
(
x
)
]
T
(4)
Remark 1
The condition of Lemma 1 is equivalent to
2
(
h
◦
f
n
(
x
))
(
h
◦
f
i
(
x
))
(
h
◦
f
j
(
x
))
=
0
(
i
,
j
=
0
,...,
n
−
1
,
i
=
j
)
It should be noted that a nonlinear discretetime systemmight not be able to be transformed to the observerform with its present output by a state transformation,but might be transformable with a transformed output,as shown by the following example.
Example 1
Consider a nonlinear discretetime system deﬁned onan open subset
U
0
={
(
x
1
,
x
2
)
:
x
1

<
1
,

x
2

<
1
}
x
1
(
k
+
1
)
=
x
2
(
k
)
x
2
(
k
+
1
)
=
1
/
2
(
x
1
(
k
)
+
x
22
(
k
))
y
(
k
)
=
sin
(
x
1
(
k
))
(5)By means of Lemma 1, the system is not transformableinto an observer form (3) under state transformation,because of
2
(
y
(
k
+
2
))
(
y
(
k
+
1
))
(
y
(
k
))
=
0
Copyright
q
2009 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control
2010;
20
:504–514DOI: 10.1002/rnc
OBSERVER DESIGN FOR NONLINEAR DISCRETETIME SYSTEMS
507Yet, applying both local output and state transformation
z
1
(
k
)
=
x
1
(
k
)
z
2
(
k
)
=
x
2
(
k
)
−
1
/
2
x
21
(
k
)
¯
y
(
k
)
=
arcsin
y
(
k
)
(6)system (5) is equivalent to
z
1
(
k
+
1
)
=
z
2
(
k
)
+
1
/
2
¯
y
2
(
k
)
z
2
(
k
+
1
)
=
1
/
2
¯
y
(
k
)
¯
y
(
k
)
=
z
1
(
k
)
(7)which is exactly in the observer form (3).3. MAIN RESULTSIn this section, we give two main results of this work.
3.1. An immersion approach to nonlinear discretetime observer design
Let us ﬁrst consider a motivating example.
Example 2
Consider the discretetime system
x
1
(
k
+
1
)
= −
1
/
4
x
1
(
k
)
+
x
2
(
k
)
+
x
22
(
k
)
x
2
(
k
+
1
)
=
1
/
2
x
2
(
k
)
y
(
k
)
=
x
1
(
k
)
(8)It can be checked that it is not transformable intotwodimensional observer form even if output transformation and state diffeomorphism are all employed.However, this system is immersible into threedimensional observer form. In fact, we deﬁne animmersion transformation
z
=
(
x
)
:
R
2
→
R
3
as
z
1
(
k
)
=
x
1
(
k
)
z
2
(
k
)
= −
34
x
1
(
k
)
+
x
2
(
k
)
+
x
22
(
k
)
z
3
(
k
)
=
x
1
(
k
)
8
−
14
x
2
(
k
)
−
12
x
22
(
k
)
(9)Then the system is immersible into the observer form
z
1
(
k
+
1
)
=
z
2
(
k
)
+
1
/
2
y
(
k
)
z
2
(
k
+
1
)
=
z
3
(
k
)
+
1
/
16
y
(
k
)
z
3
(
k
+
1
)
= −
1
/
32
y
(
k
)
y
(
k
)
=
z
1
(
k
)
(10)In this case, the twodimensional discretetime systemadmits the following threedimensional observer:
ˆ
z
(
k
+
1
)
=
(
A
−
KC
)
ˆ
z
(
k
)
+
(
y
(
k
))
ˆ
x
1
(
k
)
= ˆ
z
1
(
k
),
ˆ
x
2
(
k
)
=ˆ
z
1
(
k
)
+
2
ˆ
z
2
(
k
)
+
4
ˆ
z
3
(
k
)
(11)where
A
=
0 1 00 0 10 0 0
,
C
=[
1 0 0
]
=
12
y
(
k
),
116
y
(
k
),
−
132
y
(
k
)
T
and
K
is a design parameter column vector such that
A
−
KC
have desired prescribed eigenvalues.Now we introduce a deﬁnition on immersion analogous to continuoustime systems.
Deﬁnition 3
System (1) is said to be immersible into
n
+
d
dimensional observer form (3) if there exists a
C
∞
mapping
:
D
→
(
D
),
D
⊂
R
n
,
(
D
)
⊂
R
n
+
d
and alocal diffeomorphism
:
I
o
→
(
I
o
),
I
o
⊂
R
,
(
I
o
)
⊂
R
such that for every initial condition
x
0
and
z
0
with
z
0
=
(
x
0
)
then
◦
h
(
x
k
(
x
0
))
=
Cz
k
(
z
0
)
for every
k
,0
k
K
x
. Here
K
x
=
sup
{
k
0

x
k
(
x
0
)
∈
D
}
.
Remark 2
Deﬁnition 3 is the counterpart of the continuoustimeversion for discretetime systems. For the detaileddiscussion, refer to Deﬁnition 2.2 and Theorem 2.3in
[
21
]
.
Theorem 1
System (1) can be immersed into the observer form (3)if and only if there exist a diffeomorphism
, an integer
Copyright
q
2009 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control
2010;
20
:504–514DOI: 10.1002/rnc