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The Polynomial Extended Kalman Filter as an
Exponential Observer for Nonlinear Discrete-Time Systems
Alfredo Germani Costanzo Manes
Abstract— This paper presents some results on the local
exponential convergence of the Polynomial Extended Kalman
Filter (PEKF, see [14]) used as a state observer for deterministic
nonlinear discrete-time systems (Polynomial Extended Kalman
Observer, PEKO). A new compact formalism is introduced for
the representation of the so called Carleman linearization of
nonlinear discrete time systems, that allows for the derivation
of the observation error dynamics in a concise form, similar to
the one of the classical Extended Kalman Filter. The stability
analysis performed in this paper is also important in the
stochastic framework, in that the exponential stability of the
error dynamics can be used to prove that the moments of the
estimation error, up to a given order, remain bounded over time
(stability of the PEKF).
I. INTRODUCTION
This paper considers the state observation problem for
nonlinear discrete-time systems of the type
xt+1 =f(xt,u
t),(1)
yt=h(xt,u
t),(2)
where xt∈Rnis the system state, yt∈Rqis the
measured output, ut∈Rpis the sequence of known inputs.
f(·,·)and h(·,·)are analytic functions of the first argument
(the state). Many approaches have been explored in the
literature for the derivation of asymptotic state observers,
and many types of solutions exist for classes of systems. An
approach widely investigated is to find a nonlinear change of
coordinates and, if necessary, an output transformation, that
transform the system into some canonical form suitable for
the observer design using linear methodologies. Some papers
on the subject are [9],[21],[22], [26],[27], for autonomous
systems, and [6] for nonautonomous systems. These papers
study the conditions for the existence of the coordinate trans-
formation that allows the observer design with linearizable
error dynamics. The drawback of this approach is that in
general the computation of the coordinate transformation,
when existing, is a very difficult task. Another approach
consists in designing observers in the original coordinates,
finding iterative algorithms, typically based on the Newton
method, that asymptotically solve a suitable extension of
the state-output map, [10], [11], [23]. Sufficient conditions
of local convergence are provided, in general, under the
assumption of Lipschitz nonlinearities. Many authors restrict
This work is supported by MiUR (Italian Ministry of University and
Research) and by CNR (National Research Council of Italy).
A. Germani and C. Manes are with the Department of Electrical and
Information Engineering, University of L’Aquila, Poggio di Roio, 67040
L’Aquila, Italy, germani@ing.univaq.it, manes@ing.univaq.it. They are also
with the Istituto di Analisi dei Sistemi ed Informatica del CNR “A. Ruberti”,
Viale Manzoni 30, 00185 Roma, Italy.
the attention to the class of systems characterized by nonlin-
ear dynamics and linear output map (see, e.g. [1], [5], [28]).
The cases of bilinear dynamics and polynomial and rational
output maps are considered in [15],[17].
Although so many different approaches have been studied,
the most popular algorithm of state estimation for nonlinear
system is the Extended Kalman Filter (EKF), see e.g. [2].
The main reasons of the popularity of the EKF is its
simplicity of implementation and its good behavior in most
applications. The use of the Extended Kalman Filter as
a local observer in the deterministic framework has been
investigated [3], [4], [24] and [25].
This paper aims to extend the existing results of local
convergence of the EKF to the Polynomial Extended Kalman
Filter (PEKF) presented in [14]. The same approach used in
[24] for the study of the exponential error convergence of
the EKF has been used in this paper.
The paper is organized as follows. In section II the
EKF equations and the convergence theorem of [24] are
briefly recalled. The PEKO is presented in section III and
the convergence property is discussed in section IV. Some
elements of Kronecher algebra, used throughout the paper,
are briefly reported in the Appendix.
II. THE EKF AS AN OBSERVER
Before to proceed with the construction of a PEKO (Poly-
nomial Extended Kalman Observer), let us briefly recall the
standard form of the EKF and its use as an Observer (EKO:
Extended Kalman Observer), and discuss the convergence
properties following the approach of [24]. From now on, the
following more compact notation will be used for the system
(1)–(2):
xt+1 =fut(xt),(3)
yt=hut(xt).(4)
Global or local assumptions on the uniform boundedness
of the derivatives of the functions fu(x)and hu(x)can
be made. Local assumptions are sufficient in the proof of
convergence of the EKO if the system (3)–(4) is input-state
stable. For this reason the following assumption is made:
Assumption A0. There exist a compact set U⊂Rpand
bounded open sets Ω0and Ω, with Ω0⊂Ω⊂Rn, such that
for any input sequence with ut∈U,∀t≥t0,ifxt0∈Ω0
then xt∈Ω,∀t≥t0. Moreover, ∀u∈U,fu(x)and hu(x)
are analytical functions in Ω.
Let Ωdenote the closure of Ω.
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In order to apply the standard Kalman Filter to the
nonlinear system (3)–(4), it is useful to represent the state
transition map fu(x)and the output map hu(x)using a first
order Taylor expansion around the best estimates available:
the observation ˆxtfor fu(x)and the prediction ˜xtfor hu(x):
xt+1 =fut(ˆxt)+At(xt−ˆxt)+ϕf(xt,ˆxt),(5)
yt=hut(˜xt)+Ct(xt−˜xt)+ϕh(xt,˜xt),(6)
where Ctand Atare the Jacobians of hut(x)and fut(x)
computed at the predicted and estimated state, rspectively.
Using the notation introduced in the Appendix, eq. (88),
At=∇x⊗fut(x)|ˆxt,C
t=∇x⊗hut(x)|˜xt.(7)
The remainders ϕfand ϕhare such that there exist positive
f,h,γf,γhsuch that
ϕf(x, ¯x)≤γfx−¯x2,∀x−¯x≤f,
ϕh(x, ¯x)≤γhx−¯x2,∀x−¯x≤h.(8)
(In [24] the inequalities (8) are assumed to hold for all
¯x∈Rnand u∈Rp.) The EKO approach consists in
neglecting the remainders in the representation (5)–(6), so
that it appears as a linear system with known forcing terms,
and in applying the standard Kalman Filter equations. An
initial state estimation ¯x0is needed as a starting value of the
EKO. Also a positive definite (PD) matrix P0is needed to
inizialize the Riccati equations that provide the Kalman gain
Kt. The EKO algorithm is reported below. The symbol In
denotes the identity matrix of dimension n.
Extended Kalman Observer (EKO)
Starting values: ˜xt0=¯x0,
Pt0=P0,t=t0,
˜yt=hut(˜xt),output prediction (9)
Ct=∇x⊗hut˜xt,(10)
Kt=
PtCT
t(Ct
PtCT
t+Rt)−1
,(11)
ˆxt=˜xt+Kt(yt−˜yt),state observation (12)
Pt=(In−KtCt)
Pt,(13)
˜xt+1 =fut(ˆxt),state prediction (14)
At=∇x⊗futˆxt,(15)
Pt+1 =α2AtPtAT
t+Qt.(16)
Qtand Rtare known sequences of PD matrices that act as
forcing terms in the Riccati equations, and must be chosen
uniformly upper and lower bounded over t∈Z(in fact, in
[24] they are chosen constant). In the EKO such sequences
are free design parameters, while in the stochastic framework
(EKF) they are the covariances of the state and output
noises. The constant coefficient α≥1has the meaning of
aforgetting factor and provides exponential data weighting
when α>1.
The convergence analysis of the EKO in [24] has been
pursued by studying the stability of the recursive equation
that governs the prediction error xt−˜xt. This equation is
obtained subtracting the prediction ˜xt+1 given by (14) from
xt+1 as given by (5):
xt+1 −˜xt+1 =At(xt−ˆxt)+ϕf(xt,ˆxt),(17)
and then finding a suitable expression for the estimation error
xt−ˆxt. Considering the two identities below, obtained using
(12) and (9),
xt−ˆxt=xt−˜xt−Kt(yt−˜yt),(18)
yt−˜yt=Ct(xt−˜xt)+ϕh(xt,˜xt),(19)
the following recursion can be easily obtained
xt+1 −˜xt+1 =At(In−KtCt)(xt−˜xt)+ϕ0,(20)
where ϕ0=ϕf(xt,ˆxt)−Ktϕh(xt,˜xt).(21)
The convergence result in [24] is based on the proof of
asymptotic stability of equation (20), and is summarized
below:
Theorem 1. Consider the EKO equations (9)–(16), and let
the following assumptions hold
i) There are positive numbers a, ¯c, ¯p, p such that for all
t≥t0At≤a, Ct≤¯c, (22)
pIn≤
Pt≤¯pInpIn≤
Pt≤¯pIn.(23)
ii) Atis nonsingular ∀t≥t0.
iii) There are positive real numbers f,h,γf,γh, such
that inequalities (8) hold.
Then, there exist positive real numbers η,0,θ, with θ>α,
such that, if xt0−˜xt0<
0, then
xt−˜xt≤ηxt0−˜xt0θ−(t−t0),(24)
that means that the EKO is a local exponential observer.
Remark 1.The parameter α≥1, that appears in equation
(16), can be tuned to assign the error convergence rate.
However, in the proof of Theorem 1 in [24] it appears that
the larger is chosen α, the smaller is the convergence region
0.
Remark 2.The existence of lower and upper bounds for the
PD matrices
Ptand Ptcan only checked on line, and is
ensured if the pair (At,C
t)satisfies a uniform observability
condition (see e.g. [12]).
Remark 3.It is important to stress that the proof reported in
[24], based on the bounds (8) on the norms of the remainders
ϕfand ϕh, can be easily modified to deal with bounds of
higher order, i.e. of the type
ϕf≤γfx−¯xk,ϕh≤γhx−¯xk,(25)
for k>2.
Remark 4.The bounds (8) assumed in Theorem 1, in [24] are
formulated in a global form, i.e. the inequalities are assumed
to hold for all ¯x∈Rnand for all u∈Rp. This can be a too
strong assumption. However, the proof of Theorem 1 can be
suitably modified when inequalities (8) hold only on bounded
sets. In this case the additional assumption A0is required,
so that it is sufficient that properties (8) on the remainders
are true for all ¯x∈Ω.
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III. POLYNOMIAL EXTENDED KALMAN OBSERVER
The PEKF algorithm presented in [14] is based on the
polynomial approximation of the state transition map fu(x)
and of the output map hu(x)of the system (3)–(4). The use
of the filter in [14] as an Observer for deterministic systems
is denoted PEKO in this paper.
The formalism of the Kronecker algebra is used in this
paper for the efficient manipulation of multivariate polyno-
mials. The definition of the Kronecker product ⊗of matrices
is given in the Appendix, together with other relevant defini-
tions and properties used throughout this work. The symbol
v[k]denotes the Kronecker power of a vector v∈Rn. For the
kind of computations carried out in this paper, it is extremely
useful the definition of a symbol for the vector that collects
all the Kronecker powers of a given vector from 1 up to a
given degree m. The symbol chosen is [·]m, and operates
on vectors v∈Rnas follows
[v]m=⎡
⎢
⎢
⎢
⎣
v
v[2]
.
.
.
v[m]
⎤
⎥
⎥
⎥
⎦,v[0] =1,
v[k+1] =v[k]⊗v. (26)
Recalling that v[k]∈Rnk, then [v]m∈Rnm, with nm=
m
k=1 nk. A property of the symbol [·]mrepeatedly used
throughout the paper is the following (see Lemma (3)) in
the Appendix)
[v−¯v]m=Im(−¯v)[v]m−[¯v]m,∀v, ¯v∈Rn,(27)
where the matrix Im(−¯v)is defined in the Appendix,
eq. (81).
The PEKF in [14] is based on the Taylor polynomial
approximation of a chosen degree m>1of both maps
fut(x)and hut(x). The convergence of the PEKF used as
an observer (PEKO) requires the assumption A0, in order to
ensure the existence of uniform upper bounds on the norms
of the remainders of the Taylor approximation. The Taylor
expansion of degree mof the output map hut(xt)around
the prediction ˜xtis
hut(xt)=
m
j=0
1
j!∇[j]
x⊗hut(x)˜xt(xt−˜xt)[j]
+ϕh(xt,˜xt).
(28)
where the differential symbol ∇[j]
x⊗is defined in the Ap-
pendix, eq. (88). Based on the Lagrange remainder formula,
the following bound can be given, for all (x, ¯x, u)∈Ω×
Ω×U
ϕh(x, ¯x)≤γhx−¯xm+1,(29)
where γh= sup
(x,u)∈Ω×U
∇[m+1]
x⊗hu(x)
(m+ 1)! .(30)
Using the symbol [·]m, the Taylor formula (28) can be
written in the compact form
hut(xt)=hut(˜xt)+Hm(˜xt)[xt−˜xt]m+ϕh,(31)
where matrix Hm(˜xt)∈Rq×nmhas a row-block structure
Hm(x)=Hm(x)1··· Hm(x)m(32)
where Hm(x)j=1
j!∇[j]
x⊗hut(x),(33)
(for a simpler notation, the dependence of Hmon utis not
shown). Using the identity (27), equation (31) can be written
as
hut(xt)=hut(˜xt)+Ct[xt]m−[˜xt]m+ϕh,(34)
where Ct=Hm(˜xt)Im(−˜xt).(35)
Now define the polynomial extended state Xt∈Rnmand
the selection matrix Σ∈Rn×nmas follows
Xt=[xt]m,Σ=In0n×(nm−n),(36)
so that x=Σ[x]m,∀x∈Rn, and in particular
xt=ΣXt.(37)
The use of Xtin (34), allows to write the output equation
(4) as
yt=hut(˜xt)+CtXt−[˜xt]m+ϕh,(38)
where ϕh=ϕh(ΣXt,˜xt). In this form the output equation
depends linearly on the extended state Xt. In order to use
this linear form in a linear filter, a linear transition function
is needed for the extended state. Consider the transition map
of the extended state
Xt+1 =[xt+1 ]m=[fut(xt)]m=⎡
⎢
⎣
fut(xt)
.
.
.
f[m]
ut(xt)
⎤
⎥
⎦.(39)
The Taylor formula for the component f[k]
ut(xt)around the
current estimate ˆxt, is the following
f[k]
ut(xt)=
m
j=0
1
j!∇[j]
x⊗f[k]
ut(x)ˆxt(xt−ˆxt)[j]
+ϕf,k(xt,ˆxt).
(40)
The remainder is such that, ∀(x, ¯x, u)∈Ω×Ω×U,
ϕf,k(x, ¯x)≤γf,kx−¯xm+1.(41)
where γf,k = sup
(x,u)∈Ω×U
∇[m+1]
x⊗f[k]
u(x)
(m+ 1)! .(42)
Using the symbol [·]m, the extended transition map
[fut(xt)]mcan be written in the compact form
[fut(xt)]m=fut(ˆxt)m+Fm(ˆxt)[xt−ˆxt]m+ϕf(xt,ˆxt),
(43)
where the matrix Fm(ˆxt)∈Rnm×nmis made of m×m
blocks, defined, for k=1,...,m, j =1,...,m, as
Fm(x)k,j =1
j!∇[j]
x⊗f[k]
ut(x)∈Rnk×nj(44)
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(the dependence of Fmon utis not shown). For (x, ¯x, u)∈
Ω×Ω×U, the remainder ϕfobeys the inequality
ϕf(x, ¯x)≤γfx−¯xm+1,(45)
where γf= sup
(x,u)∈Ω×U
∇[m+1]
x⊗[fu(x)]m
(m+ 1)! .(46)
Now the extended state transition step Xt+1 =[fut(xt)]m
can be written using the representation (43) with the substi-
tutions xt−ˆxt]m=Im(−ˆxt)[xt]m−[ˆxt]m
=Im(−ˆxt)Xt−[ˆxt]m,(47)
obtaining
Xt+1 =fut(ˆxt)m+AtXt−[ˆxt]m+ϕf(ΣXt,ˆxt),
(48)
where At=Fm(ˆxt)Im(−ˆxt).(49)
The equations (48) and (38) describing system (3)–(4), can
be written as
Xt+1 =AtXt+fut(ˆxt)m−At[ˆxt]m+ϕf,(50)
yt=CtXt+hut(˜xt)−Ct[˜xt]m+ϕh,(51)
where ϕf=ϕf(ΣXt,ˆxt),ϕ
h=ϕh(ΣXt,˜xt).(52)
System (3)–(4) is said to be immersed into the system (50)–
(51), which has higher dimension, in the sense that if at a
given t0it is Xt0=[xt0]m, then Xt=[xt]m, for all t≥t0,
and therefore xt=ΣXt.
Note that in the derivation of (50)–(51) the sequences
ˆxtand ˜xtcan be any sequences in Ω(they may even be
constant).
The Carleman linearization of (3)–(4) around the se-
quences ˆxtand ˜xtconsists in neglecting the remainders ϕf
and ϕhin equations (50)–(51), and in replacing the poly-
nomial extended state Xt=[xt]mwith an approximating
vector Xt∈Rnm, to obtain
Xt+1 =AtXt+fut(ˆxt)m−At[ˆxt]m,(53)
y
t=AtXt+hut(˜xt)−Ct[˜xt]m.(54)
The Carleman linearization (53)–(54) is an approximation of
system (3)–(4) if the differences [xt]m−X
tand yt−y
tcan
be made as small as desired, at least in a finite time interval,
by increasing the degree m, provided that both xtand Xt
are consistently initialized at time t0(i.e., Xt0=[xt0]m) and
both systems are forced by the same input sequence.
Remark 5.Note that, by definition, Xtevolves on the
consistency manifold Mm, defined in Appendix, eq. (80),
while, in general, Xtis not consistent (i.e., Xt∈M
m).
Equations (53)–(54) have the appearance of a time-varing
linear system, with known system matrices and forcing
terms. Thus, the construction of an observer with the standard
Kalman Filter structure is straightforward. Such an observer,
denoted here PEKO, has the same prediction-correction
structure of the EKO, where the correction gain Ktis the
output of Riccati equations forced by two sequences of
PD matrices, Qt∈Rnm×nmand Rt∈Rq×q, uniformly
lower and upper bounded over t∈Z. The equations of the
PEKO need to be initializated using an a priori state estimate
¯x0∈Rnat time t0. The Riccati equations require a PD
matrix P0∈Rnm×nmfor the initialization, representing the
uncertainty on the initial estimate ¯x0.
The sequences of state observations ˆxtand predictions
˜xtin the PEKO are obtained as subvectors of the extended
state observations
Xtand predictions
Xtproduced by the
algorithm.
Polynomial Extended Kalman Observer
(PEKO)
Starting values: ˜xt0=¯x0,
Xt0=[˜xt0]m,
Pt0=P0,t=t0.
Ct=Hm(˜xt)Im(−˜xt),(55)
˜yt=hut(˜xt)−Ct
Xt−[˜xt]m,output pred. (56)
Kt=
PtCT
tCt
PtCT
t+Rt−1,(57)
Xt=
Xt+Kt(yt−˜yt),ext. state estim. (58)
Pt=In−KtCt
Pt,(59)
ˆxt=Σ
Xt,state estim. (60)
At=Fm(ˆxt)Im(−ˆxt),(61)
Pt+1 =α2AtPtAT
t+Qt,(62)
Xt+1 =[fut(ˆxt)]m−At
Xt−[ˆxt]m
,ext. state pred. (63)
˜xt+1 =Σ
Xt+1.state prediction (64)
Remark 6.As in the EKO, a constant coefficient α≥1
(forgetting factor) has been considered in equation (62), so
that exponential data weighting is achieved when α>1.
Remark 7.The vector
Xtis an estimate of [˜xt]m, and, by
construction, see eq. (64), it is such that Σ
Xt−[˜xt]m)=0,
although in general
Xt=[˜xt]m. Stated in other words,
Xt
in general do not belong to the consistency manifold Mm⊂
Rnm. The same considerations can be made for
Xt, that is
an estimate of [ˆx]msuch that Σ
Xt−[ˆxt]m)=0, but in
general
Xt−[ˆxt]m=0. Note that the difference
Xt−[˜xt]m
appear as a forcing term in the output prediction equation
(56), while the mismatch
Xt−[ˆxt]mappear as a forcing
term in the extended state prediction (63).
IV. CONVERGENCE ANALYSIS OF THE PEKO
Following the approach in [24], the convergence analysis
of the PEKO is addressed in this section by deriving and
studying the recursive equation that governs the dynamics
of the prediction error. Note that the PEKO provides a
sequence of estimates and predictions of the extended state
Xt=[xt]m. The following relationship exists between the
state prediction error xt−˜xtand the extended state prediction
error Xt−
Xt:
xt−˜xt=Σ(Xt−
Xt)≤Xt−
Xt.(65)
This inequality implies that the convergence of the extended
prediction implies the convergence of the state prediction (if
Xt−
Xt→0, then xt−˜xt→0).
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Thus, the convergence analysis can proceed by deriving
a recursive equation for the extended state prediction error
Xt−
Xt. Subtracting (63) from (50) yields
Xt+1 −
Xt+1 =AtXt−
Xt+ϕf.(66)
The estimation error of the extended state Xt−
Xtis
computed subtracting (58) from Xt
Xt−
Xt=Xt−
Xt−Kt(yt−˜yt).(67)
The output prediction error yt−˜ytis computed by subtracting
(56) from (51)
yt−˜yt=CtXt−
Xt+ϕh.(68)
Substitution of this into (67) yields
Xt−
Xt=Inm−KtCt(Xt−
Xt)−Ktϕh.(69)
Substitution of (69) into (66) gives
Xt+1 −
Xt+1 =AtInm−KtCt(Xt−
Xt)+ϕI,(70)
where ϕI=ϕf(xt,ˆxt)−AtKtϕh(xt,˜xt).(71)
Consider the bounds (45) and (29) on ϕfand ϕh. Using
the inequalities xt−˜xt≤Xt−
Xtand xt−ˆxt≤
Xt−
Xt, it follows
ϕf(xt,ˆxt)≤γfXt−
Xtm+1,
ϕh(xt,˜xt)≤γhXt−
Xtm+1.(72)
The following theorem can be proved following the same
lines of Theorem 1:
Theorem 2. Consider system (3)–(4), with assumption A0,
and the PEKO equations (56)–(64), for a given degree m,
and let the following assumptions hold
i) There exist positive numbers a, ¯c, ¯p, p such that for all
t≥t0
At≤a, Ct≤¯c, (73)
pIn≤
Pt≤¯pInpIn≤
Pt≤¯pIn.(74)
ii) Atis nonsingular ∀t≥t0.
iii) There exist positive real numbers γf,γh, such that
inequalities (72) hold, for (xt,ˆxt,u
t)∈Ω×Ω×U.
Then, there exist positive real numbers η,0,θ, with θ>α,
such that, if [xt0]m−[˜xt0]m<
0, then
[xt]m−[˜xt]m≤η[xt0]m−[˜xt0]mθ−(t−t0),(75)
that means that the PEKO is a local exponential observer
(recall that xt−˜xt≤[xt]m−[˜xt]m, see (65)).
V. F INAL REMARKS AND CONCLUSIONS
The local stability of the Polynomial Extended Kalman
Filter used as an asymptotic state observer (PEKO, Poly-
nomial Extended Kalman Observer) has been investigated.
The analysis is performed following the approach used in
[24] to study the convergence properties of the Extended
Kalman Filter used as an observer. A new compact formal-
ism is introduced for the representation of the Carleman
linearization of nonlinear discrete time systems, that allows
for the derivation of the state prediction error dynamics in
a form similar to the one developed in [24] for the classical
Extended Kalman Filter. It follows that the conditions that
ensure the exponential convergence of the observation error
of the PEKO are formally similar to those given in [24].
The stability analysis performed in this paper is also
important in the stochastic framework, when both state
and output noises are present. In this case the Polynomial
Extended Kalman Filter [14] should be applied, where the
sequences of matrices Qtand Rtin the Riccati equations are
not free design parameters. The conditions of exponential
stability of the error dynamics in the deterministic setting
ensure that in the stochastic setting the moments of the
estimation error, up to a given order, remain bounded over
time (stability of the PEKF).
An interesting issue to investigate in future work will be
whether higher order PEKO’s provide better convergence
properties than lower order ones, in terms of basin of
attraction and rate of convergence.
APPENDIX
USEFUL FORMULAS OF THE KRONECKER ALGEBRA
The Kronecker product of two matrices Mand Nof
dimensions p×qand r×srespectively, is the (p·r)×(q·s)
matrix
M⊗N=⎡
⎢
⎣
m11N ... m
1qN
.
.
.....
.
.
mp1N ... m
pqN
⎤
⎥
⎦,(76)
where the mij are the entries of M. The Kronecker power
of a matrix Mis recursively defined as
M[0] =1,M
[i]=M⊗M[i−1],i≥1.(77)
Note that if M∈Rp×q, then M[i]∈Rpi×qi. A quick survey
on the Kronecker algebra can be found in the Appendix of
[8]. See [18] for more properties.
The symbol [x]k, with k∈N, defined in equation (26),
can also be recursively defined as
[x]1=x, [x]k+1 =[x]k
x[k+1],k≥1.(78)
Then x∈Rnimplies [x]m∈Rnm, with nm=m
k=1 nk.
Let Σdenote the following matrix in Rn×nm
Σ=In0n×(nm−n).(79)
Σis called selection matrix because it selects the first n
component of a vector of dimension nm. It is such that x=
Σ[x]m,∀x∈Rn. Note that the vector [x]mbelongs to a
submanifold Mm⊂Rnmof dimension n, defined as
Mm=X∈Rnm:X=[ΣX]m.(80)
Mmis called consistency manifold, and if X∈M
m, then
Xis said to be consistent, because in this case x=ΣXis
such that X=[x]m.
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Let Im(v), where m∈Nbe a nm×nmmatrix defined
as
Im(v)=Inm+Sm(v),(81)
where matrix Sm(v)is a strongly lower block-triangular
matrix, whose blocks Sh,k =Sm(v)h,k, are defined as
Sh,k =0
nh×nk,for h≤k,(strongly lower diagonal)
S2,1=v⊗In+In⊗v,
Sh,1=v[h−1] ⊗In+Sh−1,1⊗v, 2<h≤m
Sh,k =Sh−1,k−1⊗In+Sh−1,k ⊗v, 1<k<h.
(82)
From the definition it easy to see that Sm(0) = 0, and
therefore
Im(0) = Inm.(83)
Lemma 3. For any given vand ¯vin Rnand m∈N, the
following hold
[v+¯v]m=Im(¯v)[v]m+[¯v]m,(84)
[v−¯v]m=Im(−¯v)[v]m−[¯v]m.(85)
[v]m=−Im(v)[−v]m,(86)
I−1
m(v)=Im(−v),(87)
The Kronecker formalism can be used also to represent
differential operators. Matrices of derivatives of any order
with respect to a vector variable x∈Rncan be represented
defining the operator ∇[i]
x⊗. Let ψ:Rn→ Rqbe a
differentiable function. The operator ∇[i]
x⊗formally acts as
a Kronecker product as follows:
∇[0]
x⊗ψ=ψ,
∇[i+1]
x⊗ψ=∇x⊗∇[i]
x⊗ψ,i≥1.(88)
with ∇x=[∂/∂x1··· ∂/∂xn]. Note that ∇x⊗ψis the
standard Jacobian of the vector function ψ.
REFERENCES
[1] V.C. Aitken and H.M. Schwartz, “On the exponential stability of
discrete-time systems with applications in observer design.” IEEE
Trans. A. C., Vol. 39, No. 9, pp. 1959–1962, 1994.
[2] B.D.O. Anderson and J.B. Moore, Optimal filtering, Prentice-Hall,
Englewood Cliffs, New Jersey, 1979.
[3] M. Boutayeb, H. Rafaralahy, and M. Darouach, “Convergence analysis
of the Extended Kalman Filter used as an observer for nonlinear
deterministic discrete-time Systems,” IEEE Trans. A. C., Vol. 42,
No. 4, pp. 581–586, 1997.
[4] M. Boutayeb and D. Aubry, “A strong tracking extended Kalman
observer for nonlinear discrete-time systems.” IEEE Trans. A. C.,
Vol. 44, No. 8, pp. 1550 - 1556, 1999.
[5] M. Boutayeb and M. Darouach, “A reduced-order observer for non-
linear discrete-time systems,” Syst. Contr. Lett., Vol. 39, No. 2, pp.
141–151, 2000.
[6] C. Califano, S. Monaco, and D. Normand-Cyrot, “On the observer
design in discrete-time,” Syst. Contr. Lett., Vol. 49, No. 4, pp. 255–
265, 2003.
[7] F. Carravetta, A. Germani, M. Raimondi, “Polynomial filtering for
linear discrete-time non-Gaussian systems,” SIAM J. Contr. Optim.,
Vol. 34, No. 5, pp. 1666–1690, 1996.
[8] F. Carravetta, A. Germani, M. Raimondi, “Polynomial filtering of
discrete-time stochastic linear systems with multiplicative state noise,”
IEEE Trans. A. C., Vol. 42, NO. 8, pp. 1106–1126, 1997.
[9] S.T. Chung and J.W. Grizzle, “Sampled-data observer error lineariza-
tion,” Automatica, Vol. 26, pp. 997–1007, 1990.
[10] G. Ciccarella, M. Dalla Mora, and A. Germani, “Observers for
discrete-time nonlinear systems,” Syst. Contr. Lett., Vol. 20, No. 5,
pp. 373–382, 1993.
[11] G. Ciccarella, M. Dalla Mora, and A. Germani, “A robust observer
for discrete time nonlinear systems,” Syst. Contr. Lett., Vol. 24, No. 4,
pp. 291–300, 1995.
[12] J.J. Deyst, Jr., and C.F. Price, “Conditions for asymptotic stability of
the discrete minimum-variance linear estimator,” IEEE Trans. A. C.,
Vol. 13, No. 6, pp. 702–705, 1968.
[13] A. Germani, C. Manes, P. Palumbo, M. Sciandrone, “A higher order
method for the solution of a nonlinear scalar equations,” J. of Opti-
mization Theory and Applications, Vol. 131., No. 3, pp. 347+-364,
2006.
[14] A. Germani, C. Manes, P. Palumbo, “Polynomial Extended Kalman
Filter,” IEEE Trans. A. C., Vol. 50, No. 12, pp. 2059–2064, dec. 2005.
[15] D. Di Martino, A. Germani, C. Manes, P. Palumbo, ”Design of
observers for systems with rational output function,” 45th IEEE Conf.
on Decision and Control (CDC06), pp. 1641+-1646, San Diego,
California, dec. 2006.
[16] A. Germani, C. Manes, P. Palumbo, “Filtering of Stochastic Nonlinear
Differential Systems via a Carleman Approximation Approach,” IEEE
Trans. A. C., Vol. 51, No. 11, pp. 2166–2172, 2007.
[17] A. Germani, C. Manes, “An exponential observer for discrete-time
systems with bilinear drift and rational output functions.” European
Control Conf. (ECC07), Kos, Greece, pp. 5777–5782, July 2007.
[18] A. Graham, Kronecker Products and Matrix Calculus with Applica-
tions, Ellis Horwood ltd. Publishers, Chichester, 1981.
[19] A.H. Jazwinski, Stochastic Processes and Filtering Theory, Academic
Press, New York, 1970.
[20] N. Kazantzis and C. Kravaris, “Discrete-time nonlinear observer
design using functional equations,” Syst. Contr. Lett., Vol. 42, pp. 81–
94, 2001.
[21] W. Lee and K. Nam, “Observer design for autonomous discrete-time
nonlinear systems,” Syst. Contr. Lett., Vol. 17, No. 1, pp. 49–58, 1991.
[22] W. Lin and C.I. Byrnes, “Remarks on linearization of discrete-time
autonomous systems and nonlinear observer design,” Syst. Contr. Lett.,
Vol. 25, No. 1, pp. 31–40, 1995.
[23] P.E. Moraal, J.E. Grizzle, “Observer design for nonlinear systems with
discrete-time measurements,” IEEE Trans. A. C., Vol. 40, No. 3, pp.
395–404, 1995.
[24] K. Reif, R. Unbehauen, “The extended Kalman filter as an exponential
observer for nonlinear systems,” IEEE Trans. Signal Process., Vol. 47,
pp. 2324–2328, 1999.
[25] Y. Song, J.W. Grizzle, “The extended Kalman filter as a local asymp-
totic observer for discrete-time nonlinear systems,” J. Math. Systems
Estimate and Control, Vol. 5, pp. 59–78, 1995.
[26] M. Xiao, N. Kazantzis, C. Kravaris, and A.J. Krener, “Nonlinear
discrete-time observer design with linearizable error dynamics,” IEEE
Trans. A. C., vol. 48, No. 4, pp. 622-626, 2003.
[27] M. Xiao, “A direct method for the construction of nonlinear discrete-
time observer with linearizable error dynamics,” IEEE Trans. A. C.,
Vol. 51, No. 1, pp. 128–135, 2006.
[28] A. Zemouche and M. Boutayeb, “Observer design for lipschitz nonlin-
ear systems: the discrete-time case,” IEEE Trans. on Circ. and Systems
II: Express Briefs, Vol. 53, No. 8, pp. 777–781, 2006.
47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008 ThC02.5
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