Minimax recursive state estimation for linear discrete-time descriptor systems
This paper describes an approach to the online state estimation of systems described by a general class of linear noncausal time-varying difference descriptor equations subject to uncertainties. An approach is based on the notions of a linear minimax estimation and an index of causality introduced h...
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| Опубліковано в: : | Системні дослідження та інформаційні технології |
|---|---|
| Дата: | 2010 |
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| Формат: | Стаття |
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Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
2010
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Minimax recursive state estimation for linear discrete-time descriptor systems / S. Zhuk // Систем. дослідж. та інформ. технології. — 2010. — № 2. — С. 94-105. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859976044812959744 |
|---|---|
| author | Zhuk, S. |
| author_facet | Zhuk, S. |
| citation_txt | Minimax recursive state estimation for linear discrete-time descriptor systems / S. Zhuk // Систем. дослідж. та інформ. технології. — 2010. — № 2. — С. 94-105. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Системні дослідження та інформаційні технології |
| description | This paper describes an approach to the online state estimation of systems described by a general class of linear noncausal time-varying difference descriptor equations subject to uncertainties. An approach is based on the notions of a linear minimax estimation and an index of causality introduced here for singular difference equations. The online minimax observer is derived by the application of the dynamical programming and Moore's pseudoinverse theory to the minimax estimation problem.
Розглянуто підхід до оцінювання стану системи, що описується дескрипторним рівнянням із дискретним часом за спостереженнями, що надходять у реальному часі. Підхід базується на понятті лінійної мінімаксної оцінки та індексу причинності, що вводяться у статті для сингулярних різницевих рівнянь. Рекурсивний оцінювач стану будується шляхом застосування методу «Київського віника» та теорії псевдоінверсних матриць до проблеми мінімаксного оцінювання.
Рассмотрен подход к оцениванию состояния системы, описываемой дескрипторным уравнением с дискретным временем по наблюдениям, поступающим в реальном времени. Подход основан на понятии линейной минимаксной оценки и индекса причинности, введенных в статье для сингулярных разностных уравнений. Рекурсивный оцениватель строится путем применения метода «Киевского веника» и теории псевдообратных матриц к проблеме минимаксного оценивания.
|
| first_indexed | 2025-12-07T16:24:22Z |
| format | Article |
| fulltext |
© S. Zhuk, 2010
94 ISSN 1681–6048 System Research & Information Technologies, 2010, № 2
УДК 517.926:681.518.2
MINIMAX RECURSIVE STATE ESTIMATION FOR LINEAR
DISCRETE-TIME DESCRIPTOR SYSTEMS
S. ZHUK
This paper describes an approach to the online state estimation of systems described
by a general class of linear noncausal time-varying difference descriptor equations
subject to uncertainties. An approach is based on the notions of a linear minimax
estimation and an index of causality introduced here for singular difference
equations. The online minimax observer is derived by the application of the
dynamical programming and Moore's pseudoinverse theory to the minimax
estimation problem.
INTRODUCTION
There is a number of physical and engineering objects most naturally modelled as
systems of differential and algebraic equations (DAEs) or descriptor systems:
microwave circuits [1], flexible-link planar parallel platforms [2] and image
recognition problems (noncasual image modeling) [3]. DAEs arise in economics
[4]. Also nonlinear differential-algebraic systems are studied with help of linear
DAEs by linearization: a batch chemical reactor model [5].
On the other hand there are many papers devoted to the mathematical
processing of data, obtained from the measuring device during an experiment. In
particular, the problem of observer design for continious-time DAEs was
considered in [7] and discrete-time case was studied in [8]–[9]. The minimax state
estimation for uncertain linear dynamical systems was investigated in [10]. Other
approaches to state estimation with set-membership description of uncertainty
were discussed in [12]–[14].
In [6] authors derive a so-called «3-block» form for the optimal filter and a
corresponding 3-block Riccati equation using a maximum likelihood approach.
A filter is obtained for a general class of time-varying descriptor models.
Measurements are supposed to contain a noise with Gaussian distribution. The
obtained recursion is stated in terms of the 3-block matrix pseudoinverse.
In [8] the filter recursion is represented in terms of a deterministic data
fitting problem solution. The authors introduce an explicit form of the 3-block
matrix pseudoinverse for a descriptor system with a special structure, so their
filter coincides with obtained in [6].
In this paper we study an observer design problem for a general class of
linear noncasual time-varying descriptor models with no restrictions on system
structure. Suppose we are given an exact mathematical model of some real
process and the vector kx describes the system output at the moment k in the
corresponding state space of the system. Also successive measurements
…… kyy0 of the system output kx are supposed to be available with the noise
…… kgg0 of an uncertain nature. (For instance we do not have a-priory infor-
Minimax recursive state estimation for linear discrete-time descriptor systems
Системні дослідження та інформаційні технології, 2010, № 2 95
mation about its distribution.) Further assume that the system input kf , start point
q and noise kg are arbitrary elements of the given set G . The aim of this paper
is to design a minimax observer kxk ˆ that gives an online guaranteed
estimation of the output kx on the basis of measurements ky and the structure
of G . In [9] minimax estimations were derived from the 2-point boundary value
problem with conditions at 0=i (start point) and ki = (end point). Hence a
recalculation of the whole history kxx ˆˆ0… is required if the moment k changes.
Here we derive the observer kk xyk ˆ),( by applying dynamical programming
methods to the minimax estimation problem similar to the posed one in [9]. We
construct a map x̂ that takes ),( kyk to kx̂ making it possible to assign a unique
sequence of estimations …… kxx ˆˆ0 to given sequence of observations
…… kyy0 in the real time. A resulting filter recursion is stated in terms of
pseudoinverse of positive semi-defined nn× - matrices.
Minimax estimation problem
Assume that n
kx R∈ is described by the equation
,0,1,=,=11 …kfxCxF kkkkk −++ (1)
with initial condition
qxF =00 , (2)
and ky is given by
,0,1,=,= …kgxHy kkkk + (3)
where kk CF , are nm× -matrices, kH is np× -matrix. Since we deal with
descriptor system we see that for any k there is a set of vectors 00
1 kxx …
satisfying (1) while 0=0,= qfi . Thus the undefined inner influence caused, by
00
1 kxx … , may appear in the system’s output. Also we suppose the initial
condition q , input }{ kf and noise }{ kg to be unknown elements of the given
set. (Here and after ),( ⋅⋅ denotes an inner product in an appropriate Euclidean
space, 2/1),(= xxx .)
⎪⎩
⎪
⎨
⎧
==Γ }){},{,(:}){},{,( kkkk gfqGgfq
⎪⎭
⎪
⎬
⎫
≤++= ∑
∞
1),(),(),(
0
kkkkkk ggRffSqSq , (4)
where kk RSS ,, are some symmetric positive-defined weight matrices with
appropriate dimensions. The trick is to fix any N -partial sum of (4) so that
}){},{,( kk gfq belongs to
:}){},{,{(=: kk
N gfqG 1}),(),(),(
0=
1
0=
≤++ ∑∑
−
kkk
N
k
kkk
N
k
ggRffSqSq . (5)
S. Zhuk
ISSN 1681–6048 System Research & Information Technologies, 2010, № 2 96
Then we derive the estimation )ˆ,,(=ˆ 1−NNN xyNvx considering a minimax
estimation problem for NG . Let us denote by N a set of all }){,},({ kk fqx such
that (1) is held. The set N
yG is said to be a-posteriori set, where
}}){},{,(,}){,},({:}{{=: N
kkkkkkk
N
y xHyfqfqxx GNG ∈−∈ . (6)
It follows from the definition that N
yG consists of all possible }{ kx , causing
an output }{ ky , while }){},{,( kk gfq runs through NG . Thus, it’s naturally to
look for estimation Nx of only among the elements of )( N
yNP G , where NP
denotes the projection that takes }{ 0 Nxx … to Nx .
Definition 1. A linear function )ˆ,( Nx is called a minimax a-posteriori
estimation if the following condition holds:
|)ˆ,(),(|sup|)~,(),(|supinf
}{}{}~{
NN
N
ykx
NN
N
ykxN
ykx
xxxx −
∈
=−
∈∈ GGG
.
The non-negative number
|)ˆ,(),(|sup=),(ˆ
}{
NN
N
ykx
xxN −
∈G
σ
is called a minimax a-posteriori error in the direction . A map
}<),(ˆ:{dim= +∞∈ NIN n
N σR
is called an index of causality for the pair of systems (1)–(3).
Now we say that a minimax estimation problem is to construct an a-
posteriori linear minimax estimation )ˆ,( Nx for the system (1) on the basis of the
measurements (3) and a-posteriori set N
yG . A solution of the minimax estimation
problem in the form of a recursive map )ˆ,( Nxk is presented in the next
section.
Minimax online observer
Denote by kQk a recursive map that takes each natural number k to the
matrix kQ , where
kkkkkkkkkkkk FSCWCSSFHRHQ ][= 111111 −−
+
−−−− ′−′+′ ,
kkkkk CSCQWHRHSFFQ ′+′+′ =,= 000000 . (7)
Let krk be a recursive map that takes each natural number k to the
vector n
kr R∈ , where
,= 1111 kkkkkkkkk yRHrWCSFr ′+′ −
+
−−− 0000 = yRHr ′ (8)
and to each natural number ∈i assign a number iα , where
),,(),(= 1111 −−
+
−− −+ iiiiiiii rrWyyRαα ),(),(= 0000 yyRgSg +α . (9)
Minimax recursive state estimation for linear discrete-time descriptor systems
Системні дослідження та інформаційні технології, 2010, № 2 97
The main result of this paper is formulated in the next theorem.
Theorema (minimax recursive estimation). Suppose we are given a
natural number N and a vector nR∈ . Then a necessary and sufficient condition
for a minimax a-posteriori error ),(ˆ Nσ to be finite is that
.=NN QQ+ (10)
Under this condition we have
2
1
2
1
),()],([1=),(ˆ +++− NNNNN QrrQN ασ (11)
and
.),(=)ˆ,( NNN rQx + (12)
Corollary 1. The index of causality NI for the pair of systems (1)–(3) can
be represented as )(rank= NN QI .
Corollary 2 (minimax obsever). The online minimax observer is given by
kkk rQxk +=ˆ and (we assume here that +∞=0/1 .)
=−
∈∈
2
}~{}{
~maxmin=)(ˆ NN
N
ykxN
ykx
xxN
GG
ρ
)}({min
)]ˆ,ˆ([1
=ˆmax
2
}{ N
xxQ
xx
i
i
NNNN
NN
N
ykx λ
α +−
−
∈
=
G
(13)
where )(Niλ are eigenvalues of .NQ In this case all possible realisations of the
state vector Nx of (1) fill the ellipsoid nN
yNP RG ⊂)( , where
1}),ˆ2(),(:{=)( ≤+− NNNN
N
yN xxQxxQxP αG . (14)
Remark 1. If )(min kkk HRH ′λ grows for …1,,= +iik then the minimax
estimation error )(ˆ kρ becomes smaller causing kx̂ to get closer to the real state
vector kx .
In [8] Kalman’s filtering problem for descriptor systems was investigated
from the deterministic point of view. Authors recover Kalman’s recursion to the
time-variant descriptor system by a deterministic least square fitting problem over
the entire trajectory: find a sequence }ˆ,,ˆ{ ||0 kkk xx … that minimises the following
fitting error cost
+−+−
2
|000
2
|000| =)}({ kk
k
kik xHygxFxJ
2
|
2
|11|
1=
kiiikiikii
k
i
xHyxCxF −+−+ −−∑
assuming that the .rank nH k
kF ≡⎥⎦
⎤
⎢⎣
⎡ According to [8] the successive optimal
estimates }ˆ,,ˆ{ ||0 kkk xx … resulting from the minimisation of kJ can be found
from the recursive algorithm
S. Zhuk
ISSN 1681–6048 System Research & Information Technologies, 2010, № 2 98
+′+′ −−−
−
−−−− 1|11
1
11|11|| ˆ)(=ˆ kkkkkkkkkkkk xCCPCEFPx
)(=ˆ, 0000|00|0| yHqFPxyRHP kkkkk ′+′′+ ,
11
11|11| ))((= −−
−−−− ′+′+′ kkkkkkkkkk HHFCPCEFP ,
1
00000|0 )(= −′+′ HHFFP . (15)
Corollary 3 (Kalman’s filter recursion). Suppose the ,rank nH k
kF ≡⎥⎦
⎤
⎢⎣
⎡ and
let krk be a recursive map that takes each natural number k to the vector
n
kr R∈ , where
,)(= 111111 −
+
−−−−− +′′+′ kkkkkkkkkk rQCCCFyHr
0000 = yHqFr ′+′ . (16)
Then kkkk xrQ |ˆ=+ for each N∈k , where kkx |ˆ is given by (15) and nI k = .
Acknowledgements. It is a pleasure to thank Prof. A.Nakonechniy and
Dr. V. Pichkur for insightful discussions about the key ideas presented in this
paper.
Proof of Theorem. By definition, put
⎟⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
−
−
−
N
Nmnmnmn
mn
mnmn
mn
mnmn
mn
mnmnmn
F
C
FC
FC
F
1
21
10
0
000
0
00
0
00
0
000
=
…
…
…
…
…
…
…
…
F ,
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
Npnpn
pnpn
pnpn
H
H
H
…
…
…
…
00
00
00
= 1
0
H ,
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
Nx
x
x
1
0
=X ,
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
Ny
y
y
y
2
1
0
Y ,
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
−1
1
0
Nf
f
f
q
F ,
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
Ng
g
g
g
2
1
0
G .
By direct calculation we obtain ),,(=),( XLNx
1}):{= 2
2
2
1 ≤−+ HXYXFXG N
y ,
where ),(),(= 1
0
2
1 kkk
N ffSqSq ∑ −+F , 2⋅ is indused by kR on the same way.
This implies
Minimax recursive state estimation for linear discrete-time descriptor systems
Системні дослідження та інформаційні технології, 2010, № 2 99
|)~,(),(|sup|=)~,(|sup
X}{
XLXL
GG
−−
∈ ∈ N
y
NN
N
ykx
xx .
Denote by M the set ]''[ HFR . We obviously get
+∞−⇔∈
∈
|<)~,(),(|sup XLXLML
GX N
y
.
The application of Corollary 4 yields (10). Consider a vector ML∈ . Clearly
N
y
N
y
N
y
GXXLXLXL
GXGX
∈≤≤
∈∈
),,(sup),(),(inf .
Let c denotes )),(inf),(sup(
2
1 XLXL GXGX N
yN
y ∈∈ + . Therefore
|)~,(c|))|(s()|((s
2
1|)~,(),(|sup XLGLGLXLXL
GX
−+−+=−
∈
N
y
N
y
N
y
hence
))|(s)|(s(
2
1=),(ˆ N
y
N
yN GLGL −+σ , ))|(s)|(s(
2
1=)ˆ,( N
y
N
yNx GLGL −− , (17)
where )|(s N
yG⋅ denotes the support function of .N
yG Clearly, N
yG is a convex
closed set. Hence the equality )ˆ,(=)~,( NxXL is held for some .~ N
yGX∈ Thus,
to conclude the proof we have to calculate .),(s N
yGL Let
},:{= 22
0 N
N β≤+ HXFXXG (18)
where 0),(1= ≥+− +
NNNNN rrQαβ .
Lemma 1.
.)|(s),(=),(s 0
N
NN
N
y rQ GLGL ++ (19)
It follows from the definition of N
0G that )|(s=)|(s 00
NN GLGL − hence
(17) implies
)|(s=)(ˆ),,(=)ˆ,( 0
N
NNN rQx GLσ+ .
The application of Lemma 2 completes the proof.
Lemma 2.
⎪⎩
⎪
⎨
⎧
≠−∞+
−
+
++
.0][,
0,=][,),(=)|(s 2
1
0
NN
NNNN
N
QQE
QQEQβGL (20)
Let kr denote nR — valued recursive map
kkkkkkkkkkkkkkkkk yRHrWCSFfSCPCSSFr ′+′+′−′ −
+
−−−−−−
+
−−−− 11111111111 )(= ,
kkkkk QCSCPyRHSqFr +′′+′ =,= 00000 , (21)
S. Zhuk
ISSN 1681–6048 System Research & Information Technologies, 2010, № 2 100
and set
+−+− 2
0000
2
00=})({ xHygxFx SkJ
22
1111
1=
kkkkkkkkkk
N
k
xHyfxCxF −+−−+ −−−−∑ ,
where ),(=2 gSgg S , ),(=2
kkkkk ffSf , ),(=2
iiiii yyRy .
Lemma 3. Let kxx ˆ be a recursive map that takes any natural∈k to
n
kx R∈ˆ , where
),)ˆ((=ˆ 11 kkkkkkkk rfxFSCPx +−′ ++
+ NNN rQx +=ˆ . (22)
Then
.})ˆ({=})({min
}{
kk
kx
xx JJ
Proof. By definition put 2
0000
2
000 =:)( xHygxFx S −+−Φ ,
2
1111
2
111 :),( +++++++ −+−−=Φ iiiiiiiiiiii xHyfxCxFxx .
Then we obviously get
.),()(=})({ 1
1
0=
0 +
−
Φ+Φ ∑ iii
N
i
k xxxxJ (23)
Let us apply a modification of Bellman’s method (so-called «Kyivsky vi-
nyk» method) to the nonlinear programming task
min})({
}{ kx
kx →J .
By definition put
)},()({min:=)( 1000
0
11 xxxx
x
Φ+Φ .
Using (7) and (21) one can get
2
00
2
00000000 :=0,),2(),(=)( ygxrxxQx S +≥+−Φ αα .
On the other hand it’s clear that
0,),2(),(=),ˆ()ˆ(=)( 111111100011 ≥+−Φ+Φ αxrxxQxxxx
where ))((=ˆ 01100000 fxFSCrPx −′++
)),((:= 000000000
2
00
2
1101 fSCrfSCrPfy ′−′−−++ +αα .
Considering )( 11 x as an induction base and assuming that
=+Φ −−−−−
−
−− )}(),({min=)( 22122
2
11 iiiii
ix
ii xxxx
111111 ),2(),( −−−−−− +−= iiiiii xrxxQ α
now we are going to prove that
Minimax recursive state estimation for linear discrete-time descriptor systems
Системні дослідження та інформаційні технології, 2010, № 2 101
.),2(),()}(),({min=)( 1111
1
iiiiiiiiiii
ix
ii xrxxQxxxx α+−=+Φ −−−−
−
(24)
Note that [11] for any convex function ),(),( yxfyx
bbaPyyxPyxyxfy =),(},=),(:),(|),({min
is convex. Thus taking into account the definition of )( 11 x one can prove by
induction that 1−i is convex and
0)(),( 1111 ≥+Φ −−−− iiiii xxx .
Hence (the function cqxxAxx +− ),2(),( is convex if 0= ≥′AA )
01 ≥−iQ , the set of global minimums 1−Ψi of the quadratic function.
111111111 ),2(),(),( −−−−−−−−− +−+Φ iiiiiiiiii xrxxQxxx α
is non-empty and iix Ψ∈−1ˆ , where (The vector 1ˆ −ix has the smallest norm among
other points of the minimum.)
))(()(=ˆ 111111111 −−−−
+
−−−−− +−′′+ iiiiiiiiiii rfxFSCCSCQx .
This implies
iiiiiiiiiiiii xrxxQxxxx α+−=+Φ −−−− ),2(),()ˆ(),ˆ(=)( 1111 ,
where
−++ −−−− ),(),(= 1111 iiiiiiii ffSyyRαα
)),(( 111111111 −−−−−−−−
+
− ′−′−− iiiiiiiii fSCrfSCrP .
Therefore, we obtain
NNNNNNNNNNN
Nx
rQxrQrxx ++− =ˆ),,(=)ˆ(=)(min α
so that })ˆ({=})({min }{ kkkx xx JJ .
Corollary 4. Suppose ][0= …L ; then
[ ] 0=]['' NN QQE +−⇔∈ HFRL
and
[ ] ),(=''
2 ++
NQLHF .
Proof. Suppose ERES kk =,= for a simplicity. If [ ]''HFRL∈ then
(*),0=,= 1+′−′+′′+′ kkkkkkNNNN zCuHzFuHzF
for some p
k
m
k uz RR ∈∈ , . Let’s find the projection N
kkk uz 0=)}ˆ,ˆ{( of the vector
N
kkk uz 0=)},{( onto the range of the matrix ][
H
F . Lemma 3 implies
(**),ˆ=ˆ,ˆˆ=ˆ,ˆ=ˆ 11000 kkkkkkkk xHuxCxFzxFz −−−
where
S. Zhuk
ISSN 1681–6048 System Research & Information Technologies, 2010, № 2 102
NNNkkkkkkkk rQxzCrxFCPx +
+++
+ ′−+′ =ˆ),ˆ(=ˆ 111 ,
+′−′+′ −
+
−−−
+
−− kkkkkkkkkk zCPCEFrPCFr )(= 111111
kkkkkk QCCPuHzFruH +′′+′′+ =,=, 00000
(*) implies 1= +′ kkk zCr , 1,0,= −Nk … , =Nr thus +
NN Qx =ˆ , =kx̂
11 ˆ ++
+ ′= kkkk xFCP or +Φ Nk QNkx ),(=ˆ ,
EssNkFCPNk kkk =),(),1,(=),( 1 Φ+Φ′Φ +
+ .
Combining this with (**) we obtain
,))1,(),((=ˆ 1
+
− −Φ−Φ Nkkk QNkCNkFz
++ ΦΦ NNkk QNFzQNkHu )(0,=ˆ,),(=ˆ 00 . (25)
By definition, put 0=(0) QU ,
kkkkkkk FCPCEFHHkkkUkkkU 2
111 )()1,(1)()1,(=)( −
+
−− ′−+′+−Φ−−Φ′ .
It now follows that
[ ] ),)((ˆˆ'' 22
0
2 +++ =+=∑ NNNN
N
QQNUuzLHF .
It’s easy to prove by induction that )(= kUQk .
Since
[ ]''HFL R∈
we obtain by substituting kk uz ˆ,ˆ into (*)
=ˆˆ NNNN uHzF ′+′ .
On the other hand (7) and (25) imply
0=][=ˆˆ NNNNNN QQEuHzF +−⇒′+′ .
Suppose that 0=][ NN QQE +− . To conclude the proof we have to show that
[ ]''][0,=),(=),( 0 HFN∈∀+
NNNNN xxxQQx … .
By induction, fix 0=N . If 0=0,= 0000 xHxF , then 0=00 xQ . We say
that [ ]''][ 0 HFN∈kxx … if
0=,=0,=0,= 110000 ssssss xHxCxFxHxF −− .
Suppose [ ]''][0,= 1011 HFN∈∀ −−− kkk xxxQ … and fix any ∈][ 0 kxx …
[ ]''HFN∈ . Then 0=,=
11 kkkkkk xHxCxF − . Combining this with (7) we obtain
(*))(= 11111 −−−
+
−− ′−′ kkkkkkkk xCCPCEFxQ .
We show that 0≥kQ in the proof of Theorem 1. One can see that
Minimax recursive state estimation for linear discrete-time descriptor systems
Системні дослідження та інформаційні технології, 2010, № 2 103
])(,)[( 2
1
11111111
2
1
1
1
−
+
−−−−
+
−−−
+
−
−
+′′+′=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
kkkkkkkk
k
kC
QQCCCQCC
Q
.
Since
1
2
1
1
1
1
2
1
1
1
2
1
1
1
2
1
1
1
−
−
−
−
−
−
+
−
−
−
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
k
k
k
k
k
k
k
k
k
k
x
Q
с
x
Q
с
Q
с
Q
с
we obviously get
0==)( 111111111 kkkkkkkkkkk xQxCxCCQCCC ⇒′+′ −−−−−
+
−−−−
as it follows from (*) . This completes the proof.
Proof of Lemma 1. Taking into account the definitions of the matrices HF,
and (6) we clearly have
1}:{= 22 ≤−+ HXYFXXG N
y .
Let X̂ be a minimum of the quadratic function 22 HXYFXX −+ . It
now follows that
)|(s)ˆ,(=)|(sˆ= 000
NNNN
y GLXLGLGXG +⇒+ .
The application of Lemma 3 yields
),(=)ˆ,( NN rQ+XL .
This completes the proof.
Proof of Lemma 2. Suppose the function 1: Rf n →R is convex and
closed. Then [11] the support function 0}))(:{|( ≤⋅ xfxs of the set
0})(:{ ≤xfx is given by
⎭
⎬
⎫
⎩
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛≤
≥ λ
λ
λ
zfxfxzs *
0
infcl=0}))(:{|( .
To conclude the proof it remains to compute the support function of N
0G
according to this rule and then apply Corollary 4.
Proof of Corollary 3. The proof is by induction on k . For 0=k , there is
nothing to prove. The induction hypothesis is 1
11|1 = −
−−− kkk QP . Suppose S is
nn× -matrix such that 0>= SS ′ , A is nm× -matrix; then
ASAASEAASA 111 )(=)( −−− ′+′+ . (26)
Using (26) we get
ASAAAAASEAAS ′+′′+′ −− 11][][= . (27)
Combining (27) with the induction assumption we get the following
×′++=′+ −−−−−−−− ][ 11|1111|11 kkkkkkkk CPCEECPCE
S. Zhuk
ISSN 1681–6048 System Research & Information Technologies, 2010, № 2 104
1
1
1111 ][ −
−
−−−− ′′+× kkkkk CCCQC .
By simple calculation from the previous equality follows
1
11|111
1
1111 )()( −
−−−−−
−
−−−− ′+=′′+− kkkkkkkkk CPCECCCQCE .
Using this and (7), 15) we obviously get kkk PQ |
1 =− .
It follows from the definitions that 0|00
1
0 x̂rQ =− . Suppose that =−
−
− 1
1
1 kk rQ
1|1ˆ −−= kkx . The induction hypothesis and (26) imply
1
1
111111|11
1
11|11 )(ˆ)( −
−
−−−−−−−−
−
−−−− +′=′+ kkkkkkkkkkkkk rQCCCxCCPCE .
Combining this with (15), (16) and using kkk PQ |
1 =− we obtain
))((=ˆ 111111
1
| kkkkkkkkkkkk yHrQCCCFQx ′++′′ −
+
−−−−−
− .
This concludes the proof.
Proof of Corollary 2. If nI k < then nQ <)(rank hence .0=)( kmin Qλ In
this case there is a direction nR∈ such that +∞=),(ˆ kσ . So +∞=)(ˆ kρ .
If nI k = then it follows from formula (11) that
=−
∈∈
=−
∈∈
2
1=}~{}{
2
}~{}{
|})~,(|max{maxmin~maxmin NN
lN
ykxN
ykx
NN
N
ykxN
ykx
xxxx
GGGG
≥−
∈
= 2
}~{1=
|})~,(|maxmaxmin{ NN
N
ykxlN
y
xx
GG
=−
∈∈
≥ 2
}~{}{1=
|})~,(|maxminmax{ NN
N
ykxN
ykxl
xx
GG
)}({min
)],([1
),(max)],([1
1= N
rrQ
QrrQ
i
i
NNNN
N
l
NNNN λ
α
α
+
++ +−
=+−= . (28)
On the other hand formula (11) implies
=−
∈
=−
∈
2
}~{1=
2
}~{
|})~,(|maxmax{~ˆmax NN
N
ykxl
NN
N
ykx
xxxx
GG
.}),()],([1max{ 22
1
2
1
1=
+++−= NNNNN
l
QrrQα (29)
Using (28)–(29), we get (13).
Since (29) we see that the condition nI N = implies N
yG is a bounded set.
On the other hand nI N = implies 0][ =− +
NN QQE for the given N . It follows
from Lemmas 1, 2 that
Minimax recursive state estimation for linear discrete-time descriptor systems
Системні дослідження та інформаційні технології, 2010, № 2 105
2
1
),(),()|()|())(|( ++ +==′= NNNN
N
y
N
yN
N
yN QrQsPsPs βGLGG , (30)
for any nR∈ . By Young’s theorem [11], (30), so that
=∈∀≤∈ })),(|(),(:{=)( nN
yN
nN
yN PsxxP RGRG
=≤−−∈= + 0}}),()ˆ,(),{(sup:{ 2
1
NNN
n Qxxx βR
1}),ˆ(2),(:{ ≤+−∈= NNNN
n xxQxxQx αR .
This completes the proof.
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Received 12.02.2008
From the Editorial Board: the article corresponds completely to submitted manuscript.
|
| id | nasplib_isofts_kiev_ua-123456789-50047 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1681–6048 |
| language | English |
| last_indexed | 2025-12-07T16:24:22Z |
| publishDate | 2010 |
| publisher | Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
| record_format | dspace |
| spelling | Zhuk, S. 2013-10-03T20:05:32Z 2013-10-03T20:05:32Z 2010 Minimax recursive state estimation for linear discrete-time descriptor systems / S. Zhuk // Систем. дослідж. та інформ. технології. — 2010. — № 2. — С. 94-105. — Бібліогр.: 14 назв. — англ. 1681–6048 https://nasplib.isofts.kiev.ua/handle/123456789/50047 517.926:681.518.2 This paper describes an approach to the online state estimation of systems described by a general class of linear noncausal time-varying difference descriptor equations subject to uncertainties. An approach is based on the notions of a linear minimax estimation and an index of causality introduced here for singular difference equations. The online minimax observer is derived by the application of the dynamical programming and Moore's pseudoinverse theory to the minimax estimation problem. Розглянуто підхід до оцінювання стану системи, що описується дескрипторним рівнянням із дискретним часом за спостереженнями, що надходять у реальному часі. Підхід базується на понятті лінійної мінімаксної оцінки та індексу причинності, що вводяться у статті для сингулярних різницевих рівнянь. Рекурсивний оцінювач стану будується шляхом застосування методу «Київського віника» та теорії псевдоінверсних матриць до проблеми мінімаксного оцінювання. Рассмотрен подход к оцениванию состояния системы, описываемой дескрипторным уравнением с дискретным временем по наблюдениям, поступающим в реальном времени. Подход основан на понятии линейной минимаксной оценки и индекса причинности, введенных в статье для сингулярных разностных уравнений. Рекурсивный оцениватель строится путем применения метода «Киевского веника» и теории псевдообратных матриц к проблеме минимаксного оценивания. en Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України Системні дослідження та інформаційні технології Методи аналізу та управління системами в умовах ризику і невизначеності Minimax recursive state estimation for linear discrete-time descriptor systems Мінімаксна рекурсивна оцінка стану лінійних дескрипторних систем із дискретним часом Минимаксная рекурсивная оценка состояния линейных дескрипторных систем с дискретным временем Article published earlier |
| spellingShingle | Minimax recursive state estimation for linear discrete-time descriptor systems Zhuk, S. Методи аналізу та управління системами в умовах ризику і невизначеності |
| title | Minimax recursive state estimation for linear discrete-time descriptor systems |
| title_alt | Мінімаксна рекурсивна оцінка стану лінійних дескрипторних систем із дискретним часом Минимаксная рекурсивная оценка состояния линейных дескрипторных систем с дискретным временем |
| title_full | Minimax recursive state estimation for linear discrete-time descriptor systems |
| title_fullStr | Minimax recursive state estimation for linear discrete-time descriptor systems |
| title_full_unstemmed | Minimax recursive state estimation for linear discrete-time descriptor systems |
| title_short | Minimax recursive state estimation for linear discrete-time descriptor systems |
| title_sort | minimax recursive state estimation for linear discrete-time descriptor systems |
| topic | Методи аналізу та управління системами в умовах ризику і невизначеності |
| topic_facet | Методи аналізу та управління системами в умовах ризику і невизначеності |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/50047 |
| work_keys_str_mv | AT zhuks minimaxrecursivestateestimationforlineardiscretetimedescriptorsystems AT zhuks mínímaksnarekursivnaocínkastanulíníinihdeskriptornihsistemízdiskretnimčasom AT zhuks minimaksnaârekursivnaâocenkasostoâniâlineinyhdeskriptornyhsistemsdiskretnymvremenem |