Framework for state and unknown input estimation of linear
time-varying systems ?
Peng Lu a , Erik-Jan van Kampen a , Cornelis C. de Visser a , Qiping Chu a

arXiv:1606.08090v1 [] 26 Jun 2016

a

Delft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands

Abstract
The design of unknown-input decoupled observers and filters requires the assumption of an existence condition in the literature.
This paper addresses an unknown input filtering problem where the existence condition is not satisfied. Instead of designing a
traditional unknown input decoupled filter, a Double-Model Adaptive Estimation approach is extended to solve the unknown
input filtering problem. It is proved that the state and the unknown inputs can be estimated and decoupled using the extended
Double-Model Adaptive Estimation approach without satisfying the existence condition. Numerical examples are presented
in which the performance of the proposed approach is compared to methods from literature.
Key words: Kalman filtering; state estimation; unknown input filtering; fault estimation; Double-Model Adaptive Estimation.

1

Introduction

Faults and model uncertainties such as disturbances can
be represented as unknown inputs. The problem of filtering in the presence of unknown inputs has received
intensive attention in the past three decades.
It is common to treat the unknown inputs as part of
the system state and then estimate the unknown inputs
as well as the system state [18]. This is an augmented
Kalman filter, whose computational load may become
excessive when the number of the unknown inputs is
comparable to the states of the original system [10].
Friedland [10] derived a two-stage Kalman filter which
decomposes the augmented filter into two reduced-order
filters. However, Friedland’s approach is only optimal in
the presence of a constant bias [18]. Hsieh and Chen derived an optimal two-stage Kalman filter which performance is also optimal for the case of a random bias [18].
On the other hand, unknown input filtering can be
achieved by making use of unbiased minimum-variance
? This paper was not presented at any IFAC meeting. Corresponding author Peng Lu. Tel. +31 152783466. Fax +31
152786480.
Email addresses: P.Lu-1@tudelft.nl (Peng Lu),
E.vanKampen@tudelft.nl (Erik-Jan van Kampen),
c.c.devisser@tudelft.nl (Cornelis C. de Visser),
q.p.chu@tudelft.nl (Qiping Chu).

Preprint submitted to Automatica

estimation [16,21,5,14,15,3]. Kitanidis [21] first developed an unbiased recursive filter based on the assumption that no prior information about the unknown input
is available [12]. Hou and Patton [14] used an unknowninput decoupling technique and the innovation filtering
technique to derive a general form of unknown-input
decoupled filters [14,15]. Darouach, Zasadzinski and
Boutayeb [7] extended Kitanidis’ method using a parameterizing technique to derive an optimal estimator
filter. The problem of joint input and state estimation,
when the unknown inputs only appear in the system
equation, was addressed by Hsieh [15] and Gillijns and
De Moor [11]. Gillijns and De Moor [12] further proposed a recursive three-step filter for the case when the
unknown inputs also appear in the measurement equation. However, their approach requires the assumption
that the distribution matrix of the unknown inputs in
the measurement equation is of full rank. Cheng et al.
[4] proposed a global optimal filter which removed this
assumption, but this filter is limited to state estimation
[1]. Later, Hsieh [17] presented a unified approach to
design a specific globally optimal state estimator which
is based on the desired form of the distribution matrix
of the unknown input in the measurement equation [17].
However, all the above-mentioned filters require the assumption that an existence condition is satisfied. This
necessary condition is given by Hou and Patton [14] and
Darouach, Zasadzinski and Boutayeb [7], in the form
of rank condition (5). Hsieh [17] presents different decoupling approaches for different special cases. However,

28 June 2016

these approaches also have to satisfy the existence condition (5). In some applications, such as that presented
in the current paper, the existence condition is not satisfied. Therefore, a traditional unknown input decoupled
filter can not be designed.

2.1

Consider the following linear time-varying system:
xk+1 = Ak xk + Bk uk + Ek dk + wk
yk = Hk xk + Fk fk + vk

Recently, particle filters are also applied to unknown input estimation [13,8,28]. These filters can cope with systems with non-Gaussian noise and have a number of applications such as for robot fault detection [2,9,30]. In
this paper, the performance of unknown input estimation using particle filters will be compared with that of
our approach.

(1)
(2)

where xk ∈ Rn represents the system states, yk ∈ Rm
the measurements, dk and fk are the unknown inputs.
Specifically, dk ∈ Rnd the disturbances, fk ∈ Rnf are
the output faults. wk and vk are assumed to be uncorrelated zero-mean white noise sequences with covariance Qk and Rk respectively. uk , the known inputs,
is omitted in the following discussion because it does
not affect the filter design [14]. Without loss of generality, we consider the case: n = m = nd = nf and
rank Hk = rank Ek = rank Fk = m, which implies all
the states are influenced by dk and fk . It should be noted
that the approach proposed in this paper can be readily
extended to the case when n 6= m or rank Hk 6= rank Ek .

This paper proposes an extended Double-Model Adaptive Estimation (DMAE) approach, which can cope with
the unknown input filtering problem when a traditional
unknown input filter can not be designed. The original
DMAE approach, which was proposed by Lu et al. [22]
for the estimation of unknown inputs in the measurement equation, is extended to allow estimation of the
unknown inputs which appear both in the system equation and the measurement equation. The unknown inputs are augmented as system states and are modeled
as random walk processes. The unknown inputs in the
system equation are assumed to be Gaussian random
processes of which covariances are estimated on-line. It
is proved that the state and unknown inputs can be
estimated and decoupled while not requiring the existence condition. Two illustrative examples are given to
demonstrate the effectiveness of the proposed approach
with comparison to other methods from literature such
as the Robust Three-Step Kalman Filter (RTSKF) [12],
the Optimal Two-Stage Kalman Filter (OTSKF) [18]
and the particle filters [13,8].

The unknown inputs are denoted as d0k , i.e., d0k =

"

dk

#

∈
fk
Rnd0 . Then, model (1) and (2) can be reformulated into
the general form as given in Hou and Patton [14] and
Darouach, Zasadzinski and Boutayeb [7]:
xk+1 = Ak xk + Ek0 d0k + wk
yk = Hk xk + Fk0 d0k + vk

(3)
(4)

In this paper, Ek0 = [Ek 0], Fk0 = [0 Fk ]. The existence
of an unknown-input decoupled filter must satisfy the
following existence condition [14,7]:

The structure of the paper is as follows: the preliminaries of the paper are given in Section 2, formulating the
filtering problem when the existence condition for a traditional unknown input decoupled filter is not satisfied
and generalizing the DMAE approach. In Section 3, the
extension of the DMAE approach to the filtering problem when the unknown inputs appear both in the system
equation and the measurement equation is presented.
Furthermore, the on-line estimation of the covariance
matrix of the unknown inputs is introduced. It is proved
that the state and the unknown inputs can still be estimated and decoupled in Section 4. In Section 5, two illustrative examples are given to show the performance of
the proposed approach with comparison to some existing unknown-input decoupled filters. Finally, Section 6
concludes the paper.
2

Problem formulation

rank

"

Fk0 Hk Ek0
0

Fk0

#

= rank

[Fk0 ]

+ rank

" #
Ek0
Fk0

(5)

In our case, since rank Hk = m, the left-hand side of condition (5) is 2m while the right-hand side is 3m. Therefore, the above existence condition does not hold, which
means that all the unknown-input filters mentioned in
the introduction can not be directly implemented.
In this paper, we consider the consecutive bias fault estimation of a system subjected to disturbances, as described in Eqs. (1) and (2). Although the existence condition of designing a traditional unknown input decoupled filter is not satisfied, it will be shown that the unknown inputs can still be decoupled using an extended
DMAE approach.
Remark 1. The model described by Eqs. (1) and (2) is
useful for applications where the disturbances appear in
the system equation and the faults appear in the measurement equation, such as bias fault estimation in aircraft air data sensors [22].

The DMAE approach

This section presents the problem formulation and the
DMAE approach.

2

x̂nf (k)

0
Filter based on x̂nf

no fault
Y
u
x̂af (k)

then the conditional probability of the two filters can be
updated recursively using the following equation:

x̂nf (k + 1)

γnf

Selective
Reinitialization x̂af (k + 1)

Filter based on

x̂0af

fault

γaf

pi (k) =

j=1

Probability evaluator

paf

Fig. 1. Block diagram for the DMAE approach

2.2

,

i = 1, 2

fyk |a,Yk−1 (yk |aj , Yk−1 )pj (k − 1)

(8)
where Yk−1 is the measurement history vector which is
defined as Yk−1 = {y(1), y(2), .., y(k − 1)}.
fyk |a,Yk−1 (yk |ai , Yk−1 ) is the probability density function
which is given by the following Gaussian form [24]:

pnf
Hypothesis Conditional
•

fyk |a,Yk−1 (yk |ai , Yk−1 )pi (k − 1)

2
P

The DMAE approach

fy(k)|a,Yk−1 (y(k)|ai , Yk−1 )
=βi (k) exp{−γiT (k)Ci−1 (k)γi (k)/2}

(9)

The DMAE1 approach proposed in Lu et al. [22] considers the model (1) and (2) for dk = 0 (nd = 0). It is
referred to as the DMAE approach in this paper, which
is generalized in the following.

where

The DMAE [22], which is a modified approach of
multiple-model-based approach [23,24], is composed of
two Kalman Filters (KFs) operating in parallel: a nofault (or fault-free) filter and an augmented fault filter.
These two filters are based on two modes of the system:
fault-free (fk = 0) and faulty (fk 6= 0). The two filters
use the same vector of measurements Y and vector of
input u, and are based on the same equations of motion,
while each hypothesizes a different fault scenario. The
state vector of the no-fault filter xnf and that of the
augmented fault filter xaf are as follows:

In Eq. (10), |•| denotes the determinant of the covariance
matrix Ci (k) which is computed by the KF at time step
k. The filter which matches the fault scenario produces
the smallest innovation which is the difference between
the estimated measurement and the true measurement.
Therefore, the conditional probability of the filter which
matches the true fault scenario is the highest between
the two filters. After the computation of the conditional
probability, the state estimate of the nonlinear system
x̂(k) can be generated by the weighted state estimate
x̂i (k) of the two filters:

xnf,k = xk , xaf,k =

"

xnf,k
fk

#

βi (k) =

x̂(k) =

(6)

(10)

x̂i (k)pi (k)

= x̂nf (k)pnf (k) + x̂af (k)paf (k).

(11)

The fault is only estimated by the augmented fault filter
and the estimate is denoted as fˆ(k). The probabilityweighted fault estimate of the DMAE approach f¯(k) is
calculated as follows:

At time step k, each of the filters produces a state estimate x̂0i (k) and a vector of innovations γi (k). The
principle is that the KF which produces the most wellbehaved innovations, contains the model which matches
the true faulty model best [23,24]. The block diagram of
the DMAE is given in Fig. 1.

f¯(k) = fˆ(k)paf (k)

(12)

The core of the DMAE approach is selective reinitialization. The flow chart of the selective reinitialization
algorithm is presented in Fig. 2.
In the algorithm, x̂0nf (x̂0af ) and x̂nf (x̂af ) denote the state
estimate of the no-fault (augmented fault) filter before
0
0
and after the reinitialization, respectively. Pnf
(Paf
) and
Pnf (Paf ) denote the covariance of state estimate error of
the no-fault (augmented fault) filter before and after the
reinitialization, respectively. x̂t , pt and Pt are the vectors
which contain the state estimate, model probability and
the covariance matrix of state estimation error of the nofault filter and the fault filter respectively. imax,k is the
index of the model with the maximum model probability

A hypothesis test uses the innovation γi (k) and the innovation covariance matrix Ci (k) of the filters in order
to assign a conditional probability to each of the filters.
Let a denote the fault scenarios of the system. If we define the hypothesis conditional probability pi (k) as the
probability that a is assigned ai for i = 1, 2 (a1 = nf ,
a2 = af ), conditioned on the measurement history up to
time step k:
i = 1, 2

1/2
i (k)|

i=1

where “nf ” means no fault and “af ” means augmented
fault. It can be noted that the state vector of the augmented fault filter is the state vector of the no-fault filter
with augmentation of the fault vector fk .

pi (k) = Pr[a = ai |Y (k) = Yk ],

2
X

1
(2π)m/2 |C

(7)

3

where wd,k is a white noise sequence with covariance:
E{wd,k (wd,l )T } = Qdk δkl . fk is also modeled as a random
walk process as:

pt,1 = pnf ; pt,2 = paf
x̂t,1 = x̂0nf,j ; x̂t,2 = x̂0af,j , j = 1 : n
0
0
Pt,1 = Pnf,jj
; Pt,2 = Paf,jj
,j = 1 : n

fk+1 = fk + wf,k ,

(15)

where wf,k is a white noise sequence with covariance:
E{wf,k (wf,l )T } = Qfk δkl . Then, the system model and
measurement model of the no-fault filter can be described as follows:

Find imax,k with
pt,imax,k = max (pt )

Yes

x̄nf,k+1 = Ānf,k x̄nf,k + w̄nf,k
yk = H̄nf,k x̄nf,k + vk

No
imax,k = 1?

(16)
(17)

where

x̂af =

#
"
x̂t,imax,k
xf0

, Paf =

#
"
Pt,imax,k 0
0

Ānf,k =

x̂nf = x̂t,imax,k

P0f

Pnf = Pt,imax,k

Āaf,k =

The DMAE approach can achieve an unbiased estimation of xk and fk when dk = 0 [22]. However, when
dk 6= 0, the unknown-input filtering problem becomes
more challenging. Since the existence condition (5) is
no longer satisfied, traditional unknown-input decoupled
filters can not be designed.

x̄nf,k =

dk

, x̄af,k =

fk

#

(19)
(20)

"

Ānf,k 0
0

I

#

, H̄af,k = [H̄nf,k Fk ], w̄af,k =

"

w̄nf,k

#

wf,k
(21)

This paper proposes a method to adapt Qdk by making
use of the augmented fault filter of the DMAE approach.
To compensate for the effect of a bad choice of Qdk on
the estimation of xk , the system noise vector w̄nf,k in
Eqs.(16), (18) and (21) is modified to:

(13)

where x̄nf,k ∈ Rn+nd and x̄af,k ∈ Rn+nd +nf . The state
vector of the augmented fault filter is that of the nofault filter augmented with the fault vector. Therefore,
the state vector of the no-fault filter can be inferred from
that of the augmented fault filter and vice versa.

w̄nf,k =

"
#
wk + wk0
wd,k

(22)

where wk0 is the noise used to compensate for the effect of
a bad choice of Qdk on the estimation of xk . In this paper,
we approximate wk0 by Ek wd,k . Therefore, w̄nf,k is

The random walk process provides a useful and general
tool for the modeling of unknown time-varying processes
[10,27,15]. dk can be modeled by a random walk process
[27,15] as:
dk+1 = dk + wd,k ,

wd,k

#

Since the difference from the DMAE in Lu et al. [22] is
the augmentation of dk , only the covariance related to
wd,k , i.e., Qdk is discussed below. It should be noted that
Qdk is usually unknown, the optimality of the filter can
be compromised by a poor choice of Qdk [21,15]. If Qdk is
not properly chosen, it can influence the estimation of
dk as well as xk .

In this section, the DMAE is extended to the case when
dk 6= 0. In order to achieve this, the state vectors of the
no-fault filter and augmented fault filter are changed to:
x̄nf,k

, H̄nf,k = [Hk 0], w̄nf,k =

wk

where

Extension of the DMAE approach

"

I

"

x̄af,k+1 = Āaf,k x̄af,k + w̄af,k
yk = H̄af,k x̄af,k + vk

P0f

" #
xk

0

#

The model of the augmented fault filter is as follows:

at time step k.
and
are the parameters which are
used for the initialization of the fault filter.
3

Ak Ek

(18)

Fig. 2. Flow chart of the Selective Reinitialization algorithm.
Note n refers to the dimension of x̂nf .

xf0

"

w̄nf,k =

(14)

4

"
#
wk + Ek wd,k
wd,k

(23)

ˆaf,k−1|k−1 denote the unbiased estimate of x̄af,k−1
Let x̄
given measurements up to time k −1. x̂k−1|k−1 , dˆk−1|k−1
and fˆk−1|k−1 denote the estimates of xk−1 , dk−1 and
fk−1 , respectively. The innovation of the augmented
fault filter is:

where Q̂k,jj , j = 1, 2, ..., m is the jth diagonal element
of Q̂k which is denoted as:

ˆaf,k|k−1
γaf,k = yk − H̄af,k x̄

The restriction Q̃k,jj ≥ 0, j = 1, 2, ..., m in Eq. (31) is to
preserve the properties of a variance [19].

Q̂k = (Ĉaf,k − Hk Qk−1 HkT − Fk Qfk FkT − Rk )

= Hk Ak−1 x̃k−1|k−1 + Hk Ek−1 d˜k−1|k−1 + Fk f˜k−1|k−1
+ Hk wk−1 + Hk Ek−1 wd,k−1 + Fk wf,k−1 + vk
(24)

with
x̃k−1|k−1 := xk−1 − x̂k−1|k−1
d˜k−1|k−1 := dk−1 − dˆk−1|k−1
f˜k−1|k−1 := fk−1 − fˆk−1|k−1

4

(32)

Unknown input decoupled filtering

This section proves that the unknown input decoupled
filtering can be achieved using the extended DMAE approach which does not need to satisfy the existence condition (5). Let l (l ≥ 1) denote the time step when the
first fault occurs and le denote the time step when the
first fault is removed, which means fk = 0 when k < l
and fk 6= 0 when l ≤ k ≤ le . Without loss of generality,
it will be proven that fk can be estimated when k ≤ le .

(25)
(26)
(27)

Therefore, the innovation covariance of the augmented
fault filter is:
T
Caf,k = E{γaf,k γaf,k
}

4.1

x
= Hk Ak−1 Pk−1|k−1
ATk−1 HkT

f
d
T
+ Hk Ek−1 Pk−1|k−1
Ek−1
HkT + Fk Pk−1|k−1
FkT
xf
xd
T
+ Hk Ak−1 Pk−1|k−1
Ek−1
HkT + Hk Ak−1 Pk−1|k−1
FkT
df
dx
T
+ Hk Ek−1 Pk−1|k−1
ATk−1 HkT + Hk Ek−1 Pk−1|k−1
Fk−1

Unknown input estimation during k < l

Theorem 1 During k < l, an unbiased estimate of dk
can be achieved by the fault-free filter of the extended
DMAE approach.

fx
fd
T
+ Fk−1 Pk−1|k−1
ATk−1 HkT + Fk Pk−1|k−1
Ek−1
HkT + Rk
T
+ Hk Qk−1 HkT + Hk Ek−1 Qdk−1 Ek−1
HkT + Fk Qfk FkT
(28)

PROOF. When k < l, fk = 0. The fault-free model
matches the true fault scenario while the augmented
fault filter does not. Therefore, according to the DMAE
approach, imax,k = 1 during this time period.

where the covariance matrices are defined as follows:
x
d
Pk|k
:= E[x̃k|k x̃Tk|k ], Pk|k
:= E[d˜k|k d˜Tk|k ]
f
T
xd
Pk|k
:= E[f˜k|k f˜k|k
], Pk|k
:= E[x̃k|k d˜Tk|k ]

The system model during this period is as follows:
xk+1 = Ak xk + Ek dk + wk
yk = Hk xk + vk

xf
dx
T
Pk|k
:= E[d˜k|k x̃Tk|k ], Pk|k
:= E[x̃k|k f˜k|k
]
fx
df
T
Pk|k
:= E[f˜k|k x̃Tk|k ], Pk|k
:= E[d˜k|k f˜k|k
]

Under this situation, dk can be estimated using the faultfree filter whose convergence condition will be discussed
later.
2

fd
Pk|k
:= E[f˜k|k d˜Tk|k ].

The actual Caf,k is approximated as follows [26,29]:
Ĉaf,k =

1
N

k
X

T
γaf,j γaf,j

(33)
(34)

(29)

The estimation of dk and fk when l ≤ k ≤ le will be
discussed in the following.

j=k−N +1

Qdk can be approximated by the main diagonal of
−1
T
Ek−1
Hk−1 Q̃k (HkT )−1 (Ek−1
)−1

4.2
(30)

For the sake of readability, the subscript “af ” will be
discarded for the remainder of the section. All the variables with a bar on top in the remainder of this section
refer to the augmented fault filter.

with Q̃k is a diagonal matrix defined as:
Q̃k := diag(max{0, Q̂k,11 }, ..., max{0, Q̂k,mm })

Unknown input estimation at k = l

(31)

5

Consequently, the expectation of fˆl|l

Using the DMAE approach, the Kalman gain K̄l can be
partitioned as follows:


Klx

E[fˆl|l ] = E[Klf Fl fl ].



 
d
K̄l = 
Kl 

(35)

Therefore, it can concluded that fl can be estimated if
and only if Klf satisfies

Klf

where Klx , Kld and Klf are the Kalman gains associated
with xk , dk and fk , respectively.

Klf Fl = I.

(36)

˜l−1|l−1 x̄
˜Tl−1|l−1 ]
P̄l−1|l−1 := E[x̄
˜l−1|l−1 = x̄l−1 − x̄
ˆl−1|l−1 .
where x̄

(37)

Due to the selective reinitialization algorithm given in
f
Fig. 2, Pl−1|l−1
= P0f . Therefore, the covariance of the
state prediction error P̄l|l−1 can be computed and partitioned as follows:


x
xd
Pl−1|l−1
Pl−1|l−1
0

 T
dx
d

P̄l|l−1 = Āl−1 
Pl−1|l−1 Pl−1|l−1 0  Āl−1
0
0
P0f


T
Ql−1 + El−1 Qdl−1 El−1
El−1 Qdl−1 0


d
T
d

+
Q
E
Q
0
l−1
l−1
l−1


0
0
Qfl−1
(45)


Px
P xd
0
 l|l−1 l|l−1

dx
d

= Pl|l−1 Pl|l−1 0 
(46)

f
0
0 Pl|l−1

where el is defined as
el := Hl Al−1 x̃l−1|l−1 + Hl El−1 d˜l−1|l−1
+ Hl wl−1 + Hl El−1 wd,l−1 + vl

(38)

Since x̂l−1|l−1 and dˆl−1|l−1 are unbiased (this can be
achieved by the DMAE1 in Lu et. al [22] since fk = 0
when k < l), E[el ] = 0.
Consequently, the expectation of γ̄l is:
E[γ̄l ] = Fl fl .

(39)

The estimation of the fault can be given by
fˆl|l = fˆl|l−1 + Klf γ̄l
= fˆl−1|l−1 + K f γ̄l
l

(44)

PROOF. Define the following covariance matrix:

PROOF. The innovation of the augmented filter is
γ̄l = el + Fl fl

2

Theorem 3 Let x̂l−1|l−1 and dˆl−1|l−1 be unbiased, then
fl can be estimated by the augmented fault filter of the
DMAE approach by choosing a sufficiently large P0f and
a sufficiently small xf0 .

Lemma 2 Let x̂l−1|l−1 and dˆl−1|l−1 be unbiased, if xf0
is chosen to be 0 or sufficiently small, then fl can be
estimated by the augmented fault filter if and only if Klf
satisfies
Klf Fl = I.

(43)

(40)

where
Since imax,k = 1 when k < l, according to the flow chart
of the selective reinitialization algorithm given in Fig. 2,
Eq. (40) can be further written into
fˆl|l = xf0 + Klf γ̄l

x
x
d
T
ATl−1 + El−1 Pl−1|l−1
El−1
Pl|l−1
:= Al−1 Pl−1|l−1
xd
T
dx
+ Al−1 Pl−1|l−1
El−1
+ El−1 Pl−1|l−1
ATl−1
T
+ Ql−1 + El−1 Qdl−1 El−1

(41)

d
d
Pl|l−1
:= Pl−1|l−1
+ Qdl−1
xd
xd
d
Pl|l−1
:= Al−1 Pl−1|l−1
+ El−1 Pl−1|l−1
+ El−1 Qdl−1

Substituting (37) into (41), yields
fˆl|l = Klf Fl fl + Klf el

dx
dx
d
T
T
Pl|l−1
:= Pl−1|l−1
ATl−1 + Pl−1|l−1
El−1
+ Qdl−1 El−1
f
Pl|l−1
:= P0f + Qfl−1

(42)

6

where

Define
C̄l := H̄l P̄l|l−1 H̄lT + Rl .

(47)

x̂∗k|k−1

Substituting Eqs. (21) and (46) into the above equation,
it follows that
f
x
HlT + Fl Pl|l−1
FlT + Rl
C̄l = Hl Pl|l−1

x̂∗k|k

k|k

(48)

f
f
fˆk|k = fˆl|l , Pk|k
= Pl|l
, l < k ≤ le

Fl−1 .

Therefore,
=
can be estimated.

xk+1 = Ak xk + Ek dk + wk
yk = Hk xk + Fk fˆl|l + vk

(50)

It follows from Lemma 2 that fl
2

Error analysis

In the previous sections, it is assumed that x̂l−1|l−1 and
dˆl−1|l−1 are unbiased. We analyze the estimation error
of fl when x̂l−1|l−1 and dˆl−1|l−1 are biased.

Theorem 4 Provided that fk has been estimated at k =
l, dk can be estimated by the augmented fault filter of the
extended DMAE approach.

Through Eq. (44), Eq. (42) can be further rewritten into
fˆl|l = fl + Fl−1 el

PROOF. During this period, the augmented fault
model matches the true fault scenario. Therefore,
imax,k = 2, which means that the fault-free filter is reinitialized by the fault filter during this period. Since this
paper considers bias fault, fk is constant for l < k ≤ le .
Therefore, during this period, we can set:

ˆk|k
x̄

(55)

Therefore, dk can be estimated using the augmented
fault filter under the same condition as for the model
described by Eqs. (33) and (34).
2

Unknown input estimation during l < k ≤ le

ˆk|k−1
x̄

(54)

As can be seen, the only unknown input is dk since the
fault filter treats fk as a known input during this period.
Since a known input does not affect the design of a filter
[14], the convergence condition of this fault filter is the
same as that of the fault-free filter based on Eqs. (33)
and (34).

4.4
4.3

(53)

It can be inferred that the model of the fault filter is
equivalent to:

(49)

C̄l ≈ Fl P0f FlT . It follows that

Klf

k

are updated by the normal Kalman filtering procedure.
It can be seen that during this period, the estimation of
the fault and the covariance are:

f
If P0f is chosen sufficiently large, then Pl|l−1
≈ P0f and



x
Pl|l−1
HlT C̄l−1


dx
T −1 
K̄l = 
Pl|l−1 Hl C̄l 
Fl−1

k|k

(52)

Consequently, the Kalman gain of the augmented filter
can be calculated and partitioned as follows:
K̄l = P̄l|l−1 H̄lT C̄l−1


x
Pl|l−1
HlT


dx
T  −1
=
Pl|l−1 Hl  C̄l
f
Pl|l−1
FlT

"
"
#
#
x
xd
P
P
x̂k|k−1
k|k−1
k|k−1
∗
,
:=
:=
, Pk|k−1
dx
d
Pk|k−1
Pk|k−1
dˆk|k−1
"
"
#
#
" #
x
xd
P
P
x̂k|k
Kkx
k|k
k|k
∗
:=
:=
, Pk|k
, Kk∗ :=
dˆk|k
P dx P d
Kd

(56)

Substitute Eq. (38) into Eq. (56), it follows
fˆl|l = fl + Fl−1 (Hl Al−1 x̃l−1|l−1 + Hl El−1 d˜l−1|l−1
+ Hl wl−1 + Hl El−1 wd,l−1 + vl )
(57)

"

#
"
#
∗
Pk|k−1
0
x̂∗k|k−1
=
, P̄k|k−1 =
f
0
Pl|l
fˆl|l
"
#
"
#
" #
∗
Pk|k
0
x̂∗k|k
Kk∗
=
, K̄k =
,
, P̄k|k =
f
0 P
0
fˆl|l

The estimation error of fl as a function of x̃l−1|l−1 and
d˜l−1|l−1 can be obtained as follows:
f˜l|l = fl − fˆl|l

(58)
˜
=
+ Hl El−1 dl−1|l−1
+ Hl wl−1 + Hl El−1 wd,l−1 + vl )
(59)
Fl−1 (Hl Al−1 x̃l−1|l−1

l|l

(51)

7

If x̃l−1|l−1 and d˜l−1|l−1 are unbiased, the expectation of
f˜l|l is zero, which means the fault estimate is unbiased.
If x̃l−1|l−1 and d˜l−1|l−1 are biased, assume

The system is described by model (1) and (2) where

A=

≤ f¯I,

aI ≤ Al−1 ≤ āI, f I ≤ Fl−1
¯
¯
hI ≤ Hl−1 ≤ h̄I, eI ≤ El−1 ≤ ēI,
¯
¯
ex I ≤ x̃l−1|l−1 ≤ ēx I, ed I ≤ d˜l−1|l−1 ≤ ēd I,
¯
¯
wI ≤ wl−1 ≤ w̄I, wd I ≤ wd,l−1 ≤ w̄d I,
¯
¯
vI ≤ vl−1 ≤ v̄I.
¯

"
#
−0.0005 −0.0084

,
(67)
1.7902
"
" #
" #
0.629
1 0
1 0
E=
,H =
,F =
, (68)
0 −0.52504
0 1
0 1
"
#
"
#
0.0022
0
0.012 0
Q=
,R =
(69)
0
0.0022
0 0.012

(60)
(61)
(62)
(63)
(64)

0.0517

0.8069
#
0

,B =

"
#
0.1815

Then it follows that the fault estimation error is bounded
by the following:

The input uk is: uk = −0.5 when 200 < k ≤ 300, otherwise uk = 0.5. fk is given by the red solid lines in
Fig. 3(c). It can be noted that the number of unknown
 h(aex + eed + w + ewd ) + v h̄(āēx + ēēd + w̄ + ēw̄d ) + v̄  inputs in [27], [7] and [16] is n (n = 2) while this paper
d
d
¯ ¯¯
¯¯ ¯¯ ¯ ¯
¯,
f
deals with 2nd unknown inputs.
f
¯
(65)

4.5

In both examples, since Ek0 = [Ek 0], Fk0 = [0 Fk ], condition (5) is not satisfied. In addition, rank yk < rank
d0k . Consequently, all the unknown input decoupled filters in the introduction are not applicable to solve the
problem, except for special cases when dk = 0 or fk = 0.
N in Eq. (29) is set to be 10. In both examples, Qfk = 0,
Qdk is updated by the main diagonal of the matrix given
in (30), xf0 = [10−3 , 10−3 ]T , P0f = 102 I.

Discussion

For the model given in Eqs. (33) and (34), the convergence condition for time-invariant case has been given
by Darouach et al. [6], which is given as follows:

rank

"
#
zI − A −E
H

0

= n + nd , ∀z ∈ C, |z| ≥ 1

Example 1. In this example, dk is a constant bias vector, which is shown by the red solid lines in Fig. 3(b).
The condition (5) is not satisfied. Therefore, traditional
unknown input filters, which require the satisfaction of
condition (5), can not be implemented.

(66)

This convergence condition is also required by traditional unknown input filters such as those in Darouach,
Zasadzinski and Boutayeb [7] and Cheng et al. [4].

The extended DMAE approach is implemented. The
true and estimated pnf and paf using the extended
DMAE approach are well matched. The probabilityweighted estimates of xk , dk , which are calculated using
Eq. (11), are shown in Fig. 3(a) and 3(b), respectively.
The probability-weighted estimate of fk (calculated using Eq. (12)) is shown in Fig. 3(c). As can be seen, xk ,
dk and fk can all be estimated.

The system considered in this paper is linear and the
noise is assumed to be Gaussian. If the system is nonlinear, the DMAE should be extended using Unscented
Kalman Filters [20,22] or particle filters [13,8,28]. If the
system noise is non-Gaussian, then it should be extended
by making use of particle filters [13,8,28]. However, this
is out of the scope of the present paper.

5

Example 2. In this example 1 , the disturbances,
"
# which
d1,k
are taken from [25], are stochastic. dk =
is gend2,k

Illustrative examples with comparison to existing methods

In this section, two examples similar to that in [27], [7]
and [16] are provided to demonstrate the performance
of the extended DMAE approach. Note that both E and
F are of full rank in this example.

1

The implementation of this work is available at:
https://www.researchgate.net/profile/Peng_Lu15/
publications?pubType=dataset

8

True
DMAE

0.9

0.02
∆ f1

1

0.8
x

0.7

0
−0.02

0.6
0.5
0

RTSKF
DMAE

0.04

100

200

300

400

−0.04
0

500

100

200

100

200

300

400

500

300

400

500

0.04

0

∆ f2

2

0.02
x

−5

0
−0.02

−10
0

100

200

time (s)

300

400

−0.04
0

500

(a) True and estimated states, example 1

1

d

Fig. 4. Errors of estimation of f1 and f2 using the RTSKF
and the DMAE approach, case 1, example 2

True
DMAE

1.5

erated using the following model [25]:
#
"
di,k

1

0

di,k

0.5
0

100

200

300

400

500

2.5

d

2

2

1
100

200

time (s)

300

400

500

(b) True and estimated disturbances, example 1

"
#
di,k−1
 0
=  V2
− L2 −2 LVgi
di,k−1
gi
q


σi L3Vgi
0
 wd,k
q
+
, i = 1, 2
√
(1 − 2 3)σi ( LVgi )3
0

1

(70)

Three cases are considered for this example. The first
two cases are special cases. In these two cases, the existence condition (5) is satisfied. Therefore, some of the
approaches mentioned in the introduction can still be
used.

True
DMAE

4



where V = 35, σ1 = 0.5, σ2 = 0.8, Lg1 = 2500, Lg2 =
0
1500 and wd,k
∼ N (0, 1). The generated dk is shown by
the red solid lines in Fig. 5(b). It should be noted that
the DMAE approach still models dk as a random walk
process since dk is treated as an unknown input.

1.5

0.5
0

time (s)

f1

2

Case 1 dk = 0, fk 6= 0

0
−2
0

100

200

100

200

300

400

500

300

400

500

In this case, Ek is a zero matrix. Therefore, condition (5)
is satisfied. The probability-weighted estimate of fk using the extended DMAE is the same as in Fig. 3(c). The
RTSKF in Gillijns and De Moor [12] is also applied and
the errors of estimation of fk compared to the DMAE
are shown in Fig. 4. In addition, particle filters [13,8] are
also applied. The model used for estimation of fk is also
the random walk. 100 particles are used. The root mean
square errors (RMSEs) of estimation of f1 and f2 using
the RTSKF, the particle filter [13,8] and the extended
DMAE are shown in Table 1.

3

f2

2
1
0
−1
0

time (s)

(c) True and estimated faults, example 1

Case 2 dk 6= 0, fk = 0

Fig. 3. Results of the DMAE approach, example 1

9

Table 1
RMSEs of the fault and disturbance estimation for Example
2

Case 2

Case 3

d1

d2

f1

f2

RTSKF [12]

-

-

0.0103

0.0102

PF [13,8]

-

-

0.1549

0.1496

DMAE

-

-

0.0060

0.0047

OTSKF [18]

0.0697

0.1442

-

-

PF [13,8]

0.1088

0.2035

-

-

DMAE

0.0709

0.1459

-

-

[12,11,18,13,8,15]

N/A

N/A

N/A

N/A

DMAE

0.0845

0.1655

0.0230

0.0283

pnf

Methods

100

200

100

200

300

400

500

300

400

500

1

0.5

0
0

In this case, Fk is a zero matrix. Therefore, condition (5)
is also satisfied. The true and estimated pnf and paf using
the extended DMAE approach are shown in Fig. 5(a).
The probability-weighted estimate of dk is presented in
Fig. 5(b). The results using the methods in Heish [15],
Heish and Chen [18], and Gillijns and De Moor [11], are
similar to that of the DMAE. Particle filter is also applied. The model used for estimation of dk is the random
walk. The RMSEs of estimation of d1 and d2 using the
OTSKF in Heish [18], the particle filter [13,8] and the
extended DMAE are shown in Table 1.

True
DMAE

0.5

0
0

paf

Case 1

1

time (s)

(a) True and estimated model probabilities, case 2, example 2
True
DMAE

1

d1

0
−1
−2
0

Case 3 dk 6= 0, fk 6= 0

100

200

100

200

300

400

500

300

400

500

4

In this case, condition (5) is not satisfied. Thus, all the
conventional filters mentioned in the introduction are
not applicable.

d2

2
0

The true and estimated pnf and paf using the extended DMAE approach are also well matched. The
probability-weighted estimates of xk , is shown in Fig. 6.
The probability-weighted estimates of dk and fk are the
same as in Figs. 5(b) and 3(c) respectively. It can be
seen that despite the fact that the existence condition
for traditional unknown-input decoupled filters is not
satisfied, xk , dk and fk can all be estimated using the
extended DMAE approach. The RMSEs of the estimation of dk and fk using the extended DMAE approach
are shown in Table 1.

−2
0

time (s)

(b) True and estimated disturbances, case 2, example 2
Fig. 5. Results of the DMAE approach, case 2, example 2

However, it is also noted that the extended DMAE approach is more sensitive to Rk errors. The RMSE of the
fault estimation increases to 0.063 when Qk is multiplied
with 103 and increases to 1.79 when Rk is multiplied
with 103 . This is expected since in section 3, the process
noise w̄nf,k is adapted while the output noise vk is not
adapted. Therefore, selection of Rk should be performed
with more caution.

Finally, the sensitivity of the DMAE with respect to
errors in Qk and Rk is discussed. To demonstrate the
sensitivity with respect to errors in Qk , Rk is fixed and
Qk is multiplied with a coefficient kQ . The sensitivity
result of the RMSE of fault estimation with kQ ranging
from 10−3 to 103 is shown in Fig. 7(a). To show the
sensitivity with respect to Rk errors, Qk is fixed and Rk
is multiplied with a coefficient kR . The sensitivity result
of the RMSE of fault estimation with kR ranging from
10−3 to 103 is shown in Fig. 7(b).

6

Conclusion

In this paper, the unknown input decoupling problem
is extended to the case when the existence condition of
traditional unknown input filters is not satisfied. It is
proved that the states, disturbances and faults can be estimated using an extended DMAE approach which does
not require the existence condition. Therefore, it can

It can be seen from Fig. 7(a) and 7(b) that the minimum RMSEs are obtained when kQ = 1 or kR = 1.

10

True
DMAE

0.1

0.5

0.09

0

0.08

−0.5

0.07

−1
0

0.06

100

200

300

400

RMSE

x

1

1

500

x

2

10

0.05
0.04

5

0.03

0

0.02

−5

0.01

−10
0

0

100

200

time (s)

300

400

500

−2

10

0

10

2

kQ

10

(a) Sensitivity with respect to Qk errors, case 3, example 2

Fig. 6. True and estimated states, case 3, example 2

be applied to a wider class of systems and applications.
Two illustrative examples demonstrate the effectiveness
of the extended DMAE approach. Future work would
consider extending the DMAE to deal with systems with
non-Gaussian noise.

2
1.8
1.6
1.4

RMSE

1.2

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1
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