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Extended high gain observer design for state and
parameter estimation
M. Farza1,T.M´enard1,A.Ltaief1,2,T.Maatoug
2,M.M’Saad
1, Y. Koubaa2
1GREYC, UMR 6072 CNRS, Universit´e de Caen, ENSICAEN, 6 Bd Mar´echal Juin, 14050 Caen Cedex, France
2Lab-STA, ENIS Sfax, Route de Soukra km 4 Sfax - B.P W.3038 Sfax, Tunisie
Abstract—This paper investigates the design of an extended
high gain observer to estimate the state together with some un-
known constant parameters for a class of nonlinear systems. We
firstly consider the case where all the parameters are known.
A high gain state observer is proposed and its exponential
convergence is carried out under a well defined persistent
excitation condition. Then, the case where some parameters
involved in the system equations are unknown is considered
using a suitable augmentation of the nonlinear system with
the parameters dynamics. Thanks to the introduction of the
notion of characteristic indices associated to the unknown
parameters, an extended high gain observer is designed to
simultaneously estimate the state together with the unknown
parameters. It is shown that the exponential convergence of the
extended high gain observer is ensured under the persistent
excitation condition that has been already considered for the
state observer together with an additional persistent excitation
to handle the parameter estimation process.
Keywords Nonlinear system, High gain observer, Char-
acteristic indices, Adaptive observer, Persistent excitation,
Nonlinear parametrization.
I. INTRODUCTION
Over the last decades, adaptive observers design has be-
come a wide and active research field as pointed out by
its underlying literature. The adaptive observers perform
a simultaneous estimation of the state variables and the
unknown parameters. Several adaptive observer designs for
nonlinear systems have been proposed up to the underlying
modeling assumptions and design concepts [2, 14, 15, 7,
5, 9, 3]. Though most contributions on adaptive observer
design deal with linear parametrization, some results dealing
with nonlinear parameterizations are nevertheless available
[1, 9, 12, 13, 16, 17].
There are two approaches that could be used to derive
adaptive observers. The first approach is essentially inspired
by a comprehensive convergence analysis and has been
commonly pursued in the available adaptive observer con-
tributions. We shall refer to as the adaptive approach as
the underlying adaptive observer share the same structure as
the state observer up to the certainty equivalence principle
which consists in replacing the unknown parameters by their
estimates provided by a suitable parameter adaptation. The
second approach is a rather standard state observer design
from a suitable observation model obtained by augment-
ing the system model with the dynamics of its unknown
(constant) parameters, i.e. ˙ρ=0where ρis the unknown
parameter vector. That is why, we shall refer to as the
augmented model approach. This allows to take advantage
of the state observer designs that are available for the
class of systems including the augmented systems with their
unknown parameter dynamics.
In [3], the authors proved the equivalence between two
adaptive observers that have been respectively derived using
the adaptive and augmented model approaches for state
affine systems where the nonlinearities only depend on the
inputs and outputs. By equivalence, we mean the same
equations of the observers as well as the same assumptions
required for the observer designs. The equivalence between
these approaches has also been established in [6] for a class
for uniformly observable systems where the nonlinearity
involves some unknown parameters intervening with a linear
parametrization.
In this paper, we shall generalize the results proposed in
[9, 6] through the design of an adaptive observer for a
class of nonlinear systems including the ones considered
in [9, 6]. More specifically, an adaptive observer is firstly
derived using the augmented model approach. Then, it is
shown that the proposed observer can also be derived under
the same assumptions using the adaptive approach.
The paper is organized as follows. The problem statement
is given in section 2 with a particular emphasis on the
considered class of systems and the usual high gain as-
sumptions. In section 3, we consider the case where all the
system parameters are known to emphasize the state observer
design process. The exponential convergence analysis of the
proposed observer is provided under a well defined persistent
excitation condition. Section 4 is devoted to the observer
design in the case where some parameters are assumed to
be unknown. A particular emphasis is put on the observer
design model and the key feature of the design. The former is
the original nonlinear system augmented with the unknown
parameters dynamics, while the latter is provided by the
characteristic indices associated to the unknown parameter-
s. These features allow to design an extended high gain
observer which simultaneously estimate the state together
with the unknown parameters. The exponential convergence
of the extended high gain observer is established under
the persistent excitation condition that has been already
considered for the state observer together with an additional
persistent excitation for the parameter estimation process. In
section 5, the equations of the proposed extended observer
are rewritten under under an adaptive observer form showing
Proceedings of the 4th International Conference on
Systems and Control, Sousse, Tunisia, April 28-30, 2015
ThAA.1
978-1-4799-8318-6/15/$31.00 ©2015 IEEE 345
thereby that the observer design can be achieved using an
adaptive approach. Some concluding remarks are given in
section 7.
Throughout the paper, Ipand 0pdenote the p-dimensional
identity and zero matrices respectively, · denotes the
euclidian norm and λM(·)(resp. λm(·)) will be used to
denote the largest (resp. the smallest) eigenvalue of (·).
II. PROBLEM STATEMENT
Consider the following class of MIMO dynamical systems:
˙x(t)=F(u, x)x(t)+Ψ(u, x)ρ+g(u, x)
y(t)=Cx(t)+Ψ
o(u)ρ=x1(t)(1)
F(u, x)=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0F1(u, x)0... ... 0
00F2(u, x)0... 0
.
.
...........
.
.
.
.
....0
0... ... 0Fq−1(u, x)
0... ... 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
C=Ip0... 0(2)
where the state x(t)=(x1T... x
qT )T∈IR n,xk∈
IR p,ρ∈Rmis a constant vector with real compo-
nents ρi∈IR; each Fk(u, x)is a p×psquare matrix
which is triangular with respect to xi.e. Fk(u, x)=
F(u, x1,...,x
k)for k=1,...,q −1,u∈Ua compact
set of IRsand y∈Rpare respectively the input and
the output of the system; gis a nonlinear vector func-
tion that has a triangular structure with respect to x,i.e.
g(u, x)=g1T(u, x1)g2T(u, x1,x
2)... g
qT (u, x)T,Ψ(u, x)=
(Ψ1(u, x)Ψ
2(u, x)... Ψm(u, x)) is a nonlinear matrix func-
tion of dimension n×meach column of which is a vector
function, Ψj(u, x), with a triangular structure with respect
to xi.e. Ψj(u, x)=Ψ1T
j(u, x1)Ψ
2T
j(u, x1,x
2)... ΨqT
j(u, x)T
for j=1,...,m. Finally, Ψo(u)is a p×mmatrix which
entries are continuous with respect to u.
When each matrix Fkis equal to the identity matrix Ipand
the parameter vector ρis known, system (1) belongs to the
class of systems considered in [11] which characterizes a
subclass of systems which are observable for any input and
for which the authors proposed a high gain observer with a
constant gain. A similar class of systems has been considered
in [8] where the matrices Fkare reduced to positive bounded
real-valued functions and for which a high gain observer
with a time-varying gain issued from the resolution a Riccatti
Ordinary Differential Equation (RODE) has been proposed.
In the next sections, we shall consider two situations de-
pending on whether the vector of constant parameters ρis
known or not. In the first case, the objective is to design a
state observer for the on line estimation of the full state. In
the second case, we shall propose an extended high observer
for the or joint estimation of the state and the unknown
parameters. It is worth mentioning that an adaptive observer
has been proposed in [19] for class of systems (1) with
Fk(u, x)=Ipfork=1,...,q−1and Ψ(u, x)=0.
III. THE STATE OBSERVER DESIGN
In this section, we suppose that the vector ρis known and the
objective is to design a state observer. For clarity purposes,
we set ϕ(u, x)=Ψ(u, x)ρ+g(u, x). Moreover, we shall
consider in this section that Ψo(u)=0without loss of
generality. Thus, the class of systems subject to the state
observer design can be written as follows
˙x(t)=F(u, x)x(t)+ϕ(u, x)
y(t)=Cx(t)=x1(t)(3)
The observer design requires some assumptions that will
be stated later. Now standard high gain observer design
assumptions are provided.
A1 The state x(t)and the control u(t)are bounded, i.e.
x(t)∈Xand u(t)∈Uwhere X⊂IR nand U⊂IR sare
compacts sets.
A2 The function Fkand ϕare Lipschitz with respect to x
uniformly in uwhere (u, x)∈U×X. Their Lipschitz
constants will be denoted by LFand Lϕ, respectively.
Since the state is confined to the bounded set X, one can
assume that Lipschitz prolongations of the nonlinearities,
using smooth saturation functions, have been carried out
and that the functions Fkand ϕare provided from these
prolongations. This allows to conclude that for any bounded
input u∈U, the functions Fkand ϕare globally Lipschitz
with respect to xand are bounded for all x∈IR n(see [10]
and references therein for more details). We shall denote
throughout this paper by xMthe upper bound of xi.e.
xM=sup
t≥0x(t)(4)
Before providing our candidate observer, let us introduce
some useful notations and definitions related to high gain
observer design.
•The diagonal matrix Δθdefined by:
Δθ=diag Ip,1/θIp,...,1/θq−1Ip(5)
where θis a positive scalar. One can easily check that the
following identities hold:
ΔθF(u, x)Δ−1
θ=θF(u, x),CΔ−1
θ=C(6)
Now, let us consider the following dynamical system:
˙
ˆx=F(u, ˆx)ˆx+ϕ(u, ˆx)−θΔ−1
θS−1(t)CT(Cˆx−y)(7)
where ˆx=(ˆx1T... ˆxqT )T∈IR nwith ˆxk∈IR p,uand y
are respectively the input and the output of system (3) and
Sis a square symmetric matrix governed by the following
Lyapunov ODE:
˙
S(t)=θS(t)+FT(u, ˆx)S(t)+S(t)F(u, ˆx)−CTC(8)
with S(0) = ST(0) >0and θ>0is a scalar design parameter.
Before adopting an additional hypothesis required for the
observer design, we define the following state transition
matrix Φ(t, τ ):
d
dt Φ(t, τ )=F(u(t),ˆx(t))Φ(t, τ )
Φ(τ, τ)=In
(9)
where uis the input of system (3), ˆxis the state of the
dynamical system (7) and Fis defined as in (2).
One will show that the dynamical system (7) is actually
a state observer provided that the following assumption is
satisfied.
[A3] The input uis such that for any trajectory ˆxof system
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(7) starting from ˆx(0) ∈X, the following persistent excitation
condition is satisfied [4]:
∃α0>0; ∃θ0>0; ∀θ≥θ0;∀t≥1
θ:
t
t−1
θ
ΦT(t, τ )CTCΦ(t, τ )dτ ≥α0
θΔ−2
θ
(10)
Such an assumption is crucial for the following result.
Theorem 3.1: Consider system (3) subject to assump-
tions A1 to A2. Then, for every bounded input satisfying
assumption A3, there exists a constant θ0such that for
every θ>θ
0, system (7) is a state observer for system
(3) with an exponential error convergence to the origin for
sufficiently high values of θ, i.e. for any initial conditions
(x(t0),ˆx(t0)) ∈X2, the observation error ˆx(t)−x(t)tends to
zero exponentially when t→∞.
Proof: By proceeding as in [4], one can show that the
matrix S(t)governed by (8) is SPD and is bounded below
by e−1αoInwhere αois given by Assumption A3. Then,
taking into account the anti-shift structure of Fand the
particular structure of Cand from the fact that the matrices
Fkare upperly bounded, one can show recursively that
each (bloc) entry Sij of the matrix S, located at the row i
and the column j, is bounded and the corresponding upper
bound is independent of θ. The detailed proof is omitted
for the sake of space limitations.
Now, set ¯x=Δ
θ˜xwhere ˜x(t)=ˆx−xis the observation error;
using the identities (6), one obtains:
˙
¯x=θ(F(u, ˆx)−S−1CTC)¯x+Δ
θ˜
F(u, ˆx, x)x
+˜ϕ(u, ˆx, x)) (11)
where ˜
F(u, ˆx, x)=F(u, ˆx)−F(u, x)and ˜ϕ(u, ˆx, x)=
ϕ(u, ˆx)−ϕ(u, x).
Let V(¯x(t)) = ¯xT(t)S(t)¯x(t)be the Lyapunov candidate
function. Using (8), one gets:
˙
V(t)=¯xT˙
S−1¯x+2¯xTS˙
¯x
=−θV −¯xTCTC¯x+2¯xSΔθ˜
F(u, ˆx, x)x+˜ϕ(u, ˆx, x)(12)
Proceeding as in [10, 9], one can show that for θ≥1:
2¯xTSΔθ˜
F(u, ˆx, x)x≤2√n¯
λM(S)V(¯x)LFxM¯x
≤2√n√σSLFxMV(¯x)
2¯xTSΔθ˜ϕ(u, ˆx, x)≤2√n¯
λM(S)V(¯x)Lϕ¯x
≤2√n√σSLϕV(¯x)
(13)
where
σS=¯
λM(S)/λm(S),¯
λM(S)=max
t≥0λM(S(t)),
λm(S)=min
t≥0λM(S(t))
where LFand Lϕare the respective Lipschitz constants of
Fand ϕas stated in assumption A2 and xMis the upper
bound of xgiven by (4).
Combining (12) with (13), one gets
˙
V(t)≤−θV −¯xTCTC¯x+2
√nσ(S)(LFxM+Lϕ)V
≤−(θ−c)V(14)
where c=2
√nσ(S−1)(LFxM+Lϕ). Now, it suffices to
choose θ=max(1,c). This ends the proof.
IV. THE EXTENDED OBSERVER DESIGN
In this section, we shall assume that the vector of parameters
ρis unknown and our objective is to design an observer for
system (1) that allows a simultaneous estimation of the state
xand the parameters vector ρ. The observer design requires
the following assumptions:
A1The state x(t), the control u(t)and the unknown param-
eters ρare bounded, i.e. x(t)∈X,u(t)∈Uand ρ∈Ω
where X⊂IR n,U⊂IR sand Ω∈IR mare compacts sets.
A2The matrix Ψ(u, x)is continuous on U×X.
A3The functions gand Ψare Lipschitz with respect to x
uniformly in uwhere (u, x)∈U×X. Their Lipschitz
constants will be denoted by Lgand LΨ, respectively.
As in the case where the vector ρis known and since the
state is confined to the bounded set X, one can assume that
Lipschitz prolongations of the nonlinearities, using smooth
saturation functions, have been carried out and that the
functions gand ψare provided from these prolongations.
This allows to conclude that for any bounded input u∈U,
the functions gand Ψare globally Lipschitz with respect to
xand are bounded for all x∈IR n(see [10] and references
therein for more details). We shall denote throughout this
paper by ΨMand ρMthe upper bounds of Ψ(t)and ρ,
respectively i.e.
ΨM=sup
t≥0Ψ(t),ρ
M=max
1≤i≤M|ρi|(15)
The bound of state still be denoted by xMas in the
previous section.
We now define the characteristic indices associated to the
unknown parameters. These definitions extend those given
in [9] where the expression of the output did not depend on
the unknown parameters. Indeed, one associates to to each
unknown parameter ρja characteristic index, denoted νj
which is defined as follows. If the expression of the output
ydepends on the parameter ρj,thenνj=0.Otherwise,
the characteristic index νjis equal to the smallest positive
integer ksuch that ∂˙xk
∂ρj=0
p×1. That is, one has ∂˙xk
∂ρj
=0
p×1
for k=1,...,ν
j−1and ∂˙xνj
∂ρj=0
p×1.
Let Ψk
j(u, x)(resp. Ψ0
j(u))bethe(p×1column sub-vector)
entry located at the (bloc) row kand column jof the matrix
Ψ(u, x)(resp. of the matrix Ψo(u)). According to system (1)
and from the definition of the characteristic indices, one has
∂y
∂ρj
=Ψ
0
j(u)and ∂˙xk
∂ρj
=Ψ
k
j(u, x). Thus, the entries of the
matrix Ψ(u, x)satisfy the following properties
Ψk
j(u, x)=0
p×1if 1≤k≤νj−1and Ψνj
j(u, x)=0
p×1(16)
Now, let Ωθbe the following m×mdiagonal matrix
Ωθ=diag 1
θν1,1
θν2,..., 1
θνm(17)
and set Γ(u, x)Δ
=Δ
θΨ(u, x)Ω−1
θ. According to the structures
of Δθand Ωθrespectively given by (5) and (17), one has
Γk
j(u, x)=θ−(k−1)Ψk
j(u, x)θνj=θνj−k+1Ψk
j(u, x)and hence,
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one can check that the matrix Γ(u, x)can be decomposed
as follows
ΔθΨ(u, x)Ω−1
θ=θG(u, x)+Ru, x, 1
θ(18)
where G(u, x)and Ru, x, 1
θare n×mrectangular matrices
whose respective entries Gk
j(u, x)and Rk
ju, x, 1
θ,for(k, j)∈
[1,q]×[1,m], are defined as follows:
Gk
j(u, x)=0if k=νjand Gνj
j(u, x)=Ψ
νj
j(u, x)(19)
Rk
j(u, x, 1
θ)=0if k≤νjand
Rk
ju, x, 1
θ=1
θk−1−νjΨk
j(u, x)otherwise
Notice that the matrix G(u, x)does not depend on the
parameter θand moreover this parameter appears with non
positive powers in the entries of the matrix Ru, x, 1
θ. Hence,
from the definition of Gand Rgiven above, the following
properties hold:
ΔθG(u, x)Ω−1
θ=θG(u, x)and R1
ju, x, 1
θ=0
for j =1,...,m (20)
Similarly, one can check that:
Ψo(u)Ω−1
θ=Ψ
0(u)(21)
Indeed, the j’th column of the matrix Ψo(u)Ω−1
θis equal
to Ψ0
j(u)θνj. This column is equal to zero if Ψ0
j(u)=0.
Otherwise, the expression of ydepends on ρjand νj=0
(θνj=1); so we get Ψ0
j(u)θνj=Ψ
0
j(u).
In the following, we shall propose a high gain observer
which performs simultaneous estimation of the state and
unknown parameters thanks to augmentation of the system
model by the unknown parameter dynamics using the
following definitions and notations.
•Let A(u, x)and C(u)be the following (n+m)×(n+m)and
p×(n+p)matrices:
A(u, x)=F(u, x)G(u, x)
0m×n0m×m,C(u)=(CΨo(u)) (22)
where F(u, x),C and G(u, x)are defined by (2) and (19),
respectively.
•Let Λθbe the following (bloc) diagonal matrix:
Λθ=diag(Δθ,Ωθ)(23)
where Δθand Ωθare given by (5) and (17), respectively.
•Let S(t)=S1(t)S2(t)
ST
2(t)S3(t)where S1(t)and S3(t)are
square symmetric matrices of dimensions n×nand m×m,
respectively and S2(t)is a n×mrectangular matrix, be the
symmetric matrix solution of the following Lyapunov ODE
˙
S(t)=−θS+AT(u, ˆx)S+SA(u, ˆx)−CT(u)C(u)(24)
with S(0) = ST(0) >0where A(·)and Care given by (22).
The following dynamical system is a suitable candidate for
the extended observer as it will be shown later.
˙
ˆx(t)
˙
ˆρ(t)=⎛
⎝
F(u(t),ˆx(t))ˆx(t)
+Ψ(u(t),ˆx(t)) ˆρ+g(u(t),ˆx(t))
0⎞
⎠(25)
−θΛ−1
θS−1(t)CT(u(t))(Cˆx(t)+Ψ
0(u(t))ˆρ(t)−y(t))
where ˆx∈IR n,ˆρ∈IR m,uand yare respectively the input and
the output of system (1), S,A, C ,Cand Λθ, are respectively
given by (24), (2), (22) and (23).
Before giving our main result, we need the following
additional assumptions:
A4The input uof system (1) is such that the persistent
condition given by (10) is satisfied where ˆxinvolved in
(9) is the state of system (25).
A5Let Υ(t)be the time-varying matrix governed by the
following ODE:
˙
Υ(t)=θ(F(u, ˆx)−S−1
1(t)CTC)Υ(t)
+G(u, ˆx)−S−1
1(t)CTΨo(u)(26)
The input uis such that for any trajectory ˆxof system
(25) starting from ˆx(0) ∈X, the matrix (CΥ(t)+Ψ
o(u))
is persistently exciting in the following sense
∃δo>0; ∃θ1>0; ∀θ≥θ1;∀t≥1
θ:(27)
t
t−1
θ
ΓT(τ)Γ(τ)dτ ≥δoImwith Γ(τ)=CΥ(τ)+Ψ
0(u(τ))
The dynamical system (25) is actually an observer for
system (1) as pointed out by the following fundamental
result.
Theorem 4.1: Consider system (1) subject to
assumptions A1to A3. Then, for every bounded input
satisfying assumptions A4and A5, there exists a constant
θ0such that for every θ>θ
0, system (25) is an observer
for system (1) with an exponential error convergence to the
origin for sufficiently high values of θ, i.e. for any initial
conditions (x(t0),ˆx(t0)) ∈X2,(ρ, ˆρ(t0)) ∈Ω2, the errors,
ˆx(t)−x(t)and ˆρ(t)−ρtend to zero exponentially when t→∞.
Proof of Theorem 4.1: Before giving the main outlines of
this proof, we shall put forward some properties that shall
be used throughout the proof.
•The matrix Υ(t)is bounded and its corresponding upper
bound does not depend on θ. Indeed, set ¯
Υ(t)=Υ(t/θ);then
one has
˙
¯
Υ(t)=(A−S−1CTC)¯
Υ+F(u(t/θ),ˆx(t/θ)
Since the matrix A−S−1CTCis Hurwitz and F(u, ˆx)
is bounded with bounds independent of θ, one naturally
concludes that ¯
Υ(t)is bounded with an upper bound inde-
pendent of θ.
•It should be emphasized that the persistent excitation con-
dition given by Assumption A5is not made on the system
but on the observer since the latter provides the estimate of
the unknown state. This makes it possible to check this as-
sumption on-line by simply computing the eigenvalues of the
symmetric matrix t
t−T(CΥ(s)+Ψ
o(u))T(CΥ(s)+Ψ
o(u)) ds.
•The matrix P(t)governed by the Ordinary Differential
Equation (ODE) (7) is SPD and bounded. Moreover, its
eigenvalues are independent of θ. Indeed, let M(t)be a
symmetric m×mmatrix governed by the following ODE:
˙
M=−θM +θΥTCTCΥwith M(0) = MT(0) >0
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one has:
M(t)=e−θtM(0) + θt
0
e−θ(t−τ)ΥT(τ)CTCΥ(τ)dτ
≥θt
0
e−θ(t−τ)ΥT(τ)CTCΥ(τ)dτ
≥θt
t−1
θ
e−θ(t−τ)ΥT(τ)CTCΥ(τ)dτ for t≥1
θ
≥e−1δoImaccording to A4 (28)
Hence M(t)isSPDandsoisM−1(t). Besides, it is easy
to check that M−1satisfies the same differential equation as
P(t)and as result P(t)is SPD and one has P−1(t)=M(t)
as soon as P−1(0) = M(0).
Let us show that under assumption A4and A5, the symmet-
ric matrix S(t)solution of the Lypunov ODE (24) is positive
definite and bounded from below and above with bounds that
are independent of θ. Indeed, from (24), one has:
˙
S1(t)=−θS1(t)+FT(u, ˆx)S1(t)+S1(t)F(u, ˆx)−CTC(29)
˙
S2(t)=−θS2(t)+FT(u, ˆx)S2(t)+S1(t)G(u, ˆx)−CTΨ0(30)
˙
S3(t)= −θS3(t)−GT(u, ˆx)S3(t)+S3(t)G(u, ˆx)
−Ψ0T(u)Ψ0(u)(31)
One notes that the matrix S1(t)satisfies the same ODE as
S(t)(compare equations (29) and (8)) with the difference
that ˆxis the state of system (7) for Swhereas it corresponds
to the sub-state of system (25) when dealing with S1. Thus,
using the same reasoning as for S, it is straighforward that
under assumption A4, the matrix S1(t)is symmetric pos-
itive definite. Let us denote by X(t)the Schur complement
of S1(t)in S(t),thatis:
X(t)=S3(t)−ST
2(t)S−1
1(t)S2(t)(32)
Since S1(t)is SPD, a necessary and sufficient condition for
S(t)to be SPD is that the Schur Complement X(t)is also
SPD [18]. So, let us derive the first time derivative of X(t).
Indeed, from (29), (30) and (31), one can show that:
d
dt X(t)= −θX(t)+CS−1
1(t)S2(t)+Ψ
o(u)T
×CS−1
1(t)S2(t)+Ψ
o(u) (33)
Now, one can show that S−1
1(t)S2(t)=Υ(t)for all
t≥0as soon as the initial conditions are chosen such
that S−1
1(0)S2(0) = Υ(0), the matrix Υ(t)being the matrix
introduced by Assumption A5. To this end, one can show
that the matrix S−1
1(t)S2(t)satisfies the same ODE as Υ(t).
For writing convenience, the variable tshall be omitted. One
has
d
dt −S−1
1S2=θF(u, ˆx)−S−1
1CTC−S−1
1S2
+θG(u, ˆx)−CTΨ0(u)
It is clear by comparing equations (26) and (34) that Υand
S−1
1S2satisfies the same ODE. As a result, the ODE (33)
that governs the dynamics of Xcan be written as follows:
˙
X(t)= −θX(t)+(CΥ(t)+Ψ
o(u))T(CΥ(t)+Ψ
o(u))(34)
Now, according to assumption A5, the matrix
(CΥ(t)+Ψ
o(u)) is persistently exciting. Moreover,
since S(0) is SPD, it’s the same for X(0). These facts
imply the positive definiteness of the matrix X(t)(see e.g.
[20]). Moreover, the smallest and largest eigenvalues of
X(t)do not depend on the design parameter θ[9].
Let us now prove the exponential convergence to zero of the
observation error related to the observer (25).
Set ˜x=ˆx−xand ˜ρ=ˆρ−ρ. From (1) and (25), one gets
⎛
⎝
˙
˜x(t)
˙
˜ρ(t)⎞
⎠=⎛
⎝
F(u(t),ˆx(t))˜x(t)+Ψ(u(t),ˆx(t))˜ρ
+˜
Ψ(u(t),ˆx(t),x(t))ρ+˜g(u(t),ˆx(t),x(t))
0⎞
⎠
−θΛ−1
θS−1(t)CT(u)(C˜x(t)+Ψ
o(u)˜ρ(t)) (35)
We now introduce the following change of coordinates:
¯x=Δ
θ˜xand ¯ρ=Ω
θρ. One gets:
⎛
⎝
˙
¯x(t)
˙
¯ρ(t)⎞
⎠=⎛
⎜
⎝
θF(u(t),ˆx(t))¯x(t)+Δ
θΨ(u(t),ˆx(t))Ω−1
θ¯ρ
+Δθ˜
Ψ(u(t),ˆx(t),x(t))ρ+˜g(u(t),ˆx(t),x(t))
0
⎞
⎟
⎠
−θS−1(t)CT(u)C¯x(t)+Ψ
o(u)Ω−1
θ¯ρ(36)
Using the decomposition of ΔθΨ(u(t),ˆx(t))Ω−1
θunder the
form (18), and the identity (21), one gets
⎛
⎝
˙
¯x(t)
˙
¯ρ(t)⎞
⎠=⎛
⎜
⎝
θ(F(u, ˆx)¯x+G(u, ˆx)) ¯ρ+R(u, ˆx, 1/θ)¯ρ
+Δθ˜
Ψ(u, ˆx, x)ρ+˜g(u, ˆx, x)
0
⎞
⎟
⎠
−θS−1(t)CT(u)(C¯x(t)+Ψ
o(u)¯ρ)(37)
In order to rewrite the error equation (38) in a condensed
form, we introduce the following variable ξ=¯x
¯ρ.
Indeed, equation (38) can be rewritten as follows:
˙
ξ(t)=θA(u(t),ˆx(t))ξ(t)+G(u(t),ˆx(t),x(t),¯ρ, ρ, 1/θ)
−θS−1(t)CT(u)C(u)ξ(t)(38)
where
G(u(t),ˆx, x, ¯ρ, ρ, 1/θ)=
R(u, ˆx, 1/θ)¯ρ+Δ
θ˜
Ψ(u, ˆx, x)ρ+˜g(u, ˆx, x)
0
Let us now consider the following candidate Lyapunov
function: V(ξ)=ξTSξwhere Sis the SPD matrix given by
(24). Then, one has
˙
V(ξ)=ξT˙
Sξ+2ξTS˙
ξ=−θV (ξ)−2θξTCTCξ
+2ξTSG(u(t),ˆx(t),x(t),¯ρ, ρ, 1/θ)(39)
Proceeding as in [9], one can show that for θ≥1:
Δθ˜
Ψ(u(t),ˆx(t),x(t))ρ≤√nLΨρM¯x≤√nLΨρMξ
Δθ˜g(u(t),ˆx(t),x(t)) ≤√nLg¯x≤√nLgξ
R(u(t),ˆx(t),1/θ)¯ρ≤√nΨM¯ρ≤√nΨMξ
The above inequalities lead to
2ξTSG(u(t),ˆx(t),x(t),¯ρ, ρ, 1/θ)
≤2¯
λM(S)V(ξ)√n(LΨρM+Lg+Ψ
M)ξ
≤2¯
λM(S)
λm(S)√n(LΨρM+Lg+Ψ
M)V(ξ)=θV(ξ)
(40)
with θ=2
¯
λmax(S)
λmin(S)√n(LΨρM+Lg+Ψ
M),
λm(S)=min
t≥0λm(S(t)),¯
λM(S)=max
t≥0λM(S(t))
From (39) and (40), one gets: ˙
V(ξ)≤−(θ−θ)ξ.Now, it
suffices to take θ>θ
. This ends the proof.
978-1-4799-8318-6/15/$31.00 ©2015 IEEE 349
V. T HE ADAPTIVE OBSERVER
Let P(t)=X−1(t)where X(t)is the Shur complement of
S1(t)in Sand is governed by the ODE (34). Then, the SPD
matrix S(t)can be decomposed under the following form
which puts forward the Schur complement of the invertible
matrix S1in S:
S=S1S2
ST
2S3=InS−1
1S2
0ImT
(41)
S10
0S3−ST
2S−1
1S2 InS−1
1S2
0Im
Δ
=In−Υ
0ImTS10
0P−1 In−Υ
0Im
From the above decomposition, one can easily compute the
inverse of S(t)which indeed can be explained as follows:
S−1=InΥ
0Im S−1
10
0P In0
ΥTIm
=S−1
1(t)+Υ(t)P(t)ΥT(t)Υ(t)P(t)
P(t)ΥT(t)P(t)(42)
The gain S−1(t)C(u)can then be obtained:
S−1(t)CT(u)=S−1(t)CT
Ψ0T(u)
=S−1
1(t)+Υ(t)P(t)ΥT(t)CT+Υ(t)P(t)Ψ0T(u)
P(t)ΥT(t)CT+P(t)Ψ0T(u)
=S−1
1(t)CT+Υ(t)P(t)(CΥ(t)+Ψ
o(u))T
P(t)(CΥ(t)+Ψ
o(u))T(43)
According to the developments detailed in the previous
section, the extended high gain observer (25) can be written
as follows:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙
ˆx(t)=F(u, ˆx)ˆx+g(u, ˆx)+Ψ(u, ˆx)ˆρ(t)
−θΔ−1
θS−1
1CT+ΥP(CΥ+Ψ
o(u))T
(Cˆx(t)+Ψ
o(u)ˆρ(t)−y(t))
˙
ˆρ(t)=−θΩ−1
θP(CΥ+Ψ
o(u))T
(Cˆx(t)+Ψ
o(u)ˆρ(t)−y(t))
˙
S1(t)=−θS1+FT(u, ˆx)S1+S1F(u, ˆx)−CTC
with S1(0) = ST
1(0) >0
˙
Υ(t)=θ(F(u, ˆx)−S−1
1CTC)Υ + G(u, ˆx)
−S−1
1CTΨo(u)
˙
P(t)=−θP (CΥ(t)+Ψ
o(u))T(CΥ+Ψ
o(u)) P
+θP with P(0) = PT(0) >0
(44)
Notice that in the case where Ψo(u)=0and
Fk(u, x)=Ip,k=1,...,q, system (1) as well as the
observer (44) become identical to the class of system and
associated observer considered in [9], respectively.
VI. CONCLUSION
An extended high gain observer has been proposed for
a nonlinear class of systems to jointly estimate the state
and some unknown parameters. The design of this observer
is achieved thanks to the notion of characteristic indices
associated to the unknown parameters using the augment-
ed model approach. Besides a comprehensive convergence
analysis of the proposed observer, it has been shown that the
equations of the provided extended high gain observer can be
written under an adquate form to recover the the equations
of available adaptive observers that haven been proposed for
systems belonging to the considered class of systems. This
approach equivalence allows to get insights on the persistent
excitation requirement.
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