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
Автор: Zhuk, S.
Формат: Стаття
Мова:Англійська
Опубліковано: Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України 2010
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Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/50047
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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
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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.
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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. REFERENCES 1. Favini A., Vlasenko L. On solvability of degenerate nonstationary differential- difference equations in Banach spaces.— Journal of Differential and Integral Equations. — 2001. — 14, № 7. — Р. 83–896. 2. James K. Mills Dynamic modelling of a a flexible-link planar parallel platform using a substructuring approach. — Mechanism and Machine Theory. — 2006. — № 41. — Р. 671–687. 3. Hasan M.A. Noncausal image modelling using descriptor approach // IEEE Transactions on Circuits and Systems II. — 1995. — 2, № 42. — P. 36–540. 4. Luenberger D., Arbel A. Singular dynamic Leontief systems // Econometrica. — 1977. — 45, № 4. — C. 12–24. 5. Becerra V.M., Roberts P.D., Griffiths G.W. Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations // Control Engineering Practice 9 (2001). — Р. 267–281. 6. Nikoukhah R., Campbell S.L. and Delebecque F. Kalman filtering for general discrete-time linear systems // IEEE Transactions on Automatic Control. — 1999. — № 44. — Р. 1829–1839. 7. Biehn N., Campbell S., Nikoukhah R., Delebecque F. Numerically constructible observers for linear time-varying descriptor systems // Automatica. — 2001. — № 37. — Р. 445–452. 8. Ishihara J.Y., Terra M.H., Campos J.C.T. Optimal recursive estimation for discrete- time descriptor systems // International Journal of System Science. — 2005. — 36, № 10. — P. 1–22. 9. Zhuk S. Minimax estimations for linear descriptor difference equations. — http://arxiv.org/abs/math/ 0609709, 2006. 10. Grygorov A., Nakonechniy A. State estimation for discrete-time systems with inner noise // Journal of applied and computational mathematics. — 1973. — № 2 — Р. 20–26. 11. Rockafellar R. Convex analysis // Princeton University Press. — 1970. — 465 c. 12. Bakan G. Analytical synthesis of guaranteed estimation algorithms of dynamic process states // Journal of automation and information science. — 2003. — № 35(5). — Р. 12–20. 13. Kurzhanski A., Valyi I. Ellipsoidal calculus for estimation and control birkhauser, 1997. — 190 c. 14. Kuntsevich V., Lychak M. Guaranteed estimates, adaptation and robustness in control system // Springer-Verlag, 1992. — 250 c. 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