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On structural approaches to H-infinity observer design

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H-infinity norm-based minimax estimators using the mu-synthesis are developed to estimate the state of an LTI system when subject to structured parameter uncertainty in the model. To this end, an estimator first published in [4], [5] is revisited, including its particular spectral factorization, minimax optimization, and mu-synthesis. Here we show that this approach is based on a problematic design assumption. We relax this assumption so as to obtain feasible results. Using structural properties of the transformed plant in spectral factorization, we introduce a lower-order estimator and devise an estimator with reduced dynamics. Simulations show that H-infinity estimators may remarkably outperform the Kalman filter in view of robustness.
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On structural approaches to Hobserver design
Daipeng Zhang, Johann Reger
AbstractHnorm-based minimax estimators using the
µsynthesis are developed to estimate the state of an LTI
system when subject to structured parameter uncertainty in
the model. To this end, an estimator first published in [4],
[5] is revisited, including its particular spectral factorization,
minimax optimization, and µsynthesis. Here we show that
this approach is based on a problematic design assumption.
We relax this assumption so as to obtain feasible results.
Using structural properties of the transformed plant in spectral
factorization, we introduce a lower-order estimator and devise
an estimator with reduced dynamics. Simulations show that
Hestimators may remarkably outperform the Kalman filter
in view of robustness.
I. INTRODUCTION
In industrial practice the use of Kalman filters is still the
standard approach for state estimation [15], [16], [17]. It is
known as the best H2estimator minimizing the expectation
of the integral (squared) error when exciting the system
with Gaussian white process and measurement noise [7].
Among other reasons, the Kalman filter is popular for its
easy implementation, with only one Riccati equation to solve
offline. But it is prone to both dynamic uncertainty and struc-
tured, non-aleatory input instead of white noise. Whenever
the structure of the uncertain dynamics is available, less
conservative results may be obtained [4], [5], [10], [14].
In robust control, Houtput feedback control is a well-
studied field; for a recent review e.g. see that contained in
[13]. From the viewpoint of minimax optimization theory,
the respective control strategy may be extracted from a
minimizing player while the input or disturbance is governed
by a maximizing player. For such saddle point problems,
an optimal control strategy minimizes the Hnorm of
the closed-loop transfer function even for worst case dis-
turbances [13]. Encompassing also structural uncertainty,
the so-called µsynthesis, i.e. D-K iteration [3], has been
established for the design of robust controllers.
The development of Hstate estimators, however, has
received less attention [10], [11], [12], [14]. Such observers
may yield robust estimates of some or all internal plant
states by processing measurement data. Robust observers
are increasingly demanded in industry as they may provide
state and parameter estimates for monitoring and diagnosis
purposes even in harsh environments and large model uncer-
tainty. It is there where Kalman filters may tend to fail.
Against this background, in this paper we study parametric
uncertainty and combine minimax optimization theory with
µsynthesis to the end of devising more robust estimators. We
Authors are with the Control Engineering Group, Technische Universit¨
at
Ilmenau, P.O. Box 10 05 65, D-98684, Ilmenau, Germany
Corresponding author: johann.reger@tu-ilmenau.de
strive to improve the small gain estimator and the so-called
¯µestimator, introduced in [4], [5]. The ¯µestimator is the
first approach reported for combining a Hobserver design
with µsynthesis, involving a minimax optimization and a
particular spectral factorization, see [6], [8] and citations
in [10], [14]. Therefore, we analyze the ¯µestimator more
deeply. In this analysis, we detected that results for the small
gain estimator and ¯µestimator in [4], [5] are erroneous. We
challenge the idea of spectral factorization, resolve the faulty
assumption in order to build a new µsynthesis minimax
estimator that helps simplify and improve the algorithm.
II. PRO BL EM DESCRIPTION
A. System model
We study LTI systems with parametric uncertainty as per
˙xp=A+Pk
i=1 Aiδixp+B+Pk
i=1 Biδiw
y=C+Pk
i=1 Ciδixp+D+Pk
i=1 Diδiw(1)
z=E xp
where xpis plant state, ymeasured output, zsome estimate,
wdisturbance and noise combined. Let |δi|<1represent
a particular, normalized parameter range with uncertainty
weights captured in Ai,Bi,Ciand Di.
In [5] it is shown that each parametric uncertainty Ai,
Bi,Ciand Dimay be factorized along
AiBi
CiDi=QiSiQiTi
RiSiRiTi=Qi
RiSiTi.(2)
These are integrated into the model by introducing intermit-
tent parameters η= diag{δiIi},such that we have
˙xp=Axp++Bw
=Sxp+T w
y=Cxp+Rη +Dw
z=Exp
(3)
with nominal matrices A,B,Cand D, i.e. for δi= 0, and
Q,Q1. . . Qk, R ,R1. . . Rk,(4)
S,S>
1. . . S>
k>, T ,T>
1. . . T >
k>.(5)
By such decomposition, the uncertainty δiis a real number
and may be arranged in diagonal block form of ∆ =
diag{δiIi}as shown in Fig. 1. Disturbance to dynamics and
noise to measurement are assumed to be independent.
B. Kalman filter
The simplified plant model for a Kalman filter reads [7]
˙xp=Axp+d1
y=Cxp+d2
(6)
assuming d1and d2are zero mean, white noise process
outputs with covariance ¯
B>¯
Band ¯
D>¯
D, respectively.
Normalizing both d1and d2to unit density and stacking
them into w=d>
1d>
2>, we may rewrite
˙xp=Axp+¯
B0w
y=Cxp+0¯
Dw(7)
and define B=¯
B0and D=0¯
Dthat now mimics
(3) without uncertainty. Note that whenever d1and d2are
non-white, i.e. have dynamics, enlarging the system with a
respective exo-system with white noise input of unit density
may help make the Kalman filter design feasible again.
The Kalman filter has the structure [5], [7]
˙
ˆx=Aˆx+L(yˆy)
ˆy=Cˆx(8)
where L= ΣC>
(DD>)1and Σis the positive semi-
definite solution of the differential Riccati equation (DRE)
˙
Σ = AΣ+ΣA>+BB>ΣC>
(DD>)1CΣ
Σ(0) = E[e(0) e>
(0)].(9)
s.t. the estimation error dynamics AL C is stable [5].
The Kalman filter is the best squared error estimator
wrt. error xe,xpˆxin the sense that it minimizes the
H2norm of the transfer function from white noise input w
to xe. Since the minimization of the LTI system is done over
an infinite horizon, the algebraic Riccati equation (ARE) may
replace the DRE. This is called Method 1 in Section VII.
C. Hestimator structure
In this paper the Hestimator has the shape
˙
ˆx=Afˆx+Ly
ˆz=Eˆx(10)
where matrix Econsiders that some zmight have no physical
meaning. Thus, only states that matter are included via E,
granting also the possibility to weight states that are more
important by assigning larger coefficients to corresponding
columns of E. Without restriction, we may safely assume
that the estimator shares the same Ewith the plant.
Define e,zˆz=E(xpˆx) = Exe. Then using upper
and lower linear fractional transformations (LFT) Fuand Fl,
respectively, the Hproblem to be solved is
min
Fmax
kFu(Fl(P, F ),∆)k= min
Fmax
max
kwk26=0
kek2
kwk2
(11)
whose value, for suboptimality, shall be less than some α.
We use k.kto denote the Hnorm of a transfer function,
and k.k2for the L2norm of a signal.
III. MIMO ANALYS IS AND µSYNTHESIS
A. MIMO analysis
Combining plant and estimator in a partially closed-loop,
see Fig. 1, the transfer function N=Fl(P, F )obeys
e=N11 N12
N21 N22η
w=Nc1
Nc2r. (12)
P
F
w
η
z
y
ˆz
eN
w
η
e
Fig. 1. Overall loop and reformulation with Nand in an upper LFT.
1
αN12 N21
P
N11
N22
w ηe
wα
Fig. 2. RP and extended RS by disturbance scaling and fictitious P.
Note that the estimator Fcannot interfere with the plant [5],
as shown in Fig. 1. Therefore, nominal stability (NS) has to
be assumed. Furthermore, by the upper LFT we have
Fu(N, ∆) = N22 +N21 ∆ (IN11∆)1N12 .(13)
Under the assumption of NS, all Nij are stable. Thus, robust
stability (RS) is determined by stability of (IN11∆)1.
By the small gain theorem, assuming kk1then RS
requires kN11k<1. However, Fcannot stabilize N11.
Finally, robust performance (RP) requires to solve problem
(11). Introducing fictitious kPk1and scaling wby
1, as in Fig. 2, the RP problem can be merged with the
RS problem and form an extended RS problem. Hereinafter,
the matrices B,Tand Din (3) will be scaled by 1,
and all notations will remain unchanged. In light of this,
the combined perturbation may be arranged as per
¯
∆ = ∆ 0
0 ∆p(14)
where is real and diagonal and Plives in a complex
unit ball. In view of the small gain theorem, the extended
RS requires that kNk<1with k¯
k1, see [4].
B. Structured singular value
Stability necessitates that kN¯
k<1. In view of the
small gain theorem kNkk¯
k<1implies kN¯
k<1.
So we might demand kNk<1/k¯
k= 1. But this is
conservative if ¯
has some structure other than being a full
complex unity ball. Against this background, in [1] Doyle
has introduced the structured singular value
µ(N),1
min
{¯σ(∆) : det(IN∆) = 0,}(15)
where is a set of admissible structured uncertainty. If no
makes INsingular then µ(N),0. Moreover,
N
DT
D1
T
P
D1
T
DT
ηd
w
d
e
η
Fig. 3. D-scaling with extended ¯
.
if is the full complex unit ball then µ(N) = kNk,
which is the same result as with the small gain theorem.
Extended RS demands kN¯
k<1which is equivalent
to µ¯
(N)<1. For this reason, all methods in this paper
are checked for µ¯
(N)<1wrt. the disturbance scaled
plant. Since µ¯
(N)is quite hard to search we use the upper
bound provided in [1]. For DT∈ DTthat commute with
real, complex or mixed elements from the given bound is
µ(N)min
DT∈DT
¯σD1
TNDT.(16)
Note that in ¯
,Pis from a fictitious complex unit ball.
So, just the real block diagonal ∆ = diag{δ1I1, . . . , δkIk}
with identity matrix Iiof dimension nineeds scaling. In the
respective commuting DT= diag{DT1, . . . , DTk}matrices
DTican be full matrices. For ¯
DTassociated to ¯
we have
¯σ(¯
D1
TN¯
DT(ω))
=D1
T(ω) 0
0IN11 (ω)N12(ω)
N21(ω)N22 (ω)DT(ω) 0
0I
=D1
T(ω)N11(ω)DT(ω)D1
T(ω)N12(ω)
N21(ω)DT(ω)N22 (ω)(17)
and the effect of D-scaling is redistributing the gain between
N12(ω)and N21 (ω), and also within N11(ω).
C. µsynthesis
The µsynthesis minimax estimator results from the so-
called µsynthesis or D-K iteration, see [3], [4]. This involves
a gradient search wrt. disturbance scaling αin each iteration:
1) Set initial D-scaling, e.g. DT(ω) = Iand form Pd.
2) Solve for minFkFl(Pd, F )k.
3) Use Ffrom step 2 to find minimizing DTin prob-
lem minDT¯σD1
T0
0IFl(P, F )DT0
0Iwhose
value is taken as µ¯
(N).
4) Use this DTto form Pdand carry out step 2 and 3
again until µ¯
(N)converges. If µ¯
(N)<1then the
current attenuation level αis accepted.
Note that in view of Fig. 3, DTshall have the realization
DT
ss
=ADBD
CDDD(18)
and the model of the D-scaled plant Pdin ηdand dreads
˙xpd
d
e
y
ss
=
AdQd1
αBd0
Sd01
αD1
DT0
Ed0 0 I
CdRDD1
αD0
xpd
ηd
w
ˆz
(19)
with
Ad=
A QCD0
0AD0
BDD1
DS0ADBDD1
DCD
Bd=hB>0T>D1
D
>B>
Di>, Qd=D>
DQ>B>
D0>
Sd=D1
DS0D1
DCD, Ed=E0 0, Cd=C RCD0
where subscript “d” is used for a D-scaled transfer function.
IV. MINIMAX OPTIMIZATION DESIGN
We expose the minimax design in the spirit of [13]. As
shown in Fig. 1, the estimator cannot affect Ndc1, which is
the corresponding D-scaled transfer matrix of Nc1 in (12),
such that in step 2 solving minFkFl(Pd, F )kamounts to
kNdc2k= min
Fmax
ηd
max
w
kek2
kwk2+kηdk2
γ(20)
for some given bound γ(suboptimality).
Introduce the cost in L2norms, given as
Lcost(xe, w, ηd) = kek2
2γ2kwk2
2γ2kηdk2
2(21)
that is, we wish Lcost(xe, w, ηd)0even under worst w
and ηd. Also introduce the value function
V(t) = x>
eP xe(22)
with P>=P0,xe(0) = 0 for V(0) = 0, and define
J(t) = ˙
V(t) + e>eγ2w>wγ2η>
dηd(23)
such that whenever J(t)0we have
ZT
0
J(t)dt =ZT
0
˙
V(t)dt +ZT
0e>eγ2w>wγ2η>
dηddt
=V(T)− V(0) + Lcost 0.(24)
Thus, J(t)0implies Lcost 0which is equivalent to
kNdc2kγ. After some derivation, we have
J(t) = x>
e˙
P+P(AdLCd)+(AdLCd)>P
+1
γ2P(QdLRDD)Q>
dD>
DR>L>P
+1
α2γ2P(BdLD)B>
dD>
L>P+E>
dEdxe
+ 2x>
eP(AdLCdAfx
γ2
ηd1
γ2Q>
dD>
DR>L>P xe
2
γ2
w1
αγ2B>
dD>L>P xe
2.(25)
For J(t)negative semi-definite, let Af=AdLCdto
cancel the indefinite cross term of xeand ˆx. Thus, Afis
the error dynamics wrt. state and output matrix of the D-
scaled plant. To render the quadratic form of xe, comprising
the variables Pand L, also negative semi-definite, we set L
to its extremum. Hence, L=γ2P1C>
d+VW1where
Pis the solution of the differential Riccati inequality (DRI)
˙
P+PAdV W 1Cd+AdV W 1Cd>P
+1
γ2PQdQ>
d+1
α2BdB>
dV W 1V>P
+E>
dEdγ2C>
dW1Cd0
(26)
and V=QdD>
DR>+1
α2BdD>
,W=RDDD>
DR>+1
α2DD>
.
Without D-scaling V= 0 by independence of disturbance
and noise and W=RR>+1
α2DD>, inequality (26)
resembles (9) with ¯
Σ = γ2P1as γgoes to infinity, i.e.
˙
¯
Σ + A¯
Σ + ¯
ΣA>+QQ>+1
α2BB>
+¯
Σ1
γ2E>EC>W1C¯
Σ0.(27)
Recall that for LTI plants, as considered here, we may solve
the algebraic Riccati equation (ARE) instead of the DRI.
This shall be applied for all the following methods.
The minimax optimization applied to N=Fl(P, F )
without µsynthesis shall later in the simulations be called
“minimax estimator” or Method 2. The µsynthesis minimax
estimator built according to Section III-C will then be called
µestimator” or Method 3.
V. SMA LL GAIN ES TIMATOR AND ¯µESTIMATOR
A. Idea of small gain estimator
The authors of [4], [5] developed a small gain estimator
and applied µsynthesis to obtain a so-called ¯µestimator. In
Section IV, we have seen that the µestimator cannot affect
Ndc1, but achieved kNdc2kγ. So, without transfer of the
gain within Nby µsynthesis, see Section III-B, the authors
introduced a transformed plant that folds the structure of Nc1
onto Nc2 to be solved by minimax optimization.
Taking notice of (12), small gain theorem implies extended
RS if kNk<1. Taking Nvas transfer function of the
conjugate system wrt. transfer function N, it follows
kNk<1Nc1
Nc2r, Nc1
Nc2r<hr, ri
r, Nv
c1Nv
c2Nc1
Nc2r<hr, ri
Nv
c1Nc1 +Nv
c2Nc2 < I . (28)
Any stable transfer function Gwhere kGk< ρ admits a
special spectral factorization Nsf in Nv
sf Nsf =ρ2IGvG
with stable inverse [5], [9]. Nc1 is devoid of F, so let Nsf
be spectral factorization of Nc1 with ρ= 1. Then (28) gives
Nv
c2Nc2 < I Nv
c1Nc1 =Nv
sf Nsf
Nv1
sf Nv
c2Nc2 N1
sf < I (29)
and leads to the problem kNc2N1
sf k<1. The transformed
plant has folded Nc1 onto Nc2 by postmultiplying N1
sf . The
authors of [5] argued that solution Fof the minimax problem
minFkNc2N1
sf kmight be better than for minFkNc2k.
B. Problem of small gain estimator
In [4], [5] the authors show good numerical results for the
small gain estimator. However, we could not reproduce these
results. Firstly, the spectral factorization of Nsf in (29) does
not exist for small scaling of αwith ρ= 1 because small α
enlarges the N12 part in Nc1. So, we relax the condition as
Nv
c2Nc2 < ρ2INv
c1Nc1 =Nv
sf Nsf (30)
and solve for the minimal ρ, say ρmin. By doing so, instead
of kNk<1we require kNk< ρ. Simulations reveal
that ρmin is much greater than 1 under initial D-scaling.
Secondly, the authors of [5] claimed that solving the
transformed plant may already provide acceptable results.
Since we have shown that µ¯
(N)<1is equivalent to a
combination of the original RS and RP, we examine whether
(28) is in line with it. Regarding Section III-A and Fig. 2,
RS requires kN11k<1, which in L2norm means
kN11k2
2<kk2
2(31)
and because η= ∆and =N11η+N12 wthat
kN11ηk2
2<kN11η+N12 wk2
2=kNc1rk2
2.(32)
Thus, Nv
c1Nc1 is lower bounded. The disturbance scaled RP
indicates kek2
2<kwk2
2, which translates into
kN21η+N22 wk2
2<kwk2
2⇔ kNc2rk2
2<kwk2
2(33)
and Nv
c2Nc2 is upper bounded. Note that Nv
c1Nc1 and
Nv
c2Nc2 are not on the same side of the inequality. So (28)
does not meet the final goal µ¯
(N)<1.
We remark that N1
sf introduces a spike in the frequency
response with amplitude ρmin. Relaxing ρmin yields much
smoother spikes. Simulations shown that ρ= 1.2·ρmin
results in an estimator closer to the methods with µsynthesis,
but the attenuation level is not comparable. The ¯µestimator
uses µsynthesis as described in Section III-C, except for that
step 2 involves solving the D-scaled transformed plant.
C. State space model of transformed plant
To build the ¯µestimator the D-scaled transformed plant
Ndc2N1
sf , where N1
sf corresponds to the spectral factoriza-
tion of Ndc1, has the realization [5]
˙xpd
˙x
e
y
ss
=
AdM1
ρQd1
αBdY1
20
0Ad+M1
ρQd1
αBdY1
20
Ed0 0 0 I
CdN1
ρRDD1
αDY 1
20
xpd
x
ηd
w
ˆz
(34)
with notation of (19) and abbreviations
M=1
ρ2QdQ>
dX+1
α2BdY1B>
dX+1
α2BdY1T>D>
D
1Sd
N=1
ρ2RDDQ>
dX+1
α2DY 1B>
dX+1
α2DY 1T>D>
D
1Sd
where Y=ρ2I1
α2T>D>
D
1D1
DTand Xis the symmetric
positive semi-definite solution of the ARE
Ad+1
α2BdY1T>D>
D
1Sd>X
+XAd+1
α2BdY1T>D>
D
1Sd
+X1
ρ2QdQ>
d+1
α2BdY1B>
dX
+S>
dI+1
α2D1
DT Y 1T>D>
D
1Sd= 0 .
(35)
D. Computation of minimax optimization
In contrast to the µestimator designed in Section IV, now
we deal with a transformed plant such that
kNdc2N1
sf k= min
Fmax
ηd
max
w
kek2
kwk2+kηdk2
γ(36)
for some given γ.
The derivation of the solution is similar to Section IV. To
this end, we use the simplified plant model from (34), i.e.
˙xpt
e
y
ss
=
At1
ρQt1
αBtY1
20
Et0 0 I
Ct1
ρRDD1
αDY 1
20
xpt
ηd
w
ˆz
.(37)
The estimator gain results in L=γ2P1C>
t+VW1
such that Af=AtLCtwhere Pis the solution of DRI
˙
P+PAtV W 1Ct+AtV W 1Ct>P
+1
γ2P1
α2BtY1B>
t+1
ρ2QtQ>
tV W 1V>(38)
+E>
tEtγ2C>
tW1Ct0
and V=1
α2BtY1D>+1
ρ2QtD>
DR>
,W=1
α2DY 1D>+
1
ρ2RDDD>
DR>are used. This will be called “¯µestimator”
or Method 4.
VI. LOW ER OR DE R ¯µEST IM ATOR
Consider the following state transform x∆2 ,xpd x
so as to replace xwith x∆2. Then (34) takes the form
˙xpd
˙x∆2
e
y
ss
=
Ad+MM1
ρQd1
αBdY1
20
0Ad0 0 0
Ed0 0 0 I
Cd+NN1
ρRDD1
αDY 1
20
xpd
x∆2
ηd
w
ˆz
(39)
in which x∆2 refers to a stable subsystem without input. So
we may take x∆2 as an external input and let the minimax
optimization consider its worst case. Taking this worst case
of x∆2 as input, in general this method is more conservative.
The result is gain L=γ2P1Cd+N
>
+VW1to be
inserted in Af=Ad+ML(Cd+N).Psolves the DRI
˙
P+PAd+MV W 1(Cd+N)
+Ad+MV W 1(Cd+N)>P
+1
γ2P1
ρ2QdQ>
d+1
α2BdY1B>
d+MM>V W 1V>P
+E>
dEdγ2(Cd+N)>W1(Cd+N)0(40)
with V=1
ρ2QdD>
DR>+1
α2BdY1D>+MN>and W=
1
ρ2RDDD>
DR>+1
α2DY 1D>+N N >— called Method 5.
In view of stable matrix Ad, state x∆2 decays to zero
exponentially. Thus, we also might ignore x∆2 completely.
Then the results is L=γ2P1Cd+N>
+VW1to be
inserted in Af=Ad+ML(Cd+N)and Psolves
˙
P+PAd+MV W 1(Cd+N)
+Ad+MV W 1(Cd+N)>P
+1
γ2P1
ρ2QdQ>
d+1
α2BdY1B>
dV W 1V>P
+E>
dEdγ2(Cd+N)>W1(Cd+N)0(41)
where then V=1
ρ2QdD>
DR>+1
α2BdY1D>and W=
1
ρ2RDDD>
DR>+1
α2DY 1D>— to be called Method 6.
VII. SIMULATION
In the simulations we resort to examples from [2], [4]. For
numerical reasons we solved all AREs for ρmin and γmin
100101102
30
20
10
0
10
optimal system
Magnitude (dB)
100101102
perturbed system
M 1
M 2
M 3
M 4
M 5
M 6
Fig. 4. Magnitude of µvalue for all estimators applied on Example 1.
using the Hamilton matrix. This way we could guarantee the
stability of N1
sf and the closed-loop stability of e.
In Example 1, the matrices for state space model (3) are
A=
0 1 0 0 0
1.01 1.414 0 1.01 0
0 0 0 1 0
0 0 99 2 0
0 0 0 1990 198.9975
, S =
2 0
1.414 0
0 0
2 10
0 1
>
C=10000, D =0 0.01, R =0 1
BR5×2, B(4,1) = 2, Q R5×2, Q(2,1) = 0.1, T R2×2, T = 0.
All the other non-mentioned values of B,Qare all zero. This
denotation also applies to Example 2. The dynamics matrix
of the DThas dimension 2. The optimal attenuation level
αis 0.1315 for Method 2, 0.1229 for Methods 3, 5 and 6,
0.1230 for Method 4 and it needs relaxation of ρ= 1.2·ρmin.
However, it turns out that the optimal solutions from Methods
2 up to 6 have a very small gain L. To further probe the
methods, we add a term ξI to the minimax ARE and use
this suboptimal solution [13], shown in Table I.
For Example 1, Fig. 4 shows similar magnitudes for µof
Methods 2 up to 6 that involve minimax optimization. Value
ρmin for existence of Nsf is above 5.2, far above 1.
Example 2 is a linearized ship model. Its state space model
(3) is given by
A=
A11 06×606×4
A21 A22 06×4
04×6A32 A33
, E =
01×12 1 0 34 127
01×12 0 0 1 0
01×12 0 0 0 1
,
C=
01×12 1000
01×12 0010
01×12 0001
, D =
00.1 0 0
0 0 0.0002 0
0 0 0 0.0002
,
BR16×4, B(6,1) = 2.6517, Q R16×4, Q((2,2),(4,4),(6,6),(6,1)) = 1,
RR3×4, R = 0, T R4×4, T (1,1) = 1.3258, T (4,1) = 0.2190,
S=
00000001×10
1.4283 0.9025 0 1.4283 0 0 01×10
0 0 1.4283 0.9025 0 1.4283 01×10
00001.4283 0.9025 01×10
,
and the non-zero entries in the dynamics matrix are
A11 =
010000
1.6593 1.5821 0 1.6593 0 0
000100
0 0 1.6593 1.5821 0 1.6593
000001
00001.6593 1.5821
,
A21 R6×6, A21(2,2) = 57.2472, A21 (4,2) = 686.3073, A21(6,2) = 6486,
A22 R6×6, A22 = diag{Asub, Asub , Asub}, Asub =0 1
0.2612 0.5204 ,
A32 =
2369.2 0 0 0 4.35 0
1.9011 1.7602 0 8.1487 0.01 0.1881
0 0 0 0 0 0
4.1986 0 0 0 0.13325 0
·106,
A33 =
18691 254912 142930 2125486
5.3781 19683 23336 2651.6
0 1060 0
152.249 23468 590.28 21996
·106.
In Example 2, the dynamics matrix of the DThas dimension
1. Methods 3, 5 and 6 show comparable magnitudes for
µ. Method 4 performs poor which may be caused by non-
optimal D-scaling, see Fig. 5. Value ρmin for existence of
Fig. 5. Magnitude of µvalue for all estimators applied on Example 2.
Method 1 2 3 4
Example 1 1.5156 0.14530.14440.1438∗†
Example 2 1.5513 0.6206 0.4266 0.5078
Method 5 6 small gain small gain relaxed
Example 1 0.1493∗† 0.1553∗† 0.7727 0.1570
Example 2 0.4262 0.4301 0.6074 0.6176
in spectral factorization ARE, ρ= 1.2·ρmin is used
suboptimal minimax ARE with ξ= 0.5is used
TABLE I
FINA L ATTE NUATI ON LE VE L αFOR E ACH ME TH OD.
Nsf is above 6.2, so it is far above 1.
In Example 2, we are able to suppress the attenuation level
of the Kalman filter by using 100·Band D/100 in (9), shown
in Fig. 6. The final attenuation level is 0.5908, which is still
not better than Method 3 up to 6, but surprisingly quite robust
in the perturbed system. Yet, it has enormously large gain in
Ldue to shrinking the noise covariance. Also in Example 1,
the Kalman filter achieves 0.1330 as attenuation level when
using B/100 and 100 ·D, close to the optimal attenuation
level of Method 3 up to 6, similarly it also results in a very
small gain L.
Note that Fig. 4, 5 and 6 show the magnitude of the µ
value at no disturbance scaling. Table I compiles the final
achieved attenuation level under disturbance scaling of which
the magnitude of µreaches 0dB somewhere.
In time simulations, not shown for lack of space, assuming
no knowledge on the initial state, xe(0) 6= 0, the H
estimators generally perform better than the Kalman filter.
Even ˆx(0) = 0.99xp(0) will make the L2gain kek2/kwk2
of the Kalman filter worse than that of the Hestimators.
Finally, in Example 1, we observe that N1
sf introduced
a big spike in the transformed plant. The reason that the
authors of [5] did not detect it might be that too few sampling
points have been used in the frequency analysis. We took
notice of that when calculating the precise solution of the
ARE for the spectral factorization.
VIII. CONCLUSION
We have investigated on various methods for robust state
estimation. Two examples, taken from literature, serve to as-
sess these approaches. Simulations show that the investigated
small gain estimator from [5], relaxed or not, does not pro-
vide convincingly better result in terms of attenuation level
than the simpler minimax estimator. The small gain estimator
combined with µsynthesis (¯µestimator in [5]), may yield
slightly better results than the minimax estimator, but only
with the proposed relaxing of the spectral factorization.
Fig. 6. Magnitude of µvalue for all estimators applied on Example 2 with
modified Kalman filter.
The most consistent and robust results are yielded by the µ
estimator. The two lower order ¯µestimators perform almost
the same, yet they require to solve one more ARE. Thus,
solving the problem on the transformed plant can hardly
justify the effort. Furthermore we notice that modifying the
covariance of the Kalman filter to improve the attenuation
level may result in a very large observer gain without getting
close to the results obtained with the previous estimators.
ACKNOWLEDGMENT
Johann Reger kindly acknowledges support by the Euro-
pean Union Horizon 2020 research and innovation program
under Marie Skłodowska-Curie grant agreement No. 734832.
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