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Transactions on Signal Processing
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. XX, XX XXXX 1
Comparing Robustness of the Kalman, H∞, and
UFIR Filters
Yuriy S. Shmaliy, Fellow, IEEE, Fr´ed´eric Lehmann, Shunyi Zhao, Member, IEEE, and Choon K. Ahn, Senior
Member, IEEE,
Abstract—This paper provides a comparative analysis for
robustness of the Kalman filter (KF), H∞filter derived using the
game theory, and unbiased finite impulse response (UFIR) filter,
which ignores the noise statistics and initial values. A comparison
is provided for Gaussian models by studying effects of errors
and disturbing factors on the bias correction gain. It is shown
that the rule of thumb of optimal filtering in terms of accuracy,
UFIR < H∞= KF, typically does not hold in the real world
implying errors in the noise statistics, mismodeling, temporary
uncertainties, and difficulties in filter tuning to optimal mode.
Under such conditions, the filters are related to each other as
KF ≶H∞<UFIR. A justification of this statement is provided
analytically and confirmed by simulations and experimentally
based on two-state polynomial and harmonic models.
Index Terms—Kalman filter, H∞filter, unbiased FIR filter,
robustness.
I. INTRODUCTION
Robustness is required from an estimator [1] when noise
environments are uncertain, model errors occur owing to
mismodeling, and a system (process) undergoes temporary
unpredictable changes such as jumps in velocity, phase, fre-
quency, etc. The Kalman filter (KF) is poorly protected against
such factors [2]. Therefore, many efforts were made during
decades to “robustify” the performance.
The minimax approach is most popular in designing robust
estimators [3]–[6]. It deals with incomplete or absent infor-
mation of the process and noise [7]. Shown that a saddle-
point property holds, the minimax estimator designed to have
worst-case optimal performance is referred to as minimax
robust [8]. Huber showed in [9] that such an estimator can be
treated as the maximum likelihood estimator (MLE) for the
least favorable member of the class [8]. During decades, there
were developed several models, which are typically exploited
to derive the minimax robust estimators.
A standard approach presumes that an uncertain matrix Υ
is ¯
Υ + ∆Υ, where ¯
Υis an undisturbed part and ∆Υ is
completely or partly unknown being either deterministic or
stochastic [10]. Minimizing errors for a set of maximized
uncertain parameters, such as ∆Υ with bounded norms, results
Y. S. Shmaliy is with Department of Electronics Engineering, Universidad
de Guanajuato, 36885, Salamanca, Mexico (e-mail:shmaliy@ugto.mx).
F. Lehmann is with Department of Communications, Image, and Informa-
tion Processing, TELECOM SudParis, 91011, Evry Cedex, France (e-mail:
frederic.lehmann@it-sudparis.eu).
S. Zhao with the Key Laboratory of Advanced Process Control for Light
Industry (Ministry of Education), Jiangnan University,Wuxi 214122, China
(e-mail: shunyi.s.y@gmail.com).
C. K. Ahn is with the School of Electrical Engineering, Korea University,
Seoul 136-701, Korea (e-mail: hironaka@korea.ac.kr).
in different kinds of robust filters. In [11], the problem of H∞
filtering was solved by finding the bounded error covariance
matrix for the Kalman gain assuming uncertainties in 1)
noise statistics and 2) model matrices. The same problem was
solved in [12] for uncertain models with known multiplicative
white noise, in [13] for uncertain models with known white
noise pursuing the H2performance criterion, and in [8] for
exact models with non-Gaussian noise having outliers. For
statistically known uncertainties, the proposed intrinsically
Bayesian robust KF was shown in [14] to perform optimally
relative to an uncertainty class of state-space models.
Another model implies that an ith uncertain matrix Υiis
coupled with bounded uncertainty Ωas Υi=¯
Υi+HiΩFi,
where ΩTΩ6I,¯
Υiis an undisturbed part of Υi, and Hiand
Fiare some known matrices. The minimization problem is
then solved for maximized norm-bounded uncertain block. In
particular, the H∞problem was solved in [15] using integral
quadratic constraints for a given bound factor. An optimal
robust linear filter was derived in [16] using the linear matrix
inequality instead of solving the Riccati equation for convex
bounded Ω. On finite horizons, a solution was found in [17] for
norm-bounded noise and unknown-but-bounded Ω, in [18] for
norm-bounded Ωwith known noise, and in [19] for a more
general case when Ωexists in model parameters and noise
covariances. A robust solution was shown in [20] for known
noise and initial values and in [21] for bilinear complex state
equation. In [22], the problem of network-based robust H∞
filtering was investigated for uncertain systems with disturbed
state and observation and in [23] for systems with delays.
One more approach invokes the game theory. An existence
of game-theoretic solutions for minimax robust linear filters
and predictors was examined in [24]. The results include
recursive Kalman-like realizations for robust procedures. A
generalization of the Huber approach to the minimax robust
Bayes statistical estimation problem of location with respect
to least favorable a priori distributions has been provided in
[7]. A series of interesting works were devoted to solving
minimax H∞problem on finite horizons for the game cost
function considered as a ratio of the filter error norm, which
must be minimized, and the sum of the initial error norm and
bounded error covariance norms, which must be maximized
[25]–[28]. The designed H∞filter was shown to have Kalman
recursions and become KF in the absence of uncertainties.
Most recently, a dynamic minimax game was employed in
[29] to provide robust filtering for a nominal Gaussian state-
space model, when a relative entropy tolerance is applied to
each time increment of a dynamic model.
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Setting aside methods of robustification of the Bayesian
KF, which has an infinite impulse response (IIR), Jazwinski
formulated in [30] a breakthrough remedy – using estimators
with finite memory. He also stated that such filters, having
the finite impulse response (FIR), are not prone to divergence
as much as the KF and have thus higher robustness. In
the subsequent decades, methods of FIR filtering have been
studied extensively and this statement was validated. The batch
optimal FIR (OFIR) filter was derived in [31], [32] and rep-
resented iteratively in [33]. Bias-constrained OFIR solutions
producing ML estimates were found in [34]–[36]. Unbiased
FIR (UFIR) algorithms were derived and investigated in [2],
[37], [38]. Several robust batch FIR solutions were found
in [39]–[42]. It was also shown that the UFIR filter, which
ignores the noise statistics and initial values, is most robust
among other FIR filters.
The H∞filter may outperform the KF, if the model errors
are properly maximized and efforts made to tune the H∞filter
properly [43]. Otherwise, the KF may perform better, while
the H∞filter may even diverge. The UFIR filter does not
require any information about noise and initial values and is
thus generally more robust by design [44]. However, this filter
does not guarantee optimality. A practical question thus arises
of how these filters measure up to each other in the real world
of uncertainties that motivates our present work.
In this paper, we test the KF, H∞filter [45], and UFIR filter
[44] for robustness by investigating effects of errors on the bias
correction gain. The test we apply assumes that any matrix ¯
Υ
may experience an unpredictable addition ∆Υ = (γ−1) ¯
Υ,
where a scaling factor γis due to mismodeling, errors in
the noise statistics, and temporary uncertainties. A filter with
lowest sensitivity to ∆Υ is considered as most robust. We
show that, under the above conditions, the norm-bounded H∞
filter may diverge when a tuning factor is not set properly,
while the UFIR filter is not prone to divergence and may
outperform both the KF and N∞filter. Verification is provided
by simulation and experimentally for two-state polynomial
and harmonic models. The rest of the paper is organized
as follows. In Section II, we discuss a model along with
filtering algorithms and formulate the problem. Effects of
errors in the noise statistics are investigated in Section III.
Errors caused by mismodeling are studied in Section IV
and by temporary uncertainties in Section V. Experimental
verification is provided in Section VI and concluding remarks
are drawn in Section VII.
II. MODEL, FILTERING ALGORITHMS,AND PROBLEM
FORMULATION
To compare filters for robustness, we will consider a linear
discrete time-invariant state-space model
xn=Axn−1+wn,(1)
yn=Cxn+vn,(2)
in which nis the discrete time index, xn∈RKis the state
vector, yn∈RMis the observation vector, A∈RK×K, and
C∈RM×K. The noise vectors, wn∈RKand vn∈RM, are
white Gaussian with known covariances, Q=E{wnwT
n}and
R=E{vnvT
n}, and the property E{wnvT
k}=0for all nand
k. We will denote ˆ
xn|kas an estimate of xnfound at nfrom
the past up to and including at k. The following variables will
be used: a priori state estimate ˆ
x−
n,ˆ
xn|n−1,a priori estimate
covariance P−
n,Pn|n−1=E{(xn−ˆ
x−
n)(xn−ˆ
x−
n)T},a
posteriori state estimate ˆ
xn,ˆ
xn|n, and a posteriori error
covariance matrix Pn,Pn|n=E{(xn−ˆ
xn)(xn−ˆ
xn)T}.
We will compare the following filtering algorithms.
1) Kalman Filter: For our purposes, we will employ an
alternative KF algorithm given in [45], for
P−
n=APn−1AT+Q(3)
computed once at n= 1 with given P0, as
Pn= (P−
n)−1+CTR−1C,(4)
KKF
n=P−1
nCTR−1,(5)
ˆ
xn=Aˆ
xn−1+KKF
n(yn−CAˆ
xn−1),(6)
P−
n+1 =AP−1
nAT+Q.(7)
2) H∞Filter: The H∞filter version used in this paper has
been derived in [25], [28] based on the game theory1as
¯
Pn= ( ¯
P−
n)−1−θSn+CT¯
R−1C,(8)
K∞
n=¯
P−1
nCT¯
R−1,(9)
ˆ
xn=Aˆ
xn−1+K∞
n(yn−CAˆ
xn−1),(10)
¯
P−
n+1 =A¯
P−1
nAT+¯
Q,(11)
where symmetric positive definite matrices ¯
P0,¯
Q, and ¯
R
chosen by the designer have different meanings than in the
KF and ¯
P−
ncan be computed via ¯
P0using (3). Matrix
Sn∈RK×Kis constrained by a positive definite matrix
(¯
P−
n)−1−θSn+CT¯
R−1C>0in order to keep (8) positive
definite at each n. It is a user choice to assign Sn, which is
introduced in the cost function Jto weight the estimation error.
If a goal is to weight all error components equally [46], [47],
one must set Sn=I. Because the squared norm error-to-error
ratio (cost J) is guaranteed in the H∞filter to be J < 1/θ,
a scalar bound θ > 0must be small enough. For Gaussian
noise with no disturbances, θ= 0 makes the H∞filter KF.
Otherwise, a small θ > 0results in better robustness of the
H∞filter. But when θapproaches the boundary specified by
the constraint, the H∞filter goes to instability. That means that
θmust be chosen carefully and its value optimized to achieve
the best possible effect. In this paper, we will consider the
case of Sn=I.
3) UFIR Filter: The UFIR filter [38] operates on a horizon
[m, n]of Nmost recent points, from m=n−N+ 1 to n.
It computes the initial generalized noise power gain (GNPG)
Gsand the estimate ˆ
xsat s=m+K−1as
Gs= (CT
m,sCm,s )−1,(12)
ˆ
xs=GsCT
m,sYm,s ,(13)
1Note that matrix Pin the H∞algorithm, (11.89) in [45], is equivalent
to the prior error covariance matrix P−
kin the KF. We save this notation in
(8)–(11).
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where
Cm,s =
C(As−m)−1
.
.
.
CA−1
C
,(14)
Ym,s =yT
myT
m+1 . . . yT
sT,(15)
and then updates iteratively both Gnand ˆ
xnby changing an
auxiliary variable lfrom s+ 1 to nusing recursions,
Gl= [CTC+ (AGl−1AT)−1]−1,(16)
KUFIR
l=GlCT,(17)
ˆ
xl=Alˆ
xl−1+KUFIR
l(yl−CAˆ
xl−1),(18)
and taking the output when l=n. An optimal horizon Nopt
for the UFIR filter can be estimated with no reference signal
by minimizing by Nthe derivative of the trace of E{ǫnǫT
n},
where ǫn(N) = yn−CAˆ
xn−1(N)is the measurement
residual [48]. Note that errors of <30 % in Nopt do not affect
essentially the UFIR estimate [44].
If noise is white Gaussian and uncorrelated and ˆ
x0,P0,
Q, and Rare exactly known, then the KF will be optimal.
The H∞filter cannot be tuned to produce a better estimate,
because ¯
P0=P0,¯
Q=Q,¯
R=R, and θn= 0 turn it to KF
and no further progress is available. On the other hand, the
UFIR filter is not optimal. In terms of the estimation accuracy
we thus have the following relation:
UFIR < H∞= KF .(19)
In the real world, it often happens that the model does not
fit a process exactly, noise is not Gaussian, the noise statis-
tics are not fully known, and some short-time unpredictable
changes occur. The H∞filter was designed to overcome such
problems. However, it will perform better only if all of the
norm-bounded model errors are properly maximized and θ
set properly. Otherwise, the UFIR filter may produce smaller
errors, because it does not require any tuning for given N.
A comparison for robustness is thus required that will be
provided in this paper for the Gaussian case, which is most
favorable to the KF.
Robustness can be viewed as an ability of a filter not to
respond to undesirable factors such as errors in the noise
statistics, model errors, and temporary uncertainties. A simple
scaling test for robustness can be organized, if to represent an
uncertain matrix as Υ = ¯
Υ+∆Υ = γ¯
Υ, in which an increment
∆Υ = (γ−1) ¯
Υholds for a positive-valued scaling factor
γ > 0such that γ= 1 means an undisturbed matrix. We will
apply this test using the following substitutions: Q←α2Q,
¯
Q←α2¯
Q,R←β2R,¯
R←β2¯
R,A←ηA, and C←µC,
where a set of scaling factors {α, β, η , µ}>0may either vary
matrix components or not when {α, β, η , µ}= 1. The product
of ηAwill be denoted as
Fg
r←
(ηA)r−g+1 g6r
ηIg=r+ 1
0g > r + 1
.(20)
Simplicity is definitely an advantage of the scaling test,
although arbitrary increments ∆Υ cannot be considered. Pur-
suing the aim, we will learn effects of {α, β, η , µ}on the
bias correction gains (5), (9), and (17). If {α, β, η , µ}cause
KKF
l,K∞
l, and KUFIR
lto decrease, the bias errors will grow.
Otherwise, the random errors will dominate. In what follows,
we will refer to this rule.
For the sake of a correct comparison, all the way we will
base our investigations on the two-state polynomial model,
which is widely used in tracking, localization, timekeeping,
and smart sensing. We will then verify findings by testing
filters by the two-state harmonic model. The increments ∆K
in the bias correction gains caused by {α, β, η, µ}and θwill
be computed for the Kalman, H∞, and UFIR filters as
∆KKF(α, β , η, µ) = KKF(α, β, η, µ)−KKF(1,1,1,1) ,
∆K∞(α, β, η, µ, θ) = K∞(α, β , η, µ, θ)−K∞(1,1,1,1,0) ,
∆KUFIR(η, µ) = KUFIR(η, µ)−KUFIR(1,1) ,
and we notice that smaller ∆Kmeans higher robustness.
The problem now formulates as follows. Given Gaussian
model (1) and (2), we would like to test the KF, H∞filter, and
UFIR filter for robustness by {α, β, η, µ}, both analytically
and experimentally. We also wish to relate the filters to each
other in terms of robustness under the real-world operation
conditions implying model errors and uncertainties in not well
specified noise environments.
III. ERRORS IN THE NOISE STATISTICS AND WEIGHTING
MATRICES
Extra errors produced by filters applied to stochastic models
often occur owing to practical inability to collect accurate
estimates of the noise covariance matrices and norm-bounded
weights, especially for time-varying systems. In this sense, the
UFIR filter ignoring noise is a robust estimator. Effects of α
and βon the KF and H∞estimates for η=µ= 1 will be
discussed next.
A. Kalman Filter
Errors caused by α6= 1 and β6= 1 in the KF can be learned
in a stationary mode by letting P−
n=P−
n−1=P−. Referring
to and substituting (3) into (4) and then (4) into (7) lead to
the discrete algebraic Riccati equation (DARE),
AP−AT−P−+α2Q−AP−(I+EP−)−1EP−AT=0,
(21)
where E=β−2CTR−1C. A solution to (21) does not exist
in a simple form. However, if to accept Pn−1≈P−
n=P−,
which is valid for small measurement noise with R≈0, then
the DARE can be substituted with the discrete-time algebraic
Lyapunov equation [49]
AP−AT−P−+α2Q=0,(22)
which is soluble. Although the approximation (22) may be
rough, especially when Ris not small, it may serve to compare
the KF and H∞filter for robustness as will be shown next.
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Lemma 1: Given α > 0and kAk6a, where a=ρ(A) +
ǫ > 0,ρ(A)is the spectral radius of A, and ε > 0, then the
solution to (22) can be represented for a < 1and α2/(1 −
a2)>1with the negative definiteness as
P−6Qα2
1−a2.(23)
Proof: Consider the Lyapunov equation (22) and repre-
sent its solution with an infinite sum as [50],
P−=α2
∞
X
i=0
AiQAiT.(24)
Transform the p-norm of (24) as
kP−kp=α2k
∞
X
i=0
AiQAiTkp,
6α2
∞
X
i=0 kAikpkQkpkAiTkp,
6α2kQkp
∞
X
i=0
a2i(25a)
=kQkp
α2
1−a2.(25b)
For negligible measurement noise, R≈0, associated with
(22), an optimal estimator can be supposed to track the state
with high accuracy and one can let Pn−1≈0. Setting this
value to (3) gives P−
n≈Q. Next, (4) yields Pn≈CTR−1C,
(7) produces P−
n+1 ≈Q, and we conclude that P−=P−
n≈
P−
n+1 ≈Q. This relation agrees with an equality in (25b) in
an isolated case of α= 1 and a= 0. By choosing the gain
factor such that α2/(1 −a2)>1, we go to (24) and complete
the proof.
Referring to lemma 1, the gain KKF (5) can be approxi-
mated for a < 1with
KKF 6CTR−1C+ ΥQ−1−1CTR−1,(26)
where Υ = β2(1−a2)
α2. Note that such an approximation does
not exist when a>1.
B. H∞filter
Considering K∞(9) for equally weighted estimation errors
with Sn=I, we approximate gain K∞for a < 1with
K∞6CT¯
R−1C+ Υ ¯
Q−1−β2θI−1CT¯
R−1.(27)
In follows from (27) that θ= 0 makes no difference with
the KF gain (26) for Gaussian models implying Q=¯
Qand
R=¯
R. Otherwise, the KF is expected to be less accurate and
the H∞filter to perform better, provided that both ¯
Qand ¯
R
are properly maximized. In fact, if to admit α6= 1 and β6= 1,
then one can try finding some small θ > 0to compensate the
effect. Because the approximations (26) and (27) are restricted
to a < 1, in the following examples we will compute each gain
Krigorously, by (5) and (9).
Example 1: Consider a two-state polynomial model (1) and
(2) represented with A=1τ
0 1 for τ= 0.1 s and C=
[ 1 0 ]. Actual noise covariances Q=σ2
wτ2/2τ/2
τ/2 1 ,
where σw= 0.2 s−2, and R=σ2
v= 1 are supposed to be not
known exactly and we substitute them in the KF algorithm
with α2Qand β2R. We also allow ¯
Q=Qand ¯
R=R. For
this model, the optimal horizon for the UFIR filter is defined
by Nopt =q12σv
τ σw∼
=24 [48].
For the ideal conditions of {α, β, η , µ}= 1 and θ= 0, the
typical root MSEs (RMSEs) computed via the trace of Pas
√tr P, were found to have the following values.
Typical RMSEs for {α, β, η , µ}= 1 and θ= 0:
0.952 (KF) = 0.952 (H∞)<0.981 (UFIR) .(28)
As expected, the H∞filter and KF perform here equally and
the UFIR filter is less accurate that is supported by (19).
We next learn effect of {α, β}and θon the bias correction
gains K1for the first state. Figure 1 sketches ∆K1for all
filters and one notices that the UFIR filter with its ∆K1= 0
is a robust estimator. It is also seen in Fig. 1a and Fig. 1b how
{α, β}and θaffect ∆K1of the KF and H∞filter. The KF is
not protected against αand β, while the H∞filter is able to
diminish ∆K1to zero (cirles) with some small θ > 0. It also
follows from Fig. 1b that ∆K1reduction is available with the
same θfor two different values of β. Admitting that the noise
standard deviation in the KF algorithm can be twice as bad as
in the process (α= 0.5), we illustrate in Fig. 1c an ability of
the H∞filter to overcome this issue with a properly chosen
θ. A picture similar to Fig. 1c can also be shown for α= 1
and β > 1.
Example 2: Consider a two-state harmonic model (1) and
(2) with A=cos φsin φ
−sin φcos φfor φ=π/128 and
C= [ 1 0 ]. Substitute Q=σ2
w1 0
0 1 , where σw= 0.2,
and R=σ2
v= 1 with α2Qand β2Rand allow ¯
Q=Qand
¯
R=R. Find Nopt =√12 3
qσ2
v
σ2
w∼
=10 using [48] for the UFIR
filter. Generate a process, apply filters, and make sure that, for
{α, β, η, µ}= 1 and θ= 0, the filters in terms of accuracy
relate to each other as in (19). Repeat the investigations for
{α, β, η, µ} 6= 1 and observe that the results qualitatively
fit Fig. 1. However, factor θhas here much narrower range
of tuning and the H∞filter is much more sensitive to θ.
Furthermore, in a wide range of θ, gain K∞demonstrates
multiple excursions that may cause ambiguities in tuning and
lead to divergence.
What follows behind these examples is that the H∞filter is
more robust than KF [45], at least against errors in the noise
covariances. However, practically it may be difficult to find
a proper θ, especially for time-varying systems. If so, then
the H∞filter may go to divergence (see, for example, rapidly
growing ∆K1with θ= 0.04 in Fig. 1b). We thus conclude
that, in terms of robustness to errors in the noise statistics, the
relation is
KF ≶H∞<UFIR ,(29)
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.2
-
0.1
-
0
0.1
0.2
a
16.0=q
0.12
0.08
0.04
KF
)0( =
¥qH
UFIR
1=b
)(
1aKD
(a)
1.0 1.6 2.2 2.8 3.4 4.0
0.1
-
0
0.1
0.2
b
KF
)0( =
¥qH
UFIR
04.0=q
0.03
0.02
0.01
1=a
)(
1bKD
(b)
0 0.1 0.2
5
-
0
5
10
5.0=a
1=b
¥
H
UFIR
KF
0.3
θ
2
110)( -
´
DqK
(c)
Fig. 1. Typical effect of α,β, and θon ∆K1(first state) of the KF and H∞filter (Example 1) for θ > 0: (a) α < 1and β= 1, (b) β > 1and α= 1,
and (c) α= 0.5and β= 1. At some points (circles), the H∞filter diminishes ∆K1to zero by properly chosen θ. The UFIR filter is {α, β, θ }-invariant
and is thus robust to these factors. A picture similar to (c) can be shown for α= 1 and β > 1.
where KF < H∞holds for properly maximized ¯
Qand ¯
R
and properly set θ. Otherwise, the KF may perform better,
especially in the harmonic model case.
IV. MISMODELING
Permanent errors occur when a model does not tolerate a
process over all time [51]. If no improvement is achievable, the
best is using robust filters. However, none of the above filters
was designed to ignore errors in the system and observation
matrices. Therefore, testing becomes an essential subject in
performance appraisal.
A. Kalman filter
Effect of model errors on the Kalman gain can be studied
by letting η6= 1 and µ6= 1 during all time. Because {η, µ}
are typically accompanied with errors in the noise covariances,
{α, β} 6= 1, we start with the Lyapunov equation (22), which
solution can be approximated using lemma 2.
Lemma 2: Given {α, η}>0and kAk6a, where a=
ρ(A) + ǫ > 0,ρ(A)is the spectral radius of A, and ε > 0,
then the prior error covariance matrix P−can be approximated
for χ= (ηa)2<1as
P−6Qα2
1−χ.(30)
Proof: Reasoning along similar lines as in lemma 1,
transform the p-norm of P−as
kP−kp=α2k
∞
X
i=0
η2iAiQAiTkp
6α2kQkp
∞
X
i=0
η2ia2i(31a)
=kQkp
α2
1−(ηa)2,(31b)
which leads to (30) and completes the proof.
Lemma 3: Given {α, η}>0and kAk6a, where a=
ρ(A) + ǫ > 0,ρ(A)is the spectral radius of A, and ε > 0,
then the prior error covariance matrix P−can be represented
as
P−6(∞I,if χ>1, a ≶1
Qα2
1−χ,if χ < 1, a < 1
η
.(32)
Proof: To prove (32), rewrite (31a) as
kP−kp6α2kQkp
∞
X
i=0
χi=α2¯η2kQkp
∞
X
i=0
a2i(33)
and show that
¯η2=
∞,if χ>1, a < 1
any positive ,if χ>1, a > 1
1−a2
1−χ,if χ < 1, a < 1
0,if χ < 1,1< a < 1
η
.(34)
To this end, consider an equality following from (33),
∞
X
i=0
χi= ¯η2
∞
X
i=0
a2i.(35)
Suppose that χ>1. If also a < 1, then the sum in the
left-hand side of (35) will equal ∞and the sum in the right-
hand side will be finite. Thus ¯η=∞and the proof of the first
statement made in (34) is complete. Otherwise, when a > 1,
both sums in (35) are infinite and ¯ηmay take any positive
value that proves the second statement in (34).
Now suppose that χ < 1. If also a < 1, then both sums in
(35) will be finite and ¯ηobeys
¯η2=1−a2
1−χ(36)
that proves the third statement in (34). Equation (35) has two
special points, ¯η2=η2= 1 and ¯η2=η2=1−a2
a2=bto mean
that ¯η < η for 1−a2
a2< η < 1and ¯η > η, if 0< η < 1−a2
a2
and η > 1. Otherwise, if a > 1, represent the infinite sum
in (33) as P∞
i=0 a2i= limh=∞Ph
i=0 a2i=1−a2(h+1)
1−a2and
find ¯η2= lim
h=∞
1−a2
(1−a2(h+1))(1−χ)= 0 that proves the fourth
statement in (34). By (35), the uncertainty ¯η2P∞
i=0 a2i= 0∞
results in 1
1−χ. Finally, substituting (34) into (33) leads to (32)
that completes the proof.
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Now, referring to lemma 2 and lemma 3, gain KKF given
by (5) can be rewritten as
KKF 61
µCTR−1C+ ΦQ−1−1CTR−1,(37)
if χ < 1,
61
µ(CTR−1C)−1CTR−1,(38)
if χ>1,
where Φ = β2(1−χ)
α2µ2, and we notice that (38) may not exist for
some models.
B. H∞filter
The game theory-based H∞filter [45] was not derived
to maximize errors in the system matrix Aand observation
matrix C. For η < 1/a and µ6= 1, gain K∞can be
approximated with
K∞61
µCTR−1C+ ΦQ−1−β2
µ2θI−1
CTR−1
if χ < 1(39)
61
µCTR−1C−β2
µ2θI−1
CTR−1→0,
if χ>1,(40)
and we notice that it goes to zero if χ>1(40). As can be seen,
θstill is able to compensate errors coursed by {α, β, η , µ} 6= 1.
C. UFIR filter
In the UFIR filter, permanent errors caused by {η, µ}act
on a horizon [m, n]. Accordingly, using (12), (14), and (17),
gain KUFIR can be expressed as
KUFIR = (CT
m,nCm,n )−1CT
=1
µ"N−1
X
i=0
η−2(N−2−i)(A−N+2+i)T
×CTCA−N+2+i−1CT.(41)
Alternatively, the following lemma can be used to represent
KUFIR similarly to (37) and (39).
Lemma 4: Given the time-invariant model (1) and (2) with
{η, µ}>0, and kAk6a, where a=ρ(A) + ǫ > 0,ρ(A)is
the spectral radius of A, and ε > 0, then the GNPG (12) of
the UFIR filter is represented at n−1with
G−1
n−16(µ21−χN−1
(1−χ)χN−2CTCχ6= 1
µ2(N−1)CTCχ= 1 ,(42)
where CTCcan be singular.
Proof: Consider (12), (14), and (20) on [m, n −1] and
represent the product CT
m,n−1Cm,n−1with a finite sum, in
which let A←ηAand C←µC,
G−1
n−1=µ2
N−2
X
i=0
η−2(N−2−i)(A−N+2+i)T
×CTCA−N+2+i.(43)
Using the triangle inequality of norms and norms properties,
transform the p-norm of (43) as
kG−1
n−1kp=µ2k
N−2
X
i=0
η−2(N−2−i)(A−N+2+i)T
×CTCA−N+2+ikp
6µ2
N−2
X
i=0
η−2(N−2−i)k(A−N+2+i)Tkp
×kCTCkpkA−N+2+ikp
6µ2kCTCkp
N−2
X
i=0
χ−N+2+i.(44)
Now transform (44) using the geometrical sum PN−2
i=0 ri=
1−rN−1
1−rif χ6= 1, substitute the result into (44) as N−1if
χ= 1, and arrive at (42). The proof is complete.
Referring to (42), KUFIR =GCTcan finally be written as
KUFIR 61
µCTC+ρ1A−TCTCA−1−1CT,(45)
if χ6= 1 ,
61
µCTC+ρ2A−TCTCA−1−1CT,(46)
if χ= 1 ,
where ρ1=a2(1−χN−1)
χN(1−χ)and ρ2=N−1
η2.
Example 3: Consider a model discussed in Example 1, refer
to (29), and notice that the H∞filter is more robust than the
KF and the UFIR filter is more robust than the H∞filter
against errors in the noise statistics and weighting matrices.
Now assume that, beginning with n= 0, the model
undergoes permanent errors caused by {η, µ} 6= 1 in the
presence of {α, β } 6= 1. Figure 2 demonstrates variations in
∆K1for the KF and UFIR filter. As can be seen, neither
the KF nor the UFIR filter are advantageous against η6= 1
in terms of robustness. However, the KF performs worse if
also α6= 1 (Fig. 2a) or β6= 1. In the case of µ6= 1, the
KF performs better for µ < 1and both filters perform near
equally if µ > 1. Even so, any variation in αand βaffects
the KF, while the UFIR filter remains unaltered. Note that the
KF will be more vulnerable if αand βundergo simultaneous
changes in opposite directions. Nevertheless, the KF and UFIR
filters have smooth and consistent increments ∆K1over all
reasonable values of ηand µ.
Quite another conclusion can be made regarding the H∞
filter (Fig. 3). In fact, if to suppose that α= 0.7and η= 1,
then ∆K1can be compensated in the H∞filter with θ= 0.08
and, for α= 0.7and η= 0.98, with θ= 0.18 (Fig. 3a).
However, if to keep constant θover a wide range of η, then the
H∞filter becomes inefficient 1) for η < 1, because it requires
unacceptably large θand 2) for η > 1, because it diverges (Fig.
3c). Similar conclusions can be made by analysing the case
of µ6= 1 illustrated in Fig. 3b and Fig. 3d.
Example 4: Repeat for a harmonic model (Example 2) and
infer that the results are qualitatively the same as in Example
3. The difference is in a much higher sensitivity of K∞to θ
and in a much narrower range for θwith no ambiguities.
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0.1 1.0 10
0.5
-
0
0.5
1.0
h
m= 1
b= 1
a= 10
5
0.1
0.5
1.0
UFIR
)(
1hKD
(a)
0.1 1.0 10
0.5
-
0
0.5
1.0
1.5
2.0
m
UFIR
a= 10
5
1
0.3
0.1
h= 1
b= 1
)(
1mKD
(b)
Fig. 2. Typical effect of ηand µon ∆K1of the KF and UFIR filter in the presence of errors in the noise statistics (α6= 1 or β6= 1) for Nopt = 24
(Example 3): (a) β=µ= 1 and (b) β=η= 1. A picture similar to (a) can be shown for α=µ= 1 and to (b) for α=η= 1.
0.9 0.95 1.0 1.05
0.2
-
0.1
-
0
0.1
0.2
h
UFIR
7.0=a
KF
18.0,7.0, ==
¥qaH
08.0,7.0, ==
¥qaH
1.1
)(
1hKD
1== mb
(a)
0.8 1.0 1.2 1.4
0.1
-
0
0.1
0.2
09.0,2, ==
¥qbH
045.0,2, ==
¥qbH
2=b
KF
UFIR
1==ha
m
)(
1mKD
(b)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
5
-
4
-
3
-
2
-
1
-
0
1
2
3
4
5
h
08.0,7.0, ==
¥qaH
1== mb
18.0,7.0, ==
¥qaH
Fig. 3a
)(
1hKD
(c)
KF, UFIR
7.0=a
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
5
-
4
-
3
-
2
-
1
-
0
1
2
3
4
5
09.0,2, ==
¥qbH
045.0,2, ==
¥qbH
1==ha
UFIR
2=b
KF Fig. 3b
m
)(
1mKD
(d)
Fig. 3. Correction of ∆K1caused by ηand µin the H∞filter in the presence of errors in the noise statistics, α6= 1 or β6= 1, (Example 3): (a) β=µ= 1,
(b) α=η= 1, (c) β=µ= 1, and (d) α=η= 1.
It now follows that the KF performance with {η , µ} 6= 1
may be deteriorated dramatically by {α, β} 6= 1 (Fig. 2). In
turn, the H∞filter [45] is less protected against errors in ¯
A
and ¯
Cin view of its ability to diverge (Fig. 3c and Fig. 3d).
Therefore, in terms of robustness against mismodeling, the
relation between the filters generally would be the following,
H∞≶KF <UFIR ,(47)
where H∞>KF holds for properly maximized errors in
Aand Cin the presence of properly maximized ¯
Qand ¯
R
and properly set θ(circled points in Fig. 3a and Fig. 3b).
Otherwise, the H∞filter may diverge (case η > 1in Fig. 3c
and case µ < 1in Fig. 3d) and KF perform better, especially
for the harmonic model.
V. TEMPORARY UNCERTAINTIES
Although permanent errors may severely deteriorate the
performance, they may also be diminished by improving the
model. On the contrary, temporary uncertainties such as jumps
in phase, frequency,and velocity are not easy to handle in view
of unpredictable and short-time nature. Such uncertainties
exist in different forms and a universal model can hardly be
assumed. One way is to assume that the model is affected by
errors in the noise statistics over all time and that errors caused
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by η6= 1 and µ6= 1 occur only at n. Supposing that model
is known at n−1and studying the filter sensitivity to η6= 1
and µ6= 1 at nmay give an idea about the robustness.
A. Kalman Filter
Let the model be affected by {α, β }≶1over all time
and Pn−1known at n−1. At the next time index n, a system
experiences an unpredictable impact caused by η6= 1 and µ6=
1. To learn how much it affects the Kalman gain, substitute
(3) into (4) and (4) into (5) and go to
KKF
n=1
µCTR−1C+β2
µ2η2
×APn−1AT+α2
η2Q−1#−1
CTR−1.(48)
Observe that KKF
nis affected by µdirectly and also through
the term in the parentheses. In turn, ηscales both Qand the
term in the parentheses. An overall effect can be forced by α
and β, although it does not bring KF to divergence.
B. H∞Filter
Reasoning similarly, K∞
ncan be represented as
K∞
n=1
µCT¯
R−1C+β2
µ2η2
×"A¯
Pn−1AT+α2
η2¯
Q−1
−η2θI#)−1
×CT¯
R−1.(49)
Effect of {α, β, η, µ} 6= 1 on K∞
nis more complex, but a
definitive similarity with (48) can be seen, except for the case
when the term in brackets approaches zero.
C. UFIR Filter
Provided that Gn−1is known at n−1, gain KUFIR
ncan be
transformed invoking (16) and (17) to
KUFIR
n=1
µCTC+1
η2µ2(AGn−1AT)−1−1
CT.(50)
As can be seen, (50) is not affected by αand β, while ηand
µproduce effects similar to (49). By virtue of this, the UFIR
filter does not demonstrate an ability to diverge.
Example 5: Let us take a look again at the model discussed
in Example 1 and suppose that it is affected by known
{α, β}≶1over all time. Let us also assume that {η, µ}= 1
until nand {η, µ} 6= 1 occur only at n.
Figure 4 sketches ∆K1as function of η(Fig. 4a) and µ
(Fig. 4b) for the KF and UFIR filter. The first point to notice
is that filters are almost equally sensitive to unpredictable
changes in ηand µwhen {α, β }= 1 and perform equally at
points corresponding to {α, β, η, µ}= 1 (circled). However,
variations in αand βmay dramatically deteriorate the KF
performance.
Now assume that errors are corrected in the H∞filter at
n−1with a properly set θ(circled points in in Fig. 5) and
that ηand µundergo unpredictable changes at n. As can be
seen in Fig. 5a and Fig. 5b, variations with η > 1may bring
the H∞filter to divergence, similarly to Fig. 3c and Fig. 3d,
while all filters perform stably and near equally when η < 1
(Fig. 5b). Note that the KF and H∞filter have almost equal
sensitivities to unpredictable changes in µ(Fig. 5c).
Example 6: Consider the harmonic model discussed in Ex-
ample 2 and conclude that the results for temporary uncertain-
ties are qualitatively the same as for the polynomial model
(Example 5). However, the H∞filter is still much more
sensitive to θthat makes it less robust.
Again we see that the H∞filter may go to divergence,
which is now caused by unpredictable jumps in η > 1(Fig.
5b). On the other hand, the UFIR filter generally outperforms
both the KF (Fig. 4) and the H∞filter (Fig. 5) in the
presence of errors in the noise statistics. We thus conclude
that, in terms of robustness against temporary uncertainties,
the relation between the filters would be the following:
H∞<KF <UFIR ,(51)
since unpredictable changes cannot be held back with θand
the H∞filter may diverge.
Note that findings made above for the two-state polynomial
and harmonic models may not hold exactly for other models
that require further investigations. Below, we will test the
filters experimentally.
VI. EXPERIMENTAL VERIFICATION
In this section, we test the Kalman, H∞, and UFIR filters
by measurements of temperature available from [52]. The two-
state space polynomial model is taken from Example 1. One
week averaging [53] is examined on a long time scale and
temperature tracking is provided on a short time scale.
A. One-Week Temperature Averaging
Because measurements in [52] are provided each hour, a
one-week horizon for the UFIR filter becomes N= 24 ×7 =
168. Having no information about the temperature process and
sensor specifications, we suppose that the standard deviation
in measurements is σv= 0.5◦C. For the two-state polynomial
model, Nopt is approximately connected to σwand σvas
Nopt ∼
=q12σv
τ σw[48] that, for the normalized time step τ= 1,
gives σw∼
=12υv
N2τ∼
=2.1×10−4◦C/s. We save this value of
σwand overrate the observation noise with σv= 2.5◦Cand
underrate with σv= 0.1◦Cto test filters for robustness.
From the database [52], we select two parts of measure-
ments with missing data as shown in Fig. 6. To deal with it,
we set all lost points to zero, model such a mode with µ= 0,
and notice that this is the worst case for the 1) H∞and UFIR
filters assuming mismodeling (Fig. 2c and Fig. 3c) and 2)
KF assuming temporary uncertainty (Fig. 4c). Because a finite
number of missing data do not fit exactly neither mismodeling
nor a one-step temporary uncertainty, we expect for some
deviations from the results obtained in above examples. To
bridge gaps of missing data, we employ the predictive UFIR
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0.1 1.0 10
0.2
-
0
0.2
0.4
0.6
0.8
1.0
UFIR
h
a= 10
3
0.3
0.1
1
KF
µ = 1
b = 1
)(
1hKD
(a)
0.1 1.0 10
0.2
-
0.1
-
0
0.1
0.2
0.3
UFIR
KF
a= 10
3
0.3
1
0.1
h= 1
b= 1
µ
)(
1mKD
(b)
Fig. 4. Typical effects of temporary unpredictable changes in ηand µon ∆K1of the KF and UFIR filter in the presence of errors in the noise statistics,
α6= 1 or β6= 1, (Example 5): (a) β=µ= 1 and (b) β=η= 1. A picture similar to (a) can be sketched for α=µ= 1 and to (b) for α=η= 1.
00.5 1.0 1.5 2.0
0.2
-
0.1
-
0
0.1
0.2
0.3
h
)(
1hKD
18.0,5.0, ==
¥qaH
5.0=a
KF
UFIR
1=a
KF
µ = 1
b = 1
(a)
0.1 1.0 10
4
-
2
-
0
2
4
h
)(
1hKD
18.0,5.0, ==
¥qaH
5.0=a
KF
KF
1=a
UFIR
µ = 1
b = 1
(b)
Fig. 5a
0.1 1.0 10
0.2
-
0.1
-
0
0.1
m
)(
1mKD
48.0,2.0, ==
¥qaH
2.0=a
KF
UFIR
1=a
KF
h = 1
b = 1
(c)
Fig. 5. Typical effect of temporary unpredictable changes in ηand µon ∆K1of the H∞filter in the presence of errors in the noise statistics, {α, β }≶1,
(Example 5): (a) short η-scale for β=µ= 1, (b) long η-scale for β=µ= 1, and (c) β=η= 1. A picture similar to (c) can be sketched for α=η= 1.
filter [54] and use its output as a benchmark. Finally, we run
filters and observe the estimates.
Figure 6a exhibits the results for data taken from a time
span of 10.5...12.5 weeks, in which missing data are observed
during 12 points (half a day). In Fig. 6b, we see similar results
for a span of 14...16 weeks with 36 lost data points (1.5 days).
Instantly we indicate that, for the underrated σv= 0.1◦C,
the H∞filter is not able to provide corrections with θ > 0,
because errors are not maximized. However, for the overrated
σv= 2.5◦C, the H∞filter operates quite good. In fact, while
the KF with σv= 2.5◦Cbecomes highly biased (dotted), the
H∞filter with θ= 0.116 (solid) behaves very close to the KF
with σv= 0.5◦C(dashed) and to the UFIR filter. Even so,
the H∞filter becomes unstable and may diverge with larger
θvalues. One can also notice that, at the first lost point, the
transient rates of the KF and H∞filter are a bit larger than
of the UFIR filter. We see the same correspondence between
the function rates at the circled points in Fig. 5.
B. Temperature Tracking
We next track the state of temperature by filtering out
measurement noise for preliminary evaluated σv= 3.5◦C,
σw= 0.1◦C/s, and N= 4. Filters were run on a time interval
of 14.0...14.2 weeks (Fig. 7). It then turned out that N= 4
serves well for the UFIR filter, but the noise covariances were
estimated incorrectly that has biased KF estimates. To remove
the bias, we next run the H∞filter with different tuning factors
θand observed the following. By θopt ∼
=0.066, the H∞
estimate has become very consistent to the UFIR estimate,
which tracks the mean value. Smaller values θ < θopt made
the H∞estimate biased, but less biased than in the KF. Finally,
θ > θopt turned the H∞filter to instability.
Several critical observations follow behind this experiment:
•For exactly known Gaussian models, the KF estimates
cannot be improved using norm-bounded H∞filters.
•Provided that errors are well maximized in uncertain
models, the H∞filter can perform better than the KF.
Otherwise, the performance cannot be improved.
•Incorrectly set θmay bring the H∞filter to divergence.
•Smaller efforts are required to find Nopt for the UFIR
filter, which is low sensitive to Nand is thus a better
choice for robust estimation.
VII. CONCLUSIONS
A comparison for robustness of the KF, norm-bounded
H∞filter, and UFIR filter has revealed the following. When
the requirements for optimal filtering are completely obeyed,
filters in terms of accuracy relate to each other as UFIR <
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10.5 11.5 12.5
10
15
20
25
30
35
116.0,5.2, ==
¥qs v
H
KF, 5.0=
v
s
KF, 5.2=
v
s
Data UFIR
Time, in weeks
Temperature, oC
UFIR Prediction
Lost
Data (a)
14 15 16
0
5
10
15
20
25
30
35
40
45
KF, 5.2=
v
s
116.0,5.2, ==
¥qs v
H
KF, 5.0=
v
s
Data UFIR
Time, in weeks
Temperature, oC
UFIR Prediction
Lost
Data
(b)
Fig. 6. Robustness of the Kalman, H∞, and UFIR filters against uncertainties
caused by missing data in the measurements of temperature for one week
averaging with N= 168 and σw= 2.1×10−4◦C: (a) 12 lost data
points and (b) 36 lost data points. The UFIR prediction is obtained using
the algorithm designed in [54].
H∞= KF. Otherwise, under the temporary or permanent
model errors, all filters become prone to extra errors. Further-
more, the H∞filter may go to instability and diverge. That
means that the folk theorem about better robustness of the H∞
filter is not fully confirmed in the real world of uncertainties.
The H∞filter may improve the performance, but it may also
dramatically worsen it, if the tuning factor is set incorrectly.
On the other hand, the UFIR filter is invariant to errors in
the noise statistics, which may essentially affect the KF and
H∞filter outputs, and low sensitive to the horizon length N.
Therefore, in terms of robustness, these filters must be related
to each other as H∞≶KF <UFIR.
In other words, the UFIR filter, which is affected by two
factors {η, µ}and low sensitive to Nshould be more robust
than both the KF, which is affected by four factors {α, β, η, µ},
and the H∞filter, which tuning factor θmay not always be
set and kept correctly to compensate effect of {α, β , η, µ}.
We have justified this rule analytically and tested both by
simulation and experimentally based on two-state polynomial
and harmonic models.
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14.0 14.1 14.2
18
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046.0, =
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