H∞ smoothing for continuous uncertain systems

Abstract

An H_∞ smoother has been developed (Blanco et al.,
2000) and it gives good results for noise uncertainties. Nevertheless,
when uncertain parameters appear, its performance decreases
significantly. Therefore, an estimator robust to noise uncertain
properties and parameter uncertainties is presented. The robust
H_∞ smoother for uncertain systems is developed as a
combination of two robust H_∞ filters. The robust
performance, for both noise and parameter uncertainties, of this new
approach is evaluated on a simple example

Full text

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H∞ ∞ ∞ ∞ SMOOTHING FOR CONTINUOUS UNCERTAINS SYSTEMS
E. Blanco1, Ph. Neveux2, G. Thomas1
1Université Claude Bernard Lyon I, LAGEP, U.M.R. CNRS Q 5007
43 Bd du 11 Novembre 1918. CPE-Lyon, Bât 308G, 69622 Villeurbanne cedex. France.
Tel: +33 (0) 4 72 43 18 75
e-mail: blanco@lagep.univ-lyon1.fr

²Université d’Avignon et des Pays du Vaucluse
Laboratoire d’Hydrodynamique Complexe
Faculté des Sciences – 33 rue Louis Pasteur. 84000 Avignon. France.
e-mail: philippe.neveux@univ-avignon.fr

Fax: +33 (0) 4 72 43 16 99

ABSTRACT
Recently [1], a H∞ smoother has been developed and it gives
good results for noise uncertainties. Nevertheless, when
appear uncertain parameters, its performances decrease
significantly. That’s why, in this paper, an estimator robust to
noise uncertain properties and parameter uncertainties is
presented. As in [1, 4, 6], the robust H∞ smoother for uncertain
systems is developed as a combination of two robust H∞
filters. The robust performances, for both noise and parameter
uncertainties, of this new approach are evaluated on a simple
example.
1. INTRODUCTION
Optimal filtering problem have widely been treated over the
last decades [5, 7, 9, 10]. Recently, problems involving noise
uncertain statistics have been tackled. Various approaches
have been developed in the continuous time domain [7, 10]
and the discrete time domain as well [7, 10]. As noise
statistics uncertainty can be considered as a model uncertainty,
the aspect of model uncertainty has been of interest, recently.
In the state space framework, these model uncertainties are
related to the dynamic matrix and the output matrix of the
system under consideration [3, 11]. In this situation, the
uncertainty can be modelled as an exogenous noise [3] and the
problem is handled as noise uncertain statistics problem. In
the present paper, such an approach is adopted in order to
derive an optimal H∞ smoother robust to both noise uncertain
statistics, dynamic matrix and output matrix uncertainties for
continuous time systems represented by state space equations.
2. STATEMENT OF THE PROBLEM
Let us consider the following linear continuous time uncertain
system:
( ) [] ( )( )
( ) [ ] ( )( )
[]
Tt withtxLtz
, 0)()(
∈=
where:
!"
x
ℜ∈
,
y
ℜ∈
,
z
ℜ∈
stationary zero mean white noises;
!"A, B, C, L are constant matrices with suitable
dimensions.
The following assumptions are made:
(A1). (A,C) detectable;
(A2). (A,B) controllable;
tvtxtCCty
twBtxtAAtx
!
)(
)(
+∆+=
+∆+=
Eq. 2.1
n
p
m
, w and v are uncorrelated
(A3). The noise covariance are defined as:
[]
[]
) ( ').(
=τ
IvtvE
with I denotes the identity matrix;
(A4). ∆A(t) and ∆C(t) represent the parameter uncertainties.
They are modelled as:
( )
( )
)(
tH
tC
c
∆
with:
!" ( )
tF
ℜ∈
an unknown matrix with Lebesgue
mesurable elements, satisfying
!"Ha(t), Hc(t) and E(t) are known continuous
bounded matrices with real values.

An H∞ optimal smoother estimating the signal z(t) from
measurement y(t) and for all acceptable uncertainties is sought
considering the quadratic optimisation of the ratio between the
energies of noises and estimation error respectively.
The estimation error is defined by:
) ( ˆ)()(
tztzte
−=

Finally, the problem can be stated as follows:
Given γ, a strictly positive real, indicating the level of noise
attenuation, a performance criterion is defined such as
2
2
L,
+
∈
vw
vw
Furthermore, the system is supposed to be quadratically
stable; i.e., there exists a definite positive symetric matrix such
that:
[][
()(
++++
tAAPPPtAA
∆∆
!
)(
)(
t
) ( ').(
τ−δ
τ−δ=τ
tIwtwE

Eq. 2.2
)()(
)(
tEtF
tH
tA
a




=




∆

Eq. 2.3
ji
×
ItFtF
≤
′
)()(
;
Eq. 2.4
2
2
2
2
2
2
sup
γ<
=
e
J
Eq. 2.5
] 0
<
)
′

Eq. 2.6
Remark 1: In this paper,
2
2
v
is the norm L2 [0, T] and A’ is
the transpose matrix of A.

3. OPTIMAL H∞ ∞ ∞ ∞ SMOOTHER
3.1. Definition of an auxiliary problem

In order to design an H∞ smoother for uncertain continuous
time systems, the combined use of a forward and a backward
filter will be done. The construction of such a smoother is

Page 2

based on the result presented in [1]. According to [4], an
optimal smoother can be obtained as the combination:
(
(
bf
with
filter and the backward filter respectively.
These two filters are derived considering the following
representation :
1
)()(

ε
)
)
1
11
11
)(
ˆ
x
)(
ˆ
x
)(
ˆ
x
−
−−
−
b
−
f
+=
+=
bfl
QQQ
tQtQQt

Eq. 3.1
f x ˆ and
b x ˆ being the optimal estimates of the forward
[
∈
][
, 0
ℜ
]
Tt withtELtz
tvtwHtCty
with
twHBtAt
m
cm
a
)()(
)()(
1
ε
0)()(
0) 0 (
η
)(
0
∈η
′
εγ
n
′
=
+




+η=
=η=



+η=η !

Eq. 3.2
with
stricly positive real number and:


ε
ℜη
is the state,
jm
m
z
+
∈
the signal to restore, ε a
yy
x
xEF
w
w
HDHBB
m
ca
=
=η




ε
=




ε
=


=
1
0
1

Eq. 3.3
Furthermore, the initial conditions are assumed to be null.

An optimal estimate
m
z ˆ of zm is sought minimising the
criterion :
2
2
2
2
2
2
2
2
,
sup
∈
v
γ<
+
=
vw
e
J
m
Lw
m

Eq. 3.4
with:
mmm
z
ˆ
ze
−=

]

The H∞ optimisation problem can be handled as a minimax
one as follows :

−
γ
Lvw
m
This criterion can be rewritten as :
1
maxmin
2
,
ˆ
2



considering :
- the definite positive symmetric equation solution to the
DRE (Differential Riccati Equation):
PBBPAPPAP
+
′
++
′
=−
!
- the following Lyapunov function :
()
0
dt
[[] ( ) tx
ˆ
Lz
ˆ
tzm
ˆ
00)(
′
′
=
′
′
=

0
1
max
,
∈
min
e
2
2
2
2
2
2
2
2
<







−=
vweJ
m
m

Eq. 3.5
(
η
)
0
ˆ
x
2
2
2
2
2
2
<






−
′
−−−=+=
′
∈
vPBwLVJJ
l
Lvw
x
mm
m
η
γ

Eq. 3.6
E
′
E
ε
2

Eq. 3.7
2
2
2
2
2
2
2
0
wP
′
BwE
dt P
d
V
T
+η−−ηε−=
=ηη′ =∫

Eq. 3.8

Assuming that :
one can consider the auxiliary problem:
(
2
2
with the associate model:
η
′
+=
PBw
ˆ
w

)
2
2
2
2
2
2
, ˆ
w
ˆ
ˆ
x
sup
∈
v
γ<
+
−η
=
′
m
vw
L
J
l
L

Eq. 3.9



′
+=
′
+=
++η=
+η=η
!
PBDCC
PBBAA
with
vw
ˆ
DCy
w
ˆ
BA
m

Eq. 3.10

Consequently, the H∞ smoothing problem for uncertain
systems can be considered as a classical H∞ problem for noise
uncertain statistics. Furthermore, the estimates
derived considering the criteria:
(
2
22
()
2
22

γ must satisfy the two criteria defined by the equations (3.11)
and (3.12). γ is defined as:
(
,,,max
fbf
with
f x ˆ and
b x ˆ are
)
2
f
2
2
2
2
ˆ
ˆ
x
L, ˆ
w
sup
v
∈
f
f
vw
L
J
γ<
+
−η
=
Eq. 3.11
2
b
2
2
2
2
ˆ
ˆ
x
L , ˆ
w
sup
v
∈
b
b
vw
L
J
γ<
+
−η
=
Eq. 3.12
)
0
>
γγγγγ=γ
b
Eq. 3.13
3.2. Forward Filter
The problem is to find a filter minimising Jf. The criterion can
be written as a minimax criterion as follows:

−η
γ
∈
2
2
ˆ
Lvw
x
f


The optimal H∞ filter minimising the criterion (3.11) is given
by the following result:

Theorem 1: Let us consider the linear continuous time -
system defined by (2.1).The filter minimising (3.11) is given
by:
(
(
mfff
yRDBCQxAx
if and only if there exists a positive definite symmetric
differentiable matrix Qf(t) satisfying:

−
′
−
′
+=
CRCQAQQAQ
ffff







−−=
2
2
2
2
2
2
2
ˆ
)
ˆ
x
(
11
, ˆmax
min
vwLJ
ff
f
Eq. 3.14
)
)
ˆ
x
ˆˆ
!
f
C
−
′
+
′
+=

Eq. 3.15
(
(
)
)1
1
2
f
0) 0 (
~
A
'
~~
−
−
′
+=
′
+=
=
′
−=
′
+







γ
DDI
DDIR
Γ
and
Q
CDBA
BBQ
LL
f
f
Γ
Γ
!
Eq. 3.16
❏❏❏

Sketch of the proof:
Let the Hamiltonian:
(
xH
f
2


)()
()( )
[]
w
ˆ
BA
w
ˆ
DCyw
ˆ
DCyw
ˆ
w
ˆ
x
ˆ
L
2
L
f
mmf
f
f
ˆ
1
+ Ψ′+








−−
′
−−−
′
−−
′
′
−=
η
ηηη
γ
η

One can easily derive the expression of the adjoint:
(
yRCx
f
f
=
′
Γ−=
Eq. 3.17
)()
C
′
DBAA andDDIR
AC
L
′
2
L
γ
fmf
Γ−
Ψ
′
−η−
′
−−η−=Ψ
~
~
ˆ

Eq. 3.18

Page 3

using the Riccati transformation:
where Qf is a symmetric definite positive matrix, the following
expression can be obtained after some calculation:

−
′
−
′
+=
!
fff
Q Ψ
+η
ˆ
=η

()()
ff
f
ffmfff
f
f
ffff
x
ˆ
L
2
L
γ
QCyRDBCQA
BBQ
L
2
L
γ
CRCQAQQAQ
ˆ
η
)
ˆ
η
(
ˆ
η
ˆ
η
'
~~
−
′
+−
′
+
′
+=
′
+







!
Γ

Considering the previous relation, the criterion can rewritten
as:
1
max
ˆ
2
2
γ
∈
f
m
f
Eq. 3.19
()
()
0)
ˆ
x
ˆ (ˆ
η
2
ˆ
η
1
2
)
ˆ
x
ˆ (
L
min
x
0
2
f
2
2
2
1
2
2
2
1
2
2
2
1
2
2
2
2
<Ψ










γ
′
′
−η+
′
−+
Ψ−Ψ
′
Γ−−−
Ψ
γ
+−η=
∫
T
fffffm
ffffm
ff
f
fff
dtQ
LL
CRCy
QCRBCyR
LQJ
Ly

Eq. 3.20
Considering the null term:
(
∫
dt
0
The criterion can be expressed as:
)
=Ψ Ψ′=
T
ffff
dtQ
d
W
0
Eq. 3.21
()
0
ˆ
η
)
ˆ
x
ˆ (
L
1
2
max
∈
m
ˆ
x
min
2
2
2
1
2
2
2
<−−−η
γ
=
fmff
f
f
f
CyRJ
Ly

In order to minimise this expression, the first term is equated
to zero choosing
ff
x ˆ
ˆ
=η

Eq. 3.22
. This completes the proof. ♣♣♣
3.3. Backward filter
The estimation is done processing the data from T to 0. The
optimal H∞ backward filter minimising criterion (3.12) is
given by the following result:

Theorem 2: Let us consider the linear system time invariant
defined by (2.1). The a filter minimising (3.3) is given by the
expressions:
(
bbb
ymRDBCQxAx
d
τ
if and only if there exists a positive definite symmetric matrix
Qb(t), continue, differentiable and such as:

′
+
′
++=
τ
) ()
b
x
ˆ
C
d
ˆˆ
−
′
−
′
+−=

Eq. 3.23
BBQ
L
2
L
γ
CRCQAQQAQ
d
d
b
b
bbbb
′
Γ−







−−
'
~~

Qb(T)=0
Eq. 3.24
❏❏❏

The result can be prooved using the same technique as in the
previous section. However, caution should be taken handling
the backward differential equations arising in the optimisation.

3.4. H∞ ∞ ∞ ∞ optimal smoother
The synthesis of the smoother using the technique presented in
section 3, gives rise the following result:
Theorem 3: There is a H∞ optimal smoother such that
if there exists two H∞ optimal filters such that:
Forward filter:
(
CyRDBCQxAx
mfff

−
′′
2
γ<
J
,
)
BBQ


L
2
L
γ
CRCQAQQAQ
!
x
ˆ
fffff
f
′
Γ+





−+=
−
′
+
′
+=
'
~~
)(
ˆˆ
!
Eq. 3.25
Backward filter:
d
b
−=
τ
() (
R
)
BBQ
L
2
L
γ
CRCQAQQAQ
d
d
x
ˆ
C ymDBCQx
ˆ
Ax
ˆ
d
bbbbb
bbb
′
Γ−










−
′
+
′
+=
τ
−
−
′
−
′
+
'
~~


H∞ ∞ ∞ ∞ optimal smoother:
(
f
Eq. 3.26
) (
(
)
)
1
11
11
1
11
) ˆ
x
(
)(
ˆ
x
)(
ˆ
x
)( ˆ
x
−
−
b
−
f
−
b
−
f
−
−
b
−
+=−=
++=
bf
QQx CovQ
tQtQQQt

Eq. 3.27
with Qb and Qb defined has in Theorem 1 and 2.
❏❏❏
4. APPLICATION
Let us consider the nominal expression for linear continuous
time system:
( )( )( )
( ) ( )( )
[]
TtwithtxLtz
, 0)()(
∈=
{}
3 , 2, 1
−−−= diagA

[] 121
−=
C

As no H∞ smoother for both uncertain noise statistics and
plant parameters do exist, the performance of the new tool will
be compared to the H2 smoother. The evaluation of the H∞
smoother will be processed in two steps.
Firstly, in order to show the robustness regarding uncertain
measurement noise statistics, a study will be done assuming no
model parameter uncertainty. In the example, both H2 and H∞
are designed with the same nominal value of the measurement
noise statistics. However, the real value is evolving in a ratio
of 1/5 and 5 with regard to the nominal value. Figure 1 shows
the efficiency of the H∞ smoother.
tvtwDtxCty
twB
+
txAtx
!
)(
++=
+=

Eq. 4.1
[
1
]' 1
−
1
and D =1
25
1
=
[
1
B
=

] 1
L
0
0,1
0,2
0,3
0,4
0,5
0,6
1/5 1/4 1/3 1/212345
EMQr
H2H infinity for uncertain system

Page 4

FIG. 1: Robust H∞ smoother, and H2 smoother. Variation of
noise properties between 1/5 to 5 of the nominal
value.
Secondly, as the H∞ smoother has been developed for
uncertain plant parameters, parametric robustness is studied
considering no noise uncertainty.
Consequently, the following parameter variation is
considered:
{
3,2, 1
−δ+−−= diagA
}
with δ = -0.2 to 0.2
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
-0,20-0,12-0,040,040,120,20
δ
EMQr
H infinity H infinity for uncertain system

FIG. 2: Robust H∞ smoother, and H∞ smoother.
The figure 2 shows the robustness property of the H∞
smoother with respect to the uncertain parameter.
5. CONCLUSION
The incertitude issue of modelisation has been extended to
modelisation of noise as well as for system. Since today, for
the non causal estimation topic for linear continuous time
systems, we found in litterature a H∞ smoother [1] respect to
noise statistical properties. The problem is that one doesn’t
give good performance when we’re confronted to uncertain
parameters. So a robust H∞ smoother has been developed in
this paper in order to fill this lack and offer good performance
regarding to modelisation uncertainties. The simulation has
shown the performance increase that offers such a technique.
When uncertainties disappear, smoothers performances tend
to join. The H∞ smoother appeared to be a sub-optimal version
a sub-optimal version of the robust H∞ smoother defined in
this paper.




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