A general approach to constructing parameter identification algorithms in the class of square root filters with orthogonal and J-orthogonal tranformations

We study modern implementations of the discrete Kalman filter, namely array square-root algorithms. An important feature of such algorithms is the use of orthogonal and J -orthogonal transformations on each filtering step. For the first time, we develop for this class of algorithms a simple universa...

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Veröffentlicht in:	Automation and remote control Jg. 75; H. 8; S. 1402 - 1419
Hauptverfasser:	Kulikova, M. V., Tsyganova, Yu. V.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Moscow Pleiades Publishing 01.08.2014
Schlagworte:	CAE) and Design Calculus of Variations and Optimal Control; Optimization Computer-Aided Engineering (CAD Control Mathematics Mathematics and Statistics Mechanical Engineering Mechatronics Queueing Systems Robotics Stochastic Systems Systems Theory Fisher Information Matrix Orthogonal Transformation Triangular Form Remote Control Sensitivity Equation
ISSN:	0005-1179, 1608-3032
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Abstract	We study modern implementations of the discrete Kalman filter, namely array square-root algorithms. An important feature of such algorithms is the use of orthogonal and J -orthogonal transformations on each filtering step. For the first time, we develop for this class of algorithms a simple universal approach that lets us generalize any numerically stable implementation of this type to the case of updates in sensitivity equations of the filter with respect to unknown system model parameters. An advantage of the resulting adaptive schemes is their numerical stability with respect to machine rounding errors. Estimation of the noisy state vector of the system and identification of unknown system parameters occur simultaneously. The proposed approach can be used for parameter identification problems, adaptive control problems, experiment planning, and others.
AbstractList	We study modern implementations of the discrete Kalman filter, namely array square-root algorithms. An important feature of such algorithms is the use of orthogonal and J -orthogonal transformations on each filtering step. For the first time, we develop for this class of algorithms a simple universal approach that lets us generalize any numerically stable implementation of this type to the case of updates in sensitivity equations of the filter with respect to unknown system model parameters. An advantage of the resulting adaptive schemes is their numerical stability with respect to machine rounding errors. Estimation of the noisy state vector of the system and identification of unknown system parameters occur simultaneously. The proposed approach can be used for parameter identification problems, adaptive control problems, experiment planning, and others.
Author	Kulikova, M. V. Tsyganova, Yu. V.
Author_xml	– sequence: 1 givenname: M. V. surname: Kulikova fullname: Kulikova, M. V. email: Kulikova.Maria@yahoo.com organization: CEMAT, Instituto Superior Tecnico, Universidade de Lisboa – sequence: 2 givenname: Yu. V. surname: Tsyganova fullname: Tsyganova, Yu. V. organization: Ulyanovsk State University
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References	KaminskiPGBrysonAESchmidtSFDiscrete Square-Root Filtering: A Survey of Current TechniquesIEEE Trans. Automat. Control1971AC-16672773510.1109/TAC.1971.1099816 KhazenEMMetody optimal’nykh statisticheskikh reshenii i zadachi optimal’nogo upravleniya1968MoscowSovetskoe Radio(Methods of Optimal Statistical Decisions, Optimal Control, and Stochastic Differential Equations) HighamNJJ-Orthogonal Matrices: Properties and GeneralizationSIAM Rev.200345350451910.1137/S00361445024149301034.650262046506 TsyganovaJVKulikovaMVState Sensitivity Evaluation within UD Based Array Covariance FilterIEEE Trans. Automat. Control201358112944295010.1109/TAC.2013.22590933126006 DenisovVIChubichVMChernikovaOSActive Parametric Identification of Stochastic Linear Discrete Systems in the Time DomainSib. J. Ind. Math.20036370871075.935432038220 ParkPKailathTNew Square-Root Algorithms for Kalman FilteringIEEE Trans. Automat. Control199540589589910.1109/9.3842250829.930751328086 SärkkäSNummenmaaARecursive Noise Adaptive Kalman Filtering by Variational Bayesian ApproximationIEEE Trans. Automat. Control200954359660010.1109/TAC.2008.20083482191552 GrewalMSAndrewsAPKalman Filtering: Theory and Practice2001New JerseyPrentice Hall MorfMKailathTSquare-Root Algorithms for Least-Squares EstimationIEEE Trans. Automat. Control1975AC-20448749710.1109/TAC.1975.1100994416718 HassibiBSayedAHArray Algorithms for H∞ EstimationIEEE Trans. Automat. Control200045470270610.1109/9.8471050980.930771764854 KulikovaMVPachecoAKalman Filter Sensitivity Evaluation with Orthogonal and J-Orthogonal TransformationsIEEE Trans. Automat. Control20135871798180410.1109/TAC.2012.22315723072863 TsyganovaYuVComputing the Gradient of the Auxiliary Quality Functional in the Parametric Identification Problem for Stochastic SystemsAutom. Remote Control20117291925194010.1134/S00051179110901411230.930962896160 MorfMSidhuGSKailathTSome New Algorithms for Recursive Estimation in Constant, Linear Discrete-Time SystemsIEEE Trans. Automat. Control1974AC-19431532310.1109/TAC.1974.1100576 LjungLSystem Identification: Theory for the User1987Englewood CliffsPrentice Hall0615.93004Translated under the title Identifikatsiya sistem. Teoriya dlya pol’zovatelya, Moscow: Nauka, 1991 DyerPMcReynoldsSExtension of Square-Root Filtering to Include Process NoiseJ. Optim. Theory Appl.1969444459 SayedAHKailathTExtended Chandrasekhar RecursionsIEEE Trans. Automat. Control199439361962210.1109/9.2807730816.930881268307 GuptaNKMehraRKComputational Aspects of Maximum Likelihood Estimation and Reduction in Sensitivity Function CalculationsIEEE Trans. Automat. Control1974AC-19677478310.1109/TAC.1974.1100714455190 VerhaegenMVan DoorenPNumerical Aspects of Different Kalman Filter ImplementationsIEEE Trans. Automat. Control1986AC-311090791710.1109/TAC.1986.1104128 BiermanGJFactorization Methods for Discrete Sequential Estimation1977New YorkAcademic0372.93001 KulikovaMVLikelihood Gradient Evaluation Using Square-Root Covariance FiltersIEEE Trans. Automat. Control200954364665110.1109/TAC.2008.20109892500657 TsyganovaYuVKulikovaMVOn Efficient Parametric Identification Methods for Linear Discrete Stochastic SystemsAutom. Remote Control201273696297510.1134/S0005117912060033 BiermanGJBelzerMRVandergraftJSPorterDWMaximum Likelihood Estimation Using Square Root Information FiltersIEEE Trans. Automat. Control199035121293129910.1109/9.610040723.930711078141 KailathTSome New Algorithms for Recursive Estimation in Constant Linear SystemsIEEE Trans. Inform. Theory1973IT-191175076010.1109/TIT.1973.1055104363565 KailathTSayedAHHassibiBLinear Estimation2000New JerseyPrentice Hall HighamNJAccuracy and Stability of Numerical Algorithms20022nd ed.PhiladelphiaSIAM10.1137/1.97808987180271011.65010 MehraRKOptimal Input Signals for Parameter Estimation in Dynamic Systems—Survey and New ResultsIEEE Trans. Automat. Control1974AC-19675376810.1109/TAC.1974.1100701437196 OgarkovMAMetody statisticheskogo otsenivaniya parametrov sluchainykh protsessov1990MoscowEnergoatomizdat(Statistic Estimation Methods for Parameters of Random Processes) KulikovaMVMaximum Likelihood Estimation of Linear Stochastic Systems in the Class of Sequential Square-Root Orthogonal Filtering MethodsAutom. Remote Control201172476678610.1134/S00051179110400841235.932422841653 AströmKJMaximum Likelihood and Prediction Error MethodsAutomatica19801655157410.1016/0005-1098(80)90078-30441.93036 GibbsRGSquare Root Modified Bryson-Frazier SmootherIEEE Trans. Automat. Control201156242545610.1109/TAC.2010.20897532761120 GantmaherFRTeoriya matrits19884th ed.MoscowNauka(Theory of Matrices) YuV Tsyganova (118_CR23) 2011; 72 KJ Aström (118_CR4) 1980; 16 B Hassibi (118_CR19) 2000; 45 M Morf (118_CR16) 1975; AC-20 NJ Higham (118_CR26) 2002 P Dyer (118_CR9) 1969 T Kailath (118_CR14) 1973; IT-19 YuV Tsyganova (118_CR24) 2012; 73 RG Gibbs (118_CR18) 2011; 56 GJ Bierman (118_CR20) 1990; 35 L Ljung (118_CR3) 1987 MV Kulikova (118_CR29) 2013; 58 NJ Higham (118_CR27) 2003; 45 JV Tsyganova (118_CR31) 2013; 58 PG Kaminski (118_CR10) 1971; AC-16 T Kailath (118_CR13) 2000 M Morf (118_CR15) 1974; AC-19 P Park (118_CR12) 1995; 40 MV Kulikova (118_CR21) 2009; 54 RK Mehra (118_CR6) 1974; AC-19 VI Denisov (118_CR7) 2003; 6 MA Ogarkov (118_CR2) 1990 MV Kulikova (118_CR22) 2011; 72 FR Gantmaher (118_CR28) 1988 M Verhaegen (118_CR8) 1986; AC-31 GJ Bierman (118_CR11) 1977 AH Sayed (118_CR17) 1994; 39 MS Grewal (118_CR25) 2001 S Särkkä (118_CR30) 2009; 54 NK Gupta (118_CR5) 1974; AC-19 EM Khazen (118_CR1) 1968
References_xml	– reference: BiermanGJFactorization Methods for Discrete Sequential Estimation1977New YorkAcademic0372.93001 – reference: AströmKJMaximum Likelihood and Prediction Error MethodsAutomatica19801655157410.1016/0005-1098(80)90078-30441.93036 – reference: HassibiBSayedAHArray Algorithms for H∞ EstimationIEEE Trans. Automat. Control200045470270610.1109/9.8471050980.930771764854 – reference: GantmaherFRTeoriya matrits19884th ed.MoscowNauka(Theory of Matrices) – reference: KailathTSome New Algorithms for Recursive Estimation in Constant Linear SystemsIEEE Trans. Inform. Theory1973IT-191175076010.1109/TIT.1973.1055104363565 – reference: HighamNJAccuracy and Stability of Numerical Algorithms20022nd ed.PhiladelphiaSIAM10.1137/1.97808987180271011.65010 – reference: GuptaNKMehraRKComputational Aspects of Maximum Likelihood Estimation and Reduction in Sensitivity Function CalculationsIEEE Trans. Automat. Control1974AC-19677478310.1109/TAC.1974.1100714455190 – reference: MorfMKailathTSquare-Root Algorithms for Least-Squares EstimationIEEE Trans. Automat. Control1975AC-20448749710.1109/TAC.1975.1100994416718 – reference: GibbsRGSquare Root Modified Bryson-Frazier SmootherIEEE Trans. Automat. Control201156242545610.1109/TAC.2010.20897532761120 – reference: ParkPKailathTNew Square-Root Algorithms for Kalman FilteringIEEE Trans. Automat. Control199540589589910.1109/9.3842250829.930751328086 – reference: VerhaegenMVan DoorenPNumerical Aspects of Different Kalman Filter ImplementationsIEEE Trans. Automat. Control1986AC-311090791710.1109/TAC.1986.1104128 – reference: SärkkäSNummenmaaARecursive Noise Adaptive Kalman Filtering by Variational Bayesian ApproximationIEEE Trans. Automat. Control200954359660010.1109/TAC.2008.20083482191552 – reference: KulikovaMVPachecoAKalman Filter Sensitivity Evaluation with Orthogonal and J-Orthogonal TransformationsIEEE Trans. Automat. Control20135871798180410.1109/TAC.2012.22315723072863 – reference: DenisovVIChubichVMChernikovaOSActive Parametric Identification of Stochastic Linear Discrete Systems in the Time DomainSib. J. Ind. Math.20036370871075.935432038220 – reference: KailathTSayedAHHassibiBLinear Estimation2000New JerseyPrentice Hall – reference: KulikovaMVLikelihood Gradient Evaluation Using Square-Root Covariance FiltersIEEE Trans. Automat. Control200954364665110.1109/TAC.2008.20109892500657 – reference: HighamNJJ-Orthogonal Matrices: Properties and GeneralizationSIAM Rev.200345350451910.1137/S00361445024149301034.650262046506 – reference: MehraRKOptimal Input Signals for Parameter Estimation in Dynamic Systems—Survey and New ResultsIEEE Trans. Automat. Control1974AC-19675376810.1109/TAC.1974.1100701437196 – reference: LjungLSystem Identification: Theory for the User1987Englewood CliffsPrentice Hall0615.93004Translated under the title Identifikatsiya sistem. Teoriya dlya pol’zovatelya, Moscow: Nauka, 1991 – reference: OgarkovMAMetody statisticheskogo otsenivaniya parametrov sluchainykh protsessov1990MoscowEnergoatomizdat(Statistic Estimation Methods for Parameters of Random Processes) – reference: SayedAHKailathTExtended Chandrasekhar RecursionsIEEE Trans. Automat. Control199439361962210.1109/9.2807730816.930881268307 – reference: GrewalMSAndrewsAPKalman Filtering: Theory and Practice2001New JerseyPrentice Hall – reference: TsyganovaYuVComputing the Gradient of the Auxiliary Quality Functional in the Parametric Identification Problem for Stochastic SystemsAutom. Remote Control20117291925194010.1134/S00051179110901411230.930962896160 – reference: KaminskiPGBrysonAESchmidtSFDiscrete Square-Root Filtering: A Survey of Current TechniquesIEEE Trans. Automat. Control1971AC-16672773510.1109/TAC.1971.1099816 – reference: DyerPMcReynoldsSExtension of Square-Root Filtering to Include Process NoiseJ. Optim. Theory Appl.1969444459 – reference: MorfMSidhuGSKailathTSome New Algorithms for Recursive Estimation in Constant, Linear Discrete-Time SystemsIEEE Trans. Automat. Control1974AC-19431532310.1109/TAC.1974.1100576 – reference: BiermanGJBelzerMRVandergraftJSPorterDWMaximum Likelihood Estimation Using Square Root Information FiltersIEEE Trans. Automat. Control199035121293129910.1109/9.610040723.930711078141 – reference: TsyganovaJVKulikovaMVState Sensitivity Evaluation within UD Based Array Covariance FilterIEEE Trans. Automat. Control201358112944295010.1109/TAC.2013.22590933126006 – reference: KulikovaMVMaximum Likelihood Estimation of Linear Stochastic Systems in the Class of Sequential Square-Root Orthogonal Filtering MethodsAutom. Remote Control201172476678610.1134/S00051179110400841235.932422841653 – reference: KhazenEMMetody optimal’nykh statisticheskikh reshenii i zadachi optimal’nogo upravleniya1968MoscowSovetskoe Radio(Methods of Optimal Statistical Decisions, Optimal Control, and Stochastic Differential Equations) – reference: TsyganovaYuVKulikovaMVOn Efficient Parametric Identification Methods for Linear Discrete Stochastic SystemsAutom. Remote Control201273696297510.1134/S0005117912060033 – volume-title: Linear Estimation year: 2000 ident: 118_CR13 – volume: 45 start-page: 702 issue: 4 year: 2000 ident: 118_CR19 publication-title: IEEE Trans. Automat. Control doi: 10.1109/9.847105 – volume: 72 start-page: 766 issue: 4 year: 2011 ident: 118_CR22 publication-title: Autom. Remote Control doi: 10.1134/S0005117911040084 – volume: AC-19 start-page: 315 issue: 4 year: 1974 ident: 118_CR15 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.1974.1100576 – start-page: 444 volume-title: J. Optim. Theory Appl. year: 1969 ident: 118_CR9 – volume: 56 start-page: 425 issue: 2 year: 2011 ident: 118_CR18 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.2010.2089753 – volume: 45 start-page: 504 issue: 3 year: 2003 ident: 118_CR27 publication-title: SIAM Rev. doi: 10.1137/S0036144502414930 – volume-title: System Identification: Theory for the User year: 1987 ident: 118_CR3 – volume: 72 start-page: 1925 issue: 9 year: 2011 ident: 118_CR23 publication-title: Autom. Remote Control doi: 10.1134/S0005117911090141 – volume: AC-31 start-page: 907 issue: 10 year: 1986 ident: 118_CR8 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.1986.1104128 – volume: IT-19 start-page: 750 issue: 11 year: 1973 ident: 118_CR14 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.1973.1055104 – volume: 35 start-page: 1293 issue: 12 year: 1990 ident: 118_CR20 publication-title: IEEE Trans. Automat. Control doi: 10.1109/9.61004 – volume-title: Factorization Methods for Discrete Sequential Estimation year: 1977 ident: 118_CR11 – volume: 58 start-page: 1798 issue: 7 year: 2013 ident: 118_CR29 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.2012.2231572 – volume: 16 start-page: 551 year: 1980 ident: 118_CR4 publication-title: Automatica doi: 10.1016/0005-1098(80)90078-3 – volume: AC-19 start-page: 774 issue: 6 year: 1974 ident: 118_CR5 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.1974.1100714 – volume-title: Teoriya matrits year: 1988 ident: 118_CR28 – volume: AC-16 start-page: 727 issue: 6 year: 1971 ident: 118_CR10 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.1971.1099816 – volume-title: Metody optimal’nykh statisticheskikh reshenii i zadachi optimal’nogo upravleniya year: 1968 ident: 118_CR1 – volume: AC-19 start-page: 753 issue: 6 year: 1974 ident: 118_CR6 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.1974.1100701 – volume: 39 start-page: 619 issue: 3 year: 1994 ident: 118_CR17 publication-title: IEEE Trans. Automat. Control doi: 10.1109/9.280773 – volume: 54 start-page: 596 issue: 3 year: 2009 ident: 118_CR30 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.2008.2008348 – volume: 40 start-page: 895 issue: 5 year: 1995 ident: 118_CR12 publication-title: IEEE Trans. Automat. Control doi: 10.1109/9.384225 – volume: 73 start-page: 962 issue: 6 year: 2012 ident: 118_CR24 publication-title: Autom. Remote Control doi: 10.1134/S0005117912060033 – volume-title: Kalman Filtering: Theory and Practice year: 2001 ident: 118_CR25 – volume: 6 start-page: 70 issue: 3 year: 2003 ident: 118_CR7 publication-title: Sib. J. Ind. Math. – volume-title: Accuracy and Stability of Numerical Algorithms year: 2002 ident: 118_CR26 doi: 10.1137/1.9780898718027 – volume: 58 start-page: 2944 issue: 11 year: 2013 ident: 118_CR31 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.2013.2259093 – volume: AC-20 start-page: 487 issue: 4 year: 1975 ident: 118_CR16 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.1975.1100994 – volume-title: Metody statisticheskogo otsenivaniya parametrov sluchainykh protsessov year: 1990 ident: 118_CR2 – volume: 54 start-page: 646 issue: 3 year: 2009 ident: 118_CR21 publication-title: IEEE Trans. Automat. Control doi: 10.1109/TAC.2008.2010989
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SubjectTerms	CAE) and Design Calculus of Variations and Optimal Control; Optimization Computer-Aided Engineering (CAD Control Mathematics Mathematics and Statistics Mechanical Engineering Mechatronics Queueing Systems Robotics Stochastic Systems Systems Theory
Title	A general approach to constructing parameter identification algorithms in the class of square root filters with orthogonal and J-orthogonal tranformations
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